Singular Thinker
What is a Hilbert space, really? It is just a fancy name for a Cauchy complete inner product vector space. Wait what? I'll explain. First, why do we need inner product vector space? Inner product vector spaces have physical meaning and usefulness more…
What is Bra in Quantum Mechanics?
As you might notice in quantum mechanics, they use funny notations <x|, |y>.
The Bra-ket notation was first used by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. But, why?
Let's not blame the physicists for being showy. There is an important theorem, the Riesz representation theorem, in functional analysis that might describe using of this notation. Let (X,<.,.>) be a Hilbert Space. Then for each continuous linear map L: X→F there is exactly one vector in X, x_L, such that L(x) = <x_L,x> and the norm of operator L, L = x_L .
Thus, in Hilbert space, each vector can be considered a continuous linear map. Thus, when we use Bra, <c|, we mean a linear map from the Hilbert space X to F such that the final result corresponds to an inner product of c and that vector.
#math
@SingularThinker
As you might notice in quantum mechanics, they use funny notations <x|, |y>.
The Bra-ket notation was first used by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. But, why?
Let's not blame the physicists for being showy. There is an important theorem, the Riesz representation theorem, in functional analysis that might describe using of this notation. Let (X,<.,.>) be a Hilbert Space. Then for each continuous linear map L: X→F there is exactly one vector in X, x_L, such that L(x) = <x_L,x> and the norm of operator L,
Thus, in Hilbert space, each vector can be considered a continuous linear map. Thus, when we use Bra, <c|, we mean a linear map from the Hilbert space X to F such that the final result corresponds to an inner product of c and that vector.
#math
@SingularThinker
🔥2
Today I encountered this amazing website, which is a medium for presenting research in a new way. It is actually similar to the Medium website but just for the machine learning articles and the blog posts are peer-reviewed. When I checked the steering committee I found out why this is so amazing. Yoshua Bengio, Andrej Karpathy, Ian Goodfellow, … are members of the committee. Unfortunately, they didn’t continue the project but it still can help a lot.
So for instance I found this introduction about Gaussian Process very interesting. Check it out.
@SingularThinker
So for instance I found this introduction about Gaussian Process very interesting. Check it out.
@SingularThinker
Distill
A Visual Exploration of Gaussian Processes
How to turn a collection of small building blocks into a versatile tool for solving regression problems.
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The Simplest Math Problem No One Can Solve - Collatz Conjecture
We are stupid, aren't we?
This simple problem has some sort of magic and as always Vertasium narrated it aweosmely.
Watch the video here.
#video #math
@SingualrThinker
We are stupid, aren't we?
This simple problem has some sort of magic and as always Vertasium narrated it aweosmely.
Watch the video here.
#video #math
@SingualrThinker
YouTube
The Simplest Math Problem No One Can Solve - Collatz Conjecture
The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium…