Global supersymmetry on curved spaces in various dimensions

We propose methods towards a systematic determination of d  -dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity point of view. In particular, by using appropriate gauged supergravities in various dimensions we show that supersymmetry can be defined in conformally flat spaces, such as non-compact hyperboloids Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and compact spheres Sⁿ${\mathbb{S}}^n$ or – by turning on appropriate Wilson lines corresponding to R-symmetry vector fields – on S¹×Sⁿ${\mathbb{S}}^1\times {\mathbb{S}}^n$, with n<6$n<6$. By group theory arguments we show that Euclidean field theories with rigid supersymmetry cannot be consistently defined on round spheres S^d${\mathbb{S}}^d$ if d>5$d>5$ (despite the existence of Killing spinors). We also show that distorted spheres and certain orbifolds are also allowed by the group theory classification.     

Co-registration of the distribution of wall shear stress and 140 complex plaques of the aorta

Previous studies provide evidence that atherosclerosis develops in vascular regions exposed to low wall shear stress (WSS) and high oscillatory shear index (OSI). 4D flow MRI was analyzed in 70 stroke patients with complex plaques (≥ 4 mm thickness, ulcerated or superimposed thrombi) and in 12 young healthy volunteers. The segmental distribution of peak systolic WSS_systole and OSI was quantified in analysis planes positioned directly at the location of 140 complex plaques found in the 70 patients. In addition, WSS_systole and OSI were evaluated in 8 standard analysis planes distributed along the aorta. Complex plaques were predominantly found at the inner curvature of the aortic arch and of the descending aorta. High OSI was co-located with the segments mostly affected by complex plaque while WSS_systole demonstrated a homogenous distribution. In standard analysis planes, patients demonstrated significantly (p < 0.01) altered distribution of wall parameters compared to volunteers (reduced WSS_systole in 91% of aortic wall segments, increased OSI in 48% of segments). OSI distribution was asymmetric with higher values at the inner curvature of the aorta. While WSS and OSI showed expected changes in patients compared to healthy controls, their distribution pattern at complex plaques indicated a more complex and heterogeneous relationship than previously anticipated.     

Feasibility of Q-Band EPR Dosimetry in Biopsy Samples of Dental Enamel, Dentine and Bone

Electron paramagnetic resonance (EPR) dosimetry of tooth enamel in X-band has been established as a suitable method for individual reconstruction of doses 0.1 Gy and higher. The objective was to demonstrate the feasibility of using Q-band EPR in small biopsy tooth enamel samples to provide accurate measurements of radiation doses. Q-band spectra of small (<10 mg) irradiated samples of dentine and bone were studied to investigate the possibility of using Q-band EPR for dose measurements in those materials if there are limited amounts of enamel available, and there is no time for the chemical sample preparation required for accurate X-band measurements in dental enamel. Our results have shown that Q-band provides accurate measurements of radiation doses higher than 0.5 Gy in tooth enamel biopsy samples as small as 2 mg. Q-band EPR spectra in powdered dentine and bone demonstrated significantly higher resolution and sensitivity than in conventional X-band measurements.     

Christoffel Words and Markoff Triples: An Algebraic Approach

We introduce a family of modules, called Markoff modules, generated by a cluster-mutation-like iterative process. We show that these modules are combinatorially similar to Christoffel words. Furthermore, we construct a bijective map between the set of Markoff module triples and the set of proper Markoff triples. This allows us to interpret the uniqueness conjecture for Markoff numbers within an algebraic framework.     

Fibonacci-like growth of numerical semigroups of a given genus

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n _g is the number of numerical semigroups of genus g, we prove that $$\lim_{g \rightarrow \infty} n_g \varphi^{-g} = S $$ where $\varphi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of n _g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number.     

Characteristic Formulas of Partial Heyting Algebras

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras.     

NP-hardness of deciding convexity of quartic polynomials and related problems

We show that unless P = NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.     

Volume functions of linear series

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole Néron–Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, inspired by the work of Lazarsfeld and Musta?? (Ann Inst Fourier (Grenoble) 56(6):1701–1734, 2006) on Okounkov bodies, we show that any continuous, homogeneous, and log-concave function appears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case.     

Zariski decomposition of b-divisors

Based on a recent work of Thomas Bauer’s (J. Algebr. Geom., to appear), reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.