In this work we study the configuration of two perfectly conducting spherical shells. This is a problem of basic importance to make possible development of experimental apparatuses that they make possible to measure the spherical Casimir effect, an open subject. We apply the mode sum method via cutoff exponential function regularization with two independent parameters: one to regularize the infinite order sum of the Bessel functions; other, to regularize the integral that becomes related, due to the argument theorem, with the infinite zero sum of the Bessel functions. We obtain a general expression of the Casimir energy as a quadrature sum. We investigate two immediate limit cases as a consistency test of the expression obtained: that of a spherical shell and that of two parallel plates. In the approximation of a thin spherical shell we obtain an expression that allows to relate our result with that of the proximity-force approximation, supplying a correction to this result.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Alternative and Explicit Derivation of the Lattice Boltzmann Equation for the...ijceronline
A lattice Boltzmann equation for fully incompressible flows is derived through the utilization of appropriate ansatzes. The result is a singular equilibrium distribution function which clarifies the algorithm for general implementation, and ensures correct steady and unsteady behavior. Through the Chapman-Enskog expansion, the exact incompressible Navier-Stokes equations are recovered. With 2D and 3D canonical numerical simulations, the application, accuracy, and workable boundary conditions are shown. Several unique benefits over the standard equation and alternative forms presented in literature are found, including faster convergence rate and greater stability.
CR-Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold with Semi-Symmet...iosrjce
We consider a nearly hyperbolic cosymplectic manifold and we study some properties of CRsubmanifolds
of a nearly cosymplectic manifold with a semi-symmetric semi-metric connection. We also obtain
some results on 휉−horizontal and 휉 −vertical CR- submanifolds of a nearly cosymplectic manifold with a semisymmetric
semi-metric connection and study parallel distributions on nearly hyperbolic cosymplectic manifold
with a semi-symmetric semi-metric connection.
EVALUATING STRUCTURAL, OPTICAL & ELECTRICAL CHARACTERIZATION OF ZINC CHALCOGE...Editor IJCATR
To evaluate the structural, optical & electrical properties of the zinc chalcogenides (ZnO, ZnS, ZnSe & ZnTe), the Full
Potential Linearized – Augumented Plane Wave plus Local Orbits (FP – LAPW+lo) method. For the purpose of exchange-correlation
energy (Exc) determination in Kohn–Sham calculation, the standard local density approximation (LDA) formalism has been utilized.
Murnaghan’s equation of state (EOS) has been used for volume optimization by minimizing the total energy with respect to the unit
cell volume. With the result of electronic density of states (DOS), the structural, optical and electrical properties of Zinc chalcogenides
have been calculated. The second derivative of energy, as a function of lattice strain has been successfully used to estimate the elastic
constants of these binary compounds. The results are in good agreement with other theoretical calculations as well as available
experimental data.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Alternative and Explicit Derivation of the Lattice Boltzmann Equation for the...ijceronline
A lattice Boltzmann equation for fully incompressible flows is derived through the utilization of appropriate ansatzes. The result is a singular equilibrium distribution function which clarifies the algorithm for general implementation, and ensures correct steady and unsteady behavior. Through the Chapman-Enskog expansion, the exact incompressible Navier-Stokes equations are recovered. With 2D and 3D canonical numerical simulations, the application, accuracy, and workable boundary conditions are shown. Several unique benefits over the standard equation and alternative forms presented in literature are found, including faster convergence rate and greater stability.
CR-Submanifolds of a Nearly Hyperbolic Cosymplectic Manifold with Semi-Symmet...iosrjce
We consider a nearly hyperbolic cosymplectic manifold and we study some properties of CRsubmanifolds
of a nearly cosymplectic manifold with a semi-symmetric semi-metric connection. We also obtain
some results on 휉−horizontal and 휉 −vertical CR- submanifolds of a nearly cosymplectic manifold with a semisymmetric
semi-metric connection and study parallel distributions on nearly hyperbolic cosymplectic manifold
with a semi-symmetric semi-metric connection.
EVALUATING STRUCTURAL, OPTICAL & ELECTRICAL CHARACTERIZATION OF ZINC CHALCOGE...Editor IJCATR
To evaluate the structural, optical & electrical properties of the zinc chalcogenides (ZnO, ZnS, ZnSe & ZnTe), the Full
Potential Linearized – Augumented Plane Wave plus Local Orbits (FP – LAPW+lo) method. For the purpose of exchange-correlation
energy (Exc) determination in Kohn–Sham calculation, the standard local density approximation (LDA) formalism has been utilized.
Murnaghan’s equation of state (EOS) has been used for volume optimization by minimizing the total energy with respect to the unit
cell volume. With the result of electronic density of states (DOS), the structural, optical and electrical properties of Zinc chalcogenides
have been calculated. The second derivative of energy, as a function of lattice strain has been successfully used to estimate the elastic
constants of these binary compounds. The results are in good agreement with other theoretical calculations as well as available
experimental data.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
A Global Approach with Cutoff Exponential Function, Mathematically Well Defin...Miltão Ribeiro
A global approach with cutoff exponential functions is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method, mathematically well defined at the outset, makes use of two regulators, one of them to make the sum of the orders
of Bessel functions finite and the other to regularize the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy and allows a major comprehension on the regularization process when nontrivial symmetries are under consideration. In particular, we determine the Casimir energy of a scalar field, showing all kinds of divergences. We consider separately the contributions of the inner and outer regions of a spherical shell and show that the results obtained are in agreement with those known in the literature, and this gives a confirmation for the consistence of the proposed approach. The choice of the scalar field was due to its simplicity in terms of physical quantity spin.
Publication Name: Advances in High Energy Physics.
Author: M.S.R. Miltão and Franz A. Farias.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
A Global Approach with Cutoff Exponential Function, Mathematically Well Defin...Miltão Ribeiro
A global approach with cutoff exponential functions is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method, mathematically well defined at the outset, makes use of two regulators, one of them to make the sum of the orders
of Bessel functions finite and the other to regularize the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy and allows a major comprehension on the regularization process when nontrivial symmetries are under consideration. In particular, we determine the Casimir energy of a scalar field, showing all kinds of divergences. We consider separately the contributions of the inner and outer regions of a spherical shell and show that the results obtained are in agreement with those known in the literature, and this gives a confirmation for the consistence of the proposed approach. The choice of the scalar field was due to its simplicity in terms of physical quantity spin.
Publication Name: Advances in High Energy Physics.
Author: M.S.R. Miltão and Franz A. Farias.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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Analytical, Numerical and Experimental Validation of Coil Voltage in Inductio...ijeljournal
This paper presents, mathematical model of induction heating process by using analytical and numerical methods. In analytical method, series equivalent circuit (SEC) is used to represent induction coil and work piece. Induction coil and workpiece parameters (resistance and reactance) are calculated by standard formulas along with Nagaoka correction factors and Bessel functions. In Numerical method, magnetic vector potential formulation is done and finite element method (FEM) is used to solve the field equations. Analytically and numerically computed parameters such as equivalent coil resistance, reactance, coil voltage, work piece power are compared and found that they are in good agreement. Analytically and numerically obtained coil voltages at different frequencies are validated by experimental results. This mathematical model is useful for coil design and optimization of induction heating process.
Analytical, Numerical and Experimental Validation of Coil Voltage in Inductio...ijeljournal
This paper presents, mathematical model of induction heating process by using analytical and numerical methods. In analytical method, series equivalent circuit (SEC) is used to represent induction coil and work piece. Induction coil and workpiece parameters (resistance and reactance) are calculated by standard formulas along with Nagaoka correction factors and Bessel functions. In Numerical method, magnetic vector potential formulation is done and finite element method (FEM) is used to solve the field equations. Analytically and numerically computed parameters such as equivalent coil resistance, reactance, coil voltage, work piece power are compared and found that they are in good agreement. Analytically and numerically obtained coil voltages at different frequencies are validated by experimental results. This mathematical model is useful for coil design and optimization of induction heating process.
ANALYTICAL, NUMERICAL AND EXPERIMENTAL VALIDATION OF COIL VOLTAGE IN INDUCTIO...ijeljournal
This paper presents, mathematical model of induction heating process by using analytical and numerical
methods. In analytical method, series equivalent circuit (SEC) is used to represent induction coil and work
piece. Induction coil and workpiece parameters (resistance and reactance) are calculated by standard
formulas along with Nagaoka correction factors and Bessel functions. In Numerical method, magnetic
vector potential formulation is done and finite element method (FEM) is used to solve the field equations.
Analytically and numerically computed parameters such as equivalent coil resistance, reactance, coil
voltage, work piece power are compared and found that they are in good agreement. Analytically and
numerically obtained coil voltages at different frequencies are validated by experimental results. This
mathematical model is useful for coil design and optimization of induction heating process.
ANALYTICAL, NUMERICAL AND EXPERIMENTAL VALIDATION OF COIL VOLTAGE IN INDUCTIO...ijeljournal
This paper presents, mathematical model of induction heating process by using analytical and numerical
methods. In analytical method, series equivalent circuit (SEC) is used to represent induction coil and work
piece. Induction coil and workpiece parameters (resistance and reactance) are calculated by standard
formulas along with Nagaoka correction factors and Bessel functions. In Numerical method, magnetic
vector potential formulation is done and finite element method (FEM) is used to solve the field equations.
Analytically and numerically computed parameters such as equivalent coil resistance, reactance, coil
voltage, work piece power are compared and found that they are in good agreement. Analytically and
numerically obtained coil voltages at different frequencies are validated by experimental results. This
mathematical model is useful for coil design and optimization of induction heating process.
Similar to Casimir energy for a double spherical shell: A global mode sum approach (20)
Temperatura do Universo: uma proposta de conteúdo para estudantes do nível fu...Miltão Ribeiro
Temos conhecimento, através das atuais pesquisas na área de Ensino de Física, que não é necessário os docentes se concentrarem em memorizações de fórmulas durante suas aulas, eles devem interagir com seus estudantes de forma criativa, com experimentações, melhores recursos metodológicos e boas estratégias avaliativas, a fim de garantir o possível aprendizado do aluno, fazendo-o associar os assuntos dados em sala de aula com o seu cotidiano. Os mapas conceituais, por exemplo, podem ajudar tanto o docente como o estudante a organizar melhor os conteúdos abordados e facilitar na aprendizagem. Conciliando os mapas com conceitos e curiosidades de Astronomia (especificamente a Temperatura do Universo, que engloba quase todos os conteúdos da Física) pode ser uma forma criativa de atrair os estudantes para as aulas de Física, tornando-as mais interessante. Apesar da complexidade dos assuntos de Astronomia, eles podem ser apresentados para uma turma de nível fundamental com uma linguagem mais apropriada e de forma conceitual. O importante é mostrar analogias/comparações com o cotidiano. Para tanto, nesse trabalho foi criado um esquema conceitual, que serviu como ponta-pé inicial para a construção de mapas conceituais para o ensino fundamental, sendo estes construídos com o auxílio de desenhos e figuras, para despertar o interesse do público alvo pela disciplina. Após todas as construções e re-análises, os mapas foram aplicados em duas escolas do município de Feira de Santana -Bahia, Brasil, possibilitando com que os estudantes da 9º ano, que estavam tendo o primeiro contato com assuntos dessa disciplina, pudessem também construir seus próprios mapas de acordo com os conteúdos abordados pelo docente. Assim, esses estudantes puderam buscar nos mapas conceituais uma forma interativa de produzir conhecimentos.
Publication Name: Experiências em Ensino de Ciências.
Author: Tamila Marques Silveira, M. S. R. Miltão.
Model for Analysis of Biaxial and Triaxial Stresses by X-ray Diffraction Assu...Miltão Ribeiro
In this work we aim to develop expressions for the calculation of biaxial and triaxial stresses in polycrystalline anisotropic materials, and to determine their elastic constants using the theory of elasticity for continuum isochoric deformations; thus, we also derive a model to determine residual stress. The constitutive relation between strain and stress in these models must be assumed to be orthotropic, obeying the generalized Hooke’s law. One technique that can be applied with our models is that of X-ray diffraction, because the experimental conditions are similar to the assumptions in the models, that is, it measures small deformations compared with the sample sizes and the magnitude of the tensions involved, and is insufficient to change the volume (isochoric deformation). Therefore, from the equations obtained, it is possible to use the sin^{2}\psi technique for materials with texture or anisotropy by first characterizing the texture through the pole figures to determine possible angles \psi that can be used in the equation, and then determining the deformation for each diffraction peak with the angles \psi obtained from the pole figures.
Publication Name: Japanese Journal of Applied Physics.
Author: Edson M. Santos, Marcos T. D. Orlando, M.S.R. Miltão, Luis G. Martinez, Álvaro S. Alves, and Carlos A. Passos.
Uma Proposta de Estudo Filosófico do Ser Social do Movimento AmbientalMiltão Ribeiro
A lógica da compreensão sócio-ambiental requer o diálogo entre os diferentes campos do saber e uma visão sistêmica deles, pois acreditamos que o problema socioambiental se reflete, em diferentes nuances, em todos os fenômenos. Este trabalho visa trazer ao debate a possibilidade de formação de sujeitos sociais do movimento ambiental em bases filosóficas, apresentando as necessidades objetivas de natureza coletiva que definem esse ser social. Inicialmente, apresentaremos os pressupostos filosóficos, ontológico e epistemológico, do conhecimento humano. Depois, apresentaremos a relação entre a questão socioambiental e a filosofia, bem como as necessidades objetivas de natureza coletiva. Estabeleceremos a necessidade de uma visão de conhecimento que seja cosmológica/holística/histórica/construtivista/dialógica para ter em conta as questões epistemológicas e ontológicas do conhecimento humano, e compreendermos a identidade social do movimento ambiental. Com isso concluímos pela necessidade de uma Educação Ambiental filosófico-crítica que contribua para a formação do ser social desse movimento.
Publication Name: Anais da 35ª Reunião Anual da ANPED.
Author: M.S.R. Miltão.
Ciências Físicas e Popularização da Astronomia na Chapada Diamantina – Bahia....Miltão Ribeiro
O Departamento de Física da UEFS desenvolve um programa de popularização de Ciências, com ênfase em várias áreas das Ciências Físicas, particularmente em Astronomia, na região da Chapada Diamantina Bahia. O trabalho tem como objetivo a apresentação, na forma de palestras, minicursos e realizações experimentais, de conceitos e teorias das Ciências Físicas e, em particular, de suas relações com o cotidiano. Neste artigo descrevemos as atividades realizadas até o momento bem como faremos algumas considerações sobre alguns desdobramentos delas.
Publication Name: Revista Ciência em Extensão.
Author: M. S. R. Miltão, R. K. Madejsky, A. V. Andrade-Neto, P. C. Araújo, J. B. Santos.
Algumas Considerações sobre a Formação em Física dos Sujeitos das EFAs, consi...Miltão Ribeiro
We investigated the Physical Sciences in the Agricultural Families Schools considering the Pedagogy of the Alternation. In this sense, we take into account the lessons of Ethnophysics that seeks to comprehend, from the own social groups, their world view to be presented the physical academic knowledge. We use field trips, staying in school for three days to start the research process. The results show that the philosophical underpinnings of the PA are not well settled and the transdisciplinarity still not processed properly.
Publication Name: Caderno Multidisciplinar da RESAB.
Author: Carla Suely Correia Santana, M.S.R. Miltão.
Rolamento e atrito de rolamento ou por que um corpo que rola páraMiltão Ribeiro
The dynamics of the rolling motion of an object on a horizontal plane is studied. We use the laws of Newton to analyze the rolling of a rigid body and of a deformable body. The analytical solutions and their discussions in various physical situations are presented. The results allow us to understand the physical basis of why the rolling motion of a body stops after a certain time interval. Also, these results should help undergraduate physics students to investigate this type of motion.
Publication Name: Revista Brasileira de Ensino de Física.
Author: A.V. Andrade-Neto, J.A. Cruz, M.S.R. Miltão, E.S. Ferreira.
Reconstruction of magnetic source images using the Wiener filter and a multic...Miltão Ribeiro
A system for imaging magnetic surfaces using a magnetoresistive sensor array is developed. The experimental setup is composed of a linear array of 12 sensors uniformly spaced, with sensitivity of 150 pT∗Hz^{−1/2} at 1 Hz, and it is able to scan an area of (16 × 18) cm^{2} from a separation of 0.8 cm of the sources with a resolution of 0.3 cm. Moreover, the point spread function of the multi-sensor system is also studied, in order to characterize its transference function and to improve the quality in the restoration of images. Furthermore, the images are generated by mapping the response of the sensors due to the presence of phantoms constructed of iron oxide, which are magnetized by a pulse
of 80 mT. The magnetized phantoms are linearly scanned through the sensor array and the remanent magnetic field is acquired and displayed in gray levels using a PC. The images of the magnetic sources are reconstructed using two-dimensional generalized parametric Wiener filtering. Our results exhibit a very good capability to determine the spatial distribution of magnetic field sources, which produce magnetic fields of low intensity.
Publication Name: Review of Scientific Instruments.
Author: J. A. Leyva-Cruz, E. S. Ferreira, M. S. R. Miltão, A. V. Andrade-Neto, A. S. Alves, J. C. Estrada, and M. E. Cano.
Philosophical-Critical Environmental Education: a proposal in a search for a ...Miltão Ribeiro
This paper aims to develop a study on environmental education from philosophical and practical bases. Philosophical considerations being established after critical analysis of some philosophical schools who have taken the environment or the Individuals as a matter of primary concern; practical considerations arising from our experience in the university environmental movement. Thus, we intend to express our thinking towards the discussion about critical Environmental Education in a philosophical perspective called philosophical-critical Environmental Education, which aims to seek a harmony, a balance between subject and object, from a philosophical view-point, and as a consequence, between society and environment, from a socio-political perspective, in addressing the socio-environmental issue.
Publication Name: Journal of Social Sciences (COES&RJ-JSS).
Author: M. S. R. Miltão
O Ensino de Física e a Educação do Campo: uma relação que precisa ser efetivadaMiltão Ribeiro
Nesse capítulo objetivamos discutir como o Ensino de Física pode contribuir nas pesquisas em Educação do Campo. Mostramos que a Educação do Campo tem uma concepção diferenciada da Educação Rural, na medida em que advém de ações desenvolvidas pelos movimentos populares defensores da reforma agrária e de uma defesa de educação que leve em consideração o contexto dos camponeses.
Publication Name: Ensino de Física: reflexões, abordagens & práticas, Edition: 1ª, Chapter: 11, pp.167-196.
Author: M.S.R. MILTÃO; SANTANA, C. S. C.; BARRETO, A. L. V.; CARDOSO, G. K. R.
Global approach with cut-off exponential function to spherical Casimir effectMiltão Ribeiro
We presented a method to calculate the spherical electromagnetic Casimir effect through the use of a regularization via the cut-off exponential function in a non ambiguous way from the start. We propose the use of two cut-off parameters: one to regularize the sum of orders of the Bessel function, and the other to regularize the integral related to Bessel function zeros. Both the interior and exterior contributions calculated have revealed all cut-off parameters dependency and give the results previously obtained in literature.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
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Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Casimir energy for a double spherical shell: A global mode sum approach
1. Casimir energy for a double spherical shell: A global mode sum approach
M. S. R. Milta˜o*
Departamento de Fı´sica, Universidade Estadual de Feira de Santana, Av. Universita´ria KM03 BR116 Campus Universita´rio,
Feira de Santana, BA, BR
(Received 21 May 2008; published 12 September 2008)
In this work we study the configuration of two perfectly conducting spherical shells. This is a problem
of basic importance to make possible development of experimental apparatuses that they make possible to
measure the spherical Casimir effect, an open subject. We apply the mode sum method via cutoff
exponential function regularization with two independent parameters: one to regularize the infinite order
sum of the Bessel functions; other, to regularize the integral that becomes related, due to the argument
theorem, with the infinite zero sum of the Bessel functions. We obtain a general expression of the Casimir
energy as a quadrature sum. We investigate two immediate limit cases as a consistency test of the
expression obtained: that of a spherical shell and that of two parallel plates. In the approximation of a thin
spherical shell we obtain an expression that allows to relate our result with that of the proximity-force
approximation, supplying a correction to this result.
DOI: 10.1103/PhysRevD.78.065023 PACS numbers: 12.20.Ds
I. INTRODUCTION
The Casimir attractive force between conducting paral-
lel plates [1] is one of the most striking demonstrations of
the quantum nature of the electromagnetic field. Such
effect was experimentally confirmed by Sparnaay in 1958
[2] and with great accuracy by Lamoreaux [3] and
Mohideen and Roy [4,5]. Casimir himself proposed to
build a model for the electron as an application of this
effect [6]. He suggested two models: a solid sphere and a
thin spherical shell. If the Casimir pressure turned out to
point inwards (as could be guessed from a crude analogy
with the parallel plates setup), it would stabilize the elec-
tron against the electrostatic repulsion, and it would pro-
vide a theoretical value for the fine structure constant.
Boyer was the first to actually calculate the Casimir pres-
sure for a spherical shell [7]. Contrary to the expectations,
the pressure is repulsive (i.e., it points outwards), and so,
invalidating Casimir’s electron model.
Since Boyer’s pioneer work, independent calculations
have confirmed that the Casimir pressure is repulsive for a
perfectly reflecting spherical shell. Different methods have
been employed: the Schwinger source theory [8], the
Green function technique [9], the multiple scattering ap-
proach [10], and the zeta function formalism [11].
However, a simple physical explanation of why the pres-
sure is repulsive is still lacking.
Being the first realistic approach to investigate the physi-
cal systems of the nature, the spherical symmetry can be
applied for the study of cosmological systems or bag
models of hadrons, to quote two examples of great rele-
vance [12–15].
In this paper, we compute the Casimir energy for two
concentric perfectly reflecting spherical shells (internal
and external radii are a and b). This type of setup was
analyzed with the help of the Green function technique for
material media with particular electric and magnetic prop-
erties (satisfying the so-called uniform velocity of light
condition) [16–19] and for more general dieletric media
[20]. A quantum statistical approach was employed for
ideal metals [21] as well as for dieletric media [22].
Recently, the measurement of the Casimir force between
two concentric spherical surfaces was proposed [23].
Here, we compute separately the contributions of field
modes from each of the three spatial regions (internal,
between the shells and external) directly from the field
zero-point energy. This ‘‘mode sum’’ approach [24–26],
when combined with the introduction of suitable cutoff
functions [27], allows for a physical interpretation of the
several terms contributing to the Casimir energy, which
correspond to different field polarizations and spatial re-
gions. All series and integrals are properly regularized by
exponential cutoff functions, so that we only deal with
well-defined quantities. We keep the cutoff dependent
terms and analyze their dependence on radius, area, and
internal volume of the spherical shells, before considering
the cancellations resulting from the sum over the three
spatial regions. As suggested by Barton [28], these cutoff
dependent terms might be physically relevant and provide
for a net attractive pressure in the context of a more realist
model for the material medium.
We are particularly interested in the limit of a thin
intershell region ðb À aÞ=a ( 1: In this case, we should
recover the Casimir attractive force between parallel plates
according to the proximity-force approximation (PFA)
[29], which replace the surfaces by tangent planes.*miltaaao@ig.com.br
PHYSICAL REVIEW D 78, 065023 (2008)
1550-7998=2008=78(6)=065023(10) 065023-1 Ó 2008 The American Physical Society
2. Hence, there must be a crossover between the single shell
repulsive regime and the thin intershell attractive regime.
Our model also allows for evaluation of the accuracy of
PFA in a problem for which an exact solution is available
[30].
The paper is organized in the following way: section II
presents the derivation of the formal results for the Casimir
energy in the form of a multipole series. The limit ðb À
aÞ=a ( 1 is considered in Sec. III, where corrections to
the PFA result are obtained. In Sec. IV, we present some
final remarks and a conclusion.
II. CASIMIR EFFECT BETWEEN TWO
SPHERICAL SHELLS
Since we consider perflectly reflecting shells, each elec-
tromagnetic field mode is confined in one of the three
spatial regions, which are labeled by the index , with ¼
1, 2, 3 denoting the inner, intershell and outer spatial
regions, respectively. Moreover, due to spherical symme-
try, we may decompose the vectorial boundary value prob-
lem into two independent problems by defining the usual
transverse electric (TE, electric field perpendicular to the
radial direction) and transverse magnetic (TM) polariza-
tions. Hence, we have 6 independent classes of field
modes, which we represent by the indexes and p ¼
TE; TM for polarization.
The Casimir energy for the double shell is given by the
modification of the zero-point energy due to the boundary
conditions
Eða;bÞ ¼
X1
n¼1
X1
j¼1
X3
¼1
X
p
ðj þ 1=2Þ@½!ðÞ
njpða;bÞ À !ðÞ
njpðrefÞŠ;
(1)
where !
njpða; bÞ is a mode frequency for a given angular
momentum j, polarization p, and spatial region . We have
taken into account the degeneracy factor 2j þ 1 and sub-
tracted the reference frequencies !
jnðrefÞ, corresponding
to the free-space limit.
In order to have discrete spectra for the outer region, we
consider a third ‘‘auxiliary’’ spherical surface of radius R,
and take the limit R ! 1 (see Fig. 1).
To obtain the zero-point energy for the free-space case,
we consider a similar configuration [7], taking the two
innermost shells to have radius R= and R=, with
1. We evaluate the sum over n in Eq. (1) with the help
of Cauchy’s theorem for analytic functions in the complex
plane of frequency [31,32], taking the contour CðÃ; Þ
indicated in Fig. 2.
We define analytic functions fðÞ
jp ða; b; zÞ such that their
zeros (when considered as functions of z) correspond to the
eigenfrequencies !ðÞ
njpða; bÞ, n ¼ 1; 2; . . . They are all con-
tained within the contour CðÃ; ’Þ in the limit à ! 1:
We find, by using the prescription proposed in [27], with
¼ j þ 1=2,
Eða; bÞ ¼ lim
Ä!Ä0
@c
2i
X1
j¼1
expðÀÞ lim
Ã!1
I
CðÃ;’Þ
dzz
 expðÀzÞ
d
dz
X3
¼1
X
p
log½fðÞ
jp ða; b; zÞŠ
À log
fðÞ
jp
R
;
R
; z
; (2)
where Ä denotes the set R, , together with the cutoff
parameters and . Note that we employ two exponential
cutoff functions to regularize our expression (including one
for the sum over angular momentum).
Equation (2) is an application of the approach proposed
in [27], which presents a cutoff method of calculating the
Casimir effect for a spherical shell including both the
interior and exterior modes in a new calculation method
that has not been used in this problem before. Basically, as
it is known in the literature, generally, the j and n sum do
not converge, and the procedure to regularize them uses
one cutoff exponential function eÀ!
jn . However, when we
FIG. 1. Boundary conditions of two spherical shells.
FIG. 2. Path integration in complex plan.
M. S. R. MILTA˜O PHYSICAL REVIEW D 78, 065023 (2008)
065023-2
3. use the argument theorem to bypass the problem of eval-
uating the implicity frequency, the cutoff exponential func-
tion eÀ!
jn renounces to protect the j series of the usual
divergence. To prevent this divergence we have used an
additional cutoff exponential function eÀ
. Thus, we can
write the equality in (2) that represents a well-defined
expression of the mathematical point of view. Therefore,
taking this prescription, the expression to Casimir energy
becomes well defined, and the manipulations can be done
without difficulty before the limits are realized.
The positive aspect of the approach proposed rests on the
fact that (i) as remarkable in literature [24,25], the mode
sum method presents a great advantage in relation to other
techniques due its simplicity and visualization of the di-
verse stages involved in calculation, as its global version
determines without difficulty the total Casimir energy;
(ii) we work exclusively with regularized expressions,
preventing cancellation of any possible divergence without
previous justification—a procedure that is not explicit in
regularizations as the generalized zeta function which, in
general, does not use the renormalization process [33], as
well as in procedures similar to those used in Refs. [34,35]
due to the appearance of divergent series in intermediate
steps. In relation to the subtraction procedure, we followed
the usual one initially considered by Casimir [1] for par-
allel plates and by Boyer [7] for spherical effect.
The limit in Eq. (2) is taken in the following sequence:
we first take R ! 1, then ! 1, ! 1, and finally
; ! 0.
The functions fð1Þ
jp actually do not depend on b, because
they correspond to the modes in the inner region r a
fð1Þ
jTEða; zÞ ¼ SjðazÞ; (3)
where SjðxÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
x=2
p
Jjþ1=2ðxÞ is the Riccati-Bessel func-
tion of the first kind [36,37], and
fð1Þ
jTMða; zÞ ¼ S0
jðazÞ; (4)
where the prime denotes the derivative.
Likewise, the functions fð3Þ
jp do not depend on a, because
they correspond to the outer region
fð3Þ
jTEðb; zÞ ¼ CjðRzÞSjðbzÞ À SjðRzÞCjðbzÞ; (5)
where CjðxÞ ¼ À
ffiffiffiffiffiffiffiffiffiffiffiffi
x=2
p
Njþ1=2ðxÞ is the Riccati-Bessel of
the second kind, and
fð3Þ
jTMðb; zÞ ¼ C0
jðRzÞS0
jðbzÞ À S0
jðRzÞC0
jðbzÞ: (6)
The functions fð2Þ
jp corresponding to the intershell region
have the same form, with the replacements b ! a and R !
b
fð2Þ
jTEða; b; zÞ ¼ CjðbzÞSjðazÞ À SjðbzÞCjðazÞ; (7)
fð2Þ
jTMða; b; zÞ ¼ C0
jðbzÞS0
jðazÞ À S0
jðbzÞC0
jðazÞ: (8)
Since we take the limit R ! 1, we may replace SjðRzÞ,
CjðRzÞ, and their derivatives by the corresponding asymp-
totic expansions for large arguments. We find
fð3Þ
jTEðb; zÞ ¼ cos½jðzÞŠSjðbzÞ À tan½jðzÞŠCjðbzÞ; (9)
with jðzÞ ¼ Rz À j
2 , and
fð3Þ
jTMðb; zÞ ¼ sin½jðzÞŠS0
jðbzÞ þ cos½jðzÞŠC0
jðbzÞ: (10)
We may also take the asymptotic expansions for large
arguments when computing fðÞ
jp ðR=; R=; zÞ in Eq. (2),
which account for the free-space zero-point energy.
In the contour of Fig. 2, only the segments À1 and À2
contribute in the limit à ! 1. For convenience, we take
’ ! 0, allowing us to take z ¼ i (with real ) every-
where in Eq. (2) except for the exponential cutoff
expðÀzÞ ¼ exp½Çi expðÇi’ÞŠ for segment À1;2,
which provides a damping term expðÀ sin’Þ [31].
The integrand is then written in terms of the modified
Bessel functions of the first and second kinds I and K.
It is also convenient to rescale the integration variable by
multiplying by .
As expected, the R dependent terms are canceled due to
the subtraction of the free-space energy in Eq. (2). The
resulting expression is written as
E ða; bÞ ¼ EðaÞ þ EðbÞ þ Ecða; bÞ; (11)
where EðaÞ [and likewise for EðbÞ] is the Casimir energy
for a single spherical shell of radius a. Besides the single-
shell energies, the Casimir energy for the double shell
configuration contains a nontrivial term representing the
joint effect of the two shells:
Ecða; bÞ ¼ À
@c
Re
X1
j¼1
2
Z 1
0
d
d
d
log
1 À
KðbÞ=IðbÞ
KðaÞ=IðaÞ
þ log
1 À
½1
2 KðbÞ þ bK0
ðbÞŠ=½1
2 IðbÞ þ bI0
ðbÞŠ
½1
2 KðaÞ þ aK0
ðaÞŠ=½1
2 IðaÞ þ aI0
ðaÞŠ
: (12)
CASIMIR ENERGY FOR A DOUBLE SPHERICAL SHELL: . . . PHYSICAL REVIEW D 78, 065023 (2008)
065023-3
4. In expression (11) of the Casimir energy, the terms EðaÞ
and EðbÞ are free of regularization and, of course, are well
defined. In the case of the remaining contribution, it is also
well defined without regularization. This fact results from
the sum-integration functional form as can be seen when
using the Debye expansion of Bessel functions that appears
there. The result of this expansion produces attenuation
exponential functions that are inherent to the physical
system and dispense the regularization functions. So we
can take the limits ! 0, ! 0, ’ ! 0, R ! 1, ! 1,
and ! 1 understood in (2).
Let us consider the ‘‘interference’’ parcel up to the
second term of the Debye expansion [36] in the parame-
ter of the functions in the argument of the two logarithms in
(12). We will be considering the expansion until the 1=
order because it will be enough to make a small annular
region approximation. In the expansion up to the 1= order
for the TE mode parcel we have
log
2
41 À
KðbÞ
IðbÞ
KðaÞ
IðaÞ
3
5 ffi log
1 À expðÀ2ð2 À 1ÞÞ
1 þ
ÀðTEÞ
1 ðtÞ
¼ À
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
Â
1 þ
Xn
m¼1
n!
m!ðn À mÞ!
ÀðTEÞ
1 ðtÞ
m
ffi À
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTEÞð1Þ
; (13)
where in the last line we only consider the first term m ¼ 1.
The 1, 2, t1, and t2 amounts associated to the Debye
expansion are given by [36]
1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a2
2
q
þ log
a
1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a2
2
p
; (14)
2 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ b2
2
q
þ log
b
1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ b2
2
p
; (15)
t1 ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a22
p ; (16)
t2 ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ b2
2
p ; (17)
and ÀðTEÞ
1 ðtÞ has the form
ÀðTEÞ
1 ðtÞ ¼
5
12
t3
2 þ
1
4
t1 À
1
4
t2 À
5
12
t3
1
; (18)
so that
OðTEÞð1Þ ¼ n
5
12
t3
2 þ
1
4
t1 À
1
4
t2 À
5
12
t3
1
: (19)
In the expansion up to the 1= order for the TM mode
parcel we have
log
2
641 À
½ð1=2ÞKðbÞþbK0
ðbÞŠ
½ð1=2ÞIðbÞþbI0
ðbÞŠ
½ð1=2ÞKðaÞþaK0
ðaÞŠ
½ð1=2ÞIðaÞþaI0
ðaÞŠ
3
75 ffi log
1 À expðÀ2ð2 À 1ÞÞ
1 þ
ÀðTMÞ
1 ðtÞ
¼ À
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
Â
1 þ
Xn
m¼1
n!
m!ðn À mÞ!
ÀðTMÞ
1 ðtÞ
m
ffi À
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTMÞð1Þ
; (20)
where ÀðTMÞ
1 ðtÞ and OðTMÞ
ð1Þ amounts have the forms
ÀðTMÞ
1 ðtÞ ¼
À
7
12
t3
2 þ
1
4
t1 À
1
4
t2 þ
7
12
t3
1
; (21)
OðTMÞð1Þ ¼ n
À
7
12
t3
2 þ
1
4
t1 À
1
4
t2 þ
7
12
t3
1
: (22)
Emphasize that, in the limit where ! 0, the loga-
rithms in (12) are given by
log
2
641 À
KðbÞ
IðbÞ
KðaÞ
IðaÞ
3
75 ffi log
2
641 À
½ð1=2ÞKðbÞþbK0
ðbÞŠ
½ð1=2ÞIðbÞþbI0
ðbÞŠ
½ð1=2ÞKðaÞþaK0
ðaÞŠ
½ð1=2ÞIðaÞþaI0
ðaÞŠ
3
75
ffi log
1 À
a
b
2
; (23)
and do not cause problems in the calculation of that ex-
pression in the inferior limit of the integral that appears in
(12).
M. S. R. MILTA˜O PHYSICAL REVIEW D 78, 065023 (2008)
065023-4
5. In order to obtain an expression equivalent to (11), however, more appropriate to approximations and numerical
calculations we can add and subtract the approximate quantities (13) and (20) in the integrand of (12) of the Casimir energy
getting
Eða; bÞ ¼ EðaÞ þ EðbÞ þ
@c
X1
j¼1
2
Z 1
0
d
8
:
log
2
641 À
KðbÞ
IðbÞ
KðaÞ
IðaÞ
3
75 þ
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTEÞ
ð1Þ
9
=
;
þ
@c
X1
j¼1
2
Z 1
0
d
8
:
log
2
641 À
½ð1=2ÞKðbÞþbK0
ðbÞŠ
½ð1=2ÞIðbÞþbI0
ðbÞŠ
½ð1=2ÞKðaÞþaK0
ðaÞŠ
½ð1=2ÞIðaÞþaI0
ðaÞŠ
3
75 þ
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTMÞð1Þ
9
=
;
À
@c
X1
j¼1
2
Z 1
0
d
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTEÞ
ð1Þ
À
@c
X1
j¼1
2
Z 1
0
d
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTMÞ
ð1Þ
; (24)
where we have realized partial integration.
We rewrite (24) in the form
E ða; bÞ ¼ EðaÞ þ EðbÞ þ EðTE interfÞ
ð01Þ þ EðTM interfÞ
ð01Þ þ EðinterfÞ
ðnumÞ ; (25)
where we have used the definitions
E ðTE interfÞ
ð01Þ ¼ À
@c
X1
j¼1
2
Z 1
0
d
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTEÞ
ð1Þ
; (26)
E ðTM interfÞ
ð01Þ ¼ À
@c
X1
j¼1
2
Z 1
0
d
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTMÞ
ð1Þ
; (27)
and
EðinterfÞ
ðnumÞ ¼
@c
X1
j¼1
2
Z 1
0
d
8
:
log
2
641 À
KðbÞ
IðbÞ
KðaÞ
IðaÞ
3
75 þ
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTEÞð1Þ
9
=
;
þ
@c
X1
j¼1
2
Z 1
0
d
8
:
log
2
641 À
½ð1=2ÞKðbÞþbK0
ðbÞŠ
½ð1=2ÞIðbÞþbI0
ðbÞŠ
½ð1=2ÞKðaÞþaK0
ðaÞŠ
½ð1=2ÞIðaÞþaI0
ðaÞŠ
3
75 þ
X1
n¼1
expðÀ2nð2 À 1ÞÞ
n
1 þ
OðTMÞ
ð1Þ
9
=
;
: (28)
In expression (25), the Casimir energy is given by the
sum of the Casimir energies that would have the shells
separately, with two contributions EðTE interfÞ
ð01Þ and
EðTM interfÞ
ð01Þ , of TE and TM mode from the region between
the two shells, plus a remaining energy EðinterfÞ
ðnumÞ . Since
EðTE interfÞ
ð01Þ and EðTM interfÞ
ð01Þ are contributions proceeding
from the two first terms in the Debye expansion, the zero
and first orders, the remaining energy EðinterfÞ
ðnumÞ must take
into consideration all the other terms of the Debye expan-
sion and will be in such a way lesser, as the convergence of
this expansion increases. In this way, we can characterize
EðinterfÞ
ðnumÞ as the remaining portion of the Casimir energy’s
expansion up to the second term of the Debye expansion.
We know that the Debye expansion is of fast conver-
gence in the case of great angular moment waves. As we
are dealing with contributions proceeding from the region
between the two shells of a and b radii (a b), this
condition of great angular moment is given by k wave
numbers with ak ) 1. But in the annular region, we
have wave lengths of the order of d ¼ b À a separation
between the shells or minors, that is, k $ ð1=dÞ (or
greater); soon, we must have a ) d, or, the fast conver-
gence of the Debye expansion occurs for small separations
between the shells relative to the radii of the shells, which
is small annular region. In this case, the remaining portion
EðinterfÞ
ðnumÞ of the expansion (11) is small.
CASIMIR ENERGY FOR A DOUBLE SPHERICAL SHELL: . . . PHYSICAL REVIEW D 78, 065023 (2008)
065023-5
6. We then got a general expression (24) for the Casimir
effect between two concentric conducting spherical shells
of radii a and bða bÞ that is accurate as the original
expression (11), and is ready to be applied to the case of the
small annular region. It will be used in two applications.
The first one consists of carrying through the waited veri-
fication of consistency that occurs in the limit where the
radius of the external shell goes to the infinite, and so the
Casimir energy of the two shells tends to the Casimir
energy of one shell, the intern. The second application
consists in obtaining the Casimir energy of the two shells
when the annular region is small, and for consistency, the
dominant term of this approximation must be the Casimir
energy of two parallel plates. This second application is the
subject of the next section.
To study the limit case b ! 1 let us consider the
expression of the Casimir energy given by (24) and rewrit-
ten in (25). In the limit where b ! 1, we desire that it
remain only as the first term EðaÞ, the Casimir energy of the
internal spherical shell. We must, therefore, demonstrate
that the limit of the other terms is null.
The second term in (25) is the Casimir energy of only
one spherical shell of radius b. We have evidently
lim
b!1
E ðbÞ ¼ lim
b!1
0; 09234738972
@c
2b
¼ 0: (29)
The third and fourth terms in (25), the ‘‘interference’’
contributions of TE and TM modes are given by (26) and
(27). In these expressions, the exponential in the integrand
are given by
exp½À2ð2 À 1ÞŠ
¼ exp
À2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ b2
2
q
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a2
2
q
þ log
b
a
1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a22
p
1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ b2
2
p
; (30)
where we use the definitions in (14) and (15). This expo-
nential goes to zero in the limit where b ! 1. Moreover,
in this limit the exponential (30) dominates on the OðTEÞ
ð1Þ
and OðTMÞ
ð1Þ terms, as it is easy to verify though its
definitions (19) and (22), given in terms of (16) and (17).
Therefore, we conclude that
lim
b!1
EðTE interfÞ
ð01Þ ¼ 0; (31)
lim
b!1
EðTM interfÞ
ð01Þ ¼ 0: (32)
In the last term in (25), in its expression given for (28)
we must observe that the terms of integrand involving the
exponential ones (30) are null in the limit b ! 1 for the
same reasons for which we demonstrate that the contribu-
tions (26) and (27) are null. As the further terms of the
integrand have logarithms, we can use the properties of the
Bessel functions [36]
lim
b!1
KðbÞ ¼ 0; lim
b!1
K0
ðbÞ ¼ 0; (33)
lim
b!1
IðbÞ ¼ 1 and lim
b!1
I0
ðbÞ ¼ 1; (34)
and conclude that all the logarithms tend to zero in the limit
b ! 1. With this we get of (28),
lim
b!1
EðinterfÞ
ðnumÞ ¼ 0: (35)
Using the results (29), (31), (32), and (35) in (25), we
obtain
lim
b!1
Eða; bÞ ¼ EðaÞ ¼ 0; 0923473
@c
2a
; (36)
that is, the Casimir energy of a spherical shell of radius a,
as we waited for consistency reasons.
III. CASIMIR ENERGY TO SMALL ANNULAR
REGION
To get the approximate Casimir energy to the small
annular region, let us consider formula (24). Being the
radii of the internal and external spherical shells given,
respectively, for a and b, we have that the annular region
between the shells has thickness equal to d ¼ b À a, so
that the excellent parameter to get the approximate energy
is given by
¼
d
a
¼
b À a
a
: (37)
With this parameter, we can express the radius of the
external shell in terms of the internal one
b ¼ að1 þ
Þ; (38)
and the condition to small annular region is given by
(
1.
To first order in
the amounts t2 and 2, defined in (15)
and (17) are given by
t2 ¼ t1 À
a2
z2
t3
1; (39)
2 ¼ 1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a2
z2
p
; (40)
where t1 and 1 are given by (14) and (16).
Considering negligible the interference term EðinterfÞ
ðnumÞ (28)
for
sufficiently small and taking into account the approx-
imations (39) and (40) and considering EðaÞ þ EðbÞ ¼
ð2 À
ÞEðaÞ, the Casimir energy expression (24) reduces to
E ða; bÞ ¼ ð2 À
ÞEðaÞ þ EðTE;interfÞ
ð01Þ þ EðTM;interfÞ
ð01Þ ; (41)
where
M. S. R. MILTA˜O PHYSICAL REVIEW D 78, 065023 (2008)
065023-6
7. EðTE;interfÞ
ð01Þ ¼ À
@c
X1
j¼1
2
Z 1
0
d
Â
X1
n¼1
expðÀ2n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a22
p
Þ
n
1 þ
OðTEÞð1Þ
;
(42)
EðTM;interfÞ
ð01Þ ¼ À
@c
X1
j¼1
2
Z 1
0
d
Â
X1
n¼1
expðÀ2n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a22
p
Þ
n
1 þ
OðTMÞð1Þ
;
(43)
and OðTEÞð1Þ and OðTMÞð1Þ, defined by (19) and (22), are
now given by
OðTEÞ
ð1Þ ¼
À
5
4
na2
2
t1
5
þ
1
4
na2
2
t1
3
; (44)
OðTMÞ
ð1Þ ¼
7
4
na2
2
t1
5
þ
1
4
na2
2
t1
3
: (45)
We get for the zero order term in (42)
E ðTE;interfÞ
ð0Þ ¼ À
@c
X1
j¼1
2
Z 1
0
d
X1
n¼1
expðÀ2n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ a22
p
Þ
n
¼ À
@c
X1
j¼1
2
X1
n¼1
1
n
Z 1
1
dyy
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y2
À 1
p expðÀ2n
yÞ
¼ À
@c
a
X1
j¼1
2
X1
n¼1
1
n
K1ð2n
Þ ¼ À
@c
a
X1
n¼1
1
n4
1
4
3
À
1
n2
11
48
; (46)
where we use an integral representation formula to K1 [38] and the Euler-Maclaurin formula in the j sum. The series in n in
the final expression are given by ð4Þ and ð2Þ, so that we get
E ðTE;interfÞ
ð0Þ ¼
@c
a
À
3
360
3
þ
11
288
: (47)
The 1= order term in (42) provides
E ðTE;interfÞ
ð1Þ ¼ À
@c
a
X1
j¼1
X1
n¼1
Z 1
0
dy expðÀ2n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ y2
q
Þ
À
5
4
y2ð1 þ y2ÞÀ5=2
þ
1
4
y2ð1 þ y2ÞÀ3=2
; (48)
where the expression (44) of OðTEÞ
ð1Þ was used and changed the integration variable to y ¼ a. To realize the integrations
in the above expression various integrations by parts are utilized with the objective of getting integrals to be given in terms
of Bessel functions K [38]. By the end of this procedure, we got the following expression:
E ðTE;interfÞ
ð1Þ ¼ À
@c
a
X1
j¼1
X1
n¼1
5ð2n
Þ4
24
À
2ð2n
Þ2
3
Z 1
0
dy
eÀ2n
ffiffiffiffiffiffiffiffi
1þy2
p
y2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ y2
p þ
À
5ð2n
Þ4
24
þ
7ð2n
Þ2
8
Â
Z 1
0
dy
eÀ2n
ffiffiffiffiffiffiffiffi
1þy2
p
y arctanðyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ y2
p þ
5ð2n
Þ3
24
À
ð2n
Þ
4
Z 1
0
dy
eÀ2n
ffiffiffiffiffiffiffiffi
1þy2
p
y arcsinhðyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ y2
p
: (49)
Resolving the remaining integrals, the expression above can be written in terms of Bessel functions as follows:
EðTE;interfÞ
ð1Þ ¼ À
@c
a
X1
n¼1
X1
j¼1
À
5
3
4K0ð2n
Þn3
3 þ
5
3
4K1ð2n
Þn3
3 À
5
6
3K0ð2n
Þn2
2 þ
7
4
2K0ð2n
Þn
À
4
3
2
K1ð2n
Þn
þ
1
4
K0ð2n
Þ
: (50)
To calculate the sums in j of each one of the six parcels of this expression we use the Euler-Maclaurin formula. Holding
back only the dominant term of order
À1 in which appears the n series that is given by ð2Þ, we get
CASIMIR ENERGY FOR A DOUBLE SPHERICAL SHELL: . . . PHYSICAL REVIEW D 78, 065023 (2008)
065023-7
8. E ðTE;interfÞ
ð1Þ ¼ À
@c
a
17
288
À
48
: (51)
Adding the contributions of the terms of order 0
(47)
and of order À1 (51), we get the ‘‘interference’’ energy
(42) in the TE mode
E ðTE;interfÞ
ð01Þ ¼
@c
a
À
3
360
1
3
À
1
48
À
48
: (52)
The contributions to orders 0
and À1
that correspond
to TM mode (43) are obtained by analogous procedures
E ðTM;interfÞ
ð0Þ ¼
@c
a
À
3
360
3
þ
11
288
; (53)
E ðTM;interfÞ
ð1Þ ¼ À
@c
a
À
31
288
þ
24
: (54)
Adding these two contributions, we get the interference
energy (43) in TM mode
E ðTM;interfÞ
ð01Þ ¼
@c
a
À
3
360
1
3
þ
7
48
À
24
: (55)
Let us notice that accomplishment of the calculations
separately for TE and TM modes allows us to evaluate, to
each order of approximation, what mode supplies the
preponderant contribution. Also, it allows to collect all
the contributions in the TE mode to join to the j ¼ 0
contribution of the scalar field and to get the Casimir
energy of a scalar field subject to Dirichlet conditions in
two concentric spherical surfaces.
Finally, substituting the ‘‘interference’’ contributions
(52) to the TE mode and (55) to TM one in (41), and
discarding the
positive power in the ð2 À
ÞEðaÞ contri-
bution, we get the Casimir energy of concentric spherical
shells in the case of small annular region
E ða; bÞ ¼
@c
a
À
3
180
1
3
þ
1
8
À
48
þ 0:09234739002
: (56)
To verify the consistency of the developed formalism let
us consider the limit where the radii of the concentric shells
go to the infinite, keeping constant the separation between
them. In this limit the Casimir energy density must become
equal to the case of two parallel plates. Dividing both the
members of Eq. (56) by the area A ¼ 4a2 of the internal
shell and remembering that
¼ d=a, we get
Eða; bÞ
A
¼ À
2
@c
720
1
d3
þ
1
8
À
48
@c
4a2
d
þ 0:09234739002
@c
4a3
: (57)
In the limit where the radii of the spherical shells go to the
infinite, with constant d separation, we get the previous
equality
lima!1
bÀa¼d
Eða; bÞ
A
¼ À
2
@c
720
1
d3
: (58)
When this limit is accurately the Casimir energy of the
electromagnetic field in the presence of two conducting
parallel plates, we have that this verification of consistency
supplies a satisfactory result to the formalism of the two
spherical shells here considered.
To relate our result given in (56) or (57) with that of
proximity-force approximation (PFA), we consider the
PFA result for the case of two concentric spherical shells.
Following [39], the PFA for the Casimir energy EC of
two arbitrary smooth surfaces is given by the surface
integral over the Casimir energy per area, which belongs
to an equivalent parallel-plate system that locally follows
the two surfaces [39]
E PFA ¼
ZZ
A
d½zðÞŠ; (59)
where A is the area of one of the opposing surfaces, which
are locally separated by the surface-dependence distance
zðÞ, and ½zðÞŠ is the corresponding Casimir energy per
area.
Considering that, in general, the plate segment d is
tangential to only one of the surfaces, and therefore, the
local distance vector ~zðÞ is perpendicular only to this
surface and not to the other [39], in the case of two
concentric spherical shells, where the local distance vector
~zðÞ is perpendicular to both spheres and have constant
modulus d, we have for the ‘‘inner-sphere-based PFA’’ that
E 5
innerÀsphere PFA ¼ À
ZZ
halfÀsphere
2
@c
720
dA
j~zðÞj3
¼ À
2@c
720d3
ZZ
halfÀsphere
dA ¼ À
2@c
720d3
A:
(60)
The choice for concentric-spheres PFA and not for plate-
sphere PFA has been made, because in this paper we
calculated the Casimir effect for thin spherical shell.
Moreover, nowadays the possibility of a measurement of
spherical Casimir effect is concrete [23], which justifies
this choice.
Comparing the result (57) with the result of the PFA (60)
we have
Eða; bÞ
A
¼
E5
innerÀsphere PFA
A
þ
1
8
À
48
@c
4a2d
þ 0:09234739002
@c
4a3
: (61)
We see by (60) that the proximity-force approximation
for the Casimir energy of two concentric spherical shells
does not allow to evaluate the error committed when using
M. S. R. MILTA˜O PHYSICAL REVIEW D 78, 065023 (2008)
065023-8
9. it. This approach supplies the dominant term, the energy of
two parallel plates and nothing more. In the approximation
(61) that we call small annular region approximation, we
have the same dominant term as that in the PFA, and more
terms that supply corrections to this dominant term.
Supposedly, the expression of the energy with these cor-
rections better approaches the energy in the case of two
concentric shells. Let us notice that the first correction in
(61) to the term in
À3
that gives the energy of two parallel
plates is a term in
À1
that originates a term with the
ða2
dÞÀ1
dependence; to follow comes the term in
0
, which
originates a term with the ða3ÞÀ1 dependence. It is noticed,
then, the absence of the term in
À2. This absence is not a
decurrent accident of the considered approaches, as it can
be verified in a tedious inspection of higher order in the
Debye expansion.
IV. FINAL CONSIDERATIONS
The expression (61) also allows to evaluate the error that
occurs by using the PFA for the energy of two concentric
spherical shells. The first term reveals that for each sepa-
ration d between the shells, it has an error that diminishes
with the square of the radius of the internal shell, that is,
with its area. The second term shows that it has an error
that diminishes with the cube of its radius, that is, with its
volume. It is important to note the absence of a term in
À2
that would originate a term with ðad2ÞÀ1 dependence and
therefore, an error that would diminish with the proper
radius of the internal spherical shell.
Deriving the expression from the energy (61) in relation
to the separation d, we get the Casimir pressure in small
annular region approximation
P ða; bÞ ¼ P5
innerÀsphere PFAðdÞ þ
1
8
À
48
@c
2a2
d2
; (62)
where P5
inner-sphere PFAðdÞ is the proximity-force approxi-
mation for the pressure between the spherical shells
P 5
inner-sphere PFAðdÞ ¼ À
2@c
240
1
d4
; (63)
that is, the pressure between the parallel plates.
For the calculation of the relative correction to the PFA
pressure, we have
25. ¼
1
8
À
48
120
2
d2
a2
¼ 0:7240
d
a
2
:
(64)
For example, for d=a ¼ 0:1, we have a relative error of
0.72%. For a ! 1, we have a null relative error, corre-
sponding to the situation of parallel plates. Fixing one
determined radius a, the relative error (64) decreases
with the quadratic power when the parameter d diminishes.
The pressure Pða; bÞ also was obtained previously by
Brevik, Skurdal, and Sollie [20], using the Green function
formalism. However, our result not only differs from that
one in the coefficients’ values for each term, as also for the
terms itself. In fact, in Brevik and collaborators’ result, it
appears the dependence in ðad3
ÞÀ1
proceeding from the
contribution of the
À2
order that is absent in our result. In
the formalism that we have adopted, we could verify that
this contribution does not appear even if we consider
superior orders in 1= (that is, 1=2 or above) in the
Debye expansion.
Concerning the qualitative different result obtained rela-
tive Ref. [20], which was obtained by a different method,
after all, only when measurement of the energy can be
performed will a definite answer be found.
ACKNOWLEDGMENTS
We would like to thank F. A. Farias, Y. H. Ribeiro, M. V.
Cougo-Pinto, and P. A. Maia-Neto for the previous reading
and commenting on this work. We also would like to thank
Dra. Ludmila Oliveira H. Cavalcante for the suggestions
that left the text more clear and readable.
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