The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems known to the author for the integral lattice Z² are reduced to some special cases of this generalization. Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35.     

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.     

A Discrete Theorem of Goursat

In order to describe stress and displacement fields in the neightborhood of singularities in fracture mechanics the so-called theory of complex stress function based on Kolosov and Muskhelishvili can be used. The relation between linear elasticity and complex function theory is based on the Theorem of Goursat. In this paper a discrete version of the Theorem is proved.     

Normal families of covering maps

Suppose that (f _n)n∈N is a sequence of meromorphic covering maps which is uniformly convergent in a neighbourhood of a pointx∈Ĉ such that the limit function is non-constant. It is proved that the convergence extends to the largest domain where the sequence eventually is defined and that the limit function again is a covering map. As a consequence of this result, we obtain a rescaling lemma for holomorphic covering maps, a version of the Carathéodory Kernel Theorem for arbitrary domains in the sphere, and an elementary access to the Riemann Uniformization Theorem for arbitrary domains in the sphere. An application to complex dynamics of transcendental entire functions provides that the existence of an invariant Baker domain implies a certain frequency of singularities of the inverse function.     

A generalization of the Serre-Swan-Mallios Theorem

In this paper we prove a generalized version of Serre-Swan-Mallios Theorem for arbitrary Waelbroeck algebras.     

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.     

Non-crossed products over function fields

Noncrossed product division algebras are constructed over all function fields and iterated power series fields over global fields, using Hilbert's Irreducibility Theorem and the construction of [B]. Minimum indexes obtained are p ² for odd p and 2³ otherwise. Examples are obtained with large index to exponent ratio. Received: 12 February 2001 / Revised version: 26 November 2001     

Corrigendum to “Morita theory for coring extensions and cleft bicomodules” [Adv. Math. 209 (2) (2007) 611-648]

The results in our paper heavily rely on the journal version of [T. Brzeziński, A note on coring extensions, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005) 15-27; a corrected version is available at http://arxiv.org/abs/math/0410020v3, Theorem 2.6]. Since it turned out recently that in the proof of the quoted theorem there are some assumptions missing, our derived results are not expected to hold at the stated level of generality either. Here we supplement the constructions in our article with the missing assumptions and show that they hold in most of our examples. In order to handle also the non-fitting case of cleft extensions by arbitrary Hopf algebroids, Morita contexts are constructed that do not necessarily correspond to coring extensions. They are used to prove a Strong Structure Theorem for cleft extensions by arbitrary Hopf algebroids. In this way we obtain in particular a corrected form of the journal version of [G. Böhm, Integral theory for Hopf algebroids, Algebr. Represent. Theory 8 (4) (2005) 563-599; Corrigendum, to be published; see also http://arxiv.org/abs/math/0403195v4, Theorem 4.2], whose original proof contains a similar error.     

Greenberg Approximation and the Geometry of Arc Spaces

We study the differential properties of generalized arc schemes and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.     

Typical Recurrence for Lifts of mean Rotation Zero Annulus Homeomorphisms

This paper is about typical (uniform topology dense G) propertiesof homeomorphisms of the torus or annulus which preserve a fixedmeasure and have mean rotation zero. We first show that ergodicityis typical (Theorem 1). We then show that the lift (to the universalcovering space) of such a homeomorphism of the annulus is theskew product of the annulus homeomorphism with respect to askewing function of mean zero. Hence Atkinson's Theorem on skewproducts, together with Theorem 1, implies that it is typicalfor an annulus homeomorphism of mean rotation zero to have arecurrent lift (Theorem 3). Standard arguments then give thePoincaré-Birkhoff Fixed Point Theorem as a corollary.     

Separate Necessary and Sufficient Conditions for the Local Minimum of a Function

This paper develops two necessary conditions for a local minimum of an arbitrary extended real-valued function (Theorem 2.2) and, quite separately, two conditions, each sufficient for such a function to have a strict local minimum (Theorem 2.3).This work was supported by the National Science Foundation of China, Grant 10171034, and the Natural Science Foundation of the Department of Education of Guangdong Province, Grant 820138.The author thanks the referees for helpful comments.     

Markov structures and decay of correlations for non-uniformly expanding dynamical systems

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure. Moreover, the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.     

Homogenization of divergence-form operators with lower-order terms in random media

The probabilistic machinery (Central Limit Theorem, Feynman-Kac formula and Girsanov Theorem) is used to study the homogenization property for PDE with second-order partial differential operator in divergence-form whose coefficients are stationary, ergodic random fields. Furthermore, we use the theory of Dirichlet forms, so that the only conditions required on the coefficients are non-degeneracy and boundedness. Received: 27 August 1999 / Revised version: 27 October 2000 / Published online: 26 April 2001     

An effective Schmidt's subspace theorem for non-linear forms over function fields

We deduce an effective version of Schmidt's subspace theorem over function fields of characteristic zero for arbitrary homogeneous polynomials in place of linear forms. We will then apply this result to study the S-integral points of a general Thue's equation.     

A growth gap for diffeomorphisms of the interval

Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of itsn-th iteration. We get a function ofn called the growth sequence. Its asymptotic behaviour is an interesting invariant, which naturally appears both in geometry of the diffeomorphism groups and in smooth dynamics. Our main result is the following Gap Theorem: the growth rate of this sequence is either exponential or at most quadratic withn. Further, we construct diffeomorphisms whose growth sequence has quite irregular behaviour. This construction easily extends to arbitrary manifolds.     

On Existence Theorems for Solutions of Non‐Linear Systems

An equivalent formulation of a recent result in [1] states that if the conditions of the Theorem of Newton‐Kantorovich are satisfied in a slightly modified (but equivalent) form in the maximum norm for a function g : G → ℝⁿ, G ⊆ ℝⁿ, guaranteeing the existence of a zero in D, then the conditions of Miranda's Theorem are automatically satisfied. We prove that this result holds for arbitrary norms if the conditions of the Theorem of Newton‐Kantorovich are suitable strengthened and Miranda's Theorem is suitably generalized.     

Bornes effectives des fonctions d'approximation des solutions formelles d'équations binomiales

The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the Weierstrass Preparation Theorem, we apply this effective result to binomial equations. We prove that the Artin function of a system of binomial equations is bounded by a doubly exponential function in general and that it is bounded by an affine function if the order of the approximated solutions is bounded.     

The Finite Horizon Optimal Multi-Modes Switching Problem: The Viscosity Solution Approach

In this paper we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is in relation with the valuation of firms in a financial market.     

UNCONDITIONAL CAUCHY SERIES AND UNIFORM CONVERGENCE ON MATRICES

The authors obtain new characterizations of unconditional Cauchy series in termsof separation properties of subfamilies of P(N), and a generalization of the Orlicz-PettisTheorem is also obtained. New results on the uniform convergence on matrices anda new version of the Hahn-Schur summation theorem are proved. For matrices whoserows define unconditional Cauchy series, a better sufficient condition for the basicMatrix Theorem of Antosik and Swartz, new necessary conditions and a new proof ofthat theorem are given.     

Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fⁿ there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.