Full Generality Research Articles

Stokes, Gibbs, and Volume Computation of Semi-Algebraic Sets

We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discontinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes’ theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a refined version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this refined LP has an optimal solution which is a continuous function. Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves recent results on Poisson’s partial differential equation (PDE).

Unified treatment of Artin-type problems

Since Hooley’s seminal 1967 resolution of Artin’s primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH).

A convolutional neural network model of multi-scale feature fusion: MFF-Net

MFF-Net (a multi-scale feature fusion convolutional neural network) was designed to improve the recognition rate of handwritten digits. The low-level, middle-level and high-level features of the image were first extracted through the convolution operation, and then the low-level and intermediate features were further extracted through different convolutional layers, later directly fused with the high-level features of the image with a certain weight, and then processed by the full connection layer. By adding a batch normalization layer before the activation layer, and a dropout layer between the full connection layers, the accuracy and generalization capacity of the network are improved. At the same time, a dynamic learning rate algorithm was designed, with which, the trained network accuracy was significantly improved as shown in the experiments on the MNIST data set. The accurate rate could reach 99.66% through only 30 epochs training. The comparison indicated that the accuracy of the network model is significantly higher than that of others.

A Note on Humphreys’ Conjecture on Blocks

Humphreys’ conjecture on blocks parametrises the blocks of a reduced enveloping algebra of the Lie algebras of a reductive algebraic group over an algebraically closed field of positive characteristic. It is well-known to hold under Jantzen’s standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result in type G2 in characteristic 3, a case in which the previously mentioned weaker assumptions do not hold. This approach requires some dimensional calculations for certain centralisers, which we conduct in the Appendix for all the exceptional Lie algebras in bad characteristic.

A full generalization of the Gini index for bearing condition monitoring

The classic Gini index (GI) is generalized recently by using nonlinear weight sequences as sparsity measures for sparse quantification and machine condition monitoring. The generalized GIs with different weight parameters are more robust to random transients. However, they show insufficient performance in discriminating repetitive transients under noise contamination. To overcome this shortage, this paper proposes a two-parameter generalization method to tune not only the weight parameter but also the norm order, allowing for a full generalization of the classic GI to quantify transient features and leading to new statistical indicators which are named fully generalized GIs (FGGIs). Mathematical derivations show that FGGIs satisfy at least four of the six typical attributes of sparsity measures and that those with weight parameter equal to one satisfy at least five sparse attributes, proving that they are a new family of sparsity measures. Numerical simulations demonstrate that FGGIs can monotonically evaluate the sparseness of the signals and that the FGGIs with appropriate parameters exhibit improved performance in resisting random transient interferences and discriminating noise-contaminated repetitive transients compared to traditional sparsity measures. The performance of FGGIs in the condition monitoring of rolling element bearings is validated using two different run-to-failure experiment datasets, including a gradual failure and a sudden failure. The results show that increasing the norm order can improve the capability of FGGIs to characterize transient fault features, allowing more accurate trending of bearing health conditions, and therefore achieving better condition monitoring performance than the traditional sparsity measures.

On the classification of extremely primitive affine groups

Let G be a finite non-regular primitive permutation group on a set Ω with point stabiliser Gα. Then G is said to be extremely primitive if Gα acts primitively on each of its orbits in Ω \ {α}, which is a notion dating back to work of Manning in the 1920s. By a theorem of Mann, Praeger and Seress, it is known that every extremely primitive group is either almost simple or affine, and all the almost simple examples have subsequently been determined. Similarly, Mann et al. have classified all of the affine extremely primitive groups up to a finite, but undetermined, collection of groups. Moreover, if one assumes Wall’s conjecture on the number of maximal subgroups of an almost simple group, then there is an explicit list of candidates, each of which has been eliminated in a recent paper by Burness and Thomas. So, modulo Wall’s conjecture, the classification of extremely primitive groups is complete. In this paper we adopt a different approach, which allows us to complete this classification in full generality, independent of the veracity or otherwise of Wall’s conjecture in the almost simple setting. Our method relies on recent work of Fawcett, Lee and others on the existence of regular orbits of almost simple groups acting on irreducible modules.

Enochs’ conjecture for small precovering classes of modules

Enochs’ conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this short paper, we prove the validity of the conjecture for small precovering classes, i.e., the classes of the form Add(M) where M is any module, under a mild additional set-theoretic assumption which ensures that there are enough non-reflecting stationary sets. We even show that M has a perfect decomposition if Add(M) is a covering class. Finally, the additional set-theoretic assumption is shown to be redundant if there exists an n < ω such that M decomposes into a direct sum of אn-generated mo dules.

Limits of semistatic trading strategies

AbstractWe show that pointwise limits of semistatic trading strategies in discrete time are again semistatic strategies. The analysis is carried out in full generality for a two‐period model, and under a probabilistic condition for multiperiod, multistock models. Our result contrasts with a counterexample of Acciaio, Larsson, and Schachermayer, and shows that their observation is due to a failure of integrability rather than instability of the semistatic form. Mathematically, our results relate to the decomposability of functions as studied in the context of Schrödinger bridges.

On a question of Kazhdan and Yom Din

Studying a question of Kazhdan and Yom Din we first prove some fixed point theorems for linear actions of a discrete countable group G on a dual Banach space V*, and then show that the answer to the question in full generality is in the negative. In an appendix it is shown that the answer is negative also for the Banach space \({\cal B}\left( H \right)\) of bounded linear operators on an infinite-dimensional Hilbert space.

Weakly non-collapsed RCD spaces are strongly non-collapsed

AbstractWe prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the propertiestr⁡(Hess⁡f)=Δ⁢f\operatorname{tr}(\operatorname{Hess}f)=\Delta fonU⊆XU\subseteq{\mathsf{X}}for every 𝑓 sufficiently regular,m=c⁢Hn\mathfrak{m}=c\mathscr{H}^{n}onU⊆XU\subseteq{\mathsf{X}}for somec&gt;0c&gt;0,whereU⊆XU\subseteq{\mathsf{X}}is open and 𝖷 is a – possibly collapsed – RCD space of essential dimension 𝑛.

Constant rank factorisations of smooth maps, with applications to sonar

AbstractSonar systems are frequently used to classify objects at a distance by using the structure of the echoes of acoustic waves as a proxy for the object’s shape and composition. Traditional synthetic aperture processing is highly effective in solving classification problems when the conditions are favourable but relies on accurate knowledge of the sensor’s trajectory relative to the object being measured. This article provides several new theoretical tools that decouple object classification performance from trajectory estimation in synthetic aperture sonar processing. The key insight is that decoupling the trajectory from classification-relevant information involves factoring a function into the composition of two functions. The article presents several new general topological invariants for smooth functions based on their factorisations over function composition. These invariants specialise to the case when a sonar platform trajectory is deformed by a non-small perturbation. The mathematical results exhibited in this article apply well beyond sonar classification problems. This article is written in a way that supports full mathematical generality.

Exhaustive existence and non-existence results for Hardy–Hénon equations in {{,mathrm{{textbf{R}}},}}^n

This paper concerns solutions to the Hardy–Hénon equation -Δu=|x|σup\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\Delta u = |x|^\\sigma u^p \\end{aligned}$$\\end{document}in {{,mathrm{{textbf{R}}},}}^n with n ge 1 and arbitrary p, sigma in {{,mathrm{{textbf{R}}},}}. This equation was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions p>1,sigma ge -2, and n ge 3 is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.

The impacts of training on change deafness and build-up in a flicker task.

Performance on auditory change detection tasks can be improved by training. We examined the stimulus specificity of these training effects in behavior and ERPs. A flicker change detection task was employed in which spatialized auditory scenes were alternated until a "change" or "same" response was made. For half of the trials, scenes were identical. The other half contained changes in the spatial locations of objects from scene to scene. On Day 1, participants were either trained on this auditory change detection task (trained group), or trained on a non-auditory change detection task (control group). On Day 2, all participants were tested on the flicker task while EEG was recorded. The trained group showed greater change detection accuracy than the control group. They were less biased to respond "same" and showed full generalization of learning from trained to novel auditory objects. ERPs for "change" compared to "same" trials showed more negative going P1, N1, and P2 amplitudes, as well as a larger P3b amplitude. The P3b amplitude also differed between the trained and control group, with larger amplitudes for the trained group. Analysis of ERPs to scenes viewed prior to a decision revealed build-up of a difference between "change" and "same" trials in N1 and P2. Results demonstrate that training has an impact early in the "same" versus "change" decision-making process, and that the flicker paradigm combined with the ERP method can be used to study the build-up of change detection in auditory scenes.

Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in \(A\times B\times C\), a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses roughly \(O(n^{28/15})\) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether \(A\times B\times C\) contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. (2017) and by Chan (2018), where all three sets A, B, and C are assumed to be one-dimensional. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple \((a,b,c) \in A\times B\times C\) that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly \(O(n^{24/13})\) constant-degree polynomial sign tests.

Quantum interpolating ensemble: Bi-orthogonal polynomials and average entropies

The density matrix formalism is a fundamental tool in studying various problems in quantum information processing. In the space of density matrices, the most well-known measures are the Hilbert–Schmidt and Bures–Hall ensembles. In this work, the averages of quantum purity and von Neumann entropy for an ensemble that interpolates between these two major ensembles are explicitly calculated for finite-dimensional systems. The proposed interpolating ensemble is a specialization of the [Formula: see text]-deformed Cauchy–Laguerre two-matrix model and new results for this latter ensemble are given in full generality, including the recurrence relations satisfied by their associated bi-orthogonal polynomials when [Formula: see text] assumes positive integer values.

Adsorption of Lattice Polymers with Quenched Topologies

We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our "topology" refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption.

The Kronig–Penney model in a quadratic channel with δ interactions: II. Scattering approach

The main purpose of the present paper is to introduce a scattering approach to the study of the Kronig–Penney model in a quadratic channel with δ interactions, which was discussed in full generality in the first paper of the present series. In particular, a secular equation whose zeros determine the spectrum will be written in terms of the scattering matrix from a single δ. The advantages of this approach will be demonstrated in addressing the domain with total energy E∈[0,12) , namely, the energy interval where, for under critical interaction strength, a discrete spectrum is known to exist for the single δ case. Extending this to the study of the periodic case reveals quite surprising behavior of the Floquet spectra and the corresponding spectral bands. The computation of these bands can be carried out numerically, and the main features can be qualitatively explained in terms of a semi-classical framework which is developed for the purpose.

Tame and strongly étale cohomology of curves

For a curve C over a perfect field k of characteristic p > 0 we study the tame cohomology of $$\mathcal{C}$$ = Spa(C, k) introduced in [Hüb21]. We prove that the tame cohomology groups of $$\mathcal{C}$$ with p-torsion coefficients satisfy cohomological purity (which is not true in full generality for the étale cohomology). Using purity we show Poincaré duality for the tame cohomology of $$\mathcal{C}$$ with p-torsion coefficients.

Numerical solutions to the Reactive Graetz Problem for CO2 Capture

Hollow Fiber Membrane Contactor Modules comprise a technology that is considered promising for mobile carbon capture due to its high surface area per unit volume and low pressure drop. Although experimental, theoretical, and numerical studies have been devoted to understanding and improving performance, no study has so far used numerical simulations to show how the system scales with flow rate, reactivity, solubility, etc. in full generality. The present study addresses this scaling by solving the so called reactive Graetz problem: a canonical partial differential equation system coupling gas and liquid flow domains, including solubility and reactivity effects unlike the original Graetz problem. It is found that the Peclet number of the liquid, the aspect ratio of the fiber, and the Damkohler number of the liquid solvent, all play decisive roles in the mass transfer of CO2 and the work exerted per molecule captured.

The Concept of Motion in Ancient Greek Thought: Foundations in Logic, Method, and Mathematics

<i>The Concept of Motion in Ancient Greek Thought: Foundations in Logic, Method, and Mathematics</i>