Arbitrary Time Scale Research Articles

Some Spectral Properties of p-Laplacian Diffusion Boundary Value Problem on Time Scales

In this paper, we considerp-Laplacian type Diffusion boundary value problem on an arbitrary time scale. We generalize some spectral properties ofp-Laplacian Diffusion problem to an arbitrary time scale. Moreover, we prove Picone's identity for this problem.

Impulsive Diffusion Equation on Time Scales

Application of boundary value problems (BVP’s) on an arbitrary time scale T is a fairly new and important subject in mathematics. In this study, we deal with an eigenvalue problem for impulsive diffusion equation with boundary conditions on T. We generalize some noteworthy results about spectral theory of classical diffusion equation into T. Also, some eigenfunction estimates of the impulsive diffusion eigenvalue problem are established on T.

Time Scale Inequalities of the Ostrowski Type for Functions Differentiable on the Coordinates

In 2016, some inequalities of the Ostrowski type for functions (of two variables) differentiable on the coordinates were established. In this paper, we extend these results to an arbitrary time scale by means of a parameter λ∈0,1. The aforementioned results are regained for the case when the time scale T=R. Besides extension, our results are employed to the continuous and discrete calculus to get some new inequalities in this direction.

Linear programming problems on time scales

In this work, we study linear programming problems on time scales. This approach unifies discrete and continuous linear programming models and extends them to other cases ?in between?. After a brief introduction to time scales, we formulate the primal as well as the dual time scales linear programming models. Next, we establish and prove the weak duality theorem and the optimality conditions theorem for arbitrary time scales, while the strong duality theorem is established for isolated time scales. Finally, examples are given in order to illustrate the effectiveness of the presented results.

OSCILLATORY AND ASYMPTOTIC CRITERIA OF THIRD ORDER NONLINEAR DELAY DYNAMIC EQUATIONS WITH DAMPING TERM ON TIME SCALES

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.

FreMEn: Frequency Map Enhancement for Long-Term Mobile Robot Autonomy in Changing Environments

We present a new approach to long-term mobile robot mapping in dynamic indoor environments. Unlike traditional world models that are tailored to represent static scenes, our approach explicitly models environmental dynamics. We assume that some of the hidden processes that influence the dynamic environment states are periodic and model the uncertainty of the estimated state variables by their frequency spectra. The spectral model can represent arbitrary timescales of environment dynamics with low memory requirements. Transformation of the spectral model to the time domain allows for the prediction of the future environment states, which improves the robot's long-term performance in changing environments. Experiments performed over time periods of months to years demonstrate that the approach can efficiently represent large numbers of observations and reliably predict future environment states. The experiments indicate that the model's predictive capabilities improve mobile robot localization and navigation in changing environments.

Monotone nonlinear realisations of response maps

ABSTRACTThe problem of realisation of a monotone response map as a monotone initialised system on a finite dimensional Euclidean state space is investigated. The dynamical part of the system is described by a delta differential equation on an arbitrary time scale. This incorporates continuous- and discrete-time systems. The main result states necessary and sufficient conditions for existence of a monotone realisation of a particular class. The criterion is expressed in the language of skew differential global universes.

Conformable fractional Dirac system on time scales

We study the conformable fractional (CF) Dirac system with separated boundary conditions on an arbitrary time scale mathbb{T}. Then we extend some basic spectral properties of the classical Dirac system to the CF case. Eventually, some asymptotic estimates for the eigenfunction of the CF Dirac eigenvalue problem are obtained on mathbb{T} . So, we provide a constructive procedure for the solution of this problem. These results are important steps to consolidate the link between fractional calculus and time scale calculus in spectral theory.

Finite-time generalization of the thermodynamic uncertainty relation.

For fluctuating currents in nonequilibrium steady states, the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving. We show that this relation holds not only for the long-time limit of fluctuations, as described by large deviation theory, but also for fluctuations on arbitrary finite time scales. This generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments, for which infinite time scales are not accessible. Importantly, often this finite-time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long-time limit. We illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle.

Positive Reachability and Observability of Linear Positive Time-Variant Systems on Non-Uniform Time Domains

Positive reachability and positive observability of positive linear time-variant systems on non-uniform time domains are studied. Such systems are treated as particular cases of linear systems on arbitrary time scales. Differential and integral calculus on time scales allows for unified treatment of continuous- and discrete-time systems. Positive reachability and observability are characterized with the aid of associated reachability and observability matrices. It is shown that the properties are dual to each other. Two types of discretizations of continuous-time systems are considered: Euler discretization and sampling. It is shown that both discretizations preserve positive reachability.

An alternative statistical method for characterizing low-frequency environments in sensitive laboratory settings

Many practitioners are familiar with the collection and expression of complex noise data as Ln statistical spectra. Given suitable instrumentation, it is easy to develop Ln statistics for low-frequency sound and vibration environments, as well. However, the traditional (instrument-generated) Ln datasets have some shortcomings when it comes to understanding some environments, especially in highly sensitive settings like research laboratories. The emergence of mass data storage and manipulation tools has provided an opportunity to employ a somewhat subtler method to characterize these environments. In this presentation, we will describe this analytical technique, which provides statistical representations similar to the Ln system. However, the technique further allows an explicit invocation of timescale, allowing the user to choose an arbitrary analytical timescale relevant to sensitive operations or expected environmental transients or other forcings. We will also show examples that demonstrate why the approach might find particular use in technical settings like nanotechnology laboratories and other environments supporting research in the physical sciences.

Partition-free theory of time-dependent current correlations in nanojunctions in response to an arbitrary time-dependent bias

Working within the Nonequilibrium Green's Function (NEGF) formalism, a formula for the two-time current correlation function is derived for the case of transport through a nanojunction in response to an arbitrary time-dependent bias. The one-particle Hamiltonian and the Wide Band Limit Approximation (WBLA) are assumed, enabling us to extract all necessary Green's functions and self energies for the system, extending the analytic work presented previously [Ridley et al. Phys. Rev. B (2015)]. We show that our new expression for the two-time correlation function generalises the B\"uttiker theory of shot and thermal noise on the current through a nanojunction to the time-dependent bias case including the transient regime following the switch-on. Transient terms in the correlation function arise from an initial state that does not assume (as is usually done) that the system is initially uncoupled, i.e. our approach is partition-free. We show that when the bias loses its time-dependence, the long time-limit of the current correlation function depends on the time difference only, as in this case an ideal steady state is reached. This enables derivation of known results for the single frequency power spectrum and for the zero frequency limit of this power spectrum. In addition, we present a technique which for the first time facilitates fast calculations of the transient quantum noise, valid for arbitrary temperature, time and voltage scales. We apply this to the quantum dot and molecular wire systems for both DC and AC biases, and find a novel signature of the traversal time for electrons crossing the wire in the time-dependent cross-lead current correlations.

Oscillation criteria for second order superlinear dynamic equations with oscillating coefficients

This paper is concerned with oscillation of second order superlinear dynamic equations with oscillating coefficients. By using generalized Riccati transformations, oscillation theorems are obtained on an arbitrary time scale. We illustrate the versatility of our results by means of examples.

Local non-integer order dynamic problems on time scales revisited

This paper deals with a non-integer order calculus on time scales in a local sense. Taking advantage of a new field structure over real and complex numbers, the researchers have normalized the proposed derivative. Non-integer order mean value theorem on time scales is stated. In addition to a hypothesis on the existence of a H\( \ddot{o} \)lder continuous function h, a given time scale \( \mathbb {T} \) is explored. This consideration provided an extension to a recently proposed local fractional differentiation theory and it paved the ground to a local fractional integration on time scales. Also, making use of the presented non-integer order calculus, a class of dynamic initial value problems of fractional orders on an arbitrary time scale is studied. Eventually, an approximate solution to the potential-free Schrdinger equation of time-fractional order is obtained and illustrated.

Virtual prototyping of embedded control software in mechatronic systems: A case study

The large majority of technological devices can be seen as the sum of components of heterogeneous nature (electrical, mechanical, thermal, software, etc.) so that their analysis and design calls for the use of tools from different engineering disciplines. Integration among the different tools is particularly relevant at control system design level, since it is at this stage that it is required to analyze the behavior of both each single component and the system as a whole, with different levels of detail. In this paper mechatronic systems are considered, that exhibit a strong interplay between mechanical and electrical components, and the issue of modeling and designing embedded control algorithms and architectures for such systems is addressed. In particular, an integrated virtual prototyping approach for analyzing the system behavior down to embedded software level is proposed, that can be used in a wide number of situations that can represent the actual real-world operational conditions of consumer products. This approach can be used for system design (in particular at control systems level), embedded software design, and virtual testing so as to optimize and reduce the costs of late stage software development, physical prototypes, and their testing. The case study is based on some recently derived software tools that perform the co-simulation of the firmware execution and the multi-physical controlled system dynamics. The actual control software implemented in the final product can be entirely developed and tested inside the virtual prototype. To prove the validity and potentialities of the proposed approach, a real case study is presented, regarding a very common, though complex to simulate, mechatronic system such as an electric sliding gate. It turns out that the proposed environment goes beyond hardware-in-the-loop tools, since it does not require the use of specific hardware (the hardware itself is simulated in detail) but allows to analyze in detail the status of the microprocessors and peripherals at arbitrary time-scales and allows the designer to study at the earliest design stages the dynamics between the multi-physical controlled system and the firmware, without committing to a given hardware structure.

Existence and uniqueness results for a fractional Riemann–Liouville nonlocal thermistor problem on arbitrary time scales

Using a fixed point theorem in a proper Banach space, we prove existence and uniqueness results of positive solutions for a fractional Riemann–Liouville nonlocal thermistor problem on arbitrary nonempty closed subsets of the real numbers.

Chain rules and inequalities for the BHT fractional calculus on arbitrary timescales

We develop the Benkhettou–Hassani–Torres fractional (noninteger order) calculus on timescales by proving two chain rules for the alpha -fractional derivative and five inequalities for the alpha -fractional integral. The results coincide with well-known classical results when the operators are of (integer) order alpha = 1 and the timescale coincides with the set of real numbers.

Digital holographic microscopy of phase separation in multicomponent lipid membranes.

Lateral in-homogeneities in lipid compositions cause microdomains formation and change in the physical properties of biological membranes. With the presence of cholesterol and mixed species of lipids, phospholipid membranes segregate into lateral domains of liquid-ordered and liquid-disordered phases. Coupling of two-dimensional intralayer phase separations and interlayer liquid-crystalline ordering in multicomponent membranes has been previously demonstrated. By the use of digital holographic microscopy (DHMicroscopy), we quantitatively analyzed the volumetric dynamical behavior of such membranes. The specimens are lipid mixtures composed of sphingomyelin, cholesterol, and unsaturated phospholipid, 1,2-dioleoyl-sn-glycero-3-phosphocholine. DHMicroscopy in a transmission mode is an effective tool for quantitative visualization of phase objects. By deriving the associated phase changes, three-dimensional information on the morphology variation of lipid stacks at arbitrary time scales is obtained. Moreover, the thickness distribution of the object at demanded axial planes can be obtained by numerical focusing. Our results show that the volume evolution of lipid domains follows approximately the same universal growth law of previously reported area evolution. However, the thickness of the domains does not alter significantly by time; therefore, the volume evolution is mostly attributed to the changes in area dynamics. These results might be useful in the field of membrane-based functional materials.

Oscillation criteria for second order sublinear dynamic equations with oscillating coefficients

This paper is concerned with oscillation of second order sublinear dynamic equations with oscillating coefficients. By using generalized Riccati transformations, oscillation theorems are obtained on an arbitrary time scale. Our results provide new oscillation criteria for arbitrary time scales.

Nonoscillation and oscillation of second-order linear dynamic equations via the sequence of functions technique

We study nonoscillation/oscillation of the dynamic equation $${(rx^\Delta)}^{\Delta}(t) + p(t)x(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$ where \({t_0 \in \mathbb{T}}\), \({{\rm sup} \mathbb{T} = \infty}\), \({r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}\), \({p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}\). By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equation $${(rx^\Delta)}^{\Delta}(t) + p(t)x^\sigma(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$ which is extensively discussed on time scales.