solution of the crow kimura model with a periodically changing two season fitness function since the origin of life both evolutionary dynamics and rhythms have played a key role in the functioning of living systems the crow kimura model of periodically changing fitness function has been solved exactly using integral equation with time ordered exponent we also found a simple approximate solution for the two season case the evolutionary dynamics accompanied by the rhythms provide important insights into the properties of certain biological systems and processes

complex adhm equations and sheaves on p 3 abstract we establish a 1 1 correspondence between certain classes of sheaves on p 3 and solutions of a complex version of the atiyah hitchin drinfeld manin equations we also show how relaxing of the locally free condition on sheaves is reflected on partial completions of the space of regular solutions we also introduce a new class of sheaves on p 3 closely related to ideal sheaves for curves within p 3

random catalytic reaction networks abstract we study networks that are a generalization of replicator or lotka volterra equations they model the dynamics of a population of object types whose binary interactions determine the specific type of interaction product such a system always reduces its dimension to a subset that contains production pathways for all of its members the network equation can be rewritten at a level of collectives in terms of two basic interaction patterns replicator sets and cyclic transformation pathways among sets although the system contains well known cases that exhibit very complicated dynamics the generic behavior of randomly generated systems is found numerically to be extremely robust convergence to a globally stable rest point it is easy to tailor networks that display replicator interactions where the replicators are entire self sustaining subsystems rather than structureless units a numerical scan of random systems highlights the special properties of elementary replicators they reduce the effective interconnectedness of the system resulting in enhanced competition and strong correlations between the concentrations

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

twisted constant scalar curvature kahler metrics and kahler slope stability on a compact kahler manifold we introduce a cohomological obstruction to the solvability of the constant scalar curvature csck equation twisted by a semipositive form appearing in works of fine and song tian as a special case we find an obstruction for a manifold to be the base of a holomorphic submersion carrying a csck metric in certain adiabatic classes we apply this to find new examples of general type threefolds with classes which do not admit a csck representative when the twist vanishes our obstruction extends the slope stability of ross thomas to effective divisors on a kahler manifold thus we find examples of non projective slope unstable manifolds

convexity of the k energy on the space of kahler metrics and uniqueness of extremal metrics we establish the convexity of mabuchi s k energy functional along weak geodesics in the space of kahler potentials on a compact kahler manifold thus confirming a conjecture of chen and give some applications in kahler geometry including a proof of the uniqueness of constant scalar curvature metrics or more generally extremal metrics modulo automorphisms the key ingredient is a new local positivity property of weak solutions to the homogeneous monge ampere equation on a product domain whose proof uses plurisubharmonic variation of bergman kernels

nonlinear diffusion effects on biological population spatial patterns motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations we investigate its implications in a paradigmatic model for the evolution of a single species density u x t the standard model includes growth and competition in a logistic expression and spreading is modeled through normal diffusion moreover the competition term is nonlocal which has been shown to give rise to spatial patterns we generalize the diffusion term through the nonlinear form tu x t d xxu x t with d 0 encompassing the cases where the state dependent diffusion coefficient either increases 1 or decreases 1 with the density yielding subdiffusion or superdiffusion respectively by means of numerical simulations and analytical considerations we display how that nonlinearity alters the phase diagram the type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape inducing fragmentation in the subdiffusive case the detection of the main persistent mode allows analytical prediction of the critical thresholds

mathematical instanton bundles on p2n 1 soit e un fibre algebrique de rang 2n sur p c 2n 1 on dit e est un fibre instanton mathematique de nombre quantique k si e est simple a un polynome de chern c t e 1 1 t 2 k et une cohomologie naturelle h 9 e l dans 2n 1 l 0 on demontre que l ensemble mip 2n 1 k des classes d isomorphismes des fibres instanton mathematiques de nombre quantique k sur p c 2n 1 peuvent s identifier a un quotient mi p2n 1 k sk 2n 2k gl k c ou sk 2n 2k est la variete des modules de kronecker non degenerees symetriques simples de rang 2n 2k

moran model as a dynamical process on networks and its implications for neutral speciation in population genetics the moran model describes the neutral evolution of a biallelic gene in a population of haploid individuals subjected to mutations we show in this paper that this model can be mapped into an influence dynamical process on networks subjected to external influences the panmictic case considered by moran corresponds to fully connected networks and can be completely solved in terms of hypergeometric functions other types of networks correspond to structured populations for which approximate solutions are also available this approach to the classic moran model leads to a relation between regular networks based on spatial grids and the mechanism of isolation by distance we discuss the consequences of this connection for topopatric speciation and the theory of neutral speciation and biodiversity we show that the effect of mutations in structured populations where individuals can mate only with neighbors is greatly enhanced with respect to the panmictic case if mating is further constrained by genetic proximity between individuals a balance of opposing tendencies takes place increasing diversity promoted by enhanced effective mutations versus decreasing diversity promoted by similarity between mates resolution of large enough opposing tendencies occurs through speciation via pattern formation we derive an explicit expression that indicates when speciation is possible involving the parameters characterizing the population we also show that the time to speciation is greatly reduced in comparison with the panmictic case

okounkov bodies and restricted volumes along very general curves abstract given a big divisor d on a normal complex projective variety x we show that the restricted volume of d along a very general complete intersection curve c x can be read off from the okounkov body of d with respect to an admissible flag containing c from this we deduce that if two big divisors d 1 and d 2 on x have the same okounkov body with respect to every admissible flag then d 1 and d 2 are numerically equivalent

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

phase diagram for the eigen quasispecies theory with a truncated fitness landscape using methods of statistical physics we present rigorous theoretical calculations of eigen s quasispecies theory with the truncated fitness landscape which dramatically limits the available sequence space of information carriers as the mutation rate is increased from small values to large values one can observe three phases the first i selective also known as ferromagnetic phase the second ii intermediate phase with some residual order and the third iii completely randomized also known as paramagnetic phase we calculate the phase diagram for these phases and the concentration of information carriers in the master sequence also known as peak configuration x 0 and other classes of information carriers as the phase point moves across the boundary between phase i and phase ii x 0 changes continuously as the phase point moves across the boundary between phase ii and phase iii x 0 has a large change our results are applicable for the general case of a fitness landscape

nondegenerate multidimensional matrices and instanton bundles in this paper we prove that the moduli space of rank 2n symplectic instanton bundles on p 2n 1 defined from the well known monad condition is affine this result was not known even in the case n 1 where by atiyah drinfeld hitchin and manin in 1978 the real instanton bundles correspond to self dual yang mills sp 1 connections over the 4 dimensional sphere the result is proved as a consequence of the existence of an invariant of the multi dimensional matrices representing the instanton bundles

maximum minimum and optimal mutation rates in dynamic environments we analyze the dynamics of the parallel mutation selection quasispecies model with a changing environment for an environment with the sharp peak fitness function in which the most fit sequence changes by k spin flips every period t we find analytical expressions for the minimum and maximum mutation rates for which a quasispecies can survive valid in the limit of large sequence size we find an asymptotic solution in which the quasispecies population changes periodically according to the periodic environmental change in this state we compute the mutation rate that gives the optimal mean fitness over a period we find that the optimal mutation rate per genome k t is independent of genome size a relationship which is observed across broad groups of real organisms

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

evolution equation of phenotype distribution general formulation and application to error catastrophe an equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics the equation is solved by using the singular perturbation method and assuming that the number of bases in the genetic sequence is large applying the equation to the mutation selection model by eigen provides the critical mutation rate for the error catastrophe phenotypic fluctuation of clones individuals sharing the same gene is introduced into this evolution equation with this formalism it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations i e noise can enhance robustness of a fitted state to mutation our formalism is systematic and general while approximations to derive more tractable evolution equations are also discussed

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation selection scheme in a recent paper phys rev e 69 046121 2004 we used the suzuki trottere formalism to study a quasispecies biological evolution model in a parallel mutation selection scheme with a single peak fitness function and a point mutation in the present paper we extend such a study to evolution models with more general fitness functions or multiple mutations in the parallel mutation selection scheme we give some analytical equations to define the error thresholds for some general cases of mean field like or symmetric mutation schemes and fitness functions we derive some equations for the dynamics in the case of a point mutation and polynomial fitness functions we derive exact dynamics for two point mutations asymmetric mutations and the four value spin model with a single peak fitness function the same method is applied for the model with a royal road fitness function we derive the steady state distribution for the single peak fitness function

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

rank two stable sheaves with odd determinant on fano threefolds of genus nine according to mukai and iliev a smooth prime fano threefold x of genus 9 is associated with a surface mathbb p mathcal v ruled over a smooth plane quartic vargamma and the derived category of vargamma embeds into that of x by a theorem of kuznetsov we use this setup to study the moduli spaces of rank 2 stable sheaves on x with odd determinant for each c 2 ge 7 we prove that a component of their moduli space mathsf m x 2 1 c 2 is birational to a brill noether locus of vector bundles with fixed rank and degree on vargamma having enough sections when twisted by mathcal v for c 2 7 we prove that mathsf m x 2 1 7 is isomorphic to the blow up of the picard variety text pic 2 vargamma along the curve parametrizing lines contained in x

hamiltonian 2 forms in kahler geometry iii extremal metrics and stability this paper concerns the existence and explicit construction of extremal kahler metrics on total spaces of projective bundles which have been studied in many places we present a unified approach motivated by the theory of hamiltonian 2 forms as introduced and studied in previous papers in the series but this paper is largely independent of that theory

dynamics of the chemical master equation a strip of chains of equations in d dimensional space we investigate the multichain version of the chemical master equation when there are transitions between different states inside the long chains as well as transitions between a few different chains in the discrete version such a model can describe the connected diffusion processes with jumps between different types we apply the hamilton jacobi equation to solve some aspects of the model we derive exact in the limit of infinite number of particles results for the dynamic of the maximum of the distribution and the variance of distribution

recombination in one and two dimensional fitness landscapes we consider many site mutation recombination models of molecular evolution where fitness is a function of a hamming distance from one one dimensional case or two two dimensional case sequences for the one dimensional case we calculate the population distribution dynamics for a model with zero fitness and an arbitrary symmetric initial distribution and find an error threshold transition point in the single peak fitness model for a given initial symmetric distribution we calculate the recombination period in the case of a single peak fitness function when the original population is located at one sequence at some hamming distance from the peak configuration steady state fitness is calculated with finite genome length corrections we derive analytical equations for the two dimensional mutation recombination model

evolutionary dynamics on strongly correlated fitness landscapes we study the evolutionary dynamics of a maladapted population of self replicating sequences on strongly correlated fitness landscapes each sequence is assumed to be composed of blocks of equal length and its fitness is given by a linear combination of four independent block fitnesses a mutation affects the fitness contribution of a single block leaving the other blocks unchanged and hence inducing correlations between the parent and mutant fitness on such strongly correlated fitness landscapes we calculate the dynamical properties such as the number of jumps in the most populated sequence and the temporal distribution of the last jump which is shown to exhibit an inverse square dependence as in evolution on uncorrelated fitness landscapes we also obtain exact results for the distribution of records and extremes for correlated random variables

quasispecies model of evolution with migration we consider the model of asexual evolution with migration which was proposed by waclaw et al phys rev lett 105 268101 2010 prltao0031 900710 1103 physrevlett 105 268101 this model setting is based on the standard mutation scheme from the quasispecies theory but with replicators moving from one habitat to another the primary goal is to solve exactly the infinite population genome length version of the model for the independent random distribution of fitnesses considered in the original paper moreover we propose the analytical solution for the single peak and the symmetric fitness landscape our analytical solution is exact at the limit of large n we found two phases the correlated phase with the identical distributions of mutations in both habitats and the uncorrelated phase where the second habitat is choosing another peak of distribution in sequence space compared to the peak in the first habitat

smooth p adic analytic spaces are locally contractible 0 introduction 293 1 piecewise rs linear spaces 298 2 r colored polysimplicial sets 308 3 r colored polysimplicial sets of length l 313 4 the skeleton of a nondegenerate pluri stable formal scheme 327 5 a colored polysimplicial set associated with a nondegenerate poly stable fibration 336 6 p adic analytic and piecewise linear spaces 346 7 strong local contractibility of smooth analytic spaces 355 8 cohomology with coefficients in the sheaf of constant functions 362

convergence of the j flow on toric manifolds we show that on a kahler manifold whether the j flow converges or not is independent of the chosen background metric in its kahler class on toric manifolds we give a numerical characterization of when the j flow converges verifying a conjecture in 19 m lejmi and g szekelyhidi the j flow and stability in this case we also strengthen existing results on more general inverse sigma k equations on kahler manifolds

solvable biological evolution model with a parallel mutation selection scheme based on the connection between a quantum spin model and an asexual biological evolution model with a single peak fitness function in parallel mutation selection scheme we solve exactly both static and dynamics of the evolution model we find that relaxation in such a parallel scheme is faster than that in a connected scheme of eigen model our method can also be extended to other fitness functions

k stability of constant scalar curvature kahler manifolds abstract we show that a polarised manifold with a constant scalar curvature kahler metric and discrete automorphisms is k stable this refines the k semistability proved by s k donaldson

phase transition in random catalytic networks the notion of auto catalytic networks has become a cornerstone in understanding the possibility of a sudden dramatic increase of diversity in biological evolution as well as in the evolution of social and economical systems here we study catalytic random networks with respect to the final outcome diversity of products we show that an analytical treatment of this long standing problem is possible by mapping the problem onto a set of nonlinear recurrence equations the solution of these equations shows a crucial dependence of the final number of products on the initial number of products and the density of catalytic production rules for a fixed density of rules we can demonstrate the existence of a phase transition from a practically unpopulated regime to a fully populated and diverse one the order parameter is the number of final products we are able to fully understand the origin of this phase transition as a crossover from one set of solutions from a quadratic equation to the other we observe a remarkable similarity of the solution of the system and the pvt diagrams in standard thermodynamics

the theorem of grauert mulich spindler the boundedness of semi stable sheaves on p is a simple corollary to the above see 13 if one looks into the proof of this theorem by spindler and the techniques by van de ven 14 and barth 1 he can generalize immediately the theorem in the following way let e be a torsion free coherent sheaf on a non singular projective variety x over an algebraically closed field k and ox 1 a very ample invertible sheaf on x if g l y n dimx are sufficiently general member of a very ample linear subsystem l of iox 1 l then y yl i1 is a non singular subvariety of codimension n in x and elf is torsion free

a modified crow kimura evolution model with reduced fitness for the smooth distribution of population abstract we investigate an evolution model in which the fitness depends on the steepness of population distribution via hamming classes the absolute value of the logarithm of the probability ratios for the neighbor hamming classes the model has a rather rich phase structure with observed oscillations of the mean fitness in the dynamics specifically the fitness is reduced compared to the standard crow kimura model when the steepness is less than a certain value d we compare the mean fitness of our model with the mean fitness of crow kimura model with the same parameters our work reveals that there is a threshold value d c for which d d c there are no oscillations of the mean fitness and it is less than the corresponding fitness in the crow kimura model with the same parameters for d d c the mean fitness oscillates with time the time averaged mean fitness is identical to the mean fitness in crow kimura model and there is a wider distribution than in the case of crow kimura model this is a significant advantage in dynamic environment

an obstruction to the existence of constant scalar curvature kahler metrics we prove that polarised manifolds that admit a constant scalar curvature kahler csck metric satisfy a condition we call slope semistability that is we define the slope for a projective manifold and for each of its subschemes and show that if x is csck then z x for all subschemes z this gives many examples of manifolds with kahler classes which do not admit csck metrics such as del pezzo surfaces and projective bundles if p e b is a projective bundle which admits a csck metric in a rational kahler class with sufficiently small fibres then e is a slope semistable bundle and b is a slope semistable polarised manifold the same is true for all rational kahler classes if the base b is a curve we also show that the slope inequality holds automatically for smooth curves canonically polarised and calabi yau manifolds and manifolds with c1 x 0 and l close to the canonical polarisation

existence of twisted constant scalar curvature kahler metrics with a large twist suppose that there exist two kahler metrics omega and alpha such that the metric contraction of alpha with respect to omega is constant i e varlambda omega alpha text const we prove that for all large enough r 0 there exists a twisted constant scalar curvature kahler metric omega in the cohomology class omega satisfying s omega r varlambda omega alpha text const we discuss its implication to k stability and the continuity method recently proposed by chen

towards a physics of evolution critical diversity dynamics at the edges of collapse and bursts of diversification systems governed by the standard mechanisms of biological or technological evolution are often described by catalytic evolution equations we study the structure of these equations and find an analogy with classical thermodynamic systems in particular we can demonstrate the existence of several distinct phases of evolutionary dynamics a phase of fast growing diversity one of stationary finite diversity and one of rapidly decaying diversity while the first two phases have been subject to previous work here we focus on the destructive aspects in particular the phase diagram of evolutionary dynamics the main message is that within a critical region massive loss of diversity can be triggered by very small external fluctuations we further propose a dynamical model of diversity which captures spontaneous creation and destruction processes fully respecting the phase diagrams of evolutionary systems the emergent time series show rich diversity dynamics including power laws as observed in actual economical data e g firm bankruptcy data we believe the present model presents a possibility to cast the famous qualitative picture of schumpeterian economic evolution into a quantifiable and testable framework

adaptive evolution on a continuous lattice model in the current work we investigate the evolutionary dynamics of a spatially structured population model defined on a continuous lattice in the model individuals disperse at a constant rate v and competition is local and delimited by the competition radius r due to dispersal the neighborhood size number of individuals competing for reproduction fluctuates over time here we address how these new variables affect the adaptive process while the fixation probabilities of beneficial mutations are roughly the same as in a panmitic population for small fitness effects s a dependence on v and r becomes more evident for large s these quantities also strongly influence fixation times but their dependencies on s are well approximated by s ensuremath 1 2 which means that the speed of the genetic wave front is proportional to sqrt s most important is the observation that the model exhibits a dual behavior displaying a power law growth for the fixation rate and speed of adaptation with the beneficial mutation rate as observed in other spatially structured population models while simultaneously showing a nonsaturating behavior for the speed of adaptation with the population size n as in homogeneous populations

four state quantum chain as a model of sequence evolution a variety of selection mutation models for dna or rna sequences well known in molecular evolution can be translated into a model of coupled ising quantum chains this correspondence is used to investigate the genetic variability and error threshold behaviour in dependence of possible fitness landscapes in contrast to the two state models treated hitherto the model explicitly takes the four state nature of the nucleotide alphabet into account and allows for the distinction of mutation rates for the different base substitutions as given by standard mutation schemes of molecular phylogeny as a consequence of this refined treatment new phase diagrams for the error threshold behaviour are obtained with appearance of a novel phase in which the nucleotide ordering of the wildtype sequence is only partially conserved explicit analytical and numerical results are presented for evolution dynamics and equilibrium behaviour in a number of accessible situations such as quadratic fitness landscapes and the kimura 2 parameter mutation scheme

exact results in the large system size limit for the dynamics of the chemical master equation a one dimensional chain of equations cme we found exact analytical expressions in large system size limit for the probability distribution including explicit expression for the dynamics of variance of distribution we also give the solution for some simple cases of the model with time dependent rates we derived the results of the van kampen method from the hje approach using a special ansatz using the van kampen method we give a system of ordinary differential equations odes to define the variance in a two dimensional case we performed numerics for the cme with stationary noise we give analytical criteria for the disappearance of bistability in the case of stationary noise in one dimensional cmes

physics of evolution selection without fitness abstract traditionally evolution is seen as a process where from a pool of possible variations of a population e g biological species or industrial goods a few variations get selected which survive and proliferate whereas the others vanish survival probabilities and proliferation rates are typically associated with the fitness of particular variations in this paper we argue that the notion of fitness is an a posteriori concept in the sense that one can assign higher fitness to species that survive but one can generally not derive or predict fitness per se proliferation rates can be measured whereas fitness landscapes i e the inter dependence of proliferation rates cannot for this reason we think that in a physical theory of evolution such notions should be avoided in this spirit here we propose a random matrix model of evolution where selection mechanisms are encoded in interaction matrices of species thereby extending the previous work of ours by a control parameter describing suppressors in the system we are able to recover some key facts of evolution dynamics endogenously such as punctuated equilibrium i e the existence of intrinsic large extinction events and at the same time periods of dramatic diversification as known e g from the fossil record further we comment on two fundamental technical problems of a physics of evolution the non closedness of its phase space and the problem of co evolving boundary conditions apparent in all systems subject to evolution

the kahler ricci flow on surfaces of positive kodaira dimension 4 estimates 9 4 1 the zeroth order and volume estimates 9 4 2 a partial second order estimate 13 4 3 gradient estimates 14 4 4 the second order estimates 18

heterotic string compactification and new vector bundles we propose a construction of kahler and non kahler calabi yau manifolds by branched double covers of twistor spaces in this construction we use the twistor spaces of four manifolds with self dual conformal structures with the examples of connected sum of n mathbb p 2 s we also construct k3 fibered calabi yau manifolds from the branched double covers of the blow ups of the twistor spaces these manifolds can be used in heterotic string compactifications to four dimensions we also construct stable and polystable vector bundles some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions

full characterization of a strange attractor chaotic dynamics in low dimensional replicator system abstract two chaotic attractors observed in lotka volterra equations of dimension n 3 are shown to represent two different cross sections of one and the same chaotic regime the strange attractor is studied in the equivalent four dimensional catalytic replicator network analytical expressions are derived for the lyapunov exponents of the flow in the centre of the chaotic regime the strange attractor was characterized by the lyapunov dimension dl 2 06 0 02 and the renyi fractal dimensions the set corresponding to the attractor represents a multifractal its singularity spectrum is computed one route in parameter space leading into the chaotic regime and crossing it was studied in detail it leads to a feigenbaum type cascade of bifurcations before the chaos is fully developed the dynamic system passes an internal crisis through a period two tangent bifurcation then a second sequence of period doubling bifurcations leading to another chaotic regime is observed which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type a series of one dimensional maps is derived from a properly chosen poincare cross section which illustrates structural changes in the attractor mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on perturbation theory the most striking result is the gradual disappearance of complex dynamics with increasing mutation rates the parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet

mean field approximation for two and three person prisoner s dilemmas abstract we consider the mean field approximation mfa to the system of interacting agents playing two and three person prisoner s dilemma games 2 pd and 3 pd the agents have three available strategies all c all d and a third one which we choose in various ways long time distribution of the strategies is compared for both games the latter game admits coexistence of larger number of strategies in the long run in the case of external pressure for cooperation more pressure is in general necessary for 3 pd than for 2 pd to guarantee cooperation for almost all initial compositions of the strategies results for different types of tit for tat strategies for 3 pd are discussed unlike the 2 pd case the more forgiving tit for tat strategy can survive in the long run in 3 pd

special test configuration and k stability of fano varieties for any at projective family x l c such that the generic bre x is a klt q fano variety and ljx q kx we use the techniques from the minimal model program mmp to modify the total family the end product is a family such that every ber is a klt q fano variety moreover we can prove that the donaldson futaki invariants of the appearing models decrease when the family is a test conguration of a xed fano variety x kx this implies tian s conjecture given x a fano manifold to test its k semi poly stability we only need to test on the special test congurations

evolutionary dynamics from a variational principle we demonstrate with a thought experiment that fitness based population dynamical approaches to evolution are not able to make quantitative falsifiable predictions about the long term behavior of some evolutionary systems a key characteristic of evolutionary systems is the ongoing endogenous production of new species these novel entities change the conditions for already existing species even darwin s demon a hypothetical entity with exact knowledge of the abundance of all species and their fitness functions at a given time could not prestate the impact of these novelties on established populations we argue that fitness is always a posteriori knowledge it measures but does not explain why a species has reproductive success or not to overcome these conceptual limitations a variational principle is proposed in a spin model like setup of evolutionary systems we derive a functional which is minimized under the most general evolutionary formulation of a dynamical system i e evolutionary trajectories causally emerge as a minimization of a functional this functional allows the derivation of analytic solutions of the asymptotic diversity for stochastic evolutionary systems within a mean field approximation we test these approximations by numerical simulations of the corresponding model and find good agreement in the position of phase transitions in diversity curves the model is further able to reproduce stylized facts of timeseries from several man made and natural evolutionary systems light will be thrown on how species and their fitness landscapes dynamically coevolve

dynamics of the eigen and the crow kimura models for molecular evolution we introduce an alternative way to study molecular evolution within well established hamilton jacobi formalism showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1n accuracy where n is the genome length for a smooth and monotonic fitness function this approach gives two dynamical phases smooth dynamics and discontinuous dynamics the latter phase arises naturally with no explicite singular fitness function counterintuitively the hamilton jacobi method yields straightforward analytical results for the models that utilize fitness as a function of hamming distance from a reference genome sequence we also show the way in which this method gives dynamical phase structure for multipeak fitness

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

eigen model as a quantum spin chain exact dynamics we map the eigen model of biological evolution naturwissenschaften 58 465 1971 into a quantum spin model with non hermitian hamiltonian based on such a connection we derive exact relaxation periods for the eigen model to approach static energy landscape from various initial conditions we also study a simple case of dynamic fitness function

the speed of evolution in large asexual populations we consider an asexual biological population of constant size n evolving in discrete time under the influence of selection and mutation beneficial mutations appear at rate u and their selective effects s are drawn from a distribution g s after introducing the required models and concepts of mathematical population genetics we review different approaches to computing the speed of logarithmic fitness increase as a function of n u and g s we present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid we then discuss approximate approaches to the finite population problem distinguishing between the case of a single selection coefficient g s s s b and a continuous distribution of selection coefficients analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10300

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

representation theory and adhm construction on quaternion symmetric spaces we determine all irreducible homogeneous bundles with anti selfdual canonical connections on compact quaternion symmetric spaces to deform the canonical connections we give a relation between the representation theory and the theory of monads on the twistor space the moduli spaces are described via the bott borel weil thereom the horrocks bundle is also generalized to higher dimensional projective spaces

biological evolution in a multidimensional fitness landscape we considered a multiblock molecular model of biological evolution in which fitness is a function of the mean types of alleles located at different parts blocks of the genome we formulated an infinite population model with selection and mutation and calculated the mean fitness for the case of recombination we formulated a model with a multidimensional fitness landscape the dimension of the space is equal to the number of blocks and derived a theorem about the dynamics of initially narrow distribution we also considered the case of lethal mutations we also formulated the finite population version of the model in the case of lethal mutations our models derived for the virus evolution are interesting also for the statistical mechanics and the hamilton jacobi equation as well

a new method for the solution of models of biological evolution derivation of exact steady state distributions we investigate well known models of biological evolution and address the open problem of how construct a correct continuous analog of mutations in discrete sequence space we deal with models where the fitness is a function of a hamming distance from the reference sequence the mutation selection master equation in the discrete sequence space is replaced by a hamilton jacobi equation for the logarithm of relative frequencies of different sequences the steady state distribution mean fitness and the variance of fitness are derived all our results are asymptotic in the large genome limit a variety of important biological and biochemical models can be solved by this new approach

filtrations and test configurations we introduce a strengthening of k stability based on filtrations of the homogeneous coordinate ring this allows for considering certain limits of families of test configurations which arise naturally in several settings we prove that if a manifold with no automorphisms admits a csck metric then it satisfies this stronger stability notion we also discuss the relation with the birational transformations in the definition of b stability

the j flow and stability we study the j flow from the point of view of an algebro geometric stability condition in terms of this we give a lower bound for the natural associated energy functional and we show that the blowup behavior found by fang and lai 11 is reflected by the optimal destabilizer finally we prove a general existence result on complex tori