Trace forms of abelian extensions of number fields of type (1, 0)

This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types.     

A characterization of a class of hyperplanes of DW(2n−1,F)

A hyperplane of the symplectic dual polar space $DW(2n?1,\mathbb{F})$, $n\ge 2$, is said to be of subspace-type if it consists of all maximal singular subspaces of $W(2n?1,\mathbb{F})$ meeting a given $(n?1)$-dimensional subspace of $\text{PG}(2n?1,\mathbb{F})$. We show that a hyperplane of $DW(2n?1,\mathbb{F})$ is of subspace-type if and only if every hex $F$ of $DW(2n?1,\mathbb{F})$ intersects it in either $F$, a singular hyperplane of $F$ or the extension of a full subgrid of a quad. In the case $\mathbb{F}$ is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane $H$ of $DW(2n?1,\mathbb{F})$ is of subspace-type or arises from the spin-embedding of $DW(2n?1,\mathbb{F})?DQ(2n,\mathbb{F})$ if and only if every hex $F$ intersects it in either $F$, a singular hyperplane of $F$, a hexagonal hyperplane of $F$ or the extension of a full subgrid of a quad.     

Unusual interactions of (1/2, 0) ⊕ (0, 1/2) particles

We continue to study the ‘fermion — 4-vector potential’ interactions in the framework of the McLennan-Case construct which is a reformulation of the Majorana theory of the neutrino. This theory is shown after applying Majorana-likeanzatzen to give rise to appearance of unusual terms as σ [A×A^*], which were recently discussed in non-linear optics.     

A small minimal blocking set in PG(n, p t ), spanning a (t − 1)-space, is linear

In this paper, we show that a small minimal blocking set with exponent e in PG(n, p ^t ), p prime, spanning a (t/e ? 1)-dimensional space, is an ${\mathbb{F}_{p^e}}$ -linear set, provided that p > 5(t/e)?11. As a corollary, we get that all small minimal blocking sets in PG(n, p ^t ), p prime, p > 5t ? 11, spanning a (t ? 1)-dimensional space, are ${\mathbb{F}_p}$ -linear, hence confirming the linearity conjecture for blocking sets in this particular case.     

On the diophantine equation X 2 − (1 + a 2)Y 4 = −2a

Let a≥1 be an integer.In this paper,we will prove the equation in the title has at most three positive integer solutions.     

On (p a,p, p a,p a−1 )-relative difference sets

Abelian relative difference sets of parameters (m, n, k, )=(p ^a , p, p ^a , p ^a–1 )are studied in this paper. In particular, we show that for an abelian groupG of orderp ^2c+1 and a subgroupN ofG of orderp, a (p ^2c , p, p ^2c , p ^2c–1 )-relative difference set exists inG relative toN if and only if exp (G)p ^c+1 .Furthermore, we have some structural results on (p ^2c p, p ^2c , p ^2c–1 )-relative difference sets in abelian groups of exponentp ^c+1 . We also show that for an abelian groupG of order 2^2c+2 and a subgroupN ofG of order 2, a (2^2c+1, 2, 2^2c+1, 2^2c )-relative difference set exists inG relative toN if and only if exp(G)2 ^c+2 andN is contained in a cyclic subgroup ofG of order 4. New constructions of (p ^2c+1 , p, p ^2c+1 , p ^2c )-relative difference sets, wherep is an odd prime, are given. However, we cannot find the necessary and sufficient condition for this case.     

Modular forms of weight 8 for Γ g (1, 2)

We complete the program indicated by the Ansatz of D’Hoker and Phong in genus 4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Ansatz. We prove dim[Γ₄(1, 2), 8]₀ = 2 and dim[Γ₄(1, 2), 8] = 7. In each genus, we classify the linear relations among the self-dual lattices of rank 16. We extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the Ansatz.     

二分(0,mf-m+1)-图的正交(0,f)-因子分解

设G=(X,Y,E(G))是一个二分图,分别用V(G)=XUY和E(G)表示G的顶点集和边集.设f是定义在V(G)上的整数值函数且对(A)x∈V(G)有f(x)≥k.设H_1,H_2,…,H_k是G的k个顶点不相交的子图,且|E(H_i)|=m,1≤i≤k.本文证明了每个二分(0,mf-m+1)-图G有一个(0,f)-因子分解正交于Hi(i=1,2,…,k).     

Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.     

Zur graphischen Integration der linearen Differentialgleichung n. Ordnung (an D + 1) (an−1 D + 1) ..... (a1 D + 1) y = s (x)(a1 D + 1) y = s (x)

Ohne Zusammenfassung     

Sets of Minimality of (1 − 1)-Rational Functions

We describe dynamical systems associated to (1 ? 1)-rational functions on the field of p-adic numbers.We focus on sets of minimality of such systems.     

A new proof of the existence of (q 2−q+1)-arcs inPG(2,q 2)

Journal of Geometry -