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[This post](https://math.stackexchange.com/a/4841216/348586) about characteristic functions asks about absolutely integrable $\phi(t)$. [This answer](https://math.stackexchange.com/a/4841216/348586) mentions "we can find $R$ such that $\int_{\mathbb R\setminus[-R,R]}\lvert\phi(t)\rvert dt<2\pi\varepsilon$".
The argu... |
Question:Is it possible to produce with 3 elements the group G=F2*(ZxZ)?
I figured that G=<a,b,c,d|[c,d]=1>.
I also know that it is impossible to produce a free group of order n with less than n elements(even though I don't know if this proposition can help in this situation).
It is an exercise that i found in som... |
Question: Is it possible to produce with $3$ elements the group $G=F_2 \ast (\mathbb{Z} \times \mathbb Z)$?
I figured that $G= \langle a,b,c,d \mid [c,d]=1 \rangle$.
I also know that it is impossible to produce a free group of order $n$ with less than $n$ elements (even though I don't know if this proposition can ... |
Do we have
$$
7^k\mid 3^n+5^n-1\Longleftrightarrow n=t\cdot 7^{k-1}\quad\text{with}\quad t\equiv 1,5\,(\operatorname{mod}6),
$$
where $k$ is an arbitrary positive integer?
The part "$\Leftarrow$" is easy. Let $a=3^{t\cdot 7^{k-1}}$, then $5^{t\cdot 7^{k-1}}\equiv a^{-1}\,(\operatorname{mod}7^k)$, and $$7^k\mid a... |
This construction comes from the chapter VII of "the local Langlands conjecture for $GL(2)$":
Let $F$ be a local field. We know that for any $n \in \mathbb{N}$ there exists a unique unramified extension of degree $n$. We can denote by $F_{\infty}$ the composite of all these fields that is clearly the maximal unramif... |
I am somewhat familiar with using RK4 for coupled ODE's, I found a very elegant way (in my opinion) to utilize it in Python, like so:
```
def RungeKutta4_2(x, y, z, dx, dydx, dzdx):
k1 = dx*dydx(x, y, z)
h1 = dx*dzdx(x, y, z)
k2 = dx*dydx(x+dx/2., y+k1/2., z+h1/2.)
h2 = dx*dzdx(x+dx/2.... |
Standard way to do Runge-Kutta (4th order) for coupled ODE's in Python? |
Can anyone help with this geometry problem? |
Suppose I have a sequence of stochastic processes $(X_n(t))_{t\in [0,1]}$ such that $X_n(t) \in H$, where $H$ is some separable Hilbert space and $X_n \in C([0,1,], H)$.
> **Goal.** I want to show that $X_n$ converges weakly to a stochastic process $X$, i.e. $\mathrm{Law}(X_n) \rightharpoonup \mathrm{Law}(X)$ as pro... |
Is $\lim_{t\to \infty} \frac{f(t+o(t))}{f(t)}=1$ for bounded continuous function $f$? |
Let $A$ be an upper triangular matrix with $A_{i,j}=A_{i+1,j+1}$ for all $i,j$. How can I show that $A^{-1}$ has the same property? That, it is (an upper triangular) matrix with $A^{-1}_{i,j}=A^{-1}_{i+1,j+1}$ for all $i,j$. I have verified this using computations, but is there a simple proof? Thanks. |
Let $A$ be an invertible upper triangular matrix with $A_{i,j}=A_{i+1,j+1}$ for all $i,j$. How can I show that $A^{-1}$ has the same property? That, it is (an upper triangular) matrix with $A^{-1}_{i,j}=A^{-1}_{i+1,j+1}$ for all $i,j$. I have verified this using computations, but is there a simple proof? Thanks. |
I conjecture, and have verified on Mathematica, that
$$\sum_{m=2}^L \,\,\sum_{l=\max(M+m-L,1)}^{\min(m-1,M)}{L-m \choose M-l}={L \choose M}-1\,.$$
How I arrived at this conjecture is a long-but-interesting story related to my quantum computing research.
I understand that the min/max are enforcing the requirement ... |
Box A contains 7 blue balls and 5 yellow balls. Box B contains 3 blue balls and 7 yellow balls. One ball is removed at random from Box A and placed in Box B. After thoroughly mixing the balls, a ball is drawn at random from box B and placed back into A.
Find the probability that at the end of the experiment, box A... |
Since $$\tan^{-1}u =1-u+u^{3}/3-..., u =\tan x $$
we have
$$x= 1- \tan x +\tan^{3}x/3..\tag 1 $$
What is the interval of convergence?
Can $x$ be summed up from powers of any circular trigonometric function ?
Is it valid for *any* inverse function whose series expansion is known? |
This is a somewhat technical question about a line in Sharpe's book **Differential Geometry: Cartan's Generalization of Klein's Erlangen Program** in the proof of the structure theorem, Theorem 8.3. We have assumed that the leaf space of a foliation $M/\sim$ is Hausdorff. Because of this, each leaf $\mathcal{L}$ is clo... |
Suppose I have a sequence of stochastic processes $(X_n(t))_{t\in [0,1]}$ such that $X_n(t) \in H$, where $H$ is some separable Hilbert space and $X_n \in C([0,1,], H)$.
> **Goal.** I want to show that $X_n$ converges weakly to a stochastic process $X$, i.e. $\mathrm{Law}(X_n) \rightharpoonup \mathrm{Law}(X)$ as pro... |
Box A contains 7 blue balls and 5 yellow balls. Box B contains 3 blue balls and 7 yellow balls. One ball is removed at random from Box A and placed in Box B. After thoroughly mixing the balls, a ball is drawn at random from box B and placed back into A.
Find the probability that at the end of the experiment, box A... |
For any set $A\subset \mathbb{N}$, define a set function $\mu(A)=\sum_{n\in A}2^{-n}$ if $A$ is finite subset otherwise $\mu(A)=\infty$.
Is $\mu$ finitely additive, countably additive, or countably subadditive?
---
It is finitely additive because for two disjoint subsets $A, B$, we have $A\cup B$ is finite and... |
Is $\mu$ finitely additive, countably additive, or countably subadditive? |
For instance, can a number like 0.1111111.... in base 3 be represented as 0.23515613.... (non-repeating) in base 8? I imagine the answer would be a resounding NO but it would be interesting to see a proof of why. |
Can a decimal that is infinitely repeating in one base be nonrepeating in another? |
Others have commented on the technical details of random walks and brownian motions, but IMHO the OP's main confusion seems to be in this sentence:
> Thus, each of these infinitely long paths are weighted with infinitely small probabilities - and since there are an infinite number of these paths : the expected value... |
In Appendix A of Tao's book *Nonlinear Dispersive Equations*, a Bernstein inequality has puzzled me a long time:
Suppose that $P_{\geq N}$ is smoothed out projection to the region $|\xi|\geq N$, where $N\in 2^{\mathbb{Z}}$ is a dyadic number, $\hat{\varphi}(\xi)$($|\hat{\varphi}(\xi)|\leq1$) is a real valued radial... |
FYI I am an engineering student and would like to refrain from very complex notations. I am simply looking for a more accurate understanding of the laplace transform which is not an algebraic definition.
From my understanding for any function f(t) we can express it via int(f(t)*e^-st))|[infinity-0]. This provides so... |
Too long for a comment, this answer addresses just this sub-question:
> would it be a completely different problem in an environment where the step sizes are irrational numbers?
Not sure I'd call it "completely" different but it would be quite different. E.g. the solution technique in @leonbloy's answer (recurre... |
For a polynomial $p(z)$, where its coefficients are complex, what is the necessary & sufficient condition that satisfy $p(\mathbb{R}) \subset \mathbb{R}$? My guess is that it should have real coefficients, but I did not find a way to rigorously prove that result.
Meanwhile, the problem also have 2 alternative versio... |
I have made it a personal long-term goal of mine to numerically solve the 1D transient Navier-Stokes equations for simple incompressible and compressible flow scenarios. I want to do it for a few reasons: to better understand the numerics behind CFD calculations, to prove to myself that I can do it, and to eventually ... |
FYI I am an engineering student and would like to refrain from very complex notations. I am simply looking for a more accurate understanding of the laplace transform which is not an algebraic definition.
From my understanding for any function $f(t)$ we can express it via $\int(f(t)*e^{-st}))|[\infty,0]$. This provid... |
As in comment, I'd assume $n \neq 0$. There are some rational points on the curve (for any $m$ and $n$): $(0, \pm 3n)$ and $(\pm (m^2 + 1), \pm 3n)$. Now, to show that the curve as infinitely many rational points (i.e. rank is at least one), it is enough to show that at least one of these points is not a torsion. By [N... |
The Axiom of Completeness can be formulated as:
There exists a set $R$ such that:
1. $R$ is an ordered Archimedean field
2. Any nonempty subset of $R$ with an upper bound has a least upper bound.
Recently, I read something that suggested this is logically equivalent to the following:
There exists a set $R$ s... |
Is the Axiom of Completeness logically equivalent to "There is no proper superset of $\mathbb R$ that is an ordered Archimedean field"? |
A multiset $S$ of numbers is considered *satisfactory* if $\left|\left\{\sum U \bmod n \mid U\subseteq S\right\}\right| = n$, that is, all remainders mod $n$ can be constructed by summing subsets of $S$.
If we start a random process with an empty multiset, and keep adding uniformly random integers $[0, n)$ (with rep... |
Average number of random integers needed to satisfy any modular subset sum? |
For each $n\in\mathbb{N}$, consider real $n$ numbers $x_{i,n}$, $1\le i \le n$, given by
$$\sum_{i=1}^{n}x_{i,n}=0,\\-1/n \le x_{i,n} \le 1-1/n.$$
I am trying to find value of
$$\lim_{n\to \infty} \sum_{l=1}^{\lfloor n/p \rfloor}\textbf{1}_{lp\le n}x_{lp,n},$$
where $p$ denotes a prime and $\textbf{1}_{\lbra... |
The Axiom of Completeness can be formulated as:
There exists a set $R$ such that:
1. $R$ is an ordered Archimedean field
2. Any nonempty subset of $R$ with an upper bound has a least upper bound.
Recently, I read something that suggested this is logically equivalent to the following:
There exists a set $R$ s... |
**I. Question**
We have a quartic in $v$ to be made a square,
$D^2 = 4(-6 - 2u + u^2)(2 - 2u + u^2) - 8(-2 - 4u + u^2)(2 - 2u + u^2)v - 16u(4 - 3u + u^2)v^2 - 4(4 - 12u + 4u^2 - 2u^3 + u^4)v^3 + (4 - 8u - 4u^3 + u^4)v^4$
If there is rational $(u,v)$ such that $D$ is also rational, then the quartic is biration... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/rrmkS.jpg
So based on the diagram here's what I've attempted:
> Note: $\bar{g}$ is the Earth's gravitational constant and $T$ is the tension force acting on the Blocks.
> $a_1,a_2,a_3$ will be the acceleration for Block A, B, and... |
Others have commented on the technical details of random walks and brownian motions, but IMHO the OP's main confusion seems to be in this sentence:
> Thus, each of these infinitely long paths are weighted with infinitely small probabilities - and since there are an infinite number of these paths : the expected value... |
I'm reading notes from commutative algebra by Pete Clark.
In one of the examples of Galois connections he introduces notation $\text{RSpec}(K)$ for the set of all total orders on a field $K$, and gives it a topology coming from identifying such order with a function $K^\times\to \{-1, 1\}$, where value $1$ correspon... |
The definition of a 2-chain comes from Fayer's paper at: https://qmro.qmul.ac.uk/xmlui/bitstream/handle/123456789/64468/Fayers%202-chains%3A%20an%20interesting%202020%20Accepted.pdf?sequence=2&isAllowed=y
In the paper he defined 2-chains to bear the following 2 conditions: 1. There is a unique way to write P as the ... |
Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
The character of the permutation representation... |
Suppose a continuous random variable $$L \sim F_{L}$$ where $$Pr[L = 0] = 0$$ even though $$L = 0$$ exists for some countable subset of the outcome space.
I am considering the existence of a variable
$$C \sim \text{Poisson}(L)$$
which would not be defined if $L=0$.
We could just condition that $L\neq 0$: $... |
Is my construction almost-surely a Poisson random variable? |
HINT: Take logarithms, using the easy to see inequality $$M^M \ge 2^M (M!).$$
IF YOU NEED MORE HELP:
First note that, taking logs of the LHS of the OP, we note: $$\log(n^{n^n}) = n^n \times \log (n).$$
On the other hand, taking logs of $(n!)!$ on the RHS of the OP:
$$\log((n!)!)$$ $$\le \log((n!)^{n!})$$
... |
Suppose a continuous random variable $$L \sim F_{L}$$ where $$Pr[L = 0] = 0$$ even though $$L = 0$$ exists for some countable subset of the outcome space.
I am considering the existence of a variable
$$C \sim \text{Poisson}(L)$$
which would not be defined if $L=0$.
We could just condition that $L\neq 0$: $... |
Is my construction almost-surely a conditional Poisson random variable? |
**HINT:** Take logarithms to conclude $$n^{n^n} \ge ((n!)!)^4,$$ and note that this gives you what you need.
En route, use the easy to see inequality $$M^M \ge 2^M (M!).$$
**********
**IF YOU NEED MORE HELP:**
First note that, taking logs of the LHS of the OP, we note: $$\log(n^{n^n}) = n^n \times \log (n)... |
Can geodesics on any surface ( embedded $\mathbb R ^3)$ be mapped as straight lines on the plane? Please provide examples.
Are there some other properties of surfaces that may need to be satisfied for a plane straight line representation ?
Stereographic projection of great circles clearly do not map as straight l... |
In space, there exists a smooth plane, and additionally, there are two points, denoted as A (x_a, y_a, z_a) and B (x_b, y_b, z_b). A beam of light is emitted from point A, reflects off the mirror surface of the smooth plane at a reflection point P, and finally arrives at point B. Now, we can obtain the mirror image poi... |
**Context**
I looked through a book called "Problems in Low-Dimensional Topology," where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and describes any progress on the problems. The book was published in 1995, so I don't know if some of the problems that Kirby stated a... |
**HINT:** Take logarithms to conclude $$n^{n^n} \ge ((n!)!)^4,$$ and note that this gives you what you need.
En route, use the easy to see inequality $$M^M \ge 2^M (M!).$$
**********
**IF YOU NEED MORE ELABORATION:**
First note that, taking logs of the LHS of the OP, we note: $$\log(n^{n^n}) = n^n \times \... |
Is there a curve such that it's parametrization r(t) has κ → ∞ as t → ∞? |
The question states:
For $0 < x < 1,\ let\ f(x) = (1+x)(1+x^4)(1+x^{16})(1+x^{64})(1+x^{256}).... $
Find $f(x).$
I've tried noticing that it looks similar to the telescoping series $(1+x)(1+x^2)(1+x^4)...$
but applying the same trick of multiplying by $(1-x)$ doesn't work here.
Can someone please gui... |
In space, there exists a smooth plane, and additionally, there are two points, denoted as $A (x_a, y_a, z_a)$ and $B (x_b, y_b, z_b)$. A beam of light is emitted from point A, reflects off the mirror surface of the smooth plane at a reflection point P, and finally arrives at point B. Now, we can obtain the mirror image... |
The question states:
For $0 < x < 1$, let $f(x) = (1+x)(1+x^4)(1+x^{16})(1+x^{64})(1+x^{256}).... $
Find $f(x).$
I've tried noticing that it looks similar to the telescoping series $(1+x)(1+x^2)(1+x^4)...$
but applying the same trick of multiplying by $(1-x)$ doesn't work here.
Can someone please gui... |
Perform the closed line integral $\int \vec{F}\cdot d\vec{s}$ for the given field along the specifcied path. Let $\hat F =r\hat{\theta}$ starting and finishing at $(2,0)$ along the counter-clockwise circle $r=2$.
$$\int_CF \cdot d\vec{s}=\int_0^{2\pi}F(\vec{r}(t))\cdot r'(t) dt = \int_0^{2\pi}2 \vec{\theta}\cdot \t... |
I am trying to solve a problem related to computational fluid dynamics. However, I got stuck on a mathematical operation and am unsure how to tackle it. Here is it.
Let $\boldsymbol{n}$ be a unit vector in 3D space with known components. Given the following rotational matrix:
$$
\begin{equation}\begin{gathered}... |
There are two complex numbers $z_1$ and $z_2$ related by $\frac{z_1}{z_2} = i$, which implies that they have an angle of $\frac{\pi}{2}$ between them ($e^{i\frac{\pi}{2}} = i$).
Does the way in which we measure the angle matter? To clarify what I mean, let me assume $z_1 = 1+i$ and $z_2 = 1-i$. This gives us two com... |
Does the direction from which we're taking the angle matter? |
I came across this definition while reading Jaap van Oosten's [Topos Theory lecture notes][1] (pg 29, def 1.3).
In a category with finite limits, an **equivalence relation** on
an object $X$ is a subobject $R$ of $X \times X$ for which the following statements
hold:
- The diagonal embedding $X \to X \times X$... |
I understand the derivation leading to the memoryless property, but I cant seem to understand its application in a real scenario.
Let say we have a period $P$, in which no event occurs, if the memoryless property is true, it seem to me that the probability that an event occur at any instance $i$ in $P$ is a constan... |
if $$\tan(\cot(x))=\cot(\tan(x))$$ find $$\sin(2x)$$
so to start
$\cot\left(\frac{\pi}{2}-\cot \left(x\right)\right)=\cot\left(\tan\left(x\right)\right)$
or $\frac{\pi}{2}=\tan\left(x\right)+\frac{1}{\tan\left(x\right)}$
which is $\frac{\sec^2\left(x\right)}{1+\tan\left(x\right)}$
now using the sin(2x) ... |
The question states:
For $0 < x < 1$, let $f(x) = (1+x)(1+x^4)(1+x^{16})(1+x^{64})(1+x^{256}).... $
Find $f(x).$
$f(x)=
\prod_{n=0}^{\infty}(1+x^{(4^n)})$
I've tried noticing that it looks similar to the telescoping series $(1+x)(1+x^2)(1+x^4)...$
but applying the same trick of multiplying by $(1-x)... |
> If $\tan(\cot(x))=\cot(\tan(x))$, find $\sin(2x)$.
so to start
$\cot\left(\frac{\pi}{2}-\cot \left(x\right)\right)=\cot\left(\tan\left(x\right)\right)$
or $\frac{\pi}{2}=\tan\left(x\right)+\frac{1}{\tan\left(x\right)}$
which is $\frac{\sec^2\left(x\right)}{1+\tan\left(x\right)}$
now using the sin(2x) ... |
I understand the derivation leading to the memoryless property, but I cant seem to understand its application in a real scenario.
Let say we have a period $P$, in which no event occurs, if the memoryless property is true, it seem to me that the probability that an event occur at any instance $i$ in $P$ is a constan... |
When I investigate the attractor of Sine-Gordon equation, a question puzzled me, that is the number of solution of equilibrium of Sine-Gordon equation:
What is the solution of $$-\Delta u+\sin u=0,x\in\mathbb{R}^{n}.$$ We know that $u(x)=k\pi,k\in\mathbb{Z}$ are constant solutions, does there exist nontrivial solut... |
How to evaluate $\int_0^1\int_0^1 x^py^q \sin(\pi xy) (xy)^{xy} (1-xy)^{1-xy} \, dx \, dy$ where $p,q$ are integers? |
I've seen a bunch of questions showing how to prove a point is a limit point, but almost nothing showing that a point is NOT a limit point.
My definition of a limit point $x$ is that $\forall \epsilon >0, (B(x,\epsilon)\backslash\{x\})\cap E \neq \emptyset$ where $B(x,\epsilon)$ is the ball centered in $x$ with radi... |
How to prove point is NOT a limit point? $E=\{\frac{4n-3}{2n-1}:n\in\mathbb{N}\}$ |
In the essay [Lockhart's Lament][1] (page 4), the author describes a proof for the standard area of triangle $(bh)/2$ by enclosing the triangle in a rectangle and chopping the rectangle into two (perhaps unequal parts) using its altitude (which is the perpendicular from the top vertex to the base). Of the four triangle... |
Let $\mu=\sum_{i=1}^n \lambda_i\delta_{w_i}$, where $\delta$ is the Dirac measure, and $\lambda_i,w_i\in\mathbb{C}$ and $\lambda_i$ is non-zero. Let $\sigma_k$ be Fourier coefficients of $f$:
$$
\sigma_k=\dfrac{1}{2T}\int_{-T/2}^{T/2} f(x)e^{-2\pi i k x}d\mu.
$$
Then why we have the following equality:
$$
\sigma... |
If A and B are positive definite matrix, $(A+B)^2||B^{-1}||-A$ is also positive definite?
When A and B are real numbers, then $(A+B)^2||B^{-1}||-A$ is obviously positive definite. When A and B are matrix, I can use orthonormal matrix $O$ to transform $B$ an diagonal matrix $diag\{\lambda_1,\cdots,\lambda_n\}$. Then ... |
$(A+B)^2||B^{-1}||-A$ is positive definite? |
Hints: Since $\frac {4n-3} {2n-2} \to 2$ we expect that $\frac {4n-3} {2n-2} >1.9$ for $n$ sufficiently large. Verify that this inequality is indeed true if $n>5$. Next note that $1.9 >\frac 5 3$. Can you now find an interval around $\frac 5 3$ which contains at most $5$ points of the given sequence? |
How many nontrivial solutions of $-\Delta u+\sin u=0$? |
I am learning the theory of posets and lattices which will eventually lead to Boolean Algebra. I am stuck with the proper understanding of the concept of duality. Followings are what I have gathered so far:
1. If $ (P,\preccurlyeq ) $ is a poset, then $ (P,\succcurlyeq) $ is also a poset where $ \succcurlyeq $ is d... |
If A and B are positive definite matrix, $(A+B)^2||B^{-1}||-A$ is also positive definite?
When A and B are real numbers, then $(A+B)^2||B^{-1}||-A$ is obviously positive definite. When A and B are matrix, I can use orthonormal matrix $O$ to transform $B$ an diagonal matrix $diag\{\lambda_1,\cdots,\lambda_n\}$. Then ... |
Definition of conjugates: Conjugate [wiki link][1].
Suppose $G$ is a group. Two elements $a$ and $b$ of $G$ are called conjugate if there exists an element $g$ in $G$ with $g*a*g^{−1} = b$. Here $*$ is operation on group.
Question: If $g * g^{−1}$ will give us identity. Then equation will become $a * e = b$
he... |
I am trying to identify bifurcation of my $3D$ system near $a=6.58$. I am getting trajectories as shown in the picture and I am guessing it is Saddle-Node periodic orbit bifurcation since I am getting periodic trajectories around a saddle point. Is my guess alright?[enter image description here][1]
[1]: https:/... |
Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$ (relative to the ground plane in the world). Assume we apply a rigid transformation to $F$, sense a new gravity vector $g'$, and compute new Euler angles $p'$ and $r'$. Can we predict the yaw... |
I have currently opted for a course titled Differential Geometry of Curves and Surfaces for which my instructor is referring to Montiel and Ros' Curves and Surfaces. Having finished Chapter 1, I am still not getting how to solve the exercises given. For example:
(18) Let $\alpha:I \rightarrow \mathbb{R}^3$ a curve p... |
I stumbled across the following problem: for a general non-degenerate 3D vector $\vec{r} = (x,y,z)$ find a perpendicular vector $\vec{v}$ (so $\vec{v} \cdot \vec{r} = 0$). For simplicity, since we assume that $\vec{r}$ is non-degenerate, we might as well assume that $|\vec{r}| = 1$.
Actually, this problem can be sim... |
Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$ (relative to the ground plane in the world). Assume we apply a rigid transformation to $F$, sense a new gravity vector $g'$, and compute new Euler angles $p'$ and $r'$. Can we predict the yaw... |
In my lecture notes, I came across the statement: 'Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix.'
Can someone explain why this is the case? |
Why is easier to get inverse of mass matrix? |
Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$. Assume we apply a rigid transformation to $F$, sense a new gravity vector $g'$, and compute new Euler angles $p'$ and $r'$. Can we predict the yaw angles $y$ or $y'$ given $(p, r, p', r')$? |
If $A$ and $B$ are positive definite matrix, $(A+B)^2\|B^{-1}\|-A$ is also positive definite?
When $A$ and $B$ are real numbers, then $(A+B)^2\|B^{-1}\|-A$ is obviously positive definite. When $A$ and $B$ are matrix, I can use orthonormal matrix $O$ to transform $B$ an diagonal matrix $\operatorname{diag}\{\lambda_1... |
$(A+B)^2\|B^{-1}\|-A$ is positive definite? |
Given a gravity vector $g$ in a 3D coordinate frame $F$ we can find pitch $p$ and roll $r$ (Euler angles) of $F$ relative to $g$. Assume we apply a rigid transformation to $F$, sense a new gravity vector $g'$, and compute new Euler angles $p'$ and $r'$. Can we predict the yaw angles $y$ or $y'$ given $(p, r, p', r')$? ... |
As mentioned in the comments, they are simply referring to the [quadratic variation][1]
$$[X]_{t}=\lim_{|P|\to 0}\sum_{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2},$$
where $P$ ranges over partitions of the interval $[0,t]$. For Brownian motion this converges to $t$ eg. https://math.stackexchange.com/questions/3192106/is-... |
[enter image description here][1]
[1]: https://i.stack.imgur.com/3xB7x.png
I recently came across this variation of linear programming problem where given a set of 2-D points, we would like to find 1 point which maximizes certain objective function. Let's say the objective function is to maximize $y = m(x+A) ... |
[Image for visualization][1]
[1]: https://i.stack.imgur.com/3xB7x.png
I recently came across this variation of linear programming problem where given a set of 2-D points, we would like to find 1 point which maximizes certain objective function. Let's say the objective function is to maximize $y = m(x+A) + B$ ... |
Is this an adequate definition of the natural numbers?
The term *number* is synonymous with *natural number* in this context. By *mutually distinct ordered pair* I intend that the front element is not the back element. The terms *front* and *back* are used in order to avoid using numbers *first* and *second* to de... |
Is this an adequate definition of the natural numbers in terms of elementary set theory? |
Last year, Terence Tao published a paper entitled ["Almost All Orbits of the Collatz Map Attain Almost Bounded Values" (via arXiv)](https://arxiv.org/abs/1909.03562).
In layperson's terms, could somebody please explain what this means?
In particular,
1. "Almost All" — Does that mean with the possible except... |
I have a linear transformation $F: U \text{->} V$. Given that the vectors $\vec{v_1}...\vec{v_n} \in V$ are linearly independent I want to show that $F$ is one-to-one.
Also $F(\vec{u_i}) = \vec{v_i}$ and $\vec{u_i} \in U$ $\forall i = 1,2,...,n$.
My approach was
$$s_1\vec{v_1}+...+s_n\vec{v_n} = \vec{0} \iff s_... |
Let $\mu=\sum_{i=1}^n \lambda_i\delta_{w_i}$, where $\delta$ is the Dirac measure, and $\lambda_i,w_i\in\mathbb{C}$ and $\lambda_i$ is non-zero. Let $\sigma_k$ be Fourier coefficients of $f$:
$$
\sigma_k=\dfrac{1}{2T}\int_{-T/2}^{T/2} f(x)e^{-2\pi i k x}d\mu.
$$
Then why do we have the following equality:
$$
\si... |
$M$ is a continuous local martingale. Show that:<br>
>a) If $M_0=0$ and $\mathbb{E} (< M >) _t <\infty$ for $t\ge 0$ then $M \in \mathcal M^{2,c}$<br>
We will show that finite quadratic variation implies that M is L2-bounded martingale. We follow Revuz-Yor (1.23) Proposition.
**L2-bounded.**
Since $M-[M]$ i... |
How to reasonably estimate the probability of your father being exactly 12222 days older than you? |
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