Related papers: The Statistic $\mathtt{pinv}$ for Number System
The Stieltjes (or sometimes called the Cauchy) transform is a fundamental object associated with probability measures, corresponding to the generating function of the moments. In certain applications such as free probability it is essential…
A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of…
A bitset is a set that encodes for a binary number. Bitsets are at the basis of a beautiful theory of recombination with n-loci and here we begin from scratch and advance to include the derivation of the fundamental results about the…
Statistical inference is the science of drawing conclusions about some system from data. In modern signal processing and machine learning, inference is done in very high dimension: very many unknown characteristics about the system have to…
The theory of grey systems plays an important role in science,engineering and in the everyday life in general for handling approximate data. In the present paper grey numbers are used as a tool for assessing with linguistic expressions the…
A permutiple is a natural number that is a nontrivial multiple of a permutation of its digits in some base. Special cases of permutiples include cyclic numbers (multiples of cyclic permutations of their digits) and palintiple numbers…
Mathematics is a limited component of solutions to real-world problems, as it expresses only what is expected to be true if all our assumptions are correct, including implicit assumptions that are omnipresent and often incorrect.…
The technique known as group averaging provides powerful machinery for the study of constrained systems. However, it is likely to be well defined only in a limited set of cases. Here, we investigate the possibility of using a `renormalized'…
The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…
We consider so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular we…
Model checking is the process of deciding whether a system satisfies a given specification. Often, when the setting comprises multiple processes, the specifications are over sets of input and output signals that correspond to individual…
In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such behavior in terms of…
The present article is devoted to the investigation of some properties of the generalized shift operator of numbers represented in terms of numeral systems with a variable alphabet.
We present a formulation of the deformed oscillator algebra which leads to intermediate statistics as a continuous interpolation between the Bose-Einstein and Fermi-Dirac statistics. It is deduced that a generalized permutation or exchange…
Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example,…
We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial…
We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most…
In this article, we study a non-uniform distribution on permutations biased by their number of records that we call \emph{record-biased permutations}. We give several generative processes for record-biased permutations, explaining also how…
A general link between geometry and intermittency in passive scalar turbulence is established. Intermittency is qualitatively traced back to events where tracer particles stay for anomalousy long times in degenerate geometries characterized…
Spectral statistics of systems that undergo many--body localization transition are studied. An analysis of the gap ratio statistics from the perspective of inter- and intra-sample randomness allows us to pin point differences between…