This document summarizes research on generalized β-closure spaces and β-connectedness. It defines key concepts such as β-closure functions, β-separated sets, and β-connectedness. Several important results are proven, including: (1) A set is β-connected if its subsets satisfy the Hausdorff-Lennes condition; (2) If a function between neighborhood spaces is β-continuous, the preimage of a β-connected set is β-connected; (3) The β-closure of a β-connected set is β-connected. The document establishes relationships between β-closure, β-separation, and β-connectedness in generalized closure spaces.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
If X be a topological space and A subspace of X, then a point x E X is said to be a cluster point of A if every open ball centered at x contains at least one point of A different from X. In the preliminary sections, review of the interior of the set X was discussed before the major work of section three was implemented.
Energy Awareness and the Role of “Critical Mass” In Smart Citiesirjes
A Smart City could be depicted as a place, logical and physical, in which a crowd of heterogeneous
entities is related in time and space through different types of interactions. Any type of entity, whether it is a
device or a person, clustered in communities, becomes a source of context-based data.
Energy awareness is able to drive the process of bringing our society to limit energy waste and to optimize
usage of available resources, causing a strong environmental and social impact. Then, following social network
analysis methodologies related to the dynamics of complex systems, it is possible to find out, emergent and
sometimes hidden new habits of electricity usage. Through an initial Critical Mass, involving a multitude of
consumers, each related to more contexts, we evaluate the triggering and spreading of a collective attitude. To
this aim, in this paper, we propose a novel analytical model defining a new concept of critical mass, which
includes centrality measures both in a single layer and in a multilayer social network.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
Comparative Study of Pre-Engineered and Conventional Steel Frames for Differe...irjes
In this paper, the conventional steel frames having triangular Pratt truss as a roofing system of 60 m
length, span 30m and varying bay spacing 4m, 5m and 6m respectively having eaves level for all the portals is at
10m and the EOT crane is supported at the height of 8m from ground level and pre-engineered steel frames of
same dimensions are analyzed and designed for wind zones (wind zone 2, wind zone 3, wind zone 4 and wind
zone 5) by using STAAD Pro V8i. The study deals with the comparative study of both conventional and preengineered
with respect to the amount of structural steel required, reduction in dead load of the structure.
Possible limits of accuracy in measurement of fundamental physical constantsirjes
The measurement uncertainties of Fundamental Physical Constants should take into account all
possible and most influencing factors. One from them is the finiteness of the model that causes the existence of
a-priori error. The proposed formula for calculation of this error provides a comparison of its value with the
actual experimental measurement error that cannot be done an arbitrarily small. According to the suggested
approach, the error of the researched Fundamental Physical Constant, measured in conventional field studies,
will always be higher than the error caused by the finite number of dimensional recorded variables of physicalmathematical
models. Examples of practical application of the considered concept for measurement of fine
structure constant, speed of light and Newtonian constant of gravitation are discussed.
Abstract: In this paper, we define and study about a new type of generalized closed set called, g∗s-closed set.Its relationship with already defined generalized closed sets are also studied
(𝛕𝐢, 𝛕𝐣)− RGB Closed Sets in Bitopological SpacesIOSR Journals
In this paper we introduce and study the concept of a new class of closed sets called (𝜏𝑖, 𝜏𝑗)− regular generalized b- closed sets (briefly(𝜏𝑖, 𝜏𝑗)− rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (𝜏𝑖, 𝜏𝑗)− rgb- neighbourhood (briefly(𝜏𝑖, 𝜏𝑗)− rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
If X be a topological space and A subspace of X, then a point x E X is said to be a cluster point of A if every open ball centered at x contains at least one point of A different from X. In the preliminary sections, review of the interior of the set X was discussed before the major work of section three was implemented.
Energy Awareness and the Role of “Critical Mass” In Smart Citiesirjes
A Smart City could be depicted as a place, logical and physical, in which a crowd of heterogeneous
entities is related in time and space through different types of interactions. Any type of entity, whether it is a
device or a person, clustered in communities, becomes a source of context-based data.
Energy awareness is able to drive the process of bringing our society to limit energy waste and to optimize
usage of available resources, causing a strong environmental and social impact. Then, following social network
analysis methodologies related to the dynamics of complex systems, it is possible to find out, emergent and
sometimes hidden new habits of electricity usage. Through an initial Critical Mass, involving a multitude of
consumers, each related to more contexts, we evaluate the triggering and spreading of a collective attitude. To
this aim, in this paper, we propose a novel analytical model defining a new concept of critical mass, which
includes centrality measures both in a single layer and in a multilayer social network.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
Comparative Study of Pre-Engineered and Conventional Steel Frames for Differe...irjes
In this paper, the conventional steel frames having triangular Pratt truss as a roofing system of 60 m
length, span 30m and varying bay spacing 4m, 5m and 6m respectively having eaves level for all the portals is at
10m and the EOT crane is supported at the height of 8m from ground level and pre-engineered steel frames of
same dimensions are analyzed and designed for wind zones (wind zone 2, wind zone 3, wind zone 4 and wind
zone 5) by using STAAD Pro V8i. The study deals with the comparative study of both conventional and preengineered
with respect to the amount of structural steel required, reduction in dead load of the structure.
Possible limits of accuracy in measurement of fundamental physical constantsirjes
The measurement uncertainties of Fundamental Physical Constants should take into account all
possible and most influencing factors. One from them is the finiteness of the model that causes the existence of
a-priori error. The proposed formula for calculation of this error provides a comparison of its value with the
actual experimental measurement error that cannot be done an arbitrarily small. According to the suggested
approach, the error of the researched Fundamental Physical Constant, measured in conventional field studies,
will always be higher than the error caused by the finite number of dimensional recorded variables of physicalmathematical
models. Examples of practical application of the considered concept for measurement of fine
structure constant, speed of light and Newtonian constant of gravitation are discussed.
Performance Comparison of Energy Detection Based Spectrum Sensing for Cogniti...irjes
With the rapid deployment of new wireless devices and applications, the last decade has witnessed a growing
demand for wireless radio spectrum. However, the policy of fixed spectrum assignment produces a bottleneck for more
efficient spectrum utilization, such that a great portion of the licensed spectrum is severely under-utilized. So the concept of
cognitive radio was introduced to address this issue.The inefficient usage of the limited spectrum necessitates the
development of dynamic spectrum access techniques, where users who have no spectrum licenses, also known as secondary
users, are allowed to use the temporarily unused licensed spectrum. For this purpose we have to know the presence or
absence of primary users for spectrum usage. So spectrums sensing is one of the major requirements of cognitive radio.Many
spectrum sensing techniques have been developed to sense the presence or absence of a licensed user. This paper evaluates
the performance of the energy detection based spectrum sensing technique in noisy and fading environments.The
performance of the energy detection technique will be evaluated by use of Receiver Operating Characteristics (ROC) curves
over additive white Gaussian noise (AWGN) and fading channels.
Effects of Cutting Tool Parameters on Surface Roughnessirjes
This paper presents of the influence on surface roughness of Co28Cr6Mo medical alloy machined
on a CNC lathe based on cutting parameters (rotational speed, feed rate, depth of cut and nose radius).The
influences of cutting parameters have been presented in graphical form for understanding. To achieve the
minimum surface roughness, the optimum values obtained for rpm, feed rate, depth of cut and nose radius were
respectively, 318 rpm, 0,1 mm/rev, 0,7 mm and 0,8 mm. Maximum surface roughness has been revealed the
values obtained for rpm, feed rate, depth of cut and nose radius were respectively, 318 rpm, 0,25 mm/rev, 0,9
mm and 0,4 mm.
Analysis of Agile and Multi-Agent Based Process Scheduling Modelirjes
As an answer of long growing frustration of waterfall Software development life cycle concepts,
agile software development concept was evolved in 90’s. The most popular agile methodologies is the Extreme
Programming (XP). Most software companies nowadays aim to produce efficient, flexible and valuable
Software in short time period with minimal costs, and within unstable, changing environments. This complex
problem can be modeled as a multi-agent based system, where agents negotiate resources. Agents can be used to
represent projects and resources. Crucial for the multi-agent based system in project scheduling model, is the
availability of an effective algorithm for prioritizing and scheduling of task. To evaluate the models, simulations
were carried out with real life and several generated data sets. The developed model (Multi-agent based System)
provides an optimized and flexible agile process scheduling and reduces overheads in the software process as it
responds quickly to changing requirements without excessive work in project scheduling.
Extrusion can be defined as the process of subjecting a material to compression so that it is forced to
flow through an opening of a die and takes the shape of the hole. Multi-hole extrusion is the process of
extruding the products through a die having more than one hole. Multi-hole extrusion increases the production
rate and reduces the cost of production. In this study the ram force has calculated experimentally for single hole
and multi-hole extrusion. The comparison of ram forces between the single hole and multi-hole extrusion
provides the inverse relation between the numbers of holes in a die and ram force. The experimental lengths of
the extruded products through the various holes of multi-hole die are different. It indicates that the flow pattern
is dependent on the material behavior. The micro-hardness test has done for the extruded products of lead
through multi-hole die. It is observed that the hardness of the extruded lead products from the central hole is
found to be more than that of the products extruded from other holes. The study suggests that multi-hole
extrusion can be used for obtaining the extruded products of lead with varying hardness. The micro-structure
study has done for the lead material before and after extrusion. It is observed that the size of grains of lead
material after extrusion is smaller than the original lead.
Cars are a very important part of this modern world because they give luxury and comfort. Even
though they are comfortable, some problems always keep arising on the safety side. After a lot of research they
rectified certain problems using air bags, auto parking, turbo charger, pedal shift…, etc.
And now we are going to discuss about one such problem that arises on the safety side. An unsuspected
accident occurs when people smash their fingers in between the car doors. Due to this kind of accident around
120,000 people are injured every year. But this was not taken as a very major safety concern for the customer.
To avoid this kind accident due to car doors, we are introducing “SAFETY DOOR LOCK SYSTEM”
with the help of “HYDRAULIC PISTON AND IR SENSORS”.
The major working process of the “SAFETY DOOR LOCK SYSTEM”is, when a person places his/her
hand or fingers in the gap between the door and the outer panel, at the time when the closing action of the door
takes place, the Sensors start to transmit the Infra Red Rays to the Receivers at the
other end, and so even if someone closes the door without anybody‟s knowledge the hydraulic piston will
automatically come out and stop the door from closing and prevent the person from the unsuspected accident
and minor injuries by the car door and ensure maximum safety to the customer.
Notions via β*-open sets in topological spacesIOSR Journals
In this paper, first we define β*-open sets and β*-interior in topological spaces.J.Antony Rex Rodrigo[3] has studied the topological properties of 𝜂 * -derived, 𝜂 * -border, 𝜂 * -frontier and 𝜂 * exterior of a set using the concept of 𝜂 * -open following M.Caldas,S.Jafari and T.Noiri[5]. By the same technique the concept of β*-derived, β*-border, β*-frontier and β*exterior of a set using the concept of β*-open sets are introduced.Some interesting results that shows the relationships between these concepts are brought about
In this paper, we define and study about a new type of generalized closed set called, 𝑔∗s-closed set.Its relationship with already defined generalized closed sets are also studied.
Congruence Lattices of Isoform LatticesIOSR Journals
A congruence of a lattice L is said to be isoform, if any two congruence classes of are isomorphic as lattices. The lattice L is said to be isoform, if all congruence's of L are isoform. We prove that every finite distributive lattice D can be represented as the congruence lattice of a finite isoform lattice.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
In 2001 and 2003, Nakaok and Oda [3] and [4] introduced the notation of maximal open sets and minimal open sets in topological spaces. In 2010, Miguel, Saeid and Seithuti [2], introduced the notation of maximal θ-open sets and minimal θ -closed sets in topological spaces. In this paper, we introduce new classes of sets called maximal α-open sets and minimal α -open sets and investigate some of their fundamental properties.
On some locally closed sets and spaces in Ideal Topological SpacesIJMER
In this paper we introduce and characterize some new generalized locally closed sets
known as
δ
ˆ
s-locally closed sets and spaces are known as
δ
ˆ
s-normal space and
δ
ˆ
s-connected space and
discussed some of their properties
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Firefly Algorithm for Optimizing Spur Gear Parameters Under Non-Lubricated ...irjes
Firefly algorithm is one of the emerging evolutionary approaches for complex and non-linear
optimization problems. It is inspired by natural firefly‟s behavior such as movement of fireflies based on
brightness and by overcoming the constraints such as light absorption, obstacles, distance, etc. In this research,
firefly‟s movement had been simulated computationally to identify the best parameters for spur gear pair by
considering the design and manufacturing constraints. The proposed algorithm was tested with the traditional
design parameters and found the results are at par in less computational time by satisfying the constraints.
The Effect of Orientation of Vortex Generators on Aerodynamic Drag Reduction ...irjes
One of the main reasons for the aerodynamic drag in automotive vehicles is the flow separation
near the vehicle’s rear end. To delay this flow separation, vortex generators are used in recent vehicles. The
vortex generators are commonly used in aircrafts to prevent flow separation. Even though vortex generators
themselves create drag, but they also reduce drag by delaying flow separation at downstream. The overall effect
of vortex generators is more beneficial and proved by experimentation. The effect depends on the shape,size and
orientation of vortex generators. Hence optimized shape with proper orientation is essential for getting better
results.This paper presents the effect of vortex generators at different orientation to the flow field and the
mechanism by which these effects takes place.
An Assessment of The Relationship Between The Availability of Financial Resou...irjes
The availability of financial resources is an important element in impacting the success of a planning
process for an effective physical planning. The extent to which however, they are articulated in the process
remained elusive both in scholarly and public discourse. The objective of this study wastherefore, to examine
the extent to which financial resources affect physical planning. In doing so, the study examinedwhether
financial resources were adequate or not to facilitate planning processes in Paidha. According to the study
findings,budget prioritization and ceilings are still a challenge in Paidha Town Council. This is partly due
limited level of knowledge of physical planning among the officials of Paidha Town Council. As a result, there
were no dedicated budget line for routine inspection of physical development plan compliance and enforcement
tools in Paidha. In conclusion, in addressing uncoordinated patterns of physical development that characterize
Uganda‟s urban centres, a critical starting point ought to be the analysis of physical planning process. The
research of this kind is not only significant to other emerging urban centres facing poor a road network,
mushrooming informal settlements and poor social services including poor pattern of residential and commercial
developments but also to all institutions that are involved in planning these towns. Knowing the extent of need
for financial influences in planning may assist local authorities to take the processes of planning seriously which
will help enhance the sustainable development of emerging urban centres including Paidha.
The Choice of Antenatal Care and Delivery Place in Surabaya (Based on Prefere...irjes
- Person's desire to do a pregnancy examination is determined by the service place that suits the tastes
and facilities owned by it. Until now, the utilization of antenatal care by pregnant women is still low (Mardiana,
2014). The purpose of the study is to analyze factors affecting the utilization of antenatal care and delivery place
in Surabaya city based on the preferences and choice theory.
Type of survey research is cross sectional approach, the population is mothers who have children aged 1-
12 months in Surabaya. The large sample of 250 mothers who have children aged 1-12 months in 2013 is taken
by simple random sampling technique. Variables of the research are the preference elements and steps, choice
elements and steps, utilization of antenatal care and delivery place. Data were collected through questionnaires
and secondary data were then analyzed with descriptive statistics in the form of a frequency distribution, shown
by the schematic diagram.
The result showed that the preference elements and steps showed almost half (42.9%) desire to give birth
in a health care because of information got from someone else, while the choice element and step shows the
bulk (57.1%) of the criteria of delivery place chosen is a safe, comfortable and cheap delivery place, the labor
place which is the main choice most (57.1%) is cheap, comfortable, close.
Conclusion of the research based on the preferences and choice theory can be found three (3) new
theories, they are preferences become choice, preferences do not become choice, choice is preceded by
preferences
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
UiPath Test Automation using UiPath Test Suite series, part 6
D223034
1. International Refereed Journal of Engineering and Science (IRJES)
ISSN (Online) 2319-183X, (Print) 2319-1821
Volume 2, Issue 2(February 2013), PP.30-34
www.irjes.com
On Generalised β-Connectedness In Isotonic Spaces
A.Francina Shalini and I.Arockiarani
Department of Mathematics, Nirmala College For Women, Coimbatore
Abstract : The purpose of this paper is to define and study β connectedness and β separated sets in generalized
β closure spaces and discuss their properties. The notion of Z connectedness and strongly connectedness in
isotonic spaces are also analysed.
Keywords: Generalized β closure spaces, β connectedness, β separated sets, Z connectedness, strongly
connectedness.
I. Introduction & Preliminaries
The structure of closure spaces is more general than that of topological spaces. Hammer studied closure
spaces extensively and a recent study on these spaces can be found in Stadler [8,9], Harris[5],Habil and Elzenati
[4].The following definition of a generalized closure space can be found in [4] and [9].Let X be a set, ℘( X) be
its power set and cl: ℘( X)→ ℘( X) be any arbitrary set valued – function, called a closure function. We call cl
(A), A⊂X, the closure of A and we call the pair ( X,cl) a generalized closure space.
An operator cl : P(X) →P(X) is called grounded if cl(ϕ)= ϕ, isotonic if A B X implies
cl(A) cl(B), expansive if A cl(A )for every A X, idempotent if cl (cl(A)) = cl(A) for every
AX and additive if cl(A B ) cl(A) cl(B) for all subsets A and B of X.
Definition: 1.1.
(i) The space ( X, cl) is said to be isotonic if cl is grounded and isotonic.
(ii) The space ( X, cl) is said to be a neighborhood space if cl is grounded, expansive
and isotonic.
(iii) The space ( X, cl) is said to be a closure space if cl is grounded, expansive, isotonic
and idempotent.
(iv) The space ( X, cl) is said to be a Cech closure space if cl is grounded, expansive,
isotonic and additive.
(v) A subset A of X is said to be closed if cl(A) = A. It is open if its complement is closed.
II. Generalized β-Closure Space
Definition 2.1:
(1) A generalized β-closure space is a pair (X,βcl) consisting of a set X and a β-closure function βcl, a
function from the power set of X to itself.
(2) The β-closure of a subset A of X, denoted βcl, is the image of A under βcl.
(3) The β-exterior of A is βExt(A) = X βcl(A), and the β-Interior of A is βInt(A) = X
βcl(XA).
(4) A is β -closed if A = β cl(A), A is β -open if A = β Int(A)and N is a β -neighborhood of
x if x ε βInt(N).
Definition 2.2: We say that a β -closure function β cl defined on X is:
(1) β -grounded if βcl(ϕ) = ϕ
(2) β -isotonic if βcl(A) βcl(B) whenever A B
(3) β -enlarging if A βcl(A) for each subset A of X
(4) β -idempotent if βcl(A) = βcl(βcl(A)) for each subset A of X
(5) β -sub-linear if βcl(A B) βcl(A) βcl(B) for all A,B X
(6) β -additive if iεI β cl(Ai) = β cl( iεI Ai) for Ai X
III. β – Separated Sets
Definition 3.1:In an generalized β-closure space (X, βcl), two subsets A, B X are called β -
separated if β cl(A) B = A β cl(B) = ϕ.
Proposition 3.2: In an generalized β-closure space (X, βcl), the following two conditions are
equivalent for all A, B X :
1. A, B are β - separated.
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2. On Generalised β-Connectedness In Isotonic Spaces
2. There are U N(A) and V N(B) such that A V = U B = ϕ.
Proof: By the definition of neighborhood if there exists U, V such that A β int(U ),
B β int (V ), A V = ϕ, and U B = ϕ. We know that A V = ϕ implies V X A and by
isotonic property B βint(V ) βint(X A). Then X β int(X A) X βint(V) implies βcl(A) X
βint(V) XB and thus βcl(A) B = ϕ. Similarly A βcl(B) = ϕ.
Now suppose A and B are β- separated. Then βcl(A) B = ϕ if and only if B X βcl(A) = βint(X
A), i.e.,X A N (B). Thus there is V N (B) such that A V = ϕ.
Similarly we have A βcl(B) = ϕ if and only if X B N (A), i.e., there is U N (A) with
U B = ϕ.
Proposition 3.3: Let (X, βcl), be an generalized β-closure space and let A, B Y X . Then A and B are
β -separated in (X, β cl) if and only if A and B are β-separated in (Y, βclY ).
Proof. Suppose A, B Y and they are β- separated in X. Then we have A βcl(B) = ϕ implies A
βcl(B) Y = A βclY (B) = ϕ. and βcl(A) B = ϕ implies βclY(A) B = ϕ.
Conversely, I f A a n d B a r e β- separated in Y. Assume A βclY (B) = ϕ.
We have A βclY (B) = A Y βcl(B) = A βcl(B) = ϕ since A Y .Hence Aβcl(B) = βcl(A) B
= ϕ.
Proposition 3.4: In neighborhood spaces let f : (X, βcl) → (Y, βcl) be β- continuous and s u p p o s e
−1 −1
A, B Y are β -separated. Then f (A) and f (B) are β -separated.
Proof. Suppose A a nd B a r e β-separated in Y then we have β cl(A) B= ϕ and Aβ cl (B) = ϕ we now
−1 −1
prove that f (A) and f (B) are β separated in X. Let us take β cl(A) B = ϕ . Then
f −1 (β cl(A)) f −1 (B) = ϕ. Since f is β continuous, βcl(f-1(A)) f-1(βcl (A)), hence
βcl(f-1(A)) f-1(B) f-1(βcl (A)) f −1 (B) implies βcl(f-1(A)) f-1(B) = ϕ.
Similarly β cl(B) A = ϕ implies β cl(f-1B) f-1(A) = ϕ. Hence f-1(A) and f-1(B) are β-separated.
IV. β connectedness
Definition 4.1: A set Y P(X) is β-connected in a generalized β closure space (X, βcl) if
it is not a disjoint union of a nontrivial β-separated pair of sets A, Y A, A ≠ ϕ , Y.
Definition 4.2: Let (X, β cl) be a space and xX . The component C (x) of x
in X is the union of all β-connected subsets of X containing x.
Theorem 4.3: A set Z P (X ) is β- connected in an g e n e r a l i z e d β closure space (X, β cl) if and only if
for each proper subset A Z holds [β cl(A) (Z A)] [β cl(Z A) A] ≠ ϕ. This is known as Hausdorff-
Lennes condition.
Definition 4.4 : Connectedness is closely related to separation .Two sets A, B P (X ) are β - separated if
t h e r e a r e β- n e i g h b o r h o o d s N' N (A) and N″ N (B) such that A N' = N″ B = ϕ; they are β
separated if there are β- neighborhoods N' N (A) and N″ N (B) such that N' N″ = ϕ.
Theorem 4.5: A neighborhood space (X, βcl) is β connected if and only if there are no
nonempty disjoint β-open (closed) sets H and K in X with X = H K .
Proof : Suppose that X is β-disconnected. Then X = H K , where H , K are β- separated,
disjoint sets. Since H β cl(K ) = ϕ , we have β cl(K ) X H K and then, by definition of
expanding β cl(K ) = K . Since β cl(H ) K = ϕ, a similarl y H is a β- closed set.
Conversely, suppose that H and K are disjoint β -open sets such that X = H K .
Now K = X H , and H is an β-open set, hence K is a β-closed set. Thus H βcl(K ) =
H K = ϕ. Similarly β cl(H ) K = ϕ . Thus H and K are β -separated and therefore X is β
disconnected.
Proposition 4.6: If X and Y are β-connected in an generalized β closure space (X, β cl) and
X Y = ϕ ,then X Y is β -connected.
Proof: W e use t he Hausdorff-Lennes co nd i tio n
[β cl(A) (Y Z ) A] [A β cl((Y Z ) A)] =
[β cl(A) (Y A)] [β cl(A) (Z A)] [A β cl((Y A) (Z A))]
{[ β cl(A) (Y A)] [A β cl(Y A)]} {[A β cl(Z A)] β cl(A) (Z A)]}
If A Y or A Z is a proper subset of Y or Z, then one of the expression is non empty .Both the expressions
are empty if and only if either A=Z and A Y= ϕ o r A Z= ϕ a n d A = Y . T h i s i s i m p o s s i b l e i f
YZ ϕ
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3. On Generalised β-Connectedness In Isotonic Spaces
Theorem 4.7: If (X, βcl) is a neighborhood space then βcl(Z ) is β-connected whenever Z is
β-connected.
Proof: Set A' = Z A and A″ = AZ. We then use the Hausdorff-Lennes condition
[βcl(A) (βcl(Z ) A)] [βcl(βcl(Z ) A) A]
[(βcl(A') βcl(A″)) (Z A')] [βcl(Z − A') (A' A″)]
{[ βcl(A') (Z A')] [ βcl(Z A' ) A' ]} [βcl(Z − A') A″]
Here we have used Z − A (β cl(Z ) A') A' which is true only if A β cl (A) holds. If A' ϕ then the term in the
braces is nonempty because Z is β connected. By assumption if A' = ϕ then A″ β cl(Z ) Z is nonempty and
hence βcl(Z A') A″= βcl(Z ) A″ = ϕ.Thus βcl(Z ) is β -connected.
Theorem 4.8: If f : (X, β cl) → (Y, β cl) is a β- continuous function between neighborhood spaces, and A is β
connected in X , then f (A) is β-connected in Y .
Proof : Suppose f(A) is not β- connected. Then f(A) = U V , where U and V are -separated and non-empty.
Thus f-1(U) and f-1(V) are -separated. Clearly A' = A f-1(U) and A″ = A f-1(V) are both non-empty and also
-separated. Further A' A″ = A and hence A is not β-connected.
Theorem 4.9:Let (X, βcl) be a generalized β-closure space with β-grounded β-isotonic β-enlarging
βC l. Then, the following are equivalent:
(1) (X, βcl) is β -connected,
(2) X cannot be a union of nonempty disjoint β open sets.
Proof :. (1) (2): Let X be a union of nonempty disjoint β -open sets A and B. Then,
X = A B and this implies that B = X A and A is a β -open set. Thus, B is β -closed and hence A β
cl(B) = A B = ϕ. Likewise, we obtain β cl(A) B = ϕ. Hence, A and B are β -closure-separated
and hence X is not β-connected. This is a contradiction.
(2) (1): Suppose that X is not β - connected. Then X = A B, where A, B are disjoint β - separated
sets, i.e Aβ cl(B) = β cl(A) B = ϕ. We have β cl(B) X A B. Since β cl is β -enlarging, we get β cl(B)
= B and hence, B is β - closed. By using βcl(A) B = ϕ. and similarly, it is obvious that A is β -closed.
This is a contradiction.
Definition 4.10: Let (X, β cl) be a generalized β -closure space with β–grounded β–isotonic βc l. Then, (X,
βcl) is called a T1- β-grounded β-isotonic space if βcl({x}) {x} for all x X .
Theorem 4.11: Let (X, βcl) be a generalized β-closure space with β-grounded β-isotonic
βcl. Then, the following are equivalent:
(1) (X, βcl) is β-connected,
(2) Any β - continuous function f: X → Y is constant for all T 1 - β -grounded β -isotonic spaces Y
= {0, 1}.
Proof :. (1)(2): Let X be β - connected. Suppose that f: X → Y is β - continuous and it is not
constant. Then there exists a set U X such that U = f −1 ({0}) and X U = f −1 ({1}). Since f
is β continuous and Y is T1 - β - grounded β -isotonic space, then we have β cl(U ) = β cl(f −1 ({0})) f
−1 (β cl{0}) f −1
({0}) = U and hence β cl(U ) (X U ) = ϕ.Similarly we have U βcl(X U ) = ϕ.
This is a contradiction. Thus, f is constant.
(2) (1): Suppose that X is not β connected. Then there exist β -closure- separated sets U and V
such that U V = X . We have β cl(U )U and β cl(V ) V and X U V . Since β cl is β -isotonic
and U and V are β - closure-separated, then βcl(X U ) βcl(V ) X U . If we consider the space (Y, β
cl) by Y = {0, 1}, βcl(ϕ) = ϕ ,βcl({0}) = {0}, βcl ({1}) = {1} and βcl (Y ) = Y , then
the space (Y, β cl) is a T 1 - β -grounded β -isotonic space. We define the function f: X → Y as f (U ) =
{0} and f ( X U ) = {1}. Let Aϕ and A Y . If A = Y , then f−1 (A) = X and hence β c l(X )
= βcl (f −1 (A)) X = f −1 (A) = f −1 ( βcl(A)).
If A = {0}, then f −1 (A) = U and hence βcl(U ) = βcl(f −1 (A)) U = f −1 (A)
= f −1 ( βcl(A)). If A = {1}, then f −1 (A) = X U and hence
β cl(X U ) = β cl(f −1 (A)) X U = f −1 (A) = f −1 ( β c l(A)). Hence, f is βcontinuous. Since f is not
constant, this is a contradiction.
Theorem 4.12: Let f : (X, βcl) → (Y, βcl) and g : (Y, βcl) → (Z, βcl) be β continuous functions.
Then, gof : X → Z is β continuous.
Proof :. Suppose that f and g are β continuous. For all A Z we have
βcl(gof )−1 (A) = βcl(f −1 (g −1 (A))) f −1 (βcl(g −1 (A))) f −1 (g −1 (βcl(A)))
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4. On Generalised β-Connectedness In Isotonic Spaces
= (gof )−1 (βcl(A)). Hence, gof : X → Z is β continuous.
Theorem 4.13: Let (X, βcl) and (Y, βcl) be generalized β-closure spaces with β-grounded β-isotonic βcl
and f : (X, βcl) → (Y, βcl) be a β continuous function onto Y . If X is β connected, then Y is β
connected.
Proof : Let us suppose that {0, 1} is a generalized β-closure spaces with β-grounded β-isotonic βC l
and g : Y → {0, 1} is a β continuous function. Since f is β continuous,
gof : X → {0, 1} is β continuous. Since X is β-connected, gof is constant and hence g is
constant. Therefore Y is β connected.
Definition 4.14: Let (Y, βcl) be a generalized β-closure space with β-grounded β-isotonic βcl and more
than one element. A generalized β-closure space (X, βcl) with β-grounded β-isotonic βcl is called Y – β
connected if any β- continuous function f : X → Y is constant.
Theorem 4.15: Let (Y, βcl) be a generalized β -closure space with β -grounded β -isotonic β – enlarging
βcl and more than one element. Then every Y - β connected generalized β -closure space with β -
grounded β -isotonic is β connected.
Proof : Let (X, βcl) be a Y - β connected generalized β -closure space with β - grounded β -isotonic βcl.
Suppose that f : X → {0, 1} is a β continuous function, where {0, 1} is a T1- β -grounded β -isotonic
space. Since Y is a generalized β -closure space with β -grounded β -isotonic β -enlarging βcl and more
than one element, then there exists a β continuous injection
g : {0, 1} → Y .gof : X → Y is β continuous. Since X is Y - β connected, then gof is constant. Thus,
f is constant and hence, b y theo r em 5 .1 1 X is β connected.
Theorem 4.16: Let (X, βcl) and (Y, βcl) be generalized β -closure spaces with β -grounded β -isotonic βcl
and f : (X, βcl) → (Y, βcl) be a β continuous function onto Y . If X is Z - β connected, then Y is Z -
β connected.
Proof : Suppose that g : Y → Z is a β continuous function. Then gof : X → Z is β continuous. Since X
is Z - β connected, then gof is constant. This implies that g is constant. Thus, Y is Z - β connected.
5.Strongly βConnected Spaces
Definition 5.1: A generalized β -closure space (X, βcl) is strongly β connected if there is no countable
collection of pairwise β -closure-separated sets {An } such that X = An .
Theorem 5.2: Every strongly β connected generalized β closure space with β -grounded β -isotonic βcl is
β connected.
Theorem 5.3: Let (X, βcl) and (Y, βcl) be generalized β -closure spaces with β -grounded β -isotonic βcl
and f : (X, βcl) → (Y, βcl) be a β- continuous function onto Y . If X is strongly β connected, then Y
is strongly β- connected.
Proof : Suppose that Y is not strongly β- connected. Then, there exists a countable collection of pairwise
β- closure-separated sets {An } such that Y = An . Since f−1 (An) βcl(f−1 (Am))
f −1 (A )f −1 (β cl(A )) = ϕ for all n m, then the collection {f−1 (A )} is pairwise β –closure-
n m n
separated. This is a contradiction. Hence, Y is strongly β-connected.
Theorem 5.4: Let (X, βclX ) and (Y,βcly ) be generalized β -closure spaces. Then, the following are
equivalent for a function f : X → Y
(1) f is β -continuous,
(2) f −1 ( β I nt(B)) β I nt(f −1 (B)) for each B Y .
Theorem 5.5: Let (X,βcl) be a generalized β -closure space with β -grounded β –isotonic β -additive βcl.
Then (X,βcl) is strongly β- connected if and only if (X, βcl) Y – β- connected for any countable T1- β -
grounded β -isotonic space (Y, βcl).
Proof : (): Let (X,βcl) be strongly β- connected. Suppose that (X, βcl) is not Y – β- connected for
some countable T1- β -grounded β -isotonic space (Y, βcl). There exists a β continuous function
f: X → Y which is not constant and hence K= f(X) is a countable set with more than one element. For
−1
each yn K , there exists UnX such that Un =f ({yn }) and hence Y = Un .
Since f is β- continuous and Y is β -grounded, then for each n m, Un βcl(Um ) =
f −1 ({yn }) βcl(f −1 ({ym })) f −1 ({yn })f −1 (β C l({ym })) f −1 ({yn })
f −1 ({ym })= ϕ. This contradicts the strong β- connectedness of X . Thus, X is
Y - β connected.
() :Let X be Y - β connected for any countable T1- β -grounded β -isotonic space (Y, βcl). Suppose that
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5. On Generalised β-Connectedness In Isotonic Spaces
X is not strongly β- connected. There exists a countable collection of pairwise β -closure-separated sets
{Un } such that X = Un .Consider the space (Z, βcl), where Z is the set of integers and βcl : P(Z )
→ P(Z ) is defined by βcl(K ) =K for each K Z .Clearly (Z, βcl) is a countable T1- β -grounded β -
isotonic space. Put Uk {Un }.We define a function
f: X → Z by f (Uk ) = {x} and f (X Uk ) = {y} where x, y Z and x y. Since βcl(Uk )
Un= ϕ for all n k, then βcl(Uk ) U n k Un= ϕ and hence βcl(Uk ) Uk . Let
−1
ϕ K Z. If x, y K then f (K )= X and βcl(f −1 (K )) = βcl(X ) X = f −1 (K )
−1 −1
= f −1 ( βcl(K )). If x K and y K ,then f (K ) = Uk and βcl(f (K )) = βcl(Uk )Uk
= f −1 (K ) = f −1 (βcl(K )).If y K and x K then f −1 (K ) = X Uk . On the other hand, for all n
k, Uk β cl(Un ) = ϕ and hence Uk U n k βcl (Un)= ϕ This implies that
Uk βcl( U n k Un)= ϕ .Thus, βcl (X Uk )X Uk . Since βcl(K ) = K for each K Z , we have βcl(f
−1 (K )) = −1 −1
βcl(X Uk )X Uk = f (K ) = f ( βcl(K )).Hence we obtain that f is β- continuous.
Since f is not constant, this is a contradiction with the Z- β connectedness of X . Hence, X is strongly
β- connected.
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