First-order Lagrangian Research Articles

Symmetry and conservation laws for tuberculosis model

In this paper, three-dimensional system of the tuberculosis (TB) model is reduced into a two-dimensional first-order and one-dimensional second-order differential equations. We use the method of Jacobi last multiplier to construct linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation. The Noether's theorem is used for determining conservation laws. We apply the techniques of symmetry analysis to a model to identify the combinations of parameters which lead to the possibility of the linearization of the system and provide the corresponding solutions.

Iteration-complexity of first-order augmented Lagrangian methods for convex programming

This paper considers a special class of convex programming (CP) problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold. We present first-order me...

Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets

In this paper, “chance optimization” problems are introduced, where one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally hard. With the objective of developing systematic numerical procedures to solve such problems, a sequence of convex relaxations based on the theory of measures and moments is provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. Indeed, we provide a sequence of semidefinite programs of increasing dimension which can arbitrarily approximate the solution of the original problem. To be able to efficiently solve the resulting large-scale semidefinite relaxations, a first-order augmented Lagrangian algorithm is implemented. Numerical examples are presented to illustrate the computational performance of the proposed approach.

Optimisation of mine ventilation networks using the Lagrangian algorithm for equality constraints

This work presents a new method of mine ventilation network optimisation as standard non-linear programming problem and discusses the use of a novel first-order Lagrangian (FOL) algorithm for equality constraints as a solution tool for these problems. Slack variables have been defined to transform such inequality constraints into their corresponding equality forms. The problem is then converted to non-linear problem with equality constrains. The methodology adopted in this paper is capable of dealing with the non-linear convex model with significant savings on computational efforts due to its use of only first derivatives. A MATLAB programme has been developed based on the FOL method to solve a generalised mine ventilation network optimisation problem. To study the validity and the viability of the FOL programme, the programme has been applied to already published network problems and both results are identical.

Spinor and twistor formulations of massless particles with rigidity

The 4-dimensional model of a massless particle with rigidity whose Lagrangian is proportional to its world-line curvature is reformulated in terms of spinor and twistor variables. We begin with a first-order Lagrangian that is equivalent to the original Lagrangian proportional to the extrinsic curvature of a particle world-line. The first-order Lagrangian is written in terms of spacetime and spinor variables, leading to a spinor representation of the Lagrangian. Then its corresponding action is expressed in terms of twistor variables, leading to the gauged Shirafuji action.

Breaking generalized covariance, classical renormalization, and boundary conditions from superpotentials

Superpotentials offer a direct means of calculating conserved charges associated with the asymptotic symmetries of space-time. Yet superpotentials have been plagued with inconsistencies, resulting in nonphysical or incongruent values for the mass, angular momentum, and energy loss due to radiation. The approach of Regge and Teitelboim, aimed at a clear Hamiltonian formulation with a boundary, and its extension to the Lagrangian formulation by Julia and Silva have resolved these issues, and have resulted in a consistent, well-defined and unique variational equation for the superpotential, thereby placing it on a firm footing. A hallmark solution of this equation is the KBL superpotential obtained from the first-order Lovelock Lagrangian. Nevertheless, here we show that these formulations are still insufficient for Lovelock Lagrangians of higher orders. We present a paradox, whereby the choice of fields affects the superpotential for equivalent on-shell dynamics. We offer two solutions to this paradox: either the original Lagrangian must be effectively renormalized, or that boundary conditions must be imposed, so that space-time be asymptotically maximally symmetric. Non-metricity is central to this paradox, and we show how quadratic non-metricity in the bulk of space-time contributes to the conserved charges on the boundary, where it vanishes identically. This is a realization of the gravitational Higgs mechanism, proposed by Percacci, where the non-metricity is the analogue of the Goldstone boson.

A fixed-point augmented Lagrangian method for total variation minimization problems

In this paper, we propose a fixed-point augmented Lagrangian method (FPALM) for general convex problems arising in image processing. We can easily obtain the alternating minimization algorithm (AMA) referred to [1] from the proposed FPALM. The proof for the convergence of the FPALM is provided under some mild assumptions. We present two kinds of first-order augmented Lagrangian schemes and show their connections to first-order primal–dual algorithms [2]. Furthermore, we apply an acceleration rule to both the FPALM and AMA to achieve better convergence rates. Numerical examples on different image denosing models including the ROF model, the vectorial TVmodel, high order models and the TV-L1 model are provided to demonstrate the efficiency of the proposed algorithms.

De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity

The De Donder-Weyl (DW) covariant Hamiltonian formulation of the Palatini first-order Lagrangian of vielbein (tetrad) gravity and its precanonical quantization are presented. No splitting into space and time is required in this formulation. Our recent generalization of Dirac brackets is used to treat the second class primary constraints appearing in the DW Hamiltonian formulation and to find the fundamental brackets. Quantization of the latter yields the representation of vielbeins as differential operators with respect to the spin connection coefficients and the Dirac-like precanonical Schrödinger equation on the space of spin connection coefficients and space time variables. The transition amplitudes on this space describe the quantum geometry of space-time. We also discuss the Hilbert space of the theory, the invariant measure on the spin connection coefficients, and point to the possible quantum singularity avoidance built in in the natural choice of the boundary conditions of the wave functions on the space of spin connection coefficients.

Conservation laws for a generalized coupled bidimensional Lane–Emden system

This paper aims to perform Noether symmetry classification of a generalized coupled bidimensional Lane–Emden system and computes the Noether operators corresponding to its first-order Lagrangian. In addition conservation laws of the various cases which admit Noether point symmetries are constructed for the underlying system.

A First-Order Augmented Lagrangian Method for Compressed Sensing

We propose a first-order augmented Lagrangian (FAL) algorithm for solving the basis pursuit problem. FAL computes a solution to this problem by inexactly solving a sequence of $\ell_1$-regularized least squares subproblems. These subproblems are solved using an infinite memory proximal gradient algorithm wherein each update reduces to “shrinkage” or constrained “shrinkage.” We show that FAL converges to an optimal solution of the basis pursuit problem whenever the solution is unique, which is the case with very high probability for compressed sensing problems. We construct a parameter sequence such that the corresponding FAL iterates are $\epsilon$-feasible and $\epsilon$-optimal for all $\epsilon>0$ within $\mathcal{O}\left(\log\left(\epsilon^{-1}\right)\right)$ FAL iterations. Moreover, FAL requires at most $\mathcal{O}(\epsilon^{-1})$ matrix-vector multiplications of the form $Ax$ or $A^Ty$ to compute an $\epsilon$-feasible, $\epsilon$-optimal solution. We show that FAL can be easily extended to solve the basis pursuit denoising problem when there is a nontrivial level of noise on the measurements. We report the results of numerical experiments comparing FAL with the state-of-the-art solvers for both noisy and noiseless compressed sensing problems. A striking property of FAL that we observed in the numerical experiments with randomly generated instances when there is no measurement noise was that FAL always correctly identifies the support of the target signal without any thresholding or postprocessing, for moderately small error tolerance values.

LAGRANGIANS FOR BIOLOGICAL MODELS

We show that a method presented in [S. L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974) 299–324] and later in [G. H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749–760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation that can be derived from them, even in the case where those authors failed such as the host-parasite model. Also we show that the Lagrangians of certain second-order ordinary differential equations derived by Volterra in [V. Volterra, Calculus of variations and the logistic curve, Hum. Biol. 11 (1939) 173–178] are particular cases of the Lagrangians that can be obtained by means of the Jacobi Last Multiplier. Actually we provide more than one Lagrangian for those Volterra's equations.

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.

Noncommutative mechanics and exotic Galilean symmetry

In terms of the Lagrange-Souriau 2-form formalism, we describe a wide set of Hamiltonian dynamical systems with first-order Lagrangians. We consider a wide class of systems derived in different phenomenological contexts. The noncommutativity of the particle position coordinates are a natural consequence. We present various explicit examples.

The fundamental form of a homogeneous Lagrangian in two independent variables

We construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the ‘fundamental Lepage equivalent’ associated with first-order Lagrangians defined on jets of sections of a fibred manifold.

THE QUANTIZATION OF A FOURTH-ORDER EQUATION WITHOUT A SECOND-ORDER LAGRANGIAN

We present an equation of the fourth-order which does not possess a second-order Lagrangian and demonstrate by means of the method of reduction of order that one can obtain a first-order Lagrangian for it. This opens the way to quantization through the construction of an Hamiltonian which is suitable to be quantized according to the procedure of Dirac with the correct physical attributes.

The method of Ostrogradsky, quantization, and a move toward a ghost-free future

The method of Ostrogradsky has been used to construct a first-order Lagrangian, hence Hamiltonian, for the fourth-order field-theoretical model of Pais–Uhlenbeck with unfortunate results when quantization is undertaken since states with negative norm, commonly called “ghosts,” appear. We propose an alternative route based on the preservation of symmetry and this leads to a ghost-free quantization.

Position-dependent noncommutativity in quantum mechanics

The model of the position-dependent noncommutativity in quantum mechanics is proposed. We start with given commutation relations between the operators of coordinates $[{\stackrel{^}{x}}^{i},{\stackrel{^}{x}}^{j}]={\ensuremath{\omega}}^{ij}(\stackrel{^}{x})$, and construct the complete algebra of commutation relations, including the operators of momenta. The constructed algebra is a deformation of a standard Heisenberg algebra and obeys the Jacobi identity. The key point of our construction is a proposed first-order Lagrangian, which after quantization reproduces the desired commutation relations. Also we study the possibility to localize the noncommutativity.

PERTURBATIVE EXPANSION FOR THE t-J MODEL CONSTRUCTED FROM THE GENERATORS OF THE SUPERSYMMETRIC HUBBARD ALGEBRA

By using the Hubbard [Formula: see text]-operators as field variables along with the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In order to satisfy the Hubbard [Formula: see text]-operator commutation rules satisfying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for this model are presented. The model is also analyzed via path-integral formalism, where the correlation-generating functional and the effective Lagrangian are constructed. In this context, the introduction of a proper ghost field is needed to render the model renormalizable. The perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator arising in the present formalism is discussed. As an example, the thermal softening of the magnon frequency is computed.

On the variational integrating matrix for hyperbolic systems

We obtain a necessary and sufficient condition for a hyperbolic system to be an Euler-Lagrange system with a first-order Lagrangian up to multiplication by some matrix. If this condition is satisfied and an integral of the system is known to us, then we can construct a family of higher symmetries that depend on an arbitrary function. Also, we consider the systems that satisfy the above criterion and that possess a sequence of the generalized Laplace invariants with respect to one of the characteristics; then we prove that the generalized Laplace invariants with respect to the other characteristic are uniquely defined.

THE HUBBARD OPERATORS AND THE SLAVE-PARTICLES PATH-INTEGRAL REPRESENTATIONS DESCRIBING THE t-J MODEL

In the present work it is shown that the family of first-order Lagrangians for the t-J model and the corresponding correlation generating functional previously found can be exactly mapped into the slave-fermion decoupled representation. Next, by means of the Faddeev-Jackiw symplectic method, a different family of Lagrangians is constructed and it is shown how the corresponding correlation generating functional can be mapped into the slave-boson decoupled representation.