Mathematics *

This world needs a new theory — a theory that could describe all the theories on the planet. A theory that could easily describe philosophy, mathematics, physics, and psychology. The one that makes all kinds of sciences computable.

This is exactly what we are working on. If we succeed, this theory will become the unified meta-theory of everything.

A year has passed since our last publication, and our task is to share the progress with our the English-speaking audience. This is still not a stable version; it’s a draft. Therefore, we welcome any feedback, as well as your participation in the development of the links theory.

As with everything we have done before, the links theory is published and released into the public domain — it belongs to humanity, that means, it is yours. This work has many authors, but the work itself is far more important than any specific authorship. We hope that today it can become useful to more people.

We invite you to become a part of this exciting adventure.

This article explores the concept of equivalence classes from the perspective of mathematical analysis and their application in QA testing. The author explains how properly defining equivalence classes helps optimize test design, reducing the number of test cases while maintaining thorough verification.

Using the example of currency conversion from rubles to euros, the article demonstrates how to construct equivalence classes, verify their compliance with mathematical properties (reflexivity, symmetry, transitivity), and identify errors in data partitioning.

This article is useful for QA engineers, developers, and analysts who want to gain a deeper understanding of logical testing principles and improve the efficiency of their test strategies.

/Sandbox 23.12.2024/

On 17.10.2024, the article “A new inherent approach to solving the Collatz 3n+1 problem and its analogues” was published on the Academia.edu website [1]. The second link is for those who find it easier to read in Russian “Новый внутренне присущий подход к решению проблемы Коллатца 3n+1 и ее аналогов” [2].

The English version was originally intended for the arXiv preprint platform, but they suggested first publishing in a peer-reviewed mathematical journal. Attempts to access other platforms HAL, Qeios, and ResearchGate were met with the requirement for affiliation, which an independent researcher does not have.

The process took almost two months — more than the research itself from idea to text. As a result, the article ended up on the Academia site, which is free from “face control”. I think it will be useful for anyone interested in the Collatz conjecture to read it. Exclusively for Habr, this short text summarizing the content and meaning of the publication.

The explicit reparameterization trick is often used to train various latent variable models due to the ease of calculating gradients of continuous random variables. However, due to its peculiarities, explicit reparameterization trick is not applicable to several important continuous standard distributions, such as mixture, Gamma, Beta and Dirichlet.

An alternative method for calculating reparameterization gradients relies on implicit differentiation of cumulative distribution functions. The implicit reparameterization trick is much more expressive and applicable to a wider class of distributions

This article provides an overview of various reparameterization tricks and announces a new Python library, irt.distributions, for sampling from various distributions using the implicit reparameterization trick.

S.B. Pshenichnikov

The article outlines a new mathematical apparatus for verbal calculations in NLP (natural language processing). Words are embedded not in a real vector space, but in an algebra of extremely sparse matrix units. Calculations become evidence-based and transparent. The example shows forks in calculations that go unnoticed when using traditional approaches, and the result may be unexpected.

The use of IT in Natural Language Processing (NLP) requires standardization of texts, for example, tokenization or lemmatization.

After this, you can try to use mathematics, since it is the highest form of standardization and turns the objects under study into ideal ones, for example, data tables into matrices of elements. Only in the language of matrices can one search for general patterns in data (numbers and texts).

If text is turned into numbers, then in NLP these are first natural numbers for numbering words, which are then embedded into real vectors is irreversible ed in a real vector space.

Perhaps we should not rush to do this but come up with a new type of numbers that is more suitable for NLP than numbers for studying physical phenomena. These are matrix hyperbinary numbers. Hyperbinary numbers are one of the types of hypercomplex numbers.

Hyperbinary numbers have their  own  arithmetic,  and  if  you get used to  it,  it  will  seem  more  familiar  and  simpler  than  Pythagorean arithmetic.

In Decision Support Systems (DSS), the texts are value judgments and a numbered verbal rating scale. Next (as in NLP), the numbers are turned into vectors of real numbers and used as sets of weighted arithmetic average coefficients.

Sergey Pshenichnikov, Tatiana Sotnikova

ALGEBRA OF MUSICAL TEXT

Sergey Pshenichnikov, Tatiana Sotnikova

Trio Sapiens

Musical text can be represented using matrix units, like the description of verbal texts and other symbolic sequences. In the future, mathematical recognition, and creation of musical sense with substantive justification for intermediate calculations (as opposed to AI) may become possible.

Sound has four properties: pitch, duration, volume, and timbre. Timbre is not considered yet. The dictionary of the algebra of musical texts is built on the basis of musical notation for the piano.

The duration here, for the sake of brevity of the first presentation, is considered as «absolute». «Relative» is not considered, although intervals are very well studied, and their features will be needed to categorize composers.

The complexity of the musical text for the application of mathematics is explained by the desire to simplify the reading of musical notes by musicians and to minimize the use of lower and upper additional lines.

To apply text algebra to musical symbolic sequences there is no need to use a five-line staff. What is useful and familiar to musicians is «unbearably harmful» for the use of algebra. It seems advisable to use a one-line staff. In this case, the musical text becomes like the verbal text.

To solve the problem, you need to find a transformation of the canonical musical text into a «thread». And as always, for a new application of algebra, correct coordination of the subject area is necessary. In this case, each used musical notation and symbol of modern musical notation must be assigned its own serial number (natural number).

Instead of a sign, you can use the names of each note symbol - then it will be a verbal notation of musical texts written in one line «thread»).

Since the musical scale is completely represented by piano keys, the first section in height of the dictionary of musical texts consists of 88 numbered white and black keys (of which 52 are white). This eliminates the need for an octave division of the scale, octave transfer signs, keys, five alteration signs (key and random), diatonic and chromatic semitones.

All notes of the scale became fundamental in algebraic musical notation. There is an order of magnitude more of them of them than the main stages of Guido Aretinsky, but the alteration signs and names of octaves disappeared, the use of which made musical texts algebraically incompatible with verbal texts. Numbers from 1 to 88 in algebraic notation constitute a fragment of the pitch dictionary for the «thread» one-line staff.

Numbering (coordination) of notes is needed to become in the future indices of mathematical objects (matrix units), which will replace the signs of notes or their names. These matrix units are binary generalizations of integers (hyperbinary numbers). The operation of division with remainder is defined for them, as for integers. The operation will allow you to divide musical texts and their f

In this third part, we will discuss Machine Learning, specifically the prediction task in the context of information theory.

The concept of Mutual Information (MI) is related to the prediction task. In fact, the prediction task can be viewed as the problem of extracting information about the signal from the factors. Some part of the information about the signal is contained in the factors. If you write a function that calculates a value close to the signal based on the factors, then this will demonstrate that you have been able to extract MI between the signal and the factors.

The traditional definition for the operation of exponentiation to a natural power (or a positive integer) had introduced approximately as follows:

Exponentiation is an arithmetic operation originally defined as the result of multiple multiplications a number by itself.

But the more precise formulation is still different:

Raising a number X to an integer power N is an arithmetic operation defined as the result of multiple [N by mod times] multiplications or divisions one by number X.

How to design an economy for your game? The answer to this question might require a series of lectures or articles. The fundamental difference in the approach is based, first of all, on monetization model: F2P or B2P. The second thing that defines the approach to developing an economy system is game genre. This article reviews the case of designing the game economy for a B2P (premium) game, which doesn’t involve earning on microtransactions.

Surprisingly, there are strict mathematical methods that literally allow to hear visual geometric forms and, conversely, to see the beauty of musical harmonies...

[Read on Russian]

In Part 1, we became familiar with the concept of entropy.

In this part, we will delve into the concept of Mutual Information, which opens doors to error-resistant coding, compression algorithms, and offers a fresh perspective on regression and Machine Learning tasks.

It is an essential component that will pave the way, in the next section, for tackling Machine Learning problems as tasks of extracting mutual information between features and the predicted variable.

Here, there will be three interesting and crucial visualizations.

The first one will visualize entropy for two random variables and their mutual information.
The second one will shed light on the very concept of dependency between two random variables, emphasizing that zero correlation does not imply independence.
The third one will demonstrate that the bandwidth of an information channel has a straightforward geometric interpretation through the convexity measure of the entropy function.

In the meantime, we will prove a simplified version of the Shannon-Hartley theorem regarding the maximum bandwidth of a noisy channel. Let's dive in!

I've long wanted to create educational materials on the topic of Information Theory + Machine Learning. I found some old drafts and decided to polish them up here, on Habr.

Information Theory and Machine Learning seem to me like an interesting pair of fields, the deep connection between which is often unknown to ML engineers, and whose synergy has not yet been fully revealed.

Let's start with basic concepts like Entropy, Information in a message, Mutual Information, and channel capacity. Next, there will be materials on the similarity between tasks of maximizing Mutual Information and minimizing Loss in regression problems. Then there will be a section on Information Geometry: Fisher metric, geodesics, gradient methods, and their connection to Gaussian processes (moving along the gradient using SGD is moving along the geodesic with noise).

It's also necessary to touch upon AIC, Information Bottleneck, and discuss how information flows in neural networks – Mutual Information between layers (Information Theory of Deep Learning, Naftali Tishby), and much more. It's not certain that I'll be able to cover everything listed, but I'll try to get started.

1 Playing Diplomacy

1.1 What this work is about

1.2 A puzzle for the talented and brave (Eccentrics are welcome: ?)

Experiments on the Torus

1. About this series of articles

1.1 Central result

The life of Langton's Ant seems sad and lonely, but, as we'll soon discover, he is not ready to put up with such an outrageous situation and is trying his best to escape. American scientist Christopher Langton invented his ant back in 1986. Since then, no one has been able to explain the strange behavior of this mysterious model...

On the Internet and in non-fiction literature you can often find various mathematical tricks. The Collatz conjecture leaves all such tricks behind. At first glance, it may seem like some kind of a trick with a catch. However, there is no catch. You think of a number and repeat one of two arithmetic operations for it several times. Surprisingly, the result of these actions will always be the same. Or, may be not always?

I just started to learn Game Development, and decided to run write my personal blog about it. So there you can find information(resources, blogs, courses, books) that i've gathered and my personal problems with learning)

Rust's type system is quite powerful as it allows to encode complex relationships between user-defined types using recursive rules that are automatically applied by the compiler. Idea behind this post is to use some of those rules to encode properties of our domain. Here we take a look at Peano axioms defined for natural numbers and try to derive some of them using traits, trait bounds and recursive impl blocks. We want to make the compiler work for us by verifying facts about our domain, so that we could invoke the compiler to check whether a particular statement holds or not. Our end goal is to encode natural numbers as types and their relationships as traits such that only valid relationships would compile. (e.g. in case we define types for 1 and 3 and relationship of less than, 1 < 3 should compile but 3 < 1 shouldn't, that all would be encoded using Rust's language syntax of course)

Let's define some natural numbers on the type level first.