The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

Feature-invariant image registration method for quantification of surgical outcomes in patients with craniosynostosis: a preliminary study

Craniosynostosis consists of a premature fusion of the sutures in an infant skull that restricts skull and brain growth. During the last decades, there has been a rapid increase of fundamentally diverse surgical treatment methods. At present, the surgical outcome has been assessed using global variables such as cephalic index, head circumference, and intracranial volume. However, these variables have failed in describing the local deformations and morphological changes that may have a role in the neurologic disorders observed in the patients. This report describes a rigid image registration-based method to evaluate outcomes of craniosynostosis surgical treatments, local quantification of head growth, and indirect intracranial volume change measurements. The developed semiautomatic analysis method was applied to computed tomography data sets of a 5-month-old boy with sagittal craniosynostosis who underwent expansion of the posterior skull with cranioplasty. Quantification of the local changes between pre- and postoperative images was quantified by mapping the minimum distance of individual points from the preoperative to the postoperative surface meshes, and indirect intracranial volume changes were estimated. The proposed methodology can provide the surgeon a tool for the quantitative evaluation of surgical procedures and detection of abnormalities of the infant skull and its development.

Natural variations at position 93 of the invariant Vα24-Jα18 α chain of human iNKT-cell TCRs strongly impact on CD1d binding

Human invariant natural killer T (NKT) cell TCRs bind to CD1d via an "invariant" Vα24-Jα18 chain (iNKTα) paired to semi-invariant Vβ11 chains (iNKTβ). Single-amino acid variations at position 93 (p93) of iNKTα, immediately upstream of the "invariant" CDR3α region, have been reported in a substantial proportion of human iNKT-cell clones (4-30%). Although p93, a serine in most human iNKT-cell TCRs, makes no contact with CD1d, it could affect CD1d binding by altering the conformation of the crucial CDR3α loop. By generating recombinant refolded iNKT-cell TCRs, we show that natural single-nucleotide variations in iNKTα, translating to serine, threonine, asparagine or isoleucine at p93, exert a powerful effect on CD1d binding, with up to 28-fold differences in affinity between these variants. This effect was observed with CD1d loaded with either the artificial α-galactosylceramide antigens KRN7000 or OCH, or the endogenous glycolipid β-galactosylceramide, and its importance for autoreactive recognition of endogenous lipids was demonstrated by the binding of variant iNKT-cell TCR tetramers to cell surface expressed CD1d. The serine-containing variant showed the strongest CD1d binding, offering an explanation for its predominance in vivo. Complementary molecular dynamics modeling studies were consistent with an impact of p93 on the conformation of the CDR3α loop.

Intestinal microbes affect phenotypes and functions of invariant natural killer T cells in mice

Invariant natural killer T (iNKT) cells undergo canonical, Vα14-Jα18 rearrangement of the T-cell receptor (TCR) in mice; this form of the TCR recognizes glycolipids presented by CD1d. iNKT cells mediate many different immune reactions. Their constitutive activated and memory phenotype and rapid initiation of effector functions after stimulation indicate previous antigen-specific stimulation. However, little is known about this process. We investigated whether symbiotic microbes can determine the activated phenotype and function of iNKT cells.

Kernels and Designs for Modelling Invariant Functions: From Group Invariance to Additivity

We focus on kernels incorporating different kinds of prior knowledge on functions to be approximated by Kriging. A recent result on random fields with paths invariant under a group action is generalised to combinations of composition operators, and a characterisation of kernels leading to random fields with additive paths is obtained as a corollary. A discussion follows on some implications on design of experiments, and it is shown in the case of additive kernels that the so-called class of “axis designs” outperforms Latin hypercubes in terms of the IMSE criterion.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

The SU(3)-invariant sector of new maximal supergravity

We investigate the SU(3)-invariant sector of the one-parameter family of SO(8) gauged maximal supergravities that has been recently discovered. To this end, we construct the N=2 truncation of this theory and analyse its full vacuum structure. The number of critical point is doubled and includes new N=0 and N=1 branches. We numerically exhibit the parameter dependence of the location and cosmological constant of all extrema. Moreover, we provide their analytic expressions for cases of special interest. Finally, while the mass spectra are found to be parameter independent in most cases, we show that the novel non-supersymmetric branch with SU(3) invariance provides the first counterexample to this.