Linear operators preserving invariants of nonbinary boolean matrices

Embeddings of metric Boolean algebras in [formula omitted

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. As a consequence, the tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.

A Characterization of Perfect Matrices

We consider combinatorial programming problems of the form (LP): max {cx|Ax ≤e, xj=0 or 1, vj}, where A is a mxn matrix of zeroes and ones, e is a column vector of m ones and c is an arbitrary (non-negative) vector of reals. Applications of this general problem include crew scheduling, political districting and others. In this paper we first summarize (without proofs) the results of a companion paper that completely characterize matrices A for Which.(IP) can be solved as an ordinary linear programming problem, i.e. where the relaxed linear program (LP): max{cx|Ax ≤e, xj≥0,vj} produces an integral solution no matter what linear form cx is maximized. (Zero-one matrices with this property are termed “perfect”). Some additional concepts and results are stated. It is shown that every totally unimodular zero-one matrix as well as every “balanced” zero-one matrix is perfect. Finally, in the concluding remarks, a reformulation of the strong perfect graph due to C. BERGE is given and some recent trends in zezo-one programming are delineated.

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

Complexity of Boolean algebras

In this chapter, we are going to study linear operators on finite-dimensional inner product spaces. In the last chapter, we introduced adjoints of linear maps between possibly different inner product spaces. Here we shall see how the adjoint can be used to understand linear operators on a fixed inner product space. The important operators we study here are the self-adjoint, skew-adjoint, normal, orthogonal, and unitary operators. We shall spend several sections on the existence of eigenvalues, diagonalizability, and canonical forms for these special but important linear operators. Having done that, we go back to the study of general linear maps and operators and establish the singular value and polar decompositions. We also show Schur’s theorem to the effect that complex linear operators have upper triangular matrix representations. This result does not depend on the spectral theorem. It is also possible to give a quick proof of the spectral theorem using only the material covered in Sect. 4.1. The chapter finishes with a section on quadratic forms and how they tie in with the theory of self-adjoint operators.

The necessity of non-binary constraint satisfaction algorithms is increasing because many real problems are inherently nonbinary. Considering overconstrained problems (and Partial Forward Checking as the solving algorithm), we analyze several lower bounds proposed in the binary case, extending them for the non-binary case. We show that techniques initially developed in the context of reversible DAC can be applied in the general case, to deal with constraints of any arity. We discuss some of the issues raised for non-binary lower bounds, and we study their computational complexity. We provide experimental results of the use of the new lower bounds on overconstrained random problems, including constraints with different weights.KeywordsConstraint Satisfaction ProblemSoft ConstraintLimited VersionFuture VariableCurrent Partial AssignmentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This article presents a new theory of subjective probability according to which different descriptions of the same event can give rise to different judgments. The experimental evidence confirms the major predictions of the theory. First, judged probability increases by unpacking the focal hypothesis and decreases by unpacking the alternative hypothesis. Second, judged probabilities are complementary in the binary case and subadditive in the general case, contrary to both classical and revisionist models of belief. Third, subadditivity is more pronounced for probability judgments than for frequency judgments and is enhanced by compatible evidence. The theory provides a unified treatment of a wide range of empirical findings. It is extended to ordinal judgments and to the assessment of upper and lower probabilities. Both laypeople and experts are often called upon to evaluate the probability of uncertain events such as the outcome of a trial, the result of a medical operation, the success of a business venture, or the winner of a football game. Such assessments play an important role in deciding, respectively, whether to go to court, undergo surgery, invest in the venture, or bet on the home team. Uncertainty is usually expressed in verbal terms (e.g., unlikely or probable), but numerical estimates are also common. Weather forecasters, for example, often report the probability of rain (Murphy, 1985), and economists are sometimes required to estimate the chances of recession (Zarnowitz, 1985). The theoretical and practical significance of subjective probability has inspired psychologists, philosophers, and statisticians to investigate this notion from both descriptive and prescriptive standpoints.

The Forward Problem (FB) of Boolean equations consists of finding solutions of a system of Boolean equations, or equivalently, a single Boolean equation of the form f(X) = 0 where f(X): Bn → B and B is an arbitrary Boolean algebra. By contrast, the Inverse Problem (IB) of Boolean equations aims to reconstruct the equation f (X) = 0 given the set of solutions and hence to verify the correctness of this set. This study derives methods that handle this inverse problem for the main types of solutions of Boolean equations. These include: (a) Subsumptive general solutions, in which each of the variables is expressed as an interval by deriving successive conjunctive or disjunctive eliminants of the original function, (b) Parametric general solutions, in which each of the variables is expressed via arbitrary parameters which are freely chosen elements of the underlying Boolean algebra and (c) Particular solutions, each of which is an assignment from the underlying Boolean algebra to every pertinent variable that makes the Boolean equation an identity. The reconstructed function f(X) in every case is set in a canonical form, such as the complete-sum form, to facilitate proving its equivalence to the original function. The methods presented herein are demonstrated with carefully-chosen illustrative examples over big Boolean algebras of various sizes. Among the methods utilized in handling the inverse problem for Boolean equations, the ones utilizing the variable-entered Karnaugh map offered pictorial insight and exhibited an efficient divide-and-conquer strategy.

Background: Using Boolean algebra for differential diagnosis of neurological disorders involves treating neurologic findings and disease entities such as Boolean variables and combining them via OR (∨) or other logical operations. Aim: To demonstrate how Boolean algebra (mathematical logic) and Boolean matrices can be applied to support differential diagnosis in neurology. Results: Each neurologic finding was encoded as a Boolean variable (1 = present, 0 = absent). Boolean variables and Boolean matrices reflect mutual interdependencies of neurologic findings across diseases, enabling reverse inference (i.e., inferring possible diseases from a pattern of findings). Conclusion: Boolean algebra offers a formal, rigorous framework to represent and manipulate differential diagnostic relationships in neurology; it may complement clinical reasoning and computational decision support.

Boolean algebra as algebra is value indifferent. Although it is very important, the classical two-valued realization of the Boolean algebra is only a special case. The real valued realization of the Boolean algebra is a frame for the Boolean consistent fuzzy logic in wider sense, such as the two valued realization is a frame for the classical case. This paper presents the brief review of the real-valued realization of the finite (atomic) Boolean algebra and its application. The real-valued implication is a Boolean consistent generalization of the classical binary implication. The real-valued implication plays important roles in the real-valued set theory as a generalization of the classical set theory, as well as, in many applications such as morphology in image processing, association rules in data mining and decision making generally.

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.

On constructing new expansions of functions using linear operators

On the number of mutually disjoint pairs of [formula omitted]-permutation matrices