Search Results (47)

We study functions satisfying the composition law $F\left(xy\right)+F(x/y)=P\left(F\right(x),F(y\left)\right)$ with a symmetric polynomial combiner P. We prove that symmetry together with a quadratic degree bound on P forces a composition law of d’Alembert type. We establish a degree mismatch exclusion criterion showing that symmetric polynomial combiners with $degP(u,v)\ge 3$ do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 1). For continuous nonconstant functions $F:{\mathbb{R}}_{>0}\to \mathbb{R}$ with $F\left(1\right)=0$ satisfying the composition law with a symmetric polynomial P of degree at most two, the combiner is necessarily of the form $P(u,v)=2u+2v+cuv$, $c\in \mathbb{R}$ (Theorem 3). The equation reduces in logarithmic coordinates to the classical d’Alembert functional equation. For $c\ne 0$, one obtains hyperbolic or trigonometric branches, while $c=0$ yields the squared-logarithm family. Under the cost-function assumptions $F\ge 0$ and convexity, only the hyperbolic branch with $c>0$ remains. A unit log-curvature calibration selects the canonical value $c=2$, which yields the canonical reciprocal cost $F\left(x\right)=\frac{1}{2}(x+x^{-1})-1$. For $c\ne 0$, the result extends to ${\mathbb{R}}_{>0}^n$: every solution depends only on a single linear combination of coordinate logarithms; for $c=0$, the solution is a general quadratic form ${\sum }_{i,j}{a}_{ij}ln{x}_{i}ln{x}_j$. In either case, nontrivial coordinate-wise separable costs are excluded. Full article

Symmetry transformations are defined by operators in quantum mechanics that preserve the modulus of the scalar product between Hilbert space vectors. According to Wigner’s theorem, any such transformation is represented by either a unitary linear operator or an antiunitary (isometric conjugate-linear) operator. Although antiunitary symmetries—most notably time reversal and charge conjugation—are encountered less frequently than unitary ones, they are fundamental to the description of non-conservative and reversible systems. The most frequently treated antiunitary operators are the involutive ones, called conjugations. Any antiunitary operator can be written as a product of a conjugation and a unitary operator. Considering general scattering problems defined by a scattering potential and separating conjugation from the symmetry operator, one can find the role of complex symmetric (in other words self-transpose) unitary operators in physical problems. This approach provides a robust framework for analyzing the role of non-Hermitian symmetries in wave scattering and dissipative dynamics. To demonstrate the practical applicability of these theoretical concepts, we analyze the case of polarized neutron reflectometry (PNR). We show that the scattering potential in PNR, comprising nuclear and magnetic terms, can satisfy the condition of being unitarily equivalent to its transpose, thereby guaranteeing reciprocity under specific orientations of the magnetic field. Full article

The Virial theorem has been applied with considerable success in various fields of natural sciences. This work proposes an extension of the theorem applied to discrete data series. This application will be called the Virial theorem extension and can be applied to the numerical solution of nonlinear dynamic systems represented by difference equations, such as logistic, discubic and random number generators, the numerical solution of differential equations like the nonlinear double pendulum and a series of pseudorandom numbers and its reciprocals. For this purpose, a coefficient was derived from the discrete Virial formalism. This coefficient can be used to detect when a time series is obtained as the solution of a differential equation, in which case the coefficient is close to 1, and when the data come from other sources, in which case it takes different values. With reference to chaotic dynamic systems, the discrete Virial coefficient shows the feasibility in the detection of a change in behavior, as an alternative to the traditional calculation of Lyapunov exponents, and it is a thousand times faster. The convergence speed of the final value of the discrete Virial coefficient of a dynamic system in a non-chaotic regime is between one and five orders of magnitude greater than in the chaotic regime, thus extending results in non-Hamiltonian systems, previously found by another author in Hamiltonian systems. The results obtained show that the proposal characterizes and distinguishes different types of behavior from the series under study. It also shows great sensitivity to the evolution of the series, even anticipating critical points. The proposed method to construct the discrete Virial extension does not require the existence of a Hamiltonian, which allows its application to a series obtained experimentally or from any differential equation. From a general point of view, this research shows a series of properties that can be reinterpreted in light of the discrete Virial coefficient, providing a novel and versatile tool, given its minimal applicability requirements. For pseudorandom number series, the extension reveals a consistent, quasi-mirror behavior between its kinetic and potential factors, suggesting an underlying structural property. Full article

This paper presents an efficient algorithm for Explicit Model-Following (EMF) control using an Output-derivative Feedback Control (OFC) scheme within the Reciprocal State Space (RSS) framework, aimed at overcoming the performance limitations associated with state-derivative dependence. For Lipschitz Nonlinear Systems (LNS), two approaches are proposed: a linear EMF (LEMF) strategy, which transforms the system into a Linear Parameter-Varying (LPV) representation via the Differential Mean Value Theorem (DMVT) to facilitate controller design, and a nonlinear EMF (NEMF) scheme, which enables the direct tracking of a nonlinear reference model. The stability of the closed-loop system is ensured by deriving control gains through Linear Quadratic Regulator (LQR) optimization. The proposed algorithms are validated through Real-Time Implementation (RTI) on an Arduino DUE platform, demonstrating their effectiveness and practical feasibility. Full article

In this paper, for the considered multi-cost variational problem (P), we associate a dual model (D) in order to study and state the connections between the solution sets of these control problems. Thus, under S-type I assumptions associated with the integral functionals involved, we formulate and prove various reciprocal results, such as weak, strong, and converse-type dualities. Full article

Safety control based on barrier functions has gradually become one of the emerging and more important directions in the field of safety. Scholars are attempting to apply barrier functions to integer-order dynamical systems, such as general nonlinear systems, hybrid systems, linear systems, etc. Moreover, the introduction of barrier functions has even expanded the research approaches on safe reinforcement learning. However, there is very little research on the safety control problem of fractional-order dynamical systems. Based on our previous work, this article further explores, in depth, the problem of the transfer and adaptability of barrier functions for integer-order systems in fractional-order systems, and it also proposes the Caputo reciprocal barrier function and Caputo zeroing barrier function. And we established two theorems, which proved that we can also achieve uniform asymptotic stability or exponential stability with guaranteed safety. In the end, we created a new description for the definition of input-to-state safety under Caputo’s fractional-order systems, and we used this description and the above two Caputo barrier functions to construct two criteria of the Caputo input-to-state safety. Thus, we, finally, established the embryonic form of the theoretical framework of safety control based on barrier functions for fractional-order systems. Full article

This paper introduces two concepts, subcompatibility and subsequential continuity, which are, respectively, weaker than the existing concepts of occasionally weak compatibility and reciprocal continuity. These concepts are studied within the framework of neutrosophic metric spaces. Using these ideas, a common fixed point theorem is developed for a system involving four maps. Furthermore, the results are applied to solve the Volterra integral equation, demonstrating the practical use of these findings in neutrosophic metric spaces. Full article

Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response. This paper presents the integral equation formulation for the size-dependent thermoelastic response of solids under steady-state conditions in three dimensions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couple-stress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this theory. Then, an integral equation formulation for three-dimensional size-dependent thermoelastic isotropic materials is derived, along with the corresponding singular infinite-space fundamental solutions or kernel functions. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter $l$, while the thermal coupling is defined in terms of the thermal expansion coefficient $\alpha$, as in the classical theory of steady-state isotropic thermoelasticity. Full article

In this paper, we present an array-fed Fabry–Perot cavity antenna (FPCA) based on a partially reflecting sheet (PRS) capable of generating a circularly polarized (CP), highly directive, far-field radiation pattern in the 27–28.5 GHz frequency range. The PRS, the cavity, and the array of feeders serve to different purposes in this original structure. The PRS is engineered to produce a circular polarization from a linearly polarized source placed inside the cavity. The cavity is optimized to obtain a directive conical beam from the dipole-like pattern of the simple source, and allows for a frequency scan of the beam along the elevation plane. The array of feeders is designed to obtain a pencil beam whose azimuthal pointing direction can be controlled by properly phasing the sources. The radiation performance is studied with a specific application of the reciprocity theorem in a full-wave solver along with the pattern multiplication principle. A number of array-pattern configurations in terms of operation frequency and phase shift are investigated and presented to show the potential of the proposed solution in terms of design flexibility and radiation performance. Full article

We propose a perturbative method to compute electromagnetic scattering from slightly rough dielectric surfaces, which leads to the same result as the usual Small Perturbation Method (SPM) in a surprisingly simple way. The proposed method is based on three pillars: the volumetric perturbative approach, the reciprocity theorem, and a proper approximation of the electric field within the perturbation volume, that we name Internal Field Approximation (IFA). The proposed new mathematical derivation of the SPM turns out to be much simpler and more concise than the classical one. In addition, being based on a volumetric perturbation approach, it has the potential of dealing in future with surface and volume scattering within a unitary framework, which is useful in modelling scattering from, e.g., vegetated soil, snow-covered terrain, and inhomogeneous soils. Therefore, although the presented result is mainly theoretical, it can have important applications in remote sensing. Full article

Calculating the hydro-mechanical coupling stress-intensity factor (SIF) is an important basis for conducting safety evaluations in geotechnical engineering. The current methods used to calculate hydro-mechanical coupling multi-crack SIFs have difficulties concerning their complicated solution processes and unsuitable stress field expressions. In this paper, a new semi-analytical method is proposed based on a new hydro-mechanical coupling stress function and the extended reciprocal theorem of the work integral formula to calculate hydro-mechanical coupling multi-crack SIFs, which can be verified by comparison with the results available in the literature. The new semi-analytical method is applicable to an arbitrary number of cracks under arbitrary hydro-mechanical coupling loading and facilitates a more effective representation of the water pressure effect on the stress field. Moreover, the influence of the integral path and loading conditions is also discussed, and the results revealed an integral path radius of r₂ < 0.75 mm when the crack spacing b is 1.5 mm. When σ_y and P_h are constant at 15 MPa, the SIFs are almost the same for different σ_y/P_h, while the maximum circumferential stresses at r = 0.25 mm are 15.79 MPa, 20.83 MPa, and 25.78 MPa. Full article

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems. Full article

Since the development of Brans–Dicke gravity, it has become well-known that a conformal transformation of the metric can reformulate this theory, transferring the coupling of the scalar field from the Ricci scalar to the matter sector. Specifically, in this new frame, known as the Einstein frame, Brans–Dicke gravity is reformulated as General Relativity supplemented by an additional scalar field. In 1959, Hans Adolf Buchdahl utilized an elegant technique to derive a set of solutions for the vacuum field equations within this gravitational framework. In this paper, we extend Buchdahl’s method to incorporate the cosmological constant and to the scalar-tensor cases beyond the Brans–Dicke archetypal theory, thereby, with a conformal transformation of the metric, obtaining solutions for a version of Brans–Dicke theory that includes a quadratic potential. More specifically, we obtain synchronous solutions in the following contexts: in scalar-tensor gravity with massless scalar fields, Brans–Dicke theory with a quadratic potential, where we obtain specific synchronous metrics to the Schwarzschild–de Sitter metric, the Nariai solution, and a hyperbolically foliated solution. Full article

Deflection measurements are usually used as a key index in civil engineering for performing structural assessments of bridge safety. However, owing to technical or cost issues, it may be difficult to implement long-term monitoring of bridge deflection, especially for short- or medium-span bridges. Therefore, this study presents a novel method for measuring the deflection of simply supported girder bridges. In the proposed method, the strain measurement was implemented under traffic loading at only one position, such as middle span, and then the strain distribution along the girder was reconstructed to calculate the girder deflection with basic structural mechanical theory. To implement the method, the theory was constructed based on the displacement reciprocal theorem at first to assess the strain distribution along the girder from the strain measurement at some position during traffic loads passing across the bridge. Second, a strain measurement method, namely long-gauge fibre Bragg grating (FBG) sensing technology, was introduced to take strain measurements for a concrete bridge. Third, various finite element (FE) bridge models were developed to validate the proposed method’s accuracy, the results from which indicated that the method accurately implemented deflection measurement with an approximately 5% calculation error. In addition, the influence of some key parameters, such as vehicle type, vehicle speed, and structural damage, was investigated. The simulation results revealed that damage to the hinge joint in the middle location could significantly influence the proposed method’s accuracy such that the error may exceed 10%. Finally, on-site experiments were conducted on a simply supported girder bridge to further validate the proposed method’s accuracy, and an approximately 8% deflection assessment error was found. Considering the additional advantages of FBG sensing technology, the proposed method can also be effective for long-term deflection measurements of short- or medium-span bridges. Full article

A numerical optimization technique of a three-dimensional (3D) SERS substrate with finite element analysis is proposed. Using the optical reciprocity theorem, we have shown that instead of the well-known local field enhancement criterion, it is more correct to use the Purcell factor as an objective function that determines the quality of the SERS substrate. This allows us to take into account the detail inhomogeneity of local fields in an arbitrary three-dimensional structure containing multiple emitters. We have theoretically shown that employment of a 3D CNT structure as a nanoparticle substrate instead of a nanoparticle monolayer allows one to achieve the enhancement of the SERS signal. Full article