On the geometry of a generalized quadric

We describe the set of diametral hyperplanes of a generalized hyperquadric in the space E^m and solve a problem posed by Du Val. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 127–131.     

Real hypersurfaces in the complex quadric whose structure Jacobi operator is of Codazzi type

First we introduce the notion of structure Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in with structure Jacobi operator of Codazzi type.     

On semirings whose additive reduct is a semilattice

A semiring S whose additive reduct is a semilattice is called a k-regular semiring if for every a∈S there is x∈S such that a+axa=axa. For a semigroup F, the power semiring P(F) is a k-regular semiring if and only if F is a regular semigroup. An element e∈S is a k-idempotent if e+e ²=e ². Basic properties of k-regular semirings whose k-idempotents are commutative have been studied.     

On functions whose graph is a Hamel basis

We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.

     

On mappings whose sum is monotone

Proposition 1 of this article points at a gap in the proof of Kuhlmann’s characterization of algebraically maximal valued fields [2]. The author [1] showed how to fill this gap. His arguments involved tools of mathematical logic and infinite combinatorics. Proposition 2 of this article provides us with a simple proof of the key fact of Kuhlmann’s characterization.     

On the functions whose inverse is jet-absolutely monotonic

The infinitely differentiable real valued functions whose inverse have all the Taylor coefficients nonnegative at the fixed point of the axis are considered. An application to the theory of the inverse pair of the Radon measures is given.     

On modules whose endomorphism ring is local

The paper is a study of modules with a local endomorphism ring. We consider decompositions of modules and give a simple proof of Azumaya’s theorem. We also define an equivalence relation on the family of direct summands of a module, and show that the properties of a decomposition are closely related to the properties of this equivalence relation. Supported by the Norwegian Research Council for Science and the Humanities.     

On submanifolds whose shape operator is unipotent

The object of this article is to characterize submanifolds of the Euclidean space whose shape operator A^ξ satisfies the equation (A^ξ)² = k||ξ ||²Id, where k > 0 is constant.     

On the integral invariants of a closed ruled surface

Summary In this paper, it is shown that the dual integral invariant of a closed ruled surface, the dual angle of pitch, corresponds to the dual spherical surface area described by the dual spherical indicatrix of the closed ruled surface. So, new geometric interpretations of the real angle of pitch and the real pitch of a closed ruled surface, and some results are given.     

On the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree

A different approach is given to recent results due mainly to R. C. Johnson and A. Leal Duarte on the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree. The techniques developed are based on some results of matching polynomials and used a work by O. L. Heilmann and E. H. Lieb on an apparently unrelated topic.      

On locally finite groups with a four-subgroup whose centralizer is small

Let $G$ be a locally finite group which contains a non-cyclic subgroup $V$ of order four such that $C_{G}\left( V\right) $ is finite and $C_{G}\left( \phi \right)$ has finite exponent for some $\phi \in V$ . We show that $[G,\phi ]^{\prime }$ has finite exponent. This enables us to deduce that $G$ has a normal series $1\le G_1\le G_2\le G_3\le G$ such that $G_1$ and $G/G_2$ have finite exponents while $G_2/G_1$ is abelian. Moreover $G_3$ is hyperabelian and has finite index in $G$ .     

Birational geometry of the moduli space of pure sheaves on quadric surface

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m+1$ and $c_1=(2,3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.