chaos control in economical model by time delayed feedback method abstract a two dimensional map describing chaotic behaviour of an economic model has been stabilized on various periodic orbits by the use of pyragas time delayed feedback control the method avoids fancy data processing used in the ott grebogi yorke approach and is based solely on the plain measurement and time lag of a scalar signal which in our case is a value of sales of a firm following an active investment strategy behrens feichtinger model we show that the application of this control method is very straightforward and one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the system s economic properties

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

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

polynomials with the half plane property and matroid theory a polynomial f is said to have the half plane property if there is an open half plane h subset of c whose boundary contains the origin such that f is non zero whenever all the variables are in h

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

high action orbits for tonelli lagrangians and superlinear hamiltonians on compact configuration spaces abstract multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds given by lagrangians or hamiltonians which have quadratic growth in the velocities or in the momenta such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions in this note we show how these results can be extended to the classical setting of tonelli lagrangians lagrangians which are c 2 convex and superlinear in the velocities or to hamiltonians which are superlinear in the momenta and have a coercive action integrand

relating homogeneous cones and positive definite cones via t algebras t algebras are nonassociative algebras defined by vinberg in the early 1960s for the purpose of studying homogeneous cones vinberg defined a cone k cal a for each t algebra cal a and proved that every homogeneous cone is isomorphic to one such k cal a we relate each t algebra cal a with a space of linear operators in such a way that k cal a is isomorphic to the cone of positive definite self adjoint operators together with vinberg s result we conclude that every homogeneous cone is isomorphic to a slice of a cone of positive definite matrices

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

non representable hyperbolic matroids the generalized lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices hyperbolic polynomials give rise to a class of hyperbolic matroids which properly contains the class of matroids representable over the complex numbers this connection was used by the first author to construct counterexamples to algebraic stronger versions of the generalized lax conjecture by considering a non representable hyperbolic matroid the va mos matroid and a generalization of it are to this day the only known instances of non representable hyperbolic matroids we prove that the non pappus and non desargues matroids are non representable hyperbolic matroids by exploiting a connection due to jordan between euclidean jordan algebras and projective geometries we further identify a large class of hyperbolic matroids that are parametrized by uniform hypergraphs and prove that many of them are non representable finally we explore consequences to algebraic versions of the generalized lax conjecture

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

homogeneous nucleation of a solid solid dilatational phase transformation abstract the problem of homogeneous coherent nucleation of a spherical particle in an infinite matrix is considered a landau ginzburg type potential is used to describe the material response the underlying landau potential is taken to be of the 2 3 4 type and the gradient energy contribution is taken to be quadratic the nucleation problem is posed as an energy extremum problem and the finite element method in conjunction with a perturbed lagrangian algorithm is used to obtain solutions with nucleus structure the present nonlinear model spans the range from classical to nonclassical nucleation and exhibits many of the physical phenomena associated with nonclassical nucleation including divergence in radius and interface thickness of the critical nucleus and vanishing of the nucleation energy as the instability temperature is approached

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

hyperbolic polynomials interlacers and sums of squares hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids the innermost of which is convex these polynomials appear in many areas of mathematics including optimization combinatorics and differential equations here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it this set of interlacers is a convex cone which we write as a linear slice of the cone of nonnegative polynomials in particular this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials using a sums of squares relaxation we then approximate a hyperbolicity cone by the projection of a spectrahedron a multiaffine example coming from the vamos matroid shows that this relaxation is not always exact using this theory we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

determinantal representations and bezoutians a major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral i e the solution sets of linear matrix inequalities we will use sum of squares decompositions of certain bilinear forms called bezoutians to approach this problem more precisely we show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that q cdot h has a definite determinantal representation besides commutative algebra the proof relies on results from real algebraic geometry

on the positivity of symmetric polynomial functions part i general results abstract we prove that a real symmetric polynomial inequality of degree d 2 holds on r n if and only if it holds for elements with at most d 2 distinct non zero components which may have multiplicities we establish this result by solving a cauchy problem for ordinary differential equations involving the symmetric power sums this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function in the final section extensions of our results to the whole space r n are outlined the main results are theorems 5 1 and 5 2 with corollaries 2 1 and 5 2 and the corresponding results for r n from the last subsection part ii will contain a discussion on the ordered vector space h d n in general as well as on the particular cases of degrees d 4 and d 5 finite test sets for positivity in the homogeneous case and other sufficient criteria

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

bifurcation in model reference adaptive control systems abstract we study the effect of unmodeled dynamics on the performance of a simple model reference adaptive control mechanism exact stability boundaries are computed and different routes to chaotic behavior and global instability are described typically we observed a region of period doubling and hopf bifurcations similar to those observed in the ziegler nichols approach to pid control design the analysis applies to higher order and certain nonlinear systems as well

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

quantum entanglement symmetric nonnegative quadratic polynomials and moment problems quantum states are represented by positive semidefinite hermitian operators with unit trace known as density matrices an important subset of quantum states is that of separable states the complement of which is the subset of entangled states we show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis only a small number of such moments are accessible experimentally and so in practice the question of quantum entanglement of a many body system e g a system consisting of several atoms can be reduced to a truncated moment problem by considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in mathbb r d we work with moments up to degree 2 only since these are most readily available experimentally we derive necessary and sufficient conditions for belonging to the moment cone which can be expressed via a linear matrix inequality of size at most 2d 2 which is independent of n the linear matrix inequalities can be converted into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone and show that the two conditions approach each other in the limit of large n the inequalities are derived via considering the dual cone of nonnegative polynomials and its sum of squares relaxation we show that the sum of squares relaxation of the dual cone is asymptotically exact and using symmetry reduction techniques blekherman and riener symmetric nonnegative forms and sums of squares arxiv 1205 3102 2012 gatermann and parrilo j pure appl algebra 192 1 3 95 128 https doi org 10 1016 j jpaa 2003 12 011 2004 it can be written as a small linear matrix inequality of size at most 2d 2 which is independent of n for the cone of symmetric nonnegative polynomials with the relevant support we also prove an analogue of the half degree principle for globally nonnegative symmetric polynomials riener j pure appl algebra 216 4 850 856 https doi org 10 1016 j jpaa 2011 08 012 2012 timofte j math anal appl 284 1 174 190 https doi org 10 1016 s0022 247x 03 00301 9 2003

multivariate polya schur classification problems in the weyl algebra a multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts we classify all linear partial differential operators in the weyl algebra a n that preserve stability an important tool that we develop in the process is the higher dimensional generalization of polya schur s notion of multiplier sequence we characterize all multivariate multiplier sequences as well as those of finite order next we establish a multivariate extension of the cauchy poincare interlacing theorem and prove a natural analog of the lax conjecture for real stable polynomials in two variables using the latter we describe all operators in a 1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in a n preserves stability if and only if its fischer fock adjoint does these are powerful multivariate extensions of the classical hermite poulain jensen theorem polya s curve theorem and schur malo szegocomposition theorems examples and applications to strict stability preservers are also discussed

on the positivity of symmetric polynomial functions part ii lattice general results and positivity criteria for degrees 4 and 5 abstract we prove that homogeneous symmetric polynomial inequalities of degree p 4 5 in n positive 1 variables can be algorithmically tested on a finite set depending on the given inequality theorem 13 the test set can be obtained by solving a finite number of equations of degree not exceeding p 2 section 3 discusses the structure of the ordered vector spaces h p n and h p n in section 4 positivity criteria for degrees 4 and 5 are stated and proved the main results are theorems 10 14 part iii of this work will be concerned with the construction of extremal homogeneous symmetric polynomials best inequalities of degree 4 in n positive variables

discriminants of cyclic homogeneous inequalities of three variables abstract in this article we study the structure of the cone of semidefinite forms it is a closed semialgebraic set but usually is not basic closed semialgebraic set a discriminant is a defining equation of an irreducible component of algebraic boundary of this cone we calculate discriminants using new tools characteristic variety and local cones a characteristic variety is a semialgebraic subset of a real projective variety on which the family of inequalities is essentially defined as linear functions local cone is a subcone of the psd cone which corresponds to a maximal ideal this theory works well for a family of polynomials which are invariant under an action of a finite group after we construct an abstract general theory we apply it to a family of cyclic homogeneous polynomials of three real variables of degree d we calculate some discriminants for d 3 4 5 and 6 and we show that this theory derives many new results

on the positivity of symmetric polynomial functions part iii extremal polynomials of degree 4 abstract in this paper which is a continuation of v timofte on the positivity of symmetric polynomial functions part i general results j math anal appl 284 2003 174 190 and v timofte on the positivity of symmetric polynomial functions part ii lattice general results and positivity criteria for degrees 4 and 5 j math anal appl in press we study properties of extremal polynomials of degree 4 and we give the construction of some of them the main results are theorems 9 13 15 16 and 18

drift instabilities and chaos in forecasting and adaptive decision theory abstract bifurcation theory shows that policy adaptation and the rational expectations hypothesis of macro economics can be used to explain unpredictability rapid changes in solution structure and chaos in decision problems with uncertainty structural errors lead to catastrophic instability and forecasts become irrelevant short evaluation horizon and the application of measures designed to give quick response give multiperiodicity and chaos finally wrong interpretations of the context lead to global bifurcations laminar drift complacency and chaotic bursting the discrete map representing these dynamics is of interest in its own right it is non invertible and displays bifurcation behaviour not commonly seen in systems derived from physical considerations

small amplitude chaos and ergodicity in adaptive control abstract the theory for continuous iterated maps on the real line is used to study the dynamics of a continuous family of one parameter adaptive systems we establish sufficient and necessary conditions for uniform asymptotic stability and ultimate boundedness exact bifurcation boundaries and sufficient conditions for the existence of chaotic attractors and ergodicity of the estimated parameter it is argued that the chaotic dynamics are structurally stable that they persist in higher dimensional adaptive control systems and that they do not necessarily lead to poor performance when the controller is modified to deal with the infinite drift and small divisor problems a novel modification involving a sign change in the adaptive law is proposed

symmetric determinantal representations in characteristic 2 abstract this paper studies symmetric determinantal representations sdr in characteristic 2 that is the representation of a multivariate polynomial p by a symmetric matrix m such that p det m and where each entry of m is either a constant or a variable we first give some sufficient conditions for a polynomial to have an sdr we then give a non trivial necessary condition which implies that some polynomials have no sdr answering a question of grenet et al a large part of the paper is then devoted to the case of multilinear polynomials we prove that the existence of an sdr for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings we develop some algorithms to test the factorizability in these rings and use them to find sdrs when they exist altogether this gives us polynomial time algorithms to factorize the polynomials in the quotient rings and to build sdrs we conclude by describing the case of alternating determinantal representations in any characteristic

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

a particular minimization property implies c0 integrability abstract the manifold m being compact and connected and h being a tonelli hamiltonian such that t m is equal to the dual tiered mane set we prove that there is a partition of t m into invariant c 0 lagrangian graphs moreover among these graphs those that are c 1 cover a dense g subset of t m the dynamic restricted to each of these sets is non wandering

tonelli hamiltonians without conjugate points and c 0 integrability we prove that all the tonelli hamiltonians defined on the cotangent bundle t t n of the n dimensional torus that have no conjugate points are c 0 integrable i e t t n is c 0 foliated by a family fc of invariant c 0 lagrangian graphs assuming that the hamiltonian is c infty we prove that there exists a g delta subset gc of fc such that the dynamics restricted to every element of gc is strictly ergodic moreover we prove that the lyapunov exponents of every c 0 integrable tonelli hamiltonian are zero and deduce that the metric and topological entropies vanish

smooth hyperbolicity cones are spectrahedral shadows hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials a well understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral in this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone that is a spectrahedral shadow

exactness of sums of squares relaxations involving 3x3 matrices and lorentz cones we consider a family pm n of cones of positive maps and a semidefinite relaxation of these cones the cone pm n can be described as the set of those linear mappings from the space rm into the space of real symmetric n n matrices which map the m dimensional lorentz cone into the cone of real symmetric positive semidefinite matrices we describe the cone pm n as a cone of nonnegative polynomials in several variables we show that the considered semidefinite relaxation is in fact a sums of squares relaxation corresponding to this description of pm n our main result is that for n 3 the relaxation is exact hence it yields the exact result for optimisation problems over the cones pm 3 in particular the matrix ellipsoid problem for real symmetric 3 3 matrices can be rewritten as feasibility problem of a linear matrix inequality for m 4 n 4 there exist points in pm n which do not lie in the semidefinite set corresponding to the relaxation hence the relaxation is exact if and only if min n m 3

polynomials with and without determinantal representations abstract the problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention helton and vinnikov 9 have proved that any real zero polynomial in two variables has a determinantal representation branden 2 has shown that the result does not extend to arbitrary numbers of variables disproving the generalized lax conjecture we prove that in fact almost no real zero polynomial admits a determinantal representation there are dimensional differences between the two sets the result follows from a general upper bound on the size of linear matrix polynomials we then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation we finally characterize polynomials of which some power has a determinantal representation in terms of an algebra with involution having a finite dimensional representation we use the characterization to prove that any quadratic real zero polynomial has a determinantal representation after taking a high enough power taking powers is thereby really necessary in general the representations emerge explicitly and we characterize them up to unitary equivalence

bianalytic maps between free spectrahedra linear matrix inequalities lmis i d sum j 1 g a jx j sum j 1 g a j x j succeq 0 play a role in many areas of applications the set of solutions of an lmi is a spectrahedron lmis in dimension free matrix variables model most problems in linear systems engineering and their solution sets are called free spectrahedra free spectrahedra are exactly the free semialgebraic convex sets this paper studies free analytic maps between free spectrahedra and under certain generically valid irreducibility assumptions classifies all those that are bianalytic the foundation of such maps turns out to be a very small class of birational maps we call convexotonic the convexotonic maps in g variables sit in correspondence with g dimensional algebras if two bounded free spectrahedra mathcal d a and mathcal d b meeting our irreducibility assumptions are free bianalytic with map denoted p then p must after possibly an affine linear transform extend to a convexotonic map corresponding to a g dimensional algebra spanned by u i a 1 ldots u i a g for some unitary u furthermore b and ua are unitarily equivalent the article also establishes a positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from mathcal d a to mathcal d b not necessarily bianalytic another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron these theorems are needed for classifying free bianalytic maps

a complete characterization of determinantal quadratic polynomials abstract the problem of expressing a multivariate polynomial as the determinant of a monic definite symmetric or hermitian linear matrix polynomial lmp has drawn a huge amount of attention due to its connection with optimization problems in this paper we provide a necessary and sufficient condition for the existence of monic hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial further we propose a method to construct such a monic determinantal representation mdr of size 2 if it exists it is known that a quadratic polynomial f x x t a x b t x 1 has a symmetric mdr of size n 1 if a is negative semidefinite we prove that if a quadratic polynomial f x with a which is not negative semidefinite has an mdr of size greater than 2 then it has an mdr of size 2 too finally we characterize all quadratic polynomials that exhibit mdrs of any size

multivariate stable polynomials theory and applications univariate polynomials with only real roots while special do occur often enough that their properties can lead to interesting conclusions in diverse areas due mainly to the recent work of two young mathematicians julius borcea and petter branden a very successful multivariate generalization of this method has been developed the first part of this paper surveys some of the main results of this theory of multivariate stable polynomials the most central of these results is the characterization of linear transformations preserving stability of polynomials the second part presents various applications of this theory in complex analysis matrix theory probability and statistical mechanics and combinatorics

real even symmetric ternary forms let sen m denote the set of all real symmetric forms of degree m 2d let psen m and sen m denote the cones of positive semidefinite psd and sum of squares sos elements of sen m respectively for m 2 or 4 these cones coincide for m 6 they do not and were analyzed in even symmetric sextics by m d choi t y lam and b reznick 1987 math z 195 pp 559 580 we present an easily checked necessary and sufficient condition for an even symmetric n ary octic to be in psen 8 and for an even symmetric ternary decic to be in pse3 10 we also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10 we proceed to discuss the extremal elements of the cones pse3 8 and pse3 10 this leads to the question how many of these extremal forms have sos representations we prove that pse3 8 se3 8 and demonstrate that pse3 10 se3 10 is nonempty providing many new examples of psd forms which are not sos we give a graphic representation of ternary forms which also indicates whether or not an element of se3 8 or se3 10 is psd we interpret elements of pse3 m as inequalities in particular we give all symmetric polynomial inequalities of degree 5 satisfied by the sides of a triangle

bianalytic free maps between spectrahedra and spectraballs abstract linear matrix inequalities lmis are ubiquitous in real algebraic geometry semidefinite programming control theory and signal processing lmis with dimension free matrix unknowns are central to the theories of completely positive maps and operator algebras operator systems and spaces and serve as the paradigm for matrix convex sets the matricial feasibility set of an lmi is called a free spectrahedron in this article the bianalytic maps between a very general class of ball like free spectrahedra examples of which include row or column contractions and tuples of contractions and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form they are all highly structured rational maps in the case that both the domain and codomain are ball like these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point in particular this result leads to a classification of automorphism groups of ball like free spectrahedra the proofs depend on a novel free nullstellensatz established only after new tools in free analysis are developed and applied to obtain fine detail geometric in nature locally and algebraic in nature globally about the boundary of ball like free spectrahedra

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2