Dimensional Vector Subspace Research Articles

The Bloch–Nevanlinna Problem Under Lineability

The original version of the celebrated Bloch–Nevanlinna problem asked whether there exists or not a holomorphic function in the unit disc of bounded characteristic whose derivative is not of bounded characteristic. This problem has been solved in the affirmative by a number of mathematicians. Starting from a result on topological genericity of this class of functions due to Hahn, our work intends—under an algebraic as well as a topological point of view—to contribute to this topic. Specifically, the family of solutions of the Bloch–Nevanlinna problem is proved to be residual in appropriate topological spaces, and it contains, except for the zero function, dense maximal dimensional vector subspaces, large linear algebras and infinite-dimensional Banach spaces.

A note on the height of the third Stiefel--Whitney class of the canonical vector bundles over some Grassmann manifolds

We compute the height of the third Stiefel--Whitney characteristic class of the canonical bundles over some infinite classes of Grassmann manifolds of five dimensional vector subspaces of real vector spaces.

Structure of tangencies to distributions via the implicit function theorem

We investigate the structure and the dimension of the tangency set to a C^1 smooth distribution of n -dimensional vector subspaces of \mathbb R^{n+m} , by an argument based on the implicit function theorem.

Implicitization of tensor product surfaces in the presence of a generic set of basepoints

Given a [Formula: see text]-dimensional vector subspace [Formula: see text] of [Formula: see text], a tensor product surface, denoted by [Formula: see text], is the closure of the image of the rational map [Formula: see text] determined by [Formula: see text]. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of [Formula: see text] in [Formula: see text]. In this paper, we show that if [Formula: see text] has a finite set of [Formula: see text] basepoints in generic position, then the implicit equation of [Formula: see text] is determined by two syzygies of [Formula: see text] in bidegrees [Formula: see text] and [Formula: see text]. This result is proved by understanding the geometry of the basepoints of [Formula: see text] in [Formula: see text]. The proof techniques for the main theorem also apply when [Formula: see text] is basepoint free.

Locally 2-fold symmetric manifolds are locally symmetric

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that for $k \ge 2$, Riemannian, pseudoriemannian and Finslerian locally $k$-fold symmetric manifolds are locally symmetric.

Lineability and algebrability of the set of holomorphic functions with a given domain of existence

Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.

Lineability criteria, with applications

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.

On the Krein–Milman Property and the Bade Property

Throughout this paper a study on the Krein–Milmam Property and the Bade Property is entailed reaching the following conclusions: If a real topological vector space satisfies the Krein–Milmam Property, then it is Hausdorff; if a real topological vector space satisfies the Krein–Milmam Property and is locally convex and metrizable, then all of its closed infinite dimensional vector subspaces have uncountable dimension; if a real pseudo-normed space has the Bade Property, then it is Hausdorff as well but could allow closed infinite dimensional vector subspaces with countable dimension. On other hand, we show the existence of infinite dimensional closed subspaces of ℓ∞ with the Bade Property that are not the space of convergence associated to any series in a real topological vector space. Finally, we characterize unconditionally convergent series in real Banach spaces by means of a new concept called uniform convergence of series.

New quotients of the d-dimensional Veronesean dual hyperoval in PG(2d + 1,2)

Let d ≥ 3 . For each e ≥ 1 , Thas and Van Maldeghem constructed a d -dimensional dual hyperoval in PG ( d ( d + 3 ) ∕ 2 , q ) with q = 2 e , called the Veronesean dual hyperoval. A quotient of the Veronesean dual hyperoval with ambient space PG ( 2 d + 1 , q ) , denoted S σ , is constructed by Taniguchi, using a generator σ of the Galois group Gal ( GF ( q d + 1 ) ∕ GF ( q ) ) . In this note, using the above generator σ for q = 2 and a d -dimensional vector subspace H of GF ( 2 d + 1 ) over GF ( 2 ) , we construct a quotient S σ , H of the Veronesean dual hyperoval in PG ( 2 d + 1 , 2 ) in case d is even. Moreover, we prove the following: for generators σ and τ of the Galois group Gal ( GF ( 2 d + 1 ) ∕ GF ( 2 ) ) , S σ above (for q = 2 ) is not isomorphic to S τ , H , S σ , H is isomorphic to S σ , H ′ for any d -dimensional vector subspaces H and H ′ of GF ( 2 d + 1 ) , and S σ , H is isomorphic to S τ , H if and only if σ = τ or σ = τ − 1 . Hence, we construct many new non-isomorphic quotients of the Veronesean dual hyperoval in PG ( 2 d + 1 , 2 ) .

UNIFORMIZERS FOR ELLIPTIC SHEAVES

In this paper we associate to t-motives with level structures a finite dimensional vector subspace (subspace of uniformizers). We study the particular case of elliptic sheaves finding some relations with some objects from conformal field theory.

Subspace subcodes of Reed-Solomon codes

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed /spl nu/-dimensional vector subspace S of GF (2/sup m/). SSRS codes are constructed using properties of the Galois field GF(2/sup m/). They are not linear over the field GF(2/sup /spl nu//), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2/sup /spl nu//) is a subfield of GF(2/sup m/), it may not be the best /spl nu/-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems.

A Characterization of Binary Bent Functions

A recent paper by Carlet introduces a general class of binary bent functions on (GF(2))n(neven) whose elements are expressed by means of characteristic functions (indicators) of (n/2)-dimensional vector-subspaces of (GF(2))n. An extended version of this class is introduced in the same paper; it is conjectured that this version is equal to the whole class of bent functions. In the present paper, we prove that this conjecture is true.

Generalized partial spreads

We exhibit a simple condition under which the sum (modulo 2) of characteristic functions of (n/2)-dimensional vector subspaces of (GF(2))/sup n/ (n even) is a Bent function. The "Fourier" transform of such a Bent function is the sum of the characteristic functions of the duals of these spaces. The class of Bent functions that we obtain contains the whole partial spreads class. Any element of Maiorana-McFarland's class or of class D is equivalent to one of its elements. Thus this new class gives a unified insight of both general classes of Bent functions studied by Dillon (1974) in his thesis. We deduce a way to construct new classes of Bent functions and exhibit an example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modular elements of lattices and topological fibration

An l-arrangement is a finite collection of (I 1 )-dimensional vector subspaces of an I-dimensional vector space V over a field K. We restrict our attention to the case that K =C for a while. The complement M of the hyperplanes is an open manifold. In [3], Falk and Randell studied a libration of M: M is called strictly linearly jibered if after a suitable linear change of coordinates the restriction of the projection to the first (I 1) coordinates is a fiber bundle projection with base N the complement of an arrangement in C? ’ and fiber a complex plane C with finitely many points removed. The arrangement is called fiber-type when it allows a successive strictly linearly fibrations finally reaching C* as base. In this case they investigated several topological properties of M. Here we are interested in a combinatorial characterization of a strictly linear fibration in lattice theoretic terms. For this, the concept of modular elements defined by Stanley [S] is crucial: For example, concerning a geometric lattice arising from an essential (= the intersection of all the elements is the origin) arrangement defined over C, the existence of l-dimensional modular element is equivalent to a strictly linear libration of the complement M (2.15). This is a corollary of Theorem (2.9). As another corollary we have a characterization (2.17) that the arrangement is fiber-type if and only if it gives rise to an SS (= supersolvable) lattice, which is defined also by Stanley [9] to be a geometric lattice, which is defined also by Stanley [9] to be a geometric lattice with an ascending maximal chain consisting of modular elements. These are done in Section 2. In Section 3, we state Theorem (3.8), asserting that the algebra A associated with a finite geometric lattice studied in [S, 63 factors as a tensor product of graded subspaces by a modular element, which is Theorem (3.8). This has a topological interpretation (3.9) when the lattice originates from an arrangement defined over C: the factorization of the cohomological ring of M into that of the base and that of the fiber. Since 135 OoOl-8708/86 $7.50

Realizing characteristic classes by manifolds

Let BG be the classifying space for stable spherical fibrations, and let V be a finite dimensional vector subspace of the cohomology algebra H ∗( BG; Z 2). We prove that V may be realized by a Poincaré duality space P, which means that if v: P→ BG is the Spivak fibration, then v ∗ maps V isomorphically onto its image in H ∗( P; Z 2). By construction, P is the product of a certain Grassman manifold and a spherical fiber space over a closed smooth manifold.