geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

a non commutative positivstellensatz on isometries a symmetric non commutative polynomial p when evalu ated at a tuple of operators on a flnite dimensional real hilbert space h has a value which is a symmetric operator we show that any such polynomial which takes positive semideflnite values on the variety z of spherical isometries is represented as a sum of squares of polynomials plus a residual part vanishing on z here by spherical isometries we mean tuples a a 1 a2 an of operators on h such that a t a1 a tan i this observation improves prior theorems known only for strictly pos itive polynomials it is known that for commutative polynomials the result is false

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

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

circular free spectrahedra abstract this paper considers matrix convex sets invariant under several types of rotations it is known that matrix convex sets that are free semialgebraic are solution sets of linear matrix inequalities lmis they are called free spectrahedra we classify all free spectrahedra that are circular that is closed under multiplication by e i t up to unitary equivalence the coefficients of a minimal lmi defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal a matrix convex set is called free circular if it is closed under left multiplication by unitary matrices as a consequence of a hahn banach separation theorem for free circular matrix convex sets we show the coefficients of a minimal lmi defining a free circular free spectrahedron have up to unitary equivalence a block decomposition as above with only two blocks this paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials

extending the threshold and frequency range for phase based frequency estimation standard phase based frequency estimation has a threshold that is high is frequency dependent and does not decrease with increasing signal length these problems are solved by processing the signal with a highly overlapped filter bank before applying phase based frequency estimation by exploiting decimation a closed form matrix inversion and cascades of simple filter banks the computational complexity is kept low

non negative hereditary polynomials in a free algebra we prove a non negative stellensatz and a null stellensatz for a class of polynomials called hereditary polynomials in a free algebra

an effective matrix geometric mean satisfying the ando li mathias properties we propose a new matrix geometric mean satisfying the ten properties given by ando li and mathias linear alg appl 2004 this mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by ando li and mathias is the limit of a sequence having order of convergence 1 this makes this new mean very easily computable we provide a geometric interpretation and a generalization which includes as special cases our mean and the ando li mathias mean

strong majorization in a free algebra we study in the spirit of modern real algebra the interplay between left ideals of the free algebra mathbb f with n generators and their suitably defined zero sets and similarly between quadratic submodules of mathbb f and their positivity sets

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

free holomorphic functions and interpolation in this paper we obtain a noncommutative multivariable analogue of the classical nevanlinna pick interpolation problem for analytic functions with positive real parts on the open unit disc given a function f lambda to mathbb c where lambda is an arbitrary subset of the open unit ball mathbb b n z in mathbb c n z 1 we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball b mathcal h n 1 such that rm re g geq 0 quad rm and quad g z f z quad z in lambda where b mathcal h is the algebra of all bounded linear operators on a hilbert space mathcal h the proof employs several results from noncommutative multivariable operator theory and a noncommutative cayley transform introduced and studied in the present paper acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions all the results of this paper are obtained in the more general setting of free holomorphic functions with operator valued coefficients as consequences we deduce some results concerning operator valued analytic interpolation on the unit ball mathbb b n

free pluriharmonic majorants and commutant lifting in this paper we initiate the study of sub pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball b h n 1 x1 xn b h n x1x1 xnxn 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h several classical results from complex analysis have analogues in this noncommutative multivariable setting we present basic properties for sub pluriharmonic curves characterize the class of sub pluriharmonic curves that admit free pluriharmonic majorants and find in this case the least free pluriharmonic majorants we show that for any free holomorphic function on b h n 1 the map 0 1 c r1 rn r rr1 rrn rr1 rrn is a sub pluriharmonic curve in the cuntz toeplitz algebra generated by the right creation operators r1 rn on the full fock space with n generators we prove that is in the noncommutative hardy space hball2 if and only if has a free pluriharmonic majorant in this case we find herglotz riesz and poisson type representations for the least pluriharmonic majorant of moreover we obtain a characterization of the unit ball of hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of when is in the unit ball of hball2 in the second part of this paper we introduce a generalized noncommutative commutant lifting gncl problem which extends to our noncommutative multivariable setting several lifting problems including the classical sz nagy foias commutant lifting problem and the extensions obtained by treil volberg foias frazho kaashoek and biswas foias frazho as well as their multivariable noncommutative versions we solve the gncl problem and using the results regarding sub pluriharmonic curves and free pluriharmonic majorants on noncommutative balls we provide a complete description of all solutions in particular we obtain a concrete schur type description of all solutions in the noncommutative commutant lifting theorem

on fixed points of self maps of the free ball abstract in this paper we study the structure of the fixed point sets of noncommutative self maps of the free ball we show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace we provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete pick spaces

quasi minimal surfaces of pseudo riemannian space forms with positive relative nullity abstract in this paper we consider surfaces in 4 dimensional pseudo riemannian space forms with index 2 first we obtain some of geometrical properties of such surfaces considering their relative null space then we get classifications of quasi minimal surfaces with positive relative nullity

the noncommutative lowner theorem for matrix monotone functions over operator systems abstract given a function f a b r lowner s theorem states f is monotone when extended to self adjoint matrices via the functional calculus if and only if f extends to a self map of the complex upper half plane in recent years several generalizations of lowner s theorem have been proven in several variables we use the relaxed agler m c carthy and young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case specifically we show that a real free function defined over an operator system must analytically continue to a noncommutative upper half plane as map into another noncommutative upper half plane

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

analysis of the variance threshold of kay s weighted linear predictor frequency estimator a theoretical approximation for the variance of kay s weighted linear predictor frequency estimator is derived from this expression an inequality describing the variance threshold of the estimator is found the window weights are then optimized to improve the variance numerical simulations demonstrate that the variance approximations are valid for medium to high signal to noise ratios or for large numbers of samples

spatial and lorentzian surfaces in robertson walker space times let l14 f c i fs gfc be a robertson walker space time which does not contain any open subset of constant curvature in this paper we provide a general study of nondegenerate surfaces in l14 f c first we prove the nonexistence of marginally trapped surfaces with positive relative nullity then we classify totally geodesic submanifolds finally we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector

frequency estimation from proper sets of correlations as a complement to the periodogram low complexity frequency estimators are of interest one such estimator is based on prony s method and rely on phase information of the auto correlations without prior knowledge of the frequency e g a given frequency interval the frequency cannot be unambiguously estimated from a single correlation only we introduce a new method of phase unwrapping using an arbitrary number more than one of correlations from this arbitrary set of correlations we propose a weighted average estimator we derive the asymptotic performance and show how the correlation lags should be properly chosen from a design aspect there is often a restriction of using a fixed number of computations in addition we therefore propose a strategy to find a proper set of correlation lags subject to a given computational complexity finally simulation results that lend support to the theoretical findings are included

an improved feedforward frequency offset estimator in this correspondence we propose a new feedforward frequency offset or single frequency estimator which is based on a single weighted sample correlation with lag k computed from n noisy observations the new estimator is a simple generalization of the unweighted lank reed pollon lrp estimator introduced in g w lank i s reed and g e pollon a semicoherent detection and doppler estimation statistic ieee transactions on aerospace and electronic systems vol 9 no 2 pp 151 165 march 1973 it is shown that by relaxing the requirement for frequency estimation range an estimator with improved accuracy and lower threshold may be designed in particular the estimator is able to provide a 20 log10 k db threshold reduction when n is an integer multiple of k closed form expressions are derived for the weighting coefficients and also for the minimum variance analytical and simulation results are presented which assess and demonstrate the estimator performance

classification of spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension spatial surfaces with parallel mean curvature vector play some important roles in general relativity theory of harmonic maps as well as in differential geometry recently spatial surfaces in four dimensional minkowski space time with parallel mean curvature vector were classified by chen and van der veken complete classification of parallel surfaces in 4 dimensional lorentzian space forms tohoku math j 61 1 2009 in this article we completely classify spatial surfaces with parallel mean curvature vector in pseudo euclidean spaces of arbitrary dimension the main result states that there exist 16 families of such surfaces as by product we achieve the complete classification of spatial surfaces with parallel mean curvature vector in minkowski space times of arbitrary dimension

pick interpolation for free holomorphic functions we give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron as an application we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function

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

on the performance of the weighted linear predictor frequency estimator a statistical analysis of kay s 1989 weighted linear predictor frequency estimator was carried out using a different analysis technique to that employed in that paper the asymptotic variance of the weighted linear predictor frequency estimator is derived

limitations of semidefinite programs for separable states and entangled games semidefinite programs sdps are a framework for exact or approximate optimization with widespread application in quantum information theory we introduce a new method for using reductions to construct integrality gaps for sdps meaning instances where the sdp value is far from the true optimum these are based on new limitations on the sum of squares sos hierarchy in approximating two particularly important sets in quantum information theory where previously no omega 1 round integrality gaps were known integrality gaps for the 2 rightarrow 4 norm had previously been sought due to its connection to small set expansion sse and unique games ug in both cases no go theorems were previously known based on computational assumptions such as the exponential time hypothesis eth which asserts that 3 sat requires exponential time to solve our unconditional results achieve the same parameters as all of these previous results for separable states or as some of the previous results for quantum correlations in some cases we can make use of the framework of lee raghavendra steurer lrs to establish integrality gaps for any sdp extended formulation not only the sos hierarchy our hardness result on separable states also yields a dimension lower bound of approximate disentanglers answering a question of watrous and aaronson et al these results can be viewed as limitations on the monogamy principle the ppt test the ability of tsirelson type bounds to restrict quantum correlations as well as the sdp hierarchies of doherty parrilo spedalieri navascues pironio acin and berta fawzi scholz indeed a wide range of past work in quantum information can be described as using an sdp on one of the above two problems and our results put broad limits on these lines of argument

positivstellensatze for algebras of matrices abstract noncommutative positivstellensatze express positive elements of algebras in terms of sums of squares here positive elements can be defined by means of representations point evaluations or abstract orderings squares are elements of the form a a we prove various types of noncommutative positivstellensatze for matrix algebras m n a where a is an algebra with involution and subalgebras of m n a such as path algebras crossed product algebras and cyclic algebras the notion of a noncommutative sum of squares is proposed and new versions of positivstellensatze are proved

convergent relaxations of polynomial optimization problems with noncommuting variables we consider optimization problems with polynomial inequality constraints in noncommuting variables these noncommuting variables are viewed as bounded operators on a hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints such problems arise naturally in quantum theory and quantum information science to solve them we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution we also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem

minimizer extraction in polynomial optimization is robust in this article we present a robustness analysis of the extraction of optimizers in polynomial optimization optimizers can be extracted by solving moment problems using flatness and the gelfand n

null and positivstellensatze for rationally resolvable ideals abstract hilbert s nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in c x in the free algebra c x the vanishing set of a two sided ideal i is defined in a dimension free way using images in finite dimensional representations of c x i in this article nullstellensatze for a simple but important class of ideals in the free algebra called tentatively rationally resolvable here are presented an ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the x variables using noncommutative rational functions in the remaining variables whether such an ideal i satisfies the nullstellensatz is intimately related to embeddability of c x i into free skew fields these notions are also extended to free algebras with involution for instance it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two sided ideal i generated by 1 j x j x j this is then applied to free real algebraic geometry polynomials positive semidefinite on spherical isometries are sums of hermitian squares modulo i similar results are obtained for nc unitary groups

homothetic and conformal symmetries of solutions to einstein s equations we present several results about the nonexistence of solutions of einstein s equations with homothetic or conformal symmetry we show that the only spatially compact globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature and also admitting an infinitesimal proper homothetic symmetry are everywhere locally flat this assumes that the matter fields either obey certain energy conditions or are the yang mills or massless klein gordon fields we find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions we show that if the dominant energy condition is assumed then minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation in other words with the exceptions cited homothetic or conformal killing fields are in fact killing in spatially compact or asymptotically flat spactimes in the conformal procedure for solving the initial value problem we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry

frequency rate estimation at high snr the problem of estimating the frequency rate of change of complex valued frequency modulated signals from noisy observations is considered the performance of four related estimators is studied both analytically and by means of simulations and their relationship to the estimators proposed by djuric and kay 1990 and lang and musicus 1989 is established

free pick functions representations asymptotic behavior and matrix monotonicity in several noncommuting variables r nevanlinna showed that elements of the pick class the set of complex analytic functions taking the upper half plane into itself have certain integral representations which reflect their asymptotic behavior at infinity later lowner connected the pick class to matrix monotone functions we generalize the nevanlinna representation theorems and lowner s theorem on matrix monotone functions to the free pick class the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

screw invariant marginally trapped surfaces in minkowski 4 space a spacelike surface in a lorentzian manifold whose mean curvature vector is lightlike everywhere is called marginally trapped the classification of marginally trapped surfaces in minkowski 4 space which are invariant under a subgroup of the lorentz group that leaves invariant a lightlike direction i e the so called screw invariant surfaces is obtained as corollaries the screw invariant marginally trapped surfaces with harmonic mean curvature vector and with prescribed gaussian curvature are explicitly described

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

addendum to classification of marginally trapped lorentzian flat surfaces in e24 and its application to biharmonic surfaces abstract due to omission an additional family shall be added to the list of marginally trapped lorentzian flat surfaces in e 2 4 given in theorem 4 1 of b y chen classification of marginally trapped lorentzian flat surfaces in e 2 4 and its application to biharmonic surfaces j math anal appl 340 2008 861 875

operator monotone functions and lowner functions of several variables we prove generalizations of lowner s results on matrix monotone functions to several variables we give a characterization of when a function of d variables is locally monotone on d tuples of commuting self adjoint n by n matrices we prove a generalization to several variables of nevanlinna s theorem describing analytic functions that map the upper half plane to itself and satisfy a growth condition we use this to characterize all rational functions of two variables that are operator monotone

noncommutative ball maps 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 in this paper we characterize 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 we find that up to normalization an nc ball map is the direct sum of the identity map with an nc analytic map of the ball into the ball that is nc ball maps are very simple in contrast to the classical result of d angelo on such analytic maps in c another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary but on these our results are limited we shall be interested in several types of noncommutative balls conventional ones but also balls defined by constraints called linear matrix inequalities lmi what we do here is a small piece of the bigger puzzle of understanding how lmis behave with respect to noncommutative change of variables

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

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

a fast and accurate single frequency estimator a novel frequency estimator for a single complex sinusoid in complex white gaussian noise is proposed the estimator is more computationally efficient that the optimal maximum likelihood estimator yet attains equally good performance at moderately high signal to noise ratios the estimator is shown to be related to the linear prediction estimator this relationship is used to reveal why the linear prediction estimator does not attain the cramer rao bound even at high signal to noise ratios

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

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

fast algorithms for single frequency estimation in this paper some new estimators of the frequency of a single complex sinusoid are presented the rotate add decimate rad method of crozier is first refined to more closely approach the cramer rao bound crb an additional modification yields an unbiased estimator erad that essentially achieves the crb above a signal to noise ratio snr threshold comparable to that of rad in addition this estimator is proven to achieve the crb for high snr the erad method requires approximately 2n complex multiply adds and log sub 2 n arctangents a modified erad merad is proposed that matches the snr threshold and computational complexity of the rad method approximately 3n complex multiply adds and log sub 2 n arctangents but achieves the crb for high snr

constrained polynomial optimization problems with noncommuting variables in this paper we study constrained eigenvalue optimization of noncommutative nc polynomials focusing on the polydisc and the ball our three main results are as follows 1 an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition 2 eigenvalue optima for nc polynomials can be computed using a single semidefinite program sdp this sharply contrasts with the commutative case where sequences of sdps are needed 3 the dual solution to this single sdp can be exploited to extract eigenvalue optimizers with an algorithm based on two ingredients solution to a truncated nc moment problem via flat extensions and gelfand naimark segal construction the implementation of these procedures in our computer algebra system ncsostools is presented and several examples pertaining to matrix inequalities are given to illustrate our results

parallel surfaces in lorentzian three manifolds admitting a parallel null vector field we classify parallel surfaces in lorentzian three manifolds admitting a parallel null vector field some families of semi parallel surfaces are also described

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

structure of fermionic density matrices complete n representability conditions we present a constructive solution to the n representability problem a full characterization of the conditions for constraining the two electron reduced density matrix to represent an n electron density matrix previously known conditions while rigorous were incomplete here we derive a hierarchy of constraints built upon i the bipolar theorem and ii tensor decompositions of model hamiltonians existing conditions d q g t1 and t2 known classical conditions and new conditions appear naturally subsets of the conditions are amenable to polynomial time computations of strongly correlated systems

connes embedding conjecture and sums of hermitian squares we show that connes embedding conjecture on von neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition the trace is nonnegative whenever self adjoint contraction matrices of the same size are substituted for the variables these algebraic certificates involve sums of hermitian squares and commutators we prove that they always exist for a similar nonnegativity condition where elements of separable ii1 factors are considered instead of matrices under the presence of connes conjecture we derive degree bounds for the certificates

loewner s theorem in several variables abstract in this paper we establish a multivariable non commutative generalization of loewner s theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half plane into itself the non commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples not a priori assumed to be analytic or continuous as a schur complement of a positive linear matrix pencil to achieve this we establish the abstract integral formula for these functions using only matrix convexity and lmis that represents operator monotone and operator concave free functions as a conditional expectation of a schur complement of a linear matrix pencil on a tensor product operator algebra the results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so including the karcher mean to obtain an explicit closed formula for these operator means

the matricial relaxation of a linear matrix inequality given linear matrix inequalities lmis l 1 and l 2 it is natural to ask q1 when does one dominate the other that is does l 1 x succeq 0 imply l 2 x succeq 0 q2 when are they mutually dominant that is when do they have the same solution set the matrix cube problem of ben tal and nemirovski siam j optim 12 811 833 2002 is an example of lmi domination hence such problems can be np hard this paper describes a natural relaxation of an lmi based on substituting matrices for the variables x j with this relaxation the domination questions q1 and q2 have elegant answers indeed reduce to constructible semidefinite programs as an example to test the strength of this relaxation we specialize it to the matrix cube problem and obtain essentially the relaxation given in ben tal and nemirovski siam j optim 12 811 833 2002 thus our relaxation could be viewed as generalizing it assume there is an x such that l 1 x and l 2 x are both positive definite and suppose the positivity domain of l 1 is bounded for our matrix variable relaxation a positive answer to q1 is equivalent to the existence of matrices v j such that begin array ll l 2 x v 1 l 1 x v 1 cdots v mu l 1 x v mu quad quad quad rm a 1 end array as for q2 we show that l 1 and l 2 are mutually dominant if and only if up to certain redundancies described in the paper l 1 and l 2 are unitarily equivalent algebraic certificates for positivity such as a1 for linear polynomials are typically called positivstellensatze the paper goes on to derive a putinar type positivstellensatz for polynomials with a cleaner and more powerful conclusion under the stronger hypothesis of positivity on an underlying bounded domain of the form x mid l x succeq0 an observation at the core of the paper is that the relaxed lmi domination problem is equivalent to a classical problem namely the problem of determining if a linear map from a subspace of matrices to a matrix algebra is completely positive complete positivity is one of the main techniques of modern operator theory and the theory of operator algebras on one hand it provides tools for studying lmis and on the other hand since completely positive maps are not so far from representations and generally are more tractable than their merely positive counterparts the theory of completely positive maps provides perspective on the difficulties in solving lmi domination problems

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

every convex free basic semi algebraic set has an lmi representation the matricial solution set of a linear matrix inequality lmi is a convex free basic open semi algebraic set the main theorem of this paper is a converse each such set arises from some lmi the result has implications for semi definite programming and systems engineering as well as for free semi algebraic geometry

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples