on stability criterion of complete intersections

For protective varieties, it is known that Chow stable implies N-th Hilbert-Mumford stable for N sufficiently large, which follows from the works of J. Fogarty [2, 6]. In this article, we firstly shall provide a simple criterion for Chow stability of complete intersections. The criterion for Chow stability was previously provided by Mumford [5], but our calculation is different from Mumford’s in that ours is based on the results of Zhang’s article [10]. Applying it, we secondly shall give an elementary proof of the above implications in a complete intersections case.     

Old and new moving-knife schemes

In general, moving-knife schemes seem to be easier to come by than pure existence results (like Neyman’s [N] theorem) but harder to come by than discrete algorithms (like the Dubins-Spanier [DS] last-diminisher method). For envy-free allocations for four or more people, however, the order of difficulty might actually be reversed. Neyman’s existence proof (for anyn) goes back to 1946, the discovery of a discrete algorithm for alln ≥ 4 is quite recent [BT1, BT2, BT3], and a moving-knife solution forn = 4 was found only as this article was being prepared (see [BTZ]). We are left with this unanswered question: Is there a moving-knife scheme that yields an envyfree division for five (or more) players?     

Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent. We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.     

An analogue of the rauch variational formula for prym differentials

In [5], H. E. Rauch discovered a formula for the first variation of an abelian differential on a Riemann surface and its periods with respect to the change of complex structure induced by a Beltrami differential. R. S. Hamilton, in [3], and discussed by C. Earle in [1], found an elegant proof of the formula using only first principles and not requiring uniformization theory. His proof uses a small amount of Hodge theory, the Riemann bilinear period relations, and a simple operator construction. In this article, we find an analogue of Rauch’s formula for the Prym differentials using some of Hamilton’s techniques, the Hodge theorem for vector bundles, and the “Prym version” of the Riemann bilinear relations. We discover a complicated set of formulas for the variation of the Prym differentials, with different specific solutions depending to the make-up of the Prym character. We conclude that the variation of the Prym periods with a given character depends on the differentials for the character and the differentials for its inverse. This explains the simplicity of the classical case, where the character is its inverse.     

Gap series constructions for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

Highly oscillatory bounded solutions of div(∇u|∇u| ^p−2) = 0 are constructed when p > 2. Fatou’s theorem is shown to fail for this equation. Tom Wolff wrote this paper in 1984, but he never published it. With his family’s permission, we have edited it for publication here. Except for the shorter proof of Lemma 2.1 and the citations of [1] and [12], our alterations to the paper have mostly been typographical. We thank Juan Manfredi for help on Section 3.     

Comment on “Duality theory in fuzzy linear programming problems with fuzzy coefficients”

This note provides a counterexample to illustrate the incorrectness of the proof of Proposition 3.3 that was presented by Wu (Fuzzy Optim Decis Mak 2:61–73, 2003). The original proof of Proposition 3.3 by Wu can only be correct when the extra assumption \(\mu _{\widetilde{y}_i}(0)= 1\) is added. The correct proof of Proposition 3.3 is also presented in this note.     

Corrigendum to “Belief functions contextual discounting and canonical decompositions” [International Journal of Approximate Reasoning 53 (2012) 146–158]

Proposition 4 and Theorem 1 of the article “Belief functions contextual discounting and canonical decompositions” [International Journal of Approximate Reasoning 53 (2012) 146–158] provide an erroneous result. We give here the true result with a correct proof.     

Sequences of 0’s and 1’s: Special sequence spaces with the separable Hahn property

As pointed out in [4] the paper [2], authored by G. Bennett, J. Boos and T. Leiger, contains a nontrivial gap in the argumentation of the proof of Theorem 5.2 which is one of main results of that paper and has been applied three times. Till now neither the gap is closed nor a counterexample has been stated. That is why the authors have examined in [4] the situation around the ‘gap’ aiming to a better understanding for the gap. The aim of this paper is to prove the mentioned applications of the theorem in doubt by using gliding hump arguments (quite similar to the classical proofs of the Theorems of Schur and Hahn in the first case (cf. [3]) and a very technical and artful construction, being of independent mathematical interest, in the second case). Research of T. Leiger supported by Estonian Science Foundation Grant 5376.     

Semigroups satisfying variable identities

The purpose of this note is to generalize a theorem of Tamura’s [3] providing a self-contained and, we think, more elementary proof than Tamura’s in that it avoids using the theory of contents. Tamura’s result states that a semigroup S satisfies an identify xy=f(x,y) with f(x,y) a word of length greater than 2 which starts with y and ends in x if and only if S is an inflation of a semilattice of groups satisfying the same identity. We investigate semigroups as in Tamura’s Theorem, except that we permit f(x,y) to vary with x and y.     

An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality

We give a short alternative proof of Berg and Nikolaev’s recent theorem on a characterization of CAT(0)-spaces via the quadrilateral inequality.     

Kolmogorov’s 1954 paper on nearly-integrable Hamiltonian systems

Following closely Kolmogorov’s original paper [1], we give a complete proof of his celebrated Theorem on perturbations of integrable Hamiltonian systems by including few “straightforward” estimates.      

Tverberg’s theorem via number fields

We show that Tverberg’s theorem follows easily from a theorem of which Bárány [1] has given a very short proof.     

A metric variant of Frobenius's theorem and some other remarks on positive matrices

The theory of positive (=nonnegative) finite square matrices continues, three quarters of a century after the pioneering and well-known papers of Perron and Frobenius [4], to present a multitude of different aspects. This is evidenced, for example, by the recent papers [1] and [2], as well as by the vast literature concerned with extensions to operators on infinite dimensional spaces (see [5]). Supposing A to be a positive n × n matrix with spectral radius r(A) = 1, the main purpose of this note is to display the role of λ = 1 as a root of the minimal polynomial of A (or equivalently, of certain norm conditions on A, for the lattice structure of the space M spanned by the unimodular eigenvectors of A as well as for the permutational character of A on M. Proposition 1 can thus be viewed as a variant of Frobenius's theorem on the peripheral spectrum of indecomposable square matrices, and we hope that the proof of Proposition 2 will clarify to what extent indecomposability is responsible for the main results available in that special case. The remaining remarks (Propositions 3 and 4) are concerned with the spectral characterization of permutation matrices and with finite groups of positive matrices. Some of that material is undoubtedly known, but we give simple, transparent proofs.     

Towards an Algebraic Proof of Deligne’s Regularity Criterion. An Informal Survey of Open Problems

We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4]. Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle. We report on ongoing correspondence with Y. André and N. Tsuzuki, leading to partial results and provide examples indicating the subtlety of the problem. Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005     

Fefferman’s SAK principle and a priori estimates for second order operators

Abstract We prove in details the higher codimensional version of Theorem 1.1 [11]. This provides a complete proof of Fefferman’s SAK Principle for a class of PDO’s with symplectic characteristic manifold. Keywords: A priori estimates, General theory of PDO’s     

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 changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

A note on monotone covering properties

In this note, we show that a monotonically normal space that is monotonically countably metacompact (monotonically meta-Lindelöf) must be hereditarily paracompact. This answers a question of H.R. Bennett, K.P. Hart and D.J. Lutzer. We also show that any compact monotonically meta-Lindelöf T₂-space is first countable. In the last part of the note, we point out that there is a gap in Proposition 3.8 which appears in [H.R. Bennett, K.P. Hart, D.J. Lutzer, A note on monotonically metacompact spaces, Topology Appl. 157 (2) (2010) 456-465]. We finally give a detailed proof of how to overcome the gap.     

A note on the stability of the Wulff shape

We give a new proof of Palmer’s result [6] that theWulff shapes are the only closed, oriented, stable hypersurfaces with constant anisotropic mean curvature. Our approach is based on the construction of a suitable testfunction in the anisotropic index form, thus generalizing the original proof of Barbosa, do Carmo [1]. Received: 3 August 2005     

Half-flat structures on decomposable lie groups

Half-at SU(3)-structures are the natural initial values for Hitchin’s evolution equations whose solutions define parallel G₂-structures. Together with the results of [SH], the results of this article completely solve the existence problem of left-invariant half-at SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras, which refines and clarifies the existing classification of five-dimensional Lie algebras.