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> Evaluate:- $$\lim_{x\to\infty}\left[ x\left(1+\frac1x\right)^x-kx^2\ln\left(1+\frac1x\right)\right]$$
I tried calculating the limits of the two terms separately. By applying L'Hopital, the 2nd limit can be solved by writing $x^2\ln\left(1+\frac1x\right)=\frac{\ln\left(1+\frac1x\right)}{\frac{1}{x^2}}$. Now, L'Hopi... |
From $h(t):= f(x_0+t)-f(x_0^{+})$, since $f(x_0^{+})$ is constant, it suffices to show that $f(x_0 +t)$ is Riemann integrable on $[0, \pi]$. We will use answer of https://math.stackexchange.com/questions/3029512/integrability-of-composite-functions?noredirect=1&lq=1 :
> Lemma. If $f$ is a Riemann integrable function... |
**Prove that for every simple graph G with maximum degree at most 3 and at least 4 vertices, $k'(G) = k(G)$.**
I am still relatively new to the definitions of $k(G)$ and $k'(G)$. So far my understanding is that $k(G)$ is the number of vertices, $|s|$, removed from $G$ so that $G - s$ is disconnected, and $k'(G)$ is ... |
*PIE = inclusion-exclusion
> How many ways are there to distribute 10 balls into 5 distinct boxes such that no two adjacent boxes are empty?
Note: the same [question statement.](https://math.stackexchange.com/questions/2471469/how-many-ways-are-there-to-distribute-10-balls-into-5-distinct-boxes-that-no-two)
I ... |
In a group of children, each child has a certain number of pencils from 1 to n. If the number of children who have i or more pencils is $2^{n-i}$, for i=1,2,3....,n and the total number of pencils is 511, find the maximum number of pencils with any child.
----------
**My attempt:-**
Number of children with ... |
I have a set of residueclasses mod $m$, e.g. with $m=11$ one could have the set
$M=\left\{[2]_{11}, [5]_{11}, [6]_{11}, [10]_{11}\right\}$.
Now I define a ratio $\operatorname{d} :=\frac{\#M}{m} = \frac{4}{11}$.
So in this set I find all the number $2,5,6,10,13,16,17,21,24,27,28,32\dots$ and infinitely more.
Now... |
The surfaces of a gaming dice are labeled with 1, 2, 3, 4, 5, 6. How many different gaming dice exist, if each face of the dice is marked with a different number from 1 to 6, and no two faces can have the same number (consider rotations)?
I come across one forum with this explanation: "When the symbols are unique an... |
Im new to topology, and am trying to prove what conditions results in the union of a closed set with an open set is closed.
Also, just want to check that the union of an open set with it's complement open, and maybe why it is? |
Let $P(x)$ be a polynomial with integer coefficients, and $p$ be a large prime. I want to find the number of solutions to the congruence equation $$P(x)\equiv 0(\mathrm{mod}\ p^\alpha).$$
In my practice the degree of $P(x)$ is $4$. We can write
$$ P(x) = \sum_{j=0}^4 a_j x^j.$$
I wonder that whether the number of ... |
For simplicity lets consider finite probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and two random variables $\xi, \eta$ on it. I wonder is there any connection between statistical and functional dependence of random variables? For example, if I have $\xi = f\circ \eta$ for some $f$, does it imply that $\xi$ and ... |
I am looking for references in discrete differential geometry for a concept I've been interested in.
It is very common to approximate smooth surfaces using discrete triangulations. I am interested in the opposite problem. I start off with some finite collection of triangles, and I want to know which (closed) discret... |
What are the possible surfaces that one can construct from a finite set ot triangels? |
Each vertex of a unit sphere is conveniently also its own tangent. I am starting with a unit sphere and then multiplying each vertex by a scale vector to create an ellipsoid. I would like to know the tangent vector at each vertex. Is there some simple math to get the tangent vector given the unit sphere vector and the ... |
Find the tangent vector or an ellipsoid given the scale vector? |
HINT: being surjective means that $\dim(\operatorname{null}T)=2$ as $\dim(\operatorname{img}T)=\dim V_2=5$. Now, as $T$ is surjective, the map $\tilde T:V_1/\operatorname{null}T \to V_2$ given by $\tilde T(\operatorname{v+null}T)=Tv$ is an isomorphism.
Thus $H:=\tilde T^{-1}(W)$ have dimension $2$ and observe that $... |
How to solve the equation $3a^4=8a^3-16$?, where a is a positive number
I have tried using the quadric formula, but got to nothing useful.
I know the solution is 2, but I don't know how to solve it. This is a pat of a problem and I need it to solve the whole problem.
Hope one of you can help me! Thank you! |
How to solve the equation $3a^4=8a^3-16$? |
Each vertex of a unit sphere is conveniently also its own normal. I am starting with a unit sphere and then multiplying each vertex by a scale vector to create an ellipsoid. I would like to know the normal vector at each vertex. Is there some simple math to get the normal vector given the unit sphere vector and the sca... |
Find the normal vector or an ellipsoid given the scale vector? |
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
\Delta(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
f_{yx} & f_{yy} & f_{yz}\\
f_{zx} & f_{zy} & f_{zz}\\
\end{bmatrix},
$$
which is real symmetric. Ther... |
I am currently engaging with the following hypergeometric function as a result of attempting to find [a solution for this probability problem for $n$ number of dice][1]:
$$_2F_1\left (\frac{n+k}{2}, \frac{n+k+1}{2};k+1;\frac{4b(d-e+1)}{d^2}\right ); \{n,k,b,d,e\}\in \mathbb{N}^+, z\in \mathbb{R},b < e \leq d$$
I'... |
Determine the minimum and maximum values of the function $\frac{x^4}{(x-1)(x-3)^3}$.
By equating the first derivative to $0$,its fairly simple to deduce that the extreme points are $x=0,\frac{6}{5}$. However,to determine minimum or maxumum,we need to analyse the 2nd derivative as well which seems to be awfully long. I... |
Let $N$ be a group. I read somewhere that
$$ Z(N) \subset [N,N] \iff N \text{ cannot be written as a direct product of groups } N = A \times B \text{ where }A \text{ is non-trivial abelian.}$$
One implication is clear to me: $\implies$. I prove it by contraposition: assume $N$ can be decomposed as $A\times B$ where... |
I am currently engaging with the following hypergeometric function as a result of attempting to find [a solution for this probability problem for $n$ number of dice][1]:
$$_2F_1\left (\frac{n+k}{2}, \frac{n+k+1}{2};k+1;\frac{4b(d-e+1)}{d^2}\right ); \{n,k,b,d,e\}\in \mathbb{N}^+, z\in \mathbb{R},b < e \leq d$$
I'... |
Consider the unitary group $U(2^n)$ and let $G$ be the subgroup isomorphic to $U(2^{n-1})$ embedded as:
\begin{pmatrix}
U & 0\\
0 & U
\end{pmatrix}
where $U \in U(2^{n-1})$ (the same matrix in both blocks). How to describe the homogeneous space $U(2^n) / G$? Do we get a Stiefel manifold as in the case of embedding... |
Why do we have different set theories like Zermelo Fraenkel Set Theory , Von Neumann Bernays Godel Set Theory and Morse Kelly set theory?
|
In the question <https://math.stackexchange.com/questions/2738732/multivariate-mean-value-theorem-reference> was written the following statement for $x,y\in \mathbb{R}^{n}$
\begin{equation}
||f(x) - f(y)||_q \leq \sup_{z\in[x,y]}||f'(z)||_{(q,p)}||x-y||_p,
\end{equation}
where $z∈[x,y]$ denotes a vector $z$ cont... |
How to solve the equation $3a^4=8a^3-16$?, where a is a positive number
I have tried using the quadric formula, but got to nothing useful.
I know the solution is $2$, but I don't know how to solve it. This is a part of a problem and I need it to solve the whole problem.
Hope one of you can help me! Thank you! |
Let $\mathcal{B}(x, r) = \{x \in \mathbb{R}^n | \|x\| \leq r, r \geq 0\}$ be a ball associated with an arbitrary norm $\|\cdot\|$.
Is it possible to express any ball as a polyhedron
$$\mathcal{P} = \{x | Ax \leq b, Bx = g\}?$$
I cannot see an obvious way when I substituting in the definition of the norm, so ... |
Is it possible to express any norm balls into a polyhedron $\{x | Ax \leq b, Bx = g\}$? |
*PIE = inclusion-exclusion
> How many ways are there to distribute 10 balls into 5 distinct boxes such that no two adjacent boxes are empty?
Note: the same [question statement.](https://math.stackexchange.com/questions/2471469/how-many-ways-are-there-to-distribute-10-balls-into-5-distinct-boxes-that-no-two)
I ... |
**Under what conditions does the expectation of this random variable $\log(1-|XY|) $exist(where $X,Y$ are iid,$|X|,|Y|<1$)?**
I wish to prove the following expectation inequality
$$E[\log(1+XY)]\geq E[\log(1-XY)]$$
, where $X,Y:\Omega \to (-1,1)$ are independently and identically distributed random variables. Ho... |
Subset Product- Given $N$ and a list of positive divisors $S$, decide if there is a product combination equal to $N$
- We will sort $S$ in numerical order from smallest to largest in order
to find out the maximum combination size to avoid redundancy.
- We will remove non-divisors in $S$. This does not impact ... |
Subset Product has a pseudo-polynomial algorithm? |
HINT: being surjective means that $\dim(\operatorname{null}T)=2$ as $\dim(\operatorname{img}T)=\dim V_2=5$. Now, as $T$ is surjective, the map $\tilde T:V_1/\operatorname{null}T \to V_2$ given by $\tilde T(\operatorname{v+null}T)=Tv$ is an isomorphism.
Thus $H:=\tilde T^{-1}(W)$ have dimension $2$ and observe that $... |
**Context**
With some effort we can show that Ramanujan's fast series implies:
\begin{align}
\frac{8E(k)K(k)}{\pi^2}-\frac{aK^2(k)}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)$ and $k=(13\sqrt{... |
Subset Product- Given $N$ and a list of positive divisors $S$, decide if there is a product combination equal to $N$
- We will sort $S$ in numerical order from smallest to largest in order
to find out the maximum combination size to avoid redundancy.
- We will remove non-divisors in $S$. This does not impact ... |
I am reading Bollobás' Linear Analysis. Chapter 8. Theorem 6., as the title says:
> $B(X^{**})$ is the $w^*$-closure of $B(X)$ in $X^{**}$
The proof starts by saying that i) $B(X^{**}$) is $w^*$-closed, and ii) $B \subset B(X^{**})$ where $B$ is the $w^*$-closure of $B(X)$ in $X^{**}$. Then, suppose that there is... |
For some time now, I have been able to use the Conditional Proof rule in Natural Deduction. We assume A, derive B within the scope of the assumption and discharge the assumption to get that A implies B. This is, I believe, the syntactic approach to the proof, pure rule-based. However, what is the motivation behind this... |
A composite $n=2^sd+1$ is a strong pseudoprime to base a if n satisfies either $a^d\equiv 1 \pmod n$ or $a^{2^{r}d}\equiv -1 \pmod n$ for some $0\le r\lt s$.
It can be shown that\
1.By considering $m=\frac{a^{2p}-1}{a^{2}-1}$, where $p$ is an odd prime not dividing $a^{2}-1$, there are infinitely many Fermat pseudo... |
Number of elements in set $S = \{(x,y,z): x,y,z \in \mathbb{Z}, x+2y+3z=42, x,y,z \ge 0\}$ is?
My solution:
This is equal to the coefficient of $t^{42}$ in
$(1+t+t^2+t^3+...+t^{43})(1+t^2+t^4+...+t^{42})(1+t^3+t^6+...+t^{42})$
= $\frac{(1-t^{43})}{1-t} \times \frac{(1-t^{44})}{1-t^2} \times \frac{(1-t^{... |
Let $F(n,p)=\sum_{m=p}^{\infty} \frac{(n+m)!}{n!\ m!\ 2^m}$. What is the upper bound for $F(n,p)$, where $p\geq n+1$ and $n\geq 0$? Does there exist a tight bound where the value of $p$ could be easily estimated under a constraint $F(n,p)\leq 2^{-k}$, where $k>1$?
Note that $F(n,p)=\frac{(n+p)!}{n!\ p!}\left(\frac{1... |
is this form useful for to detecting prime numbers, or can it be discarded?
f(x)=|sin((π*x)/P_1 )|+|sin((π*x)/P_2 )|+⋯+|sin((π*x)/P_n )|
Here is an example:
P_1=2; P_2=3
f(x)=|sin((π*x)/2 )|+|sin((π*x)/3)|
[enter image description here][1]
The result is two new prime numbers, 5 and 7. for
f... |
Is this formula for calculating prime numbers suitable for n^2 digits? f(x)=|sin((π*x)/P_1 )|+|sin((π*x)/P_2 )|+⋯+|sin((π*x)/P_n )| |
While solving the Brachistochrone problem, I setup a variable $T$,
$$T=\int\dfrac{\sqrt{1+(y')^2}}{\sqrt{2gy}}$$
Then employing the Euler-Lagrange equations (Beltrami's identity):
$$\dfrac{dy}{dx}=\sqrt{\dfrac{1}{c^2y}-1}$$
Whose parametric solution is
$$x=\dfrac{1}{2c^2}[\theta-\sin(\theta)]+k$$
$$y=\dfrac{1}{2... |
In a group of children, each child has a certain number of pencils from $1$ to $n$. If the number of children who have i or more pencils is $2^{n-i}$, for $i=1,2,3, \ldots,n$ and the total number of pencils is $511$, find the maximum number of pencils with any child.
----------
**My attempt:-**
Number of ch... |
Is this quadratic form positive definit? |
I am reading some notes on representation theory, and I have found myself confused by the following proposition:
*Suppose $G$ is a finite abelian group, then every complex representation $V$ of $G$ has a unique isotypical decomposition*
I have also read that this is true in the case of $G$ non-abelian, and if the... |
When is the isotypical decomposition of a representation not unique? |
Define the set
$$A=\{\langle x,y\rangle : x\in [-1,1]^d, \: y\in \mathbb{R}^d, \:||y||_1\leq c\}$$
for some $c\in \mathbb{R}$. I want to show that is is convex. Take $\langle x,y\rangle, \langle x',y'\rangle\in A$. I want to show that if $\lambda \in [0,1]$ then
$$\lambda\langle x,y\rangle+(1-\lambda) \langle x',y'\... |
I recently encountered the following riddle:
Let a fair coin be flipped $N$ times. Player A gets a point every time there are two heads in a row, and player B gets a point every time a tail is followed by a head, the player with more points wins. Thus, for example, for the sequence `THHH`, player A wins with 2 vs. 1... |
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$.
I was wondering if this information is enough to conclude that $$\sup_{n\in\mathbb N}\|f_n\|_H <+\infty.$$
I'd say that the answer is yes because being bounded in $H$ means that there exists $R>0$ such that $\|f_n\|_H \le R$. ... |
Let $M$ be an oriented $n$-dimensional manifold and $\omega\in \Omega^n(M)$ with compact support, $U\subseteq M$ open (if necessary you can assume that $U$ is the domain of a chart) and
$$\operatorname{supp}(\omega)\subseteq U.$$
Since $U$ can be viewed as another oriented $n$-dimensional manifold and $\iota^*\omega\... |
$\int_M\omega=\int_U\iota^*\omega$ if $\operatorname{supp}(\omega)\subseteq U\subseteq M$? |
For optimizing the calculations instead of the
$$
D(n)=(2[n/5]∗D([n/5])∗D(n \mod5)) \mod10
$$
we can calculate
$$
D(n)=(LastDigitOf(2⌊n/5⌋)∗D(⌊n/5⌋)∗D(n \mod5)) \mod10
$$
Modified formula is:
$$
D(0) = 1 \\
D(1) = 1 \\
D(2) = 2 \\
D(3) = 6 \\
D(4) = 4 \\
D(5) = 2 \\
D(6) = 2 \\
D(7) = 4 \\
D... |
I'm trying to search for as many examples of non-IBN free modules as possible. But so far I can only find two examples. One is $End(V)$, where $V$ is a vector space of *countable* dimensions. Another is a similar one, mentioned in this [question][1]. They both involve an underlying vector space/module of *infinite* dim... |
$\lim_{x\to0+}(\lfloor x \rfloor + \lfloor1-x\rfloor)$
$\lim_{x\to0-}(\lfloor x \rfloor + \lfloor1-x\rfloor)$
I tried to solve by graphing $(\lfloor x \rfloor + \lfloor1-x\rfloor)$
Graph of the function $(\lfloor x \rfloor + \lfloor1-x\rfloor)$ is straight line $y=0$
[https://www.desmos.com/calculator][1]
... |
What are the possible surfaces that one can construct from a finite set ot triangles? |
The surfaces of a gaming dice are labeled with 1, 2, 3, 4, 5, 6. How many different gaming dice exist, if each face of the dice is marked with a different number from 1 to 6, and no two faces can have the same number (consider rotations)?
Is this explanation correct? Let's just take the first symbol. It must be some... |
Let $M$ be an oriented $n$-dimensional manifold and $\omega\in \Omega^n(M)$ with compact support, $U\subseteq M$ open (if necessary you can assume that $U$ is the domain of a chart) and
$$\operatorname{supp}(\omega)\subseteq U.$$
Since $U$ can be viewed as another oriented $n$-dimensional manifold and $\iota^*\omega\... |
For optimizing the calculations instead of the
$$
D(n)=(2[n/5]∗D([n/5])∗D(n \mod5)) \mod10
$$
we can calculate
$$
D(n)=(LastDigitOf(2⌊n/5⌋)∗D(⌊n/5⌋)∗D(n \mod5)) \mod10
$$
Modified formula is:
$$
D(0) = 1 \\
D(1) = 1 \\
D(2) = 2 \\
D(3) = 6 \\
D(4) = 4 \\
D(5) = 2 \\
D(6) = 2 \\
D(7) = 4 \\
D... |
In fact, LP have mostly used different tools from mathematics; for example, see [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
However, LP and its well-known extensions such as convex optimization and integer programming, are also used in different branc... |
Let $M_k \in \mathbb{R}^{n\times n}$ and $k \in \mathbb{N}$. Suppose that $M_kx_n$ converges to $M_k x$ in $L^2([0,1], \mathbb{R}^n)$ for $n \to \infty$ and $M_k x_n \in H_0^k([0,1], \mathbb{R}^n)$, $k \in \mathbb{N}$. The space $H_0^k([0,1], \mathbb{R}^n)$ denotes the Sobolev space $W_0^{k,2}([0,1], \mathbb{R}^n)$ end... |
In PMA (baby Rudin), Rudin states that:
> Suppose $$$$ (a) the partial sums $A_n$ of $\Sigma a_n$ form a bounded sequence; $$$$ (b) $b_0 \geq b_1 \geq b_2 \geq \dots$ ;$$$$ (c) $\lim_{n\to\infty}{bn} = 0$. $$$$ Then $\Sigma a_n b_n$ converges.
In proof, he says that:
>Choose M such that $\lvert A_n \rvert \leq... |
Theorem 3.42 Principles of Mathematical Analysis (baby Rudin), How am I wrong? |
Suppose that $Mx_n$ converges to $M x$ in $L^2([0,1], \mathbb{R}^m)$ for $n \to \infty$ and $M x_n \in H_0^k([0,1], \mathbb{R}^m)$, for $M \in \mathbb{R}^{m\times m}$ and a $k \in \mathbb{N}$. The space $H_0^k([0,1], \mathbb{R}^m)$ denotes the Sobolev space $W_0^{k,2}([0,1], \mathbb{R}^m)$ endowed with the standard Sob... |
**Background**
In the <a href="https://math.stackexchange.com/questions/170747/sum-equals-integral-identities-similar-to-int-01-fracdxxx-sum-n%20">following question</a> and references therein, a number of "Sum equals integral" identities are described.
For instance, we have $$\int_{-\infty}^{+\infty} {\rm sinc}... |
I recently encountered the following riddle:
Let a fair coin be flipped $N$ times. Player A gets a point every time there are two heads in a row, and player B gets a point every time a tail is followed by a head, the player with more points wins. Thus, for example, for the sequence `THHH`, player A wins with 2 vs. 1... |
I recently encountered the following riddle:
Let a fair coin be flipped $N > 2$ times. Player A gets a point every time there are two heads in a row, and player B gets a point every time a tail is followed by a head, the player with more points wins. Thus, for example, for the sequence `THHH`, player A wins with 2 v... |
Finding an antiderivative is a non-starter, but there are a lot of levers to work with in an integral like this.
First idea: a series. $\frac1{e^x+1}=e^{-x}-e^{-2x}+e^{-3x}-e^{-4x}+\cdots$ for $x>0$. We have to be careful with the behavior near zero, and then we need $\int_0^{\infty} e^{-nx}\ln x\,dx$, which again r... |
I recently encountered the following riddle:
Let a fair coin be flipped $N > 2$ times. Player A gets a point every time there are two heads in a row, and player B gets a point every time a tail is followed by a head, the player with more points wins. Thus, for example, for the sequence `THHH`, player A wins with 2 v... |
Given $(X, \mathcal{A}, \mu)$ be a finite measure space and $f_n \in L^p(X, \mu)$ where $f_n(x) \rightarrow f(x)$ almost everywhere as $n \rightarrow \infty$ and $1 \leq p < \infty$.
Now suppose $||f_n||_q \leq M < \infty \ \forall n \ $and $M \in \mathbb{R}^+$ is a constant, plus $q > p $.
Then show that $||f... |
**Background**
In the <a href="https://math.stackexchange.com/questions/170747/sum-equals-integral-identities-similar-to-int-01-fracdxxx-sum-n%20">following question</a> and references therein, a number of "Sum equals integral" identities are described.
For instance, we have $$\int_{-\infty}^{+\infty} {\rm sinc}... |
Given $(X, \mathcal{A}, \mu)$ be a finite measure space and $f_n \in L^p(X, \mu)$ where $f_n(x) \rightarrow f(x)$ almost everywhere as $n \rightarrow \infty$ and $1 \leq p < \infty$.
Now suppose $||f_n||_q \leq M < \infty \ \forall n \ $and $M \in \mathbb{R}^+$ is a constant, plus $q > p $.
Then show that $||f... |
In my current work on stochastic dynamics, I have stumbled upon the following problem, which I will present in a simplified form. I have two (or more) different Fokker-Planck equations, all of which I can solve individually. The system should switch stochastically (in simple cases with a unrelated Poisson-process) betw... |
In my current work on stochastic dynamics, I have stumbled upon the following problem, which I will present in a simplified form. I have two (or more) different Fokker-Planck equations, all of which I can solve individually. The system should switch stochastically (in simple cases with a unrelated Poisson-process) betw... |
Given two vectors $\mathbf{a}(t) = (r(t), s(t), -v(t), -u(t))$ and $\mathbf{b}(t) = (p(t), q(t), w(t), z(t))$, along with accompanying conditions:
1. $r(t) + s(t) - v(t) - u(t) \leq 0$
2. $r(t)p'(t) + s(t)q'(t) -v(t)w'(t) - u(t)z'(t) \leq 0$ (Dot product of $\mathbf{a}(t)$ and derivative of $\mathbf{b}(t)$)
... |
I'm given variance `V`, expectation `E` and number `n` - amount of values random variable takes. Is there any general algorithm to construct discrete random variable with such variance and expectation? (probabilies of each outcome is a rational number)
Code implementation is allowed too :) |
as an exercise I am trying to answer questions from previous year's exams on Artificial Intelligence but I am stuck with the following one, for which I thought FOL would be a good candidate, and I answered 1. and 2. with it already, but I can't figure out how to answer point 3.
Any suggestions on how to do that in F... |
A basic question, but it's a little bit confusing for me. Given languages $L \subset L'$, an $L'$ formula $\phi$ and an $L$-structure $M$ and its $L'$-reduct $M'$ how do I define satisfaction in a case, where I don't have an interpretation of a given symbol contained in a formula, e.g.
symbol $0$ is in $L'$ but not ... |
For the following equation, I don't understand how the terms $df(X_t)$ and $f'(X_t)dX_t$ differ.
$$df(X_t) = f'(X_t)dX_t + 1/2 f''(X_t)(dX_t)^2$$
As I understand, both terms refer to the change in the function $f$ with respect to $X_t$. Where am I going wrong?
Thanks in advance. |
I am referring to the ISS definition by Sontag of
${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$
I understand that 0-GAS is a necessary condition for ISS. But is GAS for all constant u a sufficient condition?
*Update:*
For clarification, I mean the system is GAS for an arbitrary... |
Let $X$ be a continuous random variable taking non-negative values only. To prove: $\mathbb{E}(X)=\int_0^\infty (1-F_X(x)) dx$ where $F_X$ is the distribution function of $X$.
I have derived the following expression: $\mathbb{E}(X)=\lim_{t\to\infty} \int_0^t (1-\frac{1}{F_X(t)} F_X(x))dx$.
My question is: can I repla... |
**Context**
With some effort we can show that Ramanujan's fast series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)$ a... |
Let
\begin{align}
A &= \{(x,y) \in \mathbb R^2 : - 1 \leq x \leq 1 \land -1 \leq y \leq 1\}, \\
B &= \{(x,y) \in \mathbb R^2 : x^2 + y^2 = 1\}, \text{ and} \\
C &= \{(x,y) \in A : |x| = 1 \lor |y| = 1\}.
\end{align}
How can i find the Hausdorff distance $h(B,C)$? I'm completely confused with the definition... |
**Context**
With some effort we can show that Ramanujan's fast series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)$ a... |
I'm trying to evaluate an expression in sympy using a function $f(x)$. The function is unknown, but its value at $0$ is known $f(0)=0$. If it's any help, specifically I want to evaluate the Taylor series of f(x)'s inverse using Lagrange's Inversion Theorem.
This means I want to evaluate $$\frac{d^n}{dx^n} \Big(\frac... |
Recently, I was exposed to SDEs, though I still did not fully understand the concepts. I want to solve a nonlinear undamped oscillator
$$\frac{d^2x}{dt^2}=-\alpha^2 g(x) + f(t)$$
where $g(x)= \sin(x), \cos(x), \exp(ix), \dots$ and $f(t)$ is the force due to diffusion, where $\langle f(t) \rangle = 0$ and $\langle... |
Define the set
$$A=\{\langle x,y\rangle : x\in [-1,1]^d, \: y\in \mathbb{R}^d, \:\|y\|_1\leq c\}$$
for some $c\in \mathbb{R}$. I want to show that is is convex. Take $\langle x,y\rangle, \langle x',y'\rangle\in A$. I want to show that if $\lambda \in [0,1]$ then
$$\lambda\langle x,y\rangle+(1-\lambda) \langle x',y'\... |
So let (X,d) be a metric space and A $\subset$ X. We have a metric space (A,$D_A$).
What does the following notation mean ? $cl_A(B)$
I think it means the closure of B with respect to the set A, i.e its the closure of B in the metric space (X,d) intersection with the set A. |
What does the following notion mean in terms of metric spaces? |
I'm trying to evaluate an expression in sympy using a function $f(x)$. The function is unknown, but its value at $0$ is known $f(0)=0$. If it's any help, specifically I want to evaluate the Taylor series of f(x)'s inverse using Lagrange's Inversion Theorem.
This means I want to evaluate $$\frac{d^n}{dx^n} \Big(\frac... |
Let $N$ be a group. **Assume that $N$ is torsion-free, finitely generated and nilpotent.** I read somewhere that
$$ Z(N) \subset [N,N] \iff N \text{ cannot be written as a direct product of groups } N = A \times B \text{ where }A \text{ is non-trivial abelian.}$$
One implication is clear to me: $\implies$. I prove i... |
Let's consider a tree $T$. $P(u, v)$ represents a path between $u$ and $v$ (set of edges).
My task is to find $a$, $b$ and $c$ such that the set:
$$\left\{ e \in E: e \in P(a, b) \cup P(a,c) \cup P(b,c) \right\}$$
is maximal.
**My hypotheses:** The set is maximal when the paths created by these vertices re... |
I have a function:
$$f(A)=cond(AB)=\left \|{AB}\right \|_2 \left \| {(AB)^{-1}} \right \|_2 $$
We know that:
$$\left \| X \right \|_2= \sqrt{\lambda_{max}(X^TX)}$$
So:
$$f(A)=\sqrt{\dfrac {\lambda_{max}((AB)^T(AB))} {\lambda_{min}((AB)^T(AB))}}$$
How should I take the derivation of $f(A)$ with respect to matri... |
$\lim_{x\to0+}(\lfloor x \rfloor + \lfloor1-x\rfloor)$
$\lim_{x\to0-}(\lfloor x \rfloor + \lfloor1-x\rfloor)$
I tried to solve by graphing $(\lfloor x \rfloor + \lfloor1-x\rfloor)$
Graph of the function $(\lfloor x \rfloor + \lfloor1-x\rfloor)$ is straight line $y=0$:
[https://www.desmos.com/calculator/la... |
Now I want to figure out the following problem: Suppose that $f\in L^1(-\pi,\pi)$. Define $F(x)$ as
\begin{align}
F(x) = \int_{-\pi}^x (f(t)-c_0(f)) dt\,,~x\in[-\pi,\pi]
\end{align}
where $c_n(f)$ is
\begin{align}
c_n(f) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x) e^{-inx}dx\,.
\end{align}
Then evaluate $c_0(F)$.
... |
I'm learning Radon-Nikydom theorem and I have some doubts now. Let $\mu(B(x,r))=r^2$ for each $x\in\mathbb{R}$ and $r>0$, then $\mu$ is a radon measure on $\mathbb{R}$ and $\mu<<\mathscr{H}^1$. By Radon-Nikydom theorem, we have $\mu(E)=\int_E f d\mathscr{H}^1$ for some $f$. We can caculate that $f(x)=\lim_{r\to 0}\frac... |
***Background:***
-----------------
*I recently got an idea. My idea was analyzing primes whose digit sum is prime.*
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
I will call a special prime like this $P$.
For primes in the range $1<p<100$, these are all $P$:
$$... |
$f$ is holomorphic: $f'(z)=kz^{k-1}.$ And it's nonconstant. Therefore by the *open mapping theorem* it is open.
Btw it is not a homeomorphism. What about $k$th roots of unity, for instance? |
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