Mackey Formula in Type A

The statement (and the proof) of Theorem 4.1.1 of the paper‘Mackey formula in type A’ [Proc. London. Math.Soc. (3) 80 (2000) 545–574] is false. In this corrigendumwe provide a correct statement (and a correct proof). As a consequence,we see that Corollary 4.1.2 of the paper holds. So all the otherstatements in the paper are correct (up to minor misprints...).2000 Mathematical Subject Classification: 20G05, 20G40.     

Corrigendum to: “Fractional Functional Differential Inclusions with Finite Delay” [Nonlinear Anal. 70(5) (2009) 2091–2105]

We correct an error made in the paper (J. Henderson and A. Ouahab, Fractional Functional Differential Inclusions with Finite Delay [Nonlinear Anal. 70(5) (2009) 2091–2015]), in an application of Theorem 4.1 in the proof of Theorem 4.2. We have revised the statement of Theorem 4.2 so that Theorem 4.1 is now applicable in its proof.     

A remark on the incomparability of two criteria for a uniform convergence of probability measures

Summary This paper connects with Theorem 3 of the author’s paper [1], in which two criteria for type (B) _d convergence ([3]) are shown to be incomparable to each other by presenting two examples. However, the statement of the theorem is not complete. In the present paper, we shall modify the statement of the theorem and give a proof by presenting a new example.     

A Frobenius Theorem on Convenient Manifolds

 A Frobenius Theorem for finite dimensional, involutive subbundles of the tangent bundle of a convenient manifold is proved. As first key applications Lie’s second fundamental theorem and Nelson’s theorem are treated in the convenient case.     

A Frobenius Theorem on Convenient Manifolds

 A Frobenius Theorem for finite dimensional, involutive subbundles of the tangent bundle of a convenient manifold is proved. As first key applications Lie’s second fundamental theorem and Nelson’s theorem are treated in the convenient case. (Received 2 February 2001; in revised form 29 May 2001)     

有关高阶奇异积分的Bertrand-Poincaré型换序公式

本文把关于Cauchy核奇异积分的Bertrand-Poincaré换序公式推广到高阶奇异积分的情况。主要结果见定理1至3,其中定理1已见于文献,但这里的表达方式和证法均是新的且较简。     

On the convergence of circle packings to the Riemann map

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary. Oblatum 15-V-1995 & 13-XI-1995     

A fundamental theorem of modular arithmetic

Periodica Mathematica Hungarica - The Fundamental Theorem of Arithmetic is a statement about the uniqueness of factorization in the ring of integers. The notation and proof easily generalize to...     

Random trees under CH

We extend Jensen’s Theorem that Souslin’s Hypothesis is consistent with CH, by showing that the statement Souslin’s Hypothesis holds in any forcing extension by a measure algebra is consistent with CH. We also formulate a variation of the principle (*) (see [AT97], [Tod00]) for closed sets of ordinals, and show its consistency relative to the appropriate large cardinal hypothesis. Its consistency with CH would extend Silver’s Theorem that, assuming the existence of an inaccessible cardinal, the failure of Kurepa’s Hypothesis is consistent with CH, by its implication that the statement Kurepa’s Hypothesis fails in any forcing extension by a measure algebra is consistent with CH.     

Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties

El Kaoutit and Gómez-Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.

     

Corrigendum to: When is a sum of annihilator ideals an annihilator ideal?

In this corrigendum, we correct the statement and proof of Lemma 3.2, revise the proofs of Lemma 3.3 and Theorem 3.5, retract Proposition 3.10 and correct several typographical errors.     

Corrigendum to Cleft Extensions of Hopf Algebroids

Theorem 2.2 stated a monoidal isomorphism between the comodule categories of two bialgebroids in a Hopf algebroid. The proof of Theorem 2.2 was based on the journal version of Brzeziński (Ann Univ Ferrara Sez VII (NS) 51:15–27, 2005, Theorem 2.6), whose proof turned out to contain an unjustified step. Here we show that all other results in our paper remain valid if we drop unverified Theorem 2.2, and return to an earlier definition of a comodule of a Hopf algebroid that distinguishes between comodules of the two constituent bialgebroids.     

Corrections: on a question of boyle and handelman concerning eigenvalues of nonnegative matrices

Professor Raphale Leowy, of the Technion-Israel Institute of Technology in Haifa, has informed us that the assumptions that we made in the statement of Theorem I (which appeared in Volume 36. 1993, 125–140) concerning the case when n≥5 are weaker that those than we made use of in the proof of the theorem. Thus without a change in the proof, only the following result is correctly proved in the theorem:     

Geometric proof of R?dseth’s formula for Frobenius numbers

Using a geometric interpretation of continued fractions, we give a new proof of R?dseth’s formula for Frobenius numbers.     

Corrections: on a question of boyle and handelman concerning eigenvalues of nonnegative matrices

Professor Raphale Leowy, of the Technion-Israel Institute of Technology in Haifa, has informed us that the assumptions that we made in the statement of Theorem I (which appeared in Volume 36. 1993, 125-140) concerning the case when n≥5 are weaker that those than we made use of in the proof of the theorem. Thus without a change in the proof, only the following result is correctly proved in the theorem:     

Graph factors

This exposition is concerned with the main theorems of graph-factor theory, Hall’s and Ore’s Theorems in the bipartite case, and in the general case Petersen’s Theorem, the 1-Factor Theorem and thef-Factor Theorem. Some published extensions of these theorems are discussed and are shown to be consequences rather than generalizations of thef-Factor Theorem. The bipartite case is dealt with in Section 2. For the proper presentation of the general case a preliminary theory of “G-triples” and “f-barriers” is needed, and this is set out in the next three Sections. Thef-Factor Theorem is then proved by an argument of T. Gallai in a generalized form. Gallai’s original proof derives the 1-Factor Theorem from Hall’s Theorem. The generalization proceeds analogously from Ore’s Theorem to thef-Factor Theorem.     

Hard Lefschetz theorem for simple polytopes

McMullen’s proof of the Hard Lefschetz Theorem for simple polytopes is studied, and a new proof of this theorem that uses conewise polynomial functions on a simplicial fan is provided.     

A COUNTEREXAMPLE TO THE -CONJECTURE

There was an error in the statement of Corollary 1 in [Comm. Alg. 2000, 28, 3973–3982]. So that corollary, as well as the proof of Theorem 2 in loc. cit., should be replaced.     

The universal partition theorem for oriented matroids

We consider realization spaces of a family of oriented matroids of rank three as point configurations in the affine plane. The fundamental problem arises as to which way these realization spaces partition their embedding space. The Universal Partition Theorem roughly states that such a partition can be as complicated as any partition of ℝ ⁿ into elementary semialgebraic sets induced by an arbitrary finite set of polynomials in ℤ[X]. We present the first proof of the Universal Partition Theorem. In particular, it includes the first complete proof of the so-called Universality Theorem. This work was supported by the Deutsche Forschungsgemeinschaft, Graduiertenkolleg “Analyse und Konstruktion in der Mathematik”.     

Corrigendum to “Orbit closures in the enhanced nilpotent cone” [Adv. Math. 219 (1) (2008) 27–62]

In this note, we point out an error in the proof of Theorem 4.7 of [P.N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (1) (2008) 27–62], a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure. We also give independent proofs of later results that depend on that statement, so all other results of that paper remain valid.