An alternative approach to the structure theory of PI-rings

We expose a rather simple and direct approach to the structure theory of prime PI-rings (“Posner’s theorem”), based on fundamental properties of the extended centroid of a prime ring.     

The tame dimension vectors for trees

We consider representations of quivers over an algebraically closed field K. A dimension vector of a quiver is called hypercritical, if there is an m-parameter family of indecomposable representations for the dimension vector with m?2, but every family of representations for all smaller dimension vectors depends on a single parameter. We characterise the hypercritical dimension vectors for trees via their Tits forms and those of their decompositions and present the complete list of the hypercritical dimension vectors.Finally, this leads to a combinatorial classification of the tame dimension vectors for trees which is also given by the Tits forms.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

Ideal intrinsic extensions with connections to PI-rings

In this paper, we extend various classical results by Armendariz and Steinberg, Fisher, Kaplansky, Martindale, Posner, and Rowen on semiprime PI-rings. We do this by introducing several new generalizations of the class of semiprime PI-rings. For these new classes, some structure theorems are obtained, and connections to arbitrary semiprime rings are made (e.g., a semiprime ring has a largest essentially closed ideal from some of these classes). Numerous examples are provided to illustrate and delimit our results.     

On the Dirichlet problem

A particular case of the Dirichlet problem is solved using the Convergence Theorem for discrete-time martingales and the mean value property of harmonic functions as the main tools.     

Tilting theory and cluster combinatorics

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.     

An application of Guillemin-Abreu theory to a non-abelian group action

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin and Abreu to the case of the standard linear SU(n) action on Cn?{0}. Using this framework we (re)construct certain Kähler metrics from data on moment polytopes.     

Zero cycles on homogeneous varieties

In this paper we study the group A₀(X) of zero-dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety X. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show A₀(X)=0 for certain classes of homogeneous varieties for groups of each of the classical types, extending previous results of Swan/Karpenko, of Merkurjev, and of Panin.     

Algebraic prime subalgebras in simple Artinian algebras

We prove a Wedderburn-Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599-613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring.     

An addition and a correction to my paper “Differential inequalities for evolution equations”

This paper provides a generalization of the main lemma and a correct proof of an extended version of Theorem 2, in [G. Vidossich, Differential inequalities for evolution equations, Nonlinear Anal. TMA 25 (1995) 1063-1069].     

Renewal theory with a trend

We prove some analogs of results from renewal theory for random walks in the case when there is a drift, more precisely when the mean of the kth summand equals k^γμ, k≥1, for some μ>0 and 0<γ≤1.     

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.     

QUASIREGULAR TORSION RINGS HAVING ISOMORPHIC ADDITIVE AND CIRCLE COMPOSITION GROUPS

Abstract

We investigate the role played by torsion properties in determining whether or not a commutative quasiregular ring has its additive and circle composition (or adjoint) groups isomorphic. We clarify and extend some results for nil rings, showing, in particular, that an arbitrary torsion nil ring has the isomorphic groups property if and only if the components from its primary decomposition into p-rings do too.

We look at the more specific case of finite rings, extending the work of others to show that a non-trivial ring with the isomorphic groups property can be constructed if the additive group has one of the following groups in its decomposition into cyclic groups: Z₂ ⁿ (for n ≥ 3), Z₂ ⊕ Z₂ ⊕ Z₂, Z₂ ⊕ Z₄, Z₄ ⊕ Z₄, Z _p ⊕ Z _p (for odd primes, p), or Z _p ⁿ (for odd primes, p, and n ≥ 2).

We consider, also, an example of a ring constructed on an infinite torsion group and use a specific case of this to show that the isomorphic groups property is not hereditary.     

Almost sure explosion of solutions to stochastic differential equations

This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given.     

On the DFR property of the compound geometric distribution with applications in risk theory

In 1988, Shanthikumar proved that the sum of a geometrically distributed number of i.i.d. DFR random variables is also DFR. In this paper, motivated by the inverse problem, we study monotonicity properties related to defective renewal equations, and obtain that if a compound geometric distribution is DFR, then the random variables of the sums are NWU (a class that contains DFR). Furthermore, we investigate some applications of risk theory and give a characterization of the exponential distribution.     

An empirical likelihood approach to data analysis under two-stage sampling designs

A new empirical likelihood approach is developed to analyze data from two-stage sampling designs, in which a primary sample of rough or proxy measures for the variables of interest and a validation subsample of exact information are available. The validation sample is assumed to be a simple random subsample from the primary one. The proposed empirical likelihood approach is capable of utilizing all the information from both the specific models and the two available samples flexibly. It maintains some nice features of the empirical likelihood method and improves the asymptotic efficiency of the existing inferential procedures. The asymptotic properties are derived for the new approach. Some numerical studies are carried out to assess the finite sample performance.     

Dirichlet forms and diffusion processes for fermion random point fields

We construct the Dirichlet forms and the associated diffusion processes on the configuration space of particles moving on the Euclidean space , for which certain fermion random point fields are invariant.     

Homogeneous nilradicals over semigroup graded rings

In this paper we study the homogeneity of radicals defined by nilpotence or primality conditions, in rings graded by a semigroup S. When S is a unique product semigroup, we show that the right (and left) strongly prime and uniformly strongly prime radicals are homogeneous, and an even stronger result holds for the generalized nilradical. We further prove that rings graded by torsion-free, nilpotent groups have homogeneous upper nilradical. We conclude by showing that non-semiprime rings graded by a large class of semigroups must always contain nonzero homogeneous nilpotent ideals.