entropy and data compression schemes some new ways of defining the entropy of a process by observing a single typical output sequence as well as a new kind of shannon mcmillan breiman theorem are presented this provides a new and conceptually very simple ways of estimating the entropy of an ergodic stationary source as well as new insight into the workings of such well known data compression schemes as the lempel ziv algorithm

the multifractal spectra of certain planar recurrence sets in the continued fraction dynamical system abstract in this paper we will introduce and study the planar recurrence level set s x y 0 1 2 r x y r x y 0 where r x y and r x y are the lower and upper quantitative hitting time indicators of y by x respectively in the continued fraction dynamical system it is shown that the multifractal spectrum is always degenerate in a sense that it is constant independent of the choice of the level set

on projection matrices cal p k rightarrow cal p 2 k 3 ldots 6 and their applications in computer vision projection matrices from projective spaces cal p 3 to cal p 2 have long been used in multiple view geometry to model the perspective projection created by the pin hole camera in this work we introduce higher dimensional mappings cal p k rightarrow cal p 2 k 3 4 5 6 for the representation of various applications in which the world we view is no longer rigid we also describe the multi view constraints from these new projection matrices where k 3 and methods for extracting the non rigid structure and motion for each application

proof theory for linear lattices abstract we make a study of lattices representable by commuting equivalence relations which we call linear lattices we develop a proof theory for implications valid in linear lattices which differs from classical proof theories in that its deductions transform representative graphs rather than well formed formulas using graphtheoretic arguments we establish a duality theorem and a normal form theorem for this proof theory

the multifractal spectra for the recurrence rates of beta transformations abstract in this paper we show a handy approximate approach to provide a lower bound of the hausdorff dimension of a given subset in 0 1 related to transformation dynamical system here approximation means from special class with shift satisfying the specification property or being subshift of finite type to general 1 as an application we obtain the multifractal spectra for the recurrence rate of the first return time of transformation including the cases returning to the ball and cylinder

normal forms for binary relations we consider the representable equational theory of binary relations in a language expressing composition converse and lattice operations by working directly with a presentation of relation expressions as graphs we are able to define a notion of reduction which is confluent and strongly normalizing and induces a notion of computable normal form for terms this notion of reduction thus leads to a computational interpretation of the representable theory

discrete bipolar pseudo integrals abstract in the authors paper strboja et al 2016 42 the notion of the bipolar pseudo integral has been introduced this paper extends investigations related to this new type of bipolar integrals defined with respect to decomposable bi capacities we consider the main properties of the discrete bipolar pseudo integral a characterization theorem for the discrete bipolar pseudo integral is proven monotone convergence theorems and the fatou type lemma for the discrete bipolar pseudo integral are considered in order to illustrate obtained results many interesting examples are provided

arguesian identities in the congruence variety of abelian groups abstract a class of identities in the grassmann cayley algebra was found by m j hawrylycz 1994 geometric identities in invariant theory ph d thesis massachusetts institute of technology which yields a large number of geometric theorems on the incidence of subspaces of projective spaces in a previous paper we established a link between such identities in the grassmann cayley algebra and a family of inequalities in the class of linear lattices i e the lattices of commuting equivalence relations we proved that a subclass of identities found by hawrylycz namely the arguesian identities of order 2 can be systematically translated into inequalities holding in linear lattices however it is not known whether the arguesian identities of higher orders have such latticial extensions in this paper we give an affirmative answer to the above question in the congruence variety of abelian groups we prove that every arguesian identity regardless of the order can be systematically translated into a lattice inequality holding in the congruence variety of abelian groups in particular such a lattice inequality holds in the lattices of subspaces of vector spaces which are characteristic free and independent of dimensions as a consequence many classical theorems of projective geometry including desargues bricard fontene and their higher dimensional generalizations can be extended to lattice inequalities in the general projective spaces with the variables representing subspaces of arbitrary dimensions

non monotonic fuzzy measures and the choquet integral abstract this paper discusses non monotonic fuzzy measures which are set functions without monotonicity and the choquet integral with respect to non monotonic fuzzy measures a concrete example shows that the choquet integral is meaningful in the case of non monotonic fuzzy measures as well as in the case of ordinary fuzzy measures every comonotonically additive positively homogeneous functional of bounded variation can be represented as a choquet integral with respect to a non monotonic fuzzy measure of bounded variation the space of such functionals is a real banach space isometrically isomorphic to the space of non monotonic fuzzy measures of bounded variation

a grassmann integral equation the present study introduces and investigates a new type of equation which is called grassmann integral equation in analogy to integral equations studied in real analysis a grassmann integral equation is an equation which involves grassmann berezin integrations and which is to be obeyed by an unknown function over a finite dimensional grassmann algebra gm i e a sought after element of the grassmann algebra gm a particular type of grassmann integral equations is explicitly studied for certain low dimensional grassmann algebras the choice of the equation under investigation is motivated by the effective action formalism of lattice quantum field theory in a very general setting for the grassmann algebras g2n n 2 3 4 the finite dimensional analogues of the generating functionals of the green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations finally by imposing the condition g g0 const 0 r k k k 1 n are the generat

sugeno integral based on absolutely monotone real set functions the class of all absolutely monotone and sign stable set functions with m a 0 denoted by amss is introduced necessary and sufficient conditions are obtained for a set function m m a 0 to be a member of amss a sugeno type integral of an a measurable real valued function is defined with respect to an absolutely monotone and sign stable set function and some of its properties are shown a representation of a comonotone cosigned additive functional by sugeno integral based on m amss is obtained

arguesian identities in invariant theory abstract having been motivated by an example of doubilet rota and stein stud appl math 56 1976 185 216 we present a technique for constructing geometric identities in a grassmann cayley algebra each identity represents a projective invariant closely related to the theorem of desargues in the plane and its generalizations to higher dimensional projective space the construction employs certain combinatorial properties of matchings in bipartite graphs we also prove a dimension independence result for arguesian identities thereby connecting the identities with lattice theory

generalization of the jensen inequality for pseudo integral generalizations of the jensen integral inequality for the pseudo integral are proven there are considered two cases of the real semiring with pseudo operations one when pseudo operations are defined by monotone and continuous function g then the pseudo integral reduces on g integral and the second semiring a b max where the pseudo multiplication is generated

on the exterior calculus of invariant theory the full extent of the tragedy of hermann grassmann which has been unfolding since his death by a succession of misadventures and misunderstandings of his work unique in the history of modern mathematics is yet to be fully realized it began with a deliberate ambiguity on the part of the author of the ausdehnungslehre whereby one and the same notation was chosen to designate two different algebraic operations originally known as the progressive and the regressive product this choice was not an oversight on grassmann s part as we shall see but was rather dictated by the author s deep understanding of the philosophical reach of his discovery an understanding which lay well beyond the powers of the

the absolute line quadric and camera autocalibration we introduce a geometrical object providing the same information as the absolute conic the absolute line quadric alq after the introduction of the necessary exterior algebra and grassmannian geometry tools we analyze the grassmannian of lines of p from both the projective and euclidean points of view the exterior algebra setting allows then to introduce the alq as a quadric arising very naturally from the dual absolute quadric we fully characterize the alq and provide clean relationships to solve the inverse problem i e recovering the euclidean structure of space from the alq finally we show how the alq turns out to be particularly suitable to address the euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters providing new linear and non linear algorithms for this problem we also provide experimental results showing the good performance of the techniques

arguesian identities in linear lattices abstract a class of identities in the grassmann cayley algebra which yields a large number of geometric theorems on the incidence of subspaces of projective spaces was found by hawrylycz geometric identities in invariant theory ph d thesis massachusetts institute of technology 1994 in this paper we establish a link between such identities in the grassmann cayley algebra and a class of inequalities in the class of linear lattices i e the lattices of commuting equivalence relations we prove that a subclass of identities found by hawrylycz namely the arguesian identities of order 2 can be systematically translated into inequalities holding in linear lattices as a consequence we obtain a family of geometric theorems on the incidence of subspaces that are characteristic free and independent of dimensions

generalized choquet like aggregation functions for handling bipolar scales we are interested in modeling interaction between criteria in multi criteria decision making when underlying scales are bipolar interacting phenomena involving behavioral bias between attractive and repulsive values are in particular considered here we show in an example that both the choquet integral and the cumulative prospect theory cpt model fail to represent these interacting phenomena axioms that enable the construction of the preferences of the decision maker over each attribute and the representation of his preferences about aggregation of criteria are introduced and justified we show there is a unique aggregation operator that fits with these axioms it is based on the notion of em bi capacity and generalizes both the choquet integral and the cpt model

pseudo riemann stieltjes integral an extension of the classical riemann stieltjes integral to the field of pseudo analysis is being investigated through this paper core of the construction presented here consists of generalized pseudo operations given by monotone generating function

invariants of sets of lines in projective 3 space abstract we compute a set of generators of the ring of invariants for a set of straight lines in 3 dimensional projective space using the symbolic method introduced by f grosshans g c rota and j a stein in invariant theory and superalgebras amer math soc providence ri 1987 a geometric interpretation of the invariants is given

idempotent integral as limit of g integrals we show that sup and inf decomposable measures can be obtained as limits of families of pseudo additive measures with respect to generated pseudo additions the corresponding integrals with respect to sup or inf decomposable measures can be obtained as limits of families of g integrals

b expansions and symbolic dynamics given a real number p i it is easy to expand x e o i in a canonical way into a series of negative powers of p with integer coefficients less than or equal to p the form of the set of such p expansions which is a matter of symbolic dynamics or language theory varies according to arithmetical properties of p but there are many common features known facts which are few and open questions much more numerous are reviewed in this paper

dimensions of recurrent sets in β symbolic dynamics abstract given 1 let 0 1 t be the dynamical system moreover for any n n and x 0 1 let n x inf j 1 t j x i n x where i n x is the n th cylinder containing x then given any a b 0 with a b the hausdorff dimension of the following recurrent set e a b x 0 1 lim inf n log n x log n a lim sup n log n x log n b admits a dichotomy it is either one or zero according as a 1 or not

grobner bases and invariant theory abstract in this paper we study the relationship between buchberger s grobner basis method and the straightening algorithm in the bracket algebra these methods will be introduced in a self contained overview on the relevant areas from computational algebraic geometry and invariant theory we prove that a certain class of van der waerden syzygies forms a grobner basis for the syzygy ideal in the bracket ring we also give a description of a reduced grobner basis in terms of standard and non standard tableaux some possible applications of straightening for symbolic computations in projective geometry are indicated

varieties of boolean inverse semigroups in an earlier work the author observed that boolean inverse semigroups with semigroup homomorphisms preserving finite orthogonal joins form a congruence permutable variety of algebras called biases we give a full description of varieties of biases in terms of varieties of groups 1 every free bias is residually finite in particular the word problem for free biases is decidable 2 every proper variety of biases contains a largest finite symmetric inverse semigroup and it is generated by its members that are monoids of generalized rook matrices over groups with zero 3 there is an order preserving one to one correspondence between proper varieties of biases and certain finite sequences of varieties of groups descending in a strong sense defined in terms of wreath products by finite symmetric groups

an axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria the most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean however in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted we show that under rather natural conditions the discrete choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria the axiomatic that supports the choquet integral is presented and an intuitive approach is proposed as well

projective invariants of four subspaces abstract we compute a set of generators of the ring of invariants for a set of 4 subspaces in a projective space more precisely let k gr 4 be the ring of regular functions on the product of four grassmannians we present a set of generators for k gr 4 sl v where v is the underlying vector space we also study syzygies among the generators in most cases generators are algebraically independent only in the case where the affine dimensions of the four subspaces are given by the quadruple k k k k and k k dim v there is one and only one syzygy the precise form of which is given

asymmetric integral as a limit of generated choquet integrals based on absolutely monotone real set functions we introduce a generated choquet integral with respect to absolutely monotone and sign stable set functions another type of integrals defined with respect to such a set function is obtained as a limit of an appropriate sequence of generated choquet integrals

fuzzy measures and integrals in evaluation of strategies artificial intelligence ai techniques have been applied to games for a long time in this paper we will explore the use of fuzzy measures and integrals to evaluate strategies in games we will show their use in decision making when modeling auctions the paper describes the role of fuzzy measures and integrals in decision making and reviews some computational aspects which are important for building real applications

the coordinatization of arguesian lattices we show that the auxiliary planar ternary ring of an n frame in an arguesian lattice n 3 is indeed an associative ring with unit the addition of two weak necessary conditions allows us to coordinatize a hyperplane of this n frame this generalizes the classical work of von neumann baer inaba j6nsson and j6nsson monk

choquet like integrals abstract integrals based on a pseudo addition and a pseudo multiplication with properties similar to those of the choquet and the sugeno integral respectively are introduced the structure of convenient pseudo arithmetical operations leading to a choquet like integral is studied

jensen type inequality for the bipolar pseudo integrals abstract the main purpose of this paper is to establish conditions under which the jensen type inequality for the discrete bipolar pseudo integral is valid besides we extend investigations of the properties of the bipolar pseudo integral the observations concern the discrete bipolar pseudo integrals based on the following three canonical cases of two binary symmetric operations in the first case they are generated by an odd strictly increasing and continuous function in the remaining two cases a symmetric addition is the symmetric maximum while in the second case the corresponding pseudo multiplication is a non idempotent operation and in the third case it is the symmetric minimum

a unified approach to the monotone integral based premium principles under the cpt theory abstract we consider the relationship between different types of the monotone integral based premium principles as a common frame for the cpt premium principle and the mean premium principle the cpt like integral based premium principle is introduced and characterized the jensen type inequalities for the choquet integral and the cpt like premium principle on the class of non negative risks are proven we establish conditions under which the jensen steffensen type inequalities for the cpt premium principle and the cpt like premium principle hold for risks with finite range

quantitative recurrence properties for beta dynamical system abstract this note is concerned with the quantitative recurrence properties of beta dynamical system 0 1 t for general 1 the size of points with the prescribed recurrence rate is determined more precisely hausdorff dimensions of the sets x 0 1 t n x x n for infinitely many n n and x 0 1 t n x x e j 0 n 1 f t j x for infinitely many n n are obtained completely where is a positive function defined on n and f is a positive continuous function on 0 1 besides the pressure function p f t with a continuous potential f is proven to be continuous with respect to

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

multiple view geometry of general algebraic curves we introduce a number of new results in the context of multi view geometry from general algebraic curves we start with the recovery of camera geometry from matching curves we first show how one can compute without any knowledge on the camera the homography induced by a single planar curve then we continue with the derivation of the extended kruppa s equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve as part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus we then establish new results on the reconstruction of general algebraic curves from multiple views we address three different representations of curves i the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions one of degree d and the other of degree d d 1 moreover using this representation we address the problem of homography recovery for planar curves ii dual space representation tangents for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus and iii a new representation to computer vision based on the set of lines meeting the curve which does not require any curve fitting in image space for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone