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...
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.
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.
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.
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.
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.
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.
I thought my previous post rather funny, and was surprised seeing it initially receive so few views. I thought the entertainment flopped, but fortunately I was wrong. I therefore feel it my duty before my readers to address the subject of the Landau & Lifschitz proof of the invariance of the interval.
You can find the summary of it in Wikipedia. Making their starting point the light-like interval always being equal to zero, Landau & Lifschitz seem to make a great fuss about it. The Wikipedia article even says: ‘This is the immediate mathematical consequence of the invariance of the speed of light.’ No, it is not.
I beg everyone’s pardon, but the light-like interval always being equal to zero is nothing else but the following statement: ‘The length of a ray of light will always be equal to the length of this ray of light’. Sounds like a cool story, bros and sis, but I cannot see what further inferences can be drawn from it. The ‘proof’ of this truism cannot fail under any circumstances whatever – whether you keep the speed of light invariant, or keep or change the metric of space or time or both – or make both metric and speed of light change – the light-like interval will remain equal to zero. I am okay with anyone wanting to prove it if they feel like it, but you cannot make it an ‘immediate mathematical consequence of the invariance of the speed of light’. Neither is it possible to make the constancy of the speed of light a consequence of the invariance of the light-like interval for the reason already mentioned: this is a truism. It does not prove anything, nor can it be a consequence of anything. When Landau & Lifschitz insist that this is a consequence of the constancy of the speed of light, that is either an error or a downright subterfuge, a means employed to create a spectre of logical connection between two unconnected notions, and charge this ghostly connection with pretended significance. And, since the following proof of invariance of an arbitrary interval hangs on the invariance of the light-like interval, we can altogether dismiss it: the necessity of introduction of such a measure as interval cannot be derived from the statement that a length of something will be equal to itself in whatever frame of reference it is measured.
The previous discussion done, we have surmounted the difficult waters and are now sailing into something much more pleasure-like and hopefully even entertaining.
As I promised, we will be discussing the invariance of the interval, that is to say, the following relation:
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.
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?
