Introduction to logical entropy and its relationship to Shannon entropy

We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an “amount of information,” but, as he pointed out, “no concept of information itself was defined.” Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set-just as the Boole–Laplace logical probability is the normalized quantitative measure of the elements of a subset of a set. And partitions and subsets are mathematically dual concepts – so the logic of partitions is dual in that sense to the usual Boolean logic of subsets, and hence the name “logical entropy.” The logical entropy of a partition has a simple interpretation as the probability that a distinction or dit (elements in different blocks) is obtained in two independent draws from the underlying set. The Shannon entropy is shown to also be based on this notion of information-as-distinctions; it is the average minimum number of binary partitions (bits) that need to be joined to make all the same distinctions of the given partition. Hence all the concepts of simple, joint, conditional, and mutual logical entropy can be transformed into the corresponding concepts of Shannon entropy by a uniform non-linear dit-bit transform. And finally logical entropy linearizes naturally to the corresponding quantum concept. The quantum logical entropy of an observable applied to a state is the probability that two different eigenvalues are obtained in two independent projective measurements of that observable on that state.

The Causal Interaction between Complex Subsystems

Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems and is expected to have applications in many real world problems in different disciplines such as climate science, fluid dynamics, neuroscience, financial economics, etc.

Quantification of geometric errors made simple: application to main-group molecular structures

Nearly all electronic structure simulations begin with obtaining approximate geometries, making a systematic quantification of errors in approximate molecular structures of key importance. Recently, the geometric energy offset (GEO) framework based on a single and natural measure for quantifying and analysing these errors has been proposed [J. Phys. Chem. Lett. 2020, 11, 99579964]. An accurate and way less costly approximation to GEO is utilized here to readily quantify errors in main-group structures and analyze them in a chemically intuitive way. The use of semiexperimental geometries as a reference further simplifies the analysis. The analysis reveals new insights into the geometric performance of methods, new rankings, as well as patterns across different classes of methods and basis sets that arise from the analysis.

Comparative Morphology of the Stinger in Social Wasps (Hymenoptera: Vespidae)

The physical features of the stinger are compared in 51 species of vespid wasps: 4 eumenines and zethines, 2 stenogastrines, 16 independent-founding polistines, 13 swarm-founding New World polistines, and 16 vespines. The overall structure of the stinger is remarkably uniform within the family. Although the wasps show a broad range in body size and social habits, the central part of the venom-delivery apparatus—the sting shaft—varies only to a modest extent in length relative to overall body size. What variation there is shows no apparent correlation with social habits. This is consistent with the hypothesis that stinger size is constrained by the demands of a flight-worthy body. The sting lancets bear distinct, acute barbs in all examined species except in members of the Stenogastrinae. Barbs vary considerably among species in number, their summed lengths, and the relative degree of serration (summed length relative to lancet width). Where they are numerous and strong, it increases the likelihood of the stinger remaining fatally embedded in the skin of a vertebrate adversary (sting autotomy). Although an index that combines the number and strength of barbs is a more natural measure of overall serration, the number of barbs alone is almost as good a predictor of the likelihood of sting autotomy. Across the family as a whole, the tendency to sting autotomy is concentrated in the swarm-founding New World polistines.

Equidistribution Results for Self-Similar Measures

A well-known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one. In this paper, we give sufficient conditions for an analogue of this theorem to hold for a self-similar measure. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty }$ where $(f_n)_{n=1}^{\infty }$ is a sequence of sufficiently smooth real-valued functions satisfying some nonlinearity conditions. As a corollary of our main result, we show that if $C$ is equal to the middle 3rd Cantor set and $t\geq 1$, then with respect to the natural measure on $C+t,$ for almost every $x$, the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one.

What Could a “Natural Measure” Be?

The invocation of probabilistic considerations in physics often involves, implicitly or explicitly, some notion of relative sizes, or measures, of sets of possibilities. In equilibrium statistical mechanics, certain standard measures are introduced explicitly. It is often said that these measures are “natural,” in some sense. This chapter explores what that could mean. It does so by means of a toy example, a fictitious machine that I call the parabola gadget. The dynamics of the parabola gadget pick out a measure on the space of states of the gadget that other measures converge towards. In this sense, that measure is a natural one to use for systems that have been evolving freely long enough for the requisite washing-out of disagreements among input distributions to have taken place. We have good reason to think that the standard measures evoked in equilibrium statistical mechanics are of this sort One upshot of this is that this notion of standard measure is of no use for making judgments about probability or improbability of conditions in the early universe.

Restricted mean value theorems and the metric theory of restricted Weyl sums

We study the behaviour of Weyl sums on a subset ${\mathcal X}\subseteq [0,1)^d$ with a natural measure µ on ${\mathcal X}$. For certain measure spaces $({\mathcal X}, \mu),$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for µ-almost all points of ${\mathcal X}$ the Weyl sums satisfy the square root cancellation law. Moreover, we characterize the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $({\mathcal X}, \mu)$. We obtain general results, which we refine for some special spaces ${\mathcal X}$ such as spheres, moment curves and line segments.

Effective Cylindrical Cell Decompositions for Restricted Sub-Pfaffian Sets

The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format ${{\mathcal{F}}}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree $D$, recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.

Quantifying Noninvertibility in Discrete Dynamical Systems

Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.

Quantifying and understanding errors in molecular geometries

Electronic structure calculations are ubiquitous in most branches of chemistry, but all have errors in both energies and equilibrium geometries. Quantifying errors in possibly dozens of bond angles and bond lengths is a Herculean task. A single natural measure of geometric error is introduced, the geometry energy offset (GEO). GEO links many disparate aspects of geometry errors: a new ranking of different methods, quantitative insight into errors in specific geometric parameters, and insight into trends with different methods. GEO can also reduce the cost of high-level geometry optimizations and shows when geometric errors distort the overall error of a method. Results, including some surprises, are given for both covalent and weak interactions.