instruction stringlengths 12 30k |
|---|
Given a discrete-time LTI system as $$x_{i+1} = A x_{i} + B u_{i}$$ and suppose the feedback law $u_{i} = \mathcal{K}(x_{i})$ assymtotically stabilizes the closed-loop system. Now, consider the modified (i.e. delayed) input law $u_{i} = \tilde{\mathcal{K}}(x_{i-I})$ for some constant (but possibly unknown) $I \gg 1$ (e... |
May I affirm
$$\frac{d}{dx}\left(\int_{x_0}^{x} f(t)\cdot g(t) dt\right)=f(x)\cdot g(x)$$
if $f$ and $g$ are well definied? |
Let's take an input-output stochastic system with a given noise. First of all we should define the noise process so we will have:
$$d x_t=-\theta x_t d t+\sigma d W_t$$
where the $d W_t$ is a standard wiener process(Brownian motion).
Multiply $e^{\theta t}$ to both sides so we will have:
$$e^{\theta t}*d x_t+... |
Say $E$ is a smooth vector bundle, then is it necessarily true that $E$ has smooth sections that aren't constantly zero, namely $\Gamma^\infty(E) \neq \{0\}$ for $0$ the zero section of $E$.
My thinking is that perhaps we could look to extend local sections to global ones. In particular, if $\sigma$ is a local secti... |
In special relativity, the underlying manifold is the vector space $\mathbb R^4$ equipped with the Minkowski metric. The position of a (point) particle located at $p\in\mathbb R^4$ is the point $p$ itself, at least with respect to the standard basis. A reference frame moving with constant speed $v$ is then another basi... |
I am trying to get the average rate of growth across a series of numbers that will lead to the same sum as the discrete rates of growth.
For example, if I have 1 unit at t = 1 that grows at 2%, then 3%, then 2% again the resulting sum of units would be 1 + 1*(1.02) + 1*(1.02)*(1.03) + 1*(1.02)*(1.03)*(1.02) = 4.142... |
Ordinarily, you can get the correct Jacobian directly by taking the partial derivatives as you had done for $u,v$. This will yield a polar Jacobian matrix of
$$
J=\begin{pmatrix}
b & 0 \\
0 & 0
\end{pmatrix}
$$
This leads to your question about invertibility. Ordinarily, if the rank of the Jacobian should re... |
You can get the result from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH].
... |
I want to find a way to solve $x^9=3^x$ analytically, for two roots. one of them can be found below $$x^9=3^x\\(x^9)^{\dfrac {1}{9x}}=(3^x)^{\dfrac {1}{9x}}\\x^ { \ \frac 1x}=3^{ \ \frac 19}\\x^ { \ \frac 1x}=(3)^{ \ \frac 3{27}}\\x^ { \ \frac 1x}=(3^3)^{ \ \frac 1{27}}\\
x^ { \ \frac 1x}=(27)^{ \ \frac 1{27}}\\\to x... |
Recent linear state-space model papers like Mamba often use matrix exponential to discretize the system. They initialize the system in a continuous-time regime, and discretize it to run it like a vanilla RNN / CNN.
Consider the simplest continuous-time autonomous system $d\mathbf{h(t)}/dt = A\mathbf{h(t)}$ (assuming... |
You toss a fair coin unlimited
number of times.
How many throws on average do you need to make before the sequence НTHT appears?
The correct answer is the expectation is the sum of the inverse probabilities
of each prefix in a word, which is also a suffix
of the same word. In the case of HTHT, such suffixes-prefix... |
I want to show $$ e^{W(n+1)}-e^{W(n)} = o(1) $$ for $n\to\infty$. Since I'm not that used to the Little-o notation I'm wondering if my reasoning is correct. I tried to prove it in the following way with the mean value theorem for $n_0\in[n,n+1]$:
\begin{align}
e^{W(n+1)}-e^{W(n)} &= \frac{e^{W(n+1)}-e^{W(n)}}{(n+1)-n... |
The *fibre of an* $S$-scheme $X$ *over* $s\in S$ is defined as the fibre product $X\times_S \{s\}$ where the one point subscheme $\{s\}$ is given a local ring $k(s)$ equal to the residue field at $s$. This makes $\{s\}$ into an $S$-scheme $Spec(k(s))$.
My question is: is there any special reason for choosing this pa... |
I am trying to formulate a much bigger problem as an integer linear program, but I am stuck with one particular constraint and am not sure how to formulate it. To put it shortly, the problem deals with finding a feasible packing of packages (of different length but the same height) into a truck. Truck has three layers ... |
Consider a finite group $G$ with subgroups $N$ and $M$, where $N$ is normal in $G=MN$, $N \cap M = 1$, and $[M, N] \leq \Phi(N)$, where $[M, N]$ denotes the commutator subgroup of $M$ and $N$, and $\Phi(N)$ represents the Frattini subgroup of $N$. The question arises: is $M$ necessarily normal in $G$?
To explore thi... |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
I am trying to understand the hatcher's proof of $\pi_1(S^n)=0$ if $n\geq2$.
but it is challenging for me to understand each step.
>We can express $S^n$
as the union of two open sets $A_1$
and $A_2$ each homeomorphic to $\mathbb{R}^n$
such that $A_1 ∩ A_2$ is homeomorphic to $S^{n−1} \times\mathbb{R},$ for example... |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
I am trying to understand the hatcher's proof of $\pi_1(S^n)=0$ if $n\geq2$.
but it is challenging for me to understand each step.
>We can express $S^n$
as the union of two open sets $A_1$
and $A_2$ each homeomorphic to $\mathbb{R}^n$
such that $A_1 ∩ A_2$ is homeomorphic to $S^{n−1} \times\mathbb{R},$ for example... |
I am presently reading the course notes of Monash University's FIT-5197 (Statistical Data Modeling) for the semester "S2" of the year 2020. While reading the statement for the Central Limit Theorem here, I conjured up a (potentially inaccurate) definition for "normal distribution" in the context of real, continuous ran... |
I started three weeks ago to study wavelets with the intention of later applying them to some deep learning architectures. As a Ph.D. student in mathematics my interest lies mainly in the theoretical construction of the objects in question, I love to understand the deep reason behind things and not just understand the ... |
> **Question:** What is the closed form of this following integral? $$
\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx
$$
We have $$\displaystyle a_{k}:=\int_{0}^{\pi/2}{x \cos^k x dx}$$ then posit from the Beppo-Levi theorem we have $$\displaystyle \int_{0}^{\pi/2}{x\ln\left(1-2\cos x \right)}dx=-\sum_{k=1}^{\inf... |
I saw a very interesting task to solve and got stuck. Does anyone has some hints how to solve this? Thank you very much in advance, further I wrote what I did. There is a complex number in its algebraic form: $$z=\frac{1}{2} + i\frac{\sqrt{3}}{2}$$
----------
I rewrote it in exponential form in order to raise... |
Task on propositional logic:
"Suppose you create a truth table for A and B, both formulas in propositional calculus, and have a look at the columns below the main connectives of A and B. When do we know for sure that A ≡ B is true?"
My understanding is that this option should be correct:
"If there is a T in e... |
Why is having T in every row under two propositions is not sufficient to say that they are equivalent? |
Let $g(x)$ be:
\begin{cases}
0 \space\space \text{if}\space\space x \in \mathbb Q \\
x\space\space \text{if} \space\space x \in \mathbb R - \mathbb Q \\
\end{cases}
I know this function is continuous at $x = 0$ only, so I want to prove, using the negation of $\varepsilon - \delta$ definition, that... |
The system of ODEs
$$\dot{u} = bu - v + au(u^2 + v^2)$$
$$\dot{v} = u + bv + av(u^2 + v^2)$$
can be written in polar coordinates as
$$\dot{r} = br + ar^3$$
$$\dot{\phi} = 1$$
I know that in euclidean coordiantes I can calcualte the jacobian and evaluate it at the equilibrium (0,0) to get
$$ \begin{pm... |
Until Wiles' proof of Fermat's last theorem all proposed proofs have been erroneous. It is not known which proof Fermat himself had in mind - but it is assumed that it was erroneous, too. Have there been proofs suggested as possibly Fermat's: which were in the mathematical style of Fermat, which contained errors that F... |
The system of ODEs
$$\dot{u} = bu - v + au(u^2 + v^2)$$
$$\dot{v} = u + bv + av(u^2 + v^2)$$
can be written in polar coordinates as
$$\dot{r} = br + ar^3$$
$$\dot{\phi} = 1$$
I know that in Euclidean coordinates I can calculate the Jacobian and evaluate it at the equilibrium (0,0) to get
$$ \begin{pm... |
I am trying to understand the character table of $S_4$. I have obtained the trivial, signature and standard representations. The fourth one is the product of signature and standard.
[![enter image description here][1]][1]
Now for the last representation, I have tried two things.
1. I took the standard basis for... |
A school has 150 pupils and it is offering 2 language classes: one in Spanish & one in French. There are 80 in the Spanish class and 70 in the French class. These figures include 15 students that are taking both Spanish and French.
1. Probability that they are not taking any languages?
2. Probability they are onl... |
What's the probability of only one of the two students takes a certain class? |
I'm having trouble with this problem.
Given two vectors
$ \overrightarrow{v} = \left( \begin{matrix} v_1 \\ v_2 \end{matrix} \right) $
$ \overrightarrow{v'} = \left( \begin{matrix} v'_1 \\ v'_2 \end{matrix} \right) $
Given that T is a linear transformation
$ T:\mathbb{R}^2 \rightarrow \mathbb{R}^2 $
... |
In a Banach $*$-algebra positive elements can be defined as finite sums of the sort $\sum\limits_k b_k^*b_k$, is it true then that, for a self-adjoint $a$, the expression $\|a\|\mathbb{1} -a$ is a positive element in a Banach $*$-algebra, or is it true just for $C^*$-algebras, where arguments can be made about the spec... |
For a positive element $a$ in a unital Banach $*$-algebra, is it true that $\|a\| \mathbb{1} -a \geq 0$? |
Is there a name for the (Zariski like) topology whose closed sets are generated by degree $1$ polynomials? |
Defining the bit-length function over natural numbers as $BL(n) = \left\lceil \log_2(n+1) \right\rceil $
What are common notations for such bit-length function? It is frequently denoted as $|n|$ in cryptography, but this would collide with the absolute value function $|x| = \left\{\begin{matrix}
x \text{ if } x \ge... |
If there are $150$ pupils and $80$ take Spanish and $70$ take French, that does not automatically imply that every pupil takes one of the two language classes. You are also told that $15$ pupils are taking both. This means $80 - 15 = 65$ pupils are taking *only* Spanish and not also French; and $70 - 15 = 55$ pupils ... |
A school has 150 pupils and it is offering 2 language classes: one in Spanish & one in French. There are 80 in the Spanish class and 70 in the French class. These figures include 15 students that are taking both Spanish and French.
1. Probability that they are not taking any languages?
2. Probability they are onl... |
I am trying to prove two formulations of the Laplace-Beltrami operator $\Delta$ are equivelant:
$$\Delta f = \frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^i}(\sqrt{|g|} g^{ij} \frac{\partial f}{\partial x^j}) = g^{ij} (\frac{\partial^2 f}{\partial x^i \partial x^j} - \Gamma^k_{ij}\frac{\partial f}{\partial x^k}) $$... |
Given $f,g$ two continuous functions, let's suppose that we know $f(x)$ but we don't know $g(x,n)$. Can you solve for $g(x, n)$ from the following equation $$f(x)=\int_{-\infty}^{\infty} g(x, n) dn$$ or just simplify the integral using $g(x, n)$? I tried using laplace transform and I arrived at this: $$\mathcal L_n\{g(... |
Im trying to find values of c, when I can use Ritz method for this differential equation.$ L_cu=-u^{\prime \prime}+c u=f(x) ; \quad u^{\prime}(0)=0 ; \quad u(1)=b $ for $ u \in C^{2}([0 ; 1]) $ (this is differential formulation).Here is my try:
at first I show that $(L_cu,v)-(u,L_cv)=b(u(1)-v(1))$ and I find out that ... |
Poincaré–Miranda theorem in $\mathbb{R}^2$: $f,g:[-1,1]\times[-1,1]\to \mathbb{R}$ continuous if $f(-1,y)<0,f(1,y)>0, g(x,-1)<0$ and $g(x,1)>0$ then there exists $(x_0,y_0)$ such that $f(x_0,y_0)=g(x_0,y_0)=0$
my question:1) i am trying to modify $f(-1,y)=0, g(x,-1)=0$ instead $f(-1,y)<0, g(x,-1)<0$ and added condi... |
Poincaré–Miranda theorem in $\mathbb{R}^2$: $f,g:[-1,1]\times[-1,1]\to \mathbb{R}$ continuous if $f(-1,y)<0,f(1,y)>0, g(x,-1)<0$ and $g(x,1)>0$ then there exists $(x_0,y_0)$ such that $f(x_0,y_0)=g(x_0,y_0)=0$
my question:1) i am trying to modify $f(-1,y)=0, g(x,-1)=0$ instead $f(-1,y)<0, g(x,-1)<0$ and added condi... |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
What smooth convex functions $f \colon (0, \infty) \to (0, \infty)$ satisfy $f(1) = 0$ and $f(x) = x f\left(\frac{1}{x}\right)$? |
**Question:**
>Let $a_{i}\in\mathbb N^{+}$, and the set $A=\{a_{1},a_{2},\dots,a_{n}\}$. Show that there exists a set $B\subset A$, such that $\mathrm{card}(B)>\dfrac{n}{3}$, and that for any $x,y\in B$, then $x+y\notin B$.
This problem is from China Nankai university math competition today. How prove this? Thank... |
How to prove there exists a set $B$ such $\mathrm{card}(B)>\dfrac{n}{3}$? |
**Question**
Given a sequence $a_n$, expressed in closed form, when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition w... |
I am trying to prove two formulations of the Laplace-Beltrami operator $\Delta$ are equivelant:
$$\Delta f = \frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^i}\left(\sqrt{|g|} g^{ij} \frac{\partial f}{\partial x^j}\right) = g^{ij} \left(\frac{\partial^2 f}{\partial x^i \partial x^j} - \Gamma^k_{ij}\frac{\partial f}{\... |
>Given $A = \{1, 2, 3,\dots , 111\}$ and $B = \{1, 2, 3,\dots , 2021\}$. How many functions $f\colon A\to B$ with the following property
$$f (x + y) = f (x) + f (y)$$
for every $x, y \in A$?
All I can think of is substituting $x = y$
$$f(2x)=2f(x)$$
After that I was confused what should I do.
|
**Given:**
* A line segment in 3D space which can be represented as: \
$\begin{pmatrix}x_0 \\y_0 \\z_0 \end{pmatrix} + t\begin{pmatrix}d_{x} \\d_{y} \\d_{z} \end{pmatrix}$, where $0\le t\le1$
* A point in 3D space about which the line segment above is to be rotated (center of rotation): \
$\begin{pmatrix}x_c \\y_... |
When finding the asymptotes of function b, is minus infinity obtained? |
I was curious about how to define a linear function in a vector space whose basis is not countable. In this case, to have a concrete example, I though about the sequence space in $\mathbb{R}$, that is,
$$\mathbb{R}^\mathbb{N}=\prod^{\infty}\mathbb{R}$$
Imagine that we wanted to define a linear function $A_{i,j}$ ... |
Assuming $\rho$ is a single-qubit state. Let $\rho=\sum_{i=1}^2 p_i |i\rangle\!\langle i|$ be the eigendecomposition of $\rho$.
Obviously, a decomposition of $\rho$ as a convex combination of three pure states cannot be an eigendecomposition, as an eigendecomposition is a convex combination of *two* such states.
... |
I was curious about how to define a linear function in a vector space whose basis is not countable. In this case, to have a concrete example, I though about the sequence space in $\mathbb{R}$, that is,
$$\mathbb{R}^\mathbb{N}=\prod^{\infty}\mathbb{R}$$
Imagine that we wanted to define a linear function $A_{i,j}$ ... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
It seems to me obvious. What about the formal following proof $$E(e^{sY+tX_3})=E[E(e^{sY+tX_3}|X_3)]=E[E(e^{sY}|X_3)\times e^{tX_3}]=e^{s^2/2}\times e^{t^2/2}?$$ |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
Is there anything currently that generates rejection from the mathematical community, as happened with the complex roots of algebraic equations? |
Is there anything currently that generates rejection from the mathematical community? |
Good evening,
I was interested in the third integral from the finals of the MIT Integration Bee 2024 :
$$I = \int_{-\infty}^{\infty} \frac{1}{x^4+x^3+x^2+x+1} \hspace{0.1cm} \mathrm{d}x$$
One way to solve this is to identify that the denominator is the fifth cyclotomic polynomial, so we can write :
$$I = \d... |
Question: What is the closed form of this following integral? $
\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx.$
We have $ a_{k}:=\int_{0}^{\pi/2}{x \cos^k x dx},$ then posit from the Beppo-Levi theorem, we have $ \int_{0}^{\pi/2}{x\ln\left(1-2\cos x \right)}dx=-\sum_{k=1}^{\infty}{\frac{a_{k}}{k}}.$
$ a_{k}=\i... |
If $A$ and $B$ have a $T$ in each row, then both of them are tautologies, and any two tautologies are logically equivalent. So, both being always True is certainly a *sufficient* condition for them to be equivalent. However, it is not necessary: two statements that are False in every row are also logically equivalent. ... |
> Let $R_1,\ldots, R_n$ be (commutative) rings, and $M_1 , \ldots, M_n$ be modules. What are the minimal assumpsions about the rings and modules needed in order to define a tensor product
$M_1 \otimes_{R_1} M_2 \otimes _{R_2} \otimes\dots\otimes_{R_n} M_n$?
I am aware of an existing [similar question](https://math.... |
Let $(M, g)$ be a Riemannian manifold. The length of an admissible
curve $\gamma : [a, b] \to M$ is defined by $L(γ) = \int_a^b ∥\dot \gamma (t)∥dt$.
1)Compute the Euler–Langrange equations for the variation of the length functional.
2)I want to show that these equations are the geodesic equation if and only if the... |
Whats the smallest number $N$ where for a prime $p$ there are primes $p_a$ where $p_a = kp+a < N$ for all a $\in \{1,2 \dots p-1\}$
For example for $p = 13$, $p_1 = 53, p_2 = 41, p_3 = 3, p_4 = 17, p_5 = 5, p_6 = 19,p_7 = 7, p_8 = 47, p_9 = 61, p_{10} = 23, p_{11} = 11, p_{12} = 103$
In this case $N = 104$
I ... |
Given a matrix $B = exp(A)$, how do we express $B_{ml}$ using $A_{ij}$? If $A$ is a diagonal matrix, this is very straightforward: $B_{ii} = exp(A_{ii})$. Can we generalize it to a real square matrix $A \in \mathbb{R}^{N \times N}$?
Motivation: I am struggling to find an expression of $\frac{\partial B}{\partial A_{... |
Element-to-Element Expression of a Matrix Exponential? |
> Let $R_1,\ldots, R_n$ be (commutative) rings, and $M_1 , \ldots, M_n$ be modules. What are the minimal assumpsions about the rings and modules needed in order to define a tensor product
$M_1 \otimes_{R_1} M_2 \otimes _{R_2} \otimes\dots\otimes_{R_n} M_n$?
I am aware of an existing [similar question](https://math.... |
For an absolute Function F (meaning that the formula y=F(x) is absolute), is there a proof that $F(x) \in z$ is an absolute Formula?
This Lemma from Kunen applied for $\Phi(w,z)$ being $w \in z$ and $G_1=F$ says there should be:
[![Lemma from Kunen][1]][1]
However the proof of the lemma is not convincing so i trie... |
Recent linear state-space model papers like Mamba often use matrix exponential to discretize the system. They initialize the system in a continuous-time regime, and discretize it to run it like a vanilla RNN / CNN.
Consider the simplest continuous-time autonomous system $d\mathbf{h(t)}/dt = A\mathbf{h(t)}$ (assuming... |
On a recent assignment, I had a question where I had to prove a certain statement to be true for all $n\in\mathbb{Z}$. The format of my proof looked like this:
- Statement is true when $n=0$
- "Assume statement is true for some $k\in\mathbb{Z}$"
- Statement must be true for $k+1$
- Statement must be true for ... |
Is it possible to perform induction on the integers? |
For a self-adjoint element $a$ in a unital Banach $*$-algebra, is it true that $\|a\| \mathbb{1} -a \geq 0$? |
Let $(G, \star)$ and $(H, \blacksquare)$ be groups, and let $\varphi:G\rightarrow H$ be a group homomorphism. For any $h\in H$, we define $\varphi ^{-1}(h) = \{g\in G|\varphi (g)=h\}$.
How can I prove that for $h_1,h_2\in H$, $|\varphi ^{-1}(h_1)|=|\varphi ^{-1}(h_2)|$ ? That is, the number of elements of $G$ that $... |
Let $(G, \star)$ and $(H, \blacksquare)$ be groups, and let $\varphi:G\rightarrow H$ be a group homomorphism. Let's also assume $\varphi$ is surjective. For any $h\in H$, we define $\varphi ^{-1}(h) = \{g\in G|\varphi (g)=h\}$.
How can I prove that for $h_1,h_2\in H$, $|\varphi ^{-1}(h_1)|=|\varphi ^{-1}(h_2)|$ ? Th... |
# Setup
Assume that the random variable sequence $\{Y_n\}_{n\in\mathbb{N}}$ tends in distribution to $Y$, the random variable sequence $\{Z_n\}_{n\in\mathbb{N}}$ tends in distribution to $Z$, and $Y$ and $Z$ are independent. Also, the following limits hold.
$$
P(Y_n > y, Z_n >z) \to P(Y>y)\cdot P(Z>z).
$$
# Ques... |
Consider the following pair of *a priori* independent problems...
> A manufacturer's revenue is $\$100s^{\frac{1}{3}}h^{\frac{2}{3}}$, where $s$ is the number of tons of steel they purchase and $h$ is the number of hours of labor they hire. Steel costs $\$2000$ per ton, labor costs $\$20$ per hour, and their budget... |
How does this pair of Multivariable Calculus problems relate to the concept of duality? |
How about $$\mathbb P\left(Y_n \le y, Z_n \le \right) = \mathbb P\left(Y_n \le y\right)\mathbb P\left(Z_n \le \right) \to ?$$
Can you finish the proof? |
I need to find the first 3 terms of the Taylor expansion around $z=0$ of the complex function:
$$f(z)=\frac{\log(1+z^2)}{(1-z)\sinh z}$$
I tried the following way (obviously the radius of convergence is $R=1$ so we can consider $|z|<1$):
$$\log(1+z^2) \sim z^2$$
$$\frac{1}{1-z} \sim 1+z+z^2+z^3$$
$$\sinh... |
How about
\begin{align}
\mathbb P\left(Y_n \le y, Z_n \le z\right) &= 1 - \mathbb P\left(Y_n > y \cup Z_n > z\right)\\
&= 1 - \mathbb P\left(Y_n > y\right) - \mathbb P\left(Z_n > z\right) + \mathbb P\left(Y_n > y, Z_n > z\right) \to ?
\end{align}
Can you finish the proof? |
Does there exist a function $f$ holomorphic in $|z|<1$ that satisfies $f\left(\displaystyle\frac{1}{n}\right)=\displaystyle\frac{n}{n+2}$? I have been trying to create a proof using the uniqueness/identity theorem but only obtain $f(z)=\displaystyle\frac{1}{1+2z}$ which appears to not be holomorphic in $|z|<1$ because ... |
Can a holomorphic function satisfy f(1/n)=n/(n+2)? |
How about
\begin{align}
\mathbb P\left(Y_n \le y, Z_n \le z\right) &= 1 - \mathbb P\left(Y_n > y \cup Z_n > z\right)\\
&= 1 - \mathbb P\left(Y_n > y\right) - \mathbb P\left(Z_n > z\right) + \mathbb P\left(Y_n > y, Z_n > z\right)\\
&= \mathbb P\left(Y_n \le y\right) + \mathbb P \left(Z_n \le z\right) - 1 + \mathbb... |
Given sequence:
$$
\begin{cases}
x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 1\\
x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2
\end{cases}
$$
Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$
$A$ is actually the area bounded by the zig-zag loop between $y=\frac{1}{2\cos(\frac{\pi}{m})-x}$ and $y=x$. I tr... |
**Given:**
* A line segment in 3D space which can be represented as: \
$\begin{pmatrix}x_0 \\y_0 \\z_0 \end{pmatrix} + t\begin{pmatrix}d_{x} \\d_{y} \\d_{z} \end{pmatrix}$, where $0\le t\le1$
* A point in 3D space about which the line segment above is to be rotated (center of rotation): \
$\begin{pmatrix}x_c \\y_... |
**Question**
Given a sequence $a_n$ when it is possible to express such series as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition with the use of the exponent... |
$\log(1+z^2)=\operatorname{ln}(1+z^2)$, I assume? If so, the first three terms of its Taylor expansion about $z=0$ are $z^2-\frac{z^4}{2}+\frac{z^6}{3}$. The first three terms of $\frac{1}{1-z}$ are $1+z+z^2$ and the first three terms of $\frac{1}{\operatorname{sinh}z}$ are $\frac{1}{z}-\frac{z}{6}+\frac{7z^3}{360}$
... |
I'm trying to show that
$$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$
for an absolute constant $C > 0$, where $\delta \in (0,1]$ and $\mu > 0$.
Using the idea in [this answer][1], I wrote
$$\log(1+\delta) = 0 + \... |
If we have a $2$-state model (i.e. the simplest non-trivial example) in a hidden markov model, and some generated observation-data $\mathcal{O}$ from the algorithm for generating observations. Is it then always the case that argument $\theta_i$ of a given family $\Theta = \{\theta_i\}_{i \in I}$ of parameters (where $... |
I have a matrix of the form $$\mathbb{A}_{i,j}(\omega) = \sum_n \frac{\sin(nx_i)\sin(nx_j)}{-\omega^2+C(n)}+\sum_{r_x}\sum_{r_y}\frac{\sin(r_xx_i)\sin(r_xx_j)\sin(r_y\alpha)}{-\omega^2+D(r_x,r_y)}.$$ In truth, the matrix formulation is much more complex that this, but I have tried to make things simpler by introducing ... |
**Question**
Given a sequence $a_n$ when it is possible to express such sequence as a linear recurrence relation?
I'm looking for a characterization for such sequences, but also some necessary condition will be fine.
___
**My work**
I managed to prove some sufficient condition with the use of the expone... |
> Question: What is the closed form of this following integral? $
\int_{0}^{\frac{\pi}{2}} x \log(1-\cos x) \,dx.$
We have $ a_{k}:=\int_{0}^{\pi/2}{x \cos^k x dx},$ then posit from the Beppo-Levi theorem, we have $ \int_{0}^{\pi/2}{x\ln\left(1-2\cos x \right)}dx=-\sum_{k=1}^{\infty}{\frac{a_{k}}{k}}.$
$ a_{k}=... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/898fd.jpg
Do I use binomial distribution for part b)? |
I was trying to make an IBP problem for an integration bee.
I came up with the simple arctan integral with the square roots just for an easy round.
However, WolframAlpha challenged me a suggestion, which was adding the bounds -1 to 1.
I was so confused because I thought it was undefined, divergent, or complex. But i... |
Circular metal tokens are used to operate a washing machine in a laundromat. The diameters of the tokens are known to be normally distributed. Only tokens with diameters between 1.94 cm and 2.06 cm will operate the machine.
Find the probability that less than two tokens out of 20 will not operate the machine.
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.