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Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring S = k [ x 1 , ... , x n ]. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential S-modules , which are natural generalizations of chain complexes. We prove several results that lend evidence in support of this conjecture, including a categorical pairing between the derived categories of graded differential S -modules and coherent sheaves on P n − 1 and a proof of the conjecture in the case where S = k [ t ]. [ABSTRACT FROM AUTHOR]

Three-dimensional Direct Numerical Simulations (DNS) of turbulent premixed lean H 2 -air flames are performed for different equivalence ratios and initial turbulence intensities to investigate the influence of differential diffusion on the invariants of the velocity gradient tensor and local flow topologies. It is found that the effects of differential diffusion strengthen with a decrease in equivalence ratio. This difference results in more frequent super-adiabatic temperatures and higher positive dilatation rates for smaller values of equivalence ratio, having a significant impact on the flow topology distributions and their contributions to the scalar-turbulence interaction and vortex-stretching terms. The topologies associated with positive dilatation rates tend to occur at the curvatures associated with super-adiabatic temperatures for highly lean H 2 -air premixed flames. However, this tendency weakens with increasing equivalence ratio. These findings suggest that the chemical timescale must be accounted for in the modelling of viscous and scalar dissipation rates in lean hydrogen-air premixed flames. • Statistically planar flame DNS of equivalence ratios of 0.4 and 0.7 considered. • Effects of equivalence ratio on flow topology distributions analysed. • The non-unity Lewis number effects strengthen with deceasing equivalence ratio. • Dilatation effects strengthen with deceasing equivalence ratio. • Topologies specific to positive dilatation strengthen for small equivalence ratio. [ABSTRACT FROM AUTHOR]

We work within the framework of the variational bicomplex: we define A ∞ -algebra structures on horizontal and vertical cohomologies of (formally integrable) partial differential equations with the help of Merkulov's theorem. Since higher order A ∞ -algebra operations are related to Massey products, our observation implies the existence of invariants for differential equations that go beyond conservation laws. We also propose notions of formality for PDEs, and we present examples of formal equations. [ABSTRACT FROM AUTHOR]

In this paper, we give a description of differential operators on tensor products A ⊗ 핂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 핂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 핂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B, respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , ퟎ ) = ( V ⁢ ( f ) , ퟎ ) ⊆ ( ℂ r , ퟎ ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n. [ABSTRACT FROM AUTHOR]

In this paper, we provided, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e., Gauss curvature and mean curvature. In practice, the algorithm works well for sparse parametrizations (e.g., toric surfaces) and PN surfaces. Additionally, we provided a specific, and symbolic, algorithm for computing the symmetries of ruled surfaces. This algorithm works extremely well in practice, since the problem is reduced to that of rational space curves, which can be efficiently solved by using existing methods. The algorithm for ruled surfaces is based on the fact, proven in the paper, that every symmetry of a rational surface must also be a symmetry of its line of striction , which is a rational space curve. The algorithms have been implemented in the computer algebra system Maple, and the implementations have been made public. Evidence of their performance is given in the paper. [ABSTRACT FROM AUTHOR]

The Ames room illusion is one of the best-known geometrical illusions but its geometrical properties are often misunderstood. This study discusses the differences in the geometrical properties between the original Ames room and what have been often referred to as "Ames rooms" in recent studies. [ABSTRACT FROM AUTHOR]

The Bézier curves submitted by Pierre Bézier in the mid-1950s are splendid differential geometric structures. The paper compares a mechanical–theoretical curve (classical-conventional) with a computer-based artificial neural network (modern). While the reader witnesses hourly TEC (TECU) modeling by the class C0 Bézier structure for the first time, (s)he has the opportunity to evaluate the reliable results of segmented-continuous curves with network. The Bézier structure is established by operating the differential geometrical invariants. The yielding and accuracy of the models are evaluated with the R correlation coefficient and the mean squared error. The discussion demonstrates the harmony of the two different approaches by modeling the 365-day TEC map of 2017. Then, in particular, the outputs of the intense geomagnetic storms of 28 May (Dst = − 125 nT) and 8 September (Dst = − 122 nT) are modeled and compared. Bézier curves are read segmental-continuous every 12-h TEC map in all discussions. The network, on the other hand, employs solar wind parameters to model the TEC atlas. The models obey rigorously the causality principle. The results of the discussion display that the R score reaches around 93% and 98.8% for Bézier curves and the neural network, respectively. Again, it is noticed that the mean squared error of the network model decreases to 1.1308 TECU. The curve model emerges to the reader as an alternative to neural networks in projections of space climate conditions. [ABSTRACT FROM AUTHOR]

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function ζ (s) = Tr L − s and the heat trace Θ (t) = Tr exp (− t L). The kernel U (t ; x , x ′) of the heat semigroup exp (− t L) , called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as Tr f (t L) , that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as Tr exp (− t L +) exp (− s L −) , that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods. [ABSTRACT FROM AUTHOR]

We study a generalization of the Harish-Chandra--Itzykson--Zuber integral to tensors and its expansion in terms of trace invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which count certain families of constellations. We find an expression of these numbers in terms of monotone simple Hurwitz numbers, thereby also providing expressions for monotone double Hurwitz numbers of arbitrary genus in terms of the single ones. We give an interpretation of the different combinatorial quantities at play in terms of enumeration of nodal surfaces. In particular, our generalization of Hurwitz numbers is shown to count certain isomorphism classes of branched coverings of a bouquet of D 2-spheres that touch at one common non-branch node. [ABSTRACT FROM AUTHOR]

We present a new and efficient method to detect whether or not two given rational plane curves are similar. If both curves are the same, the method finds the symmetries of the curve. The method relies on the introduction of a complex differential invariant that has a nice behavior with respect to Möbius transformations, which are the mappings lying behind the similarities in the parameter space. From a computational point of view, our algorithm only requires bivariate gcds and factoring. The algorithm is implemented in Maple™ [Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario (2021)]. An extensive study of its practical performance is also provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mass-critical nonlinear Schrödinger equation in one dimension. Ogawa-Tsutsumi [Proc. Amer. Math. Soc. 111 (1991), no. 2, 487-496] proved a blow-up result for negative energy solution by using a scaling argument for initial data. In general, a equation with a linear potential does not have a scale invariant, so the method by Ogawa-Tsutsumi cannot be used directly to that. In this paper, we prove a blow-up result for the equation with the linear potential by modifying the argument of Ogawa-Tsutsumi. [ABSTRACT FROM AUTHOR]

The Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau–Ginzburg (LG) models — and thus, in particular, as periods. We also construct an Apéry motive, whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard–Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG‐models mirror to several Fano threefolds. By describing the "elementary" Apéry numbers in terms of regulators of higher cycles (i.e., algebraic K$K$‐theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG‐models are modular families of K3$K3$ surfaces, and the distinction between multiples of ζ(2)$\zeta (2)$ and ζ(3)$\zeta (3)$ (or (2πi)3$(2\pi \mathbf {i})^3$) translates ultimately into one between algebraic K1$K_1$ and K3$K_3$ of the family. [ABSTRACT FROM AUTHOR]

The concept of the Jacobi multiplier for ordinary differential equations up to the second order is reviewed and its connection with classical methods of canonical variables and differential invariants is established. We express, for equations of the second order, the Jacobi multiplier in terms of integrating factors for reduced equations of the first order. We also investigate, from a symmetry point of view, how two different systems with the same Jacobi multiplier are interrelated. As a result, we determine the conditions when such systems admit the same two-dimensional Lie algebra of symmetries. Several illustrative examples are given. [ABSTRACT FROM AUTHOR]

This article pays tribute to Professor Mihail Popa, a highly regarded mathematician and member of ROMAI. It provides a concise overview of his life and career, focusing on his contributions to the study of differential equations. Professor Popa's research centered on the qualitative analysis of differential equations, utilizing the theory of invariants developed by his mentor, Academician Constantin Sibirschi, and mathematician Sophus Lie. He made significant advancements in this field, including solving the "center and focus" problem and determining the upper bound of Poincaré-Lyapunov constants. Additionally, Professor Popa mentored numerous successful mathematicians and has been recognized for his work through various academic honors. The article offers a respectful and informative summary of Professor Popa's achievements in mathematics. [Extracted from the article]

In this paper, we study the algebraic structure of differential invariants of a fifth-order KdV equation, known as the Kawahara KdV equation. Using the moving frames method, we locate a finite generating set of differential invariants, recurrence relations, and syzygies among the differential invariants generators of the equation. We prove that the differential invariant algebra of the equation can be generated by two first-order differential invariants. [ABSTRACT FROM AUTHOR]

In this paper, we present a general conservative eighth-order compact finite difference scheme for solving the coupled Schrödinger-KdV equations numerically. The proposed scheme is decoupled and preserves several physical invariants in discrete sense. The matrices obtained in the eighth-order compact scheme are all circulant symmetric positive definite so that it can be used to solve other similar equations. Numerical experiments on model problems show the better performance of the scheme compared with other numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the one-dimensional nonlinear Klein-Gordon equation. Noether's First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the Klein-Gordon equation, we first present a Lagrangian whose Euler-Lagrange equation is the Klein-Gordon equation, and then according to Goncalves and Mansfield's method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action. [ABSTRACT FROM AUTHOR]

The A ^ 2 n (2) -hierarchy can be constructed from a splitting of the Kac–Moody algebra of type A ^ 2 n (1) by an involution. By choosing certain cross section of the gauge action, we obtain the A ^ 2 n (2) -KdV hierarchy. They are the equations for geometric invariants of isotropic curve flows of type A, which gives a geometric interpretation of the soliton hierarchy. In this paper, we construct Darboux and Bäcklund transformations for the A ^ 2 n (2) -hierarchy, and use it the construct Darboux transformations for the A ^ 2 n (2) -KdV hierarchy and isotropic curve flows of type A. Moreover, explicit soliton solutions for these hierarchies are given. [ABSTRACT FROM AUTHOR]

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action and the size of a curve's symmetry group. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to different types of noise. [ABSTRACT FROM AUTHOR]

The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon–Heiles system and a Lotka–Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network. [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]

Higher-order time integration methods that unconditionally preserve the positivity and linear invariants of the underlying differential equation system cannot belong to the class of general linear methods. This poses a major challenge for the stability analysis of such methods since the new iterate depends nonlinearly on the current iterate. Moreover, for linear systems, the existence of linear invariants is always associated with zero eigenvalues, so that steady states of the continuous problem become nonhyperbolic fixed points of the numerical time integration scheme. Altogether, the stability analysis of such methods requires the investigation of nonhyperbolic fixed points for general nonlinear iterations. Based on the center manifold theory for maps we present a theorem for the analysis of the stability of nonhyperbolic fixed points of time integration schemes applied to problems whose steady states form a subspace. This theorem provides sufficient conditions for both the stability of the method and the local convergence of the iterates to the steady state of the underlying initial value problem. This theorem is then used to prove the unconditional stability of the MPRK22(ff)-family of modified Patankar-Runge-Kutta schemes when applied to arbitrary positive and conservative linear systems of differential equations. The theoretical results are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

For a polarized abelian variety, Z. Jiang and G. Pareschi introduced an invariant and showed that the polarization is basepoint-free or projectively normal if the invariant is small. Their result was generalized to higher syzygies by F. Caucci; that is, the polarization satisfies property (Np) if the invariant is small. In this paper, we study a relation between the invariant and degrees of abelian subvarieties with respect to the polarization. For abelian threefolds, we give an upper bound of the invariant using degrees of abelian subvarieties. In particular, we affirmatively answer some questions on abelian varieties asked by the author, V. Lozovanu and F. Caucci in the three-dimensional case. [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]

In this paper, we study the decomposition of the direct image of π+(OX), the polynomial ring OX as a D-module, under the map π: specOX ! specOG(r;n) X, where OG(r;n) X is the ring of invariant polynomials under the action of the wreath product G(r, p):= Z/rZ o Sn. We first describe the generators of the simple components of π+(OX) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of π. Furthermore, we study the action of invariants differential operators on the higher Specht polynomials. [ABSTRACT FROM AUTHOR]

Let M be a smooth manifold of dimension 2n, and let OM be the dense open subbundle in ∧2T*M of 2-covectors of maximal rank. The algebra of Diff M -invariant smooth functions of first order on OM is proved to be isomorphic to the algebra of smooth Sp(Ωx)-invariant functions on , x being a fixed point in M , and Ωx a fixed element in (OM)x. The maximum number of functionally independent invariants is computed. [ABSTRACT FROM AUTHOR]

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.

Firstly, applying Lie's elementary theory for appropriate prolongations to jet spaces of orders 1and 2, we show that any real-analytic 2-nondegenerate constant Levi rank 1hypersurface M5 C³ carries two sorts of Cartan-Moser chains, that are of orders 1and 2. Secondly, integrating and straightening any given order 2 chain γ passing through any point p ∈ M to be the v-axis in coordinates (z, ζ, w = u + iv), centered at p, without setting up the formal theory in advance, we show that there exists a convergent change of complex coordinates (z, ζ, w) → (z', ζ', w') fixing the origin in which γ is the v-axis and in which M has Poincaré-Moser reduced equation of a specific shape. Thirdly, starting from an M having preliminary normalized equation ... assigning weights [z]: =1, [ζ] : = 0, [w] : = 2, we show that a normalizing biholomorphism exists and is unique when assumed of the form ... The values at the origin of Pocchiola's two primary Cartan-type relative differential invariants are W0 = 4 F3,0,0,2(0) and J0 = 20 F5,0,0,1(0). The proofs are detailed, accessible to non-experts. The computer-generated aspects (forthcoming) have been reduced to a minimum here. [ABSTRACT FROM AUTHOR]

We present a survey of development of the concept of hidden symmetry in the field of partial differential equations, including a series of results previously obtained by the author. We also add new examples of the classes of equations with hidden symmetry of type II and explain the nature of the earlier established nonclassical symmetry of some equations. We suggest a constructive algorithm for the description of the classes of equations, which have specified conditional or hidden symmetry and/or can be reduced to equations with smaller number of independent variables by using a specific ansatz. We consider reductions existing due to the presence of Lie and conditional symmetry and also of the hidden symmetry of type II. We also discuss relationships between the concepts of hidden and conditional symmetry. It is shown that the hidden symmetry of type II earlier regarded as a separate type of non-Lie symmetry is caused, in fact, by the nontrivial Q-conditional symmetry of the reduced equations. The proposed approach enables us not only to find hidden symmetry and new reductions of the well-known equations but also to describe a general form of equations with given Q-conditional and type-II hidden symmetry. As an example, we describe the general classes of equations with hidden and conditional symmetry under rotations in the Lorentz and Euclid groups for which the corresponding hidden and conditional symmetry allows their reduction to radial equations with smaller number of independent variables. [ABSTRACT FROM AUTHOR]

In this paper, differential invariants of null Cartan curves are studied in (n+2) dimensional Lorentzian similarity geometry. The fundamental theorem for a null Cartan curve in similarity geometry is investigated and the characterization of all self-similar null Cartan curves parameterized by de Sitter parameter in Minkowski space-time is given. [ABSTRACT FROM AUTHOR]

The classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web. In the present paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [5], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants. [ABSTRACT FROM AUTHOR]

In 2010, B. Balsam and A. Kirillov, Jr. showed that the Turaev-Viro invariants defined for a spherical fusion category A extends to invariants of 3-manifolds with corners. In 2021, A. Kirillov, Jr. described an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) using the space of "stringnets with boundary conditions" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory by L. Crane and D. Yetter (1993). [ABSTRACT FROM AUTHOR]

In this paper, using the compatible theory of differential invariants, a class of exact solutions is obtained for nonhomogeneous quasilinear hyperbolic system of partial differential equations (PDEs) describing rate type materials; these solutions exhibit genuine nonlinearity that leads to the formation of discontinuities such as shocks and rarefaction waves. For certain nonconstant and smooth initial data, the solution to the Riemann problem is presented providing a complete characterization of the solutions. [ABSTRACT FROM AUTHOR]

To any completely integrable second-order system of real or complex partial differential equations: y x k 1 x k 2 = F k 1 , k 2 x 1 , ⋯ , x n , y , y x 1 , ... , y x n with 1 ⩽ k 1 , k 2 ⩽ n and with F k 1 , k 2 = F k 2 , k 1 in n ⩾ 2 ̲ independent variables (x 1 , ... , x n) and in one dependent variable y, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials to the associated submanifold of solutions, what the vanishing of the Hachtroudi curvature gives can be precisely translated to characterize when both families of Segre varieties and of conjugate Segre varieties associated to a Levi nondegenerate real analytic hypersurface M in C n ( n ⩾ 3 ) can be straightened to be affine complex (conjugate) lines. In continuation to a previous paper devoted to the quite distinct C 2 -case, this then characterizes in an effective way those hypersurfaces of C n + 1 in higher complex dimension n + 1 ⩾ 3 that are locally biholomorphic to a piece of the (2 n + 1) -dimensional Heisenberg quadric, without any special assumption on their defining equations. [ABSTRACT FROM AUTHOR]

Singularity is one of the important features in invariant structures in several physical phenomena reflected often in the associated invariant differential equations. The classification problem for singular differential invariants in (1+3)-dimensional space associated with Lie algebras of dimension 4 is investigated. The formulation of singular invariants for a Lie algebra of dimension 4 possessed by the underlying system of three second-order ordinary differential equations is studied in detail and the corresponding canonical forms for these systems are deduced. Furthermore, the categorization of singular invariants on the basis of conditional singularity, weak uncoupling, weak linearization, partial uncoupling and partial linearization are described for the underlying canonical forms. In addition, those cases of classified canonical forms are also mentioned which do not lead to singular invariant systems of three second-order ODEs for a Lie algebra of dimension 4. The integrability aspect of these classified singular-invariant systems in (1+3)-dimensional space is discussed in a detailed manner for a Lie algebra of dimension 4. Finally, two physical systems from mechanics are presented to illustrate the utilization of the physical aspect of these singular invariants. [ABSTRACT FROM AUTHOR]

In this article, we study the generalized (2 + 1)-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie's invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the (1 + 1)-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method. [ABSTRACT FROM AUTHOR]

We find generators for the algebra of rational differential invariants for general and degenerate Kundt spacetimes and relate this to other approaches to the equivalence problem for Lorentzian metrics. Special attention is given to dimensions three and four. [ABSTRACT FROM AUTHOR]