empirical processes and random projections in this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same l2 norm we present several geometric applications of this result the most important of which is a sharpening of the johnson lindenstrauss embedding lemma our results apply to a large class of random matrices as we only require that the matrix entries have a subgaussian tail

characterization of talagrand s transport entropy inequalities in metric spaces we give a characterization of transport entropy inequalities in metric spaces as an application we deduce that such inequalities are stable under bounded perturbation holley stroock perturbation lemma

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

structured total least squares and l2 approximation problems abstract it is shown how structured and weighted total least squares and l 2 approximation problems lead to a nonlinear generalized singular value decomposition an inverse iteration scheme to find a local minimum is proposed the emphasis of the paper is not on the convergence analysis of the algorithm rather the purpose is to illustrate its use in numerous applications in systems and control including total least squares with relative errors and or fixed elements inverse singular value problems an errors in variables variant of the kalman filter impulse response realization from noisy data h 2 model reduction h 2 system identification and calculating the largest stability radius of uncertain linear systems several numerical examples are given

local tail bounds for functions of independent random variables it is shown that functions defined on 0 1 r 1 n satisfying certain conditions of bounded differences that guarantee sub gaussian tail behavior also satisfy a much stronger local sub gaussian property for self bounding and configuration functions we derive analogous locally subexponential behavior the key tool is talagrand s ann probab 22 1994 1576 1587 variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on 0 1 r 1 n for r 2

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

a characterization of dimension free concentration in terms of transportation inequalities the aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies talagrand s t 2 transportation cost inequality this theorem permits us to give a new and very short proof of a result of otto and villani generalizations to other types of concentration are also considered in particular one shows that the poincare inequality is equivalent to a certain form of dimension free exponential concentration the proofs of these results rely on simple large deviations techniques

laplacian eigenvalues and distances between subsets of a manifold in this paper we give a new method to convert results dealing with graph theoretic or markov chain laplacians into results concerning laplacians in analysis such as on riemannian manifolds we illustrate this method by using the results of cgy97 to prove 1 1 dist x y cosh 1 xc y c x y 2 for 1 the first positive neumann eigenvalue on a connected compact riemannian manifold and x y any two disjoint sets and where xc is the complement of x this inequality has a version for the k th positive eigenvalue involving k 1 disjoint sets and holds more generally for all analytic laplacians described in cgy97 we show that this inequality is optimal to first order in that it is impossible to obtain an inequality of this form with the right hand side divided by 1 for any fixed constant 0

characterizations of l n and l1n and their stabilities we characterize l n and l1n by introducing two geometric constants in n dimensional banach spaces we also prove that these characterizations are stable with respect to the banach mazur distance

on the sensitivity to noise of a boolean function in this paper we generate upper and lower bounds for the sensitivity to noise of a boolean function using relaxed assumptions on input choices and noise the robustness of a boolean network to noisy inputs is related to the average sensitivity of that function the average sensitivity measures how sensitive to changes in the inputs the output of the function is the average sensitivity of boolean functions can indicate whether a specific random boolean network constructed from those functions is ordered chaotic or in critical phase we give an exact formula relating the sensitivity to noise and the average sensitivity of a boolean function the analytic approach is supplemented by numerical results that illustrate the overall behavior of the sensitivities as various boolean functions are considered it is observed that for certain parameter combinations the upper estimates in this paper are sharper than other estimates in the literature and that the lower estimates are very close to the actual values of the sensitivity to noise of the selected boolean functions

chaos game for ifss on topological spaces abstract we explore the chaos game on continuous ifss on topological spaces we prove that the existence of an attractor implies that the chaos game can be used to construct approximations of the attractor the essential role of basin of attraction is discussed

modified log sobolev inequality for a compact pure jump markov process with degenerate jumps we study the modified log sobolev inequality for a class of pure jump markov processes that describe the interactions between brain neurons as a result we obtain concentration properties for empirical approximations of the process in particular we focus on a finite and compact process with degenerate jumps inspired by the model introduced by galves and locherbach j stat phys 151 896 921 2013

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

on extracting common random bits from correlated sources on large alphabets suppose alice and bob receive strings x x1 xn and y y1 yn each uniformly random in s n but so that x and y are correlated for each symbol i we have that yi xi with probability 1 e and otherwise yi is chosen independently and uniformly from s alice and bob wish to use their respective strings to extract a uniformly chosen common sequence from s k but without communicating how well can they do the trivial strategy of outputting the first k symbols yields an agreement probability of 1 e e s k in a recent work by bogdanov and mossel it was shown that in the binary case where s 2 and k k e is large enough then it is possible to extract k bits with a better agreement probability rate in particular it is possible to achieve agreement probability ke 1 2 2 ke 2 1 e 2 using a random construction based on hamming balls and this is optimal up to lower order terms in this paper we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows in particular we show no strategy can achieve agreement probability better than 1 e k 1 s k where s 0 as s we also show that hamming ball based constructions have much lower agreement probability rate than the trivial algorithm as s our proofs and results are intimately related to subtle properties of hypercontractive inequalities

isoperimetric constants for product probability measures a dimension free lower bound is found for isoperimetric constants of product probability measures from this some analytic inequalities are derived

almost euclidean sections in symmetric spaces and concentration of order statistics abstract we prove that for any symmetric n dimensional normed space e and any e 0 e contains a 1 e euclidean subspace of dimension at least c ln n ln 1 e where c is an absolute constant the proof is based on a concentration property of order statistics of random vectors with i i d coordinates

characterizations of matrix and operator valued φ entropies and operator efron stein inequalities we derive new characterizations of the matrix entropy functionals introduced in chen tropp chen tropp 2014 electron j prob 19 1 30 doi 10 1214 ejp v19 2964 1 these characterizations help us to better understand the properties of matrix entropies and are a powerful tool for establishing matrix concentration inequalities for random matrices then we propose an operator valued generalization of matrix entropy functionals and prove the subadditivity under lowner partial ordering our results demonstrate that the subadditivity of operator valued entropies is equivalent to the convexity as an application we derive the operator efron stein inequality 1 http dx doi org 10 1214 ejp v19 2964

dichotomies structure and concentration in normed spaces abstract we use probabilistic topological and combinatorial methods to establish the following deviation inequality for any normed space x r n there exists an invertible linear map t r n r n with p t g e t g e e t g c exp c max e 2 e log n e 0 where g is the standard n dimensional gaussian vector and c c 0 are universal constants it follows that for every e 0 1 and for every normed space x r n there exists a k dimensional subspace of x which is 1 e euclidean and k c e log n log 1 e this improves by a logarithmic on e term the best previously known result due to g schechtman

a new characterization of talagrand s transport entropy inequalities and applications we show that talagrand s transport inequality is equivalent to a restricted logarithmic sobolev inequality this result clarifies the links between these two important functional inequalities as an application we give the first proof of the fact that talagrand s inequality is stable under bounded perturbations

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

oded schramm s contributions to noise sensitivity we survey in this paper the main contributions of oded schramm related to noise sensitivity we will describe in particular his various works which focused on the spectral analysis of critical percolation and more generally of boolean functions his work on the shape fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation

submean variance bound for effective resistance of random electric networks we study a model of random electric networks with bernoulli resistances in the case of the lattice mathbb z 2 we show that the point to point effective resistance between 0 and a vertex v has a variance of order at most rm log v frac 2 3 whereas its expected value is of order log v when v goes to infinity when d 2 expectation and variance are of the same order similar results are obtained in the context of p resistance the proofs rely on a modified poincare inequality due to falik and samorodnitsky 7

on dvoretzky s theorem for subspaces of lp abstract we prove that for any 2 p and for every n dimensional subspace x of l p represented on r n whose unit ball b x is in lewis position one has the following two level gaussian concentration inequality p z e z e e z c exp c min p e 2 n e n 2 p 0 e 1 where z is the standard n dimensional gaussian vector p 0 is a constant depending only on p and c c 0 are absolute constants as a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces in particular for any 2 p and every n dimensional subspace x of l p the euclidean space l 2 k can be 1 e embedded into x with k c p min e 2 n e n 2 p where c p 0 is a constant depending only on p this improves upon the previously known estimate due to figiel lindenstrauss and milman

averages of norms and quasi norms we compute the number of summands in q averages of norms needed to approximate an euclidean norm it turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the euclidean norm particular attention is given to the case q in which the average is replaced with the maxima this is closely connected with the behavior of certain families of projective caps on the sphere

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

robust stability of state space models with structured uncertainties a method for robust eigenvalue location analysis of linear state space models affected by structured real parametric perturbations is proposed the approach based on algebraic matrix properties deals with state space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters a method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed the method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space this problem which can be reduced to finding the real solutions of a system of polynomial equations simplifies considerably when cases with one or two uncertain parameters are considered for these cases necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems

high dimensional inference with random maximum a posteriori perturbations this paper presents a new approach called perturb max for high dimensional statistical inference in graphical models that is based on applying random perturbations followed by optimization this framework injects randomness into maximum a posteriori map predictors by randomly perturbing the potential function for the input a classic result from extreme value statistics asserts that perturb max operations generate unbiased samples from the gibbs distribution using high dimensional perturbations unfortunately the computational cost of generating so many high dimensional random variables can be prohibitive however when the perturbations are of low dimension sampling the perturb max prediction is as efficient as map optimization this paper shows that the expected value of perturb max inference with low dimensional perturbations can be used sequentially to generate unbiased samples from the gibbs distribution furthermore the expected value of the maximal perturbations is a natural bound on the entropy of such perturb max models a measure concentration result for perturb max values shows that the deviation of their sampled average from its expectation decays exponentially in the number of samples allowing effective approximation of the expectation

explicit euclidean embeddings in permutation invariant normed spaces abstract let x be a real normed space of dimension n n with a basis e i 1 n such that the norm is invariant under coordinate permutations assume for simplicity that the basis constant is at most 2 consider any n n and 0 e 1 4 such that n c log e 1 1 log n we provide an explicit construction of a matrix that generates a 1 e embedding of l 2 n into x

concentration inequalities for non lipschitz functions with bounded derivatives of higher order building on the inequalities for homogeneous tetrahedral polynomials in independent gaussian variables due to r latala we provide a concentration inequality for not necessarily lipschitz functions f mathbb r n rightarrow mathbb r with bounded derivatives of higher orders which holds when the underlying measure satisfies a family of sobolev type inequalities begin aligned vert g mathbb eg vert p le c p vert nabla g vert p end aligned such sobolev type inequalities hold e g if the underlying measure satisfies the log sobolev inequality in which case c p le c sqrt p or the poincare inequality then c p le cp our concentration estimates are expressed in terms of tensor product norms of the derivatives of f when the underlying measure is gaussian and f is a polynomial not necessarily tetrahedral or homogeneous our estimates can be reversed up to a constant depending only on the degree of the polynomial we also show that for polynomial functions analogous estimates hold for arbitrary random vectors with independent sub gaussian coordinates we apply our inequalities to general additive functionals of random vectors in particular linear eigenvalue statistics of random matrices and the problem of counting cycles of fixed length in erd s renyi random graphs obtaining new estimates optimal in a certain range of parameters

degree mixing in multilayer networks impedes the evolution of cooperation traditionally the evolution of cooperation has been studied on single isolated networks yet a player especially in human societies will typically be a member of many different networks and those networks will play different roles in the evolutionary process multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society with this motivation we here consider two layer scale free networks with all possible combinations of degree mixing wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating we find that breaking the symmetry through assortative mixing in one layer and or disassortative mixing in the other layer as well as preserving the symmetry by means of assortative mixing in both layers impedes the evolution of cooperation we use degree dependent distributions of strategies and cluster size analysis to explain these results which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas

metric and classical fidelity uncertainty relations for random unitary matrices we analyze uncertainty relations on finite dimensional hilbert spaces expressed in terms of classical fidelity which are stronger than metric uncertainty relations introduced by fawzi hayden and sen we establish the validity of fidelity uncertainty relations for random unitary matrices with optimal parameters up to universal constants which improves upon known results for the weaker notion of metric uncertainty this result is then applied to locking classical information in quantum states and allows to obtain optimal locking in hellinger distance improving upon previous results on locking in the total variation distance both by strengthening the metric used and by improving the dependence on parameters we also show that general probabilistic estimates behind the main theorem can be used to prove existence of data hiding schemes with bayesian type guarantees as a byproduct of our approach we obtain existence of almost euclidean subspaces of the matrix spaces with a better dimension distortion dependence than allowed in previously known constructions

regularity of gaussian processes on donne la validite de la conjecture de fernique 1974 sur l existence de mesures de majoration pouvant caracteriser la limitation des processus gaussiens

two population three player prisoner s dilemma game we present a two population three player prisoner s dilemma finite state automata are used different types of tit for tat strategies all possible payoffs are computed due to the computational advantage in symmetric games most researches have focused on the symmetric games instead of the asymmetric ones which need more computations in this paper we present prisoner s dilemma game involving three players and suppose that two players among them agree against the third player by choosing either to cooperate together or to defect together at each round according to that assumption the game is transformed from the symmetric three player model to asymmetric two player model such that the identities of the players cannot be interchanged without interchanging the payoff of the strategies each strategy in the resulting model is expressed with two state automata we determine the payoff matrix corresponding to the all possible strategies we noticed that for some strategies it is better to be a player of the first type independent player than being of the second type allies

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

upper bounds on fourier entropy given a function f 0 1 n rightarrow mathbb r its fourier entropy is defined to be sum s widehat f 2 s log widehat f 2 s where hat f denotes the fourier transform of f this quantity arises in a number of applications especially in the study of boolean functions an outstanding open question is a conjecture of friedgut and kalai 1996 called the fourier entropy influence fei conjecture asserting that the fourier entropy of any boolean function f is bounded above up to a constant factor by the total influence average sensitivity of f

ultrametric subsets with large hausdorff dimension it is shown that for every e 0 1 every compact metric space x d has a compact subset s x that embeds into an ultrametric space with distortion o 1 e and dim h s geqslant 1 varepsilon dim h x where dimh denotes hausdorff dimension the above o 1 e distortion estimate is shown to be sharp via a construction based on sequences of expander graphs

influence and sharp threshold theorems for monotonic measures the influence theorem for product measures on the discrete space 0 1 n may be extended to probability measures with the property of monotonicity which is equivalent to strong positive association corresponding results are valid for probability measures on the cube 0 1 n that are absolutely continuous with respect to lebesgue measure these results lead to a sharp threshold theorem for measures of random cluster type and this may be applied to box crossings in the two dimensional random cluster model

majorizing measures the generic chaining majorizing measures provide bounds for the supremum of stochastic processes they represent the most general possible form of the chaining argument going back to kolmogorov majorizing measures arose from the theory of gaussian processes but they now have applications far beyond this setting the fundamental question is the construction of these measures this paper focuses on the tools that have been developed for this purpose and in particular the use of geometric ideas applications are given to several natural problems where entropy methods are powerless

matrix games with payoffs of belief structures imprecise matrix games such as interval valued matrix games and fuzzy matrix games have attracted much interest for a long time most of the previous studies on imprecise matrix games mainly focus on the fuzzy uncertainty of payoffs however the uncertainties of nonspecificity and discord involved in payoffs are not well addressed so far the purpose of this paper is to study the matrix game with such types of uncertainties in order to achieve that purpose we present a matrix game model with payoffs of belief structures so as to integrate discord and nonspecificity the proposed model can be used to express more imprecise interactions between players in the reality besides as another main contribution of the study an effective method for solving matrix games with belief structures payoffs is developed to help us find the equilibrium points of the kind of games at last an example is given to illustrate the proposed model and method

characterization of talagrand s like transportation cost inequalities on the real line abstract in this paper we give necessary and sufficient conditions for talagrand s like transportation cost inequalities on the real line this brings a new wide class of examples of probability measures enjoying a dimension free concentration of measure property another byproduct is the characterization of modified log sobolev inequalities for log concave probability measures on r

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

uncovering cooperative behaviors with sparse historical behavior data in the spatial games for past decades the main attention of the evolutionary games has been focused on cooperation mechanism with the assumption that the strategy information of all players are known however it is difficult for observers to obtain the global information of players strategies in the real world and some players even hide their strategy information to confuse their opponents here we try to solve the problem to predicate the hidden strategies with sparse historical behavior data in the evolutionary games to quantify the similarity of strategies among the players in our method the euclidean distance of players is defined from the strategies of the players in the few past rounds then the hidden strategy of a player will be determined from the tendency that players with minimum euclidean distance will adopt similar strategies the method has good performance on determining hidden strategy of human beings in both the prisoner s dilemma game and the public goods game where strategies of twenty five percent players are hidden and the success rate to determine hidden strategy reaches up to 0 9 it is also found that the success rate to determine hidden strategy depends on both length of historical behavior data and tempting payoff b the prisoner s dilemma game or multiple factor r the public goods game

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

hitting times functional inequalities lyapunov conditions and uniform ergodicity the use of lyapunov conditions for proving functional inequalities was initiated in 5 it was shown in 4 30 that there is an equivalence between a poincare inequality the existence of some lyapunov function and the exponential integrability of hitting times in the present paper we close the scheme of the interplay between lyapunov conditions and functional inequalities by showing that strong functional inequalities are equivalent to lyapunov type conditions showing that these lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times we also give some complement concerning the link between lyapunov conditions and in tegrability property of the invariant probability measure and as such transportation inequalities and we show that some unbounded lyapunov conditions can lead to uniform ergodicity and coming down from infinity property

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

superconcentration and randomized dvoretzky s theorem for spaces with 1 unconditional bases abstract let n be a sufficiently large natural number and let b be an origin symmetric convex body in r n in the l position and such that the space r n b admits a 1 unconditional basis then for any e 0 1 2 and for random c e log n log 1 e dimensional subspace e distributed according to the rotation invariant haar measure the section b e is 1 e euclidean with probability close to one this shows that the worst case dependence on e in the randomized dvoretzky theorem in the l position is significantly better than in john s position it is a previously unexplored feature which has strong connections with the concept of superconcentration introduced by s chatterjee in fact our main result follows from the next theorem let b be as before and assume additionally that b has a smooth boundary and e n b n c e n grad b 2 for a small universal constant c 0 where grad b is the gradient of b and n is the standard gaussian measure in r n then for any p 1 c log n the p th power of the norm b p is c log n superconcentrated in the gauss space

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

properties of isoperimetric functional and transport entropy inequalities via concentration various properties of isoperimetric functional transport entropy and concentration inequalities are studied on a riemannian manifold equipped with a measure whose generalized ricci curvature is bounded from below first stability of these inequalities with respect to perturbation of the measure is obtained the extent of the perturbation is measured using several different distances between perturbed and original measure such as a one sided l bound on the ratio between their densities wasserstein distances and kullback leibler divergence in particular an extension of the holley stroock perturbation lemma for the log sobolev inequality is obtained and the dependence on the perturbation parameter is improved from linear to logarithmic second the equivalence of transport entropy inequalities with different cost functions is verified by obtaining a reverse jensen type inequality the main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting of independent interest is a new dimension independent characterization of transport entropy inequalities with respect to the 1 wasserstein distance which does not assume any curvature lower bound

on the hardness of approximating minimum vertex cover we prove the minimum vertex cover problem to be np hard to approximate to within a factor of 1 3606 extending on previous pcp and hardness of approximation technique to that end one needs to develop a new proof framework and to borrow and extend ideas from several fields

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

sharpness of the phase transition for continuum percolation in mathbb r 2 r 2 we study the phase transition of random radii poisson boolean percolation around each point of a planar poisson point process we draw a disc of random radius independently for each point the behavior of this process is well understood when the radii are uniformly bounded from above in this article we investigate this process for unbounded and possibly heavy tailed radii distributions under mild assumptions on the radius distribution we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter lambda c moreover the techniques we develop in this article can be applied to other models such as the poisson voronoi and confetti percolation

fast computation of the largest stability radius for a two parameter linear system the author describes a fast algorithm to compute the minimum distance to instability or parameter margin for a linear time invariant system subject to rational variations in two real parameters both the l sub infinity and the euclidean norms are considered for measuring parameter deviations

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

evolutionary mixed games in structured populations cooperation and the benefits of heterogeneity evolutionary games on networks traditionally involve the same game at each interaction here we depart from this assumption by considering mixed games where the game played at each interaction is drawn uniformly at random from a set of two different games while in well mixed populations the random mixture of the two games is always equivalent to the average single game in structured populations this is not always the case we show that the outcome is in fact strongly dependent on the distance of separation of the two games in the parameter space effectively this distance introduces payoff heterogeneity and the average game is returned only if the heterogeneity is small for higher levels of heterogeneity the distance to the average game grows which often involves the promotion of cooperation the presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin and they also emphasize the importance of the population structure in amplifying facilitators of cooperation

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

modified logarithmic sobolev inequalities and transportation inequalities we present a class of modified logarithmic sobolev inequality interpolating between poincare and logarithmic sobolev inequalities suitable for measures of the type exp x or exp x log 2 x 1 2 and which lead to new concentration inequalities these modified inequalities share common properties with usual logarithmic sobolev inequalities as tensorisation or perturbation and imply as well poincare inequality we also study the link between these new modified logarithmic sobolev inequalities and transportation inequalities

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

the phase transition for dyadic tilings a dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts let each dyadic tile of order n be available with probability p independent of the others we prove that for p sufficiently close to 1 there exists a set of pairwise disjoint available tiles whose union is the unit square with probability tending to 1 as n infinity as conjectured by joel spencer in 1999 in particular we prove that if p 7 8 such a tiling exists with probability at least 1 3 4 n the proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling

small width low distortions quantized random embeddings of low complexity sets under which conditions and with which distortions can we preserve the pairwise distances of low complexity vectors e g for structured sets such as the set of sparse vectors or the one of low rank matrices when these are mapped or embedded in a finite set of vectors this work addresses this general question through the specific use of a quantized and dithered random linear mapping which combines in the following order a sub gaussian random projection in mathbb r m of vectors in mathbb r n a random translation or dither of the projected vectors and a uniform scalar quantizer of resolution delta 0 applied componentwise thanks to this quantized mapping we are first able to show that with high probability an embedding of a bounded set mathcal k subset mathbb r n in delta mathbb z m can be achieved when distances in the quantized and in the original domains are measured with the ell 1 and ell 2 norm respectively and provided the number of quantized observations m is large before the square of the gaussian mean width of mathcal k in this case we show that the embedding is actually quasi isometric and only suffers from both multiplicative and additive distortions whose magnitudes decrease as m 1 5 for general sets and as m 1 2 for structured set when m increases second when one is only interested in characterizing the maximal distance separating two elements of mathcal k mapped to the same quantized vector i e the consistency width of the mapping we show that for a similar number of measurements and with high probability this width decays as m 1 4 for general sets and as 1 m for structured ones when m increases finally as an important aspect of this paper we also establish how the non gaussianity of sub gaussian random projections inserted in the quantized mapping e g for bernoulli random matrices impacts the class of vectors that can be embedded or whose consistency width provably decays when m increases

strategy updating depending on local environment enhances cooperation in prisoners dilemma game a new method of strategy updating is proposed in the prisoners dilemma game some players apply the replicator dynamic rule for strategy updating while the others utilize the comprehensive method the intelligent players would consider not only the neighbors payoff but also its environment the results indicate that with the increase of intelligent players the fraction of cooperation increases an evolutionary prisoners dilemma game is studied on a square lattice when local environment is considered and a new strategy updating rule is utilized in this game first individuals are classified as two classes stochastic players and intelligent players the stochastic players are the ones who adopt the rule of replicator dynamics to update strategies and the intelligent players are those who make a decision through comprehensive considerations of the local environment namely the information of different strategy neighbors environment second the intelligent players evaluation is calculated from payoffs and local environment by dempstershafer theory the results indicate that the new strategy updating method promotes cooperation when intelligent individuals utilize the information of different strategy neighborhoods moreover the increasing number of intelligent player promotes cooperation level furthermore it is noteworthy that cooperators will finally coexist with defectors no matter what the proportion of intelligence is which is worthy to think about

a sublinear variance bound for solutions of a random hamilton jacobi equation we estimate the variance of the value function for a random optimal control problem the value function is the solution w of a hamilton jacobi equation with random hamiltonian h p x k p v x in dimension d 2 it is known that homogenization occurs as 0 but little is known about the statistical fluctuations of w our main result shows that the variance of the solution w is bounded by o log the proof relies on a modified poincare inequality of talagrand

approximate zero one laws and sharpness of the percolation transition in a class of models including two dimensional ising percolation one of the most well known classical results for site percolation on the square lattice is the equation p c p c 1 in words this equation means that for all values p c of the parameter p the following holds either a s there is an infinite open cluster or a s there is an infinite closed star cluster this result is closely related to the percolation transition being sharp below p c the size of the open cluster of a given vertex is not only a s finite but has a distribution with an exponential tail the analog of this result has been proven by higuchi in 1993 for two dimensional ising percolation at fixed inverse temperature c with external field h the parameter of the model using sharp threshold results approximate zero one laws and a modification of an rsw like result by bollobas and riordan we show that these results hold for a large class of percolation models where the vertex values can be nicely represented in a sense which will be defined precisely by i i d random variables we point out that the ordinary percolation model obviously belongs to this class and we also show that the ising model mentioned above belongs to it