Subset Of Set Research Articles

On algebraic properties of soft real points

In this paper, we introduce and discuss soft single points, which proceed towards soft real points by using real numbers and subsets of set of real numbers. We also define the basic operations on soft real points and explore the algebraic properties. We observe that the set of all soft real points forms a ring. Moreover, we study the soft real point metric using soft real point and explore some of its properties. We then establish a soft real point contraction fixed point theorem using soft real point metric space. It is interesting to mention that these concepts may be helpful for researchers to navigate the ideas put forth in a soft metric extension of several important fixed point theorems for metric spaces deduced from comparable existing results.

Maximal c consensus meets

Given a set S of subsets of a reference set X, we define the problem of finding c subsets of X that maximize the size of the intersection among the included subsets. Maximizing the size of the intersection means that they are subsets of the sets in S and they are as large as possible.We can understand the result of this problem as c consensus sets of S, or c consensus representatives of S. From the perspective of lattice theory, each representative will be a meet of some sets in S.In this paper we define formally this problem, and present heuristic algorithms to solve it. We also discuss the relationship with other established problems in the literature.

Assessing efficacy of day 3 embryo time-lapse algorithms retrospectively: impacts of dataset type and confounding factors

This study investigated the efficacy of four published day 3 embryo time-lapse algorithms based on different types of datasets (known implantation data [KID] and single embryo transfer [SET]), and the confounding effect of female age and conventional embryo morphology. Four algorithms were retrospectively applied to three types of datasets generated at Fertility North between February 2013 and December 2014: (a) KID dataset (n = 270), (b) a subset of SET (n = 144, end-point = implantation), and (c) SET (n = 144, end-point = live birth), respectively. All four algorithms showed progressively reduced predictive power (expressed as area under the receiver operating characteristics curve and 95% confidence interval [CI]) after application to the three datasets (a–c): Liu (0.762 [0.701–0.824] vs. 0.724 [0.641–0.807] vs. 0.707 [0.620–0.793]), KIDScore (0.614 [0.539–0.688] vs. 0.548 [0.451–0.645] vs. 0.536 [0.434–0.637]), Meseguer (0.585 [0.508–0.663] vs. 0.56 [0.462–0.658] vs. 0.549 [0.445–0.652]), and Basile (0.582 [0.505–0.659] vs. 0.519 [0.421–0.618] vs. 0.509 [0.406–0.612]). Furthermore, using KID dataset, the association (expressed as odds ratio and 95% CI) between time-lapse algorithms and implantation outcomes lost statistical significance after adjusting for conventional embryo morphology and female age in 3 of the 4 algorithms (KIDScore 1.832 [1.118–3.004] vs. 1.063 [0.659–1.715], Meseguer 1.150 [1.021–1.295] vs. 1.122 [0.981–1.284] and Basile 1.122 [1.008–1.249] vs. 1.038 [0.919–1.172]). In conclusion, SET is a preferred dataset to KID when developing or validating time-lapse algorithms, and day 3 conventional embryo morphology and female age should be considered as confounding factors.

COHERENT AND INCOHERENT CONTENTS OF CORRELATIONS

We examine bipartite and multipartite correlations within the construct of unitary orbits. We show that the set of product states is a very small subset of set of all possible states, while all unitary orbits contain classically correlated states. Using this we give meaning to degeneration of quantum correlations due to a unitary interactions, which we call coherent correlations. The remaining classical correlations are called incoherent correlations and quantified in terms of the distance of the joint probability distributions to its marginals. Finally, we look at how entanglement looks in this picture for the two-qubit case.

Abstract SY34-01: Genome-wide association studies in cancer: A step in the right direction

Abstract SY34-01: Genome-wide association studies in cancer: A step in the right direction

A mixed measure of content on the set of real numbers

The counting measure applies only on countable subsets of the set of real numbers. On the other hand, the Lebesgue measure applies on any countable union of intervals but ignores countable subsets since it assigns to them a null weight indiscriminately. This paper proposes a measure of content which applies on finite unions of intervals and enables to differentiate finite sets. This measure of content is shown to be a Choquet capacity. Furthermore, extension onto the system of all subsets of the real number set is discussed and ideas for generalization to the multidimensional space are presented. A class of content-based measures of comparison is also suggested, along with a discussion of some of their basic properties.

The spectrum of partitions of a Boolean algebra

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion. Some of the main results of the paper are: • (1) If there is a maximal family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ. • (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products. • (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA. • (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras.

Generic Partial Two-Point Sets Are Extendable

AbstractIt is shown that under ZFC almost all planar compacta that meet every line in at most two points are subsets of sets that meet every line in exactly two points. This result was previously obtained by the author jointly with K. Kunen and J. vanMill under the assumption that Martin’s Axiom is valid.

Meager sets on the hyperfinite time line

In this paper we study notions of a “meager subset” of a hyper-finite set. We work within an ω-saturated nonstandard universe and fix a hyperfinite natural number Є *N∖N. We shall consider subsets of the set = {1, 2, …,H}.By analogy with the meager subsets of the real interval [0, 1], a notion of meager subset of should have the following properties.1. Finite sets, countable unions of meager sets, subsets of meager sets, and translates of meager sets should be meager.2. The Baire Category Theorem should hold; that is, should not be meager.3. The internal analogue of the Cantor set should be meager.4. The notion of a meager set should be with respect to a natural topology on .5. There should exist meager subsets of of Loeb measure one.6. Sierpiński and Lusin sets should have hyperfinite counterparts with properties similar to the classical case.Property 5 is desirable so that, as with Lebesgue measure and Baire category on [0, 1], the topological and measure-theoretic notions of “large” and “small” sets are incomparable.Property 6, while not as necessary as the other five, is desirable because of the strong interplay between measure and category in the classical results about Lusin and Sierpinski sets. Our objective is to find notions of meager set which have a relationship to Loeb measure similar to the classical relationship between meager sets and Lebesgue measure.

Design of crosspoint-irredundant PLAs using minimal number of control inputs

The problem of determining a minimal number of control inputs for converting a programmable logic array (PLA) with undetectable faults to crosspoint-irredundant PLA for testing has been formulated as a nonstandard set covering problem. By representing subsets of sets as cubes, this problem has been reformulated as familiar problems. It is noted that this result has significance because a crosspoint-irredundant PLA can be converted to a completely testable PLA in a straightforward fashion, thus achieving very good fault coverage and easy testability. >

Studies on counting the echo pattern of individual fish by pattern analysis method. III. Analysis of elements of echo data.

In order to design the mask and to count echo patterns by field data, the structure and characteristics of the group of element “SET” and the relationship between these charac-teristics and several factors were analyzed from field data obtained during the cruise of the Hakuho Maru (KH-69-4) in 1969. 1) Several groups of elements were extracted from the digital echo data by applying the linkage of elements. Furthermore, several SUBSETs where the method for decision of the next base point is different from that of SET were extracted from SET by the linkage map. 2) SETs are classified into three types (A, B2 and BM) by the number of SUBSET in SET (one, two and more than two). 3) The number of SETs of TYPE-B (B2+BM) increases in accordance to slice level decreases, and the number of elements of SET for TYPE-B was found to be more than that of TYPE-A. Especially, when the slice level is low the number of elements in SET increased. 4) From tests for determining the ratio of the number of TYPE-BM to TYPE-A+B2, the number of SETs of the latter were found to be more than that of the former in all cases. 5) The effect of ship speed on the number of SETs in 6 knots was not as considerable as that in 4 knots. 6) Moreover, two undesirable types of SETs, SET ELEMENT OVER and SUBSET OVER, were counted in each case, because these SETs are not suitable for application to design echo masks. The former is SET which has more than 118 elements, and the latter is SET which has more than 16 SUBSETs. These special SETs increased abruptly when the slice level becomes low. 7) From these analyses, it seems that 0.4 or 0.5 for slice level and 6 knots for ship speeds are optimum levels for designing of the mask and echo pattern counting.