multi element least square hdmr methods and their applications for stochastic multiscale model reduction stochastic multiscale modeling has become a necessary approach to quantify uncertainty and characterize multiscale phenomena for many practical problems such as flows in stochastic porous media the numerical treatment of the stochastic multiscale models can be very challengeable as the existence of complex uncertainty and multiple physical scales in the models to efficiently take care of the difficulty we construct a computational reduced model to this end we propose a multi element least square high dimensional model representation hdmr method through which the random domain is adaptively decomposed into a few subdomains and a local least square hdmr is constructed in each subdomain these local hdmrs are represented by a finite number of orthogonal basis functions defined in low dimensional random spaces the coefficients in the local hdmrs are determined using least square methods we paste all the local hdmr approximations together to form a global hdmr approximation to further reduce computational cost we present a multi element reduced least square hdmr which improves both efficiency and approximation accuracy in certain conditions to effectively treat heterogeneity properties and multiscale features in the models we integrate multiscale finite element methods with multi element least square hdmr for stochastic multiscale model reduction this approach significantly reduces the original model s complexity in both the resolution of the physical space and the high dimensional stochastic space we analyze the proposed approach and provide a set of numerical experiments to demonstrate the performance of the presented model reduction techniques multi element least square hdmr is proposed to treat stochastic models random domain is adaptively decomposed into some subdomains to obtain adaptive multi element hdmr least square reduced hdmr is proposed to enhance computation efficiency and approximation accuracy in certain conditions integrating msfem and multi element least square hdmr can significantly reduce computation complexity

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

on decoding concatenated codes the correcting properties of concatenated codes with parallel decoding over an additive channel are investigated the ith inner decoder s output is a codeword if the euclidean distance between the received vector and some codeword is less than delta sub i and an erasure otherwise the outer decoders correct errors and erasures the error correcting capability which is taken to be the minimum length of any noise vector that can cause an error is obtained for a bank of z inner and outer decoders as a function of the thresholds used the set of thresholds that maximize the error correcting capability is also found it is shown that for a small number of branches the error correcting capability is nearly as large as any decoder

free holomorphic functions on the unit ball of b h n abstract in this paper we continue the study of free holomorphic functions on the noncommutative ball b h n 1 x 1 x n b h n x 1 x 1 x n x n 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h and n 1 2 or n several classical results from complex analysis have free analogues in our noncommutative setting we prove a maximum principle a naimark type representation theorem and a vitali convergence theorem for free holomorphic functions with operator valued coefficients we introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by schwarz and harnack the borel caratheodory theorem is extended to our noncommutative setting using a noncommutative generalization of schwarz s lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball b h n 1 we obtain an analogue of julia s lemma for free holomorphic functions f b h n 1 b h m 1 we also obtain pick julia theorems for free holomorphic functions with operator valued coefficients we provide a noncommutative generalization of a classical inequality due to lindelof which turns out to be sharper then the noncommutative von neumann inequality finally we introduce a pseudohyperbolic metric on b h n 1 which is invariant under the action of the free holomorphic automorphism group of b h n 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on b n the open unit ball of c n in this setting we obtain a schwarz pick type lemma we also provide commutative versions of these results for operator valued multipliers of the drury arveson space

operator valued chordal loewner chains and non commutative probability abstract we adapt the theory of chordal loewner chains to the operator valued matricial upper half plane over a c algebra a we define an a valued chordal loewner chain as a subordination chain of analytic self maps of the a valued upper half plane such that each f t is the reciprocal cauchy transform of an a valued law t such that the mean and variance of t are continuous functions of t we relate a valued loewner chains to processes with a valued free or monotone independent increments just as was done in the scalar case by bauer 8 and schleisinger 48 we show that the loewner equation t f t z d f t z v t z when interpreted in a certain distributional sense defines a bijection between lipschitz mean zero loewner chains f t and vector fields v t z of the form v t z g t z where t is a generalized a valued law based on the loewner equation we derive a combinatorial expression for the moments of t in terms of t we also construct non commutative random variables on an operator valued monotone fock space which realize the laws t finally we prove a version of the monotone central limit theorem which describes the behavior of f t as t when t has uniformly bounded support

low complexity erasure insertion in rs coded sfh spread spectrum communications with partial band interference and nakagami m fading we propose two novel low complexity low delay erasure insertion schemes namely the output threshold test ott and joint maximum output and ratio threshold test mo rtt the employment of the ott and mo rtt is beneficial in the context of the errors and erasures reed solomon rs decoding in a slow frequency hopping spread spectrum sfh ss system using m ary frequency shift keying mfsk the statistics of the erasure insertion related decision variables associated with the ott mo rtt as well as with the ratio threshold test rtt are investigated when the channel of each frequency hopping fh slot is modeled as flat nakagami m fading the transmitted signals also experience both additive white gaussian noise awgn as well as partial band gaussian interference pbgi the properties of these erasure insertion schemes are investigated with the aid of their statistics the performance of the proposed erasure insertion schemes and that of the erasure insertion scheme using the rtt is investigated and compared in the context of rs coded sfh ss systems using mfsk furthermore the performance of the rs coded sfh ss systems is also compared both with and without side information concerning the pbgi

a positivstellensatz for non commutative polynomials a non commutative polynomial which is positive on a bounded semi algebraic set of operators has a weighted sum of squares representation this positivstellensatz parallels similar results in the commutative case a broader issue is to what extent does real semi algebraic geometry extend to non commutative polynomials our strict positivstellensatz is positive news on the opposite extreme from strict positivity would be a real nullstellensatz we give an example which shows that there is no non commutative real nullstellensatz along certain lines however we include a successful type of non commutative nullstellensatz proved by george bergman

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

singularities of rational functions and minimal factorizations the noncommutative and the commutative setting we show that the singularities of a matrix valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization the proof uses a new notion of noncommutative backward shifts as an application we establish the commutative counterpart of the singularities theorem the singularities of a matrix valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its fornasini marchesini realizations with the minimal possible state space dimension the singularities results imply the absence of zero pole cancellations in a minimal factorization both in the noncommutative and in the commutative setting

toeplitz corona and the douglas property for free functions abstract the well known douglas lemma says that for operators a b on hilbert space that a a b b 0 implies b a c for some contraction operator c the result carries over directly to classical operator valued toeplitz operators simply by replacement of operator by toeplitz operator throughout free functions generalize the notion of free polynomials and formal power series and trace back to the work of j taylor in the 1970s they are of current interest in part because of their connections with free probability and engineering systems theory in this article for given free functions a and b with noncommutative domain k defined by free polynomial inequalities we obtain a sufficient condition in terms of the difference a a b b for the existence of a free function c taking contractive values on k such that b a c the connection to recent work of agler and mccarthy and their free toeplitz corona theorem is expounded

clipped diversity combining for channels with partial band interference ii ratio statistic combining for pt i see ibid vol com 3 no 12 p 1320 1987 ratio statistic combining is proposed for mitigating partial band interference in systems with diversity transmission and frequency hop signaling systems with noncoherent demodulation and binary orthogonal signaling are covered the partial band interference is gaussian and gaussian quiescent noise is included in the analysis to account for wideband noise sources the exact probability of error is found for a receiver using ratio statistic combining and this is compared to the exact error probabilities for receivers with optimum combining with perfect side information clipped linear combining the ratio threshold test with majority logic decoding and self normalization diversity combining numerical results are also given for a frequency hop system which uses ratio statistic combining for channels with rayleigh fading and partial band interference it is determined that ratio statistic combining is an excellent diversity combining scheme for systems with partial band interference and fading

channels with block interference a new class of channel models with memory is presented in order to study various kinds of interference phenomena it is shown among other things that when all other parameters are held fixed channel capacity c is an em increasing function of the memory length while the cutoff rate r 0 generally is a em decreasing function calculations with various explicit coding schemes indicate that c is better than r 0 as a performance measure for these channel models as a partial resolution of this c versus r 0 paradox the conjecture is offered that r 0 is more properly a measure of coding delay rather than of coding complexity

on the capacity of channels with unknown interference the process of communicating in the presence of interference that is unknown or hostile is modeled as a two person zero sum game with the communicator and the jammer as the players the objective function considered is the rate of reliable communication the communicator s strategies are encoders and distributions on a set of quantizers the jammer s strategies are distributions on the noise power subject to certain constraints various conditions are considered on the jammer s strategy set and on the communicator s knowledge for the case where the decoder is uninformed of the actual quantizer chosen it is shown that from the communicator s perspective the worst case jamming strategy is a distribution concentrated on a finite number of points thereby converting a functional optimization problem into a nonlinear programming problem moreover the worst case distributions can be characterized by means of necessary and sufficient conditions which are easy to verify for the case where the decoder is informed of the actual quantizer chosen the existence of saddle point strategies is demonstrated the analysis is also seen to be valid for a number of situations where the jammer is adaptive

the convex positivstellensatz in a free algebra abstract given a monic linear pencil l in g variables let p l p l n n n where p l n x s n g l x 0 and s n g is the set of g tuples of symmetric n n matrices because l is a monic linear pencil each p l n is convex with interior and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form p l the main result of this paper establishes a perfect noncommutative nichtnegativstellensatz on a convex semialgebraic set namely a noncommutative matrix valued polynomial p is positive semidefinite on p l if and only if it has a weighted sum of squares representation with optimal degree bounds p s s j finite f j l f j where s f j are matrices of noncommutative polynomials of degree no greater than deg p 2 this noncommutative result contrasts sharply with the commutative setting where there is no control on the degrees of s f j and assuming only p nonnegative as opposed to p strictly positive yields a clean positivstellensatz so seldom that such cases are noteworthy

proper analytic free maps this paper concerns analytic free maps these maps are free analogs of classical analytic functions in several complex variables and are defined in terms of non commuting variables amongst which there are no relations they are free variables analytic free maps include vector valued polynomials in free non commuting variables and form a canonical class of mappings from one non commutative domain d in say g variables to another non commutative domain d in g variables as a natural extension of the usual notion an analytic free map is proper if it maps the boundary of d into the boundary of d assuming that both domains contain 0 we show that if f d d is a proper analytic free map and f 0 0 then f is one to one moreover if also g g then f is invertible and f 1 is also an analytic free map these conclusions on the map f are the strongest possible without additional assumptions on the domains d and d

implicit inverse function theorems for free noncommutative functions abstract we prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices we apply these results to study dependence of the solution of the initial value problem for odes in noncommutative spaces on the initial data and to extremal problems with noncommutative constraints

asymptotic performance of m ary orthogonal signals in worst case partial band interference and rayleigh fading it is shown that for worst case partial band jamming the error probability performance for fixed e sub b n sub i becomes worse with increasing m for m 16 the asymptotic probability of error is not zero for any e sub b n sub i ln 2 but decreases inverse linearly with respect to it in the fading case the error probability performance for fixed e sub b n sub 0 improves with m for noncoherent detection but worsens with m for coherent detection for large e sub b n sub 0 the performance of the rayleigh fading channel asymptotically approaches the same limit as the worst case partial band jammed channel however for values of m at least up to 4096 the partial band jammed channel does better while it is unlikely that an m ary orthogonal signal set with m 1024 will be used in a practical situation these results suggest an important theoretical problem namely what signal set achieves reliable communication

fixed point theorems for noncommutative functions abstract we establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices the conclusions are stronger if such a mapping also respects matrix similarities i e is a noncommutative function as a special case we prove the corresponding contractive mapping theorem which can be viewed as a new version of the banach fixed point theorem this result is then applied to prove the existence and uniqueness of a solution of the initial value problem for odes in noncommutative spaces as a by product of the ideas developed in this paper we establish a noncommutative version of the principle of nested closed sets

analytic mappings between noncommutative pencil balls abstract in this paper we analyze problems involving matrix variables for which we use a noncommutative algebra setting to be more specific we use a class of functions called nc analytic functions defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions we call such situations dimension free these types of functions have recently been used in the study of dimension free linear system engineering problems helton et al 2009 10 de oliviera et al 2009 8 in the earlier paper helton et al 2009 9 we characterized nc analytic maps that send dimension free matrix balls to dimension free matrix balls and carry the boundary to the boundary such maps we call nc ball maps in this paper we turn to a more general dimension free ball b l called a pencil ball associated with a homogeneous linear pencil l x a 1 x 1 a g x g a j c d d for x col x 1 x g c n n g define l x a j x j and let b l x c n n g l x 1 n n we study the generalization of nc ball maps to these pencil balls b l and call them pencil ball maps we show that every b l has a minimal dimensional in a certain sense defining pencil l up to normalization a pencil ball map is the direct sum of l with an nc analytic map of the pencil ball into the ball that is pencil ball maps are simple in contrast to the classical result of d angelo 1993 7 chapter 5 showing there is a great variety of such analytic maps from c g to c m when g m to prove our main theorem this paper uses the results of our previous paper helton et al 2009 9 plus entirely different techniques namely those of completely contractive maps what we do here is a small piece of the bigger puzzle of understanding how linear matrix inequalities lmis behave with respect to noncommutative change of variables

coding for frequency hopped spread spectrum communication with partial band interference part i capacity and cutoff rate the performance of optimal codes on frequency hopped channels with partial band interference is investigated the performance measures considered are channel capacity and cutoff rate worst case partial band gaussian noise interference is assumed with the interference independent of the transmitted signal the capacity and cutoff rate are calculated as a function of the signal to noise ratio we consider soft decision receivers and hard decision receivers with and without side information optimal code rates are found for each of the above cases the required signal to noise ratio for reliable communication when codes are used is determined as a function of the code rate

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares