The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On a Generalized 휷푹 − Birecurrent Affinely Connected Spaceinventionjournals
In the present paper, we introduced a Finsler space whose Cartan's third curvature tensor 푅푗푘 ℎ 푖 satisfies the generalized 훽푅 −birecurrence property which posses the properties of affinely connected space will be characterized by ℬ푚 ℬ푛푅푗푘 ℎ 푖 = 푎푚푛 푅푗푘 ℎ 푖 +푏푚푛 훿푘 푖 푔푗ℎ − 훿ℎ 푖 푔푗푘 , where 푎푚푛 and 푏푚푛 are non-zero covariant tensors field of second order called recurrence tensors field, such space is called as a generalized 훽푅 −birecurrent affinely connected space. Ricci tensors 퐻푗푘 and 푅푗푘 , the curvature vector 퐻푘 and the curvature scalars 퐻 and 푅 of such space are non-vanishing. Some conditions have been pointed out which reduce a generalized 훽푅 −birecurrent affinely connected space 퐹푛 (푛 > 2) into Finsler space of scalar curvature.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
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.
Second Order Parallel Tensors and Ricci Solitons in S-space forminventionjournals
In this paper, we prove that a symmetric parallel second order covariant tensor in (2m+s)- dimensional S-space form is a constant multiple of the associated metric tensor. Then we apply this result to study Ricci solitons for S-space form and Sasakian space form of dimension 3
We make use of the conformal compactification of Minkowski spacetime M# to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M#]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M# with the projective light cone in (4+2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors - one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus
modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy
eigen value and its associated total wave function . This potential with some suitable conditions reduces to
two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical
results for energy eigen value with different values of q as dimensionless parameter. The result shows that
the values of the energies for different quantum number(n) is negative(bound state condition) and increases
with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1)
shows the different energy levels for a particular quantum number.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
On a Generalized 휷푹 − Birecurrent Affinely Connected Spaceinventionjournals
In the present paper, we introduced a Finsler space whose Cartan's third curvature tensor 푅푗푘 ℎ 푖 satisfies the generalized 훽푅 −birecurrence property which posses the properties of affinely connected space will be characterized by ℬ푚 ℬ푛푅푗푘 ℎ 푖 = 푎푚푛 푅푗푘 ℎ 푖 +푏푚푛 훿푘 푖 푔푗ℎ − 훿ℎ 푖 푔푗푘 , where 푎푚푛 and 푏푚푛 are non-zero covariant tensors field of second order called recurrence tensors field, such space is called as a generalized 훽푅 −birecurrent affinely connected space. Ricci tensors 퐻푗푘 and 푅푗푘 , the curvature vector 퐻푘 and the curvature scalars 퐻 and 푅 of such space are non-vanishing. Some conditions have been pointed out which reduce a generalized 훽푅 −birecurrent affinely connected space 퐹푛 (푛 > 2) into Finsler space of scalar curvature.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
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.
Second Order Parallel Tensors and Ricci Solitons in S-space forminventionjournals
In this paper, we prove that a symmetric parallel second order covariant tensor in (2m+s)- dimensional S-space form is a constant multiple of the associated metric tensor. Then we apply this result to study Ricci solitons for S-space form and Sasakian space form of dimension 3
We make use of the conformal compactification of Minkowski spacetime M# to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M#]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M# with the projective light cone in (4+2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors - one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus
modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy
eigen value and its associated total wave function . This potential with some suitable conditions reduces to
two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical
results for energy eigen value with different values of q as dimensionless parameter. The result shows that
the values of the energies for different quantum number(n) is negative(bound state condition) and increases
with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1)
shows the different energy levels for a particular quantum number.
Exact Bound State Solution of Qdeformed Woods-Saxon Plus Modified Coulomb Pot...ijrap
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method. We also obtained the energy eigen value and its associated total wave function . This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential. Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter. The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant). The graph in figure (1) shows the different energy levels for a particular quantum number.
A Coupled Thermoelastic Problem of A Half – Space Due To Thermal Shock on the...iosrjce
This paper is concerned with the determination of temperature and displacement of a half space
bounding surface due to thermal shock. This paper deals with the place boundary of the half-space is free of
stress and is subjected to a thermal shock. Moreover , the perturbation method is employed with the
thermoelastic coupling facter ԑ as the perturbation parameter. The Laplace transform and its inverse with very
small thermoelastic coupling facter ԑ are used. The deformation field is obtained for small values of time.
푃푎푟푖푎
7
has formulated different types of thermal boundary condition problems
Effect of Michell’s Function in Stress Analysis Due to Axisymmetric Heat Supp...IJERA Editor
The present paper deals with the determination of quasi static thermal stresses in a limiting thick circular plate
subjected to arbitrary heat flux on upper and lower surface and the fixed circular edge is thermally insulated.
Initially the limiting thick circular plate is at zero temperature. Here we modify Kulkarni (2009) and compute
the effects of Michell function on the limiting thickness of circular plate by using stress analysis with internal
heat generation and axisymmetric heat supply in terms of stresses along radial direction. The governing heat
conduction equation has been solved by the method of integral transform technique. The results are obtained in
a series form in terms of Bessel’s functions. The results for stresses have been computed numerically and
illustrated graphically.
Differential geometry three dimensional spaceSolo Hermelin
This presentation describes the mathematics of curves and surfaces in a 3 dimensional (Euclidean) space.
The presentation is at an Undergraduate in Science (Math, Physics, Engineering) level.
Plee send comments and suggestions to improvements to solo.hermelin@gmail.com. Thanks/
More presentations can be found at my website http://www.solohermelin.com.
EXACT SOLUTIONS OF SCHRÖDINGER EQUATION WITH SOLVABLE POTENTIALS FOR NON PT/P...ijrap
We have obtained explicitly the exact solutions of the Schrodinger equation with Non PT/PT symmetric
Rosen Morse II, Scarf II and Coulomb potentials. Energy eigenvalues and the corresponding
unnormalized wave functions for these systems for both Non PT and PT symmetric are also obtained using
the Nikiforov-Uvarov (NU) method.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized E...ijrap
We apply an approximation to centrifugal term to find bound state solutions to Schrodinger equation with
Hulthen Plus generalized exponential Coulomb potential Using Nikiforov-Uvarov Method. Using this
method, we obtained the energy-eigen value and the total wave function. We implement C++ algorithm, to
obtained the numerical values of the energy for different quantum state starting from the first excited state
for different values of the screening parameter.
Applied Mathematics Multiple Integration by Mrs. Geetanjali P.Kale.pdfGeetanjaliRao6
In mathematics (specifically multivariable calculus), a Multiple Integral is a Definite Integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in (the real-number plane) are called Double Integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called Triple Integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at
the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of
vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by
diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions
(single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
Modeling of Redistribution of Infused Dopant in a Multilayer Structure Dopant...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions (single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Similar to Crack problems concerning boundaries of convex lens like forms (20)
‘Six Sigma Technique’ A Journey Through its Implementationijtsrd
The manufacturing industries all over the world are facing tough challenges for growth, development and sustainability in today’s competitive environment. They have to achieve apex position by adapting with the global competitive environment by delivering goods and services at low cost, prime quality and better price to increase wealth and consumer satisfaction. Cost Management ensures profit, growth and sustainability of the business with implementation of Continuous Improvement Technique like Six Sigma. This leads to optimize Business performance. The method drives for customer satisfaction, low variation, reduction in waste and cycle time resulting into a competitive advantage over other industries which did not implement it. The main objective of this paper ‘Six Sigma Technique A Journey Through Its Implementation’ is to conceptualize the effectiveness of Six Sigma Technique through the journey of its implementation. Aditi Sunilkumar Ghosalkar "‘Six Sigma Technique’: A Journey Through its Implementation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64546.pdf Paper Url: https://www.ijtsrd.com/other-scientific-research-area/other/64546/‘six-sigma-technique’-a-journey-through-its-implementation/aditi-sunilkumar-ghosalkar
Edge Computing in Space Enhancing Data Processing and Communication for Space...ijtsrd
Edge computing, a paradigm that involves processing data closer to its source, has gained significant attention for its potential to revolutionize data processing and communication in space missions. With the increasing complexity and data volume generated by modern space missions, traditional centralized computing approaches face challenges related to latency, bandwidth, and security. Edge computing in space, involving on board processing and analysis of data, offers promising solutions to these challenges. This paper explores the concept of edge computing in space, its benefits, applications, and future prospects in enhancing space missions. Manish Verma "Edge Computing in Space: Enhancing Data Processing and Communication for Space Missions" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64541.pdf Paper Url: https://www.ijtsrd.com/computer-science/artificial-intelligence/64541/edge-computing-in-space-enhancing-data-processing-and-communication-for-space-missions/manish-verma
Dynamics of Communal Politics in 21st Century India Challenges and Prospectsijtsrd
Communal politics in India has evolved through centuries, weaving a complex tapestry shaped by historical legacies, colonial influences, and contemporary socio political transformations. This research comprehensively examines the dynamics of communal politics in 21st century India, emphasizing its historical roots, socio political dynamics, economic implications, challenges, and prospects for mitigation. The historical perspective unravels the intricate interplay of religious identities and power dynamics from ancient civilizations to the impact of colonial rule, providing insights into the evolution of communalism. The socio political dynamics section delves into the contemporary manifestations, exploring the roles of identity politics, socio economic disparities, and globalization. The economic implications section highlights how communal politics intersects with economic issues, perpetuating disparities and influencing resource allocation. Challenges posed by communal politics are scrutinized, revealing multifaceted issues ranging from social fragmentation to threats against democratic values. The prospects for mitigation present a multifaceted approach, incorporating policy interventions, community engagement, and educational initiatives. The paper conducts a comparative analysis with international examples, identifying common patterns such as identity politics and economic disparities. It also examines unique challenges, emphasizing Indias diverse religious landscape, historical legacy, and secular framework. Lessons for effective strategies are drawn from international experiences, offering insights into inclusive policies, interfaith dialogue, media regulation, and global cooperation. By scrutinizing historical epochs, contemporary dynamics, economic implications, and international comparisons, this research provides a comprehensive understanding of communal politics in India. The proposed strategies for mitigation underscore the importance of a holistic approach to foster social harmony, inclusivity, and democratic values. Rose Hossain "Dynamics of Communal Politics in 21st Century India: Challenges and Prospects" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64528.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/history/64528/dynamics-of-communal-politics-in-21st-century-india-challenges-and-prospects/rose-hossain
Assess Perspective and Knowledge of Healthcare Providers Towards Elehealth in...ijtsrd
Background and Objective Telehealth has become a well known tool for the delivery of health care in Saudi Arabia, and the perspective and knowledge of healthcare providers are influential in the implementation, adoption and advancement of the method. This systematic review was conducted to examine the current literature base regarding telehealth and the related healthcare professional perspective and knowledge in the Kingdom of Saudi Arabia. Materials and Methods This systematic review was conducted by searching 7 databases including, MEDLINE, CINHAL, Web of Science, Scopus, PubMed, PsycINFO, and ProQuest Central. Studies on healthcare practitioners telehealth knowledge and perspectives published in English in Saudi Arabia from 2000 to 2023 were included. Boland directed this comprehensive review. The researchers examined each connected study using the AXIS tool, which evaluates cross sectional systematic reviews. Narrative synthesis was used to summarise and convey the data. Results Out of 1840 search results, 10 studies were included. Positive outlook and limited knowledge among providers were seen across trials. Healthcare professionals like telehealth for its ability to improve quality, access, and delivery, save time and money, and be successful. Age, gender, occupation, and work experience also affect health workers knowledge. In Saudi Arabia, healthcare professionals face inadequate expert assistance, patient privacy, internet connection concerns, lack of training courses, lack of telehealth understanding, and high costs while performing telemedicine. Conclusions Healthcare practitioners telehealth perceptions and knowledge were examined in this systematic study. Its collection of concerned experts different personal attitudes and expertise would help enhance telehealths implementation in Saudi Arabia, develop its healthcare delivery alternative, and eliminate frequent problems. Badriah Mousa I Mulayhi | Dr. Jomin George | Judy Jenkins "Assess Perspective and Knowledge of Healthcare Providers Towards Elehealth in Saudi Arabia: A Systematic Review" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64535.pdf Paper Url: https://www.ijtsrd.com/medicine/other/64535/assess-perspective-and-knowledge-of-healthcare-providers-towards-elehealth-in-saudi-arabia-a-systematic-review/badriah-mousa-i-mulayhi
The Impact of Digital Media on the Decentralization of Power and the Erosion ...ijtsrd
The impact of digital media on the distribution of power and the weakening of traditional gatekeepers has gained considerable attention in recent years. The adoption of digital technologies and the internet has resulted in declining influence and power for traditional gatekeepers such as publishing houses and news organizations. Simultaneously, digital media has facilitated the emergence of new voices and players in the media industry. Digital medias impact on power decentralization and gatekeeper erosion is visible in several ways. One significant aspect is the democratization of information, which enables anyone with an internet connection to publish and share content globally, leading to citizen journalism and bypassing traditional gatekeepers. Another aspect is the disruption of conventional media industry business models, as traditional organizations struggle to adjust to the decrease in advertising revenue and the rise of digital platforms. Alternative business models, such as subscription models and crowdfunding, have become more prevalent, leading to the emergence of new players. Overall, the impact of digital media on the distribution of power and the weakening of traditional gatekeepers has brought about significant changes in the media landscape and the way information is shared. Further research is required to fully comprehend the implications of these changes and their impact on society. Dr. Kusum Lata "The Impact of Digital Media on the Decentralization of Power and the Erosion of Traditional Gatekeepers" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64544.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/political-science/64544/the-impact-of-digital-media-on-the-decentralization-of-power-and-the-erosion-of-traditional-gatekeepers/dr-kusum-lata
Online Voices, Offline Impact Ambedkars Ideals and Socio Political Inclusion ...ijtsrd
This research investigates the nexus between online discussions on Dr. B.R. Ambedkars ideals and their impact on social inclusion among college students in Gurugram, Haryana. Surveying 240 students from 12 government colleges, findings indicate that 65 actively engage in online discussions, with 80 demonstrating moderate to high awareness of Ambedkars ideals. Statistically significant correlations reveal that higher online engagement correlates with increased awareness p 0.05 and perceived social inclusion. Variations across colleges and a notable effect of college type on perceived social inclusion highlight the influence of contextual factors. Furthermore, the intersectional analysis underscores nuanced differences based on gender, caste, and socio economic status. Dr. Kusum Lata "Online Voices, Offline Impact: Ambedkar's Ideals and Socio-Political Inclusion - A Study of Gurugram District" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64543.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/political-science/64543/online-voices-offline-impact-ambedkars-ideals-and-sociopolitical-inclusion--a-study-of-gurugram-district/dr-kusum-lata
Problems and Challenges of Agro Entreprenurship A Studyijtsrd
Noting calls for contextualizing Agro entrepreneurs problems and challenges of the agro entrepreneurs and for greater attention to the Role of entrepreneurs in agro entrepreneurship research, we conduct a systematic literature review of extent research in agriculture entrepreneurship to overcome the study objectives of complications of agro entrepreneurs through various factors, Development of agriculture products is a key factor for the overall economic growth of agro entrepreneurs Agro Entrepreneurs produces firsthand large scale employment, utilizes the labor and natural resources, This research outlines the problems of Weather and Soil Erosions, Market price fluctuation, stimulates labor cost problems, reduces concentration of Price volatility, Dependency on Intermediaries, induces Limited Bargaining Power, and Storage and Transportation Costs. This paper mainly devoted to highlight Problems and challenges faced for the sustainable of Agro Entrepreneurs in India. Vinay Prasad B "Problems and Challenges of Agro Entreprenurship - A Study" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64540.pdf Paper Url: https://www.ijtsrd.com/other-scientific-research-area/other/64540/problems-and-challenges-of-agro-entreprenurship--a-study/vinay-prasad-b
Comparative Analysis of Total Corporate Disclosure of Selected IT Companies o...ijtsrd
Disclosure is a process through which a business enterprise communicates with external parties. A corporate disclosure is communication of financial and non financial information of the activities of a business enterprise to the interested entities. Corporate disclosure is done through publishing annual reports. So corporate disclosure through annual reports plays a vital role in the life of all the companies and provides valuable information to investors. The basic objectives of corporate disclosure is to give a true and fair view of companies to the parties related either directly or indirectly like owner, government, creditors, shareholders etc. in the companies act, provisions have been made about mandatory and voluntary disclosure. The IT sector in India is rapidly growing, the trend to invest in the IT sector is rising and employment opportunities in IT sectors are also increasing. Therefore the IT sector is expected to have fair, full and adequate disclosure of all information. Unfair and incomplete disclosure may adversely affect the entire economy. A research study on disclosure practices of IT companies could play an important role in this regard. Hence, the present research study has been done to study and review comparative analysis of total corporate disclosure of selected IT companies of India and to put forward overall findings and suggestions with a view to increase disclosure score of these companies. The researcher hopes that the present research study will be helpful to all selected Companies for improving level of corporate disclosure through annual reports as well as the government, creditors, investors, all business organizations and upcoming researcher for comparative analyses of level of corporate disclosure with special reference to selected IT companies. Dr. Vaibhavi D. Thaker "Comparative Analysis of Total Corporate Disclosure of Selected IT Companies of India" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64539.pdf Paper Url: https://www.ijtsrd.com/other-scientific-research-area/other/64539/comparative-analysis-of-total-corporate-disclosure-of-selected-it-companies-of-india/dr-vaibhavi-d-thaker
The Impact of Educational Background and Professional Training on Human Right...ijtsrd
This study investigated the impact of educational background and professional training on human rights awareness among secondary school teachers in the Marathwada region of Maharashtra, India. The key findings reveal that higher levels of education, particularly a master’s degree, and fields of study related to education, humanities, or social sciences are associated with greater human rights awareness among teachers. Additionally, both pre service teacher training and in service professional development programs focused on human rights education significantly enhance teacher’s knowledge, skills, and competencies in promoting human rights principles in their classrooms. Baig Ameer Bee Mirza Abdul Aziz | Dr. Syed Azaz Ali Amjad Ali "The Impact of Educational Background and Professional Training on Human Rights Awareness among Secondary School Teachers" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64529.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/education/64529/the-impact-of-educational-background-and-professional-training-on-human-rights-awareness-among-secondary-school-teachers/baig-ameer-bee-mirza-abdul-aziz
A Study on the Effective Teaching Learning Process in English Curriculum at t...ijtsrd
“One Language sets you in a corridor for life. Two languages open every door along the way” Frank Smith English as a foreign language or as a second language has been ruling in India since the period of Lord Macaulay. But the question is how much we teach or learn English properly in our culture. Is there any scope to use English as a language rather than a subject How much we learn or teach English without any interference of mother language specially in the classroom teaching learning scenario in West Bengal By considering all these issues the researcher has attempted in this article to focus on the effective teaching learning process comparing to other traditional strategies in the field of English curriculum at the secondary level to investigate whether they fulfill the present teaching learning requirements or not by examining the validity of the present curriculum of English. The purpose of this study is to focus on the effectiveness of the systematic, scientific, sequential and logical transaction of the course between the teachers and the learners in the perspective of the 5Es programme that is engage, explore, explain, extend and evaluate. Sanchali Mondal | Santinath Sarkar "A Study on the Effective Teaching Learning Process in English Curriculum at the Secondary Level of West Bengal" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd62412.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/education/62412/a-study-on-the-effective-teaching-learning-process-in-english-curriculum-at-the-secondary-level-of-west-bengal/sanchali-mondal
The Role of Mentoring and Its Influence on the Effectiveness of the Teaching ...ijtsrd
This paper reports on a study which was conducted to investigate the role of mentoring and its influence on the effectiveness of the teaching of Physics in secondary schools in the South West Region of Cameroon. The study adopted the convergent parallel mixed methods design, focusing on respondents in secondary schools in the South West Region of Cameroon. Both quantitative and qualitative data were collected, analysed separately, and the results were compared to see if the findings confirm or disconfirm each other. The quantitative analysis found that majority of the respondents 72 of Physics teachers affirmed that they had more experienced colleagues as mentors to help build their confidence, improve their teaching, and help them improve their effectiveness and efficiency in guiding learners’ achievements. Only 28 of the respondents disagreed with these statements. With majority respondents 72 agreeing with the statements, it implies that in most secondary schools, experienced Physics teachers act as mentors to build teachers’ confidence in teaching and improving students’ learning. The interview qualitative data analysis summarized how secondary school Principals use meetings with mentors and mentees to promote mentorship in the school milieu. This has helped strengthen teachers’ classroom practices in secondary schools in the South West Region of Cameroon. With the results confirming each other, the study recommends that mentoring should focus on helping teachers employ social interactions and instructional practices feedback and clarity in teaching that have direct measurable impact on students’ learning achievements. Andrew Ngeim Sumba | Frederick Ebot Ashu | Peter Agborbechem Tambi "The Role of Mentoring and Its Influence on the Effectiveness of the Teaching of Physics in Secondary Schools in the South West Region of Cameroon" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64524.pdf Paper Url: https://www.ijtsrd.com/management/management-development/64524/the-role-of-mentoring-and-its-influence-on-the-effectiveness-of-the-teaching-of-physics-in-secondary-schools-in-the-south-west-region-of-cameroon/andrew-ngeim-sumba
Design Simulation and Hardware Construction of an Arduino Microcontroller Bas...ijtsrd
This study primarily focuses on the design of a high side buck converter using an Arduino microcontroller. The converter is specifically intended for use in DC DC applications, particularly in standalone solar PV systems where the PV output voltage exceeds the load or battery voltage. To evaluate the performance of the converter, simulation experiments are conducted using Proteus Software. These simulations provide insights into the input and output voltages, currents, powers, and efficiency under different state of charge SoC conditions of a 12V,70Ah rechargeable lead acid battery. Additionally, the hardware design of the converter is implemented, and practical data is collected through operation, monitoring, and recording. By comparing the simulation results with the practical results, the efficiency and performance of the designed converter are assessed. The findings indicate that while the buck converter is suitable for practical use in standalone PV systems, its efficiency is compromised due to a lower output current. Chan Myae Aung | Dr. Ei Mon "Design Simulation and Hardware Construction of an Arduino-Microcontroller Based DC-DC High-Side Buck Converter for Standalone PV System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64518.pdf Paper Url: https://www.ijtsrd.com/engineering/mechanical-engineering/64518/design-simulation-and-hardware-construction-of-an-arduinomicrocontroller-based-dcdc-highside-buck-converter-for-standalone-pv-system/chan-myae-aung
Sustainable Energy by Paul A. Adekunte | Matthew N. O. Sadiku | Janet O. Sadikuijtsrd
Energy becomes sustainable if it meets the needs of the present without compromising the ability of future generations to meet their own needs. Some of the definitions of sustainable energy include the considerations of environmental aspects such as greenhouse gas emissions, social, and economic aspects such as energy poverty. Generally far more sustainable than fossil fuel are renewable energy sources such as wind, hydroelectric power, solar, and geothermal energy sources. Worthy of note is that some renewable energy projects, like the clearing of forests to produce biofuels, can cause severe environmental damage. The sustainability of nuclear power which is a low carbon source is highly debated because of concerns about radioactive waste, nuclear proliferation, and accidents. The switching from coal to natural gas has environmental benefits, including a lower climate impact, but could lead to delay in switching to more sustainable options. “Carbon capture and storage” can be built into power plants to remove the carbon dioxide CO2 emissions, but this technology is expensive and has rarely been implemented. Leading non renewable energy sources around the world is fossil fuels, coal, petroleum, and natural gas. Nuclear energy is usually considered another non renewable energy source, although nuclear energy itself is a renewable energy source, but the material used in nuclear power plants is not. The paper addresses the issue of sustainable energy, its attendant benefits to the future generation, and humanity in general. Paul A. Adekunte | Matthew N. O. Sadiku | Janet O. Sadiku "Sustainable Energy" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64534.pdf Paper Url: https://www.ijtsrd.com/engineering/electrical-engineering/64534/sustainable-energy/paul-a-adekunte
Concepts for Sudan Survey Act Implementations Executive Regulations and Stand...ijtsrd
This paper aims to outline the executive regulations, survey standards, and specifications required for the implementation of the Sudan Survey Act, and for regulating and organizing all surveying work activities in Sudan. The act has been discussed for more than 5 years. The Land Survey Act was initiated by the Sudan Survey Authority and all official legislations were headed by the Sudan Ministry of Justice till it was issued in 2022. The paper presents conceptual guidelines to be used for the Survey Act implementation and to regulate the survey work practice, standardizing the field surveys, processing, quality control, procedures, and the processes related to survey work carried out by the stakeholders and relevant authorities in Sudan. The conceptual guidelines are meant to improve the quality and harmonization of geospatial data and to aid decision making processes as well as geospatial information systems. The established comprehensive executive regulations will govern and regulate the implementation of the Sudan Survey Geomatics Act in all surveying and mapping practices undertaken by the Sudan Survey Authority SSA and state local survey departments for public or private sector organizations. The targeted standards and specifications include the reference frame, projection, coordinate systems, and the guidelines and specifications that must be followed in the field of survey work, processes, and mapping products. In the last few decades, there has been a growing awareness of the importance of geomatics activities and measurements on the Earths surface in space and time, together with observing and mapping the changes. In such cases, data must be captured promptly, standardized, and obtained with more accuracy and specified in much detail. The paper will also highlight the current situation in Sudan, the degree to which survey standards are used, the problems encountered, and the errors that arise from not using the standards and survey specifications. Kamal A. A. Sami "Concepts for Sudan Survey Act Implementations - Executive Regulations and Standards" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63484.pdf Paper Url: https://www.ijtsrd.com/engineering/civil-engineering/63484/concepts-for-sudan-survey-act-implementations--executive-regulations-and-standards/kamal-a-a-sami
Towards the Implementation of the Sudan Interpolated Geoid Model Khartoum Sta...ijtsrd
The discussions between ellipsoid and geoid have invoked many researchers during the recent decades, especially during the GNSS technology era, which had witnessed a great deal of development but still geoid undulation requires more investigations. To figure out a solution for Sudans local geoid, this research has tried to intake the possibility of determining the geoid model by following two approaches, gravimetric and geometrical geoid model determination, by making use of GNSS leveling benchmarks at Khartoum state. The Benchmarks are well distributed in the study area, in which, the horizontal coordinates and the height above the ellipsoid have been observed by GNSS while orthometric heights were carried out using precise leveling. The Global Geopotential Model GGM represented in EGM2008 has been exploited to figure out the geoid undulation at the benchmarks in the study area. This is followed by a fitting process, that has been done to suit the geoid undulation data which has been computed using GNSS leveling data and geoid undulation inspired by the EGM2008. Two geoid surfaces were created after the fitting process to ensure that they are identical and both of them could be counted for getting the same geoid undulation with an acceptable accuracy. In this respect, statistical operation played an important role in ensuring the consistency and integrity of the model by applying cross validation techniques splitting the data into training and testing datasets for building the geoid model and testing its eligibility. The geometrical solution for geoid undulation computation has been utilized by applying straightforward equations that facilitate the calculation of the geoid undulation directly through applying statistical techniques for the GNSS leveling data of the study area to get the common equation parameters values that could be utilized to calculate geoid undulation of any position in the study area within the claimed accuracy. Both systems were checked and proved eligible to be used within the study area with acceptable accuracy which may contribute to solving the geoid undulation problem in the Khartoum area, and be further generalized to determine the geoid model over the entire country, and this could be considered in the future, for regional and continental geoid model. Ahmed M. A. Mohammed. | Kamal A. A. Sami "Towards the Implementation of the Sudan Interpolated Geoid Model (Khartoum State Case Study)" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63483.pdf Paper Url: https://www.ijtsrd.com/engineering/civil-engineering/63483/towards-the-implementation-of-the-sudan-interpolated-geoid-model-khartoum-state-case-study/ahmed-m-a-mohammed
Activating Geospatial Information for Sudans Sustainable Investment Mapijtsrd
Sudan is witnessing an acceleration in the processes of development and transformation in the performance of government institutions to raise the productivity and investment efficiency of the government sector. The development plans and investment opportunities have focused on achieving national goals in various sectors. This paper aims to illuminate the path to the future and provide geospatial data and information to develop the investment climate and environment for all sized businesses, and to bridge the development gap between the Sudan states. The Sudan Survey Authority SSA is the main advisor to the Sudan Government in conducting surveying, mappings, designing, and developing systems related to geospatial data and information. In recent years, SSA made a strategic partnership with the Ministry of Investment to activate Geospatial Information for Sudans Sustainable Investment and in particular, for the preparation and implementation of the Sudan investment map, based on the directives and objectives of the Ministry of Investment MI in Sudan. This paper comes within the framework of activating the efforts of the Ministry of Investment to develop technical investment services by applying techniques adopted by the Ministry and its strategic partners for advancing investment processes in the country. Kamal A. A. Sami "Activating Geospatial Information for Sudan's Sustainable Investment Map" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63482.pdf Paper Url: https://www.ijtsrd.com/engineering/information-technology/63482/activating-geospatial-information-for-sudans-sustainable-investment-map/kamal-a-a-sami
Educational Unity Embracing Diversity for a Stronger Societyijtsrd
In a rapidly changing global landscape, the importance of education as a unifying force cannot be overstated. This paper explores the crucial role of educational unity in fostering a stronger and more inclusive society through the embrace of diversity. By examining the benefits of diverse learning environments, the paper aims to highlight the positive impact on societal strength. The discussion encompasses various dimensions, from curriculum design to classroom dynamics, and emphasizes the need for educational institutions to become catalysts for unity in diversity. It highlights the need for a paradigm shift in educational policies, curricula, and pedagogical approaches to ensure that they are reflective of the diverse fabric of society. This paper also addresses the challenges associated with implementing inclusive educational practices and offers practical strategies for overcoming barriers. It advocates for collaborative efforts between educational institutions, policymakers, and communities to create a supportive ecosystem that promotes diversity and unity. Mr. Amit Adhikari | Madhumita Teli | Gopal Adhikari "Educational Unity: Embracing Diversity for a Stronger Society" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64525.pdf Paper Url: https://www.ijtsrd.com/humanities-and-the-arts/education/64525/educational-unity-embracing-diversity-for-a-stronger-society/mr-amit-adhikari
Integration of Indian Indigenous Knowledge System in Management Prospects and...ijtsrd
The diversity of indigenous knowledge systems in India is vast and can vary significantly between different communities and regions. Preserving and respecting these knowledge systems is crucial for maintaining cultural heritage, promoting sustainable practices, and fostering cross cultural understanding. In this paper, an overview of the prospects and challenges associated with incorporating Indian indigenous knowledge into management is explored. It is found that IIKS helps in management in many areas like sustainable development, tourism, food security, natural resource management, cultural preservation and innovation, etc. However, IIKS integration with management faces some challenges in the form of a lack of documentation, cultural sensitivity, language barriers legal framework, etc. Savita Lathwal "Integration of Indian Indigenous Knowledge System in Management: Prospects and Challenges" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63500.pdf Paper Url: https://www.ijtsrd.com/management/accounting-and-finance/63500/integration-of-indian-indigenous-knowledge-system-in-management-prospects-and-challenges/savita-lathwal
DeepMask Transforming Face Mask Identification for Better Pandemic Control in...ijtsrd
The COVID 19 pandemic has highlighted the crucial need of preventive measures, with widespread use of face masks being a key method for slowing the viruss spread. This research investigates face mask identification using deep learning as a technological solution to be reducing the risk of coronavirus transmission. The proposed method uses state of the art convolutional neural networks CNNs and transfer learning to automatically recognize persons who are not wearing masks in a variety of circumstances. We discuss how this strategy improves public health and safety by providing an efficient manner of enforcing mask wearing standards. The report also discusses the obstacles, ethical concerns, and prospective applications of face mask detection systems in the ongoing fight against the pandemic. Dilip Kumar Sharma | Aaditya Yadav "DeepMask: Transforming Face Mask Identification for Better Pandemic Control in the COVID-19 Era" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd64522.pdf Paper Url: https://www.ijtsrd.com/engineering/electronics-and-communication-engineering/64522/deepmask-transforming-face-mask-identification-for-better-pandemic-control-in-the-covid19-era/dilip-kumar-sharma
Streamlining Data Collection eCRF Design and Machine Learningijtsrd
Efficient and accurate data collection is paramount in clinical trials, and the design of Electronic Case Report Forms eCRFs plays a pivotal role in streamlining this process. This paper explores the integration of machine learning techniques in the design and implementation of eCRFs to enhance data collection efficiency. We delve into the synergies between eCRF design principles and machine learning algorithms, aiming to optimize data quality, reduce errors, and expedite the overall data collection process. The application of machine learning in eCRF design brings forth innovative approaches to data validation, anomaly detection, and real time adaptability. This paper discusses the benefits, challenges, and future prospects of leveraging machine learning in eCRF design for streamlined and advanced data collection in clinical trials. Dhanalakshmi D | Vijaya Lakshmi Kannareddy "Streamlining Data Collection: eCRF Design and Machine Learning" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-8 | Issue-1 , February 2024, URL: https://www.ijtsrd.com/papers/ijtsrd63515.pdf Paper Url: https://www.ijtsrd.com/biological-science/biotechnology/63515/streamlining-data-collection-ecrf-design-and-machine-learning/dhanalakshmi-d
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2024.06.01 Introducing a competency framework for languag learning materials ...
Crack problems concerning boundaries of convex lens like forms
1. International Journal of Trend in Scientific Research and Development(IJTSRD)
ISSN:2456-6470 —www.ijtsrd.com—Volume -2—Issue-3
CRACK PROBLEMS CONCERNING BOUNDARIES OF CONVEX LENS LIKE
FORMS
DOO-SUNG LEE
Department of Mathematics
College of Education, Konkuk University
120 Neungdong-Ro, Kwangjin-Gu, Seoul, Korea
e-mail address: dslee@kumail.konkuk.ac.kr
Abstract
The singular stress problem of a peripheral edge
crack around a cavity of spherical portion in an
infinite elastic medium when the crack is sub-
jected to a known pressure is investigated. The
problem is solved by using integral transforms
and is reduced to the solution of a singular in-
tegral equation of the first kind. The solution
of this equation is obtained numerically by the
method due to Erdogan, Gupta , and Cook, and
the stress intensity factors are displayed graphi-
cally.
Also investigated in this paper is the penny-
shaped crack situated symmetrically on the cen-
tral plane of a convex lens shaped elastic mate-
rial.
Key words: cavity of spherical portion/ periph-
eral edge crack/penny-shaped crack /SIF.
1.Introduction.
The problem of determining the distribution
of stress in an elastic medium containing a cir-
cumferential edge crack has been investigated by
several researchers including the present author.
Among these investigations, the notable ones
are Keer et al.[1,2],Atsumi and Shindo[3,4], and
Lee[5,6]. Keer et al.[1] considered a circumfer-
ential edge crack in an extended elastic medium
with a cylindrical cavity the analysis of which
provides immediate application to the study of
cracking of pipes and nozzles if the crack is small.
Another important problem involving a cir-
cumferential edge crack is that concerned with a
spherical cavity. Atsumi and Shindo[4] investi-
gated the singular stress problem of a peripheral
edge crack around a spherical cavity under uni-
axial tension field. In more recent years, Wan
et al.[7] obtained the solution for cracks emanat-
ing from surface semi-spherical cavity in finite
body using energy release rate theory. In previ-
ous studies concerning the spherical cavity with
the circumferential edge crack, the cavity was a
full spherical shape. In this present analysis, we
are concerned with a cavity of a spherical por-
tion, rather than a full spherical cavity. More
briefly describing it, the cavity looks like a con-
vex lens. Here, we employ the known methods of
previous investigators to derive a singular inte-
gral equation of the first kind which was solved
numerically, and obtained the s.i.f. for various
spherical portions. It is also shown that when
this spherical portion becomes a full sphere, the
present solution completely agrees with the al-
ready known solution.
2.Formulation of problem and reduction to
singular integral equation. 0
We employ cylindrical coordinates (r, φ, z)
with the plane z = 0 coinciding the plane of pe-
ripheral edge crack. The spherical coordinates
0
@IJTSRD— Available Online@www.ijtsrd.com—Volume-2—Issue-3—Mar-Apr 2018 Page:982
1
2. 2INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT(IJTSRD)ISSN:2456-6470
(ρ, θ, φ) are connected with the cylindrical coor-
dinates by
z = ρ cos θ, r = ρ sin θ.
Spherical coordinates (ζ, ϑ, φ) whose origin is
located at z = −δ, r = 0, and is the center of
the upper spherical surface, are also used. The
cavity is symmetrical with respect to the plane
z = 0.
The crack occupies the region z = 0, 1 ≤
r ≤ γ. So the radius of the spherical cavity is
ζ0 =
√
1 + δ2.
The boundary conditions are:
On the plane z = 0, we want the continuity of
the shear stress, and the normal displacement:
uz(r, 0+
)−uz(r, 0−
) = 0, γ ≤ r < ∞, (2.1)
σrz(r, 0+
) − σrz(r, 0−
) = 0, 1 ≤ r < ∞.
(2.2)
And the crack is subjected to a known pressure
p(r), i.e.,
σzz(r, 0+
) = −p(r), 1 ≤ r ≤ γ. (2.3)
On the surface of the spherical cavity, stresses
are zero:
σζζ(ζ0, ϑ) = 0, (2.4)
σζϑ(ζ0, ϑ) = 0. (2.5)
We can make use of the axially symmetric so-
lution of the equations of elastic equilibrium due
to Green and Zerna [8] which states that if ϕ(r, z)
and ψ(r, z) are axisymmetric solutions of Laplace
equation, then the equations
2µur =
∂ϕ
∂r
+ z
∂ψ
∂r
, (2.6)
2µuz =
∂ϕ
∂z
+ z
∂ψ
∂z
− (3 − 4ν)ψ, (2.7)
where µ is the modulus of rigidity and ν is Pois-
son’s ratio, provide a possible displacement field.
The needed components of stress tensor are given
by the equations
σrz =
∂2φ
∂r∂z
+ z
∂2ψ
∂r∂z
− (1 − 2ν)
∂ψ
∂r
, (2.8)
σzz =
∂2φ
∂z2
+ z
∂2ψ
∂z2
− 2(1 − ν)
∂ψ
∂z
. (2.9)
The functions φ(1) and φ(2) for the regions z > 0
and z < 0, respectively, are chosen as follows:
φ(1)
(r, z) = (2ν − 1)
∞
0
ξ−1
A(ξ)J0(ξr)e−ξz
dξ
+
∞
n=0
an
Pn(cos θ)
ρn+1
, (2.10)
φ(2)
(r, z) = (2ν − 1)
∞
0
ξ−1
A(ξ)J0(ξr)eξz
dξ
−
∞
n=0
an(−1)n Pn(cos θ)
ρn+1
. (2.11)
Here the superscripts (1) and (2) are taken for
the region z > 0 and z < 0, respectively. The
functions ψ(1) and ψ(2) are chosen as follows:
ψ(1)
(r, z) =
∞
0
A(ξ)J0(ξr)e−ξz
dξ
+
∞
n=0
bn
Pn(cos θ)
ρn+1
, (2.12)
ψ(2)
(r, z) = −
∞
0
A(ξ)J0(ξr)eξz
dξ
+
∞
n=0
bn(−1)n Pn(cos θ)
ρn+1
. (2.13)
Then we can immediately satisfy condition (2.2)
by this choice of functions (2.10)-(2.13).
Now the condition (2.1) requires
∞
0
A(ξ)J0(ξr)dξ = 0, r > γ. (2.14)
0
Equation (2.14) is automatically satisfied by
setting
A(ξ) =
γ
1
tg(t)J1(ξt)dt. (2.15)
0@IJTSRD— Available Online@www.ijtsrd.com—Volume-2—Issue-3—Mar-Apr 2018 Page:983
3. INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT(IJTSRD)ISSN:2456-64703
Then from the boundary condition (2.3), we
obtain
∞
0
ξA(ξ)J0(ξr)dξ −
∞
n=0
a2n
(2n + 1)2P2n(0)
r2n+3
−2(1 − ν)
∞
n=0
b2n+1
P2n+1(0)
r2n+3
= −p(r),
1 ≤ r ≤ γ, (2.16)
where prime indicates the differentiation with re-
spect to the argument.
By substituting (2.15) into (2.16), it reduces
to
−
2
π
γ
1
tg(t)R(r, t)dt −
∞
n=0
r−(2n+3)
{P2n(0)
×(2n + 1)2
a2n + αb2n+1P2n+1(0)} = −p(r),
1 ≤ r ≤ γ, (2.17)
where
R(r, t) =
1
r2 − t2
E
r
t
, t > r,
=
r
t
1
r2 − t2
E
t
r
−
1
rt
K
t
r
, r > t.
(2.18)
K and E in (2.18) are complete elliptic integrals
of the first and the second kind, respectively and
α = 2(1 − ν).
The solution will be complete, if the condi-
tions on the surface of the spherical cavity are
satisfied.
3.Conditions on the surface of the spheri-
cal cavity.
Equation (2.17) gives one relation connecting
unknown coefficients an and bn. The stress com-
ponents besides (2.8) and (2.9) which are needed
for the present analysis are given by the following
equations
σζζ =
∂2φ
∂ζ2
+ ζ cos ϑ
∂2ψ
∂ζ2
− 2(1 − ν) cos ϑ
∂ψ
∂ζ
+2ν
sin ϑ
ζ
∂ψ
∂ϑ
, (3.1)
σζϑ =
1
ζ
∂2φ
∂ζ∂ϑ
−
1
ζ2
∂φ
∂ϑ
+ cos ϑ
∂2ψ
∂ζ∂ϑ
+(1 − 2ν) sin ϑ
∂ψ
∂ζ
− 2(1 − ν)
cos ϑ
ζ
∂ψ
∂ϑ
. (3.2)
To satisfy boundary conditions on the spherical
surface, it is needed to represent φ, ψ in (2.10)-
(2.13) in terms of ζ, ϑ variables. To do so we uti-
lize the following formula whose validity is shown
in the Appendix 1. An expression useful for the
present analysis is the following
Pn(cos θ)
ρn+1
=
∞
k=0
n + k
k
Pn+k(cos ϑ)
ζn+k+1
δk
.
(3.3)
Thus
∞
n=0
an
Pn(cos θ)
ρn+1
=
∞
n=0
an
∞
k=0
n + k
k
×
Pn+k(cos ϑ)
ζn+k+1
δk
=
∞
j=0
Pj(cos ϑ)
ζj+1
Aj, (3.4)
where
Aj =
j
n=0
j!
(j − n)!n!
anδj−n
. (3.5)
Also
∞
0
ξ−1
A(ξ)J0(ξr)e−ξz
dξ =
γ
1
tg(t)
×
∞
0
ξ−1
J0(ξr)J1(ξt)e−ξz
dξdt. (3.6)
If we make use of the formula in Whittaker and
Watson[9,pp.395-396]
π
−π
exp{−ξ(z+ix cos u+iy sin u)}du = 2πe−ξz
J0(ξr),
0
to the inner integral of (3.6), it can be written
as, if we are using the shortened notation
β = z + ix cos u + iy sin u,
then
∞
0
ξ−1
J0(ξr)J1(ξt)e−ξz
dξ
=
1
2π
π
−π
∞
0
ξ−1
J1(ξt)e−ξβ
dξdu
0@IJTSRD— Available Online@www.ijtsrd.com—Volume-2—Issue-3—Mar-Apr 2018 Page:984
4. 4INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT(IJTSRD)ISSN:2456-6470
=
1
2π
π
−π
t
β + β2 + t2
du
= −
1
2π
π
−π
β
t
− 1 +
β2
t2
du
= −
1
2π
π
−π
β
t
−
∞
n=0
(−1
2 )n(−1)n
n!
β
t
2n
du.
(3.7)
As |β| < |t|, the above series expansion is valid.
Next,
β2n
= (−δ+Z+ix cos u+iy sin u)2n
= (−δ+β )2n
=
2n
k=0
2n
k
(−δ)2n−k
(β )k
, (3.8)
where (r, φ, Z) is the cylindrical coordinates sys-
tem centered at (r, z) = (0, −δ). We have
the following formula from Whittaker and Wat-
son[9,p.392]
π
−π
(β )n
du =
π
−π
(Z + ix cos u + iy sin u)n
du
= 2πζn
Pn(cos ϑ). (3.9)
If we utilize (3.8) and (3.9) in (3.7), it can be
written as
∞
0
ξ−1
J0(ξr)J1(ξt)e−ξz
dξ
=
∞
n=0
(−1
2 )n(−1)n
n!
1
t2n
×
2n
k=0
2n
k
(−δ)2n−k
ζk
Pn(cos ϑ)
+δ −
ζ
t
P1(cos ϑ). (3.10)
Thus finally, from (3.4), (3.6) and (3.10), φ(1)
can be written in terms of spherical coordinates
(ζ, ϑ) as
φ(1)
=
∞
k=0
Pk(cos ϑ)
Ak
ζk+1
+ Φkζk
+δ
γ
1
tg(t)dt − ζP1(cos ϑ)
γ
1
g(t)dt,
where
Φk = −(α − 1)
∞
n=[(k+1)/2]
(−1
2 )n(−1)n
n!
×
(2n)!
(2n − k)!k!
(−δ)2n−k
γ
1
g(t)
t2n−1
dt, (3.11)
and [(k+1)/2] is the greatest integer ≤ (k+1)/2.
It is also necessary to express ψ(1) in terms of
spherical coordinates (ζ, ϑ). Now, as in (3.4)
∞
n=0
bn
Pn(cos θ)
ρn+1
=
∞
j=0
Pj(cos ϑ)
ζj+1
Bj, (3.12)
where
Bj =
∞
n=0
j!
(j − n)!n!
bnδj−n
.
We first express following integral in (ρ, θ) co-
ordinates
∞
0
A(ξ)J0(ξr)e−ξz
dξ =
γ
1
tg(t)dt
×
∞
0
J1(ξt)J0(ξr)e−ξz
dξ. (3.13)
The inner integral on the right-hand side of
(3.13) is
−
1
t
∞
0
J0(ξr)e−ξz ∂
∂ξ
J0(ξt)dξ =
1
t
−
1
t
∞
0
{rJ1(ξr) + zJ0(ξr)}e−ξz
J0(ξt)dξ.
(3.14)
Using the equation in Erd´elyi et al.[10] 0
J0(ξt) =
2
π
∞
t
sin(ξx)dx
√
x2 − t2
,
equation (3.14) is equal to
1
t
+
1
t
2
π
∞
t
dx
√
x2 − t2
×
∞
0
{rJ1(ξr) + zJ0(ξr)}e−ξ(z+ix)
dξ
=
1
t
+
1
t
2
π
∞
t
1
√
x2 − t2
−ix
r2 + (z + ix)2
dx
=
1
t
−
1
t
2
π
∞
t
1
√
x2 − t2
dx
1 − 2ρ
xi cos θ + (ρ
x i)2
=
1
t
−
1
t
∞
n=0
ρ
t
2n+1
(−1)n
P2n+1(cos θ)
(1
2 )n
n!
,
(3.15)
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5. INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT(IJTSRD)ISSN:2456-64705
where we used the generating function
1
√
1 − 2x cos θ + x2
=
∞
n=0
Pn(cos θ)xn
.
To express (3.15) in the (ζ, ϑ) spherical coordi-
nates, we use the following formula whose valid-
ity is shown in the Appendix 2.
ρ2n+1
P2n+1(cos θ) =
2n+1
k=0
(2n + 1)!
k!(2n + 1 − k)!
×(−δ)2n+1−k
ζk
Pk(cos ϑ). (3.16)
0
If we use (3.16) in (3.15), then with (3.12), we
get ψ(1) in spherical coordinates (ζ, ϑ)
ψ(1)
=
∞
k=0
Pk(cos ϑ)
Bk
ζk+1
+Ψkζk
+
γ
1
g(t)dt,
(3.17)
where
Ψk = −
∞
n=[k/2]
(−1)n(1
2 )n
n!
(2n + 1)!
k!(2n + 1 − k)!
×(−δ)2n+1−k
γ
1
g(t)
t2n+1
dt.
Then if we substitute these values of φ(1) and
ψ(1) given by (3.11) and (3.17) into (3.2), and
simplify the results by using the properties of
Legendre polynomials, from the condition that
on the spherical surface ζ = ζ0, the shear stress
σζϑ = 0, we obtain the following equation
−Ak+1
k + 3
ζk+4
0
+
Bk
ζk+2
0
α − (k + 1)2
2k + 1
−
Bk+2
ζk+4
0
(k + 3)(2α + k + 2)
2k + 5
+kζk−1
0 Φk+1 + ζk−1
0 Ψk
k(k + 1 − 2α)
2k + 1
−ζk+1
0 Ψk+2
k + 2 + α − (k + 2)(k + 3)
2k + 5
= 0.
(3.18)
Likewise, from the condition σζζ(ζ0, ϑ) = 0,
we get following two equations,
−A0
2
ζ3
0
−
B1
ζ3
0
2(2α + 1)
3
−
1
3
Ψ1(α − 4) = 0,
(3.19a)
−Ak+1
(k + 2)(k + 3)
ζk+4
0
+
Bk
ζk+2
0
(k + 1){2 − α − (k + 1)(k + 4)}
2k + 1
−
Bk+2
ζk+4
0
(k + 2)(k + 3)(2α + k + 2)
2k + 5
−k(k+1)ζk−1
0 Φk+1−ζk−1
0 Ψk
k(k + 1)(k + 1 − 2α)
2k + 1
−ζk+1
0 Ψk+2
(k + 2){(k + 2)(k − 1) + α − 2}
2k + 5
= 0.
(3.19b)
Thus if we multiply (3.18) by k + 2 and subtract
(3.19b) from the resulting equation we find
Bk = −
ζk+2
0 (2k + 1)
α(2k + 3) + 2k(k + 1)
k(2k+3)ζk−1
0 Φk+1
+ζk−1
0 Ψk
(2k + 3)k(k + 1 − 2α)
2k + 1
+ζk+1
0 Ψk+2k(k+2) .
(3.20)
If we solve (3.20) for bi using the theorem
which is in the Appendix 3 and the relation,
Φk+1 =
α − 1
k + 1
Ψk,
we find
bi
δi
=
i
k=0
i!(−1)i−k
(i − k)!k!
1
δk
[N1(k)Ψk + N2(k)Ψk+2],
(3.21)
where
N1(k) = −
ζ2k+1
0 (2k + 3)k
α(2k + 3) + 2k(k + 1)
−α + k2
k + 1
,
N2(k) = −
ζ2k+3
0 (2k + 1)k(k + 2)
α(2k + 3) + 2k(k + 1)
.
Equation (3.21) can be written as
bi = −
i
k=0
i!(−1)i−k
(i − k)!k!
δi
δk
γ
1
g(t)
t
×
1
n=0
[Nn+1(k)fk+2n
1
2
,
δ
t
dt, (3.22)
0@IJTSRD— Available Online@www.ijtsrd.com—Volume-2—Issue-3—Mar-Apr 2018 Page:986
6. 6INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT(IJTSRD)ISSN:2456-6470
where
fk(c, x) =
(−δ)1−k
k!
∞
=[k/2]
(−1) (c) (2 + 1)!x2
(2 + 1 − k)! !
with (c) = c(c + 1)(c + 2) · · · (c + − 1).
Now from (3.19a,b) we have
Ak+1 = M1(k)Ψk + M2(k)Ψk+2 + M3(k)Ψk+4,
A0 = ˜A1Ψ1 + ˜A2Ψ3,
where 0
M1(k) = −
ζ2k+3
0 k(−α + k2)
(k + 2)(k + 3)(2k + 1)
×
(2k + 3)˜g(k)
h(k)
+ 1 ,
M2(k) =
ζ2k+5
0
k + 3
−
˜g(k)
h(k)
k(k+1)+
˜g(k − 2)
2k + 5
+
(k + 2)(2k + 7)(2α + k + 2){−α + (k + 2)2}
(2k + 5)h(k + 2)
,
M3(k) =
ζ2k+7
0 (k + 2)(k + 4)(2α + k + 2)
h(k + 2)
,
˜A1 = −
ζ3
0
6
α − 4 −
5(2α + 1)(−α + 1)
h(1)
,
˜A2 =
3(2α + 1)ζ5
0
h(1)
,
with
˜g(k) = 2 − α − (k + 1)(k + 4),
h(k) = α(2k + 3) + 2k(k + 1).
In a similar way ais can be found from these
equations as follows:
a2i = ( ˜A1Ψ1 + ˜A2Ψ3)δ2i
+
2i−1
k=0
(2i)!(−δ)2i−k−1
(2i − 1 − k)!(k + 1)!
˜F(k),
where
˜F(k) = −
γ
1
g(t)
t
M1(k)fk
1
2
,
δ
t
+ M2(k)
×fk+2
1
2
,
δ
t
+ M3(k)fk+4
1
2
,
δ
t
dt.
Thus
∞
i=0
a2i
(2i + 1)2P2i(0)
r2i+3
=
a0
r3
+
∞
i=1
( ˜A1Ψ1 + ˜A2Ψ3)
×δ2i (2i + 1)2(−1)i(1
2)i
r2i+3i!
+
∞
i=1
(2i + 1)2(−1)i(1
2 )i
r2i+3i!
×
2i−1
k=0
(2i)!(−δ)2i−k−1
(2i − 1 − k)!(k + 1)!
˜F(k).
If we interchange the order of summation in the
last term of the above equation, and use
Ψ1 = −
γ
1
g(t)
t
f1
1
2
,
δ
t
dt,
Ψ3 = −
γ
1
g(t)
t
f3
1
2
,
δ
t
dt,
we finally obtain
∞
i=0
a2i
(2i + 1)2P2i(0)
r2i+3
= −
γ
1
tg(t)T1(r, t)dt,
where
T1(r, t) =
1
r3t2
×
∞
k=0
2
m=0
Mm+1(k)fk+2m
1
2
,
δ
t
hk
δ
r
+ ˜A1f1
1
2
,
δ
t
+ ˜A2f3
1
2
,
δ
t
1 + j
δ
r
,
with
hk
δ
r
=
(−δ)−k−1
(k + 1)!
×
∞
i=[k/2+1]
(2i + 1)2(−1)i(1
2 )i(2i)!
(2i − 1 − k)!i!
δ
r
2i
,
j
δ
r
=
∞
i=1
(2i + 1)2(−1)i(1
2 )i
i!
δ
r
2i
.
Also,
∞
i=0
b2i+1
P2i+1(0)
r2i+3
= −
∞
i=0
(−1)i(3
2 )i
i!r2i+3
×
2i+1
k=0
(2i + 1)!(−δ)2i−k+1
(2i + 1 − k)!k!
0@IJTSRD— Available Online@www.ijtsrd.com—Volume-2—Issue-3—Mar-Apr 2018 Page:987
8. 8INTERNATIONAL JOURNAL OF TREND IN SCIENTIFIC RESEARCH AND DEVELOPMENT(IJTSRD)ISSN:2456-6470
×2k
(1
2 )k
k!
2
1
(rt)2k−2
+
α
2H(1)
−5F(1) +
9
t2
.
Using
˜A1f1
1
2
, 0 + ˜A2f3
1
2
, 0 = ˜A1 − ˜A2
1
2t2
,
we finally obtain
−
2
π
S(r, t) =
1 + ν
3
1
r3t2
+
∞
k=1
2
H(k)
2(2k + 1)2
(k + 1)r2
× k(k + 1)F(k) −
1
t2
1
4(4k + 3)
{(2k − 1)(4k + 3)
×2kG(k) − H(k)G(k − 1)} + kF(k) (2k + 2)
×(2k − 1)
1
t2
+ F(k)
4k + 1
4k − 1
×
(2k − 1)!!
(2k)!!
2
1
(rt)2kr
.
The above equation completely agrees with the
result by Atsumi and Shindo.
A quantity of physical interest is the stress in-
tensity factor which is given as
K = lim
r→γ+
2(r − γ)σzz(r, 0).
We choose p(r) = p0 and put γ − 1 = a. We let
r =
a
2
(s + 1) + 1, t =
a
2
(τ + 1) + 1, (3.24)
and in order to facilitate numerical analysis, as-
sume g(t) to have the following form:
g(t) = p0(t − 1)
1
2 (γ − t)−1
2 ˜G(t). (3.25)
With the aid of (3.24), g(τ) can be rewritten as
g(τ) = p0
˜G(τ)
1 + τ
1 − τ
1
2
. (3.26)
The stress intensity factors K can therefore be
expressed in terms of ˜G(t) as
K/p0 =
√
2a ˜G(γ), (3.27)
or in terms of the quantity actually calculated
K/p0
√
a =
√
2 ˜G(γ). (3.28)
4.Numerical analysis.
In order to obtain numerical solution of (3.23),
substitutions are made by the application of
(3.24) and (3.26) to obtain the following expres-
sion:
a
π
1
−1
1 + τ
1 − τ
1
2
˜G(τ)
a
2
(τ + 1) + 1 [R(s, τ)
+S(s, τ)]dτ = 1, −1 < s < 1. (4.1)
The numerical solution technique is based on
the collocation scheme for the solution of singu-
lar integral equations given by Erdogan, Gupta,
and Cook [11]. 0
This amounts to applying a
Gaussian quadrature formula for approximating
the integral of a function f(τ) with weight func-
tion [(1+τ)/(1−τ)]
1
2 on the interval [-1,1]. Thus,
letting n be the number of quadrature points,
1
−1
1 + τ
1 − τ
1
2
f(τ)dτ
2π
2n + 1
n
k=1
(1 + τk)f(τk),
(4.2)
where
τk = cos
2k − 1
2n + 1
π, k = 1, . . . n. (4.3)
The solution of the integral equation is ob-
tained by choosing the collocation points:
si = cos
2iπ
2n + 1
, i = 1, . . . , n, (4.4)
and solving the matrix system for G∗(τk) :
n
k=1
[R(sj, τk) + S(sj, τk)]G∗
(τk) =
2n + 1
2a
,
j = 1, . . . , n, (4.5)
where
˜G(τk) =
G∗(τk)
(1 + τk)[a
2 (τk + 1) + 1]
. (4.6)
5.Numerical results and consideration.
Numerical calculations have been carried out
for ν = 0.3. The values of normalized stress
intensity factor K/p0
√
a versus a are shown in
Fig.1-3 for various values of δ.
Fig.1 shows the variation of K/p0
√
a with re-
spect to a when δ = 0.
This figure shows that as a increases, S.I.F.
decreases steadily.
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Fig.2 and 3 deal with the cases when δ = 0.3
and δ = 0.5, respectively. Here we omit units.
We can see that the trend is similar. Theoreti-
cally the infinite series involved converges when
δ < 1 by comparison test. However, because of
the overflow, computations could not be accom-
plished beyond the values δ > 0.5. And we found
that the variation of SIF is very small with re-
spect to the variation of δ.
0
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6.Penny-shaped crack.
In this section we are concerned with a penny-
shaped crack in a convex lens shaped elastic ma-
terial. The problem of determining the distri-
bution of stress in an elastic sphere containing
a penny-shaped crack or the mixed boundary
value problems concerning a spherical boundary
has been investigated by several researchers. Sri-
vastava and Dwivedi [12] considered the prob-
lem of a penny-shaped crack in an elastic sphere,
whereas Dhaliwal et al.[13] solved the problem of
a penny-shaped crack in a sphere embedded in an
infinite medium. On the other hand, Srivastav
and Narain [14] investigated the mixed boundary
value problem of torsion of a hemisphere.
7.Formulation of problem and reduction to
a Fredholm integral equation of the sec-
ond kind. 0
We employ cylindrical coordinates
(r, φ, z) with the plane z = 0 coinciding the plane
of the penny shaped crack. The center of the
crack is located at (r,z)=(0,0). As before, spher-
ical coordinates (ρ, θ, φ) are connected with the
cylindrical coordinates by
z = ρ cos θ, r = ρ sin θ.
Spherical coordinates (ζ, ϑ, φ) whose origin is at
z = −δ, r = 0, and is the center of the upper
spherical surface of the convex elastic body, is
also used. The elastic body is symmetrical with
respect to the plane z = 0.
The crack occupies the region z = 0, 0 ≤
r ≤ 1. The radius of the spherical portion is
ζ0 = γ2 + δ2. The boundary conditions are:
On the plane z = 0, we want the continuity of
the shear stress, and the normal displacement:
uz(r, 0+
) − uz(r, 0−
) = 0, 1 ≤ r ≤ γ, (7.1)
σrz(r, 0+
)−σrz(r, 0−
) = 0, 0 ≤ r ≤ γ. (7.2)
And the crack is subjected to a known pressure
p(r), i.e.,
σzz(r, 0+
) = −p(r), 0 ≤ r ≤ 1. (7.3)
On the surface of the spherical portion, stresses
are zero:
σζζ(ζ0, ϑ) = 0, (7.4)
σζϑ(ζ0, ϑ) = 0. (7.5)
We can make use of (2.6)-(2.9) also, for the
present case. The functions φ(1) and φ(2) for the
regions z > 0 and z < 0, respectively, are chosen
as follows:
φ(1)
(r, z) = (2ν − 1)
∞
0
ξ−1
A(ξ)J0(ξr)e−ξz
dξ
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+
∞
n=0
anρn
Pn(cos θ), (7.6)
φ(2)
(r, z) = (2ν − 1)
∞
0
ξ−1
A(ξ)J0(ξr)eξz
dξ
−
∞
n=0
an(−1)n
ρn
Pn(cos θ). (7.7)
Here the superscripts (1) and (2) are taken for
the region z > 0 and z < 0, respectively. The
functions ψ(1) and ψ(2) are chosen as follows:
ψ(1)
(r, z) =
∞
0
A(ξ)J0(ξr)e−ξz
dξ
+
∞
n=0
bnρn
Pn(cos θ), (7.8)
ψ(2)
(r, z) = −
∞
0
A(ξ)J0(ξr)eξz
dξ
+
∞
n=0
bn(−1)n
ρn
Pn(cos θ). (7.9)
Then we can immediately satisfy condition (7.2)
by these choice of functions (7.6)-(7.9).
Now the condition (7.1) requires
∞
0
A(ξ)J0(ξr)dξ = 0, r > 1. (7.10)
Equation (7.10) is automatically satisfied by
setting
A(ξ) =
1
0
g(t) sin(ξt)dt, g(0) = 0. (7.11)
Then from the boundary condition (7.3), we ob-
tain
∞
0
ξA(ξ)J0(ξr)dξ −
∞
n=0
a2n(2n)2
P2n(0)r2n−2
−2(1 − ν)
∞
n=0
b2n+1P2n+1(0)r2n
= −p(r),
0 ≤ r ≤ 1, (7.12)
where the prime indicates the differentiation
with respect to the argument. 0
By substituting (7.11) into (7.12), it reduces
to
g(t) −
2
π
∞
n=0
(−1)n
{2nt2n−1
a2n
+αb2n+1t2n+1
} = h(t), 0 ≤ r ≤ 1, (7.13)
where
h(t) = −
2
π
t
0
rp(r)
√
t2 − r2
dr.
The solution will be complete, if the condi-
tions on the surface of the spherical portion are
satisfied.
8.Conditions on the surface of the sphere.
Equation (7.13) gives one relation connecting
unknown coefficients an and bn. The stress com-
ponents besides (2.8) and (2.9) which are needed
for the present analysis are given by (3.1) and
(3.2). To satisfy boundary conditions on the
spherical surface, it is needed to represent φ, ψ
in (7.6)-(7.9) in terms of ζ, ϑ variables. To do so
we utilize the following formula in the Appendix
2. An expression useful for the present analysis
is the following
Pn(cos θ)ρn
=
n
k=0
n
k
Pk(cos ϑ)ζk
(−δ)n−k
.
(8.1)
Thus
∞
n=0
anPn(cos θ)ρn
=
∞
n=0
an
∞
k=0
n
k
Pk(cos ϑ)
×ζk
(−δ)n−k
=
∞
j=0
Pj(cos ϑ)ζj
Aj, (8.2)
where
Aj =
∞
n=j
n!
(n − j)!j!
an(−δ)n−j
. (8.3)
Also
−
∂
∂t
∞
0
ξ−1
A(ξ)J0(ξr)e−ξz
dξ
= −
1
0
g(t)
∞
0
J0(ξr) cos(ξt)e−ξz
dξdt. (8.4)
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The inner integral of the above equation is
∞
0
J0(ξr) cos(ξt)e−ξz
dξ
=
∞
0
J0(ξr)e−ξ(z+it)
dξ
=
1
r2 + (z + it)2
=
1
√
r2 + z2 + 2zit − t2
=
1
ρ 1 − 2 cos θ(−it
ρ ) + (−it
ρ )2
=
1
ρ
∞
n=0
t
ρ
2n
(−1)n
P2n(cos θ). (8.5)
Thus finally, from (8.2), and using the for-
mula in Appendix, φ(1) can be written in terms
of spherical coordinates (ζ, ϑ) as
φ(1)
=
∞
k=0
Pk(cos ϑ)
Φk
ζk+1
+ Akζk
, (8.6)
where
Φk = −(α − 1)
[k/2]
n=0
k!
(2n)!(k − 2n)!
δk−2n
×(−1)n
1
0
g(t)t2n+1
2n + 1
dt. (8.7)
It is also necessary to express ψ(1) in terms of
spherical coordinates (ζ, ϑ). Now, as in (8.2)
∞
n=0
bnPn(cos θ)ρn
=
∞
j=0
Pj(cos ϑ)ζj
Bj, (8.8)
where
Bj =
∞
n=j
n!
j!(n − j)!
bn(−δ)n−j
. (8.9)
0
We first express following integral in (ρ, θ) co-
ordinates
∞
0
A(ξ)J0(ξr)e−ξz
dξ =
1
0
g(t)dt
×
∞
0
sin(ξt)J0(ξr)e−ξz
dξ. (8.10)
The inner integral on the right-hand side of
(8.10) is
−
∞
0
J0(ξr)e−ξ(z+it)
dξ
=
∞
n=0
P2n+1(cos θ)
ρ2n+2
(−1)n
t2n+1
. (8.11)
To express (8.11) in the (ζ, ϑ) spherical co-
ordinates, we use the following formula in the
Appendix 1.
P2n+1(cos θ)
ρ2n+1
=
∞
k=0
(2n + 1 + k)!
k!(2n + 1)!
×δk P2n+1+k(cos ϑ)
ζ2n+2+k
. (8.12)
If we use (8.8) and (8.12), we get ψ(1) in spherical
coordinates (ζ, ϑ) as
ψ(1)
=
∞
k=0
Pk(cos ϑ)
Ψk
ζk+1
+ Bkζk
+ B0,
(8.13)
where
Ψk =
[(k−1)/2]
n=0
k!
(2n + 1)!(k − 2n − 1)!
δk−2n−1
×(−1)n
1
0
g(t)t2n+1
dt. (8.14)
Then if we substitute these values of φ(1) and
ψ(1) given by (8.6) and (8.13) into (3.2), and
simplify the results by using the properties of
Legendre polynomials, from the condition that
on the spherical surface ζ = ζ0, σζϑ = 0, we
obtain the following equation
Ak+1kζk−1
0 − Bk+2ζk+1
0
α − (k + 2)2
2k + 5
−Bkζk−1
0
k(2α − k − 1)
2k + 1
−
k + 3
ζk+4
0
Φk+1
−
Ψk+2
ζk+4
0
(k + 3)(k + 2 + 2α)
2k + 5
+
Ψk
ζk+2
0
α − (k + 1)2
2k + 1
= 0. (8.15)
Likewise, from the condition σζζ(ζ0, ϑ) = 0,
we get the following equation,
−Ak+1k(k + 1)ζk−1
0
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+Bk+2ζk+1
0
(k + 2){2 − α − (k − 1)(k + 2)}
2k + 5
+Bkζk−1
0
k(k + 1)(2α − k − 1)
(2k + 1)
−
(k + 2)(k + 3)
ζk+4
0
Φk+1
−Ψk+2
(k + 2)(k + 3)(k + 2 + 2α)
ζk+4
0 (2k + 5)
−Ψk
(k + 1){(k + 4)(k + 1) + α − 2}
ζk+2
0 (2k + 1)
= 0. (8.16)
Thus if we multiply (8.15) by k + 1 and add
(8.16), from the resulting equation we find
Bk+2 =
(2k + 5)
ζk+1
0 {−α(2k + 3) + 2(k + 2)(k + 3)}
×
(k + 3)(2k + 3)
ζk+4
0
Φk+1
+Ψk+2
(2k + 3)(k + 3)(k + 2 + 2α)
ζk+4
0 (2k + 5)
+
Ψk
ζk+2
0
(k + 1)(k + 3) . (8.17)
0
Using the relation
Φk+1 = −
α − 1
k + 2
Ψk+2,
Bk+2 is
Bk+2 =
(2k + 5)
ζk+1
0 {−α(2k + 3) + 2(k + 2)(k + 3)}
× Ψk+2
(2k + 3)(k + 3)((k + 3)2 − α)
ζk+4
0 (2k + 5)(k + 2)
+
Ψk
ζk+2
0
(k + 1)(k + 3) . (8.18)
From (8.15) we have
Ak+3 =
Ψk
ζ2k+3
0
L(k) +
Ψk+2
ζ2k+5
0
M(k) +
Ψk+4
ζ2k+7
0
N(k),
where
L(k) = (k + 1)(k + 3)
(2α − k − 3)
H(k)
,
M(k) =
1
k + 2
(k + 3)
(2k + 3)(2α − k − 3)I(k)
(2k + 5)H(k)
+(k + 4)(k + 5)
G(k)
H(k + 2)
+
G(k + 2)
2k + 5
,
N(k) =
G(k)
H(k + 2)
(2k + 7) − 1
×
(k + 5)I(k + 2)
(2k + 9)(k + 2)(k + 3)
,
with
H(k) = −α(2k + 3) + 2(k + 2)(k + 3),
G(k) = 2−α−(k+1)(k+4), I(k) = −α+(k+3)2
.
Therefore using the formula in Theorem B of Ap-
pendix 4, we have
A =
∞
n=1
(−1)n
2nt2n−1
a2n
=
∞
n=1
(−1)n
2nt2n−1 1
δ2n(2n)!
∞
k=2n
k!Akδk
(k − 2n)!
=
∞
k=2
[k/2]
n=1
k!(−1)n
(2n − 1)!(k − 2n)!
t
δ
2n
1
t
×Akδk
=
∞
k=−1
fk(t)Ak+3 =
∞
k=−1
fk(t)
×
Ψk
ζ2k+3
0
L(k) +
Ψk+2
ζ2k+5
0
M(k) +
Ψk+4
ζ2k+7
0
N(k) .
(8.19)
In the above equation Ψk = 0, if k ≤ 0 and
fk(t) = δk+3
[(k+3)/2]
n=1
(k + 3)!(−1)n
(2n − 1)!(k + 3 − 2n)!
×
t
δ
2n
1
t
.
If we substitute the values of Ψk into (8.19),it
reduces to
A =
1
0
g(u)Ω1(t, u)du,
where
Ω1(t, u) =
∞
k=−1
fk(t)
hk(u)
ζ2k+3
0
L(k)
+
hk+2(u)
ζ2k+5
0
M(k) +
hk+4(u)
ζ2k+7
0
N(k) .
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In the above equation hk(u) = 0, if k ≤ 0 and
hk(u) = δk
[(k−1)/2]
n=0
(−1)nk!
(2n + 1)!(k − 2n − 1)!
×
u
δ
2n+1
.
Also, using Theorem B of Appendix 4, we get
∞
n=0
(−1)n
b2n+1t2n+1
=
∞
n=0
(−1)n t2n+1
(2n + 1)!δ2n+1
×
∞
k=2n+1
k!Bkδk
(k − 2n − 1)!
=
∞
k=1
[ k−1
2
]
n=0
(−1)nk!
(2n + 1)!(k − 2n − 1)!
t
δ
2n+1
Bkδk
=
∞
k=−1
hk+2(t)Bk+2
=
∞
k=−1
hk+2(t)
2k + 5
ζk+1
0 H(k)
Ψk
ζk+2
0
(k + 1)(k + 3)
+
Ψk+2
ζk+4
0
(k + 3)(2k + 3)
2k + 5
I(k)
=
1
0
g(u)Ω2(t, u)du,
0
where
Ω2(t, u) =
∞
k=−1
hk+2(t)
2k + 5
ζk+1
0 H(k)
hk(u)
ζk+2
0
(k + 1)
×(k + 3) +
hk+2(u)
ζk+4
0
(k + 3)(2k + 3)
2k + 5
I(k) .
In the above equation hk(u) = 0, if k ≤ 0. Thus
(7.13) reduces to the following Fredholm integral
equation of the second kind
g(t) +
1
0
g(u)K(t, u)du = h(t),
where
K(t, u) = −
2
π
{Ω1(t, u) + αΩ2(t, u)}.
Appendix 1. Proof of (3.3). Since
ρ2
= (ζ cos ϑ − δ)2
+ (ζ sin ϑ)2
= ζ2
1 − 2 cos ϑ
δ
ζ
+
δ
ζ
2
,
1
ρ
=
1
ζ 1 − 2 cos ϑδ
ζ + (δ
ζ )2
=
∞
n=0
Pn(cos ϑ)
δn
ζn+1
,
Pn(cos θ)
ρn+1
=
(−1)n
n!
∂n
∂zn
1
ρ
=
∞
k=0
(−1)n
n!
∂n
∂zn
Pk(cos ϑ)
ζk+1
δk
=
∞
k=0
(−1)n
n!
∂n+k
∂zn+k
1
ζ
(−1)k
k!
δk
=
∞
k=0
n + k
k
Pn+k(cos ϑ)
ζn+k+1
δk
.
Appendix 2. Proof of (3.16).
2πρn
Pn(cos θ) =
π
−π
(z + ix cos u + iy sin u)n
du
=
π
−π
(−δ + Z + ix cos u + iy sin u)n
du
=
n
k=0
n!
(n − k)!k!
(−δ)n−k
×
π
−π
(Z + ix cos u + iy sin u)k
du
= 2π
n
k=0
n!
(n − k)!k!
(−δ)n−k
ζk
Pk(ϑ).
Appendix 3. Theorem A. If the equation
Bk =
k
i=0
k!
(k − i)!i!
ai,
is solved for ais, it will be written as
ai =
i
k=0
i!
(i − k)!k!
Bk(−1)i−k
.
Proof. We prove it by mathematical induction.
Suppose it is true for i, we will show that it is
also true for i + 1. Suppose Bi+1 is given by
Bi+1 =
i+1
k=0
(i + 1)!
(i + 1 − k)!k!
ak = ai+1
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+
i
k=0
(i + 1)!
(i + 1 − k)!k!
ak.
Thus
ai+1 = Bi+1 −
i
k=0
(i + 1)!
(i + 1 − k)!k!
×
k
j=0
Bj
k!
j!(k − j)!
(−1)k−j
= Bi+1 −
i
j=0
Bj
(i + 1)!
j!
i
k=j
(−1)k−j
(i + 1 − k)!(k − j)!
.
(A.1)
The inner summation in the above equation can
written as, by changing the variable k − j = m
i
k=j
(−1)k−j
(i + 1 − k)!(k − j)!
=
i−j
m=0
(−1)m
(i + 1 − m − j)!m!
= −
(−1)i−j+1
(i − j + 1)!
,
0
since
i−j+1
m=0
(−1)m(i − j + 1)!
(i + 1 − m − j)!m!
= (1 − 1)i−j+1
= 0.
Then (A.1) is equal to
ai+1 = Bi+1 +
i
j=0
Bi(i + 1)!(−1)i+1−j
j!(i + 1 − j)!
=
i+1
j=0
Bi(i + 1)!(−1)i+1−j
j!(i + 1 − j)!
.
Appendix 4. Theorem B. If the equation
Bk =
∞
n=k
n
k
(−δ)n−k
an,
is solved for ans, it will be written as
an =
∞
k=n
k
n
δk−n
Bk.
Proof. Let
∞
n=0
an(−δ)n
xn
= f(x),
then
an(−δ)n
n! = f(n)
(0),
and
f(k)
(1) =
∞
n=k
an(−δ)n n!
(n − k)!
= k!Bk(−δ)k
.
From Taylor’s series
f(x) =
∞
k=0
f(k)(1)
k!
(x − 1)k
.
Thus
an(−δ)n
n! = f(n)
(0) =
∞
k=0
f(k)(1)
k!
dn
dxn
(x−1)k
x=0
=
∞
k=n
f(k)(1)
(k − n)!
(−1)k−n
=
∞
k=n
Bk(−1)k−n(−δ)kk!
(k − n)!
.
Therefore
an =
∞
k=n
k!
n!(k − n)!
Bkδk−n
.
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