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Let’s Discuss the Lorentz Transforms – Part 1: Einstein’s 1905 Derivation

Mathematics *Physics

Even as I am posting this, I can see that my previous post received a hundred and twenty plus views, but no comments yet. I am saying again that my pursuit is not to give an answer, but to ask a question. I only wonder if there is in fact no answer to the questions I am asking – but anyway, I will continue asking them. If you know how to deal with the problems I am setting – or happen to understand they are not problems at all, I will be most grateful for a constructive input in the comments section. I am sorry to say I was unable to make this post sound as light and unpretentious as the previous one. This one deals with harder questions, is a little wordy, and requires at least elementary knowledge of calculus to be read properly.

In my previous post we discussed the ‘Galilean’ velocity composition used for introduction or substantiation of relative simultaneity. It is not the only point where Einstein resorts to sums c + v or c – v: he does that actually to deduce the Lorentz transforms, notwithstanding the fact that a corollary of the Lorentz transforms is a different velocity composition which makes the above sums null and void. It looks like the conclusions of this deduction negate its premises – but this is not the only strange thing about Einstein’s deduction of the Lorentz transforms undertaken by him in his famous 1905 article.

In Paragraph 3 of that paper Einstein is considering the linear function τ (the time of the reference frame in motion) of the four variables x′ = x – vt, y, z, and t (the three spatial coordinates and time of the frame of reference at rest) and eventually derives a relation between the coefficients of this linear function.

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Total votes 3: ↑3 and ↓0 +3
Views 222
Comments 0

Let’s Discuss Relativity of Simultaneity

Mathematics *Physics

There is one only too obvious problem with relativity of simultaneity in the way it is normally introduced, and I have never found an answer to it – what’s more, I never read or heard anyone formulate it. I will be grateful for an enlightening discussion.

The framework of the thought experiment introducing relativity of simultaneity is this. Two rays of light travel in opposite directions and reach their destination simultaneously in one frame of reference and at different moments in the other.

For example, in the Wikipedia article on the subject you can read:

‘A flash of light is given off at the center of the traincar just as the two observers pass each other. For the observer on board the train, the front and back of the traincar are at fixed distances from the light source and as such, according to this observer, the light will reach the front and back of the traincar at the same time.

‘For the observer standing on the platform, on the other hand, the rear of the traincar is moving (catching up) toward the point at which the flash was given off, and the front of the traincar is moving away from it. As the speed of light is finite and the same in all directions for all observers, the light headed for the back of the train will have less distance to cover than the light headed for the front. Thus, the flashes of light will strike the ends of the traincar at different times’.

I am always not a little surprised at the modesty displayed by the authors of such illustrations. If we grant the statement ‘the light headed for the back of the train will have less distance to cover than the light headed for the front’ to be true – how then do we evaluate the magnitude of the effect? Or, in other words, how much longer is one distance in comparison to the other?

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Total votes 3: ↑2 and ↓1 +1
Views 219
Comments 0

FL_PyTorch is publicly available on GitHub

Mathematics *Machine learning *Artificial Intelligence

FL_PyTorch: Optimization Research Simulator for Federated Learning is publicly available on GitHub.

FL_PyTorch is a suite of open-source software written in python that builds on top of one of the most popular research Deep Learning (DL) frameworks PyTorch. We built FL_PyTorch as a research simulator for FL to enable fast development, prototyping, and experimenting with new and existing FL optimization algorithms. Our system supports abstractions that provide researchers with sufficient flexibility to experiment with existing and novel approaches to advance the state-of-the-art. The work is in proceedings of the 2nd International Workshop on Distributed Machine Learning DistributedML 2021. The paper, presentation, and appendix are available in DistributedML’21 Proceedings (https://dl.acm.org/doi/abs/10.1145/3488659.3493775).

The project is distributed in open source form under Apache License Version 2.0. Code Repository: https://github.com/burlachenkok/flpytorch.

To become familiar with that tool, I recommend the following sequence of steps:

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Total votes 1: ↑0 and ↓1 -1
Views 664
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What are neural networks and what do we need them for?

Mathematics *Machine learning *Data Engineering *

Explaining through simple examples

For a long time, people have been thinking on how to create a computer that could think like a person. The advent of artificial neural networks is a significant step in this direction. Our brain consists of neurons that receive information from sensory organs and process it: we recognize people we know by their faces, and we feel hungry when we see delicious food. All of this is the result of brain neurons working and interacting with each other. This is also the principle that artificial neural networks are based on, simulating the processes occurring in the human brain.

What are neural networks

Artificial neural networks are a software code that imitates the work of a brain and is capable of self-learning. Like a biological network, an artificial network also consists of neurons, but they have a simpler structure.

If you connect neurons into a sufficiently large network with controlled interaction, they will be able to perform quite complex tasks. For example, determining what is shown in a picture, or independently creating a photorealistic image based on a text description.

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Total votes 1: ↑1 and ↓0 +1
Views 865
Comments 1

Riddles of the fast Fourier transform

Programming *Algorithms *Mathematics *Sound Visual programming *

• The method of phase-magnitude interpolation (PMI)

• Accurate measure of frequency, magnitude and phase of signal harmonics

• Detection of resonances

The Fast Fourier Transform (FFT) algorithm is an important tool for analyzing and processing signals of various nature.

It allows to reconstruct magnitude and phase spectrum of a signal into the frequency domain by magnitude sample into the time domain, while the method is computationally optimized with modest memory consumption.

Although there is not losing of any information about the signal during the conversion process (calculations are reversible up to rounding), the algorithm has some peculiarities, which hinder high-precision analysis and fine processing of results further.

The article presents an effective way to overcome such "inconvenient" features of the algorithm.

Read in Russian

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Rating 0
Views 771
Comments 0

One does not simply calculate the absolute value

Programming *Java *Mathematics *

It seems that the problem of calculating the absolute value of a number is completely trivial. If the number is negative, change the sign. Otherwise, just leave it as it is. In Java, it may look something like this:

public static double abs(double value) {
  if (value < 0) {
    return -value;
  return value;

It seems to be too easy even for a junior interview question. Are there any pitfalls here?

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Total votes 11: ↑10 and ↓1 +9
Views 31K
Comments 4

Measuring Traffic Rate by Means of U-models

Qrator Labs corporate blog Algorithms *Mathematics *
stream rate art
Measuring of stream rate in an artist's impression.

In one of our previous publications, we talked about a way to measure event stream rate using a counter based on exponential decay. It turns out that the idea of such a counter has an interesting generalization. This paper by Artem Shvorin and Dmitry Kamaldinov, Qrator Labs, reveals it.
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Total votes 4: ↑4 and ↓0 +4
Views 1.1K
Comments 2

AngouriMath 1.3 update

Open source *.NET *C# *Mathematics *F# *

Four months of awesome work together with a few new contributors finally result in a new major release, which I'm happy to announce about.

Now we get completely new matrices, improved parser, a lot of new functions, almost rewritten interactive package (for working in Jupyter) and many more.

This article about a big update in a FOSS symbolic algebra library for .NET, I hope it may be interesting for someone!

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Total votes 5: ↑5 and ↓0 +5
Views 3.5K
Comments 0

Overview of Morris's counters

Qrator Labs corporate blog High performance *Algorithms *Mathematics *

On implementing streaming algorithms, counting of events often occurs, where an event means something like a packet arrival or a connection establishment. Since the number of events is large, the available memory can become a bottleneck: an ordinary n-bit counter allows to take into account no more than 2^n - 1events.
One way to handle a larger range of values using the same amount of memory would be approximate counting. This article provides an overview of the well-known Morris algorithm and some generalizations of it.

Another way to reduce the number of bits required for counting mass events is to use decay. We discuss such an approach here [3], and we are going to publish another blog post on this particular topic shortly.

In the beginning of this article, we analyse one straightforward probabilistic calculation algorithm and highlight its shortcomings (Section 2). Then (Section 3), we describe the algorithm proposed by Robert Morris in 1978 and indicate its most essential properties and advantages. For most non-trivial formulas and statements, the text contains our proofs, the demanding reader can find them in the inserts. In the following three sections, we outline valuable extensions of the classic algorithm: you can learn what Morris's counters and exponential decay have in common, how to improve the accuracy by sacrificing the maximum value, and how to handle weighted events efficiently.

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Total votes 12: ↑12 and ↓0 +12
Views 776
Comments 0

Compilation of math functions into Linq.Expression

Programming *.NET *Algorithms *C# *Mathematics *

Here I am going to cover my own approach to compilation of mathematical functions into Linq.Expression. What we are going to have implemented at the end:

1. Arithmetical operations, trigonometry, and other numerical functions

2. Boolean algebra (logic), less/greater and other operators

3. Arbitrary types as the function's input, output, and those intermediate

Hope it's going to be interesting!

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Total votes 4: ↑4 and ↓0 +4
Views 4.6K
Comments 1

Doing «Data Science» even if you have never heard the words before

Python *Algorithms *Mathematics *Machine learning *Artificial Intelligence

There’s a lot of talk about machine learning nowadays. A big topic – but, for a lot of people, covered by this terrible layer of mystery. Like black magic – the chosen ones’ art, above the mere mortal for sure. One keeps hearing the words “numpy”, “pandas”, “scikit-learn” - and looking each up produces an equivalent of a three-tome work in documentation.

I’d like to shatter some of this mystery today. Let’s do some machine learning, find some patterns in our data – perhaps even make some predictions. With good old Python only – no 2-gigabyte library, and no arcane knowledge needed beforehand.

Interested? Come join us.

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Rating 0
Views 1.1K
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Jupyter for .NET. «Like Python»

.NET *C# *Mathematics *F# *
A few months ago Microsoft announced about the creation of Jupyter for .NET. However, people are barely interested in it despite how attractive the topic is. I decided to make a LaTeX wrapper for the Entity class from a symbolic algebra library:

Looks awesome. Is simple. Very enjoyable. Let's see more!
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Total votes 2: ↑2 and ↓0 +2
Views 1.6K
Comments 0

Objects Representations for Machine Learning system based on Lattice Theory

Mathematics *Machine learning *

This is a fourth article in the series of works (see also first one, second one, and third one) describing Machine Learning system based on Lattice Theory named 'VKF-system'. The program uses Markov chain algorithms to generate causes of the target property through computing random subset of similarities between some subsets of training objects. This article describes bitset representations of objects to compute these similarities as bit-wise multiplications of corresponding encodings. Objects with discrete attributes require some technique from Formal Concept Analysis. The case of objects with continuous attributes asks for logistic regression, entropy-based separation of their ranges into subintervals, and a presentation corresponding to the convex envelope for subintervals those similarity is computed.

got idea!

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Total votes 5: ↑5 and ↓0 +5
Views 1.1K
Comments 0

Mathematics of Machine Learning based on Lattice Theory

Mathematics *Machine learning *

This is a third article in the series of works (see also first one and second one) describing Machine Learning system based on Lattice Theory named 'VKF-system'. It uses structural (lattice theoretic) approach to representing training objects and their fragments considered to be causes of the target property. The system computes these fragments as similarities between some subsets of training objects. There exists the algebraic theory for such representations, called Formal Concept Analysis (FCA). However the system uses randomized algorithms to remove drawbacks of the unrestricted approach. The details follow…
Areas of Formal Concept Analysis

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Views 1.5K
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MEMS accelerometers, magnetometers and orientation angles

Global Positioning Systems *Algorithms *Mathematics *Robotics

When it's necessary to evaluate the orientation angles of an object you may have the question — which MEMS sensor to choose. Sensors manufacturers provide a great amount of different parameters and it may be hard to understand if the sensor fit your needs.

Brief: this article is the description of the Octave/Matlab script which allows to estimate the orientation angles evaluation errors, derived from MEMS accelerometers and magnetometers measurements. The input data for the script are datasheet parameters for the sensors. Article can be useful for those who start using MEMS sensors in their devices. You can find the project on GitHub.
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Total votes 5: ↑5 and ↓0 +5
Views 7.6K
Comments 0

Developing a symbolic-expression library with C#. Differentiation, simplification, equation solving and many more

Open source *Programming *.NET *C# *Mathematics *

[UPD from 12.06.2021: if you're looking for a symbolic algebra library, AngouriMath is actively developed. It's on Github and has a website. Discord for questions]

Why does programming a calculator seem to be a task, which every beginner undertakes? History might have the answer — computers were created for this exact purpose. Unlike the beginners, we will develop a smart calculator, which, although won't reach the complexity of SymPy, will be able to perform such algebraic operations as differentiation, simplification, and equations solving, will have built-in latex support, and have implemented features such as compilation to speed up the computations.

What are the articles about?
It will superficially tell about assembling an expression, parsing from a string, variable substitution, analytic derivative, equation numerical solving, and definite integration, rendering to LaTeX format, complex numbers, compiling functions, simplifying, expanding brackets, and blah blah blah.
For those who urgently need to clone something, repository link.

Let's do it!
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Total votes 6: ↑5 and ↓1 +4
Views 5.5K
Comments 0

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