The relevance of Freiman’s theorem for combinatorial commutative algebra

Mathematische Zeitschrift - Freiman’s theorem gives a lower bound for the cardinality of the doubling of a finite set in $${mathbb R}^n$$ . In this paper we give an interpretation of his...     

An Assmus–Mattson theorem for codes over commutative association schemes

We prove an Assmus–Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with s classes). This in particular generalizes the Assmus–Mattson-type theorems for \(\mathbb {Z}_4\)-linear codes due to Tanabe (Des Codes Cryptogr 30:169–185, 2003) and Shin et al. (Des Codes Cryptogr 31:75–92, 2004), as well as the original theorem by Assmus and Mattson (J Comb Theory 6:122–151, 1969). The weights of a code are s-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining t-designs from the code involve concepts from polynomial interpolation in s variables. The Terwilliger algebra is the main tool to establish our results.     

An effective Schmidt’s subspace theorem over function fields

We establish an effective version of the Schmidts subspace theorem for higher dimensional function fields of characteristic zero.Received: 30 January 2001     

Sturm’s theorem on zeros of linear combinations of eigenfunctions

Motivated by recent questions about the extension of Courant’s nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm–Liouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the twentieth-century. Although not specialists in History of Sciences, we have tried to put this theorem into the context of nineteenth century mathematics.     

Vaught’s conjecture for modules over a commutative Prüfer ring

We prove that Vaught’s conjecture is true for modules over a commutative Prüfer ring. It is shown that a positive solution to Vaught’s conjecture for modules over 1-dimensional Noetherian domains would imply the same for modules over finitely presented algebras. This article was written during the visit of the second author to the University of Manchester supported by EPSRC grant GR/L68827. She would like to thank the University for hospitality. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 419–435, July–August, 1999.     

Berkson’s theorem

We give a new proof of the theorem that Amitsur’s complex for purely inseparable field extensions has vanishing homology in dimensions higher than 2. This is accomplished by computing the kernel and cokernel of the logarithmic derivativet →Dt/t mapping the multiplicative Amitsur complex to the acyclic additive one (D is a derivation of the extension field). This research was supported by National Science Foundation grant NSF GP 1649.     

Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

A refinement of Nesterenko’s linear independence criterion with applications to zeta values

We refine (and give a new proof of) Nesterenko’s famous linear independence criterion from 1985, by making use of the fact that some coefficients of linear forms may have large common divisors. This is a typical situation appearing in the context of hypergeometric constructions of mathbbQ{mathbb{Q}}-linear forms involving zeta values or their q-analogs. We apply our criterion to sharpen previously known results in this direction.     

Rosenberg’s reconstruction theorem

Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber. Moreover, we determine the automorphism class group of the category of quasi-coherent sheaves.     

On Jordan’s theorem

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

Funayama’s theorem revisited

Funayama’s theorem states that there is an embedding e of a lattice L into a complete Boolean algebra B such that e preserves all existing joins and meets in L iff L satisfies the join infinite distributive law (JID) and the meet infinite distributive law (MID). More generally, there is a lattice embedding e: L → B preserving all existing joins in L iff L satisfies (JID), and there is a lattice embedding e: L → B preserving all existing meets in L iff L satisfies (MID). Funayama’s original proof is quite involved. There are two more accessible proofs in case L is complete. One was given by Grätzer by means of free Boolean extensions and MacNeille completions, and the other by Johnstone by means of nuclei and Booleanization. We show that Grätzer’s proof has an obvious generalization to the non-complete case, and that in the complete case the complete Boolean algebras produced by Grätzer and Johnstone are isomorphic. We prove that in the non-complete case, the class of lattices satisfying (JID) properly contains the class of Heyting algebras, and we characterize lattices satisfying (JID) and (MID) by means of their Priestley duals. Utilizing duality theory, we give alternative proofs of Funayama’s theorem and of the isomorphism between the complete Boolean algebras produced by Grätzer and Johnstone. We also show that unlike Grätzer’s proof, there is no obvious way to generalize Johnstone’s proof to the non-complete case.     

Extending Kotzig’s theorem

The weight of a graphG is the minimum sum of the two degrees of the end points of edges ofG. Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grünbaum and Shephard proved that every graph triangulating the torus has weight at most 15. We extend these results for graphs, multigraphs and pseudographs “triangulating” the sphere withg handlesS _g ,g≧1, showing that the corresponding weights are at most about and 24g−9, respectively; if a (multi, pseudo) graph triangulatesS _g and it is big enough, then its weight is at most 15.     

Connected Baranyai’s theorem

Let K _n ^h = (V, ( _h ^V )) be the complete h-uniform hypergraph on vertex set V with ¦V¦ = n. Baranyai showed that K _n ^h can be expressed as the union of edge-disjoint r-regular factors if and only if h divides rn and r divides \((_{h - 1}^{n - 1} )\) . Using a new proof technique, in this paper we prove that λK _n ^h can be expressed as the union \(\mathcal{G}_1 \cup ... \cup \mathcal{G}_k \) of k edge-disjoint factors, where for 1≤i≤k, \(\mathcal{G}_i \) is r _i -regular, if and only if (i) h divides r _i n for 1≤i≤k, and (ii) \(\sum\nolimits_{i = 1}^k {r_i = \lambda (_{h - 1}^{n - 1} )} \) . Moreover, for any i (1≤i≤k) for which r _i ≥2, this new technique allows us to guarantee that \(\mathcal{G}_i \) is connected, generalizing Baranyai’s theorem, and answering a question by Katona.