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In [11], Hausel introduced a commutative algebra — the multiplicity algebra — associated to a fixed point of the C ∗ -action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in the isomorphism class of the algebra. [ABSTRACT FROM AUTHOR]

• We propose the µ-basis formulas for the implicit monoid curves/surfaces. • New µ-basis formulas for the parametric monoid curves/surfaces in general expression. • Approximate µ-bases are computed for the monoid curves/surfaces as well as the error estimations. [Display omitted] The μ -basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ -basis from an implicit form. In this paper, we derive the explicit forms of μ -basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ -basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing "approximate μ -basis" as well as the error estimation. [ABSTRACT FROM AUTHOR]

We prove that if the focus of a conic on the sphere is not the pole of the conic's directrix, then the conic can only be quadratic if it is parabolic, and it can not be symmetric. [ABSTRACT FROM AUTHOR]

In this paper we study the subgroup of the Picard group of Voevodsky's category of geometric motives DMgm(k;Z/2) generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [ On the invertibility of motives of affine quadrics , Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over k. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics. [ABSTRACT FROM AUTHOR]

• Characteristic roots detect relative positions between ellipsoids and paraboloids. • Positions between quadrics is detected by the roots of a fourth degree polynomial. • Coefficients of the polynomial suffice to detect relative positions. • The computational cost is low and allows continuous collision detection. We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid (small meaning that the two surfaces cannot be tangent at two points simultaneously when one is moved with respect to the other). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial. [ABSTRACT FROM AUTHOR]

This is a survey on various characteristic properties of ellipsoids and convex quadrics in the family of convex hypersurfaces in Rn. The topics under consideration include planar sections and projections, planarity conditions on midsurfaces and shadow-boundaries, intersections of homothetic copies, projective centers, and invariant mappings. [ABSTRACT FROM AUTHOR]

Abstract We prove that the Feynman quadrics-motive of the massive sunset graph is "generic" not mixed-Tate. Moreover, we explicitly describe its "extra" complexity in terms of a Prym variety. [ABSTRACT FROM AUTHOR]

We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measure are well known and have been used in the literature to prove Poncelet's and the zigzag theorems. Some further generalizations are also obtained by applying the new measure. [ABSTRACT FROM AUTHOR]

We characterize all possible relative positions between a circular hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. As an application, this provides a method to detect contact between the 2 surfaces by a simple calculation in many real world applications. [ABSTRACT FROM AUTHOR]

New lower degree rational parametrizations of regular quadric offsets in the 3-dimensional Euclidean space are explicitly derived. Offsets of ellipsoids, elliptic paraboloids, and doubly ruled quadrics are parametrized with bidegrees ( 4 , 8 ) , ( 4 , 6 ) , and ( 4 , 4 ) , respectively. These degree bounds are expected to be minimal. [ABSTRACT FROM AUTHOR]

Objectives Neandertal femora are particularly known for having a marked sagittal femoral curvature. This study examined femoral curvature in Neandertals in comparison to a modern human population from Belgium by the use of three-dimensional (3D) quadric surfaces modeled from the bone surface. 3D models provide detailed information and enabled femoral curvature to be analyzed in conjunction with other morphological parameters. Materials and Methods 3D models were created from CT scans of 75 modern human femora and 7 Neandertal femora. Quadric surfaces (QS) were created from the triangulated surface vertices in all areas of interest (neck, head, diaphyseal shaft, condyles) extracted from previously placed anatomical landmarks. The diaphyseal shaft was divided into five QS shapes and curvature was measured by degrees of difference between QS shapes. Each bone was placed in a local coordinate system enabling each bone to be analyzed in the same way. Results The use of 3D quadric surface fitting allowed the distribution of curvature with similarly curved femora to be analyzed and the different patterns of curvature between the two groups to be determined. The Neandertals were shown to have a higher degree of femoral curvature and a more distal point of femoral curvature than the modern human population from Belgium. Conclusions Morphological aspects of the Neandertal femur are different from this modern human population although mainly seem unrelated to femoral curvature. The relative lack of correlations with other femoral bony morphological factors suggests femoral curvature variations may be related to other aspects. [ABSTRACT FROM AUTHOR]

We construct a family of ( q − 1 ) 2 -ovoids of Q ( 4 , q ) , the parabolic quadric of PG ( 4 , q ) , for q ≡ 3 ( mod 4 ) . The existence of ( q − 1 ) 2 -ovoids of Q ( 4 , q ) was only known for q = 3 , 7 , or 11. Our construction provides the first infinite family of ( q − 1 ) 2 -ovoids of Q ( 4 , q ) . Along the way, we also give a construction of ( q + 1 ) 2 -ovoids in Q ( 4 , q ) for q ≡ 1 ( mod 4 ) . [ABSTRACT FROM AUTHOR]

• A condition on the size and the shape of an ellipsoid and another quadric allows simple contact detection between them. • Contact between the quadrics is detected in terms of the real or non-real nature of the roots of a characteristic polynomial. • A method to detect contact between a small ellipsoid and a combination of quadrics is proposed. We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition , is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given. [ABSTRACT FROM AUTHOR]

Given the equations of the first and the second order manifolds in R n , we construct the distance equation , i.e. a univariate algebraic equation one of the zeros of which (generically minimal positive) coincides with the square of the distance between these manifolds. To achieve this goal we employ Elimination Theory methods. In the frame of this approach we also deduce the necessary and sufficient algebraic conditions under which the manifolds intersect and propose an algorithm for finding the coordinates of their nearest points. The case of parameter dependent manifolds is also considered. [ABSTRACT FROM AUTHOR]

A two-character set in a finite projective space is a set of points with the property that the intersection number with any hyperplanes only takes two values. In this paper constructions of some two-character sets are given. In particular, infinite families of tight sets of the symplectic generalized quadrangle W ( 3 , q 2 ) and the Hermitian surface H ( 3 , q 2 ) are provided. A quasi-Hermitian variety H in PG ( r , q 2 ) is a combinatorial generalization of the (non-degenerate) Hermitian variety H ( r , q 2 ) so that H and H ( r , q 2 ) have the same number of points and the same intersection numbers with hyperplanes. Here we construct two families of quasi-Hermitian varieties, for r , q both odd, admitting P Γ O + ( r + 1 , q ) and P Γ O − ( r + 1 , q ) as automorphisms group. [ABSTRACT FROM AUTHOR]

In PG(3,q2), with q odd, we determine the possible intersection sizes of a Hermitian surface H and an irreducible quadric Q having the same tangent plane at a common point P∈Q∩H. [ABSTRACT FROM AUTHOR]

A certain Diophantine problem and 2D crystallography are linked through the notion of standard realizations which was introduced originally in the study of random walks. In the discussion, a complex projective quadric defined over ℚ is associated with a finite graph. "Rational points" on this quadric turns out to be related to standard realizations of 2D crystal structures. [ABSTRACT FROM AUTHOR]

Abstract: The authors have recently developed methods for cutter orientation and tool path generation in 5-axis sculptured surface machining, where the design surface is approximated locally by a quadric. This paper presents, from a purely geometric perspective, the fundamental theory for optimising the cutter orientation on a quadric, which maximises the machined strip width whilst avoiding local and rear gouging. The analysis focuses on the flat-end cutter which is modelled by a circular cylinder but can be generalised for any fillet-end cutter using an appropriate offset of the design surface and the concept of geometric equivalency. The theory is illustrated by three examples. [Copyright &y& Elsevier]

Abstract: Let be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools—a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting—to study semiorthogonal decompositions of the bounded derived category . Together with results in the theory of quadratic forms, we apply these tools in the case where and has relative dimension 1, 2, or 3, in which case the fibers are curves of genus one, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X. [Copyright &y& Elsevier]

Abstract: We study the existence of invariant quadrics for a class of systems of difference equations in defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix A and we prove that there is a correspondence between non-degenerate invariant quadrics and solutions to a certain matrix equation involving A. We show that if A is semisimple and the corresponding system admits non-degenerate quadrics, then every orbit of the dynamical system is contained either in an invariant affine variety or in an invariant quadric. [Copyright &y& Elsevier]

Abstract: The conchoid surface of a surface F with respect to a fixed reference point O is a surface obtained by increasing the distance function with respect to O by a constant d. This contribution studies conchoid surfaces of quadrics in Euclidean and shows that these surfaces admit real rational parameterizations. We present an algorithm to compute these parameterizations and discuss several configurations of the position of O with respect to F where the computation simplifies significantly. [Copyright &y& Elsevier]

Quasi-Hermitian varieties [ABSTRACT FROM AUTHOR]

Abstract: The analysis of large volumes of unordered multidimensional data is a problem confronted by scientists and data analysts every day. Often, it involves searching for data alignments that emerge as well-defined structures or geometric patterns in datasets. For example, straight lines, circles, and ellipses represent meaningful structures in data collected from electron backscatter diffraction, particle accelerators, and clonogenic assays. Also, customers with similar behavior describe linear correlations in e-commerce databases. We describe a general approach for detecting data alignments in large unordered noisy multidimensional datasets. In contrast to classical techniques such as the Hough transforms, which are designed for detecting a specific type of alignment on a given type of input, our approach is independent of the geometric properties of the alignments to be detected, as well as independent of the type of input data. Thus, it allows concurrent detection of multiple kinds of data alignments, in datasets containing multiple types of data. Given its general nature, optimizations developed for our technique immediately benefit all its applications, regardless the type of input data. [Copyright &y& Elsevier]

Abstract: Let be a point set in the -dimensional affine space over the finite field of elements and . In this paper we extend the definition of directions determined by : a -dimensional subspace at infinity is determined by if there is an affine -dimensional subspace through such that spans . We examine the extremal case , and classify point sets not determining every -subspace in certain cases. [Copyright &y& Elsevier]

Abstract: We develop a rational biquadratic analogue of the non-uniform B-spline paradigm. These splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure. This enables a design process that starts with basic shapes, re-represents them in spline form and uses the spline form to provide shape handles for localized free-form modification that can preserve, in the large, the initial fair, basic shapes. [Copyright &y& Elsevier]

The paper deals with rigid isotopy classes of three-dimensional real quadric line complexes and associated Kummer surfaces. We prove that there exist twelve rigid isotopy classes of real quadric line complexes and seven rigid isotopy classes of associated Kummer surfaces. Characteristics determining these rigid isotopy classes are given. [ABSTRACT FROM AUTHOR]

Abstract: Existing orientation strategies in 5-axis sculptured surface machining mostly limit the cutter to incline in one direction and lack an effective approach to assess the orientation. This paper presents the quadric method (QM) that exploits fully the two orientation angles to maximise the machining efficiency at a cutter contact point. The geometric construction assumes that the local shape of a sculptured surface can be suitably approximated by two quadrics. An upper quadric lying above the surface is used to orient the cutter, and a lower quadric lying below the surface is used to evaluate machined strip width. Then, the cutter orientation is optimised with respect to the width and is guaranteed to give no gouging. Both flat-end cutter and fillet-end cutter are considered. Simulated examples demonstrate the improved machining efficiency of the QM over current published methods. [Copyright &y& Elsevier]

Abstract: We study the functional codes defined by Lachaud in [G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, in: Arithmetic, Geometry, and Coding Theory, Luminy, France, 1993, de Gruyter, Berlin, 1996, pp. 77–104] where is an algebraic projective variety of degree d and dimension m. When X is a Hermitian surface in , Sørensen in [A.B. Sørensen, Rational points on hypersurfaces, Reed–Muller codes and algebraic-geometric codes, PhD thesis, Aarhus, Denmark, 1991], has conjectured for (where ) the following result: which should give the exact value of the minimum distance of the functional code . In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. ), we show the geometrical structure of the minimum weight codewords and their number; we also estimate the second weight and the geometrical structure of the codewords reaching this second weight. [Copyright &y& Elsevier]

The article discusses the morphisms in quadrics. It states that x, y must be a smooth projective varieties and f : X → Y a surjective morphism. It relates that it is natural to ask under what conditions the family of possible f is bounded, that is, whether the parameterizing space for such morphisms has only finitely many irreducible components. It adds that it is interesting to check the boundedness when Y is an n-dimensional quadric (n > 2). Computations are presented.

Abstract: Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square-type graphs and whose underlying set is a nonelementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer''s result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes. [Copyright &y& Elsevier]

Abstract: A Tallini set in a semilinear space is a set B of points, such that each line not contained in B intersects B in at most two points. In this paper, the following notion of a tangential Tallini set in the Grassmannian , q odd, is investigated: a Tallini set is called tangential when it meets every ruled plane (i.e. the set of lines contained in a plane of ) in either or elements. A Tallini set in can be associated with each tangential Tallini set B in . Each is a line of intersecting in either 0, or 1, or points; when and B is covered by -dimensional projective subspaces of the first case does not occur. If B is a tangential Tallini set in covered by -dimensional subspaces, any of which is in the set of all lines through a point and in a hyperplane, then either is a quadric, and B is the set of all lines contained in, or tangent to, , or B is a linear complex. [Copyright &y& Elsevier]

Abstract: In this paper, we study a sweeping algorithm for computing the arrangement of a set of quadrics in . We define a “trapezoidal” decomposition in the sweeping plane, and we study the evolution of this subdivision during the sweep. A key point of this algorithm is the manipulation of algebraic numbers. In this perspective, we put a large emphasis on the use of algebraic tools, needed to compute the arrangement, including Sturm sequences and Rational Univariate Representation of the roots of a multivariate polynomial system. [Copyright &y& Elsevier]

An algorithm for constructing arbitrary parametric quadratic quadric triangles in rational Bézier form is presented. The algorithm does not require the knowledge of the underlying quadric, an important property in view of applying this method for the interpolation of triangulated 3D data points. The algorithm consists of four steps starting with the arbitrary choice of the three corner points and corner weights of the patch, by then constructing a certain triangle and a tetrahedron by means of which the remaining inner control points and weights are obtained guaranteeing the resulting patch to lie on a quadric surface. [ABSTRACT FROM AUTHOR]

Infinite faces of perfect Voronoi polyhedra are studied. The result substantially varies depending on whether the perfect Voronoi polyhedron is considered as the closed or nonclosed convex hull of the set of Voronoi points (see Theorem 1 in Sec. 5 and Theorem 2 in Sec. 7). [ABSTRACT FROM AUTHOR]

We introduce a new construction method for semipartial geometries, called point derivation, which is a generalization of the known theory of constructing partial quadrangles from generalized quadrangles. [Copyright &y& Elsevier]

Debroey and Thas introduced semipartial geometries and determined the full embeddings of semipartial geometries in AG(n,q) for n=2 and 3. For n>3 there is no such classification. A model of a semipartial geometry fully embedded in AG(4,q), q even, due to Hirschfeld and Thas, is the spg(q−1,q2,2,2q(q−1)) constructed by projecting the quadric Q−(5,q) from a point of PG(5,q)&z.drule;Q−(5,q). In this paper this semipartial geometry is characterized amongst the spg(q−1,q2,2,2q(q−1)) (of which there is an infinite family of non-classical examples due to Brown) by its full embedding in AG(4,q). [Copyright &y& Elsevier]

The aim of this paper is to study what happens when a slight perturbation affects the coefficients of a quadratic equation defining a variety (a quadric) in Rn. Structurally stable quadrics are those such that a small perturbation on the coefficients of the equation defining them does not give rise to a “different” (in some sense) set of points. In particular, we characterize structurally stable quadrics and give the “bifurcation diagrams” of the non-stable ones (showing which quadrics meet all of their neighbourhoods), when dealing with the “affine” and “metric” equivalence relations. This study can be applied to the case where a set of points, which constitute the set of solutions of a problem, is defined by a quadratic equation whose coefficients are given with parameter uncertainty. [Copyright &y& Elsevier]

In this note strongly regular graphs with new parameters are constructed using nested “blown up” quadrics in projective spaces. [Copyright &y& Elsevier]

Bruen proved that if A is a set of points in AG(n,q) which intersects every hyperplane in at least t points, then |A|⩾(n+t−1)(q−1)+1, leaving as an open question how good such bound is. Here we prove that, up to a trivial case, if t>((n−1)(q−1)+1)/2, then Bruen''s bound can be improved. If t is equal to the integer part of ((n−1)(q−1)+1)/2, then there are some examples which attain such a lower bound. Somehow, this suggests the following combinatorial characterization: if a set S of points in PG(3,q) meets every affine plane in at least q−1 points and is of minimum size with respect to this property, then S is a hyperbolic quadric. [Copyright &y& Elsevier]

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion "degree of neighborhood" is used for the geometric description of polar subspaces of quadrics. The polarity induced by a quadric is also studied. [ABSTRACT FROM AUTHOR]