May 7, 2017 · 4 min read

I will try to explain a few basic concepts which will be incorporated in the applications developed inside Notary Platform. However, first we need to describe some basics from mathematics and engineering.

According to the definition in Wikipedia [https://en.wikipedia.org/wiki/Mathematical_model], a mathematical model is a description of a system/phenomenon using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling.

Nowadays, mathematical models are used in all fields of life, most intensively in physics, engineering, statistics, information technology, economy, etc. The basic idea is that the mathematical model may help to explain a system under observation — if one is able to make robust mathematical model about corresponding phenomenon, then it is easy to study the influence of effects of different variables and easy to predict the behavior of associated system. Therefore, ability to make applicable mathematical models and ability to use them give us a competitive advantage over others and helps to build the better future for all mankind.

However, a reliable treatment of different phenomena/systems is based on measurements and the description based on relations between the observed results. From the theoretical point of view, the relations are most appropriately specified in terms of abstract mathematical models representing mathematical laws. But from the practical point of view, simulated analog models based on electronic devices are sometimes more convenient [http://www.springer.com/us/book/9783642643590]. Furthermore, the use of two basic assumptions, namely (1) discretization, which allows us to describe very complex, usually continuous phenomena by system of partial differential equations, and (2) linearization, which allow us to describe observed phenomena by system of ordinary differential equations, enables practical solutions by using standard mathematical procedures.

The term mapping in mathematics [https://en.wikipedia.org/wiki/Map_(mathematics)] refers to either a function, or a morphism in category theory, which generalizes the idea of a function. In the computer field (which we refer to), a map often refers to the binary higher-order function that takes a function f and list [x0, x1, …, xn] as input parameters and returns [f(x0), f(x1), …, f(xn)], where n ≥ 0. So we can use functions (note that there are different variants and generalizations of them) to reliable treat different phenomena. This procedure can be presented as follows in the figure below.

We can generalize above diagram for two main procedures which determine our technology: encryption used in the distributed ledger technology (DLT) and artificial neural networks (ANNs) as a tool for simulation of real life phenomena.

While the first one (encryption) is deterministic, the second one (ANN) is generally probabilistic (ANN learning can produce different results for the same input; also, variables do not have specific values, but instead have probability distributions). Encryption must assure that same input always produce the same results (e.g. corresponding timestamp in the blockchain). ANNs are suitable for different kind of tasks. Instead of developing explicit functions or procedures, ANNs are capable of learning from data. Under certain conditions even ANNs (some special types) can always produce the same output for the same input. It is obvious that combination of both, on the same or different levels, can significantly improve applications of DLT. One very good example of using ANNs can be found here [https://www.armstrongeconomics.com/socrates/].

Real life phenomena are complex. Linear thinking and crowd behavior is not enough to master our lives. We need to get insight into non-linear behavior and here the science and technology can help. Many real life phenomena are dynamic and look very chaotic. However, each phenomenon has its inherent characteristics which are hardly revealed without math and modern information and computer technology.

Just to get an impression about the complexity of the real life systems the pictures below show such inherent characteristics. Only one single parameter govern these characteristics (e.g. N=2 — left and N=4 — right), but despite the similarity, the response of both systems is significantly different.

The Notary Platform team will try to simplify the presented ideas and concepts, and incorporate them into useful products.

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## Notary Platform

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