modular groups of quantum fields in thermal states for a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half sided translations the modular flows corresponding to the algebras of the forward light cone and a space like wedge admit a simple geometric description in two dimensional models that factorize in light cone coordinates at large distances from the domain boundary compared to the inverse temperature the flow pattern is essentially the same as time translations whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone for each domain there is also a one parameter group with a positive generator for which the thermal state is a ground state formally this may be regarded as a certain converse of the unruh effect

modular intersections of von neumann algebras in quantum field theory we show that modular intersections of von neumann algebras occur naturally in quantum field theory an example are local observable algebras associated with wedge regions which have a lightray in common see also bo 2 wi 3 conversely starting from a set of four algebras lying in a specified modular position relative to each other we construct a net of local observables of a 2 1 dimensional quantum field theory

on the structure of some families of fuzzy measures the generation of fuzzy measures is an important question arising in the practical use of these operators in this paper we deal with the problem of developing a random generator of fuzzy measures more concretely we study some of the properties that any random generator should satisfy these properties lead to some theoretical problems concerning the group of isometries that we tackle in this paper for some subfamilies of fuzzy measures

computation of minimal rank and path cover number for certain graphs for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertexdisjoint paths occurring as induced subgraphs of g that cover all the vertices of g for trees the relationship between minimum rank and path cover number is completely understood however for non trees only sporadic results are known we derive formulae for the minimum rank and path cover number for graphs obtained from edge sums and formulae for minimum rank of vertex sums of graphs in addition we examine previously identified special types of vertices and attempt to unify the theory behind them

modular hamiltonians on the null plane and the markov property of the vacuum state we compute the modular hamiltonians of regions having the future horizon lying on a null plane for a cft this is equivalent to regions with a boundary of arbitrary shape lying on the null cone these hamiltonians have a local expression on the horizon formed by integrals of the stress tensor we prove this result in two different ways and show that the modular hamiltonians of these regions form an infinite dimensional lie algebra the corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities but act non locally outside the null plane we regain this result in greater generality using more abstract tools on the algebraic quantum field theory finally we show that modular hamiltonians on the null surface satisfy a markov property that leads to the saturation of the strong sub additive inequality for the entropies and to the strong super additivity of the relative entropy

k maxitive fuzzy measures a scalable approach to model interactions abstract fuzzy measures are powerful at modeling interactions between elements unfortunately they use a number of coefficients that exponentially grows with the number of elements beyond the computational complexity assigning a value to any coalition of a large set of elements does not make sense k order measures model interactions involving at most k elements the number of coefficients to identify is reduced and their modeling capacity is preserved in real problems where the number of interacting elements is limited in extreme situations of full redundancy or complementariness it is mathematically proven that the complete fuzzy measure is both k additive and k maxitive a learning algorithm to identify k maxitive measures from labeled data is designed on the basis of hlms heuristic least mean squares in a classification context the study of synthetic data with partial redundancy or complementariness supports the idea that the difference between full and partial interaction is a matter of degree not of kind dealing with two real world datasets a comparison of the complete fuzzy measure and a k maxitive one shows the number of interacting elements is limited and the k maxitive measures yield the same characterization of interactions and a comparable classification accuracy

conformal quantum field theory and half sided modular inclusions of von neumann algebras letn be von neumann algebras on a hilbert space a comon cyclic and separarting vector assume to be cyclic and separating also forn denote byj jn the modular conjugations to and n the associated modular operators if open image in new window and open image in new window these data define in a canonical way a conformal quantum field theory in a cricle conversely the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way

nuclearity and thermal states in conformal field theory we introduce a new type of spectral density condition that we call l2 nuclearity one formulation concerns lowest weight unitary representations of sl 2 mathbb r and turns out to be equivalent to the existence of characters a second formulation concerns inclusions of local observable von neumann algebras in quantum field theory we show the two formulations to agree in chiral conformal qft and starting from the trace class condition rm tr e beta l 0 0 we include further discussions on higher dimensional spacetimes in particular we verify that l2 nuclearity is satisfied for the scalar massless klein gordon field

scale and mobius covariance in two dimensional haag kastler net given a two dimensional haag kastler net which is poincare dilation covariant with additional properties we prove that it can be extended to a mobius covariant net additional properties are either a certain condition on modular covariance or a variant of strong additivity the proof relies neither on the existence of stress energy tensor nor any assumption on scaling dimensions we exhibit some examples of poincare dilation covariant net which cannot be extended to a mobius covariant net and discuss the obstructions

an algebraic characterization of vacuum states in minkowski space an algebraic characterization of vacuum states on nets ofc algebras over minkowski space is given and space time translations are reconstructed with the help of the modular structures associated with such states the result suggests that a condition of geometrical modular action might hold in quantum field theories on a wider class of spacetime manifolds

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 defect of complementarity of fuzzy measures it is well known that the fuzzy measures are non additive this means that the sum between the measure of a set and the measure of its complement is not equal with the measure of space as a consequence in this paper we introduce the concept of defect of complementarity also for large classes of fuzzy measures this defect of complementarity is calculated or estimated finally some applications and interpretations are presented

natural energy bounds in quantum thermodynamics given a stationary state for a noncommutative flow we study a boundedness condition depending on a parameter 0 which is weaker than the kms equilibrium condition at inverse temperature this condition is equivalent to a holomorphic property closely related to the one recently considered by ruelle and d antoni zsido and shared by a natural class of non equilibrium steady states our holomorphic property is stronger than ruelle s one and thus selects a restricted class of non equilibrium steady states we also introduce the complete boundedness condition and show this notion to be equivalent to the pusz woronowicz complete passivity property hence to the kms condition

modular theory and the reconstruction of four dimensional quantum field theories in this letter we construct a representation of the 3 1 dimensional poincare group by modular groups of von neumann algebras lying in a specified modular position with respect to each other combining this new result with an old one of bisognano wichmann j math phys 16 985 1975 we obtain a net of local observables of a 3 1 dimensional quantum field theory out of a finite set of algebras

the minimum rank of matrices and the equivalence class graph abstract for a given connected undirected graph g the minimum rank of g v g e g is defined to be the smallest possible rank over all hermitian matrices a whose i j th entry is non zero if and only if i j and i j is an edge in g i j e g for each vertex x in g x v g n x is the set of all neighbors of x let r be the equivalence relation on v g such that x y v g xry n x n y our aim is find classes of connected graphs g v g e g such that the minimum rank of g is equal to the number of equivalence classes for the relation r on v g

generalizations of k order additive discrete fuzzy measures abstract two types of generalizations of k order additive discrete fuzzy measures recently introduced by grabisch are shown first a special type of discrete fuzzy measures generalizing the pseudo additive measures with respect to a pan addition is introduced following the ideas of grabisch a mobius like transformation of discrete fuzzy measures is presented and an evaluation formula for the corresponding choquet like integral is given including the sugeno integral next k order additive fuzzy measures on general spaces are introduced combination of both generalizations is discussed and several open problems are addressed

classification of subsystems for graded local nets with trivial superselection structure we classify haag dual poincare covariant subsystems open image in new window of a graded local net open image in new window on 4d minkowski spacetime which satisfies standard assumptions and has trivial superselection structure the result applies to the canonical field net open image in new window of a net open image in new window of local observables satisfying natural assumptions as a consequence provided that it has no nontrivial internal symmetries such an observable net open image in new window is generated by the abstract versions of the local energy momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum noether theorem moreover for a net open image in new window of local observables as above we also classify the poincare covariant local extensions open image in new window which preserve the dynamics

learning k maxitive fuzzy measures from data by mixed integer programming abstract fuzzy measures model interactions between the inputs in aggregation problems their complexity grows exponentially with the dimensionality of the problem and elicitation of fuzzy measure coefficients either from domain experts or from empirical data poses a significant challenge the notions of k additivity and k maxitivity simplify the fuzzy measures by limiting interactions to subsets of up to k elements learning fuzzy measures from data is an important elicitation technique which relies on solving an optimisation problem a heuristic learning algorithm to identify k maxitive fuzzy measures from the data on the basis of hlms heuristic least mean squares was recently presented in murillo et al 2017 11 we present an alternative formulation of the fitting problem which delivers a globally optimal solution through the solution of a mixed integer programming mip problem to deal with high computational cost of mip in moderate to large dimensions we also propose a simple mip relaxation technique which involves solving two related linear programming problems we also provide a linear programming formulation for fitting k tolerant fuzzy measures we discuss implementations of the fitting methods and present the results of numerical experiments

k maxitive sugeno integrals as aggregation models for ordinal preferences we consider an order variant of k additivity so called k maxitivity and present an axiomatization of the class of k maxitive sugeno integrals over distributive lattices to this goal we characterize the class of lattice polynomial functions with degree at most k and show that k maxitive sugeno integrals coincide exactly with idempotent lattice polynomial functions whose degree is at most k we also discuss the use of this parametrized notion in preference aggregation and learning in particular we address the question of determining optimal values of k through a case study on empirical data

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

spectral multiplicity and splitting results for a class of qualitative matrices abstract this paper presents some results concerning the location and multiplicity of eigenvalues of sign symmetric matrices whose associated graphs are trees in particular it extends previous spectral multiplicity and splitting results proved by others

particle physics and quantum field theory at the turn of the century old principles with new concepts the present state of quantum field theory qft is analyzed from a new viewpoint whose mathematical basis is the modular theory of von neumann algebras its physical consequences suggest new ways of dealing with interactions symmetries hawking unruh thermal properties and possibly also extensions of the scheme of renormalized perturbation theory interactions are incorporated by using the fact that the s matrix is a relative modular invariant of the interacting relative to the incoming net of wedge algebras this new point of view allows many interesting comparisions with the standard quantization approach to qft and is shown to be firmly rooted in the history of qft its radical change of paradigm aspect becomes particularly visible in the quantum measurement problem

pseudo optimal measures based on a pseudo addition on 0 pseudo optimal measures related to non negative set functions vanishing at the empty set non negative games are introduced for a fixed pseudo addition an equivalence on non negative set functions is studied a special stress is given to pure pseudo optimal measures admitting no other equivalent set function

on the maximum multiplicity of an eigenvalue in a matrix whose graph contains exactly one cycle we consider the general problem of determining the maximum possible multiplicity of an eigenvalue in a hermitian matrix whose graph contains exactly one cycle for some cases we express that maximum multiplicity in terms of certain parameters associated with the graph

half sided modular inclusions of von neumann algebras letn m be von neumann algebras on a hilbert spaceh a common cyclic and separating vector denote m n resp jm jn the associated modular operators and conjugations assume m it n it n fort 0 we call such an inclusion half sided modular then we prove the existence of a oneparameter unitary groupu a onh a r with generator frac 1 2 pi ln delta cal n ln delta m mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x 0 and relations 1 itu a it it nit u a nit u e 2 ta for alla t r 2 jmjn u 2 3 nit u 1 it u 1 for allt r 4 n u 1 u 1

k additivity and c decomposability of bi capacities and its integral k additivity is a convenient way to have less complex bi capacities this paper gives a new characterization of k additivity introduced by grabisch and labreuche of bi capacities and contrasts between the existing characterization and the new one that differs from the one of saminger and mesiar besides in the same way for capacities a concept of c decomposability distinct from the proposal of saminger and mesiar but closely linked to k additivity is introduced for bi capacities moreover the concept of c decomposability applies to the choquet integral with respect to bi capacities

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

super quantum measures on effect algebras with the riesz decomposition properties we give one basis of the space of super quantum measures on finite effect algebras with the riesz decomposition properties rdp for short then we prove that the super quantum measures and quantum interference functions on finite effect algebras with the rdp are determined each other at last we investigate the relationships between the super quantum measures and the diagonally positive signed measures on finite effect algebras with the rdp in detail

nuclear maps and modular structures i general properties abstract we establish a correspondence between the split property of inclusions a b of von neumann algebras and nuclearity properties of the natural embeddings p l a lp b p 1 2 given by the modular theory

on the structure of the k additive fuzzy measures in this paper we present some results concerning the vertices of the set of fuzzy measures being at most k additive we provide an algorithm to compute them we give some examples of the results obtained with this algorithm and give lower bounds on the number of vertices for the n 1 additive case proving that it grows much faster than the number of vertices of the general fuzzy measures the results in the paper suggest that the structure of k additive measures might be more complex than expected from their definition and in particular that they are more complex than general fuzzy measures

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

pan operations structure abstract the structure of pan addition and pan multiplication of a commutative isotonic semiring r is analyzed we show that if v supremum then r is a g semiring i e a b g 1 g a g b and d a b g 1 g a g b conclusions for pan integrals with respect to a measure are shown

the number of p vertices in a matrix with maximum nullity abstract let t be a tree with n 2 vertices set s t for the set of all real symmetric matrices whose graph is t let a s t and i 1 n we denote by a i the principal submatrix of a obtained after deleting the row and column i we set m a 0 for the multiplicity of the eigenvalue zero in a the nullity of a when m a i 0 m a 0 1 we say that i is a p vertex of a as usual m t denotes the maximum nullity occurring of b s t in this paper we determine an upper bound and a lower bound for the number of p vertices in a matrix a s t with nullity m t we also prove that if the integer b is between these two bounds then there is a matrix e s t with b p vertices and maximum nullity