Showing papers on "Lie group published in 2018"

Circuit complexity for free fermions

Abstract: We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.

A micro Lie theory for state estimation in robotics

Abstract: A Lie group is an old mathematical abstract object dating back to the XIX century, when mathematician Sophus Lie laid the foundations of the theory of continuous transformation groups. As it often happens, its usage has spread over diverse areas of science and technology many years later. In robotics, we are recently experiencing an important trend in its usage, at least in the fields of estimation, and particularly in motion estimation for navigation. Yet for a vast majority of roboticians, Lie groups are highly abstract constructions and therefore difficult to understand and to use. This may be due to the fact that most of the literature on Lie theory is written by and for mathematicians and physicists, who might be more used than us to the deep abstractions this theory deals with. In estimation for robotics it is often not necessary to exploit the full capacity of the theory, and therefore an effort of selection of materials is required. In this paper, we will walk through the most basic principles of the Lie theory, with the aim of conveying clear and useful ideas, and leave a significant corpus of the Lie theory behind. Even with this mutilation, the material included here has proven to be extremely useful in modern estimation algorithms for robotics, especially in the fields of SLAM, visual odometry, and the like. Alongside this micro Lie theory, we provide a chapter with a few application examples, and a vast reference of formulas for the major Lie groups used in robotics, including most jacobian matrices and the way to easily manipulate them. We also present a new C++ template-only library implementing all the functionality described here.

Geometric Group Theory

Abstract: Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups Coarse geometry Coarse topology Ultralimits of metric spaces Gromov-hyperbolic spaces and groups Lattices in Lie groups Solvable groups Geometric aspects of solvable groups The Tits alternative Gromov's theorem The Banach-Tarski paradox Amenability and paradoxical decomposition Ultralimits, fixed point properties, proper actions Stallings's theorem and accessibility Proof of Stallings's theorem using harmonic functions Quasiconformal mappings Groups quasiisometric to $\mathbb{H}^n$ Quasiisometries of nonuniform lattices in $\mathbb{H}^n$ A survey of quasiisometric rigidity Appendix: Three theorems on linear groups Bibliography Index

Explorations in Homeomorphic Variational Auto-Encoding

Abstract: The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous en-coder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds with the Gaussian prior assumption typically made in Variational Auto-Encoders (VAEs), because the density of a Gaussian concentrates near a blob-like manifold. In this paper we investigate the use of manifold-valued latent variables. Specifically, we focus on the important case of continuously differen-tiable symmetry groups (Lie groups), such as the group of 3D rotations SO(3). We show how a VAE with SO(3)-valued latent variables can be constructed, by extending the reparameterization trick to compact connected Lie groups. Our exper-iments show that choosing manifold-valued latent variables that match the topology of the latent data manifold, is crucial to preserve the topological structure and learn a well-behaved latent space.

Seifert fibering operators in 3d $$ \mathcal{N}=2 $$ theories

Abstract: We study 3d $$ \mathcal{N}=2 $$ supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q)b with rational squashing parameter b2 ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

Consistent EKF-Based Visual-Inertial Odometry on Matrix Lie Group

Abstract: In this paper, we present a novel visual-inertial navigation algorithm for low-cost and computationally constrained vehicle in global positioning system denied environments by modeling the state space as the matrix Lie group (LG), based on the recent theory of the invariant Kalman filter. The multistate constraint Kalman filter (MSCKF) is a well-known visual-inertial odometry algorithm that performs the fusion of the visual and inertial information by constraining each other through the stochastically cloned pose within a sliding window. However, conventional MSCKF (MSCKF-Conv) suffers from the inconsistent state estimates caused by the spurious gain along the unobservable directions, resulting in large estimation errors. To tackle this problem, we extend the concepts of the state and noise of the MSCKF from Euclidean space to matrix LG. We model the state of the MSCKF as the element of the specially customized matrix LG and use the noise uncertainty modeling with the corresponding Lie algebra. The detailed derivation and observability analysis of the proposed filter are provided to prove that the proposed filter is more consistent than the MSCKF-Conv. The proposed MSCKF on matrix LG naturally enforces the state vector to exist in the state space that maintains the unobservability characteristics without any artificial remedies. The performance of the proposed filter is validated through the Monte-Carlo simulation and the real-world experimental dataset.

Screw and Lie group theory in multibody dynamics

Abstract: After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user-friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies and the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive $O ( n ) $ algorithms, for which the so-called “spatial operator algebra” is one example, and allows for use of readily available geometric data. In this paper, three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, that is, joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit–Hartenberg parameters, redundant. Four different definitions of twists are recalled, and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.

Geomstats: A Python Package for Riemannian Geometry in Machine Learning

Abstract: We introduce geomstats, a python package that performs computations on manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations. We provide efficient and extensively unit-tested implementations of these manifolds, together with useful Riemannian metrics and associated Exponential and Logarithm maps. The corresponding geodesic distances provide a range of intuitive choices of Machine Learning loss functions. We also give the corresponding Riemannian gradients. The operations implemented in geomstats are available with different computing backends such as numpy, tensorflow and keras. We have enabled GPU implementation and integrated geomstats manifold computations into keras deep learning framework. This paper also presents a review of manifolds in machine learning and an overview of the geomstats package with examples demonstrating its use for efficient and user-friendly Riemannian geometry.

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Abstract: Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups $$\mathrm {PO}(p,q)$$ by considering their action on the associated pseudo-Riemannian hyperbolic space $$\mathbb {H}^{p,q-1}$$ in place of the Riemannian symmetric space. Following work of Barbot and Merigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

Positivity and higher Teichmüller theory

Abstract: We introduce $\Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $\Theta$-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p

Deformation Based Curved Shape Representation

Abstract: In this paper, we introduce a deformation based representation space for curved shapes in $\mathbb {R}^{n}$ Given an ordered set of points sampled from a curved shape, the proposed method represents the set as an element of a finite dimensional matrix Lie group Variation due to scale and location are filtered in a preprocessing stage, while shapes that vary only in rotation are identified by an equivalence relationship The use of a finite dimensional matrix Lie group leads to a similarity metric with an explicit geodesic solution Subsequently, we discuss some of the properties of the metric and its relationship with a deformation by least action Furthermore, invariance to reparametrization or estimation of point correspondence between shapes is formulated as an estimation of sampling function Thereafter, two possible approaches are presented to solve the point correspondence estimation problem Finally, we propose an adaptation of k-means clustering for shape analysis in the proposed representation space Experimental results show that the proposed representation is robust to uninformative cues, eg, local shape perturbation and displacement In comparison to state of the art methods, it achieves a high precision on the Swedish and the Flavia leaf datasets and a comparable result on MPEG-7, Kimia99 and Kimia216 datasets

Spatially isotropic homogeneous spacetimes

Abstract: We classify simply-connected homogeneous ($D+1$)-dimensional spacetimes for kinematical and aristotelian Lie groups with $D$-dimensional space isotropy for all $D\geq 0$. Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for $D=1,2$. These geometries share the same amount of symmetry (spatial rotations, boosts and spatio-temporal translations) as the maximally symmetric spacetimes, but unlike them they do not necessarily admit an invariant metric. We determine the possible limits between the spacetimes and interpret them in terms of contractions of the corresponding transitive Lie algebras. We investigate geometrical properties of the spacetimes such as whether they are reductive or symmetric as well as the existence of invariant structures (riemannian, lorentzian, galilean, carrollian, aristotelian) and, when appropriate, discuss the torsion and curvature of the canonical invariant connection as a means of characterising the different spacetimes.

A (2+1)-dimensional breaking soliton equation: Solutions and conservation laws

Abstract: In this paper, we consider a (2+1)-dimensional breaking soliton equation which describe the (2+1)-dimensional interaction of the Riemann wave propagating along the y-axis with a long wave along the x-axis. By the Lie group analysis, the Lie point symmetry generators and symmetry reductions were deduced. From the viewpoint of exact solutions, we have performed two distinct methods to the equation for getting some exact solutions. Kudryashov’s simplest methods and ansatz method with the assistance of Maple were carried out. The local conservation laws are also constructed by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.

Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

Abstract: We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kahler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension 4, while in the latter case we provide examples of such Lie groups admitting lattices in any even dimension.

Equivariant basic cohomology of Riemannian foliations

Abstract: The basic cohomology of a Riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse-Bott function with critical set equal to C we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino's observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.

Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups

Abstract: We establish an exponential error term for the renewal theorem in the context of products of random matrices, which is surprising compared with classical abelian cases. A key tool is the Fourier decay of the Furstenberg measures on the projective spaces, which is a higher dimensional generalization of a recent work of Bourgain-Dyatlov.

On Lie symmetries and soliton solutions of $$(2+1)$$ ( 2 + 1 ) -dimensional Bogoyavlenskii equations

Abstract: The present article is devoted to find some invariant solutions of the $$(2+1)$$ -dimensional Bogoyavlenskii equations using similarity transformations method. The system describes $$(2+1)$$ -dimensional interaction of a Riemann wave propagating along y-axis with a long wave along x-axis. All possible vector fields, commutative relations and symmetry reductions are obtained by using invariance property of Lie group. Meanwhile, the method reduces the number of independent variables by one, which leads to the reduction of Bogoyavlenskii equations into a system of ordinary differential equations. The system so obtained is solved under some parametric restrictions and provides invariant solutions. The derived solutions are much efficient to explain the several physical properties depending upon various existing arbitrary constants and functions. Moreover, some of them are more general than previously established results (Peng and Shen in Pramana 67:449–456, 2006; Malik et al. in Comput Math Appl 64:2850–2859, 2012; Zahran and Khater in Appl Math Model 40:1769–1775, 2016; Zayed and Al-Nowehy in Opt Quant Electron 49(359):1–23, 2017). In order to provide rich physical structures, the solutions are supplemented by numerical simulation, which yield some positons, negatons, kinks, wavefront, multisoliton and asymptotic nature.

The structure theory of Nilspaces II: Representation as nilmanifolds

Abstract: This paper forms the second part of a series by the authors [GMV1,GMV3] concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy. A nilspace is a compact space $X$ together with closed collections of cubes $C_n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$ satisfying some natural axioms. From these axioms it follows that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. Our main result is a new proof of a result due to Antol\'in Camarena and Szegedy [CS12], stating that if each of these groups is a torus then $X$ is isomorphic (in a strong sense) to a nilmanifold $G/\Gamma$. We also extend the theorem to a setting where the nilspace arises from a dynamical system $(X,T)$. These theorems are a key stepping stone towards the general structure theorem in [GMV3] (which again closely resembles the main theorem of [CS12]). The main technical tool, enabling us to deduce algebraic information from topological data, consists of existence and uniqueness results for solutions of certain natural functional equations, again modelled on the theory in [CS12].

Priors with Coupled First and Second Order Differences for Manifold-Valued Image Processing

Abstract: We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore, we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds, our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models, we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.

Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics

Abstract: Part I Mathematical foundations1 Lie groups and Lie algebras: Basic concepts1.1 Topological groups and Lie groups1.2 Linear groups and symmetry groups of vector spaces1.3 Homomorphisms of Lie groups1.4 Lie algebras1.5 From Lie groups to Lie algebras1.6 From Lie subalgebras to Lie subgroups1.7 The exponential map1.8 Cartan's Theorem on closed subgroups1.9 Exercises for Chapter 12 Lie groups and Lie algebras: Representations and structure theory2.1 Representations2.2 Invariant metrics on Lie groups2.3 The Killing form2.4 Semisimple and compact Lie algebras2.5 Ad-invariant scalar products on compact Lie groups2.6 Homotopy groups of Lie groups2.7 Exercises for Chapter 23 Group actions3.1 Transformation groups3.2 Definition and first properties of group actions3.3 Examples of group actions3.4 Fundamental vector fields3.5 The Maurer-Cartan form and the differential of a smooth group action3.6 Left or right actions?3.7 Quotient spaces3.8 Homogeneous spaces3.9 Stiefel and Grassmann manifolds3.10 The exceptional Lie group G23.11 Godement's Theorem on the manifold structure of quotient spaces3.12 Exercises for Chapter 34 Fibre bundles4.1 General fibre bundles4.2 Principal fibre bundles4.3 Formal bundle atlases4.4 Frame bundles4.5 Vector bundles4.6 The clutching construction4.7 Associated vector bundles4.8 Exercises for Chapter 45 Connections and curvature5.1 Distributions and connections5.2 Connection 1-forms5.3 Gauge transformations5.4 Local connection 1-forms and gauge transformations5.5 Curvature5.6 Local curvature 2-forms5.7 Generalized electric and magnetic fields on Minkowski spacetime of dimension 45.8 Parallel transport5.9 The covariant derivative on associated vector bundles5.10 Parallel transport and path-ordered exponentials5.11 Holonomy and Wilson loops5.12 The exterior covariant derivative5.13 Forms with values in Ad(P)5.14 A second and third version of the Bianchi identity5.15 Exercises for Chapter 56 Spinors6.1 The pseudo-orthogonal group O(s t) of indefinite scalar products6.2 Clifford algebras6.3 The Clifford algebras for the standard symmetric bilinear forms6.4 The spinor representation6.5 The spin groups6.6 Majorana spinors6.7 Spin invariant scalar products6.8 Explicit formulas for Minkowski spacetime of dimension 46.9 Spin structures and spinor bundles6.10 The spin covariant derivative6.11 Twisted spinor bundles6.12 Twisted chiral spinors6.13 Exercises for Chapter 6Part II The Standard Model of elementary particle physics7 The classical Lagrangians of gauge theories7.1 Restrictions on the set of Lagrangians7.2 The Hodge star and the codifferential7.3 The Yang-Mills Lagrangian7.4 Mathematical and physical conventions for gauge theories7.5 The Klein-Gordon and Higgs Lagrangians7.6 The Dirac Lagrangian7.7 Yukawa couplings7.8 Dirac and Majorana mass terms7.9 Exercises for Chapter 78 The Higgs mechanism and the Standard Model8.1 The Higgs field and symmetry breaking8.2 Mass generation for gauge bosons8.3 Massive gauge bosons in the SU(2)U(1)-theory of the electroweak interaction8.4 The SU(3)-theory of the strong interaction (QCD)8.5 The particle content of the Standard Model8.6 Interactions between fermions and gauge bosons8.7 Interactions between Higgs bosons and gauge bosons8.8 Mass generation for fermions in the Standard Model8.9 The complete Lagrangian of the Standard Model8.10 Lepton and baryon numbers8.11 Exercises for Chapter 89 Modern developments and topics beyond the Standard Model9.1 Flavour and chiral symmetry9.2 Massive neutrinos9.3 C, P and CP violation9.4 Vacuum polarization and running coupling constants9.5 Grand Unified Theories9.6 A short introduction to the Minimal Supersymmetric Standard Model (MSSM)9.7 Exercises for Chapter 9Part III AppendixA Background on differentiable manifoldsA.1 ManifoldsA.2 Tensors and formsB Background on special relativity and quantum field theoryB.1 Basics of special relativityB.2 A short introduction to quantum field theoryReferencesIndex

Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations

Abstract: This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

Lie symmetry analysis and conservation law of variable-coefficient Davey–Stewartson equation

Abstract: A variable-coefficient Davey–Stewartson (vcDS) equation is investigated in this paper. Infinitesimal generators and symmetry groups are presented by the Lie group method, and the optimal system is presented with adjoint representation. Based on the optimal system, similarity reductions to partial differential equations (PDEs) are obtained, then some PDEs are reduced to ordinary differential equations (ODEs) by two-dimensional subalgebras, and the similarity solutions are provided, including periodic solutions and elliptic function solutions. With Lagrangian, it is shown that vcDS is nonlinearly self-adjoint. Furthermore, based on nonlinear self-adjointness, conservation laws for vcDS equation are derived.

The decomposition of almost paracontact metric manifolds in eleven classes revisited

Abstract: This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds become twelve. Also, we determine the classes of $$\alpha $$ -para-Sasakian, $$\alpha $$ -para-Kenmotsu, normal, paracontact metric, para-Sasakian, K-paracontact and quasi-para-Sasakian manifolds. Moreover, we study 3-dimensional almost paracontact metric manifolds and show that they belong to four basic classes from the considered classification. We define an almost paracontact metric structure on any 3-dimensional Lie group and give concrete examples of Lie groups belonging to each of the four basic classes, characterized by commutators on the corresponding Lie algebras.

Quantitative null-cobordism

Abstract: For a given null-cobordant Riemannian n n -manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n n . In the appendix the bound is improved to one that is O ( L 1 + e ) O(L^{1+\varepsilon }) for every e > 0 \varepsilon >0 . This construction relies on another of independent interest. Take X X and Y Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y Y is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic L L -Lipschitz maps f , g : X → Y f,g:X \to Y are homotopic via a C L CL -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y Y .

Numerical Methods for Stochastic Differential Equations in Matrix Lie Groups Made Simple

Abstract: A large number of significant applications involve numerical solution of stochastic differential equations (SDE's) evolving in Lie groups such as $SO(3)$ . In the engineering literature, the proper formulation of numerical schemes has largely been ignored so that many schemes are flawed, i.e., do not guarantee the solution stays in the Lie group. There is a small mathematics literature but it is not easily accessible. With this in mind, we give a directly accessible derivation of numerical schemes for solving SDE's that do not rely on differential geometry or advanced random process theory. In doing so, we develop some new results. We illustrate the numerical schemes with simulations.

Random Lie symmetries of Itô stochastic differential equations

Abstract: Lie point symmetries of stochastic differential equations (SDEs) which include transformations depending on the Brownian motion are considered. The corresponding Lie group transformations, acting in the space of the independent variable (time), the dependent variables and the Brownian motion, preserve the differential form of SDEs and the properties of the Brownian motion. Such symmetries are called random symmetries. The focus of the paper is a definition of the random symmetries and their properties. It is shown that the random symmetries form a Lie algebra. We also investigate relations between the random symmetries and the conserved quantities of SDEs. For several stochastic differential equations we find their random symmetries and compare them with the deterministic symmetries, which correspond to Lie group transformations in the space of the independent and dependent variables.

On some invariant solutions of (2+1)-dimensional Korteweg–de Vries equations

Abstract: In the present research, similarity transformation method is proposed to obtain some more general invariant solutions of (2+1)-dimensional Korteweg–de Vries equations This system of equations describes nonlinear waves propagation on the surface of shallow water The method reduces the number of independent variables by one using invariance property of Lie group theory Thus, Korteweg–de Vries equations are reduced into a system of ordinary differential equations employing twice of similarity transformation method This system of ordinary differential equations is solved under some parametric restrictions and provides invariant solutions The obtained results are supplemented by numerical simulation taking suitable choice of arbitrary constants and functions Eventually, the elastic behavior of multisoliton, compacton, negaton, positon, kink wave solution and dromion annihilation profiles are shown to make this research more admirable