Arbitrary Terms Research Articles

Techniques for solving static Klein-Gordon equation with self-interaction λϕ4 and arbitrary spherical source terms

The Klein-Gordon equation for a scalar field sourced by a static spherically symmetric background is an interesting second-order differential equation with applications in particle physics, astrophysics, and elsewhere. Here we present static solutions for generic source density profiles in the case where the scalar field has no interactions or a mass term. For a λϕ4 self-interaction term, we develop the techniques that are necessary numerical computation. We also provide code to perform the numerical calculations that can be adapted for arbitrary density profiles. ▪.

A Volterra edge dislocation problem in second-order elasticity theory

A displacement function technique has been developed to study plane strain problems in second-order elasticity theory for incompressible elastic materials. The formulation culminates in the development of a single inhomogeneous biharmonic equation of the form [Formula: see text] for the second-order displacement function [Formula: see text]. In the second-order, the isotropic stress associated with constitutive behaviour is governed by an inhomogeneous harmonic equation of the form [Formula: see text]. Both functions [Formula: see text] and [Formula: see text] depend only on the solution to the analogous problem in classical elasticity. While there are similarities to the Airy stress function, the displacement function approach enables the evaluation of the first- and second-order displacement fields purely through its derivatives, thus eliminating the introduction of arbitrary rigid body terms normally associated with formulations where the strains derived from a stress function approach need to be integrated. The displacement function approach is applied to develop a second-order elasticity solution to the problem of a Volterra edge dislocation.

GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations

Traditional machine learning is generally treated as a black-box optimization problem and does not typically produce interpretable functions that connect inputs and outputs. However, the ability to discover such interpretable functions is desirable. In this work, we propose GINN-LP, an interpretable neural network to discover the form and coefficients of the underlying equation of a dataset, when the equation is assumed to take the form of a multivariate Laurent Polynomial. This is facilitated by a new type of interpretable neural network block, named the “power-term approximator block”, consisting of logarithmic and exponential activation functions. GINN-LP is end-to-end differentiable, making it possible to use backpropagation for training. We propose a neural network growth strategy that will enable finding the suitable number of terms in the Laurent polynomial that represents the data, along with sparsity regularization to promote the discovery of concise equations. To the best of our knowledge, this is the first model that can discover arbitrary multivariate Laurent polynomial terms without any prior information on the order. Our approach is first evaluated on a subset of data used in SRBench, a benchmark for symbolic regression. We first show that GINN-LP outperforms the state-of-the-art symbolic regression methods on datasets generated using 48 real-world equations in the form of multivariate Laurent polynomials. Next, we propose an ensemble method that combines our method with a high-performing symbolic regression method, enabling us to discover non-Laurent polynomial equations. We achieve state-of-the-art results in equation discovery, showing an absolute improvement of 7.1% over the best contender, by applying this ensemble method to 113 datasets within SRBench with known ground-truth equations.

Rewriting and Completeness of Sum-Over-Paths in Dyadic Fragments of Quantum Computing

The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems. We give here a new set of rewrite rules for the formalism, and show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We show that the rewriting is terminating, but not confluent (which is expected from the universality of the fragment). We do so using the connection between Sum-over-Paths and graphical language ZH-calculus, and also show how the axiomatisation translates into the latter. We provide generalisations of the presented rewrite rules, that can prove useful when trying to reduce terms in practice, and we show how to graphically make sense of these new rules. We show how to enrich the rewrite system to reach completeness for the dyadic fragments of quantum computation, used in particular in the Quantum Fourier Transform, and obtained by adding phase gates with dyadic multiples of $\pi$ to the Toffoli-Hadamard gate-set. Finally, we show how to perform sums and concatenation of arbitrary terms, something which is not native in a system designed for analysing gate-based quantum computation, but necessary when considering Hamiltonian-based quantum computation.

Exact and Approximate Solutions for Some Classes of the Inhomogeneous Pantograph Equation

The standard pantograph delay equation (SPDDE) is one of the famous delay models. This standard model is basically homogeneous in nature and it has been extensively studied in the literature. However, the studies on the general inhomogeneous form of such a model seem rare. This paper considers the inhomogeneous pantograph delay equation (IPDDE) with a kind of arbitrary inhomogeneous term. This arbitrary inhomogeneous term is used in different forms to generate various classes of IPDDEs. The solutions of the present classes are obtained in closed series forms which satisfy the criteria of convergence. Also, the exact solutions are determined for these classes under a certain relation between the given initial condition of the model and the initial value of the inhomogeneous term. Several classes are generated and solved when the inhomogeneous term takes the form of trigonometric, exponential, and hyperbolic functions. Some existing results in the literature are recovered as special cases of the present ones. Moreover, the behaviors of the obtained solutions are demonstrated through graphs for various kinds of IPDDEs.

High-order phase reduction for coupled 2D oscillators.

Phase reduction is a general approach to describe coupled oscillatory units in terms of their phases, assuming that the amplitudes are enslaved. The coupling should be small for such reduction, but one also expects the reduction to be valid for finite coupling. This paper presents a general framework, allowing us to obtain coupling terms in higher orders of the coupling parameter for generic two-dimensional oscillators and arbitrary coupling terms. The theory is illustrated with an accurate prediction of Arnold's tongue for the van der Pol oscillator exploiting higher-order phase reduction.

Improving and Extending STERGM Approximations Based on Cross-Sectional Data and Tie Durations

Temporal exponential-family random graph models (TERGMs) are a flexible class of models for network ties that change over time. Separable TERGMs (STERGMs) are a subclass of TERGMs in which the dynamics of tie formation and dissolution can be separated within each discrete time step and may depend on different factors. The Carnegie et al. approximation improves estimation efficiency for a subclass of STERGMs, allowing them to be reliably estimated from inexpensive cross-sectional study designs. This approximation adapts to cross-sectional data by attempting to construct a STERGM with two specific properties: a cross-sectional equilibrium distribution defined by an exponential-family random graph model (ERGM) for the network structure, and geometric tie duration distributions defined by constant hazards for tie dissolution. In this article we focus on approaches for improving the behavior of the Carnegie et al. approximation and increasing its scope of application. We begin with Carnegie et al.’s observation that the exact result is tractable when the ERGM is dyad-independent, and then show that taking the sparse limit of the exact result leads to a different approximation than the one they presented. We show that the new approximation outperforms theirs for sparse, dyad-independent models, and observe that the errors tend to increase with the strength of dependence for dyad-dependent models. We then develop theoretical results in the dyad-dependent case, showing that when the ERGM is allowed to have arbitrary dyad-dependent terms and some dyad-dependent constraints, both the old and new approximations are asymptotically exact as the size of the STERGM time step goes to zero. We note that the continuous-time limit of the discrete-time approximations has the desired cross-sectional equilibrium distribution and exponential tie duration distributions with the desired means. We show that our results extend to hypergraphs, and we propose an extension of the Carnegie et al. framework to dissolution hazards that depend on tie age. Supplementary materials for this article are available online.

Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich–Frank–Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.

Parity Quantum Optimization: Encoding Constraints

Constraints make hard optimization problems even harder to solve on quantum devices because they are implemented with large energy penalties and additional qubit overhead. The parity mapping, which has been introduced as an alternative to the spin encoding, translates the problem to a representation using only parity variables that encodes products of spin variables. In combining exchange interaction and single spin flip terms in the parity representation, constraints on sums and products of arbitrary k-body terms can be implemented without additional overhead in two-dimensional quantum systems.

Parity Quantum Optimization: Compiler

We introduce parity quantum optimization with the aim of solving optimization problems consisting of arbitrary k-body interactions and side conditions using planar quantum chip architectures. The method introduces a decomposition of the problem graph with arbitrary k-body terms using generalized closed cycles of a hypergraph. Side conditions of the optimization problem in form of hard constraints can be included as open cycles containing the terms involved in the side conditions. The generalized parity mapping thus circumvents the need to translate optimization problems to a quadratic unconstrained binary optimization problem (QUBO) and allows for the direct encoding of higher-order constrained binary optimization problems (HCBO) on a square lattice and full parallelizability of gates.

Arbitrary Order Total Variation for Deformable Image Registration

In this work, we investigate image registration in a variational framework and focus on regularization generality and solver efficiency. We first propose a variational model combining the state-of-the-art sum of absolute differences (SAD) and a new arbitrary order total variation regularization term. The main advantage is that this variational model preserves discontinuities in the resultant deformation while being robust to outlier noise. It is however non-trivial to optimize the model due to its non-convexity, non-differentiabilities, and generality in the derivative order. To tackle these, we propose to first apply linearization to the model to formulate a convex objective function and then break down the resultant convex optimization into several point-wise, closed-form subproblems using a fast, over-relaxed alternating direction method of multipliers (ADMM). With this proposed algorithm, we show that solving higher-order variational formulations is similar to solving their lower-order counterparts. Extensive experiments show that our ADMM is significantly more efficient than both the subgradient and primal-dual algorithms particularly when higher-order derivatives are used, and that our new models outperform state-of-the-art methods based on deep learning and free-form deformation. Our code implemented in both Matlab and Pytorch is publicly available at https://github.com/j-duan/AOTV.

Symmetry analysis of the (3+1) dimensional Kadomtsev-Petviashvili equation with variable coefficients and an arbitrary nonlinear term

Symmetry analysis of the (3+1) dimensional Kadomtsev-Petviashvili equation with variable coefficients and an arbitrary nonlinear term

Symmetry analysis of the (3+1) dimensional Kadomtsev-Petviashvili equation with variable coefficients and an arbitrary nonlinear term

Symmetry analysis of the (3+1) dimensional Kadomtsev-Petviashvili equation with variable coefficients and an arbitrary nonlinear term

Qade: solving differential equations on quantum annealers

We present a general method, called Qade, for solving differential equations using a quantum annealer. One of the main advantages of this method is its flexibility and reliability. On current devices, Qade can solve systems of coupled partial differential equations that depend linearly on the solution and its derivatives, with non-linear variable coefficients and arbitrary inhomogeneous terms. We test this through several examples that we implement in state-of-the-art quantum annealers. The examples include a partial differential equation and a system of coupled equations. This is the first time that equations of these types have been solved in such devices. We find that the solution can be obtained accurately for problems requiring a small enough function basis. We provide a Python package implementing the method at gitlab.com/jccriado/qade.

Application of Generalized Logistic Function to Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

The generalized Logistic function that solves a first-order nonlinear ODE with an arbitrary positive power term of the dependent variable is introduced in this paper, by means of which the traveling wave solutions of a class of nonlinear evolution equations, including the generalized Fisher equation, the generalized Nagumo equation, the generalized Burgers-Fisher equation, the generalized Gardner equation, the generalized KdV-Burgers equation, and the generalized Benney equation, are obtained successfully. In these particular cases, traveling wave solutions of several important model PDEs in mathematical physics are also discovered.

Inverse Scale Space Iterations for Non-Convex Variational Problems: The Continuous and Discrete Case

Nonlinear filtering approaches allow to obtain decomposition of images with respect to a non-classical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The associated inverse scale space flow can be obtained using the classical Bregman iteration with quadratic data term. We apply the Bregman iteration to lifted, i.e., higher-dimensional and convex, functionals in order to extend the scope of these approaches to functionals with arbitrary data term. We provide conditions for the subgradients of the regularizer – in the continuous and discrete setting– under which this lifted iteration reduces to the standard Bregman iteration. We show experimental results for the convex and non-convex case.

The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein’s general relativity. We construct a proof of the zeroth law of black hole thermo-dynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove that surface gravity remains constant on its horizon when computed for such stationary black holes, which is the zeroth law. We argue that the constancy of surface gravity on the horizon is related to specific components of the equations of motion in such theories. We further use a specific boost symmetry of the near horizon space-time of the stationary black hole to constrain the off-shell structure of the equations of motion. Our proof for the zeroth law is valid up to arbitrary order in the expansion in the higher derivative couplings.

Universality in Anisotropic Linear Anelasticity

In linear elasticity, universal displacements for a given symmetry class are those displacements that can be maintained by only applying boundary tractions (no body forces) and for arbitrary elastic constants in the symmetry class. In a previous work, we showed that the larger the symmetry group, the larger the space of universal displacements. Here, we generalize these ideas to the case of anelasticity. In linear anelasticity, the total strain is additively decomposed into elastic strain and anelastic strain, often referred to as an eigenstrain. We show that the universality constraints (equilibrium equations and arbitrariness of the elastic constants) completely specify the universal elastic strains for each of the eight anisotropy symmetry classes. The corresponding universal eigenstrains are the set of solutions to a system of second-order linear PDEs that ensure compatibility of the total strains. We show that for three symmetry classes, namely triclinic, monoclinic, and trigonal, only compatible (impotent) eigenstrains are universal. For the remaining five classes universal eigenstrains (up to the impotent ones) are the set of solutions to a system of linear second-order PDEs with certain arbitrary forcing terms that depend on the symmetry class.

On Spencer’s displacement function approach for problems in second-order elasticity theory

The paper describes the displacement function approach first proposed by AJM Spencer for the formulation and solution of problems in second-order elasticity theory. The displacement function approach for the second-order problem results in a single inhomogeneous partial differential equation of the form , where is Stokes’ operator and depends only on the first-order or the classical elasticity solution. The second-order isotropic stress is governed by an inhomogeneous partial differential equation of the form , where is Laplace’s operator and depends only on the first-order or classical elasticity solution. The introduction of the displacement function enables the evaluation of the second-order displacement field purely through its derivatives and avoids the introduction of arbitrary rigid body terms normally associated with formulations where the strains need to be integrated. In principle, the displacement function approach can be systematically applied to examine higher-order effects, but such formulations entail considerable algebraic manipulations, which can be facilitated through the use of computer-aided symbolic mathematical operations. The paper describes the advances that have been made in the application of Spencer’s fundamental contribution and applies it to the solution of Kelvin’s concentrated force, Love’s doublet, and Boussinesq’s problems in second-order elasticity theory.

Global existence results for solutions of general conservative advection-diffusion equations in [formula omitted

In this paper a rigorous study concerning global existence results and estimates for the sup norm of non-negative bounded solutions for one-dimensional advection-diffusion equations ut+(b(x,t)uk+1)x=μ(t)uxx, with 0≤k<2 and initial data u0∈L1(R)∩L∞(R) is provided using a technique based on energy methods. In respect of the arbitrary advective speed term, it is only assumed that b(x,t) and ∂b(x,t)/∂x are limited.