Nontrivial Measure Research Articles

Compton scattering from proton pairs – illustrating weak and strong measurements

Abstract Neutrons interacting with protons in Compton scattering is here chosen as an example of a non-trivial quantum measurement. It illustrates a situation where a probe particle makes an observation (a weak measurement) of the studied protons, preparing for a final state in which one of them takes up the recoil in the collision. The proton is ejected during a strong (projective) measurement that selects one particular momentum value out of the proton’s initial distribution of momentum states. This course of events in the measurement process is evidenced by the observation that destructive interference appears during the weak interaction stage which leads to a strongly reduced neutron cross section for the proton system.

Badly approximable grids and k$k$‐divergent lattices

AbstractLet be a matrix. In this paper, we investigate the set of badly approximable targets for , where is the ‐torus. It is well known that is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of and Diophantine properties of . On the one hand, we give the first examples of a nonsingular such that has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on that slightly strengthens nonsingularity, and show that under the assumption that satisfies this condition, is a null‐set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.

Probe thermometry with continuous measurements

Temperature estimation plays a vital role across natural sciences. A standard approach is provided by probe thermometry, where a probe is brought into contact with the sample and examined after a certain amount of time has passed. In situations where, for example, preparation of the probe is non-trivial or total measurement time of the experiment is the main resource that must be optimized, continuously monitoring the probe may be preferred. Here, we consider a minimal model, where the probe is provided by a two-level system coupled to a thermal reservoir. Monitoring thermally activated transitions enables real-time estimation of temperature with increasing accuracy over time. Within this framework we comprehensively investigate thermometry in both bosonic and fermionic environments employing a Bayesian approach. Furthermore, we explore adaptive strategies and find a significant improvement on the precision. Additionally, we examine the impact of noise and find that adaptive strategies may suffer more than non-adaptive ones for short observation times. While our main focus is on thermometry, our results are easily extended to the estimation of other environmental parameters, such as chemical potentials and transition rates.

Proof-of-Monitoring (PoM): A Novel Consensus Mechanism for Blockchain-Based Secure Service Level Agreement Management

In the current 5th Generation (5G) networking paradigm, the enforcement of Service Level Agreements (SLAs) is a non-trivial measure to ensure the scope and the quality of services and standards between tenants and service providers (SPs). On top of this, Secure Service Level Agreements (SSLA) are introduced to ensure that SPs deliver the most critical and required security-related standards defined in the contract, such as integrity, confidentiality, availability, non-repudiation, and privacy assurance. However, with the tendency for more distributed and multi-stakeholder networking architectures in next-generation networks, the management process of such SSLAs will be challenging due to the diversified security vulnerabilities and complexity of underlying technologies. Although blockchain is emerging as a platform to facilitate such distributed SSLA/SLA management frameworks, its currently available consensus mechanisms are more generic. Still, they need to improve in terms of applying in multi-stakeholder networks. Therefore, this paper presents a novel consensus mechanism called Proof-of-Monitoring (PoM) for a blockchain-based novel SSLA management framework. Moreover, we provide details about the prototype implementation of our proposed consensus algorithm and SSLA management framework. It is proven by comparing our proposal with the other existing solutions that our solution outperforms in many aspects, such as energy consumption, computation cost, and security features.

Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity

Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ > 0 {\kappa }^{\ast }\gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -\Delta u+u={u}^{p}+\kappa \mu in R N {{\bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 < κ ≤ κ ∗ 0\lt \kappa \le {\kappa }^{\ast } . Here, N ≥ 2 N\ge 2 , μ \mu is a nontrivial nonnegative Radon measure in R N {{\bf{R}}}^{N} with a compact support, and p > 1 p\gt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.

An inner product for 4D quantum gravity and the Chern–Simons–Kodama state

We demonstrate that reality conditions for the Ashtekar connection imply a non-trivial measure for the inner product of gravitational states in the polarization where the Ashtekar connection is diagonal, and we express the measure as the determinant of a certain first-order differential operator. This result opens the possibility to perform a non-perturbative analysis of the quantum gravity scalar product. In this polarization, the Chern–Simons–Kodama state, which solves the constraints of quantum gravity for a certain factor ordering, and which has de Sitter space as a semiclassical limit, is perturbatively non-normalizable with respect to the naïve inner product. Our work reopens the question of whether this state might be normalizable when the correct non-perturbative inner product and choice of integration contour are taken into account. As a first step, we perform a semi-classical treatment of the measure by evaluating it on the round three-sphere, viewed as a closed spatial slice of de Sitter. The result is a simple, albeit divergent, infinite product that might serve as a regulator for a more complete treatment of the problem. Additionally, our results suggest deep connections between the problem of computing the norm of the CSK state in quantum gravity and computing the Chern–Simons partition function for a complex group.

Hierarchy of Multipartite Nonlocality and Device-Independent Effect Witnesses.

According to recent new definitions, a multiparty behavior is genuinely multipartite nonlocal (GMNL) if it cannot be modeled by measurements on an underlying network of bipartite-only nonlocal resources, possibly supplemented with local (classical) resources shared by all parties. The new definitions differ on whether to allow entangled measurements upon, and/or superquantum behaviors among, the underlying bipartite resources. Here, we categorize the full hierarchy of these new candidate definitions of GMNL in three-party quantum networks, highlighting the intimate link to device-independent witnesses of network effects. A key finding is the existence of a behavior in the simplest nontrivial multipartite measurement scenario (three parties, two measurement settings, and two outcomes) that cannot be simulated in a bipartite network prohibiting entangled measurements and superquantum resources-thus witnessing the most general form of GMNL-but can be simulated with bipartite-only quantum states with an entangled measurement, indicating an approach to device-independent certification of entangled measurements with fewer settings than in previous protocols. Surprisingly, we also find that this (3,2,2) behavior, as well as the others previously studied as device-independent witnesses of entangled measurements, can all be simulated at a higher echelon of the GMNL hierarchy that allows superquantum bipartite resources while still prohibiting entangled measurements. This poses a challenge to a theory-independent understanding of entangled measurements as an observable phenomenon distinct from bipartite nonlocality.

Historic and physical wandering domains for wild blender-horseshoes

We present diffeomorphisms of wild blender-horseshoes which belong to closures of two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure supported by saddle periodic orbit. It is a non-trivial extension of Colli–Vargas’ model (Colli and Vargas 2001 Ergod. Theory Dyn. Syst. 21 1657–81) to the higher dimensional dynamics with the use of wild blender-horseshoes.

Semantic inconsistency measures using 3-valued logics

AI systems often need to deal with inconsistencies. One way of getting information about inconsistencies is by measuring the amount of information in the knowledgebase. In the past 20 years numerous inconsistency measures have been proposed. Many of these measures are syntactic measures, that is, they are based in some way on the minimal inconsistent subsets of the knowledgebase. Very little attention has been given to semantic inconsistency measures, that is, ones that are based on the models of the knowledgebase where the notion of a model is generalized to allow an atom to be assigned a truth value that denotes contradiction. In fact, only one nontrivial semantic inconsistency measure, the contension measure, has been in wide use. The purpose of this paper is to define a class of semantic inconsistency measures based on 3-valued logics. First, we show which 3-valued logics are useful for this purpose. Then we show that the class of semantic inconsistency measures can be developed using a graphical framework similar to the way that syntactic inconsistency measures have been studied. We give several examples of semantic inconsistency measures and show how they apply to three useful 3-valued logics. We also investigate the properties of these inconsistency measures and show their computation for several knowledgebases.

Interpretable Machine Learning Approach for Identifying the Tip Sharpness in Atomic Force Microscopy

Atomic force microscopy (AFM) is routinely used with indentation techniques to characterize the plastic deformation of materials. The accurate quantification of the features associated with the indent, which is used to quantify the hardness and indentation deformation mechanisms, depends on the sharpness of the AFM tip used for imaging. However, identifying the tip-sharpness of an atomic force microscope requires non-trivial measurements. Here, using machine learning, we develop a model to predict the tip sharpness of the AFM cantilever directly from the indent images. Further, we employ explainable machine learning models, such as integrated gradients and gradient shap, to interpret the features learned by the model. Altogether, we show that machine learning approaches can accelerate experiments by providing non-trivial information about the instrument performance, thereby enabling researchers to perform better quality experiments.

Supermeasured: Violating Bell-Statistical Independence Without Violating Physical Statistical Independence

Bell’s theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as “fine-tuning” or a “conspiracy”. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call “supermeasured”. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell’s theorem without requiring correlations between hidden variables and measurement settings (physical statistical independence).

Measurement disturbance tradeoffs in three-qubit unsupervised quantum classification

We consider measurement disturbance tradeoffs in quantum machine learning protocols which seek to learn about quantum data. We study the simplest example of a binary classification task in the unsupervised regime. Specifically, we investigate how a classification of two qubits, that can each be in one of two unknown states, affects our ability to perform a subsequent classification on three qubits when a third is added. Surprisingly, we find a range of strategies in which a nontrivial first classification does not affect the success rate of the second classification. There is, however, a nontrivial measurement disturbance tradeoff between the success rate of the first and second classifications, and we fully characterize this tradeoff analytically.

Regularity for graphs with bounded anisotropic mean curvature

We prove that \(m\)-dimensional Lipschitz graphs with anisotropic mean curvature bounded in \(L^p\), \(p>m\), are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a \(C^{1,1}\) neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a \(C^{1,1}\) (resp. \(C^1\)) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.

L2 estimates and existence theorems for the [formula omitted] operators in infinite dimensions, I

The classical L2 method for the ∂‾ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this method to infinite dimensional complex analysis, but this is a longstanding unsolved problem, due to the essential difficulty that there exists no nontrivial translation invariance measure in the setting of infinite dimensions. The main purpose in this series of work is to give an affirmative solution to the above problem. In this first part, we focus on the simplest case, i.e., L2 estimates and existence theorems for the ∂‾ equations on the whole space of ℓp for p∈[1,∞). The key of our approach is to introduce a suitable working space, i.e., a Hilbert space for (s,t)-forms on ℓp (for each nonnegative integers s and t), via which we define the ∂‾ operator from (s,t)-forms to (s,t+1)-forms and establish the exactness of these operators, and therefore in this case we solve a problem which has been open for forty years.

Optimal fidelity witnesses for gravitational entanglement

Optomechanical systems open new possibilities in fundamental research at the interface between quantum information and gravity. Recently, an ambitious experimental proposal was suggested by Bose et al. to measure the entanglement between two optomechanical systems generated by their gravitational interaction. The scheme relies on witnessing entanglement between the two systems. Here we develop a general framework to study the quality of bipartite entanglement witnesses using fidelity witnesses. We then apply this framework to the gravitational entanglement proposal, optimizing for the detection of entanglement. We construct a witness consisting of only 5 non-trivial spin measurements, which we compare with other proposed witnesses. With post-processing our witness can detect entanglement for any choice of phases in the setup, up to a set of measure zero.

On a 1D model with nonlocal interactions and mass concentrations: an analytical-numerical approach* *Abreu was partially supported by CNPq 306385/2019-8, BR and Ferreira was partially supported by FAPESP and CNPq, BR.

We study global well-posedness and finite time blow-up of solutions for a nonlinear one-dimensional transport equation with nonlocal velocity , ν > 0, and measure initial data. Such model arises in fluid mechanics in vortex-sheet problems and its nonlocal feature comes from the presence of a singular integral operator (Hilbert transform ) in the velocity field. In the viscous case ν > 0, we analytically obtain an explicit condition on the size of the initial data for the global well-posedness in the framework of pseudomeasure spaces. In fact, we can give the condition depending on the initial-mass and analyze how the flow evolves from singular measures. Also, we numerically study blow-up of concentration type and global diffusion-smooth behavior of solutions. We obtain numerics that indicate the threshold value 2π for the initial-data mass that decides between blow-up or global smoothness of solutions. Such value is the same obtained for regular initial-data and by means of entropy methods. Thus, it seems to be intrinsic to the nonlocal PDE and independent of a particular framework, approach and initial-data regularity. The inviscid case ν = 0 is remarkable: simulations for model , evidence that the solution presents blow-up of concentration type with mass-preserving, while an attenuation effect is observed for the model with opposite sign in the nonlinearity , for any nontrivial (positive) measure as initial data.

Non-existence of bi-infinite polymers

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.

A driven tagged particle in asymmetric exclusion processes

We consider the asymmetric exclusion process with a driven tagged particle on Z which has different jump rates from other particles. When the non-tagged particles have non-nearest-neighbor jump rates , we show that the tagged particle can have a speed which has a different sign from the mean derived from its jump rates. We also show the existence of some non-trivial invariant measures for the environment process viewed from the tagged particle. Our arguments are based on coupling, martingale methods, and analyzing currents through fixed bonds.

Displacement Estimation for Nonlinear Structures Using Seismic Acceleration Response Data

ABSTRACT The displacements estimated using seismic accelerations are useful for post-earthquake inspection of structural damage. In previous studies, displacements were usually obtained through non-trivial measurement or double integration of the measured acceleration response and then an additional correction process. However, from the signal processing perspective, double integration induces substantial numerical errors. In this study, a displacement estimation approach is developed based on the extended Kalman filter (EKF) with an embedded Bayesian noise-parameters updating technique. The displacement is estimated by EKF with updated time-varying noise parameters based on a Bouc–Wen model. Numerical simulations and a shaking‐table test are presented to demonstrate the proposed method.

Extension of Haar’s theorem

Haar’s theorem ensures a unique nontrivial regular Borel measure on a locally compact Hausdorff topological group, up to multiplication by a positive constant. In this article, we extend Haar’s theorem to the case of locally compact Hausdorff strongly topological gyrogroups. We simultaneously prove the existence and uniqueness of a Haar measure on a locally compact Hausdorff strongly topological gyrogroup, using the method of Steinlage. We then find a natural relationship between Haar measures on gyrogroups and on their related groups. As an application of this result, we study some properties of a convolution-like operation on the space of Haar integrable functions defined on a locally compact Hausdorff strongly topological gyrogroup