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Using a two‐dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non‐singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t)$\mathbb {F}_q(t)$, provided char(Fq)>3$\operatorname{char}(\mathbb {F}_q)>3$. Under the same hypotheses, we also verify weak approximation. [ABSTRACT FROM AUTHOR]

Quadrics are important in finite geometry and can be used to construct binary codes. In this paper, we first define an incidence matrix M based on points and non-degenerate quadrics in the classical projective space PG (n − 1 , q) , where q is a prime power. As a consequence, we establish a binary code C (M) with the generator matrix M and determine the dimension of C (M) when q and n are both odd. In particular, we study the minimum distances of C (M) and C ⊥ (M) in PG (2 , q) and give their upper bounds. [ABSTRACT FROM AUTHOR]

AbstractUsing the Buchsbaum-Eisenbud structure theorem for a minimal free resolution of an arithmetically Gorenstein variety, we describe a structure theorem for the highest linear syzygies among quadrics defining a del Pezzo variety. Indeed, such syzygies can be represented as columns of a skew-symmetric matrix whose entries are wedge products of linear forms. [ABSTRACT FROM AUTHOR]

We establish connections between the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices. [ABSTRACT FROM AUTHOR]

We obtain geometric characterizations of the dual functions for quadratic bent and vectorial bent functions in terms of quadrics. Additionally, using the zeros of the polynomial X q + 1 + X + a which have been studied recently in the literature, we provide some examples of binomial quadratic bent functions on F q 4 and F q 6 , where q is a power of 2. [ABSTRACT FROM AUTHOR]

If the small spheres of a Finslerian projective metric are quadrics, then it is a Riemannian projective metric of constant curvature. [ABSTRACT FROM AUTHOR]

We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier–Mukai functors with well-behaved target and source spaces. To accomplish this, we first study the characteristic morphism introduced in Buchweitz and Flenner [Adv. Math. 217 (2008), 205–242] in order to control it for tilting bundles. Then, we continue by applying twisted Hodge diamonds of hypersurfaces embedded in projective space to compute the Hochschild dimension of these spaces. This allows us to compute the kernel of the embedding into the projective space in Hochschild cohomology. Finally, we use the above computations to apply the construction in Rizzardo, Van den Bergh, and Neeman [Invent. Math. 216 (2019), 927–1004] of non-Fourier–Mukai functors and verify that the constructed functors indeed cannot be Fourier–Mukai for odd-dimensional quadrics. Using this approach, we prove that there are a large number of Hochschild cohomology classes that can be used for the construction of Rizzardo, Van den Bergh, and Neeman [Invent. Math. 216 (2019), 927–1004]. Furthermore, our results allow the application of computer-based calculations to construct candidate functors for arbitrary degree hypersurfaces in arbitrary high dimensions. Verifying that these are not Fourier–Mukai still requires the existence of a tilting bundle. In particular, we prove that there is at least one non-Fourier–Mukai functor for every odddimensional smooth quadric. [ABSTRACT FROM AUTHOR]

In this article, we construct a novel self-referential fractal multiquadric function which is symmetric about the origin. The scaling factors are suitably restricted to preserve the differentiability and the convexity of the underlying classical multiquadric function. Based on the translates of a fractal multiquadric function defined on a grid, we propose two fractal quasi-interpolants L C α f and L D α f to approximate smooth and irregular functions. We study the convergence of L C α f and L D α f to f using uniform error estimates. We investigate the linear polynomial reproducing property, convexity/concavity and monotonicity features of these quasi-interpolation operators. The advantages of fractal quasi-interpolants over the classical quasi-interpolants are demonstrated by various examples. [ABSTRACT FROM AUTHOR]

The quasi-interpolation operator is widely used in numerical approximation and numerical solutions of differential equations. This paper proposes a new multiquadric(MQ) quasi-interpolate and formulates a meshfree method for the Korteweg-de Vries(KdV) equation based on the proposed multiquadric quasi-interpolate. More specifically, based on the multiquadric function, a new univariate multiquadric(MQ) quasi-interpolation scheme is structured, which possesses high accuracy, simple structure, and ease of programming. Moreover, the error estimation of the new quasi-interpolate is shown in detail. Next, a meshfree method for the Korteweg-de Vries (KdV) is proposed by using the novel multiquadric(MQ) quasi-interpolation operator. In the spatial direction, the derivative is approximated by the proposed multiquadric quasi-interpolate, and the forward divided difference approximates the temporal derivative. Several numerical examples are presented at the end of the paper to verify the expected approximation capability, and the experiment results show that the meshfree method (based on the new multiquadric(MQ) quasi-interpolation operator) is valid. [ABSTRACT FROM AUTHOR]

A projective space of dimension 3 over a finite Galois field GF(q) is denoted as 𝑃𝐺(3, 𝑞). It is defined as the set of all one-dimensional subspaces of 4-dimensional vector space over this Galois field. Klein transformation maps a projective plane of 𝑃𝐺(3,2) to a Greek plane of the Klein quadric. This paper introduces the fuzzification of Greek planes passing through the base point, any point on the base line different from the base point, and any point not on the base line of the base plane of 5-dimensional fuzzy projective space. [ABSTRACT FROM AUTHOR]

We study local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, we develop a semi-integral version of the Brauer–Manin obstruction interpolating between Manin's classical version for rational points and the integral version developed by Colliot-Thélène and Xu. We determine the status of local-global principles, and obstructions to them, in two families of orbifolds naturally associated to quadric hypersurfaces. Further, we establish a quantitative result measuring the failure of the semi-integral Brauer–Manin obstruction to account for its integral counterpart for affine quadrics. [ABSTRACT FROM AUTHOR]

The block graph of a quasi-symmetric design is strongly regular. It is a challenging problem to decide which strongly regular graphs are block graphs of quasi-symmetric designs. We rule out the possibility of quasi-symmetric designs with non-zero intersection numbers whose block graphs have the parameters as certain pseudo-geometric graphs.

In this paper, we analyze the Hessian locus associated to a general cubic hypersurface, by describing its singular locus and its desingularization for every dimension. The strategy is based on strong connections between the Hessian and the quadrics defined as partial derivatives of the cubic polynomial. In particular, we focus our attention on the singularities of the Hessian hypersurface associated to the general cubic four-fold. It turns out to be a minimal surface of general type: its analysis is developed by exploiting the nature of this surface as a degeneracy locus of a symmetric vector bundle map and by describing an unramified double cover, which is constructed in a more general setting. [ABSTRACT FROM AUTHOR]

We show that generalized multiquadric radial basis functions (RBFs) on ℝd have a mean dimension that is 1 + O(1/d) as d → ∞ with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches 1. As a consequence, these RBFs become essentially additive as their dimension increases. Gaussian RBFs by contrast can attain any mean dimension between 1 and d. We also find that a test integrand due to Keister that has been influential in quasi-Monte Carlo theory has a mean dimension that oscillates between approximately 1 and approximately 2 as the nominal dimension d increases. [ABSTRACT FROM AUTHOR]

In this note we construct an example of a smooth projective threefold that is irrational over Q but is rational at all places. Our example is a complete intersection of two quadrics in P 5 , and we show it has the desired rationality behavior by constructing an explicit element of order 4 in the Tate–Shafarevich group of the Jacobian of an associated genus 2 curve. [ABSTRACT FROM AUTHOR]

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup. [ABSTRACT FROM AUTHOR]

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical 55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space. [ABSTRACT FROM AUTHOR]

This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona–Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group S5. From this construction, we are able to describe the Chow ring explicitly [ABSTRACT FROM AUTHOR]

In the realm of ballistic target analysis, micro-motion attributes, such as warhead precession, nutation, and decoy oscillations, play a pivotal role. This paper addresses these critical aspects by introducing an advanced analytical model for assessing the Doppler power spectra of convex quadric revolution bodies during precession. Our model is instrumental in calculating the Doppler shifts pertinent to both precession and swing cones. Additionally, it extends to delineate the Doppler power spectra for configurations involving cones and sphere–cone combinations. A key aspect of our study is the exploration of the effects exerted by geometric parameters and observation angles on the Doppler spectra, offering a comparative perspective of various micro-motion forms. The simulations distinctly demonstrate how different micro-motion patterns of a cone influence the Doppler power spectra and underscore the significance of geometric parameters and observational angles in shaping these spectra. This research not only contributes to enhancing LIDAR target identification methodologies but also lays a groundwork for future explorations into complex micro-motions like nutation. [ABSTRACT FROM AUTHOR]

A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z 2 -graded algebra E (θ) , which corresponds to a noncommutative quadric hypersurface E ! / (z) for some central regular element z ∈ E 2 !. It turns out that the bounded derived category D b (gr Z 2 E (θ)) is equivalent to the stable category of the maximal Cohen-Macaulay modules over E ! / (z) provided that E ! is noetherian. As a consequence, E ! / (z) is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E (θ) is a semisimple Z 2 -graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces. As an application, we recover Knörrer's periodicity theorem without using matrix factorizations. [ABSTRACT FROM AUTHOR]

Passive location based on TDOA (time difference of arrival) and FDOA (frequency difference of arrival) is the mainstream method for target localization. This paper proposes a fast time–frequency difference positioning method to address issues such as low accuracy, large computational resource utilization, and limited suitability for real-time signal processing in the conventional CAF (cross-ambiguity function)-based approach, aiming to complete the processing of the target radiation source to obtain the target parameters within a short timeframe. In the mixing product operation step of the CAF, a frequency-domain approach replaces the time-domain convolution operation in PW-ZFFT (pre-weighted Zoom-FFT) to reduce the computational load of the CAF. Additionally, a quadratic surface fitting method is used to enhance the accuracy of TDOA and FDOA. The localization solution is obtained using Newton's method, which can provide more accurate results compared to analytical methods. Next, a signal processing platform is designed with FPGA (field-programmable gate array) and multi-core DSP (digital signal processor), and works by dividing and mapping the algorithm functional modules according to the hardware's characteristics. We analyze the architectural advantages of multi-core DSP and design methods to improve program performance, such as EDMA transfer optimization, inline function optimization, and cache optimization. Finally, this paper constructs simulation tests in typical positioning scenarios and compares them to hardware measurement results, thus confirming the correctness and real-time capability of the program. [ABSTRACT FROM AUTHOR]

In Part I of this study, it was shown that all the three known types of monohedral hexagonal tessellations of the plane, those composed of equal irregular hexagons, have just a single deformation mode when tiles are considered as rigid bodies hinged to each other along the edges. A gallery of tessellated plates was simulated numerically to demonstrate the range of achievable deformed shapes. In Part II, the displacement field was first derived and a continuous interpolant for each type of tessellated plate. It turns out that all corresponding metasurfaces are described by quadrics. Afterwards, a parametric analysis was carried out to determine the effect of varying angles and edge lengths on the curvature, and the values of the geometric Poisson ratio of the plates. Finally, a method of fabrication is proposed based on the additive manufacturing of stiff tiles of negligible deformability and flexible connectors. Using this modular technique, it is possible to join together different monohedral tessellated plates able to deform into piece-wise quadrics. The nodal positions in the deformed configuration of the realized plates are measured after enforcing one principal curvature to assume a chosen value. The estimate of the other principal curvature confirms the analytical predictions. The presented tessellated plates permit to realize doubly curved shape-morphing metasurfaces with assorted shapes, which also can feature a certain surface roughness, and they can be employed in all applications demanding high surface accuracy and few actuators or just one. [ABSTRACT FROM AUTHOR]

Let f : Y → X be a finite morphism between Fano manifolds Y and X such that the Fano index of X is greater than 1. On the one hand, when both X and Y are fourfolds of Picard number 1, we show that the degree of f is bounded in terms of X and Y unless X ≅ P 4 ; hence, such X does not admit any non-isomorphic surjective endomorphism. On the other hand, when X = Y is either a fourfold or a del Pezzo manifold, we prove that, if f is an int-amplified endomorphism, then X is toric. Moreover, we classify all the singular quadrics admitting non-isomorphic endomorphisms. [ABSTRACT FROM AUTHOR]

The control of the shape of complex metasurfaces is a challenging task often addressed in the literature. This work presents a class of tessellated plates able to deform into surfaces of preprogrammed shape upon activation by any flexural load and that can be controlled by a single actuator. Quadric metasurfaces are obtained from infinitesimal origami maps of monohedral hexagonal tessellations of the plane, that is pavings in which all tiles are congruent to each other. Monohedral tessellated portions can be joined together to obtain more complex shapes, which can be locally synclastic or anticlastic and can have a certain roughness. We broaden previous work by providing a complete characterization of all the three known types of monohedral tessellations composed by irregular hexagons. The proposed two-dimensional structures may have applications in prosthetics, tissue engineering, wearable devices, energy harvesting devices, tunable focus mirrors and adaptive facades. The study is divided in two parts. In Part I, after introducing the discrete kinematics of tessellated plates, it is proved analytically that essentially each type of monohedral hexagonal tessellation possesses only one deformation mode. Afterwards, several numerical examples are provided to demonstrate the variety of achievable surface shapes. In Part II, first the metasurfaces corresponding to assigned tile geometries are given a continuum description, which establishes that the continuous interpolant is always a quadric. Then, experimental results on fabrication, assembly and surface accuracy are reported. [ABSTRACT FROM AUTHOR]

The paper describes the effect of the local surface modelling methods known as plane, quadric and triangulation on the generation of the 3D mesh models of the historical objects with rock-based building structures using Cloudcompare, an open-source software for point cloud data processing. The methodology begins with data collection of the 3D test objects using terrestrial laser scanning technology, the pre-processing of the point cloud data, the point cloud subsampling process, the process of generating of the local surface modelling data, the process of generating the 3D mesh models and finally, the analysis, which involves the 3D surface deviation analysis between the generated 3D mesh models. The overall results shows that there are no significant differences between the 3D mesh models that was generated from all the three local surface modelling methods. The histogram analysis shows that the plane and the quadric local surface modelling methods is the best methods to be used in the research where the 3D test object to be modelled contains curvy surfaces without sharp edges and corners. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of homogeneous totally real minimal two-spheres in a complex hyperquadric Q n . Let f be a totally real minimal immersion from two-sphere in Q n , and τ X Y ,   τ X c (see Sec. 2) are globally defined invariants relative to the first and second fundamental forms. We prove that if its Gauss curvature K and τ X Y are constants, and τ X c vanishes identically, then f is congruent to F 2 k , 2 l constructed by the Boruvka spheres with n = 2 (k + l). [ABSTRACT FROM AUTHOR]

This study focuses on the development of a multi-point technique for future constellation missions, aiming to measure gradients at various orders, in particular the linear and quadratic gradients, of a general field. It is well established that, in order to estimate linear gradients, the spacecraft must not lie on a plane. Through analytical exploration within the framework of least squares, it is demonstrated that at least 10 spacecraft that do not lie on any quadric surface are required to estimate both linear and quadratic gradients. The spatial arrangement of the spacecraft can be characterized by a set of quality factors. In cases where there is poor temporal synchronization among the spacecraft leading to non-simultaneous measurements, temporal gradients must be included. If the spacecraft have multiple velocities, by incorporating temporal gradients it is possible to reduce the number of required spacecraft. Furthermore, it is proved that the accuracy of the linear gradient is of second order and that of the quadratic gradient is of first order. Additionally, a method for estimating errors in the calculation is also illustrated. [ABSTRACT FROM AUTHOR]

Soit k un corps p-adique. On montre que toute intersection de deux quadriques dans l'espace projectif ℙ k 4 contient un point sur une extension quadratique, ce qui généralise un résultat de Creutz et Viray pour le cas lisse. La preuve utilise un théorème de Lichtenbaum sur les courbes de genre 1 sur un corps p-adique. On déduit de ce résultat que toute intersection lisse de deux quadriques X ⊂ ℙ k 5 sur un corps de nombres k possède un point sur une extension quadratique. On déduit aussi de ce résultat une démonstration relativement courte d'un théorème de Heath-Brown : le principe de Hasse vaut pour les intersections complètes lisses X ⊂ ℙ k 7 de deux quadriques sur un corps de nombres. On donne aussi une démonstration alternative d'un principe de Hasse pour certaines intersections de deux quadriques dans ℙ k 5 , dû à Iyer et Parimala. Lichtenbaum proved that index and period coincide for a curve of genus one over a p-adic field. Salberger proved that the Hasse principle holds for a smooth complete intersection of two quadrics X ⊂ ℙ n over a number field, if n ≥ 5 and X contains a conic. Building upon these two results, we extend recent results of Creutz and Viray (2021) on the existence of a quadratic point on intersections of two quadrics over p-adic fields and over number fields. We then recover Heath-Brown's theorem (2018) that the Hasse principle holds for smooth complete intersections of two quadrics in ℙ 7 . We also give an alternate proof of a theorem of Iyer and Parimala (2022) on the local-global principle in the case n = 5 . [ABSTRACT FROM AUTHOR]

We propose a conjecture relevant to Galkin's lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture O holds in these two cases. [ABSTRACT FROM AUTHOR]

Some ten years ago we managed to take a first step in the classification of nonsingular 2 × 2 × 2 × 2 hypercubes over a finite field by resolving the special case where the hypercubes can be written as a product of two 2 × 2 × 2 hypercubes, i.e., nonsingular 2 × 2 × 2 × 2 hypercubes of 12-rank two. We have now been able to extend this classification to hypercubes of 12-rank three, based on the connection between nonsingular hypercubes and bundles of nonintersecting quadrics in 3 dimensions. A bit surprisingly, the number of inequivalent nonsingular hypercubes of 12-rank three is only of the same order of magnitude as in the case of 12-rank two. We also made some headway into the remaining case of 12-rank four. In particular, we prove that there are essentially only 2 nonsingular 2 × 2 × 2 × 2 hypercubes that correspond to a hyperbolic fibration of 3-dimensional projective space. [ABSTRACT FROM AUTHOR]

Local CR-diffeomorphisms between standard Hermitian quadrics of codimension 2 in C m are treated using a method based on the fundamental theorem of projective geometry. The cases when such diffeomorphisms must be projective are distinguished. [ABSTRACT FROM AUTHOR]

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question "which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?", we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases. The extremal varieties of dimension n , codimension e , and degree d are exactly characterized by the following two types: (i) Varieties with d = e + 2 , depth X = n , and Green-Lazarsfeld index a (X) = 0 , (ii) Arithmetically Cohen-Macaulay varieties with d = e + 3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]). In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective quadric defined over a field of characteristic 2. We prove that in the Chow group of codimension 2 or 3 of X the torsion subgroup has at most two elements. In codimension 2, we determine precisely when this torsion subgroup is nontrivial. In codimension 3, we show that there is no torsion if dim ⁡ X ≥ 11. This extends the analogous results in characteristic different from 2, obtained by Karpenko in the nineteen-nineties. [ABSTRACT FROM AUTHOR]

There are two main problems in finding the higher genus superstring measure. The first one is that for g ≥ 5 the super moduli space is not projected. Furthermore, the supermeasure is regular for g ≤ 11, a bound related to the source of singularities due to the divisor in the moduli space of Riemann surfaces with even spin structure having holomorphic sections, such a divisor is called the θ-null divisor. A result of this paper is the characterization of such a divisor. This is done by first extending the Dirac propagator, that is the Szegö kernel, to the case of an arbitrary number of zero modes, that leads to a modification of the Fay trisecant identity, where the determinant of the Dirac propagators is replaced by the product of two determinants of the Dirac zero modes. By taking suitable limits of points on the Riemann surface, this holomorphic Fay trisecant identity leads to identities that include points dependent rank 3 quadrics in ℙg−1. Furthermore, integrating over the homological cycles gives relations for the Riemann period matrix which are satisfied in the presence of Dirac zero modes. Such identities characterize the θ-null divisor. Finally, we provide the geometrical interpretation of the above points dependent quadrics and show, via a new θ-identity, its relation with the Andreotti-Mayer quadric. [ABSTRACT FROM AUTHOR]

In this article, we study the influence of certain parameters on the stability conditions of the reaction front in a liquid medium. The mathematical model consists of the heat equation, the concentration equation and the Navier–Stockes equation under the Boussinesq approximation. An asymptotic analysis was performed using the approximation proposed by Zeldovich and Frank–Kamentskii to obtain the interface problem. A stability analysis was carried out to obtain a linearized problem which will be solved numerically using a multiquadric radial basis function method to find the convective threshold. This will allow us to conclude the effect of each parameter on the stability of the front, in particular the amplitude and the resonance frequency. [ABSTRACT FROM AUTHOR]

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in P 5 . We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of I ( Sec (C 6)) . We study the images C p and C p q of a sextic C 6 under the projection from a general point P ∈ P 5 and a general line P Q ¯ ⊂ P 5 . In particular, we show that C p is k-normal for all k ≥ 2 and I (C p) is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that C p q is k-normal for all k ≥ 3 and I (C p q) is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4. [ABSTRACT FROM AUTHOR]

Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples of quasi-quadrics have been constructed using a technique called {\em pivoting} in a paper by De Clerck, Hamilton, O'Keefe and Penttila. We introduce a more general notion of pivoting, called switching, and also extend this notion to Hermitian polar spaces. The main result of this paper studies the switching technique in detail by showing that, for \(q\geq 4\), if we modify the points of a hyperplane of a polar space to create a quasi-polar space, the only thing that can be done is pivoting. The cases \(q=2\) and \(q=3\) play a special role for parabolic quadrics and are investigated in detail. Furthermore, we give a construction for quasi-polar spaces obtained from pivoting multiple times. Finally, we focus on the case of parabolic quadrics in even characteristic and determine under which hypotheses the existence of a nucleus (which was included in the definition given in the De Clerck-Hamilton-O'Keefe-Penttila paper) is guaranteed.Mathematics Subject Classifications: 51E20Keywords: Projective geometry, quadrics, hyperplanes, quasi-quadrics, intersection numbers

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors. [ABSTRACT FROM AUTHOR]

In this paper, we present a new key exchange protocol in which Alice and Bob have secret keys given by quadric surfaces embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded quadrics. Both of them reconstruct the isomorphism class of the intersection which is a curve of genus 1, and is uniquely determined by the j-invariant. An eavesdropper, to find this j-invariant, has to solve problems which are conjecturally quantum-resistant. [ABSTRACT FROM AUTHOR]

We refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular 3‐manifolds in R6$\mathbb {R}^6$ and singular corank one 3‐manifolds in R5$\mathbb {R}^5$. For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by four ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3‐manifold can go to infinity when the manifold is projected orthogonally in a tangent direction. [ABSTRACT FROM AUTHOR]

We show that the birationality class of a quadric surface bundle over ℙ 2 is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality of very general quadric bundles over ℙ 2 with discriminant curves of fixed degree. In particular, we construct explicit models of these bundles for some discriminant data. Among others, we obtain various birational models of a nodal Gushel–Mukai fourfold, as well as of a cubic fourfold containing a plane. Finally, we prove stable irrationality of several types of quadric surface bundles. [ABSTRACT FROM AUTHOR]

We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces OP 2 = E 6 sc / P 6 and E 7 sc / P 7 , known respectively as the Cayley plane and Freudenthal variety. These models are defined on the complement X can ∨ of an anti-canonical divisor of the "Langlands dual homogeneous spaces" X ∨ = P ∨ ﹨ G ∨ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in [20,25,23]. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in [26] using a restriction to an algebraic torus, also known as the Lusztig torus , as proven in [28]. We also give a cluster structure on C [ X ∨ ] , prove that the Plücker coordinates form a Khovankii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families. [ABSTRACT FROM AUTHOR]

We construct wonderful compactifications of the spaces of linear maps and symmetric linear maps of a given rank as blowups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and their relations with some spaces of degree two stable maps. [ABSTRACT FROM AUTHOR]

In computer vision, camera pose estimation from correspondences between 3D geometric entities and their projections into the image has been a widely investigated problem. Although most state-of-the-art methods exploit low-level primitives such as points or lines, the emergence of very effective CNN-based object detectors in the recent years has paved the way to the use of higher-level features carrying semantically meaningful information. Pioneering works in that direction have shown that modelling 3D objects by ellipsoids and 2D detections by ellipses offers a convenient manner to link 2D and 3D data. However, the mathematical formalism most often used in the related litterature does not enable to easily distinguish ellipsoids and ellipses from other quadrics and conics, leading to a loss of specificity potentially detrimental in some developments. Moreover, the linearization process of the projection equation creates an over-representation of the camera parameters, also possibly causing an efficiency loss. In this paper, we therefore introduce an ellipsoid-specific theoretical framework and demonstrate its beneficial properties in the context of pose estimation. More precisely, we first show that the proposed formalism enables to reduce the pose estimation problem to a position or orientation-only estimation problem in which the remaining unknowns can be derived in closed-form. Then, we demonstrate that it can be further reduced to a 1 Degree-of-Freedom (1DoF) problem and provide the analytical derivations of the pose as a function of that unique scalar unknown. We illustrate our theoretical considerations by visual examples and include a discussion on the practical aspects. Finally, we release this paper along with the corresponding source code in order to contribute towards more efficient resolutions of ellipsoid-related pose estimation problems. The source code is available here: https://gitlab.inria.fr/vgaudill/p1e. [ABSTRACT FROM AUTHOR]