error threshold in optimal coding numerical criteria and classes of universalities for complexity the free energy of the random energy model at the transition point between the ferromagnetic and spin glass phases is calculated at this point equivalent to the decoding error threshold in optimal codes the free energy has finite size corrections proportional to the square root of the number of degrees the response of the magnetization to an external ferromagnetic phase is maximal at values of magnetization equal to one half we give several criteria of complexity and define different universality classes according to our classification at the lowest class of complexity are random graphs markov models and hidden markov models at the next level is the sherrington kirkpatrick spin glass connected to neuron network models on a higher level are critical theories the spin glass phase of the random energy model percolation and self organized criticality the top level class involves highly optimized tolerance design error thresholds in optimal coding language and maybe financial markets living systems are also related to the last class the concept of antiresonance is suggested for complex systems

homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

non linear residually finite groups abstract we give the first example of a non linear residually finite 1 related group a t a t 2 a 2

homological representations of the hecke algebra in this paper a topological construction of representations of thean 1 series of hecke algebras associated with 2 row young diagrams will be given this construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection the fibres of the vector bundle are homology spaces of configuration spaces of points in c with a suitable twisted local coefficient system it is also shown that there is a close correspondence between this construction and the work of tsuchiya and kanie who constructed hecke algebra representations from the monodromy ofn point functions in a conformal field theory onp1 this work has significance in relation to the one variable jones polynomial which can be expressed in terms of characters of the iwahori hecke algebras associated with 2 row young diagrams it gives rise to a topological description of the jones polynomial which will be discussed elsewhere l2

the burau representation of the braid group b_n is unfaithful for large n two fundamental theorems of classical knot theory by j alexander are that every knot is a closed braid and second that a certain procedure see 1 2 for both results assigns to each knot k a polynomial a k in t which only depends on the topological type of the knot once normalized by a power of t to take a positive value at t 0 with this normalization take t to be a transcendental number since 1926 2 it has been known that the function a knots c is not one to one however it has not become clear whether the composite

a lower bound for the balanced truncation error for mimo systems we show that for a class of systems which includes state space symmetric systems the balanced truncation error is bounded from below by twice the sum of the tail of the hankel singular values including multiplicities divided by the dimension of the input space

a note on words in braid monoids abstract the purpose of this note is to give a self contained apart from simple facts about coxeter groups and we hope a bit shorter and more understandable account of some results of c1 c2 on normal forms of braids which are themselves based on the papers d1 t in particular a motivation was to give a proof of proposition 5 1 that we use in b m some proofs and results from section 2 onwards seem to be new i thank several people for improvements from earlier versions of the manuscript m geck for pointing out some errors f digne for pointing out that some results don t need the braid group to be of finite type and j y hee for suggesting and providing further improvements in that direction

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on twisted lawrence krammer representations abstract lawrence krammer representations are an important family of linear representations of artin tits groups of small type which are known under some assumptions on the parameters to be faithful when the type is spherical or more generally when they are restricted to the artin tits monoid and irreducible when the type is connected here we investigate an analogue of these representations introduced by digne in the spherical cases for every artin tits monoid that appears as the submonoid of fixed points of an artin tits monoid of small type under a group of graph automorphisms and for the corresponding artin tits group under the same assumptions on the parameters as in the small type cases we first show that these so called twisted lawrence krammer representations are faithful and we then prove by computing their formulas when the group of graph automorphisms is of order two or three their irreducibility in all the spherical and connected cases but one

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

the braid group b4 is linear we study a linear representation b n gl m z q 1 t 1 with m n n 1 2 we will show that for n 4 this representation is faithful we prove a relation with the new charney length function we formulate a conjecture implying that is faithful for all n

optimal design robustness and risk aversion in a series of recent papers carlson and doyle 1 3 have proposed a model for designed systems which they call highly optimized tolerance or hot the fundamental idea behind hot is that systems designed for high performance naturally organize into highly structured statistically unlikely states that are robust to perturbations they were designed to handle yet fragile to rare perturbations and design flaws as an example they consider an idealized model of forest fires 2 in this model a forester is charged with finding the optimal distribution of the trees on a grid so as to maximize tree harvest in the face of occasional fires that burn complete connected clusters of trees and are started by sparks that arrive with a given spatial distribution they find that optimizing the harvest or yield for the model gives rise to a segmented forest consisting of contiguous patches of trees separated by firebreaks and that the resulting distribution of fire sizes usually follows a power law while this type of configuration does typically achieve very good yields the system is also fragile in the sense that perturbations to the firebreaks or changes in the spark distribution can lead to substantially suboptimal performance they argue that these are pervasive phenomena high performance engineering leads to systems that are robust to stresses for which they were designed but fragile to errors or unforeseen events in this paper we argue that simple yield maximization is problematic even if there are no errors in firebreaks or changes in the spark distribution because the power law distributions generated by yield maximization have fat tails disastrously large forest fires occur with nonnegligible frequency far greater frequency than one would expect from intuition based on normal distributions this idea that yield optimization can lead to ruinous outcomes is not new for the classic problem of gambler s ruin for example it is well known that optimizing total return leads to ruin with probability one by contrast if one is willing to accept suboptimal returns it is possible to construct gambling strategies that are immune to ruin 4 applying similar ideas in the present context we show that a risk averse engineer who is willing to accept some loss in average system performance can effectively limit the large deviations in the event size distribution so that disasters are rare we call this variation on the hot theme constrained optimization with limited deviations or cold by avoiding total ruin a cold design is more robust than a hot one even in a world of perfect error free optimization to demonstrate the difference between hot and cold we first revisit the hot forest fire model we give an analytic solution for the model which shows that the distribution of fire sizes does indeed follow a power law the cumulative distribution having an exponent 21 1 1d where d is the dimensionality of the system using a fast percolation algorithm 5 we perform numerical simulations that confirm the value of this exponent we then generalize our solution to include risk aversion in the design thereby breaking the power law scaling and dramatically reducing the frequency of disastrously large events following refs 1 2 then we consider a forest divided into a large number of regions or patches with firebreaks between them that prevent the spread of fire from one patch to another although the original forest fire model was based on a lattice the model we consider is a continuum one since this makes the mathematical treatment more tractable for large system sizes we expect the behavior of this continuum model to converge to that of the lattice model it is assumed that during the lifetime of the forest a single spark lands at a random position r and starts a fire that burns the surrounding patch let us denote the area of this patch by sr the forest is then harvested giving a yield equal to the area of the remaining forest in units where the total area of the forest is one the yield is 1 2 sr 2 f where f is the cost in terms of yield of constructing the firebreaks

braid groups are linear in a previous work 11 the author considered a representation of the braid group p bn glm z ql1 t l m n n 1 2 and proved it to be faithful for n 4 bigelow 3 then proved the same representation to be faithful for all n by a beautiful topological argument the present paper gives a different proof of the faithfulness for all n we establish a relation between the charney length in the braid group and exponents of t a certain bn invariant subset of the module is constructed whose properties resemble those of convex cones we relate line segments in this set with the thurston normal form of a braid

highly optimized tolerance and power laws in dense and sparse resource regimes power law cumulative frequency p versus event size l distributions p l similar to l alpha are frequently cited as evidence for complexity and serve as a starting point for linking theoretical models and mechanisms with observed data systems exhibiting this behavior present fundamental mathematical challenges in probability and statistics the broad span of length and time scales associated with heavy tailed processes often require special sensitivity to distinctions between discrete and continuous phenomena a discrete highly optimized tolerance hot model referred to as the probability loss resource plr model gives the exponent alpha 1 d as a function of the dimension d of the underlying substrate in the sparse resource regime this agrees well with data for wildfires web file sizes and electric power outages however another hot model based on a continuous dense distribution of resources predicts alpha 1 1 d in this paper we describe and analyze a third model the cuts model which exhibits both behaviors but in different regimes we use the cuts model to show all three models agree in the dense resource limit in the sparse resource regime the continuum model breaks down but in this case the cuts and plr models are described by the same exponent

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

the infimum supremum and geodesic length of a braid conjugacy class abstract algorithmic solutions to the conjugacy problem in the braid groups bn n 2 3 4 were given in earlier work this note concerns the computation of two integer class invariants known as inf and sup a key issue in both algorithms is the number m of times one must cycle resp decycle in order to either increase inf resp decrease sup or to be sure that it is already maximal resp minimal for the class our main result is to prove that m is bounded above by n2 n 2 1 in the situation stated by e a elrifai and h r morton 1994 quart j math oxford45 479 497 and by n 2 in the situation stated by authors 1998 adv math 139 322 353 it follows immediately that the computation of inf and sup is polynomial in both word length and braid index in both algorithms the integers inf and sup determine but are not determined by the shortest geodesic length for elements in a conjugacy class and so we also obtain a polynomial time algorithm for computing this length

on lawrence krammer representations lawrence krammer representations or simply lk representations for short are linear representations of artin tits groups of small type over the ring z x 1 y 1 which are of particular interest since they are known to be faithful when the type is spherical and irreducible when the type is spherical and connected here we define and study lk representations over an arbitrary commutative ring r under some conditions on r and on the parameters of an lk representation which generalize the classical settings we recover the faithfulness property and give a new proof of irreducibility that works for every non necessarily spherical connected type we then study the equivalence of distinct lk representations of a given artin tits group for distinct choices of parameters finally we generalize to our setting the known classification of lk representations in the spherical cases and proceed with their classification in the affine cases

reading the dual garside length of braids from homological and quantum representations we show that lawrence s representation and linear representations of the braid groups from generic highest weight representations of u q mathfrak sl 2 detect the dual garside length of braids in a simple and natural way that is by expressing a representation as a matrix over a laurent polynomial ring using a certain natural basis the span of the variable is equal to a constant multiple of the dual garside length

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

krammer representations for complex braid groups abstract let b be the generalized braid group associated to some finite complex reflection group w we define a representation of b of dimension the number of reflections of the corresponding reflection group which generalizes the krammer representation of the classical braid groups and is thus a good candidate in view of proving the linearity of b we decompose this representation in irreducible components and compute its zariski closure as well as its restriction to parabolic subgroups we prove that it is faithful when w is a coxeter group of type ade and odd dihedral types and conjecture its faithfulness when w has a single class of reflections if true this conjecture would imply various group theoretic properties for these groups that we prove separately to be true for the other groups