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We prove a chain lemma for inner product spaces over commutative local rings R with residue field other than F 2 and use this to show that the usual presentation of the Grothendieck–Witt group of symmetric bilinear forms over R as the zeroth Milnor–Witt K-group holds provided the residue field of R is not F 2 . [ABSTRACT FROM AUTHOR]

For the given signature operator H = I r ⊕ − I n − r , a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form 〈 ⋅ , ⋅ 〉 H , and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n × n pseudo-Jacobi matrix from a prescribed n -tuple of distinct real numbers and a Jacobi matrix of order not less than ⌊ n 2 ⌋ , such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm. [ABSTRACT FROM AUTHOR]

For a projective variety $ X $ in $ {\mathbb{P}}^{m} $ of dimension $ n $, an additive action on $ X $ is an effective action of $ {\mathbb{G}}_{a}^{n} $ on $ {\mathbb{P}}^{m} $ such that $ X $ is $ {\mathbb{G}}_{a}^{n} $-invariant and the induced action on $ X $ 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 $ {\mathbb{G}}_{a}^{n} $-action.

In this study, the main purpose is to obtain pseudo quaternionic Lorentzian evolute-involute curves on 𝐿𝑄 4 . In fact the pseudo quaternion is a quaternion which has 3-dimension and its temporal part is ziro. So it has only vectoral part of the quaternions. Thus, if defined a pseudo-quaternionic curve on 𝐿𝑄 4, then its quaternionic part is pseudo quaternion. What is meant to do here, to write out Lorentzian evolute-involute curves of the quaternionic time-like curves in 𝐿𝑄 4 using quaternions and to reveal some relationships between these curves. [ABSTRACT FROM AUTHOR]

For $n\geq 2,$ we show that every extreme point of the unit ball of ${\mathcal L}_s(^2l_{\infty}^n)$ is extreme in ${\mathcal L}_s(^2l_{\infty}^{n+1})$, which answers the question in [Period. Math. Hungar. 2018, 77 (2), 274-290]. As a corollary we show that every extreme point of the unit ball of ${\mathcal L}_s(^2l_{\infty}^n)$ is extreme in ${\mathcal L}_s(^2l_{\infty})$. We also show that every extreme point of the unit ball of ${\mathcal P}(^2l_{\infty}^2)$ is extreme in ${\mathcal P}(^2l_{\infty}^n).$ As a corollary we show that every extreme point of the unit ball of ${\mathcal P}(^2l_{\infty}^2)$ is extreme in ${\mathcal P}(^2l_{\infty})$.

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]

We say that a field extension L / F has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over F that become isometric (resp. similar) over L are already isometric (resp. similar) over F. The famous Artin–Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic 2. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic 2, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic 2. [ABSTRACT FROM AUTHOR]

For n ≥ 2, we show that every extreme point of the unit ball of Ls( ² l n ∞) is extreme in Ls( ² l n+1 ∞ ), which answers the question in [Period. Math. Hungar. 2018, 77 (2), 274–290]. As a corollary we show that every extreme point of the unit ball of Ls( ² l n ∞) is extreme in Ls( ² l∞). We also show that every extreme point of the unit ball of P( ² l ² ∞) is extreme in P( ² l n ∞). As a corollary we show that every extreme point of the unit ball of P( ² l ² ∞) is extreme in P( ² l∞). [ABSTRACT FROM AUTHOR]

Let S m (F) denote the set of symmetric matrices over a finite field F . Let C be an additive subset of S m (F) which has minimum rank-distance d and | C | meets the upper bound. We call C a maximum additive d-code. In particular, when d = m , C is equivalent to a symplectic semifield spreadset. For a given invertible matrix P, S 0 ∈ S m (F) and nonzero a ∈ F , it is easy to see that the map M ↦ a P T M σ P + S 0 defines an isometry on S m (F) with respect to the rank-metric. However, by examples of symplectic semifields, it is already known that there exist other maps preserving the rank-distance of maximum additive m-codes. In this paper, we prove that this can never happen when d < m . We also present a new construction of maximum additive 2-codes in S m (F) for odd m which are not t-designs. [ABSTRACT FROM AUTHOR]

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.