Equivalence Theorem Research Articles

Conditional convergence modes for random sequences and Lévy's equivalence theorem in the conditional framework

Almost sure convergence, convergence in probability and convergence in distribution are three fundamental convergence modes for sequences of random variables. The purpose of this article is to define the corresponding conditional versions of these three convergence modes and derive rigorously the relationship among these three conditional convergence modes. Specifically, Lévy's equivalence theorem for sequences of independent random variables is generalized to the conditional setup.

An efficient method of moments for analysis of electromagnetic scattering from a multilayered arbitrary‐shape anisotropic dielectric object

AbstractThis paper introduces an efficient method of moments (MoM) designed to explore electromagnetic scattering in multilayered anisotropic structures. Each layer is made up of a dielectric anisotropic material characterised by a generalised tensor for permittivity, which is unrestricted in its geometrical configuration. The authors’ approach analyses each layer independently, employing the surface equivalence theorem to substitute the interfaces between layers with suitable equivalent electric and magnetic surface current densities. The authors derive the necessary surface integral equations (SIEs) for each interface by implementing the proper boundary conditions. The analysis utilises rotated dyadic Green's functions that populate the infinite space with the material properties specific to each anisotropic layer. The rotation angle corresponds to the deviation between the local principal coordinate system of the material and the global coordinate system, which is determined by diagonalising the full dielectric tensor of the respective anisotropic material given in the global coordinate system. To address the SIEs for determining the unknown equivalent electric and magnetic surface current densities, Galerkin's MoM is applied. This involves expanding the unknown surface currents using suitable basis functions, simplifying the issue to a matrix equation solved through the inversion of a block‐tridiagonal impedance matrix. The diagonal nature and sparse structure of the impedance matrix, along with an effective block‐inversion method, significantly boost computational efficiency and reduce memory demands. To demonstrate the feasibility of the proposed method, the authors present a detailed derivation of the impedance matrix for the case of non‐magnetic uniaxial anisotropic media for which the Green's functions are available in closed form. The validity and efficiency of the proposed SIE‐MoM scheme are demonstrated by comparing the results of several case studies against those found in literature and results obtained via commercial numerical codes.

Low-virtuality splitting in the Standard Model

When the available collision energy is much above the mass of the particles involved, scattering amplitudes feature kinematic configurations that are enhanced by the much lower virtuality of some intermediate particle. Such configurations generally factorise in terms of a hard scattering amplitude with exactly on-shell intermediate particle, times universal factors. In the case of real radiation emission, such factors are splitting amplitudes that describe the creation or the annihilation — for initial or final state splittings — of the low-virtuality particle and the creation of the real radiation particles. We compute at tree-level the amplitudes describing all the splittings that take place in the Standard Model when the collision energy is much above the electroweak scale. Unlike previous results, our splitting amplitudes fully describe the low-virtuality kinematic regime, which includes the region of collinear splitting, of soft emission, and combinations thereof. The splitting amplitudes are compactly represented as little-group tensors in an improved bi-spinor formalism for massive spin-1 particles that automatically incorporates the Goldstone Boson Equivalence Theorem. Simple explicit expressions are obtained using a suitably defined infinite-momentum helicity basis representation of the spinor variables. Our results, combined with the known virtual contributions, could enable systematic predictions of the leading electroweak radiation effects in high-energy scattering processes, with particularly promising phenomenological applications to the physics of future colliders with very high energy such as a muon collider.

Optimal designs for linear regression models with skew-normal errors

This paper investigates optimal designs for linear regression models with skew-normal errors which provide an alternative to traditional models with symmetric normal errors and eliminate the need for data transformation. Using the maximum likelihood estimation approach, we consider various design criteria, including D-, A-, A s -optimality. Equivalence theorems are provided to determine the optimality of the designs. Optimal designs for a centered parametrization model are also considered. Two examples are provided to illustrate the theoretical results. In addition, the efficiencies of the optimal designs based on skew-normal errors are compared with those based on normal errors.

Topological structure of the solution sets to neutral evolution inclusions driven by measures

Abstract This study is concerned with topological structure of the solution sets to evolution inclusions of neutral type involving measures on compact intervals. By using Górniewicz-Lassonde fixed-point theorem, the existence of solutions and the compactness of solution sets for neutral measure differential inclusions are obtained. Second, based on the R δ {R}_{\delta } -structure equivalence theorem, by constructing a continuous function that can make the solution set homotopic at a single point, the R δ {R}_{\delta } -type structure of the solution sets of this kind of differential inclusion is obtained.

On α-Fractional Bregman Divergence to study α-Fractional Minty’s Lemma

In this paper fractional variational inequality problems (FVIP) and dual fractional variational inequality problems (DFVIP), Fractional minimization problems are defined with the help of fractional Caputo differential operator and Reimann-Liouville differential operator. The equivalence theorem of Caputo FVIP and fractional minimization problem is established. The equivalence between FVIP and DFVIP is studied with the help of fractional Bregman divergence to establish the fractional Minty’s lemma. Introduction: The theory of variational inequality problems is one of the best tool to solve the equilibrium and non-equilibrium problems arises in the branches of Applied Mathematics, Applied Physics, Engineering, Medical, Financial and many more. On the other hand, fractional calculus manages to study various types of dynamic problems arises in real and complex analysis It is certain for the development of fractional calculus because the fractional derivative has global correlation, which can reflect the historical process of the systematic function.With regard to the definition of fractional derivatives, many mathematicians studied fractional derivatives from different aspects and advanced different definitions for them, for example Riemann-Liouville (RL) derivative, Caputo derivative and other derivatives. Conclusions: The concept of -fractionally differential inequality problem is defined and studied its existence theorem with the help of -fractionally convex function. The concept of -fractionally Bregman divergence is developed and using it the existence of -fractionally minimization problem and -fractionally differential inequality problem are studied. Equivalence theorem in between -fractionally differential inequality problem and -fractionally dual differential inequality problem is studied to establish the Minty’s lemma in fractional calculus.

Gaussian estimates vs. elliptic regularity on open sets

AbstractGiven an elliptic operator $$L= - {{\,\textrm{div}\,}}(A \nabla \cdot )$$ L = - div ( A ∇ · ) subject to mixed boundary conditions on an open subset of $$\mathbb {R}^d$$ R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Construction of weighted efficiency optimal designs for experiments with mixtures

This paper presents a weighted efficiency optimality criterion to obtain a compound optimal design that balances two different optimality objectives. An equivalence theorem and a search algorithm for a concave–concave combination of criteria for finding the weighted efficiency optimal design are given and applied to the Scheffé mixture model.

Interconnection between Polarization-Detected and Population-Detected Signals: Theoretical Results and Ab Initio Simulations.

Most of spectroscopic signals are specified by the nonlinear laser-induced polarization. In recent years, population-detection of signals becomes a trend in femtosecond spectroscopy. Polarization-detected (PD) and population-detected signals are fundamentally different, because they are determined by photoinduced processes acting on disparate time scales. In this work, we consider the fluorescence-detected (FD) N-wave-mixing (NWM) signal as a representative example of population-detected signals, derive a rigorous expression for this signal, and discuss its approximate variants suitable for numerical simulations. This leads us to the definition of the phenomenological FD (PFD) signal, which contains as a special case all definitions of FD signals available in the literature. Then we formulate and prove the population-polarization equivalence (PPE) theorem, which states that PFD NWM signals produced by (possibly strong) laser pulses can be evaluated as conventional PD signals in which the effective polarization is determined by the PFD transition dipole moment operator. We use the PPE theorem for the construction of the ab initio protocol for the simulation of PFD 4WM signals. As an example, we calculate electronic two-dimensional (2D) PFD spectra of the gas-phase pyrazine and compare them with the corresponding PD 2D spectra.

Embedding of Neutrosophic Graphs on Topological Surfaces

Background: A planar graph (PG) is a graph with no intersecting edges. Particular to both crisp and neutrosophic graphs (NG) is the planar graph, in contrast to crisp planar graphs. NPGs allow for the intersection of neutrosophic edge NEs, since the value of planarity in these graphs is the degree of planarity of the intersected NEs. The NPGs are often represented on a flat surface. Materials and Methods: This study discusses how to embed NGs on surfaces such as spheres and m-toruses by defining the degree of intersection of the neutrosophic edges of NGs with finding the faces on the given graph structures using Euler's theorems. Here, the proofs of Euler's theorems help us find, given the total NFV of G, the interval containing that value. Result: As result of this work obtained that for any two isomorphic planer graphs, they have the same planarity value. For any neutrosophic planer graph with f = (1,1,1) can be embedded in the plane if it can be embedded in the sphere and according to NPGs, for planar and spherical surfaces, equivalent theorems to Euler's formula are proved and shown. Conclusion: It concludes that by using neutrosophic sets and crisp graphs to construct neutrosophic graphs with the benefit of Euler’s theorem, it can provide the concept of embedding neutrosophic graphs in different topological surfaces such as a plane, sphere, and m-torus.

Optimal designs of accelerated degradation tests with random shock failures and measurement errors

AbstractAccelerated degradation tests (ADTs) are widely used for assessing the reliability of long‐life products. During an ADT, accelerated stresses not only expedite the degradation of test products but also increase the likelihood of encountering traumatic shocks. Moreover, it is important to acknowledge that measurement errors can be inevitable during the observation process of an ADT. Unfortunately, these errors are often overlooked in the optimal design of the ADT, especially when multiple competing failure modes are present. In this article, we propose a new approach to design ADTs when measurement errors exist and test products suffer from degradation failures and random shock failures. We utilize the Wiener process to model the degradation path, incorporating normally distributed measurement errors, and an exponential distribution to fit the time between random shock failures. Given the number of test products and the termination time, we optimize the ADT plans under three common design criteria. The equivalence theorem is used to verify the optimality of the optimal ADT plans. A real‐life example and sensitivity analysis are provided to illustrate our proposed method. The results demonstrate that when competing failure modes are present, the optimal ADT plans, which account for measurement errors, differ significantly from those that do not consider such errors.

On energy and particle production in cosmology: the particular case of the gravitino

It is well-known that the number of particles produced in cosmology, commonly defined in the literature from the Fock space of the instantaneous hamiltonian of the canonically normalized fields, is ambiguous. On the other hand, the energy computed from the energy-momentum tensor should be physical. We compare the corresponding Fock spaces and relate them through a Bogolyubov transformation. We find that for particles of spin 0, 1 and 3/2 the two Fock spaces are different, whereas they are the same for spin 1/2 fermions. For spin 0 and 1, for particles of high momenta the two Fock spaces align, as intuitively expected. For the spin 3/2, one finds two puzzles. The first one is that the two corresponding Fock spaces do not match even in the limit of high momenta. The second is that whereas we provide evidence for the equivalence theorem between longitudinal gravitinos and the goldstino in terms of an exact matching between the lagrangians and the instantaneous hamiltonians for the canonically normalized fields, the energy operator computed from the Rarita-Schwinger action does not seem to be captured in a simple way by the goldstino action. Our results suggest a re-analysis of non-thermal gravitino production in cosmology.

Stability equivalence between regime‐switching jump diffusion delayed systems and corresponding systems with piecewise continuous arguments and application to discrete‐time feedback control

AbstractIn this paper, we mainly study the equivalence of exponential stability for regime‐switching jump diffusion delayed systems (RSJDDSs) and RSJDDSs with piecewise continuous arguments (RSJDDSs‐PCA). Our results show that if one of the RSJDDS and the RSJDDS‐PCA is th moment exponentially stable, then another system is also th moment exponentially stable when time delay and segment step size have a common upper bound, while both equations are almost surely exponentially stable, and we also provided a method to calculate this upper bound. In addition, as an application of the stability equivalence theorem, we design discrete‐time state and mode observations feedback control to stabilize unstable RSJDDSs and investigate that controllers of the drift, diffusion, and jump terms are all able to play a stabilizing effect on the controlled system.

Estimates for the second Hankel Clifford transform and Titchmarsh equivalence theorem

Estimates for the second Hankel Clifford transform and Titchmarsh equivalence theorem

A general mirror equivalence theorem for coset vertex operator algebras

A general mirror equivalence theorem for coset vertex operator algebras

Holography of broken U(1) symmetry

We examine the Abelian Higgs model in (d + 1)-dimensional anti-de Sitter space with an ultraviolet brane. The gauge symmetry is broken by a bulk Higgs vacuum expectation value triggered on the brane. We propose two separate Goldstone boson equivalence theorems for the boundary and bulk degrees of freedom. We compute the holographic self-energy of the gauge field and show that its spectrum is either a continuum, gapped continuum, or a discretuum as a function of the Higgs bulk mass. When the Higgs has no bulk mass, the AdS isometries are unbroken. We find in that case that the dual CFT has a non-conserved U(1) current whose anomalous dimension is proportional to the square of the Higgs vacuum expectation value. When the Higgs background weakly breaks the AdS isometries, we present an adapted WKB method to solve the gauge field equations. We show that the U(1) current dimension runs logarithmically with the energy scale in accordance with a nearly-marginal U(1)-breaking deformation of the CFT.

R-optimal designs for Becker’s models for mixture experiments

This paper investigates the R-optimal designs for Becker’s models H 2 and H 3. We derive the R-optimal allocations for Becker’s models H 2 and H 3 using optimal design theory. Moreover, we verify that the saturated R-optimal allocations cannot satisfy the R-optimality equivalence theorem, and the non-saturated R-optimal designs are discussed by increasing the number of support points. In addition, we prove the equivalence of the R-optimal designs for the two models H 2 and H 3 for the {q, q} weighted simplex-centroid design. Numerical results have verified the necessary and sufficient conditions of R-optimality of the non-saturated weighted simplex-centroid design.

Enhancing decision-making in breast cancer diagnosis for women through the application of nano beta open sets

Nano topology is a modern type of topology based on rough sets, introducing nano open sets and their properties. This study explores the new properties of nano-β-open sets and nano-closures in nano topological space, providing some equivalent definitions, theorems, and lemmas for illustration. The effectiveness of incorporating nano β-open sets in medical decision-making is demonstrated in diagnosing breast cancer in two cases for women. Results indicated that utilizing nano open sets enhanced the accuracy, sensitivity, and specificity of cancer diagnosis. Additionally, an algorithm has been developed to aid clinicians in diagnosing breast cancer infections. The overarching objective is to provide physicians with a valuable tool to make informed decisions and reach optimal conclusions in patient care.

A light scattering model for large particles with surface roughness

A physical-optics hybrid method designed for the computation of single-scattering properties of particles with complex shapes, including surface roughness, is presented. The method applies geometric optics using a novel ray backtracing algorithm to compute the scattered field on the particle surface. A surface integral equation based on the equivalence theorem is used to compute the scattered far-field, which yields the full Mueller matrix and integrated single-scattering parameters. The accuracy is tested against the discrete dipole approximation for fixed orientation smooth and roughened compact hexagonal columns for 3 values of refractive index. The method is found to compute asymmetry parameter, and scattering and extinction efficiencies with mean errors of −1.0%, −1.4%, −1.2%, respectively, in a computation time reduced by 3 orders of magnitude. The work represents a key step forwards for modelling particles with physical surface roughness within the framework of physical-optics and provides a versatile tool for the fast and quantitative study of light scattering from non-spherical particles with size much larger than the wavelength.

Upwind difference scheme for fuzzy convection–diffusion equations

In this paper, our main objective is to build fuzzy upwind difference scheme for fuzzy convection–diffusion equations based on generalized Hukuhara derivative with new Taylor expansions. The truncation error and consistency of this scheme are discussed in the sequel. Stability of this scheme is proved by fuzzy Fourier transform, and convergence of this scheme is derived by Lax equivalence theorem. Finally, a numerical example is given and shows that our methods are efficient and stable for solving fuzzy convection–diffusion problems.