Polar Cremona Transformations and Monodromy of Polynomials

Consider the gradient map associated to any non-constant homogeneous polynomial f\in \C[x_0,...,x_n] of degree $d$, defined by \phi_f=grad(f): D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x)) where D(f)=\{x\in \CP^n; f(x)\neq 0\} is the principal open set associated to $f$ and $f_i=\frac{\partial f}{\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \ref{p1} we give a new lower bound for the degree $d(f)$ of $\phi_f$ under the assumption that the projective hypersurface $V:f=0$ has only isolated singularities. When $d(f)=1$, Theorem \ref{t4} yields very strong conditions on the singularities of $V$.Comment: 8 page

Bondi-Metzner-Sachs symmetry, holography on null-surfaces and area proportionality of "light-slice" entropy

It is shown that certain kinds of behavior, which hitherto were expected to be characteristic for classical gravity and quantum field theory in curved spacetime, as the infinite dimensional Bondi-Metzner-Sachs symmetry, holography on event horizons and an area proportionality of entropy, have in fact an unnoticed presence in Minkowski QFT. This casts new light on the fundamental question whether the volume propotionality of heat bath entropy and the (logarithmically corrected) dimensionless area law obeyed by localization-induced thermal behavior are different geometric parametrizations which share a common primordeal algebraic origin. Strong arguments are presented that these two different thermal manifestations can be directly related, this is in fact the main aim of this paper. It will be demonstrated that QFT beyond the Lagrangian quantization setting receives crucial new impulses from holography onto horizons. The present paper is part of a project aimed at elucidating the enormous physical range of "modular localization". The latter does not only extend from standard Hamitonian heat bath thermal states to thermal aspects of causal- or event- horizons addressed in this paper. It also includes the recent understanding of the crossing property of formfactors whose intriguing similarity with thermal properties was, although sometimes noticed, only sufficiently understood in the modular llocalization setting.Comment: 42 pages, changes, addition of new results and new references, in this form the paper will appear in Foundations of Physic

A critical look at 50 years particle theory from the perspective of the crossing property

The crossing property is perhaps the most subtle aspect of the particle-field relation. Although it is not difficult to state its content in terms of certain analytic properties relating different matrixelements of the S-matrix or formfactors, its relation to the localization- and positive energy spectral principles requires a level of insight into the inner workings of QFT which goes beyond anything which can be found in typical textbooks on QFT. This paper presents a recent account based on new ideas derived from "modular localization" including a mathematic appendix on this subject. Its main novel achievement is the proof of the crossing property of formfactors from a two-algebra generalization of the KMS condition. The main content of this article is the presentation of the derailments of particle theory during more than 4 decades: the S-matrix bootstrap, the dual model and its string theoretic extension. Rather than being related to crossing, string theory is the (only known) realization of a dynamic infinite component one-particle wave function space and its associated infinite component field. Here "dynamic" means that, unlike a mere collection of infinitely many irreducible unitary Poincar\'e group representation or free fields, the formalism contains also operators which communicate between the different irreducible Poincar\'e represenations (the levels of the "infinite tower") and set the mass/spin spectrum. Wheras in pre-string times there were unsuccessful attempts to achieve this in analogy to the O(4,2) hydrogen spectrum by the use of higher noncompact groups, the superstring in d=9+1, which uses instead (bosonic/fermionic) oscillators obtained from multicomponent chiral currents is the only known unitary positive energy solution of the dynamical infinite component pointlike localized field project.Comment: 66 pages, addition of new results, addition of references, will appear in this form in Foundations of Physic

Genetic polymorphisms associated with the inflammatory response in bacterial meningitis

BACKGROUND Bacterial meningitis (BM) is an infectious disease that results in high mortality and morbidity. Despite efficacious antibiotic therapy, neurological sequelae are often observed in patients after disease. Currently, the main challenge in BM treatment is to develop adjuvant therapies that reduce the occurrence of sequelae. In recent papers published by our group, we described the associations between the single nucleotide polymorphisms (SNPs) AADAT +401C > T, APEX1 Asn148Glu, OGG1 Ser326Cys and PARP1 Val762Ala and BM. In this study, we analyzed the associations between the SNPs TNF -308G > A, TNF -857C > T, IL-8 -251A > T and BM and investigated gene-gene interactions, including the SNPs that we published previously. METHODS The study was conducted with 54 BM patients and 110 healthy volunteers (as the control group). The genotypes were investigated via primer-introduced restriction analysis-polymerase chain reaction (PIRA-PCR) or polymerase chain reaction-based restriction fragment length polymorphism (PCR-RFLP) analysis. Allelic and genotypic frequencies were also associated with cytokine and chemokine levels, as measured with the x-MAP method, and cell counts. We analyzed gene-gene interactions among SNPs using the generalized multifactor dimensionality reduction (GMDR) method. RESULTS We did not find significant association between the SNPs TNF -857C > T and IL-8 -251A > T and the disease. However, a higher frequency of the variant allele TNF -308A was observed in the control group, associated with changes in cytokine levels compared to individuals with wild type genotypes, suggesting a possible protective role. In addition, combined inter-gene interaction analysis indicated a significant association between certain genotypes and BM, mainly involving the alleles APEX1 148Glu, IL8 -251 T and AADAT +401 T. These genotypic combinations were shown to affect cyto/chemokine levels and cell counts in CSF samples from BM patients. CONCLUSIONS In conclusion, this study revealed a significant association between genetic variability and altered inflammatory responses, involving important pathways that are activated during BM. This knowledge may be useful for a better understanding of BM pathogenesis and the development of new therapeutic approaches

On the order of the automorphism group of foliations

Let F be a holomorphic foliation with ample canonical bundle on a smooth projective surface X. We obtain an upper bound on the order of its automorphism group which depends only on KF2 and KFKX provided this group is finite. Here, KF and KX are the canonical bundles of F and X, respectively