Proof Of Equivalence Research Articles

A proof of convergence and equivalence for 1D finite element methods and ReLU neural networks

This paper investigates the convergence and equivalence properties of the Finite Element Method (FEM) and Rectified Linear Unit Neural Networks (ReLU NNs) in solving differential equations. We provide a detailed comparison of the two approaches, highlighting their mutual capabilities in function space approximation. Our analysis proves the subset and superset inclusions that establish the equivalence between FEM and ReLU NNs for approximating piecewise linear functions. Furthermore, a comprehensive numerical evaluation is presented, demonstrating the error convergence behavior of ReLU NNs as the number of neurons per basis function varies. Our results show that while increasing the number of neurons improves approximation accuracy, this benefit diminishes beyond a certain threshold. The maximum observed error between FEM and ReLU NNs is 10-4, reflecting excellent accuracy in solving partial differential equations (PDEs). These findings lay the groundwork for integrating FEM and ReLU NNs, with important implications for computational mathematics and engineering applications.

Proven Distributed Memory Parallelization of Particle Methods

We provide a mathematically proven parallelization scheme for particle methods on distributed-memory computer systems. Particle methods are a versatile and widely used class of algorithms for computer simulations and numerical predictions in various applications, ranging from continuum fluid dynamics to discrete granular flows and molecular dynamics simulations. Particle methods naturally lend themselves to implementation on parallel computing systems. So far, however, a mathematical proof of correctness and equivalence to sequential implementations was only available for shared-memory parallelism. Here, we leverage a formal definition of the algorithmic class of particle methods to provide a proven parallelization scheme for distributed-memory computers. We prove that thus parallelized particle methods on distributed-memory computers are formally equivalent to their sequential counterpart for a well-defined class of particle methods, and we provide analytical expressions for the speed-up and scalability bounds of this class of algorithms in function of their parameters. The parallelization scheme analyzed here is the basis of many real-world software designs for parallel particle methods. The present analysis is, therefore, of direct relevance to existing and new parallel implementations of particle methods and places them on solid theoretical grounds, rationalizing best practices and providing useful scalability and speedup bounds for benchmarking.

Quantization of two- and three-player cooperative games based on QRA

Abstract In this paper, a novel quantization scheme for cooperative games is proposed. The circuit is inspired by the Eisert-Wilkens-Lewenstein protocol, which was modified to represent cooperation between players and extended to 3--qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. The expected value of the Shapley value concerning quantum probabilities is employed to distribute players' payoffs after the measurement. We study how entanglement, representing the level of pre-agreement between players, affects the final utility distribution. The paper also demonstrates how the Quantum Register Algebra and GAALOP software can automate all necessary calculations.

CCLemma: E-Graph Guided Lemma Discovery for Inductive Equational Proofs

The problem of automatically proving the equality of terms over recursive functions and inductive data types is challenging, as such proofs often require auxiliary lemmas which must themselves be proven. Previous attempts at lemma discovery compromise on either efficiency or efficacy. Goal-directed approaches are fast but limited in expressiveness, as they can only discover auxiliary lemmas which entail their goals. Theory exploration approaches are expressive but inefficient, as they exhaustively enumerate candidate lemmas. We introduce e-graph guided lemma discovery , a new approach to finding equational proofs that makes theory exploration goal-directed. We accomplish this by using e-graphs and equality saturation to efficiently construct and compactly represent the space of all goal-oriented proofs. This allows us to explore only those auxiliary lemmas guaranteed to help make progress on some of these proofs. We implemented our method in a new prover called CCLemma and compared it with three state-of-the-art provers across a variety of benchmarks. CCLemma performs consistently well on two standard benchmarks and additionally solves 50% more problems than the next best tool on a new challenging set.

Quantum mechanics, radiation, and the equivalence proof

This paper re-evaluates the formative year of quantum mechanics—from Heisenberg’s first paper on matrix mechanics to Schrödinger’s equivalence proof—by focusing on the role of radiation in the emerging theory. We argue that the radiation problem played a key role in early quantum mechanics, a role that has not been taken into account in the standard histories. Radiation was perceived by the main protagonists of matrix and wave mechanics as a central lacuna in these emerging theories and continued to contribute to the theoretical development and conceptual clarification of quantum mechanics. Studying the interplay between quantum mechanics and radiation, the paper provides an account of (a) how quantum mechanics was able to connect to its empirical basis in spectroscopy and (b) how Schrödinger’s equivalence proof emerged from his explorative calculations on the emission of radiation.

Proof of equivalence of semantic methods for a selected domain-specific language

Proof of equivalence of semantic methods for a selected domain-specific language

Study of the Accuracy of the Ronen Method at Material Interfaces

The Ronen Method (RM) is a nonlinear iterative scheme that demands successive resolutions of the diffusion equation, where local diffusion constants are modified to reproduce more accurate estimates of the neutron currents by a transport operator. The methodology is currently formulated using the formalism of the collision probability method for evaluation of the current. The RM was recently tested on a complete suite of one-dimensional (1-D) multigroup benchmark problems. Small differences in the flux (less than 2%) were reported at material interfaces and close to the vacuum boundary with respect to the reference solution from transport. This work investigates first a possible numerical equivalence between transport and diffusion in some representative 1-D problems from the same benchmark test suite. The equivalence is sought with optimal diffusion coefficients computed using reference transport solutions that allow for adjusting the diffusion model. The RM, which attempts to obtain such equivalent diffusion coefficients without knowing the reference solution, is then compared to the optimal coefficients. The accuracy of the flux distribution at material interfaces is investigated for different approximations of the vacuum boundary and by decreasing progressively the RM convergence tolerance set in the iterative scheme. Using tighter convergence criteria, the RM calculates more accurate flux distributions at all material interfaces, regardless of the value of the diffusion coefficient and the extrapolated distance set at the beginning of the iterative scheme. Maximum flux deviations are remarkably reduced when the RM convergence tolerance is set to eight or more significant digits, leading to improvements in the flux deviation of two orders in magnitude and providing numerical proof for equivalence with transport in the tested configurations.

Master actions and helicity decomposition for spin-4 models in [formula omitted]

The present work introduces a master action which interpolates between four self-dual models, SD(i), describing massive spin-4 particles in D=2+1 dimensions. These models are designated by i=1,2,3 and 4, representing the order in derivatives. Our results show that the four descriptions are quantum equivalents by comparing their correlation functions, up to contact terms. This is an original result since that a proof of quantum equivalence among these models have not been demonstrated in the literature. Besides, a geometrical approach is demonstrated to be a useful tool in order to describe the third and fourth order in derivatives models. On the other hand, the construction of the master action relies on the introduction of mixing terms, which must be free of particle content. Here, we demonstrate how the helicity decomposition method can be used in order to verify the absence of particle content of such terms, ensuring the proper usability of the master action technique. This kind of result can be very useful in situations where we do not have access to the higher spin-projection basis which would allow the analysis to be handle by explicitly calculating the propagator and subsequently analysing its poles.

Euler and a Proof of the Functional Equation for the Riemann Zeta-Function He Could Have Given

Euler and a Proof of the Functional Equation for the Riemann Zeta-Function He Could Have Given

PASDA: A partition-based semantic differencing approach with best effort classification of undecided cases

Equivalence checking is used to verify whether two programs produce equivalent outputs when given equivalent inputs. Research in this field mainly focused on improving equivalence checking accuracy and runtime performance. However, for program pairs that cannot be proven to be either equivalent or non-equivalent, existing approaches only report a classification result of unknown, which provides no information regarding the programs’ non-/equivalence.In this paper, we introduce PASDA, our partition-based semantic differencing approach with best effort classification of undecided cases. While PASDA aims to formally prove non-/equivalence of analyzed program pairs using a variant of differential symbolic execution, its main novelty lies in its handling of cases for which no formal non-/equivalence proof can be found. For such cases, PASDA provides a best effort equivalence classification based on a set of classification heuristics.We evaluated PASDA with an existing benchmark consisting of 141 non-/equivalent program pairs. PASDA correctly classified 61%–74% of these cases at timeouts from 10 s to 3600 s. Thus, PASDA achieved equivalence checking accuracies that are 3%–7% higher than the best results achieved by three existing tools. Furthermore, PASDA’s best effort classifications were correct for 70%–75% of equivalent and 55%–85% of non-equivalent cases across the different timeouts.

Weighted, Circular and Semi-Algebraic Proofs

In recent years there has been an increasing interest in studying proof systems stronger than Resolution, with the aim of building more efficient SAT solvers based on them. In defining these proof systems, we try to find a balance between the power of the proof system (the size of the proofs required to refute a formula) and the difficulty of finding the proofs. In this paper we consider the proof systems circular Resolution, Sherali-Adams, Nullstellensatz and Weighted Resolution and we study their relative power from a theoretical perspective. We prove that circular Resolution, Sherali-Adams and Weighted Resolution are polynomially equivalent proof systems. We also prove that Nullstellensatz is polynomially equivalent to a restricted version of Weighted Resolution. The equivalences carry on also for versions of the systems where the coefficients/weights are expressed in unary. The practical interest in these systems comes from the fact that they admit efficient algorithms to find proofs in case these have small width/degree.

Hankel transform, [formula omitted]-Bessel functions and zeta distributions in the Dunkl setting

We study analytic properties of a Hankel transform for the type A Dunkl setting with arbitrary multiplicity parameter k≥0 which goes back to Baker and Forrester and, in an earlier symmetrized version, to Macdonald. Moreover, we introduce a Dunkl analogue of the Bessel functions and K-Bessel functions, generalizing those of a symmetric cone which occur when the multiplicity parameter k is chosen to be twice the Peirce dimension constant of the cone. In particular, we show that our K-Bessel function is a solution of a type B system of Dunkl operators which is a generalization of the Bessel system on a symmetric cone. Furthermore, we define zeta distributions and prove a functional equation relating them with their type B Dunkl transform. In the case of symmetric cones the aforementioned functional equation reduces to the known functional equation between zeta distributions and their Fourier transform. For the proof of this functional equation, the K-Bessel function and its analytic properties turn out to be crucial. Finally, we examine which of the zeta distributions are given by a positive measure.

Two-axis multiple masses resonator with frequency and Q-factor matching

This paper reports a design and experimental results of a two-axis symmetric dynamically balanced resonator based on a triple mass system, whose frequency and Q-factor can be independently tuned. Two dynamically equivalent proof masses are connected by four coupling springs, and four coupling proof masses are implemented between them. The frequency and Q-factor of the anti-phase modes can be tuned independently of each other. The frequency is tuned by electrostatically softening the suspensions stiffness, and the Q-factor is adjusted by controlling the squeeze film damping through the mode coupling between the anti-phase and in-phase modes. The changes of frequency and mode coupling under different DC bias tunings are studied by FEA, which reveals the modification of the suspension stiffness only decreases the frequency and the modification of the inner stiffness of the coupling proof masses exerts a large effect on the mode shape, i.e. Q-factor, while a minor effect on the frequency. The as-fabricated frequency and Q-factor mismatches were evaluated as 5% and 7.7% in a fabricated device, respectively. A large adjustment of the Q-factor by 21% was observed by adding a 27 V DC voltage on the tuning electrodes inside the coupling proof masses, while the frequency was only decreased by as small as 810 ppm. A mode matching was achieved by electrostatic softening the suspensions of both main proof masses. Thanks to the decoupling between frequency and Q-factor, a frequency matching under 10 ppm and a Q-factor matching under 650 ppm were experimentally proved.

An eigenvalue-free implementation of the log-conformation formulation

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows.In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.

New and simple proofs of Ramanujan’s modular equations of degree 11

On pp. 243 and 244 of his second notebook, Ramanujan recorded seven modular equations of degree 11 followed by two more which he had crossed out. The first modular equation in the list was proved by Berndt using theta function identities. The same was also proved by Venkatachaliengar in a different way. To the best of our knowledge, the only available proofs to the other six modular equations in the list are by Berndt. These proofs employ the theory of modular forms. Interestingly these are the first set of modular equations, the proofs to which by Berndt take a shift from the elementary algebraic methods to the theory of modular forms. In this paper, we reprove all these modular equations of degree 11 (except the first one in the list) by elementary algebraic techniques. First, we use two series identities due to Ye and Liu to prove one of the modular equations and its reciprocal. The remaining modular equations are proved by the method of parametrization.

Interchangeability between factor analysis, logistic IRT, and normal ogive IRT.

In existing studies, it has been argued that factor analysis (FA) is equivalent to item response theory (IRT) and that IRT models that use different functions (i.e., logistic and normal ogive) are also interchangeable. However, these arguments have weak links. The proof of equivalence between FA and normal ogive IRT assumes a normal distribution. The interchangeability between the logistic and normal ogive IRT models depends on a scaling constant, but few scholars have examined whether the usual values of 1.7 or 1.702 maximize interchangeability. This study addresses these issues through Monte Carlo simulations. First, the FA model produces almost identical results to those of the normal ogive model even under severe nonnormality. Second, no single scaling constant maximizes the interchangeability between logistic and normal ogive models. Instead, users should choose different scaling constants depending on their purpose in using a model and the number of response categories (i.e., dichotomous or polytomous). Third, the interchangeability between logistic and normal ogive models is determined by several conditions. The interchangeability is high if the data are dichotomous or if the latent variables follow a symmetric distribution, and vice versa. In summary, the interchangeability between FA and normal ogive models is greater than expected, but that between logistic and normal ogive models is not.

Synergetic learning for unknown nonlinear [formula omitted] control using neural networks

The well-known H∞ control design gives robustness to a controller by rejecting perturbations from the external environment, which is difficult to do for completely unknown affine nonlinear systems. Accordingly, the immediate objective of this paper is to develop an on-line real-time synergetic learning algorithm, so that a data-driven H∞ controller can be received. By converting the H∞ control problem into a two-player zero-sum game, a model-free Hamilton–Jacobi–Isaacs equation (MF-HJIE) is first derived using off-policy reinforcement learning, followed by a proof of equivalence between the MF-HJIE and the conventional HJIE. Next, by applying the temporal difference to the MF-HJIE, a synergetic evolutionary rule with experience replay is designed to learn the optimal value function, the optimal control, and the worst perturbation, that can be performed on-line and in real-time along the system state trajectory. It is proven that the synergistic learning system constructed by the system plant and the evolutionary rule is uniformly ultimately bounded. Finally, simulation results on an F16 aircraft system and a nonlinear system back up the tractability of the proposed method.

Graph-based approximate message passing iterations

Abstract Approximate message passing (AMP) algorithms have become an important element of high-dimensional statistical inference, mostly due to their adaptability and concentration properties, the state evolution (SE) equations. This is demonstrated by the growing number of new iterations proposed for increasingly complex problems, ranging from multi-layer inference to low-rank matrix estimation with elaborate priors. In this paper, we address the following questions: is there a structure underlying all AMP iterations that unifies them in a common framework? Can we use such a structure to give a modular proof of state evolution equations, adaptable to new AMP iterations without reproducing each time the full argument? We propose an answer to both questions, showing that AMP instances can be generically indexed by an oriented graph. This enables to give a unified interpretation of these iterations, independent from the problem they solve, and a way of composing them arbitrarily. We then show that all AMP iterations indexed by such a graph verify rigorous SE equations, extending the reach of previous proofs and proving a number of recent heuristic derivations of those equations. Our proof naturally includes non-separable functions and we show how existing refinements, such as spatial coupling or matrix-valued variables, can be combined with our framework.

Tropical descendant invariants with line constraints

AbstractVia correspondence theorems, rational log Gromov–Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalised) lattice path counts and floor diagram techniques. So far, the cases for which there exist algorithms do not extend to non‐stationary rational descendant log Gromov–Witten invariants, that is, those where Psi‐conditions do not have to be matched up with the evaluation of a point. The case of rational descendant log Gromov–Witten invariants satisfying point conditions (without Psi‐conditions) and one Psi‐condition of any power combined with a line plays a particularly important role, because it shows up in mirror symmetry as contributions to coefficients of the ‐function. We provide recursive formulas to compute those numbers via tropical methods. Our method is inspired by the tropical proof of the WDVV equations. We also extend our study to counts involving two lines, both paired up with a Psi‐condition, appearing with power 1.

Operationally-based program equivalence proofs using LCTRSs

We propose an operationally-based framework for deductive proofs of program equivalence. It is based on encoding the language semantics as logically constrained term rewriting systems (LCTRSs) and the two programs as terms. As a novelty of our method, we show that it enables relational reasoning about programs in various settings, which are encoded in the operational semantics. For example, we show how our method can be used to prove programs that are equivalent when considering an unbounded stack, but where the equivalence fails to hold if the stack is bounded. We also show how to formalize read-sets and write-sets of symbolic expressions and statements by extending the operational semantics in a conservative way. This enables the relational verification of program schemas, which we exploit to prove compiler optimizations that cannot be handled by other tools. Our method requires an extension of standard LCTRSs with axiomatized symbols, which generate new research questions. We also present a prototype implementation that proves the practical feasibility of our approach.