Arbitrary Time Scale Research Articles

Stability and [formula omitted]-gain characterization of positive linear systems on time scales

The stability and Lp-gain (p∈{1,∞}) analysis problems are investigated for positive linear systems on arbitrary time scales in this paper. Firstly, by virtue of the linear copositive Lyapunov function, the uniform stability and uniformly exponential stability conditions are presented for positive linear time-varying systems on time scales. Then, we conclude that the uniformly exponential stability condition of positive linear time-invariant systems is independent of time scales. Meanwhile, it is further proved that the Lp-gain (p∈{1,∞}) conditions of these systems are also independent of time scales and can be fully characterized by system matrices. Finally, a numerical example is given to demonstrate the validity of our conclusions.

Hochstadt-Lieberman type theorems for a diffusion operator on a time scale

In this paper, we consider a half inverse problem for a diffusion operator on a time scale which is the union of an interval and another arbitrary time scale such as . We give a Hochstadt-Lieberman type theorem for this problem and some appropriate examples.

A sparse multiscale nonlinear autoregressive model for seizure prediction

Objectives. Accurate seizure prediction is highly desirable for medical interventions such as responsive electrical stimulation. We aim to develop a classification model that can predict seizures by identifying preictal states, i.e. the precursor of a seizure, based on multi-channel intracranial electroencephalography (iEEG) signals. Approach. A two-level sparse multiscale classification model was developed to classify interictal and preictal states from iEEG data. In the first level, short time-scale linear dynamical features were extracted as autoregressive (AR) model coefficients; arbitrary (usually long) time-scale linear and nonlinear dynamical features were extracted as Laguerre–Volterra AR model coefficients; root-mean-square error of model prediction was used as a feature representing model unpredictability. In the second level, all features were fed into a sparse classifier to discriminate the iEEG data between interictal and preictal states. Main results. The two-level model can accurately classify seizure states using iEEG data recorded from ten canine and human subjects. Adding arbitrary (usually long) time-scale and nonlinear features significantly improves model performance compared with the conventional AR modeling approach. There is a high degree of variability in the types of features contributing to seizure prediction across different subjects. Significance. This study suggests that seizure generation may involve distinct linear/nonlinear dynamical processes caused by different underlying neurobiological mechanisms. It is necessary to build patient-specific classification models with a wide range of dynamical features.

Time scale state feedback h-stabilisation of linear systems under Lipschitz-type disturbances

This paper studies the h-stabilisation problem of certain classes of perturbed systems on time scales. Sufficient conditions for the control law design are proposed to ensure the h-stability of the closed-loop dynamical system under Lipschitz-type disturbances. Using the Gronwall inequality approach with time scale theory, the h-stability of the closed-loop system is investigated in non-uniform time domains with bounded graininess. Some numerical examples are provided to show the feasibility of the obtained results using the proposed approach for systems evolving on some arbitrary time scales.

Ultrafast Transient Holographic Microscopy.

Nanotechnology is increasingly being applied in many emerging technologies, ranging from metamaterials to next-generation nanodrugs. A key ingredient for its success is the ability to specifically tailor ultrafast nanoscale light–matter interactions over very large areas. Unfortunately, dynamic imaging by ultrafast nanoscopy so far remains limited to very small 2D areas. This shortcoming prevents connecting single-particle observations with large-scale functionality. Here, we address this experimental challenge by combining concepts of ultrafast spectroscopy, wide-field nanoscopy, and digital holography. We introduce an ultrafast holographic transient microscope for wide-field transient nanoscale imaging with high frequency all-optical signal demodulation. We simultaneously record ultrafast transient dynamics of many individual nano-objects and demonstrate time-resolved spectroscopy of gold nanoparticles over a large volume irrespective of their x–y–z position. Our results pave the way to single-shot 3D microscopy of 2D and 3D materials on arbitrary time scales from femtosecond carrier dynamics in optoelectronic materials to millisecond dynamics in complex tissues.

SPECTRAL PROPERTIES OF A CONFORMABLE BOUNDARY VALUE PROBLEM ON TIME SCALES

We study a self-adjoint conformable dynamic equation of second order on an arbitrary time scale $\mathbb{T}$. We state an existence and uniqueness theorem for the solutions of this equation. We prove the conformable Lagrange identity on time scales. Then, we consider a conformable eigenvalue problem generated by the above-mentioned dynamic equation of second order and we examine some of the spectral properties of this boundary value problem.

A multivalued logarithm on time scales

A new definition of a multivalued logarithm on time scales is introduced for delta-differentiable functions that never vanish. This new logarithm arises naturally from the definition of the cylinder transformation that is also the wellspring of the definition of exponential functions on time scales. This definition will lead to a logarithm function on arbitrary time scales with familiar and useful properties that previous definitions in the literature lacked.

Oscillation criteria for solution of hyperbolic delay dynamic equations with time and spatial variables on arbitrary time scales

Oscillatory behavior of a hyperbolic delay partial dynamic equation with time and spatial variables defined on arbitrary time scales is studied in this article. The Green’s identity on an arbitrary time scale is presented. Using that identity and Riccati transformation, several oscillation criteria for the concern dynamic equation with Neumann boundary condition is established. Examples are provided to illustrate our results.

Bennett–Leindler Type Inequalities for Nabla Time Scale Calculus

In this study, we generalize the converse of Hardy and Copson inequalities, which are known as Bennett and Leindler type inequalities, for nabla time scale calculus. This generalization allows us not only to unify all the related results existing in the literature for an arbitrary time scale but also to obtain new results which are analogous to the results of the delta time scale calculus.

Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scale

The Mei symmetry and its corresponding conserved quantities for non-migrated Birkhoffian systems on a time scale are proposed and studied. Firstly, the dynamic equations of non-migrated Birkhoffian systems (including free Birkhoffian systems, generalized Birkhoffian systems and constrained Birkhoffian systems) on a time scale are derived based on the time-scale Pfaff-Birkhoff principle and time-scale generalized Birkhoff principle. Secondly, based on the fact that the dynamical functions in the non-migrated Birkhoff’s equations still satisfy the original equations after they have been transformed, the definitions of Mei symmetry on an arbitrary time scale are given, and the corresponding criterion equations are derived. Thirdly, Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scales are established and proved, and Mei conserved quantities of Birkhoffian systems on a time scale are obtained. The results are illustrated by three examples.

Hilbert-type inequalities for time scale nabla calculus

This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

On Dynamical Systems with Nabla Half Derivative on Time Scales

This paper is devoted to study of dynamical systems involving nabla half derivative on an arbitrary time scale. We prove existence and uniqueness of the solution of such system supplied with a suitable initial condition. Both Riemann–Liouville and Caputo approaches to noninteger-order derivatives are covered. Under special conditions we present an explicit form of the solution involving a time scales analogue of Mittag–Leffler function. Also an algorithm for solving of such problems on isolated time scales is established. Moreover, we show that half power functions are positive and decreasing with respect to $$t-s$$ on an arbitrary time scale.

Factorization Theorems of Cesàro and Copson Spaces on Time Scales

In this paper, we prove some factorization theorems of Cesaro and Copson spaces on an arbitrary time scale $$\mathbb{T}$$ , which offer enhancements of dynamic Copson’s and Hardy’s inequalities. Our results enhance, among others, the best-known forms of dynamic Hardy’s inequality. The main results will be proved by employing the time scales Holder’s inequality and the derived time scales power rule of integration.

A New Dynamic Scheme via Fractional Operators on Time Scale

The present work investigates the applicability and effectiveness of the generalized Riemann-Liouville fractional integral operator integral method to obtain new Minkowski, Gr\"{u}ss type and several other associated dynamic variants on an arbitrary time scale, which are communicated as a combination of delta and fractional integrals. These inequalities extend some dynamic variants on time scales, and tie together and expand some integral inequalities. The present method is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractional differential equations applied in mathematical physics.

SOME CLASSES OF CONVEX FUNCTIONS ON TIME SCALES

We have introduced diamond $\phi_{h-s, \mathbb{T}}$ derivative and diamond $\phi_{h-s,\mathbb{T}}$ integral on an arbitrary time scale. Moreover, various interconnections with the notion of classes of convex functions about these new concepts are also discussed.

Gamma-Nets: Generalizing Value Estimation over Timescale

Temporal abstraction is a key requirement for agents making decisions over long time horizons—a fundamental challenge in reinforcement learning. There are many reasons why value estimates at multiple timescales might be useful; recent work has shown that value estimates at different time scales can be the basis for creating more advanced discounting functions and for driving representation learning. Further, predictions at many different timescales serve to broaden an agent's model of its environment. One predictive approach of interest within an online learning setting is general value function (GVFs), which represent models of an agent's world as a collection of predictive questions each defined by a policy, a signal to be predicted, and a prediction timescale. In this paper we present Γ-nets, a method for generalizing value function estimation over timescale, allowing a given GVF to be trained and queried for arbitrary timescales so as to greatly increase the predictive ability and scalability of a GVF-based model. The key to our approach is to use timescale as one of the value estimator's inputs. As a result, the prediction target for any timescale is available at every timestep and we are free to train on any number of timescales. We first provide two demonstrations by 1) predicting a square wave and 2) predicting sensorimotor signals on a robot arm using a linear function approximator. Next, we empirically evaluate Γ-nets in the deep reinforcement learning setting using policy evaluation on a set of Atari video games. Our results show that Γ-nets can be effective for predicting arbitrary timescales, with only a small cost in accuracy as compared to learning estimators for fixed timescales. Γ-nets provide a method for accurately and compactly making predictions at many timescales without requiring a priori knowledge of the task, making it a valuable contribution to ongoing work on model-based planning, representation learning, and lifelong learning algorithms.

Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

A family of effective equations for wave propagation in periodic media for arbitrary timescales mathcal {O}(varepsilon ^{-alpha }), where varepsilon ll 1 is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017, arXiv:1701.08600; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, arXiv:1803.09455). The effective solutions in the family are proved to be varepsilon close to the original wave in a norm equivalent to the {mathrm {L}^{infty }}(0,varepsilon ^{-alpha }T;{{mathrm {L}^{2}}(varOmega )}) norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.

Critical time windows for renewable resource complementarity assessment

This paper proposes a framework to assess the complementarity between geographically dispersed variable renewable energy resources over arbitrary time scales. More precisely, the framework relies on the concept of critical time windows, which offers an accurate time-domain description of low probability power production events impacting power system operation and planning. A scalar criticality indicator is derived to quantify the spatiotemporal complementarity that renewable generation sites may exhibit, and it is leveraged to propose optimisation models seeking to identify deployment patterns with maximum complementarity. The usefulness of the framework is shown in a case study investigating the complementarity between wind regimes in continental western Europe and southern Greenland, using roughly 300 candidate locations and 10 years of reanalysis and simulated data with hourly resolution. Besides showing that the occurrence of low wind power production events can be reduced on a regional scale by exploiting diversity in local wind patterns, results highlight the fact that aggregating wind power production sites located on different continents may result in a lower occurrence of system-wide low wind power production events and point to potential benefits of intercontinental electricity interconnections.

Entanglement negativity as a universal non-Markovianity witness

In order to engineer an open quantum system and its evolution, it is essential to identify and control the memory effects. These are formally attributed to the non-Markovianity of dynamics that manifests itself by the evolution being indivisible in time, a property which can be witnessed by a non-monotonic behavior of contractive functions or correlation measures. We show that by monitoring directly the entanglement behavior of a system in a tripartite setting it is possible to witness all invertible non-Markovian dynamics, as well as all (also non-invertible) qubit evolutions. This is achieved by using negativity, a computable measure of entanglement, which in the usual bipartite setting is not a universal non-Markovianity witness. We emphasize further the importance of multipartite states by showing that non-Markovianity cannot be faithfully witnessed by any contractive function of single qubits. We support our statements by an explicit example of eternally non-Markovian qubit dynamics, for which negativity can witness non-Markovianity at arbitrary time scales.