257.15

# Mathematics *

Mother of all sciences

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## VERBAL CALCULATION (VC) IN EVIDENCE-BASED DSS AND NLP

Medium
14 min
235

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.

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## ALGEBRA OF MUSICAL TEXT

Medium
5 min
252

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

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## 3. Information theory and ML. Forecast

31 min
545

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.

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## Why x^0 = 1 visually

3 min
1.1K

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.

Let's figure it out under the cut! >>
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## Game economy design of Premium games through the example of a 4X strategy on PC

Medium
9 min
1.1K

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.

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## GEOMETRY OF SOUND

5 min
1.2K

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

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## 2. Information Theory + ML. Mutual Information

11 min
799

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!

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## 1. Information theory + ML. Entropy

10 min
1K

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.

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## Affordable as a Bus, Comfortable as a Taxi: A Promising Type of Public Transport for Large and Medium-Sized Cities.Part3

Medium
32 min
1.4K

Translation provided by ChatGPT, link to the original article in Russian

Link to Part 1: «Preliminary Analysis» (ру / eng )
Link to Part 2: «Experiments on a Torus» (ру / eng )
Link to Part 3: «Practically Significant Solutions» (ру / eng )
Link to «Summary» (ру / eng )

### 1 Playing Diplomacy

#### 1.1 What this work is about

You're reading the third and final article in a series dedicated to minibus route schemes that would allow you to travel reasonably quickly, inexpensively, and most importantly, without any transfers, from any intersection to any other within a large city. You'll see many graphs, formulas, and figures below, but before we get to the technical part, I'd like to discuss the challenge of implementing this idea and invite you to participate in solving it.

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

I propose a game,
I propose that you become part of a positive change in the lifestyle of almost a billion people around the planet,
I can't do this alone.
To start, I need your help with the following:
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## Affordable as a Bus, Comfortable as a Taxi: A Promising Type of Public Transport for Large and Medium-Sized Cities.Part2

Medium
56 min
1.1K

(Jean-Claude Mézières)

Translation provided by ChatGPT, link to the original article in Russian

Link to Part 1: «Preliminary Analysis» (ру / eng )
Link to Part 2: «Experiments on a Torus» (ру / eng )
Link to Part 3: «Practically Significant Solutions» (ру / eng )
Link to «Summary» (ру / eng )

## Experiments on the Torus

This is the second part of a study dedicated to exploring new public transportation movement schemes. In the first part, we examined the simplest non-stop scheme and a single-transfer scheme based on it, which can be implemented in a grid city on a plane. In this part, our city model will be a grid city on a «flat» torus. Unlike a rectangle, a torus has no edge, and the positions of all points on it are absolutely equivalent. Due to the absence of an edge and (transitive) symmetry, calculations for a toroidal city are simpler, and numerical results are nearly identical to those for a rectangular city on a plane. These two conditions make a toroidal grid city an ideal testing ground for new passenger transportation movement schemes. In this article, we will explore two such schemes on the torus, and in the next one, we will return to the plane and adapt the results obtained here for use under the realistic conditions of a rectangular city.

The content of this study is not standalone and presupposes familiarity with the first part of the article. To understand Chapter 2, you will need a level of mathematics that corresponds roughly to the first two years of university; for everything else, high school level should suffice. It can be helpful to have a pencil and a piece of paper at hand while reading. If your browser displays formulas incorrectly, try refreshing the page a few times.
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## Affordable as a Bus, Comfortable as a Taxi: A Promising Type of Public Transport for Large and Medium-Sized Cities.Part1

Medium
40 min
1.9K

(Jean-Claude Mézières)

Translation provided by ChatGPT, link to the original article in Russian

Link to Part 1: «Preliminary Analysis» (ру / eng )
Link to Part 2: «Experiments on a Torus» (ру / eng )
Link to Part 3: «Practically Significant Solutions» (ру / eng )
Link to «Summary» (ру / eng )

#### 1.1 Central result

If I haven't made a critical mistake, I have discovered an astonishing passenger transportation scheme with unique characteristics. Imagine this scenario: you are in a big city and need to get from point A to point B. All you need to do is walk to the nearest intersection and indicate the destination on your smartphone or a special terminal installed there. In a few minutes, a small but spacious bus will arrive for you. The bus is designed for easy entry without bending, and you can bring a stroller, bicycle, or even a cello inside. It provides comfortable seating where you can stretch your legs. This bus will take you to the nearest intersection to point B, and you will reach your destination without any transfers. The entire journey, including waiting at the stop, will take only 25-50% more time than if you were traveling by private car. Based on my estimation, in modern metropolises, this type of transportation will be widely adopted, and the cost of a trip on such buses will be similar to the fare of a regular city bus.

Surprisingly, the reasoning behind these findings is based on relatively simple mathematics, and perhaps even a talented high school student, under fortunate circumstances, could have guessed them on their own. The practical significance of the topic and the modest level of mathematical requirements prompted me to make an effort to write the article in such a way that the reader could follow the path of discoveries, learn some research techniques, and gain a successful example to explain to their children the purpose of mathematics and how it can be applied in everyday life.
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## «Promising Public Transportation for Large and Medium-Sized Cities» — the main idea in a brief summary

Easy
9 min
1.5K

(source)

Translation provided by ChatGPT, link to the original article.

I recently published a series of articles titled 'As Cheap as a Bus, as Convenient as a Taxi...':

Link to Part 1: «Preliminary Analysis» (ру / eng )
Link to Part 2: «Experiments on a Torus» (ру / eng )
Link to Part 3: «Practically Significant Solutions» (ру / eng )

dedicated to making public transportation in large cities completely seamless, without the need for transfers. In the last article of the series, I extensively described a microbus movement scheme that allows them to operate almost like taxis while accommodating 5-10 passengers at once. Such a transportation system would enable city residents to travel from any intersection to another without any transfers, comparable in time to a personal car journey, and at a cost similar to a regular city bus ticket. However, the feedback from readers indicated that I chose an extremely ineffective way to convey the information, resulting in a failure to effectively communicate the essence of the matter.

I must admit that the previous three articles were written in a way that allowed readers to apply the acquired knowledge in practice or continue the research I started on their own. Unfortunately, my desire to 'teach' resulted in nearly 100 pages of complex mathematical text, which is clearly excessive for readers who simply wanted to familiarize themselves with the idea. Here, I will attempt to rectify this mistake and briefly, yet simply, explain the bus taxi technology.
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Medium
1 min
1K

Task: To provide automation for transfer of large number of files.

Source - computer with autotest codebase.

Receiver - gateway for industrial data processing.

Test receiver - second PC with installed vsftpd service.

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## Langton's ant: a mystery cellular automaton

4 min
2.5K

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...

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## The Collatz conjecture is the greatest math trick of all time

4 min
3.2K

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?

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## Let's start in GameDev

2 min
1.9K

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)

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## Proof's by induction using Rust's type-system

5 min
1.9K

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.

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## The On-Line Encyclopedia of Integer Sequences today

14 min
1.5K

You can encounter integer sequences all around combinatorics, number theory, and recreational mathematics. And if there is a multitude of objects of the similar form, then one can create an index for these objects. The On-Line Encyclopedia of Integer Sequences, OEIS, is such an index.

This is a translation of my article The On-Line Encyclopedia of Integer Sequences in 2021, published in Mat. Pros. Ser. 3 28, 199–212 (2021).

This article covers the On-Line Encyclopedia inclusion criteria, its editorial process, its role in mathematics, and its future.

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## Methodology for calculating results of a task set: taking into account its level of difficulty

3 min
2.4K

In the world of academic knowledge evaluation, objective calculation of large data presents a serious problem. Can a student studying in an Advanced Maths class and getting B-marks be evaluated equally with another student, getting B-marks in a General Maths class? Can we create a system that would take into account the level of difficulty those students face?

This article will describe a system of independent evaluation we have been using for school olympics in five subjects (Mathematics, English Language, Russian Language, Tatar Language, Social Science) for students grades 1 to 11. In each academic year we organise six qualification tournaments, with about 15,000 students from different regions of Russia. Then we select the top ten participants in each subject and each grade for their future participation in the final (seventh) tournament, where only the best of the best are chosen. It means that 550 participants compete in the final tournament, which is about 5.5% of all participants in the academic year.

It is obvious that those multiple tournaments cannot be absolutely homogenous, and inevitably the levels of difficulty for each set of tasks vary. Therefore, it is critical for us to take into consideration those variations of difficulty and calculate the results in the most objective manner.

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## Let’s Discuss the Lorentz Transforms – Part the Last: The Real Derivation, or The Nail in the Casket

9 min
948

In this post there are a lot of references to the previous one – it is essential that you read it before getting down to this.

In my previous posts (see the list below below) I tried to express my doubts whether there is a real physical substrate to the Lorentz transforms. The assumptions about the constancy of the speed of light, the homogeneity of space-time, and the principle of relativity do not and cannot lead to the deduction of the Lorentz transforms – Einstein himself, for one, gets quite different transforms, and from those he goes over directly to the Lorentz transforms obviously missing a logical link (see Einstein p. 7, and also Part 1 of this discussion). As for the light-like interval being equal to zero, we saw that it can be attached to such assumptions only in error and cannot in itself be a foundation of a theory. I have to conclude that all that fine, intricately latticed construction of scientifictitious, physics-like arguments with the air of being profound is nothing but a smokescreen creating the appearance of a physical foundation while there is none.

What is then the real foundation of the Lorentz transforms? Let’s start from the rear end, the Minkowski mathematics. Historically, this appeared later than special relativity as a non-contradictory model of the Lorentz mathematical world; previously mentioned Varićak was among those who took part in its creation. Notwithstanding its coming later in history, it can be used as the starting point for derivation of the Lorentz transforms.