Bipolar Hypersoft Sets

Hypersoft set theory is an extension of soft set theory and is a new mathematical tool for dealing with fuzzy problems; however, it still suffers from the parametric tools’ inadequacies. In order to boost decision-making accuracy even more, a new mixed mathematical model called the bipolar hypersoft set is created by merging hypersoft sets and bipolarity. It is characterized by two hypersoft sets, one of which provides positive information and the other provides negative information. Moreover, some fundamental properties relative to it such as subset, superset, equal set, complement, difference, relative (absolute) null set and relative (absolute) whole set are defined. Furthermore, some set-theoretic operations such as the extended intersection, the restricted union, intersection, union, AND-operation and OR-operation of two bipolar hypersoft sets with their properties are discussed and supported by examples. Finally, tabular representations for the purposes of storing bipolar hypersoft sets in computer memory are used.

Mathematical Proof on the Existential Necessity of the World

The "best of all possible worlds" theory (hereinafter the "possible world theory") was presented by Leibniz many centuries ago, and provided an opportunity for metaphysics to understand the ontological meaning of the world in an in-depth way. Furthermore, the theory has multiple impacts on the multiverse theory of modern physics. However, considering the ontological questions of the world solely on linguistics logic leads to basing the possible world theory or the multiverse theory on relatively uncertain inferential reasons. In this study, symbolic logic was used to consider the possible world theory on a more solid foundation. The entire world was made into oneness, and tracking the existential process at the beginning of the world, realized the existential necessity of the world, even though the initial causal point of the world was assumed to be completely void (null set). The existential system in this study, was generalized without specific elements, apart from existence and non-existence. It was discovered that existential possibility and existential necessity are logically equivalent.

A piecewise smooth Fermi–Ulam pingpong with potential

In this paper we study a Fermi–Ulam model where a pingpong ball bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is 1-periodic and piecewise $C^3$ with a singularity, $\dot {f}(0+)\ne \dot {f}(1-)$ . If the second derivative $\ddot {f}(t)$ of the platform motion is either always positive or always less than $-g$ , where g is the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits co-exist with bounded orbits at arbitrarily high energy levels.

Remarks on some generalization of the notion of microscopic sets

We consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ-ideal of classic microscopic sets.

The N-Integral

In this paper, we introduced a Henstock-type integral named N-integral of a real valued function f on a closed and bounded interval [a,b]. The set N-integrable functions lie entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function f on [a,b], the following are equivalent:The function f is N-integrable;There exists a null set S for which given epsilon 0 there exists a gauge delta such that for any delta-fine partial division D={(xi,[u,v])} of [a,b] we have [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u|epsilon] where phi_S(D)={(xi,[u,v])in D:xi not in S} and [Gamma_epsilon={(xi,[u,v]): |f(v)-f(u)|= epsilon}] andThe function f is continuous almost everywhere. A characterization of continuous almost everywhere functions was also given.

Presentations of left-c.e. reals

This chapter examines presentations of left–c.e. reals, proving Theorem 1.4. One of the main ideas of this book is unifying the combinatorics of constructions in various subareas of computability theory. The chapter looks at one such subarea: algorithmic randomness. It provides a brief account of the basics of algorithmic randomness, and includes the basic definitions required in the chapter. While algorithmic randomness has a history going back to the early work of Borel on normal numbers, von Mises, and even Turing, the key concept in the modern incarnation of algorithmic information theory is Martin-Löf randomness. A notion of randomness is determined by a countable collection of null sets, with each null set considered a statistical test. Elements of the null sets are those which have failed the test; they are atypical, in the sense of measure. One of the reasons the notion of ML-randomness is central is that it is robust.

AN APPLICATION OF RECURSION THEORY TO ANALYSIS

Mauldin [15] proved that there is an analytic set, which cannot be represented by $B\cup X$ for some Borel set B and a subset X of a $\boldsymbol{\Sigma }^0_2$-null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated $\sigma $-ideals having approximation property under the assumption that every real is constructible, answering Mauldin’s question raised in [15].

VOTE: Verifiable Auditing for Outsourced Database with Token Enforced Cloud Storage

Database deploying is one of the remarkable utilities in cloud computing where the Information Proprietor (IP) assigns the database administration to the Cloud Service Provider (CSP) in order to lower the administration overhead and preservation expenditures of the database. Regardless of its overwhelming advantages, it experiences few security problems such as confidentiality of deployed database and auditability of search outcome. In recent past, survey has been carried out on the auditability of search outcome of deployed database that gives preciseness and intactness of search outcome. But in the prevailing schemes, since there is flow of data between IP and the clients repeatedly, huge communication cost is incurred at the Information Proprietor side. To address this challenge, we introduce Verifiable Auditing of Outsourced Database with Token Enforced Cloud Storage (VOTE) mechanism based on Merkle Hash Tree (MHT), Invertible Bloom Filter(IBF) and Counting Bloom Filter(CBF). The proposed scheme reduces the huge communication cost at the Information Proprietor side and achieves preciseness and intactness of the search outcome. Experimental analysis show that the proposed scheme has totally reduced the huge communication cost at the Information Proprietor side, and simultaneously achieves the preciseness and intactness of search outcome though the semi- trusted CSP deliberately sends a null set

Normed Interval Space and Its Topological Structure

Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in R cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies.