ricci curvature of finite markov chains via convexity of the entropy we study a new notion of ricci curvature that applies to markov chains on discrete spaces this notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott sturm and villani for geodesic measure spaces in order to apply to the discrete setting the role of the wasserstein metric is taken over by a different metric having the property that continuous time markov chains are gradient flows of the entropy using this notion of ricci curvature we prove discrete analogues of fundamental results by bakry emery and otto villani further we show that ricci curvature bounds are preserved under tensorisation as a special case we obtain the sharp ricci curvature lower bound for the discrete hypercube

impossibility of local state transformation via hypercontractivity local state transformation is the problem of transforming an arbitrary number of copies of a bipartite resource state to a bipartite target state under local operations that is given two bipartite states is it possible to transform an arbitrary number of copies of one of them into one copy of the other state under local operations only this problem is a hard one in general since we assume that the number of copies of the resource state is arbitrarily large in this paper we prove some bounds on this problem using the hypercontractivity properties of some super operators corresponding to bipartite states we measure hypercontractivity in terms of both the usual super operator norms as well as completely bounded norms

exact recovery in the ising blockmodel we consider the problem associated to recovering the block structure of an ising model given independent observations on the binary hypercube this new model called the ising blockmodel is a perturbation of the mean field approximation of the ising model known as the curie weiss model the sites are partitioned into two blocks of equal size and the interaction between those of the same block is stronger than across blocks to account for more order within each block we study probabilistic statistical and computational aspects of this model in the high dimensional case when the number of sites may be much larger than the sample size

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

non commutative calculus optimal transport and functional inequalities in dissipative quantum systems we study dynamical optimal transport metrics between density matrices associated to symmetric dirichlet forms on finite dimensional c algebras our setting covers arbitrary skew derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for markov chains lindblad equations and the fermi ornstein uhlenbeck semigroup we develop a non nommutative differential calculus that allows us to obtain non commutative ricci curvature bounds logarithmic sobolev inequalities transport entropy inequalities and spectral gap estimates

reconstruction of markov random fields from samples some observations and algorithms markov random fields are used to model high dimensional distributions in a number of applied areas much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the markov random fields we analyze a simple algorithm for reconstructing the underlying graph defining a markov random field on n nodes and maximum degree d given observations we show that under mild nondegeneracy conditions it reconstructs the generating graph with high probability using theta d epsilon 2 delta 4 log n samples where epsilon delta depend on the local interactions for most local interactions epsilon delta are of order exp o d our results are optimal as a function of n up to a multiplicative constant depending on d and the strength of the local interactions our results seem to be the first results for general models that guarantee that the generating model is reconstructed furthermore we provide explicit o n d 2 epsilon 2 delta

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

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

contractivity of positive and trace preserving maps under lp norms we provide a complete picture of contractivity of trace preserving positive maps with respect to p norms we show that for p 1 contractivity holds in general if and only if the map is unital when the domain is restricted to the traceless subspace of hermitian matrices then contractivity is shown to hold in the case of qubits for arbitrary p 1 and in the case of qutrits if and only if p 1 in all noncontractive cases best possible bounds on the p norms are derived

the χ2 divergence and mixing times of quantum markov processes we introduce quantum versions of the 2 divergence provide a detailed analysis of their properties and apply them in the investigation of mixing times of quantum markov processes an approach similar to the one presented in the literature for classical markov chains is taken to bound the trace distance from the steady state of a quantum processes a strict spectral bound to the convergence rate can be given for time discrete as well as for time continuous quantum markov processes furthermore the contractive behavior of the 2 divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm in this context we analyze different versions of quantum detailed balance and finally give a geometric conductance bound to the convergence rate for unital quantum markov processes

the hopfield model on a sparse erdos renyi graph we analyze the storage capacity of the hopfield model on a sparse g n p random graph we show that it is proportional to pn in the entire regime where the corresponding random graph is asymptotically connected and for all value of c 0 03

invariant helical subspaces for the navier stokes equations three dimensional solutions with helical symmetry are shown to form an invariant subspace for the navier stokes equations uniqueness of weak helical solutions in the sense of leray is proved and these weak solutions are shown to be regular strong solutions existing for arbitrary time t the global universal attractor for the infinite dimensional dynamical system generated by the corresponding semi group of helical flows is shown to be compact and finite dimensional the hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class with the most detailed results obtained for rotating hagen poiseuille pipe flow in this case the dimension either hausdorff or fractal up to an absolute constant is bounded from above by frac operatorname re sqrt alpha 2 n 2 where is the axial wavenumber n is the azimuthal wavenumber and re is the reynolds number based on the radius of the pipe these upper bounds are independent of the rotation rate

noncommutative martingale deviation and poincare type inequalities with applications we prove a deviation inequality for noncommutative martingales by extending oliveira s argument for random matrices by integration we obtain a burkholder type inequality with satisfactory constant using continuous time we establish noncommutative poincare type inequalities for nice semigroups with a positive curvature condition these results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to bobkov and gotze in the commutative case to demonstrate our setting is general enough we give various examples including certain group von neumann algebras random matrices and classical diffusion processes among others

rigorous results on the bipartite mean field model we consider a bipartite mean field model in which interaction coefficients and magnetic fields depend only on the groups the particles belong to we rigorously compute the value of the limiting pressure per particle using tail estimation techniques we study the phase space of the model in the symmetric regime without an external field and when the interaction coefficients within the two groups are identical magnetic field perturbations are considered

a statistical mechanics approach to granovetter theory abstract in this paper we try to bridge breakthroughs in quantitative sociology econometrics pioneered during the last decades by mac fadden brock durlauf granovetter and watts strogatz by introducing a minimal model able to reproduce essentially all the features of social behavior highlighted by these authors our model relies on a pairwise hamiltonian for decision maker interactions which naturally extends the multi populations approaches by shifting and biasing the pattern definitions of a hopfield model of neural networks once introduced the model is investigated through graph theory to recover granovetter and watts strogatz results and statistical mechanics to recover mac fadden and brock durlauf results due to the internal symmetries of our model the latter is obtained as the relaxation of a proper markov process allowing even to study its out of equilibrium properties the method used to solve its equilibrium is an adaptation of the hamilton jacobi technique recently introduced by guerra in the spin glass scenario and the picture obtained is the following shifting the patterns from 1 1 0 1 implies that the larger the amount of similarities among decision makers the stronger their relative influence and this is enough to explain both the different role of strong and weak ties in the social network as well as its small world properties as a result imitative interaction strengths seem essentially a robust request enough to break the gauge symmetry in the couplings furthermore this naturally leads to a discrete choice modelization when dealing with the external influences and to imitative behavior a la curie weiss as the one introduced by brock and durlauf

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

dynamics of influence on hierarchical structures dichotomous spin dynamics on a pyramidal hierarchical structure the bethe lattice are studied the system embodies a number of classes where a class comprises nodes that are equidistant from the root head node weighted links exist between nodes from the same and different classes the spin hereafter state of the head node is fixed we solve for the dynamics of the system for different boundary conditions we find necessary conditions so that the classes eventually repudiate or acquiesce in the state imposed by the head node the results indicate that to reach unanimity across the hierarchy it suffices that the bottommost class adopts the same state as the head node then the rest of the hierarchy will inevitably comply this also sheds light on the importance of mass media as a means of synchronization between the topmost and bottommost classes surprisingly in the case of discord between the head node and the bottommost classes the average state over all nodes inclines towards that of the bottommost class regardless of the link weights and intraclass configurations hence the role of the bottommost class is signified

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

the limit of vanishing viscosity for the incompressible 3d navier stokes equations with helical symmetry abstract in this paper we are concerned with the vanishing viscosity problem for the three dimensional navier stokes equations with helical symmetry in the whole space we choose viscosity dependent initial u 0 with helical swirl an analogue of the swirl component of axisymmetric flow of magnitude o in the l 2 norm we assume u 0 u 0 in h 1 the new ingredient in our analysis is a decomposition of helical vector fields through which we obtain the required estimates

global regularity for some classes of large solutions to the navier stokes equations in previous works by the first two authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the main feature of the initial data considered in one of those studies is that it varies slowly in one direction though in some sense it is well prepared its norm is large but does not depend on the slow parameter the aim of this article is to generalize that setting to an ill prepared situation the norm blows up as the small parameter goes to zero as in those works the proof uses the special structure of the nonlinear term of the equation

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

quantum logarithmic sobolev inequalities and rapid mixing a family of logarithmic sobolev inequalities on finite dimensional quantum state spaces is introduced the framework of non commutative lp spaces is reviewed and the relationship between quantum logarithmic sobolev inequalities and the hypercontractivity of quantum semigroups is discussed this relationship is central for the derivation of lower bounds for the logarithmic sobolev ls constants essential results for the family of inequalities are proved and we show an upper bound to the generalized ls constant in terms of the spectral gap of the generator of the semigroup these inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups when the ls constant and the spectral gap are of the same order convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory we provide a number of examples where improved bounds on the mixing time of several semigroups are obtained incl

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

families of completely positive maps associated with monotone metrics abstract an operator convex function on 0 which satisfies the symmetry condition k x 1 x k x can be used to define a type of non commutative multiplication by a positive definite matrix or its inverse using the primitive concepts of left and right multiplication and the functional calculus the operators for the inverse can be used to define quadratic forms associated with riemannian metrics which contract under the action of completely positive trace preserving maps we study the question of when these operators define maps which are also completely positive cp although a d 1 2 a d 1 2 is the only case for which both the map and its inverse are cp there are several well known one parameter families for which either the map or its inverse is cp we present a complete analysis of the behavior of these families as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points our primary tool is an order relation based on the concept of positive definite functions although some results can be obtained from known properties we also prove new results based on the positivity of the fourier transforms of certain functions concrete computations of certain fourier transforms not only yield new examples of positive definite functions but also examples in the much stronger class of infinitely divisible functions

percolation on correlated random networks we consider a class of random weighted networks obtained through a redefinition of patterns in an hopfield like model and by performing percolation processes we get information about topology and resilience properties of the networks themselves given the weighted nature of the graphs different kinds of bond percolation can be studied stochastic deleting links randomly and deterministic deleting links based on rank weights each mimicking a different physical process the evolution of the network is accordingly different as evidenced by the behavior of the largest component size and of the distribution of cluster sizes in particular we can derive that weak ties are crucial in order to maintain the graph connected and that when they are the most prone to failure the giant component typically shrinks without abruptly breaking apart these results have been recently evidenced in several kinds of social networks

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

bistable monostable transition in the ising model on two connected complex networks in this paper we investigate the behavior of the ising model on two sparsely connected complex networks the networks have the topology of a random graph or barabasi and albert scale free networks we extend our previous analysis and show that a bistable monostable phase transition occurs in such systems during this transition the magnetization undergoes a discontinuous jump we calculate the critical temperature analytically for regular random graphs and study a more general case using an iterative map corresponding to mean field dynamics the calculations are confirmed by numeric simulations based on monte carlo approach

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

relative entropy convergence for depolarizing channels we study the convergence of states under continuous time depolarizing channels with full rank fixed points in terms of the relative entropy the optimal exponent of an upper bound on the relative entropy in this case is given by the log sobolev 1 constant our main result is the computation of this constant as an application we use the log sobolev 1 constant of the depolarizing channels to improve the concavity inequality of the von neumann entropy this result is compared to similar bounds obtained recently by kim and we show a version of pinsker s inequality which is optimal and tight if we fix the second argument of the relative entropy finally we consider the log sobolev 1 constant of tensor powers of the completely depolarizing channel and use a quantum version of shearer s inequality to prove a uniform lower bound

large global solutions to the navier stokes equations slowly varying in one direction in to previous papers by the authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions in the whole space contrary to the previous examples the initial data has no particular oscillatory properties but varies slowly in one direction the proof uses the special structure of the nonlinear term of the equation

majority model on a network with communities we focus on the majority model in a topology consisting of two coupled fully connected networks thereby mimicking the existence of communities in social networks we show that a transition takes place at a value of the interconnectivity parameter above this value only symmetric solutions prevail where both communities agree with each other and reach consensus below this value in contrast the communities can reach opposite opinions and an asymmetric state is attained the importance of the interface between the subnetworks is shown

hypercontractivity for a quantum ornstein uhlenbeck semigroup we prove hypercontractivity for a quantum ornstein uhlenbeck semigroup on the entire algebra mathcal b h of bounded operators on a separable hilbert space h we exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain then we deduce a logarithmic sobolev inequality for the semigroup and gain an elementary estimate of the best constant

hypercontractivity and the logarithmic sobolev inequality for the completely bounded norm we develop the notions of hypercontractivity hc and the log sobolev ls inequality for completely bounded norms of one parameter semigroups of super operators acting on matrix algebras we prove the equivalence of the completely bounded versions of hc and ls under suitable hypotheses we also prove a version of the gross lemma which allows ls at general q to be deduced from ls at q 2

multivariate normal approximation with stein s method of exchangeable pairs under a general linearity condition in this paper we establish a multivariate exchangeable pairs approach within the framework of stein s method to assess distributional distances to potentially singular multivariate normal distributions by extending the statistics into a higher dimensional space we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to stein s exchangeable pairs approach to illustrate the method we provide the examples of runs on the line as well as double indexed permutation statistics

hypercontractivity in noncommutative lpspaces we introduce and study a setup for hypercontractivity in the interpolating family of noncommutative lpspaces associated to a locally normal state on an inductive limitc algebra

on the regularity of the axisymmetric solutions of the navier stokes equations we obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional navier stokes equations with nonzero swirl in particular we prove that the integrability of single component of vorticity or velocity fields in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions to obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the navier stokes equations

on semidefinite relaxations for the block model the stochastic block model sbm is a popular tool for community detection in networks but fitting it by maximum likelihood mle involves a computationally infeasible optimization problem we propose a new semidefinite programming sdp solution to the problem of fitting the sbm derived as a relaxation of the mle we put ours and previously proposed sdps in a unified framework as relaxations of the mle over various subclasses of the sbm which also reveals a connection to the well known problem of sparse pca our main relaxation which we call sdp 1 is tighter than other recently proposed sdp relaxations and thus previously established theoretical guarantees carry over however we show that sdp 1 exactly recovers true communities over a wider class of sbms than those covered by current results in particular the assumption of strong assortativity of the sbm implicit in consistency conditions for previously proposed sdps can be relaxed to weak assortativity for our approach thus significantly broadening the class of sbms covered by the consistency results we also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed sdp approaches and not an artifact of the proofs our analysis of sdps is based on primal dual witness constructions which provides some insight into the nature of the solutions of various sdps in particular we show how to combine features from sdp 1 and already available sdps to achieve the most flexibility in terms of both assortativity and block size constraints as our relaxation has the tendency to produce communities of similar sizes this tendency makes it the ideal tool for fitting network histograms a method gaining popularity in the graphon estimation literature as we illustrate on an example of a social networks of dolphins we also provide empirical evidence that sdps outperform spectral methods for fitting sbms with a large number of blocks

multi group binary choice with social interaction and a random communication structure a random graph approach abstract we construct and analyze a random graph model for discrete choice with social interaction and several groups of equal size we concentrate on the case of two groups of equal sizes and we allow the interaction strength within a group to differ from the interaction strength between the two groups given that the resulting graph is sufficiently dense we show that with probability 1 the average decision in each of the two groups is the same as in the fully connected model in particular we show that there is a phase transition if the interaction among a group and between the groups is strong enough the average decision per group will either be positive or negative and the decision of the two groups will be correlated we also compute the free energy per particle in our model

the fluctuations of the overlap in the hopfield model with finitely many patterns at the critical temperature we investigate the limiting fluctuations of the order parameter in the hopfield model of spin glasses and neural networks with finitely many patterns at the critical temperature 1 c 1 at the critical temperature the measure valued random variables given by the distribution of the appropriately scaled order parameter under the gibbs measure converge weakly towards a random measure which is non gaussian in the sense that it is not given by a dirac measure concentrated in a gaussian distribution this remains true in the case of n c 1 as n provided n converges to c 1 fast enough i e at speed 1 the limiting distribution is explicitly given by its random density

quantum reverse hypercontractivity we develop reverse versions of hypercontractive inequalities for quantum channels by generalizing classical techniques we prove a reverse hypercontractive inequality for tensor products of qubit depolarizing channels we apply this to obtain a rapid mixing result for depolarizing noise applied to large subspaces and to prove bounds on a quantum generalization of non interactive correlation distillation

helically symmetric solutions to the 3 d navier stokes equations for compressible isentropic fluids abstract we prove the existence of global weak solutions to the navier stokes equations for compressible isentropic fluids for any 1 when the cauchy data are helically symmetric where the constant is the specific heat ratio moreover a new integrability estimate of the density in any neighborhood of the symmetry axis the singularity axis is obtained

fluctuation results for general block spin ising models we study a block spin mean field ising model i e a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices and where pair interactions are given according to their blocks for the vector of block magnetizations we prove large deviation principles and central limit theorems under general assumptions for the block interaction matrix using the exchangeable pair approach of stein s method we establish a rate of convergence in the central limit theorem for the block magnetization vector in the high temperature regime

inverse problem robustness for multi species mean field spin models the inverse problem method is tested for a class of mean field statistical mechanics models representing a mixture of particles of different species the robustness of the inversion is investigated for different values of the physical parameters systemsizesandindependentsamples weshowhowtoreconstruct the parameter values with a precision of a few per cent

global solutions of the navier stokes equations in thin three dimensional domains this paper is concerned with the spatially periodic navier stokes equations in a thin three dimensional domain it is shown that the classical solution exists globally when the initial velocityu0and the external forcefare chosen in the set significantly larger than those presented in earlier works what is more it is shown that these global solutions which seem large in a function space are small in nature

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

global existence of weak solutions to the three dimensional euler equations with helical symmetry abstract in this paper we mainly investigate the weak solutions of the three dimensional incompressible euler equations with helical symmetry in the whole space when the helical swirl vanishes specifically we establish the global existence of weak solutions when the initial vorticity lies in l 1 l p with p 1 our result extends the previous work 2 where the initial vorticity is compactly supported and belongs to l p with p 4 3 the key ingredient in this paper involves the explicit analysis of biot savart law with helical symmetry in domain r 2 via the theories of singular integral operators and second order elliptic equations

hypercontractivity of quasi free quantum semigroups hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate a logarithmic sobolev inequality and the corresponding logarithmic sobolev constant can be inferred from the semigroup s hypercontractive norm bound we consider completely positive quantum mechanical semigroups described by a lindblad master equation to prove the norm bound we follow an approach which has its roots in the study of classical rate equations we use interpolation theorems for non commutative spaces to obtain a general hypercontractive inequality from a particular norm bound then we derive a bound on the norm from an analysis of the block diagonal structure of the semigroup s spectrum we show that the dynamics of an n qubit graph state hamiltonian weakly coupled to a thermal environment is hypercontractive as a consequence this allows for the efficient preparation of graph states in time by coupling at sufficiently low temperature furthermore we extend our results to gapped liouvillians arising from a weak linear coupling of a free fermion systems

resonant triad interactions in symmetric systems abstract we analyse resonant triad interactions in symmetric systems the case of coupling at quadratic order in amplitudes and the case of coupling at cubic order are considered we study modulated travelling waves and heteroclinic cycles in amplitude equations and we discuss their stability types we describe possible dynamical regimes and spatial patterns arising from triad interactions of long waves in rapidly rotating hagen poiseuille flow

organization and evolution of synthetic idiotypic networks we introduce a class of weighted graphs whose properties are meant to mimic the topological features of idiotypic networks namely the interaction networks involving the b core of the immune system each node is endowed with a bit string representing the idiotypic specificity of the corresponding b cell and the proper distance between any couple of bit strings provides the coupling strength between the two nodes we show that a biased distribution of the entries in bit strings can yield fringes in the weighted degree distribution small world features and scaling laws in agreement with experimental findings we also investigate the role of aging thought of as a progressive increase in the degree of bias in bit strings and we show that it can possibly induce mild percolation phenomena which are investigated too

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

on blow up space jets for the navier stokes equations in r3 with convergence to euler equations formation of blow up singularities for the navier stokes equations ut u u p u and div u 0 in r3 0 t is studied in cylindrical polar coordinates r z in r3 their restriction to the linear subspace w2 span 1 z is shown to be consistent using links with blow up theory for nonlinear reaction diffusion partial differential equations the following questions are under scrutiny ii introducing a self similar blow up swirl mechanism with the angular swirl divergences t ln t t and t t t as t t where r is a parameter ii existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional global blow up of the z component of the velocity iii as an intrinsic part of the construction convergence of the above rescaled patterns as t t to smooth blow up self similar solutions of the corresponding three dimensional euler equations ut u u p and div u 0 in r3 0 t which iv are also shown to admit single point blow up in

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

on the global regularity of a helical decimated version of the 3d navier stokes equations we study the global regularity for all time and all initial data in h 1 2 o f ar e cently introduced decimated version of the incompressible 3d navier stokes dns equa tions the model is based on a projection of the dynamical evolution of navier stokes ns equations into the subspace where helicity the l 2 scalar product of velocity and vorticity is sign definite the presence of a second beside energy sign definite inviscid conserved quadratic quantity which is equivalent to the h 1 2 sobolev norm allows us to demonstrate global existence and uniqueness of space periodic solutions together with continuity with respect to the initial conditions for this decimated 3d model this is achieved thanks to the establishment of two new estimates for this 3d model which show that the h 1 2 and the time average of the square of the h 3 2 norms of the velocity field remain finite such two additional bounds are known in the spirit of the work of h fujita and t kato arch ration mech anal 16 269 315 1964 rend semin mat univ padova 32 243 260 1962 to be sufficient for showing well posedness for the 3d ns equations furthermore they are di rectly linked to the helicity evolution for the dns model and therefore with a clear physical meaning and consequences

scaling limits for multi species statistical mechanics mean field models we study the limiting thermodynamic behavior of the normalized sums of spins in multi species curie weiss models we find sufficient conditions for the limiting random vector to be gaussian or to have an exponential distribution of higher order and compute the covariance matrix in terms of model parameters

rescaled magnetization for critical bipartite curie weiss models we consider a bipartite generalization of the curie weiss model in a critical regime in order to study the asymptotic behavior of the random vector of the total magnetization we apply the change of variables that diagonalizes the hessian matrix of the pressure functional associated to the model we obtain a new vector that suitably rescaled weakly converges to the product of a gaussian distribution and a distribution proportional to exp xi x 4 where the positive constant xi can be computed from the pressure functional

the thermodynamics of the curie weiss model with random couplings we study the curie weiss version of an ising spin system with random positively biased couplings in particular the case where the couplings ij take the values one with probabilityp and zero with probability 1 p which describes the ising model on a random graph is considered we prove that ifp is allowed to decrease with the system sizen in such a way thatnp n asn then the free energy converges after trivial rescaling to that of the standard curie weiss model almost surely similarly the induced measures on the mean magnetizations converge to those of the curie weiss model generalizations of this result to a wide class of distributions are detailed

effects of mixing in threshold models of social behavior we consider the dynamics of an extension of the influential granovetter model of social behavior where individuals are affected by their personal preferences and observation of the neighbors behavior individuals are arranged in a network usually the square lattice and each has a state and a fixed threshold for behavior changes we simulate the system asynchronously by picking a random individual and we either update its state or exchange it with another randomly chosen individual mixing we describe the dynamics analytically in the fast mixing limit by using the mean field approximation and investigate it mainly numerically in the case of finite mixing we show that the dynamics converge to a manifold in state space which determines the possible equilibria and show how to estimate the projection of this manifold by using simulated trajectories emitted from different initial points we show that the effects of considering the network can be decomposed into finite neighborhood effects and finite mixing rate effects which have qualitatively similar effects both of these effects increase the tendency of the system to move from a less desired equilibrium to the ground state our findings can be used to probe shifts in behavioral norms and have implications for the role of information flow in determining when social norms that have become unpopular in particular communities such as foot binding or female genital cutting persist or vanish

non commutative nash inequalities a set of functional inequalities called nash inequalities are introduced and analyzed in the context of quantum markov process mixing the basic theory of nash inequalities is extended to the setting of non commutative l p spaces where their relationship to poincare and log sobolev inequalities is fleshed out we prove nash inequalities for a number of unital reversible semigroups

two quantum analogues of fisher information from a large deviation viewpoint of quantum estimation we discuss two quantum analogues of the fisher information the symmetric logarithmic derivative fisher information and kubo mori bogoljubov fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators as another comparison it is shown that the difference between them is characterized by the change of the order of limits

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

large deviation estimates for a conditional probability distribution applications to random interaction gibbs measures let x i y i d be independent identically distributed random variables with arbitrary distribution we show that for almost every y i i the conditional law of the empirical field given y i i satisfies to large deviation inequalities this applies to the study of gibbs measures with random interaction in the case of some mean field models as well as of short range summable interaction we show that the pressure is nonrandom and is given by a variational formula these random gibbs measures have the same large deviation rate which does not depend on the particular realization of the interaction their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula

a conditional curie weiss model for stylized multi group binary choice with social interaction this paper proposes a conditional curie weiss model as a model for decision making in a stylized society made up of binary decision makers that face a particular dichotomous choice between two options following brock and durlauf discrete choice with social interaction i theory 1955 we set up both socio economic and statistical mechanical models for the choice problem we point out when both the socio economic and statistical mechanical models give rise to the same self consistent equilibrium mean choice level s phase diagram of the associated statistical mechanical model and its socio economic implications are discussed