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# Event Calendar

Mar
6
Fri
Graduate Seminar in Fluid Mechanics @ 213 Hodson Hall
Mar 6 @ 4:00 pm – 5:00 pm

4:10 pm Presentation

A Vortex Sheet as an Interface Retaining Coarsed Model for Multiphase Flow

Presented by XIANYANG CHEN (Adviser: Prof. Tryggvason)

Multiphase flows are characterized by sharp moving phase boundaries, separating different fluids or phases. In many cases the dynamics of the interfaces determines the behavior of the flow. In a coarse, or reduced order model, either an averaged two-fluid model or a large-eddy-simulation like one, it is therefore critical to retain a sharp interface for the resolved scales. To develop a more general strategy we are experimenting with a particularly simple approach, modeling the interface for a weakly stratified flow using an inviscid vortex sheet with baroclinically generated vorticity, advected using  the Biot-Savart law, and adjustable model parameters to match the results from a full resolved simulation. We start by filtering the fully resolved results, using a filter that we call weighted coordinates smoothing (WCS), that simplifies the interface and concentrates the vorticity at the interface. The model parameters (“vortex blob’’ sizes) are found by optimization, using a cost function that compares the velocity of the filtered interface with the model prediction. Once the model parameters have been found, we plan to use machine learning to develop a strategy to evolve them in time. We discuss the overall work plan, the filtering strategy, and show preliminary results.

4:35 pm Presentation

“Optimal Prediction of Wall Shear Stress from Filtered Velocity Datas

Presented by MENGZE WANG (Adviser: Prof. Zaki)

Resolving near-wall turbulence is difficult in simulations of high-Reynolds-number flows. To mitigate the near-wall resolution requirements and reduce computational cost, wall models are indispensable in practical numerical solvers such as large-eddy simulations (LES). Conventional wall models adopt instantaneous LES velocity as their input and predict a corresponding instantaneous wall shear-stress which is fed back to LES as a boundary condition. Despite the numerous existing wall models, a fundamental question remains unaddressed: given instantaneous filtered off-wall velocity data, what is the highest possible accuracy of predicted wall shear stress? This problem is investigated using an adjoint variational data assimilation approach. By combining a full Navier-Stokes solver with off-wall filtered-velocity data, we reconstruct the initial condition that tracks the full flow state and optimally predicts the spatio-temporal evolution of the wall shear stress. We demonstrate that the optimal prediction is robust to the filter width. As the height of the first available velocity grid point increases, the correlation between the true and optimally predicted wall stresses deteriorates, but remains at least twice as large as that of the true and equilibrium wall-model stresses. Finally, the Reynolds number effect on the optimal prediction is explored.

Mar
13
Fri
Graduate Seminar in Fluid Mechanics @ 213 Hodson Hall
Mar 13 @ 4:00 pm – 5:00 pm

4:10 pm Presentation

Effect of Axial Casing Groove Geometry on Rotor-Groove Interactions in the Tip Region of a Compressor

Presented by SUBHRA SHANKHA KOLEY (Adviser: Prof. Katz)

The present experimental study expands an ongoing effort to characterize the interactions of axial casing grooves (ACGs) with the flow in the tip region of an axial turbomachine. In recent work, we have tested a series of grooves with the same inlet geometry but with different exit directions. Two geometries have stood out: The U grooves, which have an outflow in the negative circumferential direction (opposing the blade motion) are the most effective in suppressing stall, but cause a decrease in efficiency around the best efficiency point (BEP). In contrast, the S grooves, which have an outflow in the positive circumferential direction, achieve a milder improvement in stall suppression but do not degrade the performance near BEP. Stereo-Particle Image Velocimetry (SPIV) measurements is carried out at various planes to elucidate the flow in the tip region and within the U and S grooves. At low flow rates, the inflow into both grooves peaks periodically when the blade pressure side (PS) faces the entrance (downstream side) to the grooves. This inflow rolls up into a large vortex that remains and lingers within the groove long after the blade clears the groove. The outflow depends on the shape of the groove. For the S groove, the outflow exits at the upstream end of the groove in the positive circumferential direction, as designed. In contrast, for the U grooves, the fast radially and circumferentially negative outflow peaks at the base of the U. The resulting jet causes substantial periodic variations in the flow angle near the leading edge of the rotor blade. Close to the BEP, the chordwise location of primary blade loading moves downstream, as expected. The inflow into the grooves occurs for a small fraction of the blade passing period, and most of the tip leakage vortex remains in the main flow passage. For the S grooves, the rotor-groove interactions seem to be minimal, with little (but not zero) inflow or outflow at both ends, and minimal changes to the flow angle in the passage. In contrast, for the U groove, the inflow into and outflow from the groove reverses direction (compared to the low flowrate trends). The resulting entrainment of secondary flows from the groove into the passage are likely contributors to the reduced efficiency at BEP for the U grooves.

4:35 pm Presentation

“An Input-Output Inspired Method for Permissible Perturbation Amplitude of Transitional Wall-Bounded Shear Flows

Presented by CHANG LIU (Adviser: Prof. Gayme)

Determining the permissible level of perturbation to maintain a laminar flow state is of critical importance in a wide range of applications. Historical approaches to this problem in wall-bounded shear flows focused on linear analysis that is only provably valid in a small neighborhood of the laminar solution and precludes the evaluation of the role of nonlinearity. Extensive simulations or experimental studies allow exploration of the full dynamics, but a finite set cannot provide a definitive bound. This work takes an input-output approach that partitions the dynamics into a feedback interconnection of the linear and nonlinear dynamics (i.e., a Luré system in which the nonlinearity is static feedback). We construct a model of the nonlinear term that is constrained by system physics to be energy conserving (passive) and to have bounded input-output energy. We then formulate computation of the region of attraction of the laminar state (set of safe perturbations) and permissible perturbation amplitude as Linear Matrix Inequalities (LMI), which allows a more computationally efficient solution than prevailing nonlinear approaches based on Sum of Squares (SOS) programming. We apply our approach to low dimensional nonlinear shear flow models for a range of Reynolds numbers. The results from our analytically derived bounds are consistent with the bounds identified through exhaustive simulations. However, our results are obtained at a much lower computational cost and have the benefit of providing a provable guarantee that a certain level of perturbation is permissible.

Apr
2
Thu
Online Mechanical Engineering Spring Seminar Series: Class 530.804 @ Zoom: https://wse.zoom.us/j/560809286
Apr 2 @ 3:00 pm – 4:00 pm

via Zoom: https://wse.zoom.us/j/560809286

“Neuroscience in The Matrix: Closing the loop around the brain to understand how it controls the body”

Presented by Professor Noah Cowan
Department of Mechanical Engineering, Johns Hopkins University

The nervous system is a sophisticated control system, controlling equally sophisticated biomechanical “plant”. Understanding how the nervous system (1) encodes and processes sensory information, (2) transforms it into meaningful intermediate representations in the brain, and (3) computes motor output, therefore, involves decoding a complex closed-loop control system.

This talk will present a research program devoted to developing and applying ideas in engineering to decode closed-loop neuromechanical control in animals (including humans). Central to this program is the ability to wrap artificial feedback systems around freely behaving animals—in some cases by “reading their minds,” i.e. decoding neural activity in real time. Using this approach, the direct neural output is used to adjust sensory inputs on the fly in an effort to disentangle the interactions between the nervous system and the biomechanical plant that it controls.

Noah J. Cowan received a B.S. degree from the Ohio State University, Columbus, in 1995, and M.S. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1997 and 2001 – all in electrical engineering. Following his Ph.D., he was a Postdoctoral Fellow in Integrative Biology at the University of California, Berkeley for 2 years. In 2003, he joined the mechanical engineering department at Johns Hopkins University, Baltimore, MD, where he is now a Professor. Prof. Cowan’s research interests include mechanics and multisensory control in animals and machines. Prof. Cowan received the NSF PECASE award in 2010, the James S. McDonnell Foundation Scholar Award in Complex Systems in 2012, and two Johns Hopkins Discovery Awards in 2015 and 2016.  In addition, Prof. Cowan received the William H. Huggins Award for excellence in teaching in 2004, and the Dunn Family Award in 2014, conferred for having “. . . an extraordinarily positive impact upon the lives of one or more undergraduate students.”

Apr
3
Fri
Online Graduate Seminar in Fluid Mechanics @ Join on-line via Zoom: https://wse.zoom.us/j/435449376
Apr 3 @ 4:00 pm – 5:00 pm

Join on-line via Zoom: https://wse.zoom.us/j/435449376

“Effect of Axial Casing Groove Geometry on Rotor-Groove Interactions in the Tip Region of a Compressor”

Presented by SUBHRA SHANKHA KOLEY
(Adviser: Prof. Katz)

The present experimental study expands an ongoing effort to characterize the interactions of axial casing grooves (ACGs) with the flow in the tip region of an axial turbomachine. In recent work, we have tested a series of grooves with the same inlet geometry but with different exit directions. Two geometries have stood out: The U grooves, which have an outflow in the negative circumferential direction (opposing the blade motion) are the most effective in suppressing stall, but cause a decrease in efficiency around the best efficiency point (BEP). In contrast, the S grooves, which have an outflow in the positive circumferential direction, achieve a milder improvement in stall suppression but do not degrade the performance near BEP. Stereo-Particle Image Velocimetry (SPIV) measurements is carried out at various planes to elucidate the flow in the tip region and within the U and S grooves. At low flow rates, the inflow into both grooves peaks periodically when the blade pressure side (PS) faces the entrance (downstream side) to the grooves. This inflow rolls up into a large vortex that remains and lingers within the groove long after the blade clears the groove. The outflow depends on the shape of the groove. For the S groove, the outflow exits at the upstream end of the groove in the positive circumferential direction, as designed. In contrast, for the U grooves, the fast radially and circumferentially negative outflow peaks at the base of the U. The resulting jet causes substantial periodic variations in the flow angle near the leading edge of the rotor blade. Close to the BEP, the chord-wise location of primary blade loading moves downstream, as expected. The inflow into the grooves occurs for a small fraction of the blade passing period, and most of the tip leakage vortex remains in the main flow passage. For the S grooves, the rotor-groove interactions seem to be minimal, with little (but not zero) inflow or outflow at both ends, and minimal changes to the flow angle in the passage. In contrast, for the U groove, the inflow into and outflow from the groove reverses direction (compared to the low flowrate trends). The resulting entrainment of secondary flows from the groove into the passage are likely contributors to the reduced efficiency at BEP for the U grooves.

“An Input-Output Inspired Method for Permissible Perturbation Amplitude of Transitional Wall-Bounded Shear Flows”

Presented by CHANG LIU
(Adviser: Prof. Gayme)

Determining the permissible level of perturbation to maintain a laminar flow state is of critical importance in a wide range of applications. Historical approaches to this problem in wall-bounded shear flows focused on linear analysis that is only provably valid in a small neighborhood of the laminar solution and precludes the evaluation of the role of nonlinearity. Extensive simulations or experimental studies allow exploration of the full dynamics, but a finite set cannot provide a definitive bound. This work takes an input-output approach that partitions the dynamics into a feedback interconnection of the linear and nonlinear dynamics (i.e., a Luré system in which the nonlinearity is static feedback). We construct a model of the nonlinear term that is constrained by system physics to be energy conserving (passive) and to have bounded input-output energy. We then formulate computation of the region of attraction of the laminar state (set of safe perturbations) and permissible perturbation amplitude as Linear Matrix Inequalities (LMI), which allows a more computationally efficient solution than prevailing nonlinear approaches based on Sum of Squares (SOS) programming. We apply our approach to low dimensional nonlinear shear flow models for a range of Reynolds numbers. The results from our analytically derived bounds are consistent with the bounds identified through exhaustive simulations. However, our results are obtained at a much lower computational cost and have the benefit of providing a provable guarantee that a certain level of perturbation is permissible.

Apr
9
Thu
Online Mechanical Engineering Spring Seminar Series: Class 530.804 @ https://wse.zoom.us/j/145648770
Apr 9 @ 3:00 pm – 4:00 pm

Via Zoom: https://wse.zoom.us/j/145648770

#### “Mechanical Regulation of Cell Growth and Proliferation”

Presented by Professor Sean Sun
Department of Mechanical Engineering, Johns Hopkins University

From the point-of-view of engineering, the cell is an extraordinary autonomous system capable of self regulation in a changing environment. One of the critical processes of the cell that require extensive regulation is cell growth and proliferation. Indeed, failure in proper regulation of cell growth is the root cause of many diseases, including cancer. However, studying cell growth control turns out to be difficult, due to measurement problems at the cell single level and inherent noisiness of the growth process. In this talk, I will describe a new AI-based microscopy method to quantitatively measure cell size in real time. Using this method, we can observe single cell growth over multiple rounds of cell division. We find novel signatures of cell growth control and the presence of a cell size checkpoint during the cell cycle. The data clearly shows that cancer cells and normal cells regulate their cell size and growth differently. Moreover, normal cell growth is sensitive to mechanical tension and forces applied to the cell, whereas cancer cells seems to be insensitive to force application. These results suggest that there are redundant control mechanisms of cell growth and cell size regulation in normal cells, and cancer cells can escape normal growth regulation by altering their mechanosensitivity. The implications of these results are important for understanding disease mechanisms, and our method of studying cell growth and proliferation will be useful for further revealing molecular-level control algorithms.

Sean Sun is a faculty member in the Department of Mechanical Engineering and Institute of NanoBioTechnology at JHU. He is interested in the mechanical behavior of the cell and the mechanisms behind cell movement, cell growth and cell size regulation. He uses a combination of microfluidic methods, microscopy, genetic and biophysical techniques, and mathematical modeling to reveal novel mechanisms in cell biomechanics. He is a member of the JHU Physical Sciences in Oncology Center, and a Fellow of the American Physical Society (APS) and the American Institute for Medical and Biological Engineering (AIMBE).

Apr
10
Fri
Graduate Seminar in Fluid Mechanics @ Join on-line via Zoom: https://wse.zoom.us/j/435449376
Apr 10 @ 4:00 pm – 5:00 pm

#### Join online via Zoom: https://wse.zoom.us/j/435449376

“Scalar Transport in Restricted Nonlinear Wall-Turbulence”

Presented by BENJAMIN MINNICK
(Adviser: Prof. Gayme)

Streamwise coherent structures are inherent in wall-bounded turbulence and their study has given insight into structural features and dynamics of the flow. For example, streamwise vortices have been attributed with redistributing the momentum and developing the blunted mean velocity profile. Numerical and experimental studies of scalars, such as heat or chemical species, transported by wall-bounded turbulent flow have shown scalar fluctuations are highly correlated to streamwise velocity fluctuations, therefore it is likely that the dynamics of streamwise coherent structures also play a key role in scalar transport. We test this hypothesis by simulating scalar transport in the restricted nonlinear (RNL) model, a quasi-two-dimensional representation of wall-turbulence shown to accurately predict low-order statistics and cross-plane features of the momentum field. We have found that restricting both the momentum and scalar fields to streamwise constant mean dynamics interacting with a single non-zero wavenumber perturbation is capable of accurately predicting the expected time-averaged scalar log-law profile for a range of Prandtl numbers. Furthermore, this RNL model is shown to predict similar cross-plane structures in the scalar field to direct numerical simulation (DNS), suggesting streamwise vortices are indeed the structures largely responsible for mixing scalars in wall-bounded turbulent flow. After applying this model to study passive scalar transport, we conclude by investigating buoyancy effects in stably-stratified turbulence.

“Towards a Reduced-Order Model for Finite-Sized Bubbles in Turbulence”

Presented by ASHIK U. M. MASUK
(Adviser: Prof. Ni)

In both natural and industrial turbulent multiphase flows, bubbles are often present as the dispersed second phase controlling many important processes such as ocean-atmosphere gas exchange, heat transfer, and mixing in chemical reactors. For better predictive modeling of such processes, it is essential to understand the contribution of different flow components in determining the shape evolution of bubbles. The shape and relative orientation of a bubble directly control the forces that it experiences in a flow field which eventually determines its dynamics in the flow. In this work, from our unique simultaneous 3D measurements of bubbles and their surrounding flow, we identify the mechanisms that are responsible for the deformation of finite-size bubbles; and based on these mechanisms, we develop a phenomenological model to predict the deformation and orientation of bubbles in turbulence. Finally, the model predictions and experimental measurements are compared.

Apr
16
Thu
Mechanical Engineering Spring Seminar Series: Class 530.804 @ via Zoom: https://wse.zoom.us/j/348258728
Apr 16 @ 3:00 pm – 4:00 pm

3:00pm via Zoom: https://wse.zoom.us/j/348258728

“Mechanics of Granular Materials at “Micro” Length and Time Scales from in-situ X-ray Measurements”

Presented by Professor Ryan Hurley
Department of Mechanical Engineering, Johns Hopkins University

Granular materials are ubiquitous in nature and technology. They include the soils we build structures on, the raw materials we compress and sinter into a vast number of solid products, the foods we love (rice and coffee), and the powders we use in pharmaceuticals and beauty products. Despite their ubiquity, granular materials exhibit very complex behavior that eludes our mechanical models. They size-segregate during flow, exhibit complicated creep behavior when not flowing, demonstrate long-range correlated behavior when compressed, and dissipate an enormous amount of energy due to friction. How can a granular material containing particles that only interact mechanically with their contacting neighbors exhibit such complexity?

In this talk, I will first highlight some interesting behaviors of granular materials with both academic and real-world applications. I will then motivate a need to better experimentally understand granular material behavior: to study their internal structure and response to mechanical loading. I will then describe the portion of my research program that aims to quantitatively measure the behavior of granular materials down to a “microscopic” length scale (the scale of individual particles and their contacts with neighboring particles) and “microscopic” time scales (nanoseconds during dynamic impact). Central to this research program is the use of multiple in-situ X-ray techniques, including X-ray computed tomography (XRCT), 3D X-ray diffraction (3DXRD), and dynamic X-ray phase contrast imaging and radiography. I will describe how we use these techniques and analysis to study how opaque, 3D granular materials develop spatially-correlated inter-particle forces, dissipate energy due to slipping and fracturing, transport wave energy, and dynamically compress due to shock. I will discuss some of our exciting ongoing and future research directions.

Ryan Hurley is an Assistant Professor in the Department of Mechanical Engineering and a Fellow of the Hopkins Extreme Materials Institute, with a Secondary Appointment in the Department of Civil and Systems Engineering, at the Johns Hopkins University (JHU). He received his B.S. in Civil Engineering from the University of Maryland, College Park (2011) and his M.S. (2012) and Ph.D. (2015) in Applied Mechanics from the California Institute of Technology. From 2015 – 2017, Ryan was a postdoctoral researcher in the Computational Geosciences Group at Lawrence Livermore National Laboratory in Livermore, California and an Assistant Research Professor at JHU. He joined JHU full-time as an Assistant Professor in January 2018. He is the recipient of a 2017 Secretary’s Appreciation Award from the U.S. Department of Energy and a 2019 CAREER Award from the U.S. National Science Foundation. In his research, Ryan seeks to see, understand, and predict the mechanical behavior governing a variety of geologic, structural, and composite materials using advanced experimental and numerical techniques.

Apr
17
Fri
Graduate Seminar in Fluid Mechanics @ Join on-line via Zoom: https://wse.zoom.us/j/435449376
Apr 17 @ 4:00 pm – 5:00 pm

Join on-line via Zoom: https://wse.zoom.us/j/435449376

“Generalizing the Coupled Wake Boundary Layer Model for Wind Farm Power Prediction”

Presented by GENEVIEVE STARKE
(Advisers: Profs. Meneveau & Gayme)

During recent years, wind has continued to grow as an electricity provider in the United States. As interest in wind has grown, a need has arisen to be able to accurately predict the effect of turbines on the wind and how this impacts the overall power of the wind farm. Large-scale simulations can predict this accurately, however, they are too expensive and too slow to be used in real-time applications. To achieve faster results, we are working on developing and applying reduced-order models of wind farms that accurately capture key dynamics in the wind farm. Here we present a model that combines two reduced-order models that depend on different scales to improve the overall prediction of the power of the wind farm. The first model is called a wake model, and represents the individual turbines and their effect on the wind. The second model, called the top-down model, works on a larger scale and represents the wind farm as an impediment to the wind in the atmospheric boundary layer, which describes the interaction between the wind in the atmosphere and the ground. This model gives average quantities over the wind farm that are then matched to those obtained from the wake model to obtain a combined prediction of the power. To determine the area over which the average quantities are calculated, the model uses Voronoi tessellation to establish regions of the flow that belong to each turbine. This enables the models to be applied to each turbine and the area around that turbine individually, which allows application to nonuniform wind farms. As a result of our average treatment of the farm, the boundary layer caused by the wind farm in the atmospheric boundary layer is defined to start at the initial line of freestream turbines, rather than on an individual turbine basis. The coupled wake boundary layer model is validated using data from LES simulations for various wind directions from the National Renewable Energy Laboratory, and found to reproduce trends in overall power produced as a function of the inflow direction of the wind.

“LES of Laminar-Turbulent Transition with Turbulent/Non-Turbulent Classification”

Presented by GHANESH NARASIMHAN
(Advisers: Profs. Meneveau & Zaki)

The laminar-turbulent transition in boundary layers and channel flow is usually classified as either orderly (natural) or bypass transition. Natural transition starts with amplification of Tollmien-Schlichting (TS) instability waves which subsequently undergo secondary instability and ultimately break down to turbulence. Often, however, when background disturbances are present, bypass transition takes place instead. It involves initial linear amplification of streamwise perturbations through lift-up mechanism, followed by non-linear evolution and break down into turbulence spots. Direct Numerical Simulation (DNS) of these transition processes is computationally expensive. Therefore, Large Eddy Simulation (LES) is used as an alternative to simulate the laminar-turbulent transition. Wall-Resolved LES (WRLES) using dynamic Smagorinsky Sub-Grid Scale (SGS) model simulates this transition which is comparable to the DNS. However, resolving the inner layer in WRLES demands more grid points thereby making the WRLES of transition computationally expensive. Hence, LES with wall-modeling which models the inner layer rather than resolving it, is used to further decrease the computational cost. Using Wall-Modeled LES (WMLES) to simulate transition is again difficult as the wall model assumes the flow is fully developed. Since a transitional flow has features from both laminar and turbulent states, LES equations with wall-modeling could still be applied in the turbulent regions. Therefore, it is essential to identify the Turbulent/Non-Turbulent (T/NT) regions in the flow. To this end, self-organizing map (SOM), an unsupervised machine learning algorithm is used for the T/NT classification in a transitional channel flow. Firstly, the natural transition is simulated by the WMLES with SOM of the time evolution of channel flow with 2D TS wave and oblique 3D waves as the initial perturbation. Secondly, the case of bypass transition in channel flow is simulated by performing WMLES with SOM of the time evolution of a three-dimensional localized perturbation. The time evolution of the friction Reynolds number from the LES is compared to the DNS. Both the transition scenarios are well predicted by the LES with significantly less computational resources. Thus, the WMLES with T/NT classification using SOM is proposed as a new technique for modeling transitional flows.

Apr
23
Thu
Mechanical Engineering Spring Seminar Series: Class 530.804 @ Join online via Zoom
Apr 23 @ 3:00 pm – 4:00 pm

Zoom: https://wse.zoom.us/j/361732230; password 779781

“On the Breakup and Transport of Crude Oil by Surface Waves and Subsurface Plumes”

Presented by Professor Joseph Katz
Department of Mechanical Engineering, Johns Hopkins University

Nearly a decade after the Deepwater Horizon massive oil spill, this presentation summaries a series of laboratory studies aimed at characterizing interfacial phenomena affecting the breakup of surface slicks and subsurface plumes of crude oil. After entrainment of slicks by surface waves, the droplet size distributions agree with classical turbulence-based scaling, and their subsequent temporal evolution can be modeled by combining effects of turbulent diffusion and buoyant rise.  Pre-mixing the crude oil with dispersant, which drastically reduces the oil-water interfacial tension, causes tip streaming, an interfacial phenomenon that decreases the droplet sizes to the micron range. Aerosolization of oil is caused by the initial splash and by subsequent bubble bursting. Premixing the oil with dispersant increases the concentration of airborne nano-droplets by one to two orders of magnitude, raising health concern. In contrast, the dispersant causes a reduction in concentration of volatile organic compounds. Extended mixing of oil with seawater generates poly-dispersed water-in-oil emulsions with two-orders-of-magnitude higher viscosity. Dispersant partially separates the water, but the viscosity of the remaining sparser emulsion is still higher than that of the original oil. Micro emulsions also form as oil droplets rises and cross an oil-water interface. These droplets do not mix with the bulk oil since they remain coated by submicron water films that persist long after crossing. These films eventually break up owing to droplet deformation induced by electrostatic forces. Below the surface, fragmentation of a vertical buoyant oil jet is elucidated by refractive index matching. Compound oil droplets containing water droplets, some with smaller oil droplets, form regularly. Their fraction increases with droplet diameter, reaching 78% for 2mm droplets. While the exterior surfaces of the oil droplets are deformed by the high shear field, the interior interfaces remain spherical, indicating quiescent domains. In the presence of cross flow, entrainment of small droplets into the core of the counter-rotating vortex pair defines the lower boundary of the plume while large droplets escape and define the upper boundary. Hence, reduction of droplet sizes by dispersant increases the fraction of oil entrained into the vortex pair and lowers the upper boundary of the plume.

Joseph Katz received his B.S. degree from Tel Aviv University, and his M.S. and Ph.D. from California Institute of Technology, all in mechanical engineering. He is the William F. Ward Sr. Distinguished Professor of Engineering, and the director and co-founder of the Center for Environmental and Applied Fluid Mechanics at Johns Hopkins University. He is a Member of the National Academy of Engineering, as well as a Fellow of the American Society of Mechanical Engineers (ASME) and the American Physical Society. He has served as the Editor of the Journal of Fluids Engineering, and as the Chair of the board of journal Editors of ASME. He has co-authored more than 380 journal and conference papers. Dr. Katz research extends over a wide range of fields, with a common theme involving experimental fluid mechanics, and development of advanced optical diagnostics techniques for laboratory and field applications. His group has studied laboratory and oceanic boundary layers, flows in turbomachines, flow-structure interactions, swimming behavior of marine plankton in the laboratory and in the ocean, as well as cavitation, bubble, and droplet dynamics, the latter focusing on interfacial phenomena associated with oil spills.

Apr
24
Fri
Graduate Seminar in Fluid Mechanics @ Join online via Zoom
Apr 24 @ 4:00 pm – 5:00 pm

Join online via Zoom: https://wse.zoom.us/j/435449376

“Large Eddy Simulations of a Developing Turbulent Boundary Layer over Cube Roughened Plate”

Presented by SAMVIT KUMAR
(Advisers: Profs. Meneveau & Mittal)

Turbulent boundary layers are present whenever there is fluid flow over a solid surface or wall. More often than not, these surfaces are rough. The roughness strongly influences flow physics and the drag at the wall. Therefore, it is important to understand the fundamental nature of turbulent boundary layers over rough walls. We first present an improved method for generation of turbulent inflow for simulations of developing boundary layers. The approach is based on prior recycling methods for flow over smooth (Lund et al., 1998) and rough (Yang and Meneveau, 2015) surfaces. In the recycling method, mean and fluctuation velocities on a sample plane are rescaled, combined and recycled back to the inlet, as the inflow velocity. A roughness-length related scale is chosen for rescaling of the inner layer, depending on the surface geometry and the displacement thickness is chosen instead of $\delta_{99}$ as the length scale to rescale the outer layer. The blending function, dependent on both the inner and the outer length scales, is used to combine the two profiles, to obtain the inflow velocity.

We then find an appropriate grid size and use the integral wall model for Large Eddy Simulations of flow over a staggered array of cubes. Results obtained are compared with Direct Numerical Simulations carried out by (Lee et al., 2011) and good agreement is shown. We also set up simulations for flow over multi scale roughness elements.

“The Pressure Field and its Relation to Cavitation Inception in a Turbulent Shear Layer”

Presented by KARUNA AGARWAL
(Adviser: Prof. Katz)

We investigate the unsteady pressure field associated with quasi-streamwise vortices in the shear layer behind a backward facing step. The primary objective is to understand the conditions for cavitation inception, which occurs in the core of these vortices, and appears as 1-2 mm wide and 5-7 mm long cavities. The Reynolds numbers based on the step height are 1.6×10^4 and 5.8×10^4. High speed tomographic imaging followed by 3D particle tracking using the Shake-the-Box method is used for calculating the instantaneous velocity and acceleration fields. The data are interpolated using a constrained cost minimization technique, which establishes a divergence free velocity and curl-free material acceleration fields at a spatial resolution of 200µm. The pressure is then calculated by spatially integrating the material acceleration. Statistical analysis provides the probability density functions of the pressure, strength, size, and straining of the quasi-streamwise vortices. Effects of spatial resolution of the measurements are also discussed. With increasing Reynolds number, the pressure minima are more preferentially located within the quasi streamwise vortices, for longer durations, and appear to be strongly influenced by vortex stretching.


text:
# Volume of A Triangular Prism Calculator

Enter Length (l)

Enter Base (b)

Enter Height (h)

Volume of a Triangular Prism :

The Volume of a Triangular Prism Calculator an online tool which shows Volume of a Triangular Prism for the given input. Byju's Volume of a Triangular Prism Calculator is a tool
which makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number.

#### Practise This Question

Select the correct diagram for the π 2py - orbital:

format: plain


text:
# Modelling resource dependency in the assignment problem

The assignment problem is well-studied and has a nice polynomial time algorithm. I'm interested in an extension of this problem where all edges are in a certain group and taking multiple edges from the same group can end up with diminishing returns due to dependence on a shared resource. This means the edges are no longer independent and therefore it cannot be directly modeled as an assignment problem.

More formally, the problem is:

Given:

• A weighted bipartite graph $$G=(V,E)$$ with weight function $$w:E\rightarrow \mathbb{Q}$$,

• a "type set" $$\mathcal{T}$$ which partitions $$E$$ into groups that require the same resource. $$T:E\rightarrow \mathcal{T}$$ shows which type is associated with an edge.

• A resource function $$R: \mathcal{T}\times \mathbb{N}\rightarrow \mathbb{Q}$$, decreasing in the second argument

Find a matching $$M$$ of $$G$$ and index $$I:M\rightarrow \mathbb{N}$$ that maximizes $$\sum_{e\in M} \min\{w(e), R(T(e),I(e))\},$$ where $$I$$ can use each value only once for each type (i.e. if $$I(e)=I(e')$$ and $$e\neq e'$$, then $$T(e)\neq T(e')$$).

Can this problem still be solved in polynomial time? If not, what would be a good alternative to model this, and why? Does this change if the number of types is a small constant (e.g. $$<10$$)?

Some observations:

I think it is possible to model this as an MILP, but that could take long even when the number of types is small. Still, it is possible that is the best we can do.

Also note that this is a special case of the 3-dimensional (weighted) matching problem, but that problem is very hard and modelling this in that way seems less promising than the MILP.

This question is inspired by an answer I gave here, which can be seen as an application of this problem.

• If you only ask for a matching, the problem is trivial in case $R$ and $w$ are positive. In the vanilla assignment problem, the task is to find a perfect matching of minimal total weight (or in its rectangular version, what is sometimes referred to as a full matching; i.e. one covering at least one of the partitions of the graph). Is that what you want here as well? – fuglede Aug 8 '19 at 17:52
• @fuglede $R,w$ have positive range, yes. I'm not sure why you think the problem is trivial, can you elaborate? I am indeed looking for any matching, not necessarily a perfect one, but do note that I want to maximize the sum in this formulation. It could be that the usual assignment problem has minimization, I mostly "inherited" the maximisation formulation from the inspiration problem in the linked answer. – Discrete lizard Aug 8 '19 at 18:02
• Gotcha; I did indeed erroneously read your "maximize" as a "minimize". – fuglede Aug 8 '19 at 18:05


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### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In problems 3-6, use the figure to solve each inequality.a) $f\left(x\right)>0$b) $f\left(x\right)\le 0$

a)$\left(-\infty ,-2\right)$ or $\left(2,\infty \right)$

b) $\left[-2,2\right]$

See the step by step solution

## Part (a) Step 1. Given Information

It is given that, $f\left(x\right)>0$ and the graph is,

## Part (a) Step 2. Observe the graph

• From the graph, it is observed that the value of $y$ or $f\left(x\right)$ is positive for role="math" localid="1646049546322" $x<-2$ or $x>2$.
• In set builder form, for $f\left(x\right)>0$, $\left\{x|x<-2orx>2\right\}$. Also, in interval notation it is denoted as $\left(-\infty ,-2\right)or\left(2,\infty \right)$.

## Part (b) Step 1. Given Information

It is given that, $f\left(x\right)\le 0$ and the graph is,

## Part (b) Step 2. Observe the graph

• From the graph, it is observed that the value of $y=f\left(x\right)$ is negative or equal to 0 for $-2\le x\le 2$.
• In set builder form, for $f\left(x\right)\le 0$, $\left\{x|-2\le x\le 2\right\}$. Also, in interval notation it is denoted as $\left[-2,2\right]$.


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# Math Help - Maximum Triangles

1. ## Maximum Triangles

What is the maximum number of distinct triangles that can be formed if m<A = 30, b = 8, and a = 5
(1) 1 (3) 3
(2) 2 (4) 0

Can I use law of sines to answer such questions about how many different triangles?

2. Originally Posted by magentarita
What is the maximum number of distinct triangles that can be formed if m<A = 30, b = 8, and a = 5
(1) 1 (3) 3
(2) 2 (4) 0

Can I use law of sines to answer such questions about how many different triangles?
Hello Magentarita,

Let's look at both cases: Recall that the sine of an angle is equal to the sine of its supplement.

Case I (diagram 1)

$\frac{\sin A}{A}=\frac{\sin B}{B}$

$\frac{\sin 30}{5}=\frac{\sin B}{8}$

$\sin B = .8$

$B \approx 53.1^{\circ}$

Case II (diagram 2)

$\triangle B'BC$ is isosceles.

$\angle B$ is supplementary to $\angle AB'C$

$\angle AB'C=180-53.1 \Rightarrow 126.9$

$\frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}$

So, looks like 2 triangles can have the characteristics you describe.

$\triangle ABC \ \ and \ \ \triangle AB'C$

3. ## Tell me......

Originally Posted by masters
Hello Magentarita,

Let's look at both cases: Recall that the sine of an angle is equal to the sine of its supplement.

Case I (diagram 1)

$\frac{\sin A}{A}=\frac{\sin B}{B}$

$\frac{\sin 30}{5}=\frac{\sin B}{8}$

$\sin B = .8$

$B \approx 53.1^{\circ}$

Case II (diagram 2)

$\triangle B'BC$ is isosceles.

$\angle B$ is supplementary to $\angle AB'C$

$\angle AB'C=180-53.1 \Rightarrow 126.9$

$\frac{\sin 30}{5}=\frac{\sin 126.9}{8}=\frac{\sin 53.1}{8}$

So, looks like 2 triangles can have the characteristics you describe.

$\triangle ABC \ \ and \ \ \triangle AB'C$
Is this the ambiguous case?

4. Originally Posted by magentarita
Is this the ambiguous case?
Yes, it is.

5. ## by the way...

Originally Posted by masters
Yes, it is.
By the way, I like playing with ambiguous case trig questions.


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Process fixed-length text files by using Data Factory mapping data flows. K Factor in sheet metal bending is a constant used to calculate sheet metal flat length or Flat-pattern. n: an integer. How to create an empty vector of length zero in the R programming language. Check if you have put an equal number of arguments in all c() functions that you assign to the vectors and that you have indicated strings of words with "".. Also, note that when you use the data.frame() function, character variables are imported as factors or categorical variables. The length of levels(x), which is zero if x has no levels. Like all functions, we can define our factory with parameters which change the shape of the returned object. However, it’s a list with vector structures of the same length. This syntax gives a count rather than a sum. It can store both character and integer types of data. For expressions and But this is not always the case! length<-has a "factor" method. Andrie de Vries is a leading R expert and Business Services Director for Revolution Analytics. C. Unchanged. The replacement form can be used to reset the length of a vector. a vector is shortened, extra values are discarded and when a vector is . The length of a wire is doubled. (v <- runif(10)) ACI 318 (Section 10) allows the columns to be considered as component of nonsway (braced) frame if the additional moments induced by second-order effects are not greater than 5% of the initial end moments. The effective length factor for braced columns varies from 0.5 to 1.0, whereas for unbraced columns, it can vary from 1 to infinity. R factors variable is a vector of categorical data. The below script will create and save the pie chart in the current R working directory. R --vanilla - Allows you to start a clean session of R. 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We can check if a variable is a factor or not using class() function. length(as.name("xyz")) APPLIES TO: Azure Data Factory Azure Synapse Analytics . pairlists (including language objects and dotlists) it is the levels, factor. The length of an atomic vector is simply the number of elements in it. x: a vector or list. classes of objects, see InternalMethods. Let’s start the process of creating factors, accessing its components, generating factor level and modification with the help of examples.  Browser’s URL length limits The appropriate page of its URL length is 75 characters long. length(d) Example: Convert Factor to Numeric in R. To convert a factor to numeric in R can be a tricky task. The value of condition factor was found to be more than 1 in all sub-populations of the … We know, R = A ρ × L Hence, R R ′ = L L ′ × A ′ A = L 2 L × 4 A A = 2 1 Hence, R ′ = 2 R Answered By . The original H&R firm was in business for over a century from 1871 to 1986. For vectors (including lists) and factors the length is the number of Solution. The factor function is used to create a factor. This is usually applied to a factor, but other objects can have levels. perc <- function(x, n){ 100*length((which(x == n))) / length(x) } Note the syntax involved in setting up a function in R. Now let’s use the count function to count the threes in the vector b. count(b, 3) [1] 4. perc(b, 4) [1] 7.692308. In the following, I’m therefore going to explain how to convert a factor vector to numeric properly without a loss of information. all use the effective length factor) actually buckles. Factor variables: summary() gives you a table with frequencies. {r} Ordered factors. – Typeset by FoilTEX – 14 Data Frames A list with possible heterogeneous vector elements of the same length. See Also. {r} (l <- list(a = 1, b = 2:4, c = 5:9, d = 10:16, e = 17:25)) You’ll find all relevant information for the conversion of R factors to numeric in the following tutorial. Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) length(a) length<-has a "factor" method. In theory, you could use parameterized factories with hundreds of arguments to return very specific and deeply nested objects, but as we’ll see, that’s not at all in the spirit of composition. The same is true of matrices. [R] linear mixed model with nested factors [R] count of factors [R] Mixed-effects model for overdispersed count data? Details. decimate creates a lowpass filter. Value . R factor is used to store categorical data as levels. string: Input vector. So, why waste time? names, which are adjusted (and if necessary extended by ""). n: an integer. Hence we use length(x). 2 times. The function factor is used to encode a vector as a factor (the terms ‘category’ and ‘enumerated type’ are also used for factors). Topography: LS-Factor The Universal Soil Loss Equation (USLE) model is the most frequently used model for soil erosion risk estimation. Length and width of the sepal and petal are numeric variables and the species is a factor with 3 levels (indicated by num and Factor w/ 3 levels after the name of the variables). You want to change the order in which the levels of a factor appear. sessionInfo() Print version information about R and attached or loaded packages. {r} Package authors have written methods that return a result of length nchar for counting the number of characters in character In the case of materials, it is often expressed in terms of R-value per unit length (e.g. Length-length relationship showed both lengths are highly correlated to each other (r=0.99). L ′ = 2 L r ′ = 2 r A ′ = 4 A. [R] Count of unique factors within another factor [R] Counting two factors at the same time [R] lattice barchart [R] Counting observations of a combined factor The solution is to index the levels by the factor itself, and then to convert to numeric: > as.numeric(as.character(numbers)) [1] 9 8 10 8 9. length(myList[[1]]) ?  Matrices are vectors with more then one dimension, therefore, matrices has rows and columns. length(cl) They are useful in the colu length(c) >is.factor, is.ordered, as.factor and as.ordered are the membership … ; Use str_length() on boy_names to find the length of each name and save the result to boy_length. Factors in R are stored as a vector of integer values with a corresponding set of character values to use when the factor is displayed. (Again, coerce to list to see why.) Summary: At this point you should have learned how to initialize a vector with a fixed length of zero in R. Let me know in the comments below, in case you have any additional questions. In the worst case scenario you can use varchar(max) b < 7 [1] FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE FALSE … By using mapping data flows in Microsoft Azure Data Factory, you can transform data from fixed-width text files. length(ex) Example: Convert Factor to Numeric in R. To convert a factor to numeric in R can be a tricky task. str_length.Rd. have length one: note that for functions this differs from S. The replacement form removes all the attributes of x except its a non-negative integer or double (which will be rounded down). Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) Unlike the default implementation of length, this method retains the class and levels attribute of the factor object. Sometimes you will also deal with factors that do have a natural ordering between its categories. This is usually applied to a factor, but other objects can have levels. History. length(myList[[1]]) ? For example, an u with a umlaut might be represented as a single character or as the combination a u and an umlaut. Source: R/length.r. You can use the merge function to do a cross-product join:. 1.Define data type as varchar(a) , where a is ball park figure of the length . lengthened, it is padded out to its new length with NAs  For lists, the length is the number of top-level elements Functions always have length 1. any other R object for which a method has been defined. Similarly, levels of a factor can be checked using the levels() function. The length of levels(x), which is zero if x has no levels. For an environment it is the number of objects in the Package authors have written methods that return a result of length elements. The length of levels(x), which is zero if x has no levels. This index can be said as tool to diagnose the physiological con-dition of fish in terms of its interaction to the biotic and abiotic factors. "factor" method. If the factor object is lengthened, missing values are placed at the end of the resulting vector. 1:5 These factors are created with the help of factor() functions, by taking a vector as input. Since refinement programs aim at minimizing the difference between observed and calculated amplitudes ... (~1 T). If you pass multiple objects in braces, they only count as one item. str_length.Rd. The length of a function call is 1 plus the number of arguments passed to the function.  In this tutorial, we will learn how to find string length in R programming. (earlier versions of sets). $$2^{31}-1$$ elements, when it returns a double. Table of contents. When to treat a variable like a factor depends a bit on the situation, but, as a general rule, avoid more than ten different levels in a factor and try to have at least five values per level. You want to change the order in which the levels of a factor appear. Try entering b <- 7 at the keyboard. (nul for raw vectors). Each factor level is a character string which in some cases can be many thousands of characters long. Before we can start, we need to create an example factor vector in R: Coerce the formulae to lists to see why. levels, factor. names, which are adjusted (and if necessary extended by ""). Andrie de Vries is a leading R expert and Business Services Director for Revolution Analytics. The New S Language. Upvote(0) How satisfied are you with the answer? Combining the length() and which() ... R PROVIDES ANOTHER ALTERNATIVE THAT NOT EVERYONE KNOWS ABOUT. The length of an expression is the number of calls, symbols or constants that you pass into it.  In this article we will discuss sheetmetal bend deduction, bend allowance, K-factor, Y-factor and sheet metal flat pattern calculations. xptr <- xml2::read_xml("")\$node length<- has a length(x) 3 Matrices. Details. elements. A. For example, an u with a umlaut might be represented as a single character or as the combination a u and an umlaut. … Get Instant Solutions, 24x7. Mit length(Vektor) kann die Anzahl der Elemente eines Vektors ermittelt werden. Examples. Numerical and factor variables: summary() gives you the number of missing values, if there are any. To find the length of a String in R, use nchar() function. df1 <- data.frame(x = 1:3) df2 <- data.frame(y = c("a", "b", "c")) merge(df1, df2, by = NULL) # x y # 1 1 a # 2 … About the Book Author. identical(length(d), ncol(d)) Source: R/length.r. It doesn't matter the length.we will bring it to your doorstep. known as Fulton’s condition factor, or length-weight factor, or Ponderal index). Changing the order of levels of a factor Problem. environment, and NULL has length 0. Where a single double value is other than one (Formula) and that return a vector of type Example. The replacement form can be used to reset the length of a vector. The replacement form can be used to reset the length of a vector. Solution. Effective Length (K) Factor Explained: The K factor approximates the length that a column (steel column's, concrete column's, aluminum column's, etc. ; Use str_length() on boy_names to find the length of each name and save the result to boy_length. The effective length factor for braced columns varies from 0.5 to 1.0, whereas for unbraced columns, it can vary from 1 to infinity. pairlists (including language objects and dotlists) it is the The replacement form can be used to reset the length of a vector. If you pass multiple objects in braces, they only count as one item.  Get or set the length of vectors (including lists). tutorial series, visit our R … length of the pairlist chain. length(if) R factor Function. ACI 318 (Section 10) allows the columns to be considered as component of nonsway (braced) frame if the additional moments induced by second-order effects are not greater than 5% of the initial end moments. #freedelivery #mattfinish #6mlength 0111 050 700 0710 111 700 As seen before, R attaches an equal value to the levels for such factors. This will help us to improve better. Both functions are generic: you can write methods to handle specific Take a look at the boy_names vector, it's long, so use head() to see the first few elements. per inch or metre of thickness). The length Function in R; The R Programming Language . {y <- x ^ 2; y + 2}, Usage length(x) length(x) <- n Arguments. a non-negative integer or double (which will be rounded down). Get the length of the resulting vector using length(). So, why waste time? Among the six input layers, the combined slope length and slope angle (LS-factor) has the greatest influence on soil loss at the European scale. The use of 'within' here performs this task without leaving copies of x and y as separate objects. (nul for raw vectors). toppr. integer of length 1, except for vectors of more than Both work on logical vectors whose elements are either TRUE or FALSE. If you want more information or if you just want to review and take a look at a comparison of the five general data structures in R, watch the small video below: A quick look at the table on the right displays the multiples of half wavelengths. x: a vector or list. When using rate-R PI, the length of the readout can also be shortened by a factor R. This is the result of the extent of k-space being traversed R times faster, leading to a factor R decrease in the steepness of the aforementioned phase gradient. The default is a Chebyshev Type I filter designed using cheby1.This filter has a normalized cutoff frequency of 0.8/r and a passband ripple of 0.05 dB. – Typeset by FoilTEX – 14 Data Frames A list with possible heterogeneous vector elements of the same length. Following is the syntax of nchar function. length(~ x1 + x2 + x3 + x4) {r} Hi Is there a maximum length for the character string representing a level of a factor? The code counts the number of times the values match, so I don't think primary keys would be relevant.. That said, it's hard to be confident without a full reprex. Since "Male" and "Female" are unordered (or nominal) factor levels, R returns a warning message, telling you that the greater than operator is not meaningful. all.equal(length(a), prod(dim(a))) Be careful: for data.frames, the length refers to the number of columns, not the total number of elements. One code point usually corresponds to one character, but not always. {r} The replacement form can be used to reset the length of a vector. length(xptr) The R help at ?factor describes a solution to this problem. This is usually applied to a factor, but other objects can have levels. "factor" method. MEDIUM. [R] how to quote "factors" in a function? mean(1:10, na.rm = TRUE), length(m) {r} {r} length(as.name("xyz"))  The length of a function call is 1 plus the number of arguments passed to the function. They are better than using simple integer labels because factors are self describing: "low", "medium", and "high"" is … Value. sum(b < 7) [1] 9. )) {r} [R] Comparing each level of a factor to the global mean [R] change factor to mtrix [R] Ploting different values with different colors [R] ggplot2 and facet_wrap help [R] CAPER pgls() anova not working with variable as factor [R] problems with coercing a factor to be numeric [R] Making an aggregated factor variable from an existing one factor() function creates a factor variable, and calculates the categorical distribution of a vector data. The New S Language. As for indexing documents with long URLs (over 75-120 characters), they’re generally indexed quite well, although there are also difficulties within the ranking. as.list(~ x1 + x2 + x3 + x4) Both functions are generic: you can write methods to handle specific Factors in R come in two varieties: ordered and unordered, e.g., {small, medium, large} and {pen, brush, pencil}.For most analyses, it will not matter whether a factor … returned that can be represented as an integer it is returned as a length(var) For compatibility with S there is also a function ordered . Slope length and steepness (LS) factor for USLE slope_steepness=name Name for output slope steepness raster map Slope steepness (S) factor for USLE convergence=integer Convergence factor for MFD (1-10) 1 = most diverging flow, 10 = most converging flow. All other objects (including functions) returned that can be represented as an integer it is returned as a How to fix a problem. When we execute the above code, it produces the following result − Factors in R come in two varieties: ordered and unordered, e.g., {small, medium, large} and {pen, brush, pencil}.For most analyses, it will not matter whether a factor … These factors are created with the help of factor() functions, by taking a vector as input. 8/18/2019; 2 minutes to read; b; k; j; d; K; In this article. The actual factor levels (if they exist) can be obtained with the levels function. URL Length for SEO are an important, though often overlooked, element of optimization that can benefit your website’s rankings. About the Book Author. [R] max length of a factor variable; Richard Mott. Factors in R are stored as a vector of integer values with a corresponding set of character values to use when the factor is displayed. (m <- matrix(1:30, 5, 6)) Did you see the weird values for the variable cyl? str_length (string) Arguments. {r} (cl <- quote(mean(1:10, na.rm = TRUE))) length(cl)  The length of an expression is the number of calls, symbols or constants that you pass into it. string: Input vector. as.list(y ~ x1 + x2 + x3 + x4) (earlier versions of sets). Recommended: 5 Default: 5 memory=integer Maximum memory to be used with -m flag (in MB) Default: 300. Take a look at the boy_names vector, it's long, so use head() to see the first few elements. Mathematically k factor value is equal to the ratio of position of neutral axis and sheet thickness. Value. To defined number of columns and rows you use the functions nrow and ncol, respectively.Similarly to vectors, matrices can have numbers, characters and logical values. You can see … classes of objects, see InternalMethods. For replacement, a vector or factor. Answer . length(l) str_length (string) Arguments. length-one integer vector. The replacement form can be used to reset the length of a vector. vectors, lengths for getting the length of every element an R object. Vectors are generally created using the c() function.Since, a vector must have elements of the same type, this function will try and coerce elements to the same type, if they are different.Coercion is from lower to higher types from logical to integer to double to character.If we want to create a vector of consecutive numbers, the : operator is very helpful. (ex <- expression( {r} double (Matrix), even with non-integer values {r} (ex <- expression( x, {y <- x ^ 2; y + 2}, … When to treat a variable like a factor depends a bit on the situation, but, as a general rule, avoid more than ten different levels in a factor and try to have at least five values per level. When a value is assigned to the length of a factor object, the factor object is shortened or lengthened to the specified length. Each factor level is a character string which in some cases can be many thousands of characters long. In R’s memory, these factors are represented by numbers (1, 2, 3). R - Factors - Factors are the data objects which are used to categorize the data and store it as levels. The default method for length currently returns a non-negative Following is an example of factor in R. > x [1] single married married single Levels: married single Here, we can see that factor x has four elements and two levels. The actual factor levels (if they exist) can be obtained with the levels function. All other objects (including functions) See the different variables types in R if you need a refresh. levels, factor. (d <- data.frame(x = letters[1:7], y = runif(7) > 0.5)) This is an example of nested functions in R, in which you pass the results of one function to a second function. I don't believe so. The default method for length currently returns a non-negative It is best to assemble a data frame of x and y data, to keep these two vectors associated in a single object; subsequent fitting & plotting of the data can then reference this data frame. x, Technically this returns the number of "code points", in a string. I have a set of several million variables, each a factor of length 19. We've pulled out just the names from 2014, and created the vectors boy_names and girl_names for you. 4 times. Le Cren (1951) proposed the relative condition factor in preference to condition factor which is denoted by ‘Kn’. If an R object. Be aware of the meaning of syntax like sum(b < 7). Examples nlevels(gl(3, 7)) # = 3 See the different variables types in R if you need a refresh. {r} have length one: note that for functions this differs from S. The replacement form removes all the attributes of x except its integer of length 1, except for vectors of more than vectors, lengths for getting the length of every element Wadsworth & Brooks/Cole. The S-factor measures the effect of slope steepness, and the L-factor defines the impact of slope length. Both numeric and character variables can be made into factors, but a factor's levels will always be character values. Let’s start the process of creating factors, accessing its components, generating factor level and modification with the help of examples. environment, and NULL has length 0. factor(x,levels=)encodes a vector x as a factor gl(n,k,length=n*k,labels=1:n)generate levels (factors) by spec-ifying the pattern of their levels; k is the number of levels, and n is the number of replications expand.grid()a data frame from all combinations of the supplied vec-tors or factors rbind(...) combine arguments by rows for matrices, data frames, and others cbind(...)id. Examples. To see the rest of the R is Not So Hard! For expressions and Sep 27, 2010 at 9:15 am: Hi Is there a maximum length for the character string representing a level of a factor? Get the length of the resulting vector using length(). any other R object for which a method has been defined. Click hereto get an answer to your question  5. The actual factor levels (if they exist) can be obtained with the levels function. See Also. length-one integer vector. Changing the order of levels of a factor Problem. {r} Both are primitive functions. By what factor does the resistance change \begin{array}{ll}{\text { (a) } 4 \text { time as large }} & {\text { (b) twice as large }} \\ {\text { (c) unchanged }} & {\text { (d) half as large }}\end{array} I am trying to find out why my analysis fails - I just wanted to rule out the possibility that the internal factor conversion has a problem parsing long strings. For an environment it is the number of objects in the  One code point usually corresponds to one character, but not always. It can store both character and integer types of data. dim() Retrieve or set the dimension of an object. (If you want to learn about the filter() function, take the Data Manipulation in R with dplyr course!).. This approach is recommended, to avoid cluttering up the R workspace with unnecessary objects. D. 0.5 times. Each factor level is a character string which in some cases can be many thousands of characters long. a vector is shortened, extra values are discarded and when a vector is formulae also have a length: 3 if there is a left-hand side, and 2 if not. The factor function is used to create a factor.The only required argument to factor is a vector of values which will be returned as a vector of factor values. For vectors (including lists) and factors the length is the number of Get or set the length of vectors (including lists). As such, they can actually be seen as special types of lists and can be accessed as either a matrix or a list. In the following, I’m therefore going to explain how to convert a factor vector to numeric properly without a loss of information. So do names. They can store both strings and integers. For replacement, a vector or factor. The effective length can be longer, shorter, or exactly the actual length depending on the rigidity of the supports. The length of the pallet should be same as the number of values we have for the chart. length of the pairlist chain. See Also. The elements of a data frame can be numeric vectors, factor vectors, and logical vectors, but they must all be of the same length.  … and arrays. Value. (If you want to learn about the filter() function, take the Data Manipulation in R with dplyr course!).. double (Matrix), even with non-integer values Sometimes, the specified filter order produces passband distortion due to round-off errors accumulated from the convolutions needed to create the transfer function. {r} Where a single double value is If in a list. Get or set the length of vectors (including lists) and factors, and ofany other Robject for which a method has been defined. length<- has a If a vector is shortened, extra values are discarded and when a vector is lengthened, it is padded out to its new length with NAs (nul for raw vectors). {r} Remember that this type of data structure requires variables of the same length. Character variables: summary() doesn’t give you any information at all apart from the length and the class (which is ‘character’). nchar for counting the number of characters in character (a <- array(1:24, 2:4)) External pointers always have length 1. On creating any data frame with a column of text data, R treats the text column as categorical data and creates factors on it. Technically this returns the number of "code points", in a string. If argument ordered is TRUE , the factor levels are assumed to be ordered. Wadsworth & Brooks/Cole. identical(length(m), nrow(m) * ncol(m)) The only required argument to factor is a vector of values which will be returned as a vector of factor values. Get or set the length of vectors (including lists) and factors, and of Hi @srini,. other than one (Formula) and that return a vector of type I have a set of several million variables, each a factor of length 19. in a list. length(v) R-values are additive for layers of materials, and the higher the R-value the better the performance. Both are primitive functions.  Get or set the length of vectors (including lists) and factors, and of Value . length() Retrieve or set the dimension of an object. I have a set of several million variables, each a factor of length 19. length {base} R Documentation: Length of a Vector or List Description. Length and width of the sepal and petal are numeric variables and the species is a factor with 3 levels (indicated by num and Factor w/ 3 levels after the name of the variables). answr. B.  By what factor does the resistance change. ` Usage length(x) length(x) <- n Arguments. (cl <- quote(mean(1:10, na.rm = TRUE))) In the example below, the frequency given is 50.150 and the chosen coax velocity factor is 0.87. You can see the calculator has worked out the electrical lengths which should be used. length(y ~ x1 + x2 + x3 + x4) Free R-factor. 2^31 - 1 elements, when it returns a double.

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## Error in .bbl File

### LaTeX Forum: BibTeX, biblatex and biber

bibliographiesBibTeXAdd tags Information and discussion about BiBTeX - the bibliography tool for LaTeX documents.
Using Kile 2.1.0 in Linux Mint 13 and writing a document (named proposal.tex) in standard article format. For the references, I am using a separate bib file using the commands
Code: Select all  •  Open in online LaTeX editor
`\bibliographystyle{unsrt}\bibliography{library}`

where library.bib is generated by my reference management software (Mendeley). When I compile the document without the above commands, it gives no error at all. But when I compile with the above commands active, it gives several errors in the proposal.bbl file (which is produced at some stage) and refuses to generate the dvi. I am attaching the log file and the bbl file if they are necessary. Please help.

Correction: I have discovered more specific pattern in the error. It happens specifically when I need to include non-cited items from the bib files in the reference list and use the
Code: Select all  •  Open in online LaTeX editor
`\nocite{Key}`

command in conjunction with the above commands.
Attachments
proposal.bbl
proposal.log
Last edited by swagatopablo on Wed Jul 10th, 2013, edited 1 time in total.
swagatopablo

Posts: 16
Joined: Sun Mar 25th, 2012

The provided files are quite useless without the corresponding source code (*.tex) and the bibliography database (*.bib). All this information belongs to an adequate problem description by a self-contained and minimal example.

Thorsten
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¹ System: openSUSE 13.1 (Linux 3.11.10), TeX Live 2013 (vanilla), TeXworks 0.5 (r1351)
² Posting stopped indefinitely due to offenses

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text:
# √c+√c+√c+⋯\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}, or the limit of the sequence xn+1=√c+xnx_{n+1} = \sqrt{c+x_n}

(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)

For $c \gt 0$, consider the quadratic equation
$x^2 - x - c = 0, x > 0$.

Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining

Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.

Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.

This is being repurposed in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions.

and here: List of abstract duplicates.

Assuming that you know that a monotone, bounded sequence converges, you want to do two things. First, show that $\langle x_n:n\in\mathbb{Z}^+\rangle$ is monotone and bounded, and then show that its limit is the positive root of $x^2-x-c=0$.

If $c=x_1=1$, $x_2=\sqrt2>x_1$, while if $c=1$ and $x_1=2$, $x_2=\sqrt3, so if the sequence is monotonic, the direction in which it’s monotonic must depend on $c$ and $x_1$. A good first step would be to try to figure out how this dependence works.

The positive root of the quadratic is $\frac12(1+\sqrt{1+4c})$, which I’ll denote by $r$. If $x_n\to r$, as claimed, and does so monotonically, it must be the case that the sequence increases monotonically if $x_1 and decreases monotonically if $x_1>r$. In the examples in the last paragraph, $r=\frac12(1+\sqrt5)\approx 1.618$, so they behave as predicted.

This suggests that your first step should be to show that if $x_n, then $x_n, while if $x_n>r$, $x_n>x_{n+1}>r$; that would be enough to show that $\langle x_n:n\in\mathbb{Z}^+\rangle$ is both monotone and bounded and hence that it has a limit.

Suppose that $0\le x_n; you can easily check that $x_n^2-x_n-c<0$, i.e., that $x_n^2. On the other hand, $x_{n+1}^2=c+x_n$, so $x_{n+1}^2>x_n^2$, and therefore $x_{n+1}>x_n$. Is it possible that $x_{n+1}\ge r$? That would require that $x_{n+1}^2-x_{n+1}-c\ge 0$ (why?) and hence that which is clearly impossible. Thus, if $0\le x_n, we must have $x_n, as desired. I leave the case $x_n>r$ to you.

Once this is done, you still have to show that the limit of the sequence really is $r$. Let $f(x)=\sqrt{c+x}$; clearly $f$ is continuous, so if the sequence converges to $L$, we have and from there it’s trivial to check that $L=r$.

Added: Note that although the problem gave us $x_1>0$, this isn’t actually necessary: all that’s needed is that $x_1\ge -c$, so that $x_2$ is defined, since $x_2=\sqrt{c+x_1}\ge 0$ automatically.


text:
1. Compact NIZKs from Standard Assumptions on Bilinear Maps
Shuichi Katsumata, Ryo Nishimaki, Shota Yamada and Takashi Yamakawa
[View PDF on eprint.iacr.org]

@misc{cryptoeprint:2020:223,
author = {Shuichi Katsumata and Ryo Nishimaki and Shota Yamada and Takashi Yamakawa},
title = {Compact NIZKs from Standard Assumptions on Bilinear Maps},
howpublished = {Cryptology ePrint Archive, Report 2020/223},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/223}},
}

A non-interactive zero-knowledge (NIZK) protocol enables a prover to convince a verifier of the truth of a statement without leaking any other information by sending a single message. The main focus of this work is on exploring short pairing-based NIZKs for all NP languages based on standard assumptions. In this regime, the seminal work of Groth, Ostrovsky, and Sahai (J.ACM’12) (GOS-NIZK) is still considered to be the state-of-the-art. Although fairly efficient, one drawback of GOS-NIZK is that the proof size is multiplicative in the circuit size computing the NP relation. That is, the proof size grows by O(|C|λ), where C is the circuit for the NP relation and λ is the security parameter. By now, there have been numerous follow-up works focusing on shortening the proof size of pairing-based NIZKs, however, thus far, all works come at the cost of relying either on a non-standard knowledge-type assumption or a non-static q-type assumption. Specifically, improving the proof size of the original GOS-NIZK under the same standard assumption has remained as an open problem.

Our main result is a construction of a pairing-based NIZK for all of NP whose proof size is additive in |C|, that is, the proof size only grows by |C|+\poly(λ), based on the decisional linear (DLIN) assumption. Since the DLIN assumption is the same assumption underlying GOS-NIZK, our NIZK is a strict improvement on their proof size.

As by-products of our main result, we also obtain the following two results: (1) We construct a perfectly zero-knowledge NIZK (NIPZK) for NP relations computable in NC1 with proof size |w|⋅\poly(λ) where |w| is the witness length based on the DLIN assumption. This is the first pairing-based NIPZK for a non-trivial class of NP languages whose proof size is independent of |C| based on a standard assumption. (2)~We construct a universally composable (UC) NIZK for NP relations computable in NC1 in the erasure-free adaptive setting whose proof size is |w|⋅\poly(λ) from the DLIN assumption. This is an improvement over the recent result of Katsumata, Nishimaki, Yamada, and Yamakawa (CRYPTO’19), which gave a similar result based on a non-static q-type assumption.

The main building block for all of our NIZKs is a constrained signature scheme with decomposable online-offline efficiency. This is a property which we newly introduce in this paper and construct from the DLIN assumption. We believe this construction is of an independent interest.

1.


text:
# Thread: Challenging Number placement puzzle

1. ## Challenging Number placement puzzle

Imagine you have a 5 x 5 grid. You have to place the numbers 1-25 on the grid so that each row, column and diagonal add up the the same number, can it be done?

For example, here's what I mean with a 3x3 version.

4 9 2
3 5 7
8 1 6

All columns, rows and diagonals all add up to 15.

I would love someone to work this out, been annoying me for ages.

2. Originally Posted by craig
Imagine you have a 5 x 5 grid. You have to place the numbers 1-25 on the grid so that each row, column and diagonal add up the the same number, can it be done?

For example, here's what I mean with a 3x3 version.

4 9 2
3 5 7
8 1 6

All columns, rows and diagonals all add up to 15.

I would love someone to work this out, been annoying me for ages.
If you have Matlab, just type:
Code:
magic(5)
I think that's what you're looking for.

3. Originally Posted by janvdl
If you have Matlab, just type:
Code:
magic(5)
I think that's what you're looking for.
Do you have the corresponding Maple command by any chance ?

4. Originally Posted by craig
Do you have the corresponding Maple command by any chance ?
I'm afraid not (I've only done a Matlab course), but just give me a moment, I'm installing Matlab now, then I'll just post the matrix here for you.

5. Originally Posted by janvdl
I'm afraid not (I've only done a Matlab course), but just give me a moment, I'm installing Matlab now, then I'll just post the matrix here for you.
Thankyou! Someone gave me this problem & I've been stumped ever since

6. I'm sorry, the bloody thing refuses to activate my LEGAL copy... I'll get back to you on this one in the next hour.

7. Originally Posted by janvdl
I'm sorry, the bloody thing refuses to activate my LEGAL copy... I'll get back to you on this one in the next hour.
Just found some on the internet.

19 8 22 11 5
12 1 20 9 23
12 24 13 2 16
3 17 6 25 14
21 15 4 18 7

Thanks for the help anyway

8. Originally Posted by craig
Just found some on the internet.

19 8 22 11 5
12 1 20 9 23
12 24 13 2 16
3 17 6 25 14
21 15 4 18 7

Thanks for the help anyway

9. There are hundreds of 5-by-5 Magic Squares
and there are formulas for generating them.

Here's one of them:

. . $\begin{array}{ccccc}17 & 24 & 1 & 8 & 15 \\ 23 & 5 & 7 & 14 & 16 \\ 4 & 6 & 13 & 20 & 22 \\ 10 & 12 & 19 & 21 & 3 \\ 11 & 18 & 25 & 2 & 9 \end{array}$

10. Originally Posted by janvdl


text:
# INIT¶

INIT(model, weights, tol, runtime, logfile, epsilon)[source]

Use the INIT algorithm (Agren et al., 2012) to extract a context specific model using data. INIT algorithm find the optimal trade-off between inluding and removing reactions based on their given weights. If desired, accumulation of certain metabolites can be allowed or even forced.

Usage

tissueModel = INIT(model, weights, tol, runtime, logfile)

Inputs

• model – input model (COBRA model structure)
• weights – column with positive and negative weights for each reaction positive weights are reactions with high expression, negative weigths for reaction with low expression (must be same length as model.rxns)

Optional inputs

• tol – minimum flux threshold for “expressed” reactions (default 1e-8)
• logfile – name of the file to save the MILP log (string)
• runtime – maximum solve time for the MILP (default value - 7200s)
• epsilon – value added/subtracted to upper/lower bounds (default 1)

Output

• tissueModel – extracted model

Agren et al. (2012). Reconstruction of genome-scale active metabolic networks for 69 human cell types and 16 cancer types using INIT. PLoS Comput. Biol. 8, e1002518.


text:
A bit of everything

# Particle effects

Particle effects are a graphical effect involving the simulation of many small objects with some physics involved for them. They can be useful to give an impression of many such physical systems, such as fire, smoke, fluids, dust, or other phenomenon, such as magical effects.

The basic idea behind a particle effect is the generation of a system of small objects (the so-called particles), usually in a somewhat random manner and for a finite amount of time, and making them evolve according to some set of rules to simulate their behaviour.

The simplest particle effect is simply the generation of radial particles : every frame, there is some probability to generate $n$ particles, each one with a uniform probability to have its momentum in any direction. And every frame, each particle advances one step in this direction.

To generate a single particle, we will need to generate its speed, its lifespan (before deletion), and an angle for its direction. For now we will assume that all particles are generated at the center of the frame.
stuff function generateParticle(x, y) { var lifespan = rndInt(minLifespan1, maxLifespan1); var speed = (Math.random() * maxSpeed1) + minSpeed1; var vx = speed * Math.cos(Math.random() * 2 * Math.PI); var vy = speed * Math.sin(Math.random() * 2 * Math.PI); return {x : x, y : y, lifespan : lifespan, vx : vx, vy :vy}; }
To avoid the particles going off in random directions, we can restrict the angle generated. For instance we may choose to only direct the particles upward, using
stuff var angle = rndFloat(-Math.PI / 4, -3 * Math.PI / 4);
We can also make the distribution non-uniform for added effect, such as a normal distribution, using the Box-Muller transform. For instance here's a normal distribution centered around $\pi / 2$ with a standard deviation of $0.3$.
Not clearing the screen

Adding the effects of gravity to the particles is fairly straightforward, at least approximately : every turn, the vertical momentum is accelerated downward (due to the orientation of the screen, this means increasing it)

Last updated : 2017-08-30 15:04:45


text:
• 主管单位：
• 上海市教育委员会
• 主办单位：
• 上海理工大学
• 主  编：
• 庄松林
• 地  址：
• 上海市军工路516号
• 邮政编码：
• 200093
• 联系电话：
• 021-55277251
• 电子邮件：
• xbzrb@usst.edu.cn
• 国际标准刊号：
• 1007-6735
• 国内统一刊号：
• 31-1739/T
• 邮发代号：
• 4-401
• 单  价：
• 15.00
• 定  价：
• 90.00

Normal criteria relating to derived curves and shared hyperplanes

DOI：10.13255/j.cnki.jusst.20210707001

 作者 单位 E-mail 范楚君 上海理工大学 理学院 上海 200093 刘晓俊 上海理工大学 理学院 上海 200093 xiaojunliu2007@hotmail.com

基于值分布和正规族理论以及高等代数相关知识，研究了全纯曲线族及其导曲线分担处于$t$次一般位置的超平面的正规定则。设$\mathcal{F}$是一族从区域$D \subset \mathbb{C}$到${\mathbb{P}}^{N}(\mathbb{C})$的全纯曲线，${H_\ell } = \rhbr \left\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \right.\left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = {\text{0}}} \right\}$是${\mathbb{P}^N}(\mathbb{C})$中处于$t$次一般位置的超平面，${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}},{\text{ }}\ell = 1,2, \cdots ,3t + 1$，${H_0} = \left\{ {{x_0} = {\text{0}}} \right\}$，$t\geqslant N$。假定对任意的$f \in \mathcal{F}$满足条件：若$f(z) \in {H_\ell }$，则$\nabla f(z) \in {H_\ell }$，$\ell = 1,2, \cdots ,3t + 1$；若$f(z) \in \displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$，则$\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant\delta$，其中，$\delta \in \left(0，1\right)$且为常数。那么，$\mathcal{F}$在$D$上正规。对于$N = 3$，$t = 3,4,5$的特殊情形，本文有效降低了所分担超平面的个数。

Based on the theories of value distribution, normal family and the knowledge of advanced algebra, the normal criteria for the holomorphic curves and their derived curves sharing hyperplanes located in t-subgeneral position was studied. Let $\mathcal{F}$ be a family of holomorphic curves of a domain $D \subset \mathbb{C}$ to ${\mathbb{P}^N}(\mathbb{C})$, ${H_\ell } =\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = 0} \right\}$ be hyperplanes in ${\mathbb{P}^N}(\mathbb{C})$ located in t-subgeneral position, where ${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}}$, $\ell = 1,2, \cdots ,3t + 1$, ${H_0} = \left\{ {{x_0} = 0} \right\}$, $t\geqslant N$. Assume the following conditions hold for every $f \in \mathcal{F}$: If $f(z) \in {H_\ell }$, then $\nabla f(z) \in {H_\ell }$, $\ell = 1,2, \cdots ,3t + 1$; If $f(z) \in\displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$, then $\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant \delta$, where $\delta \in \left(0，1\right)$ is a constant. Then $\mathcal{F}$ is normal on $D$. For the special case of $N = 3$ and $t = 3,4,5$, the number of shared hyperplanes can be effectively reduced through this research.
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text:
# Determining whether the unit circle group is a cyclic group

1. Jul 27, 2013

### mahler1

1. The problem statement, all variables and given/known data

Let S be the set of complex numbers z such that |z|=1. Is S a cyclic group?

3. The attempt at a solution

I think this group isn't cyclic but I don't know how to prove it. My only idea is:
If G is a cyclic group, then there is an element x in G such that for all g in G, g can be written as g=x^j (j integer) and x is called a generator of the group G. Now, in this case, if S is not cyclic, I could prove it by contradiction, that means: I assume S is cyclic, then I use the definition of cyclic, that is, there is an element x in S such that if g is in S then g is of the form g=x ^j for some integer j. If I want to show that this cannot be always true, I must find a (generic) complex number z whose absolute value is 1 but z is different from x^j for all integer j. Ok, this may sound stupid but I don't know how to actually do this last part and I got stuck here.

I edit the post to add a new idea: I "know" that the set of complex numbers z with |z|=1 is an uncountable set. Now, if S was a cyclic group then it would have a generator, lets call it x. By definition, for all element s in S, s=x^j for some integer j. Now, the set of integers is countable and so is the set X={x^j} with j integer. As I can't have a surjective function from a countable set to an uncountable one, then, I could find an element s in S that can't be related to any element of the set X.
Is this idea correct? If it is, then I have some doubts on the following points: 1) How can I show that the unit circle is an uncountable set? 2) I automatically assumed that I can't have a surjective function from a countable set to an uncountable set, this can seem obvious but I don't know how to prove it.

Sorry if I've made any grammar or spelling mistakes, english is not my native language

Last edited: Jul 27, 2013
2. Jul 27, 2013

### micromass

The idea to use countable is very good.

Find a bijection with $\mathbb{R}$ or some other set you know to be uncountable.

Well, what's your definition of countable? Try to argue by contradiction. Assume you have a surjective function,...

3. Jul 27, 2013

### mahler1

Well, I still don't know how to do the first part, I mean, showing that the unit circle is uncountable. I thought of a bijective function between this set and another known uncountable set (the real numbers or the interval [0 1]) but I don't know how to explicitly construct a function between these sets. If I come up with any ideas, I'll post and ask again.

4. Jul 27, 2013

### micromass

Do something with sine and cosine functions to parametrize the circle.

5. Jul 27, 2013

### mahler1

Ok, with all the things you've said I came up with the following "proof":

Suppose S is a cyclic group. By definition, there is an element in S, call it x, such that for all s in S, s can be written as s=x^j for some integer j. Let X={x^j} be all the elements in S written in that way. Consider the map between X and a subset J of the integers simply by letting f(x^j)=j. The set Z of all integers is a countable set, then, in particular, J is a countable set and this means that S is also countable.

Every complex number z is of the form z=a+bi for a,b real numbers. So S can be defined as S={a,b in R such that a^2+b^2=1}. Now, consider the functions a=cos(t) and b=sin(t) with t in [0 2π); this is a bijective parameterization (parametrization?) of the unit circle. So, there exists a bijective map from [0 2π) to the set S. It follows that S has the same cardinality as the set [o 2π), which is just the interval [0 1) multiplied by 2π. But [0 1) is an uncountable set, then S must be uncountable. But this is absurd since S is countable. The contradiction comes from assuming that there is an element x in S such that for all s in S, s=x^j with j integer. In conclusion, S is not a cyclic group.

Is something wrong with the proof or is it ok?

6. Jul 27, 2013

### micromass

One small detail that we can't ignore. You defined the map $f(x^j) = j$. But is this well-defined? That is, what if $x^2 = x^4$ (for example). Then $2 = f(x^2)= f(x^4) = 4$. This is clearly false.

So, how do we know that $x^n = x^m$ can't happen for distinct $m$ and $n$?

7. Jul 27, 2013

### pasmith

I would state and prove, as a separate proposition, that if a group is cyclic then it is countable. It is more natural, and avoids micromass's objection, to define a map from the natural numbers to a cyclic group rather than the reverse. Recall that to show that a non-empty set is countable, it is sufficient to show that there exists a surjection from the natural numbers to that set.

It then follows directly from that proposition that if S is uncountable then S cannot be cyclic, and you will show that S is uncountable.

This is really the heart of the proof, so it's not enough to simply state the fact that $t \mapsto \cos (t) + \mathrm{i}\sin (t)$ is a bijection from $[0, 2\pi)$ to $S$. You have actually to prove it.

(Actually I'd save a step and find a bijection directly from $[0,1)$ to S.)

8. Jul 27, 2013

### mahler1

You're right on this point, what I can do is to separate in the only two possible cases: 1) S is a finite cyclic group. 2) S is an infinite cyclic group

Case 1) Suppose S is a finite cyclic group of order n, I want to show that if <x> generates S, then the powers of the set {1, x, ..., x^(n-1)} are all distinct. As S is finite cyclic of order n, then there is an element x in S such that x^k is different from 1 for all 0<k<n. Consider the set {1, x, ..., x^(n-1)} Lets assume x^j=x^i where, without loss of generality,0<= i<j<=n-1. Then, multiplying both sides of the equation by x^i (I can do this because all the elements in a group have inverses)yields x^(j-i)=1. But this is absurd since 0<j-i<n. So, all the powers of that set must be distinct. From the preceding observations we can also see that <x> generates S, as the set {1, x, ..., x^(n-1)} has n different elements, all of which belong to a group of order n.

Case 2) Suppose S is infinite cyclic and S is generated by <x>. Lets show that if j is different from i, where j and i are integers, then x^j is different from x^i. Suppose, without loss of generality, that j>i and that x^j=x^i. As S is a group, I can multiply both sides of the equation by x^i inverse to get
x^(j-i)=1, but this means that S is finite, which is absurd. Then, if j is different from i, it can't happen that x^j=x^i.

Sorry for not writing in latex, I'll try to learn how to use it as soon as I can.

Last edited: Jul 27, 2013
9. Jul 27, 2013

### mahler1

To pasmith: Yes, what you propose seems to be simpler, but I cannot think immediately how to prove all of that, I'll try to follow your suggestions and write the proof. Thanks

10. Jul 27, 2013

### micromass

OK, but how does this help? Isn't just saying that such a group is finite enough? After all, if the circle is uncountable, it cannot be finite.

11. Jul 27, 2013

### mahler1

Sorry, I don't get what you're saying. I assumed S was cyclic and invented that bijective map. If it is well defined, it cannot happen, as you said, that x^n=x^m for n different to m. So, I proved that whenever n is different to m, x^n can't be equal to x^m. To prove this, I assumed that S is cyclic, and if S is cyclic, it can be finite or infinite. In the last part of the proof, I invented another mapping between S and another set to arrive to the conclusion that S is uncountable, which is absurd since from the first mapping it follows that S is a countable set or a finite set.

Last edited: Jul 27, 2013
12. Jul 27, 2013

### micromass

You basically have it. Either the group is finite or it is countably infinite. Either way, it can't be uncountable.

13. Jul 27, 2013

### mahler1

micromass: I edited my last reply because then I reread your answer and realized that i didn't get what I was doing wrong. But, putting all the ideas together, I think I can rewrite and change my original "proof", I'll do it.

14. Jul 28, 2013

### Boorglar

I can suggest another proof that avoids using the uncountability of the unit circle, since this property is not necessary (there are in fact countable subsets of the unit circle that also can't be cyclic groups).

Using the polar coordinate representation of complex numbers, the unit circle is the set of all numbers of the form $cis(2\pi t), t \in [0,1]$. Suppose the unit circle were cyclic. Then let $cis(2\pi t_{0})$ be a generator.

The unit circle can therefore be described as $\left\{cis(2n\pi t_0), n \in Z \right\}$.
Now there are two cases to consider:

1)
If $t_0$ is rational, then $t_0 = \frac{a}{b}$ for some $a, b \in Z, b ≠ 0$. Then the unit circle consists of $\left\{cis(2\pi \frac{na}{b} ), n \in Z \right\}$. But we can easily find a number in the unit circle that is not in the set (which one?) and thus we reach a contradiction.

2)
If $t_0$ is irrational, then $cis(\pi) = -1$ is not in the set (why?). Again, we reached a contradiction.

Therefore, the unit circle is not a cyclic group.
Keep in mind that the set of all solutions to the equation $x^n = 1, n \in N$ is a countable subset of the unit circle, and yet it is not cyclic either.

15. Aug 1, 2013

### mahler1

Thanks for your suggestion and for the counterexample of the countable subset, it's much more simpler to prove the statement using your ideas.
In 1), if we choose t=1/2, then it's easy to see that cis(1/2) is not in the set because in that case we would have 2πna/b=1/ iff π=b/(4na) which is absurd since π is irrational and clearly b/(4na) is rational.
In 2), cis(π) is not in the set, as you said, because if this is true, then we would have that π=2nπt0 iff 1/2n = t0 which is absurd since 1/2n is a rational number and t0 is irrational.


text:
# Union with Complement

## Theorem

The union of a set and its complement is the universe:

$S \cup \map \complement S = \mathbb U$

## Proof

Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $T \cup \relcomp S T = S$ from Union with Relative Complement.

$\blacksquare$

## Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Union with Relative Complement.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.


text:
# Heat in a resistor (circuit with inductor and capacitor)

1. Sep 1, 2014

### Rugile

1. The problem statement, all variables and given/known data
A source of ε = 10V, capacitor of C = 5 μf, inductor of L = 15 mH and r = 10 Ω and a resistor of R = 100 Ω are connected as in the schematic (attached). How much heat will dissipate in the resistor after the switch is opened?

2. Relevant equations

dQ = I2Rdt
KVL

3. The attempt at a solution

First of all I was quite confused by the question, so I just assumed that the switch is opened after a long time, when the capacitor and the inductor are fully charged. Then the voltage across the capacitor is U0 = ε and the current in the inductor is I0 = ε / (R+r). We can write the equation (KVL):
$L\frac{dI}{dt} + I(R+r) + U_C = 0$, where UC is the voltage across the capacitor. I get stuck here - I suppose I could write second order differential equation involving the function of q: $L \frac{d^2 q}{dt^2} + (R+r) \frac{dq}{dt} + \frac{C}{q} = 0$, but I'm not sure what to do with that either. Any hints?

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2. Sep 1, 2014

### td21

Step 1: After switch is opened, LC oscillation occurs as described by your equation.
Step 2: Heat loss through R and r. Your task is to find out the heat loss thorugh R. At the end, all energy is dissipated through heat loss. So I suggest you find the ratio between heat loss through R: heat loss through r. I also suggest you find the initial energy.
Step 3: The ratio can be found from the ratio of R:r.

3. Sep 1, 2014

### Rugile

Okay, so it is enough to find the initial energy of the system and then calculate the heat loss in R using the ratio? Can I say that the initial energy of the system (after opening the switch) was $E = \frac{CU^2}{2} + \frac{LI^2}{2}$, where $I = \frac{\epsilon}{R+r}$ and U = epsilon = EMF ?

4. Sep 2, 2014

What switch?

5. Sep 3, 2014

### Staff: Mentor

The one marked with letter 'J'.

6. Sep 3, 2014

### Rugile

Yes, exactly.

Does anyone have any idea if the equation of initial energy written above is correct? I have doubts about the current in the inductor, since when we open the switch it disappears (gradually), doesn't it?

7. Sep 3, 2014

### BvU

O2: Very subtle...
Rugile's post #3 seems pretty good to me: keep going !

And post #6: yes, it dampens out quickly. But the exercise only wants the total, so no need to solve the D.E.

8. Sep 3, 2014

### Staff: Mentor

That looks right. That's the stored energy before the jumper to the battery is removed.

After the jumper is removed, the inductor current becomes the capacitor current, the two now being in series.