On the secant varieties to the osculating variety of a Veronese surface

In this paper we study the k-th osculating variety of the order d Veronese embedding of P ⁿ . In particular, for k=n=2 we show that the corresponding secant varieties have the expected dimension except in one case.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

The birational type of the moduli space of even spin curves

We determine the Kodaira dimension of the moduli space Sg of even spin curves for all g. Precisely, we show that Sg is of general type for g>8 and has negative Kodaira dimension for g<8.     

Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule

We prove the representation of a fractal crumpled structure of a strongly collapsed unknotted polymer chain. In this representation, topological considerations result in the chain developing a densely packed system of folds, which are mutually segregated at all scales. We investigate topological correlations in randomly generated knots on rectangular lattices (strips) of fixed widths. We find the probability of the spontaneous formation of a trivial knot and the probability that each finite part of a trivial knot becomes a trivial knot itself after joining its ends in a natural way. The complexity of a knot is characterized by the highest degree of the Jones–Kauffman polynomial topological invariant. We show that the knot complexity is proportional to the strip length in the case of long strips. Simultaneously, the typical complexity of a quasi-knot, which is a part of a trivial knot, is substantially less. Our analysis shows that the latter complexity is proportional to the square root of the strip length. The results obtained clearly indicate that the topological state of any part of a trivial knot densely filling the lattice is also close to the trivial state.     

A new characterization of projections of quadrics in finite projective spaces of even characteristic

We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L} of the geometry there are either 0, α, or q lines through p intersecting L. An example of such a geometry with α=2 is the following well known geometry . Let Q_n+1 be a nonsingular quadric in a finite projective space , n≥3, q even. We project Q_n+1 from a point r∉Q_n+1, distinct from its nucleus if n+1 is even, on a hyperplane not through r. This yields a partial linear space whose points are the points p of , such that the line 〈p,r〉 is a secant to Q_n+1, and whose lines are the lines of which contain q such points. This geometry is fully embedded in an affine subspace of and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry.     

Translation invariant valuations in the space of planes

The paper is related to the problem of measure generation in the space IE of planes in IR ³ by combinatorial, translation invariant valuations. General results concerning that problem have been derived in 1994 and 1996 (this journal vol. 31, no. 4. and vol. 33, no. 4, respectively). The purpose of the present article is to give a proof of two geometrical identities on which the theorem on valuations in the space IE can be based. The article consists of the motivational first part that contains the basic concepts from the theory of combinatorial valuations and measure generation in IE, and the second that gives a proof of the identities in question.     

Low-type submanifolds of real space forms via the immersions by projectors

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n5. For example, the complete minimal hypersurfaces of the unit sphere Sⁿ⁺¹ which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface and the Clifford minimal hypersurfaces M_k,n−k for n≠2k. An interesting characterization of horospheres in is also obtained.     

Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form where is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.

     

On the generic splitting of quadratic forms in characteristic 2

In [9] and [10] Knebusch established the basic facts of generic splitting theory of quadratic forms over a field of characteristic different from 2. This paper is related to [11] and [13] where Knebusch and Rehmann generalized partially this theory to a field of characteristic 2. More precisely, we begin with a complete characterization of quadratic forms of height 1 (we don't exclude anisotropic quadratic forms with quasi-linear part of dimension at least 1). This allows us to extend the notion of degree to characteristic 2. We prove some results on excellent forms and splitting tower of a quadratic form. Some results on quadratic forms of height 2 and degree 1 or 2 are given. Received: 6 March 2000; in final form: 5 October 2001 / Published online: 17 June 2002     

Linear coloring of graphs embeddable in a surface of nonnegative characteristic

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every graph G with girth g(G) and maximum degree Δ(G) that can be embedded in a surface of nonnegative characteristic has lc(G) = Δ(2G )+ 1 if there is a pair (Δ, g) ∈ {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G s...     

Birational invariants in the case¶of prime characteristic

The Frobenius automorphism of the function field allows to define some discrete birational invariants of algebraic manifolds using p ^s -th powers of differentials. Examples of algebraic hypersurfaces are sufficient to show the independence of the familiar birational invariants. Received: Received: 6 March 1997     

利用有限域上伪辛几何中一类2维非迷向子空间构作PBIB设计

设Ｆｑ是特征为２的有限域，本文利用Ｆｑ上２ｖ＋２维伪辛几何中包含固定的１维非迷向子空间的一类的２维非迷向子空间作处理，构作了具有２（ｑ－１）个结合类的结合方法和ＰＢＩＢ设计，并计算了相应的参数。     

The edge-face coloring of graphs embedded in a surface of characteristic zero

Let G be a graph embedded in a surface of characteristic zero with maximum degree Δ. The edge-face chromatic number χ_ef(G) of G is the least number of colors such that any two adjacent edges, adjacent faces, incident edge and face have different colors. In this paper, we prove that χ_ef(G)≤Δ+1 if Δ≥13, χ_ef(G)≤Δ+2 if Δ≥12, χ_ef(G)≤Δ+3 if Δ≥4, and χ_ef(G)≤7 if Δ≤3.     

Explicit Classifications of Orbits in Jordan Algebra and Freudenthal Vector Space Over the Exceptional Lie Groups

ABSTRACT

We give the explicit classifications of orbits in the Jordan algebra 𝔍 over the group E _6(?26) and the Freudenthal, R -vector space 𝔓 over the group E _7(?25).

Communicated by E. Zelmanov     

A note on Wronskians and the ABC theorem in function fields of prime characteristic

We provide a sharp bound for the order sequence of Wronskians. We also give another proof of the truncated second main theorem over function fields which is a generalization of the ABC theorem due to Mason, Voloch, Brownawell and Masser, Noguchi and the author. Received: 9 June 1998 / Revised version: 24 September 1998