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The existence of a non-defective stationary disc attached to a non-degenerate model quadric in C N is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs [6]. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tumanov [14]. Our proof is based on the explicit expression of stationary discs, and opens up a conjecture for the unique 1-jet determination to hold when the model is not necessarily strongly pseudoconvex. [ABSTRACT FROM AUTHOR]

This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for a set of two-dimensional quadrics, including ellipsoid, elliptic hyperboloid, elliptic paraboloid, hyperbolic paraboloid, torus, one-sheeted hyperboloid, Moëbius strips, as well as on a generic surface. The estimated geodesic solutions for the tested surfaces are obtained with residuals at the machine-error level. In principle, the proposed approach can be applied to solve boundary value problems in more complex scenarios, such as on Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Let |$p$| and |$q$| be anisotropic non-degenerate quadratic forms of dimension |$\geq 2$| over an arbitrary field |$F$|⁠ , let |$s$| be the unique non-negative integer for which |$2^s<{\textrm{dim}(p)} \leq 2^{s+1}$|⁠ , and let |$k$| be the dimension of the anisotropic part of |$q$| after extension to |$F(p)$|⁠. A recent conjecture of the author then asserts that |${\textrm{dim}(q)}$| must lie within |$k$| of an integer multiple of |$2^{s+1}$|⁠. This statement, which holds trivially if |$k \geq 2^s -1$|⁠ , represents a natural generalization of the well-known separation theorem of Hoffmann, bridging a gap between it and certain classical results on the Witt kernels of function fields of quadrics. In the present article, we prove the conjecture in the case where |$\textrm{char}(F) \neq 2$| and |${\textrm{dim}(p)}> 2k - 2^{s-1}$|⁠. This implies, in particular, that the conjecture holds if |$\textrm{char}(F) \neq 2$| and either |$k \leq 2^{s-1} + 2^{s-2}$| or |${\textrm{dim}(p)} \geq 2^s + 2^{s-1} - 4$|⁠. [ABSTRACT FROM AUTHOR]

Avramov, Conca, and Iyengar ask whether β i S (R) ≤ ( g i ) for all i when R = S / I is a Koszul algebra minimally defined by g quadrics. In recent work, we give an affirmative answer to this question when g ≤ 4 by completely classifying the possible Betti tables of Koszul algebras defined by height-two ideals of four quadrics. Continuing this work, the current paper proves a structure theorem for Koszul algebras defined by four quadrics. We show that all these Koszul algebras are LG-quadratic, proving that an example of Conca of a Koszul algebra that is not LG-quadratic is minimal in terms of number of defining equations. We then characterize precisely when these rings are absolutely Koszul, and establish the equivalence of the absolutely Koszul and Backelin–Roos properties up to field extensions for such rings (in characteristic zero). The combination of the above paper with the current one provides a fairly complete picture of all Koszul algebras defined by g ≤ 4 quadrics. [ABSTRACT FROM AUTHOR]

Let ℚ 3 be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in ℚ 3 . By an isotropic curve, we mean a nonconstant holomorphic map from a Riemann surface into ℚ 3 , null with respect to the conformal structure of ℚ 3 . The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed. [ABSTRACT FROM AUTHOR]

Tracking the trajectory of a moving body is one of key issues for a multiple freedom motion joint with multi-point surface contacts. In general, the tangent space of the motion can be determined by using the contact relation of the screw theory, but it is difficult to determine which vector in the tangent space represents the tangent vector of motion joint's freedom. This is because the tangent space is a linear space generated by the tangent vector of the motion joint, which includes more elements than actual tangent vectors. Then, a second-order interference detection approach is presented to identify the tangent vector of the freedom in the motion tangent space. And recognised tangent directions can be introduced to track adjacent contact states in 3D assembly models, which support the direct creation of the configuration space of a 3D assembly model. [ABSTRACT FROM AUTHOR]

A rank 5 vector bundle G on HT ht is constructed as the homology of a certain Beilinson monad. By the double point formula, HT ht = - 4. The entries of the third row are given by I i >= 2 I i - HT ht . Substituting this value in the double point formula, we get HT ht = -12. [Extracted from the article]

How to improve the dressing method has become a key technology for high-precision flex and circular rigid splines with involute tooth profile in gear form grinding, where the machining precision mainly depends on dressing accuracy of the grinding wheel. Based on the geometric principle (i.e., a spatial straight line changes to quadric conical surface by continuously rotating around its non-coplanar coordinate axis), a method of generating hyperbola by a single degree of freedom linear dressing motion is proposed to fit the involute tooth profile. With the design of a novel diamond dresser, mechanism model and different compensation models are established to demonstrate the feasibility of the fitting principle. Meanwhile, the error models are also established according to the mapping relationship of the device, which can reveal the error transmission law and the accurate adjustment method of geometric-profiling deviations, and enhance grinding accuracy. Especially, through complementary numerical simulations and experimental method, a specific example is presented to verify that the accuracy of the straight-line feed motion can meet its precision of the fitting and dressing. Eventually, it is hoped that this systematic method can be extended to manufacture high-precision involute gears for small-module precision gear transmission system and establish a theoretical foundation for setting standard on evaluating the tooth profile deviation, which is influenced by the fitting principle errors in gear form grinding. [ABSTRACT FROM AUTHOR]

A Minkowski plane is Euclidean if and only if at least one ellipse is a quadric. We discuss the higher dimensional consequences too. [ABSTRACT FROM AUTHOR]

A Minkowski plane is Euclidean if and only if at least one hyperbola is a quadric. We discuss the higher dimensional consequences too. [ABSTRACT FROM AUTHOR]

• An effective scheme based on an improved grid fitting method is proposed for detecting higher-level symmetry patterns between surface-level features in engineering mesh models. • A progressive optimization framework based on iterative constrained fitting is devised for enhancing higher-level regular relations between parametric surfaces of surface-level features. • A method is presented for transferring the enhanced regularities between continuous parametric surfaces to discrete mesh models. Recovering geometric regularities from scanned mesh models with various types of surface features has always been a challenging task in reverse engineering. To address this problem, this paper presents a regularity detection and enhancement framework for surface mesh reconstruction. Initially, surface patches are identified by decomposing the original model into planar, quadric and freeform surface patches. Similar surface patches are aligned with each other by pairwise registration, and symmetry patterns are detected from the accumulated affine transformations using an improved grid fitting method. Regular relations between symmetry patterns and individual surface patches are enumerated and progressively strengthened by orientation, dimension and placement optimizations. Finally, the resultant model with enhanced regularities is obtained by projecting surface patches onto the optimized parametric surfaces iteratively. Comparative experiments on test models demonstrate that the proposed method outperforms existing methods in recovering both lower- and higher-level regularities of engineering models, especially those with freeform surfaces. [ABSTRACT FROM AUTHOR]

We compute the algebraic Morava K-theory ring of split special orthogonal and spin groups. In particular, we establish certain stabilization results for the Morava K-theory of special orthogonal and spin groups. Besides, we apply these results to study Morava motivic decompositions of orthogonal Grassmannians. For instance, we determine all indecomposable summands of the Morava motives of a generic quadric. [ABSTRACT FROM AUTHOR]

This paper proposes a method for the deployment of a multi-agent system of unmanned aerial vehicles (UAVs) as a shield with potential applications in the protection of infrastructures. The shield shape is modeled as a quadric surface in the 3D space. To design the desired formation (target distances between agents and interconnections), an algorithm is proposed where the input parameters are just the parametrization of the quadric and the number of agents of the system. This algorithm guarantees that the agents are almost uniformly distributed over the virtual surface and that the topology is a Delaunay triangulation. Moreover, a new method is proposed to check if the resulting triangulation meets that condition and is executed locally. Because this topology ensures that the formation is rigid, a distributed control law based on the gradient of a potential function is proposed to acquire the desired shield shape and proofs of stability are provided. Finally, simulation and experimental results illustrate the effectiveness of the proposed approach. • Development of a method with low computational cost to obtain the target formation. • A distributed method to check that the triangulation is Delaunay's. • A distributed control law for 3D formations constrained to virtual quadric surfaces. • New rigidity properties are derived to study stability. • Validation of the results over an experimental platform. [ABSTRACT FROM AUTHOR]

Interpolating trajectories of points and geometric entities is an important problem for kinematics. To describe these trajectories, several algorithms have been proposed using matrices, quaternions, dual-quaternions, and the Study quadric; the last one allows the embedding of motors as 8D vectors into projective space P 7 , where the interpolation of rotations and translations becomes a linear problem. Furthermore, conformal geometric algebra (CGA) is an effective and intuitive framework for representing and manipulating geometric entities in Euclidean spaces, and it allows the use of quaternions and dual-quaternions formulated as Motors. In this paper, a new methodology for accelerating the Study quadric Interpolation based on Conformal Geometric Algebra is presented. This methodology uses General Purpose Graphics Processing Units (GPUs) and it is applied for medical robotics, but it can also be extended to other areas such as aeronautics, robotics, and graphics processing. [ABSTRACT FROM AUTHOR]

The modeling of real-valued random fields indexed by spherical coordinates arises in different disciplines of the natural sciences, especially in environmental, atmospheric and earth sciences. However, there is currently a lack of parametric models allowing a flexible representation of the spatial correlation structure of multivariate data located on a spherical surface. To bridge this gap, we provide analytical expressions of twenty-two parametric families of isotropic p-variate covariance kernels on the d-dimensional sphere, defined for any integers p > 0 and d > 1 , together with their respective spectral representations (Schoenberg matrices) and sufficient validity conditions on the covariance parameters. These families include multiquadric, sine power, exponential, Bessel and hypergeometric kernels, and provide covariances exhibiting varied shapes, short-scale and large-scale behaviors. Our construction relies on the so-called multivariate parametric adaptation approach, where a matrix-valued covariance kernel is defined on the basis of a scalar covariance to which matrix-valued parameters are applied, coupled with matrix manipulations. [ABSTRACT FROM AUTHOR]

We determine some composition laws related to the Fricke surface and the "double" Fricke surface. This latter surface admits the squares of Markov triples as its solutions. [ABSTRACT FROM AUTHOR]

A new technique to generate smooth motion trajectories for robot manipulators using multiquadric radial basis functions (MQ-RBFs) is presented in this paper. In order to get the optimal trajectory, two objective functions are minimized that are proportional to the execution time, the integral of the squared jerk (which denotes the time derivative of the acceleration) along the whole trajectory. Also, the proposed interpolation technique is introduced for solving the trajectory planning problem in the joint space, where the interpolation of via-points takes into account boundary conditions and also satisfies kinematics limits of velocity, acceleration, and jerk. Then, the proposed approach is compared with a set of classical interpolation techniques based on radial basis function models and cubic splines. Finally, the proposed technique has been tested for the six-joint PUMA 560 manipulator in two cases (minimum time and minimum time-jerk) and results are compared with those proposed of other important trajectory planning techniques. Numerical results show the competent performances of the proposed methodology to generate trajectories in short total transfer time and with high smooth profile. [ABSTRACT FROM AUTHOR]

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of Cn × Cm. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N and the projection onto the nullspace of □b. The main application of our formulas is the critical case of codimension two quadrics in C4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups. [ABSTRACT FROM AUTHOR]

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional \operatorname {CAT}(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group \operatorname {BS}(1,2). Most of these examples satisfy a strong form of the shortcut property. The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions. [ABSTRACT FROM AUTHOR]

It is shown that the standard transfinite interpolation in quadrilateral patches may be written in terms of two sets of Bernstein polynomials. The former set is of the same degree with the corresponding blending functions while the latter is of the same degree with the Lagrange polynomials which operate like trial functions along each of the four edges as well as the additional inter-boundaries of the patch. The replacement of the Lagrange polynomials by the Bernstein ones allows the use of control points useful for design. Also it allows the determination of weights that may ensure accurate representation of quadric surfaces including those of revolution (spheres, ellipsoids, hyperboloids, etc). The presentation restricts to the standard Gordon formulation, which refers to structured stencils, similar to rectangular plates reinforced with longitudinal and transverse stiffeners. As an example, the proposed formula is applied to the geometric representation of a cylindrical and a spherical patch. In the latter case a nonlinear programming optimization technique was applied, starting with initial data pertinent to a tensor product surface, from which original weights and control points were obtained for the accurate representation of a spherical cap. In addition, the involved shape functions were successfully applied to the numerical solution of a boundary value problem. [ABSTRACT FROM AUTHOR]

The classical Jacobi–Chasles theorem states that tangent lines drawn at all points of a geodesic curve on a quadric in n-dimensional Euclidean space are tangent, in addition to the given quadric, to n – 2 other confocal quadrics, which are the same for all points of the geodesic curve. This theorem immediately implies the integrability of a geodesic flow on an ellipsoid. In this paper, we prove a generalization of this result for a geodesic flow on the intersection of several confocal quadrics. Moreover, if we add the Hooke's potential field centered at the origin to such a system, the integrability of the problem is preserved. [ABSTRACT FROM AUTHOR]

We extend the locus problems discussed in [9], [10] and [12], for a quadric surface when the fixed point is at an infinity. This paper will benefit those students who have backgrounds in Linear Algebra and Multivariable Calculus. As we shall see that the transformation from a quadric surface Σ to its locus surface Δ is a linear transformation. Consequently, how the eigenvectors are related to the position of the fixed point at an infinity will be discussed. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex quadric Q m = S O m + 2 / S O 2 S O m . In the following, as an application, we show a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on Hopf real hypersurfaces in the complex quadric Q m . [ABSTRACT FROM AUTHOR]

This research work is to study the numerical solution of three-dimensional second-order hyperbolic telegraph equations using an efficient local meshless method based on radial basis function (RBF). The model equations are used in nuclear material science and in the modeling of vibrations of structures. The explicit time integration technique is utilized to semi-discretize the model in the time direction whereas the space derivatives of the model are discretized by the proposed local meshless procedure based on multiquadric RBF. Numerical experiments are performed with the proposed numerical scheme for rectangular and non-rectangular computational domains. The proposed method solutions are converging quickly in comparison with the different existing numerical methods in the recent literature. [ABSTRACT FROM AUTHOR]

We study the problem of determining which diffeomorphism classes of Kähler manifolds admit a Hamiltonian circle action. Our main result is the following: Let M$M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension 4k$4k$, having a Hamiltonian circle action such that each component of the fixed‐point set is an isolated fixed point or has dimension 2mod4$2 \mod {4}$. Then M$M$ is diffeomorphic to CP4k$\mathbb {CP}^{4k}$, a quadric Q⊂CP4k+1$Q \subset \mathbb {CP}^{4k+1}$ or an intersection of two quadrics Q1∩Q2⊂CP4k+2$Q_1 \cap Q_2 \subset \mathbb {CP}^{4k+2}$. [ABSTRACT FROM AUTHOR]

We show that if a polyhedron in the three‐dimensional affine space with triangular faces is flexible, that is, can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and −1$-1$. We do this via elementary combinatorial considerations, made possible by a well‐known compactification of the three‐dimensional affine space as a quadric in the four‐dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one‐dimensional analog, which is trivial to solve. [ABSTRACT FROM AUTHOR]

cube can be twisted in a playful manner for visual and algebraic insights. The twisting process and the resulting ruled surfaces can be demonstrated using 3D modeling tools (e.g., GeoGebra R and Autodesk Fusion 360 R ) or elastic cords on a 3D-printable scaffold. The twisted cube is aesthetically appealing, posing interesting questions that are worthwhile at multiple levels. Algebraically, the volume of the twisted cube is shown to be two-thirds of the reference cube. The twisted faces are parts of hyperbolic paraboloids, whose implicit and parametric equations can be established from diverse perspectives in support of further dynamic explorations and discussions about the surface area. [ABSTRACT FROM AUTHOR]

The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found. [ABSTRACT FROM AUTHOR]

For a projective variety in of dimension , an additive action on is an effective action of on such that is -invariant and the induced action on has an open orbit. Arzhantsev and Popovskiy have classified additive actions on hyperquadrics of corank 0 or 1. In this paper, we give the classification of additive actions on hyperquadrics of corank 2 whose singularities are not fixed by the -action. [ABSTRACT FROM AUTHOR]

A quadratic set of a nonsingular quadric Q of Witt index at least three is defined as a set of points intersecting each subspace of Q in a possibly reducible quadric of that subspace. By using the theory of pseudo-embeddings and pseudo-hyperplanes, we show that if Q is one of the quadrics Q + (5 , 2) , Q(6, 2), Q - (7 , 2) , then the quadratic sets of Q are precisely the intersections of Q with the quadrics of the ambient projective space of Q. In order to achieve this goal, we will determine the universal pseudo-embedding of the geometry of the points and planes of Q. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric Q m ∗ = SO 2 , m o / SO 2 SO m . The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes -principal or -isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in Q m ∗ = SO 2 , m o / SO 2 SO m with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist. [ABSTRACT FROM AUTHOR]

In the paper, we give the Liouville classification of five interesting cases of topological billiards glued from two flat billiards bounded by arcs of confocal quadrics in the Minkowski plane. For each billiard, we calculate the marked Fomenko–Zieschang molecule, in other words the invariant of an integrable Hamiltonian system that completely determines the type of its Liouville foliation. [ABSTRACT FROM AUTHOR]

Augmented reality (AR) is an emerging technology that has permeated different spheres of life, one of them is education, and specifically the teaching-learning process at different educational levels and objects of study. For this reason, this paper presents the development of a learning model of quadric surfaces mediated by a mobile AR application and based on didactic engineering. The model was applied to a group of environmental engineering students of the Catholic University of Manizales. To obtain information on the use of the application and the learning results obtained, some intervention instruments were developed. The students stated that the use of AR allowed them to better understand the concepts of quadric surfaces, even more so in a time of pandemic by COVID-19, where education was highly measured by ICTs. [ABSTRACT FROM AUTHOR]

We present an efficient method to create samples directly on surfaces defined by constructive solid geometry (CSG) trees or graphs. The generated samples can be used for visualization or as an approximation to the actual surface with strong guarantees. We chose to use quadric surfaces as CSG primitives as they can model classical primitives such as planes, cubes, spheres, cylinders, and ellipsoids, but also certain saddle surfaces. More importantly, they are closed under affine transformations, a desirable property for a modeling system. We also propose a rendering method that performs local quadric ray‐tracing and clipping to achieve pixel‐perfect accuracy and hole‐free rendering. [ABSTRACT FROM AUTHOR]

We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete level 2 structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method. [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]

The mixing and segregation of granular materials are essential for industrial design and applications. In this work, the superquadric equation is used to construct spherical and non-spherical particles, and the discrete element method is used to investigate the radial segregation characteristics of a gaussian-dispersed mixture in a horizontal rotating drum. The influences of particle shape, standard deviation of the mixture, and rotating speed on the segregation behaviors are discussed by the gyration degree and mixing index. The results reveal that small and large particles are respectively distributed in the center and periphery of the drum, while the medium-sized particles are evenly distributed in the system. Larger standard deviation, smoother particle shape, and smaller rotation speed are conducive to the segregation of granular systems. When the granular system is close to the high-speed centrifugal state, the smallest particles have the largest radial velocity and are close to the drum wall. [Display omitted] • Segregation for gaussian-dispersed mixtures in rotating drums is studied by DEM. • Small and large particles accumulate respectively in the drum center and periphery. • The gyration degree of the particles first decreases and then increases as the particle diameter increases. • Spheres have more significant segregation properties than non-spherical particles due to their smoother surfaces. • Small particles are close to the wall of the drum at high rotation speed. [ABSTRACT FROM AUTHOR]

First we introduce the notion of parallel Ricci tensor ∇Ric = 0 for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 with parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

We show that the space of anti‐symplectic involutions of a monotone S2×S2 whose fixed point set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti‐symplectic involutions in that space are Hamiltonian isotopic. [ABSTRACT FROM AUTHOR]

In the present paper, two effective numerical schemes depending on a second-order Strang splitting technique are presented to obtain approximate solutions of the one-dimensional Burgers' equation utilizing the collocation technique and approximating directly the solution by multiquadric-radial basis function (MQ-RBF) method. To show the performance of both schemes, we have considered two examples of Burgers' equation. The obtained numerical results are compared with the available exact values and also those of other published methods. Moreover, the computed L 2 and L ∞ error norms have been given. It is found that the presented schemes produce better results as compared to those obtained almost all the schemes present in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the moduli space of Ulrich bundles on a smooth complete intersection of two |$4$| -dimensional quadrics in |$\mathbb P^5$|⁠. The main ingredient is the semiorthogonal decomposition by Bondal–Orlov, combined with the categorical methods pioneered by Kuznetsov and Lahoz–Macrì–Stellari. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in |$\mathbb P^5$| carries an Ulrich bundle of rank |$r$| for every |$r \ge 2$|⁠. Moreover, we provide a description of the moduli space of stable Ulrich bundles. [ABSTRACT FROM AUTHOR]

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory. [ABSTRACT FROM AUTHOR]

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. [ABSTRACT FROM AUTHOR]

We show that extended cyclic codes over F q with parameters [ q + 2 , 3 , q ] , q = 2 m , determine regular hyperovals. We also show that extended cyclic codes with parameters [ q t - q + t , 3 , q t - q ] , 1 < t < q , q is a power of t, determine (cyclic) Denniston maximal arcs. Similarly, cyclic codes with parameters [ q 2 + 1 , 4 , q 2 - q ] are equivalent to ovoid codes obtained from elliptic quadrics in PG(3, q). Finally, we give simple presentations of Denniston maximal arcs in PG(2, q) and elliptic quadrics in PG(3, q). [ABSTRACT FROM AUTHOR]

Given a monomial m in a polynomial ring and a subset L of the variables of the polynomial ring, the principal L -Borel ideal generated by m is the ideal generated by all monomials which can be obtained from m by successively replacing variables of m by those which are in L and have smaller index. Given a collection I = { I 1 , ... , I r } where I i is L i -Borel for i = 1 , ... , r (where the subsets L 1 , ... , L r may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets L 1 , ... , L r is chordal bipartite, then the defining equations of the multi-Rees algebra of I has a Gröbner basis of quadrics with squarefree lead terms under a lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm. [ABSTRACT FROM AUTHOR]

We use the results of Moser and Knörrer on relations between geodesics on quadrics and solutions of the classical Neumann system to describe explicitly the geodesic scattering on hyperboloids. We explain the relation of Knörrer's reparametrisation with projectively equivalent metrics on quadrics introduced by Tabachnikov and independently by Matveev and Topalov, giving a new proof of their result. We show that the projectively equivalent metric is regular on the projective closure of hyperboloids and extend Knörrer's map to this closure. [ABSTRACT FROM AUTHOR]

A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q n , q n) and a Laguerre plane of order q n (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas [15] asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q − (5 , q) , non-equivalent to the classical example , a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q − (5 , q) is projectively equivalent to a pseudo-conic. Thas [16] characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q − (5 , q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of Q − (5 , q) can be analysed from this viewpoint. [ABSTRACT FROM AUTHOR]