Stochastic calculus of variations

A theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes. Generalizations of the Euler equation and Noether's theorem are obtained and several conservation laws are discussed. An application to Nelson's probabilistic framework of quantum mechanics is also given.     

On Hilbert's Irreducibility Theorem

A method for obtaining very precise results along the lines of the Hilbert Irreducibility Theorem is described and then applied to a special case. In addition, the relationship of the irreducibility theorem to other tools of diophantine analysis is investigated. In particular, we give a proof of the irreducibility theorem that uses only Noether's lemma and the fact that an absolutely irreducible curve has a rational point over a finite field of large order.     

Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

Fractional Noether’s theorem in the Riesz-Caputo sense

We prove a Noether’s theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.     

Monodromy of Variations of Hodge Structure

We present a survey of the properties of the monodromy of local systems on quasi-projective varieties which underlie a variation of Hodge structure. In the last section, a less widely known version of a Noether–Lefschetz-type theorem is discussed.     

On spectral analysis in locally compact motion groups

Beurling's theorem in spectral analysis of bounded functions on the real line is generalized to a class of locally compact motion groups and to Heisenberg groups.     

Helly-type theorems for pseudoline arrangements in P2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.     

Hurwitz' theorem implies Rouché's theorem

It is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem.     

Hypergraphs and Whitney's theorem on edge-isomorphisms of graphs

This paper presents a new proof of Whitney's theorem on edge-isomorphisms of graphs and extends Whitney's theorem to hypergraphs. Whitney's theorem asserts that any two edge-isomorphic graphs of order at least 5 have their edge-isomorphism induced by a node-isomorphism isomorphism. Previous results of Gardner and of Berge and Rado are used.     

Determinants,integrality and Noether’s theorem for quantum commutative algebras

The aim of this paper is to generalize Noether’s theorem for finite groups acting on commutative algebras, to finite-dimensional triangular Hopf algebras acting on quantum commutative algebras. In the process we construct a non-commutative determinant function which yields an analogue of the Cayley-Hamilton theorem for certain endomorphisms. In memory of Professor Shimshon Amitsur The author was supported by the Plan and Budget Committee of the Israeli Government.     

Ramsey's theorem with sums or unions

Using Hindman's theorem as a strong pigeonhole principle, we prove strengthened versions of Ramsey's theorem and of various generalizations of Ramsey's theorem due to Nash-Williams, Galvin and Prikry, and Silver.     

Explicit representation of membership in polynomial ideals

We introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates. We consider simple examples as the classical Macaulay theorem as well as a quite recent result by Hickel, related to the effective Nullstellensatz. We also obtain a related result that generalizes Max Noether’s classical AF + BG theorem.     

Quelques theoremes de dualite combinatoire

The purpose of this paper is to prevent some new and unified proofs of a number of known results in combinatorial programming: generalized versions of Minty's lemma, of Ford and Fulkerson's and Hoffman's theorems the length-width inequality and the strong complementary slackness theorem. All these results are derived from a “main duality theorem” which can be deduced from the duality theorem of linear programming or from more general results of convex analysis.     

Extensions of Liouville theorems

The method of deriving Liouville's theorem for subharmonic functions in the plane from the corresponding Hadamard three-circles theorem is extended to a more general and abstract setting. Two extensions of Liouville's theorem for vector-valued holomorphic functions of several complex variables are also mentioned.     

A quantifier for matroid duality

A quantifier is introduced on the elements of a matroid which, given an element e, says “for all elements (except possibly e) of some circuit containing e,…”. The matroid dual of this quantifier is shown to be identical with its logical dual, and this provides an elegant reformulation of Minty's self-dual axiomatization of matroids.This approach also provides a practical, and in a sense optimal, means of taking a statement in terms of circuits and constructing its dual, still in terms of circuits.     

Lyapunov's center theorem for resonant equilibrium

Lyapunov's center theorem relative to the existence of families of periodic orbits emanating from an equilibrium is generalized to cases where a resonance occurs between two basic frequencies. Analytical Hamiltonian systems are considered and the theorems depend on the nonannulation of an invariant of the system.The proof is performed in two steps. In a first step the theorems are shown to be valid for some approximation of the Hamiltonian system. These results are described in a previous paper (Henrard, 1970) and are only summarized here. In a second step Poincaré's perturbation theorem is generalized in order to transfer to the original system the conclusions relatives to its approximations.In the conclusion, our results are compared with similar results published recently.