Equivalence Theorem Research Articles

K-functional related to the Deformed Hankel Transform

Abstract The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness for the Deformed Hankel Transform. Before that, we introduce the K-functional associated to the Deformed Hankel Transform.

Frames associated with an operator in spaces with an indefinite metric

<abstract><p>In the present paper, we study frames associated with an operator ($ \mathcal{W} $-frames) in Krein spaces, and we give the definition of frames associated with an operator depending on the adjoint of the operator in the Krein space (Definition 4.1). We prove that the definition given in [A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces, <italic>Ann. Funct. Anal.</italic>, <bold>14</bold> (2023), 10.], which depends on the adjoint of the operator in the associated Hilbert space, is a consequence of our definition. We prove that our definition is independent of the fundamental decomposition (Theorem 4.1) and that having $ \mathcal{W} $-frames for the Krein space necessarily gives $ \mathcal{W} $-frames for the Hilbert spaces that compose the Krein space (Theorem 4.4). We also prove that orthogonal projectors generate new operators with their respective frames (Theorem 4.2). We prove an equivalence theorem for $ \mathcal{W} $-frames (Theorem 4.3), without depending on the fundamental symmetry as usually given in Hilbert spaces.</p></abstract>

On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions

In this work, we prove the convergence of residual distribution (RD) schemes to dissipative weak solutions of the Euler equations. We need to guarantee that the RD schemes are fulfilling the underlying structure preserving methods properties such as positivity of density and internal energy. Consequently, the RD schemes lead to a consistent and stable approximation of the Euler equations. Our result can be seen as a generalization of the Lax–Richtmyer equivalence theorem to nonlinear problems that consistency plus stability is equivalent to convergence.

Equivalence of K-functional and modulus of smoothness constructed by Fourier-Bessel transform in the space L2,γ(ℝn+)

Using a generalized shift operator, we define generalized modulus of smothness in the space L2,γ(ℝn+). Based on the Laplace-Bessel differential operator we define Sobolev-type space and K-functionals. In this paper paper we prove the equivalence theorem for a K-functional and a modulus of smoothness for the Fourier-Bessel transformation on ℝn+.

Locally D-optimal designs for spline measurement error models with estimated knots

This paper is concerned with the problem of constructing the locally D-optimal designs for spline measurement error models with estimated knots, where the degree of splines is at most m in each subinterval delimited by knots and it is continuous and differentiable at any knot. Given the number of knots in advance, an equivalence theorem is established and used to check the D-optimality of designs. The characterizations of the locally D-optimal designs are provided under certain conditions. It is shown that the support points of the D-optimal designs for such models are associated with the endpoints of the design interval. The locally D-optimal designs for a class of the models can be determined explicitly. Two examples are presented for illustration.

A- and D-optimal designs in real life data using imperialist comparative algorithm

In this paper, we used Imperialist Competitive Algorithm (ICA) to get A-and D-optimal de­signs for two models; one with nonlinear and the other linear model were found for two life data. It was noted that the number of iterations for D-optimal design for both models was smaller than that of A-optimal design. This implies that getting a D-optimal design is preferable and saves time and energy compared to A-optimal since they are addressing the same issues related to the variance-covariance function. The Efficiency Lower Bound and the Sensitivity function embedded in ICA which support the general equivalent theorem in getting optimal design were also found in this study. Another advantage of using ICA algorithm is that it does not require the design space or the region of uncertainty to be discretized which means that the search for the support points of the optimal design is not restricted to the grid point. This was also confirmed here that the optimal design for A and D- designs were not restricted to the grid points.

Locally R-optimal designs for a class of nonlinear multiple regression models

This paper concerns with optimal designs for a wide class of nonlinear models with information driven by the linear predictor. The aim of this study is to generate an R-optimal design which minimizes the product of the main diagonal entries of the inverse of the Fisher information matrix at certain values of the parameters. An equivalence theorem for the locally R-optimal designs is provided in terms of the intensity function. Analytic solutions for the locally saturated R-optimal designs are derived for the models having linear predictors with and without intercept, respectively. The particle swarm optimization method has been employed to generate locally non-saturated R-optimal designs. Numerical examples are presented for illustration of the locally R-optimal designs for Poisson regression models and proportional hazards regression models.

The identical distribution hypothesis is equivalent to the parameter discrepancy hypothesis: Adversarial attacks on graph neural networks

Graph neural networks are vulnerable to adversarial attacks. Previous studies have mainly focused on the describing attack methodology, but few attempted to refine model hypotheses. This paper summarizes current attack theories as contradictory data hypotheses, highlighting the lacking vital precondition: distribution consistency between training and test sets. We propose an identical distribution hypothesis to address this issue and consequently establish a consistent attack model. Then we analyze the distinctive problem for a graph poisoning attack, derive the influence of perturbations in the training set on the test set, and delimit the attack feasible region. The attack model’s closed form solution is proposed and proven on the attack gradient, and we show that the proposed method based on the identical distribution hypothesis is equivalent to inducing significant changes for training parameters before and after an attack, establishing the equivalence between the hypotheses of identical distribution and parameter discrepancy. Numerical examples validate the attack gradient correctness and equivalence theorem, and experimental results verify the reasonableness of the identical distribution hypothesis and the effectiveness of the proposed method. The proposed attacks outperform current state-of-the-art attacks.

An equivalent surface dipoles based electric field to magnetic field transformations for estimating radiated emissions

The new generation of communication devices operating in higher frequency bands is constantly pushing the complexity of field measurements for electromagnetic compliance and the design of field probes. These compliance are to be met for both electric and magnetic fields, of which if one set is available, the other can be estimated computationally using source reconstruction algorithms. The main objective of this paper is to refine the dipole-based source reconstruction algorithms using surface equivalence principle. Compared to earlier works, the algorithm is based on a novel way of placing a single dipole per point tangentially to a fictitious surface based on surface equivalence theorem instead of multiple orthogonal dipoles per point. The numerical results are compared with the state-of-art to demonstrate the effectiveness in obtaining accurate fields and locating the regions of the Device Under Test responsible for overshooting the interference limits.

Resource Theory of Heat and Work with Non-commuting Charges

We consider a theory of quantum thermodynamics with multiple conserved quantities (or charges). To this end, we generalize the seminal results of Sparaciari et al. (Phys. Rev. A 96:052112, 2017) to the case of multiple, in general non-commuting charges, for which we formulate a resource theory of thermodynamics of asymptotically many non-interacting systems. To every state we associate the vector of its expected charge values and its entropy, forming the phase diagram of the system. Our fundamental result is the Asymptotic Equivalence Theorem, which allows us to identify the equivalence classes of states under asymptotic approximately charge-conserving unitaries with the points of the phase diagram. Using the phase diagram of a system and its bath, we analyze the first and the second laws of thermodynamics. In particular, we show that to attain the second law, an asymptotically large bath is necessary. In the case that the bath is composed of several identical copies of the same elementary bath, we quantify exactly how large the bath has to be to permit a specified work transformation of a given system, in terms of the number of copies of the “elementary bath” systems per work system (bath rate). If the bath is relatively small, we show that the analysis requires an extended phase diagram exhibiting negative entropies. This corresponds to the purely quantum effect that at the end of the process, system and bath are entangled, thus permitting classically impossible transformations (unless the bath is enlarged). For a large bath, or many copies of the same elementary bath, system and bath may be left uncorrelated and we show that the optimal bath rate, as a function of how tightly the second law is attained, can be expressed in terms of the heat capacity of the bath. Our approach solves a problem from earlier investigations about how to store the different charges under optimal work extraction protocols in physically separate batteries.

Causality constraints on nonlinear supersymmetry

It is well-known that gravitino propagation in standard supergravities is free of any causality problems. However, two issues related to gravitino propagation were recently uncovered in specific supergravities with nonlinear supersymmetry. One of them concerns potential acausality/superluminality, whereas the second one arises from the vanishing of the sound speed at specific points during inflation. The former is famously related to positivity constraints on specific EFT operators, derived from dispersion relations on the energy-growing part of scattering amplitudes, and indeed we show that subluminality constraints for the gravitino are related via the equivalence theorem to positivity bounds in low-energy goldstino actions. However, the former are stronger, in the sense that they apply to functions of the scalar fields not only in the ground state, but for any field values such as those scanned by time-dependent solutions, unlike bounds derived from 2 → 2 scattering amplitudes in the vacuum. We also argue that nontrivial causality constraints arise only in the case where nonlinear supersymmetry in the matter sector is encoded into superfield constraints which do not seem to arise from microscopic two-derivative lagrangians, in particular for the orthogonal constraint used to build minimal models of inflation in supergravity. This allows us to propose simple alternatives which maintain the minimality of the spectra and are causal in all points of the theory parameter space. We also discuss minimal supergravity models of inflation along these lines.

R-optimal designs for linear log contrast model with mixture experiments

The R-optimality criterion is proposed in the literature as an alternative to the most frequently used D-optimality criterion. This criterion can be used in the experimental design when the main objective is to construct a rectangular confidence region. The present work investigates the R-optimal designs of the linear log contrast model supported on the general q-mixture components. The necessary and sufficient conditions of R-optimality have been examined by the equivalence theorem.

An accurate and efficient local one-dimensional method for the 3D acoustic wave equation

Abstract We establish an accurate and efficient scheme with four-order accuracy for solving three-dimensional (3D) acoustic wave equation. First, the local one-dimensional method is used to transfer the 3D wave equation into three one-dimensional wave equations. Then, a new scheme is obtained by the Padé formulas for computation of spatial second derivatives and the correction of the truncation error remainder for discretization of temporal second derivative. It is compact and can be solved directly by the Thomas algorithm. Subsequently, the Fourier analysis method and the Lax equivalence theorem are employed to prove the stability and convergence of the present scheme, which shows that it is conditionally stable and convergent, and the stability condition is superior to that of most existing numerical methods of equivalent order of accuracy in the literature. It allows us to reduce computational cost with relatively large time step lengths. Finally, numerical examples have demonstrated high accuracy, stability, and efficiency of our method.

Equivalence of Competitive Equilibria, Fuzzy Cores, and Fuzzy Bargaining Sets in Finite Production Economies

For an exchange economy with a continuum of traders and a finite-dimensional commodity space under some standard assumptions, Aumann showed that the core and the set of competitive allocations (two most important solutions) coincide and Mas-Colell proved that the bargaining set and the set of competitive allocations coincide. However, in the case of exchange economies with a finite number of traders, it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we establish the equivalence of the fuzzy core, the fuzzy bargaining set (or Aubin bargaining set), and the set of competitive allocations in a finite coalition production economy with an infinite-dimensional commodity space under some standard assumptions. We first derive a continuous equivalence theorem and then discretize it to obtain the desired equivalence in finite economies.

On equivalence of decomposition integrals based on different monotone measures

This paper investigates the coincidences of decomposition integrals based on different monotone measures. We propose the concept of total balance of a monotone measure related to decomposition integral and by means of its characteristics show a version of equivalence theorem of decomposition integrals. A binary relation on the set of monotone measures is introduced, which is a preorder related to a fixed decomposition system. Another version of the equivalence theorem of decomposition integrals is shown in the framework concerning preorder of a pair of monotone measures. As special cases, we present a necessary and sufficient condition that two pan-integrals (resp. concave integrals) with respect to different monotone measures coincide. By using the characteristics of total balance of monotone measures, it is shown that Chebyshev's Inequality holds for a decomposition integral if and only if the considered decomposition system is complete.

Nonlinear public debt and resource rent nexus in highly indebted resource-rich sub-Saharan economies: Evidence from Nigeria

This paper contributes to empirical insights on nonlinear public debt and resource rent nexus in the Nigerian economy which profoundly typifies highly indebted resource-rich sub-Saharan nations. Predicated on the Ricardian equivalence theorem, results of nonlinear ARDL indicate (i) significant short-run and long-run asymmetric nexuses, (ii) negative shocks of resource rent impact more on public debt than positive shocks, and (iii) the pre-shock equilibria are never attained subsequently. These results are consistent with the robust Wald tests of asymmetries, and asymmetric dynamic multipliers estimates in explaining the persistently endemic public debt accumulation in resource-rich HISSA economies. Policy implications and options are discussed.

Optimal designs for semi-parametric dose-response models under random contamination

With the increasing popularity of personalized medicine, it is more and more crucial to capture not only the dose-effect but also the effects of the prognostic factors due to individual differences in a dose-response experiment. This paper considers the design issue for predicting semi-parametric dose-response curves in the presence of linear effects of covariates. Inspired by the Neyman-Pearson paradigm, a novel design criterion, namely bias constraint optimality, is introduced to minimize the overall prediction error. The corresponding equivalence theorems are established, the characteristics of the optimal designs are shown, and an equivalent bias compound optimality criterion is proposed for practical implementation. Based on the obtained theoretical results, efficient algorithms for searching for optimal designs are developed. Numerical simulations are given to illustrate the superior performance of the obtained optimal designs.

On the coincidence of measure-based decomposition and superdecomposition integrals

This paper introduces two types of preorders on the system of all non-empty sets of collections (i.e., the set of all decomposition systems) based on a fixed monotone measure μ. Each of them refines the previous two kinds of preorders of decomposition systems. By means of these two new preorders of decomposition systems we investigate the coincidences of decomposition integrals and that of superdecomposition integrals, respectively. The generalized integral equivalence theorem is shown in the general framework involving an ordered pair of decomposition systems. This generalized theorem includes as special cases all the previous results related to the coincidences among the Choquet integral, the concave (or convex) integral and the pan-integrals. Thus, a unified approach to the coincidences of several well-known decomposition and superdecomposition integrals is presented.

Stability of a coefficient inverse problem for the discrete Schrödinger equation and a convergence result

In this paper, we prove the stability and convergence of a coefficient inverse problem for the semi-discrete Schrödinger equation by means of an appropriate Carleman estimate. First, we establish a new discrete Carleman estimate for the semi-discrete Schrödinger equation discretized by finite differences. Then by the developed Carleman estimate, we prove the stability of the semi-discrete Schrödinger equation. The consistency between the semi-discrete inverse problem and the continuous one is also proved. With the consistency and stability, the convergence of the semi-discrete inverse problem can be obtained based on the Lax equivalence theorem.

On convergence of numerical solutions for the compressible MHD system with weakly divergence-free magnetic field

Abstract We study a general convergence theory for the analysis of numerical solutions to a magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension $d$$(=2,3)$. First, we introduce the concept of dissipative weak (DW) solutions and prove the weak–strong uniqueness property for DW solutions, meaning a DW solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the DW solution, as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax equivalence theorem. Finally, as an application of this theory, we show the convergence analysis of two numerical methods.