The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45     

Monge extensions of cooperation and communication structures

Cooperation structures without any a priori assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for marginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the Lovász extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson’s graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley’s convexity model for classical cooperative games.     

Cauchy structures in closure and proximity spaces

Research supported by Hungarian National Foundation for Scientific Research, grant no. 2114.     

关于非线性抛物型方程组Cauchy问题

本文研究了非线性抛物型方程组初值问题整体存在唯一性，拓广和加强了Ｈ．Ｆｕｊｉｔａ［１］的结果。     

On extensions of sub-Riemannian structures on Lie groups

We define the extension of a left-invariant sub-Riemannian structure in terms of an extension of the underlying Lie group and compatibility of the respective distributions and metrics. We show that geodesics of a structure can be lifted to geodesics of any extension of the structure. In the case of central extensions, we show that the normal geodesics of the minimal extension are the projection (in a sense) of the normal geodesics of any other compatible extension. Several illustrative examples are discussed.     

Some topological structures of extensions of abstract reachability problems

We consider the problem of the reachability of states that are elements of a topological space under constraints of asymptotic nature on the choice of an argument of a given objective mapping. We study constructions that have the sense of extensions of the original space and are implemented with the use of methods that are natural for applied mathematics but employ elements of extensions used in general topology. The study is oriented towards the application in the problem on the construction and investigation of properties of reachability sets for control systems.Constructions involving an approximate observation of constraints in control problems, as well as various generalized regimes, were widely used by N.N. Krasovskii and his students. In particular, this approach was applied in the proof of N.N. Krasovskii and A.I. Subbotin’s fundamental theorem of the alternative, which made it possible to establish the existence of a saddle point in a nonlinear differential game. In the investigation of impulse control problems, Krasovskii used techniques from the theory of generalized functions, which formed the basis for many studies in this direction. A number of A.B. Kurzhanski’s papers are devoted to the solution of control problems related in one way or another to the construction of reachability sets. Control problems with incomplete information, duality issues for control and observation problems, and team control problems constitute a far from exhaustive list of research areas where Kurzhanskii obtained profound results. These studies are characterized by the use of a wide range of tools and methods from applied mathematics and various constructions as well as by the combination of theoretical investigations and procedures related to the possibility of computer modeling.The research direction developed in the present paper mainly concerns the problem of constraint observation (including “asymptotic” constraints) and involves other issues. Nevertheless, the idea of constructing generalized elements of various nature (in particular, generalized controls) seems to be useful for the purpose of asymptotic analysis of control problems that do not possess stability as well as problems on the comparison of different tendencies in the choice of control in the form of dependences on a complex of factors inherent in the original real-life problem. The use of such tools as the Stone–?ech compactification and Wallman’s extension is, of course, oriented toward the study of qualitative issues. In the authors’ opinion, the combined application of the approaches to the construction of extensions used in control theory and in general topology holds promise from the point of view of both pure and applied mathematics. Apparently, the present paper can be considered as a certain step in this direction.     

Maximal small extensions of o‐minimal structures

A proper elementary extension of a model is called small if it realizes no new types over any finite set in the base model. We answer a question of Marker, and show that it is possible to have an o‐minimal structure with a maximal small extension. Our construction yields such a structure for any cardinality. We show that in some cases, notably when the base structure is countable, the maximal small extension has maximal possible cardinality (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lax entwining structures,groupoid algebras and cleft extensions

For an algebra A, a coalgebra C and a lax entwining structure (A, C, ψ), in this paper we introduce the notions of lax C-Galois extension with normal basis and lax C-cleft extension and we prove that these notions are equivalent if the functor A ? — preserve coequalizers.     

Homogeneous Riemannian structures on some solvable extensions of the Heisenberg group

Two families of four or five-dimensional Riemannian solvable Lie groups, which are extensions of the Heisenberg group, are considered. We determine all the homogeneous Riemannian structures on them, and the simply connected groups of isometries corresponding to the associated reductive decompositions. Some of these structures are homogeneous Kähler or homogeneous cosymplectic, and in these cases they are realized by the complex hyperbolic plane ?H(2) and by ?H(2)×?, respectively.     

Isomorphism problems for Hopf–Galois structures on separable field extensions

Let $L/K$ be a finite separable extension of fields whose Galois closure $E/K$ has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on $L/K$ has the form $E{[N]}^G$ for some group N of order $[L:K]$. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree $p^n$, for p an odd prime number.     

Definable group extensions in semi‐bounded o‐minimal structures

In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi‐bounded (respectively, linear) o‐minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈R^m, +〉 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relations on the period mapping giving extensions of mixed hodge structures on compact Riemann surfaces

We consider a period map from Teichmüller space to , which is a real vector bundle over the Siegel upper half space. This map lifts the Torelli map. We study the action of the mapping class group on this period map. We show that the period map from Teichmüller space modulo the Johnson kernel is generically injective. We derive relations that the quadratic periods must satisfy. These identities are generalizations of the symmetry of the Riemann period matrix. Using these higher bilinear relations, we show that the period map factors through a translation of the subbundle and is completely determined by the purely holomorphic quadratic periods. We apply this result to strengthen some theorems in the literature. One application is that the quadratic periods, along with the abelian periods, determine a generic marked compact Riemann surface up to an element of the kernel of Johnson's homomorphism. Another application is that we compute the cocycle that exhibits the mapping class group modulo the Johnson kernel as an extension of the group SP _g () by the group .