Finitely Supported Binary Relations between Infinite Atomic Sets

In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the T-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets ℘fs(Ak×Al), ℘fs(Ak×℘m(A)), ℘fs(℘n(A)×Ak) and ℘fs(℘n(A)×℘m(A)) do not contain uniformly supported infinite subsets. Moreover, the functions space ZAm does not contain a uniformly supported infinite subset whenever Z does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.

A Synopsis Based Approach for Itemset Frequency Estimation over Massive Multi-Transaction Stream

The streams where multiple transactions are associated with the same key are prevalent in practice, e.g., a customer has multiple shopping records arriving at different time. Itemset frequency estimation on such streams is very challenging since sampling based methods, such as the popularly used reservoir sampling, cannot be used. In this article, we propose a novel k -Minimum Value (KMV) synopsis based method to estimate the frequency of itemsets over multi-transaction streams. First, we extract the KMV synopses for each item from the stream. Then, we propose a novel estimator to estimate the frequency of an itemset over the KMV synopses. Comparing to the existing estimator, our method is not only more accurate and efficient to calculate but also follows the downward-closure property. These properties enable the incorporation of our new estimator with existing frequent itemset mining (FIM) algorithm (e.g., FP-Growth) to mine frequent itemsets over multi-transaction streams. To demonstrate this, we implement a KMV synopsis based FIM algorithm by integrating our estimator into existing FIM algorithms, and we prove it is capable of guaranteeing the accuracy of FIM with a bounded size of KMV synopsis. Experimental results on massive streams show our estimator can significantly improve on the accuracy for both estimating itemset frequency and FIM compared to the existing estimators.

On the Closure of Aging Classes Under the Formation of Coherent Structures

In this paper, we study the closure property of the Increasing Failure Rate (IFR) class under the formation of coherent systems. Sufficient conditions for the nonpreservation of the IFR attribute for reliability structures consisting of [Formula: see text] independent and identically distributed ([Formula: see text] components are provided. More precisely, we deal with the IFR preservation (or nonpreservation) under the formation of structures with two common failure criteria by the aid of their signature vectors.

Watson–Crick Jumping Finite Automata

Watson–Crick jumping finite automata work on tapes which are double stranded sequences of symbols similar to that of Watson–Crick automata. The double stranded sequence is scanned in a discontinuous manner. That is, after reading a double stranded string, the automata can jump over some subsequence and continue scanning depending on the rule. Some variants of such automata are 1-limited, No state, All final and Simple Watson–Crick jumping finite automata. The comparison of the languages accepted by these variants with the language classes in Chomsky hierarchy has been carried out. We investigate some closure properties. We also try to place the duplication closure of a word in Watson–Crick jumping finite automata family. We have discussed the closure property of Watson–Crick jumping finite automata family under duplication operations.

Monad combinatorics

This chapter considers how the monadic formalization deals with interactions between the phenomena from the second part of the book by examining the pairwise interaction of all three phenomena. Distributive laws are introduced to combine monads. The chapter shows that only certain combinations of the monads from the second part have definable distributive laws. These results comport with linguistic intuitions. The option of using the related category-theoretic concepts of functors and applicative functors instead of monads is also considered. The chapter shows that functors are not powerful enough and that applicative functors introduce a tracking/layering problem that is inelegant. Also, the closure property of applicative functor composition, whereby the composition of any two applicative functors is also an applicative functor, overgenerates with respect to the data. Monads are therefore argued to be empirically superior to applicative functors in this domain precisely because they lack the closure property. Some exercises are provided to aid understanding.

A channel-based perspective on conjugate priors

A desired closure property in Bayesian probability is that an updated posterior distribution be in the same class of distributions – say Gaussians – as the prior distribution. When the updating takes place via a statistical model, one calls the class of prior distributions the ‘conjugate priors’ of the model. This paper gives (1) an abstract formulation of this notion of conjugate prior, using channels, in a graphical language, (2) a simple abstract proof that such conjugate priors yield Bayesian inversions and (3) an extension to multiple updates. The theory is illustrated with several standard examples.

Projective method for spinorial techniques: Unifying calculational schemes of Dirac amplitudes

There have been numerous approaches to the calculation of spin dependent amplitudes for Dirac particles. All of them have their own advantages, particularly, the standard method of calculation, based on the multiplication by the unit, has a few shortcomings. In this work we use the closure property of the Dirac spinors to present a general method for the amplitude computation. It is shown that the massless spinor method and the helicity spinor method can be formulated through this method. Finally, we get an example of this calculation procedure computing the spin dependentamplitude for the Compton process.

An Intelligent Decision in Smart Systems Using A Weighted Frequent Itemset Mining Algorithm

Intelligent decision is the key technology of smart systems. Data mining technology has been playing an increasingly important role in decision making activities. The introduction of weight makes the weighted frequent itemsets not satisfy the downward closure property any longer. As a result, the search space of frequent itemsets cannot be narrowed according to downward closure property which leads to a poor time efficiency. In this paper, the weight judgment downward closure property for weighted frequent itemsets and the existence property of weighted frequent subsets are introduced and proved first. The Fuzzy-based WARM satisfies the downward closure property and prunes the insignificant rules by assigning the weight to the itemset. This reduces the computation time and execution time. This paper presents an Enhanced Fuzzy-based Weighted AssociationRuleMining(E-FWARM) algorithm for efficient mining of the frequent itemsets. The pre-filtering method is applied to the input dataset to remove the item having low variance. Data discretization is performed and E-FWARM is applied for mining the frequent itemsets. The experimental results show that the proposed E-FWARM algorithm yields maximum frequent items, association rules, accuracy and minimum execution time than the existing algorithms.