Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

An Alternative Section Method for Casting and Posterior Laser Welding of Metallic Frameworks for an Implant-Supported Prosthesis

Purpose: The aim of this study was to compare the accuracy of fit of three types of implant-supported frameworks cast in Ni-Cr alloy: specifically, a framework cast as one piece compared to frameworks cast separately in sections to the transverse or the diagonal axis, and later laser welded. Materials and Methods: Three sets of similar implant-supported frameworks were constructed. The first group of six 3-unit implant-supported frameworks were cast as one piece, the second group of six were sectioned in the transverse axis of the pontic region prior to casting, and the last group of six were sectioned in the diagonal axis of the pontic region prior to casting. The sectioned frameworks were positioned in the matrix (10 N(.)cm torque) and laser welded. To evaluate passive fit, readings were made with an optical microscope with both screws tightened and with only one-screw tightened. Data were submitted to ANOVA and Tukey-Kramer`s test (p < 0.05). Results: When both screws were tightened, no differences were found between the three groups (p > 0.05). In the single-screw-tightened test, with readings made opposite to the tightened side, the group cast as one piece (57.02 +/- 33.48 mu m) was significantly different (p < 0.05) from the group sectioned diagonally (18.92 +/- 4.75 mu m) but no different (p > 0.05) from the group transversally sectioned (31.42 +/- 20.68 mu m). On the tightened side, no significant differences were found between the groups (p > 0.05). Conclusions: Results of this study showed that casting diagonally sectioned frameworks lowers misfit levels of prosthetic implant-supported frameworks and also improves the levels of passivity to the same frameworks when compared to structures cast as one piece.

Modified section method for laser-welding of ill-fitting cp Ti and Ni-Cr alloy one-piece cast implant-supported frameworks

P>This study aimed to verify the effect of modified section method and laser-welding on the accuracy of fit of ill-fitting commercially pure titanium (cp Ti) and Ni-Cr alloy one-piece cast frameworks. Two sets of similar implant-supported frameworks were constructed. Both groups of six 3-unit implant-supported fixed partial dentures were cast as one-piece [I: Ni-Cr (control) and II: cp Ti] and evaluated for passive fitting in an optical microscope with both screws tightened and with only one screw tightened. All frameworks were then sectioned in the diagonal axis at the pontic region (III: Ni-Cr and IV: cp Ti). Sectioned frameworks were positioned in the matrix (10-Ncm torque) and laser-welded. Passive fitting was evaluated for the second time. Data were submitted to anova and Tukey-Kramer honestly significant difference tests (P < 0 center dot 05). With both screws tightened, one-piece cp Ti group II showed significantly higher misfit values (27 center dot 57 +/- 5 center dot 06 mu m) than other groups (I: 11 center dot 19 +/- 2 center dot 54 mu m, III: 12 center dot 88 +/- 2 center dot 93 mu m, IV: 13 center dot 77 +/- 1 center dot 51 mu m) (P < 0 center dot 05). In the single-screw-tightened test, with readings on the opposite side to the tightened side, Ni-Cr cast as one-piece (I: 58 center dot 66 +/- 14 center dot 30 mu m) was significantly different from cp Ti group after diagonal section (IV: 27 center dot 51 +/- 8 center dot 28 mu m) (P < 0 center dot 05). On the tightened side, no significant differences were found between groups (P > 0 center dot 05). Results showed that diagonally sectioning ill-fitting cp Ti frameworks lowers misfit levels of prosthetic implant-supported frameworks and also improves passivity levels of the same frameworks when compared to one-piece cast structures.

Energy spectrum and eigenfunctions through the Quantum Section Method

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Quantum section method for the soft stadium

The soft stadium is defined by a monomial potential with exponent α as a parameter, such that α → ∞ corresponds to the billiard. The practical use of the quantum section method depends only on the partial separability of the system on both sides of the section, which holds for all α's. In particular, for α = 1.0, the system becomes globally separable, allowing for a general test of the method. For various values of the parameter, we also tested the use of the asymptotic WKB-type approximation in the construction of Green's functions and asymptotic overlap integrals to obtain higher energy eigenvalues. We show these approximations to be reliable. © 2000 Elsevier Science B.V. All rights reserved.

Prosthetic misfit of implant-supported prosthesis obtained by an alternative section method

PURPOSE. Adequate passive-fitting of one-piece cast 3-element implant-supported frameworks is hard to achieve. This short communication aims to present an alternative method for section of one-piece cast frameworks and for casting implant-supported frameworks. MATERIALS AND METHODS. Three-unit implant-supported nickel-chromium (Ni-Cr) frameworks were tested for vertical misfit (n = 6). The frameworks were cast as one-piece (Group A) and later transversally sectioned through a diagonal axis (Group B) and compared to frameworks that were cast diagonally separated (Group C). All separated frameworks were laser welded. Only one side of the frameworks was screwed. RESULTS. The results on the tightened side were significantly lower in Group C (6.43 +/- 3.24 mu m) when compared to Groups A (16.50 +/- 7.55 mu m) and B (16.27 +/- 1.71 mu m) (P<.05). On the opposite side, the diagonal section of the one-piece castings for laser welding showed significant improvement in the levels of misfit of the frameworks (Group A, 58.66 +/- 14.30 mu m; Group B, 39.4.8 +/- 12.03 mu m; Group C, 23.13 +/- 8.24 mu m) (P<.05). CONCLUSION. Casting diagonally sectioned frameworks lowers the misfit levels. Lower misfit levels for the frameworks can be achieved by diagonally sectioning one-piece frameworks. [J Adv Prosthodont 2012;4:89-92]

Fiberglass-reinforced glulam beams: Mechanical properties and theoretical model

The glued-laminated lumber (glulam) technique is an efficient process for making rational use of wood. Fiber-Reinforced Polymers (FRPs) associated with glulam beams provide significant gains in terms of strength and stiffness, and also alter the mode of rupture of these structural elements. In this context, this paper presents a theoretical model for designing reinforced glulam beams. The model allows for the calculation of the bending moment, the hypothetical distribution of linear strains along the height of the beam, and considers the wood has a linear elastic fragile behavior in tension parallel to the fibers and bilinear in compression parallel to the fibers, initially elastic and subsequently inelastic, with a negative decline in the stress-strain diagram. The stiffness was calculated by the transformed section method. Twelve non-reinforced and fiberglass reinforced glulam beams were evaluated experimentally to validate the proposed theoretical model. The results obtained indicate good congruence between the experimental and theoretical values.

Theoretical model and experimental analysis of glulam beam reinforced with FRP

This work present a study of glulam beams reinforced with FRP. It was developed a theoretical model that calculates strength and stiffness of the beams. The model allows for the calculation of the bending moment, the hypothetical distribution of linear strains along the height of the beam, and considers the wood has a linear elastic fragile behavior in tension parallel to the fibers and bilinear in compression parallel to the fibers, initially elastic and subsequently inelastic, with a negative decline in the stress-strain diagram. The stiffness was calculated by the transformed section method. Twelve non-reinforced and fiberglass reinforced glulam beams were evaluated experimentally to validate the proposed theoretical model. The results obtained indicate good congruence between the experimental and theoretical values.

Chronic Pain after Cesarean Section. Influence of Anesthetic/Surgical Technique and Postoperative Analgesia

Cancado TO, Omais M, Ashmawi HA, Torres MLA - Chronic Pain after Cesarean Section. Influence of Anesthetic/Surgical Technique and Postoperative Analgesia. Background and objectives: Brazil ranks second among countries with the highest rates of cesarean section in the world. Little is known about the future consequences of this procedure on maternal health. This study investigated the influence of anesthetic/surgical technique and postoperative analgesia on the onset of chronic pain after three months of cesarean section. Method: This is a prospective randomized study of 443 patients undergoing cesarean section (elective and emergency), with different doses of hyperbaric bupivacaine 0.5% and opioids in spinal anesthesia. Patients were alocated into five groups as follow: G1 received hyperbaric bupivacaine (8 mg), sufentanil (2.5 mu g), and morphine (100 mu g); G2 received hyperbaric bupivacaine (10 mg), sufentanil (2.5 mu g), and morphine (100 mu g); G3 received hyperbaric bupivacaine (12.5 mg) and morphine (100 mu g); G4 received hyperbaric bupivacaine (15 mg) and morphine (100 mu g); G5 received hyperbaric bupivacaine (12.5 mg) and morphine (100 mu g), without perioperative anti-inflammatory. Pain at rest and in movement were evaluated in the immediate postoperative period. Phone contact was made after three months of surgery for identification of patients with chronic pain. Results: The incidence of chronic pain in the groups was G1 = 20%; G2 = 13%; G3 = 7.1%; G4 = 2.2%, and G5 = 20.3%. Patients who reported higher pain scores in the postoperative period had a higher incidence of chronic pain (p < 0.05). Conclusion: The incidence of chronic pain decreases with higher doses of local anesthetics and use of anti-inflammatory drugs. The higher pain scores in the postoperative period were associated with chronic pain development after three months of cesarean section.

Postpartum depression in Recife - Brazil: prevalence and association with bio-socio-demographic factors

OBJECTIVE:To use a semi-structured interview to detect depression in postpartum women according to the criteria proposed by the DSM in child health care clinics in the city of Recife, together with the proper association of this disorder to bio-socio-demographic data. METHODS: The study used a cross-section method and contained a convenience sample of 400 women that were between 2 and 26 weeks of postpartum in child health care clinics. A bio-socio-demographic questionnaire and the Portuguese version of the Structured Clinical Interview for DSM-IV Axis I Disorders were used. RESULTS: Twenty nine of the mothers (7.2%) were diagnosed as suffering from postpartum depression. Women with a past history of psychiatric disorders, a family history of psychiatric disorder and some sort of clinical complication presented a higher prevalence of depression. The same happened to those with a past history of spontaneous abortion, those who had a transpelvic birth and those over 8 weeks of puerperium. CONCLUSION: The rate of postpartum depression in this sample, 7.2%, was lower than that reported by other Brazilian studies. It probably occurred because the other researchers used screening scales to assess this estimate instead of a clinical interview.

As consequências econômicas e sociais da representação legislativa das mulheres

Com o aumento progressivo do percentual das mulheres nas câmaras legislativas do mundo, examinamos quais seriam suas consequências em decisões de gastos públicos, saúde e educação na pré-infância e até na aprovação de medidas polêmicas, como o aborto sob demanda. Sob a luz dos modelos de ``cidadão-candidato'' e com base em evidências empíricas de que mulheres têm preferências políticas mais voltadas ao bem-estar social, utilizamos o método do corte seccional aplicado a médias no tempo a fim de testar nossas hipóteses. De fato, a presença feminina nos congressos do mundo traz maiores gastos públicos sobre produto, direcionados principalmente a saúde e educação, além de uma redução nos gastos militares. Nas taxas de matrícula em educação pré-primária, também há influência positiva de parlamentares do gênero feminino, o mesmo não podendo ser dito sobre indicadores de saúde infantil. Em uma análise gráfica, encontramos relação positiva entre mulheres nos parlamentos e legalização do aborto e do casamento homossexual.