Word problems and ceers

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non-periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non-commutative and non-Boolean c.e. ring.     

Splitting trees at vertices

We present tight bounds on splitting trees into “small” subtrees.     

On the notion of Cohen–Macaulayness for non-Noetherian rings

There exist many characterizations of Noetherian Cohen–Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non-Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster–Eagon result on Cohen–Macaulayness of invariant rings, in the context of non-Noetherian rings.     

Cubic Bridgeless Graphs and Braces

There are many long-standing open problems on cubic bridgeless graphs, for instance, Jaeger’s directed cycle double cover conjecture. On the other hand, many structural properties of braces have been recently discovered. In this work, we bijectively map the cubic bridgeless graphs to braces which we call the hexagon graphs, and explore the structure of hexagon graphs. We show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using two types of McCuaig’s augmentations. In addition, we present a reformulation of Jaeger’s directed cycle double cover conjecture in the class of hexagon graphs.     

Differential geometric analysis of alterations in MH α‐helices

Antigen presenting cells present processed peptides via their major histocompatibility (MH) complex to the T cell receptors (TRs) of T cells. If a peptide is immunogenic, a signaling cascade can be triggered within the T cell. However, the binding of different peptides and/or different TRs to MH is also known to influence the spatial arrangement of the MH α‐helices which could itself be an additional level of T cell regulation. In this study, we introduce a new methodology based on differential geometric parameters to describe MH deformations in a detailed and comparable way. For this purpose, we represent MH α‐helices by curves. On the basis of these curves, we calculate in a first step the curvature and torsion to describe each α‐helix independently. In a second step, we calculate the distribution parameter and the conical curvature of the ruled surface to describe the relative orientation of the two α‐helices. On the basis of four different test sets, we show how these differential geometric parameters can be used to describe changes in the spatial arrangement of the MH α‐helices for different biological challenges. In the first test set, we illustrate on the basis of all available crystal structures for (TR)/pMH complexes how the binding of TRs influences the MH helices. In the second test set, we show a cross evaluation of different MH alleles with the same peptide and the same MH allele with different peptides. In the third test set, we present the spatial effects of different TRs on the same peptide/MH complex. In the fourth test set, we illustrate how a severe conformational change in an α‐helix can be described quantitatively. Taken together, we provide a novel structural methodology to numerically describe subtle and severe alterations in MH α‐helices for a broad range of applications. © 2013 Wiley Periodicals, Inc.     

On arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ; from a result by Barth and Fournier it follows that Δ(T)?4. A necessary and a sufficient condition for being an arbitrarily vertex decomposable star-like tree have been exhibited. The conditions seem to be very close to each other.