Deriving weights from pairwise comparison ratio matrices: An axiomatic approach

Every economic system is certainly based on the ideology that provides the foundation and goal on the one hand and the axioms and principles on the other. Every economic system create a framework in which a socio-economic communities can take advantage of the natural resources and human for the sake of production and distribute them for consumption. In this case, Syed NawabHaider Naqvi transform the ethical values of Islam into a set of axioms, that is tauhid (Unity), equilibrium, free will, and responsibility. A set of axioms can be used as a reference in formulating economic behavior consistent. Axiomatic approach in Islamic economics is an analytical tool used to draw up the basic elements of Islamic economics. With sourced from the Al-Qur’an and the Hadith, the Islamic ethical axiom set has been built to meet the five characteristics logically fundamental axioms of the system.

Round robin tournaments are omnipresent in sport competitions and beyond. We investigate three integer programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations, and find that the relaxation of the matching formulation is stronger than the other relaxations, while still being solvable in polynomial time. In addition, we provide an exponentially sized class of valid inequalities for the matching formulation. Complementing our theoretical assessment of the strength of the different formulations, we also experimentally show that the matching formulation is superior on a broad set of instances. Finally, we describe a branch-and-price algorithm for finding round robin tournaments that is based on the matching formulation.

Minimization of rest mismatches in round robin tournaments

Context. The complexity of understanding the concept of multivalued dependencies (when setting out their theoretical foundations, an axiomatic approach is used) in relational (table) databases leads to problems of their modeling, which in turn can cause a violation of data integrity. A partial solution to these issues is facilitated by the search for equivalent axiomatic systems of multivalued dependencies that are minimal in terms of the number and power of their components. This, in turn, requires establishing the equivalence of these axiomatic systems and proving the independence of their elements. The object of the study is the axiomatisations of multivalued dependencies.The purpose of the work is to establish the equivalence of two axiomatic systems of multivalued dependencies, one of which is minimal in terms of the number and power of components, and to prove the independence of specified axiomatics. Method. Set-theoretic and logical-algebraic methods are used in the construction of proofs. The equivalence of two axiomatic systems of multivalued dependencies is established classically: the relation of syntactic inference is considered and sequences of multivalued dependencies are constructed that are proofs of the rules of each axiomatics from the components of the other. To build proof of independence of the axiomatic system, models of relational schemes are proposed, such that for an axiom or rule whose independence is being proved, it is impossible to construct proof from set of other axioms or rules.Results. Equivalence of the axiomatisation of multivalued dependencies proposed by a group of scientists Beeri, Fagin and Howard and the Biskup’s axiomatisation of multivalued dependencies has been established. The independence of the Biskup’s axiomatic system is proved in the sense that without loss of completeness it is impossible to remove neither the only axiom nor none of the inference rules.Conclusions. The independence of the Biskup’s axiomatisation of multivalued dependencies, which has no more in number and power of components than other axiomatic systems of multivalued dependencies, is proved. Using such axiomatic system has advantages in the development of CASE-tools (Computer-Aided Software Engineering Tools), which include the implementation of reducing the relational database schema to the fourth normal form, and also, when manually designing a logical model of a relational database.

In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on $\mathbb{S}_{n}(\mathbb{F}_{q})$, the space of $n\times n$ symmetric matrices over $\mathbb{F}_{q}$. Using this method, we obtain two classes of binary LDPC codes, $\cal{C}(n,q)$ and $\cal{C}^{T}(n,q)$, both of which have grith $8$. Then both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of $\cal{C}^{T}(n,q)$ are both $2q$. As for $\cal{C}(n,q)$, we determine the minimum distance and the stopping distance for some special cases and obtain the lower bounds for other cases.

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system.

Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented.

In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

We establish uncertainty relations between information loss in general open quantum systems and the amount of non-ergodicity of the corresponding dynamics. The relations hold for arbitrary quantum systems interacting with an arbitrary quantum environment. The elements of the uncertainty relations are quantified via distance measures on the space of quantum density matrices. The relations hold for arbitrary distance measures satisfying a set of intuitively satisfactory axioms. The relations show that as the non-ergodicity of the dynamics increases, the lower bound on information loss decreases, which validates the belief that non-ergodicity plays an important role in preserving information of quantum states undergoing lossy evolution. We also consider a model of a central qubit interacting with a fermionic thermal bath and derive its reduced dynamics, to subsequently investigate the information loss and nonergodicity in such dynamics. We comment on the minimal situations that saturate the uncertainty relations.

This paper presents an inductive inferential algebra, which enable us to obtain inductive conclusions for practical problems. The algebra is formulated as a set of axioms and theorems established on the basis of Boolean algebra. Instead of complementary law in Boolean algebra, we use classificatory axiom and change denial definition for handling practical inductive problems. The algebra involves mathematical descriptions of thought laws thus proving the way of obtaining inductive conclusion by symbolic operation.

This paper describes a visual processing technique for automatic frog (Xenopus Laevis sp.) localization and identification. The problem of frog identification is to process and classify an unknown frog image to determine the identity which is recorded previously on an image database. The frog skin pattern (i.e. texture) provides a unique feature for identification. Hence, the study investigates three different kind of features (i.e. Gabor filters, granulometry, threshold set compactness) to extract texture information. The classifier is built on nearest neighbor principle; it assigns the query feature to the database feature which has the minimum distance. Hence, the study investigates different distance measures and compares their performance. The detailed results show that the most successful feature and distance measure is granulometry and weighted L1 norm for the frog identification using skin texture features.

Axiomatic approach to the cosmological constant

ABSTRACTIn many different applications of group decision-making, individual ranking agents or judges are able to rank only a small subset of all available candidates. However, as we argue in this article, the aggregation of these incomplete ordinal rankings into a group consensus has not been adequately addressed. We propose an axiomatic method to aggregate a set of incomplete rankings into a consensus ranking; the method is a generalization of an existing approach to aggregate complete rankings. More specifically, we introduce a set of natural axioms that must be satisfied by a distance between two incomplete rankings; prove the uniqueness and existence of a distance satisfying such axioms; formulate the aggregation of incomplete rankings as an optimization problem; propose and test a specific algorithm to solve a variation of this problem where the consensus ranking does not contain ties; and show that the consensus ranking obtained by our axiomatic approach is more intuitive than the consensus ranking obtained by other approaches.

Axiomatic foundation of the theory of control systems