Framework Of Calculus Research Articles

Personal Information Disclosure via Voice Assistants: The Personalization–Privacy Paradox

The purpose of this research is to examine the factors affecting the willingness to disclose personal information based upon the privacy calculus framework and its relation to the continued usage of the Voice Assistant (VA) devices. The information disclosure is analyzed from a dual channel benefit/risk perspective through the calculus lens. Particularly, the joint effects of the two independent paths of benefits and risks that can induce the desire to disclose personal information, along with their various antecedents are proposed and empirically tested. Data are collected from 427 respondents using Amazon Mechanical Turk (MTurk) as the platform and analyzed using a Maximum Likelihood Structural Equation Modelling (ML-SEM) technique. The analysis shows that personalized services, perceived enjoyment and perceived complementarity influence the perceived benefits, which in turn positively affects the personal information disclosure. Perceived severity and perceived control are found to have a positive effect, while perceived trust has a negative effect on the perceived risks, respectively, which in turn is negatively associated with the personal information disclosure. Compared to the perceived risks, the effect of perceived benefits on the information disclosure is found to be stronger, which motivates the users to continuously use the services of the VA devices. This work extends the current literatures on privacy concerns to the users’ intention of disclosing their personal information by proposing and empirically testing six hypotheses as a part of the research model. Finally, the research implications are discussed, and suggestions are provided.

Finite Mellin transform for $$(p,q)$$ and symmetric calculus

In this paper, we have introduced and studied finite $$(p,q)$$ -Mellin transform which extends the known theory of finite q-Mellin transform. Also, in this paper, we study finite $$(p,q)$$ -Mellin transform in the framework of symmetric calculus. This study is new even for the special case $$p=q$$ . In both the cases, we provide several basic properties, inversion formulas, the appropriate convolution product and the Parseval type relations.

Path Integral Quantization of Singular Lagrangians using Fractional Derivatives

In this work, a general theory is given for quantizing singular systems within framework of fractional calculus using the path integral approach. The fractional singular Lagrangians and fractional path integral representation are presented. The fractional Hamilton-Jacobi function is constructed to obtain the path integral quantization for singular systems. One physical example is examined.

Convergence analysis of symmetric dual-wind discontinuous Galerkin approximation methods for the obstacle problem

This paper formulates and analyzes symmetric dual-wind discontinuous Galerkin (DG) methods for second order elliptic obstacle problem. These new methods follow from the DG differential calculus framework that defines discrete differential operators to replace the continuous differential operators when discretizing a partial differential equation (PDE). We establish optimal a priori error estimates for both linear and quadratic elements provided the exact solution is sufficiently regular. These results are also shown to hold for some non-positive penalty parameters, with the emphasis on zero penalization across all interior and boundary edges. Numerical experiments are provided to validate the theoretical results and gauge the performance of the proposed methods.

Hereditariness and non-locality in wave propagation modeling

The classical wave equation is generalized within the framework of fractional calculus in order to account for the memory and non-local effects that might be material features. Both effects are included in the constitutive equation, while the equation of motion of the deformable body and strain are left unchanged. Memory effects in viscoelastic materials are modeled through the distributed-order fractional constitutive equation that generalizes all linear models having differentiation orders up to order one. The microlocal approach in analyzing singularity propagation is utilized in the case of viscoelastic material described by the fractional Zener model, as well as in the case of two non-local models: non-local Hookean and fractional Eringen.

Paracontrolled quasi-geostrophic equation with space-time white noise

We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.

Radio Interferometric Calibration Using a Complex Student’s t-distribution and Wirtinger Derivatives

Abstract Radio interferometric gain calibration can be biased by incomplete sky models and radio frequency interference, resulting in calibration artefacts that can restrict the dynamic range of the resulting images. It has been suggested that calibration algorithms employing heavy-tailed likelihood functions are less susceptible to this due to their robustness against outliers in the data. We present an algorithm based on a Student’s t-distribution which leverages the framework of complex optimisation and Wirtinger calculus for efficient and robust interferometric gain calibration. We integrate this algorithm as an option in the newly released calibration software package, CubiCal. We demonstrate that the algorithm can mitigate some of the biases introduced by incomplete sky models and radio frequency interference by applying it to both simulated and real data. Our results show significant improvements compared to a conventional least-squares solver which assumes a Gaussian likelihood function. Furthermore, we provide some insight into why the algorithm outperforms the conventional solver, and discuss specific scenarios (for both direction-independent and direction-dependent self-calibration) where this is expected to be the case.

Introducing the p-Laplacian spectra

In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p ∈ (1, 2). With this decomposition we can process signals of different degrees of smoothness.We first analyze solutions of scale spaces, generated by γ-homogeneous operators, γ∈R. An analytic solution is formulated when the scale space is initialized with a nonlinear eigenfunction of the respective operator. We show that the flow is extinct in finite time for γ ∈ [0, 1).A main innovation in this study is concerned with operators of fractional homogeneity, which require the mathematical framework of fractional calculus. The proposed transform rigorously defines the notions of decomposition, reconstruction, filtering and spectrum. The theory is applied to the p-Laplacian operator, where the tools developed in this framework are demonstrated.

Fractional Calculus Approach to Nonlocal Three-Dimensional Vibration Analysis of Plates

The nonlocal three-dimensional vibration analysis of rectangular plates is investigated within the framework of fractional calculus in the sense of the Caputo fractional derivative. To show the effect of the fractional derivative on nonlocality, the frequency spectra of the plates with different boundary conditions and symmetry modes are carried out for the different orders of the fractional derivative and different values of the length scale parameter . The vibration analysis is obtained by the Ritz energy method, whereas Chebyshev polynomials are used as admissible functions. The results of the frequency spectrum demonstrate that the nonlocal effect decreases and the results get closer to the values of the classical frequencies as the order of the fractional derivative approaches the classical derivative .

Darboux transformation: new identities

This letter reports some new identities for multisoliton potentials that are based on the explicit representation provided by the Darboux matrix. These identities can be used to compute the complex gradient of the energy content of the tail of the profile with respect to the discrete eigenvalues and the norming constants. The associated derivatives are well defined in the framework of the so-called Wirtinger calculus which can aid a complex variable based optimization procedure in order to generate multisolitonic signals with desired effective temporal and spectral width.

Reliability of Poisson-Nernst-Planck Anomalous Models for Impedance Spectroscopy.

We investigate possible connections between two different implementations of the Poisson-Nernst-Planck (PNP) anomalous models used to analyze the electrical response of electrolytic cells. One of them is built in the framework of the fractional calculus and considers integro-differential boundary conditions also formulated by using fractional derivatives; the other one is an extension of the standard PNP model presented by Barsoukov and Macdonald, which can also be related to equivalent circuits containing constant phase elements (CPEs). Both extensions may be related to an anomalous diffusion with subdiffusive characteristics through the electrical conductivity and are able to describe the experimental data presented here. Furthermore, we apply the Bayesian inversion method to extract the parameter of interest in the analytical formulas of impedance. To resolve the corresponding inverse problem, we use the delayed-rejection adaptive-Metropolis algorithm (DRAM) in the context of Markov-chain Monte Carlo (MCMC) algorithms to find the posterior distributions of the parameter and the corresponding confidence intervals.

Optical spatial dispersion in terms of Jones calculus

Traditionally, optical spatial dispersion (OSD) is defined as the dependence $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varepsilon}}(\stackrel{P\vec}{k})$ of the dielectric permittivity tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varepsilon}}$ on the light wave vector $\stackrel{P\vec}{k}$, similarly to the frequency ($\ensuremath{\omega}$) dispersion of the dielectric tensor $\ensuremath{\varepsilon}(\ensuremath{\omega})$. We have developed an approach for the description of the OSD phenomena in the framework of Jones calculus. In Jones calculus the differential Jones matrix (DJM) $N$ is the generalization of the light wave-vector $\stackrel{P\vec}{k}$ in the same sense that $\stackrel{P\vec}{k}$ is the generalization of the light wave number $k$. The latter inspires us to expect that there must exist a way to describe the OSD phenomena in terms of the DJM. We show that such a relation between the OSD phenomena and Jones calculus indeed exists. To prove the latter we derive a general relation between the DJM and components of the dielectric permittivity tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\varepsilon}}$. We establish the relation of the DJM approach, proposed in this paper, to the traditional OSD approach of the gyration pseudotensor as well as to that developed by Mauguin for light propagation in cholesteric liquid crystals [M. C. Mauguin, Bull. Soc. Fr. Mineral. Crystallogr. N3, 71 (1911)]. We demonstrate that both the gyration pseudotensor and Mauguin's approach can be derived as particular cases of the proposed DJM approach. In our approach the integral Jones matrix (IJM) of the medium taking into account OSD is the product of the IJM without taking into account OSD by the correction IJM, which accounts for the OSD effects. In a general case, when all components of the OSD DJM ${N}^{D}$ are nonzero, the secular equation for the refractive indices of the eigenwaves is a quartic equation. The coefficient ${a}_{3}$ at the cubic term in the secular equation is nonzero only for nonzero OSD corrections to the average refractive index. For transparent crystals at nonzero OSD correction to the average refractive index and zero to all other correction parameters in ${N}^{D}$, the secular equation has two distinct real and two complex-conjugate roots. We assign the complex-conjugate roots to the forward and backward light scattering. Therefore, taking into account the OSD effect on the refractive index, the Jones calculus becomes capable of describing light scattering. The proposed Jones calculus approach is a general tool for taking into account OSD in optically inhomogeneous media, in which several or all OSD correction parameters are simultaneously nonzero, for example, in liquid-crystal cells with a spatially nonuniform director field, including those containing defects.

Remarks on Föllmer's pathwise Itô calculus

We extend some results about Follmer's pathwise Ito calculus that have only been derived for continuous paths to cadlag paths with quadratic variation. We study some fundamental properties of pathwise Ito integrals with respect to cadlag integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise Ito integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of Follmer's calculus.

Formula omitted]-Mitigators: Improving your stochastic network calculus output bounds

Giving tight estimates for output bounds is key to an accurate network analysis using the stochastic network calculus (SNC) framework. In order to upper bound the delay for a flow of interest in the network, one typically has to calculate output bounds of cross-traffic flows several times. Thus, an improvement in the output bound calculation pays off considerably. In this paper, we propose a new output bound calculation in the SNC framework by making use of Jensen’s inequality. In consists of inserting a convex function h into the bound, the so-called h-mitigator. We prove the bound’s validity and also show that, by choosing h as the power function, that it is always at least as accurate as the state-of-the-art method. Numerical evaluations demonstrate that even in small heterogeneous two-server topologies, our approach can improve a delay bound’s violation probability by a factor of over 135. For a set of randomly generated parameters, the bound is still decreased by a factor of 1.23 on average. Furthermore, our approach can be easily integrated in existing end-to-end analyses. Last but not least, we investigated another variant for h, the exponential function and showed numerically that this approach is mostly disadvantageous.

Fully complex conjugate gradient-based neural networks using Wirtinger calculus framework: Deterministic convergence and its application

Conjugate gradient method has been verified to be one effective strategy for training neural networks due to its low memory requirements and fast convergence. In this paper, we propose an efficient conjugate gradient method to train fully complex-valued network models in terms of Wirtinger differential operator. Two ways are adopted to enhance the training performance. One is to construct a sufficient descent direction during training by designing a fine tuning conjugate coefficient. Another technique is to pursue the optimal learning rate instead of a fixed constant in each iteration which is determined by employing a generalized Armijo search. In addition, we rigorously prove its weak and strong convergence results, i.e., the gradient norms of objective function with respect to weights approach zero along with the increasing iterations and the weight sequence tends to the optimal point. To verify the effectiveness and rationality of the proposed method, four illustrated simulations have been performed on both typical regression and classification problems.

Command Filtering-Based Neural Network Control for Fractional-Order PMSM With Input Saturation

Command filtering-based neural network control is investigated in this paper for fractional-order input saturated permanent magnet synchronous motor (PMSM). First, the fractional-order command filter is introduced to cope with the “explosion of complexity” problem caused by the repeated derivatives of virtual signals in backstepping. Next, a compensation mechanism related to error is investigated to decrease the filtering errors under fractional calculus framework. Then, a neural network with its weight being updated online is accepted to eliminate restrictions on the uncertain nonlinear functions. Besides, the minimal learning parameterization technique is introduced to construct fractional-order adaptive law for the parameters of the neural network. Finally, the simulation results testify the availability and advantage of the designed approach.

Stochastic Control and Differential Games with Path-Dependent Influence of Controls on Dynamics and Running Cost

In this paper we consider the functional Itô calculus framework to find a path-dependent version of the Hamilton--Jacobi-Bellman equation for stochastic control problems that feature dynamics and running costs that depend on the path of the control. We also prove a dynamic programming principle for such problems. We apply our results to path-dependence of the delay type. We further study stochastic differential games in this context.

A note on the discrete Cauchy‐Kovalevskaya extension

In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type.

A Control Flow Graph Based Approach to Make the Verification of Cyber-Physical Systems Using KeYmaera Easier

KeYmaera is an interactive theorem prover and is used to verify safety properties of cyber-physical systems (CPSs). It implements a Dynamic Logic for Hybrid Programs (HPs), while a HP models a CPS very precisely. Verifying properties of a given system in KeYmaera can become a challenge for a user since the proof is authored in a classical sequent calculus framework and a successful proof requires from the user intimate knowledge of the available calculus rules. Another barrier for widespread application of KeYmaera is the purely textual representation of current proof goals, what requires from the user very good training, experience, and patience. In this paper, we present an alternative verification approach based on KeYmaera, which drastically improves usability and minimizes user interaction. The main idea is to let the user annotate invariants and contracts to states of the hybrid automaton. Thus, the user can employ the graphical representation of the modelled system and is not bound to the purely textual form of hybrid programs as in KeYmaera. Based on the user-provided contracts, one can generate proof obligations, which are much simpler than the original proof goal in KeYmaera. The article is published in the authors’ wording.

Fractional hyper-chaotic model with no equilibrium

This paper considers a novel four dimensional dynamical model containing hyper-chaotic attractors. The concept fractional differential based on the exponential and Mittage–Leffler kernel were used to extend the classical version in order to include into the mathematical formulation the crossover in waiting time distribution. We have presented for both models the conditions under which the existence and the uniqueness of exact solutions are reached.We have used a newly established numerical scheme, that combines the fundamental theorem of fractional calculus and the Lagrange interpolation polynomial to solve the system numerically. We presented some numerical simulations for different values of fractional order, we compared both models with the existing one under the framework of fractional calculus. Our model has shown very new chaotic features in particular with the Atangana–Baleanu fractional derivative.