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Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra in $\mathbb{R}$. Could you help me to show that
$$
\left(\int_{-\infty}^\infty \sqrt{... |
Consider the system
$$\begin{align}
\dot{u}&= v\\
147\dot{v}&=8150-588v-20000w\sin{u}\\
330\dot{w}&=-135w+85\cos{u}+61
\end{align}$$
Find all equilibria where each $u,v,w\in[-\pi,\pi]$.
I can't see a nice way to solve this since there are no common terms in the second and third equation. There is also nonlinearity pre... |
Let $\phi$ be a continuous (complex valued) function on the real interval [−1, 1] inside C, and define
$$f(z)=\int_{-1}^1\frac{\phi(t)}{t-z}dt$$
Show that f is analytic on C less the interval [−1, 1].
I thought about CR equation to prove analytic but the function is not of the form $f(z) = u(x, y) + iv(x, y)$
|
Consider the system
$$
\dot{x}=x-z+y^2,\quad\dot{y}=x-2y+z+y^2+2x^2z,\quad\dot{z}=-2x+2y+z^2-y^2.
$$
and the equilibrium $(0,0,0)$.
Now, there is used some statement that I did not know yet:
Since
$$
\frac{\partial (x-z+y^2,x-2y+z+y^2+2x^2z)}{\partial(x,z)}_{|(x,y,z)=0}=\begin{pmatrix}1 & -1\\1 & 1\end{pmatrix}~~... |
It can be proved that there are two irrational numbers $a$ and $b$ such that $a^b$ is rational (see Can an irrational number raised to an irrational power be rational?) and that for each irrational number $c$ there exists another irrational number $d$ such that $c^d$ is rational (see For each irrational number b, does ... |
The standard definition suggests that every element in the codomain should have a preimage. So, Can different elements in codomain or range have same domain?
A worst question I think. Please reply...
|
When I was reading "Pattern Recognition and Machine Learning", I come across following equation (Equation 2.134, Chapter 2):$$ - \lim_{N \rightarrow \infty } \frac{1}{N} \sum_{n=1}^{N} \frac{\partial}{\partial \theta} \ln p(x_n|\theta) = \mathbb{E}_x \left[ - \frac{\partial}{\partial \theta} \ln p(x|\theta) \right]$$
I... |
Let $X$ be an Enriques surface and $f\colon X\to \Bbb{P}^1$ an elliptic fibration. I will denote by $F$ the general fibre and by $r_1F_1,\cdots,r_kF_k$ the multiple fibres of the fibration. I wanted to prove as an exercise that
$$k=2 , \quad r_1=r_2=2$$
Here is what I tried. By the canonical bundle formula
$$K_X\sim -F... |
Let $\sigma$, $\tau$ be two permutations of $S_n$.
We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides $$\gcd(\left|\langle \sigma \rangle\right|, \left|\langle \tau \rangle\right|).$$
Do we have other results that offer us a way to ... |
Let $X \subseteq \mathbb{P}^n$ be a geometrically integral projective surface of degree $d$ defined over $\mathbb{Q}$. Does it then imply that $X$ is smooth? I was wondering if someone could provide me with a counter example. Thank you very much!
|
Consider the following setting. Let $k = 1, \ldots, n$ and define
$$y_k= \Phi^{-1}\left(\frac{k}{n+1}\right),$$
where $\Phi$ is the inverse of the CDF of a standard normal.
I noticed numerically that
$$ \sum_{k =1}^{n} y_k = 0$$
But I cannot prove it. Any idea for a proof?
|
Is there a simple formula for the integral
$$\int_a^{\infty}\!\text{erf}(\alpha x)\,\text{erf}(\beta x)\:\frac{dx}{x^2} $$
where $a, \alpha, \beta > 0$?
|
Given the metric $$d(x,y)=\text{min}\{1,|x-y|\}$$ on $\Bbb{R}$.There is a bounded sequence in $(\Bbb{R},d)$ without a convergent subsequence. Prove that a sequence in $(\Bbb{R},d)$ converges iff it converges in the standard metric on $\Bbb{R}$.
My try: Say $\{p_n\}$ is a bounded sequence in $(\Bbb{R},d)$. Assume it con... |
This is an IQ question.
"Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now?"
How to solve such questions? I think their combined age should be given but its not.
|
$\mu(E)\ge \nu(E)\ \forall E\in A\ \Rightarrow\ \mu(\cup E)\ge\nu(\cup E)$?
A. Here $\mu,\nu$ are probability measures on a $\sigma$-algebra $M$ on a set $X$. We can assume that arbitrary unions of the elements of $A$ are measurable. Of course, $A\subset M$. (This was already answered negatively below.)
B. In addition,... |
I've came accros this excersize:
Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$
suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where $\bar z$ is the complex conjugate of $z$. Prove that $f$ cannot be analytic in $D$.
I started with assuming that $f$ is indeed ... |
If the relation between sub-tangent and sub-tangent at any point on the curve $y^2 = (x+a)^3$ is $p(SN)=q(ST)^2$ then find the value of $p:q$ where $SN$ is length of sub-normal and $ST$ is length of sub-tangent respectively.
|
A signal x(t) inceases linearly to the value 2 at $t=2$, starting from $t=1$. It stays constant for $t \in [2,3]$ then decreases linearly to 0 at $t=5$. Let $y(t)=x(2t-1)$.
What is the maximum and minimum value of the slope of y(t) for $t \in [0, \infty]$?
I know $x(t)$ is a piecewise function. But now the trouble... |
Sorry about the formatting, but how would I go about this question:
$$\frac{d}{dx} \int_{\cos x}^1 \sqrt{(1 + e)^t} dt$$
What I've learned in class is that the derivative of an integral is just the function itself ($\sqrt{(1 + e)^t}$)
However, it doesn't seem to be the case here.
Thanks for your time
|
There is nothing original about this question. It was asked here.
I am just curious about an answer that is beyond my mathematical level.
In one of the simulations appearing in the comments to the OP, the number $e$ is approximated through a formula like this:
$$\Large2 + \mathrm E\left(\frac{1}{\left(\lceil \frac{1}{X... |
What tools, ways would you propose for getting the closed form of this integral?
$$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$.
When $B=0$, from Table of Integrals, Series, and Products, we can use this to calculate $$\int^\infty_0 \exp \left(-\frac{\beta}{4x}-\gamma x\right)\,dx= \sqrt{\f... |
Find the disc of convergence
$$\sum_{n=0}^\infty z^{n^{3}}$$
I have applied the ratio test but I can not seem to come up with a conclusion.
|
Here is mine computation for uniform acceleration, i.e $v'(t) = \text{const}\ a$:
$$v(t) = v(t_0) + \int_{t_0}^{t} v'(t)dt = v(t_0) + a\cdot \Delta t$$ where $$\Delta t = t- t_0$$ and coordinate is therefore $$x(t) = x(t_0) + \int_{t_0}^{t}x'(t)dt= x(t_0) + \int_{t_0}^{t}v(t)dt = \\ x(t_0) + \int_{t_0}^{t}(v(t_0) + a\c... |
Suppose that $X$ is a set and $V_{0}$, $J$, and $V_{1}$ are pairwise disjoint subsets of $X$ whose union is $X$. If the three subsets were nonempty it would be a partition of $X$. However, I wish to include the possibility that at least one of the sets, perhaps two of them, are empty.
For further context I work with ab... |
I am trying to solve part (c) of the following Representation Theory question:
Let $D$ be a division ring and let $n$ be a positive integer. For $ 1 \leq l \leq n $ let
$$C_l= \{A = (a_{ij}) \in {M_n}{(D)} : a_{ij} = 0 \text{ for } j \neq l \} .$$
(a) Show that each $C_l$ is a simple submodule of ${M_n}{(D)}$.
(b) Dedu... |
I have the following problem:
Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$.
Thank you!
|
I have trouble proving that $h(z)=\overline{h(\overline{z})}$ for all $z \in \mathbb{C}$ under the assumption that $h$ is holomorphic and the real line maps itself or in other words: $h(z) \in \mathbb{R}$ if $z \in \mathbb{R}$.
I showed that $\overline{h(\overline{z})}$ is holomorphic and thus $\overline{h'(\overline{z... |
Tao's analysis book uses following example for Russell's paradox:
$$P(x) \Longrightarrow `` x\text{ is a set, and }x \notin x"\\
\Omega := \{x : P(x)\text{ is true} \} = \{x : x\text{ is a set and }x \notin x\}$$
then conclude that $\Omega \in \Omega$ and $\Omega \notin \Omega$ happened same time. so it is a paradox... |
I have to find the volume of the region bounded by $ y= \sqrt{x-1} $, y=3, the y-axis and the x-axis rotated around y=5 I set up
$\int_1^{10} $ $\pi((5-(\sqrt{x-1}))^2 - (5-3)^2)$dx + $\int_{0}^1$ $\pi(5^2 - (5-3)^2)$
My question is do I square each part of the integral separately like ((5^-$\sqrt{x-1})^2$) or with t... |
I'm told to prove or disprove that $4^{\sqrt{n}}$ grows faster than $\sqrt{4^n}$
As n tends to infinity.
From my Previous years Calculus I know that if I take the derivative of two functions, and one is bigger, than it must grow faster.
$f(n) = 4^{\sqrt{n}}$
$g(n) = \sqrt{4^n}$
if
$f'(n)$ is > $g'(n)$ Then $f(n)$ grows... |
With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range":
$$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$
The reflection principle gives, for $a > 0$, $P(\overline{M_t} \geq a) = 2 P(B_t \geq a)$ (as stated by @A.S. as comment on this... |
Let $A_t=t-S_{N_t -1}$ with $N_t$ a renewal process
1)Show $A_t$ checks the Markov property
my proof:
$S_{N_t}=X_1+\cdots+X_{N_t}$ and the increments are independents
$$P(S_{N_t-1}=t-y\mid S_{N_{u_1}-1}=u_1-x_1,\ldots,S_{N_{u_n}-1}=u_n-x_n)=P(S_{N_t-1}-S_{N_{u_n}-1}=t-y-(u_n-x_n))=P(S_{N_t-1}=t-y\mid S_{N_{u_n}-1}=u_n-... |
I'm having a hard time understanding the proof of Lemma 17 in this article. Essentially, the assertion of the lemma boils down to replacing a constant in a sequence of random variables that satisfies a CLT (from the previous lemma) by a sequence that converges almost surely to said constant and obtaining that the CLT s... |
I'm learning some real analysis and encountered the following question:
Let $E=\left\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:\sqrt{x_2^2+x_3^2+x_4^2}\le x_1\right\}$. for which values of $t\in\mathbb{R}^4$ is the integral the$\int_E e^{-\langle x,t\rangle} \, dx$ finite? Calculate the integral.
My first attempt was:
$$\int_... |
I have this task about cards:
Consider choosing a card from a well-shuffled standard deck of 52
playing cards.
Suppose that, after the first extraction, the card
is not reinserted in the deck. What is the probability that the second
card is an ace?
Suppose that, after the first extraction, the card
is reins... |
Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to approach it. AS in title $$\log{|z|}=-2\arg(z)$$ In general I guess this could be expressed as $$\log(\sqrt{{x^2+y^2})}=-2\t... |
Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ and which is perpendicular to the plane $5x+3y-6z+8=0$
By setting $z=0$ I found a point $(-\frac{14}{3}, \frac{13}{3}, 0)$ but not able to obtain normal vector of the plane. Would finding complete equatio... |
If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator?
Here's what I tried, is this right?
$$
\begin{align}
\mathbb{E}[\hat{\theta}_1] &= \mathbb{E}[e^{-X}] \\
&= e^{\mathbb{E}[-X]}\\
&= e^{-u} \\
&= \theta
\end{align}
$$
Show that $\hat{\... |
I have to show that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty $. I spend very much time on it, and I can't conclude. How can I find a $\delta>0$ s.t. $|f(x)-f(y)|<\varepsilon$ if $|x-y|<\delta$ for all $\varepsilon>0$ ?
|
I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a convergent subsequence of $x_n$ with limit $c$. Then, we let $C$ bet the set of all cluster points. The problem is to... |
How to solve the following differential equation: $$2f'(t)+tf(t)=0$$ with $f(0)=\sqrt{\pi}$.
I tried to write $2f'(t)+tf(t)=0$ something like $(f(t)g(t))'=0$ for some function $g$ but it was unsuccesful.
I am not very good familiar with DE and can anyone show me how to solve it easily?
|
Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$.
Using the chain rule, we can differentiate $f(x)$ to get
$f'(x) = \nabla^Th(g(x))g'(x)$
My question is why do we take the transpose of the gradient of $h$? Is it just to make sure the result is a scalar, since $f(x)$ is in $R$?
If so, d... |
Given $a>1$ and
$$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$
I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq \delta \}$ but the sequence $\{f_{n}\}$ does not converge uniformly for all real numbers.
A little help...
|
Congruency question: if $17x+11y \equiv 7 \pmod {29}$ and $13x+10y \equiv 8 \pmod {29}$, we need to find $x$ and $y$. There may be more than one answer.
Not sure how to go about this; any help appreciated :)
|
Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$.
Define the diffusion-scaled partial sum as
\begin{equation}
\hat S_n(t) = \frac{1}{\sqrt{n}}\left(S( nt) - \mathbb E[X]nt\right),
\end{equation}
where $S(t) = \sum_{i=1... |
Imagine you have two dynamical systems, given by the statespace equations: $\frac{dx}{dt}=F_1(x)$ and $\frac{dx}{dt}=F_2(x)$, and you are concerned with trajectories form a point in phase space $x_0$. Where $x_{0},x\in \mathbb{R}^n$, and $F_1(x)$ and $F_2(x)$ are mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$.
Its cle... |
Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false:
The element $gag^{-1}$ is of order $2$.
$(ag)^2=g^2$
if $ag$ is of order $2$ then $gag = a$
So 1 is immediate for any abelian group, and I can't find any other group that will have an element of orde... |
Calculate:
$$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
|
Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?
|
Is y(t) = 1 a causal system?
From the definition of causal systems , a causal system is a system where the output depends on past and current inputs.
Here the system doesn't depend on any input. So, is the system causal?
|
Let $A,B \in M_n$. Is this formula true?
$$\det(A+B) = \det(A) + \det(B) + \operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB).$$
|
Proof. As $t\rightarrow x$, we have, by Theorem 4.4, (Baby Rudin, p.104)
$f\left( t\right) -f\left( x\right)$ = $\dfrac {f\left( t\right) -f\left( x\right) } {t-x}\cdot \left( t-x\right) \rightarrow f'(x).0=0.$
Note that I know it is equal to 0, yet why f is continous? My question is that I did not proof of theorem. Ca... |
On the internet there are many proofs but very summary
Definition
$|a| = $
$a$ if $a \ge 0$
$-a$ if $a < 0$
Proposition 1 $|a| = 0 \Leftarrow \Rightarrow a = 0$
$\Rightarrow$)
Suppose that $|a| = 0$
This implies that $a\ge 0$ xor $a<0$
Case $a<0$
$|a| = -a$ by hypothesis $-a= 0$
As $a<0$
$a + (-a)<0 + 0$
$0<0$ (!contr... |
Let us consider a bounded sequence $\{a_n\}$ . Now as it is a bounded sequence it must contain a convergent sub-sequence, $\{b_n\}$. Now let us filter out $\{b_n\}$ out of $\{a_n\}$. As such we are still left with a bounded sequence. But , I want to know if this sequence can again contain infinite terms so as to have ... |
I'm looking at my textbooks steps for calculating the complexity of bubble sort...and it jumps a step where I don't know what exactly they did.
I see everything up to that point using summation rules and all, but am unsure about that jump. Any help on explaining more how they got that?
|
What is the difference between Riemann-integrable and Lebesgue-integrable? Does it have anything to do with the absolute value of the integrand; something like
$\text{Lebesgue-integrable} \Leftrightarrow \int |f(x)| < \infty$?
|
I'm pretty sure the answer is out there, see this MathOverflow question, but that is unfortunately way over my head :).
I'm interested in the case that $G$ is a Lie group (e.g. $U(1)$), but I don't need to know the answer for general manifolds -- just for boring old Euclidean space.
For example, I sort of feel like... |
I would like to know how the left hand side of the equation is achieved. In particular why the $\frac{1}{2}$ is there. I don't understand how one can get from the left to the right side.
$$ \sum_{k=0}^{d} \binom{2d+1}{k} = \frac{1}{2} \cdot 2^{2d+1} $$
I have looked at this wikipedia entry, but it does not help me, si... |
Find all elements of $\mathbb{Z}_{19}^*$ of multiplicative order $18$.
I started by using Euler's Theorem and since gcd(18, 19) = 1 it implies that $a^{\phi (19)} \equiv 1 \pmod n$. Which means that the multiplicative order divides $\phi(19) = 18$, but I don't know where to go from here.
|
Suppose $X_1, X_2, \ldots, X_n$ are $IID$ normal RVs with mean $\mu$ and variance $1$. However, we observe only $Y_i$'s where $Y_i = \max (0, X_i)$. I would like to know how to write likelihood function for $\mu$ given $Y_i$'s. Since, $Y_i$ has both discrete and continuous components, I tried to write it in the followi... |
GetThere Airlines currently charges $200$ dollars per ticket,and
sells $40,000$ tickets.For every $10$ dollars they increase the
ticket price,they sell $1000$ fewer tickets.
How much should they charge to maximize their revenue ?
I am not able to model an equation for the second part of the problem,where the co... |
Let $A$, $B$ and $C$ be 3 sets. I want to show that $A=B=C$, can we use only three inclusions to do that ? For example we use $A \subset B\subset C\subset A$. Is this the only way to do this with three inclusions ? and can we do that with two inclusions ? Thank you for your help!
|
Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat.
Question. Are weakly étale ring homomorphisms of finite presentation étale?
According to Proposition 2.3.3 here, the answer is yes. Unfortunately, the proof cites some results in a monograph... |
It has been way too many years since high school. How can I simplify this equation to solve for m:
$\frac{x}{c+pm}=m$
I got to
$x = cm + pm^2$
and I don't know how to get any further. I wish this stuff stayed in my head over the last 20 years.
|
I recently proved that the product of all primitive roots of an odd prime $p$ is $\pm 1$ as an exercise. As a result, I became interested in how few distinct primitive roots need to be multiplied to guarantee a non-primitive product. Testing some small numbers, I have the following claim: "If $n$ and $m$ are two distin... |
I have a recurrence relation defined as :
$A(i, j) = A(i, j-1) + A(i+1, j)$
where both $i$ and $j$ are less than a fixed variable $N$.
Also,
$A(i,1) = 1\:\:$ for all $1 \leq i \leq N$.
$A(N, j) = 1\:\:$ for all $1 \leq j \leq N$.
How do I proceed to solve this equation?
I tried to expand this recurrence relation and g... |
I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that:
$$\phi(\mathbf{z})=\phi(-\mathbf{z})$$
It is suggest that I use connectedness.
I know both sets are connected, and that a continuous map preserv... |
Let $T$ be a linear operator on a vector space $V$, and let $W$ be an invariant subspace. If $V(\lambda)$ denotes the $\lambda$-eigenspace of $V$ and $W(\lambda)$ the eigenspace of $T$ on $W$, then does $[V(\lambda)/W(\lambda)] = [V/W](\lambda)$.
If I can show this, then it will help with something I am working on. Is ... |
I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It just comes out as follows, but some notations;
$L(z_1,z_2)$ represents the straight line that connects points $z_1,z_2 ... |
I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion:
Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), \mathcal{B} (C[0, \infty)))$ which converges weakly to a probability measure $\mathbb{P}$. Suppose, in addition, that $\{f_n\}$ is... |
I'm looking at the Boyd & Vandenberghe slides on Convex Optimization. In slide 18, it applies the rules of vector composition on an example to say that it is convex. The example given is
$\log\sum_{i=1}^{m}\exp g_i$ is convex if $g_i$ is convex.
I know the exponential (and sum of exponential) function is convex. But is... |
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers:
$A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers)
$B = (a_1 + a_2 + \cdots + a_n)^{1 / n}$
Prove that $A \geq B$.
This is what I'm trying to prove for an assignment, and I'm not sure where to start or how do do it.... |
Let $f$ be a Schwartz function. Prove that, whenever $2\le r < \infty,$
$$\| e^{it \Delta} f\|_{L^{3r}(\mathbb{R}^2_{xt})} \le c \| \widehat{f}\|_{r'},$$
Where $1/r + 1/r' = 1.$
My Attempt
My main idea was to use some Plancherel-like Theorem, either for one or for two variables. More specifically, my main idea was pro... |
I just begun studying image deblurring on my own, and I have a question.
Most books I found say that I can see the images as arrays, and that I can "vectorize" the arrays of the images by stacking the columns.
So, the image is reresented by a vector, and I have a linear system (something like $g = K * f$, where $g$ is ... |
Pillai's conjecture is a generalization of Catalan's conjecture. It's say that for fixed positive integers $A, B, C$ the equation $Ax^n - By^m = C$ has only finitely many solutions $(x,y,m,n)$ with $(m,n) ≠ (2,2)$.
But if we fix $A=B=1$ and $x=2,y=3$ then we have :
$$2^n-3^m=C$$
My question : Have this equation only... |
I am looking for classes of functions which are:
1) holomorphic
2) |f(z)|>0 for all z
3) complex differentiable (i.e. f(z)=mod(z) is not valid)
Particularly I am looking for functions whose complex derivatives are less complicated.
While conditions $(1)$ and $(3)$ may be too general, in particular I am ... |
Given a polygon in the $x-y$ plane, what is the simplest formula for expanding the polygon so that all sides lie on a grid?
The image below demonstrates the problem I am trying to solve. The filled in line is the original polygon, while the outer line shows the expected outcome.
Fill - Original shape. Line - desired s... |
The statement I'm trying to prove is:
$n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0
I eventually need to prove $(k + 1)^3 + 7(k + 1) + 3$ is divisible by 3.
I don't really understand how to deal with $k + 1$, so I'm a little lost.
I've know that the base case of P(0) is true, but I'm not sure about proving ... |
Question: Let $X$ be the Twisted Cubic in $\mathbb{P}^3$, and $\pi_p:X\rightarrow \mathbb{P}^2$, the projection of the Twisted Cubic from $p$. Find the equations of the projection of the twisted cubic from the point $p=(1:0:0:1)$ in the hyperplane $Z_3=0$.
So far I realized that the map $\pi_p(X)$ sends the coordinates... |
Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class $[N]$. Geometrically, if $N'$ is another $n$-submanifold, $[N]^\#([N'])$ counts the number of intersections of $N'$ with $N$ ... |
In triangle $ABC$, point $M$ is the midpoint of $BC$ and $N$ is on the angle bisector of $\angle A$ such that $MN \parallel AB$. Prove that $MN = \dfrac{|b − c|}{2}$.
Attempt: I drew it out and noticed that $\triangle{DMN}\sim\triangle{DBA}$. I am not sure how to use the fact that $M$ is the midpoint yet, but I think... |
Ok so we are given a Markov chain $X_n$, $P=P(ij)$ as the transition matrix and the $(\pi_1,\pi_2,\pi_3,...,\pi_n)$ as steady-state distribution of the chain. We are asked to prove that for every $i$:
$\sum_{i\neq j} \pi_iPij = \sum_{j\neq i} \pi_jPji$
Can somebody explain me what that means and why it is (obviously tr... |
Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$.
Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained here
Which of the following statements are true?
(1)$\lim_{n\rightarrow \infty} na_n=\infty \leftrightarrow \lim_{n\rightar... |
$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$.
$Z=X+Y$
$\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition matrix :
$$Q=\begin{bmatrix}0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 &\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & ... |
We have $z^{n-1} = \bar{z}\ \forall\ n > 2$ which gives us $z^n = |z|^2$, but I dont see why that means $|z| = 1$?
|
Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following:
And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question is why? Why does forming it that way give you the magnitude of an orthogonal vector and how is it related to the $\sin(\th... |
Here $\alpha>0$, $Q>0$, and $J_1$ is a Bessel function. I'm fairly certain the closed form in the title is accurate for a couple of reasons. First, I've evaluated the integral numerically in Mathematica for many $\alpha$ and $Q$ values, and it matches every time. The second reason is a little more elaborate, and involv... |
The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition that states that $\Psi$ is a subnet of $\Phi$ if for every subset $A\subset X$ holds that $\Psi$ is eventually in $A$ w... |
I'm not sure how to deal with an exponent like this. Can I simplify it into terms that are easier to work with?
I know that $2^3 · 2^4 = 2^{3+4} = 128$, but I don't know about $2^{k + 1}$
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Let $O$ be a center of a circle, circumscribed over $\triangle ABC$. Perpendicular, drown from the point $A$ on the line $CO$, cross the line $CB$ in the point $M$. Find $|CM|$, if $|CA|=a$ and $|CB|=b$.
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Let $S$ be the set of all numbers of the form $a + b\sqrt 2$ where $a$ and $b$ are rational. Let $f : S \to R$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x$ and $y$ in $S$. Then $f(x)=f(1)x$ for all $x$ in $S$.
How to prove disprove this? And can you go really slow with explaining?
Here are some steps done in ... |
I'm reading the book "Mathematical Models in Biology" by Leah Edelstein-Keshet and in page 70 the following explanation appears. Here, F(x,y) is a function with P = F(X0 + Y0) and the idea is to calculate the value of F using a Taylor series expansion. Then, the book says that, because the serie involves partial deriva... |
Somebody told me that
if i have something like this:
$x''(t) + x'(t) = -2x(t) + u$
$x(0) = 7$
and use laplace transform on it i will get
$s^2X(s) + sX(s) = -2X(s) + U(s)$
next i'm getting Transfer function form this above
$G(s) = \frac{1}{s^2+s-2}$
The main question is how it its possible when $x(0) = 7$??
The laplac... |
I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly?
Here is one of my problems:
Find $d^{\prime}$ and locate $X_C$ approximately on drawing of
distributions.
$$\begin{array}{|l|l|l|} \hline \text... |
I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds).
I want to write the following thing:
Let $X$ be a smooth manifold. Choosing local coordinates $(q^1, \dots, q^n)$
on $X$ defines natural coordinates $(q^1, \dots, q^n, p^1, \dots, p^n)$ on $T^{*}X$: the po... |
I am trying to prove this statement, but I'm not sure where to go from here. Is don't think this is sufficiently reduced to conclude the statement is true, but I'm not positive.
$k ≥ 2$
Can I further simplify this?
|
I have a 5 star rating system that outputs a recommendation.
A user inputs a rating based on a Like or Dislike.
What I would like to know is how to convert the number of likes and dislikes into a number which can be put into a 5 star rating system?
An Example:
Say if I have:
172 Likes
101 Dislikes
How would I then get ... |
In analysis class I saw a proof but I would like see another
Proof:
$\Rightarrow$
Suppose that $a \le 0$ $\land$ $\epsilon > 0$
if $a=0$ by hypothesis $a < \epsilon$
if $a<0$ $\Rightarrow$ $a < \epsilon$
$\Leftarrow$
Suppose that $\epsilon > 0$ $\land$ $a < \epsilon$
$0<a$ $\land$ $a<\epsilon$ $\Rightarrow$ $0 < \epsil... |
Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$.
Well, I've proved really easily that it is reflexsive, symmetrical and transitive.
But I'm really struggling to find the equivalence class.
Find: Equivalence class of $\{1,2,3\}$, Given $A = \{1,2,3,4\} , B = \{1,... |
I am having some difficulties figuring out how to approach "Test stability problems".
I usually test the linearization of the system (since it is very straightforward and easy), and if that doesn't work - I construct a Lyapunov function.
After I go for the obvious functinos (2nd degree polynomials, or energy\hamiltonia... |
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