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I've done so far a few courses in logic and formal verification, and I've always wondered:
mathematical logic, at least as Hilbert envisioned, tries to model mathematics.
Formally define what a "true" statement is, or why proving something (at least in a sound system) makes it true.
But, every logic course uses ma... |
How can mathematical logic try to model math, when mathematics are used to define mathematical logic? |
I am struggling to find an example with a solution for a Minimum Cost Capacitated Flow problem. My network is defined as a graph $G = (V, E)$, where each edge has a capacity $c(u, v) > 0$, a flow $f(u, v) \ge 0$, and a cost $a(u, v)$. The solution must show a minimum cost using the maximum capacity of the network edge... |
How to calculate the sum of an infinite series, such as $\frac{n(n-1)}{2^n}$? |
I am trying to solve part b) of the following problem. I solved part a here e https://math.stackexchange.com/questions/4866561/compute-a-quantum-state-rho-corresponding-to-an-ensemble/4866565#4866565
Consider the ensemble consisting of the qubit states $|0⟩⟨0|$ and $|1⟩⟨1|$ occuring with probabilities 2/3 and... |
I'm working on calculating the probability distribution of the interarrival time of the Poisson process. The method used in my textbook is very strange I don't understand how the probabilities are calculated. I am familiar with the method used eg. [here][1] but not the working below. $ N(t) $ is the Poisson process:
$... |
What rule is used in this derivation of the interarrival time for the Poisson process? |
Given an inner product on a vector space, I can always define a norm and a metric (and a topology using that metric). Is the converse true? That is, given a metric on a vector space, can I define an inner product with it? And what about a metric space that is not a vector space? |
I am trying to do some simulation of the binomial point process, and I want some help with the following problem. I am trying to find a formula for the final coordinates of a moving point on the surface of a sphere of radius $R$. For simplicity, we assume the center of the sphere is the origin (0,0,0). Assume the initi... |
I want to check the convexity of the following multivariate function:
$$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+\hat{b}\frac{s\lambda}{Q}+K\frac{\lambda}{Q} +c\lambda,$$
where $Q>0$, $s \ge 0$ and $h, b, \hat{b}, c, \lambda, K > 0$ and are known.
I want to know/prove that this function is convex.
Calcul... |
I think that an easily comprehensible explanation, the intuitive one, will kind of come after contemplation on the rate of decreasing of members of the series.
The best I could find on the topic is the way Augustin-Louis Cauchy developed his [condensation test][1]
[1]: https://en.wikipedia.org/wiki/Cauchy_c... |
I am trying to do some simulation of the binomial point process, and I want some help with the following problem. I am trying to find a formula for the final coordinates of a moving point on the surface of a sphere of radius $R$. For simplicity, we assume the center of the sphere is the origin (0,0,0). Assume the initi... |
I Found the following identity:
$\partial_k g_{ij}=\partial_j g_{ik} $ ?
or at least that is what it looks that it is being used.
g is a riemannian metric by the way.
[![enter image description here][1]][1]
**Why is it true? .I know the metric is symmetric , but this doesn't make sense to me.**
[1]:... |
Is $\partial_k g_{ij}=\partial_j g_{ik} $? |
In a certain exercise I am asked to find the probability of the event in which a dice is rolled seven times and the sum of the results is $14$, given the fact that the dize is regular (this is, the probability of each of the six possible results is the same: $\frac{1}{6}$.
I think I know the approach: let $X_i$ deno... |
A dice is rolled $7$ times. What is the probability the sum of the results is $14$? |
Let $d,n \in \mathbb{N}$. Assume we have $n$ points, $x_1,\dots,x_n$ where for every $i \in \{1,\dots,n\}$, we have $x_i \in \mathbb{R}^d$. Define the geometric median as
$$
\theta^\star \in \arg\min_{\theta \in \mathbb{R}^d} \sum_{i=1}^n \lVert \theta - x_i \rVert
$$
Fix a hyperplane $(w_0,\theta_0)$, i.e.,... |
Consider the plane with a point removed $M=\mathbb{R}^2\setminus\{0\}$. Let $p_1,p_2\in M$ be two points. I want to understand how to classify the homotopy classes of paths joining $p_1$ and $p_2$.
Let's take for example $p_1 = (a,0)$ and $p_2 = (0,a)$ with both $a>0$. One possibility of path from $p_1$ to $p_2$ is... |
The [copy of C&F that I can find on the web](https://books.google.co.uk/books?id=eI36TEsH4N8C&pg=PA14&source=gbs_toc_r&cad=2#v=onepage&q&f=false) actually has $\frac{\partial v_i}{\partial x_m}(\psi^{-1}(y))$ which is obviously wrong.
I think it should be $\frac{\partial v_i}{\partial x_m}(\psi(y))$ which would be e... |
My differential geometry course is using Do Carmo's *Differential Geometry of Curves and Surfaces*. I am struggling with the following homework problem.
**I have already shown:**\
Suppose $F:\mathbb{R}^3 \rightarrow \mathbb{R}^2 $ is a map described by $F(x,y,z) = (f(x,y,z),g(x,y,z))$ with $F(x_0,y_0,z_0) = (0,0)$.... |
Question b)
This isn't a complete answer, but a good "step forward" by a complete simplification of your expression for s(M).
Let $m$ be the affix of one of the points $M_k$ belonging to the unit circle, therefore with unit norm.
As $a,b,c$ are as well with unit norm, we can write
$$s(M)=9 + 2m\overline{(... |
There are $25$ distinguishable students in total. They can be arranged in groups of $5$ such that every group has a unique year of birth shared among all the members of the group (meaning there are $5$ years of birth in total).
The question is: In how many ways can these $25$ students form $5$ groups of $5$ so that ... |
In how many ways can 25 students form 5 groups of 5 such that no group consists of students with the same year of birth? |
Consider the plane with a point removed $M=\mathbb{R}^2\setminus\{0\}$. Let $p_1,p_2\in M$ be two points. I want to understand how to classify the homotopy classes of paths joining $p_1$ and $p_2$.
Let's take for example $p_1 = (a,0)$ and $p_2 = (0,a)$ with both $a>0$. One possibility of path from $p_1$ to $p_2$ is... |
I have encountered many types of sigma algebra generated by a random variable. I would like to clarify my understanding of them.
The simplest one is generated by a random variable. I skip it here.
Then I have sigma algebra generated by any family of random variables. i.e. Suppose $X_i$s are rvs from $(\Omega, A)$ ... |
In [this][1] thread, the convergence of coefficient for univariate dependent variable is proven. I wonder, assuming the same setup, how can the convergence be extended to multivariate as:
$$Y=XW+\epsilon$$
where $Y_{n\times d}$, $X_{n\times d}$, $\epsilon_{n\times d}$ and $W_{d\times d}$.
My issue is how to write ... |
Question b)
This isn't a complete answer, but a good "step forward" by a complete simplification of your expression for $s(M)$.
Let $m$ be the affix of one of the points $M_k$ belonging to the unit circle, therefore with unit norm.
As $a,b,c$ are as well with unit norm, we can write
$$s(M)=9 + 2m\overline... |
I'm reading chapter 3 of Stein's Real Analysis and have come across Lemma 3.5, which states:
> **Lemma 3.5**
> Suppose $G$ is real-valued and continuous on $\mathbb R$.
> Let $E$ be the set of points $x$ such that
> $$ G(x+h) > G(x) \quad \text{ for some } h = h_x > 0. $$
>
> If $E$ is non-empty, then it must ... |
If $x_i$, $1 \le i \le 7$ represents one less than the number on the $i^{\text{th}}$ die, for favorable outcomes we have
$$x_1 + x_2 + \cdots + x_7 = 7, \;\;\;0 \le x_i \le 5$$
Can you take it from here? Find the number of valid solutions ... |
How do you "orthogonally project" the shape? After I drew multiple graphs, I still had no idea.
The correct answer is 12. I guess it is 3 root 2 times 2 root 2 according to the gist of the graph that I drew (a plane with two parallel sides through four mid points each, and two shorter sides a little out of shape)... |
How to calculate the area of a projected path onto a plane? (Bee traveling problem) |
I am currently reading "[Restrictions on harmonic maps of surfaces][1]" by J. Eells and J. C. Wood and have trouble understanding the proof of the lemma on p. 265. The paper states:
> Let $X$ and $Y$ be closed orientable smooth surfaces, and $\phi: X \to Y$ a smooth map. Relative to Riemannian metrics $g,h$ on $X,Y$... |
How do you "orthogonally project" the shape? After I drew multiple graphs, I still had no idea.
The correct answer is $12$. I guess it is $3 \sqrt 2 \cdot 2 \sqrt 2$ according to the gist of the graph that I drew (a plane with two parallel sides through four mid points each, and two shorter sides a little out of ... |
I was seeing the proof that a polynomial that annihilates a linear operator is a multiple of a minimal polynomial, but I don't understand very well this demonstration:
Let $T\in L(V,V)$ a linear operator. If $p(x) \in \mathbb{P}(\mathbb{K})$ is a polynomial such that $p(T)(v)=0$, for all $v\in V$, so $p(x)$ is a mul... |
Why a polynomial that annihilates a linear operator is a multiple of a minimal polynomial? |
I know that the general word problem is undecidable, but is there an effective enumeration of presentations all finitely presented groups generated by $n$ elements in which each isomorphism class of a group appears exactly once?
I have no idea of how to tackle this problem, and I'm motivated by the fact that isomorp... |
Is there an enumeration of finitely presented groups? |
Let $\mathscr P_n[x]$ be the space of real polynomials in $x\in\mathbb R$ of degree at most $n\in\mathbb N$, and $$\mathscr P[x] := \lim_{n\to\infty} \mathscr P_n[x]$$ the set of all real polynomials in $x$.
Then consider the set $V$ whose elements $v(x)$ are defined by $$v(x) = \frac{p(x)}{q(x)} \qquad \text{for} \qu... |
Are rational functions a vector space? |
Consider a 2-manifold $(M,g)$ isometric to $(\mathbb{R}^2,\bar{g})$, where $\bar{g}$ is the Euclidean metric, and let $F:\mathbb{R}^2 \to M$ be an isometry. Let $K_0 \subset \mathbb{R}^2$ and $K \subset M$ be two compact subsets (actually, I consider geodesic triangles, if that matters), such that $K = F(K_0).$
I co... |
Let $ K $ be a field. Let $ K^n $ be an $ n $ dimensional vector space over $ K $. Let $ KP^{n-1} $ be the projective space of lines in $ K^n $. Let $ GL(n,K) $ be the group of invertible $ n \times n $ matrices over $ K $. What is the smallest subgroup of $ GL(n,K) $ that acts transitively on $ KP^{n-1} $?
Context... |
Let $a(x)$, $b(x)$, $c(x)$ be smooth coefficient on $\mathbb{R}$ and $u(x)$ be the solution to the linear ODE over $\mathbb{R}$:
$$ au''+bu'+cu=-1 $$
Suppose $a(x)>1$, can we have some estimate on $|u'|$ independent of the derivatives of $a(x)$, $b(x)$, $c(x)$? I can obtain the estimate by differentiating the equ... |
I'm not sure how to prove a statement about extended Weyl groups.
Let us define extended Weyl group $W^{ae}$ as the semidirect product $W⋉t(P^\vee)$, and
let $\tilde{w}=t(λ)w\in W^{ae}$. Suppose we have $a=[\alpha,r]$ an affine root (I use the Cherednik notation).
Write $\tilde{w}$ $s_\alpha=t(\lambda')w'$, with $w... |
By the way, "$\mathbf{span}\ U\cap W$" is a little [pleonastic](https://www.merriam-webster.com/dictionary/pleonasm), as $U\cap W$ is necessarily a subspace, and hence equal to its own span.
What you did, trying to row-reduce a "double augmented" matrix where both sides of the augmentation correspond to variables, i... |
I'm trying to understand the following easy (not for me) exercise.
Consider $\mathbb{P}^1_k$ where $k$ is an algebraically closed field.
We have closed points, namely the ones corresponding to $(a_0x_1-a_1x_0)$, and $\xi$ the generic, corresponding to the trivial ideal.\
Then we have that $\kappa(\xi)= k(t)$
... |
Assume $\bf A\in \mathbb R^{n\times n}$
If I know ${\bf A}^k$ and that it is symmetric, how can I systematically find the $\bf A$ which are not?
_____
**Own work** One approach I have considered is to assume a Jordan form:
$${\bf A}^k = ({\bf SJS}^{-1})^k$$
Now complications will arise if there are real neg... |
How can I find possible non-symmetric $A$ if $A^k$ is symmetric? |
I've been struggling with this for some time but can't prove what question asks.
Question is as follows:
If $\sin x = \sin \alpha\sin (x + y)$, prove that
$\tan x = \frac{\sin\alpha\sin y}{1 - \sin\alpha\cos y}$
I began by expanding what is given, i.e.
$$\sin x = \sin \alpha(\sin x\cos y + \cos x\sin y)$$
... |
$$ 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 $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
[Integral equation described below][1]
[1]: https://i.stack.imgur.com/13QnP.png
In my physics reading, we have the integral over $x_s$ from -L/2 to L/2 of $\frac{1}{((y^2+x_s^2)^{3/2})}$. Somehow, from this, the reading gets $\frac{x_s}{y^2*\sqrt{y^2+x_s^2}}$ from -L/2 to L/2. I'm assuming this was done via u... |
What U substitution is being used here? (direct integral calculation of charge) |
>![Integral equation described below][1]
[1]: https://i.stack.imgur.com/13QnP.png
In my physics reading, we have the integral over $x_s$ from -L/2 to L/2 of $\frac{1}{((y^2+x_s^2)^{3/2})}$. Somehow, from this, the reading gets $\frac{x_s}{y^2*\sqrt{y^2+x_s^2}}$ from -L/2 to L/2. I'm assuming this was done via... |
I am trying to do some simulation of the binomial point process, and I want some help with the following problem. I am trying to find a formula for the final coordinates of a moving point on the surface of a sphere of radius $R$. For simplicity, we assume the center of the sphere is the origin (0,0,0). Assume the initi... |
That's the correct definition of $\alpha_f$, yes. Really $A_f\to A_{\mathfrak{p}}$ is just localisation of $A_f$ at $\mathfrak{p}A_f$, though it is worth noting the homomorphisms in question need not be injective (this requires $A\setminus\mathfrak{p}$ to have no zero divisors, which might not be true) and thus they do... |
I've been struggling with this for some time but can't prove what question asks.
Question is as follows:
> If $\sin x = \sin \alpha\sin (x + y)$, prove that
> $$\tan x = \frac{\sin\alpha\sin y}{1 - \sin\alpha\cos y}$$
I began by expanding what is given, i.e.
$$\sin x = \sin \alpha(\sin x\cos y + \cos x\sin y... |
It is a common misconception many logic students pass by.
Let us start with an example: Every microprocessor we use has its own design, and we can implement it in different ways: Either physically with silicon wafers, virtually with emulators, or even [over Minecraft](https://www.youtube.com/watch?v=5nViIUfDMJg). N... |
Can someone tell me what this angle is, or if its a typo in the image?
[![enter image description here][1]][1]
It's $2^\circ24'$
I don't need help with the question. Just trying to understand what the angle is supposed to be.
[1]: https://i.stack.imgur.com/xGbG9.png |
That's the correct definition of $\alpha_f$, yes. Really $A_f\to A_{\mathfrak{p}}$ is just localisation of $A_f$ at $\mathfrak{p}A_f$, though it is worth noting the homomorphisms in question need not be injective (this requires $A\setminus\mathfrak{p}$ to have no zero divisors, which might not be true) and thus they do... |
In introducing the concept of $K(G, 1)$, Hatcher's *Algebraic Topology* proves the theorem that the homotopy type of a CW complex $K(G, 1)$ is uniquely determined
by $G$, by citing a proposition and claiming the proposition may be used to deduce the theorem. I am wondering why the bolded statement is true here:
> ... |
Hatcher Theorem 1B.8 / Proposition 1B.9 concluding that a map inducing the identity is homotopic to the identity? |
I have been reading the paper "Point Groups and Space Groups in Geometric Algebra" by David Hestenes as part of my introduction seminar to GA. On page 5 the remark is made, that to prove that $C_p$ and $H_p$ are the only point groups in 2D, one would need to prove the following: Consider the possibility of a group gene... |
Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 1/(2n+2i)!$.
Is there a formula for the inner product of $r$ and the first column of the inverse of $A$?
Or, can it be shown that the inner product afore... |
For the equation[enter image description here][1]
[1]: https://i.stack.imgur.com/zSLs8.gif
There are 5 distinct complex solutions and I noticed that all of them are colinear in the complex plane, is this a coincidence or is there a theorem behind? |
I am sure there is a similar question but I don't know how to phrase it, so I cannot find it. I'm searching for some intuition or direction to material of how to treat an expression like this:
$$
\frac{1}{1 + \partial_x}f(x)
$$
or more in general (with $L$ some operator)
$$
\frac{1}{1 + L}f(x).
$$
My first gu... |
How do I treat a derivative at the denominator? |
In the book Pattern Recognition and Machine Learning, CM Bishop has elaborated on calculating marginal, conditional and joint probabilities by creating a table with rows and columns being outcomes of individual random variable. The marginal probabilities can hence just be calculated by summing up the desired row and di... |
For the equation: m\power{6}=(m-1)\power{6}
There are 5 distinct complex solutions and I noticed that all of them are colinear in the complex plane, is this a coincidence or is there a theorem behind? |
In the book Pattern Recognition and Machine Learning, CM Bishop has elaborated on calculating marginal, conditional and joint probabilities by creating a table with rows and columns being outcomes of individual random variable. The marginal probabilities can hence just be calculated by summing up the desired row and di... |
For the equation: $m^6=(m-1)^6$
There are 5 distinct complex solutions and I noticed that all of them are colinear in the complex plane, is this a coincidence or is there a theorem behind? |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
Let $X$ be an $n \times n$ random matrix with *i.i.d. standard Gaussian entries*, i.e., the entries $X_{ij}$ are independent standard normal random variables. Given any deterministic unit vectors $u, v \in \mathbb R^n$, calculate the *mean and variance* of $Z_n := u^\top X^\top Xu + \sqrt n v^\top Xv$, and prove the ce... |
The problem comes from Volumn I, (7.4.3), Grothendieck's EGA. For a scheme $X$, denote the sheaf of rational functions by $\mathscr{R}(X)$.
**Corollary (7.4.3)**. Over an integral scheme $X$, any torsion-free, quasi-coherent and rank-$1$ $O_X$-module sheaf (denote this sheaf by $\mathscr{F}$) is isomorphic to a sub... |
In the book Pattern Recognition and Machine Learning, CM Bishop has elaborated on calculating marginal, conditional and joint probabilities by creating a table with rows and columns being outcomes of individual random variable. The marginal probabilities can hence just be calculated by summing up the desired row and di... |
I am trying to do some simulation of the binomial point process, and I want some help with the following problem. I am trying to find a formula for the final coordinates of a moving point on the surface of a sphere of radius $R$. For simplicity, we assume the center of the sphere is the origin (0,0,0). Assume the initi... |
Any $n\times n$ matrix $A$ over a field $\mathbb{k}$ admits a [rational canonical form][1] (also called a Frobenius normal form).
The standard proof that this form exists appears to be to first prove a more general structural theorem for finitely generated modules over a principal ideal domain (which is proven, for... |
Simpler / alternate proofs for the existence of the rational canonical form of a matrix? |
Informally: Every group is representable (up to an isomorhism) as a group of isomorphisms.
Formally: For every group $G$ there exists a binary relation $f$ on some set $U$ such that $G$ is isomorphic to group of all bijections $g$ on $U$ such that $g^{-1}\circ f\circ g=f$.
In other words, is every group $G$ isomo... |
Is every group isomorphic to a set of isomorphisms? |
I am trying to solve this equation with Frobenius method :
$(x^2-x)y''(x) + (1/3) y'(x) + xe^{(x-1)}y(x)=0$
I am trying to find the roots with : $r^2 + (P_0-1) + Q_0 = 0$
And, $P_0 = (1/0!)(1/3x)^{(0)}$ evaluated at $x=0$ but it gives me $1/0$!
I am able to find my $Q_0$ with : $Q_0 = (1/0!)((x-1)e^{(x-1)}... |
I assume all manifolds are closed (and hence that we're talking about finitely presented fundamental groups).
There are no restrictions for an almost complex structure. The notion of almost complex structure is very generous; for 4-manifolds it's an entirely algebraic condition that's not hard to satisfy. See [here]... |
I understand that this is a consequence of a related, more general theorem for non-unital $C^*$-algebras, where we can find approximants of a unit, but for the simple case of a unital $C^*$-algebra $\mathscr{A}$ , what is the simple argument that implies that positive linear functionals $\varphi:\mathscr{A} \rightarrow... |
Why is it that for a unital $C^*$-algebra any positive linear functional $\varphi$ must be such that $\|\varphi\| =\varphi()$? |
On a math test, for my online Honors Pre-Calculus course, that I recently took I got this question wrong and don't understand the explanation:
>Suppose $f(x) = \begin{cases} x^2-2, & x \not= 2 \\ 0, & x = 2 \end{cases}$.
>
>
>Which condition of continuity is **not** met by $f(x)$ at $x = 2$?
The correct answ... |
**How many ways are there to arrange the numbers $1-10$ in a circle such that each number is not greater than the sum of the two numbers adjacent to it?**
My Progress: Not much really. When I can solve a problem, I can solve it, and if I can't, it's usually because I don't know how to start.
I thought a little b... |
It is a well known pathological counterexample in group theory that (assuming the axiom of choice), we have an isomorphism$$\mathbb{R} \cong \mathbb{R}^2$$
, where these are additive groups. This somewhat unsettling result generally is shown by appealing to fact that both of these groups, when viewed as $\mathbb{Q}$-v... |
Let $\mathscr P_n[x]$ be the space of real polynomials in $x\in\mathbb R$ of degree at most $n\in\mathbb N$, and $$\mathscr P[x] := \lim_{n\to\infty} \mathscr P_n[x]$$ the set of all real polynomials in $x$.
Then consider the set $V$ whose elements $v(x)$ are defined by $$v(x) = \frac{p(x)}{q(x)} \qquad \text{for} \qu... |
Informally: Every group is representable (up to an isomorhism) as a group of isomorphisms.
Formally: For every group $G$ there exists a binary relation $f$ on some set $U$ such that $G$ is isomorphic to group of all bijections $g$ on $U$ such that $g^{-1}\circ f\circ g=f$.
In other words, is every group $G$ isomo... |
> How many ways are there to arrange the numbers $1$–$10$ in a circle such that each number is not greater than the sum of the two numbers adjacent to it?
My Progress: Not much really. When I can solve a problem, I can solve it, and if I can't, it's usually because I don't know how to start.
I thought a little b... |
The probability mass function of the Binomial distribution is given by
$$\begin{equation}
f(x)=\binom{n}{x} p^x (1-p)^{n-x},
\end{equation}$$
where $p \in [0,1]$ and $x=\{0,1,\dots,n\}$ (finite support).
The Mellin transform of $f$ is therefore
$$\begin{align}
\mathcal{M}\{f\}(s)&=\int_0^{\infty}x^{s-1} \bin... |
How analytically solve $ax^2-b \sqrt{x}+c=0$, where a, b, c are positive constants? mathematica gives long complex solutions but not sure if it's reliable. |
When when want to find particular solutions to a LDE of order 2 like :
$$ y'' + ay' + by = f(x) $$
I see in a lot of source that says that after finding two solutions for the homogeneous equations $y_1, y_2$, we want to find **two functions** $\lambda(x), \mu(x)$ that satisfies the system :
$$ \Biggl \{ \begin... |
I am studying the consensus problem of a linear multi-agent system, and a control gain needs to be designed to reach the consensus property. But the design procedure depends on the solution of the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$, and $A$ is not Hurwitz. Does the linear matrix inequali... |
Does the solution exist for the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$? |
I recently went back to re-learning some bits of linear algebra: change of basis.
And, as an exercise, I decided to revisit a video game mechanic: portals.
[As this youtuber puts it][1], the math behind it is that you want to know the Camera's position relative to the Blue Portal and then that position relative t... |
The statement of exercise 17 in section 14.2 of "Abstract Algebra" by Dummit & Foote says the following:
----------
[![enter image description here][1]][1]
----------
The statement assumes that there is a finite number of embeddings of $K$ into the algebraic closure of $F$, but I cannot see why this is true. ... |
Recent linear state-space model papers like Mamba often use matrix exponential to discretize the system. They initialize the system in a continuous-time regime, and discretize it to run it like a vanilla RNN / CNN.
Consider the simplest continuous-time autonomous system $d\mathbf{h(t)}/dt = A\mathbf{h(t)}$ (assuming... |
Backpropagation: Chain Rule for Matrix Exponential? |
It is a well known pathological counterexample in group theory that (assuming the axiom of choice), we have an isomorphism$$\mathbb{R} \cong \mathbb{R}^2,$$
where these are additive groups. This somewhat unsettling result generally is shown by appealing to fact that both of these groups, when viewed as $\mathbb{Q}$-ve... |
Let us consider the Euclidean space $\mathbb{R}^n$ of any $n \in \mathbb{N}$.
Then, I somehow vaguely recall that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is dense in the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$.
Here, two questions arise, on which I have not been able to find satisfacto... |
Is the Schwartz space "sequentially dense" in the space of tempered dsitributions with respect to some topology? |
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