Singularities of slice regular functions

Beginning in 2006, G. Gentili and D. C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball B(0, R) centered at 0 the set of regular functions coincides with that of quaternionic power series $\sum _{n \in {\mathbb {N}}} q^n a_n$ converging in B(0, R). In 2009 the author proposed a classification of singularities of regular functions as removable, essential or as poles and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls B(0, R). Quite recently, F. Colombo, G. Gentili and I. Sabadini (2010) and the same authors in collaboration with D. C. Struppa (2009) identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent‐type expansions at points p other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in 2009. Poles are studied, as well as essential singularities, for which a version of the Casorati‐Weierstrass Theorem is proven.     

On existence of admissible and weakly admissible limits for functions of several complex variables

Kishinev. Translated fromSibirski Matematicheski Zhurnal, Vol. 33, No. 4, pp. 212–214, July–August, 1992.     

Extended admissible functions and Gaussian limiting distributions

We consider an extension of Hayman's notion of admissibility to bivariate generating functions that have the property that the coefficients satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a large class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is also presented. We apply this concept to various combinatorial examples.

     

Canonical functions of admissible measures in the half-plane

For a γ-admissible measure λ in the upper half-plane, we introduce the notion of canonical function, generalizing the canonical Nevanlinna product for analytic functions of finite order in the half-plane. It is shown that, for any growth function γ defined by the Boutroux proximate order, the given definition and the canonical Nevanlinna product coincide.     

Uniqueness of admissible meromorphic functions in the unit disc

The uniqueness of admissible meromorphic functions in the unit disc is studied. Project supported by the National Natural Science Foundation of China (Grant No. 19601021).     

Singularities of Blaschke normal maps of convex surfaces

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.     

Singularities and normal forms of odd-dimensional Pfaff equations

Kharkov Institute of Radioelectronics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 1, pp. 70–71, January–March, 1989.     

Singularities of functions of one and several bicomplex variables

In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price (An Introduction to Multicomplex Spaces and Functions, Dekker, New York, 1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.     

Singularities of Shannon functions in the case of automation bases

It is shown that in the realization of functions of logical algebra by means of circuits in automaton bases, the asymptotic behavior of the Shannon function can depend on many parameters of an arbitrarily large number of basis elements.Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 73–82, January, 1972.In conclusion, I wish to thank O. B. Lupanov for his attention to this work.     

Stein's idea and minimax admissible estimation of a multivariate normal mean

We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator.     

Correction to “Singularities of the Weierstrass type functions”

The online version of the original article can be found at     

A class of admissible functions in the principle of invariance of wave operators

A class of admissible functions that satisfies the well-known invariance principle in the scattering theory is extended.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 227–229, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.     

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system.To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding “admissible faces”. Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.