Computing Multi-Homogeneous Bezout Numbers is Hard

The multi-homogeneous Bezout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the Bezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous Bezout number is actually NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multi-homogeneous Bezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP contains NP

Real Computational Universality: The Word Problem for a class of groups with infinite presentation

The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. Most important, the free group will be generated by an uncountable set of generators with index running over certain sets of real numbers. This allows to include many mathematically important groups which are not captured in the framework of the classical word problem. Our contribution extends computational group theory from the discrete to the Blum-Shub-Smale (BSS) model of real number computation. We believe this to be an interesting step towards applying BSS theory, in addition to semi-algebraic geometry, also to further areas of mathematics. The main result establishes the word problem for such groups to be not only semi-decidable (and thus reducible FROM) but also reducible TO the Halting Problem for such machines. It thus provides the first non-trivial example of a problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.

Semi-algebraic complexity-additive complexity of diagonalization of quadratic forms

Sin resume

Real Interactive Proofs for VPSPACE

We study interactive proofs in the framework of real number complexity as introduced by Blum, Shub, and Smale. The ultimate goal is to give a Shamir like characterization of the real counterpart IP_R of classical IP. Whereas classically Shamir\u27s result implies IP = PSPACE = PAT = PAR, in our framework a major difficulty arises from the fact that in contrast to Turing complexity theory the real number classes PAR_R and PAT_R differ and space resources considered alone are not meaningful. It is not obvious to see whether IP_R is characterized by one of them - and if so by which. In recent work the present authors established an upper bound IP_R is a subset of MA(Exists)R, where MA(Exists)R is a complexity class satisfying PAR_R is a strict subset of MA(Exists)R, which is a subset of PAT_R and conjectured to be different from PAT_R. The goal of the present paper is to complement this result and to prove interesting lower bounds for IP_R. More precisely, we design interactive real protocols for a large class of functions introduced by Koiran and Perifel and denoted by UniformVSPACE^0. As consequence, we show PAR_R is a subset of IP_R, which in particular implies co-NP_R is a subset of IP_R, and P_R^{Res} is a subset of IP_R, where Res denotes certain multivariate Resultant polynomials. Our proof techniques are guided by the question in how far Shamir\u27s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVSPACE^0 are extremely helpful

On the expressive power of CNF formulas of bounded tree- and clique-width

AbstractWe study representations of polynomials over a field K from the point of view of their expressive power. Three important examples for the paper are polynomials arising as permanents of bounded tree-width matrices, polynomials given via arithmetic formulas, and families of so called CNF polynomials. The latter arise in a canonical way from families of Boolean formulas in conjunctive normal form. To each such CNF formula there is a canonically attached incidence graph. Of particular interest to us are CNF polynomials arising from formulas with an incidence graph of bounded tree- or clique-width.We show that the class of polynomials arising from families of polynomial size CNF formulas of bounded tree-width is the same as those represented by polynomial size arithmetic formulas, or permanents of bounded tree-width matrices of polynomial size. Then, applying arguments from communication complexity we show that general permanent polynomials cannot be expressed by CNF polynomials of bounded tree-width. We give a similar result in the case where the clique-width of the incidence graph is bounded, but for this we need to rely on the widely believed complexity theoretic assumption #P⊈FP/poly

Acid water tolerance in a New Zealand native freshwater fish

Physiological experiments were conducted to measure the effects of low pH water upon Na⁺ fluxes and swimming performance in east coast neutral water banded kokopu, and west coast banded kokopu living in naturally acid water. This allowed comparisons to be made regarding the relative acid tolerance between the two population stocks. A morphological study of the gills using scanning electron microscopy and transmission electron microscopy was carried out on a limited number of fish to compare gill structures of east coast acid and non acid exposed fish with west coast fish. East coast kokopu from neutral water showed a reduced sodium influx on the first day of exposure to pH4 water. This was accompanied by a reduction in passive loss or efflux. After 4 days at pH4 influx recovered allowing the fish to approach sodium balance. West coast fish showed no significant change in influx upon introduction to pH7 water and maintained influx values in pH4 water near those measured for control fish. For east coast fish, swimming performance was significantly impaired on the 2nd swimming trial when pH was lowered from 7 to 4. On the third swim with the pH returned to 7, Ucrit was similar to the initial value. West coast banded kokopu showed a significant increase in performance between trial one and three and were unaffected by pH7 water, matching performances measured in control fish. Gills from east coast acid exposed fish resembled west coast fish in that the trailing edges of the lamellae were swollen due to the large numbers of chloride cells which were present to a point approximately half way up the lamellar edge. Microridges were present on the apical surfaces of the chloride cells of east coast control fish which contrasted with the microvilli structures observed in west coast and acid exposed east coast fish. The microvilli structures present on the chloride cells of west coast and east coast acid exposed banded kokopu appeared to assist in the anchorage of mucus to the apical surface of the cell

SCIAMACHY: The new Level 0-1 Processor

SCIAMACHY (SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY) is a scanning nadir and limb spectrometer covering the wavelength range from 212 nm to 2386 nm in 8 channels. It is a joint project of Germany, the Netherlands and Belgium and was launched in February 2002 on the ENVISAT platform. After the platform failure in April 2012, SCIAMACHY is now in the postprocessing phase F. SCIAMACHY�s originally specified in-orbit lifetime was double the planned lifetime. SCIAMACHY was designed to measure column densities and vertical profiles of trace gas species in the mesosphere, in the stratosphere and in the troposphere (Bovensmann et al., 1999). It can detect a large amount of atmospheric gases (e.g. O3 , H2CO, CHOCHO, SO2 , BrO, OClO, NO2 , H2O, CO, CH4 , among others ) and can provide information about aerosols and clouds. The operational processing of SCIAMACHY is split into Level 0-1 processing (essentially providing calibrated radiances) and Level 1-2 processing providing geophysical products. The operational Level 0-1 processor has been completely re-coded and embedded in a newly developed framework that speeds up processing considerably. In the frame of the SCIAMACHY Quality Working Group activities, ESA is continuing the improvement of the archived data sets. Currently Version 9 of the Level 0-1 processor is being implemented. It will include An updated degradation correction Several improvements in the SWIR spectral range like a better dark correction, an improved dead & bad pixel characterisation and an improved spectral calibration Improvements to the polarisation correction algorithm Improvements to the geolocation by a better pointing characterisation Additionally a new format for the Level 1b and Level 1c will be implemented. The version 9 products will be available in netCDF version 4 that is aligned with the formats of the GOME -1 and Sentinel missions. We will present the first results of the new Level 0-1 processing in this paper

Associations between abdominal adipose tissue, reproductive span, and brain characteristics in post-menopausal women

The menopause transition involves changes in oestrogens and adipose tissue distribution, which may influence female brain health post-menopause. Although increased central fat accumulation is linked to risk of cardiometabolic diseases, adipose tissue also serves as the primary biosynthesis site of oestrogens post-menopause. It is unclear whether different types of adipose tissue play diverging roles in female brain health post-menopause, and whether this depends on lifetime oestrogen exposure, which can have lasting effects on the brain and body even after menopause. Using the UK Biobank sample, we investigated associations between brain characteristics and visceral adipose tissue (VAT) and abdominal subcutaneous adipose tissue (ASAT) in 10,251 post-menopausal females, and assessed whether the relationships varied depending on length of reproductive span (age at menarche to age at menopause). To parse the effects of common genetic variation, we computed polygenic scores for reproductive span. The results showed that higher VAT and ASAT were both associated with higher grey and white matter brain age, and greater white matter hyperintensity load. The associations varied positively with reproductive span, indicating more prominent associations between adipose tissue and brain measures in females with a longer reproductive span. The effects were in general small, but could not be fully explained by genetic variation or relevant confounders. Our findings indicate that associations between abdominal adipose tissue and brain health post-menopause may partly depend on individual differences in cumulative oestrogen exposure during reproductive years, emphasising the complexity of neural and endocrine ageing processes in females