On (k⊕l∣q)(k\oplus l |q)-dimensional supermanifolds

We define a $(k\oplus l|q)$-dimensional supermanifold as a manifold having q odd coordinates and k + l even coordinates with l of them taking only nilpotent values. We show that this notion can be used to formulate superconformal field theories with different number of supersymmetries in holomorphic and antiholomorphic sectors.Comment: 19 pages, Late

The essence of quintessence and the cost of compression

Standard two-parameter compressions of the infinite dimensional dark energy model space show crippling limitations even with current SN-Ia data. Firstly they cannot cope with rapid evolution - the best-fit to the latest SN-Ia data shows late and very rapid evolution to w_0 = -2.85. However all of the standard parametrisations (incorrectly) claim that this best-fit is ruled out at more than 2-sigma, primarily because they track it well only at very low redshifts, z < 0.2. Further they incorrectly rule out the observationally acceptable region w 1. Secondly the parametrisations give wildly different estimates for the redshift of acceleration, which vary from z_{acc}=0.14 to z_{acc}=0.59. Although these failings are largely cured by including higher-order terms (3 or 4 parameters) this results in new degeneracies which open up large regions of previously ruled-out parameter space. Finally we test the parametrisations against a suite of theoretical quintessence models. The widely used linear expansion in z is generally the worst, with errors of up to 10% at z=1 and 20% at z > 2. All of this casts serious doubt on the usefulness of the standard two-parameter compressions in the coming era of high-precision dark energy cosmology and emphasises the need for decorrelated compressions with at least three parameters.Comment: 7 pages, 4 colour figures, EmulateApJ; v2: includes Bayesian evidence analysis and table that were only present in published version, because of increased interest in Bayesian model comparison (no new material beyond the one in the published ApJL of 2004

The hepta-β-glucoside elicitor-binding proteins from legumes represent a putative receptor family

The ability of legumes to recognize and respond to β-glucan elicitors by synthesizing phytoalexins is consistent with the existence of a membrane-bound β-glucan-binding site. Related proteins of approximately 75 kDa and the corresponding mRNAs were detected in various species of legumes which respond to beta-glucans. The cDNAs for the beta-glucan-binding proteins of bean and soybean were cloned. The deduced 75-kDa proteins are predominantly hydrophilic and constitute a unique class of glucan-binding proteins with no currently recognizable functional domains. Heterologous expression of the soybean beta-glucan-binding protein in tomato cells resulted in the generation of a high-affinity binding site for the elicitor-active hepta-β-glucoside conjugate (K-d = 4.5 nM). Ligand competition experiments with the recombinant binding sites demonstrated similar ligand specificities when compared with soybean. In both soybean and transgenic tomato, membrane-bound, active forms of the glucan-binding proteins coexist with immunologically detectable, soluble but inactive forms of the proteins. Reconstitution of a soluble protein fraction into lipid vesicles regained beta-glucoside-binding activity but with lower affinity (K-d = 130 nM). We conclude that the beta-glucan elicitor receptors of legumes are composed of the 75 kDa glucan-binding proteins as the critical components for ligand-recognition, and of an as yet unknown membrane anchor constituting the plasma membrane-associated receptor complex

The Loss Rank Principle for Model Selection

We introduce a new principle for model selection in regression and classification. Many regression models are controlled by some smoothness or flexibility or complexity parameter c, e.g. the number of neighbors to be averaged over in k nearest neighbor (kNN) regression or the polynomial degree in regression with polynomials. Let f_D^c be the (best) regressor of complexity c on data D. A more flexible regressor can fit more data D' well than a more rigid one. If something (here small loss) is easy to achieve it's typically worth less. We define the loss rank of f_D^c as the number of other (fictitious) data D' that are fitted better by f_D'^c than D is fitted by f_D^c. We suggest selecting the model complexity c that has minimal loss rank (LoRP). Unlike most penalized maximum likelihood variants (AIC,BIC,MDL), LoRP only depends on the regression function and loss function. It works without a stochastic noise model, and is directly applicable to any non-parametric regressor, like kNN. In this paper we formalize, discuss, and motivate LoRP, study it for specific regression problems, in particular linear ones, and compare it to other model selection schemes.Comment: 16 page

Two Component Model of Dark Energy

We consider the possibility that the dark energy is made up of two or more independent components, each having a different equation of state. We fit the model with supernova and gamma-ray burst (GRB) data from resent observations, and use the Markov Chain Monte Carlo (MCMC) technique to estimate the allowed parameter regions. We also use various model selection criteria to compare the two component model with the LCDM, one component dark energy model with static or variable w(XCDM), and with other multi-component models. We find that the two component models can give reasonably good fit to the current data. For some data sets, and depending somewhat on the model selection criteria, the two component model can give better fit to the data than XCDM with static w and XCDM with variable w parameterized by w = w_0 + w_az/(1+z).Comment: 10 pages, 8 figures, 3 tables; Version accepted by PR

Black Hole Entropy, Topological Entropy and the Baum-Connes Conjecture in K-Theory

We shall try to exhibit a relation between black hole entropy and topological entropy using the famous Baum-Connes conjecture for foliated manifolds which are particular examples of noncommutative spaces. Our argument is qualitative and it is based on the microscopic origin of the Beckenstein-Hawking area-entropy formula for black holes, provided by superstring theory, in the more general noncommutative geometric context of M-Theory following the Connes- Douglas-Schwarz article.Comment: 17 pages, Latex, contains an important paragraph in section 2 which gives a better understandin

Polarizations and Nullcone of Representations of Reductive Groups

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f_1,f_2,...,f_n be homogeneous invariant functions. Then the polarizations of f_1,f_2,...,f_n define the nullcone of k 0} h(t) x = 0 for all x in L. This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of $SL_2$ on the n-fold tensor product C^2 otimes C^2 otimes ... otimes C^2. Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n <= 3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. {bf 224} (2006), 119--140.