This document discusses the formulation of fractional supersymmetric theories in one dimension. It begins by presenting fractional superspace and fractional supersymmetry of order F=3, including the fractional supersymmetry transformations. It then derives the fractional supercharges and Euler-Lagrange equations for F=3. Finally, it generalizes the formulation to arbitrary fractional order F ≥ 3 by introducing fractional superspace and supersymmetry transformations of order F, as well as an action invariant under such transformations.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
: The tensor product G H of two graphs G and H is well-known graph product and studied in detail in the literature. This concept has been generalized by introducing 2-tensor product G H 2 and it has been discussed for special graphs like P n and Cn [5]. In this paper, we discuss G H 2 , where G and H are connected graphs. Mainly, we discuss connectedness of G H 2 and obtained distance between two vertices in it.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
: The tensor product G H of two graphs G and H is well-known graph product and studied in detail in the literature. This concept has been generalized by introducing 2-tensor product G H 2 and it has been discussed for special graphs like P n and Cn [5]. In this paper, we discuss G H 2 , where G and H are connected graphs. Mainly, we discuss connectedness of G H 2 and obtained distance between two vertices in it.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Stereographic Circular Normal Moment Distributionmathsjournal
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
Divergence measures are useful for comparing two probability distributions. Depending on the nature of
the problem, different divergence measures are suitable. So it is always desirable to develop a new
divergence measure.
Recently, Jain and Chhabra [6] introduced new series ( ( , ) m x P Q , ( , ) m z P Q and ( , ) m w P Q for
mÎN ) of information divergence measures, defined the properties and characterized, compared with
standard divergences and derived the new series ( ( ) * , m x P Q formÎN ) of metric spaces.
In this work, various important and interesting relations among divergences of these new series and other
well known divergence measures are obtained. Some intra relations among these new divergences are
evaluated as well and bounds of new divergence measure ( ( ) 1 x P,Q ) are obtained by using Csiszar’s
information inequalities. Numerical illustrations (Verification) regarding bounds are done as well.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
Radix-3 Algorithm for Realization of Discrete Fourier TransformIJERA Editor
In this paper, a new radix-3 algorithm for realization of discrete Fourier transform (DFT) of length N = 3m (m =
1, 2, 3,...) is presented. The DFT of length N can be realized from three DFT sequences, each of length N/3. If
the input signal has length N, direct calculation of DFT requires O (N
2
) complex multiplications (4N
2
real
multiplications) and some additions. This radix-3 algorithm reduces the number of multiplications required for
realizing DFT. For example, the number of complex multiplications required for realizing 9-point DFT using the
proposed radix-3 algorithm is 60. Thus, saving in time can be achieved in the realization of proposed algorithm.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Some Common Fixed Point Theorems for Multivalued Mappings in Two Metric SpacesIJMER
In this paper we prove some common fixed point theorems for multivalued mappings in two
complete metric spaces.
AMS Mathematics Subject Classification: 47H10, 54H25
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
L Inequalities Concerning Polynomials Having Zeros in Closed Interior of A Ci...IOSR Journals
Let
0
( )
n
j
i
j
p z a z
be a polynomial of degree n and p(z) be its derivative, then Zygmund [9]
proved that
1 1
2
0
2
'( ) | ( ) | , 1
0
| |
r r i r i r p e d n p e d r
In this paper we shall obtain similar type of inequalities in reverse order for the polynomials having r fold
zeros at origin and rest of the zeros in | z | k , k 1.
Mathematics Subject Classification (2010): 30A10, 30C15, 30C10
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Stereographic Circular Normal Moment Distributionmathsjournal
Minh et al (2003) and Toshihiro Abe et al (2010) proposed a new method to derive circular distributions from the existing linear models by applying Inverse stereographic projection or equivalently bilinear transformation. In this paper, a new circular model, we call it as stereographic circular normal moment distribution, is derived by inducing modified inverse stereographic projection on normal moment distribution (Akin Olosunde et al (2008)) on real line. This distribution generalizes stereographic circular normal distribution (Toshihiro Abe et al (2010)), the density and distribution functions of proposed model admit closed form. We provide explicit expressions for trigonometric moments.
Divergence measures are useful for comparing two probability distributions. Depending on the nature of
the problem, different divergence measures are suitable. So it is always desirable to develop a new
divergence measure.
Recently, Jain and Chhabra [6] introduced new series ( ( , ) m x P Q , ( , ) m z P Q and ( , ) m w P Q for
mÎN ) of information divergence measures, defined the properties and characterized, compared with
standard divergences and derived the new series ( ( ) * , m x P Q formÎN ) of metric spaces.
In this work, various important and interesting relations among divergences of these new series and other
well known divergence measures are obtained. Some intra relations among these new divergences are
evaluated as well and bounds of new divergence measure ( ( ) 1 x P,Q ) are obtained by using Csiszar’s
information inequalities. Numerical illustrations (Verification) regarding bounds are done as well.
International Journal of Research in Engineering and Science is an open access peer-reviewed international forum for scientists involved in research to publish quality and refereed papers. Papers reporting original research or experimentally proved review work are welcome. Papers for publication are selected through peer review to ensure originality, relevance, and readability.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
Radix-3 Algorithm for Realization of Discrete Fourier TransformIJERA Editor
In this paper, a new radix-3 algorithm for realization of discrete Fourier transform (DFT) of length N = 3m (m =
1, 2, 3,...) is presented. The DFT of length N can be realized from three DFT sequences, each of length N/3. If
the input signal has length N, direct calculation of DFT requires O (N
2
) complex multiplications (4N
2
real
multiplications) and some additions. This radix-3 algorithm reduces the number of multiplications required for
realizing DFT. For example, the number of complex multiplications required for realizing 9-point DFT using the
proposed radix-3 algorithm is 60. Thus, saving in time can be achieved in the realization of proposed algorithm.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Some Common Fixed Point Theorems for Multivalued Mappings in Two Metric SpacesIJMER
In this paper we prove some common fixed point theorems for multivalued mappings in two
complete metric spaces.
AMS Mathematics Subject Classification: 47H10, 54H25
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
L Inequalities Concerning Polynomials Having Zeros in Closed Interior of A Ci...IOSR Journals
Let
0
( )
n
j
i
j
p z a z
be a polynomial of degree n and p(z) be its derivative, then Zygmund [9]
proved that
1 1
2
0
2
'( ) | ( ) | , 1
0
| |
r r i r i r p e d n p e d r
In this paper we shall obtain similar type of inequalities in reverse order for the polynomials having r fold
zeros at origin and rest of the zeros in | z | k , k 1.
Mathematics Subject Classification (2010): 30A10, 30C15, 30C10
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
In this paper, we have studied various properties of the F- sturcture manifold satisfying 2 0 p F F where p is odd prime. Metric F-structure, kernel, tangent and normal vectors have also been discussed.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
Abstract
In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and
engineering applications including energy systems. We make use of a Padé Approximation method for the solution of the
Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional
differential transformation, then the numerical solution of the equation was put into Padé series form for accelerating
convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential
equation.
Key Words: Helmholtz equation, Two-dimensional differential transformation, Multivariate Padé approximation,
power series
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
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.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...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.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
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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:
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UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
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Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...
On 1 d fractional supersymmetric theory
1. Advances in Physics Theories and Applications www.iiste.org
ISSN 2224-719X (Paper) ISSN 2225-0638 (Online)
Vol.17, 2013
12
ON 1D FRACTIONAL SUPERSYMMETRIC THEORY
H. Chaqsare1
*, A. EL Boukili1
, B. Ettaki2
,
M.B. Sedra1
and J. Zerouaoui1
1
Ibn Tofail University, Faculty of Sciences
Physics Department, (LHESIR), Kénitra, Morocco
2
Ecole des Sciences de l’Information
BP 6204, Agdal, Rabat, Morocco
* E-mail of the corresponding author: h. chaqsare@gmail.com
Abstract
Following our previous work on fractional supersymmetry (FSUSY) [1,2], we focus here our contribute to the study
of the superspace formulation in D1 that is invariant under FSUSY where 3=F and defined by HQ =3
, we
extend our formulation in the end of our paper to arbitrary F with 3>F .
Key-words Fractional superspace - Fractional Supersymmetry of order F - Fractional Supercharge - Covariant
Derivative
1. Introduction
Motivating by the results founded in [1,2], the aim of this paper is to develop a superspace formulation in D1 QFT
that is invariant under fractional supersymmetry (FSS). In such construction, the Hamiltonian H is expressed as the
th
F power of a conserved fractional supercharge: HQF
= , with 0=],[ QH and 3≥F . Here, we shall reformulate
these results in fractional superspace, using generalized Grassmann variable of order F satisfying 0=F
θ .
Additionally, we construct the Noether fractional supercharges in the case where 3=F .
The presentation of this paper is as follow: In section 2, we present the Fractional Superspace and Fractional
Supersymmetry 3=F . In section 3, we will give the Fractional supercharges and Euler-Lagrange equations for
3=F . In section 4, we will generalise the FSUSY in arbitrary order and finally, we give a conclusion.
2. Fractional Superspace and Fractional Supersymmetry 3=F
The FSUSY of order 3 are generated by the Hamiltonian H generator of the time translation, and Q the generator
of FSUSY transformations . they satisfies:
HQHQ =;0=],[ 3
(1)
In fields quantum theory, this symmetry can be realized on generalized superspace ),( θt , where t is the time, and
θ is a real generalized Grassmann variable. The latter variable and his derivative ∂
∂
∂
=
θ
satisfies:
0=;0= 33
∂θ
2. Advances in Physics Theories and Applications www.iiste.org
ISSN 2224-719X (Paper) ISSN 2225-0638 (Online)
Vol.17, 2013
13
Ι∂−∂ =θθ θθ q (2)
2
θθ ∂≡∫d
The introduction of the ε (parameter of the transformation associated to Q ) et f (parameter of the transformation
associated to H ) in the case where 3=F give the following transformations [3]:
)(= 22
εθθε +−−′ qftt (3)
εθθ +′ =
where ε verify:
0=3
ε
εθθε q= (4)
where 3
2
=
πi
eq . The q-commutation relation between the two variables ε and θ ensures that:
• if 0== 33
θε then 0=)( 3
εθ + ;
• the reality of the time is not affected by the FSUSY transformation;
• the FSUSY transformations q-commute with covariant derivative;
• the FSUSY transformations satisfied the Leibnitz rules.
We now can introduce the scalar superfield Φ of order 3:
2
22
1
2
1
0=),( ϕθθϕϕθ qqt ++Φ (5)
where 10 ,ϕϕ et 2ϕ are the extension of the bosonic and the fermionic field. These fields verifies:
θϕθϕ )(=)( 00 tt
θϕθϕ )(=)( 1
2
1 tqt (6)
θϕθϕ )(=)( 22 tqt
We now can see that *
= ΦΦ . Using relations (3), we get easily the FSUSY transformations upon the fields:
1
1/2
0 = εϕδϕ q
2
1/2
1 = εϕδϕ −
−q (7)
012 = ϕεδϕ −∂−
Then, let us consider the two basic objects Q and D , which represent respectively the FSUSY generator and the
covariant derivative [1]
22
1
2
)()(=
θ
θθ
θ
θ
∂
∂
+
∂
∂
−∂− − qqQ
222
1
2
)()()(=
θ
θθ
θ
θ
∂
∂
+
∂
∂
−∂− − qD (8)
Using the equations (2) and (4), we can prove that:
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zQD ∂== 33
DQqQD 2
= (9)
ΦΦ εε δδ DD =
where
ΦΦ Qεδε = (10)
The invariant action under the FSUSY transformations in equations (7) is:
dtLDdtd
q
S =
2
= 1
2
∫∫ ΦΦ∂−θ
][
2
= 12
2
21
2
0
22
ϕϕϕϕϕθθ &&& qqdtd
q
−+−∫ (11)
][
2
1
= 12
2
21
2
0 ϕϕϕϕϕ &&& qqdt +−∫
3. Fractional supercharges and Euler-Lagrange equations
Following [3] and [6], we can introduce the generalized momenta conjugate to iϕ
0
0
0 = ϕ
ϕ
π &
&∂
∂
≡
L
2
1
1 =2 ϕ
ϕ
π q
L
−
∂
∂
≡
&
(12)
1
2
2
2 =2 ϕ
ϕ
π q
L
&∂
∂
≡
If we consider Φ& and ΦD as independent variables, we can prove that the generalized momenta conjugate are the
components of the fractional superspace momentum conjugate to ),( θtΦ
Φ
Φ∂
∂
≡Π D
L
q
t =
2
),(
&
θ (13)
which is decomposed as
)(2
2
12
2
0=
21
2
1
2
0
2
==),( i
i
i
i
qqqqt −
−
∑−+−Π πθθϕϕϕθθ & (14)
Note that ΠΠ =*
. If wish to add
0=],[ ΦΠ∫ θd
0=],[ ΦΠ∫ &θd (15)
0=],[ ΦΦ& (16)
we must require
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ijijji −≠ 3if= ϕϕϕϕ && (17)
1221 = ϕϕϕϕ q (18)
Focus on the internal-space part of the Lagrangian 12
2
21
22
= ϕϕϕϕ &&
qq
L +− . The Lagrangian variation
]..[
2
]..[
2
= 1212
2
2121 δϕϕϕϕδδϕϕϕϕδδ &&&& +++−
qq
L
]..[
2
]..[
2
= 21
2
12
2
1221 ϕδϕϕϕδϕδϕϕϕδ &&&& q
q
q
q
+++− (19)
and knowing that
1
2
2
2
1 2
=;
2
= ϕ
ϕ
ϕ
ϕ
&&
qLqL
−
∂
∂
∂
∂
1
2
2
2
1 2
=;
2
= ϕ
ϕ
ϕ
ϕ
qLqL
&& ∂
∂
−
∂
∂
(20)
Then, the Lagrangian variation will be:
)()(=
2
2
2
2
1
1
1
1
ϕ
ϕδ
ϕ
δϕ
ϕ
ϕδ
ϕ
δϕδ
&
&
&
&
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂ LLLL
L (21)
From this equation, it is easy to show that the generalized Euler-Lagrange equations which follow from a least-action
principle are:
0=)(
11 ϕϕ &∂
∂
−
∂
∂ L
dt
dL
0=)(
22 ϕϕ &∂
∂
−
∂
∂ L
dt
dL
(22)
Therefore, the quantity
0=;=
2
1=
dt
dC
X
L
C
i
i
i
−
∂
∂
∑ ϕ
δϕ
&
(23)
is a constant of motion when the lagrangian varies under a transformation iδϕ by the total derivative dtdXL /=δ
where 10
2
1
2
1
= ϕϕ&qX . The particular case of the Hamiltonian when ii ϕδϕ &= is:
2
0
2
0=
2
1
== ϕ
ϕ
ϕ &
&
& L
L
H
i
i
i
−
∂
∂
∑ (24)
For iδϕ given by (7) and corresponding X , we find the following fractional supercharge associated with the
symmetry transformations:
)
2
1
(
2
= 2
201
2
1
ϕϕϕ +&
q
Q (25)
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Note, like in [6], that QQ εε =)( *
, i.e., qQQ =*
, using (17), we can prove that Qq 2
1
−
is a real charge.
4. FSUSY of arbitrary order
In this section, we will give a generalisation of the FSUSY of order F where 3≥F . For this, we introduce the
expression of the superfield ),( θtΦ of order F .
i
i
iF
i
qt ϕθθ 2
2
1
0=
=),( ∑
−
Φ (26)
where θ is a real generalized Grassmann variable satisfied 0=F
θ and q is the F-th root of unity ( F
i
eq
π2
= ). The
superfield components verifies the following commutation relations:
θϕθϕ i
i
i q−
=
iiF
i
iFi q ϕϕϕϕ −− = (27)
the first relation in (27) implies that ΦΦ =*
. while the second relationship is used to introduce the following
commutation relation:
iF
i
iFi q ϕϕϕϕ 1= −− && (28)
for 2=F , the equations (27) and (28) reduces to the usual results of supersymmetry θϕθϕ 11 = − , 0=2
1ϕ and
1111 = ϕϕϕϕ && − while 0000 = ϕϕϕϕ && .
Les FSUSY transformations of order F are generated by the generator of the FSUSY Q whose expression is:
])()([= 1221
3
0=
1 −−−−−
−
−
∂
∂
+
∂
∂
−∂− ∑ FFiFFi
F
i
t
F
qqAQ
θ
θθ
θ
θθ (29)
where F
F
F
F
FqA
12
)1}!({)(=
−
−
−
−
−− . This implies that:
t
F
Q ∂= (30)
Acting on ),( θtΦ , we have
),(=),( θεθδ tQt ΦΦ (31)
which gives on components:
1
2
2
2
21)(
1
1}!{)(= +
−
+
−
−− i
ii
iF
i FqqAq εϕδϕ
1
2
22)(
2
21)(
21
2 1}!{)(= −
−
−
−
−−
− − F
FF
FF
F FqqA εϕδϕ (32)
0
112
21)(
1 )(= ϕεδϕ &−−
−
−
− − FF
F
F qAqq
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while 3}{0,1,..., −∈ Fi ,
q
q
F
F
−
−
−
−
1
1
=1}{
1
and ε is real infinitesimal parameter verify:
i
i
i qq εϕεϕεθθε =;= (33)
To build invariant action under FSUSY transformations need the introduction of the fractional covariant derivative
commuting with Qε .
])()([= 12211
3
0=
1 −−−−−−−
−
−
∂
∂
+
∂
∂
−∂− ∑ FFiFFiiF
F
i
t
F
qBD
θ
θθ
θ
θθ (34)
where ])1}!({)...(1)[(=
11
232
1
F
F
FFFF
F
FqqqB
−
−−
−−
−
−− . Q et D satisfies the following relations:
t
F
DqQDDQ ∂=;= (35)
DqDqQQ =;= εεεε
After defining the two generators of FSUSY, we can now give the expression of the action S invariant under the
FSUSY transformations (32)
ΦΦ
−
− −
∫ &Ddtd
FB
S F
θ1
1}!{2
1
=
iFi
iFi
iF
iF
iF
i
FF
qqqqFdtd
F
S −
−
−
−
−
−−
∑∫ −+
−
ϕϕϕθθ && )(2
2)(
2
2
2
1=
2
0
11
1}!{{
1}!2{
1
=
}1}!{ 11
2
21)(
12
1
ϕϕ &−
−
−
−− F
F
F
qqqF (36)
iFi
iFi
iF
iF
iF
i
qqqqFdtS −
−
−
−
−
∑∫ −+ ϕϕϕ && )(2
2)(
2
2
2
1=
2
0 1}!{{
2
1
=
}1}!{ 11
2
21)(
12
1
ϕϕ &−
−
−
−− F
F
F
qqqF (37)
Conclusion
In this paper, we have extended the results founded in [1] and [2] of fractional symmetry (FSUSY) from D2 to D1 ,
and following [3] and [6], we are giving the fractional supercharge in 3=F . In the last section, we gave the
generalized formulation of the generator of the FSUSY Q and fractional covariant derivative D .
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