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Let {gij(x)}n i,j=1 and {Lij(x)}n i,j=1 be the sets of all coefficients of the first and second fundamental forms of a hypersurface x in Rn+1 . For a connected open subset U Rn and a C 8-mapping x : U Rn+1 the hypersurface x is said to be d-nondegenerate, where d {1, 2, . . . n}, if Ldd(x) 1= 0 for all u U . Let M(n) = {F : Rn -1 Rn | Fx = gx + b, g O(n), b Rn}, where O(n) is the group of all real orthogonal n × n-matrices, and SM(n) = {F M(n) | g SO(n)}, where SO(n) = {g O(n) | det(g) = 1}. In the present paper, it is proved that the set {gij(x),Ldj(x), i, j = 1, 2, . . ., n} is a complete system of a SM(n + 1)-invariants of a d-non-degenerate hypersurface in Rn+1 . A similar result has obtained for the group M(n + 1). [ABSTRACT FROM AUTHOR]

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3 ≤ r ≤ n) , there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n − r) th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm. [ABSTRACT FROM AUTHOR]

Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed. [ABSTRACT FROM AUTHOR]

We study the first integral and the solution of electromagnetic field by Lie svnmietry technique and the differential invariant method. The definition and properties of differential invariants are introduced and the infinitesimal generators of Lie symmetries and the differential invariants of electromagnetic field are obtained. The first integral and the solution of electromagnetic field are given by the Lie symmetry technique and the differential invariants method. A typical example is presented to illustrate the application of our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract As we know, a ruled surface is the tangent ruled surface of a space curve if and only if its ruled distance density function vanishes identically; a ruled surface is the binormal ruled surface of a space curve if and only if its ruled translation density function vanishes identically. The ruled distance density function and translation density function are differential invariants of ruled surfaces in three dimensional Euclidean space. In this paper we give the necessary and sufficient condition of which a ruled surface is the principal normal ruled surface of a space curve using the theories of ruled invariants of ruled surface in three dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

We use moving frames to construct and classify the joint invariants and joint differential invariants of binary and ternary forms. In particular, we prove that the differential invariant algebra of ternary forms is generated by a single third order differential invariant. To connect our results with earlier analysis of Kogan, we develop a general method for relating differential invariants associated with different choices of cross-section. [ABSTRACT FROM AUTHOR]

In this paper we study differential invariants and give a local classification of the second order linear differential operators, acting in sections of line bundles, and a local classification of corresponding differential equations. [ABSTRACT FROM AUTHOR]

The class of second-order ordinary differential equations y'' = A(x, y)y' + B(x, y) is studied by methods of the geometry of jet spaces and the geometric theory of differential equations. The symmetry group of this class of equations is calculated, and the field of differential invariants of its action on equations is described. These results are used to state and prove a criterion for the local equivalence of two nondegenerate ordinary differential equations of the form y'' = A(x, y)y' + B(x, y), inwhich the coefficients A and B are rational in x and y. [ABSTRACT FROM AUTHOR]

We study geometric and algebraic approaches to classification problems of differential equations. We consider the so-called Lie problem: provide the point classification of ODEs y" = F(x, y). In the first part of the paper we consider the case of smooth right-hand side F. The symmetry group for such equations has infinite dimension, so classical constructions from the theory of differential invariants do not work. Nevertheless, we compute the algebra of differential invariants and obtain a criterion for the local equivalence of two ODEs y" = F(x, y). In the second part of the paper we develop a new approach to the study of subgroups in the Cremona group. Namely, we consider class of differential equations y" = F(x, y) with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of C², which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others. [ABSTRACT FROM AUTHOR]

Scale selection methods based on local extrema over scale of scale-normalized derivatives have been primarily developed to be applied sparsely-at image points where the magnitude of a scale- normalized differential expression additionally assumes local extrema over the domain where the data are defined. This paper presents a methodology for performing dense scale selection, so that hypotheses about local characteristic scales in images, temporal signals, and video can be computed at every image point and every time moment. A critical problem when designing mechanisms for dense scale selection is that the scale at which scale-normalized differential entities assume local extrema over scale can be strongly dependent on the local order of the locally dominant differential structure. To address this problem, we propose a methodology where local extrema over scale are detected of a quasi quadrature measure involving scale-space derivatives up to order two and pro- pose two independent mechanisms to reduce the phase dependency of the local scale estimates by (i) introducing a second layer of postsmoothing prior to the detection of local extrema over scale, and (ii) performing local phase compensation based on a model of the phase dependency of the local scale estimates depending on the relative strengths between first- and second-order differential structures. This general methodology is applied over three types of domains: (i) spatial images, (ii) temporal signals, and (iii) spatio-temporal video. Experiments demonstrate that the proposed methodology leads to intuitively reasonable results with local scale estimates that re ect variations in the characteristic scales of locally dominant structures over space and time. [ABSTRACT FROM AUTHOR]

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of order k ≥ 2 on an n -dimensional manifold, n ≥ 2 , and apply the results to the equivalence problem of such operators. [ABSTRACT FROM AUTHOR]

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms. [ABSTRACT FROM AUTHOR]

A point classification of ordinary differential equations of the form y″ = F( x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F( x, y) is calculated, and Lie's problem on the point equivalence of such equations is solved. [ABSTRACT FROM AUTHOR]

In this paper, local geometry of curves in the symplectic Grassmannian homogeneous space Sp ( 4 , R ) / ( Sp ( 2 , R ) × Sp ( 2 , R ) ) and its connection with that of the pseudo-hyperbolic space H 2 , 2 are studied. The group-based Serret–Frenet equations and the associated Maurer–Cartan differential invariants for the Grassmannian curves are obtained by using the equivariant moving frame method. The Grassmannian natural frame is also constructed by a gauge transformation from the Serret–Frenet frame, relating to the hyperbolic natural frame by the local Lie group isomorphism. Using the natural frames, invariant curve flows in the Grassmannian and the hyperbolic spaces are studied. It is shown that certain intrinsic curve flows induce the bi-Hamiltonian integrable matrix mKdV equation on the Maurer–Cartan differential invariants. [ABSTRACT FROM AUTHOR]

Second-order differential invariants of the metric field and a -form on the manifold are described using the factorization of the differential group with respect to a proper subgroup. It is proved that, all these invariants depend on the metric, its curvature tensor and covariant derivatives of the -form. [ABSTRACT FROM AUTHOR]

We describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples. [ABSTRACT FROM AUTHOR]

A pure algebraic approach to differential invariants of curves and surfaces is presented. By the use of this approach the explicit formulae for the generating differential invariants and invariant differential operators for the generic curves in n-dimensional classical geometries are constructed. [ABSTRACT FROM AUTHOR]

The article presents a mathematical analysis on general problems of contact and point classifications for functions defined on the n-jet space Jn&#211D; for an arbitrary n≿2. It describes the actions of contact and point pseudogroups on the n-jet space in the Lie alegra of the contact pseudogroup contact. It uses the methods of the theory of differential invariants in solving the problem.

The action of the pseudogroup of point transformations on the set of smooth functions on the contact 1-jet space Jℝ is considered. Differential invariants of this action are found, and the problem of local point equivalence of regular smooth functions is solved. [ABSTRACT FROM AUTHOR]

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of the second order and show their application to the equivalence problem of such operators. [ABSTRACT FROM AUTHOR]

In this paper, we show that characteristic covectors of a system of equations of 3-dimensional adiabatic gas motion generate a geometric structure on every solution of this system. This structure consists of a hyperplane and a non degenerate cone in every cotangent space to a solution. These hyperplane and cone intersect in zero point only. We construct differential invariants of this structure: a vector field, a conformal structure, a Lorentzian metric, and a linear connection. In the case of polytropic gas motion, we calculate classes of explicit solutions possessing the linear connection with zero torsion. [ABSTRACT FROM AUTHOR]

In this paper we suggest an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations. This approach is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one-dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion, which separates regular orbits. [ABSTRACT FROM AUTHOR]

In this paper we suggest an approach to the study of the action of point pseudogroup in (contact) 1-jet space J 1 R n . This approach is based on the following idea. We replace the canonic projection π 1 , 0 : J 1 R n → J 0 ( R n ) from 1-jet space to 0-jet space with projective action on the fiber by some bundle, which is called symplectization , with the linear action on the fiber. This makes it possible to apply methods of studying the linear actions of algebraic groups and to find the set of independent differential invariants. Finally, we rewrite the action of point pseudogroup in terms of these invariants and classify the orbits of point pseudogroup action. We also give two examples of application of this method. In the first example we obtain classification of contact vector fields in J 1 R n , and in the second we obtain classification of symbols of linear differential operators. [ABSTRACT FROM AUTHOR]

Abstract: We consider systems of Euler–Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lieʼs infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples. [Copyright &y& Elsevier]

We study the structure of second order natural Lagrangians on the bundle of linear coframes F* X over an n-dimensional manifold X. They are identified with the corresponding differential invariants which can be obtained by the factorization method. We give an explicit description of these differential invariants in terms of their bases. For construction of natural Lagrangians, the canonical odd n-form on F* X is also introduced. [ABSTRACT FROM AUTHOR]

Let M(n, p) be the group of all motions of an n-dimensional pseudo-Euclidean space of index p. It is proved that the complete system of M(n,p)-invariant differential rational functions of a path (curve) is a generating system of the differential field of all M(n, p) -invariant differential rational functions of a path (curve), respectively. A fundamental system of relations between elements of the complete system of M(n,p)-invariant differential rational functions of a path (curve) is described. [ABSTRACT FROM AUTHOR]

Applying Cartan's method of equivalence, this study demonstrates a classification of spacetime curves under Galilean motions. Among with trying to signify fundamental invariants which tend to new conservation laws, we also attempt to find a necessary and sufficient condition for Galilean equivalent curves via a simple and direct method. An straight forward deduction of our investigation in physics is, in addition, discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study orbits of - and -actions on the spaces of binary and ternary polynomials as well as rational forms and find criteria for their equivalence. Similar results are also valid for real forms. [Copyright &y& Elsevier]

The aim of this paper is to characterize all second order tensor-valued and scalar differential invariants of the bundle of linear frames FX over an n-dimensional manifold X. These differential invariants are obtained by factorization method and are described in terms of bases of invariants. Second order natural Lagrangians of frames have been characterized explicitly; if n = 1, 2, 3, 4, the number of functionally independent second order natural Lagrangians is N = 0, 6, 33, 104, respectively. [ABSTRACT FROM AUTHOR]

The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found. [ABSTRACT FROM AUTHOR]

This paper is devoted to differential invariants of equations w.r.t. point transformations. The natural bundle of these equations and its bundles of k-jets of sections, k=0,1,2,..., are considered. The action of the pseudogroup of all point transformations on these bundles is investigated. Tensor differential invariants distinguishing orbits of this action on jet bundles of second and third orders are constructed. A complete collection of generators and their differential syzygies is obtained for the algebra of all scalar differential invariants of a wide class of considered equations containing generic equations. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudo-group acting on the submanifolds of an analytic manifold. Under the assumption of local freeness of a suitably high order prolongation of the pseudo-group action, we develop computational algorithms for locating a finite generating set of differential invariants, a complete system of recurrence relations for the differentiated invariants, and a finite system of generating differential syzygies among the generating differential invariants. In particular, if the pseudo-group acts transitively on the base manifold, then the algebra of differential invariants is shown to form a rational differential algebra with non-commuting derivations. The essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the usual symbol module associated with the infinitesimal determining system of the pseudo-group, and a new “prolonged symbol module” constructed from the symbols of the annihilators of the prolonged pseudo-group generators. Modulo low order complications, the generating differential invariants and differential syzygies are in one-to-one correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely on combining the moving frame approach developed in earlier papers with Gröbner basis algorithms from commutative algebra. [Copyright &y& Elsevier]

We demonstrate that the invariant operators on a homogeneous space generate differential invariants and invariant differentiation operators. The coordinate-free method of this article makes it possible to simply the computations essentially, namely to reduce them to operations of linear algebra. Some examples are exhibited. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, motions of non-stretching curves in some Klein geometries, determined by the transformation groups acting on , are studied. It is shown that several 1+1-dimensional integrable equations including the KdV, mKdV, defocusing mKdV, Sawada–Kotera, Burgers equations and their hierarchies arise naturally from motions of non-stretching curves in such geometries. [Copyright &y& Elsevier]

In this paper we study the recognition of low-degree polynomial curves based on minimal tactile data. Euclidean differential and semidifferential invariants have been derived for quadratic curves and special cubic curves that are found in applications. These invariants, independent of translation and rotation, are evaluated over the differential geometry at up to three points on a curve. Their values are independent of the evaluation points. Recognition of the curve reduces to invariant verification with its canonical parametric form determined along the way. In addition, the contact locations are found on the curve, thereby localizing it relative to the touch sensor. Simulation results support the method despite numerical errors. Preliminary experiments have also been carried out with the introduction of a method for reliable curvature estimation. The presented work distinguishes itself from traditional model-based recognition in its ability to simultaneously recognize and localize a shape from one of several classes, each consisting of a continuum of shapes, by the use of local data. [ABSTRACT FROM AUTHOR]

We investigate differential invariants for homoclinic tangencies and discuss the role of these invariants in the Hausdorff dimension of invariant sets associated with the tangency and its unfoldings. Further, we give a streamlined proof of a theorem of Goncheko and Shilnikov on the case of a tangency in an area preserving map. Specifically, the invariants determine whether or not a hyperbolic invariant set is formed near the tangency. [ABSTRACT FROM AUTHOR]

The problem of finding all (higher order) differential invariants of immersions f : X → Y, where X and Y are manifolds endowed with metric fields, is investigated. The underlying jet manifolds are introduced, and the action of the corresponding differential groups on them are analysed. It is shown that the differential invariants can be described by means of the factorization of the group action with respect to a distinguished subgroup. By the orbit reduction method a basis of first-order invariants is found. In particular, this basis is used for a characterization of all first-order invariant Lagrangians depending on two metrics and an immersion. [Copyright &y& Elsevier]

We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting on line bundles over smooth manifolds with respect to groups of automorphisms. [ABSTRACT FROM AUTHOR]

We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of automorphisms. [ABSTRACT FROM AUTHOR]