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Publisher: Springer-Verlag   (Total: 2353 journals)

 Showing 1201 - 1400 of 2353 Journals sorted alphabetically J. of Clinical Geropsychology       (Followers: 1) J. of Clinical Immunology       (Followers: 14, SJR: 1.332, h-index: 75) J. of Clinical Monitoring and Computing       (Followers: 1, SJR: 0.661, h-index: 37) J. of Clinical Psychology in Medical Settings       (Followers: 13, SJR: 0.46, h-index: 34) J. of Cluster Science       (SJR: 0.416, h-index: 31) J. of Coal Science and Engineering (China)       (SJR: 0.188, h-index: 8) J. of Coastal Conservation       (Followers: 7, SJR: 0.474, h-index: 25) J. of Coatings Technology and Research       (Followers: 5, SJR: 0.425, h-index: 25) J. of Combinatorial Optimization       (Followers: 5, SJR: 1.093, h-index: 34) J. of Communications Technology and Electronics       (Followers: 2, SJR: 0.29, h-index: 16) J. of Community Genetics       (Followers: 1, SJR: 0.727, h-index: 14) J. of Community Health       (Followers: 8, SJR: 0.921, h-index: 44) J. of Comparative Physiology A: 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h-index: 65) J. of Ornithology       (Followers: 24) J. of Orofacial Orthopedics / Fortschritte der Kieferorthopädie       (SJR: 0.574, h-index: 33)
 Environmental Fluid Mechanics   [SJR: 0.585]   [H-I: 29]   [5 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1573-1510 - ISSN (Online) 1567-7419    Published by Springer-Verlag  [2353 journals]
• Flow and drag force around a free surface piercing cylinder for
environmental applications
• Authors: Thomas Ducrocq; Ludovic Cassan; Jacques Chorda; Hélène Roux
Pages: 629 - 645
Abstract: Abstract This paper investigates flows around a free surface piercing cylinder with Froude number F > 0.5 and Reynolds number around Re = 50,000. The aim of this work is to gain a better understanding of the flow behaviour in environmental systems such as fishways. The advances are based upon experimental and numerical results. Several flow discharges and slopes are tested to obtain both subcritical and supercritical flows. The drag force exerted on the cylinder is measured with the help of a torque gauge while the velocity field is obtained using particle velocimetry. For the numerical part, two URANS turbulence models are tested, the k- $$\omega$$ SST and the RNG k- $$\varepsilon$$ models using the OpenFOAM software suite for subcritical cases, and then compared with the corresponding experimental results. With fishways applications in mind, the changes in drag coefficient $$C_d$$ versus Froude number and water depth are studied and experimental correlations proposed. We conclude that the most suitable URANS turbulence model for reproducing this kind of flow is the k- $$\omega$$ SST model.
PubDate: 2017-08-01
DOI: 10.1007/s10652-016-9505-9
Issue No: Vol. 17, No. 4 (2017)

• Identification of evolving interfacial waves by spatial harmonic analysis
• Authors: Meysam Fazeli; Ammar Safaie; Mirmosadegh Jamali
Pages: 647 - 664
Abstract: Abstract This paper is concerned with an analysis of image processing data to identify interfacial waves at the interface between two fluid layers in a laboratory flume. The interfacial waves are excited through a non-linear resonant interaction with a surface wave traveling in a wave flume filled with a two-layer fluid. A spatial harmonic analysis is proposed to extract information about the nonlinear evolution of the constituent waves from the interface oscillation data. The analysis is capable of handling different stages of the resonance in the wave flume and gives accurate and consistent results on time variations of the wave amplitudes. Contrary to a temporal harmonic analysis, the proposed method does not require prescription of a functional form for the amplitude variations. The experimental measurements are compared to the theoretical predictions and available methods. The proposed method provides an efficient solution for detecting these waves and capturing their behaviors.
PubDate: 2017-08-01
DOI: 10.1007/s10652-016-9506-8
Issue No: Vol. 17, No. 4 (2017)

• Ritter’s dry-bed dam-break flows: positive and negative wave
dynamics
• Authors: Oscar Castro-Orgaz; Hubert Chanson
Pages: 665 - 694
Abstract: Abstract Dam-break flood waves are associated with major environmental disasters provoked by the sudden release of water stored in reservoirs. Ritter found in 1892 an analytical solution to the wave structure of an ideal fluid released during an instantaneous dam failure, propagating over initially dry horizontal terrain. This solution, though ideal, hence frictionless, is widely used to test numerical solutions of the Shallow Water Equations (SWE), and as educational tool in courses of fluid mechanics, given that it is a peculiar case of the Riemann problem. However, the real wave structure observed experimentally differs in a major portion of the wave profile, including the positive and negative fronts. Given the importance of an accurate prediction of the dam break wave, the positive and negative wave portions originating from the breaking of a dam with initially dry land on the tailwater reach are revisited in this work. First, the propagation features of the dry-front are investigated using an analytical boundary-layer type model (Whitham/Dressler/Chanson model) constructed matching an (outer) inviscid dynamic wave to an (inner) viscous diffusive wave. The analytical solution is evaluated using an accurate numerical solution of the SWE produced using the MUSCL-Hancock finite-volume method, which is tested independently obtaining the solution based on the discontinuous Galerkin finite-element method. The propagation features of the negative wave are poorly reproduced by the SWE during the initial stages of dam break flows, and, thus, are then investigated using the Serre–Green–Naghdi equations for weakly-dispersive fully non-linear water waves, which are solved using a finite volume-finite difference scheme.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9512-5
Issue No: Vol. 17, No. 4 (2017)

• Influence of the secondary motions on pollutant mixing in a meandering
open channel flow
• Authors: Ignacio J. Moncho-Esteve; Frederik Folke; Manuel García-Villalba; Guillermo Palau-Salvador
Pages: 695 - 714
Abstract: Abstract This paper presents large eddy simulation of turbulent flow in a meandering open channel with smooth wall and rectangular cross-section. The Reynolds number based on the channel height is 40,000 and the aspect ratio of the cross-section is 4.48. The depth-averaged mean stream-wise velocity agree well to experimental measurements. In this specific case, two interacting cells are formed that swap from one bend to the other. Transport and mixing of a pollutant is analysed using three different positions of release, e.g. on the inner bank, on the outer bank and on the centre of the cross section. The obtained depth-average mean concentration profiles are reasonably consistent with available experimental data. The role of the secondary motions in the mixing processes is the main focus of the discussion. It is found that the mixing when the scalar is released on the centre of the cross-section is stronger and faster than the mixing of the scalar released on the sides. When the position of release is close to a bank side, the mixing is weaker and a clear concentration of scalar close to the corresponding side-wall can be observed in both cases.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9513-4
Issue No: Vol. 17, No. 4 (2017)

• A stochastic Lagrangian micromixing model for the dispersion of reactive
scalars in turbulent flows: role of concentration fluctuations and
improvements to the conserved scalar theory under non-homogeneous
conditions
• Authors: Andrea Amicarelli; Giovanni Leuzzi; Paolo Monti; Stefano Alessandrini; Enrico Ferrero
Pages: 715 - 753
Abstract: Abstract Several reaction schemes, based on the conserved scalar theory, are implemented within a stochastic Lagrangian micromixing model to simulate the dispersion of reactive scalars in turbulent flows. In particular, the formulation of the reaction-dominated limit (RDL) reaction scheme is here extended to improve the model performance under non-homogeneous conditions (NHRDL scheme). The validation of the stochastic model is obtained by comparison with the available measurements of reactive pollutant concentrations in a grid-generated turbulent flow. This test case describes the dispersion of two atmospheric reactant species (NO and O3) and their reaction product (NO2) in an unbounded turbulent flow. Model inter-comparisons are also assessed, by considering the results of state-of-the-art models for pollutant dispersion. The present validation shows that RDL reaction scheme provides a systematic overestimation (relative error of ca. 85% around the centreline) in computing the local reactant consumption/production rate, whereas the NHRDL scheme drastically reduces this gap (relative error lower than 5% around the centreline). In terms of NO2 production (or reactant consumption), neglecting concentration fluctuations determines overestimations of the product mean of around 100% and a NO2 local production of one order of magnitude higher than the reference simulation. In terms of standard deviations, the concentration fluctuations of both the passive and reactive scalars are generally of the same order of magnitude or up to 1 or 2 orders of magnitudes higher than the corresponding ensemble mean values, except for the background reactant close to the plume edges. The study highlights the importance of modelling pollutant reactions depending on the instantaneous instead of the mean concentrations of the reactants, thus quantifying the role of the turbulent fluctuations of concentration, in terms of scalar statistics (mean, standard deviation, intensity of fluctuations, skewness and kurtosis of concentration, segregation coefficient, simulated reaction rate). This stochastic particle method represents an efficient numerical technique to solve the convection–diffusion equation for reactive scalars and involves several application fields: micro-scale air quality (urban and street-canyon scales), accidental releases, impact of odours, water quality and fluid flow industrial processes (e.g. combustion).
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9516-1
Issue No: Vol. 17, No. 4 (2017)

• One-dimensional modelling of sediment deposition in reservoirs with
sinking streams
• Authors: Xiaofeng Zhang; Yi Hu; Siqiang Wang; Yao Yue
Pages: 755 - 775
Abstract: Abstract It is common in karst regions that rivers are occasionally cut by mountains, resulting in the alternate appearances of open channel flow and pressurized flow. With more and more reservoirs being built in this region, the complicated sediment transport processes of such mixture flow are urgently needed to study. In this paper a one-dimensional numerical model with free surface and pressurized flows coupled together is presented. The simulated results are analyzed to explore deposition process in reservoirs with sinking streams; impacts of different hydraulic conditions on the sedimentation are also studied. To verify the computed results, a flume experiment is also conducted. The results show that deposition of sediments mainly occurs in open channel reaches where the longitudinal profile is similar to that of conventional reservoirs, i.e. typical delta has formed, demonstrating characteristics of deltaic deposition morphology in that the crest of delta moves to the downstream direction over time. The model provided by this paper is not only proved to simulate the characteristics of deposition in karst reservoirs successfully, but also reveal the impacts of hydraulic conditions in such circumstances.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9517-0
Issue No: Vol. 17, No. 4 (2017)

• Single-column model and large eddy simulation of the evening transition in
the planetary boundary layer
• Authors: G. C. Cuchiara; B. Rappenglück
Pages: 777 - 798
Abstract: Abstract In the present study, the well-known case of day 33 of the Wangara experiment is resimulated using the Weather Research and Forecasting (WRF) model in an idealized single-column mode to assess the performance of a frequently used planetary boundary layer (PBL) scheme, the Yonsei University PBL scheme. These results are compared with two large eddy simulations for the same case study imposing different surface fluxes: one using previous surface fluxes calculated for the Wangara experiment and a second one using output from the WRF model. Finally, an alternative set of eddy diffusivity equations was tested to represent the transition characteristics of a sunset period, which led to a gradual decrease of the eddy diffusivity, and replaces the instantaneous collapse of traditional diagnostics for eddy diffusivities. More appreciable changes were observed in air temperature and wind speed (up to 0.5 K, and 0.6 m s−1, respectively), whereas the changes in specific humidity were modest (up to 0.003 g kg−1). Although the representation of the convective decay in the standard parameterization did not show noticeable improvements in the simulation of state variables for the selected Wangara case study day, small changes in the eddy diffusivity over consecutive hours throughout the night can impact the simulation of distribution of trace gases in air quality models. So, this work points out the relevance of simulating the turbulent decay during sunset, which could help air quality forecast models to better represent the distribution of pollutants storage in the residual layer during the entire night.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9518-z
Issue No: Vol. 17, No. 4 (2017)

• Penetrative convection in slender containers
• Authors: Suhas U. Pol; Harindra J. S. Fernando
Pages: 799 - 814
Abstract: Abstract An experimental study was conducted to investigate the penetration of a convective mixed layer into an overlying stably (solutally) stratified layer contained in a narrow, tall vessel when the fluid is subjected to a destabilizing heat flux from below. The interest was the evolution of the bottom mixed-layer height ( $$h$$ ) with time ( $$t$$ ) in the presence of side-wall effects, but without the formation of conventional double-diffusive layers. The side-wall effects are expected at small mixed-layer aspect ratios, $$\varGamma_{h} = (W/h)$$ , where $$W$$ is the container width. This case has not been studied hitherto, although there are important environmental and industrial applications. The mixed-layer growth laws for low aspect ratio convection were formulated by assuming a balance between the vertical kinetic energy flux at the interface and the rate of change of potential energy of the fluid system due to turbulent entrainment. The effects of sidewalls were considered using similarity arguments, by taking characteristic rms velocities to be a function of $$\varGamma_{h}$$ , in addition to buoyancy flux ( $$q_{0}$$ ) and $$h$$ . In all stages of evolution, the similarity variables $$\xi = h/W$$ and $$t^{\prime } = Nt/A$$ , where $$A = N^{3} W^{2} /4q_{0}$$ and $$N$$ is the buoyancy frequency, scaled the mixed-layer evolution data remarkably well. Significant wall effects were noted when $$\varGamma_{h} < 1$$ , and for this case the interfacial vertical turbulent velocity and length scales were identified via scaling arguments and experimental data.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9522-3
Issue No: Vol. 17, No. 4 (2017)

• Hydrodynamics and bed stability of open channel flows with submerged
foliaged plants
• Authors: M. X. Bao; C. W. Li
Pages: 815 - 831
Abstract: Abstract In this work we investigate experimentally and numerically the flow structure around foliaged plants deployed in a channel with gravels on the bed under submerged and barely submerged conditions. Velocity and Reynolds stress were measured by using a NORTEK Vectrino profiler. Visual observation shows that the initial motion of gravels is easier to be triggered under the condition of flow with barely submerged vegetation. This is confirmed by the measured velocity, Reynolds stress and total kinetic energy (TKE) profiles. The velocity exhibits a speed up in the near-bed region, and the associated Reynolds stress and TKE increase there. A 3D numerical model is then verified against the experiments and used to investigate systematically the effect of degree of submergence of foliaged plants on the channel bed shear stress. The results show that the maximum bed shear stress occurs when the foliage is situated slightly below the water surface, which can enhance channel bed instability.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9524-1
Issue No: Vol. 17, No. 4 (2017)

• Influence of a flooding event discharge on accretion in wetlands
• Authors: T. Serra; A. Ros; C. Vergés; X. Casamitjana
Pages: 833 - 851
Abstract: Abstract Flooding events in wetlands transport sediment particles landwards and can increase accretion in some areas of the wetland or resuspend the sediment in other areas. In this study a flooding event with different water discharges was analyzed in a laboratory simulated wetland to determine the effect stem density has on particle trapping. The discharge that entered the simulated wetland was a particle laden barotropic current that initially produced a pulse that traveled through the wetland. After the first pulse, a baroclinic current, with a different timescale to the initial pulse, developed. Three stem densities, along with the ‘without plant case’, were considered. A semi-empirical model was formulated to explain the propagation of the water pulse. The model predicted the velocity of the pulse dampening in the presence of the simulated vegetation, by using the roughness parameter that had been found to increase with stem density. The baroclinic current propagated at a lower velocity than the pulse did, and its velocity decreased with stem density. As less sediment was found in the wetland with denser canopies, this indicates that the presence of a canopy acts as a barrier to sediment transportation. Furthermore, a greater amount of sediment was deposited in regions at the foot of the denser vegetated wetland zone and the sediment deposition also increased with the water discharge.
PubDate: 2017-08-01
DOI: 10.1007/s10652-017-9528-x
Issue No: Vol. 17, No. 4 (2017)

• Gyre formation in open and deep lacustrine embayments: the example of Lake
Geneva, Switzerland
• Authors: A. M. Razmi; U. Lemmin; D. Bouffard; A. Wüest; R. E. Uittenbogaard; D. A. Barry
Pages: 415 - 428
Abstract: Abstract Numerical simulations were carried out to investigate gyres within open lacustrine embayments subjected to parallel-to-shore currents. In such embayments, gyre formation occurs due to flow separation at the embayment’s upstream edge. High momentum fluid from the mixing layer between the embayment and offshore flows into the embayment and produces recirculating flow. Systematic numerical experiments using different synthetic embayment configurations were used to examine the impact of embayment geometry. Geometries included embayments with different aspect ratios, depths and embayment corner angles. The magnitudes of the recirculation and turbulent kinetic energy (TKE) in the embayment vary significantly for angles in the range 40°–55°. Embayments with corner angles less than 50° have much stronger recirculation and TKE, other parameters remaining the same. The numerical findings are consistent with gyre formation observed in two embayments located in Lake Geneva, Switzerland, and thus help explain flow patterns recorded in lacustrine shoreline regions.
PubDate: 2017-06-01
DOI: 10.1007/s10652-016-9494-8
Issue No: Vol. 17, No. 3 (2017)

• Impact of windage on ocean surface Lagrangian coherent structures
• Authors: Michael R. Allshouse; Gregory N. Ivey; Ryan J. Lowe; Nicole L. Jones; C. J. Beegle-Krause; Jiangtao Xu; Thomas Peacock
Pages: 473 - 483
Abstract: Abstract Windage, the additional direct, wind-induced drift of material floating at the free surface of the ocean, plays a crucial role in the surface transport of biological and contaminant material. Lagrangian coherent structures (LCS) uncover the hidden organizing structures that underlie material transport by fluid flows. Despite numerous studies in which LCS ideas have been applied to ocean surface transport scenarios, such as oil spills, debris fields and biological material, there has been no consideration of the influence of windage on LCS. Here we investigate and demonstrate the impact of windage on ocean surface LCS via a case study of the ocean surrounding the UNESCO World Heritage Ningaloo coral reef coast in Western Australia. We demonstrate that the inclusion of windage is necessary when applying LCS to the study of surface transport of any floating material in the ocean.
PubDate: 2017-06-01
DOI: 10.1007/s10652-016-9499-3
Issue No: Vol. 17, No. 3 (2017)

• Comparison of analytical descriptions of nocturnal low-level jets within
the Ekman model framework
• Authors: Elizabeth Smith; Evgeni Fedorovich; Alan Shapiro
Pages: 485 - 495
Abstract: Abstract This study focuses on the inertial oscillation aspect of the nocturnal low-level jet (NLLJ). In the context of the Ekman model solutions, conceptual NLLJ inertial oscillation analytical frameworks proposed by Blackadar in 1957 and Shapiro and Fedorovich and van de Wiel et al. in 2010 are compared. Considering a NLLJ produced via direct numerical simulation over flat terrain with no baroclinic influence as a reference case, the deficiencies of each framework in representing a realistic NLLJ are assessed. The Blackadar theory results in unrealistic wind profiles near the surface. While extensions of Blackadar’s framework by Shapiro and Fedorovich and van de Wiel et al. produce more realistic NLLJs, the simpler approach taken by van de Wiel et al. does not describe the NLLJ wind hodograph at later times sufficiently in qualitative terms.
PubDate: 2017-06-01
DOI: 10.1007/s10652-016-9502-z
Issue No: Vol. 17, No. 3 (2017)

• LES validation of urban flow, part II: eddy statistics and flow structures
• Authors: Denise Hertwig; Gopal Patnaik; Bernd Leitl
Pages: 551 - 578
Abstract: Abstract Time-dependent three-dimensional numerical simulations such as large-eddy simulation (LES) play an important role in fundamental research and practical applications in meteorology and wind engineering. Whether these simulations provide a sufficiently accurate picture of the time-dependent structure of the flow, however, is often not determined in enough detail. We propose an application-specific validation procedure for LES that focuses on the time dependent nature of mechanically induced shear-layer turbulence to derive information about strengths and limitations of the model. The validation procedure is tested for LES of turbulent flow in a complex city, for which reference data from wind-tunnel experiments are available. An initial comparison of mean flow statistics and frequency distributions was presented in part I. Part II focuses on comparing eddy statistics and flow structures. Analyses of integral time scales and auto-spectral energy densities show that the tested LES reproduces the temporal characteristics of energy-dominant and flux-carrying eddies accurately. Quadrant analysis of the vertical turbulent momentum flux reveals strong similarities between instantaneous ejection-sweep patterns in the LES and the laboratory flow, also showing comparable occurrence statistics of rare but strong flux events. A further comparison of wavelet-coefficient frequency distributions and associated high-order statistics reveals a strong agreement of location-dependent intermittency patterns induced by resolved eddies in the energy-production range. The validation concept enables wide-ranging conclusions to be drawn about the skill of turbulence-resolving simulations than the traditional approach of comparing only mean flow and turbulence statistics. Based on the accuracy levels determined, it can be stated that the tested LES is sufficiently accurate for its purpose of generating realistic urban wind fields that can be used to drive simpler dispersion models.
PubDate: 2017-06-01
DOI: 10.1007/s10652-016-9504-x
Issue No: Vol. 17, No. 3 (2017)

• Numerical and experimental study of sensitivity factors on heat island of
residence community
• Authors: Weixue Cao; Yuying Xue
Abstract: Abstract How to mitigate the effect of the urban heat island has become a hot research problem in the field of the environment, landscape engineering, and the architectural design. The heat island of the micro-climate, such as the residence community, has much more impact on the people’s life, and should be paid more attentions. In this paper, the experiment and simulation methods are combined to study the heat island of residence community. Four sensitivity factors (wind velocity, underlying and building surface, anthropogenic heating and landscape) are chosen to study their influences on the urban heat island (UHI). The results show that the correlation coefficients of the above four sensitivity factors on the UHI are −0.718, 0.521, 0.251 and −0.229, respectively. And the sequence of the above factors to reduce UHI from strong to weak is wind velocity, underlying and the building surface, anthropogenic heating and the landscape.
PubDate: 2017-07-21
DOI: 10.1007/s10652-017-9544-x

• A new model to predict the relative permeability of wetting phase for
tight sandstones
• Authors: Yuhang Guo; Baozhi Pan; Lihua Zhang; Chunhui Fang
Abstract: Abstract As the existing model is not suitable for tight sandstone reservoirs, a new model has been derived theoretically to correlate the relative permeability of wetting phase, saturation and the tortuosity ratio based on revised Kozeny–Carman equation. According to the similarity between the flow of fluid and current, we found that the wetting phase relative permeability is a function of the tortuosity ratio, saturation and the saturation index. In order to verify the new model, 11 cores were taken from tight sandstone reservoirs in northwestern and northeastern China. By comparing with the existing models, the new model is more suitable for the core which has low permeability (<10 mD) and saturation index (<1.8).
PubDate: 2017-07-20
DOI: 10.1007/s10652-017-9543-y

• On dust concentration profile above an area source in a neutral
atmospheric surface layer
• Authors: Zhengping Zhu; Ruifeng Hu; Xiaojing Zheng; Yan Wang
Abstract: Abstract The power law and logarithmic law among others are the most commonly well-known theories for equilibrium mean profile of dust concentration in a neutral atmospheric surface layer. However, these theories are based on the assumption of homogeneity in horizontal (streamwise) direction, thereby the horizontal advection term in the transport equation is neglected. In this study, with the help of a self-similar analytical solution of the two-dimensional steady-state dust transport equation (Chamecki and Meneveau, J Fluid Mech 683:1–26,  2011), we examine the applicabilities of the assumption which the one-dimensional theories are based on. In addition, we also propose an empirical fit of the theory, and good agreements with reference data have been obtained. The new approximate profile can be used as an alternative of the one-dimensional theories in practical applications.
PubDate: 2017-07-12
DOI: 10.1007/s10652-017-9542-z

• Large-eddy simulation of turbulent flow and structures within and above an
idealized building array
• Authors: M. Saeedi; B.-C. Wang
Abstract: Abstract In this research, wall-modeled LES has been performed to investigate turbulent flow in an idealized urban environment comprised of an array of rectangular buildings (MUST array). An inflow condition based on generation of grid turbulence is utilized to mimic the turbulent approaching boundary layer at the inlet of the computational domain. The numerical approach has been validated by comparing the first- and second-order statistical moments of the turbulent flow field against the available water-channel measurement data. Turbulent coherent flow structures, energy spectra, and budget balance of the resolved kinetic energy are investigated. With the aid of the dynamic nonlinear subgrid-scale (SGS) stress model, the backscatter of the kinetic energy from small to large scales of flow motions is also studied. It is found that the scatter of kinetic energy is sensitive to the presence of the building obstacles. As the elevation increases, the ratio of the SGS dissipation rate to the viscous dissipation rate remains stable under the backward scatter condition but varies significantly under the forward scatter condition.
PubDate: 2017-07-01
DOI: 10.1007/s10652-017-9540-1

• Development and validation of a Lagrangian method for 3D turbulent flows
with curvilinear free-surface
• Authors: Mitra Javan; Sharareh Mahmodinia; Hamed Hasani
Abstract: Abstract A numerical model was developed based on a moving grid method for simulating three-dimensional turbulent flows affected by curvilinear free surface. Reynolds-Averaged Navier–Stokes equations with the k–ε turbulence model were solved in non-orthogonal curvilinear coordinates. In the free surface, the kinematic boundary condition was implicitly imposed in the pressure Poisson equation derived from the momentum and continuity equations. The water surface elevation was calculated at each time step without solving additional equations. The developed numerical model was validated using the experimental data of the strongly curved channels and submerged hydraulic jump. The numerical simulation of the flow field and free surface elevation in all cases were compared with the experimental results. The agreement between the simulated and measured results was satisfactory for three-dimensional turbulent flows in the strongly curved channels and submerged hydraulic jump. The secondary flow and dip phenomena were correctly simulated in strongly curved channels. This numerical model could accurately predict steep water surface gradients in a submerged hydraulic jump. The reasonable results of the numerical simulation demonstrate the capability of the presented Lagrangian method in hydraulic engineering applications. This model provides a suitable method for simulating the free surface in engineering and environmental problems.
PubDate: 2017-06-30
DOI: 10.1007/s10652-017-9541-0

• Hydrodynamics and turbulence in emergent and sparsely vegetated open
channel flow
• Authors: Soumen Maji; Debasish Pal; Prashanth R. Hanmaiahgari; Umesh P. Gupta
Abstract: Abstract This present study reports the results of an experimental study characterizing thorough variation of turbulent hydrodynamics and flow distribution in emergent and sparsely vegetated open channel flow. An emergent and rigid sparse vegetation patch with regular spacing between stems along the flow and transverse directions was fixed in the central region of the cross-section of open channel. Experiments were conducted in subcritical flow conditions and velocity measurements were obtained with an acoustic Doppler Velocimetry system. Large variations of the turbulence intensities, Reynolds shear stress, turbulent kinetic energy and vortical motions are found in and around the vegetation patch. At any cross-section through the interior of the vegetation patch, streamwise velocity decreases with increase in streamwise length and the velocity profiles converge from the log-law to a linear profile with increasing slope. Time-averaged lateral and vertical velocities inside the vegetation patch increase with increasing streamwise distance and converge from negative values to positive values. Turbulence intensities interior of the sparse vegetation patch are more than those of without the vegetation patch. Similar to the trend of streamwise velocity profiles inside the vegetation, turbulence intensities and longitudinal-normal Reynolds shear stress profile decreases with streamwise direction. In the interior of the vegetation patch and downstream of the trailing edge, turbulent kinetic energy profiles are exhibiting irregular fluctuations and the maximum values are occurring in the outer layer. Analysis of flow distribution confirms sparse vegetation patch is inducing a serpentine flow pattern in its vicinity. At the leading edge, flow is rushing towards the right hand sidewall, and at the trailing edge, flow is turning to the left hand sidewall. In between the leading and trailing edges, the streamlines are following a zig-zag fashion at varied degree along the streamwise and lateral directions. Immediate upstream of the leading edge and in the interior of the vegetation patch, vortex motion is clearly visible and the vortices are stretched along the width of the channel with streamwise direction.
PubDate: 2017-05-12
DOI: 10.1007/s10652-017-9531-2

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# Feature Engineering Techniques

Engineering features is key to improving your LB score. Below are some ideas on how to engineer new features. Create a new feature and then evaluate it with a local validation scheme to see if it improves your model's CV (and thus LB). Keep beneficial features and discard the others.

If you create lots of new features at once, you can use forward feature selection, recursive feature elimination, LGBM importance, or permutation importance to determine which are useful.

The kernel here by Konstantin shows this procedure and demonstrates many of the following techniques

## NAN processing

If you give np.nan to LGBM, then at each tree node split, it will split the non-NAN values and then send all the NANs to either the left child or right child depending on what’s best. Therefore NANs get special treatment at every node and can become overfit. By simply converting all NAN to a negative number lower than all non-NAN values (such as - 999),

df[col].fillna(-999, inplace=True)

df[col].fillna(-999, inplace=True)

then LGBM will no longer overprocess NAN. Instead it will give it the same attention as other numbers. Try both ways and see which gives the highest CV.

## Label Encode/ Factorize/ Memory reduction

Label encoding (factorizing) converts a (string, category, object) column to integers. Afterward you can cast it to int8, int16, or int32 depending on whether max is less than 128, less than 32768, or not. Factorizing reduces memory and turns NAN into a number (i.e. -1) which affects CV and LB as described above. Factorizing also gives you the choice to treat categorical variable as numeric described below.

df[col],_ = df[col].factorize() if df[col].max()<128: df[col] = df[col].astype('int8') elif df[col].max()<32768: df[col] = df[col].astype('int16') else: df[col].astype('int32')

df[col],_ = df[col].factorize()

if df[col].max()<128: df[col] = df[col].astype('int8')
elif df[col].max()<32768: df[col] = df[col].astype('int16')
else: df[col].astype('int32')

Additionally for memory reduction, people use the popular memory_reduce function on the other columns. A simpler and safer approach is to convert all float64 to float32 and all int64 to int32. (It's best to avoid float16. You can use int8 and int16 if you like).

for col in df.columns:
if df[col].dtype=='float64': df[col] = df[col].astype('float32')
if df[col].dtype=='int64': df[col] = df[col].astype('int32')

## Categorical Features

With categorical variables, you have the choice of telling LGBM that they are categorical or you can tell LGBM to treat it as numerical (if you label encode it first). Either way, LGBM can extract the category classes. Try both ways and see which gives the highest CV. After label encoding do the following for category or leave it as int for numeric

df[col] = df[col].astype('category')

## Splitting

A single (string or numeric) column can be made into two columns by splitting. For example a string column id_30 such as “Mac OS X 10_9_5” can be split into Operating System “Mac OS X” and Version “10_9_5”. Or for example number column TransactionAmt “1230.45” can be split into Dollars “1230” and Cents “45”. LGBM cannot see these pieces on its own, you need to split them.

## Combining

Two (string or numeric) columns can be combined into one column. For example card1 and card2 can become a new column with

df['uid'] = df[‘card1’].astype(str)+’_’+df[‘card2’].astype(str)

Numeric columns can combined with adding, subtracting, multiplying, etc. This helps LGBM because by themselves card1 and card2 may not correlate with target and therefore LGBM won’t split them at a tree node. But the interaction uid = card1_card2 may correlate with target and now LGBM will split it.

## Frequency Encoding

Frequency encoding is a powerful technique that allows LGBM to see whether column values are rare or common. For example, if you want LGBM to "see" which credit cards are used infrequently, try

temp = df['card1'].value_counts.to_dict() df['card1_counts'] = df['card1'].map(temp)

## Aggregations / Group Statistics

Providing LGBM with group statistics allows LGBM to determine if a value is common or rare for a particular group. You calculate group statistics by providing pandas with 3 variables. You give it the group, variable of interest, and type of statistic. For example,

temp = df.groupby('card1')['TransactionAmt'].agg(['mean'])
.rename({'mean':'TransactionAmt_card1_mean'},axis=1)
df = pd.merge(df,temp,on='card1',how='left')

The feature here adds to each row what the average TransactionAmt is for that row's card1 group. Therefore LGBM can now tell if a row has an abnormal TransactionAmt for their card1 group.

## Normalize / Standardize

You can normalize columns against themselves. For example

df[col] = ( df[col]-df[col].mean() ) / df[col].std()

Or you can normalize one column against another column. For example if you create a Group Statistic (described above) indicating the mean value for D3 each week. Then you can remove time dependence by

df['D3_remove_time'] = df['D3'] - df['D3_week_mean']

The new variable D3_remove_time no longer increases as we advance in time because we have normalized it against the affects of time.

## Outlier Removal / Relax / Smooth / PCA

Normally you want to remove anomalies from your data because they confuse your models. However in this competition, we want to find anomalies so use smoothing techniques carefully. The idea behind these methods and to determine and remove uncommon values. For example, after frequency encoding a variable, you can remove all values that appear less than 0.1% by replacing them with a new value like -9999.

#### 'EDA Study > 캐글' 카테고리의 다른 글

 Chris의 Feature Engineering 팁  (0) 2019.09.12 2019.09.11 2019.03.02

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UR Home

Circular sets of primes of imaginary quadratic number fields

URN to cite this document:
urn:nbn:de:bvb:355-epub-5594
DOI to cite this document:
10.5283/epub.559
Vogel, Denis
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Date of publication of this fulltext: 05 Aug 2009 13:23

Abstract

Let p be an odd prime number and let K be an imaginary
quadratic number field whose class number is not divisible by p. For a set S of primes of K whose norm is congruent to 1 modulo p, we introduce the notion of strict circularity. We show that if S is strictly circular, then the group G(KS(p)=K) is of cohomological dimension 2 and give some explicit examples.

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# I Square Root in an alternating power series

1. Apr 24, 2016

### Satirical T-rex

I had a question similar to Σ0 (-1)^n (x)^(n/2) and attempted to solve it using the root test getting abs(√x)<1, but I've also seen some places answer it as √abs(x)<1 so am I skipping a step.

2. Apr 24, 2016

### BvU

Hello again,
You want to read your new thread as if you were an interested helper. At this moment
doesn't make much sense to me and I expect it doesn't to many others either. What is this about and what is your question ?

3. Apr 24, 2016

### Satirical T-rex

I'm sorry about that, I was trying to find the interval of convergence of ∑ 0, n→∞ (-1)^n*(x)^(n/2), I got √x<1 or [0,1) after using the root test but when I checked Wolfram it said the absolute value of √x or (1,1) and I tried to find a similar question with the steps involved but couldn't, I hope it makes more sense now, I tried to proofread it but thats not my strong suit.

4. Apr 24, 2016

### BvU

do the terms look familiar if I rewrite as$$\sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$

5. Apr 24, 2016

### Satirical T-rex

$$\sum_{{\rm n}=0}^\infty \left (-1 \right )^n \left (\sqrt x \right )^\left (n/2 \right ) \ \ \rm ?$$
sorry, not great at putting in equations just figuring it out hope it's less vague now, also I just realized the root test would make the -1 a moot point anyway, thanks for trying to work with my vagueness, also if I haven't made it clear I'm trying to figure out if the interval of integration is 0 to 1 or -1 to 1.

6. Apr 25, 2016

### BvU

Hold it, post #1 doesn't have $\displaystyle \sum_{{\rm n}=0}^\infty \left (-1 \right )^n \left (\sqrt x \right )^\left (n/2 \right ) \ \ \rm$ but (from $x^{n\over 2}$ ) $$\sum_{{\rm n}=0}^\infty \left (-1 \right )^n \left (\sqrt x \right )^n\ \ = \sum_{{\rm n}=0}^\infty \left (- \sqrt x \right )^n\ \ \rm !$$

7. Apr 25, 2016

### Satirical T-rex

I didn't notice the square root and copied yours to figure out the equation text so didn't notice there either really sorry about making this more complicated than it should be, used to using fractional exponents in sums not roots, looks like you had it right.


text:
# Practical Applications

• ## Reservoir Simulation

 What is Reservoir Simulation? Reservoir simulation is the art of combining physics, mathematics, reservoir engineering, and computer programming to develop a tool for predicting hydrocarbon reservoir performance via various operating strategies. It is an important decision-making tool. For example, engineers use it to obtain information pertaining to the processes that take place in oil reservoirs. Such information enables an analysis of the various recovery strategies in order to effect optimal oil recovery. The crucial part of reservoir simulations is to solve large-scale discretized PDEs (highly coupled, nonsymmetric, and indefinite) over and over again. However, this is also the most time-consuming process of any modern petroleum reservoir simulator (more than 75%). The complexity of the geometry and of the physical model, heterogeneity, and size of reservoir model are continuing to grow, which makes these linear systems more difficult to solve using standard direct or iterative solvers. A typical reservoir. The FASP Method for Reservoir Simulations The FASP method takes full advantage of the underlying physical and analytic properties of the mathematical model.           Transforms the complicated Jacobian system into three simpler auxiliary problems: an elliptic problem for the pressure variables, a hyperbolic problem for the saturation variables, and a purely algebraic problem for the well bottom-hole pressure variables. Thus, it can be used to design efficient and robust smoothers or preconditioners for each auxiliary problem. It then couples the auxiliary problems and applies the preconditioned Krylov subspace methods. Numerical Results for Reservoir Simulations SPE 10 Benchmark             Two phases (water and oil): 1.1 million cells, 5 wells. Reference: X. Hu, W. Liu, G. Qin, J. Xu, Y. Yan, and C. Zhang (2011);  X. Hu, J. Xu, and C. Zhang (2013)

• ## Fluid Structure Interaction

 What is FSI? Fluid-structure interaction (FSI) aims at understanding the interaction between moving structure and fluid and how their interaction affects the interface between them. FSI has a wide range of applications in many areas including hemodynamics and wind/hydro turbines simulation. FSI problems are computationally challenging. The computational domain of FSI consists of fluid and structure subdomains. The position of the interface between fluid domain and structure domain is time dependent. Therefore, the shape of the fluid domain is one of the unknowns, increasing the nonlinearity of the FSI problems. Examples of fuel cells: (left) cardiac simulation; (right) 2D Turek benchmark Monolithic Solver for FSI Numerical solutions of FSI are roughly classified into partitioned approaches and monolithic approaches. Partitioned approaches employ single-physics solvers to solve the fluid and structure problems separately and then couple them by the interface conditions. Monolithic approaches solve the fluid and structure problems simultaneously. Monolithic approaches are considered more stable, although it is accompanied with larger linear systems and higher computational cost.         Lagrangian coordinate is used for structure equations, and Eulerian coordinate for fluid equations. We use ALE method to update the fluid mesh. We discretize the coupled fluid and structure equations in a single linear system. In order to solve it efficiently, we develop block preconditioners for this problem, which proves to be robust with respect to varying parameters and problems sizes. Numerical Results for 2D Benchmark Turek Fluid-Structure Interaction Benchmark References: J. Xu and K. Yang (2014) Hydroelectric Generator Hydroelectric generator simulation involves the moving fluid domain, which is due to the rotation and deformation of the blade of the generator. Therefore, efficient monolithic solver for the couple system is difficult to construct. In order to simulation fluid coupled with rotating structures, like hydroturbine in hydroelectric generators, we develop a new ALE method to update the fluid mesh so that it can handle arbitrary rotation. Numerical Results for Hydroelectric Generator Moving mesh for real hydroelectric generator Artificial Heart Artificial heart is a kind of effective treatment for heart failure, which is the finial battlefield of cardiovascular disease. The artificial heart significantly changes the hemodynamics of the aorta. The main difficulty of artificial heart simulation is the complex interaction between blood, aorta and artificial heart. Artificial heart in vessel Numerical Results for Artificial Heart Hemodynamics near the artificial heart References: Q. Zhang, B. Gao, K. Gu, Y. Chang, J. Xu and P. Deuflhard (2014)
• ## Magnetohydrodynamics

 What is MHD? The magnetohydrodynamics (MHD) model describes the dynamics of charged fluids in the presence of electromagnetic fields. A principle application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics. MHD is also used to model the flow of liquid metals, for example, in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis. Tokamak Structure-preserving Discretization and Robust Solver for MHD We keep the electric field $E$ to form the mixed formulation of the MHD system, which preserves the divergence-free condition of the magnetic field exactly on the discrete level. The block lower triangular preconditioner for FGMRes is applied. When the time step size is small enough, robust number of iterations can be proved theoretically. And no such constrain required in actual tests. Numerical Results for MHD Lid-driven cavity, Re=400, Rm=400: (left) stream line of the velocity; (right) distribution of the total magnetic field       Lid-driven cavity, Re=400, Rm=400: number of iterations Reference: K. Hu, Y. Ma, and J. Xu (2015);  K. Hu, X. Hu, Y. Ma, and J. Xu (2015)

• ## Energy Storage

 Lithium Ion Battery Lithiumion batteries are rechargeable, and they are characterized by lithium ions that move from the negative electrode to the positive electrode during discharge and then back again during charging. Newton-Krylov-Multigrid Schemes for Lithium Ion Battery Simulation Finite volume method Newton’s method for the Butler-Volmer equation, and the whole nonlinear system Krylov subspace method (GMRes) with block Gauss-Seidel preconditioner Multigrid method for solving the Poisson-like problems in the preconditioner References: J. Wu, V. Srinivasan, X,  and C-Y, Wang (2002); J. Wu, J. Xu, and H. Zou (2006). Fuel Cells A fuel cell is a device that converts a fuel’s chemical energy from a fuel into electricity through a chemical reaction with oxygen or another oxidizing agent. Hydrogen is the most commonly used fuel for this purpose, but hydrocarbons such as natural gas and alcohols like methanol are sometimes used. Examples of fuel cells. Robust Methods for Fuel Cells Newton’s method for the nonlinear system Kirchhoff transformation Finite element-upwind finite volume method Nonoverlapping Schwarz domain decomposition method Overlapping domain decomposition method with non-matching grids Newton-Krylov-based solvers Numerical Results for Fuel Cells Fast convergence within 21 iterations versus oscillatory/nonconvergent iterations using commercial CFD software. References: P.Sun, G. Xue, C-Y. Wang, and Xu (2008);  P.Sun, G. Xue, C-Y. Wang, and Xu (2009); P.Sun, C-Y. Wang, and Xu (2010).
• ## Subsurface Flow Simulation

 What is subsurface flow simulation? Subsurface flow, in hydrology, is the flow of water beneath the earth’s surface that constitutes part of the water cycle. Numerical Results for Subsurface Flow Simulation Reference:  J. Cheng, X. Huang, S. Shu, J. Xu, C. Zhang, S. Zhang, and Z. Zhou (2013)


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Drawing

It's the 2nd quadrant of a unit circle, with an angle (theta) subtending it. Where the "leading edge"? (I can't remember my terminology) meets the circle, there is a horizontal dotted line leading to the y-axis, and a vertical dotted line leading to the x-axis. Where the vertical line meets the x axis there is the equation [x = cos \theta]. Where the horizontal line meets the y axis, there is the equation [y = sin \theta]. There are two more dotted lines that are red. They meet at a point slightly lower on the circle. Where each of these lines meets the axes there is y-prime and x-prime. I'm having trouble understanding how to draw in the tikz graphic package (what I've assumed I should be using based on the LaTex manual) and would appreciate any help or being pointed in the right direction! If anybody has a link to a walk-through of a similar drawing that would be AMAZING. Thanks so much for your guys' help.

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# Thread: distribution of the mean of an exponential distribution (mgf)

1. ## distribution of the mean of an exponential distribution (mgf)

Let $\overline{X}$ be the mean of a random sample from the exponential distribution, Exp( $\theta$). Using the mgf technique determine the distribution of $\overline{X}$.

$M_{\overline{X}}(t) = Ee^{t\overline{X}}$

$M_{\overline{X}}(t) = Ee^{\frac{t}{n}\sum{X_i}}$

$M_{\overline{X}}(t) = e^{\frac{1}{n}}(1-\frac{t}{\theta})^{-n}$ I'm sure this is wrong, but I don't know why.

The book has the answer in the back, saying it should be $Gamma(\alpha = n, \beta = \frac{\theta}{n})$, but I don't know how to get there. MGF problems often seem to give me difficulty.

2. Let $Y=\sum_{k=1}^nX_k$, where each $X_k\sim\Gamma(1,\theta)$.
Then by indep $Y\sim\Gamma(n,\theta)$.
Then let $Z=Y/n$ and do a calc one change of variables.

OR if you wish to use MGFs...
$E(e^{\bar X t}) = E(e^{Y(t/n)})=M_Y(t/n)$.
Now, I don't know how you're writing your exponential/gamma densities.
BUT subtitute t/n for t in a $\Gamma(n,\theta)$ MGF and it's over.


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# M09 Q7

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M09 Q7 [#permalink]

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25 Jun 2012, 18:07
On the picture below, is the area of the triangle $$ABC$$ greater than 1?

m09-7.gif

$$\angle ABC \lt 90^\circ$$
Perimeter of the triangle $$ABC$$ is greater than $$\frac{4}{a}$$

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

Given isosceles triangle ABC with $$\text{base}=AC=\frac{2}{a}$$ and $$\text{height}=a^2$$ .

$$\text{area}=\frac{1}{2}*\text{base}*\text{height}=\frac{1}{2}*\frac{2}{a}*a^2=a$$ . So, the question basically ask whether $$a \gt 1$$ ?

(1) $$\angle ABC \lt 90^\circ$$ $$\right$$ assume $$\angle ABC=90^\circ$$ then hypotenuse is $$AC=\frac{2}{a}$$ , as ABC becomes isosceles right triangle ( $$45-45-90=1-1-\sqrt{2}$$ ) then the $$\text{leg}=BC=AB=\frac{2}{a \sqrt{2}}=\frac{\sqrt{2}}{a}$$ .

But $$BC=\frac{\sqrt{2}}{a}$$ also equals to $$\sqrt{(\frac{1}{a})^2+(a^2)^2}$$ , so we have $$\frac{\sqrt{2}}{a}=\sqrt{(\frac{1}{a})^2+(a^2)^2}$$ $$\right$$ $$2=1+a^6$$ $$\right$$ $$a^6=1$$ $$\right$$ $$a=1$$ (as $$a$$ per diagram is positive). Now, if we increase $$a$$ then the base $$\frac{2}{a}$$ will decrease and the height $$a^2$$ will increase thus making angle ABC smaller than 90 and if we decrease $$a$$ then the base $$\frac{2}{a}$$ will increase and the height $$a^2$$ will decrease thus making the angle ABC greater than 90. So, as angle ABC is less than 90, $$a$$ must be more than 1. Sufficient.

(2) Perimeter of the triangle $$ABC$$ is greater than $$\frac{4}{a}$$ $$\right$$ $$P=BC+AB+AC=2BC+AC=2\sqrt{(\frac{1}{a})^2+(a^2)^2}+\frac{2}{a} \gt \frac{4}{a}$$ $$\right$$ $$\sqrt{\frac{1+a^6}{a^2}} \gt \frac{1}{a}$$ $$\right$$ $$a^6 \gt 0$$ $$\right$$ $$a \gt 0$$ . Hence we don't know whether $$a \gt 1$$ is true. Not sufficient.

I understand the above explanation. Is there some shorter and faster method of solving this question? Anyone has a good strategic guess for this one?
Kellogg MMM ThreadMaster
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Re: M09 Q7 [#permalink]

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25 Jun 2012, 20:58
Picture missing.

Regards,
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Re: M09 Q7 [#permalink]

### Show Tags

26 Jun 2012, 00:54
teal wrote:
On the picture below, is the area of the triangle $$ABC$$ greater than 1?

m09-7.gif

$$\angle ABC \lt 90^\circ$$
Perimeter of the triangle $$ABC$$ is greater than $$\frac{4}{a}$$

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

Given isosceles triangle ABC with $$\text{base}=AC=\frac{2}{a}$$ and $$\text{height}=a^2$$ .

$$\text{area}=\frac{1}{2}*\text{base}*\text{height}=\frac{1}{2}*\frac{2}{a}*a^2=a$$ . So, the question basically ask whether $$a \gt 1$$ ?

(1) $$\angle ABC \lt 90^\circ$$ $$\right$$ assume $$\angle ABC=90^\circ$$ then hypotenuse is $$AC=\frac{2}{a}$$ , as ABC becomes isosceles right triangle ( $$45-45-90=1-1-\sqrt{2}$$ ) then the $$\text{leg}=BC=AB=\frac{2}{a \sqrt{2}}=\frac{\sqrt{2}}{a}$$ .

But $$BC=\frac{\sqrt{2}}{a}$$ also equals to $$\sqrt{(\frac{1}{a})^2+(a^2)^2}$$ , so we have $$\frac{\sqrt{2}}{a}=\sqrt{(\frac{1}{a})^2+(a^2)^2}$$ $$\right$$ $$2=1+a^6$$ $$\right$$ $$a^6=1$$ $$\right$$ $$a=1$$ (as $$a$$ per diagram is positive). Now, if we increase $$a$$ then the base $$\frac{2}{a}$$ will decrease and the height $$a^2$$ will increase thus making angle ABC smaller than 90 and if we decrease $$a$$ then the base $$\frac{2}{a}$$ will increase and the height $$a^2$$ will decrease thus making the angle ABC greater than 90. So, as angle ABC is less than 90, $$a$$ must be more than 1. Sufficient.

(2) Perimeter of the triangle $$ABC$$ is greater than $$\frac{4}{a}$$ $$\right$$ $$P=BC+AB+AC=2BC+AC=2\sqrt{(\frac{1}{a})^2+(a^2)^2}+\frac{2}{a} \gt \frac{4}{a}$$ $$\right$$ $$\sqrt{\frac{1+a^6}{a^2}} \gt \frac{1}{a}$$ $$\right$$ $$a^6 \gt 0$$ $$\right$$ $$a \gt 0$$ . Hence we don't know whether $$a \gt 1$$ is true. Not sufficient.

I understand the above explanation. Is there some shorter and faster method of solving this question? Anyone has a good strategic guess for this one?

This question is discussed here: ds-triangle-m09q07-72173.html
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Re: M09 Q7   [#permalink] 26 Jun 2012, 00:54
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## FR Coveralls

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# Arc length parametrization

#### Differential geometry of curves - Wikipedia

Let (i) n N {\displaystyle n\in \mathbb {N} } {\displaystyle n\in \mathbb {N} } , (ii) r { N, Any regular curve may be parametrized by the arc length (the natural parametrization) and from the point of view of a bug on the curve that does not...

#### Moving Along a Curve with Specified Speed - Geometric Tools

, t = f(s) is referred to as reparameterization by arc length. How do you, Let X(s) be an arc-length parameterization of a curve. Let Y(t) be...

#### WolframAlpha Examples: Arc Length

Calculator to compute the arc length of a curve. Specify a curve in polar coordinates or parametrically. Compute arc length in arbitrarily many dimensions.

#### REPARAMETRIZATION OF NURBS CURVES 1 ... - Bologna

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particular parametrization called S that provides us a tangent that has unit. length. .... The arc length is the parametrization that gives us constant unit length.

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A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamilton's equations. This method can be...

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Parameterization by Arc Length. Recall that like parametric equations, vector valued function describe not just the path of the particle, but also how the particle is...

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, 3.1.2 Arc length preserving flows . . . . . . . . . . . . . . . . . 17. 3.2 Flows on discrete arc length parametrized curves . . . . . . . . . 18. 3.2.1 Tangent flow .

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Chord-length versus arc-length parametrization. in u. Compare this rubber-band construction with that of arc-length parametrization, where the parameter...

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#### Geometry of Curves - Piazza

reparametrization of r(t), and is often called an arc-length parametrization of the curve. However, if we attempt to do find a reparametrization on a non-regular...

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#### HANDOUT THREE: GEOMETRY OF SPACE CURVES 1. Unit speed ...

on the particular choice of parameterization: for instance, Kepler's second law, calling that speed is the derivative of arc-length, what this means it that we want.

#### ArcLengthWolfram Language Documentation

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#### What is a curve parametrized by arc length? Ngô Quc Anh

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text:
# What is the molar mass of sulfuric acid H_2SO_4?

Apr 8, 2017

#### Answer:

Sulfuric acid's molar mass is $\text{98.08 g/mol}$.

#### Explanation:

Step one: Find the atomic masses in each atom -- units: $\text{g/mol}$

• Hydrogen: $1.01$
• Sulfur: $32.06$
• Oxygen: $16$

Step two: Find how many of those atoms are of the formula:

• Hydrogen: $2$ atoms
• Sulfur: $1$ atom
• Oxygen: $4$ atoms

Step three: Add all the atomic masses together. (Note, if there are more than one of a particular atom, you must multiply it by how many of that atom is in that formula):

$\text{MM} = \left(1.01 \times 2\right) + 32.06 + \left(16 \times 4\right)$

$= \text{ 98.08 g/mol}$

Hope this helped!


text:
# American Institute of Mathematical Sciences

December  2018, 38(12): 6015-6027. doi: 10.3934/dcds.2018152

## The two membranes problem for fully nonlinear operators

 1 Mathematics Department, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712, USA 2 Departamento de Matemática, Universidad Nacional de Mar del Plata/Conicet, Deán Funes 3600, Mar del Plata, Buenos Aires 7600, Argentina

* Corresponding author

Received  August 2017 Revised  February 2018 Published  April 2018

We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal $C^{1, 1}$ regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.

Citation: Luis Caffarelli, Luis Duque, Hernán Vivas. The two membranes problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6015-6027. doi: 10.3934/dcds.2018152
##### References:

show all references

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### Home > CALC > Chapter 6 > Lesson 6.4.2 > Problem6-149

6-149.

elnx = x is another way of saying that y = ex and y = lnx are inverse functions.
Inverse functions have reciprocal derivatives at their corresponding (x, y) → (y, x) values.

If f(x) has coordinate point (a, b).

$\text{And }f'(a)=\frac{2}{5}$

And if g(x) is the inverse of f(x),

$\text{then }g'(b)=\frac{5}{2}$

Let f(x) = ex and g(x) = lnx.
First find f '(x) = _____. (This should be easy!)

To find g'(x), take the reciprocal of f '(x), but NOT at the
x-value...

$g'(x)=\frac{1}{f'(y)}=\frac{1}{f'(g(x))}$

Now evaluate g'(x) when f(x) = ex.