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We observed the existence of periodic orbits in immune network under transitive solvable Lie algebra. In this article, we focus to develop condition of maximal Lie algebra for immune network model and use that condition to construct a vector field of symmetry to study nonlinear pathogen model. We used two methods to obtain analytical structure of solution, namely normal generator and differential invariant function. Numerical simulation of analytical structure exhibits correlated periodic pattern growth under spatiotemporal symmetry, which is similar to the linear dynamical simulation result. We used Lie algebraic method to understand correlation between growth pattern and symmetry of dynamical system. We employ idea of using one parameter point group of transformation of variables under which linear manifold is retained. In procedure, we present the method of deriving Lie point symmetries, the calculation of the first integral and the invariant solution for the ordinary differential equation (ODE). We show the connection between symmetries and differential invariant solutions of the ODE. The analytical structure of the solution exhibits periodic behavior around attractor in local domain, same behavior obtained through dynamical analysis.

In this work, ruled surfaces in 3-dimensional Riemannian manifolds are studied. We determine the expressions for the extrinsic and sectional curvatures of a parametrized ruled surface, where the former one is shown to be non-positive. We also quantify the set of ruling vector fields along a given base curve which allows us to define a relevant reference frame that we refer to as Sannia frame. The fundamental theorem of existence and equivalence of Sannia ruled surfaces in terms of a system of invariants is given. The second part of the article tackles the concept of the striction curve, which is proven to be the set of points where the so-called Jacobi evolution function vanishes on a ruled surface. This characterisation of striction curves provides independent proof for their existence and uniqueness in space forms and disproves their existence or uniqueness in some other cases. [ABSTRACT FROM AUTHOR]

In this paper, we find necessary and sufficient conditions for existence of a transformation of independent spatial variables that transforms the Fokker–Planck equation to an equation with constant coefficients. Using these conditions, we calculate explicit solutions for two-dimensional Fokker–Planck equations. Our motivation comes from applications in image processing, where the Fokker–Planck equation typically describes blurring processes. [ABSTRACT FROM AUTHOR]

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications. [ABSTRACT FROM AUTHOR]

In the paper, we consider the problem of describing the general structure of differential invariants for transformation groups that act freely and regularly. We formulate two theorems describing the structures of differential invariants for intransitive and transitive free actions, respectively. In both cases it is shown that the differential invariants can be expressed in terms of the symbols of right-invariant vector fields. Finally, we discuss prospects for solving the problem considered for more general group actions. [ABSTRACT FROM AUTHOR]

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

In this paper, a point-set similarity model (PSM) is introduced, which provides a new elastic point set registration frame by using similarity invariant signature curves. According to the potential correspondences obtained by PSM, the similarity transformations between the point sets could be estimated. Experiments on synthetic data and real data of coronary artery show that our method offers certain advantages comparing with typical methods. [ABSTRACT FROM AUTHOR]

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]

We consider the differential equation that describes filtration of real gases obeying Darcy's law in presence of gravitation. We provide the group classification of this equation, find exact solutions for one of the equations obtained by symmetry reduction, construct conservation laws, and describe the algebras of differential invariants. [ABSTRACT FROM AUTHOR]

We construct a basis of invariants for the set of second-order matrices consisting of the original matrix and its derivatives. It is shown that the presence of derivatives imposes connections on the elements of this set and reduces the number of elements in the basis compared to the purely algebraic case. Formulas for calculating algebraic invariants of such a set are proved. We state a generalization of Fricke's formulas in terms of the traces of the product of matrices in this set. [ABSTRACT FROM AUTHOR]

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. Then for the ruled surfaces in three dimensional Euclidean space we describe their geometric structures and obtain total classifications. Since the ruled surfaces are the simplest foliated submanifolds, our effective and elementary methods can be used to reveal the properties and structures of the Riemannian foliations and foliated submanifolds. [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.

In this paper, we study scalar the forth order linear differential operators over an oriented 2-dimensional manifold. We investigate differential invariants of these operators and show their application to the equivalence problem. [ABSTRACT FROM AUTHOR]

In this paper, a variable-coefficient cubic–quintic nonlinear Schrödinger equation involving five arbitrary real functions of space and time is analyzed from the point of view of symmetry analysis by using Lie's invariance infinitesimal criterion. The infinitesimal generators of corresponding equivalence transformations are presented. The first-order differential invariants are constructed to identify when the equation can be mapped to a constant-coefficient cubic–quintic nonlinear Schrödinger equation. The constrained conditions on the variable coefficients we arrived here extend the cases discussed before and present more general results. Some brightlike and darklike solitary wave solutions for special potentials and cubic–quintic nonlinearities are obtained. [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]

In this paper, we present an explicit description of finite transcendence bases in the differential field of differential rational functions that are invariant under the action of the Galilean transformation group in a real finite-dimensional space. Necessary and sufficient conditions of the equivalence of paths in the n-dimensional Galilean space are obtained. [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]

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [11]) in 1977 that an affine maximal graph of a smooth and locally uniformly convex function on two-dimensional Euclidean space R 2 must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term "affine geometry" refers to "equi-affine geometry".) A natural problem arises: Whether the hyperbola is a general-affine maximal curve in R 2 ? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in R 2 , and show the general-affine maximal curves in R 2 are much more abundant and include the explicit curves y = x α (α is a constant and α ∉ { 0 , 1 , 1 2 , 2 }) and y = x log ⁡ x. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with GA (n) = GL (n) ⋉ R n. Moreover, in general-affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for general-affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this general-affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the general-affine heat flow is proved. [ABSTRACT FROM AUTHOR]

Single-View Geometry (SVG) studies the world-to-image mapping or warp, which is the relationship that exists between a body's model and its image. For a rigid body observed by a projective camera, the warp is described by the usual camera matrix and its properties. However, it is clear that for a body whose deformation state changes between the body's model and its image, the 'simple,' globally parameterized warp described solely by the camera matrix, breaks down. Existing work has exploited deformation to reconstruct the deformed body from its image, but did not establish the properties of the deformable warp. Studying these properties is part of deformable SVG and forms a recent research topic. Because deformations may take place anywhere in the object's body, and because they may be uncorrelated, the warp is local in nature. Using a differential framework is thus an obvious choice. We propose a differential–algebraic projective framework based on modeling the body's surface by a locally rational projective embedding and on the 1D projective camera. We show that this leads, via the study of univariate rational functions, to differential invariants that the warp must satisfy. It may seem surprising, given the generic hypothesis made on the observed body, hardly stronger than mere local smoothness, that constraints can still be found. Our framework generalizes the Schwarzian derivative, the first-order projective differential invariant, which holds under the assumption that the body's shape is locally linear. Our invariants may be used to construct regularizers to be used in warp estimation. We report experimental results of two types on simulated and real data. The first type shows that the proposed invariants hold well for an independently estimated warp. The second type shows that the proposed regularizers improve warp estimation from point correspondences compared to the classical derivative-penalizing regularizers. [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]

Let G be a subgroup in the group of all invertible linear transformations of a finite-dimensional real space X. One of the problems in differential geometry is that of finding easily verified necessary and sufficient conditions for G-equivalence of paths in X. In solving this problem, we use methods of the theory of differential invariants which give descriptions of transcendence bases of differential fields of G-invariant differential rational functions. Having explicit forms of transcendence bases, we can obtain efficient criteria for G-equivalence of paths with respect to actions of the special linear, orthogonal, pseudoorthogonal, and symplectic groups. We present a description of one finite transcendence basis in the differential field of differential rational functions invariant with respect to the action of the pseudo-Galilean group ΓO. Based on this, we establish necessary and sufficient conditions of ΓO-equivalence of paths. [ABSTRACT FROM AUTHOR]

Abstract In this paper we suggest an approach to study actions of semisimple (or reductive) algebraic groups in their algebraic complex representations. We use differential-geometric methods instead of classical algebraic constructions. Namely, according to Borel–Weil–Bott theorem, every algebraic representation of semisimple algebraic group is isomorphic to the action of this group on the module of holomorphic sections of some reductive bundle over homogeneous space. Using this, we give a complete description of the field of differential invariants for this action and obtain a criterion, which separates regular orbits. [ABSTRACT FROM AUTHOR]

In the present work we study linear ordinary differential equations of second order with rational coefficients. Such equations are very important in complex analysis (for example they include the so-called Fuchs equations, which appear in 21 Hilbert's problem) and in the theory of special functions (Bessel function, Gauss hypergeometric function, etc.). We compute the symmetry group of this class of equations, and it appears that this group includes non-rational (and even nonalgebraic) transformations. Also, the field of differential invariants is described and the effective equivalence criterion is obtained. Finally, we present some examples. [ABSTRACT FROM AUTHOR]

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kotera flow in the affine geometry, etc. Using the fact that two different curves in a given geometry are governed by the same integrable equation, we obtain Bäcklund transformations relating to these two integrable geometric flows. Some special solutions of the integrable systems are used to obtain the explicit Bäcklund transformations.

We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer. [ABSTRACT FROM AUTHOR]