Integrable Discretizations of Chiral Models

A construction of conservation laws for chiral models (generalized sigma-models on a two-dimensional space-time continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and B\"acklund transformations for the class of models considered in this work.Comment: 14 pages, Late

Is vitrification involved in depression of the phase transition temperature in dry phospholipids?

AbstractRecent literature has suggested that the depression of the phase transition temperature (Tm) in dry phospholipids by sugars may be ascribed to vitrification of the stabilizing solute, rather than by the direct interaction between sugar and phospholipid we have proposed. Koster et al. ((1994) Biochim. Biophys. Acta 1193, 14–150) claim that the only necessity is that the glass transition (Tg) for the sugar exceed Tm for the lipid. Evidence is presented in the present paper that this is not sufficient. Based on the vitrification hypothesis of Koster et al., the predicted order of effectiveness in depressing Tm in dry dipalmitoylphosphatidylcholine (DPPC) is dextran ≥ hydroxyethyl starch > stachyose > raffinose > trehalose > sucrose > glucose. In fact, the opposite order was seen. The effect of raffinose, sucrose, or trehalose on Tm in dry DPPC depends on the thermal history of the sample, as we have reported previously. When DPPC dried with trehalose is heated for the first time, Tm is about 55°C, but on the second and subsequent heating scans Tm falls to about 25°C. Koster et al. suggest that this effect is due to heating the sample above Tg rather than to melting the hydrocarbon chains. We present evidence here that all that is required is for the chains to be melted. Further, we show that retention of residual water by DPPC dried with trehalose depends on the drying temperature, but is independent of drying temperature with glucose, a finding that is consistent with direct interaction. We conclude that vitrification is not in itself sufficient to depress Tm in dry phospholipids

Low frequency of HER2 amplification and overexpression in early onset gastric cancer

BACKGROUND: The recent ToGA trial results indicated that trastuzumab is a new, effective, and well-tolerated treatment for HER2-positive gastric cancer (GC). Although GC mainly affects older patients, fewer than 10% of GC patients are considered early-onset (EOGC) (presenting at the age of 45 years or younger). These EOGC show different clinicopathological and molecular profiles compared to late onset GC suggesting that they represent a separate entity within gastric carcinogenesis. In light of potential trastuzumab benefit, subpopulations of GC such as EOGC (versus late onset) should be evaluated for their frequency of amplification and overexpression using currently available techniques. METHODS: Tissue microarray (TMA) blocks of 108 early onset GC and 91 late onset GC were stained by immunohistochemistry (IHC, Hercep test, DAKO) and chromogenic in situ hybridization (CISH, SPoT-Light, Invitrogen). RESULTS: Overall, we found only 5% HER2 high level amplification and 3% HER2 3+ overexpression (6/199). In addition, 8 patients (4%) showed a low level CISH amplification and 9 patients (4.5%) showed a 2+ IHC score. IHC and CISH showed 92% concordance and CISH showed less heterogeneity than IHC. In 2/199 cases (1%), IHC showed clinically relevant heterogeneity between TMA cores, but all cases with focal IHC 3+ expression were uniformly CISH high level amplified. Early onset GCs showed a significantly lower frequency of HER2 amplification (2%) and overexpression (0%) than late onset GCs (8% and 7% respectively) (p = 0.085 and p = 0.008 respectively). Proximal GC had more HER2 amplification (9% versus 3%) and overexpression (7% versus 2%) than distal tumours although this difference was not significant (p = 0.181 and p = 0.182 respectively). HER2 CISH showed more high level amplification in the intestinal type (7%, 16% if low-level included) compared to the mixed (5%, 5% if low-level included) and diffuse type (3%, 4% if low-level included) GCs (p = 0.029). A similar association was seen for HER2 IHC and histologic type (p = 0.008). Logistic regression indicated a significant association between HER2 expression and age, which remained significant when adjusted for both location and histological type. CONCLUSIONS: Even focal HER2 overexpression in GC points to uniform HER2 amplification by CISH. We show for the first time that early onset GC has a lower frequency of HER2 amplification and overexpression than late onset GC, and confirm that intestinal type GC shows the highest rate of HER2 amplification and overexpression

Noncommutative Geometry of Finite Groups

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).Comment: 34 pages, Late

Selfsimilarity and growth in Birkhoff sums for the golden rotation

We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean rotation number a with periodic continued fraction approximations p(n)/q(n), where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of S(q(n),a) with the existence of an experimentally established limit function f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity on the interval [0,1]. The function f satisfies a functional equation f(ax) + (1-a) f(x)= b(x) with a monotone function b. The limit lim S(q(n),a) for n going to infinity can be expressed in terms of the function f.Comment: 14 pages, 8 figure

Differential Geometry of Group Lattices

In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of "bicovariant" Cayley graphs with the property that ad(S)S is contained in S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first order calculi extend to higher orders and then allow to introduce further differential geometric structures. Furthermore, we explore the properties of "discrete" vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analogue of the lattice gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained.Comment: 51 pages, 11 figure

The Asymptotic Number of Attractors in the Random Map Model

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as being its image. We derive here explicit formulas for the statistical distribution of the number of attractors in the system. As in related results, the number of operations involved by our formulas increases exponentially with n; therefore, they are not directly applicable to study the behavior of systems where n is large. However, our formulas lend themselves to derive useful asymptotic expressions, as we show.Comment: 16 pages, 1 figure. Minor changes. To be published in Journal of Physics A: Mathematical and Genera

Programming effects of an early-life diet containing large phospholipid-coated lipid globules are transient under continuous exposure to a high-fat diet

Breastfeeding is associated with a lower risk of developing obesity during childhood and adulthood compared to feeding infant milk formula (IMF). Previous studies have shown that an experimental IMF (eIMF; comprising Nuturis®), programmed mouse pups for a lower body weight and fat mass gain in adulthood when challenged with a high-fat diet (HFD), compared to a control IMF (cIMF). Nuturis has a lipid composition and structure more similar to breastmilk. Here, the long-term effects were tested of a similar eIMF, but with an adapted lipid composition, and a cIMF, on body weight, glucose homeostasis, liver and adipose tissue. Nutrient composition was similar for the eIMF and cIMF; the lipid fractions comprised ∼50% milkfat. C57BL/6JOlaHsd mice were fed cIMF or eIMF from postnatal (PN) day 16-42 followed by a HFD until PN168. Feeding eIMF versus cIMF in early life resulted in a lower body weight (-9%) and body fat deposition (-14%) in adulthood (PN105). The effect appeared transient, as from PN126 onward, after 12 weeks HFD, eIMF-fed mice caught up on controls and body and fat weights became comparable between groups. Glucose and energy metabolism were similar between groups. At dissection (PN168), eIMF-fed mice showed larger (+27%) epididymal fat depots and a lower (-26%) liver weight without clear morphological aberrations. Our data suggest the size and coating but not the lipid composition of IMF fat globules underlies the programming effect observed. Prolonged exposure to a HFD challenge partly overrules the programming effect of early diet