pseudo arithmetical operations as a basis for the general measure and integration theory several generalizations of the classical measure and integration theory are based on some generalizations of the standard arithmetical operations the axiomatic approach to the pseudo arithmetical operations of pseudo addition and pseudo multiplication is discussed some of required properties strongly influence the structure of these operations and consequently the resulting measure and integral generalizations so e g the idempotency of the unit element u results to the idempotency of the pseudoaddition i e v sup several other properties of and and their consequences are discussed and illustrated by examples

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

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

generalized fuzzy integrals part i fundamental concepts abstract in this paper we introduce a definition of fuzzy integral which is a generalization of both s fuzzy integral defined by m sugeno and n fuzzy integral due to zhao ruhuaiand discuss the fundamen tal properties of this kind of fuzzy integral

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

fast low rank modifications of the thin singular value decomposition this paper develops an identity for additive modivations of a singular value decomposition svd to reflect updates downdates shifts and edits of the data matrix this sets the stage for fast and ememory efficient sequential algorithms for tracking singular values and subspaces in conjunction with a fast solution for the pseudo inverse of a submatrix of an orthogonal matrix we develop a scheme for computing a thin svd of streaming data in a single pass with linear time complexity a rank r think svd of a p x q matrix can be computed in o pqr time for r less than or equal sqroot min p q linear algebra and its applications

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

a note on convergence properties of interval valued capacity functionals and choquet integrals based on the classical theory of random sets feng and nguyen 2007 5 studied the convergence of capacity functionals of random sets in terms of choquet integrals in this paper we consider an interval valued capacity functional which is motivated by the goal to generalize a capacity functional and the choquet integral with respect to an interval valued capacity in particular we discuss some convergence theorems for interval valued capacity functionals and interval valued probability measures in the hausdorff metric and choquet weak sense

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

representations of lie algebras by non skewselfadjoint operators in hilbert space we study non skewselfadjoint representations of a finite dimensional real lie algebra gg to this end we embed a non skewselfadjoint representation of gg into a more complicated structure that we call a gg operator vessel and that is associated to an overdetermined linear conservative input state output system on the corresponding simply connected lie group gg we develop the frequency domain theory of the system in terms of representations of gg and introduce the joint characteristic function of a gg operator vessel which is the analogue of the classical notion of the characteristic function of a single non selfadjoint operator as the first non commutative example we apply the theory to the lie algebra of the ax bax b group the group of affine transformations of the line

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

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

on the representation of choquet integrals of set valued functions and null sets in this paper we consider the definition of set valued choquet integrals we obtain some properties and the representation of set valued choquet integrals and also we discuss some fundamental relations between null sets and set valued choquet integrals

autocontinuity convergence in measure and convergence in distribution abstract it is shown that the autocontinuity is equivalent to the property that the convergence in measure implies the convergence in distribution this result is applied to denneberg s dominated convergence theorem for the choquet integral

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

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

a note on the convergence theorem of generalized fuzzy integrals in the note we prove a new convergence theorem of generalized fuzzy integrals for the decreasing sequence of nonnegative measurable functions which improves and generalizes the corresponding results of wu et al fuzzy sets and systems 70 1995 75 wang and klir fuzzy measure theory plenum new york 1992

on the convergence of sequences of fuzzy measures and generalized convergence theorems of fuzzy integrals abstract in this paper we will investigate further fuzzy measures and fuzzy integrals given by sugeno 1974 our purpose is concentrated on the convergence of sequences of fuzzy measures or fuzzy integrals some concepts such as convergence uniform convergence convergence in integral weak convergence are introduced for the sequence of fuzzy measures then the properties and relations of them are discussed furthermore various kinds of convergence theorems of fuzzy integrals are extended these are generalized monotone convergence theorem generalized fatou s lemma etc

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

a note on the interval valued generalized fuzzy integral by means of an interval representable pseudo multiplication and their convergence properties in this paper we consider the generalized fuzzy integral which is a fuzzy integral based on the idempotent pseudo addition supremum and a pseudo multiplication studied by xie and fang in 2006 the purpose of this study is to define the interval valued generalized fuzzy integral with respect to a fuzzy measure by means of an interval representable pseudo multiplication of measurable interval valued functions and to investigate some characterizations and convergence properties of them

the complexity of primal dual fixed point methods for ridge regression abstract we study the ridge regression l 2 regularized least squares problem and its dual which is also a ridge regression problem we observe that the optimality conditions describing the primal and dual optimal solutions can be formulated in several different but equivalent ways the optimality conditions we identify form a linear system involving a structured matrix depending on a single relaxation parameter which we introduce for regularization purposes this leads to the idea of studying and comparing in theory and practice the performance of the fixed point method applied to these reformulations we compute the optimal relaxation parameters and uncover interesting connections between the complexity bounds of the variants of the fixed point scheme we consider these connections follow from a close link between the spectral properties of the associated matrices for instance some reformulations involve purely imaginary eigenvalues some involve real eigenvalues and others have all eigenvalues on the complex circle we show that the deterministic quartz method which is a special case of the randomized dual coordinate ascent method with arbitrary sampling recently developed by qu richtarik and zhang can be cast in our framework and achieves the best rate in theory and in numerical experiments among the fixed point methods we study remarkably the method achieves an accelerated convergence rate numerical experiments indicate that our main algorithm is competitive with the conjugate gradient method

some properties of choquet integrals of set valued functions abstract in this paper we will define choquet integrals of set valued functions then we show some properties of choquet integrals of set valued functions

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

an algebraic geometry study of the b c system with arbitrary twist fields and arbitrary statistics we present an analysis of the generalb c system including the system on a compact riemann surface of arbitrary genusg 0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion ope of the system studying a very in fact optimally general form of theb c system we prove rigorously that the standard practice of eliminating zero modes and even the standard lagrangian follow from the analyticity structure dictated by the ope alone we extend the analysis to consider the most general case of the presence of twist e g spin fields we then determine all the possible correlation functions of theb c system with statistics unspecified compatible with the ope on imposing fermi and bose statistics we obtain the correlation functions of the fermionicb c and systems respectively

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

fuzzy integrals of fuzzy valued functions abstract this paper based on the fuzzy measures and integrals given by sugeno first defines fuzzy integrals of set valued functions and then the fuzzy integral of fuzzy valued functions is obtained and discussed next it defines the fuzzy integrals of fuzzy mappings then pays much effort to the fuzzy integral of interval valued functions the properties and convergence theorems are obtained all these are generalizations of earlier results

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

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

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

corrections and remarks to the paper in fuzzy and systems 124 2001 117 123 the purpose of this note is to correct two convergence theorems for the generalized fuzzy integrals given in fang on the convergence theorems of generalized fuzzy integral sequence fuzzy sets and systems 124 2001 117 123

on the convergence theorems of generalized fuzzy integral sequence in this paper we give a counterexample showing that two convergence theorems of generalized fuzzy integrals given by wu et al fuzzy sets and systems 70 1995 75 87 is not valid and modify them to the right forms

interpolation problems for holomorphic functions abstract in this note we discuss the nevanlinna pick problem corona problem and caratheodory fejer problem for bounded holomorphic functions

some properties of sequences of generalized fuzzy integrable functions in this paper we continue with the investigation of the generalized fuzzy integral for short g fuzzy integral which is introduced by wu et al in 1993 generalized fuzzy integrals part 1 fundamental concepts fuzzy sets and systems 57 1993 219 226 some properties of g fuzzy integrable functions are obtained the relations between convergence in fuzzy measure and mean convergence in g fuzzy integral for a sequence of non negative measurable functions are discussed moreover we also study the uniform integrability of a sequence of g fuzzy integrable functions the uniformly absolute continuity and the uniform boundedness of a sequence of g fuzzy integrals

generalized fuzzy integrals part 3 convergent theorems abstract in this part we continue the work of preceding papers to consider all kinds of convergent theorems of g fuzzy integrals

generalized pseudo convolution in the theory of probabilistic metric spaces information fuzzy numbers optimization system theory a notion of the generalized pseudo convolution of functions based on pseudo operations is introduced it is proved that such general pseudo convolution is commutative associative operation that in some cases has a unit element we present that the generalized pseudo convolution occurs as a basic notion in many different theories as probabilistic metric spaces information theory fuzzy numbers optimization system theory

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

a note on the pseudo stolarsky type inequality for the g integral daraby 2012 proved the pseudo stolarsky type inequality for pseudo integrals and mesiar and pap 1999 defined the pseudo integrals with respect to a sup or int decomposable measure in this paper we defined the interval valued pseudo integral for short g integral in the case of a generated semiring with an interval valued generator g by using interval representable pseudo operations and prove the pseudo stolarsky type inequality for the g integrals

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra