on robust hurwitz polynomials in this note kharitonov s theorem on robust hurwitz polynomials is simplified for low order polynomials specifically for n 3 4 and 5 the number of polynomials required to check robust stability is one two and three respectively instead of four furthermore it is shown that for n geq 6 the number of polynomials for robust stability checking is necessarily four thus further simplification is not possible the same simplifications arise in robust schur polynomials by using the bilinear transformation applications of these simplifications to two dimensional polynomials as well as to robustness for single parameters are indicated

smoothness of equisingular families of curves francesco severi 1921 showed that equisingular families of plane nodal curves are t smooth i e smooth of the expected dimension whenever they are non empty for families with more complicated singularities this is no longer true given a divisor d on a smooth projective surface it thus makes sense to look for conditions which ensure that the family v irr d s 1 s r of irreducible curves in the linear system d l with precisely r singular points of types s 1 s r is t smooth considering different surfaces including the projective plane general surfaces in p 3 c products of curves and geometrically ruled surfaces we produce a sufficient condition of the type e t i 1 s i d k 2 where is some invariant of the singularity type and is some constant this generalises the results of greuel lossen and shustin 2001 for the plane case combining their methods and the method of bogomolov instability for many singularity types the invariant leads to essentially better conditions than the invariants used by greuel lossen and shustin 1997 and for most classes of geometrically ruled surfaces our results are the first known for t smoothness at all

the kharitonov theorem with degree drop the purpose of this paper is to present a proof of the kharitonov theorem based on bezoutians an interesting consequence of this proof is that it shows the validity of kharitonov s result in the presence of a degree drop

robust stability of a class of polynomials with coefficients depending multilinearly on perturbations necessary and sufficient conditions are given for robust stability of a family of polynomials each polynomial is obtained by a multilinearity perturbation structure restrictions on the multilinearity are involved but in contrast to existing literature these restrictions are derived from physical considerations stemming from analysis of a closed loop interval feedback system the main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half plane if and only if two requirements are satisfied at each frequency the first requirement is the zero exclusion condition involving four kharitonov rectangles the second requirement is that a specially constructed theta 0 parameterized set of 16 intervals must cover the positive reals for each theta epsilon 0 2 pi

linear syzygies of curves with prescribed gonality abstract we prove two statements concerning the linear strand of the minimal free resolution of a k gonal curve c of genus g firstly we show that a general curve c of genus g of non maximal gonality k g 1 2 satisfies schreyer s conjecture that is b g k 1 c c g k this statement goes beyond green s conjecture and predicts that all highest order linear syzygies in the canonical embedding of c are determined by the syzygies of the k 1 dimensional scroll containing c secondly we prove an optimal effective version of the gonality conjecture for general k gonal curves which makes more precise the asymptotic gonality conjecture proved by ein lazarsfeld and improves results of rathmann

a sufficient condition for the stability of interval matrix polynomials the root location of sets of scalar polynomials whose coefficients are confined to intervals and the associated extension to eigenvalues of sets of constant matrices whose coefficients are contained in intervals are reviewed a central result for complex scalar interval polynomials is a theorem developed by v l kharatonov 1978 which states that each member of a set of such polynomials is stable if and only if eight special polynomials from the set are stable the case of interval matrix polynomials is examined and a kharitonov like result for their strong stability is provided this in turn yields a sufficient condition for stability of a set of interval matrix polynomials

robust stability of control systems extreme point results for the stability of edges for the investigation of the robust stability of control systems with structured uncertainties many results have been presented recently that lead to stability tests of the edges of a polytope in this paper some results are discussed where the stability of an edge is guaranteed by the stability of the vertices of the edge given the characteristic polynomial of a closed loop system with structured uncertainties in the parameters for special types of uncertainties the family of polynomials is defined by a polytope in the space of parameters in the frequency domain the corresponding value set of the family for s jw fixed is a polygon whose edges are formed by the so called exposed edges of the polytope for the investigation of the stability of the polynomial family these exposed edges must be tested kraus and truol 1991 a this test can be simplified if an edge has the vertex property i e if the stability of the edge is guaranteed by the stability of the two vertices

unfoldings in knot theory in in r the link at infinity of a hypersurface v cie r defined by a good polynomial map f ty ii2 was used as an extended example f was called good if it had only isolated singularities and it was claimed that the link at infinity then always has a milnor fibration this is incorrect although we have found it to be a common misconception to correct it the definition of good must be modified as follows

the orevkov invariant of an affine plane curve we show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute it possesses a more accessible quotient which we call the orevkov invariant

a simple proof of kharitonov s theorem an alternative proof of kharitonov s theorem on the stability of linear time invariant continuous systems under parameter variations is presented

robust strict positive realness new results for interval plant controller families a frequency domain approach is used to derive several robust strict positive realness results for interval plants and interval plant plus controller families of transfer functions based on simple frequency domain properties of transfer functions the approach provides a framework for obtaining new results and constructing easy proofs of several important existing results on robust strict positive realness the main new result states that the minimum of the real part of the transfer functions belonging to an interval plant controller family is achieved on one of the 32 kharitonov segments of the interval plant the argument used in the proof is of wider interest and suggests easy ways of proving that robustness of other different frequency properties of interval plant plus controller families of transfer functions such as robust stability of h sub infinity norm computation can be deduced from a fixed number of segments of transfer functions of the family

modified mikhailov plots for robust absolute stability with non parametric perturbations and uncertain nonlinearity this paper generalizes the classical mikhailov stability result for testing robust absolute stability of linear systems with both non parametric perturbations and uncertain nonlinearity three types of non parametric perturbations are considered additive multiplicative and stable factor perturbations modified mikhailov plots are developed for simultaneously testing the stability of the nominal system and computing the maximum robust absolute stability margin

on the robust popov criterion for interval lur e system the classical popov criterion is generalized to include problems with parametric uncertainty as well as nonlinear sector bounded perturbations the parametric dependence is modeled by an interval plant description a robust popov criterion is obtained for a plant controller configuration stronger external results are obtained for a special class of controllers an example illustrating the numerical implementation of these results is presented

application of kharitonov s theorem to mechanical systems kharitonov s theorem is used to derive a robust stability condition for proportional integral derivative pid controlled multi degree of freedom mechanical systems the characteristic equation of such a system is given as a determinant of a third order polynomial with matrix coefficients from which a scalar interval polynomial is obtained a simple procedure for designing pid controllers for these mechanical systems is described and a new kharitonov like result which states roughly that a controller designed for an upper bounding inertia matrix results in stable set point regulation for all other inertias is proved

uniruledness of strata of holomorphic differentials in small genus abstract we address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials we show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on k3 and del pezzo surfaces respectively restricting to genus 3 g 6 we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g 1 hence showing in addition that these strata are unirational

invariance of a strict hurwitz property for uncertain polynomials with dependent coefficients this note is concerned with the invariance of strict hurwitz property for uncertain polynomials having dependent coefficients a feasible test criterion is provided which reduces the conservatism of the kharitonov theorem

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

stability analysis of polynomials with coefficients in disks the aim of this note is to report results on the stability of a class of polynomials from the small gain theorem point of view the authors consider families of polynomials whose coefficients lie in closed circular disks around their nominal values various measures of variation of polynomial coefficients around their nominal value are considered and in each case necessary and sufficient conditions are presented for stability of the resulting family of polynomials the stability region could be any closed region of the complex plane based on similar ideas of small gain the authors also provide sufficient conditions for testing the stability of systems with commensurate time delays and for two dimensional type systems these conditions become both necessary and sufficient in some special cases all tests are easy to implement and require checking the stability of a matrix or equivalently checking the stability of the central polynomial and evaluation of a norm

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

robustness analysis of interval matrices based on kharitonov s theorem this paper deals with the robustness analysis problem of interval matrices expansion of det a b is used to get characteristic polynomials of two corresponding systems kharitonov s theorem is applied to test hurwitz properties of the two polynomials a new sufficient condition for all eigenvalues of the interval matrix to lie in a damping cone on complex plan is derived illustrative examples are given

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

frequency response of uncertain systems with interval plants frequency domain properties of uncertain rational functions depending on two independent interval polynomials are investigated several simplifications for efficient computation of the envelope of the nyquist plots of the uncertain family are reported stronger computational reduction results are given for the bode magnitude and phase plots of an interval plant controller family of transfer functions these results allow for a clear understanding of the extremality properties of several frequency domain performance specifications commonly used in control system design

on stability of quasipolynomials with annular uncertainties and interval commensurate delays the paper concerns the hurwitz stability of a family of quasipolynomials with commensurate delays each coefficient of the quasipolynomials belongs to a prescribed annulus in the complex plane and the delay belongs to a prescribed real interval a computationally tractable robust stability criterion is the main result of the paper

quasi isometric classification of graph manifold groups we show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi isometric we also classify the quasi isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs a corollary is the quasi isometric classification of artin groups whose presentation graphs are trees in particular any two right angled artin groups whose presentation graphs are trees of diameter greater than 2 are quasi isometric further this quasi isometry class does not include any other right angled artin groups a finitely generated group can be considered geometrically when endowed with a word metric up to quasi isometric equivalence such metrics are unique in this article only finitely generated groups are considered given a collection of groups g gromov 11 proposed the fundamental questions of identifying which groups are quasi isometric to those in g rigidity and which groups in g are quasi isometric to each other classification in this article we focus on the classification question for graph manifold groups and right angled artin groups a compact 3 manifold m is called geometric if m m admits a geometric structure in the sense of thurston i e a complete locally homogeneous riemannian metric of finite volume thurston s geometrization conjecture see 29 17 21 23 22 provides that every irreducible 3 manifold of zero euler characteristic i e with boundary consisting only of tori and klein bottles admits a decomposition along tori and klein bottles into geometric pieces the minimal such decomposition being called the geometric decomposition there is a considerable literature on quasi isometric rigidity and classification of 3 manifold groups the rigidity results can be briefly summarized in the following form

robust stability of a family of disc polynomials in his well known theorem v l kharitonov established that hurwitz stability of a set f1 of interval polynomials with complex coefficients polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane is equivalent to the hurwitz stability of only eight polynomials in this set in this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fd of disc polynomials characterized by the fact that each coefficient of a typical element p s in fd can be any complex number in an arbitrary but fixed disc of the complex plane our result shows that the entire set is hurwitz stable if and only if the center polynomial is stable and the h norms of two specific stable rational functions are less than one our result can be readily extended to deal with the schur stability problem and the resulting condition is equally simple

geometry of families of nodal curves on the blown up projective plane let p2r be the projective plane blown up at r generic points denote by e0 e1 er the strict transform of a generic straight line on p2 and the exceptional divisors of the blown up points on p2r respectively we consider the variety virr of all irreducible curves c with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness irreducibility and non emptiness moreover we extend our conditions for the smoothness and the irreducibility on families of reducible curves for r 9 we give the complete answer concerning the existence of nodal curves in virr

the homflypt skein algebra of the torus and the elliptic hall algebra we give a generators and relations presentation of the homflypt skein algebra h of the torus t2 and we give an explicit description of the module corresponding to the solid torus using this presentation we show that h is isomorphic to the 1 specialization of the elliptic hall algebra of burban and schiffmann as an application for an iterated cable k of the unknot we use the elliptic hall algebra to construct a 3 variable polynomial that specializes to the colored homflypt polynomial of k we show that this polynomial also specializes to one constructed by cherednik and danilenko using the gln double affine hecke algebra this proves one of the connection conjectures in their recent work

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

a new parameterization of stable polynomials in this note we develop a new characterization of stable polynomials specifically given n positive ordered numbers frequencies we develop a procedure for constructing a stable degree n monic polynomial with real coefficients this construction can be viewed as a mapping from the space of n ordered frequencies to the space of stable degree n monic polynomials the mapping is one one and onto thereby giving a complete parameterization of all stable degree n monic polynomials we show how the result can be used to generate parameterizations of stabilizing fixed order proper controllers for unity feedback systems we apply these results in the development of stability margin lower bounds for systems with parameter uncertainty

frequency domain criteria for l sup p robust stability of continuous linear systems necessary and sufficient criteria in the frequency domain for robust stability are given under the assumption that coefficients of a characteristic polynomial belong to a transformed l sup p ball three cases are considered in detail p infinity interval uncertainty p 2 ellipsoidal uncertainty and p 1 octahedral uncertainty it is shown that frequency domain stability criteria are efficient when one deals with robust stability problems the frequency domain approach considered can be extended to various problems of robust stability

robust hurwitz l sup p stability of polynomials with complex coefficients necessary and sufficient conditions for robust stability of a family of polynomials with complex coefficients are provided this work generalizes a frequency domain criteria for robust stability of continuous linear systems reported on by y z tsypkin and b t polyak 1991

moduli of nodal curves on k3 surfaces abstract we consider modular properties of nodal curves on general k3 surfaces let k p be the moduli space of primitively polarized k3 surfaces s l of genus p 3 and v p m k p be the universal severi variety of nodal irreducible curves in m l on s l k p we find conditions on p m for the existence of an irreducible component v of v p m on which the moduli map v m g with g m 2 p 1 1 has generically maximal rank differential our results which for any p leave only finitely many cases unsolved and are optimal for m 5 except for very low values of p are summarized in theorem 1 1 in the introduction

complex algebraic plane curves via their links at infinity considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew surprisingly little is known about the topology of affine algebraic plane curves we topologically classify regular algebraic plane curves in complex affine 2 space using splice diagrams certain decorated trees that code puiseux data at infinity the regularity condition that the curve be a typical fiber of its defining polynomial can conjecturally be avoided we also show that the splice diagram determines such algebraic information as the minimal degree of the curve even in the irregular case among other things this enables algebraic classification of regular algebraic plane curves with given topology

the alexander polynomial of a plane curve singularity and the ring of functions on it we prove two formulae that express the alexander polynomial delta sp c of several variables of a plane curve singularity c in terms of the ring mathscr o sb c of germs of analytic functions on the curve one of them expresses delta sp c in terms of dimensions of some factors corresponding to a multi indexed filtration on the ring mathscr o sb c the other one gives the coefficients of the alexander polynomial delta sp c as euler characteristics of some explicitly described spaces complements to arrangements of projective hyperplanes

robustness analysis and synthesis of siso systems under both plant and controller perturbations abstract in this paper we consider the robustness analysis and synthesis problem for a class of control systems under both plant and controller perturbations necessary and sufficient conditions are given for robust stability of siso systems subject to additive norm bounded perturbations linear matrix inequality lmi based controller design method is presented robust stability of systems with both plant and controller perturbations and stability analysis of a class of multilinear disc polynomials are unified in the same framework and can be solved by computing the largest norm of a second order matrix along the imaginary axis

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

an alternative proof of kharitonov s theorem an alternative proof is presented of kharitonov s theorem for real polynomials the proof shows that if an unstable root exists in the interval family then another unstable root must also show up in what is called the kharitonov plane which is delimited by the four kharitonov polynomials this fact is proved by using a simple lemma dealing with convex combinations of polynomials then a well known result is utilized to prove that when the four kharitonov polynomials are stable the kharitonov plane must also be stable and this contradiction proves the theorem

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

nondefinite least squares and its relation to h sub infinity minimum error state estimation the problem of recursive nondefinite least squares state estimation of continuous time stationary processes is solved by applying pontryagin s maximum principle a comparison of the derived solution to the result that is obtained for the h sub infinity minimum error estimation suggests a new interpretation for the h sub infinity optimal estimation mechanism according to this interpretation the estimator tries to optimally estimate the required combination of the states in the l sub 2 norm sense against the worst disturbance signal that stems from a fictitious measurement of this combination

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling

modular properties of nodal curves on k3 surfaces in this article we are going to address the following issues 1 the first is a rigidity property for pairs s c consisting of a general projective k3 surface s and a curve c obtained as the normalization of a nodal hyperplane section hat c hookrightarrow s we prove that a non trivial deformation of a pair s c induces a non trivial deformation of c 2 the second question concerns the wahl map of curves c obtained as above we prove that the wahl map of the normalization of a nodal curve contained in a general projective k3 surface is non surjective in both cases we impose upper bounds on the number of nodes of hat c

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients