Characterizations of quadric and Hermitian Veroneseans over finite fields

In Hirschfeld and Thas [5] the most important characterizations of quadric Veroneseans are surveyed. However a few difficult cases were still open, in particular the even case. In [10, 11] Thas and Van Maldeghem not only solve all open cases, but they also generalize most of these characterizations in several ways: they do not restrict themselves to the quadric Veronesean of the plane PG, they allow ovals instead of conics, and they also characterize projections of quadric Veroneseans. Further, Cooperstein, Thas and Van Maldeghem [1] contains some properties of Hermitian Veroneseans over finite fields and also these varieties and some of their projections are characterized. All these results on Veroneseans will be surveyed here.     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24     

Characterizations of schunck classes of finite groups

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a _i b _j=0 for each coefficient a _i of f and b _j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.     

Einstein-Weyl spaces and (1 , n) -curves in the quadric surface

Following the ideas of Hitchin on the twistoral approach to 3-dimensional Einstein-Weyl geometry we construct a series of complex surfaces containing rational curves with self-intersection number 2. These mini twistor spaces are obtained by taking an n-fold covering of a neighbourhood of a (1 , n)- curve in the quadric CP₁ x CP₁ branched along the curve. We describe the corresponding Einstein-Weyl geometry on the parameter space of curves.     

Conjugate biprimitive finite groups saturated with finite simple subgroupsU 3(2 n )

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic conjugate biprimitive finite groups saturated with groups in the set {U₃(2ⁿ)}. It is proved that every such group is isomorphic to a simple group U₃(Q) over a locally finite field Q of characteristic 2. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 606–615, September–October, 1998.     

Characterizations and constructions of n-to-1 mappings over finite fields

n-to-1 mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of n-to-1 mappings over finite fields are studied. First, we provide a characterization of general n-to-1 mappings over ${\mathbb{F}}_{p^m}$ by means of the Walsh transform. Then, we completely determine 3-to-1 polynomials with degree no more than 4 over ${\mathbb{F}}_{p^m}$. Furthermore, we obtain an AGW-like criterion for characterizing some close relationship between the n-to-1 property of a mapping over finite set A and that of another mapping over a subset of A. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit n-to-1 mappings. Especially, three explicit constructions of the form $x^{r}h\left(x^s\right)$ from the cyclotomic perspective, and several classes of n-to-1 mappings of the form $g(x^{q^k}-x+\delta )+cx$ are provided.     

逆p·n·p·矩阵的表征

一个n阶实方阵A,若其各阶主子式皆非正,则称A为p.n.p.矩阵,记作A∈PNP;特别地,若A∈NP且各阶主子式皆负,则称A为p.n.矩阵,记作A∈PN进一步,若n阶实方阵A非奇异,且A^-1∈PNP,则称A为逆p.n.p.矩阵,记作A∈IPNP;特别地,若A^-1∈PN,则称A为逆p.n.矩阵,记作A∈IPN。     

Characterizations of finite classical polar spaces by intersection numbers with hyperplanes and spaces of codimension 2

In this article we show that non-singular quadrics and non-singular Hermitian varieties are completely characterized by their intersection numbers with respect to hyperplanes and spaces of codimension 2. This strongly generalizes a result by Ferri and Tallini [5] and also provides necessary and sufficient conditions for quasi-quadrics (respectively their Hermitian analogues) to be non-singular quadrics (respectively Hermitian varieties).     

Quasirecognition by prime graph of finite simple groups L n (2) and U n (2)

Let G be a finite group. The prime graph ??(G) of G is defined as follows. The vertices of ??(G) are the primes dividing the order of G and two distinct vertices p, p?? are joined by an edge if G has an element of order pp??. Let L=L _n (2) or U _n (2), where n?R17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that ??(G)=??(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of L _n (2). Also we conclude that the simple group U _n (2) is quasirecognizable by element orders.     

Characterizations of finite groups by sets of orders of their elements

For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω) stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic to S₉, S₁₁, S₁₂, S₁₃, or A₁₂, and h(ω(G))=2 if G is isomorphic to S₂(6) or to O ₈ ⁺ (2). 01 Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 37–53, January–February, 1997.     

Characterizations on 2-isometries

We consider 2-isometries, weak 2-isometries and 2-continuous mappings and investigate the relation between three mappings in linear 2-normed spaces. Also we prove that the Riesz theorem holds when X is a linear 2-normed spaces.     

Stability of quadric bundles

A decorated vector bundle on a smooth projective curve \(X\) is a pair \((E,\varphi )\) consisting of a vector bundle and a morphism \(\varphi :(E^{\otimes a})^{\oplus b}\rightarrow (\det E)^{\otimes c}\otimes \mathsf {N}\) , where \(\mathsf {N}\in \text {Pic}(X)\) . There is a suitable semistability condition for such objects which has to be checked for any weighted filtration of \(E\) . We prove, at least when \(a=2\) , that it is enough to consider filtrations of length \(\le \) 2. In this case decorated bundles are very close to quadric bundles and to check semistability condition one can just consider the former. A similar result for L-twisted Higgs bundles and quadric bundles was already proved (García-Prada et al. in The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, 2012; Schmitt in Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich, 2008). Our proof provides an explicit algorithm which requires a destabilizing filtration and ensures a destabilizing subfiltration of length at most two. Quadric bundles can be thought as a generalization of orthogonal bundles. We show that the simplified semistability condition for decorated bundles coincides with the usual semistability condition for orthogonal bundles. Finally we note that our proof can be easily generalized to decorated vector bundles on nodal curves.     

算子代数上(m,n)-Jordan导子的刻画

设R是一个环,M是一个R-双边模,m和n是两个非负整数满足m+n≠0,如果δ是一个从R到M的可加映射满足对任意A∈R,(m+n)δ(A~2)=2mAδ(A)+2nδ(A)A,则称δ是一个(m,n)-Jordan导子.本文证明了,如果R是一个单位环,M是一个单位R-双边模含有一个由R中幂等元代数生成的左(右)分离集,那么,当m,n0且m≠n时,每一个从R到M的(m,n)-Jordan导子恒等于零.还证明了,如果A和B是两个单位环,M是一个忠实的单位(A,B)-双边模(N是一个忠实的单位(B,A)-双边模),m,n0且m≠n,U=[A N M B]是一个|mn(m-n)(m+n)|-无挠的广义矩阵环,那么每一个从U到自身的(m,n)-Jordan导子恒等于零.     

On Ferri's characterization of the finite quadric Veronesean

We generalize and complete Ferri's characterization of the finite quadric Veronesean by showing that Ferri's assumptions also characterize the quadric Veroneseans in spaces of even characteristic.