Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 172–185, January, 2006.     

Centrality as a gradual notion: A new bridge between fuzzy sets and statistics

This paper aims at formalizing the intuitive idea that some points are more central in a probability distribution than others. Our proposal relies on fuzzy events to define a fuzzy set of central points for a distribution (or a family of distributions, including imprecise probability models). This framework has a natural interpretation in terms of fuzzy logic and unifies many known notions from statistics, including the mean, median and mode, interquantile intervals, the Lorenz curve, the halfspace median, the zonoid and lift zonoid, the coverage function and several expectations and medians of random sets, and the Choquet integral against an infinitely alternating or infinitely monotone capacity.     

Some implications of the union-intersection principle for tests of sphericity

The sphericity hypothesis may be expressed as an intersection of simpler hypotheses on the invariant subspaces of the variance matrix. Applying the union-intersection principle to dissections of this type establishes a link between tests of independence and tests of sphericity. We use some recent results of Bloomfield and Watson [2] and Knott [4] to derive a class of union-intersection tests for sphericity from likelihood ratio tests of independence of two sets of variates. As well, we show that the ordinary likelihood ratio test for sphericity has a natural union-intersection interpretation.     

Feature Extraction for Nonparametric Discriminant Analysis

In high-dimensional classification problems, one is often interested in finding a few important discriminant directions in order to reduce the dimensionality. Fisher's linear discriminant analysis (LDA) is a commonly used method. Although LDA is guaranteed to find the best directions when each class has a Gaussian density with a common covariance matrix, it can fail if the class densities are more general. Using a likelihood-based interpretation of Fisher's LDA criterion, we develop a general method for finding important discriminant directions without assuming the class densities belong to any particular parametric family. We also show that our method can be easily integrated with projection pursuit density estimation to produce a powerful procedure for (reduced-rank) nonparametric discriminant analysis.     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

Commutative basic algebras and non-associative fuzzy logics

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. ?ukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L _CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers and includes the class of MV-algebras. We show that the logic L _CBA is very close to the ?ukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization.     

Tuning membership functions of kernel fuzzy classifiers by maximizing margins

We propose two methods for tuning membership functions of a kernel fuzzy classifier based on the idea of SVM (support vector machine) training. We assume that in a kernel fuzzy classifier a fuzzy rule is defined for each class in the feature space. In the first method, we tune the slopes of the membership functions at the same time so that the margin between classes is maximized under the constraints that the degree of membership to which a data sample belongs is the maximum among all the classes. This method is similar to a linear all-at-once SVM. We call this AAO tuning. In the second method, we tune the membership function of a class one at a time. Namely, for a class the slope of the associated membership function is tuned so that the margin between the class and the remaining classes is maximized under the constraints that the degrees of membership for the data belonging to the class are large and those for the remaining data are small. This method is similar to a linear one-against-all SVM. This is called OAA tuning. According to the computer experiment for fuzzy classifiers based on kernel discriminant analysis and those with ellipsoidal regions, usually both methods improve classification performance by tuning membership functions and classification performance by AAO tuning is slightly better than that by OAA tuning.     

基于Kernel Discriminant Canonical Correlation(KDCC)的多观测样本分类算法

针对多观测样本分类问题,提出一种基于Kernel Discriminant CanonicalCorrelation(KDCC)来实现多观测样本分类的模型.该算法首先把原空间样本非线性的投影到高维特征空间,通过KPCA得到核子空间,然后在高维特征空间定义一个使类内核子空间的相关性最大,同时使类间核子空间的相关性最小的KDCC矩阵,通过迭代法训练出最优的KDCC矩阵,把每个核子空间投影到KDCC矩阵上得到转换核子空间,采用典型相关性作为转换核子空间之间的相似性度量,并采用最近邻准则作为多观测样本的分类决策,从而实现多观测样本的分类.在三个数据库上进行了一系列实验,实验结果表明提出的方法对于多观测样本分类具有可行性和有效性.     

Quantum logics,games, and equilibriums

We establish a natural relation between antagonistic matrix games and ortholattices (quantum logics).We show that an equilibrium in the corresponding quantum game determines the operator representation of a quantum logic. We formulate a condition for quantum equilibrium.     

Logics for belief functions on MV-algebras

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism.     

Quantum set theory: Transfer Principle and De Morgan's Laws

In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws.     

Quantum logics with the Riesz Interpolation Property

We study the quantum logics which satisfy the Riesz Interpolation Property. We call them the RIP logics. We observe that the class of RIP logics is considerable large—it contains all lattice quantum logics and, also, many (infinite) non‐lattice ones. We then find out that each RIP logic can be enlarged to an RIP logic with a preassigned centre. We continue, showing that the “nearly” Boolean RIP logics must be Boolean algebras. In a somewhat surprising contrast to this, we finally show that the attempt for the σ‐complete formulation of this result fails: We show by constructing an example that there is a non‐Boolean nearly Boolean σ‐RIP logic. As a result, there are interesting σ‐RIP logics which are intrinsically close to Boolean σ‐algebras. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral sequences in combinatorial geometry: Cheeses, inscribed sets, and Borsuk-Ulam type theorems

Algebraic topological methods are especially well suited for determining the non-existence of continuous mappings satisfying certain properties. In combinatorial problems it is sometimes possible to define a mapping from a space X of configurations to a Euclidean space Rm in which a subspace, a discriminant, often an arrangement of linear subspaces A, expresses a target condition on the configurations. Add symmetries of all these data under a group G for which the mapping is equivariant. If we remove the discriminant from Rm, we can pose the problem of the existence of an equivariant mapping from X to the complement of the discriminant in Rm. Algebraic topology may sometimes be applied to show that no such mapping exists, and hence the image of the original equivariant mapping must meet the discriminant.We introduce a general framework, based on a comparison of Leray-Serre spectral sequences. This comparison can be related to the theory of the Fadell-Husseini index. We apply the framework to:

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solve a mass partition problem (antipodal cheeses) in Rd,

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determine the existence of a class of inscribed 5-element sets on a deformed 2-sphere,

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obtain two different generalizations of the theorem of Dold for the non-existence of equivariant maps which generalizes the Borsuk-Ulam theorem.

     

An additive utility mixed integer programming model for nonlinear discriminant analysis

Mathematical programming (MP) discriminant analysis models can be used to develop classification models for assigning observations of unknown class membership to one of a number of specified classes using values of a set of features associated with each observation. Since most MP discriminant analysis models generate linear discriminant functions, these MP models are generally used to develop linear classification models. Nonlinear classifiers may, however, have better classification performance than linear classifiers. In this paper, a mixed integer programming model is developed to generate nonlinear discriminant functions composed of monotone piecewise-linear marginal utility functions for each feature and the cut-off value for class membership. It is also shown that this model can be extended for feature selection. The performance of this new MP model for two-group discriminant analysis is compared with statistical discriminant analysis and other MP discriminant analysis models using a real problem and a number of simulated problem sets.     

Imprecise stochastic orders and fuzzy rankings

We extend the notion of stochastic order to the pairwise comparison of fuzzy random variables. We consider expected utility, stochastic dominance and statistical preference, which are related to the comparisons of the expectations, distribution functions and medians of the underlying variables, and discuss how to generalize these notions to the fuzzy case, when an epistemic interpretation is given to the fuzzy random variables. In passing, we investigate to which extent the earlier extensions of stochastic dominance and expected utility to the comparison of sets of random variables can be useful as fuzzy rankings.     

模糊Hopfield网络及其模糊聚类功能研究

本文提出了一种能进行模糊逻辑计算的Ｈｏｐｆｉｌｅｄ型人工神经元网络，ＦｕｚｚｙＨＮ的神经元对应模式集合中的元素，模式间的模糊相似关系作为联结神经元的权值矩阵被存储在ＦｕｚｚｙＨＮ中。本文对ＦｕｚｚｙＨＮ的稳定性及模糊聚类功能进行研究，获得了良好的理论分析结果和实验研究结果。     

Ordered group-valued probability,positive operators,and integral representations

Probability maps are additive and normalised maps taking values in the unit interval of a lattice ordered Abelian group. They appear in theory of affine representations and they are also a semantic counterpart of Hájek’s probability logic. In this paper we obtain a correspondence between probability maps and positive operators of certain Riesz spaces, which extends the well-known representation theorem of real-valued MV-algebraic states by positive linear functionals. When the codomain algebra contains all continuous functions, the set of all probability maps is convex, and we prove that its extreme points coincide with homomorphisms. We also show that probability maps can be viewed as a collection of states indexed by maximal ideals of a codomain algebra, and we characterise this collection in special cases.     

A bipolar model of assertability and belief

Valuation pairs are introduced as a bipolar model of the assertability of propositions. These correspond to a pair of dual valuation functions, respectively, representing the strong property of definite assertability and the dual weaker property of acceptable assertability. In the case where there is uncertainty about the correct valuation pair for a language then a probability distribution is defined on possible valuation pairs. This results in two measures, μ⁺ giving the probability that a sentence is definitely assertable, and μ⁻ giving the probability that a sentence is acceptable to assert. It is shown that μ⁺ and μ⁻ can be determined directly from a two dimensional mass function m defined on pairs of sets of propositional variables. Certain natural properties of μ⁺ and μ⁻ are easily expressed in terms of m, and in particular we introduce certain consonance or nestedness assumptions. These capture qualitative information in the form of assertability orderings for both the propositional variables and the negated propositional variables. On the basis of these consonance assumptions we show that label semantics, intuitionistic fuzzy logic and max-min fuzzy logic can all be viewed as special cases of this bipolar model. We also show that bipolar belief measures can be interpreted within an interval-set model.     

Visible acyclic differential nets, Part I: Semantics

We give a geometric condition that characterizes the differential nets having a finitary interpretation in finiteness spaces: visible acyclicity. This is based on visible paths, an extension to differential nets of a class of paths we introduced in the framework of linear logic nets. The characterization is then carried out as follows: the differential nets having no visible cycles are exactly those whose interpretation is a finitary relation. Visible acyclicity discloses a new kind of correctness for the promotion rule of linear logic, which goes beyond sequent calculus correctness.