相关论文: The base-e representation of numbers and the power…
In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes…
A closed-form formula is derived for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also…
It is declared that the aim of simplifying representations of coefficients of power series of classical statistical mechanics is to simplify a process of obtaining estimates of the coefficients using their simplified representations. The…
The tree based representation described in this paper, hereditarily binary numbers, applies recursively a run-length compression mechanism that enables computations limited by the structural complexity of their operands rather than by their…
Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical…
Factor graphs are important models for succinctly representing probability distributions in machine learning, coding theory, and statistical physics. Several computational problems, such as computing marginals and partition functions, arise…
We consider a system composed of a fixed number of particles with total energy smaller or equal to some prescribed value. The particles are non-interacting, indistinguishable and distributed over fixed number of energy levels. The energy…
A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree $k$ follows a power law, decaying like $k^{-\alpha}$, often with $2 < \alpha < 3$. However,…
In discrete contexts such as the degree distribution for a graph, \emph{scale-free} has traditionally been \emph{defined} to be \emph{power-law}. We propose a reasonable interpretation of \emph{scale-free}, namely, invariance under the…
Organisms have to keep track of the information in the environment that is relevant for adaptive behaviour. Transmitting information in an economical and efficient way becomes crucial for limited-resourced agents living in high-dimensional…
Place value numbers, such as the binary or decimal numbers can be represented by the end vertices (leaf or pendant vertices) of rooted symmetrical trees. Numbers that consist of at most a fixed number of digits are represented by vertices…
We study two positional numeration systems which are known for allowing very efficient addition and multiplication of complex numbers. The first one uses the base $\beta = \imath - 1$ and the digit set $\mathcal{D} = \{ 0, \pm 1, \pm \imath…
This work studies the problem of constructing a representative workload from a given input analytical query workload where the former serves as an approximation with guarantees of the latter. We discuss our work in the context of workload…
Does the brain construct an efficient representation of the sensory world? We review progress on this question, focusing on a series of experiments in the last decade which use fly vision as a model system in which theory and experiment can…
Why does Zipf's law give a good description of data from seemingly completely unrelated phenomena? Here it is argued that the reason is that they can all be described as outcomes of a ubiquitous random group division: the elements can be…
Identifying the trade-offs between model-based and model-free methods is a central question in reinforcement learning. Value-based methods offer substantial computational advantages and are sometimes just as statistically efficient as…
In the field of decision trees, most previous studies have difficulty ensuring the statistical optimality of a prediction of new data and suffer from overfitting because trees are usually used only to represent prediction functions to be…
We first describe a general class of optimization problems that describe many natural, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances…
A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to…
Full binary trees naturally represent commutative non-associative products. There are many important examples of these products: finite-precision floating-point addition and NAND gates, among others. Balance in such a tree is highly…