On Jordan’s theorem

In the present paper, we give some remarks on the well-known Jordan theorem and Hamiltonians.     

Analogs of Steiner’s porism and Soddy’s hexlet in higher dimensions via spherical codes

In this paper we consider generalizations to higher dimensions of classical results on chains of tangent spheres.     

On Litoff’s theorem

We give a short proof of Litoff’s theorem from the viewpoint of completely reducible modules.     

On Turner’s theorem and first-order theory

A theorem of E.C. Turner states that if F is a finitely generated free group, then the test words are precisely the elements not contained in any proper retract. In this paper, we examine some ideas in model theory and logic related to Turner’s characterization of test words and introduce Turner groups, a class of groups containing all finite groups and all stably hyperbolic groups satisfying this characterization. We show that Turner’s theorem is not first-order expressible. However, we prove that every finitely generated elementary free group is a Turner group.     

λ-fold indecomposable large sets of Steiner triple systems

A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).     

On Bohl’s argument theorem

The classical Bohl argument theorem of a conditionally periodic function is generalized. Conditionally periodic motions on a torus are replaced by the solutions of a nonlinear system of differential equations with invariant measure. Cases in which this system is assumed ergodic or strictly ergodic are considered.     

On Berndtsson’s generalization of Prékopa’s theorem

We present a new approach to Berndtssons complex extension of Prékopas theorem. This approach is inspired by the recent local proof of Prékopas theorem obtained by Ball, Barthe and Naor. In the complex setting, this approach leads to a formula expressing in terms of (derivatives of) where e^–(z)= As a consequence, we obtain new conditions ensuring that is subharmonic when is plurisubharmonic.     

Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems

In this paper, we establish some new generalizations of Darbo’s fixed point theorem for multivalued mappings. Moreover, we prove the existence of solutions for a class of integral equations by Darbo’s fixed point theorem and the existence of solutions for a class of differential inclusions using a generalization of Darbo’s fixed point theorem.     

Noether’s theorem of nonholonomic systems in optimal control

In this paper, we extend Noether’s theorem to nonholonomic constraints systems in optimal control. We present a systematic way to calculate conserved quantities along the Pontryagin extremals for optimal control problems with nonholonomic constraints, which are invariant under the parameter groups of infinitesimal transformations that change all (time, state, control) variables. Meanwhile, the Noether equalities corresponding to the conservation laws are given. Then, we obtain a new version of Noether’s theorem to optimal control systems. An example is given to illustrate the application of these results.     

On dilworth’s theorem in the in finite case

The following theorem is proved: Letc be an infinite cardinal. There exists a partially ordered set of cardinalc, which contains no infinite independent subset, and which is not decomposable into less thanc chains.     

On Miquel’s theorem and inversions in normed planes

Asplund and Grünbaum proved that Miquel’s six-circles theorem holds in strictly convex, smooth normed planes if the considered circles have equal radii. We extend this result in two directions. First we prove that Miquel’s theorem for circles of equal radii (more precisely, a generalized version of it) is true in every normed plane, without the assumptions of strict convexity and smoothness, and give also some properties of the circle configuration related to this theorem. Second we clarify the situation if the circles of the corresponding configuration do not necessarily have equal radii.     

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in Rn, n≥3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in R3, while finite, can be much larger than possible in the plane.