Hyperspherical three body calculations are performed to study and review the various properties of
the hypertriton bound state nucleus
3H in the quark model using -N potentials. In these calculations we study
the different effects of the -N potentials on the hypertriton bound states as well as the separation energy B. A
combination of realistic two body N-N potentials with various - N potentials are considered. Complete
symmetric and mixed symmetric wave functions are introduced. using the renormalized Numerov method. The
agreement between the calculated
3H binding energies and the available experimental data basically depends
on the type of the -N interactions used in the calculations. It was found that the -N potentials are the most
effective part in the hypertriton binding energy as well as the separation energy B where the -N potentials is
very effective to bound or unbound the
3H hyper nucleus
Pacs numbers: 21.30. + y, 21.10.+dr,27.20.+n
This document presents a novel Copula based approach to generate critical sea states given a target reliability index based on the return period of the extreme event. Copula based approach is much more flexible and powerful when compared to conventional approaches using linear correlation coefficient.
This document presents a novel Copula based approach to generate critical sea states given a target reliability index based on the return period of the extreme event. Copula based approach is much more flexible and powerful when compared to conventional approaches using linear correlation coefficient.
Energy Grid Theorem.
«For a stable electrical network without sources and energy storage the value of its streams coincide with the solution of optimization task of energy streams across the network with minimal losses».
THE ENERGY MINIMIZATION, FOR THE STUDENTS OF M.PHARM, B.PHARM AND OTHERS USEFUL FOR ACADEMIC TOO. THE PRESENT DATA IS MOST USEFUL FOR PHARMACY PURPOSE.
An Experimental Study on Strength Properties of Concrete When Cement Is Parti...IOSR Journals
Ordinary Portland cement is recognized as major construction material throughout the world.
Cement is responsible for about 5% - 8% of global co2 emissions. Therefore the utilization of industrial waste
has been increased which is environment friendly. In this paper an attempt has been made to utilize sugarcane
bagasse ash (S.C.B.A) a residue from sugar industry as a supplementary cement replacement material.
The present study investigates with the strength and durability properties of the concrete when the
cement is replaced with S.C.B.A in different proportions. In the first stage the S.C.B.A (B.A.1) is partially
replaced in the percentages of 0 to 40% in increasing steps of 5%. Further in the second stage the S.C.B.A
(B.A.2) is heated up to 8500
c for about 8 hours in a muffle furnace and is replaced in the percentages of 0 to
40% in increasing steps of 5%. Cubes, Cylinders and Beams are casted and tests are conducted and
compressive strength, split tensile strength and flexural strength obtained for both the conditions.
Formulation of an anti-inflammatory drug as fast dissolving tabletsIOSR Journals
The demand for mouth dissolving tablets has been growing during the last decade especially for elderly and children who have difficulties in swallowing. Ketorolac tromethamine is an effective anti-inflammatory agent that has been extensively used for the prevention of pain and inflammation associated with a wide variety of reasons. This study was aimed to form Ketorolac tromethamine mouth dissolving tablets by direct compression using superdisintegratants as crospovidone (CP) ,crosscarmellose sodium (CCS), and sodium starch glycolate (SSG) at concentrations of 3%, 6%, 9%, and 12%. The physical mixtures of the drug and the used excipients were evaluated for their micromeretric properties such as angle of repose, particle size, Hausner's ratio and % compressibility. Also, FTIR spectroscopy and DSC calorimetry were performed to indicate any possible interaction between the drug with the used excipients.All the prepared tablets were evaluated for their weight variation, thickness, hardness, wetting time, and disintegration time. Also in-vitro release study was done for all the prepared tablets using distilled deionized water as dissolution medium at 37.5± 0.5 c˚. Based on in-vitro release study and stability studies, G5 (contained 3% CCS) was found to be the promising formulae and subjected to further studies.
Asif’s Equation of Charge Variation and Special RelativityIOSR Journals
The theory of special relativity plays an important role in the modern theory of classical electromagnetism. Considering deeply the effect of Special relativity in Electromagnetism, when a charge particle moves with high speed as comparable to the speed of light in vacuum tube or in space under influence of electromagnetic field, its mass varies under Lorentz transformation [1].The question arises that does its charge vary under Lorentz transformation? In this paper, Asif's equation of charge variation demonstrates the variation of electric charge under Lorentz transformation. The more sophisticated view of electromagnetism expressed by electromagnetic fields in moving inertial frame can be achieved by considering some relativistic effect including charge as well. One can easily achieve the mass-energy relation from Asif’s equation of charge variation as proved in this paper.
Energy Grid Theorem.
«For a stable electrical network without sources and energy storage the value of its streams coincide with the solution of optimization task of energy streams across the network with minimal losses».
THE ENERGY MINIMIZATION, FOR THE STUDENTS OF M.PHARM, B.PHARM AND OTHERS USEFUL FOR ACADEMIC TOO. THE PRESENT DATA IS MOST USEFUL FOR PHARMACY PURPOSE.
An Experimental Study on Strength Properties of Concrete When Cement Is Parti...IOSR Journals
Ordinary Portland cement is recognized as major construction material throughout the world.
Cement is responsible for about 5% - 8% of global co2 emissions. Therefore the utilization of industrial waste
has been increased which is environment friendly. In this paper an attempt has been made to utilize sugarcane
bagasse ash (S.C.B.A) a residue from sugar industry as a supplementary cement replacement material.
The present study investigates with the strength and durability properties of the concrete when the
cement is replaced with S.C.B.A in different proportions. In the first stage the S.C.B.A (B.A.1) is partially
replaced in the percentages of 0 to 40% in increasing steps of 5%. Further in the second stage the S.C.B.A
(B.A.2) is heated up to 8500
c for about 8 hours in a muffle furnace and is replaced in the percentages of 0 to
40% in increasing steps of 5%. Cubes, Cylinders and Beams are casted and tests are conducted and
compressive strength, split tensile strength and flexural strength obtained for both the conditions.
Formulation of an anti-inflammatory drug as fast dissolving tabletsIOSR Journals
The demand for mouth dissolving tablets has been growing during the last decade especially for elderly and children who have difficulties in swallowing. Ketorolac tromethamine is an effective anti-inflammatory agent that has been extensively used for the prevention of pain and inflammation associated with a wide variety of reasons. This study was aimed to form Ketorolac tromethamine mouth dissolving tablets by direct compression using superdisintegratants as crospovidone (CP) ,crosscarmellose sodium (CCS), and sodium starch glycolate (SSG) at concentrations of 3%, 6%, 9%, and 12%. The physical mixtures of the drug and the used excipients were evaluated for their micromeretric properties such as angle of repose, particle size, Hausner's ratio and % compressibility. Also, FTIR spectroscopy and DSC calorimetry were performed to indicate any possible interaction between the drug with the used excipients.All the prepared tablets were evaluated for their weight variation, thickness, hardness, wetting time, and disintegration time. Also in-vitro release study was done for all the prepared tablets using distilled deionized water as dissolution medium at 37.5± 0.5 c˚. Based on in-vitro release study and stability studies, G5 (contained 3% CCS) was found to be the promising formulae and subjected to further studies.
Asif’s Equation of Charge Variation and Special RelativityIOSR Journals
The theory of special relativity plays an important role in the modern theory of classical electromagnetism. Considering deeply the effect of Special relativity in Electromagnetism, when a charge particle moves with high speed as comparable to the speed of light in vacuum tube or in space under influence of electromagnetic field, its mass varies under Lorentz transformation [1].The question arises that does its charge vary under Lorentz transformation? In this paper, Asif's equation of charge variation demonstrates the variation of electric charge under Lorentz transformation. The more sophisticated view of electromagnetism expressed by electromagnetic fields in moving inertial frame can be achieved by considering some relativistic effect including charge as well. One can easily achieve the mass-energy relation from Asif’s equation of charge variation as proved in this paper.
Transient Three-dimensional Numerical Analysis of Forced Convection Flow and ...IOSR Journals
A three-dimensional transient numerical study of a constant property Newtonian fluid in curved pipe under laminar flow conditions is presented for a uniform wall temperature boundary condition. Numerical solutions were obtained using the control volume method described by Patankar for the range of. The working fluid was water. The transient flow pattern and the temperature distribution on the tube section were derived for different values of the Reynolds number. Graphical results for velocity and temperature are presented and analyzed. Results have shown that the maximum velocity in center of velocity profile increase with increasing of Reynolds number. In curved pipes, time averaged results exhibited Dean circulation and a strong velocity and temperature stratification in the radial direction. Flow and heat transfer were strongly asymmetric, with higher values near the outer pipe bend.
Abstract: The increasing amount of applications using digital multimedia technologies has accentuated the need to provide copyright protection to multimedia data. This paper reviews one of the data hiding techniques - digital image watermarking. Through this paper we will explore some basic concepts of digital image watermarking techniques.Two different methods of digital image watermarking namely spatial domain watermarking and transform domain watermarking are briefly discussed in this paper. Furthermore, two different algorithms for a digital image watermarking have also been discussed. Also the comparision between the different algorithms,tests performed for the robustness and the applications of the digital image watermarking have also been discussed.
Vibration Analysis of Cracked Rotor Using Numerical ApproachIOSR Journals
In general rotating machines have wide applications in systems, plants, vehicles, and industries. Every rotating machine uses shaft as power transforming unit. It is very dangerous to operate the machine with the presence of crack in the shaft. The growth of the crack is dangerous to operate and may lead to catastrophic failure. It is to be detected at earlier stages. In this paper relation between vibration amplitude and on the crack depth was developed, this helps in determine the depth of the crack by measuring the vibration amplitudes. To develop the relation equation strain energy density function was used. By observing the generated curves amplitude of vibration increases with respect to the depth of the crack due to reduction in stiffness of the shaft.
Enhancing Security of Multimodal Biometric Authentication System by Implement...IOSR Journals
Abstract : Conventional personal identification techniques for instance passwords, tokens, ID card and PIN
codes are prone to theft or forgery and thus biometrics isa solution thereto. Biometrics is the way of recognizing
and scrutinizing the physical traits of a person. Automated biometrics verification caters as a conducive and
legitimate method, but there must be an assurance to its cogency. Furthermore, in most of the cases unimodal
biometric recognition is not able to meet the performance requirements of the applications. According to recent
trends, recognition based on multimodal biometrics is emerging at a greater pace. Multimodal biometrics
unifies two or more biometric traits and thus the issues that emerge in unimodal recognition can be mitigated in
multimodal biometric systems. But with the rapid ontogenesis of information technology, even the biometric
data is not secure. Digital watermarking is one such technique that is implemented to secure the biometric data
from inadvertent or premeditated attacks.This paper propounds an approach that is projected in both the
directions of improving the performance of biometric identification system by going multimodal and, increasing
the security through watermarking. The biometric traits are initially transformed using Discrete Wavelet and
Discrete Cosine Transformation and then watermarked using Singular Value Decomposition. Scheme depiction
and presented outcomes justifies the effectiveness of the scheme.
Keywords: Discrete Cosine Transform (DCT), Discrete Wavelet Transform (DWT), Multimodal biometrics,
Singular Value Decomposition, Watermarking
An Efficient Approach for Requirement Traceability Integrated With Software R...IOSR Journals
Abstract: Traceability links between requirements of a system and its source code are helpful in reducing
system conception effort. During software updates and maintenance, the traceability links become invalid since
the developers may modify or remove some features of the source code. Hence, to acquire trustable links from a
system source code, a supervised link tracing approach is proposed here. In proposed approach, IR techniques
are applied on source code and requirements document to generate baseline traceability links. Concurrently,
software repositories are also mined to generate validating traceability links i.e. Histrace links which are then
called as experts. Now a trust model named as DynWing is used to rank the different types of experts. DynWing
dynamically assigns weights to different types of experts in ranking process. The top ranked experts are then fed
to the trust model named as Trumo. Trumo validates the baseline links with top ranked experts and finds the
trustable links from baseline links set. While validating the links, Trumo is capable of discarding or re-ranking
the experts and finds most traceable links. The proposed approach is able to improve the precision and recall
values of the traceability links.
Index Terms: Traceability, requirements, features, source code, repositories, experts.
Implementation of Algorithms For Multi-Channel Digital Monitoring ReceiverIOSR Journals
Abstract: Monitoring Receivers form an important constituent of the Electronic support. In Monitoring
Receiver we can monitor, demodulate or scan the multiple channels.
In this project, the Implementation of algorithm for multi channel digital monitoring receiver. The
implementation will carry out the channelization by the way of Digital down Converters (DDCs) and Digital
Base band Demodulation. The Intermediate Frequency (IF) at 10.7 MHz will be digitalized using Analog to
Digital Converter (ADC) with sampling frequency 52.5 MHz and further converted to Base band using DDCs.
Virtually all the digital receivers perform channel access using a DDC. The Base band data will be streamed to
the appropriate demodulators. Matlab Simulink will be used to simulate the logic modules before the
implementation. This system will be prototyped on an FPGA based COTS (Commercial-off-the-shelf)
development board. Xilinx System Generator will be used for the implementation of the algorithms.
Keywords: DDC, ADC, Digital Base band demodulation, IF, Monitoring Receiver.
Cataloging Of Sessions in Genuine Traffic by Packet Size Distribution and Ses...IOSR Journals
Abstract: Cataloging traffic keen on precise network applications is vital for application-aware network
organization and it turn into more taxing because modern applications incomprehensible their network
behaviors. Whereas port number-based classifiers work merely for a little renowned application and signaturebased
classifiers are not significant to encrypted packet payloads, researchers are inclined to classify network
traffic rooted in behaviors scrutinized in network applications. In this document, a session level Flood
Cataloging (SLFC) approach is proposed to organize network Floods as a session, which encompasses of
Floods in the equal discussion. SLFC initially classifies flood into the analogous applications by packet size
distribution (PSD) and subsequently faction Floods as sessions by port locality. With PSD, each Flood is
distorted into a set of points in a two-Dimension space and the remoteness among all Flood and the
representatives of preselected applications are calculated. The Flood is predicted as the application having a
least distance. Meanwhile, port locality is accustomed to cluster Floods as sessions since an application often
uses successive port statistics surrounded by a session. If flood of a session are categorized into diverse
applications, an arbitration algorithm is invoked to make the improvement.
Keywords: Flood Cataloging; session grouping; session Cataloging; packet size distribution
Synthesis, Structure Investigation and Dyeing Assessment of Novel Bisazo Disp...IOSR Journals
Novel bisazo-disperse dyes were prepared by the coupling of diazotized solutions of various aromatic diamines with 2,4-dihydroxybenzophenone. The resultant bisazo disperse dyes were characterized by elemental analysis, IR and 1H NMR spectral studies. The UV Visible absorption spectral data were investigated in dimethylformamide. and are discussed in terms of structural property relationship. Their dyeing assessment of bisazo disperse dyes has been made on polyester fabrics. The results show that a better hue was obtained on polyester fabrics and have mild to moderate fastness properties.
Prevalence of Iron Deficiency Anaemia among Pregnant Women in Calabar, Cross ...IOSR Journals
Iron is a component of a number of proteins including haemoglobin, myoglobin, cytochromes and enzymes involved in redox reactions. Inadequate iron intake can lead to varying degrees of deficiency, from low iron stores to early iron deficiency and iron-deficiency anaemia and this is dangerous to both baby and mother. The objective of this study is to assess the prevalence of iron deficiency and iron deficiency anaemia among pregnant women in Calabar, Cross River State Nigeria. Seventy pregnant women within the age range of 15-45 years from University of Calabar Teaching Hospital were recruited as subjects in this study. The control consisted of fifty age-matched apparently healthy non-pregnant women . The tests that were carried out using standard method include include full blood count (packed cell volume, haemoglobin, mean cell haemoglobin, mean cell haemoglobin concentration and red cell count), serum iron, total iron binding capacity, transferrin saturation,serum ferritin and soluble transferrin recptor. The prevalence of anaemia and iron deficiency anaemia were found to be significantly higher (p<0.05)><0.01)><0.01) increased in pregnant than non-pregnant. It was also shown that pregnant women in their third trimesters and multigravidae had the highest prevalence of iron deficiency and iron deficiency anaemia while pregnant women in their second trimester had the highest prevalence of anaemia. In conclusion the study has shown that the prevalence of anaemia, iron deficiency and iron deficiency anaemia among pregnant women in the studied area were still high and can be considered public health problem.
The influence of Rolling process on the porosity and wear behavior of Spray-f...IOSR Journals
Spray forming, also called spray casting or spray deposition, is the inert gas atomization of a liquid metal stream into variously sized droplets. These droplets are interrupted by a substrate which collects and solidifies the droplets into coherent, near fully dense perform. The present work concerns with this techniques. This technology was applied to produce Al-6%Si-20%Pb alloys. The spray droplets were deposited over a copper substrate to achieve a disc shape perform. After spray deposition samples were rolled at room temperature on two high rolling mills and total porosity and wear characteristic of spray deposits were studies. The total porosity decreases with the increase in the percentage of reduction in thickness of the samples. Thickness of the samples in both middle and peripheral region. Wear testing of spray formed and rolled Al-Si-Pb alloy were investigated on a pin on disc type wear machine. Wear rate behavior with applied load and increase in percentage of reduction in thickness of samples was observed. Wear rate increased with increasing the load and decreased with increase in percentage of reduction in thickness of the sample.
Low Power FPGA Based Elliptical Curve CryptographyIOSR Journals
Abstract: Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. The development of public-key cryptography is the greatest and perhaps the only true revolution in the entire history of cryptography. Elliptic Curve Cryptography is one of the public-key cryptosystem showing up in standardization efforts, including the IEEE P1363 Standard. The principal attraction of elliptic curve cryptography compared to RSA is that it offers equal security for a smaller key-size, thereby reducing the processing overhead. As a Public-Key Cryptosystem, ECC has many advantages such as fast speed, high security and short key. It is suitable for the hardware of implementation, so ECC has been more and more focused in recent years. The hardware implementation of ECC on FPGA uses the arithmetic unit that has small area, small storage unit and fast speed, and it is an extremely suitable system which has limited computation ability and storage space.[1][2] The modular arithmetic division operations are carried out using conditional successive subtractions, thereby reducing the area. The system is implemented on Vertex-Pro XCV1000 FPGA. Index Terms – VHDL, FSM, FPGA, Elliptic Curve Cryptography.
Electrophoretic Patterns of Esterases in Eri silkworm Samia Cynthia riciniIOSR Journals
The present study was carried out to investigate the patterns of esterase isozymes extracted from the silk gland, haemolymph and mid gut of Eri silkworm (Samia Cynthia ricini). The qualitative analysis of esterases was carried out by 7.5% of native Polyacrylamide Gel Electrophoresis (PAGE). The inhibitor sensitivity of the enzymes towards paraxon, eserine and pCMB was used to classify the individual zones of esterases. Three zones of esterases were observed in different tissues of Eri silkworm. Silk gland esterases were classified as CHsp (Cholinesterase like enzymes) esterases. The haemolymph and mid gut esterases were classified into Esdp (Enzyme inhibited by paraxon and pCMB).
“Comparitive Study of Prevalence of Hyperlactatemia in HIV / AIDS Patients re...IOSR Journals
Hyperlactatemia is one of the important metabolic abnormalities in HIV infected patients. The
prevalence of hyperlactatemia in natural course of HIV disease is approximately about 2%. Aim of this study is
to estimate the prevalence of hyperlactatemia in HIV patients receiving two antiretroviral regimens, advocated
by NACO by monitoring the plasma lactate levels. This study was taken up with 200 patients to compare the
prevalence of hyperlactatemia of two commonly used NACO regimens (zidovudine+ lamivudine+ nevirapine)
Vs (stavudine+ lamivudine+ nevirapine). The plasma lactate levels were estimated between 9th to 18thmonth
after initiation of antiretroviral therapy. The comparision and correlation between plasma lactate levels, CD4
counts and haemoglobin percentage in both regimens was done. There was statistically significant rise in the
plasma lactate levels (p<0.05) in both regimens. The increase in plasma lactate levels is more in stavudine
group compared to zidovudine group. There was low degree of positive correlation between plasma lactate and
haemoglobin in Stavudine group but negative correlation between Plasma lactate and CD4 counts in both
groups. More focus is needed on Pharmacovigilance of NRTIs induced hyperlactatemia especially Stavudine.
The need for a rapid, famous method for hair removal has led to the development of various laser for
hair removal. These include ruby, alexandrite, diode, and Nd:YAG lasers. This research discusses the basic
principles of laser hair removal, examines theoretically the properties of specific laser systems, and focuses on
patient selection and treatment protocols for the various systems designed to ensure safe and effective treatment.
Radiation dose to the eyes of readers at the least distance of distinct visio...IOSR Journals
This work reports the measurements of the radionuclide contents in some widely read daily newspapers published in Nigeria, using gamma spectrometry. The radionuclides detected in the newspapers measured consisted of the natural radionuclides belonging to the series headed by 228Ra and 226Ra as well as the singly occurring radionuclide 40K. The mean activity concentrations obtained for 40K, 226Ra and 228Ra respectively in the newspapers were 183.41±135.43, 9.06±3.64 and 6.11±1.36 Bqkg-1, 139.10±90.38, 7.58±1.87 and 5.29±1.33 Bqkg-1 and 152.10±114.32, 9.62±1.40 and 5.76±1.29 Bqkg-1 respectively from P1, P2 and P3. The doses to the eyes due to the measured activity concentrations in the newspaper samples were determined for a distance of 0.25 m (least distance of distinct vision) from the eyes. The annual effective doses to the eye resulting from the activities of the radionuclides identified with observed regularity in all the newspaper samples, obtained in this study are 0.012±0.010, 0.010±0.009, 0.010±0.009 μSv y-1 respectively for the newspapers. These values show that the doses to the lens of the eye from the Nigerian newspapers assessed in this work are very low compared to the annual dose limit of 15 mSv y
Discovery of Self-Sustained 235U Fission Causing Sunlight by Padmanabha Rao E...IOSR Journals
For the first time in solar physics, this paper reports a comprehensive study how 235Uranium fission
causes Sunlight by the atomic phenomenon, Padmanabha Rao Effect against the theory of fusion. The first major
breakthrough lies in identifying as many as 153 solar lines in the Bharat Radiation range from 12.87 to 31 nm
reported by various researchers since 1960s. The Sunlight phenomenon is explained as follows. For example, the
energy equivalence 72.48 eV of the most intense 17.107 nm emission in the middle of solar spectrum is the energy
lost by β, γ, or X-ray energy of a fission product while passing through core-Coulomb space. This energy loss is the
Bharat Radiation energy that cause EUV, UV, visible, and near infrared emissions on valence excitation. From vast
data of emissions and energies of various fission products, 606.31 keV β (Eβmax) energy of 131I was chosen as the
source of 17.107 nm emission. For the first time a typical Bharat Radiation spectrum was observed when plotted
energy loss against β, γ, or X-ray energies of fission products supposedly present in solar flare and atmosphere :
113Xe, 131I, 137Cs, 95Zr, 144Cs, 134I, 140Ba, 133I, 140La, 133In etc that caused solar lines. Consistent presence of a sharp
line for four months in AIA spectral EUV band at 335A exemplifies self-sustained uranium fission from a small site
appeared in SDO/AIA image at 304A. Sun’s dark spot is explained as a large crater formed on Sun’s core surface as
a result of fission reaction that does not show any emission since fission products would be thrown away from the
site during fission. Purely the same Sun’s core material left over at the site after fission reaction devoid of fission
products and any emission seems to be the familiar dark Matter. This could be the first report on the existence of
Sun’s Dark Matter.
Dust Interception Capacity And Alteration Of Various Biometric And Biochemica...IOSR Journals
The dust accumulation capacity of Ficus carica L. was evaluated from eight different sites in and around Multan. The impact of dust accumulation was observed via various biometric attributes (leaf area, leaf fresh and dry weights) and biochemical attributes (chlorophyll contents, carotenoids & ascorbic acid) from leaves of F. carica. The maximum dust accumulation was occurred in the plants growing at Road sides while, the minimum dust was found on plants growing at Bahauddin Zakariya University. Dust accumulation has caused a significant effect on almost all foliage and biochemical attributes of F. carica. A positive correlation was found between dust accumulation and biometric attributes in F. carica. Biochemical responses had shown an inconsistency as chlorophylls (a, b & total), carotenoids decreased and ascorbic acid contents increased with an increase in dust accumulation. A negative correlation was found between dust deposition and chlorophyll contents. Whereas, accumulation of ascorbic acid was associated with a decline in pigment contents
Bound State Solution to Schrodinger Equation With Modified Hylleraas Plus Inv...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger equation with modified
Hylleraass plus inversely quadratic potential using Supersymmetric quantum mechanics approach.
Applying perkeris approximation to the centrifugal term, we obtained the eigen-energy and the normalized
wave function using Gauss and confluent hypergeometric functions. We implement Fortran algorithm to
obtained the numerical result of the energy for the screening parameter α = 0.1,0.2,0.3, 0.4 0.5 and .
The result shows that the energy increases with an increase in the quantum state. The energy spectrum
shows increase in angular quantum state spacing as the screening parameter increases.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
BOUND STATE SOLUTION TO SCHRODINGER EQUATION WITH MODIFIED HYLLERAAS PLUS INV...ijrap
In this work, we obtained an approximate bound state solution to Schrodinger equation with modified
Hylleraass plus inversely quadratic potential using Supersymmetric quantum mechanics approach.
Applying perkeris approximation to the centrifugal term, we obtained the eigen-energy and the normalized
wave function using Gauss and confluent hypergeometric functions. We implement Fortran algorithm to
obtained the numerical result of the energy for the screening parameter α = 0.1,0.2,0.3, 0.4 0.5 and .
The result shows that the energy increases with an increase in the quantum state. The energy spectrum
shows increase in angular quantum state spacing as the screening parameter increases.
On the basis of the first law of thermodynamics, an analysis of the orientation of structural interactions was carried out. In the systems in which the interaction proceeds along the potential gradient (positive work), the resultant potential energy is found based on the principle of adding reciprocals of corresponding energies of subsystems - there is a corpuscular mechanism. In the systems in which the interactions proceed against the potential gradient (negative performance) the algebraic addition of their masses, as well as the corresponding energies of subsystems is performed - there is a wave mechanism. Act of quantum action expressed via Plank’s constant is narrowed to the energy equilibrium-exchange redistribution between the corpuscular and wave processes, that is demonstrated via the angular vector of such motion. Еnergy characteristics of functional states of bio-systems are defined basically by P-parameter values of atom first valence electron. The principles of stationary biosystem formation are similar to the conditions of wave processes in the phase. Under the condition of the minimum of such interactions, the pathological (but not stationary) biostructures containing the molecular hydrogen can be formed.
RELATIVISTIC MEAN FIELD STUDY OF SM ISOTOPES WITH FTBCS APPROACHijrap
A theoretical study of the thermal pairing correlation as a function of temperature is performed for eveneven
148-154Sm isotopes using Finite Temperature Bardeen-Cooper-Schrieffer (FTBCS) approach within the
Relativistic Mean Field (RMF) model. Numerical results obtained at T=0 are found to be consistent with
the available experimental values. Further, results show the thermal dependency of various nuclear
parameters like gap parameter, pairing energy, binding energy, deformation and density. At T≠0.0 MeV,
the destruction of Cooper pairs and the pairing phase transition as well as shape transition is observed in
148-154Sm nuclei at critical temperature Tc≠0.
Thermodynamic behavior of tetrahydrofuron in p dioxane, methylcyclohexane and...eSAT Journals
Abstract The liquid state is intermediate in its properties of solid and gas. There are many attempt to develop a theory of liquid state are based on simple consideration of molecular behaving like hard sphere having attractive forces as perturbative forces. The equation of state for Lenard Jones fluid has been derived in the formation an expression for work, obtained from partition function through perturbation approach and found faithful reproduction of ultrasonic velocity and density data , theoretically at the given temperature. It has been applied to the binary liquid mixtures of tetrahydrofuron in p-dioxane methylcyclohexane and cyclohexanol. There is a close agreement with experimental values. The thermodynamic picture build up in this formulation could be considered as a good representation of molecular cluster in liquid state. Keywords: Ultrasonic velocity, tetrahydrofuron, p-dioxane, methylcyclohexane, cyclohexanol, adiabatic compressibility, molar volume
Relativistic Mean Field Study of SM Isotopes with FTBCS Approach ijrap
A theoretical study of the thermal pairing correlation as a function of temperature is performed for eveneven
148-154Sm isotopes using Finite Temperature Bardeen-Cooper-Schrieffer (FTBCS) approach within the
Relativistic Mean Field (RMF) model. Numerical results obtained at T=0 are found to be consistent with
the available experimental values. Further, results show the thermal dependency of various nuclear
parameters like gap parameter, pairing energy, binding energy, deformation and density. At T≠0.0 MeV,
the destruction of Cooper pairs and the pairing phase transition as well as shape transition is observed in
148-154Sm nuclei at critical temperature Tc≠0.
A NEW STUDY OF TRAPEZOIDAL, SIMPSON’S1/3 AND SIMPSON’S 3/8 RULES OF NUMERICAL...mathsjournal
The main goal of this research is to give the complete conception about numerical integration including Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to determine the best method, as well as the results, are compared. It includes graphical comparisons mentioning these methods graphically. After all, it is then emphasized that the among methods considered, Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving a definite integral.
Applied Mathematics and Sciences: An International Journal (MathSJ)mathsjournal
The main goal of this research is to give the complete conception about numerical integration including
Newton-Cotes formulas and aimed at comparing the rate of performance or the rate of accuracy of
Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8. To verify the accuracy, we compare each rules
demonstrating the smallest error values among them. The software package MATLAB R2013a is applied to
determine the best method, as well as the results, are compared. It includes graphical comparisons
mentioning these methods graphically. After all, it is then emphasized that the among methods considered,
Simpson’s 1/3 is more effective and accurate when the condition of the subdivision is only even for solving
a definite integral.
International journal of engineering and mathematical modelling vol2 no1_2015_2IJEMM
This paper is devoted to the homogenization of the Maxwell equations with periodically oscillating coefficients in the bianisotropic media which represents the most general linear media. In the first time, the limiting homogeneous constitutive law is rigorously justified in the frequency domain and is found from the solution of a local problem on the unit cell. The homogenization process is based on the two-scale convergence conception. In the second time, the implementation of the homogeneous
constitutive law by using the finite element method and the introduction of the boundary conditions in the discrete problem are introduced. Finally, the numerical results associated of the perforated chiral media are presented.
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Quark Model Three Body Calculations for the Hypertriton Bound State
1. IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 4, Issue 1 (May. - Jun. 2013), PP 14-20
www.iosrjournals.org
www.iosrjournals.org 14 | Page
Quark Model Three Body Calculations for the Hypertriton
Bound State
M. A .Allam
Physics Department, Faculty of Science, Cairo University Cairo, Egypt
Abstract: Hyperspherical three body calculations are performed to study and review the various properties of
the hypertriton bound state nucleus
3
H in the quark model using -N potentials. In these calculations we study
the different effects of the -N potentials on the hypertriton bound states as well as the separation energy B. A
combination of realistic two body N-N potentials with various - N potentials are considered. Complete
symmetric and mixed symmetric wave functions are introduced. using the renormalized Numerov method. The
agreement between the calculated
3
H binding energies and the available experimental data basically depends
on the type of the -N interactions used in the calculations. It was found that the -N potentials are the most
effective part in the hypertriton binding energy as well as the separation energy B where the -N potentials is
very effective to bound or unbound the
3
H hyper nucleus
Pacs numbers: 21.30. + y, 21.10.+dr,27.20.+n
Keywords: Hyper spherical three body calculations.-N and N-N quark model potentials, Hyper nucleus
3
H .
I. Introduction
One way to get information about the hyperon nucleon interactions is to study the bound state of the hyperons
,namely the hypernuclei, The most suitable nucleus for such study would be the lightest of these hypernuclei,
the hypertriton
3
H. It is well known that the hypertriton is a bound state which consists of proton, neutron and
hyperon. Interest in the hypertriton study comes mainly from the fact that it is the lightest and the most
loosely bound hypernucleus. As the hypertriton consists of proton, neutron and hyperon, it may be considered
as deuteron and a particle bound in a state with binding energy B = 0.130.05 MeV [1]. The deuteron
binding energy Bd = 2.2246 MeV and the binding energy of the hypertriton is B = 2.35 MeV. The hypertriton
being the lightest hypernucleus is the first system in which - N potential can be tested in the nuclear medium.
This is also supported by the fact that the strength of the- N interaction is not sufficient to produce a bound
two body system where the available - N scattering data are still extremely poor. Therefore the hypertriton is
expected to play important role in hypernuclear physics similar to that of the deuteron in the conventional
nuclear physics.
The hypertriton or approximately the -deuteron system is a very loosely bound system which has a root
mean square radius of [r2
] = 0.2 fm with reduced mass = 0.780 mN is about 314 of the nucleon mass mc2
= 939
MeV. Thus the -deuteron system is 5 times larger than the root mean square radius Rd = 1.971 fm of the
deuteron. Thus the hypertriton is a very simple system but it is close to the - dripline and therefore it is very
sensitive to the - N interaction which is limited by the hypertriton properties. Because of the experimental
difficulties however, the existing information on the hypertriton is mostly from old measurements [2]-[4]. The
hypertriton study includes a combination of simplicity, sensitivity and fundamental interest in connection with
the interactions between nucleons and strange particles. Therefore the theoretical investigations of hypertriton
have attracted considerably attentions in nuclear physics study over the last four decades .The interest is
reflected in a number of investigations where different techniques and models have been used. Faddeev
calculations were performed in order to study and investigate the hypertriton using various - N and N- N
interactions [5]-[16]. In these calculations a combinations of the - N and N- N interactions which came out
with the results that the hypertriton is unbound [5],[6] ,while in using the Nijmegen potentials the calculations
led to a bound state of the hypertriton [7]. Faddeev quark model calculations were performed [8]-[14] using
quark model - N interactions combined with various N- N interactions which also came out with the result that
the they unbound the hypertriton except for one or two - N interactions which could bound the hypertriton
[12]-[15].Variational calculations have been made [16]-[19] in order to study the hypertriton bound state using
various - N and N- N interactions. Some of the calculations turned out to unbound the hypertriton and others
led to a hypertriton bound state [20].The powerful and exact hyperspherical expansion method was also used to
investigate the different properties of the hypertriton bound state using various - N and N- N interactions [22]-
[26].The common factor in all the above different methods of calculations is that they have led to considerable
2. Quark Model Three Body Calculations For The Hypertriton Bound State
www.iosrjournals.org 15 | Page
variation in the hypertriton binding energy values from the (observed) bound three body system to the (wrong)
unbound -deuteron system [6],[7].
It is the aim of our work to investigate and review the different properties of the hypertriton bound state
nucleus
3
H in the quark model using - N and N-N potentials using hyperspherical three body calculations. In
these calculations we study the different effects of the - N potentials on the bound states of the hypertriton as
well as the separation energy B. A combination of the two body N - N potentials with various - N potentials
are considered. In the hyperspherical expansion formalism, a complete symmetric and mixed symmetric wave
functions are introduced. using the renormalized Numerov method. The agreement between the calculated
3
H
binding energies and the available experimental data basically depends on the type of the -N interactions used
in the calculations. It was found that the -N potentials are the most effective part in the hypertriton binding
energy as well as the separation energy B where the -N potentials are very effective to bound or unbound the
3
H hyper nucleus. The Fabre optimal subset [27]-[29] is adopted to obtain fast and good convergence for the
calculated binding energy using the renormalized Numerov method. [30],[31]. The (HH) method is a powerful
tool for the ab initio solution of the few-body Schrodinger equation for a given set of interaction potentials
among the constituent particles. In the (HH) method, the wave function describing a system of N particles (in
the center of mass system) is expanded in terms of a complete set of orthonormal functions of 3N-4 variables.
The expansion coefficients are functions of a single variable that represents the length of 3N-3 dimensional
vector. By substituting the wave function expansion into the Schroedinger equation describing the system, one
obtains an infinite set of coupled differential equations for the expansion coefficients. The resulting set of
coupled differential equations can be solved numerically by the renormalized Numerov method [30],[31] A
multi pole potential is also obtained by expanding the two body interaction on a complete set of hyperspherical
harmonics. This multi pole potential is very helpful and useful when used in the Schroedinger equation. As for
the three-body system, the angular harmonics are functions of five angular variables. In order to determine the
potential matrix of the three-body Schroedinger equation, the matrix elements of the multi pole potentials
between a pair of such hyperspherical harmonics were calculated. The symmetry of the system under study rules
out some harmonics from appearing in the set of coupled equations. Further, the centrifugal barrier terms
occurring in the set of coupled equations grow considerably with higher harmonics. One can therefore, truncate
this infinite set [27]-[29] and work with a finite set (Fabre optimal subset ) of coupled differential equations or a
corresponding one dimensional integral equation. The (HH) method is essentially an exact one and more reliable
than other methods. It involves no approximation except for a possible truncation of the expansion basis. By
gradually expanding the expansion basis and checking the rate of convergence, any desired precision in the
binding energy can, in principle, be achieved. However, the number of coupled differential equations and,
therefore, the complexity in the numerical solution increases rapidly as the expansion basis is increased by
including larger hyper-angular-momentum quantum number. The numbers of equations that have to be retained
in any calculation using the (HH) method will, of course, depend on the nature of the potential used.
In the present work, the Fabre optimal subset [27]-[29] was used to obtain a converged set of coupled
differential equations in a single variable, namely, the hyper-radius. By numerically solving these equations, the
eigenvalues and eigenfunctions of the hyper-nucleus
3
H wave function were determined. In section 2 the (HH)
method is presented including the different equations used in our calculation. In section 3 the numerical work
and results are presented while the discussion and conclusion are given in section 4.Section 5 is devoted for
references.
II. Theoretical Work
Let the position vectors of the two nucleons denoted by r1 and r2, respectively, and their masses by mN. The
position vector of the particle is denoted by r3 and its mass by m.The total mass of the
3
H hyper-nucleus is
M = m + 2mN and that for the lambda particle is taken to be m = (6/5) m, where m is the nucleon (proton or
neutron) mass. The Jacobi coordinates set used by Clare and Levinger [32] was chosen here:
)1()( 21 rr
)2()2( 321 rrr
where
mm
m
and
mm
m
N
N
N
N
2)2(2
3
The inter-particle separation are expressed as:
3. Quark Model Three Body Calculations For The Hypertriton Bound State
www.iosrjournals.org 16 | Page
12r ,
2
1
2
1
13 r , and )3(
2
1
2
1
23
r
Now, the hyper-spherical coordinates ρ and θ are introduced, where ρ is the hyper-radius and θ is the hyper-
spherical angle. As a result, the following relations are obtained
)4(,cos,sin 222
with )5()
2
0(tan
The non relativistic Schroedinger equation for the hyper-nucleus
3
H, after separating out the motion of the
center of mass, can be written as :
)6(0),()((
2
2
)
2
2
ji
ij ErV
where μ is chosen to be μ = M/3 and V(rij) is the two-body central potential. Therefore, the Schroedinger
equation for the hyper-nucleus
3
H system expressed in terms of the hyper-spherical coordinates becomes:
)7(0)()(
2
2
2
EV
Expanding the wave function Ψ(ρ) on the HH basis, Y [L] (Ω), gives
)8()()()(
][
][][
2
5
L
LL UY
where Ω is a set of 5 angular coordinates describing the position of a point at the surface of the unit hypersphere.
U[L]() are the renormalized hyper radial partial wave functions, [L] stands for the set of quantum numbers
including spin and isospin defining the state of grand orbital L. Substituting the expansion (8) into Eq.(7), yields
an infinite set of second order coupled differential equations written as:
)9(0)()()()(}]4
1
)2(
[{ ][][
][
][][2
2
2
22
LL
L
LL UYVYUE
L
d
d
which is subsequently truncated in order to be treated numerically. This can be done by using the Fabre optimal
subset [27]-[29]. As a result, the infinite set of coupled equations is transformed into a finite set of coupled ones
to be solved numerically. For the case of central potential, the ground state of the hyper-nucleus
3
H nucleus is
described by even number of the grand orbital momentum L= 2K+l, due to the parity conservation. Then, the
finite set of coupled equations, for orbital momentum l = 0, becomes:
)10(0)()()()(][ 2
2
222
K
K
K
K
K
KK UVUET
with the kinetic energy operator and the potential matrix elements being expressed as:
)11(]4
1
)22(
[ 2
2
2
22
2
K
d
d
T K
and
)12()(),0,()(
'''
2
2
2
2
2
K
K
K
K
K
K VCKCV
respectively. The geometrical coefficients, ),0,(2
2 CKC K
K
, appearing in Eq.(12) couple the set of equations
with the main equation for which K = 0 for each component of the central components of the two-body
potentials. Explicit expressions for these coefficients are given in Ref. [27]. The multi-pole potentials, V2K′′(ρ),
given in Eq.(12) introduce the multi-poles of the central parts of the two-body potential and are expressed as :
)13()()( 22
ji
ij
KK VV
with )14())(,
2
3
,2,()()(1
16
)( 2
12
2
0
22
2
x
KKFxVx
x
dxV ij
K
4. Quark Model Three Body Calculations For The Hypertriton Bound State
www.iosrjournals.org 17 | Page
where V(x) = V(rij) is the two-body potential. In the hyper-nucleus
3
H the two body potentials refers to the –
and the -α interactions. Equation (14) may be written as:
)15(),
2
3
,2,()()1(
16
)( 2
12
1
0
22
1
2
2 uKKFuVuuduV ij
K
with V(ρu) = V(rij) , which is more useful in our numerical calculations.
III. Numerical Calculations and Results
In order to carry out the numerical calculations, the set of coupled differential equations represented by Eq. (10)
were written in matrix form as:
)16(0)(2
2
UQ
d
d
I
where [I] is the unit matrix and the column vector U(ρ) contains the partial waves U2K(ρ) as its components.
Also the matrix element [Q] is given by:
)17()()( 2
effVIEQ
The components of the effective potential matrix is given by:
)18(]4
15
)1(2
[)()( 2
2
2
2
KK
VV K
Keff
where )(2
2 K
KV
is given by Eq.(12).
The renormalized Numerov method [30],[31] was then used to solve [27]- [29] the set of coupled equations
(16). In order to obtain fast convergence of the hyper-nucleus
3
H energy eigenvalue the Fabre optimal subset
[27]-[29] have been used, using a different Gaussian forms of the realistic N-N and quark model -N
interactions. The convergence of
3
H eigenvalue have been attained by including terms up to K= K{max}.
Therefore we have solved K=20 coupled equations in order to obtain the hyper-nucleus
3
H binding energy
values for the considered realistic N-N and quark model -N interactions.
In calculating the binding energy of the
3
H hyper nucleus we have used Gaussian form potential for the
realistic two body N- N as well as the Λ-N quark model interactions. For the N- N interactions, we have
considered two different sets for N- N potentials to be combined with the suggested Λ -N potentials. The first
set is the Minnesota N-N potentials MI [33] for which parameters were determined so as to produce the low
energy N- N scattering data. The second set is the modified Volkov two body potential Vx
[34],[35] for which
the singlet and triplet parts are unequal. These two body potential sets satisfy the requirements that they both
produce reasonably well the binding energy and size for the trinucleon systems which a necessary condition for
our present calculations. For the Λ- Λ interactions: It is well known that the Λ- N interactions are constructed in
such a way that the calculations with the Λ- N interactions reproduce the experimentally known bound states of
the hypernuclei and it is difficult to perform scattering experiments with the Λ- particles because of their short
lifetimes ( 10-10
Sec ) and the extremely low intensity beams that can be obtained. It is also well known that
even when scattering data are available in full, it is impossible to construct Λ- N interactions in a unique way
namely one can always obtain different phases and equivalent potentials. According to the above requirements
we consider two different sets of quark model Λ- N interactions which reproduce the Λ- P existing scattering
data and reproduce the low energy behavior of the Λ- N phase shift obtained in the full coupled calculations.
The first one is a three terms Gaussian shape Λ- N interaction QM1 [36] and the second on is the two term
Gaussian shape QM2[37] . All the considered types of -N and N-N interactions
3
H are of Gaussian shape and
given by the following relation:
6
1
2
2
)(
)exp()(
i i
i
N
d
r
VrV
where the parameters Vi and di are given in table (1).
Table (1)
Potn. i 1 2 3 4 5 6
MI [33]
Vi MeV 200.0 –91.85 –178.0
di Fm 1.487 0.465 0.639
Vx
[35]
Vi MeV 130.00 -110.0 130.00 -65.30
di Fm 0.8 1.5 0.8 1.5
5. Quark Model Three Body Calculations For The Hypertriton Bound State
www.iosrjournals.org 18 | Page
QM1 36]
Vi MeV 200.0 -109.8 -121.3
di Fm 1.638 0.7864 0.7513
QM2[37]
Vi MeV 1015.80 -128.0 1072.8 -50.245
di Fm 5.3830 0.8908 13.740 0.75310
Table (1). Parameters for the different sets of Gaussian shape N-N and -N interactions
In studying the different properties of the
3
H hypernucleus bound state, we have considered a
combination of the sets of the N-N and the sets of the -N interactions. The calculated
3
H binding energies
and the separation energy B obtained by the combinations are given in table2. The first four partial waves of
the
3
H hypernucleus wave functions generated by combining MI and QM1 potential are presented in Fig (1.a)
and those generated by combining Vx
and QM1 potential are presented in Fig(1.b). Similarly the first three
partial waves of the
3
H hypernucleus wave functions generated by combining MI and QM2 potential are
presented in Fig (2.a) and those generated by combining Vx
and QM2 potential and are presented in Fig(2.b).
N-N MI [33] Vx
[35]
-N QM1[36] QM2[37] QM1[36] QM2[37] Exp.[ 1]
E (MeV) 2.8566 2.2207 2.6491 2.1871 2.3556
B (MeV) 0.632 0.039 0.4245 0.0375 0.1310.05
Table (2): Calculated binding energies E for the
3
H hypernucleus and separation energy B for the particle
in
3
H hypernucleus using different combinations of the realistic N-N and quark model -N interactions. The
experimental binding energy value for deuteron Ed is given as Ed = 2.2246 MeV
IV. Discussion and Conclusions
In calculating the different properties o the
3
H hypernucleus we have used the exact powerful and reliable
hyperspherical expansion method which is successfully applied in solving and calculating properties of the three
and many body problems. We have calculated the
3
H binding energy E and the separation energy of the
particle BΛ using different combinations of realistic N-N and quark model - N interactions. In order to get a
good and fast convergence for the
3
H binding energy ,Fabre optimal subset [27],[28] was considered in
solving the finite set of coupled differential equations, where up to up to Kmax = 20 coupled equations have been
solved. The calculated binding energy E and separation energy BΛ for the
3
H hypernucleus using different
combinations of realistic N-N and quark model - N interactions are presented in table (2). It is shown form
table (2) that the binding energies results for the combination of MI and Vx
interactions with quark model QM1
interactions indicate that a bound state is obtained for the
3
H hypernucleus . It is also shown from Table (2)
that the separation energy BΛ indicate that the particle is very close to the nucleon- nucleon pair (core). This
result is supported by figures(1.a,1.b) for which a bound state wave function is generated using these potentials.
As for the N-N interactions with the QM2 interaction it is clear from the table that the results unbound the
3
H
hypernucleus and the separation energy values BΛ show that the
3
H is loosely bound and the particle is far
from the nucleon-nucleon cluster, a result which is also supported by Figures (2.a,2.b) for which the generated
wave functions indicate unbound state for the
3
H hypernucleus. Finally in reviewing and studying the
properties of the
3
H using different quark model - N interactions we conclude that the different types of the
- N interactions affect the binding energies E and separation energy BΛ in the case of the
3
H trinucleon
system. In other words the - N interactions is effective to bound or unbound the bound state of the
3
H
hypernucleus a result which is expected due to the quark construction of the particle.
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Fig.(1) First four hyperradial partial waves U(2K) (with K = 0,,2,3,4) for the
3
H hypernucleus: generated by
MI[ 33] N-N and QM1[ 36 ] -N interactions Fig. (1.a), Vx
[ 35 ] N-N and QM1[ 36 ] -N interactions Fig.(1.b)
respectively.
7. Quark Model Three Body Calculations For The Hypertriton Bound State
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Fig.(2) First three hyperradial partial waves U(2K) (with K = 0,,2,3) for the
3
H hypernucleus: generated by
MI[33] N-N and QM2[37] -N interactions Fig.(2.a), Vx
[ 35] N-N and QM2[37] -N interactions Fig.(2.b)
respectively.