the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

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

stability in terms of two measures for difference equations this paper establishes new stability criteria in terms of two measures for difference equations via vector lyapunov functions and a new comparison theorem by employing the concept of the upper quasi monotone increasing of the function which connects the solutions of two finite difference equations and a scalar one an example is given to illustrate the results obtained

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

determinantal representations of closed orbits abstract in this paper we show that the orbit of a point mass under a central force f r r 2 r 3 is realized as the hyperbolic curve f a 1 x y 0 associated with a nilpotent matrix a on the contrary we show that the orbit of motion of particles of infinitesimal mass in the gravitational field described by schwarzschild geodesic metric is transcendental in this case the transcendental orbit has no determinantal representations

criteria on boundedness in terms of two measures for volterra type discrete systems abstract we consider the boundedness for the discrete system of volterra types we establish several criteria on boundedness in terms of two measures for this system employing razumikhin techniques

on the asymptotic stability for nonautonomous functional differential equations by lyapunov functionals sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous fde s whose right hand sides can be unbounded functions of the time the theorems are based upon lyapunov krasovskii functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals the functionals and their derivatives are estimated by either x t the norm of the instantaneous value of the solutions or x t 2 the l 2 norm of the phase segment of the solutions examples are given to show that the conditions are sharp and the main theorems with the two different types of estimates are independent and improve earlier results the theorems are applied to linear and nonlinear retarded fde s with one delay and with distributed delays

on lusin s theorem for non additive measures that take values in an ordered topological vector space abstract lusin s theorem was established for real valued monotone measures under an equivalent condition to egoroff s theorem recently in this paper we show that the same result remains valid for non additive measures that take values in an ordered topological vector space we apply our result to several ordered topological vector spaces

stochastic convergence uniform integrability and convergence in mean on fuzzy measure spaces the paper deals with the convergence of sequences of measurable functions on fuzzy measure spaces a classical result that relates the uniform integrability plus stochastic convergence with convergence in mean is extended to this case where the additivity property of measures is not required we also investigate the sufficient and necessary conditions that fuzzy measures must satisfy in the dominated convergence theorem

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

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

precise damping conditions for global asymptotic stability for nonlinear second order systems global asymptotic stability of the rest point for nonlinear equations has been treated by a number of authors in this paper we unify and extend previous work by obtaining various necessary and sucient conditions for the asymptotic stability as t 1 of solutions of quasi variational systems of the form rl t u u 0 0 r ul t u u 0 q t u u 0

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

the algebraic degree of semidefinite programming given a generic semidefinite program specified by matrices with rational entries each coordinate of its optimal solution is an algebraic number we study the degree of the minimal polynomials of these algebraic numbers geometrically this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices we determine this degree using methods from complex algebraic geometry such as projective duality determinantal varieties and their chern classes

on a sufficient condition of lusin s theorem for non additive measures that take values in an ordered topological vector space classical lusin s theorem is extended for real valued fuzzy measures under the weak null additivity condition recently in this paper we show similar results for non additive measures that take values in an ordered topological vector space firstly we prove lusin type theorem for weakly null additive borel measures that are continuous from above and possess an additional continuity property suggested by sun in 1994 secondly we state another one for weakly null additive fuzzy borel measures our results are applicable to several ordered topological vector spaces

the relationship between structural characteristics of fuzzy measure abstract in this paper some new concepts property s 1 property ps property ps 1 are introduced the converse problems of convergence theorems such as egoroff s theorem and riesz s theorem are discussed some new results about the relationship between structural characteristics of fuzzy measure and convergences of sequences of measurable functions are obtained

the autocontinuity of set function and the fuzzy integral abstract the autocontinuity and some other concepts of a set function are introduced and sugeno s fuzzy measure with some annexed conditions is studied on a fuzzy measure space egoroffs theorem is proved it is also proved that f mean convergence is equivalent to convergence in measure for a sequence of measurable functions finally some convergence theorems of sequence of fuzzy integrals are proved especially using the concept of autocontinuity a necessary and sufficient condition for the convergence in measure theorem is given

on egoroff s theorems on fuzzy measure spaces in this paper we show that egoroff s theorem which is well known in classical measure theory remains valid for a fuzzy measure without any additional condition previous results obtained by wang zhenyuan and others are improved egoroff s theorem on fuzzy measure spaces is formulated in full generality taylor s theorem which concerns convergence almost everywhere in classical measure theory is also extended to fuzzy measure spaces

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

convergence theorems for monotone measures in classical measure theory there are a number of convergence theorems such as the egorov the riesz and the lusin theorem among others we consider monotone measures i e monotone set functions vanishing in the empty set and defined on a measurable space and discuss how and to which extent classical convergence theorems can be carried over to this more general case

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

reduction of the c numerical range to the classical numerical range abstract let a be an n n complex matrix and c c 1 c 2 c n a real n tuple the c numerical range of a is defined as the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis for c n when c 1 0 0 w c a becomes the classical numerical range of a which is often defined as the set w a x ax x c n x x 1 we show that for any n n complex matrix a and real n tuple c there exists a complex matrix b of size at most n such that w c a w b constructions of the matrix b for some matrices a and real n tuple c are provided

stability for impulsive differential systems in terms of two measures stability properties of impulsive differential systems are investigated using two different measures and employing piecewise continuous lyapunov functions

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

on sufficient conditions for the egoroff theorem of an ordered topological vector space valued non additive measure in this paper we consider an ordered vector space endowed with a locally full topology which is called an ordered topological vector space we show that the egoroff theorem remains valid for the ordered topological vector space valued non additive measure in the following four cases the first case is that the measure is strongly order totally continuous the second case is that the measure is strongly order continuous and possesses an additional continuity property suggested by sun in 1994 when the ordered topological vector space has a certain property the third case is that the measure is continuous from above and below when the topology is locally convex the fourth case is that the measure is uniformly autocontinuous from above continuous from below and strongly order continuous when the topology is locally convex our results are applicable to several ordered topological vector spaces

convergence of sequence of measurable functions on fuzzy measure spaces abstract the purpose of this paper is to investigate the convergence of sequence of measurable functions on fuzzy measure spaces several classical results on the convergence of measurable functions such as egoroff s theorem lebesgue s theorem and riesz s theorem are extended to fuzzy measure spaces by using the pseudometric generating property and order continuity of fuzzy measures

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

characteristic polynomials of symmetric matrices over the univariate polynomial ring abstract viewing a bivariate polynomial f r x t as a family of univariate polynomials in t parametrized by real numbers x we call f real rooted if this family consists of monic polynomials with only real roots if f is the characteristic polynomial of a symmetric matrix with entries in r x it is obviously real rooted in this article the converse is established namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring as a byproduct we present a purely algebraic proof of the helton vinnikov theorem which solved the 60 year old lax conjecture on the existence of definite determinantal representation of ternary hyperbolic forms

lusin s theorem on monotone measure spaces in this paper a kind of regularity of monotone measures is shown by using the pseudometric generating property of set function for short p g p and an equivalence condition for egoroff s theorem lusin s theorem which is well known in classical measure theory is generalized to monotone measure spaces the continuity from above and below of set functions is not required in the discussions therefore the previous results obtained by jiang et al are generalized an approximation of measurable functions by continuous functions on monotone measure spaces is presented

property s of fuzzy measure and riesz s theorem abstract in this note a sufficient and necessary condition for riesz s theorem to hold in the fuzzy measure space is given

on quasi stability for impulsive differential systems on developpe une theorie de la stabilite relativement a une solution donnee de systemes differentiels impulsifs une telle etude conduit a un mecanisme complique et les notions de stabilite doivent etre modifiees

generalized convergence theorems for monotone measures abstract in this paper we propose three types of absolute continuity for monotone measures and present some of their basic properties by means of these three types of absolute continuity we establish generalized egoroff s theorem generalized riesz s theorem and generalized lebesgue s theorem in the framework involving the ordered pair of monotone measures the egoroff theorem the riesz theorem and the lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case these three generalized convergence theorems include as special cases several previous versions of egoroff like theorem riesz like theorem and lebesgue like theorem for monotone measures respectively

conditions for egoroff s theorem in non additive measure theory abstract this paper gives necessary and or sufficient conditions for egoroff s theorem in non additive measure theory a necessary and sufficient condition described without measurable functions two sufficient conditions and a necessary condition one of the two sufficient conditions is strong order total continuity continuity at measurable sets of measure zero with respect to net convergence and the other is strong order continuity sequential continuity at measurable sets of measure zero together with property s the necessary condition is strong order continuity in addition the paper shows the following continuity from above and below which is a known sufficient condition and the above mentioned two sufficient conditions are independent of each other the disjunction of these three sufficient conditions is not a necessary condition if the underlying set is at most countable then strong order continuity is necessary and sufficient and generally strong order continuity is not sufficient

asymptotic structural characteristics of fuzzy measure and their applications abstract in fuzzy measure theory as sugeno s fuzzy measures lose additivity in general the concept almost which is well known in classical measure theory splits into two different concepts almost and pseudo almost in order to replace the additivity it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called waxing and waning and to introduce some new concepts such as autocontinuity converse autocontinuity and pseudo autocontinuity these concepts describe some asymptotic structural characteristics of a fuzzy measure in this paper by means of the asymptotic structural characteristics of fuzzy measure we also give four forms of generalization for both egoroff s theorem riesz s theorem and lebesgue s theorem respectively and prove the almost everywhere pseudo almost everywhere convergence theorem the convergence in measure pseudo in measure theorem of the sequence of fuzzy integrals in the last two theorems the employed conditions are not only sufficient but also necessary

resolvent operators for integral equations in a banach space conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a banach space the resolvent operator is similar to an evolution operator for nonautonomous differential equations in a banach space as in the finite dimensional case this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

the egoroff property and the egoroff theorem in riesz space valued non additive measure theory the egoroff theorem remains valid for any riesz space valued non additive measure which is strong order continuous and possesses a form of continuity called property s in the literature whenever the riesz space has the egoroff property this version of the egoroff theorem is also valid for any non additive measure with the property of uniform autocontinuity strong order continuity and continuity from below by assuming only the weak s distributivity which is weaker than the egoroff property

garding s theory of hyperbolic polynomials this paper presents a simple self contained account of garding s theory of hyperbolic polynomials together with a recent convexity result of bauschke guler lewis sendov and an inequality of gurvits this account begins by establishing some new results the first concerns the existence of a pointwise arrangement of the eigenvalues so that they become global real analytic functions the second asserts that the associated branches are independent of the choice of hyperbolic direction 2013 wiley periodicals inc

on the stability in terms of two measures of perturbed neutral functional differential equations different concepts of stability are presented in the literature to recall a few we name the usual stability partial stability and the conditional stability the overall concepts of stability can be put forward in the notion of stability in terms of two different measures following movchan 12 many authors have successfully developed the theory of stability in terms of two measures which became important in the investigation of converse theorems see 1 3 8 9 10 13

new construction of algebro geometric solutions to the camassa holm equation and their numerical evaluation an independent derivation of solutions to the camassa holm equation in terms of multi dimensional theta functions is presented using an approach based on fay s identities reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface the solutions are studied numerically for concrete examples also in the limit where the surface degenerates to the riemann sphere and where solitons and cuspons appear

determinantal representations of invariant hyperbolic plane curves abstract we study hyperbolic polynomials with nice symmetry and express them as the determinant of a hermitian matrix with special structure the goal of this paper is to answer a question posed by chien and nakazato in 2015 3 by properly modifying a determinantal representation construction of dixon 1902 5 we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix moreover if the polynomial is invariant under the action of the dihedral group of order n the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries

projective spectrum and cyclic cohomology abstract for a tuple a a 1 a 2 a n of elements in a unital algebra b over c its projective spectrum p a or p a is the collection of z c n or respectively z p n 1 such that the multi parameter pencil a z z 1 a 1 z 2 a 2 z n a n is not invertible in b b valued 1 form a 1 z d a z contains much topological information about p c a c n p a in commutative cases invariant multi linear functionals are effective tools to extract that information this paper shows that in non commutative cases the cyclic cohomology of b does a similar job in fact a chen weil type map from the cyclic cohomology of b to the de rham cohomology h d p c a c is established as an example we prove a closed high order form of the classical jacobi s formula

convergence in measure theorems of nonlinear integrals of functions integrable to the pth power abstract one can find a subadditive measure and a sequence f n n n of nonnegative measurable functions converging in measure to such a function f with the property that for any p 1 f n p no longer converges in measure to f p therefore in nonadditive measure theory an additional and intrinsic discussion is needed to prove integral convergence theorems for functions integrable to the pth power the purpose of the paper is to propose a method for automatically deriving pth power versions of convergence in measure theorems of nonlinear integrals from already established ones the proposed method is based on the robustness of some properties of integral functional under the operation f f p raising a nonnegative function f to the pth power

perturbing families of lyapunov functions and stability in terms of two measures on considere les proprietes de stabilite non uniforme en termes de deux mesures en employant des familles perturbantes de fonctions de lyapunov

finite dimensional de branges spaces on riemann surfaces abstract we study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real riemann surface these spaces are analogues of the spaces introduced by l de branges when the riemann sphere is replaced by a compact real riemann surface of a higher genus in de branges theory an important role is played by resolvent like difference quotient operators r here we introduce generalized difference quotient operators r y for any non constant meromorphic function y on the riemann surface the spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators r y 1 i and r y 2 2 where y 1 and y 2 generate the field of meromorphic functions on the riemann surface which satisfy a supplementary identity analogous to the de branges identity for difference quotients just as the classical de branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint or nonunitary operator the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks

the smallest semicopula based universal integrals ii in this paper we continue studying the smallest universal integral i s having s as the underlying semicopula we present convergence theorems for i s integral sequences including monotone almost everywhere almost uniform in measure and in mean converging sequences of measurable functions respectively it emerges that these convergences characterize the underlying measure properties such as null additivity monotone autocontinuity and autocontinuity we provide many examples and counter examples as well as a few interesting open problems

on the pseudo autocontinuity of fuzzy measures abstract the property ps of fuzzy measures is introduced a set of characterizations of the pseudo autocontinuity of fuzzy measures are given the equivalence of pseudo autocontinuity from above and pseudo autocontinuity from below is established the convergence pseudo in measure is investigated

detecting rigid convexity of bivariate polynomials abstract given a polynomial x r n p x in n 2 variables a symbolic numerical algorithm is first described for detecting whether the connected component of the plane sublevel set p x p x 0 containing the origin is rigidly convex or equivalently whether it has a linear matrix inequality lmi representation or equivalently if polynomial p x is hyperbolic with respect to the origin the problem boils down to checking whether a univariate polynomial matrix is positive semidefinite an optimization problem that can be solved with eigenvalue decomposition when the variety c x p x 0 is an algebraic curve of genus zero a second algorithm based on bezoutians is proposed to detect whether p has an lmi representation and to build such a representation from a rational parametrization of c finally some extensions to positive genus curves and to the case n 2 are mentioned

limit shapes and the complex burgers equation in this paper we study surfaces in r 3 that arise as limit shapes in random surface models related to planar dimers these limit shapes are surface tension minimizers that is they minimize a functional of the form h dx dy among all lipschitz functions h taking given values on the boundary of the domain the surface tension has singularities and is not strictly convex which leads to formation of facets and edges in the limit shapes we find a change of variables that reduces the euler lagrange equation for the variational problem to the complex inviscid burgers equation complex hopf equation the equation can thus be solved in terms of an arbitrary holomorphic function which is somewhat similar in spirit to weierstrass parametrization of minimal surfaces we further show that for a natural dense set of boundary conditions the holomorphic function in question is in fact algebraic the tools of algebraic geometry can thus be brought in to study the minimizers and especially the formation of their singularities this is illustrated by several explicitly computed examples

asymptotic stability in functional differential equations by liapunov functionals we consider the asymptotic stability in a system of functional differential equations x t f t xt by liapunov functionals v the work generalizes some well known results in the literature in that we only require the derivative of v to be negative definite on a sequence of intervals i sn tn we also show that it is not necessary to require a uniform upper bound on v for nonuniform asymptotic stability

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

elementary transformations of determinantal representations of algebraic curves abstract we study elementary transformations first introduced by livsic and kravitsky in an operator theoretic context of determinantal representations of algebraic curves we consider determinantal representations of a smooth irreducible curve f over an algebraically closed field when regarded as acting on the corresponding vector bundle the elementary transformations are a matrix generalization of scalar linear fractional transformations they add to the class of divisors of the vector bundle a single zero and a single pole we show that given a determinantal representation of f we can build all the nonequivalent determinantal representations of f by applying to the given representation finite sequences of at most g elementary transformations where g is the genus of f

planar dimers and harnack curves in this article we study the connection between dimers and harnack curves discovered in 15 we prove that every harnack curve arises as a spectral curve of some dimer model we also prove that the space of harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates we characterize harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their ronkin function with given boundary conditions

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

stability criteria for impulsive differential equations in terms of two measures in general the stability notions of a given solution d t of an impulsive differential system cannot be transferred to the stability notions of the trivial solution by change of variables this is because of the fact that moments of impulse effects of 4 t need not be the same as that of a different solution x t and therefore demanding that the difference lx t d t 1 be small for all future time seems unnatural see 4 73 as a result the definitions of stability require suitable changes in the general situations and the corresponding theory has not yet been developed however in the case of impulsive differential systems with fixed moments of impulse effects such a problem does not arise and hence the standard definitions of stability will suffice employing suitable discontinuous lyapunov functions stability theory has recently been developed for such systems 1 2 3 51 in this paper we shall discuss stabiliy theory of impulsive differential systems with fixed moments of impulse effects in terms of two measures since such a theory enables us to unify a variety of stability results found in the literature we shall employ theory of impulsive differential inequalities as well as a direct approach to bring out the advantages of each approach

set valued lusin type theorem for null null additive set multifunctions in a previous paper we presented a set valued egoroff type theorem for monotone set multifunctions in this paper a new version of this set valued egoroff type theorem is obtained it enables a set valued lusin type theorem for null null additive monotone continuous set multifunctions to be established several applications of this lusin type theorem are given pointing out some relationships among m measurability a measurability and totally measurability of real functions we also compare our new lusin type theorem with lusin s theorem previously obtained by the second author

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

invariance and efficiency of convex representations we consider two notions for the representations of convex cones g representation and lifted g representation the former represents a convex cone as a slice of another the latter allows in addition the usage of auxiliary variables in the representation we first study the basic properties of these representations we show that some basic properties of convex cones are invariant under one notion of representation but not the other in particular we prove that lifted g representation is closed under duality when the representing cone is self dual we also prove that strict complementarity of a convex optimization problem in conic form is preserved under g representations then we move to study efficiency measures for representations we evaluate the representations of homogeneous convex cones based on the smoothness of the transformations mapping the central path of the representation to the central path of the represented optimization problem

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

stability criteria in terms of two measures for discrete systems abstract this paper studies stability problems for two kinds of discrete systems in terms of two different measures the basic idea of this approach is to discuss the qualitative behaviour of a map h along solutions of discrete systems whose initial value are measured by a second map h 0 by doing this one may deal with in a unified way several concepts and associated problems which are usually considered separately stability criteria in terms of two measures are established employing the method of lyapunov functions and razumikhin techniques