Measure Zero Research Articles

On eventually greedy best underapproximations by Egyptian fractions

Erdős and Graham found it conceivable that the best n-term Egyptian underapproximation of almost every positive number for sufficiently large n gets constructed in a greedy manner, i.e., from the best (n−1)-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero.

Hausdorff dimension of the parameters for (α,β)-shifts with the specification property

In this paper, we consider the specification property for ( α , β ) -shifts. When α = 0 , Schmeling shows that the set of β > 1 for which the β-shift has the specification property has the Lebesgue measure zero but has the full Hausdorff dimension [Schmeling. Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory Dyn. Syst. 17 (1997), pp. 675–694]. So it is natural to ask what happens when α > 0 . Buzzi shows that for fixed α the set of β > 1 for which the ( α , β ) -shift has the specification property has Lebesgue measure zero. Hence we consider the Hausdorff dimension of the parameter space of ( α , β ) -shifts.

Cenozoic Carbon Dioxide: The 66 Ma Solution

The trend in partial pressure of atmospheric CO2, P(CO2), across the 66 MYr of the Cenozoic requires elucidation and explanation. The Null Hypothesis sets sea surface temperature (SST) as the baseline driver for Cenozoic P(CO2). The crystallization and cooling of flood basalt magmas is proposed to have heated the ocean, producing the Paleocene–Eocene Thermal Maximum (PETM). Heat of fusion and heat capacity were used to calculate flood basalt magmatic Joule heating of the ocean. Each 1 million km3 of oceanic flood basaltic magma liberates ~5.4 × 1024 J, able to heat the global ocean by ~0.97 °C. Henry’s Law for CO2 plus seawater (HS) was calculated using δ18O proxy-estimated Cenozoic SSTs. HS closely parallels Cenozoic SST and predicts the gas solute partition across the sea surface. The fractional change of Henry’s Law constants, Hn−HiHn−H0 is proportional to ΔP(CO2)i, and Hn−HiHn−H0×∆P(CO2)+P(CO2)min, where ΔP(CO2) = P(CO2)max − P(CO2)min, closely reconstructs the proxy estimate of Cenozoic P(CO2) and is most consistent with a 35 °C PETM ocean. Disparities are assigned to carbonate drawdown and organic carbon sedimentation. The Null Hypothesis recovers the glacial/interglacial P(CO2) over the VOSTOK 420 ka ice core record, including the rise to the Holocene. The success of the Null Hypothesis implies that P(CO2) has been a molecular spectator of the Cenozoic climate. A generalizing conclusion is that the notion of atmospheric CO2 as the predominant driver of Cenozoic global surface temperature should be set aside.

Perturbations of exponential maps: Non-recurrent dynamics

AbstractWe study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero, which complements a result of Qiu from 1994. Moreover, we show that non-recurrent parameters can be approximated by hyperbolic ones.

Robust stabilisation of uncertain nonlinear systems with matched and unmatched uncertainties

This paper considers the problem of robust stabilisation of uncertain nonlinear systems with both matched and unmatched uncertainties. A new robust controller is proposed to guarantee practical or asymptotic stability under non-vanishing or vanishing unmatched uncertainties. First, the system uncertainties are decomposed optimally into the matched and unmatched portions with an objective of the minimum norm for the unmatched part. Next, instead of a robust control design based on only the matched part of uncertainties commonly used in the previous works, a robust controller is synthesised based on both matched and mismatched portions of the uncertainties. It is shown that in uncertain systems whose state cannot simultaneously stay in a so-called null set and diverge to infinity, the proposed controller guarantees stability no matter how large the input-unrelated unmatched uncertainties are. Numerical examples illustrate that the proposed controller is more robust against uncertainties, and offers improved damping and stabilisation features.

Heteroclinic traveling waves of two-dimensional parabolic Allen–Cahn systems

In this paper we show the existence of traveling waves w\colon [0,+\infty) \times \mathbb{R}^{2} \to \mathbb{R}^{k} ( k \geq 2 ) for the parabolic Allen–Cahn system \partial_{t} w - \Delta w = -\nabla_{u} V(w) in [0,+\infty) \times \mathbb{R}^{2} , satisfying some heteroclinic conditions at infinity. The potential V is a nonnegative and smooth multi-well potential, which means that its null set is finite and contains at least two elements. The traveling wave w propagates along the horizontal axis according to a speed c^{\star}>0 and a profile \mathfrak{U} . The profile \mathfrak{U} joins as x_{1} \to \pm \infty (in a suitable sense) two locally minimizing one-dimensional heteroclinics which have different energies, and the speed c^{\star} satisfies certain uniqueness properties. The proof is variational and, in particular, it requires the assumption of an upper bound, depending on V , on the difference between the energies of the one-dimensional heteroclinics.

Generalized set-valued Ekeland variational principles via the null set concept with applications to some set-valued optimization problems

In this paper, we establish two new versions of generalized set-valued Ekeland variational principles related to the so-called null set in a hyperspace ordered by a convex cone. Moreover, we propose some interval versions of Ekeland variational principles. As applications, the obtained theoretical results are applied to prove a set-valued fixed point theorem, to solve a set-valued optimization problem, and to explore a set-valued noncooperative game.

Recurrent and (strongly) resolvable graphs

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.

Length and Area of an Apollonian Packing

Summary We prove two theorems about the circular Apollonian packing. The first states that the circles used in the packing have an infinite total length. The second states that the largest circle is covered by the other circles used in the packing, up to an area of Lebesgue measure zero. Our proofs are elementary and only use basic coordinate geometry.

Nonlinear scalarization in set optimization based on the concept of null set

Nonlinear scalarization in set optimization based on the concept of null set

Biharmonic equation with discontinuous nonlinearities

We study the biharmonic equation with discontinuous nonlinearity and homogeneous Dirichlet type boundary conditions $$\displaylines{ \Delta^2u=H(u-a)q(u) \quad \hbox{in }\Omega,\cr u=0 \quad \hbox{on }\partial\Omega,\cr \frac{\partial u}{\partial n}=0 \quad \hbox{on }\partial\Omega, }$$ where \(\Delta\) is the Laplace operator, \(a> 0\), \(H\) denotes the Heaviside function, \(q\) is a continuous function, and \(\Omega\) is a domain in \(R^N \) with \(N\geq 3\). Adapting the method introduced by Ambrosetti and Badiale (The Dual Variational Principle), which is a modification of Clarke and Ekeland's Dual Action Principle, we prove the existence of nontrivial solutions. This method provides a differentiable functional whose critical points yield solutions despite the discontinuity of \(H(s-a)q(s)\) at \(s=a\). Considering \(\Omega\) of class \(\mathcal{C}^{4,\gamma}\) for some \(\gamma\in(0,1)\), and the function \(q\) constrained under certain conditions, we show the existence of two non-trivial solutions. Furthermore, we prove that the free boundary set \(\Omega_a=\{x\in\Omega:u(x)=a\}\) for the solution obtained through the minimizer has measure zero. For more information see https://ejde.math.txstate.edu/Volumes/2024/15/abstr.html

Automated Data Transformation and Feature Extraction for Oxygenation-Sensitive Cardiovascular Magnetic Resonance Images.

Oxygenation-sensitive cardiovascular magnetic resonance (OS-CMR) is a novel, powerful tool for assessing coronary function in vivo. The data extraction and analysis however are labor-intensive. The objective of this study was to provide an automated approach for the extraction, visualization, and biomarker selection of OS-CMR images. We created a Python-based tool to automate extraction and export of raw patient data, featuring 3336 attributes per participant, into a template compatible with common data analytics frameworks, including the functionality to select predictive features for the given disease state. Each analysis was completed in about 2min. The features selected by both ANOVA and MIC significantly outperformed (p < 0.001) the null set and complete set of features in two datasets, with mean AUROC scores of 0.89eatures f 0.94lete set of features in two datasets, with mean AUROC scores that our tool is suitable for automated data extraction and analysis of OS-CMR images.

Lie pairs

Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``$\preceq$-morphisms'' preserving a surpassing relation $\preceq$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.

STRONG MEASURE ZERO SETS ON FOR INACCESSIBLE

Abstract We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$ Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC.

Possibility Fermatean Neutrosophic Soft Set

In this paper, we introduce the concept of Possibility Fermatean Neutrosophic Soft Set and define some related concepts such as Possibility Fermatean Neutrosophic Soft subset, Possibility Fermatean Neutrosophic Soft null set, and Possibility Fermatean Neutrosophic Soft universal set. Then, we define set-theoretical operations of Possibility Fermatean Neutrosophic Soft Sets such as union, intersection, and complement, and investigate some properties of these operations. We also introduce AND-product and OR-product operations between two Possibility Fermatean Neutrosophic Soft Sets. We propose a decision-making method called the Possibility Fermatean Neutrosophic Soft decision-making method (PFNS-decision-making method) which can be applied to decision-making problems involving uncertainty based on AND-product operation. We finally give a numerical example to display the application of the method that can be successfully applied to the problems.

Solving Fuzzy Nonlinear Optimization Problems Using Null Set Concept

Solving Fuzzy Nonlinear Optimization Problems Using Null Set Concept

Measurable Sensitivity for Semi-Flows

Sensitive dependence on initial conditions is a crucial characteristic of chaos. The concept of measurable sensitivity (MS) was introduced as a measure-theoretic version of sensitive dependence on initial conditions. Their research demonstrated that MS arises from light mixing, indicates a finite number of eigenvalues for a transformation, and is not present in the case of infinite measure preservation. Unlike the traditional understanding of sensitivity, MS carries up to account for isomorphism in the sense of measure theory, which ignores the function’s behavior on null sets and eliminates dependence on the chosen metric. Inspired by the results of James on MS, this paper generalizes some of the concepts (including MS) that they used in their study of MS for conformal transformations to semi-flows, and generalizes their main results in this regard to semi-flows.

Martin–Löf reducibility and cost functions

Martin-Löf (ML)-reducibility compares the complexity of K-trivial sets of natural numbers by examining the Martin-Löf random sequences that compute them. One says that a K-trivial set A is ML-reducible to a K-trivial set B if every ML-random computing B also computes A. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets intended to capture ML-random sequences. We show that for every cost function there is a c.e. set that is ML-complete among the sets obeying it. We characterise the K-trivial sets computable from a fragment of the left-c.e. random real Ω given by a computable set of bit positions. This leads to a new characterisation of strong jump-traceability.

An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements

Consider a finite collection of affine hyperplanes in mathbb R^d. The hyperplanes dissect mathbb R^d into finitely many polyhedral chambers. For a point xin mathbb R^d and a chamber P the metric projection of x onto P is the unique point yin P minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by text {dim}(x,P). We prove that for every given kin {0,ldots , d}, the number of chambers P for which text {dim}(x,P) = k does not depend on the choice of x, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873–2887 (2010)].

The Case of the Jeffreys-Lindley-paradox as a Bayes-frequentist Compromise: A Perspective Based on the Rao-Lovric-Theorem

Testing a precise hypothesis can lead to substantially different results in the frequentist and Bayesian approach, a situation which is highlighted by the Jeffreys-Lindley paradox. While there exist various explanations why the paradox occurs, this article extends prior work by placing the less well-studied point-null-zero-probability paradox at the center of the analysis. The relationship between the two paradoxes is analyzed based on accepting or rejecting the existence of precise hypotheses. The perspective provided in this paper aims at demonstrating how the Bayesian and frequentist solutions can be reconciled when paying attention to the assumption of the point-null-zero-probability paradox. As a result, the Jeffreys-Lindley-paradox can be reinterpreted as a Bayes-frequentist compromise. The resolution shows that divergences between Bayesian and frequentist modes of inference stem from (a) accepting the existence of a precise hypothesis or not, (b) the assignment of positive measure to a null set and (c) the use of unstandardized p-values or p-values standardized to tail-area probabilities.