a topos foundation for theories of physics iv categories of systems this paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics the motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos the previous papers in this series are concerned with implementing this program for a single system in the present paper we turn to considering a collection of systems in particular we are interested in the relation between the topos representation for a composite system and the representations for its constituents we also study this problem for the disjoint sum of two systems our approach to these matters is to construct a category of systems and to find a topos representation of the entire category

mixed quasi variational inequalities in this paper we use the resolvent operator and auxiliary principle techniques to suggest and analyze several iterative algorithms for solving mixed quasi variational inequalities and related problems we study the convergence criteria of these algorithms under mild conditions we also study the global stability and existence of a unique solution of these quasi variational inequalities by using the dynamical systems approach our results represent refinement and improvement of the previously known results for variational inequalities

predictor corrector methods for general mixed quasi variational inequalities in this paper we use the auxiliary principle technique to suggest a class of predictor corrector methods for solving general mixed quasi variational inequalities the convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator which is weaker than co coercivity as special cases we obtain various known and new results for solving various classes of variational inequalities and related problems

determining the optimal level of smoothing in cortical thickness analysis a hierarchical approach based on sequential statistical thresholding article i nfo the extent of smoothing applied to cortical thickness maps critically influences sensitivity anatomical precision and resolution of statistical change detection theoretically it could be optimized by increasing the trade off between vertex wise sensitivity and specificity across several levels of smoothing but to date neither parametric nor nonparametric methods are able to control the error at the vertex level if the null hypothesis is rejected after smoothing of cortical thickness maps to overcome these drawbacks we applied sequential statistical thresholding based on a simple hierarchical model this methodology aims at controlling erroneous detections firstly at the level of clusters over smoothed statistical maps and secondly at the vertex level over unsmoothed statistical maps by applying an adaptive false discovery rate fdr procedure to clusters previously detected the superior performance of the proposed methodology over other conventional procedures was demonstrated in simulation studies as expected only the hierarchical method yielded a predictable false discovery proportion near the predefined fdr q value for any smoothing level at the same time as being as sensitive as the others at the optimal setting it was therefore the only method able to approximate the optimal size of spatial smoothing when the true change was assumed unknown the hierarchical method was further validated in a cross sectional study comparing moderate alzheimer s disease ad patients with healthy elderly subjects results suggest that the extent of cortical thinning reported in previous ad studies might be artificially inflated by the choice of inadequate smoothing in these cases interpretation should be based on the location of local maxima of suprathreshold regions rather than on the spatial extent of the detected signal in the statistical parametric map

piecewise homomorphisms and jordan maps on c algebras and factor von neumann algebras abstract we investigate maps between c algebras that are well behaved with respect to mutually commuting elements we contribute to the mackey gleason problem by showing that any continuous bijection between self adjoint parts of c algebras that preserves triple product a b a b a and is linear on commutative subspaces is already linear this allows us to describe such maps as direct differences of linear jordan isomorphisms we shall show that any weak continuous bijection between positive invertible elements of von neumann factors of dimension at least 9 that preserves products of commuting elements in both directions is of the form a e log a a c where is a linear jordan isomorphism c nonzero real number and is a hermitian continuous functional in a similar way we describe the same type of bicontinuous maps between unitary groups of von neumann factors general form of the above mentioned maps on c algebras is also presented

characterizations of categories of commutative c subalgebras we aim to characterize the category of injective homomorphisms between commutative c subalgebras of a given c algebra a we reduce this problem to finding a weakly terminal commutative subalgebra of a and solve the latter for various c algebras including all commutative ones and all type i von neumann algebras this addresses a natural generalization of the mackey piron programme which lattices are those of closed subspaces of hilbert space we also discuss the way this categorified generalization differs from the original question

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

contextuality and noncommutative geometry in quantum mechanics observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values dually states can be modelled as functions from the algebra of observables to outcome values the probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality noncommuting observables which are not assumed to be jointly measurable cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state here we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical deterministic state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry by adapting the spectral presheaf of hamilton isham butterfield a formulation of quantum state space that collates contextual data we reconstruct tools of noncommutative geometry in an explicitly geometric fashion in this way we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry a la connes et al to each unital c algebra mathcal a we associate a geometric object a diagram of topological spaces collating quotient spaces of the noncommutative space underlying mathcal a that performs the role of a generalised gel fand spectrum we show how any functor f from compact hausdorff spaces to a suitable target category mathsf c can be applied directly to these geometric objects to automatically yield an extension tilde f acting on all unital c algebras this procedure is used to give a novel formulation of the operator k0 functor via a finitary variant tilde k f of the extension tilde k of the topological k functor we then delineate a c algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital c algebra the zariski topological lattice of its primitive ideal spectrum i e its lattice of closed two sided ideals we prove the von neumann algebraic analogue of this conjecture

self adjoint operators as functions i lattices galois connections and the spectral order observables of a quantum system described by self adjoint operators in a von neumann algebra or affiliated with it in the unbounded case form a conditionally complete lattice when equipped with the spectral order using this order theoretic structure we develop a new perspective on quantum observables

a topos foundation for theories of physics i formal languages for physics this paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we discuss two different types of language that can be attached to a system s the first is a propositional language pl s the second is a higher order typed language l s both languages provide deductive systems with an intuitionistic logic the reason for introducing pl s is that as shown in paper ii of the series it is the easiest way of understanding and expanding on the earlier work on topos theory and quantum physics however the main thrust o

active lattices determine aw algebras abstract we prove that operator algebras that have enough projections are completely determined by those projections their symmetries and the action of the latter on the former this includes all von neumann algebras and all aw algebras we introduce active lattices which are formed from these three ingredients more generally we prove that the category of aw algebras is equivalent to a full subcategory of active lattices crucial ingredients are an equivalence between the category of piecewise aw algebras and that of piecewise complete boolean algebras and a refinement of the piecewise algebra structure of an aw algebra that enables recovering its total structure

linear mixed effects modeling approach to fmri group analysis conventional group analysis is usually performed with student type t test regression or standard an c ova in which the variance covariance matrix is presumed to have a simple structure some correction approaches are adopted when assumptions about the covariance structure is violated however as experiments are designed with different degrees of sophistication these traditional methods can become cumbersome or even be unable to handle the situation at hand for example most current fmri software packages have difficulty analyzing the following scenarios at group level 1 taking within subject variability into account when there are effect estimates from multiple runs or sessions 2 continuous explanatory variables covariates modeling in the presence of a within subject repeated measures factor multiple subject grouping between subjects factors or the mixture of both 3 subject specific adjustments in covariate modeling 4 group analysis with estimation of hemodynamic response hdr function by multiple basis functions 5 various cases of missing data in longitudinal studies and 6 group studies involving family members or twins here we present a linear mixed effects modeling lme methodology that extends the conventional group analysis approach to analyze many complicated cases including the six prototypes delineated above whose analyses would be otherwise either difficult or unfeasible under traditional frameworks such as an c ova and general linear model glm in addition the strength of the lme framework lies in its flexibility to model and estimate the variance covariance structures for both random effects and residuals the intraclass correlation icc values can be easily obtained with an lme model with crossed random effects even at the presence of confounding fixed effects the simulations of one prototypical scenario indicate that the lme modeling keeps a balance between the control for false positives and the sensitivity for activation detection the importance of hypothesis formulation is also illustrated in the simulations comparisons with alternative group analysis approaches and the limitations of lme are discussed in details

on random variational inequalities abstract in this paper we consider a random variational inequality an existence theorem for a unique solution of a random variational inequality is proved which includes the results of noor lions and stampacchia several special cases which can be obtained from our results are also discussed

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

a comparison of two topos theoretic approaches to quantum theory the aim of this paper is to compare the two topos theoretic approaches to quantum mechanics that may be found in the literature to date the first approach which we will call the contravariant approach was originally proposed by isham and butterfield and was later extended by doring and isham the second approach which we will call the covariant approach was developed by heunen landsman and spitters

sensitivity analysis for abstract equilibrium problems we develop the general framework of sensitivity analysis for equilibrium problems in the setting of vector topological normed space our approach does not make any recourse to geometrical properties and the obtained result can be viewed as an extension and generalization of the well known results on variational inequalities in the literature even though we have worked under arbitrary constraints k with holder property that have been decisive in our treatment we have obtained in a similar spirit of domokos j math anal appl 230 1999 382 389 the best lower bound for the continuity modulus despite of the properties of the boundary of k

a topos foundation for theories of physics ii daseinisation and the liberation of quantum theory this paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we study in depth the topos representation of the propositional language pl s for the case of quantum theory in doing so we make a direct link with and clarify the earlier work on applying topos theory to quantum physics the key step is a process we term daseinisation by which a projection operator is mapped to a subobject of the spectral presheaf the topos quantum analog of a classical state space in the second part of the paper we change gear

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

topos quantum theory reduced by context selection functors in this paper we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von neumann algebras of bounded operators on a hilbert space our aim is first to reduce presheaf based topos quantum theory via sheafification and then to import quantum probabilities to the reduced sheaf quantum theory the first is done by means of a functor that selects some expedient contexts note that since the functor defines a grothendieck topology on the category consisting of all contexts it induces a sheaf topos on which we construct a downsized quantum theory we also show that the sheaf quantum theory can be replaced by a more manageable presheaf quantum theory quantum probabilities are imported by means of a grothendieck topology that is defined on a category consisting of probabilities and that enables to regard them as intuitionistic truth values from these topologies we construct another grothendieck topology that is defined on the product of the context category and

topologies on quantum topoi induced by quantization in the present paper we consider effects of quantization in a topos approach of quantum theory a quantum system is assumed to be coded in a quantum topos by which we mean the topos of presheaves on the context category of commutative subalgebras of a von neumann algebra of bounded operators on a hilbert space a classical system is modeled by a lie algebra of classical observables it is shown that a quantization map from the classical observables to self adjoint operators on the hilbert space naturally induces geometric morphisms from presheaf topoi related to the classical system to the quantum topos by means of the geometric morphisms we give lawvere tierney topologies on the quantum topos and their equivalent grothendieck topologies on the context category we show that among them there exists a canonical one which we call a quantization topology we furthermore give an explicit expression of a sheafification functor associated with the quantization topology

some applications of variational inequalities in nonsmooth analysis abstract in this paper we apply some results on variational inequalities to the clarke generalized gradient and the generalized second order directional derivative f x u v and obtain some variational inequalities of them

isomorphisms of ordered structures of abelian c subalgebras of c algebras abstract the aim of this note is to study the interplay between the jordan structure of c algebra and the structure of its abelian c subalgebras let abel a be a system of unital c subalgebras of a unital c algebra a ordered by set theoretic inclusion we show that any order isomorphism abel a abel b can be uniquely written in the form c c s a i c s a where is a partially linear jordan isomorphism between self adjoint parts of unital c algebras a and b as a corollary we obtain that for certain class of c algebras including von neumann algebras ordered structure of abelian subalgebras completely determines the jordan structure the results extend hitherto known results for abelian c algebras and may be relevant to foundations of quantum theory

diagonalizing matrices over aw algebras abstract every commuting set of normal matrices with entries in an aw algebra can be simultaneously diagonalized to establish this a dimension theory for properly infinite projections in aw algebras is developed as a consequence passing to matrix rings is a functor on the category of aw algebras

classical and quantum probabilities as truth values we show how probabilities can be treated as truth values in suitable sheaf topoi the scheme developed in this paper is very general and applies both in classical and quantum physics on the quantum side the results naturally tie in with the topos approach to quantum theory that has been developed in the last 14 years by the authors and others earlier results on the representation of arbitrary quantum states are complemented with a purely logical perspective

mobius functions of lattices abstract we introduce the concept of a bounded below set in a lattice this can be used to give a generalization of rota s broken circuit theorem to any finite lattice we then show how this result can be used to compute and combinatorially explain the mobius function in various examples including non crossing set partitions shuffle posets and integer partitions in dominance order next we present a generalization of stanley s theorem that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers we also give some applications of this second main theorem including the tamari lattices

the unit ball of an injective operator space has an extreme point abstract we define an a w tro as an off diagonal corner of an a w algebra and show that the unit ball of an a w tro has an extreme point in particular the unit ball of an injective operator space has an extreme point which answers a question raised in 8 affirmatively we also show that an a w tro respectively an injective operator space has an ideal decomposition that is it can be decomposed into the direct sum of a left ideal a right ideal and a two sided ideal in an a w algebra respectively an injective c algebra in particular we observe that an a w tro hence an injective operator space has an algebrization which admits a quasi identity

topos logic in measurement based quantum computation we report first steps towards elucidating the relationship between contextuality measurement based quantum computation mbqc and the non classical logic of a topos associated with the computation we show that in a class of mbqc classical universality requires non classical logic which is consumed during the course of the computation thereby pinpointing another potential quantum computational resource

sensitivity analysis of generalized variational inequalities abstract dafermos studied the sensitivity properties of the solutions of a variational inequality with regard to continuity and differentiability of such solutions with respect to a parameter in the present paper we extend this analysis for a generalized variational inequality of the type introduced by noor of which the variational inequality of dafermos is a particular case our results are such that they automatically extend the regularity properties of solutions with respect to a parameter when the variational inequality is treated on a hilbert space

dye s theorem and gleason s theorem for aw algebras abstract we prove that any map between projection lattices of a w algebras a and b where a has no type i 2 direct summand that preserves orthocomplementation and suprema of arbitrary elements is a restriction of a normal jordan homomorphism between a and b this allows us to generalize dye s theorem from von neumann algebras to a w algebras we show that mackey gleason bunce wright theorem can be extended to homogeneous a w algebras of type i the interplay between dye s theorem and gleason s theorem is shown as an application we prove that jordan homomorphisms are commutatively determined another corollary says that jordan parts of a w algebras can be reconstructed from posets of their abelian subalgebras

a topos foundation for theories of physics iii the representation of physical quantities with arrows this paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system in paper ii we studied the topos representations of the propositional language pl s for the case of quantum theory and in the present paper we do the same thing for the more extensive local language l s one of the main achievements is to find a topos representation for self adjoint operators this involves showing that for any physical quantity a there is an arrow o a r where r is the quantity value object for this theory the construction of o a is an extension of the daseinisation of projection operators that was discussed in paper ii the object r is a monoid object only in the topos setsv h op of the theory and to enhance the applicability of the formalism we apply to r a topos a

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

type decomposition for von neumann algebra embeddings this paper establishes a type decomposition theory for embeddings of a von neumann algebra a into a von neumann algebra m by studying projections in a that are semifinite resp abelian relative to m an application of this theory is a complete characterization of the subalgebras a of m that have an n equipartition of the identity i e i k 1npk with projections pk a which are equivalent in m from this follows r v kadison s result amer j math 106 1984 1451 1468 that masas of m always have equipartitions and hence that normal operator valued matrices are diagonalizable a second application is an answer to a question by g k pedersen and e stormer indiana univ math j 23 1973 121 129 on the characterization of g finite projections in discrete crossed products

some aspects of variational inequalities abstract in this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence uniqueness computational properties various generalizations sensitivity analysis and their applications we also propose some open problems with sufficient information and references so that someone may attempt solution s in his her area of special interest we also include some new results which we have recently obtained

on a class of variational inequalities in this paper we introduce and study a new unified and general class of variational inequalities we derive the general error estimates for the finite element solutions of variational inequalities it has been shown that a class of contact problems with friction terms arising in elastostatics can be studied in the framework of variational inequalities several special cases which can be obtained from the general results are also discussed

the mackey gleason problem let a be a von neumann algebra with no direct summand of type i 2 and let p a be its lattice of projections let x be a banach space let m p a x be a bounded function such that m p q m p m q whenever p and q are orthogonal projections the main theorem states that m has a unique extension to a bounded linear operator from a to x in particular each bounded complex valued finitely additive quantum measure on p a has a unique extension to a bounded linear functional on a

a kochen specker theorem for integer matrices and noncommutative spectrum functors abstract we investigate the possibility of constructing kochen specker uncolorable sets of idempotent matrices whose entries lie in various rings including the rational numbers the integers and finite fields most notably we show that there is no kochen specker coloring of the n n idempotent integer matrices for n 3 thereby illustrating that kochen specker contextuality is an inherent feature of pure matrix algebra we apply this to generalize recent no go results on noncommutative spectrum functors showing that any contravariant functor from rings to sets respectively topological spaces or locales that restricts to the zariski prime spectrum functor for commutative rings must assign the empty set respectively empty space or locale to the matrix ring m n r for any integer n 3 and any ring r an appendix by alexandru chirvasitu shows that kochen specker colorings of idempotents in partial subalgebras of m 3 f for a perfect field f can be extended to partial algebra morphisms into the algebraic closure of f

a horizontal categorification of gel fand duality abstract in the setting of c categories we provide a definition of spectrum of a commutative full c category as a one dimensional unital fell bundle over a suitable groupoid equivalence relation and prove a categorical gel fand duality theorem generalizing the usual gel fand duality between the categories of commutative unital c algebras and compact hausdorff spaces although many of the individual ingredients that appear along the way are well known the somehow unconventional way we glue them together seems to shed some new light on the subject

test retest repeatability of human speech biomarkers from static and real time dynamic magnetic resonance imaging static anatomical and real time dynamic magnetic resonance imaging rt mri of the upper airway is a valuable method for studying speech production in research and clinical settings the test retest repeatability of quantitative imaging biomarkers is an important parameter since it limits the effect sizes and intragroup differences that can be studied therefore this study aims to present a framework for determining the test retest repeatability of quantitative speech biomarkers from static mri and rt mri and apply the framework to healthy volunteers subjects n 8 4 females 4 males are imaged in two scans on the same day including static images and dynamic rt mri of speech tasks the inter study agreement is quantified using intraclass correlation coefficient icc and mean within subject standard deviation e inter study agreement is strong to very strong for static measures icc min median max 0 71 0 89 0 98 e 0 90 2 20 6 72 mm poor to strong for dynamic rt mri measures of articulator

topos models for physics and topos theory what is the role of topos theory in the topos models for quantum theory as used by isham butterfield doring heunen landsman spitters and others in other words what is the interplay between physical motivation for the models and the mathematical framework used in these models concretely we show that the presheaf topos model of butterfield isham and doring resembles classical physics when viewed from the internal language of the presheaf topos similar to the copresheaf topos model of heunen landsman and spitters both the presheaf and copresheaf models provide a quantum logic in the form of a complete heyting algebra although these algebras are natural from a topos theoretic stance we seek a physical interpretation for the logical operations finally we investigate dynamics in particular we describe how an automorphism on the operator algebra induces a homeomorphism or isomorphism of locales on the associated state spaces of the topos models and how elementary propositions and truth values transform under the action of this homeomorphism also with dynamics the focus is on the internal perspective of the topos

diagonalizing matrices over c x a complete characterization of those compact hausdorff spaces is given such that for every n each normal element in the algebra c x mn of continuous functions from x to mn can be continuously diagonalized the conditions are that x be a sub stonean space with dim x 2 and carries no nontrivial g bundles over any closed subset for g a symmetric group or the circle group in particular diagonalization is assured on every totally disconnected sub stonean space but also on connected spaces of the form y y where y is a simply connected noncompact graph

a generalization of rota s nbc theorem abstract we generalize rota s theorem characterizing the mobius function of a geometric lattice in terms of subsets of atoms containing no broken circuit and give applications to the weak bruhat order of a finite coxeter group and the tamari lattices we also give a direct proof of the fact that in the geometric case any total order of the atoms can be used simple involutions are used in both proofs finally we show how involutions can be used in similar situations specifically in a special case of rota s crosscut theorem as well as in related proofs of walker on hall s theorem and reiner on characteristic and poincare polynomials

topos quantum theory on quantization induced sheaves in this paper we construct a sheaf based topos quantum theory it is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von neumann algebras of bounded operators on a hilbert space also it is already known that quantization naturally induces a lawvere tierney topology on the presheaf topos we show that a topos quantum theory akin to the presheaf based one can be constructed on sheaves defined by the quantization induced lawvere tierney topology that is starting from the spectral sheaf as a state space of a given quantum system we construct sheaf based expressions of physical propositions and truth objects and thereby give a method of truth value assignment to the propositions furthermore we clarify the relationship to the presheaf based quantum theory we give translation rules between the sheaf based ingredients and the corresponding presheaf based ones the translation rules have coarse graining effects on the spaces of the presheaf based ingredients a lot of different proposition presheaves truth presheaves and presheaf based truth values are translated to a proposition sheaf a truth sheaf and a sheaf based truth value respectively we examine the extent of the coarse graining made by translation

disjointness preservers of aw algebras abstract in this paper we study bijective linear maps a b between a w algebras preserving either zero products or range orthogonality such a map is automatically continuous and provides an algebra or a algebra isomorphism 1 1 from a onto b our results extend previous works for the case of w algebras and also cover such maps between c algebras satisfying an extra assumption on preserving abelian c subalgebras