Asked 17th Oct, 2022

How can we calculate the dimensionality of some new investigated discrete space?

How can we calculate the number of dimensions in a discrete space if we only have a complete scheme of all its points and possible transitions between them (or data about the adjacency of points)? Such a scheme can be very confusing and far from the clear two- or three-dimensional space we know. We can observe it, but it is stochastic and there are no regularities, fractals or the like in its organization. We only have access to an array of points and transitions between them.
Such computations can be resource-intensive, so I am especially looking for algorithms that can quickly approximate the dimensionality of the space based on the available data about the points of the space and their adjacencies.
I would be glad if you could help me navigate in dimensions of spaces in my computer model :-)

All Answers (3)

Anil Kumar Jain
Jain Super Mart
Applications are made with some set of conditions only, you may design or can think your own different algorithm as per need but a perfect answer or any such applications with random choices do not exist. In short from design point of view question is false or not asked as per needs.
Sergey Miroshnikov
Saint Petersburg State University
Thank you for your answer! Did I understand correctly that my question is too general and therefore cannot have a universal answer - an algorithm for all possible cases?
Can I then ask another question: do you know any approaches (algorithms) to calculating dimensionality for discrete space, or can we talk about calculating dimensionality only for linear continuous space?
Sergey Miroshnikov
Saint Petersburg State University
Anil Kumar Jain The description of discrete spaces is found in physical works, e.g. "Discrete spacetime, quantum walks and relativistic wave equations" by Leonard Mlodinow and Todd A. Brun, But I have not seen any attempt to quantify the dimensionality of such spaces. This is exactly what I am looking for.

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The original manuscript of this article is to answer some questions asked by Zhihu users, here added the second half of the content into the original manuscript, one for sharing, and the other to keep some views written casually, so as not to be lost, it may be useful in the future.
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In the entire thermodynamic theoretical system, you can hardly find such a sensation, such as delicate, rigorous, physical image clarity, similar to that in the other theoretical systems of physics, and the appeasement philosophy is all over the place.
Statistical physics cannot independently establish equations for the relationships between thermodynamic state functions, relies on thermodynamics in the theoretical system, which also inherited the problems from thermodynamic theory. Statistical physics itself also brings more problems, for instance, statistical physics cannot explain such a process, an ideal gas does work to compress a spring, the internal energy of the ideal gas is converted into the elastic potential energy of the spring. If such simple, realistic problem cannot be explained, what are the use your statistical ensemble, phase spaces, the Poincaré recurrence theorem, mathematical transformations?
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Our answer is YES. A new question (at has been answered affirmatively, confirming the YES answer in this question, with wider evidence in +12 areas.
This question continued the same question from 3 years ago, with the same name, considering new published evidence and results. The previous text of the question maybe useful and is available here:
We now can provably include DDF [1] -- the differentiation of discontinuous functions. This is not shaky, but advances knowledge. The quantum principle of Niels Bohr in physics, "all states at once", meets mathematics and quantum computing.
Without infinitesimals or epsilon-deltas, DDF is possible, allowing quantum computing [1] between discrete states, and a faster FFT [2]. The Problem of Closure was made clear in [1].
Although Weyl training was on these mythical aspects, the infinitesimal transformation and Lie algebra [4], he saw an application of groups in the many-electron atom, which must have a finite number of equations. The discrete Weyl-Heisenberg group comes from these discrete observations, and do not use infinitesimal transformations at all, with finite dimensional representations. Similarly, this is the same as someone trained in infinitesimal calculus, traditional, starts to use rational numbers in calculus, with DDF [1]. The similar previous training applies in both fields, from a "continuous" field to a discrete, quantum field. In that sense, R~Q*; the results are the same formulas -- but now, absolutely accurate.
New results have been made public [1-3], confirming the advantages of the YES answer, since this question was first asked 3 years ago. All computation is revealed to be exact in modular arithmetic, there is NO concept of approximation, no "environmental noise" when using it.
As a consequence of the facts in [1], no one can formalize the field of non-standard analysis in the use of infinitesimals in a consistent and complete way, or Cauchy epsilon-deltas, against [1], although these may have been claimed and chalk spilled.
Some branches of mathematics will have to change. New results are promised in quantum mechanics and quantum computing.
This question is closed, affirming the YES answer.
Preprint FT = FFT

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