In this article, we would like to compare the core mathematical bases of the two most popular theories and associative theory.
Mathematics *
Mother of all sciences
Riddles of the fast Fourier transform
• 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.
The Ideal Economy
I am not an economist, but in light of current events with cryptocurrencies and the economy in general, I would like to share my thoughts on some kind of ideal economy, around which everything is happening now.
One does not simply calculate the absolute value
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?
Measuring Traffic Rate by Means of U-models
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.
AngouriMath 1.3 update
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!
Overview of Morris's counters
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 -bit counter allows to take into account no more than events.
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.
Compilation of math functions into Linq.Expression
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!
What's new in AngouriMath 1.2?
After 210 days, 600 commits, tens of debugging nights, and thousands of messages in the project chat, I finally released AngouriMath 1.2.
This is an open-source symbolic algebra library for C# and F#, maybe it is interesting for someone?
Doing «Data Science» even if you have never heard the words before
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.
Jupyter for .NET. «Like Python»
Entity
class from a symbolic algebra library:Looks awesome. Is simple. Very enjoyable. Let's see more!
Objects Representations for Machine Learning system based on Lattice Theory
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.
Mathematics of Machine Learning based on Lattice Theory
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…
MEMS accelerometers, magnetometers and orientation angles
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.
Developing a symbolic-expression library with C#. Differentiation, simplification, equation solving and many more
[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.
For those who urgently need to clone something, repository link.
Let's do it!
How elliptic curve cryptography works in TLS 1.3
A couple of reader alerts:
In order to (somewhat) simplify the description process and tighten the volume of the article we are going to write, it is essential to make a significant remark and state the primary constraint right away — everything we are going to tell you today on the practical side of the problematics is viable only in terms of TLS 1.3. Meaning that while your ECDSA certificate would still work in TLS 1.2 if you wish it worked, providing backwards compatibility, the description of the actual handshake process, cipher suits and client-server benchmarks covers TLS 1.3 only. Of course, this does not relate to the mathematical description of algorithms behind modern encryption systems.
This article was written by neither a mathematician nor an engineer — although those helped to find a way around scary math and reviewed this article. Many thanks to Qrator Labs employees.
(Elliptic Curve) Diffie-Hellman (Ephemeral)
The Diffie–Hellman legacy in the 21 centuryOf course, this has started with neither Diffie nor Hellman. But to provide a correct timeline, we need to point out main dates and events.
There were several major personas in the development of modern cryptography. Most notably, Alan Turing and Claud Shannon both laid an incredible amount of work over the field of theory of computation and information theory as well as general cryptanalysis, and both Diffie and Hellman, are officially credited for coming up with the idea of public-key (or so-called asymmetric) cryptography (although it is known that in the UK there were made serious advances in cryptography that stayed under secrecy for a very long time), making those two gentlemen pioneers.
In what exactly?
A City Without Traffic Jams
Chapter 2.
(the link to Chapter 1)
The Art of Designing Road Networks
Transport problems of a city through the eyes of a Computer Scientist
If I were recommended an article with the title “The Art of Designing Road Networks,” I would immediately ask how many road networks were built with the participation of its author. I must admit, my professional activity was far from road construction and was recently associated with the design of microprocessors where I, among other responsibilities, was engaged in the resource consumption of data switching. At that time my table stood just opposite the panoramic window which opened up a beautiful view of the long section of the Volgograd Highway and part of the Third Transport Ring with their endless traffic jams from morning to evening, from horizon to horizon. One day, I had a sudden shock of recognition: “The complexities of the data switching process that I struggle with on a chip may be similar to the difficulties the cars face as they flow through the labyrinth of road network”.
Probably, this view from the outside and the application of methods that were not traditional for the area in question gave me a chance to understand the cause of traffic jams and make recommendations on how to overcome the problem in practice.
Polygonal Mesh to B-Rep Solid Conversion: Algorithm Details and C++ Code Samples
How to Catch a Cat with TLA+
Conditions
You’re in a hallway with seven doors on one side leading to seven rooms. A cat is hiding in one of these rooms. Your task is to catch the cat. Opening a door takes one step. If you guess the correct door, you catch the cat. If you do not guess the correct door, the cat runs to the next room.
Version 12 Launches Today! (And It’s a Big Jump for Wolfram Language and Mathematica)
Quick links
— The Road to Version 12
— First, Some Math
— The Calculus of Uncertainty
— Classic Math, Elementary and Advanced
— More with Polygons
— Computing with Polyhedra
— Euclid-Style Geometry Made Computable
— Going Super-Symbolic with Axiomatic Theories
— The n-Body Problem
— Language Extensions & Conveniences
— More Machine Learning Superfunctions
— The Latest in Neural Networks
— Computing with Images
— Speech Recognition & More with Audio
— Natural Language Processing
— Computational Chemistry
— Geographic Computing Extended
— Lots of Little Visualization Enhancements
— Tightening Knowledgebase Integration
— Integrating Big Data from External Databases
— RDF, SPARQL and All That
— Numerical Optimization
— Nonlinear Finite Element Analysis
— New, Sophisticated Compiler
— Calling Python & Other Languages
— More for the Wolfram “Super Shell”
— Puppeting a Web Browser
— Standalone Microcontrollers
— Calling the Wolfram Language from Python & Other Places
— Linking to the Unity Universe
— Simulated Environments for Machine Learning
— Blockchain (and CryptoKitty) Computation
— And Ordinary Crypto as Well
— Connecting to Financial Data Feeds
— Software Engineering & Platform Updates
— And a Lot Else…
Authors' contribution
-
alizar 1716.0 -
MagisterLudi 1405.4 -
haqreu 1373.0 -
varagian 1161.0 -
Sirion 1085.0 -
mkot 980.0 -
maisvendoo 941.0 -
OsipovRoman 937.4 -
Dmytro_Kikot 935.0 -
mephistopheies 812.0