Search

In this paper, we investigate invariant Ricci solitons, an important subclass in the class of homogeneous Ricci solitons. A classification of invariant Ricci solitons on three-dimensional Lie groups with left-invariant Riemannian metric and semi-symmetric connection different from the Levi-Civita connection is obtained. It is proved that in this case there exist invariant Ricci solitons with non-conformally Killing vector field. [ABSTRACT FROM AUTHOR]

Lie groups considered as three-dimensional almost paracontact almost paracomplex Riemannian manifolds are investigated. In each basic class of the classification used for the manifolds under consideration, a correspondence is established between the Lie algebra and the explicit matrix representation of its Lie group. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Advanced Mathematical Modeling (JAMM) is the property of Shahid Chamran University of Ahvaz and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras—graded central simple algebras or graded central simple algebras with involution. [ABSTRACT FROM AUTHOR]

We discuss the general Rodrigues problem and we give explicit determinant formulae for the coefficients when the eigenvalues of the matrix have double multiplicity (Theorem 5). When n = 4 explicit formulae and effective computations for the exponential map on the Lie group SO(4) are presented. [ABSTRACT FROM AUTHOR]

A subgroup H of a topological group G is called cocompact (or uniform) if the quotient space G / H ¯ is compact, where H ¯ denotes the closure of H in G. The aim of this note is to determine all topological groups with the property that every non-trivial discrete subgroup is cocompact. [ABSTRACT FROM AUTHOR]

In this work, an efficient method for constructing a complete integral of the geodesic Hamilton–Jacobi equation on pseudo-Riemannian manifolds with simply transitive groups of motions is suggested. The method is based on using a special transition to canonical coordinates on coadjoint orbits of the group of motion. As a non-trivial example, we consider the problem of constructing a complete integral of the geodesic Hamilton–Jacobi equation in the McLenaghan–Tariq–Tupper spacetime. An essential feature of this example is that the Hamilton-Jacobi equation is not separable in the corresponding configuration space. [ABSTRACT FROM AUTHOR]

In this paper we first construct a Lie group structure on n × n Hankel matrices over R + by Hadamard product and then we find its Lie algebra structure and finally calculate dimension of this manifold over R +. Moreover, we discuss topological properties of this manifold using Frobenious norm. We pointed out the relation between Lie group and Lie algebra structures of these matrices by exponential map. It is also shown that the Hadamard product on Hankel matrices over R + is not bounded by Frobenious norm. Lastly, we provide some applications of these manifolds. [ABSTRACT FROM AUTHOR]

The traditional error model for strapdown inertial navigation system (SINS)/Doppler velocity log (DVL) integrated navigation is establish for special orthogonal group Kalman filter (SOKF), which is only suitable for the case of small misalignment angle. In this article, the Lie group based nonlinear error model is investigated for SINS/DVL in inertial frame, this model is no longer limited by the size of misalignment angle like traditional SOKF, so initial alignment and integrated navigation could be combined into one stage, and the navigation solution is more accurate and faster. First, a thorough theoretical analysis is presented for state transformation extended Kalman filter (STEKF) modeling in both special orthogonal group (SO(3)) and special euclidean group (SE(3)). Then, the nonlinear error models (both left and right invariant error models) in inertial frame are derived in extended special euclidean group (SE 2 (3)). On this basis, SINS/DVL is verified as a type of left invariant observation system which is more applicable to the left invariant error model. Finally, both simulation and field test for SINS/DVL are conducted to evaluate the actual performance of the investigated methods. The results show that the proposed left invariant error model in inertial frame performs better for SINS/DVL than right invariant error model, and it also has more excellent performance than other existing integrated navigation methods, such as SOKF and STEKF. Especially compared with the current advanced invariant error model defined in earth frame, the model we investigated is more concise, efficient and accurate. • A thorough theoretical analysis is presented for STEKF modeling in both special orthogonal group (SO(3)) and special euclidean group (SE(3)). The inherent defects of STEKF are discussed from linearized modeling and Lie group theory. • A new navigation state is defined in inertial frame extended special euclidean group (SE 2 (3)), which is proved to satisfy the group affine property. The left and right invariant error models corresponding to the aforementioned state are derived for SINS/DVL integrated navigation. SINS/DVL is discussed as a type of left invariant observation system based on the analysis of transition matrices. • The simulation and field test are conducted to evaluate the actual performance of the investigated nonlinear error models. The results are explicitly presented with thorough discussion, which are consistent with theoretical analysis. The proposed left invariant error model in inertial frame outperform the common models (such as SOKF and STEKF) on the state estimation. Especially compared with the current advanced invariant error model defined in earth frame, the model we investigated is more concise, efficient, and accurate. [ABSTRACT FROM AUTHOR]

Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

Traditionally, network risk assessment uses a statistical computation method. This paper proposes Lie group kinematics to describe the feature space of the attack behavior. A matrix composed of indicators and topologies in a network system is mapped to a Lie group. The attack behavior path and the powers of the attack and defense are defined. The risk value of the attack and defense is calculated from the change in the indicators. Then, we evaluate the network security risk status using calculus. To examine the validity of the network risk assessment based on a Lie group, we conduct one experiment and utilize the CIC2017 dataset to show the applicability and efficiency of the proposed method. The experimental results show the effectiveness of the calculation method based on a Lie group, and the risk value of the attack and defense is valid compared to those of other machine learning algorithms. The calculation method based on a Lie group can quantitatively analyze network security risks..

Abstract Rossby waves, one of significant waves in the solitary wave, have important theoretical meaning in the atmosphere and ocean. However, the previous studies on Rossby waves commonly were carried out in the zonal area and could not be applied directly to the spherical earth. In order to overcome the problem, the research on ( 3+1 $3+1$)-dimensional Rossby waves in the paper is placed into the spherical area, and some new analytical solutions of ( 3+1 $3+1$)-dimensional Rossby waves are given through the classic Lie group method. Finally, the dissipation effect is analyzed in the sense of the above mentioned new analytical solutions. The new solutions on ( 3+1 $3+1$)-dimensional Rossby waves have important value for understanding the propagation of Rossby waves in the rotating earth with the influence of dissipation.

In 2009, Gyurko introduced Π-rough path which extends p-rough path. Inspired by this work we introduce the degree-(Π,N) signature which can be treated as the step-N signature for some Π. The degree-(Π,N) signature holds some algebraic properties which will be proven in this paper. [ABSTRACT FROM AUTHOR]

In this paper, a Lie coordinate-free torque based Euler–Lagrange equations of motion are developed for a 3-D link (3-DOF) robot. Intentional torque and jerky torque (non-intentional torque) are considered as the inputs to the dynamic profile of the robot. The jerky torque is modelled as a superposition of compound Poisson processes, which is a unique feature. The state vector of the robot, i.e., angular position and angular velocity vector, is thus a Markov process whose transition probability generator can be expressed in terms of the rate of the compound Poisson process that defines the jerky torque. Proof of frame invariance is provided to support the coordinate-free robot dynamics profile. Noise-free measurement is investigated as an ideal case. Angular position measurement is considered with white Gaussian noise. Further, an implementable finite-dimensional EKF approximate to Kushner–Kallianpur filter is obtained to estimate the robot state vector. Finally, the simulations are implemented on commercially available Omni bundle robot. [ABSTRACT FROM AUTHOR]

Remote sensing scene classification is used to label specific semantic categories for images. The current methods have achieved competitive performances, but they are only for Euclidean space samples. As a result, their representations are not robust for non-Euclidean space samples, which affects the classification accuracy. In this letter, we introduce the Lie group manifold into the traditional feature representation method and propose a novel intrinsic mean representation method within the Lie group. At the same time, the kernel function based on the sample of the Lie group is designed to further improve the robustness and accuracy of classification. In addition, our method achieves satisfactory performance on two public and challenging remote sensing data sets of UC Merced and NWPU-RESISC45. [ABSTRACT FROM AUTHOR]

A hom-Lie group structure is a smooth group-like multiplication on a manifold, where the structure is twisted by a isomorphism. The notion of hom-Lie group was introduced by Jiang et al. as integration of hom-Lie algebra. In this paper we want to study hom-Lie group and hom-Lie algebra from the Lie group's point of view. We show that some of important hom-Lie group issues are equal to similar types in Lie groups and then many of these issues can be studied by Lie group theory. [ABSTRACT FROM AUTHOR]

We describe the orbits of a cohomogeneity two Riemannian G-manifold M from topological point of view, under the conditions that G is nonsemisimple and M decomposes as a product of negatively curved Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Geometry modeling methods can conserve the geometry characters of a system, which helps the dynamic equations more concisely and is good for long simulations. Reduced attitude, Lie group and Lie algebra are three different expressions of geometry. Models for the dynamics of a planer pendulum and a 3D pendulum were built with these three geometry expressions. According to the variation method, the dynamics models as ordinary differential equations were transformed into nonlinear equations which are solved by Newton iteration. The simulation results show that Lie group and Lie algebra calculations can conserve the geometric structure, but have different long-time behavior. The complete Lie group expression has the best long simulation behavior and has the lowest sensitivity to the time step in both planer and 3D pendulum simulations, because it saves the complete geometry of the system in the dynamics model.

The aim of the present research work is to design a control law for a quadcopter drone based on the Virtual Attractive-Repulsive Potentials (VARP) theory. VARP theory, originally designed to enable path following by a small wheeled robot, will be tailored to control a quadcopter drone, hence allowing such device to learn flight planning. The proposed strategy combines an instance of VARP method to control a drone’s attitude (SO(3)-VARP) and an instance of VARP method to control a drone’s spatial location (R3-VARP). The resulting control strategy will be referred to as double-VARP method, which aims at making a drone follow a predefined path in space. Since the model of the drone as well as the devised control theory are formulated on a Lie group, their simulation on a computing platform is performed through a numerical analysis method specifically designed for these kinds of numerical simulations. A numerical simulation analysis is used to assess the salient features of the proposed regulation theory. In particular, resilience against shock-type disturbances are assessed numerically.

We address several problems of coordination and consensus on S O (3) and S 3 that can be formulated as minimization problems on these Lie groups. Then, gradient descent methods for minimization of the corresponding functions provide distributed algorithms for coordination and consensus in a multi-agent system. We point out main differences in convergence of algorithms on the two groups. We discuss advantages and effects of representing 3D rotations by quaternions and applications to the coordinated motion in space. In some situations (and depending on the concrete problem and goals) it is advantageous to run algorithms on S 3 and map trajectories onto S O (3) via the double cover map S 3 → S O (3) , instead of working directly on S O (3). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of Sasaki manifolds, called G-Sasaki manifolds with a reductive G-group action on their Kähler cones. By proving the properness of K-energy on such manifolds, we obtain a sufficient and necessary condition for the existence of G-Sasaki–Einstein metrics. A similar result is also obtained for G-Sasaki–Ricci solitons. As an application, we construct many new examples of G-Sasaki–Ricci solitons by an established openness theorem. [ABSTRACT FROM AUTHOR]

We show the existence of non-Einstein homogeneous critical metrics for any quadratic curvature functional in dimension three. [ABSTRACT FROM AUTHOR]

Time series prediction has attracted much attention as various issues can be formulated as such a task. It is one of the critical challenges to extract the intrinsic information for estimating future trends from historical data. Long Short-Term Memory Network (LSTM) shows excellent performance in this assignment. Probabilistic information extraction, which is demonstrated effective in object recognition in recent years, has not been introduced in time series prediction. To our knowledge, there has not been any work on plugging trainable probability distributions into LSTM as feature representation in an end-to-end manner. In this work we put forward an Adaptive Log-Euclidean Gaussian embedding Network (ALGeNet) to take one step further on solving this problem. The core of the network is capturing statistical information through the Gaussian Distribution with LSTM for end-to-end learning. As the space of Gaussian Distribution is a manifold, we try to embed Gaussian layer into LSTM through mapping Gaussian space into a linear space based on the Lie group and logarithm operation. We introduce four descriptors of Gaussian, two descriptors performing direct logarithm and the others performing indirect logarithm. All of them can extract first-order and second-order statistical features while utilizing the structures of geometry and smooth group of Gaussians. Furthermore, our adaptive mechanism merges the advantages of each descriptor and works well. Experimental results on a real-world wind speed dataset and a system-level electricity load dataset show that the proposed adaptive network outperforms some state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Let G be a regular Lie group which is a directed union of regular Lie groups Gi (all modelled on possibly infinite‐dimensional, locally convex spaces). We show that G=lim⟶Gi as a regular Lie group if G admits a so‐called direct limit chart. Notably, this allows the regular Lie group Diffc(M) of compactly supported diffeomorphisms to be interpreted as a direct limit of the regular Lie groups DiffK(M) of diffeomorphisms supported in compact sets K⊆M, even if the finite‐dimensional smooth manifold M is merely paracompact (but not necessarily σ‐compact), which is new. Similar results are obtained for the test function groups Cck(M,F) with values in a Lie group F. [ABSTRACT FROM AUTHOR]

Simultaneous Localisation and Mapping (SLAM) is the archetypal chicken and egg problem: Localisation of a robot with respect to a map requires an estimate of the map, while mapping an environment from data acquired by a robot requires an estimate of the robot localisation. The nonlinearity and co-dependence of the SLAM problem has made it an ongoing research problem for more than thirty years. The present paper details recent advances in understanding the SLAM problem, specifically the existence of an underlying geometry and symmetry structure that provides significant insight into the difficulties that have plagued many SLAM algorithms. To demonstrate the power of the geometric insight we derive a constant gain observer for the SLAM problem that; that does not depend on linearisation, has globally asymptotically stable error dynamics, is very robust, and operates in dynamic environments (estimating the landmark velocities as states in the observer). [ABSTRACT FROM AUTHOR]

In this paper, we have concentrated on a group action on the tangent bundle of some smooth/differentiable manifolds which has been built from a regular Lie group action on such smooth/differentiable manifolds. Interestingly, elements of orbit space yield smooth sections of the tangent bundle having beautiful algebraic properties. Moreover, each of those smooth sections behaves nicely as a left-invariant vector field with respect to Lie group action by G. We have explained here a simple isomorphism between the set of such smooth sections and each tangent space of that smooth/differentiable manifold. Also we have discussed more about F-relatedness and have introduced vector field relatedness by notations relX(M)(F); relDiff(M)(X), etc. which are sets based on both vector field related diffeomorphisms and diffeomorphism related vector fields. We have presented consequences based on the algebraic structure on relX(M)(F); relDiff(M)(X), etc. sets and built some related group actions. We have placed some interrelationship between the both kinds of rel operations. [ABSTRACT FROM AUTHOR]

We study almost hypercomplex structure with Hermitian–Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. All the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on one parameter are investigated. There are studied some geometrical characteristics of the respective almost hypercomplex manifolds with Hermitian–Norden metrics. [ABSTRACT FROM AUTHOR]

A metric is said to be Lie–Einstein if it is both a one-sided invariant metric on a Lie group and also an Einstein space. In this article right-invariant metrics are used thoughout. Several Lie–Einstein metrics on six-dimension Lie groups are constructed. The associated Lie algebras are necessarily solvable and are assumed to be indecomposable. They belong to two classes studied by Mubarakzyanov and Turkowski, respectively. Since the general problem of finding Lie–Einstein metrics in dimension six appears to be intractable, a certain ansatz for the form of the metric and class of Lie algebra is adopted. Nineteen Mubarakzyanov and five Turkowski Lie algebras are shown to be Lie–Einstein. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of Lie groups with bi-invariant (α,β)-metrics. We first show by an elementary proof that bi-invariant (α,β)-metrics are of BeGUKTld type. We give an explicit formula for the ag curvature of biinvariant (α,β)-metrics which improves the ag curvature formula of bi-invariant Randers metrics given in [10]. A necessary and sufficient condition that left invariant (α,β)-metrics on Lie groups are bi-invariant is given. [ABSTRACT FROM AUTHOR]

This paper proposes an improved initial alignment method for a strap-down inertial navigation system/global navigation satellite system (SINS/GNSS) integrated navigation system with large misalignment angles. Its methodology is based on the three-dimensional special Euclidean group and extended Kalman filter (SE 2 (3)/EKF) and aims to overcome the challenges of achieving fast alignment under large misalignment angles using traditional methods. To accurately characterize the state errors of attitude, velocity, and position, these elements are constructed as elements of a Lie group. The nonlinear error on the Lie group can then be well quantified. Additionally, a group vector mixed error model is developed, taking into account the zero bias errors of gyroscopes and accelerometers. Using this new error definition, a GNSS-assisted SINS dynamic initial alignment algorithm is derived, which is based on the invariance of velocity and position measurements. Simulation experiments demonstrate that the alignment method based on SE 2 (3)/EKF can achieve a higher accuracy in various scenarios with large misalignment angles, while the attitude error can be rapidly reduced to a lower level.

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based F e 3 O 4 nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of F e 3 O 4 , and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws. [ABSTRACT FROM AUTHOR]

In this paper, we shall apply symmetry analysis to some first order functional differential equations with constant coefficients. The approach used in this paper accounts for obtaining the inverse of the classification. We define the standard Lie bracket and make a complete classification of some first order linear functional differential equations with constant coefficients to solvable Lie algebras. We also classify some nonlinear functional differential equations with constant coefficients to solvable Lie algebras. [ABSTRACT FROM AUTHOR]

For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity. [ABSTRACT FROM AUTHOR]

Almost contact B -metric manifolds of the lowest dimension 3 are constructed by a three-parametric family of Lie groups. Our aim is to determine the class of considered manifolds in a classification of almost contact B -metric manifolds and their most important geometric characteristics and properties. Also, the type of the constructed Lie algebras is established in the Bianchi classification. [ABSTRACT FROM AUTHOR]

In this paper we have studied the driftless control system on a Lie group which arises due to the invariance of Black-Scholes equation by conformal transformations. These type of studies are possible as Black-Scholes equation can be mapped to one dimensional free Schrödinger equation. In particular we have studied the controllability, optimal control of the resulting dynamics as well as stability aspects of this system.We have also found out the trajectories of the states of the system through two unconventional integrators along with conventional Runge-Kutta integrator.

We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with first Betti number b1(M/T)=b1(M)−dim T.

The 6D Pose estimation is a crux in many applications, such as visual perception, autonomous navigation, and spacecraft motion. For robotic grasping, the cluttered and self-occlusion scenarios bring new challenges to the this field. Currently, society uses CNNs to solve this problem. The CNN models will suffer high uncertainty caused by the environmental factors and the object itself. These models usually maintain a Gaussian distribution, which is not suitable for the underlying manifold structure of the pose. Many works decouple rotation from the translation and quantify rotational uncertainty. Only a few works pay attention to the uncertainty of the 6D pose. This work proposes a distribution that can capture the uncertainty of the 6D pose parameterized by the dual quaternions, meanwhile, the proposed distribution takes the periodic nature of the underlying structure into account. The presented results include the normalization constant computation and parameter estimation techniques of the distribution. This work shows the benefits of the proposed distribution, which provides a more realistic explanation for the uncertainty in the 6D pose and eliminates the drawback inherited from the planar rigid motion.

The paper is dealing with stability problems for a nonlinear system on the Lie group SO(3) × R3 × R3. The approximate analytic solutions of the considered system via Optimal Homotopy Asymptotic Method are presented, too.

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.