flow invariance preserving feedback controllers for the navier stokes equation abstract in this paper we develop a concrete procedure for designing feedback controllers to ensure that the resultant dynamics of turbulence will preserve certain prescribed physical constraints examples of such constraints include in particular the level sets of well known invariants of the inviscid flow such as helicity we also bring to light a certain m accretivity property of suitable quantization of the nonlinearity in the navier stokes equation and utilize the theory of nonlinear semigroups to resolve the controlled navier stokes inclusion with a multi valued feedback term

a new entropy power inequality for integer valued random variables the entropy power inequality epi yields lower bounds on the differential entropy of the sum of two independent real valued random variables in terms of the individual entropies versions of the epi for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy but no universal inequality is known beyond trivial ones more recently the sumset theory for the entropy function yields a sharp inequality h x x prime h x geq 1 over 2 o 1 when x x prime are independent identically distributed i i d with high entropy this paper provides the inequality h x x prime h x geq g h x where x x prime are arbitrary i i d integer valued random variables and where g is a universal strictly positive function on bbr satisfying g 0 0 extensions to nonidentically distributed random variables and to conditional entropies are also obtained

monotonic convergence in an information theoretic law of small numbers an entropy increasing to the maximum result analogous to the entropic central limit theorem barron 1986 artstein 2004 is obtained in the discrete setting this involves the thinning operation and a poisson limit monotonic convergence in relative entropy is established for general discrete distributions while monotonic increase of shannon entropy is proved for the special class of ultra log concave distributions overall we extend the parallel between the information theoretic central limit theorem and law of small numbers explored by kontoyiannis 2005 and harremoeumls 2007 2008 2009 ingredients in the proofs include convexity majorization and stochastic orders

a penalized neumann control approach for solving an optimal dirichlet control problem for the navier stokes equations we introduce a penalized neumann boundary control approach for solving an optimal dirichlet boundary control problem associated with the two or three dimensional steady state navier stokes equations we prove the convergence of the solutions of the penalized neumann control problem the suboptimality of the limit and the optimality of the limit under further restrictions on the data we describe the numerical algorithm for solving the penalized neumann control problem and report some numerical results

expectation propagation detection for high order high dimensional mimo systems modern communications systems use multiple input multiple output mimo and high order qam constellations for maximizing spectral efficiency however as the number of antennas and the order of the constellation grow the design of efficient and low complexity mimo receivers possesses big technical challenges for example symbol detection can no longer rely on maximum likelihood detection or sphere decoding methods as their complexity increases exponentially with the number of transmitters receivers in this paper we propose a low complexity high accuracy mimo symbol detector based on the expectation propagation ep algorithm ep allows approximating iteratively at polynomial time the posterior distribution of the transmitted symbols we also show that our ep mimo detector outperforms classic and state of the art solutions reducing the symbol error rate at a reduced computational complexity

entropy bounds on abelian groups and the ruzsa divergence over the past few years a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by ruzsa tao and others motivated by analogous results in additive combinatorics this paper extends these earlier results to the case of random variables taking values in mathbb r n or more generally in arbitrary locally compact and polish abelian groups we isolate and study a key quantity the ruzsa divergence between two probability distributions and we show that its properties can be used to extend the earlier inequalities to the present general setting the new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other although the setting is quite general the results are already of interest and new for random vectors in mathbb r n in that special case we discuss quantitative bounds for the stability of the equality conditions in the entropy power inequality a reverse entropy power inequality for log concave random vectors an information theoretic analog of the rogers shephard inequality for convex bodies and consequences of some of our results to determinant inequalities for sums of positive definite matrices moreover by considering various multiplicative subgroups of the complex plane one obtains new inequalities for the differential entropies of products and ratios of nonzero complex valued random variables

a binary analog to the entropy power inequality let x sub n y sub n be independent stationary binary random sequences with entropy h x h y respectively let h zeta zeta log zeta 1 zeta log 1 zeta 0 or sigma x sigma y where sigma z h sup 1 h z and alpha beta alpha 1 beta beta 1 alpha when y sub n are independent identically distributed this reduces to mrs gerber s lemma from a d wyner and j ziv 1973

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

sumis near optimal soft in soft out mimo detection with low and fixed complexity the fundamental problem of interest here is soft input soft output multiple input multiple output mimo detection we propose a method referred to as subspace marginalization with interference suppression sumis that yields unprecedented performance at low and fixed deterministic complexity our method provides a well defined tradeoff between computational complexity and performance apart from an initial sorting step consisting of selecting channel matrix columns the algorithm involves no searching nor algorithmic branching hence the algorithm has a completely predictable run time and allows for a highly parallel implementation we numerically assess the performance of sumis in different practical settings full partial channel state information sequential iterative decoding and low high rate outer codes we also comment on how the sumis method performs in systems with a large number of transmit antennas

gaussian optimizers for entropic inequalities in quantum information we survey the state of the art for the proof of the quantum gaussian optimizer conjectures of quantum information theory these fundamental conjectures state that quantum gaussian input states are the solution to several optimization problems involving quantum gaussian channels these problems are the quantum counterpart of three fundamental results of functional analysis and probability the entropy power inequality the sharp young s inequality for convolutions and the theorem gaussian kernels have only gaussian maximizers quantum gaussian channels play a key role in quantum communication theory they are the quantum counterpart of gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime the quantum gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space the restriction of the quantum limited gaussian attenuator to input states diagonal in the fock basis coincide

performance analysis of maximum likelihood detection in a mimo antenna system we provide an analysis of the performance of maximum likelihood detection mld over flat fading channels in a wireless multiple input multiple output mimo antenna system a tight union bound with an asymptotic form on the probability of symbol error rate ser for mimo mld systems with two dimensional signal constellations such as qam and psk is introduced using this analytic bound the performance of the mimo antenna system is demonstrated quantitatively with respect to channel estimation constellation size and antenna configuration

layered tabu search algorithm for large mimo detection and a lower bound on ml performance in this letter we are concerned with low complexity detection in large multiple input multiple output mimo systems with tens of transmit receive antennas our new contributions in this letter are two fold first we propose a low complexity algorithm for large mimo detection based on a layered low complexity local neighborhood search second we obtain a lower bound on the maximum likelihood ml bit error performance using the local neighborhood search the advantages of the proposed ml lower bound are i it is easily obtained for mimo systems with large number of antennas because of the inherent low complexity of the search algorithm ii it is tight at moderate to high snrs and iii it can be tightened at low snrs by increasing the number of symbols in the neighborhood definition the proposed detection algorithm based on the layered local neighborhood search achieves bit error performances which are quite close to this lower bound for large number of antennas and higher order qam

existence of optimal controls for viscous flow problems a class of optimal control problems in viscous flow is studied main result is the existence theorem for optimal control three typical flow control problems are formulated within this general class

towards a theory of negative dependence the fkg theorem says that the positive lattice condition an easily checkable hypothesis which holds for many natural families of events implies positive association a very useful property thus there is a natural and useful theory of positively dependent events there is as yet no corresponding theory of negatively dependent events there is however a need for such a theory this paper unfortunately contains no substantial theorems its purpose is to present examples that motivate a need for such a theory give plausibility arguments for the existence of such a theory outline a few possible directions such a theory might take and state a number of specific conjectures which pertain to the examples and to a wish list of theorems

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

h control theory of fluid dynamics robust or h control theory has been an expanding subject in recent years because of its wide applications and its close connection to operator theory on hardy spaces we develop an hcontrol theor

on the acceleration of viscous fluid through an unbounded channel viscous incompressible time dependent flow through a two dimensional channel of finite cross section is considered the well posed theorem for this problem has remained open in the past because flux carrying velocity fields possess infinite energy we resolve this issue by constructing a suitable infinite energy vector field and seeking the solution of the navier stokes problem as a finite energy perturbation of this vector field

an optimal control problem in exterior hydrodynamics in this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure an existence theorem for the speed trajectory which corresponds to the absolute minimum is provided the results are valid for arbitrary reynolds numbers

a simple way to compute the minimum distance in multiuser cdma systems a simple method to compute the minimum distance d sub min in multiuser code division multiple access cdma systems based on the cholesky decomposition of the positive definite symmetric correlation matrix is proposed although finding d sub min is known to be np hard this decomposition allows the computation of the minimum distance and the asymptotic efficiency of optimum multiuser detection to be performed very efficiently in almost all cases of practical interest numerical results for synchronous cdma with binary random signature waveforms of length 31 are used to illustrate the method

monotonicity thinning and discrete versions of the entropy power inequality we consider the entropy of sums of independent discrete random variables in analogy with shannon s entropy power inequality where equality holds for normals in our case infinite divisibility suggests that equality should hold for poisson variables we show that some natural analogues of the epi do not in fact hold but propose an alternative formulation which does always hold the key to many proofs of shannon s epi is the behavior of entropy on scaling of continuous random variables we believe that renyi s operation of thinning discrete random variables plays a similar role to scaling and give a sharp bound on how the entropy of ultra log concave random variables behaves on thinning in the spirit of the monotonicity results established by artstein ball barthe and naor we prove a stronger version of concavity of entropy which implies a strengthened form of our discrete epi

constrained maximum likelihood detection in cdma the detection strategy usually denoted optimal multiuser detection is equivalent to the solution of a 0 1 constrained maximum likelihood ml problem a problem which is known to be np hard in contrast the unconstrained ml problem can be solved quite easily and is known as the decorrelating detector in this paper we consider the constrained ml problem where the solution vector is restricted to lie within a closed convex set ccs such a design criterion leads to detector structures which are ml under the constraint assumption a close relationship between a sphere constrained ml detector and the well known minimum mean square error detector is found and verified an iterative algorithm for solving a ccs constraint problem is derived based on results in linear variational inequality theory special cases of this algorithm subject to a box constraint are found to correspond to known nonlinear successive and parallel interference cancellation structures using a clipped soft decision for making tentative decisions while a weighted linear parallel interference canceler with signal dependent weights arises from the sphere constraint convergence issues are investigated and an efficient implementation is suggested the bit error rate performance is studied via computer simulations and the expected performance improvements over unconstrained ml are verified

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

a new modified logarithmic sobolev inequality for poisson point processes and several applications by means of the martingale representation we establish a new modified logarithmic sobolev inequality which covers the previous modified logarithmic sobolev inequalities of bobkov ledoux and the l 1 logarithmic sobolev inequality obtained in our previous work from it we derive several sharp deviation inequalities of talagrand s type by following the powerful herbst method developed recently by ledoux and al moreover this new modified logarithmic sobolev inequality is transported on the discontinuous path space with respect to the law of a levy process

a simple proof of the entropy power inequality this correspondence gives a simple proof of shannon s entropy power inequality epi using the relationship between mutual information and minimum mean square error mmse in gaussian channels

monotonic decrease of the non gaussianness of the sum of independent random variables a simple proof artstein ball barthe and naor have recently shown that the non gaussianness divergence with respect to a gaussian random variable with identical first and second moments of the sum of independent and identically distributed i i d random variables is monotonically nonincreasing we give a simplified proof using the relationship between non gaussianness and minimum mean square error mmse in gaussian channels as artstein we also deal with the more general setting of nonidentically distributed random variables

entropy power inequalities for qudits shannon s entropy power inequality epi can be viewed as a statement of concavity of an entropic function of a continuous random variable under a scaled addition rule f a x 1 a y af x 1 a f y a 0 1 here x and y are continuous random variables and the function f is either the differential entropy or the entropy power konig and smith ieee trans inf theory 60 3 1536 1548 2014 and de palma mari and giovannetti nat photonics 8 12 958 964 2014 obtained quantum analogues of these inequalities for continuous variable quantum systems where x and y are replaced by bosonic fields and the addition rule is the action of a beam splitter with transmissivity a on those fields in this paper we similarly establish a class of epi analogues for d level quantum systems i e qudits the underlying addition rule for which these inequalities hold is given by a quantum channel that depends on the parameter a 0 1 and acts like a finite dimensional analogue of a beam splitter with transmissiv

the entropy power inequality and mrs gerber s lemma for groups of order 2 n shannon s entropy power inequality can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies the entropy power inequality has played a key role in resolving a number of problems in information theory it is therefore interesting to examine the existence of a similar inequality for discrete random variables in this paper we obtain an entropy power inequality for random variables taking values in a group of order 2 n i e for such a group g we explicitly characterize the function f g x y giving the minimum entropy of the sum of two independent g valued random variables with respective entropies x and y random variables achieving the extremum in this inequality are thus the analogs of gaussians in this case and these are also determined it turns out that f g x y is convex in x for fixed y and by symmetry convex in y for fixed x this is a generalization to groups of order 2 n of the result known as mrs gerber s lemma

optimal control of thermally convected fluid flows we examine the optimal control of stationary thermally convected fluid flows from the theoretical and numerical point of view we use thermal convection as control mechanism that is control is effected through the temperature on part of the boundary control problems are formulated as constrained minimization problems existence of optimal control is given and a first order necessary condition of optimality from which optimal solutions can be obtained is established we develop numerical methods to solve the necessary condition of optimality and present computational results for control of cavity and channel type flows showing the feasibility of the proposed approach

dynamic programming of the navier stokes equations a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation

boundary value problems and optimal boundary control for the navier stokes system the two dimensional case we study optimal boundary control problems for the two dimensional navier stokes equations in an unbounded domain control is effected through the dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates an objective of interest is the drag functional we first establish three important results for inhomogeneous boundary value problems for the navier stokes equations namely we identify the trace space for the velocity fields possessing finite energy we prove the existence of a solution for the navier stokes equations with boundary data belonging to the trace space and we identify the space in which the stress vector along the boundary of admissible solutions is well defined then we prove the existence of an optimal solution over the control set finally we justify the use of lagrange multiplier principles derive an optimality system of equations in the weak sense from which optimal states and controls may be determined and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values

on renyi entropy power inequalities this paper gives improved renyi entropy power inequalities r epis consider a sum s n sum k 1 n x k of n independent continuous random vectors taking values on mathbb r d and let alpha in 1 infty an r epi provides a lower bound on the order alpha renyi entropy power of s n that up to a multiplicative constant which may depend in general on n alpha d is equal to the sum of the order alpha renyi entropy powers of the n random vectors x k k 1 n for alpha 1 the r epi coincides with the well known entropy power inequality by shannon the first improved r epi is obtained by tightening the recent r epi by bobkov and chistyakov which relies on the sharpened young s inequality a further improvement of the r epi also relies on convex optimization and results on rank one modification of a real valued diagonal matrix

thinning entropy and the law of thin numbers renyi s thinning operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables the properties of thinning are investigated in an information theoretic context especially in connection with information theoretic inequalities related to poisson approximation results the classical binomial to poisson convergence sometimes referred to as the law of small numbers is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions a rate of convergence is provided for this limit and nonasymptotic bounds are also established this development parallels in part the development of gaussian inequalities leading to the information theoretic version of the central limit theorem in particular a thinning markov chain is introduced and it is shown to play a role analogous to that of the ornstein uhlenbeck process in connection to the entropy power inequality