Finite Linear Space Research Articles

Alternating groups and point‐primitive linear spaces with number of points being squarefree

AbstractThis paper is a further contribution to the classification of point‐primitive finite regular linear spaces. Let be a nontrivial finite regular linear space whose number of points is squarefree. We prove that if is point‐primitive with an alternating socle, then is the projective space .

Optimal control and learning for cyber‐physical systems

Optimal control and learning for <scp>cyber‐physical</scp> systems

Classification of point-primitive linear spaces with $2pq$ points

This paper is a further contribution to the classification of point-primitive finite linear spaces. Let $ p,q$ be two primes. We prove that if $\mathcal{S}$ is a non-trivial finite linear space with $2pq$ points, and $G\leq Aut(\mathcal{S})$ is point-primitive, then $G $ is line-transitive and $\mathcal{S}$ is the Ree unital $U_R(3), $ or the Hermitian unital $U_H(s). $

The Levi-Civita equation in function classes

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form sum _{i=1}^np_i cdot m_i, where m_1 ,ldots ,m_n are exponentials belonging to V, and p_1 ,ldots ,p_n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra {{mathcal {A}}} of complex valued functions such that whenever an exponential m belongs to {{mathcal {A}}}, then m^{-1}in {{mathcal {A}}}. As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary sigma -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

Line-transitive point-imprimitive linear spaces with number of points being a product of two primes

The work studies the line-transitive point-imprimitive automorphism groups of finite linear spaces, and is underway on the situation when the numbers of points are products of two primes. Let [Formula: see text] be a non-trivial finite linear space with [Formula: see text] points, where [Formula: see text] and [Formula: see text] are two primes. We prove that if [Formula: see text] is line-transitive point-imprimitive, then [Formula: see text] is solvable.

Point-primitive linear spaces with number of points being a product of two primes

ABSTRACTThis work is a part of the general program to classify point-primitive finite linear spaces. Let 𝒮 be a non-trivial finite regular linear space with mp points and G≤Aut(𝒮) be point-primitive, here m and p are two primes with m≤p. In this paper, we give a classification of the pairs (G,𝒮). For a small positive integer k≥3 and a non-regular transitive group G on 𝒫, where |𝒫| = v, we also present an algorithm to sift all 2-(v,k,1) designs with point-set 𝒫, and admitting G as an automorphism group.

Another strongly exceptional collection of coherent sheaves on a Grassmannian

On the Grassmannian of 2-dimensional subspaces in a finite dimensional linear space we construct a Karoubian complete strongly exceptional PO set of coherent sheaves, parametrized by the cosets of the Weyl group of the general linear group of the linear space modulo the Weyl group of the parabolic subgroup stabilizing the subspace, from subquotients of the Frobenius direct image of the structure sheaf of the Grassmannian defined over a field of large positive characteristic. Our collection diverges from the one discovered by Kapranov. We also show in the general setting over any field of positive characteristic that the sheaf corresponding to the longest element of the cosets is a direct summand of the Frobenius direct image of the structure sheaf of the homogeneous space.

Extremely primitive groups and linear spaces

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G ≤ Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) ≥ 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group.

An algebraic approach to finite projective planes

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley---Reisner ring $$R/I_\Lambda $$R/I? and the inverse system algebra $$R/I_\Delta $$R/IΔ. We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a classification of the characteristics in which the inverse system algebra associated to a finite projective plane has the weak or strong Lefschetz Property.

Operator norm attainment and Birkhoff–James orthogonality

Recently in [Sain D., Paul K., Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013) 2448–2452] it was proved that if T is a linear operator on a finite dimensional real normed linear space X such that T attains norm only on ±D, where D is a connected closed subset of SX, then T satisfies the Bhatia–Šemrl (BŠ) property [Bhatia R., Šemrl P., Orthogonality of matrices and distance problem, Linear Algebra Appl. 287 (1999) 77–85], i.e., for A∈L(X), T⊥BA implies that there exists x∈D such that Tx⊥BAx. Here we explore the converse of the above result. We prove that in a real normed linear space X of dimension 2, a linear operator T satisfies the BŠ property if and only if the set of unit vectors on which T attains norm is connected in the corresponding projective space RP1≡SX/{x∼−x}. Motivated by the result in 2 dimensions, we conjecture that this characterization of BŠ property is true in general n-dimensional real normed linear spaces. We further prove that if the space X is strictly convex, then the set of operators in L(X) which satisfy the BŠ property is dense in L(X).

The Subspaces Cover of Linear Space

This paper mainly discusses the problem that linear space is covered by its subspaces, which is a difficult part in related books. We used the cardinality of infinite set to reveal the subspaces cover, proposed the concept of minimal cover, and analyzed the minimal cover of finite dimensional linear space.

Operator norm attainment and inner product spaces

In this paper we prove that a finite dimensional real normed linear space X is an inner product space iff for any linear operator T on X, T attains its norm at e1,e2∈SX implies T attains its norm at span{e1,e2}∩SX.

Line-transitive point-imprimitive linear spaces with Fang-Li parameter gcd(k, r) at most ten

This paper is a further contribution to the classification of linetransitive finite linear spaces. We prove that if ℐ is a non-trivial finite linear space such that the Fang-Li parameter gcd(k, r) is 9 or 10, and the group G ⩽ Aut(ℐ) is line-transitive and point-imprimitive, then ℐ is the Desarguesian projective plane PG(2, 9).

Matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields

Algebraic number fields are of particular interest and play an important role in both mathematics and engineering since an algebraic number field can be viewed as a finite dimensional linear space over the rational number field Q. Algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics. In this paper, we properly formulate the matrix splitting problem over any general subfield of it C, including an algebraic number field as a special case, and provide a simple necessary and sufficient condition for a 2×2 matrix of Laurent polynomials with symmetry to be able to be factorized by a 2×2 matrix of Laurent polynomials with certain symmetry structure. We propose an effective algorithm on how to obtain the factorization matrix step by step. As an application, we obtain a satisfactory algorithm for constructing dyadic framelet filter banks with the perfect reconstruction property and with symmetry over algebraic number fields. Several examples are provided to illustrate the algorithms proposed in this paper.

Robust Experiment Design for System Identification via Semi-Infinite Programming Techniques

Robust optimal experiment design for dynamic system identification is cast as a minmax optimization problem, which is infinite-dimensional. If the input spectrum is discretized (either by considering a Riemmann approximation, or by restricting it to the span of a finite dimensional linear space), this problem falls within the class of semi-infinite convex programs. One approach to this optimization problem of infinite constraints is the so called “scenario approach”, which is based on a probabilistic description of the uncertainty to deliver a finite program that attempts to approximate the optimal solution with a prescribed probability. In this paper, we propose as an alternative an exchange algorithm based on some recent advances in the field of semi-infinite programming to tackle the same problem. This method is compared with the scenario approach both from the aspects of accuracy and computational efficiency. Furthermore, the comparison includes the MATLAB semi-infinite solver fseminf to provide a general palette of methods approximating the robust optimal design problem.

On the Minimal Polynomial of a Vector

It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proof was published in 1878 by G. Frobenius, and numerous others have appeared since then, for example see [1,2]. From the structure theorem for finitely generated modules over a principal ideal domain it straightforwardly follows the Cayley-Hamilton theorem and the proposition that there exists a vector v in a finite dimensional linear space V such that and a linear transformation of V have the same minimal polynomial. In this note, we provide alternative proofs of these results by only utilizing the knowledge of linear algebras.

Auerbach Bases and Minimal Volume Sufficient Enlargements

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

Particle-in-wavelets scheme for the 1D Vlasov-Poisson equations

A new numerical scheme called particle-in-wavelets is proposed for the Vlasov-Poisson equations, and tested in the simplest case of one spatial dimension. The plasma distribution function is discretized using tracer particles, and the charge distribution is reconstructed using wavelet-based density estimation. The latter consists in projecting the Delta distributions corresponding to the particles onto a finite dimensional linear space spanned by a family of wavelets, which is chosen adaptively. The stability and accuracy of the scheme is supported by numerical computations of Landau damping and of the two-stream instability. By direct comparison with a reference solution obtained by a very precise semi-Lagrangian method, we show that the precision is improved roughly by a factor 3 compared to a classical PIC scheme, for a given number of particles.

Line-primitive linear spaces with Fang-Li parameter gcd(k, r) at most 12

Let G be a subgroup of the automorphism group of a non-trivial finite linear space S. In this paper we prove that if G is line-primitive with Fang-Li parameter gcd(k, r) = 11 and 12, then G is also point-primitive. This extends the results of Fang and Li (1993) and Li and Liu (2001) which combined gives the result for gcd(k, r) ≤ 10.

Concurrent and parallel chords of spheres in normed linear spaces

We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.