on the structure of specht modules in the principal block of fσ3p abstract let f be a field of characteristic p at least 5 we study the loewy structures of specht modules in the principal block of f 3 p we show that a specht module in the block has loewy length at most 4 and composition length at most 14 furthermore we classify which specht modules have loewy length 1 2 3 or 4 produce a specht module having 14 composition factors describe the second radical layer and the socle of the reducible specht modules and prove that if a specht module corresponds to a partition that is p regular and p restricted then the head of the specht module does not extend the socle

on the capacity of block multiantenna channels in this paper we consider point to point multiantenna channels with certain block distributional symmetries that do not require the entries of the channel matrix to be either gaussian or independent or identically distributed a main contribution is a capacity theorem for these channels which might be regarded as a generalization of telatar s theorem 1999 which reduces the numerical optimization domain in the capacity computation with this information theoretic result and some free probability arguments we prove an asymptotic capacity theorem that in addition to reducing the optimization domain does not depend on the dimension of the channel matrix this theorem allows us to apply free probability techniques to numerically compute the asymptotic capacity of the channels under consideration these theorems provide a very efficient method for numerically approximating both the capacity and a capacity achieving input covariance matrix of certain channels

transformations of matrix pencils and implications in linear systems theory a new transformation of matrix pencils is proposed and its theory formally developed its restriction to generalized state space system matrix pencils is shown to be an appropriate generalization of the usual state space transformation and to represent a closed form expression in system matrix terms of verghese s concept of strong equivalence of such pencils

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

asymptotics of unitary and orthogonal matrix integrals abstract in this paper we prove that in small parameter regions arbitrary unitary matrix integrals converge in the large n limit and match their formal expansion secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the hciz integral our convergence result also leads us to new results of smoothness of microstates we finally generalize our approach to integrals over the orthogonal group

stochastic integration on the cuntz algebra o abstract we develop a theory of non commutative stochastic integration with respect to the creation and annihilation process on the full fock space over l2 r our theory largely parallels the theories of non commutative stochastic ito integration on boson and fermion fock space as developed by r hudson and k r parthasarathy it provides the first example of a non commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations but it is based on the theory of reduced free products of c algebras by d voiculescu this theory shows that the creation and annihilation processes on the full fock space over l2 r which generate the cuntz algebra o can be interpreted as a generalized brownian motion we should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the c norm on o i e uniformly rather than in some state dependent strong operator topology or non commutative l2 norm

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

concavification of free entropy abstract we introduce a modification of voiculescu s free entropy which coincides with the lim inf variant of voiculescu s free entropy on extremal states but is a concave upper semi continuous function on the trace state space we also extend the orbital free entropy of hiai et al 2009 8 to non hyperfinite multivariables and prove freeness in the case of additivity of voiculescu s entropy or vanishing of our extended orbital entropy

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

canonical forms for descriptor systems under restricted system equivalence abstract the problem of canonical forms of implicit descriptor systems under restricted system equivalence is considered it is shown that the canonical form is directly related to the echelon form of the autoregressive matrix inducing the external behavior of the system the proposed canonical form is a generalization of the popov form and is also valid for singular systems under transfer equivalence

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

a jordan canonical form for reachable linear systems abstract we construct a new canonical form for reachable matrix pairs a b under the similarity action the canonical pairs are explicitly characterized by the property that a is in jordan canonical form and b is in a novel block echelon form whose block sizes are determined by the jordan structure of a the reachable pairs are classified by lists of indices and it is shown that the canonical form is continuous on the jordan strata consisting of all reachable pairs with the same index list the jordan strata are characterized topologically and explicit dimension formulas are derived finally we indicate some open problems related to the jordan decomposition of the space of reachable systems

absence of algebraic relations and of zero divisors under the assumption of full non microstates free entropy dimension abstract we show that in a tracial and finitely generated w probability space existence of conjugate variables excludes algebraic relations for the generators moreover under the assumption of maximal non microstates free entropy dimension we prove that there are no zero divisors in the sense that the product of any non commutative polynomial in the generators with any element from the von neumann algebra is zero if and only if at least one of those factors is zero in particular this shows that in this case the distribution of any non constant self adjoint non commutative polynomial in the generators does not have atoms questions on the absence of atoms for polynomials in non commuting random variables or for polynomials in random matrices have been an open problem for quite a while we solve this general problem by showing that maximality of free entropy dimension excludes atoms

on the stabilizer subgroup of a fair of matrices abstract let f g be a pair of matrices defined over an arbitrary field fn n gn m consider the natural action of gln x glm on this pair given by f g gfg 1 ggh 1 where g h gln glm this action is of interest in system theory as well as the representation theory of quivers in this paper we study the stabilizer subgroup of this action stab f g i e g h gl n x gl m gfg 1 f ggh 1 g

specht modules in the auslander reiten quiver abstract let a be a symmetric algebra over an algebraically closed field we study the position of indecomposable a modules with small socles or heads in auslander reiten components of tree class a we apply the results to specht modules when a is a block of a group algebra of a symmetric group in particular we show that if the block has weight 2 then all specht modules are quasi simple that is they lie at the ends of their components

generalized hermite matrices and complete invariants of strict system equivalence a complete list of invariants for reachable system matrices sigma s left begin array 20 c p s q s v s w s end array right with respect to strict system equivalence s s e is determined by polynomial methods the polynomial input output pairs u y for which there exists a polynomial vector z such that pz qu and y vz wu form a k s module mu sigma it is shown that the unique basis matrix of mu sigma in hermite form yields a complete set of discrete hermite indices resp continuous invariants of s s e the hermite invariants are characterized in state space terms and a realization of sigma s in hermite canonical form is presented nice orders and generalized hermite forms are introduced in order to develop a framework that encompasses hermite invariants and kronecker invariants hermite s theorem is generalized to these matrices finally nice orders are used to single out unique representatives among all minimal bas

a canonical form for controllable singular systems abstract a new canonical form for the action of restricted system equivalence on controllable singular systems is given the construction of this form is based on the weierstrass decomposition of the singular system into a slow and a fast subsystem both subsystems are transformed into hermite canonical form the resulting hermite canonical form for singular systems has a particularly simple structure and is expected to be useful for e g identification purposes continuity properties of the hermite form are investigated and the nonexistence of a globally defined continuous canonical form for controllable singular systems is shown

a jordan control canonical form for singular systems a canonical form for controllable singular systems modulo restricted system equivalence is presented the canonical form respects the decomposition of the singular system into a regular and an impulsive part the construction is based on popov s control canonical form and on the jordan canonical form recently developed by hinrichsen and pratzel wolters continuity properties are analysed and the cellular partition of the orbit space induced by the canonical form is briefly discussed

regular points in system spaces representation theory of quivers is applied to system theory as a result the regular points in system spaces are characterized moreover all prestable points in system spaces are described and another generic property of linear systems is introduced

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

a wild quiver in linear systems theory abstract we study the action t l a b t 1 at t 1 bl of the group gl n k gl m k of linear state and input transformations on the space u n m of control systems a b k n n k n m with n different eigenvalues k an algebraically closed field this action is of interest in systems theory as well as the representation theory of quivers we construct a canonical form for on u n m and determine completely its stabilizer subgroups the results are based on an analysis of the scaling action d e b d 1 be of the group of diagonal matrices d n k d m k on k n m we construct a canonical form for this action by graph theoretic methods and determine the adherence order of the underlying stratification of k n m

canonical forms for polynomial and quadratic differential operators we define the canonical representative for the equivalence class consistingof all polynomial and quadratic differential operators that take the same values on a given subspaceof c

on a homeomorphism between orbit spaces of linear systems and matrix polynomials abstract the orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact open topology is considered in the latter as a consequence the variation of the finite and left wiener hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described

feedback invariants via output injection abstract the feedback equivalence classes of matrix pairs a kc b kw obtained by output injection of matrix k on a system a b c w are completely characterized working on arbitrary fields some related problems are also solved

local limit theorems for multiplicative free convolutions abstract this paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution we show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law superconvergence to the marginal law of free multiplicative brownian motion at a specified time is also studied in the unitary case the superconvergence to free brownian motion and that to the haar measure are shown to be uniform over the entire unit circle implying further a free entropic limit theorem and a universality result for unitary free levy processes finally the method of proofs on the positive half line gives rise to a new multiplicative boolean to free bercovici pata bijection

analogues of entropy in bi free probability theory non microstate abstract in this paper we extend the notion of non microstate free entropy to the bi free setting using a diagrammatic approach involving bi non crossing diagrams bi free difference quotients are constructed as analogues of the free partial derivations adjoints of bi free difference quotients are discussed and used to define bi free conjugate variables notions of bi free fisher information non microstate bi free entropy and non microstate bi free entropy dimension are defined and known properties in the free setting are extended to the bi free setting

on tangential matrix interpolation abstract the paper presents an algebraic approach using polynomial and rational models over an arbitrary field to tangential interpolation problems both by polynomial as well as rational matrix functions appropriate extensions of scalar problems associated with the names of lagrange first order hermite high order and newton recursive are derived the relation of interpolation problems to the matrix chinese remainder theorem are clarified some two sided interpolation problems are discussed using the theory of tensored functional models

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

a note about proving non γ under a finite non microstates free fisher information assumption we prove that if x1 xn n 1 are self adjoints in a w probability space with finite non microstates free fisher information then the von neumann algebra w x1 xn they generate doesn t have property especially is not amenable this is an analog of a well known result of voiculescu for microstates free entropy we also prove factoriality under finite non microstates entropy

the spectral edge of unitary brownian motion the brownian motion u n t t ge 0 on the unitary group converges as a process to the free unitary brownian motion u t t ge 0 as n rightarrow infty in this paper we prove that it converges strongly as a process not only in distribution but also in operator norm in particular for a fixed time t 0 we prove that the unitary brownian motion has a spectral edge there are no outlier eigenvalues in the limit we also prove an extension theorem any strongly convergent collection of random matrix ensembles independent from a unitary brownian motion also converge strongly jointly with the brownian motion we give an application of this strong convergence to the jacobi process

some applications of free fisher information on frame theory we introduce the notion of operator valued free fisher information with respect to a positive map of a random variable in an operator valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations then we can apply this notion on the study of frame theory especially on the disjointness problem of modular frames arising from conditional expectations

hermite indices and the action of the feedback group the controllability indices are a complete system of invariants for the feedback equivalence relation of controllable matrix pairs the hermite indices are invariant for the similarity of matrix pairs but they are not invariant by changing bases on the input space and performing state feedback the aim of this work is to partially characterize the hermite indices of a controllable matrix pair when transformations of the feedback group are performed on it

local laws for polynomials of wigner matrices abstract we consider general self adjoint polynomials in several independent random matrices whose entries are centered and have the same variance we show that under certain conditions the local law holds up to the optimal scale i e the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory we prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i i d entries thus establishing the optimal bulk local law for these classes of ensembles in particular we generalize a similar result of anderson for anticommutator for more general polynomials our conditions are effectively checkable numerically

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

cartan matrices and morita equivalence for blocks of the symmetric groups theorem 1 the blocks of the symmetric groups with a given weight w 0 over a field of characteristic p can be collected into families according to the shapes of their cores each family consists of morita equivalent blocks the number of such families is at most flf 1 i l w 1 11 and each family contains a block of some g where n is less than or equal to p2 p 1 2 w 1 2 4 wp

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

canonical forms for linear systems this paper considers canonical forms for the similarity action of gl n on n m a b cn n cn m gl n n m n m h a b hah 1 hb those canonical forms are obtained as an application of a more general method to select canonical elements mc in the orbits om of a matrix group g acting on a set of matrices m cl p we define a total order on cl p different from the lexicographic order l 0l x x 0 but 0 x 0 for x r and consider normalized om elements with a minimal number of parameters min m om m normalized it is shown that the row and column echelon forms the jordan canonical form and nice control canonical forms for reachable a b pairs have a homogeneous interpretation as such minimal orbit elements moreover new canonical forms for the general action are determined via this method

free entropy dimension and regularity of non commutative polynomials abstract we show that the spectral measure of any non constant non commutative polynomial evaluated at a non commutative n tuple cannot have atoms if the free entropy dimension of that n tuple is n see also work of mai speicher and weber under stronger assumptions on the n tuple we prove that the spectral measure of any non constant non commutative polynomial function is not singular and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval s length

dipolynomial minimal bases and linear systems in ar representation abstract this paper deals with a module theoretic approach to the dipolynomial matrix parametrization of discrete time behavior systems as introduced by willems canonical minimal lag representations for these behaviors are constructed which are tighter than the corresponding polynomial representations

holder continuity of cumulative distribution functions for noncommutative polynomials under finite free fisher information abstract this paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory more precisely we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non microstates free fisher information highlighting the special case of lipschitz conjugate variables for the first time in this generality it is shown that the analytic distributions of those evaluations have holder continuous cumulative distribution functions with an explicit holder exponent that depends only on the degree of the considered polynomial for linear polynomials we reach in the case of finite non microstates free fisher information the optimal holder exponent 2 3 and get lipschitz continuity in the case of lipschitz conjugate variables in particular our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite non microstates free entropy which partially settles a conjecture of charlesworth and shlyakhtenko 8 we further provide a very general criterion that gives for weak approximations of measures having holder continuous cumulative distribution functions explicit rates of convergence in terms of the kolmogorov distance finally we combine these results to study the asymptotic eigenvalue distributions of polynomials in gues or matrices with more general gibbs laws for gibbs laws this extends the corresponding result obtained in 21 from convergence in distribution to convergence in kolmogorov distance in the gue case we even provide explicit rates which quantify results of 23 24 in terms of the kolmogorov distance

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

feedback canonical form for singular systems controllability and impulse controllability indices for singular systems are defined via the concept of minimal indices for modules over polynomial rings these indices are invariant under the transformations of restricted system equivalence and feedback equivalence they yield a characterization of controllability and impulse controllability of singular systems analogous to the state space situation and form a complete list of invariants for impulse controllable systems under feedback equivalence a feedback canonical form described by these invariants is constructed explicitly

intersections of schubert varieties and eigenvalue inequalities in an arbitrary finite factor the intersection ring of a complex grassmann manifold is generated by schubert varieties and its structure is governed by the littlewood richardson rule given three schubert varieties s1 s2 s3 with intersection number equal to one we show how to construct an explicit element in their intersection this element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags and is therefore well defined over an arbitrary field of scalars moreover this result also applies to appropriately defined analogues of schubert varieties in the grassmann manifolds associated with a finite von neumann algebra the arguments require the combinatorial structure of honeycombs particularly the structure of the rigid extremal honeycombs it is known that the eigenvalue distributions of self adjoint elements a b c with a b c 0 in the factor r are characterized by a system of inequalities analogous to the classical horn inequalities of linear algebra we prove that these inequalities are in fact true for elements of an arbitrary finite factor in particular if x y z are self adjoint elements of such a factor and x y z 0 then there exist self adjoint a b c r such that a b c 0 and a respectively b c has the same eigenvalue distribution as x respectively y z a complete matricial form of this result is known to imply an affirmative answer to an embedding question formulated by connes the critical point in the proof of this result is the production of elements in the intersection of three schubert varieties when the factor under consideration is the algebra of n n complex matrices our arguments provide new and elementary proofs of the horn inequalities which do not require knowledge of the structure of the cohomology of the grassmann manifolds

the single ring theorem we study the empirical measure lan of the eigenvalues of non normal square matrices of the form an untnvn with un vn independent haar distributed on the unitary group and tn real diagonal we show that when the empirical measure of the eigenvalues of tn converges and tn satisfies some technical conditions lan converges towards a rotationally invariant measure on the complex plane whose support is a single ring in particular we provide a complete proof of feinberg zee single ring theorem 6 we also consider the case where un vn are independent haar distributed on the orthogonal group

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of n n complex matrices a b under the simultaneous similarity tat 1 tbt 1 roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set m 0n 2 r mn mn mn the set of n n complex valued matrices here r and are two positive integers then we construct a finite number of rational functions o1 os in the entries of a and b whose values are constant on all pairs similar in m n 2 r to a b the values of the functions oi a b i 1 s determine a finite number at most n 2 r of similarity classes in m n 2 r let sn be the subspace of complex symmetric matrices in mn for a b sn sn we consider the similarity class tatt tbtt where t ranges over all complex orthogonal matrices then the characteristic polynomial i a xb determines a finite number of similarity classes for almost all pairs a b sn sn

the common structure of parametrizations for linear systems abstract a general framework for parametrizations of armax systems in the case of essentially affine restrictions is described for this general case topological and geometrical properties of the parametrizations are derived in a unified way our treatment includes for instance echelon canonical forms the overlapping description of the manifold of all transfer functions of order n and structural identifiability by affine restrictions also the case of polynomial zero restrictions is considered