calculation of exact bounds for the solution set of linear interval systems abstract this paper presents some topological and graph theoretical properties of the solution set of linear algebraic systems with interval coefficients based on these properties we describe a method which in a finite number of steps either calculates exact bounds for each component of the solution set or finds a singular matrix within the interval coefficients the calculation of exact bounds of the solution set is known to be np hard our method needs p calls of a polynomial time algorithm where p is the number of nonempty intersections of the solution set with the orthants frequently due to physical or economical requirements many variables do not change the sign in those cases p is small and our method works efficiently

algebraic volume density property of affine algebraic manifolds we introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it in particular that it implies the volume density property the main results of the paper are producing two big classes of examples of stein manifolds with volume density property one class consists of certain affine modifications of n equipped with a canonical volume form the other is the class of all linear algebraic groups equipped with the left invariant volume form

possibilistic linear systems and their application to the linear regression model abstract it is the purpose of this paper to illustrate possibilistic linear systems based on possibility measure and to formulate a linear regression analysis by possibilistic linear models linear regression by a possibilistic model is called possibilistic linear regression this is a new interpretation of fuzzy linear regression and also includes a new method by which interval analysis can be done in fuzzy numbers

fuzzy linear regression and its applications to forecasting in uncertain environment abstract linear regression has been used for many years in developing mathematical models for application in marketing management and sales forecasting in this paper two different sales forecasting techniques are discussed the first technique involves non fuzzy abstract methods of linear regression and econometrics a study of the international market sales of cameras done in 1968 by john scott armstrong utilized these non fuzzy forecasting techniques the second sales forecasting technique uses fuzzy linear regression introduced by h tanaka s uejima and k asai in 1980 in this paper a study of the computer and peripheral equipment sales in the united states is discussed using fuzzy linear regression moreover fuzzy linear regression is applied to forecasting in an uncertain environment finally some possible improvements and suggestions for further study are mentioned

on the solution sets of particular classes of linear interval systems we characterize the solution set s of real linear system ax b by a set of inequalities if b lies between some given bounds b b and if the n n coefficient matrix a varies similarly between two bounds a and a in addition we restrict a to a particular class of matrices for instance the class of the symmetric the skew symmetric the persymmetric the toeplitz and the hankel matrices respectively in this way we generalize the famous oettli prager criterion numer math 6 1964 405 results by hartfiel numer math 35 1980 355 and the contents of the papers in r b kearfott v kreinovich eds applications of interval computations kluwer boston ma 1996 pp 61 79 and siam j matrix anal appl 18 1997 693

maximal inner boxes in parametric ae solution sets with linear shape we consider linear systems of equations a p x b p where the parameters p are linearly dependent and come from prescribed boxes and the sets of solutions defined in various ways which have linear boundary one fundamental problem is to compute a box being inside a parametric solution set we first consider parametric tolerable solution sets being convex polyhedrons for such solution sets we prove that finding a maximal inner box is an np hard problem this justifies our exponential linear programming methods for computing maximal inner boxes we also propose a polynomial heuristic that yields a large but not necessarily the maximal inner box next we discuss how to apply the presented linear programming methods for finding large inner estimations of general parametric ae solution sets with linear shape numerical examples illustrate the properties of the methods and their application

on optimal solution of interval linear equations for interval linear algebraic systems bf a x bf b we consider the problem of component wise evaluation of the set sigma exists exists bf a bf b a 1 b mid a in bf a b in bf b formed by all solutions of ax b when a and b vary independently in bf a and bf b respectively an iterative pss algorithm is introduced that computes optimal exact componentwise estimates of sigma exists exists and its convergence is proved under fairly general conditions on the interval system we introduce the concept of a sequentially guaranteeing algorithm as a reasonable compromise between the requirements for the interval result to be guaranteed and to be obtained in a practically acceptable time

preordered forests packed words and contraction algebras abstract we introduce the notions of preordered and heap preordered forests generalizing the construction of ordered and heap ordered forests we prove that the algebras of preordered and heap preordered forests are hopf for the cut coproduct and we construct a hopf morphism to the hopf algebra of packed words moreover we define another coproduct on the preordered forests given by the contraction of edges finally we give a combinatorial description of morphism defined on hopf algebras of forests with values in the hopf algebras of shuffles or quasi shuffles

large discrete sets in stein manifolds we construct infinite discrete subsets of stein manifolds with remarkable properties these generalize results of rosay and rudin on discrete subsets of mathbb c n

solving interval linear systems with linear programming techniques abstract in this paper we show how it is possible to use convex polyhedra for solving linear interval systems without using preconditioning we first show how to derive from an enclosure of a b a polyhedron which contains the convex hull of the solution set then a simplex like method enables us to find a new outer inclusion moreover the constraints obtained may be used to compute an inner inclusion of a b

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

safe bounds in linear and mixed integer linear programming current mixed integer linear programming solvers are based on linear programming routines that use floating point arithmetic occasionally this leads to wrong solutions even for problems where all coefficients and all solution components are small integers an example is given where many state of the art milp solvers fail it is then shown how using directed rounding and interval arithmetic cheap pre and postprocessing of the linear programs arising in a branch and cut framework can guarantee that no solution is lost at least for mixed integer programs in which all variables can be bounded rigorously by bounds of reasonable size

dubrovin s duality for f manifolds with eventual identities abstract a vector field e on an f manifold m e is an eventual identity if it is invertible and the multiplication x y x y e 1 defines a new f manifold structure on m we give a characterization of such eventual identities this being a problem raised by manin 2005 12 we develop a duality between f manifolds with eventual identities and we show that this duality is compatible with the local irreducible decomposition of f manifolds and preserves the class of riemannian f manifolds we find necessary and sufficient conditions on the eventual identity which ensure that the classes of harmonic higgs bundles d c h k structures and weak cv structures are preserved by our duality examples of such structures are given in the case of a semi simple multiplication we use eventual identities to construct compatible pairs of metrics

relations between various methods for solving linear interval and parametric equations abstract in the first part of the paper we consider standard systems of linear interval equations and we focus particularly on two solution methods the bauer skeel method and the hansen bliek rohn method we show relations between these two methods and between various modifications that are based on preconditioning of the system and on the residual form we compare as well the quality of the bounds produced by the different variants and we show that for some variants the bauer skeel bounds naturally arise from other approaches such as krawczyk of jacobi iterations too in the second part of the paper we consider interval parametric linear systems with affine linear dependencies we also investigate various forms of enclosures and we not only compare them with each other but we also show relations to some already known methods as a consequence we come up with novel and interesting relations between several algorithms

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

a fresh view on the tolerance approach to sensitivity analysis in linear programming abstract the tolerance approach to sensitivity analysis in linear programming aims at finding a unique numerical value tolerance representing the maximum absolute perturbation which can be applied simultaneously and independently on each right hand side or objective coefficient without affecting the optimality of the given basis some extensions have been proposed in the literature which allow for individual tolerances for each coefficient thus enlarging the tolerance region in this paper we review the main results concerning the approach giving new and simpler proofs and we propose an efficient geometric algorithm returning a tolerance region that is maximal with respect to inclusion we compare our method with the existing ones on two examples showing how a priori information can be naturally exploited by our algorithm to further enlarge individual tolerances

a unified algebraic approach to the classical yang baxter equation in this paper the different operator forms of the classical yang baxter equation are given in the tensor expression through a unified algebraic method it is closely related to left symmetric algebras which play an important role in many fields in mathematics and mathematical physics by studying the relations between left symmetric algebras and the classical yang baxter equation we can construct left symmetric algebras from certain classical r matrices and conversely there is a natural classical r matrix constructed from a left symmetric algebra which corresponds to a parakahler structure in geometry moreover the former in a special case gives an algebraic interpretation of the left symmetry as a lie bracket left twisted by a classical r matrix

factorial algebraic group actions and categorical quotients abstract given an action of an affine algebraic group with only trivial characters on a factorial variety we ask for categorical quotients we characterize existence in the category of algebraic varieties moreover allowing constructible sets as quotients we obtain a more general existence result which for example settles the case of a finitely generated algebra of invariants as an application we provide a combinatorial git type construction of categorical quotients for actions of not necessarily reductive groups on e g complete varieties with finitely generated cox ring via lifting to the characteristic space

f manifold algebras and deformation quantization via pre lie algebras abstract the notion of an f manifold algebra is the underlying algebraic structure of an f manifold we introduce the notion of pre lie formal deformations of commutative associative algebras and show that f manifold algebras are the corresponding semi classical limits we study pre lie infinitesimal deformations and extension of pre lie n deformation to pre lie n 1 deformation of a commutative associative algebra through the cohomology groups of pre lie algebras we introduce the notions of pre f manifold algebras and dual pre f manifold algebras and show that a pre f manifold algebra gives rise to an f manifold algebra through the sub adjacent associative algebra and the sub adjacent lie algebra we use rota baxter operators more generally o operators and average operators on f manifold algebras to construct pre f manifold algebras and dual pre f manifold algebras

a new notion of tameness abstract we generalize the notion of tame discrete sets introduced by rosay and rudin from complex euclidean space to arbitrary complex manifolds and establish their basic properties we show that complex linear algebraic groups different from the complex line or the punctured complex line contain tame discrete sets

improved enclosure for some parametric solution sets with linear shape consider linear algebraic systems where the elements of the matrix and the right hand side vector depend linearly on a number of interval parameters we prove some sufficient conditions for the united parametric solution set of such a system to have linear boundary these conditions imply an equivalent representation of the parametric system where each parameter appears once in a diagonal matrix the latter representation allows us to expand the scope of applicability of the best known so far interval method developed by a neumaier and a pownuk for enclosing the parametric solution set and to generalize the method for systems where the parameter dependencies connect the matrix and the right hand side vector some examples demonstrate that parametric solution sets with linear boundary appear in various application domains the generalized method improves the solution enclosure and the proven sufficient conditions can be helpful for solving various other interval problems

extended support vector interval regression networks for interval input output data in many applications it is natural to use interval data because of uncertainty existence in the measurements variability in defining terms such as the temperature during a given day description for extremely behavior such as the maximum wind speed in a given area etc in order to handle such interval data a novel approach called the interval support vector interval regression networks isvirns is proposed the isvirns is extended from our previous work the support vector interval regression networks svirns it is easy to find that svirns can handle interval output data but for input data they must be crisp in this study the hausdorff distance is employed as the distance measure of interval data and is incorporated into the kernel functions of svirns to determine the initial structure of isvirns because the proposed approach can provide a better initial structure for isvirns it can have a faster convergent speed the experimental results with real data sets show the validity of the proposed isvirns

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

interval priorities in ahp by interval regression analysis abstract this paper proposes an interval approach for obtaining interval weights of priorities in the analytic hierarchy process ahp reflecting inconsistency of pairwise comparison ratios given by a decision maker even if pairwise comparison values with respect to preference are given as crisp values the priority weights should be estimated as intervals because of a decision maker s uncertainty of judgments we deal with two types of comparison matrices represented as crisp and interval elements this allows some uncertainty of estimations made by a decision maker the proposed approach can obtain the estimated interval priorities since interval regression analysis is based on the concept of possibility the interval weights obtained by the proposed method can be interpreted as the possible ranges estimated from the given data the sum of widths of the obtained intervals can be regarded as the degree of inconsistency included in the given data it is emphasized that interval data of preference can obtain the lower and upper models for the estimated weight intervals

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

enclosing solutions of linear equations it is shown that rump s method for enclosing solutions of linear equations can be reformulated in an interval free form and that the underlying inclusion result can be proved by elementary means without using brouwer s fixed point theorem a sufficient condition on rump s inflation parameter eps is given under which finite termination occurs also a more general modified algorithm is studied for which the number of iterations can be expressed by an explicit formula

new operator and method for solving real preconditioned interval linear equations we deal with real preconditioned interval linear systems of equations we present a new operator which generalizes the interval gauss seidel operator also based on the new operator and properties of well known methods we propose a new algorithm called the magnitude method we illustrate by numerical examples that our approach outperforms some classical methods with respect to both time and sharpness of enclosures

solving over constrained systems of non linear interval equations and its robotic application abstract this paper presents and describes in details an original method developed to solve over constrained systems of non linear interval equations that arise namely in parameter identification problems deriving from physical models and uncertain measurements our approach consists of computing an interval enclosure of the least square solution set and an inner box of tolerable solution set this method is applied in a detailed example and some interesting results obtained for the calibration of a cable driven robot are shown

a comparison of some methods for solving linear interval equations certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined we extend a proposed technique of hansen and rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix bf a underline a overline a is an h matrix when bf a is centered about a diagonal matrix these bounds are the smallest possible i e the bounds are then the solution hull hansen s scheme also computes the solution hull when the linear interval system bf a bf x bf b underline b overline b is such that bf a is inverse positive and underline b overline b neq 0 earlier results of others also imply that when bf a is an m matrix and bf b geq 0 bf b leq 0 or 0 in bf b interval gaussian elimination iga computes the hull we also give a method of computing the solution hull inexpensively in many instances when bf a is inverse positive given an outer approximation such as that obtained from iga examples are used to compare these schemes under various conditions

invariant rings and quasiaffine quotients we study hilbert s fourteenth problem from a geometric point of view nagata s celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety in this paper we will show that nevertheless it is always isomorphic to the ring of functions on a quasi affine variety

on simple novikov algebras and their irreducible modules abstract we give a complete classification of finite dimensional simple novikov algebras and their irreducible modules over an algebraically closed field with prime characteristic moreover we introduce novikov poisson algebras and their tensor theory our tensor theory enables us to understand better certain finite dimensional simple novikov algebras and their irreducible modules

the tolerance approach to sensitivity analysis in linear programming in constrast to ordinary sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right hand side terms this approach yields a maximum tolerance percentage such that as long as selected coefficients or terms are accurate to within that percentage of their estimated values the same basis is optimal in particular if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values then the same solution is optimal

on the possibilistic approach to linear regression models involving uncertain indeterminate or interval data abstract we consider linear regression models where both input data the observations of independent variables and output data the observations of the dependent variable are affected by loss of information caused by uncertainty indeterminacy rounding or censoring instead of real valued crisp data only intervals are available we study a possibilistic generalization of the least squares estimator so called ols set for the interval model investigation of the ols set allows us to quantify whether the replacement of real valued crisp data by interval values can have a significant impact on our knowledge of the value of the ols estimator we show that in the general case very elementary questions about properties of the ols set are computationally intractable assuming p np we also focus on restricted versions of the general interval linear regression model to the crisp input case taking the advantage of the fact that in the crisp input interval output model the ols set is a zonotope we design both exact and approximate methods for its description we also discuss special cases of the regression model e g a model with repeated observations

variable metric relaxation methods part ii the ellipsoid method the deepest or least shallow cut ellipsoid method is a polynomial time and space method which finds an ellipsoid representable by polynomial space integers such that the maximal ellipsoidal distance relaxation method using this fixed ellipsoid is polynomial this is equivalent to finding a linear transforming such that the maximal distance relaxation method of agmon motzkin and schoenberg in this transformed space is polynomial if perfect arithmetic is used then the sequence of ellipsoids generated by the method converges to a set of ellipsoids which share some of the properties of the classical hessian at an optimum point of a function and thus the ellipsoid method is quite analogous to a variable metric quasi newton method

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

parameterized outer estimation of ae solution sets to parametric interval linear systems abstract we consider linear algebraic equations where the elements of the matrix and of the right hand side vector are linear functions of interval parameters and their parametric ae solution sets which are defined by applying universal and existential quantifiers to the interval parameters usually interval methods find numerical interval vector that contains an ae solution set in this work we propose a method that generates an outer estimate of a parametric ae solution set in form of a linear parametric interval function called parameterized outer solution p solution parameterized outer solution is proposed for the parametric united solution set in kolev 2014 and takes precedence over the classical interval solution enclosure when the latter is part of other problems involving the same parameters the method we present generalizes the method from kolev 2016 for parametric ae solution sets it is also a parameterized analogue of a method from popova and hladik 2013 and produces the same interval enclosure as the method from the last reference

flexible varieties and automorphism groups given an irreducible affine algebraic variety x of dimension n 2 we let saut x denote the special automorphism group of x that is the subgroup of the full automorphism group aut x generated by all one parameter unipotent subgroups we show that if saut x is transitive on the smooth locus x reg then it is infinitely transitive on x reg in turn the transitivity is equivalent to the flexibility of x the latter means that for every smooth point x x reg the tangent space t x x is spanned by the velocity vectors at x of one parameter unipotent subgroups of aut x we also provide various modifications and applications

brief paper robust set membership state estimation application to underwater robotics this paper proposes a new observer for estimating the state vector of a nonlinear system this observer which is robust with respect to outliers assumes that the measurement errors as well as the number of outliers that could occur within a given time window are bounded the principle of the approach is to use interval analysis to deal properly with the nonlinearities involved in the system without any linearization or approximation and to propagate through the time in a forward and backward manner the assumptions made about outliers a test case related to the localization and control of an underwater robot is also proposed to illustrate the efficiency of the approach

construction of the largest sensitivity region for general linear programs this paper develops an alternative approach to post optimality analysis for general linear program models which provides the largest sensitivity region of any single or simultaneous change of right hand side of constraints and the coefficients of decision variables in the objective function the goal is a theoretical unification of various types of sensitivity analyses as well as advancement in the practical implementation of post optimality analysis we extend our proposed approach in the construction of sensitivity region to maintain the degenerate vertex for models with degenerate optimal solution as well as maintaining the multiple solutions for the models with non unique optimal solutions as a by product the paper resolves the paradoxical situation known as the more for less less for more situations the proposed method is based on the given optimal solution s therefore it is easy to understand easy to implement and provides useful information to the decision makers the methodology and their computational algorithms are presented and discussed in the context of some illustrative numerical examples