on the long time behavior of type iii ricci flow solutions we show that three dimensional homogeneous ricci flow solutions that admit finite volume quotients have long time limits given by expanding solitons we show that the same is true for a large class of four dimensional homogeneous solutions we give an extension of hamilton s compactness theorem that does not assume a lower injectivity radius bound in terms of riemannian groupoids using this we show that the long time behavior of type iii ricci flow solutions is governed by the dynamics of an mathbb r action on a compact space

bayesian inference in quantum systems abstract bayes criteria are explicitly applied to statistical decision problems in simple quantum mechanical systems the minimum bayes cost is calculated for systems including polarised spins and relativistic spin 1 2 particles the results suggest that in decisions for a given ensemble of particles on the average the bayes cost almost always decreases if one treats the ensemble as a single composite system instead of performing separate measurements on the individual particles the optimal average sample size in observing spin 1 2 particles is also derived these results are potentially relevant to experimental design

simulating mesoscopic reaction diffusion systems using the gillespie algorithm we examine an application of the gillespie algorithm to simulating spatially inhomogeneous reaction diffusion systems in mesoscopic volumes such as cells and microchambers the method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements these transitions are expressed in the form of a set of reactions which are added to the chemical system the derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid the diffusion coefficient of each species is allowed to be inhomogeneous in space including discontinuities the resulting system is solved by the gillespie algorithm using the fast direct method we show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction rate equation describing the cubic autocatalytic reaction

fluctuations in the lattice gas for burgers equation the langevin equation for the biased random hopping of particles onto unoccupied sites of a lattice is obtained from the master equation for finite systems by invoking a limit theorem due to kurtz for a weakly asymmetric process with diffusive scaling the continuum limit of this equation yields a deterministic equation that is equivalent to burgers equation the stochastic equation for the leading lattice corrections is derived and the approach to the continuum limit is illustrated with simulations

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

a priori error estimates for numerical methods for scalar conservation laws part ii flux splitting monotone schemes on irregular cartesian grids this paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux splitting monotone schemes to the entropy solution we obtain the optimal rate of convergence of x 1 2 in l l 1 for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution we then extend this result to less consistent schemes called p consistent schemes and prove that they converge to the entropy solution with the rate of x min 1 2 p in l l 1 again no regularity property of the approximate solution is used finally we propose a new explanation of the fact that even inconsistent schemes converge with the rate of x 1 2 in l l 1 we show that this well known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution total variation boundedness to compensate for its lack of consistency the nonlinear nature of the problem does not play any role in this mechanism all the above results hold in the multidimensional case provided the grids are cartesian products of one dimensional nonuniform grids

laplacian flow of homogeneous g2 structures and its solitons we use the bracket flow algebraic soliton approach to study the laplacian flow of g2 structures and its solitons in the homogeneous case we prove that any homogeneous laplacian soliton is equivalent to a semi algebraic soliton that is a g invariant g2 structure on a homogeneous space g k that flows by pull back of automorphisms of g up to scaling algebraic solitons are geometrically characterized among laplacian solitons as those with a diagonal evolution unlike the ricci flow case where any homogeneous ricci soliton is isometric to an algebraic soliton we have found as an application of the above characterization an example of a left invariant closed semi algebraic soliton on a nilpotent lie group which is not equivalent to any algebraic soliton the normalized bracket flow evolution of such a soliton is periodic in the context of solvable lie groups with a codimension one abelian normal subgroup we obtain long time existence for any closed laplacian flow solution furthermore the norm of the torsion is strictly decreasing and converges to zero we also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding laplacian solitons

on the moment map for the variety of lie algebras we consider the moment map m pvn iu n for the action of gl n on vn 2 cn cn the critical points of the functional fn m 2 pvn r are studied in order to understand the stratification of ln pvn defined by the negative gradient flow of fn where ln is the algebraic subset of all lie algebras we obtain a description of the critical points which lie in ln in terms of those which are nilpotent as well as the minima and maxima of fn ln r a characterization of the critical points modulo isomorphism as the union of categorical quotients of suitable actions is considered and some applications to the study of ln are given

simulation of stochastic reaction diffusion processes on unstructured meshes we model stochastic chemical systems with diffusion by a reaction diffusion master equation on a macroscopic level the governing equation is a reaction diffusion equation for the averages of the chemical species on a mesoscopic level the master equation for a well stirred chemical system is combined with a discretized brownian motion in space to obtain the reaction diffusion master equation the space is covered in our method by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the laplace operator on the macroscale the resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso or on the macroscale level the accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology

connecting the dots semi analytical and random walk numerical solutions of the diffusion reaction equation with stochastic initial conditions we study a system with bimolecular irreversible kinetic reaction a b a where the underlying transport of reactants is governed by diffusion and the local reaction term is given by the law of mass action we consider the case where the initial concentrations are given in terms of an average and a white noise perturbation our goal is to solve the diffusion reaction equation which governs the system and we tackle it with both analytical and numerical approaches to obtain an analytical solution we develop the equations of moments and solve them approximately to obtain a numerical solution we develop a grid less monte carlo particle tracking approach where diffusion is modeled by a random walk of the particles and reaction is modeled by annihilation of particles the probability of annihilation is derived analytically from the particles co location probability we rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number this enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies

applications of quantum and classical fisher information to two level complex and quaternionic and three level complex systems in the bayesian theory of statistical inference as first suggested by harold jeffreys in highly influential work one can employ the square root of the determinant of an n n fisher information matrix as a reparametrization invariant prior generally unnormalized measure over an n dimensional family riemannian manifold of probability distributions jeffreys ansatz is adopted here to the quantum context that is with regard to density matrices rather than probability distributions by computing the quantum fisher information matrices associated with helstrom and holevo for the three five and eight dimensional convex sets of two level complex two level quaternionic and three level complex systems respectively in both the two level cases the priors have been normalized to probability distributions over the 2 2 density matrices while in the much more computationally demanding three level situation no such normalization has been accomplished an argument is made for the general form in term

increased efficiency of quantum state estimation using non separable measurements we address the major open problem of evaluating how much increased efficiency in estimation is possible using non separable as opposed to separable measurements of n copies of m level quantum systems first we study the six cases m 2 n 2 7 by computing the 3 3 fisher information matrices for the corresponding optimal measurements recently devised by vidal et al we obtain simple polynomial expressions for the gill massar traces of the products of the inverse of the quantum helstrom information matrix and these fisher information matrices the six traces all have minima of 2n 1 in the pure state limit while for separable measurements the traces can equal n but not exceed it then the result of an analysis for m 3 n 2 leads us to conjecture that for optimal measurements for all m and n the gill massar trace achieves a minimum of 2n 1 m 1 in the pure state limit

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

noncompact homogeneous einstein manifolds attached to graded lie algebras in this paper we study the nilradicals of parabolic subalgebras of semisimple lie algebras and the natural one dimensional solvable extensions of them we investigate the structures curvatures and einstein conditions of the associated nilmanifolds and solvmanifolds we show that our solvmanifold is einstein if the nilradical is two step new examples of einstein solvmanifolds with three step and four step nilradicals are also given

bayesian inference for complex and quaternionic two level quantum systems abstract jeffreys approach for generating reparameterization invariant prior distributions is applied to the three dimensional convex set of complex two level quantum systems for this purpose such systems are identified with bivariate complex normal distributions over the vectors of two dimensional complex hilbert space the trivariate prior obtained is improper or non normalizable over the convex set however its three bivariate marginals are through a limiting procedure normalizable to probability distributions and are consequently suitable for the bayesian inference of two level systems analogous results hold for the five dimensional convex set of quaternionic two level systems the complex univariate and quaternionic trivariate marginals of the improper priors are uniform distributions the bivariate marginals in the two cases are opposite in character

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

asymptotic redundancies for universal quantum coding clarke and barren 1990 1994 1995 have shown that the jeffreys invariant prior of bayesian theory yields the common asymptotic minimax and maximin redundancy of universal data compression in a parametric setting we seek a possible analog of this result for the two level quantum systems we restrict our considerations to prior probability distributions belonging to a certain one parameter family q sub u spl infin u 1 within this setting we are able to compute exact redundancy formulas for which we find the asymptotic limits we compare our quantum asymptotic redundancy formulas to those derived by naively applying the nonquantum counterparts of clarke and barren and find certain common features our results are based on formulas we obtain for the eigenvalues and eigenvectors of 2 sup n spl times 2 sup n bayesian density matrices spl zeta sub n u these matrices are the weighted averages with respect to q sub u of all possible tensor products of n identical 2 spl times 2 density matrices representing the two level quantum systems we propose a form of universal coding for the situation in which the density matrix describing an ensemble of quantum signal states is unknown a sequence of n signals would be projected onto the dominant eigenspaces of spl zeta sub n u

the bayesian choice a decision theoretic motivation contents introduction decision theoretic foundations of statistical inference from prior information to prior distributions bayesian point estimation tests and confidence regions admissibility and complete classes invariance haar measures and equivariant estimators hierarchical and empirical bayes extensions bayesian calculations a defense of the bayesian choice

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

a step towards the alekseevskii conjecture we refine existing structure results for non compact homogeneous einstein manifolds and provide a reduction in the classification problem of such spaces using this work we verify the generalized alekseevskii conjecture for a large class of homogeneous spaces

information gains expected from separate and joint measurements of n identical spin 1 2 systems noninformative bayesian analyses an investigator is assumed to have n replicas of a spin 1 2 system but lacks any specific knowledge pertinent to constructing a 2 2 density matrix for it we compute within a noninformative bayesian framework the expected information gains for various measurement schemes first separate and then joint in nature in particular we obtain a result which fully accords with a certain plausibility argument of peres while the parallel result in a recent analysis of massar and popescu does not

the ricci flow on solvmanifolds of real type abstract we show that for any solvable lie group of real type any homogeneous ricci flow solution converges in cheeger gromov topology to a unique non flat solvsoliton which is independent of the initial left invariant metric as an application we obtain results on the isometry groups of non flat solvsoliton metrics and einstein solvmanifolds

optimizing the accuracy of lattice monte carlo algorithms for simulating diffusion the behavior of a lattice monte carlo lmc algorithm if it is designed correctly must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero however we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed the accuracy of the algorithm is optimal for a certain finite value of the other parameter in one dimension the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments and minimizes the error for the higher moments at large times of the particle distribution and preserves the first two moments of the first passage time distributions in two and three dimensions the same level of accuracy requires simultaneous moves along two axes diagonal moves such moves attempting to cross an impenetrable boundary should be projected along the boundary rather than simply rejected we also treat the case of absorbing boundaries we discuss the relation between optimally accurate lmc algorithms and a particular case of lattice boltzmann lb algorithms for simulating diffusion and compare the computational efficiency of optimal lmc and optimal lb algorithms

hall normalization constants for the bures volumes of the n state quantum systems we report the results of certain integrations of quantum theoretic interest relying in this regard upon recently developed parametrizations of boya et al 1998 preprint quant ph 9810084 of the n n density matrices in terms of squared components of the unit n 1 sphere and the n n unitary matrices firstly we express the normalized volume elements of the bures minimal monotone metric for n 2 and 3 thereby obtaining bures prior probability distributions over the two and three state systems then as a first step in extending these results to n 3 we determine that the hall normalization constant cn for the marginal bures prior probablity distribution over the n 1 dimensional simplex of the n eigenvalues of the n n density matrices is for n 4 equal to 71 680 2 since we also find that c3 35 it follows that c4 is simply equal to 211c3 c2 itself is known to equal 2 the constant c5 is also found it too is associated with a remarkably simple decomposition involving the product of the eight consecutive prime numbers from 3 to 23 we also preliminarily investigate several cases n 5 with the use of quasi monte carlo integration we hope that the various analyses reported will prove useful in deriving a general formula which evidence suggests will involve the bernoulli numbers for the hall normalization constant for arbitrary n this would have diverse applications including quantum inference and universal quantum coding

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

quantum fisher bures information of two level systems and a three level extension braunstein and caves have recently demonstrated that the bures metric on the mixed quantum states is equivalent up to a proportionality factor of four to the statistical distinguishability or quantum fisher information metric the volume element of these metrics can then adapting a fundamental bayesian principle of jeffreys to the quantum context serve as a reparametrization invariant prior measure over the quantum states the implications of this line of reasoning for the two level systems in general and an embedding of them into a certain set of three level systems are investigated

implications of reference priors for prior information and for sample size abstract here we use posterior densities based on relative entropy reference priors for two purposes the first purpose is to identify data implicit in the use of informative priors we represent an informative prior as the posterior from an experiment with a known likelihood and a reference prior minimizing the relative entropy distance between this posterior and the informative prior over choices of data results in a data set that can be regarded as representative of the information in the informative prior the second implication from reference priors is obtained by replacing the informative prior with a class of densities from which one might wish to make inferences for each density in this class one can obtain a data set that minimizes a relative entropy the maximum of these sample sizes as the inferential density varies over its class can be used as a guess as to how much data is required for the desired inferences we bound this sample size above and below by other techniques that permit it to

voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids natural neighbours are based on the voronoi diagram which partitions space into closest point regions the sibson and the laplace non sibsonian interpolants which are based on natural neighbours have shown promise within a galerkin framework for the solution of partial differential equations in this paper we focus on the laplace interpolant with a two fold objective first to unify the previous developments related to the laplace interpolant and to indicate its ties to some well known numerical methods and secondly to propose a voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids a conservation law in integral form is discretized on voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids the proposed scheme can also be viewed as a point collocation technique a detailed study on consistency is conducted and the satisfaction of the discrete maximum principle stability is established owing to symmetry of the laplace weight a symmetric positive definite stiffness matrix is realized which permits the use of efficient linear solvers on a regular rectangular or hexagonal grid the difference scheme reduces to the classical finite difference method numerical examples for the poisson equation with dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme copyright 2003 john wiley sons ltd

on distinguished orbits of reductive representations abstract let g be a real reductive lie group and let g gl v be a real reductive representation of g with restricted moment map m g v 0 g in this work we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a g orbit is distinguished i e it contains a critical point of the norm squared of m g we give an elementary proof of the well known convexity theorem of atiyah guillemin sternberg in our particular case and we use it to give an easy to check sufficient condition for a g orbit of an element in a nice space to be distinguished in the case where g is algebraic and is a rational representation the above condition is also necessary making heavy use of recent results of michael jablonski obtaining a generalization of nikolayevsky s nice basis criterion we also provide useful characterizations of nice spaces in terms of the weights of finally some applications to ternary forms are presented

oscillator spacetimes are ricci solitons abstract we consider the four dimensional oscillator group equipped with a well known one parameter family of left invariant lorentzian metrics which includes the bi invariant one gadea oubina 1999 in a suitable system of global coordinates the ricci soliton equation for these metrics translates into a system of partial differential equations solving such system we prove that all these metrics are ricci solitons in particular the bi invariant metric on the oscillator group gives rise to infinitely many ricci solitons and so also to yamabe solitons

solvable lie algebras and graphs abstract we define a solvable extension of the graph 2 step nilpotent lie algebras of 5 by adding elements corresponding to the 3 cliques of the graph we study some of their basic properties and we prove that two such lie algebras are isomorphic if and only if their graphs are isomorphic we also briefly discuss some metric properties providing examples of homogeneous spaces with nonpositive curvature operator and solvsolitons

coarse graining and hybrid methods for efficient simulation of stochastic multi scale models of tumour growth the development of hybrid methodologies is of current interest in both multi scale modelling and stochastic reaction diffusion systems regarding their applications to biology we formulate a hybrid method for stochastic multi scale models of cells populations that extends the remit of existing hybrid methods for reaction diffusion systems such method is developed for a stochastic multi scale model of tumour growth i e population dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics in order to formulate this method we develop a coarse grained approximation for both the full stochastic model and its mean field limit such approximation involves averaging out the age structure which accounts for the multi scale nature of the model by assuming that the age distribution of the population settles onto equilibrium very fast we then couple the coarse grained mean field model to the full stochastic multi scale model by doing so within the mean field region we are neglecting noise in both cell numbers population and their birth rates structure this implies that in addition to the issues that arise in stochastic reaction diffusion systems we need to account for the age structure of the population when attempting to couple both descriptions we exploit our coarse graining model so that within the mean field region the age distribution is in equilibrium and we know its explicit form this allows us to couple both domains consistently as upon transference of cells from the mean field to the stochastic region we sample the equilibrium age distribution furthermore our method allows us to investigate the effects of intracellular noise i e fluctuations of the birth rate on collective properties such as travelling wave velocity we show that the combination of population and birth rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front which cannot be accounted for by the coarse grained model such fluctuations have non trivial effects on the wave velocity beyond the development of a new hybrid method we thus conclude that birth rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth

the alekseevskii conjecture in low dimensions the long standing alekseevskii conjecture states that a connected homogeneous einstein space g k of negative scalar curvature must be diffeomorphic to mathbb r n this was known to be true only in dimensions up to 5 and in dimension 6 for non semisimple g in this work we prove that this is also the case in dimensions up to 10 when g is not semisimple for arbitrary g besides 5 possible exceptions we show that the conjecture holds up to dimension 8

edwards wilkinson equation from lattice transition rules continuum equations of motion for the height fluctuations of lattice growth models are derived from their transition rules by regularizing and coarse graining the associated discrete langevin equations for models with random deposition followed by instantaneous relaxation to a neighboring site based on identifying the local height minimum our methodology yields the edwards wilkinson equation the application of this procedure to other growth models is discussed

ricci flow of homogeneous manifolds we present in this paper a general approach to study the ricci flow on homogeneous manifolds our main tool is a dynamical system defined on a subset mathcal h q n of the variety of q n dimensional lie algebras parameterizing the space of all simply connected homogeneous spaces of dimension n with a q dimensional isotropy which is proved to be equivalent in a precise sense to the ricci flow the approach is useful to better visualize the possible nonflat pointed limits of ricci flow solutions under diverse rescalings as well as to determine the type of the possible singularities ancient solutions arise naturally from the qualitative analysis of the evolution equation we develop two examples in detail a 2 parameter subspace of mathcal h 1 3 reaching most of 3 dimensional geometries and a 2 parameter family in mathcal h 0 n of left invariant metrics on n dimensional compact and non compact semisimple lie groups

negative ricci curvature on some non solvable lie groups ii we construct many examples of lie groups admitting a left invariant metric of negative ricci curvature we study lie algebras which are semidirect products mathfrak l mathfrak a oplus mathfrak u imes mathfrak n and we obtain examples where mathfrak u is any semisimple compact real lie algebra mathfrak a is one dimensional and mathfrak n is a representation of mathfrak u which satisfies some conditions in particular when mathfrak u mathfrak s mathfrak u m mathfrak s mathfrak o m or mathfrak s mathfrak p m and mathfrak n is a representation of mathfrak u in some space of homogeneous polynomials we show that these conditions are indeed satisfied in the case mathfrak u mathfrak s mathfrak u 2 we get a more general construction where mathfrak n can be any nilpotent lie algebra where mathfrak s mathfrak u 2 acts by derivations we also prove a general result in the case when mathfrak u is a semisimple lie algebra of non compact type

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified