Differential Theorem Research Articles

Impulse Controllability for Singular Hybrid Coupled Systems

This study examines the concept of impulse controllability within singular hybrid coupled systems through the utilisation of decentralised proportional plus derivative (P-D) output feedback. By employing the Differential Mean Value Theorem, the nonlinear model can be converted into a linear parameter-varying large-scale system. Our analysis leads to the establishment of algebraic conditions that are both necessary and sufficient for the existence of a decentralised P-D output feedback controller that can guarantee impulse controllability in these complex systems. Moreover, we address the issue of admissibility within these systems by employing matrix trace inequalities. We present a novel sufficient condition for impulse controllability, which offers a new perspective on addressing this challenging problem. To validate our findings, we present numerical examples that demonstrate the effectiveness of the proposed methodologies in practice.

Towards the sociology of pets—Social functions of animals in contemporary societies observed from systems theoretical perspective

AbstractThis paper advocates for introducing sociology of pets as a particular field of research as the response of ever growing social relevance of pets in contemporary societies. Though other theoretical approaches are suitable for tackling these complex issues, the article offers viewpoint from Luhmann's social systems theory. Based on Luhmann's writings on ecological communication, the article departs from the system/environment distinction and conceives pets as being part of a society's environment that attained resonance of a social system. Further, entry of pets into a social system is interpreted through differentiation theorem arguing that pets perform highly relevant social functions for all functional subsystems of a society by contributing to their autopoietic reproduction. This is primarily accomplished through communication about pets, emerging specific semantics, usage of binary codes specific for different subsystems, correspondence to their mediums, programs and functions. The major part of work is presenting relations of pets to 10 functional subsystems of a society, as this delineates scope of empirical phenomena of potential interest for sociology of pets. These processes might be observed at systemic, organizational and interaction levels. Despite possessing multiple merits, also, certain limits of Luhmann's account of pets are unfolded, such as incapability to hear authentic voices of animals, but rather distorted semantic codes emerging from consciousness of human psychic systems, afterwards structurally coupled with communication as a medium of social systems. At this point, social systems theory might be complemented by theory of interspecies communication, as taking into consideration other theoretical approaches anchored at ontologies pertinent to include non‐human agents is invocated. The tentative outcome of this introductory discussion is that apparent process of anthro‐morphosization of animals is taking place while the question of relation between humans–humans in the context of building just societies of the 21st century still remains open.

Two-Sided Clifford-Valued Special Affine Fourier Transform: Properties and Associated Convolution

The primary focus of the present study is to analyse the Clifford-valued functions by introducing the notion of a two-sided Clifford-valued special affine Fourier transform in L2(Rn,Cl0,n). Firstly we propose the novel definition of the Clifford-valued special affine Fourier transform and derive its fundamental properties which include inversion formula, translation covariance, scaling covariance and Plancherel formula. Subsequently we introduce the boundedness, continuity and differentiation theorems for the proposed transform. Finally we culminate our investigation by deriving the convolution for this newly proposed transform.

Adaptive fixed-time tracking control for nonlinear systems subject to asymmetric input saturation

For nonlinear systems with asymmetric input saturation, an adaptive fixed-time control strategy via multi-dimensional Taylor network (MTN) is proposed to effectively solve the tracking control problem. Firstly, the Gaussian error function and the differential mean value theorem are used to simulate the asymmetric input saturation model, that is, the nonlinear model is transformed into a linear model with a bounded perturbation. Secondly, MTNs are applied to control the design process, which approximate the nonlinear structures with linear combination of polynomials. Thirdly, in addition to the tracking error converging to an arbitrarily small neighborhood around the origin within a fixed time, the proposed control scheme not only guarantees that any signal in the controlled system remains bounded, but also avoids the problem of finite-time control relying on the initial state. Finally, three examples are used to validate the feasibility and superiority of the scheme.

Unravelling Three Differential Mean Value Theorems in Calculus

The theoretical foundation of differential aspect in real analysis is the differential mean value theorem, which connects to both the local and total properties of the copula function. The link between the copula function's local and overall properties is differential calculus. The maximum value, minimum value, extreme value, monotonicity, and other difficulties may all be resolved with the help of these differential mean value theorems. Additionally, they aid in the computation of limits, the demonstration of inequality, and the identification of the curve's inflection point and concave-convex interval. They are also capable of mapping the function and locating the equation's root. To tackle these issues, one can apply a number of significant findings from the differential mean value theorem. This study gives the formulae for solving three distinct mean value theorems. The Cauchy, Lagrange, and Roller theorems are examples of these mean value theorems. Understanding the connections between the three mean value theorems is made easier by these studies, which also provide an explanation of the theory underlying the theorems and provide examples and visuals of their practical application.

Quantized and event-triggered modeling and fault detection for networked fuzzy systems

This article concentrates on the issue of fault detection in the networked system. Quantized output and event-triggered communication scheme are absorbed into the system analysis, and consequently, by designing a new quantizer with event triggering mechanism, a smooth communication environment without undesired system data is guaranteed. And the imperfect information interaction strategy induced asynchronous membership functions between system and fault detection observer are handled via a differential mean value theorem based approach. Combining the methods of Lyapunov stability theory and linear matrix inequality, we obtain the stability criterion of the T-S fuzzy systems closed by the predefined network channel. In the end, simulation examples demonstrate the validity of the fault detection method.

Capacitary maximal inequalities and applications

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function,MCf(x):=supr>0⁡1C(B(x,r))∫B(x,r)|f|dC, for C= the Hausdorff content or a Riesz capacity. For these maximal functions, we prove a strong-type (p,p) bound for 1<p≤+∞ on the capacitary integration spaces Lp(C) and a weak-type (1,1) bound on the capacitary integration space L1(C). We show how these estimates clarify and improve the existing literature concerning maximal function estimates on capacitary integration spaces. As a consequence, we deduce correspondingly stronger differentiation theorems of Lebesgue-type, which in turn, by classical capacitary inequalities, yield more precise estimates concerning Lebesgue points for functions in Sobolev spaces.

Differential and fuzzy differential sandwich theorems involving quantum calculus operators

The principle of subordination is useful in comparing two holomorphic functions when the range of one holomorphic function is a subset of the other and they comply at a single point. The subordination, when spoken in fuzzy set theory, becomes fuzzy subordination as the comparison between two holomorphic functions is made using the fuzzy membership function. In this article, differential and fuzzy differential Subordination, superordination, and sandwich theorems have been discussed for the classes defined by using q-derivative and symmetric q-derivative operators.

Numerical simulation and experimental study of dip angle spray film thickness under elliptical double Gaussian sum model

To address the problem of modeling the growth rate of coating film thickness when spraying at inclination angle, based on Gaussian sum model, it is proposed to use the elliptic double Gaussian sum model to establish the cumulative model of coating film thickness when spraying at static inclination angle of the spray gun. The differential geometry amplification theorem is used to establish the coating growth rate model with the spraying inclination angle as the variable; after that, the static inclination spraying experiments are carried out, and the coating thickness data of the sampling points are recorded through the spraying experiments, and the Levenberg-Maquart algorithm is used for the least-squares fitting of the model, which results in the static spraying film thickness distribution model. Finally, compared with the elliptic double β model, the fitting accuracy of the elliptic double Gaussian sum model is 6.3% higher than that of the elliptic double β model when spraying at inclination angle by comparing the R-square values, and the elliptic double Gaussian sum model is more capable of obtaining a better fitting accuracy, which further confirms the validity and practicability of the model.

Derivation of Differential Geometry Theorems in High Dimensional Spaces

Derivation of Differential Geometry Theorems in High Dimensional Spaces

Observer-based robust preview tracking control for a class of continuous-time Lipschitz nonlinear systems

&lt;p&gt;In this paper, a novel observer-based robust preview tracking controller design method is proposed for a class of continuous-time Lipschitz nonlinear systems with external disturbances and unknown states. First, a state observer is designed to reconstruct unknown system states. Second, using differentiation, the state lifting technique, the differential mean value theorem, and several ingenious mathematical manipulations, an augmented error system (AES) containing the previewable information of a reference signal is constructed, thereby transforming the tracking control problem into a robust $ H_{\infty} $ control problem. Based on linear parameter-varying (LPV) system theory, a sufficient condition for asymptotic stability of a closed-loop system with a robust $ H_{\infty} $ performance level is established in terms of the linear matrix inequality (LMI). Furthermore, a tracking controller, which includes observer-based feedback control, integral control, and preview feedforward compensation, is established for the original system. In particular, the tracking controller design is simplified by computing the observer and tracking controller gains simultaneously via only a one-step LMI algorithm. Finally, numerical simulation results demonstrate that the proposed controller leads to superior improvement in the output tracking performance compared with the existing methods.&lt;/p&gt;

Generalized application of differential mean value theorem in infinite intervals

Generalized application of differential mean value theorem in infinite intervals

Punctually presented structures I: Closure theorems

We study the primitive recursive content of various closure results in algebra and model theory, including the algebraic, the real, and the differential closure theorems. In the case of ordered fields and their real closures, our result settles a question recently raised by Selivanova and Selivanov.

High-order adaptive multiresolution wavelet upwind schemes for hyperbolic conservation laws

We present a novel system of high-order adaptive multiresolution wavelet collocation upwind schemes, implemented in a meshfree framework, for solving hyperbolic conservation laws. Our approach constructs asymmetrical wavelet bases with interpolation properties to achieve the upwind property and address nonlinearities in hyperbolic problems. An adaptive algorithm based on multiresolution analysis in wavelet theory is designed to capture moving shock waves and identify new localized steep regions. To suppress the Gibbs phenomenon, we propose an integration average reconstruction method based on the Lebesgue differentiation theorem. These numerical techniques enable the wavelet collocation upwind scheme to provide a general framework for devising satisfactory adaptive wavelet upwind methods with high-order accuracy. We perform several benchmark tests for 1D hyperbolic problems to verify the accuracy and efficiency of the proposed wavelet schemes. Notably, our wavelet collocation upwind schemes successfully solve the two interacting blast waves problem when discretizing the spatial derivatives in the physical space directly. This is in contrast to classical high-order schemes that usually involve transforming variables back and forth between physical and characteristic spaces with many local projections. Our results indicate the robustness, stability, and efficiency of our proposed schemes for strong shock wave interaction problems.

A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities.

A numerical solution for a quasi solution of the time-fractional stochastic backward parabolic equation

In the present study, we consider a time-fractional stochastic backward parabolic equation driven by standard Brownian motion. In this problem, the fractional derivative is considered in the Caputo sense. Using the minimization of a least-squares functional, stochastic variational formulation, Fréchet differentiability and utility theorems adopted directly from deterministic fractional backward equations, the existence and uniqueness theorems for a quasi solution of the proposed problem are proved. To approximate the quasi solution, a numerical technique based on 2D Chebyshev wavelets is applied. We employ the Levenberg–Marquardt regularization technique since the derived equivalent system of linear equations is ill-posed. Also, the convergence analysis for this numerical algorithm is investigated. Our results provide a new insight to find quasi solutions and apply adapted deterministic methods for some fractional stochastic backward equations. Moreover, a numerical example is provided to indicate the accuracy and efficiency of the Chebyshev wavelet method in solving the mentioned problem.

Fixed-time synchronization for quaternion-valued memristor-based neural networks with mixed delays

In this paper, the fixed-time synchronization (FXTSYN) of unilateral coefficients quaternion-valued memristor-based neural networks (UCQVMNNs) with mixed delays is investigated. A direct analytical approach is suggested to obtain FXTSYN of UCQVMNNs utilizing one-norm smoothness in place of decomposition. When dealing with drive–response system discontinuity issues, use the set-valued map and the differential inclusion theorem. To accomplish the control objective, innovative nonlinear controllers and the Lyapunov functions are designed. Furthermore, some criteria of FXTSYN for UCQVMNNs are given using inequality techniques and the novel FXTSYN theory. And the accurate settling time is obtained explicitly. Finally, in order to show that the obtained theoretical results are accurate, useful, and applicable, numerical simulations are presented at the conclusion.

Córdoba’s differentiation theorem: revisited

Córdoba’s differentiation theorem: revisited

Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves

Using the Shioda–Tate theorem and an adaptation of Silverman’s specialization theorem, we reduce the specialization of Mordell–Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields [Formula: see text] to the specialization theorem for Néron–Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface [Formula: see text], for all vertical curves [Formula: see text] of a fibration [Formula: see text] with [Formula: see text] from the complement of a sparse subset of [Formula: see text], the Mordell–Weil rank of an abelian scheme over [Formula: see text] stays the same when restricted to [Formula: see text].

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue differentiation theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon–Nikodým property.