Nondegenerate Form Research Articles

The Common Basis Complex and the Partial Decomposition Poset

Abstract For a finite-dimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper non-zero subspaces of $V$, and $\sigma $ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in \sigma $. This complex was introduced by Rognes in 1992 in connection with stable buildings. In this article, we prove that the common basis complex is homotopy equivalent to the proper part of the poset of partial direct sum decompositions of $V$. Moreover, we establish this result in a more general combinatorial context, including the case of free groups, matroids, vector spaces with non-degenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains.

Homotopy properties of the complex of frames of a unitary space

AbstractLet be a finite‐dimensional vector space equipped with a nondegenerate Hermitian form over a field . Let be the graph with vertex set the one‐dimensional nondegenerate subspaces of and adjacency relation given by orthogonality. We give a complete description of when is connected in terms of the dimension of and the size of the ground field . Furthermore, we prove that if , then the clique complex of is simply connected. For finite fields , we also compute the eigenvalues of the adjacency matrix of . Then, by Garland's method, we conclude that for all , where is a field of characteristic 0, provided that . Under these assumptions, we deduce that the barycentric subdivision of deformation retracts to the order complex of the certain rank selection of that is Cohen–Macaulay over . Finally, we apply our results to the Quillen poset of elementary abelian ‐subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of and the poset of orthogonal decompositions of .

Nontrivial t-designs in polar spaces exist for all t

AbstractA finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over $$\mathbb {F}_q$$ F q equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-$$(n,k,\lambda )$$ ( n , k , λ ) design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly $$\lambda $$ λ members of Y. Nontrivial examples are currently only known for $$t\le 2$$ t ≤ 2 . We show that t-$$(n,k,\lambda )$$ ( n , k , λ ) designs in polar spaces exist for all t and q provided that $$k>\frac{21}{2}t$$ k > 21 2 t and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

In this paper we use broken book decompositions to study Reeb flows on closed 3 -manifolds. We show that if the Liouville measure of a non-degenerate contact form can be approximated by periodic orbits, then there is a Birkhoff section for the associated Reeb flow. In view of Irie’s equidistribution theorem, this is shown to imply that the set of contact forms whose Reeb flows have a Birkhoff section contains an open and dense set in the C^{\infty} -topology. We also show that the set of contact forms whose Reeb flows have positive topological entropy is open and dense in the C^{\infty} -topology.

Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: Symplectic and hermitian forms

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace π, we find the number of non-singular subspaces that are trivially intersecting with π and span a non-singular subspace with π. Lower bounds for the quantity of such pairs where π is non-singular were first studied in “Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)”, which was later improved in “Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)” and generalised in “Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)”. In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds.

Integration of a degenerate system of ODEs

The integrability of a two-dimensional autonomous polynomial system of ordinary differential equations (ODEs) with a degenerate singular point at the origin that depends on six parameters is investigated. The integrability condition for the first quasihomogeneous approximation allows one of these parameters to be fixed on a countable set of values. The further analysis is carried out for this value and five free parameters. Using the power geometry method, the system is reduced to a non-degenerate form through the blowup process. Then, the necessary conditions for its local integrability are calculated using the method of normal forms. In other words, the conditions for the parameters under which the original system is locally integrable near the degenerate stationary point are found. By resolving these conditions, we find seven twoparameter families in the five-dimensional parametric space. For parameter values from these families, the first integrals of the system are found. The cumbersome calculations that occur in the problem under consideration are carried out using computer algebra.

The Properties of Topological Manifolds of Simplicial Polynomials

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition.

Pfister forms and a conjecture due to Colliot-Thélène in the mixed characteristic case

Let $R$ be a regular local ring of mixed characteristic $(0,p)$, where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. We fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.

Формы Пфистера и гипотеза Кольe-Телена для случая смешанной характеристики

Let $R$ be a regular local ring of a mixed characteristic $(0,p)$ where $p\neq 2$ is a prime number. Suppose that the quotient ring $R/pR$ is also regular. Fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$ and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$ if and only if it has a solution over the fraction field $K$.

Commutativity of quantization with conic reduction for torus actions on compact CR manifolds

We define conic reductions Xνred\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\ extrm{red}}_{\ u }$$\\end{document} for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} labeling a unitary irreducible representation. There is a natural residual circle action on Xνred\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\ extrm{red}}_{\ u }$$\\end{document}. We have two natural decompositions of the corresponding Hardy spaces H(X) and H(Xνred)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X^{\ extrm{red}}_{\ u })$$\\end{document}. The first one is given by the ladder of isotypes H(X)kν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X)_{k\ u }$$\\end{document}, k∈Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}$$\\end{document}; the second one is given by the k-th Fourier components H(Xνred)k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X^{\ extrm{red}}_{\ u })_k$$\\end{document} induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)-forms with L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-coefficient when X is a CR manifold with non-degenerate Levi form.

Complete description of invariant, associative pseudo-Euclidean metrics on left Leibniz algebras via quadratic Lie algebras

A pseudo-Euclidean non-associative algebra (g,•) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈,〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈u•v,w〉=〈u,v•w〉 for all u,v,w∈g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra.This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5.

Remarks on star products for non-compact CR manifolds with non-degenerate Levi form

Remarks on star products for non-compact CR manifolds with non-degenerate Levi form

Centers of multilinear forms and applications

The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.

Packings and Steiner systems in polar spaces

A finite classical polar space of rank \(n\) consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that \(n\) is the maximal dimension of such a subspace. A \(t\)-Steiner system in a finite classical polar space of rank \(n\) is a collection \(Y\) of totally isotropic \(n\)-spaces such that each totally isotropic \(t\)-space is contained in exactly one member of \(Y\). Nontrivial examples are known only for \(t=1\) and \(t=n-1\). We give an almost complete classification of such \(t\)-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.Mathematics Subject Classifications: 51E23, 05E30, 33C80Keywords: Association schemes, codes, polar spaces, Steiner systems

Affine Hypersurfaces of Arbitrary Signature with an Almost Symplectic Form

In this paper we study affine hypersurfaces with non-degenerate second fundamental form of arbitrary signature additionally equipped with an almost symplectic structure omega . We prove that if R^pomega =0 or nabla ^pomega =0 for some positive integer p then the rank of the shape operator is at most one. The results provide complete classification of affine hypersurfaces with higher order parallel almost symplectic forms and are generalization of recently obtained results for Lorentzian affine hypersurfaces.

Passivity and Immersion (P&I) Approach for Constructive Stabilization and Control of Nonlinear Systems

This letter proposes a control algorithm as a stepping stone for the stabilization and control of structured and unstructured systems discussed in the literature. The foundation behind this proposed controller design theory utilizes an invariant target manifold giving rise to a non-degenerate form, through which the definition of certain passive outputs and storage functions provide the control law for stabilizing the system. The proposed control framework is developed by integrating the notions of immersion and passivity hence labeled as the “Passivity and Immersion (P&I) approach.” Immersing the system dynamics into the user-defined lower order target dynamics and using the concept of attractivity of the manifold based upon the passivity theory results in encompassing various design paradigms under a single roof of the P&I approach. The benchmark examples are solved to show the effectiveness of the proposed P&I approach in confirming how the various design paradigms can be unified.

Toeplitz Operators on CR Manifolds and Group Actions

Let X be a connected orientable compact CR manifold with non-degenerate Levi form. In this paper, we study the algebra of Toeplitz operators on X and we establish star product for a certain class of symbols on X. In the second part of this paper, we consider a locally free action of a compact Lie group G on X and we investigate the associated algebra of G-invariant Toeplitz operators. The main application is a construction of deformation quantization of compact “quantizable” pseudo-Kähler manifolds.

On the existence of supporting broken book decompositions for contact forms in dimension 3

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the tight sphere or on a tight lens space. Moreover we get that if M is a closed oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy.

3-form Yang-Mills based on 2-crossed modules

In this paper, we study the higher Yang-Mills theory in the framework of higher gauge theory. It was shown that the 2-form electromagnetism can be generalized to the 2-form Yang-Mills theory with the group U(1) replaced by a crossed module of Lie groups. To extend this theory to even higher structure, we develop a 3-form Yang-Mills theory with a 2-crossed module of Lie groups. First, we give an explicit construction of non-degenerate symmetric G-invariant forms on the 2-crossed module of Lie algebras. Then, we derive the 3-Bianchi-Identities for 3-curvatures. Finally, we create a 3-form Yang-Mills action and obtain the corresponding field equations.

Translation Surfaces of the Third Fundamental Form in Lorentz-Minkowski Space

In this paper we study translation surfaces with the non-degenerate third fundamental form in Lorentz- Minkowski space $mathbb{L}^{3}$. As a result, we classify translation surfaces satisfying an equation in terms of the position vector field and the Laplace operator with respect to the third fundamental form $III$ on the surface.