Adjoint Map Research Articles

Invariant analysis of the multidimensional Martinez Alonso–Shabat equation

Abstract This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end.

Robust Invariance Conditions of Uncertain Linear Discrete Time Systems Based on Semidefinite Programming Duality

This article proposes a novel robust invariance condition for uncertain linear discrete-time systems with state and control constraints, utilizing a method of semidefinite programming duality. The approach involves approximating the robust invariant set for these systems by tackling the dual problem associated with semidefinite programming. Central to this method is the formulation of a dual programming through the application of adjoint mapping. From the standpoint of semidefinite programming dual optimization, the paper presents a novel linear matrix inequality (LMI) conditions pertinent to robust positive invariance. Illustrative examples are incorporated to elucidate the findings.

The adjoint map of Euclidean plane curves and curvature problems

The adjoint map of a pair of naturally parametrized curves in the Euclidean plane is studied from the point of view of the curvature. A main interest is when the given curve and its adjoint curve share the same natural parameter and the same curvature. For the general linear second order differential equation we introduce a function expressing the deformation of curvatures induced by the adjoint map.

Linearly convergent adjoint free solution of least squares problems by random descent

We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations, especially no evaluation of the adjoint map is needed. Our method is motivated by the simple observation that one can get an unbiased estimate of the application of the adjoint. We show convergence of the method and then derive a more efficient method that uses an exact linesearch. This method, called random descent, resembles known methods in other context and has the randomized coordinate descent method as special case. We provide convergence analysis of the random descent method emphasizing the dependence on the underlying distribution of the random vectors. Furthermore we investigate the applicability of the method in the context of ill-posed inverse problems and show that the method can have beneficial properties when the unknown solution is rough. We illustrate the theoretical findings in numerical examples. One particular result is that the random descent method actually outperforms established transposed-free methods (TFQMR and CGS) in examples.

Some results on the class of weakly sequentially precompact sets and operators

The paper contains some results on weakly sequentially precompact sets and operators. In particular, we establish some relationships between weakly sequentially precompat operators and those whose the adjoint map (L) sets into relatively norm compact ones. Besides, we characterize the class of weak* Dunford-Pettis operators through weakly sequentially precompact operators and deduce in the sequel a new characterization of Dunford-Pettis* property. Moreover, we generalize [9, Theorem 2.5.9] and show that order weakly compact operators carry almost order Dunford-Pettis sets into weakly sequentially precompact ones. Furthermore, we prove that the product of order weakly compact operators and b-weakly compact ones maps weakly sequentially precompact sets into relatively weakly compact ones. Finally, we present some results about the positive Schur property.

THE ADJOINT MAP AND DUAL MAP OF MULTILINEAR MAP

Let be the vector spaces and V_i^* be their dual spaces for 1≤i≤r . In this paper, (V_1×V_2×…×V_r )^*≅V_1^*×V_2^*×…×V_r^* is examined. Furthermore, the dual map F^* and the adjoint map F^' of the r-linear map F:V_1×V_2×…×V_r→W are defined and for their matrices, the equality [F^* ]=[F^' ]=[F ̅ ]^T is found. Analog, the dual map F^* of the r-variable vector valued linear map F:V_1×V_2×…×V_r→W_1×W_2×…×W_r is defined and for their matrices, the equality [F^* ]=[F ̅ ]^T is true, for the vector spaces V_i and W_i .

О свойстве конечного носителя гомеоморфизма Гулько-Хмылёвой

It is established that for the homeomorphism of the spaces c0 and s × c0 constructed by S.P. Gulko and T.E. Khmylyov, the image of the first coordinate functional under the adjoint mapping has no finite support. As a consequence, we see that this homeomorphism does not have the finite support property. In addition, it is shown that the images of all other coordinate functionals under the adjoint mapping have finite supports. The authors thank S.P. Gubko and T.E. Khmyleva, as well as A.V. Osipov for their kind attention to this work. The authors thank S.P. Gubko and T.E. Khmyleva, as well as A.V. Osipov for their kind attention to this work.

Optimal control of differential inclusions with endpoint constraints and duality

The article considers a high-order optimal control problem and its dual problems described by high-order differential inclusions. In this regard, the established Euler–Lagrange type inclusion, containing the Euler–Poisson equation of the calculus of variations, is a sufficient optimality condition for a differential inclusion of a higher order. It is shown that the adjoint inclusion for the first-order differential inclusions, defined in terms of a locally adjoint mapping, coincides with the classical Euler–Lagrange inclusion. Then the duality theorems are proved.

Gradings induced by nilpotent elements

An element a is nilpotent last-regular if it is nilpotent and its last nonzero power is von Neumann regular. In this paper we show that any nilpotent last-regular element a in an associative algebra R over a ring of scalars Φ gives rise to a complete system of orthogonal idempotents that induces a finite Z-grading on R; we also show that such element gives rise to an sl2-triple in R with semisimple adjoint map adh, and that the grading of R with respect to the complete system of orthogonal idempotents is a refinement of the Φ-grading induced by the eigenspaces of adh. These results can be adapted to nilpotent elements a with all their powers von Neumann regular, in which case the element a can be completed to an sl2-triple and a is homogeneous of degree 2 both in the Z-grading of R and in the Φ-grading given by the eigenspaces of adh.Conversely, when there are enough invertible elements in Φ, we will show that if a nilpotent element a can be completed to an sl2-triple, then all powers ak are von Neumann regular.

Supercanonical coordinates for orthosymplectic evolution operators

A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum osp(m′/n′)⊕sne(n′+1), of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra no(m′+1) elements times ne(n′+1) elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of bosons and, separately, of fermions with coefficients given in terms of the same coefficients of Π. Intertwining is invariant under transformations of Π, which leave certain binary forms of the coefficients of Π in the Hamiltonian unchanged. Alternatively, the coefficients can be interpreted as simultaneous time-dependent (super-) control parameters for both spin-statistics. Time-dependent inhomogeneous linear supercanonical transformations of wave vectors leave invariant the Heisenberg superalgebra n(m′/n′) and belong to the semidirect product Osp(m′/n′) ⋉ Ne(n′ + 1) of the orthosymplectic supergroup with the even Heisenberg group. The unitary time evolution operator is constructed using the adjoint map in canonical coordinates determined by the supercanonical transformation. The method is a generalization of an Inönu–Wigner contraction procedure and a Wei–Norman method for superalgebras with a selection of subalgebras associated with the root space decomposition of the Lie superalgebra. Analogously, this is a separation of variables method for quantum mechanical problems in systems with bosons and fermions. The standard Floquet theory leads to new results concerning stability for locally periodic coefficients. The lowest dimensional cases are explicitly computed. The intertwining of boson and fermions systems and the Hamiltonians considered here are of interest in quantum control theory for systems including fermions and bosons, in quantum optics, and quantum computation.

Optimization of Mayer functional in problems with discrete and differential inclusions and viability constraints

This paper derives the optimality conditions for a Mayer problem with discrete and differential inclusions with viable constraints. Applying necessary and sufficient conditions of problems with geometric constraints, we prove optimality conditions for second order discrete inclusions. Using locally adjoint mapping, we derive Euler-Lagrange form conditions and transversality conditions for the optimality of the discrete approximation problem. Passing to the limit, we establish sufficient conditions to the optimal problem with viable constraints. Conditions ensuring the existence of solutions to the viability problems for differential inclusions of second order have been studied in recent years. However, optimization problems of second-order differential inclusions with viable constraints considered in this paper have not been examined yet. The results presented here are motivated by practices for optimization of various fields as the mass movement model well known in traffic balance and operations research.

Geometry of Adjoint-Invariant Submanifolds of SE(3)

This article aims to extend the theory of Lie subgroups and symmetric subspaces for studying an important class of submanifolds of the special Euclidean group SE(3) whose tangent space at each point on the submanifold relates to that at the identity by an adjoint map. These submanifolds, which we call adjoint-invariant submanifolds in this article, are known in the literature as persistent submanifolds, since they are strictly related to the concept of persistent screw systems. The difference is that in this article, just as Lie subgroups and symmetric subspaces, we put forward adjoint-invariant submanifolds as independent geometric objects from mechanisms and their associated local screw systems. Adjoint invariance relaxes the strict left and right invariance of Lie subgroups and the reflective invariance of symmetric subspaces by allowing generic moving reference frame in the aforementioned adjoint map. It turns out such adjoint invariance can be studied under the framework of distributions on manifolds, which allows us to explore global geometric properties of adjoint-invariant submanifolds. We classify adjoint-invariant submanifolds into reflective-type and product-type submanifolds and derive the conditions for their adjoint invariance. We then propose geometric methods and algorithms for synthesizing the kinematic generators for reflective-type submanifolds, as demonstrated with a number of examples.

Supercanonical transformations and time-dependent Hamiltonians

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp. The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

The Non-m-Positive Dimension of a Positive Linear Map

We introduce a property of a matrix-valued linear map Φ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

Method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe

The evolution of quantum light through linear optical devices can be described by the scattering matrix $S$ of the system. For linear optical systems with $m$ possible modes, the evolution of $n$ input photons is given by a unitary matrix $U=\varphi_{m,M}(S)$ given by a known homomorphism, $\varphi_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, $M$. We present a method to decide whether a given unitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linear optics or not and the inverse transformation $\varphi_{m,M}^{-1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map bewtween the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

Tremor Estimation and Removal in Robot-Assisted Surgery Using Lie Groups and EKF

SummaryThis paper aims at estimating the tremor torque using extended Kalman filter (EKF) applied to a two-link 3-DOF robot with nonlinear dynamics modelled using Lie-group and Lie-algebra theory. Later, it is generalised to d number of links with (d + 1) -DOF. The configuration of each link at any time is described by its rotation relative to the preceding link. Using this formulation, an elegant formula for the kinetic energy of the (d + 1) -DOF system is obtained as a quadratic form in the angular velocities with coefficients being highly nonlinear trigonometric functions of the angles. Properties of the Lie algebra generators and the Lie adjoint map are used to arrive at this expression. Further, the gravitational potential energy and the torque potential energy are expressed as nonlinear trigonometrical functions of the angles using properties of the SO(3) group. The input torque comprises a nonrandom intentional torque component and a highly nonlinear tremor torque component. The tremor torque is modelled as a stochastic differential equation (sde) satisfying Ornstein–Uhlenbeck (OU) process with diffusion and damping coefficients. Further, the tremor is treated as the disturbance. The Euler–Lagrange equations for the angles are derived. These form a system of sdes, and the EKF is used to get a more accurate disturbance estimate than that provided by the usual disturbance observer. The EKF is based on noisy angle measurements and yields as a bonus the angle and angular velocity estimates on a real-time basis. The parameters in the OU process model of the tremor torque, and parameters of the Fourier components of the intentional torque have also been estimated.

Fuel-Optimal Low-Thrust Trajectory Optimization Using Indirect Method and Successive Convex Programming

Although the robustness of indirect methods is enhanced by the homotopic approach and switching detection technique when applied to fuel-optimal low-thrust trajectory optimization, the bottleneck in adjoint initialization still needs further investigation. This paper overcomes this bottleneck by the adjoint mapping between the Lagrange multipliers of direct methods to the adjoint variables. The nonconvex optimization problem deduced from direct methods is converted into a convex one by lossless convexification and successive convex programming. By combining these techniques, a framework is built to effectively solve the fuel-optimal low-thrust trajectory optimization problem.

A matrix CFT at multiple large charges

We investigate matrix models in three dimensions where the global SU(N ) symmetry acts via the adjoint map. Analyzing their ground state which is homogeneous in space and can carry either a unique or multiple fixed charges, we show the existence of at least two distinct fixed points of the renormalization group (rg) flow. In particular, the one type of those fixed points manifests itself via tractable deviations in the large-charge expansion from the known predictions in the literature. We demonstrate most of the novel features using mainly the example of the SU(4) matrix theory to compute the anomalous dimension of the lowest scalar operator with large global charge(s).

Adjoint Map Representation for Shape Analysis and Matching

AbstractIn this paper, we propose to consider the adjoint operators of functional maps, and demonstrate their utility in several tasks in geometry processing. Unlike a functional map, which represents a correspondence simply using the pull‐back of function values, the adjoint operator reflects both the map and its distortion with respect to given inner products. We argue that this property of adjoint operators and especially their relation to the map inverse under the choice of different inner products, can be useful in applications including bi‐directional shape matching, shape exploration, and pointwise map recovery among others. In particular, in this paper, we show that the adjoint operators can be used within the cycle‐consistency framework to encode and reveal the presence or lack of consistency between distortions in a collection, in a way that is complementary to the previously used purely map‐based consistency measures. We also show how the adjoint can be used for matching pairs of shapes, by accounting for maps in both directions, can help in recovering point‐to‐point maps from their functional counterparts, and describe how it can shed light on the role of functional basis selection.

Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on Lie group SE(2)

ABSTRACTThis paper studies the problem of set-point stabilisation, trajectory tracking, and formation tracking for unicycle-type vehicles by taking advantage of the exponential coordinates and the other mathematical tools provided by the Lie group setting of the special Euclidean group SE(2). Motivated by recent developments in geometric control approaches for holonomic systems, we first study the stabilisation problem for nonholonomic constrained kinematic systems by improving the logarithmic stabilising control laws in such a way that they can satisfy nonholonomic constraints too. We then extend the control design to the problem of trajectory tracking by proposing auxiliary systems and investigating the conditions for which the adjoint map in the Lie algebra preserves the velocity constraint. This leads to a global control law which is valid for general time-varying as well as non-smooth trajectories. The tracking control law is also applied to the problem of leader-follower formation tracking by constructing nonholonomic virtual leaders in the desired formation. Finally, the kinematic control law is translated to a differential drive robot using the backstepping technique. To demonstrate the effectiveness of the controllers, the numerical results are presented and the performances are compared with the other three controllers in the literature.