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In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described.

Image segmentation still represents an active area of research since no universal solution can be identified. Traditional image segmentation algorithms are problem-specific and limited in scope. On the other hand, machine learning offers an alternative paradigm where predefined features are combined into different classifiers, providing pixel-level classification and segmentation. However, machine learning only can not address the question as to which features are appropriate for a certain classification problem. The article presents an automated image segmentation and classification platform, called Active Segmentation, which is based on ImageJ. The platform integrates expert domain knowledge, providing partial ground truth, with geometrical feature extraction based on multi-scale signal processing combined with machine learning. The approach in image segmentation is exemplified on the ISBI 2012 image segmentation challenge data set. As a second application we demonstrate whole image classification functionality based on the same principles. The approach is exemplified using the HeLa and HEp-2 data sets. Obtained results indicate that feature space enrichment properly balanced with feature selection functionality can achieve performance comparable to deep learning approaches. In summary, differential geometry can substantially improve the outcome of machine learning since it can enrich the underlying feature space with new geometrical invariant objects.

Orientadores: Carlos Eduardo Durán Fernandez, Marcos Benevenutto Jardim Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica Resumo: A presente tese dá continuidade ao recente trabalho de J.C . Álvarez e C.E. Durán acerca dos invariantes geométricos de uma classe genérica de curvas em variedades de Grassmann, ditas "curvas fanning". Mais precisamente, considera-se como tais curvas de planos lagrangeanos comportam-se mediante uma redução simplética, e conclui-se a existência de dois novos invariantes que desempenham um papel fundamental neste contexto, mais notavelmente a maneira pela qual eles generalizam as bem conhecidas fórmulas de O'Neill para submersões isométricas Abstract: The present thesis gives continuity to the recent work of J.C. Álvarez e C.E. Durán about the geometric invariants of a generic class of curves in the Grassmann manifolds, called "fanning curves". More precisely, we look at how such curves of lagrangean planes behave under a symplectic reduction, and establish the existence of two new invariants which play a fundamental role in that context, more notably the way they generalize the well known O'Neill's formulas for isometric submersions Doutorado Matemática Doutor em Matemática

Orientador: Carlos Eduardo Duran Fernandez Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica Resumo: O presente trabalho tem por objetivo estudar o Método do Referencial Móvel de Cartan aplicado a curvas, através de diversos exemplos, desde problemas simples, passando por publicações dos anos 60 e 70 até artigos recentes. Embora existam teorias gerais para encontrar referenciais de Cartan, optamos por estudar uma forma um pouco mais "artesanal" de construção dos referenciais móveis; a ênfase está na absorção das variadas técnicas e intuições que se adaptam a cada geometria Abstract: The aim of this work is to present the Cartan's Moving Frame Method applied to curves, through several examples, starting with simple problems, going through publications of the 60's, 70's, and up to recent results. Although there are general theories for finding Cartan's moving frames, we chose to study a slightly more "handcraft" way of building the required moving frame; the emphasis being on the absorption of the different techniques and intuitive understanding adapted to each geometry Mestrado Mestre em Matemática

The thesis consists of 6 papers. (1) We calculate the generators of SA₃(ℝ)-invariants for parabolic surfaces. (2) We calculate rigid relative invariants for rigid constant Levi-rank 1 and 2-non-degenerate hypersurfaces in ℂ³: V₀, I₀, Q₀ having 11, 52, 824 monomials in their numerators. (3) We organize all affinely homogeneous nondegenerate surfaces in ℂ³ in inequivalent branches. (4) For a directed harmonic current near a non-hyperbolic linearized singularity which does not give mass to any of the trivial separatrices and whose trivial extension across 0 is ddc-closed, we show that the Lelong number at 0 is: 4.1) strictly positive if the eigenvalue λ>0; 4.2) zero if λ is a negative rational number; 4.3) zero if λ=2. (6) We show that circulant matrices having k ones and k+1 zeros in the first row are always nonsingular when 2k+1 is either a power of a prime, or a product of two distinct primes. For any other integer 2k+1 we exhibit a singular circulant matrix.; La thèse se compose de 6 articles. (1) Nous calculons les générateurs des SA₃(ℝ)-invariants pour les surfaces paraboliques. (2) Nous calculons les invariants rigides relatifs pour les hypersurfaces rigides 2-non-dégénérées de rang de Levi constant 1 dans ℂ³: V₀, I₀, Q₀ ayant 11, 52, 824 monômes au numérateur. (3) Nous organisons tous les modèles affinement homogènes non-dégénérés dans ℂ³ en branches inéquivalentes. (4) Pour un courant harmonique dirigé autour d'une singularité linéarisée non-hyperbolique qui ne charge pas les séparatrices triviales dont l'extension triviale à travers 0 est ddc-fermée, nous démontrons que le nombre de Lelong en 0 est : 4.1) strictement positif si λ>0 ; 4.2) nul si λ est rationnel et négatif ; 4.3) nul si λ est négatif et si T est invariant sous l'action d'un sous-groupe cofini du groupe de monodromie. (5) Nous construisons des fibrés holomorphes en droites en toute dimension n>=2 non-prolongeables au sens de Hartogs. (6) Nous montrons que les matrices circulantes ayant k entrées 1 et k+1 entrées 0 dans leur première rangée sont toujours non singulières lorsque 2k+1 est soit une puissance d'un nombre premier, soit un produit de deux nombres premiers distincts. Pour tout autre entier 2k+1, nous exhibons une matrice circulante singulière.

Due to the principle of minimal information gain, the measurement of points in an affine space V determines a Legendrian submanifold of V&times, V*&times, R. Such Legendrian submanifolds are equipped with additional geometric structures that come from the central moments of the underlying probability distributions and are invariant under the action of the group of affine transformations on V. We investigate the action of this group of affine transformations on Legendrian submanifolds of V&times, R by giving a detailed overview of the structure of the algebra of scalar differential invariants, and we show how the scalar differential invariants can be constructed from the central moments. In the end, we view the results in the context of equilibrium thermodynamics of gases, and notice that the heat capacity is one of the differential invariants.

For the generalized Rapoport-Leas equations, algebra of differential invariants is constructed with respect to point transformations, that is, transformations of independent and dependent variables. The finding of a general transformation of this type reduces to solving an extremely complicated functional equation. Therefore, following the approach of Sophus Lie, we restrict ourselves to the search for infinitesimal transformations which are generated by translations along the trajectories of vector fields. The problem of finding these vector fields reduces to the redefined system decision of linear differential equations with respect to their coefficients. The Rapoport-Leas equations arise in the study of nonlinear filtration processes in porous media, as well as in other areas of natural science: for example, these equations describe various physical phenomena: two-phase filtration in a porous medium, filtration of a polytropic gas, and propagation of heat at nuclear explosion. They are vital topic for research: in recent works of Bibikov, Lychagin, and others, the analysis of the symmetries of the generalized Rapoport-Leas equations has been carried out; finite-dimensional dynamics and conditions of attractors existence have been found. Since the generalized RapoportLeas equations are nonlinear partial differential equations of the second order with two independent variables; the methods of the geometric theory of differential equations are used to study them in this paper. According to this theory differential equations generate subvarieties in the space of jets. This makes it possible to use the apparatus of modern differential geometry to study differential equations. We introduce the concept of admissible transformations, that is, replacements of variables that do not derive equations outside the class of the Rapoport-Leas equations. Such transformations form a Lie group. For this Lie group there are differential invariants that separate its regular orbits, which allow us to classify the generalized Rapoport-Leas equations.

We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem., Comment: Minor revision of the text plus ancillary files are added in this version

Noether's First Theorem guarantees conservation laws provided that the Lagrangian is invariant under a Lie group action. In this paper, via the concept of Killing vector fields and the Minkowski metric, we first construct an important Lie group, known as Hyperbolic Rotation-Translation group. Then, according to Gonçalves and Mansfield's method, we obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for Lagrangians, which are invariant under Hyperbolic Rotation-Translation (or HRT) group action, in the case where the independent variables are not invariant.

In this paper, a family of variable-coefficient fifth-order KdV equations has been considered. By using an infinitesimal method based on the determination of the equivalence group, differential invariants and invariant equations are obtained. Invariants provide an alternative way to find equations from the family which may be equivalent to a specific subclass of the same family and the invertible transformation which maps both equivalent equations. Here, differential invariants are applied to obtain exact solutions.

We investigate a class of multi-dimensional two-component systems of Monge-Amp\`ere type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of normal forms of such systems is obtained. All two-component systems of Monge-Amp\`ere type turn out to be integrable, and can be represented as the commutativity conditions of parameter-dependent vector fields. Geometrically, systems of Monge-Amp\`ere type are associated with linear sections of the Grassmannians. This leads to an invariant differential-geometric characterisation of the Monge-Amp\`ere property., Comment: arXiv admin note: text overlap with arXiv:1503.02274

Source at https://doi.org/10.1016/j.geomphys.2016.05.017. We compute the quotient of the self-duality equation for conformal metrics by the action of the diffeomorphism group. We also determine Hilbert polynomial, counting the number of independent scalar differential invariants depending on the jet-order, and the corresponding Poincaré function. We describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action, resolving the local recognition problem for self-dual conformal structures.

We consider the most general two dimensional linear parabolic equations. Motivated by the recent work of Ibragimov et al. [1-3] we construct differential invariants, semi-invariants and invariant equations. These results are achieved with the employment of the equivalence group admitted by this class of parabolic equations. We derive those variable coefficient equations of this class of linear parabolic equations that can be mapped into constant coefficient equations. Further applications are presented. © 2012 Elsevier B.V. 17 9 3673 3681 Cited By :3

We construct the moduli space of r-jets of Riemannian metrics at a point on a smooth manifold. The construction is closely related to the problem of classification of metric jets via scalar differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratification of moduli spaces is carried out for metrics in dimension n = 2 .

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential invariants., Comment: Journal of Symbolic Computation (2008)

We study the geometry of an important class of generic curves in the Grassmann manifolds of n-dimensional subspaces and Lagrangian subspaces of R2n under the action of the linear and linear symplectic groups.

In this article we study real 2-dimensional surfaces in the Grassmannian of 2-planes in a 4-dimensional vector space. These surfaces occur naturally as the fibers of jet bundles of partial differential equations.On the Grassmannian there is an invariant conformal quadratic form and we will use the structure induced by this quadratic form to study the surfaces. We give a topological classification of compact hyperbolic surfaces similar to the classification by Gluck and Warner [Duke Math. J. 50 (1) (1983)] of compact elliptic surfaces. In contrast with elliptic surfaces there are several topological possibilities for hyperbolic surfaces. We make a calculation of the differential invariants under the action of the group of conformal isometries. Finally, we analyze a class of surfaces called geometrically flat and show that within this class there exist many examples of non-trivial compact surfaces.

In this paper, we consider the class of wave-type equations u tt = f ( x , t , u ) u xx + g ( x , t , u , u x , u t ) and two special cases of it. We derive the equivalence transformations for these equations and using these transformations, we obtain differential invariants and invariant equations. We employ these invariants or/and invariant equations to classify all equations of this general class that can be mapped into a simple linear hyperbolic or into an elliptic equation. Additional applications are presented.

In this paper we consider the general class of hyperbolic equations ux t = F (x, t, u, ux, ut). We use equivalence transformations to derive differential invariants for this class and for two subclasses. Then we employ these invariants to construct equations that can be linearized via local mappings. Further applications of the differential invariants are given. © 2007 Elsevier Inc. All rights reserved. 339 2 762 773 Cited By :8

© 2016, Pleiades Publishing, Ltd. In the present paper we establish the necessary and sufficient conditions for two generalized Abel differential equations to be locally equivalent under the action of the pseudogroup of linear transformations of the form {x ↦ f(x), y ↦ g(x) · y + h(x)}. These conditions are formulated in terms of differential invariants.

A new systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class y′′′ = f(x, y, y′, y′′) under general point equivalence transformations in the generic case.As a result, all third‐order differential invariants, relative and absolute invariant differentiation operators are determined for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative. These relative invariant differentiation operators are used to determine the fourth‐order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex planewith four infinitesimal symmetries for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative, are provided. Scopus

We consider evolution equations of the form ut = f(x, u, ux)uxx + g(x, u, ux) and ut = uxx + g(x, u, ux). In the spirit of the recent work of Ibragimov [Ibragimov NH. Laplace type invariants for parabolic equations. Nonlinear Dynam 2002 28:125-33] who adopted the infinitesimal method for calculating invariants of families of differential equations using the equivalence groups, we apply the method to these equations. We show that the first class admits one differential invariant of order two, while the second class admits three functional independent differential invariants of order three. We use these invariants to determine equations that can be transformed into the linear diffusion equation. © 2006 Elsevier B.V. All rights reserved. 12 7 1133 1145 Cited By :10

International audience; Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zero-dimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.

In this paper we investigate a local structure of geometrical quantities on the Lobachevski plane. This structure is used to describe metric differential invariants on the Lobachevski plane ([1])., В этой работе описываются локальные структуры геометрических величин на плоскости Лобачевского. Это описание содержит как линейные, например тензоры, так и нелинейные геометрические величины и существенным образом используется при нахождении базисных дифференциальных инвариантов ([1])., У цій роботі описуються локальні структури геометричних величин на плоскості Лобачевського. Цей опис містить як лінійні, наприклад тензори, так і нелінійні геометричні величини і істотним чином використовується при знаходженні базисних диференціальних інваріантів ([1]).

Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions.

The classical assumption of differential algebra, differential elimination theory and formal integrability theory is that the derivations do commute. This is the standard case arising from systems of partial differential equations written in terms of the derivations w.r.t. the independant variables. We inspect here the case where the derivations satisfy nontrivial commutation rules. Such a situation arises, for instance, when we consider a system of equations on the differential invariants of a Lie group action. We develop the algebraic foundations for such a situation. They lead to algorithms for completion to formal integrability and differential elimination.

Dette er forfatternes aksepterte versjon. We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f(x, y) and g4(x, y), ..., gd(x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g4, and d − 4 PDEs of the second order with respect to f and g4, ..., gd. For d = 4, this result confirms Blaschke’s conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage.

We consider a general class of third order evolution equations. We construct differential invariants with the employment of the infinitesimal method that using equivalence groups. We use the differential invariants to classify those equations from the class that can be mapped into a specific linear equation. © 2014 Elsevier B.V. 20 2 352 359 Cited By :1

Lagrangian curves in 4-space entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori and determine Lagrangian geodesics., Comment: (04/2013)

We determine all minimal polynomials for second order homogeneous linear differential equations with algebraic solutions decomposed into invariants and we show how easily one can recover the known conditions on differential Galois groups [10, 17, 24] using invariant theory. Applying these conditions and the differential invariants of a differential equation we deduce an alternative method to the algorithms given in [10, 18, 24] for computing Liouvillian solutions. For irreducible second order equations our method determines solutions by formulas in all but three cases.

The endpoint trajectories of human movements fulfill characteristic power laws linking velocityand curvature. The parameters of these power laws typically vary between different segments of longer action sequences. These parameters might thus be exploited for the unsupervised segmentation of actions into movement primitives. For the example of sign language we investigate whether such segments can be identified by Bayesian binning, using a Gaussian observation model whose mean has a polynomial time dependence. We show that this method yields good segmentation and correctly models ground-truth data composed of segments obeying power laws correctly models ground truth kinematics composed of consecutive segments derived from wrist trajectories recorded from users of Israeli Sign Language. Importantly, polynomial orders between 3 and 5 yield an optimal trade-off between complexity and accuracy of the trajectory approximation, in accordance with the minimum acceleration and minimum jerk models.Comparing the orders of the polynomials best approximating natural kinematics against those needed to fit the power lawground truth data suggests that kinematic properties not compatible withpower laws are also not adequately represented by low order polynomials and require higher order polynomials for a good approximation.

The endpoint trajectories of human movements fulfill characteristic power laws linking velocity and curvature. The parameters of these power laws typically vary between different segments of longer action sequences. These parameters might thus be exploited for the unsupervised segmentation of actions into movement primitives. For the example of sign language we investigate whether such segments can be identified by Bayesian binning (BB), using a Gaussian observation model whose mean has a polynomial time dependence. We show that this method yields good segmentation and correctly models ground truth kinematics composed of consecutive segments derived from wrist trajectories recorded from users of Israeli Sign Language (ISL). Importantly, polynomial orders between 3 and 5 yield an optimal trade-off between complexity and accuracy of the trajectory approximation, in accordance with the minimum acceleration and minimum jerk models. Comparing the orders of the polynomials best approximating natural kinematics against those needed to fit the power law ground truth data suggests that kinematic properties not compatible with power laws are also not adequately represented by low order polynomials and require higher order polynomials for a good approximation.

International audience; 2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are "complete" in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization

In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form y″2 + A(x, y, y′)y″ + B(x, y, y′) = 0 to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional distributions. © 2013 Pleiades Publishing, Ltd.

Includes bibliography (p. 125-131). Thesis (Ph. D.) -- University of Cyprus, Faculty of Pure and Applied Sciences, Department of Mathematics and Statistics, October 2008. The University of Cyprus Library holds the printed form of the thesis. Number of sources in the bibliography: 69 Σε αυτή την διατριβή, πρώτα θα αναφερθούμε σε βασικούς ορισμούς των ομάδων μετασχηματισμών Lie, των απειροστών μετασχηματισμών και το αναλλοίωτο των μερικών διαφορικών εξισώσεων, τα οποία είναι χρήσιμα για τα επόμενα κεφάλαια. Καταρχήν, θα ξεκινήσουμε παρουσιάζοντας γνωστά αποτελέσματα. Δηλαδή θα εφαρμόσουμε την απειροστή μέθοδο του Lie για τον υπολογισμό ομάδων ισοδύναμων μετασχηματισμών για την μη γραμμική εξίσωση διάχυσης. Επίσης, θα χρησιμοποιήσουμε την μέθοδο για τον υπολογισμό των αναλλοίωτων συναρτήσεων της γραμμικής υπερβολικής εξίσωσης. Επιπλέον. Θα περιγράψουμε τον τρόπο με τον οποίο ο Ibragimov έλυσε το πρόβλημα του Laplace. Σαν δεύτερο βήμα, ακολουθώντας την ιδέα του Ibragimov, για εύρεση ισοδύναμων μετασχηματισμών, θα υπολογίσουμε ισοδύναμους μετασχηματισμούς για δοσμένες οικογένειες μερικών διαφορικών εξισώσεων. Συγκεκριμένα, θα υπολογίσουμε τις ομάδες ισοδυναμίας για την γενική μορφή της υπερβολικής εξίσωσης και δύο ειδικών περιπτώσεων της. Επίσης, θα υπολογίσουμε τις ομάδες ισοδυναμίας για την υπερβολική εξίσωση διάστασης n, για την κυματική εξίσωση διάστασης n και τέλος για σύστημα που αποτελείται από δύο υπερβολικές εξισώσεις. Γι’ αυτές τις οικογένειες εξισώσεων, βρίσκουμε τις αναλλοίωτες συναρτήσεις πρώτης ή/και δεύτερης τάξης. Σε περισσότερες από αυτές, θα χρησιμοποιήσουμε τις αναλλοίωτες συναρτήσεις ή/και τις αναλλοίωτες εξισώσεις για να βρούμε την μορφή αυτών των εξισώσεων οι οποίες μπορούν να απεικονισθούν σε εξισώσεις συγκεκριμένης μορφής. Τέλος, θα δώσουμε κάποια γνωστά αποτελέσματα που αφορούν τους σημειακούς μετασχηματισμούς μιας μερικής διαφορικής εξίσωσης. Συγκεκριμένα, θα παρουσιάσουμε την μορφή των σημειακών μετασχηματισμών οι οποίοι συνδέουν τρεις ομάδες μερικών διαφορικών εξισώσεων που αποτελούνται από μια εξαρτημένη και δύο ανεξάρτητες μεταβλητές. Σκοπός μας είναι να γενικεύσουμε αυτά τα αποτελέσματα για συστήματα που αποτελούνται από δύο διαφορικές εξισώσεις. Δηλαδή, πρώτα θα παρουσιάσουμε αποτελέσματα που αφορούνε τις σχέσεις των μετασχηματισμών μερικών παραγώγων με τις αρχικές παραγώγους και δεύτερον χρησιμοποιώντας αυτά τα αποτελέσματα θα προσδιορίσουμε τη μορφή των σημειακών μετασχηματισμών που συνδέουν συγκεκριμένα συστήματα μερικών διαφορικών εξισώσεων. In this thesis, firstly we develop the basic concepts of Lie groups of transformations, infinitesimal transformations and invariance of partial differential equations that are necessary in the following chapters. In the beginning we start with known results. That is, we use the Lie infinitesimal method for calculating the continuous group of equivalence transformations, for the non-linear diffusion equation. Also, we apply this method to derive differential invariants for the linear hyperbolic equation in two variables. Finally, we describe the method which used by Ibragimov to solve the Laplace problem. The second step is to calculate equivalence transformations for given families of equations. In the spirit of the recent work of Ibragimov, who adopted the infinitesimal method for calculation of invariants of families of differential equations using the infinitesimal groups, we apply the method to several partial differential equations. In this thesis, we derive the equivalence group for hyperbolic equations of general class and for two special cases of it. Also, we calculate equivalence transformations for n-dimensional hyperbolic equations, for n-dimensional wave-type equations and finally for hyperbolic equations with two dependent variables. For these families of equations, we find the forms of differential invariants of first or/and second order. In certain cases, we will use the derived invariants or/and invariant equations to find the form of those equations that can be mapped into an equation with particular form. Furthermore, we work on form-preserving point transformations for partial differential equations. We present some known results for three classes of equations restricted to one dependent variable and two independent variables concerning the nature of connecting point transformations. We will generalize these results for forms of point transformations connecting two systems of two partial differential equations. The aim of this part is first to present results concerning the relation of the transformed partial derivatives to the original partial derivatives and secondly to exploit these results to reduce the general range of point transformations connecting systems of two partial differential equations belonging to restricted classes.

These notes start with an introduction to differential invariants. They continue with an algebraic treatment of the theory. The algebraic, differential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail. We expose the recent results on the topic of rational and algebraic differential invariants. Finally we give a new algebraic version of the finiteness theorem of Lie–Tresse for the case of finite dimensional algebraic groups.

This paper studies relationships between the order reductions of ordinary differential equations derived by the existence of $\lambda$-symmetries, telescopic vector fields and some nonlocal symmetries obtained by embedding the equation in an auxiliary system. The results let us connect such nonlocal symmetries with approaches that had been previously introduced: the exponential vector fields and the $\lambda$-coverings method. The $\lambda$-symmetry approach let us characterize the nonlocal symmetries that are useful to reduce the order and provides an alternative method of computation that involves less unknowns. The notion of equivalent $\lambda$-symmetries is used to decide whether or not reductions associated to two nonlocal symmetries are strictly different.

The pseudogroup defined by B. Malgrange as a generalization of the differential Galois group for nonlinear differential equation is presented. It is proved that a equation integrable by quadratures has a solvable pseudgroup. From this a new proof of a theorem of M. Singer is given.

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws. © 2010 Birkhäuser Verlag Basel/Switzerland. 61 5 793 809 Cited By :5

Includes bibliography (p. 125-131). Thesis (Ph. D.) -- University of Cyprus, Faculty of Pure and Applied Sciences, Department of Mathematics and Statistics, October 2008. The University of Cyprus Library holds the printed form of the thesis. Number of sources in the bibliography: 69 Σε αυτή την διατριβή, πρώτα θα αναφερθούμε σε βασικούς ορισμούς των ομάδων μετασχηματισμών Lie, των απειροστών μετασχηματισμών και το αναλλοίωτο των μερικών διαφορικών εξισώσεων, τα οποία είναι χρήσιμα για τα επόμενα κεφάλαια. Καταρχήν, θα ξεκινήσουμε παρουσιάζοντας γνωστά αποτελέσματα. Δηλαδή θα εφαρμόσουμε την απειροστή μέθοδο του Lie για τον υπολογισμό ομάδων ισοδύναμων μετασχηματισμών για την μη γραμμική εξίσωση διάχυσης. Επίσης, θα χρησιμοποιήσουμε την μέθοδο για τον υπολογισμό των αναλλοίωτων συναρτήσεων της γραμμικής υπερβολικής εξίσωσης. Επιπλέον. Θα περιγράψουμε τον τρόπο με τον οποίο ο Ibragimov έλυσε το πρόβλημα του Laplace. Σαν δεύτερο βήμα, ακολουθώντας την ιδέα του Ibragimov, για εύρεση ισοδύναμων μετασχηματισμών, θα υπολογίσουμε ισοδύναμους μετασχηματισμούς για δοσμένες οικογένειες μερικών διαφορικών εξισώσεων. Συγκεκριμένα, θα υπολογίσουμε τις ομάδες ισοδυναμίας για την γενική μορφή της υπερβολικής εξίσωσης και δύο ειδικών περιπτώσεων της. Επίσης, θα υπολογίσουμε τις ομάδες ισοδυναμίας για την υπερβολική εξίσωση διάστασης n, για την κυματική εξίσωση διάστασης n και τέλος για σύστημα που αποτελείται από δύο υπερβολικές εξισώσεις. Γι’ αυτές τις οικογένειες εξισώσεων, βρίσκουμε τις αναλλοίωτες συναρτήσεις πρώτης ή/και δεύτερης τάξης. Σε περισσότερες από αυτές, θα χρησιμοποιήσουμε τις αναλλοίωτες συναρτήσεις ή/και τις αναλλοίωτες εξισώσεις για να βρούμε την μορφή αυτών των εξισώσεων οι οποίες μπορούν να απεικονισθούν σε εξισώσεις συγκεκριμένης μορφής. Τέλος, θα δώσουμε κάποια γνωστά αποτελέσματα που αφορούν τους σημειακούς μετασχηματισμούς μιας μερικής διαφορικής εξίσωσης. Συγκεκριμένα, θα παρουσιάσουμε την μορφή των σημειακών μετασχηματισμών οι οποίοι συνδέουν τρεις ομάδες μερικών διαφορικών εξισώσεων που αποτελούνται από μια εξαρτημένη και δύο ανεξάρτητες μεταβλητές. Σκοπός μας είναι να γενικεύσουμε αυτά τα αποτελέσματα για συστήματα που αποτελούνται από δύο διαφορικές εξισώσεις. Δηλαδή, πρώτα θα παρουσιάσουμε αποτελέσματα που αφορούνε τις σχέσεις των μετασχηματισμών μερικών παραγώγων με τις αρχικές παραγώγους και δεύτερον χρησιμοποιώντας αυτά τα αποτελέσματα θα προσδιορίσουμε τη μορφή των σημειακών μετασχηματισμών που συνδέουν συγκεκριμένα συστήματα μερικών διαφορικών εξισώσεων. In this thesis, firstly we develop the basic concepts of Lie groups of transformations, infinitesimal transformations and invariance of partial differential equations that are necessary in the following chapters. In the beginning we start with known results. That is, we use the Lie infinitesimal method for calculating the continuous group of equivalence transformations, for the non-linear diffusion equation. Also, we apply this method to derive differential invariants for the linear hyperbolic equation in two variables. Finally, we describe the method which used by Ibragimov to solve the Laplace problem. The second step is to calculate equivalence transformations for given families of equations. In the spirit of the recent work of Ibragimov, who adopted the infinitesimal method for calculation of invariants of families of differential equations using the infinitesimal groups, we apply the method to several partial differential equations. In this thesis, we derive the equivalence group for hyperbolic equations of general class and for two special cases of it. Also, we calculate equivalence transformations for n-dimensional hyperbolic equations, for n-dimensional wave-type equations and finally for hyperbolic equations with two dependent variables. For these families of equations, we find the forms of differential invariants of first or/and second order. In certain cases, we will use the derived invariants or/and invariant equations to find the form of those equations that can be mapped into an equation with particular form. Furthermore, we work on form-preserving point transformations for partial differential equations. We present some known results for three classes of equations restricted to one dependent variable and two independent variables concerning the nature of connecting point transformations. We will generalize these results for forms of point transformations connecting two systems of two partial differential equations. The aim of this part is first to present results concerning the relation of the transformed partial derivatives to the original partial derivatives and secondly to exploit these results to reduce the general range of point transformations connecting systems of two partial differential equations belonging to restricted classes.

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern–Gauss–Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in the first two papers in this series, see Alexakis (2007, 2009) [2] , [3] . These results may be of independent interest, applicable to related problems.

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equa- tions. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered.

We consider linear hyperbolic equations of the form u t t = ∑ i = 1 n u x i x i + ∑ i = 1 n X i ( x 1 , … , x n , t ) u x i + T ( x 1 , … , x n , t ) u t + U ( x 1 , … , x n , t ) u . We derive equivalence transformations which are used to obtain differential invariants for the cases n = 2 and n = 3 . Motivated by these results, we present the general results for the n-dimensional case. It appears (at least for n = 2 ) that this class of hyperbolic equations admits differential invariants of order one, but not of order two. We employ the derived invariants to construct interesting mappings between equivalent equations.

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.

In this article we study real 2-dimensional surfaces in the Grassmannian of 2-planes in a 4-dimensional vector space. These surfaces occur naturally as the fibers of jet bundles of partial differential equations. On the Grassmannian there is an invariant conformal quadratic form and we will use the structure induced by this quadratic form to study the surfaces. We give a topological classification of compact hyperbolic surfaces similar to the classification by Gluck and Warner [Duke Math. J. 50 (1) (1983)] of compact elliptic surfaces. In contrast with elliptic surfaces there are several topological possibilities for hyperbolic surfaces. We make a calculation of the differential invariants under the action of the group of conformal isometries. Finally, we analyze a class of surfaces called geometrically flat and show that within this class there exist many examples of non-trivial compact surfaces.

In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection., Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined ''subsymbol'' of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.