Conditions For Uniqueness Research Articles

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

We study inhomogeneous Diophantine approximation over the completion Kv of a function field K (over a finite field) for a discrete valuation v, with affine algebra Rv. We obtain an effective upper bound for the Hausdorff dimension of the setBadA(ϵ)={θ∈Kvm:lim inf(p,q)∈Rvm×Rvn,‖q‖→∞lim inf(p,q)∈Rvm×Rvn,‖q‖→∞‖q‖n‖Aq−θ−p‖m⩾ϵ}, of ϵ-badly approximable targets θ in Kvm given A in Mm,n(Kv), using an effective version of entropy rigidity in homogeneous dynamics for some diagonal action on the space of Rv-grids. We characterize matrices A for which BadA(ϵ) has full Hausdorff dimension for some ϵ>0 by a Diophantine condition of singularity on average. Our methods work for the approximation using weighted ultrametric distances.

Nonlinear stabilization and reference tracking of visual servo system using TS fuzzy augmented iterative learning control: Experimental validation

To address the nonlinear stabilization problem and improve the tracking control feature of ball on plate system (BPS), this paper puts forward a novel Takagi Sugeno (TS) fuzzy control augmented with the current cycle feedback iterative learning control (CCF-ILC) scheme. According to Bode’s sensitivity integral, the performance of linear controllers is always a trade-off between reference tracking and robustness. Hence, to deal with the so-called ‘waterbed’ effect, this work exploits the capability of TS fuzzy to handle the nonlinear dynamics and synthesizes a learning control scheme based on current iteration error to capitalize the information rich error signal for enhancing the robustness and trajectory tracking features. The global asymptotic stability of the proposed TS fuzzy augmented ILC scheme is proved using the Lyapunov function and linear matrix inequalities (LMIs). Moreover, the monotonic convergence of ILC is presented based on the singular value condition. For identifying the rolling mass from the video stream, a background subtraction algorithm based on thresholding technique is implemented. Finally, the robustness and tracking features of the proposed scheme are evaluated on a two degrees of freedom (DoF) laboratory scale BPS system through hardware in loop (HIL) testing for three realistic test cases. The tracking performance quantified using the root mean square error (RMSE) and power spectral density plot corroborates that the proposed scheme can offer better setpoint tracking and robustness feature compared to state-of-the-art fuzzy and ILC control techniques implemented on BPS.

A computationally effective singularity avoidance method for redundant manipulators with non-spherical wrists

For redundant manipulators with non-spherical wrists, we propose a general and computation-effective method of singularity avoidance. Firstly, the dimension of the Jacobian of the redundant manipulator is reduced by the elementary transformation method to obtain the singular factors. Secondly, we use the damped reciprocal method to solve the inverse kinematics solutions of the redundant manipulator. Thirdly, the singularities of the redundant manipulator are divided into non-escapable and escapable singularities, and the singular conditions are determined with the singular factors. Then taking the singularity avoidance as the secondary task of the control system, we propose a singular factor optimization method to avoid both types of singularities of the redundant manipulator. The complexity analysis shows that no need to calculate the pseudo-inverses or partial derivatives of the Jacobian, the proposed method has lower computational complexity. Finally, simulations have been made on the non-escapable and escapable singularities to verify the proposed methods.

On unique tensor rank decomposition of 3-tensors

We answer to a question posed recently in reference [Lovitz B, Petrov F. A generalization of Kruskal's theorem on tensor decomposition. Available at arXiv 2103.15633; 2021], proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition.

A closed-form solution for inverse kinematics of redundant space manipulator with multiple joint offsets

This paper proposes a novel parameterization analytical method to solve the inverse kinematics of a redundant space manipulator, which has link offsets at the shoulder, elbow, and wrist parts. Due to the manipulator’s particular configuration, current arm angle concepts cannot provide the analytical solution directly. Hence, this paper defines a new arm angle parameter that describes the manipulator’s redundancy and obtains the closed-form solutions directly. This is because, unlike manipulators without offset links, existing approaches involve parameters that may cause the inverse kinematic solution to fail when applied to the manipulator type considered in this work. Specifically, we analyze the parameter interplay on the inverse kinematic computation and then investigate the relations between the singularity and arm angle based on the singularity conditions of the Jacobian matrix. Besides, a semi-analytical method is proposed to determine the feasible arm angle intervals, while avoiding the singularity and inverse kinematic computation. Finally, a simple approach is proposed to compute the arm angle according to the feasible intervals. The proposed method has obvious geometric significance, is suitable for configuration control, and can be implemented in real-time control systems.

Cosmological models in scale-independent energy-momentum squared gravity

Scale-independent EMSG is a particular model of energy–momentum squared gravity (EMSG) in which the new terms in the Einstein field equations arising from the EMSG theory enter with the same power as the usual terms from Einstein–Hilbert part of the action. However, the model violates the local energy–momentum conservation and matter-current conservation in general and hence, permits a process of matter creation/annihilation in an expanding universe. Consequently, the scale factor dependencies of the energy densities are modified by the dimensionless model parameter α. We revisit some nostalgias such as static universes and de Sitter/steady state universes. We reproduce the original ones, moreover, present some novelties, e.g., a spatially flat static universe, de Sitter expansion by negative vacuum energy, steady state universes in the presence of arbitrary fluids with constant equation of state (EoS) parameter other than dust, etc. We also investigate the possible dynamics of dust dominated and radiation dominated universes. Depending on the value of α, dust/radiation dominated universe exhibits power-law accelerated/decelerated expansion, corresponds to a steady state model or may end in a big rip. In the framework of anisotropic cosmology, we reproduce Barrow’s quiescent universe in the presence of stiff fluid and extend it to fluids with arbitrary constant EoS parameter. We also relax the condition for isotropic initial singularity (big bang) owing to that EMSG effectively allows ultra-stiff EoS parameters.

Uniform Error Estimates of the Finite Element Method for the Navier-Stokes Equations in R2 with L2 Initial Data.

In this paper, we study the finite element method of the Navier-Stokes equations with the initial data belonging to the L2 space for all time t>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t∈[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.

Title: Hip and lower limbs 3D motion tracking using a double-stage data fusion algorithm for IMU/MARG-based wearables sensors

This paper presents an algorithm to estimate the relative orientation of body segments during 3D motion tracking, either as multiple individual body segments or as an articulated body chain. For this purpose, a double-stage data fusion algorithm (algorithm-based system) was implemented for inertial measurement units (IMU)/Magnetic Angular Rate and Gravity (MARG)-based wearable sensors during gait. The methodology considers two stages of complementary filters for the data fusion of the IMU/MARG sensors and a Proportional-Integral controller (PI) to incorporate the processed data. Quaternions were employed for the orientation estimation, a Direction Cosine Matrix quaternion-based (DCM) to estimate the relative orientation between the body mobile coordinate system and the inertial reference coordinate system, and Euler angles for the orientation representation. A customized Ambulatory Sensing Motion System (ASMS), consisting of seven monitoring devices was used for the evaluation. The double-stage algorithm-based system was compared with a digital motion processor (DMP) performance. Experimental results of the double-stage algorithm-based system indicated a 0.7° RMSE (M1) and 1° RMSE (M1-M7) with mean values lower than 1° concerning the DMP outcomes. Based on the Bland-Altman analysis, the level of agreement indicated that the double-stage algorithm-based system is neither underestimated nor overestimated and is suitable for the DMP outcome. No singularity conditions nor significant levels of noise or drift were observed in the conducted experiments. The results demonstrate the feasibility of using the proposed method for 3D human motion tracking of multiple body segments, individually or as an articulated body chain. Besides, it suits different IMU/MARG sensor-based wearable sensors.

Positive solutions for a class of biharmonic equations: Existence and uniqueness

In this paper, we analyze the existence of positive solution to a biharmonic equation with Navier boundary conditions, and formulate simple uniqueness conditions for the positive solution. In addition, a necessary condition for the existence of the positive solution is also derived.

Common Fixed Point Theorem for (\(\phi\),\(\mathfrak{F}\))–Multi-Valued Mappings in Cone b-Metric Spaces over Banach Algebra

In the present paper, we introduced the concept of generalized multi-valued contraction mappings, via the class functions \(\Phi\) and \(\Psi\) Also we proved some fixed point results for (\(\phi\),\(\mathfrak{F}\))- multi-valued mappings on cone b-metric spaces over Banach algebra \(\mathfrak{V}\). The conditions for existence and uniqueness of the fixed point are investigated. We give an example to support our main result.

Assessing habitat diversity and potential areas of similarity across protected areas globally

Biophysical characterization analyses of protected areas (PA) that provide information on their ecological values and potential areas with similar characteristics are needed to make informed PA network planning and management decisions. This study combines and further develops methodologies that use remote sensing and modelling to identify habitat functional types in PAs and map similar areas at the ecoregion level. The study also develops new terrestrial habitat diversity and irreplaceability indices at habitat and PA scale that allow the comparison and ranking of PAs in terms of biophysical gradients and singular environmental conditions. Six PAs were selected to highlight and discuss the results of the proposed methodology. Both individual and composite indices should be considered when trying to compare PAs to understand the overall complexity and ecological values of each PA. Results can inform planning and management of individual and protected area networks as well as identify new areas for conservation. The information provided by the model about similar habitats outside protected areas can also help assess their representativeness and support studies to strengthen ecological connectivity. Besides systematic comparisons, detailed assessments of protected areas can also be performed using medium and high-resolution input variables. This is especially relevant for protected areas in developing countries where undertaking fieldwork is very difficult and the budget devoted to conservation is limited.

Power Efficient Design a Compliant Robotic Leg Based on Klann's Linkage

This article presents the analysis and optimization of a compliant robotic leg based on Klann's linkage. This leg is specifically designed to efficiently distribute its power requirement over its complete motion cycle, avoiding large peaks in the energy drawn from the battery. The structural compliance of the leg will be shown to be able to both provide a satisfactory walking motion and a timely energy boost to help with the gait. Klann's linkage is selected here as the basic kinematic structure of the leg to demonstrate the proposed methodology, namely to combine in a single structure both a complex trajectory generation and energy storage/release. This work is first aiming at proposing a thorough kinematic analysis of that mechanism using planar screw theory. The latter will be shown to be able to efficiently provide the velocity equations of the linkage as well as its force input–output relationship and singularity conditions. In a second part of this article, the previous kinetostatic model will be used to design and optimize a compliant version of the leg optimizing the power required for the robot to move. Finally, experiments will be shown to support the proposed approach.

Uniqueness of Clearing Payment Matrices in Financial Networks

We study bankruptcy problems in financial networks in the presence of general bankruptcy laws. The set of clearing payment matrices is shown to be a lattice, which guarantees the existence of a greatest clearing payment and a least clearing payment. Multiplicity of clearing payment matrices is both a theoretical and a practical concern. We present a new condition for uniqueness that generalizes all the existing conditions proposed in the literature. Our condition depends on the decomposition of the financial network into strongly connected components. A strongly connected component that contains more than one agent is called a cycle, and the involved agents are called cyclical agents. If there is a cycle without successors, then one of the agents in such a cycle should have a strictly positive endowment. The division rule used by a cyclical agent with a strictly positive endowment should be positive monotonic, and the rule used by a cyclical agent with a zero endowment should be strictly monotonic. Because division rules involving priorities are not positive monotonic, uniqueness of the clearing payment matrix is a much bigger concern for such division rules than for proportional ones. As a final contribution of the paper, we exhibit the relationship between the uniqueness of clearing payment matrices and the continuity of bankruptcy rules, a property that is very much desired for stability of financial systems. Funding: This research was supported by the Higher Education Institutional Excellence Program 2020 of the Ministry of Innovation and Technology in the framework of the “Financial and Public Services” research project [Grant TKP2020-IKA-02] at Corvinus University of Budapest. P. Csóka received funding from the National Research, Development and Innovation Office [Grants K-120035 and K-138826].

Fractional Burgers equation with singular initial condition

We consider the fractional Burgers equation ut=Δα/2u+b⋅∇(u|u|(α−1)/β) on Rd, d≥2, with α∈(1,2) and β>1 and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any Lp(Rd), 1≤p≤∞. Next, we apply the general results to the initial condition u0(x)=M|x|−β, 1<β<d, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.

Network Design for Silent Link User Equilibrium

Connected vehicles (CVs) are anticipated to improve road safety and travel efficiency in a transportation system. However, the deployment of CV technologies in transportation networks can lead to privacy issues, as the communication among CVs can expose vehicles' location information. To address this issue, we introduce a privacy protection method named "silent link" to transportation networks and propose a silent link user equilibrium (SLUE) framework to study the impact of privacy protection countermeasures on network flow. A theoretical analysis regarding existence and uniqueness conditions of SLUE is provided. The proposed SLUE facilitates privacy-oriented network design to achieve optimal levels of privacy for CVs. Accordingly, a bi-level network optimization problem is formulated for the design of silent links in transportation networks. Numerical examples are demonstrated using the Braess and Sioux Falls networks.

Dynamic behaviors for the graphene nano/microelectromechanical system in a fractal space

The microelectromechanical devices have triggered rocketing interest in various advanced technologies, that is, sensors, material science, and energy harvesting, and now the devices tend to a nanoscale size, making the technology much more attractive in both academic and industrial communities. This paper takes a graphene nano/microelectromechanical system as an example to study its dynamical property, which is the foundation for the optimal design and reliable operation. As the system is inextricably complex with singularity and zero conditions in a fractal space, the iteration perturbation method is used to obtain the periodic solution of the system. The results show clearly the low-frequency property of the system and pull-in instability; furthermore, the fractal dimension affects greatly the system’s dynamical property.

Global existence and regularity for a pseudo-parabolic equation with p(x,t)-Laplacian

We study the Dirichlet problem for the pseudo-parabolic equationut=div(|∇u|p(x,t)−2∇u)+Δut+f(x,t,u,∇u) in the cylinder (x,t)∈QT=Ω×(0,T), Ω⊂Rd, d≥2. It is shown that under appropriate conditions on the regularity of the data and the growth of the source f with respect to the second and third arguments, the problem has a global in time solution with the propertiesu∈L∞(0,T;H02(Ω)),ut,|∇ut|∈L2(QT),|∇u|∈L∞(0,T;Lp(⋅)(Ω))∩Lp(⋅,⋅)+δ(QT) with some δ>0. For special choices of the source f, sufficient conditions of uniqueness are derived, stability of solutions with respect to perturbations of the nonlinear structure of the equation is proven, and the rate of vanishing of ‖u‖W1,2(Ω) is found.

Coupled simultaneous analysis of vaccine and self-awareness strategies on evolutionary dilemma aspect with various immunity

On evolutionary game theory (EGT), two intervention policies: vaccination and self-awareness, are considered to account for how human attitude impacts disease spreading. Although these interventions can impose, their implementation may depend on the various immunity systems such as shield immunity, innate immunity, waning immunity, natural immunity, and artificial immunity. This framework provides an epidemic SEIRVA (susceptible-exposed-infected-removed-vaccinated-aware) model and two EGT dynamics to analyze the interplay between the immunity system and social learning interventions. The prospect of exploring the individual's strategy and social dilemma for removing a disease could assist design an effective vaccine program and self-awareness policy. Also, we evaluated the indicator of social efficiency deficit (SED) for a social dual-dilemma to measure the presence of a dilemma situation. Extensive theoretical analysis displays that stability includes the reproduction number, conditions for positivity and uniqueness, and the strength number analyzed in the equilibria, including fundamental properties validated by numerical simulation of the discretization method that appraises a variety of graphs at adjusting parameters. We present extensive numerical studies investigating the affect of controlling parameters, individual vulnerability, optimal policies, and individual costs. It turns out that, even with the affordable vaccine, individuals may have very different behaviors; self-awareness strategy plays a vital role in controlling diseases.

Observability Analysis for Orbit Determination Using Spaceborne Gradiometer

The application of a spaceborne gradiometer to satellite orbit determination has been proposed in recent years. The advantages lie in its full autonomy and its immunity against spoofing attacks. The objective of this paper is to study the observability of the spaceborne gradiometer-based orbit determination (SGOD) system. To address the problem of unknown biases in spaceborne gradiometer measurements, the joint estimation of satellite orbital states and gradiometer biases is adopted. Then, the system observability matrices are formulated based on the theory of nonlinear observability. The local weak observability is assessed numerically by singular values and condition numbers of the observability matrices. A special unobservable case is demonstrated, which is the circular orbit in the Keplerian two-body model. Different orbital parameters, attitude control modes, and gravity models are furtherly employed to analyze their effects on system observability. The observability analysis shows that low altitudes, large eccentricities, large inclinations, attitude maneuverings, and the J2 perturbation can improve the system observability. Furthermore, a numerical simulation using the extended Kalman filter (EKF) is developed to obtain accuracies and convergence rates of the SGOD system. Effects of orbital parameters, attitude control modes, and the J2 perturbation on orbit determination results are also studied. It is shown that the simulation results agree with the observability analysis. Using a spaceborne gradiometer with 0.1 E (1 E=10−9 s−2) accuracy, position accuracy of tens of meters is achieved for a low Earth orbit.

Unfolding-Aided Bootstrapped Phase Retrieval in Optical Imaging: Explainable AI reveals new imaging frontiers

Phase retrieval in optical imaging refers to the recovery of a complex signal from phaseless data acquired in the form of its diffraction patterns. These patterns are acquired through a system with a coherent light source that employs a diffractive optical element (DOE) to modulate the scene, resulting in coded diffraction patterns (CDPs) at the sensor. Recently, the hybrid approach of a model-driven network or deep unfolding has emerged as an effective alternative to conventional model- and learning-based phase-retrieval techniques because it allows for bounding the complexity of algorithms while also retaining their efficacy. Additionally, such hybrid approaches have shown promise in improving the design of DOEs that follow theoretical uniqueness conditions. There are opportunities to exploit novel experimental setups and resolve even more complex DOE phase-retrieval applications. This article presents an overview of algorithms and applications of deep unfolding for bootstrapped—regardless whether near, middle, or far zones—phase retrieval.