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https://gigaom.com/2008/11/05/californians-reject-energy-props-could-get-high-speed-rail/
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# Californians Reject Energy Props, Could Get High Speed Rail
Election results are still pouring in all around the country, and while the cleantech community is surely celebrating the new president elect, sorting out the results of California’s propositions is not as cut and dried. Californian’s have rejected Propositions 7 and 10, which would have increased renewable energy targets and given incentives for alternative fuel vehicles, respectively. Meanwhile, the prospect of a high-speed train is getting stronger, as prop 1A leads with more results still coming in. Here’s a break down on the election results for the propositions we profiled in our voter’s guide.
Proposition 1A: High-Speed Passenger Train Bond: LIKELY TO PASS: The results on this prop have yet to officially be called, but it looks like Californians have agreed to put $9 billion into a high-speed train project that would connect Los Angeles and San Francisco. Debate around this prop was pretty straightforward and centered on the burden for the taxpayer. The proposition authorizes the state of California to issues$9.95 billion worth of bonds. While the rail is far from being built and represents the largest public works undertaking in the state’s history, California has taken a large step toward constructing America’s first high-speed rail system.
Proposition 7: The Solar and Clean Energy Act: NO:
The proposition would have raised the state’s renewable energy targets to 40 percent of the total energy mix by 2020 and 50 percent by 2025. Many groups that would normally endorse higher green energy goals, including environmental groups and cleantech startups, opposed this proposition claiming the confusingly written measure would actually slow renewable energy development. “I think people in California were able to see that Prop. 7, the way it was written, was not going to work,” Jim Metropulos, senior advocate for the Sierra Club, told SFGate. California will still need to work hard to meet its existing renewable portfolio standard to generate 20 percent of its power from renewable sources by 2010.
Proposition 10 The California Renewable Energy and Clean Alternative Fuel Act: NO:
T. Boone Pickens may have poured millions
into the Yes on 10 campaign, but it was to no avail. The measure, which would have provided big rebates for consumers purchasing alternative fuel vehicles, could have propped up natural gas vehicles with billions at a time when the electric automotive revolution looks to be starting. The prop would have been paid for with \$5 billion in state bonds. This takes more wind out of the Pickens Plan’s sails.
San Francisco Proposition H: The San Francisco Clean Energy Act: NO:
San Franciscans have rejected the proposition of the city running its own public power system a dozen times now. This year the measure, which would have given the city supervisors the power to acquire PG&E’s electrical distribution grid, was wrapped in green energy mandates, including the lofty goal of getting the city to generate all of its power from renewable sources by 2040. PG&E will maintain control of its facilities for now and will still have to meet California’s renewable portfolio standard.
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https://www.springerprofessional.de/en/sensors-and-instrumentation-aircraft-aerospace-energy-harvesting/18430042
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main-content
Sensors and Instrumentation, Aircraft/Aerospace and Energy Harvesting, Volume 7: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics, 2020, the seventh volume of eight from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Shock & Vibration, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing including papers on:
Alternative Sensing & Acquisition
Active Controls
Instrumentation
Aircraft/Aerospace & Aerospace Testing Techniques
Energy Harvesting
### Chapter 1. Orion MPCV E-STA Nonlinear Dynamics Uncertainty Factors
Abstract
Matt Griebel, Adam Johnson, Brent Erickson, Andrew Doan, Chris Flanigan, Jesse Wilson, Paul Bremner, Joel Sills, Erica Bruno
### Chapter 2. The Vibration and Acoustic Effects of Prop Design and Unbalance on Small Unmanned Aircraft
Abstract
The vibration and acoustic effects due to prop design, damage, and unbalance on a popular small unmanned aircraft systems (sUAS) is presented. The use of sUAS or drones is becoming ever more popular for hobbyists, as well as in commercial and military operations, and the influence of props on the vibration and acoustic environment are of interest. Many types of props are available for use on sUAS that promise extended run times, increased performance, and quieter operation. While many systems promise these results, few studies have been conducted to measure and evaluate their true performance. Therefore, this review provides data obtained by experimentally measuring the vibration levels onboard the host aircraft as well as the acoustic levels produced. The data is analyzed to gain further understanding of the vibration and acoustic properties and make predictions on prop performance. The fundamental frequencies of the aircraft are found along with the acoustic signature. These two outcomes are compared and studied to find the correlation between them. The analysis will utilize an airframes that is commonly used in the UAS community along with frequently used props that are balanced and unbalanced and with and without damage. The data is obtained from aircraft fully powered and airborne in a hovering or level flight configuration. This study provides sUAS operators the information required for choosing the most effective prop design to effectively reduce vibration and acoustic sound levels.
William H. Semke, Djedje-Kossu Zahui, Joseph Schwalb
### Chapter 3. A Deformed Geometry Synthesis Technique for Determining Stacking and Cryogenically Induced Preloads for the Space Launch System
Abstract
The Space Launch System (SLS) stacking and Core Stage (CS) fueling induce significant preloads that contribute to the liftoff pad separation “twang”. To accurately capture this, an approach is required that can replicate the physics of all SLS physical stacking steps, CS cryogenic shrinkage, associated geometric nonlinearities, and the transient behavior and decay of the preloads with changing boundary conditions as the vehicle separates from the pad.
The Deformed Geometry Synthesis (DGS) approach presented here satisfies the above requirements. DGS determines induced preloads by modeling components in their deformed geometry states and then enforcing compatibility by closing the resulting “deadbands”. DGS seamlessly integrates into the multibody modal synthesis framework and does not require the use of artificial external loads to enforce preloads or post-processing steps to remove their influence. Since DGS iterates to solve for the deformed state inclusive of geometric nonlinearities, running linearized parametrics to exercise different potential orientations of ball jointed struts that connect the CS to Boosters for cryo-shrinkage analyses is entirely avoided.
Relative to the transient behavior and decay of stacking and cryo-induced preloads with SLS liftoff pad separation, this is an area of considerable interest to the SLS program. To capture this in the most accurate way possible, DGS algorithms are designed to work with Henkel-Mar nonlinear pad separation algorithms which operate on the separating longitudinal and lateral degrees of freedom (DoFs) between the vehicle and the pad. As the separating DoFs release, in whatever manner as dictated by the interface geometries, interface loads and interface flexibilities as well as the external loading on the vehicle, the subject preloads generate a complex twang/decay time-trace as dictated by the physics of the problem.
This paper presents DGS numerical verification against the closed-form solution for Timoshenko’s 3 ball-jointed strut preload problem. This problem is then extended by the authors to the geometric nonlinear case where DGS is compared to the Newton-Raphson solution of the nonlinear equations. Next, DGS is utilized to solve the SLS stacking and cryogenic shrinkage coupled loads analyses. Finally, Henkel-Mar pad separation simulations are executed that isolate the impact of the induced preloads’ twang and decay characteristics.
Joel Sills, Arya Majed, Edwin Henkel
### Chapter 4. End-to-End Assessment of Artemis-1 Development Flight Instrumentation
Abstract
As the space industry continues to strive for more efficient launch vehicles they must rely on increasingly accurate predictive models. Verification of models typically requires physical testing. Flight data measurements offer the most real and therefore the most accurate data for model correlation. As NASA prepares for the inaugural launch of their new Space Launch System (SLS), Artemis-1, they must rely heavily on predictive system models to ensure flight safety. Artemis-1 will be an unmanned scientific mission with the intent of blazing a trail for future manned missions. NASA has implemented a system of Development Flight Instrumentation (DFI) in the hopes of recovering useful flight data during liftoff and ascent to aid in correlating their predictive models to ensure human safety in future missions. An end-to-end assessment of the DFI system was performed to verify data acquired during Artemis-1 would be adequate for the targeted flight test objectives (FTOs). This was accomplished using a computational simulation of all sensors and Data Acquisition (DAQ) parameters to investigate any potential problem areas in the current architecture. Input nominal signals were approximated and injected into the system model. Synthesized acquired signals were recovered to verify FTO success.
Andrew Doan, Adam Johnson, Tony Loogman, Paul Bremner, Joel Sills, Erica Bruno
### Chapter 5. Space Launch System Mobile Launcher Modal Pretest Analysis
Abstract
NASA is developing an expendable heavy lift launch vehicle capability, the Space Launch System, to support lunar and deep space exploration. To support this capability, an updated ground infrastructure is required including modifying an existing Mobile Launcher system. The Mobile Launcher is a very large heavy beam/truss steel structure designed to support the Space Launch System during its buildup and integration in the Vehicle Assembly Building, transportation from the Vehicle Assembly Building out to the launch pad, and provides the launch platform at the launch pad. The previous Saturn/Apollo and Space Shuttle programs had integrated vehicle ground vibration tests of their integrated launch vehicles performed with simulated free-free boundary conditions to experimentally anchor and validate structural and flight controls analysis models. For the Space Launch System program, the Mobile Launcher will be used as the modal test fixture for the ground vibration test of the first Space Launch System flight vehicle, Artemis 1, programmatically referred to as the integrated vehicle modal test. The integrated vehicle modal test of the Artemis 1 integrated launch vehicle will have its core and second stages unfueled while mounted to the Mobile Launcher while inside the Vehicle Assembly Building, which is currently scheduled for the summer of 2020. The Space Launch System program has implemented a building block approach for dynamic model validation. The modal test of the Mobile Launcher is an important part of this building block approach in supporting the integrated vehicle modal test since the Mobile Launcher will serve as a structurally dynamic test fixture whose modes will couple with the modes of the Artemis 1 integrated vehicle. The Mobile Launcher modal test will further support understanding the structural dynamics of the Mobile Launcher and Space Launch System during rollout to the launch pad, which will play a key role in better understanding and prediction of the rollout forces acting on the launch vehicle. The Mobile Launcher modal test is currently scheduled for the summer of 2019. Due to a very tight modal testing schedule, this independent Mobile Launcher modal pretest analysis has been performed to ensure there is a high likelihood of successfully completing the modal test (i.e. identify the primary target modes) using the planned instrumentation, shakers, and excitation types. This paper will discuss this Mobile Launcher modal pretest analysis for its three test configurations and the unique challenges faced due to the Mobile Launcher’s size and weight, which are typically not faced when modal testing aerospace structures.
James C. Akers, Joel Sills
### Chapter 6. Feasibility Study to Extract Artemis-1 Fixed Base Modes While Mounted on a Dynamically Active Mobile Launch Platform
Abstract
There are several challenges associated with the scheduled integrated modal test (IMT) of the Space Launch System (SLS) Artemis-1 flight vehicle mounted on the mobile launcher (ML). While the goal of the test is to characterize the Artemis-1, the inclusion of the ML as the support stand for the test means that the entire system must be well characterized. A considerable amount of effort and schedule will have to be devoted to understanding both the test stand (the ML) and the Artemis-1 flight vehicle, and there is a risk that the effort may not be completed in time for a successful launch.
NASA has requested that alternative methods be investigated to generate test results that can remove the effects of the ML from the test. ATA Engineering, Inc., (ATA) was given a reduced model of the Artemis-1 flight vehicle’s IMT configuration containing the candidate set of accelerometers and interface degrees of freedom (DOF). The model was used to determine how well ATA’s fixed base correction technique is able to estimate fixed base modes from test data collected on the IMT. This paper presents the fixed base correction method results.
Kevin L. Napolitano
### Chapter 7. Challenges to Develop and Design Ultra-high Temperature Piezoelectric Accelerometers
Abstract
Piezoelectric accelerometer sensors are widely used for testing and monitoring vibrations in automotive, aerospace or industrial applications. The temperature limitation of most piezoelectric accelerometers is below 500 °C, which meets the requirements of typical vibration measurement applications. However, in extreme cases such as engine monitoring, measurements are required up to 900 °C, where the ultra-high temperature accelerometers are needed. This research work focuses on the development of ultra-high temperature piezoelectric accelerometers using bismuth layer-structured ferroelectric piezoceramics as sensing elements. The challenges to develop these materials and create the corresponding ultra-high temperature accelerometers will be discussed in this paper.
Chang Shu, Neill Ovenden, Sina Saremi-Yarahmadi, Bala Vaidhyanathan
### Chapter 8. Application of Quasi-Static Modal Analysis to an Orion Multi-Purpose Crew Vehicle Test
Abstract
Bolted structural joints often exhibit load-dependent stiffness and energy dissipation that leads to nonlinear, amplitude dependent frequency and damping in the structure. As an alternative to direct integration of the nonlinear equations of motion, quasi-static modal analysis (QSMA) determines the dependence of frequency and damping on response amplitude using loading behavior from nonlinear static analyses. QSMA has previously been demonstrated to substantially reduce computational cost and maintain accuracy relative to full nonlinear dynamic simulation. This work explores the applicability of QSMA to a complex, large-scale aerospace structure. QSMA is employed to analyze a nonlinear model of test hardware developed to support the Orion Multi-Purpose Crew Vehicle program, which exhibited nonlinear behavior during dynamic testing at flight-like load levels. In addition to the extraction of amplitude-dependent frequency and damping curves, a Bouc-Wen hysteresis model was used in conjunction with the quasi-static results to develop nonlinear, uncoupled, time-domain modal equations of motion for the structure. Excellent agreement was observed between the reduced and full-order nonlinear models, encouraging future employment of QSMA to support accurate and efficient model reduction of structures with bolted joint nonlinearities.
Matthew S. Allen, Joe Schoneman, Wesley Scott, Joel Sills
### Chapter 9. Using BB-gun or Acoustic Excitation to Find High Frequency Modes in Additively Manufactured Parts
Abstract
Additive manufacturing (AM) considers parts that are produced at a low volume or with complex geometries. Identifying internal defects on these parts is difficult as current testing techniques are not optimized for AM processes. The resonant frequency method can be used to find defects in AM parts as an alternative to X-ray or CT scanning. Higher frequency modes at approximately 8000 Hz and above cannot be tested with a traditional modal hammer or shaker since they do not provide enough excitation. The goal of this paper is to evaluate creative testing techniques to find internal defects in parts with high frequency modes. The two types of testing methods considered are acoustic excitation provided by two speakers and high velocity impact testing produced by a BB – gun. Although the frequency ranges of interest are part dependent, these techniques were able to reach up to 16,000 Hz, which is an additional 8000 Hz above what the traditional modal hammer is able to reach. This work was funded by the Department of Energy’s Kansas City National Security Campus which is operated and managed by Honeywell Federal Manufacturing Technologies, LLC under contract number DE-NA0002839.
Aimee Allen, Kevin Johnson, Jason R. Blough, Andrew Barnard, Troy Hartwig, Ben Brown, David Soine, Tristan Cullom, Douglas Bristow, Robert Landers, Edward Kinzel
### Chapter 10. Parametric Analysis and Voltage Generation Performance of a Multi-directional MDOF Piezoelastic Vibration Energy Harvester
Abstract
Piezoelectric vibration energy harvesting has been extensively investigated in recent years and the majority of results focus on using the cantilever beam model under base driven motion. The main focus of this paper is to perform a parametric analysis of multi degrees of freedom piezo-elastic energy harvester to optimize the capability curve exploiting crossing/veering between modes. The structure under test consists of a combination of slender beams with one or more orthogonal beam segments placed on it. The resulting combined structure exhibits bending vibration modes in orthogonal planes. The cross and veering phenomena are studied in deep, attempting to improve the resulting mechanical-electrical energy conversion of the combined structure. A numerical model of the system under investigation is developed considering also non-classical damping. A parametric analysis of the system’ s performance due to geometrical and electrical properties variations are investigated to design a broadband harvester. An experimental analysis is performed on a test rig specially built to investigate the crossing and veering phenomena effects on the resulting output voltage from the energy harvester. Numerically simulated and experimental data are compared to provide information for updating the model as well as to address the efficiency of the harvester in terms of voltage generation.
Paulo S. Varoto, Elvio Bonisoli, Domenico Lisitano
### Chapter 11. Are We Nearly There Yet? Progress Towards the Fusion of Test and Analysis for Aerospace Structural Dynamics
Abstract
The dual roles of Test and Analysis (experimentation and simulation) in Structural Dynamics have become increasingly inter-dependent for several years, and we are now approaching a level of interaction referred to as Fusion. Today, analysis and test can serve to minimise the effects of the inevitable uncertainties which arise in real engineering circumstances where assumptions, approximations and selections must be made. Here we review the status of this development and identify some specific situations where further enhancements can still be made.
David Ewins
### Chapter 12. Feasibility Study of SDAS Instrumentation’s Ability to Identify Mobile Launcher (ML)/Crawler-Transporter (CT) Modes During Rollout Operations
Abstract
The Space Launch System (SLS) and its Mobile Launcher (ML) will be transported to the launch pad via the Crawler-Transporter (CT) system. Rollout (i.e., transportation) loads produce structural loads on the integrated SLS/Orion Multi-Purpose Crew Vehicle (MPCV) launch vehicle which are of a concern with respect to fatigue. As part of the risk reduction process and in addition to the modal building block test approach that has been adopted by the SLS Program, acceleration data will be obtained during rollout for use in modal parameter estimation. There are several occurrences where the ML/CT will be transported either into the Vertical Assembly Building (VAB) or to the launch pad and back without the SLS stack as part of the Kennedy Space Center (KSC) Exploration Ground Systems (EGS) Integrated Test and Checkout (ITCO). NASA KSC EGS has instrumentation installed on both the ML and CT to record data during rollout, at the launch pad, and during liftoff. The EGS instrumentation on the ML, which includes accelerometers, is referred to as the Sensor Data Acquisition System (SDAS). The EGS instrumentation on the CT, which also includes accelerometers, is referred to as the CT Data Acquisition System (CTDAS). The forces and accelerations applied to the ML and CT during a rollout event will be higher than any of the planned building block modal tests. This can be very beneficial in helping identify nonlinear behavior in the structure. Developing modal parameters from the same test hardware in multiple boundary conditions and under multiple levels of excitation is a key step in developing a well correlated FEM.
The purpose of this study was three fold. First, determine the target modes of the ML/CT in its rollout configuration. Second, determine if the test degrees of freedom (DOF) corresponding to the layout of the SDAS/CTDAS accelerometers (i.e. position and orientation) is sufficient to identify the target modes. Third, determine if the Generic Rollout Forcing Functions (GRFF’s) (“Development of Generic Crawler/Transporter Rollout Forcing Functions for Coupled System Dynamics Analysis,” NASA Exploration Systems Directorate/Cross-Program Systems Integration Technical Assessment Report, ESD 20038, July 31, 2018) is sufficient for identifying the ML/CT target modes accounting for variations in CT speed, modal damping, and sensor/ambient background noise levels.
The finding from the first part of this study identified 28 target modes of the ML/CT rollout configuration based upon Modal Effective Mass Fractions (MEFF) and engineering judgement. The finding from the second part of this study showed that the SDAS/CTDAS accelerometers (i.e. position and orientation) are able to identify a sufficient number of the target modes to support model correlation of the ML/CT FEM. The finding from the third part of this study confirms the GRFFs sufficiently excite the ML/CT such that varying quantities of the defined target modes should be able to be extracted when utilizing an Experimental Modal Analysis (EMA) Multi-Input Multi-Output (MIMO) analysis approach. An EMA analysis approach was used because Operational Modal Analysis (OMA) tools were not available and the GRFFs were sufficiently uncorrelated. Two key findings from this third part of the study are that the CT speed does not show a significant impact on the ability to extract the modal parameters and that keeping the ambient background noise observed at each accelerometer location at or below 30 μgrms is essential to the success of this approach.
Even though this study relies heavily upon the accuracy of both uncorrelated ML and CT FEM’s and unconfirmed rollout forcing functions, all of which will most likely differ from actuality, it provides important insights into the ability to extract modal parameters from the upcoming rollout events.
James P. Winkel, James C. Akers, Erica Bruno
### Chapter 13. The Integrated Modal Test-Analysis Process (2020 Challenges)
Abstract
The prevailing approach to the integrated test analysis process (ITAP) in the U.S. aerospace community involves execution of seven successive tasks, namely (1) model development, (2) modal test planning, (3) measured data acquisition, (4) measured data analysis, (5) experimental modal analysis, (6) test analysis correlation, and (7) model reconciliation. Persistence of closely spaced body and shell breathing modes in launch vehicle and spacecraft structures presents a “many modes” challenge to selection of analytical and experimental target modes. Well established coupled loads analysis decomposition appear to provide relief to the “many modes” challenge. Strategies aimed at prioritizing the multitude of sensitivities and uncertainties offer further relief. Established government orthogonality criteria are difficult to satisfy due to the multitude of closely-spaced measured complex modes. Recent introduction of a left- hand eigenvector, experimental modal analysis strategy appears to eliminate these difficulties. Unavoidable nonlinearity in structural joints threatens to invalidate many aspects of the established ITAP scheme. Recollection of space shuttle era nonlinear coupled loads methodologies and incorporation of modern, nonlinear hysteretic joint models offers promising relief to the structural joint difficulties. All of the above noted techniques and strategies may lead to more effective test analysis correlation and reconciliation outcomes.
Robert N. Coppolino
### Chapter 14. Roadmap for a Highly Improved Modal Test Process
Abstract
Difficulties encountered in modal test planning and recent advances in experimental modal analysis of aerospace systems have resulted in a new paradigm for experimental modal analysis. The Simultaneous Frequency Domain (SFD) method, weighted complex linear least squares correlation methodology, and state space model left- and right- hand eigenvector properties combine to produce the following benefits that are independent of an explicit TAM mass matrix: (1) verification and validation of experimental modes via isolation of individual experimental modes, (2) automatic self- orthogonality of experimental modes, (3) test- FEM cross orthogonality, and (4) experimental complex mode kinetic energy distribution. The new approach directly employs complex experimental modes (rather than real mode approximations) and frees experimental data from a potentially flawed TAM mass matrix. A roadmap for incorporation of the new paradigm into NASA and USAF standards and continued progress are outlined in this paper.
Robert N. Coppolino
### Chapter 15. Using Low-Cost “Garage Band” Recording Technology for Acquiring High Resolution High-Speed Data
Abstract
The Oak Ridge National Laboratory (ORNL) has developed and tested a novel system architecture for acquiring high fidelity high-speed data. The approach uses a consumer grade audio recording device that is normally associated with “garage band” recording of music. ORNL has coupled this low-cost data acquisition hardware with computing technology running open-source software. The main advantage of this approach is per-channel cost; an instrument grade data acquisition system typically costs between $800 to$2000 per channel compared to less than \$50 per channel for these consumer grade components. Three systems, each featuring four channels, have been deployed for acquiring data from geophones and the electrical supply system that supports the High Flux Isotope Reactor (HFIR) and the Radiochemical Engineering Development Center (REDC) at ORNL. Each channel samples at 96 kHz at 24-bit resolution. The deployed systems operate continuously 24/7 and produce about 4 terabytes of data per month per system. This paper provides a technical overview of this approach, its implementation, and some preliminary results from qualification testing. This work was conducted in support of the Multi-Informatics for Nuclear Operations Scenarios (MINOS).
Randall Wetherington, Gregory Sheets, Tom Karnowski, Ryan Kerekes, Michael Vann, Michael Moore, Eva Freer
### Chapter 16. Hybrid Slab Systems in High-rises for More Sustainable Design
Abstract
Greenhouse gases trap heat within our atmosphere, leading to an unnatural increase in temperature. Carbon dioxide and its equivalent emissions have been a large focus when considering sustainability in the civil engineering field, with a reduction of global warming potential being a top priority. According to a 2017 report by the World Green Building Council, the construction and usage of buildings account for 39 percent of human carbon emissions in the United States, almost one third of which are from the extraction, manufacturing, and transportation of materials. Substituting wood for high emission materials could greatly reduce carbon if harvested and disposed of in a controlled way. To investigate this important issue, San Francisco State University and University of South Carolina partnered with Skidmore, Owings & Merrill LLP, a world leader in designing high-rise buildings, through a National Science Foundation (NSF) Research Experience for Undergraduates (REU) Site program, to investigate and quantify the embodied carbons of various slab system designs using a high-rise residential complex in San Francisco as a case study. Three concept designs were considered: a concrete building with cementitious replacement, a concrete building without cementitious replacement, and a concrete building with cementitious replacement and nail-laminated timber wood inlays inserted into various areas of the superstructure slabs. The composite structural slab system has the potential to surpass the limitations of wood-framed structures yet incorporate the carbon sequestration that makes wood a more sustainable material. The results show that wood substitution could decrease overall emissions from the aforementioned designs and reduce the environmental footprint of the construction industry.
Katherine Berger, Samuel Benzoni, Zhaoshuo Jiang, Wenshen Pong, Juan Caicedo, David Shook, Christopher Horiuchi
### Chapter 17. Ground Vibration Testing of the World’s Longest Wingspan Aircraft—Stratolaunch
Abstract
The record-setting Stratolaunch (Roc) carrier aircraft first took to the skies on April 13, 2019, staying aloft for 149 minutes before successfully landing back in Mojave, California. Since 2012, Stratolaunch Systems Corporation, a space transportation venture created in part by Scaled Composites, has been designing, building, and testing the world’s largest composite aircraft. The goal of this mobile launch system is to make orbital access to space more convenient, reliable, and routine. To achieve the first successful flight of Roc, several ground vibration tests (GVTs) were necessary to characterize the modal properties of the composite aircraft and its subassemblies. ATA Engineering, Inc., (ATA) completed two partial GVTs and a full-scale GVT to help Stratolaunch Systems Corporation engineers achieve their successful first flight. The results of the GVTs were used to update the finite element models (FEMs) used for flutter and dynamic stability predictions. Testing an aircraft of this size imposed a number of challenges not encountered in most GVT programs; to efficiently conduct the tests, a distributed data acquisition system approach was used, and seismic accelerometers characterized the aircraft’s low-frequency rigid body modes. The distribution of shakers and sensors around the aircraft was addressed by the implementation of a new sensor cable system and the adaptation of multishaker excitation methods using temporary support structures.
Douglas J. Osterholt, Timothy Kelly
### Chapter 18. Using Recorded Data to Improve SRS Test Development
Abstract
The Shock Response Spectrum (SRS) approach is an effective and widely used method for analyzing mechanical shock phenomena. Synthetic waveforms are commonly used in the development of an SRS; unfortunately, there is often no synthetic pulse with a frequency response that matches well to a given real-world transient event.
This presentation describes a unique approach, where recordings from a field environment are modified to meet or exceed a specified SRS. Comparisons are made between a modified user waveform developed from this field-based SRS and waveforms developed with commonly applied synthetic pulses.
Joel Minderhoud
### Chapter 19. Distributed Acquisition and Processing Network for Experimental Vibration Testing of Aero-Engine Structures
Abstract
Detailed vibration testing of large assembled structures, such as aeroengines, leads to significant requirements on data acquisition and processing. This can lead to high system cost and long post processing times, which often limit the amount of data that can be acquired. A novel hardware-software acquisition system combination is proposed here to overcome some of the challenges of large scale data acquisition, based on the idea to distribute the acquisition and data processing load between a network of specialized acquisition nodes. The nodes work in parallel and are independent of each other, while sharing a synchronization clock. Each node has the capability to process the data being acquired on-line. The network allows for testing of novel data analysis methods and its modular nature enables an easy expansion of the system when required.
Michal J. Szydlowski, Christoph W. Schwingshackl, Andrew Rix
### Chapter 20. Modal Test of the NASA Mobile Launcher at Kennedy Space Center
Abstract
The NASA Mobile Launcher (ML), located at Kennedy Space Center (KSC), has recently been modified to support the launch of the new NASA Space Launch System (SLS). The ML is a massive structure—consisting of a 345-foot tall tower attached to a two-story base, weighing approximately 10.5 million pounds—that will secure the SLS vehicle as it rolls to the launch pad on a Crawler Transporter, as well as provide a launch platform at the pad. The ML will also provide the boundary condition for an upcoming SLS Integrated Modal Test (IMT). To help correlate the ML math models prior to this modal test, and allow focus to remain on updating SLS vehicle models during the IMT, a ML-only experimental modal test was performed in June 2019. Excitation of the tower and platform was provided by five uniquely-designed test fixtures, each enclosing a hydraulic shaker, capable of exerting thousands of pounds of force into the structure. For modes not that were not sufficiently excited by the test fixture shakers, a specially-designed mobile drop tower provided impact excitation at additional locations of interest. The response of the ML was measured with a total of 361 accelerometers. Following the random vibration, sine sweep vibration, and modal impact testing, frequency response functions were calculated and modes were extracted for three different configurations of the ML in 0 Hz to 12 Hz frequency range. This paper will provide a case study in performing modal tests on large structures by discussing the Mobile Launcher, the test strategy, an overview of the test results, and recommendations for meeting a tight test schedule for a large-scale modal test.
Eric C. Stasiunas, Russel A. Parks, Brendan D. Sontag, Dana E. Chandler
### Chapter 21. Using Deep-Learning Approach to Detect Anomalous Vibrations of Press Working Machine
Abstract
In recent years, there has been a demand for advanced maintenance in factories. Data collection from factory equipment is being carried out, and the collected sensor data is widely used for statistical analysis in quality control and failure prediction by machine learning. For example, if it is possible to detect an abnormality using vibration data obtained from an equipment, increase in the operation rate of the plant can be expected. In this research, we aim at early detection of equipment failure by finding signs of abnormality from vibration data, using a deep-learning technique, particularly an autoencoder.
In this paper, the following two methods were tested. The first scheme is based on the reconstruction error in an autoencoder. An autoencoder is trained using normal data only. Looking at the difference between input data and reconstructed data, we can regard the data having higher difference as abnormal. In the second approach, given the input data, values of the middle layer of the autoencoder are extracted, and we calculate the degree of abnormality using a Gaussian Mixture Model (GMM), representing a data set by superposition of a mixture of Gaussian distributions. In this framework, regarding an autoencoder structure, we tested both full-connection networks and convolutional networks.
In this work, we chose a press machine. Frequency characteristics were acquired from the data in production mode of a press machine. Then using each method, we evaluated whether abnormality could be found by calculating the degree of abnormality. We employed two-day data without failure as training data, and another data set was prepared as forecast data obtained on the following days; on one of the days the machine stopped due to a sudden abnormality. Similar to time-series signal processing, we applied framing processing so that we can analyze data even in the case we can only get a small amount of data. As a result, our method succeeded in finding the day when the abnormality occurred and the machine stopped. In addition, the degree of abnormality became higher before the abnormality occurs, indicating we can detect signs of abnormality.
In conclusion, the degree of abnormality could be calculated using the reconstruction error using an autoencoder from the vibration data during production, and the method using GMM from the middle layer of autoencoder. We consequently conclude it is possible to detect a sudden abnormality in which the device stopped, from actual vibration data. These results provide new solutions for equipment failure estimation.
Kazuya Inagaki, Satoru Hayamizu, Satoshi Tamura
### Chapter 22. DAQ Evaluation and Specifications for Pyroshock Testing
Abstract
Pyroshock events contain high-amplitude, extreme rise-time accelerations that can be damaging to electronics and small structures. Due to their extreme nature, these events can be difficult to capture, exceeding the performance limits of transducers, signal conditioning, and data acquisition (DAQ) equipment. This study assesses the ability of different data acquisition systems to record quality pyroshock data. Using a function generator and voltage input, different tests were performed to characterize the data acquisition systems’ anti-alias filter, out-of-band energy attenuation, number of effective bits, in-band gain, and slew rate. These tests include a shorted-input noise test, a sine sweep test, and a high amplitude low frequency square wave test. Although the data acquisition systems evaluated have similar specifications, their ability to record quality pyroshock data varied. Some of these data acquisition systems do not appropriately handle the rapid transient content and may have inadequate fidelity to record pyroshock data. Data acquisition system performance for pyroshock testing cannot be evaluated by the specification sheet alone.
Erica M. Jacobson, Jason R. Blough, James P. DeClerck, Charles D. Van Karsen, David Soine
### Chapter 23. Optimal Replicator Dynamic Controller via Load Balancing and Neural Dynamics for Semi-Active Vibration Control of Isolated Highway Bridge Structures
Abstract
During the past few decades, major structural damages due to natural disasters like earthquakes has led bridge engineers to develop structural control systems to mitigate damage and improve vibration reduction in real-time. Among different kinds of vibration control systems, base isolation is one of the most commonly used passive control strategies for civil structures. However, base isolators have their own limitations due to the lack of real-time adaptability and lower energy dissipation. In order to overcome this limitation, semi-active damping devices are installed between the deck and piers. In the present study, a semi-active control system comprised of magneto-rheological (MR) dampers is proposed for vibration mitigation of isolated bridge structures. Recently, inspired by evolutionary game theory, a replicator dynamic control algorithm was developed to allocate the input voltage of MR dampers. In this paper, a load balancing strategy is studied to reallocate additional resources and improve the power distribution over semi-active MR dampers. In order to achieve a high-performance design of the replicator controller, a modified patented Neural Dynamic (ND) model of Adeli and Park is used to optimize the load-balanced replicator control parameters. The ND model incorporates a penalty function, the Lyapunov stability theorem, and the Karush-Kuhn-Tucker conditions to guarantee the global convergence of the solution. The objective function is then defined to minimize the dynamic response of the bridge. The proposed methodology is evaluated using a benchmark control problem that is based on Interstate 5 overcrossing California State Route 91 bridge in Southern California subjected to near-field earthquake accelerograms. The performance of the proposed controller is evaluated and compared with conventional Lyapunov and fuzzy control algorithms in terms of 16 different performance criteria describing the reductions in dynamic response of the bridge structure.
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https://www.physicsforums.com/threads/quick-simple-question-about-contraviant-and-covariant-components.594932/
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# Quick, simple question about contraviant and covariant components
1. Apr 9, 2012
### enfield
Can the covariant components of a vector, v, be thought of as v multiplied by a matrix of linearly independent vectors that span the vector space, and the contravariant components of the same vector, v, the vector v multiplied by the *inverse* of that same matrix?
thinking about it like that makes it easy to see why the covariant and contravariant components are equal when the basis is the normalized mutually orthogonal one, for example, because then the matrix is just the identity one, which is its own inverse.
that's what the definitions i read seem to imply.
Thanks!
2. Apr 9, 2012
### Mentz114
The contravariant and covariant components of a vector are linear combinations of each other, but the transformation is performed by the metric and the inverse metric. It is a change of basis between the tangent and cotangent spaces.
The zero'th component of Vμ is given by
V0= g0aVa = g00V0+g01V1+g02V2+g03V3
If the metric is the identity matrix then the components are the same.
Last edited: Apr 9, 2012
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https://www.aimsciences.org/openAccessArticles?journalId=be993534-f1c4-4d7d-ac98-82dbd65b5341
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# American Institute of Mathematical Sciences
ISSN:
1930-5346
eISSN:
1930-5338
All Issues
## Advances in Mathematics of Communications
Open Access Articles
2020 doi: 10.3934/amc.2020073 +[Abstract](393) +[HTML](234) +[PDF](438.53KB)
Abstract:
Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now showing promise in providing quantum-resistant cryptographic primitives, and may provide suitable alternatives for those looking to address known quantum attacks. In this paper, we introduce a new protocol called a Meta Key Agreement and Authentication Protocol (MKAAP) that has some characteristics of a public-key solution and some of a shared-key solution. Specifically, it has the deployment benefits of a public-key system, allowing two entities that have never met before to authenticate without requiring real-time access to a third-party, but does require secure provisioning of key material from a trusted key distribution system (similar to a symmetric system) prior to deployment. We then describe a specific MKAAP instance, the Ironwood MKAAP, discuss its security, and show how it resists certain quantum attacks such as Shor's algorithm or Grover's quantum search algorithm. We also show Ironwood implemented on several "internet of things" (IoT devices), measure its performance, and show how it performs significantly better than ECC using fewer device resources.
2020 doi: 10.3934/amc.2020053 +[Abstract](927) +[HTML](366) +[PDF](314.61KB)
Abstract:
We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret codes and disguise it by considering its shortened expanded code over the base field. Considering shortened expanded codes provides a safeguard against distinguisher attacks based on the Schur product. Moreover, without using a cyclic or a quasi-cyclic structure we obtain a key size reduction of nearly \begin{document}$45 \%$\end{document} compared to the classic McEliece cryptosystem proposed by Bernstein et al.
2019 doi: 10.3934/amc.2020045 +[Abstract](836) +[HTML](428) +[PDF](337.74KB)
Abstract:
We investigate a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. These codes have at most three weights and some of them are almost optimal so that they are suitable for applications in secret sharing schemes. This is a supplement of the results raised by Wang et al. (2017) and Kong et al. (2019).
2020, 14(2): 301-306 doi: 10.3934/amc.2020021 +[Abstract](1254) +[HTML](521) +[PDF](299.97KB)
Abstract:
McNie [8] is a code-based public key encryption scheme submitted to the NIST Post-Quantum Cryptography standardization [10] as a candidate. In this paper, we present Dual-Ouroboros, an improvement of McNie, which can be seen as a dual version of the Ouroboros-R protocol [1], another candidate to the NIST competition. This new improved protocol permits, first, to avoid an attack proposed by Gaborit [7] and second permits to benefit from a reduction security to a standard problem (as the original Ouroboros protocol).
2020, 14(2): 279-299 doi: 10.3934/amc.2020020 +[Abstract](1037) +[HTML](521) +[PDF](430.21KB)
Abstract:
This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters \begin{document}$[216,190,\geqslant 18]$\end{document} over \begin{document}$\mathbb{F}_{64}$\end{document} yields a new record.
2015, 9(1): 9-21 doi: 10.3934/amc.2015.9.9 +[Abstract](3345) +[PDF](424.1KB)
Abstract:
In this paper, a novel method for constructing complementary sequence set with zero correlation zone (ZCZ) is presented by interleaving and combining three orthogonal matrices. The constructed set can be divided into multiple sequence groups and each sequence group can be further divided into multiple sequence subgroups. In addition to ZCZ properties of sequences from the same sequence subgroup, sequences from different sequence groups are orthogonal to each other while sequences from different sequence subgroups within the same sequence group possess ideal cross-correlation properties, that is, the proposed ZCZ sequence set has inter-group orthogonal (IGO) and inter-subgroup complementary (ISC) properties. Compared with previous methods, the new construction can provide flexible choice for ZCZ width and set size, and the resultant sequences which are called IGO-ISC sequences in this paper can achieve the theoretical bound on the set size for the ZCZ width and sequence length.
2019 Impact Factor: 0.734
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http://pure.au.dk/portal/en/publications/on-shrinking-targets-for-zm-actions-on-tori(29c74310-dbd3-11dd-9710-000ea68e967b).html
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# Department of Mathematics
## On shrinking targets for Zm actions on tori
Research output: Working paper
Let $~A$ be an $~n \times m$ matrix with real entries. Consider the set $~\mathbf{Bad}_A$ of $~\mathbf{x} \in [0,1)^n$ for which there exists a constant $~c(\mathbf{x})>0$ such that for any $~\mathbf{q} \in \mathbb{Z}^m$ the distance between $~\mathbf{x}$ and the point $~\{A \mathbf{q}\}$ is at least $~c(\mathbf{x}) |\mathbf{q}|^{-m/n}$. It is shown that the intersection of $~\mathbf{Bad}_A$ with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.
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https://diffeq.sciml.ai/latest/tutorials/faster_ode_example/
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# Code Optimization for Differential Equations
Note
See this FAQ for information on common pitfalls and how to improve performance.
## Code Optimization in Julia
Before starting this tutorial, we recommend the reader to check out one of the many tutorials for optimization Julia code. The following is an incomplete list:
User-side optimizations are important because, for sufficiently difficult problems, most of the time will be spent inside of your f function, the function you are trying to solve. "Efficient" integrators are those that reduce the required number of f calls to hit the error tolerance. The main ideas for optimizing your DiffEq code, or any Julia function, are the following:
• Make it non-allocating
• Use StaticArrays for small arrays
• Make it type-stable
• Reduce redundant calculations
• Make use of BLAS calls
• Optimize algorithm choice
We'll discuss these strategies in the context of differential equations. Let's start with small systems.
## Example Accelerating a Non-Stiff Equation: The Lorenz Equation
Let's take the classic Lorenz system. Let's start by naively writing the system in its out-of-place form:
function lorenz(u,p,t)
dx = 10.0*(u[2]-u[1])
dy = u[1]*(28.0-u[3]) - u[2]
dz = u[1]*u[2] - (8/3)*u[3]
[dx,dy,dz]
end
Here, lorenz returns an object, [dx,dy,dz], which is created within the body of lorenz.
This is a common code pattern from high-level languages like MATLAB, SciPy, or R's deSolve. However, the issue with this form is that it allocates a vector, [dx,dy,dz], at each step. Let's benchmark the solution process with this choice of function:
using DifferentialEquations, BenchmarkTools
u0 = [1.0;0.0;0.0]
tspan = (0.0,100.0)
prob = ODEProblem(lorenz,u0,tspan)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 1350 samples with 1 evaluation.
Range (min … max): 2.296 ms … 13.764 ms ┊ GC (min … max): 0.00% … 67.48%
Time (median): 2.561 ms ┊ GC (median): 0.00%
Time (mean ± σ): 3.699 ms ± 2.223 ms ┊ GC (mean ± σ): 14.83% ± 17.79%
█▆▄ ▁ ▃▄▃▄▂
████▆▆█▇▇▇█▇█████▆▄▄▄▁▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▄▆▇▆▇▇▆▇▇▆█▇▇▆ █
2.3 ms Histogram: log(frequency) by time 11.3 ms <
Memory estimate: 7.82 MiB, allocs estimate: 101102.
The BenchmarkTools.jl package's @benchmark runs the code multiple times to get an accurate measurement. The minimum time is the time it takes when your OS and other background processes aren't getting in the way. Notice that in this case it takes about 5ms to solve and allocates around 11.11 MiB. However, if we were to use this inside of a real user code we'd see a lot of time spent doing garbage collection (GC) to clean up all of the arrays we made. Even if we turn off saving we have these allocations.
@benchmark solve(prob,Tsit5(),save_everystep=false)
BenchmarkTools.Trial: 1490 samples with 1 evaluation.
Range (min … max): 2.010 ms … 14.612 ms ┊ GC (min … max): 0.00% … 65.56%
Time (median): 2.313 ms ┊ GC (median): 0.00%
Time (mean ± σ): 3.350 ms ± 2.095 ms ┊ GC (mean ± σ): 14.55% ± 17.48%
█▇▅▁ ▃▅▅▄▃
█████▇▇▇▆▇███████▅▁▄▁▁▁▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▄▄▄▅▇▇▆▆█▇▇▆▆▆▆ █
2.01 ms Histogram: log(frequency) by time 11 ms <
Memory estimate: 6.83 MiB, allocs estimate: 89529.
The problem of course is that arrays are created every time our derivative function is called. This function is called multiple times per step and is thus the main source of memory usage. To fix this, we can use the in-place form to ***make our code non-allocating***:
function lorenz!(du,u,p,t)
du[1] = 10.0*(u[2]-u[1])
du[2] = u[1]*(28.0-u[3]) - u[2]
du[3] = u[1]*u[2] - (8/3)*u[3]
nothing
end
Here, instead of creating an array each time, we utilized the cache array du. When the in-place form is used, DifferentialEquations.jl takes a different internal route that minimizes the internal allocations as well.
Note
Notice that nothing is returned. When in in-place form, the ODE solver ignores the return. Instead, make sure that the original du array is mutated instead of constructing a new array
When we benchmark this function, we will see quite a difference.
u0 = [1.0;0.0;0.0]
tspan = (0.0,100.0)
prob = ODEProblem(lorenz!,u0,tspan)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 8180 samples with 1 evaluation.
Range (min … max): 415.800 μs … 12.112 ms ┊ GC (min … max): 0.00% … 93.21%
Time (median): 463.700 μs ┊ GC (median): 0.00%
Time (mean ± σ): 605.847 μs ± 695.892 μs ┊ GC (mean ± σ): 11.02% ± 9.07%
▄█▇▅▃▂▂▂▂▁▁▄▅▅▃▂▂▂▁▂▁▁ ▂
███████████████████████▇▇▇▇▇▆▆▆▅▅▄▄▅▃▅▄▅▄▅▄▁▅▁▃▄▁▃▁▁▁▁▃▄▁▁▄▁▃ █
416 μs Histogram: log(frequency) by time 1.68 ms <
Memory estimate: 1016.36 KiB, allocs estimate: 11641.
@benchmark solve(prob,Tsit5(),save_everystep=false)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
Range (min … max): 197.900 μs … 315.800 μs ┊ GC (min … max): 0.00% … 0.00%
Time (median): 206.700 μs ┊ GC (median): 0.00%
Time (mean ± σ): 207.688 μs ± 6.241 μs ┊ GC (mean ± σ): 0.00% ± 0.00%
▃▅▅▇█▅▃ ▁▁▂▁▃▂▁
▁▁▁▂▃▃▅▇█████████▇▇▇▆█████████▆▆▅▅▃▂▃▂▂▂▂▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▃
198 μs Histogram: frequency by time 227 μs <
Memory estimate: 4.94 KiB, allocs estimate: 41.
There is a 16x time difference just from that change! Notice there are still some allocations and this is due to the construction of the integration cache. But this doesn't scale with the problem size:
tspan = (0.0,500.0) # 5x longer than before
prob = ODEProblem(lorenz!,u0,tspan)
@benchmark solve(prob,Tsit5(),save_everystep=false)
BenchmarkTools.Trial: 4755 samples with 1 evaluation.
Range (min … max): 1.014 ms … 1.485 ms ┊ GC (min … max): 0.00% … 0.00%
Time (median): 1.042 ms ┊ GC (median): 0.00%
Time (mean ± σ): 1.048 ms ± 31.281 μs ┊ GC (mean ± σ): 0.00% ± 0.00%
▁▆▆▅▇█▇▆▆▂▁ ▁ ▁
▂▄█████████████▇▇▇▇▇▇█▇▇█▇███▇█▇▇▅▆▄▄▄▃▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁ ▄
1.01 ms Histogram: frequency by time 1.12 ms <
Memory estimate: 4.94 KiB, allocs estimate: 41.
Since that's all setup allocations the user-side optimization is complete.
### Further Optimizations of Small Non-Stiff ODEs with StaticArrays
Allocations are only expensive if they are "heap allocations". For a more in-depth definition of heap allocations, there are a lot of sources online. But a good working definition is that heap allocations are variable-sized slabs of memory which have to be pointed to, and this pointer indirection costs time. Additionally, the heap has to be managed and the garbage controllers has to actively keep track of what's on the heap.
However, there's an alternative to heap allocations, known as stack allocations. The stack is statically-sized (known at compile time) and thus its accesses are quick. Additionally, the exact block of memory is known in advance by the compiler, and thus re-using the memory is cheap. This means that allocating on the stack has essentially no cost!
Arrays have to be heap allocated because their size (and thus the amount of memory they take up) is determined at runtime. But there are structures in Julia which are stack-allocated. structs for example are stack-allocated "value-type"s. Tuples are a stack-allocated collection. The most useful data structure for DiffEq though is the StaticArray from the package StaticArrays.jl. These arrays have their length determined at compile-time. They are created using macros attached to normal array expressions, for example:
using StaticArrays
A = SA[2.0,3.0,5.0]
typeof(A) # SVector{3, Float64} (alias for SArray{Tuple{3}, Float64, 1, 3})
Notice that the 3 after SVector gives the size of the SVector. It cannot be changed. Additionally, SVectors are immutable, so we have to create a new SVector to change values. But remember, we don't have to worry about allocations because this data structure is stack-allocated. SArrays have a lot of extra optimizations as well: they have fast matrix multiplication, fast QR factorizations, etc. which directly make use of the information about the size of the array. Thus, when possible they should be used.
Unfortunately static arrays can only be used for sufficiently small arrays. After a certain size, they are forced to heap allocate after some instructions and their compile time balloons. Thus static arrays shouldn't be used if your system has more than ~20 variables. Additionally, only the native Julia algorithms can fully utilize static arrays.
Let's ***optimize lorenz using static arrays***. Note that in this case, we want to use the out-of-place allocating form, but this time we want to output a static array:
function lorenz_static(u,p,t)
dx = 10.0*(u[2]-u[1])
dy = u[1]*(28.0-u[3]) - u[2]
dz = u[1]*u[2] - (8/3)*u[3]
SA[dx,dy,dz]
end
To make the solver internally use static arrays, we simply give it a static array as the initial condition:
u0 = SA[1.0,0.0,0.0]
tspan = (0.0,100.0)
prob = ODEProblem(lorenz_static,u0,tspan)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
Range (min … max): 196.600 μs … 6.310 ms ┊ GC (min … max): 0.00% … 95.70%
Time (median): 220.900 μs ┊ GC (median): 0.00%
Time (mean ± σ): 265.006 μs ± 302.623 μs ┊ GC (mean ± σ): 6.91% ± 5.82%
▅█▄▄▄
▁▄██████▅▄▃▂▂▁▁▂▁▁▁▁▁▁▁▂▃▄▅▅▆▆▆▅▅▅▄▃▃▃▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
197 μs Histogram: frequency by time 374 μs <
Memory estimate: 394.50 KiB, allocs estimate: 1319.
@benchmark solve(prob,Tsit5(),save_everystep=false)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
Range (min … max): 144.100 μs … 242.600 μs ┊ GC (min … max): 0.00% … 0.00%
Time (median): 151.000 μs ┊ GC (median): 0.00%
Time (mean ± σ): 151.875 μs ± 7.502 μs ┊ GC (mean ± σ): 0.00% ± 0.00%
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144 μs Histogram: frequency by time 185 μs <
Memory estimate: 3.67 KiB, allocs estimate: 22.
And that's pretty much all there is to it. With static arrays you don't have to worry about allocating, so use operations like * and don't worry about fusing operations (discussed in the next section). Do "the vectorized code" of R/MATLAB/Python and your code in this case will be fast, or directly use the numbers/values.
## Example Accelerating a Stiff Equation: the Robertson Equation
For these next examples, let's solve the Robertson equations (also known as ROBER):
\begin{aligned} \frac{dy_1}{dt} &= -0.04y₁ + 10^4 y_2 y_3 \\ \frac{dy_2}{dt} &= 0.04 y_1 - 10^4 y_2 y_3 - 3*10^7 y_{2}^2 \\ \frac{dy_3}{dt} &= 3*10^7 y_{2}^2 \\ \end{aligned}
Given that these equations are stiff, non-stiff ODE solvers like Tsit5 or Vern9 will fail to solve these equations. The automatic algorithm will detect this and automatically switch to something more robust to handle these issues. For example:
using DifferentialEquations
function rober(du,u,p,t)
y₁,y₂,y₃ = u
k₁,k₂,k₃ = p
du[1] = -k₁*y₁+k₃*y₂*y₃
du[2] = k₁*y₁-k₂*y₂^2-k₃*y₂*y₃
du[3] = k₂*y₂^2
nothing
end
prob = ODEProblem(rober,[1.0,0.0,0.0],(0.0,1e5),[0.04,3e7,1e4])
sol = solve(prob)
plot(sol,tspan=(1e-2,1e5),xscale=:log10)
julia> using BenchmarkTools
julia> @btime solve(prob)
97.000 μs (1832 allocations: 132.30 KiB)
### Choosing a Good Solver
Choosing a good solver is required for getting top notch speed. General recommendations can be found on the solver page (for example, the ODE Solver Recommendations). The current recommendations can be simplified to a Rosenbrock method (Rosenbrock23 or Rodas5) for smaller (<50 ODEs) problems, ESDIRK methods for slightly larger (TRBDF2 or KenCarp4 for <2000 ODEs), and QNDF for even larger problems. lsoda from LSODA.jl is sometimes worth a try for the medium sized category.
More details on the solver to choose can be found by benchmarking. See the SciMLBenchmarks to compare many solvers on many problems.
From this, we try the recommendation of Rosenbrock23() for stiff ODEs at default tolerances:
@btime solve(prob,Rosenbrock23())
# 61.200 μs (918 allocations: 78.72 KiB)
### Declaring Jacobian Functions
In order to reduce the Jacobian construction cost, one can describe a Jacobian function by using the jac argument for the ODEFunction. First we have to derive the Jacobian $\frac{df_i}{du_j}$ which is J[i,j]. From this we get:
function rober_jac(J,u,p,t)
y₁,y₂,y₃ = u
k₁,k₂,k₃ = p
J[1,1] = k₁ * -1
J[2,1] = k₁
J[3,1] = 0
J[1,2] = y₃ * k₃
J[2,2] = y₂ * k₂ * -2 + y₃ * k₃ * -1
J[3,2] = y₂ * 2 * k₂
J[1,3] = k₃ * y₂
J[2,3] = k₃ * y₂ * -1
J[3,3] = 0
nothing
end
f = ODEFunction(rober, jac=rober_jac)
prob_jac = ODEProblem(f,[1.0,0.0,0.0],(0.0,1e5),(0.04,3e7,1e4))
julia> @btime solve(prob_jac,Rosenbrock23())
57.400 μs (978 allocations: 82.58 KiB)
### Automatic Derivation of Jacobian Functions
But that was hard! If you want to take the symbolic Jacobian of numerical code, we can make use of ModelingToolkit.jl to symbolic-ify the numerical code and do the symbolic calculation and return the Julia code for this.
using ModelingToolkit
de = modelingtoolkitize(prob)
We can tell it to compute the Jacobian if we want to see the code:
julia> ModelingToolkit.generate_jacobian(de)[2] # Second is in-place
:(function (ˍ₋out, ˍ₋arg1, ˍ₋arg2, t)
#= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:303 =#
#= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:304 =#
let var"x₁(t)" = #= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:190 =# @inbounds(ˍ₋arg1[1]), var"x₂(t)" = #= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:190 =# @inbounds(ˍ₋arg1[2]), var"x₃(t)" = #= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:190 =# @inbounds(ˍ₋arg1[3]), α₁ = #= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:190 =# @inbounds(ˍ₋arg2[1]), α₂ = #= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:190 =# @inbounds(ˍ₋arg2[2]), α₃ = #= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:190 =# @inbounds(ˍ₋arg2[3])
#= C:\Users\accou\.julia\dev\Symbolics\src\build_function.jl:378 =#
#= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:350 =# @inbounds begin
#= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:346 =#
ˍ₋out[1] = (*)(-1, α₁)
ˍ₋out[2] = α₁
ˍ₋out[3] = 0
ˍ₋out[4] = (*)(α₃, var"x₃(t)")
ˍ₋out[5] = (+)((*)((*)(-2, α₂), var"x₂(t)"), (*)((*)(-1, α₃), var"x₃(t)"))
ˍ₋out[6] = (*)((*)(2, α₂), var"x₂(t)")
ˍ₋out[7] = (*)(α₃, var"x₂(t)")
ˍ₋out[8] = (*)((*)(-1, α₃), var"x₂(t)")
ˍ₋out[9] = 0
#= C:\Users\accou\.julia\packages\SymbolicUtils\0KTj4\src\code.jl:348 =#
nothing
end
end
end)
Now let's use that to give the analytical solution Jacobian:
prob_jac2 = ODEProblem(de,[],(0.0,1e5),jac=true)
julia> @btime solve(prob_jac2)
122.600 μs (1425 allocations: 99.34 KiB)
See the ModelingToolkit.jl documentation for more details.
### Accelerating Small ODE Solves with Static Arrays
If the ODE is sufficiently small (<20 ODEs or so), using StaticArrays.jl for the state variables can greatly enhance the performance. This is done by making u0 a StaticArray and writing an out-of-place non-mutating dispatch for static arrays, for the ROBER problem, this looks like:
using DifferentialEquations, StaticArrays
function rober_static(u,p,t)
y₁,y₂,y₃ = u
k₁,k₂,k₃ = p
du1 = -k₁*y₁+k₃*y₂*y₃
du2 = k₁*y₁-k₂*y₂^2-k₃*y₂*y₃
du3 = k₂*y₂^2
SA[du1,du2,du3]
end
prob = ODEProblem(rober_static,SA[1.0,0.0,0.0],(0.0,1e5),SA[0.04,3e7,1e4])
sol = solve(prob,Rosenbrock23())
If we benchmark this we see a really fast solution with really low allocation counts:
@btime sol = solve(prob,Rosenbrock23())
# 12.100 μs (87 allocations: 18.53 KiB)
This version is thus very amenable to multithreading and other forms of parallelism.
## Example Accelerating Linear Algebra PDE Semi-Discretization
In this tutorial we will optimize the right-hand side definition of a PDE semi-discretization.
Note
We highly recommend looking at the Solving Large Stiff Equations tutorial for details on customizing DifferentialEquations.jl for more efficient large-scale stiff ODE solving. This section will only focus on the user-side code.
Let's optimize the solution of a Reaction-Diffusion PDE's discretization. In its discretized form, this is the ODE:
\begin{align} du &= D_1 (A_y u + u A_x) + \frac{au^2}{v} + \bar{u} - \alpha u\\ dv &= D_2 (A_y v + v A_x) + a u^2 + \beta v \end{align}
where $u$, $v$, and $A$ are matrices. Here, we will use the simplified version where $A$ is the tridiagonal stencil $[1,-2,1]$, i.e. it's the 2D discretization of the LaPlacian. The native code would be something along the lines of:
using DifferentialEquations, LinearAlgebra
# Generate the constants
p = (1.0,1.0,1.0,10.0,0.001,100.0) # a,α,ubar,β,D1,D2
N = 100
Ax = Array(Tridiagonal([1.0 for i in 1:N-1],[-2.0 for i in 1:N],[1.0 for i in 1:N-1]))
Ay = copy(Ax)
Ax[2,1] = 2.0
Ax[end-1,end] = 2.0
Ay[1,2] = 2.0
Ay[end,end-1] = 2.0
function basic_version!(dr,r,p,t)
a,α,ubar,β,D1,D2 = p
u = r[:,:,1]
v = r[:,:,2]
Du = D1*(Ay*u + u*Ax)
Dv = D2*(Ay*v + v*Ax)
dr[:,:,1] = Du .+ a.*u.*u./v .+ ubar .- α*u
dr[:,:,2] = Dv .+ a.*u.*u .- β*v
end
a,α,ubar,β,D1,D2 = p
uss = (ubar+β)/α
vss = (a/β)*uss^2
r0 = zeros(100,100,2)
r0[:,:,1] .= uss.+0.1.*rand.()
r0[:,:,2] .= vss
prob = ODEProblem(basic_version!,r0,(0.0,0.1),p)
In this version we have encoded our initial condition to be a 3-dimensional array, with u[:,:,1] being the A part and u[:,:,2] being the B part.
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 48 samples with 1 evaluation.
Range (min … max): 96.001 ms … 130.443 ms ┊ GC (min … max): 7.04% … 16.06%
Time (median): 104.225 ms ┊ GC (median): 10.48%
Time (mean ± σ): 105.063 ms ± 6.812 ms ┊ GC (mean ± σ): 9.42% ± 2.62%
▃█▃ █▃█ ▃██ ▃█ ▃▃ ▃
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96 ms Histogram: frequency by time 130 ms <
Memory estimate: 186.83 MiB, allocs estimate: 7341.
While this version isn't very efficient,
#### We recommend writing the "high-level" code first, and iteratively optimizing it!
The first thing that we can do is get rid of the slicing allocations. The operation r[:,:,1] creates a temporary array instead of a "view", i.e. a pointer to the already existing memory. To make it a view, add @view. Note that we have to be careful with views because they point to the same memory, and thus changing a view changes the original values:
A = rand(4)
@show A
B = @view A[1:3]
B[2] = 2
@show A
Notice that changing B changed A. This is something to be careful of, but at the same time we want to use this since we want to modify the output dr. Additionally, the last statement is a purely element-wise operation, and thus we can make use of broadcast fusion there. Let's rewrite basic_version! to ***avoid slicing allocations*** and to ***use broadcast fusion***:
function gm2!(dr,r,p,t)
a,α,ubar,β,D1,D2 = p
u = @view r[:,:,1]
v = @view r[:,:,2]
du = @view dr[:,:,1]
dv = @view dr[:,:,2]
Du = D1*(Ay*u + u*Ax)
Dv = D2*(Ay*v + v*Ax)
@. du = Du + a.*u.*u./v + ubar - α*u
@. dv = Dv + a.*u.*u - β*v
end
prob = ODEProblem(gm2!,r0,(0.0,0.1),p)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 58 samples with 1 evaluation.
Range (min … max): 80.456 ms … 106.830 ms ┊ GC (min … max): 0.00% … 10.74%
Time (median): 85.858 ms ┊ GC (median): 6.34%
Time (mean ± σ): 86.916 ms ± 4.214 ms ┊ GC (mean ± σ): 6.46% ± 1.75%
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80.5 ms Histogram: frequency by time 102 ms <
Memory estimate: 119.71 MiB, allocs estimate: 5871.
Now, most of the allocations are taking place in Du = D1*(Ay*u + u*Ax) since those operations are vectorized and not mutating. We should instead replace the matrix multiplications with mul!. When doing so, we will need to have cache variables to write into. This looks like:
Ayu = zeros(N,N)
uAx = zeros(N,N)
Du = zeros(N,N)
Ayv = zeros(N,N)
vAx = zeros(N,N)
Dv = zeros(N,N)
function gm3!(dr,r,p,t)
a,α,ubar,β,D1,D2 = p
u = @view r[:,:,1]
v = @view r[:,:,2]
du = @view dr[:,:,1]
dv = @view dr[:,:,2]
mul!(Ayu,Ay,u)
mul!(uAx,u,Ax)
mul!(Ayv,Ay,v)
mul!(vAx,v,Ax)
@. Du = D1*(Ayu + uAx)
@. Dv = D2*(Ayv + vAx)
@. du = Du + a*u*u./v + ubar - α*u
@. dv = Dv + a*u*u - β*v
end
prob = ODEProblem(gm3!,r0,(0.0,0.1),p)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 71 samples with 1 evaluation.
Range (min … max): 66.051 ms … 78.626 ms ┊ GC (min … max): 0.00% … 6.51%
Time (median): 69.778 ms ┊ GC (median): 0.00%
Time (mean ± σ): 70.563 ms ± 3.392 ms ┊ GC (mean ± σ): 1.68% ± 2.94%
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66.1 ms Histogram: frequency by time 77.1 ms <
Memory estimate: 29.98 MiB, allocs estimate: 4695.
But our temporary variables are global variables. We need to either declare the caches as const or localize them. We can localize them by adding them to the parameters, p. It's easier for the compiler to reason about local variables than global variables. ***Localizing variables helps to ensure type stability***.
p = (1.0,1.0,1.0,10.0,0.001,100.0,Ayu,uAx,Du,Ayv,vAx,Dv) # a,α,ubar,β,D1,D2
function gm4!(dr,r,p,t)
a,α,ubar,β,D1,D2,Ayu,uAx,Du,Ayv,vAx,Dv = p
u = @view r[:,:,1]
v = @view r[:,:,2]
du = @view dr[:,:,1]
dv = @view dr[:,:,2]
mul!(Ayu,Ay,u)
mul!(uAx,u,Ax)
mul!(Ayv,Ay,v)
mul!(vAx,v,Ax)
@. Du = D1*(Ayu + uAx)
@. Dv = D2*(Ayv + vAx)
@. du = Du + a*u*u./v + ubar - α*u
@. dv = Dv + a*u*u - β*v
end
prob = ODEProblem(gm4!,r0,(0.0,0.1),p)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 75 samples with 1 evaluation.
Range (min … max): 63.820 ms … 76.176 ms ┊ GC (min … max): 0.00% … 6.03%
Time (median): 66.711 ms ┊ GC (median): 0.00%
Time (mean ± σ): 67.396 ms ± 3.167 ms ┊ GC (mean ± σ): 1.55% ± 2.78%
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63.8 ms Histogram: frequency by time 74 ms <
Memory estimate: 29.66 MiB, allocs estimate: 1020.
We could then use the BLAS gemmv to optimize the matrix multiplications some more, but instead let's devectorize the stencil.
p = (1.0,1.0,1.0,10.0,0.001,100.0,N)
function fast_gm!(du,u,p,t)
a,α,ubar,β,D1,D2,N = p
@inbounds for j in 2:N-1, i in 2:N-1
du[i,j,1] = D1*(u[i-1,j,1] + u[i+1,j,1] + u[i,j+1,1] + u[i,j-1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
end
@inbounds for j in 2:N-1, i in 2:N-1
du[i,j,2] = D2*(u[i-1,j,2] + u[i+1,j,2] + u[i,j+1,2] + u[i,j-1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
end
@inbounds for j in 2:N-1
i = 1
du[1,j,1] = D1*(2u[i+1,j,1] + u[i,j+1,1] + u[i,j-1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
end
@inbounds for j in 2:N-1
i = 1
du[1,j,2] = D2*(2u[i+1,j,2] + u[i,j+1,2] + u[i,j-1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
end
@inbounds for j in 2:N-1
i = N
du[end,j,1] = D1*(2u[i-1,j,1] + u[i,j+1,1] + u[i,j-1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
end
@inbounds for j in 2:N-1
i = N
du[end,j,2] = D2*(2u[i-1,j,2] + u[i,j+1,2] + u[i,j-1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
end
@inbounds for i in 2:N-1
j = 1
du[i,1,1] = D1*(u[i-1,j,1] + u[i+1,j,1] + 2u[i,j+1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
end
@inbounds for i in 2:N-1
j = 1
du[i,1,2] = D2*(u[i-1,j,2] + u[i+1,j,2] + 2u[i,j+1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
end
@inbounds for i in 2:N-1
j = N
du[i,end,1] = D1*(u[i-1,j,1] + u[i+1,j,1] + 2u[i,j-1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
end
@inbounds for i in 2:N-1
j = N
du[i,end,2] = D2*(u[i-1,j,2] + u[i+1,j,2] + 2u[i,j-1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
end
@inbounds begin
i = 1; j = 1
du[1,1,1] = D1*(2u[i+1,j,1] + 2u[i,j+1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
du[1,1,2] = D2*(2u[i+1,j,2] + 2u[i,j+1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
i = 1; j = N
du[1,N,1] = D1*(2u[i+1,j,1] + 2u[i,j-1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
du[1,N,2] = D2*(2u[i+1,j,2] + 2u[i,j-1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
i = N; j = 1
du[N,1,1] = D1*(2u[i-1,j,1] + 2u[i,j+1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
du[N,1,2] = D2*(2u[i-1,j,2] + 2u[i,j+1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
i = N; j = N
du[end,end,1] = D1*(2u[i-1,j,1] + 2u[i,j-1,1] - 4u[i,j,1]) +
a*u[i,j,1]^2/u[i,j,2] + ubar - α*u[i,j,1]
du[end,end,2] = D2*(2u[i-1,j,2] + 2u[i,j-1,2] - 4u[i,j,2]) +
a*u[i,j,1]^2 - β*u[i,j,2]
end
end
prob = ODEProblem(fast_gm!,r0,(0.0,0.1),p)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 700 samples with 1 evaluation.
Range (min … max): 5.433 ms … 26.007 ms ┊ GC (min … max): 0.00% … 56.34%
Time (median): 5.878 ms ┊ GC (median): 0.00%
Time (mean ± σ): 7.138 ms ± 2.348 ms ┊ GC (mean ± σ): 11.10% ± 13.88%
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5.43 ms Histogram: log(frequency) by time 13.4 ms <
Memory estimate: 29.62 MiB, allocs estimate: 432.
Notice that in this case fusing the loops and avoiding the linear operators is a major improvement of about 10x! That's an order of magnitude faster than our original MATLAB/SciPy/R vectorized style code!
Since this is tedious to do by hand, we note that ModelingToolkit.jl's symbolic code generation can do this automatically from the basic version:
function basic_version!(dr,r,p,t)
a,α,ubar,β,D1,D2 = p
u = r[:,:,1]
v = r[:,:,2]
Du = D1*(Ay*u + u*Ax)
Dv = D2*(Ay*v + v*Ax)
dr[:,:,1] = Du .+ a.*u.*u./v .+ ubar .- α*u
dr[:,:,2] = Dv .+ a.*u.*u .- β*v
end
a,α,ubar,β,D1,D2 = p
uss = (ubar+β)/α
vss = (a/β)*uss^2
r0 = zeros(100,100,2)
r0[:,:,1] .= uss.+0.1.*rand.()
r0[:,:,2] .= vss
prob = ODEProblem(basic_version!,r0,(0.0,0.1),p)
de = modelingtoolkitize(prob)
# Note jac=true,sparse=true makes it automatically build sparse Jacobian code
# as well!
fastprob = ODEProblem(de,[],(0.0,0.1),jac=true,sparse=true)
Lastly, we can do other things like multithread the main loops. LoopVectorization.jl provides the @turbo macro for doing a lot of SIMD enhancements, and @tturbo is the multithreaded version.
### Optimizing Algorithm Choices
The last thing to do is then ***optimize our algorithm choice***. We have been using Tsit5() as our test algorithm, but in reality this problem is a stiff PDE discretization and thus one recommendation is to use CVODE_BDF(). However, instead of using the default dense Jacobian, we should make use of the sparse Jacobian afforded by the problem. The Jacobian is the matrix $\frac{df_i}{dr_j}$, where $r$ is read by the linear index (i.e. down columns). But since the $u$ variables depend on the $v$, the band size here is large, and thus this will not do well with a Banded Jacobian solver. Instead, we utilize sparse Jacobian algorithms. CVODE_BDF allows us to use a sparse Newton-Krylov solver by setting linear_solver = :GMRES.
Note
The Solving Large Stiff Equations tutorial goes through these details. This is simply to give a taste of how much optimization opportunity is left on the table!
Let's see how our fast right-hand side scales as we increase the integration time.
prob = ODEProblem(fast_gm!,r0,(0.0,10.0),p)
@benchmark solve(prob,Tsit5())
BenchmarkTools.Trial: 3 samples with 1 evaluation.
Range (min … max): 1.578 s … 2.502 s ┊ GC (min … max): 31.99% … 58.83%
Time (median): 1.580 s ┊ GC (median): 35.16%
Time (mean ± σ): 1.887 s ± 532.716 ms ┊ GC (mean ± σ): 44.74% ± 14.66%
█ ▁
█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
1.58 s Histogram: frequency by time 2.5 s <
Memory estimate: 2.76 GiB, allocs estimate: 39323.
using Sundials
@benchmark solve(prob,CVODE_BDF(linear_solver=:GMRES))
BenchmarkTools.Trial: 11 samples with 1 evaluation.
Range (min … max): 450.051 ms … 476.904 ms ┊ GC (min … max): 0.00% … 0.38%
Time (median): 460.246 ms ┊ GC (median): 0.75%
Time (mean ± σ): 461.439 ms ± 9.264 ms ┊ GC (mean ± σ): 0.56% ± 0.33%
█ █ █ █ █ █ ██ █ █ █
█▁▁▁▁█▁▁▁█▁▁█▁█▁▁▁▁▁▁▁▁█▁██▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁█▁▁▁▁█ ▁
450 ms Histogram: frequency by time 477 ms <
Memory estimate: 120.93 MiB, allocs estimate: 20000.
prob = ODEProblem(fast_gm!,r0,(0.0,100.0),p)
# Will go out of memory if we don't turn off save_everystep!
@benchmark solve(prob,Tsit5(),save_everystep=false)
BenchmarkTools.Trial: 2 samples with 1 evaluation.
Range (min … max): 3.075 s … 3.095 s ┊ GC (min … max): 0.00% … 0.00%
Time (median): 3.085 s ┊ GC (median): 0.00%
Time (mean ± σ): 3.085 s ± 14.570 ms ┊ GC (mean ± σ): 0.00% ± 0.00%
█ █
█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
3.07 s Histogram: frequency by time 3.1 s <
Memory estimate: 2.90 MiB, allocs estimate: 60.
@benchmark solve(prob,CVODE_BDF(linear_solver=:GMRES),save_everystep=false)
BenchmarkTools.Trial: 4 samples with 1 evaluation.
Range (min … max): 1.342 s … 1.386 s ┊ GC (min … max): 0.00% … 0.00%
Time (median): 1.352 s ┊ GC (median): 0.00%
Time (mean ± σ): 1.358 s ± 19.636 ms ┊ GC (mean ± σ): 0.00% ± 0.00%
█ █ █ █
█▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
1.34 s Histogram: frequency by time 1.39 s <
Memory estimate: 3.09 MiB, allocs estimate: 49880.
prob = ODEProblem(fast_gm!,r0,(0.0,500.0),p)
@benchmark solve(prob,CVODE_BDF(linear_solver=:GMRES),save_everystep=false)
BenchmarkTools.Trial: 3 samples with 1 evaluation.
Range (min … max): 1.817 s … 1.915 s ┊ GC (min … max): 0.00% … 0.00%
Time (median): 1.825 s ┊ GC (median): 0.00%
Time (mean ± σ): 1.853 s ± 54.179 ms ┊ GC (mean ± σ): 0.00% ± 0.00%
█ █ █
█▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
1.82 s Histogram: frequency by time 1.91 s <
Memory estimate: 3.83 MiB, allocs estimate: 66931.
Notice that we've eliminated almost all allocations, allowing the code to grow without hitting garbage collection and slowing down.
Why is CVODE_BDF doing well? What's happening is that, because the problem is stiff, the number of steps required by the explicit Runge-Kutta method grows rapidly, whereas CVODE_BDF is taking large steps. Additionally, the GMRES linear solver form is quite an efficient way to solve the implicit system in this case. This is problem-dependent, and in many cases using a Krylov method effectively requires a preconditioner, so you need to play around with testing other algorithms and linear solvers to find out what works best with your problem.
Now continue to the Solving Large Stiff Equations tutorial for more details on optimizing the algorithm choice for such codes.
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http://nrich.maths.org/646
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### Some(?) of the Parts
A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle
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http://www.purplemath.com/learning/viewtopic.php?f=17&t=534
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## A recent study indicated that 20% of adults exercise... Find
Standard deviation, mean, variance, z-scores, t-tests, etc.
### A recent study indicated that 20% of adults exercise... Find
A recent study indicated that twenty percent of adults exercise regularly. Suppose that five adults are selected at random. Use the binomial probability formula to find the probability that the number of people in the sample who exercise is:
a. exactly 3
b. at least 3
Also, find the:
c. mean
d. standard deviation
FWT
Posts: 107
Joined: Sat Feb 28, 2009 8:53 pm
FWT wrote:A recent study indicated that twenty percent of adults exercise regularly. Suppose that five adults are selected at random. Use the binomial probability formula to find the probability that the number of people in the sample who exercise is:
a. exactly 3
b. at least 3
The formula for the probability P that x of n results will be "good" is:
. . . . .$P(x)\, =\, C(n,\, x)\,p^x\, (1\, -\, p)^{n\, -\, x}$
In this case, n = 5 and p = 0.2. Then:
. . . . .$\mbox{a. }\, P(3)\, =\, \frac{5!}{2!3!}\, (0.2)^3\, (0.8)^2$
Evaluate to find the needed value.
For part (b), note that "at least three" means three, four, or five. Compute all three, and sum the values.
FWT wrote:Also, find the:
c. mean
d. standard deviation
The expected value (or mean) is np, or (5)(0.2). The standard deviation is given by:
. . . . .$\sigma\, =\, \sqrt{np(1\, -\, p)}$
stapel_eliz
Posts: 1729
Joined: Mon Dec 08, 2008 4:22 pm
### Re: A recent study indicated that 20% of adults exercise... Find
for (b) I did P(3)+P(4)+P(5) and got the ansewr in the back of the book. thanks!
FWT
Posts: 107
Joined: Sat Feb 28, 2009 8:53 pm
|
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https://ftp.aimsciences.org/article/doi/10.3934/amc.2019021
|
# American Institute of Mathematical Sciences
May 2019, 13(2): 313-328. doi: 10.3934/amc.2019021
## Symmetries of weight enumerators and applications to Reed-Muller codes
1 Université Paris 8, LAGA, CNRS (UMR 7539), Université Paris 13, Sorbonne Paris Cité, F-93526 Saint-Denis, France 2 Universität Bern, Mathematisches Institut (MAI), Sidlerstrasse 5, CH-3012 Bern, Switzerland
Received May 2018 Published February 2019
Fund Project: The first author was partially supported by PEPS - Jeunes Chercheur-e-s - 2017.
Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.
Citation: Martino Borello, Olivier Mila. Symmetries of weight enumerators and applications to Reed-Muller codes. Advances in Mathematics of Communications, 2019, 13 (2) : 313-328. doi: 10.3934/amc.2019021
##### References:
[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836. [2] J. Ax, Zeroes of polynomials over finite fields, Amer. J. Math., 86 (1964), 255-261. doi: 10.2307/2373163. [3] I. A. Berchenko and P. J. Olver, Symmetries of polynomials, J. Symb. Comp., 29 (2000), 485-514. doi: 10.1006/S0747-7171(99)90307-3. [4] K. Betsumiya and M. Harada, Classification of formally self-dual even codes of lengths up to 16, Des. Codes Cryptogr., 23 (2001), 325-332. doi: 10.1023/A:1011223128089. [5] H. F. Blichfeldt, Finite Collineation Groups, The Univ. Chicago Press, Chicago, 1917. [6] M. Borello, On the automorphism groups of binary linear codes, Topics in Finite Fields, Contemporary Mathematics, 632 (2015), 29-41. doi: 10.1090/conm/632/12617. [7] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbol. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. [8] N. D. Elkies, Linear codes and algebraic geometry in higher dimensions, Preprint, 2006. [9] A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3 (1971), 211–215. [10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge university press, Cambridge, 2003. doi: 10.1017/CBO9780511807077. [11] N. Kaplan, Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory, (Doctoral dissertation), Harvard University, 2013, Retrieved from: http://users.math.yale.edu/ nk354/papers/kaplanthesis.pdf. [12] G. T. Kennedy, Weight distributions of linear codes and the Gleason-Pierce theorem, J. Combin. Theory Ser. A, 67 (1994), 72-88. doi: 10.1016/0097-3165(94)90004-3. [13] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Ⅰ. North-Holland Publishing Co., Amsterdam-New York-Oxford. North-Holland Mathematical Library, 1977. [14] C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Information and Control, 22 (1973), 188-200. doi: 10.1016/S0019-9958(73)90273-8. [15] O. Mila, Invariance for Weight Enumerators of Evaluation Codes and Counting ${\mathbb{F}}_q$-rational Points on Hypersurfaces, (Master dissertation), EPFL, 2015, Retrieved from: http://archiveweb.epfl.ch/csag.epfl.ch/files/content/sites/csag/files/MasterOlivierMila.pdf. [16] G. Nebe, E. M. Rains and N. J. A. Sloane, Self-dual Codes and Invariant Theory, Vol. 17. Berlin: Springer, 2006. [17] E. M. Rains and N. J. A. Sloane, Self-dual codes, In: Pless, V.S., Huffman, W.C. (Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, (2002), 177–294. [18] N. Sloane, Is there a (72, 36) d = 16 self-dual code?, IEEE Transactions on Information Theory, 19 (1973), 251-251. doi: 10.1109/tit.1973.1054975. [19] N. J. A. Sloane, Gleason's Theorem on self-dual codes and its generalizations, preprint, arXiv: math/0612535.
show all references
##### References:
[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836. [2] J. Ax, Zeroes of polynomials over finite fields, Amer. J. Math., 86 (1964), 255-261. doi: 10.2307/2373163. [3] I. A. Berchenko and P. J. Olver, Symmetries of polynomials, J. Symb. Comp., 29 (2000), 485-514. doi: 10.1006/S0747-7171(99)90307-3. [4] K. Betsumiya and M. Harada, Classification of formally self-dual even codes of lengths up to 16, Des. Codes Cryptogr., 23 (2001), 325-332. doi: 10.1023/A:1011223128089. [5] H. F. Blichfeldt, Finite Collineation Groups, The Univ. Chicago Press, Chicago, 1917. [6] M. Borello, On the automorphism groups of binary linear codes, Topics in Finite Fields, Contemporary Mathematics, 632 (2015), 29-41. doi: 10.1090/conm/632/12617. [7] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system Ⅰ: The user language, J. Symbol. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. [8] N. D. Elkies, Linear codes and algebraic geometry in higher dimensions, Preprint, 2006. [9] A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3 (1971), 211–215. [10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge university press, Cambridge, 2003. doi: 10.1017/CBO9780511807077. [11] N. Kaplan, Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory, (Doctoral dissertation), Harvard University, 2013, Retrieved from: http://users.math.yale.edu/ nk354/papers/kaplanthesis.pdf. [12] G. T. Kennedy, Weight distributions of linear codes and the Gleason-Pierce theorem, J. Combin. Theory Ser. A, 67 (1994), 72-88. doi: 10.1016/0097-3165(94)90004-3. [13] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Ⅰ. North-Holland Publishing Co., Amsterdam-New York-Oxford. North-Holland Mathematical Library, 1977. [14] C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Information and Control, 22 (1973), 188-200. doi: 10.1016/S0019-9958(73)90273-8. [15] O. Mila, Invariance for Weight Enumerators of Evaluation Codes and Counting ${\mathbb{F}}_q$-rational Points on Hypersurfaces, (Master dissertation), EPFL, 2015, Retrieved from: http://archiveweb.epfl.ch/csag.epfl.ch/files/content/sites/csag/files/MasterOlivierMila.pdf. [16] G. Nebe, E. M. Rains and N. J. A. Sloane, Self-dual Codes and Invariant Theory, Vol. 17. Berlin: Springer, 2006. [17] E. M. Rains and N. J. A. Sloane, Self-dual codes, In: Pless, V.S., Huffman, W.C. (Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, (2002), 177–294. [18] N. Sloane, Is there a (72, 36) d = 16 self-dual code?, IEEE Transactions on Information Theory, 19 (1973), 251-251. doi: 10.1109/tit.1973.1054975. [19] N. J. A. Sloane, Gleason's Theorem on self-dual codes and its generalizations, preprint, arXiv: math/0612535.
$\bar S(w_{\mathcal{RM}_2(r,m)}(x,y))$
$r\backslash m$ 1 2 3 4 5 6 7 0 $\infty$ $D_4$ $D_8$ $D_{16}$ $D_{32}$ $D_{64}$ $D_{128}$ 1 $\infty$ $D_4$ $S_4$ $D_{8}$ $D_{16}$ $D_{32}$ $D_{64}$ 2 $\infty$ $\infty$ $D_8$ $D_{8}$ $S_4$ $D_{4}$ $D_{8}$ 3 $\infty$ $\infty$ $\infty$ $D_{16}$ $D_{16}$ $D_{4}$ $S_4$ 4 $\infty$ $\infty$ $\infty$ $\infty$ $D_{32}$ $D_{32}$ $D_{8}$ 5 $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $D_{64}$ $D_{64}$ 6 $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $D_{128}$
$r\backslash m$ 1 2 3 4 5 6 7 0 $\infty$ $D_4$ $D_8$ $D_{16}$ $D_{32}$ $D_{64}$ $D_{128}$ 1 $\infty$ $D_4$ $S_4$ $D_{8}$ $D_{16}$ $D_{32}$ $D_{64}$ 2 $\infty$ $\infty$ $D_8$ $D_{8}$ $S_4$ $D_{4}$ $D_{8}$ 3 $\infty$ $\infty$ $\infty$ $D_{16}$ $D_{16}$ $D_{4}$ $S_4$ 4 $\infty$ $\infty$ $\infty$ $\infty$ $D_{32}$ $D_{32}$ $D_{8}$ 5 $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $D_{64}$ $D_{64}$ 6 $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $\infty$ $D_{128}$
$\bar S(w_{\mathcal{RM}_3(r,m)}(x,y))$
$r\backslash m$ 1 2 3 4 0 $D_3$ $D_9$ $D_{27}$ $D_{81}$ 1 $D_3$ $C_3$ $C_9$ $C_{27}$ 2 $\infty$ $C_3$ $C_3$ $C_{3}$ 3 $\infty$ $D_9$ $C_3$ 4 $\infty$ $\infty$ $C_9$ 5 $\infty$ $\infty$ $D_{27}$ $C_{3}$ 6 $\infty$ $\infty$ $\infty$ $C_{27}$ 7 $\infty$ $\infty$ $\infty$ $D_{81}$
$r\backslash m$ 1 2 3 4 0 $D_3$ $D_9$ $D_{27}$ $D_{81}$ 1 $D_3$ $C_3$ $C_9$ $C_{27}$ 2 $\infty$ $C_3$ $C_3$ $C_{3}$ 3 $\infty$ $D_9$ $C_3$ 4 $\infty$ $\infty$ $C_9$ 5 $\infty$ $\infty$ $D_{27}$ $C_{3}$ 6 $\infty$ $\infty$ $\infty$ $C_{27}$ 7 $\infty$ $\infty$ $\infty$ $D_{81}$
$\bar S(w_{\mathcal{RM}_4(r, m)}(x, y))$
$r\backslash m$ 1 2 3 0 $D_8$ $D_{16}$ $D_{64}$ 1 $V_4$ $C_4$ $C_{16}$ 2 $D_8$ $\{{\rm Id}\}$ $C_4$ 3 $\infty$ $\{{\rm Id}\}$ $\{{\rm Id}\}$ 4 $\infty$ $C_4$ 5 $\infty$ $D_{16}$ $\{{\rm Id}\}$ 6 $\infty$ $\infty$ $C_4$ 7 $\infty$ $\infty$ $C_{16}$ 8 $\infty$ $\infty$ $D_{64}$
$r\backslash m$ 1 2 3 0 $D_8$ $D_{16}$ $D_{64}$ 1 $V_4$ $C_4$ $C_{16}$ 2 $D_8$ $\{{\rm Id}\}$ $C_4$ 3 $\infty$ $\{{\rm Id}\}$ $\{{\rm Id}\}$ 4 $\infty$ $C_4$ 5 $\infty$ $D_{16}$ $\{{\rm Id}\}$ 6 $\infty$ $\infty$ $C_4$ 7 $\infty$ $\infty$ $C_{16}$ 8 $\infty$ $\infty$ $D_{64}$
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2020 Impact Factor: 0.935
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http://tex.stackexchange.com/questions/61848/how-to-make-tikz-multipart-node-parts-have-uniform-size
|
# How to make tikz multipart node parts have uniform size?
When making boxed nodes in TikZ, a common requirement is that the nodes be of equal size. Because TikZ automatically adjusts the size of nodes to fit the contents of the node, this is usually done by setting a minimum size for the nodes, and setting the inner sep to 0.
How can I do the same thing for the parts of multipart nodes? As the following example shows, setting a minimum size for a multipart node only sets the height (as described in the manual). So how do I make each part be the same size as my non-split nodes?
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.multipart}
\usepackage[margin=1in]{geometry}
\parindent=0pt
\begin{document}
\tikzset{
every node/.style={draw, minimum size=4ex,inner sep=0pt},
split/.style={rectangle split, rectangle split parts=2,draw,inner sep=0ex,
rectangle split horizontal,rectangle split part align=base,minimum size=4ex}}
\begin{tikzpicture}
\node at (0,0) {A} ;
\node at (1,0) {a} ;
\node at (2,0) {X} ;
\node at (3,0) {j} ;
\node at (4,0) {};
\node[split] at (5,0) {};
\node[split] at (6,0) {X\nodepart{two}j};
\end{tikzpicture}
\end{document}
-
Not quite a duplicate but closely related: tex.stackexchange.com/questions/54252/… – Caramdir Jul 1 '12 at 17:52
A workaround would be something like \node (n1) at (0,0) {X}; \node[right=-0.4pt of n1] (n2) {j}; (where 0.4pt is the line width and text height/depth are set. – Caramdir Jul 1 '12 at 18:19
I updated my answer with the idea from Caramdir's comment. This method is used in the tutorial "Diagrams as Simple Graphs" of the pgfmanual.
Update With text width=4ex it's better to use Polgab'method : align=center instead of \hfil ...\hfil
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.multipart}
\usepackage[margin=1in]{geometry}
\parindent=0pt
\begin{document}
\tikzset{
every node/.style={draw, minimum size=4ex,inner sep=0pt,textstyle},
split/.style={rectangle split, rectangle split parts=2,draw,inner sep=0ex,
rectangle split horizontal,minimum size=4ex},
textstyle/.style={text height=1.5ex,text depth=.25ex}}
\begin{tikzpicture}
\node at (0,0) {A} ;
\node at (1,0) {a} ;
\node at (2,0) {X} ;
\node at (3,0) {j} ;
\node at (4,0) {};
\node[split,text width=4ex] at (6,0) {\nodepart{two}};
\node[split,text width=4ex] at (8,0) {\hfil X\hfil\nodepart{two}\hfil r\hfil};
\end{tikzpicture}
\end{document}
-
Thanks, the text width solution a much cleaner one altogether. And it's simple to make a macro to wrap the part contents in \hfils. (We can probably delete the clarifying comments.) – Alan Munn Jul 1 '12 at 17:21
There is a downside of removing the align=base part: now if you have e.g. an X and an r in the two parts they are not aligned with the same baseline. – Alan Munn Jul 1 '12 at 17:30
You have the same problem with aand A in the single nodes – Alain Matthes Jul 1 '12 at 17:34
Yes, but in that case it's much less noticeable. :) – Alan Munn Jul 1 '12 at 17:36
@Caramdir: You are right. But this is not sufficient: part two does not use text height and text depth! We must add rectangle split part align=base. – Paul Gaborit Jul 1 '12 at 18:11
show 4 more comments
Here is my solution with rectangle split part align=base (and using chains just to chain the nodes).
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.multipart,chains}
\begin{document}
\begin{tikzpicture}[start chain=going right,node distance=.5ex]
\tikzset{
simple node/.style={
draw,
text height=2.8ex,text depth=1.2ex,
inner sep=0pt,text width=4ex,align=center
},
split node/.style={
simple node,
rectangle split,rectangle split horizontal,rectangle split parts=2,
draw,inner sep=0ex,rectangle split part align=base,
},
}
\node[on chain,simple node] {j} ;
\node[on chain,split node] {j\nodepart{two}A};
\node[on chain,simple node] {A} ;
\node[on chain,simple node] {X} ;
\node[on chain,split node] {X\nodepart{two}j};
\node[on chain,simple node] {j} ;
\node[on chain,split node] {.\nodepart{two}\textbullet};
\end{tikzpicture}
\end{document}
Below the result without rectangle split part align=base (note the difference between j on left or on right part of slit node):
-
Yes you are right with text width=4ex the correct way is to use align=center. I find my problem, with text height=2.8ex,text depth=1.2ex and rectangle split part align=base, there is no problem to align the characters. – Alain Matthes Jul 2 '12 at 6:04
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http://nepalinlp.com/detail/approaches-to-predicting-part-of-speech-tags-of-unknown-words/
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by Ingroj Shrestha on Nov. 8, 2017
One of the challenges faced by statistical part-of-speech taggers is the presence of words in test datasets that do not exist in the training dataset. Such words are called unknown words. In this blog post we'll look into different ways to tag unknown words.
A statistical part-of-speech tagger uses annotated training dataset to perform tagging. The tagger obtains statistical information from the training corpus while tagging words in test data. However, the tagger gets no statistical information for the unknown words.
## Emission Probability
Emission probability is the probability that an emitted word is given a particular part-of-speech tag. The
$$b_i(o_t) = P(o_t|q_i) = \frac{C(o_t,q_i)}{C(q_i)}$$
Where,
• $b_i(o_t)$, is the probability of an observation $o_t$ being generated from a state $q_i$
• $C(o,q_i)$, is the number of times word 'o' occurs with the part-of-speech tag $q_i$
• $C(q_i)$, is the number of times the part-of-speech tag occurs in the training dataset
• Let $x_1....x_n$, be a sentence that contains an unknown word. Since, an unknown word does not occur in the training dataset, the value of $C(o_t,q_i)$ is zero for the word, i.e.,
$$C(o_t,q_i) = 0$$
Therefore, the emission probability for the word is also zero, $b_i(o_t) = 0$
So, the probability for all possible part-of-speech tag sequences for an observation sequence containing an unknown word becomes 0 and then the tagger cannot choose between the sequences.
## Tagging Unknown Words
Unknown words cause a serious problem in part-of-speech tagging, which is especially true in case of low resource language like Nepali. There are number of ways to handle unknown words. Some of them are discussed below.
### Laplace/Active Smoothing
It is a popular technique used to smooth categorical data. For a set of vocabulary(all tags), V, observation(word in a sentence), $o_t$, and state(tag), $q_i$, the Laplace smoothing technique gives:
$$P(o_t|q_i) = \frac{C(o_t,q_i) + 1}{C(q_i) + C(V) + 1}$$
The idea is to add a small probability of $\frac{1}{C(q_i) + C(V)}$ so that the possible tag sequences for an observation sequence with one or more unknown words will have non-zero probability.
### Most Frequent Part-of-Speech Tag
Assuming that the unknown word always takes the most frequent part-of-speech tag is one way of handling them. Therefore, for all unknown words $P(o_t|q_i) = 1$ for the most frequent tag and $P(o_t|q_i) = 0$ for the remaining tags. The most frequent part-of-speech tag from an annotated corpus can be obtained using FreqDist( ).
from probability import FreqDist
train_data = [('घटना', 'NN'), ('हरु', 'IH'), ('नेपाल', 'NN'), ('मा', 'IE')]
tags = []
for d in data:
tags.append(d[1])
fdist = FreqDist(tags)
for word, frequency in fdist.most_common(1):
print('{}{}'.format(word, frequency))
OUTPUT: NN 2
FreqDist( ) returns a dictionary with part-of-speech tag as keys and the number of word with that tag as values.
### Overall Part-of-Speech Distribution
The overall part-of-speech distribution from known words can used to estimate and assign part-of-speech tags for unknown words. Say if the probability of a known word having a tag $q_i$ is 'x' the probability of an unknown word having tag $q_i$ is also 'x'.
Also, we can consider the part-of-speech distribution to open class words only. This is because, it is usually possible to exhaustively list all the members in the closed class words. Therefore, the chances that an unknown word belonging to a closed class is very slim. So, we can consider the part-of-speech distribution from open class words only.
### Morphological Features
Words in Nepali are typically formed by affixation, compounding, repetition of word or a part of word and phonetic similarity. These morphological processes are useful in guessing part-of-speech tags of the unknown words.
Affixation is very common in Nepali. Prefixes and suffixes do a lot more than forming new words. They also carry information about the categories the words belong to. Thus, affixes rules and their probability distribution can be used to predict part-of-speech tag of the unknown words.
Compounding is a very productive process in in Nepali verbs. Multiple numbers of combinations are possible, which is why multiple stem morphemes are present in compound words. Nepali verb compounds are tagged according to the last verb in the compound.
The existing morphological rules for part-of-speech taggers in Nepali does not focus on words formed as a result of repetition of word or a part of word and phonetic similarity, which leaves a space for further research in Nepali linguistics.
## References
• Part-of-Speech Tagging
• Morphological features help POS tagging of unknown words across language varieties
• Tags: NLP , Pre Processing
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http://physics.stackexchange.com/questions/26826/quantum-statistics-of-branes?answertab=votes
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# Quantum statistics of branes
Quantum statistics of particles (bosons, fermions, anyons) arises due to the possible topologies of curves in D-dimensional spacetime winding around each other
What happens if we replace particles by branes? It seems like their quantum statistics should be described by something like a generalization of TQFT in which the "spacetime" (worldbrane) is equipped with an embedding into an "ambient" manifold (actual spacetime). The inclusion of non-trivial topology for the "ambient" manifold introduces additional effects, to 1st approximation describable by inclusion of k-form fluxes coupling to the brane. To 2nd approximation, however, there is probably non-trivial coupling between these fluxes and the "generalized quantum statistics"
A simple example of non-trivial "brane quantum statistics" is the multiplication of quantum amplitudes of strings by the exponential of the euler charactestic times a constant. In string theory this corresponds to changing the string coupling constant / dilaton background.
Were such generalized TQFTs studied? Which non-trivial examples are there for branes in string theory?
-
You are probably aware of this, but just for completeness: N coincident branes have a U(N) gauge symmetry, which is broken to $U(1)^N\times S_N$ when they are separated. The permutation symmetry $S_N$ is a discrete gauge symmetry which ensures branes are treated as identical particles. Your question seems related to which kind of identical particles they are (bosons, fermions, or anyons). – user566 Dec 3 '11 at 2:18
Just slightly picky, user: the product should be semidirect, not direct. ;-) – Luboš Motl Jan 30 at 14:26
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https://www.ideals.illinois.edu/browse?type=subject&value=Philosophy%20of%20Language
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# Browse by Subject "Philosophy of Language"
• (2014-01-16)
Classical logic and a given nonclassical logic are, by definition, incompatible in some sense. In some cases, this incompatibility is innocuous. In other cases, the nonclassical logic is incompatible with classical logic ...
application/pdf
PDF (460Kb)
• (2012-09-18)
This dissertation presents a novel theory of tense and tense-like constructions. It is named after a key theoretical component of the theory, the event restriction interval. In Event Restriction Interval (ERI) Theory, ...
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PDF (1Mb)
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https://brilliant.org/problems/answer-within-a-minute-iv-corrected/
|
# Answer Within A Minute IV (Corrected)
Algebra Level 1
Evaluate the following expression:
$123456789^{2} - (123456788 \times 123456790).$
If you use a calculator whose precision is not strong enough to answer this question, then you will answer this problem incorrectly.
×
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https://gateoverflow.in/1062/gate2004-68
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7k views
A hard disk with a transfer rate of $10$ Mbytes/second is constantly transferring data to memory using DMA. The processor runs at $600$ MHz, and takes $300$ and $900$ clock cycles to initiate and complete DMA transfer respectively. If the size of the transfer is $20$ Kbytes, what is the percentage of processor time consumed for the transfer operation?
1. $5.0 \%$
2. $1.0\%$
3. $0.5\%$
4. $0.1\%$
recategorized | 7k views
0
They should've written the question as "percentage of processor time consumed in the total transfer operation?" Their wording is a bit difficult to understand.
Clock cycle time = $\frac{1}{600 \times 10^6}$ [ Frequency = 1/Time]
For DMA initiation and completion = $\frac{(900+300)}{600\times10^6} = 2$ microsec .
Disk Transfer rate $= 10$ Mbytes/sec
$1$ byte $= \frac{1}{10^7}$ sec
$20$ Kbytes $= 2$ milisec $= 2000$ micro sec
Percentage $= \left (\frac{2}{2+2000} \right )\times100 =0.0999 ≃ 0.1\%$
option (D)
$\%$ of CPU time consume $\frac{x}{x+y}$
Now, when, use $x=$ Data preparation time or Total cycle time used by CPU and $y=$ Data transfer time
To calculate the fraction of CPU time to the data transfer time - we use $\frac{x}{x+y}$ it is burst mode.
edited
0
@Anirudh Sir...Hard disk is given in the question so We are using burst mode of transfer...If keyboard or printer would be given then we have to use cycle stealing mode.
Is is right?
0
Y arent we taking MB as 2^20 and instead taking it as 10^6 ?
@Arjun sir
@habibkhan ?
+2
Please correct the above answer at % of CPU time consume =(2/2002)*100 not (2/200)+2*100..
+3
DURING DATA STORING 2^20
and during transfering the data 10^6
+1
Only % of processor time consumed for transfer operation.
Then why is DMA initialization also considered in transfer operation?
In a way this whole process can be called transfer operation. Thing is in initialization CPU is busy in initialization. During data preparation time CPU is busy doing other usefull task. And during transfer CPU is idle. So what times will be taken to say as processor time consumed in transfer operation??
is this like the duration when CPU is engaged in transfer operation will be considered?
0
transfer takes 2000 and intialization 2units.
then percentage of transfer should be 2000/(2+2000) ??
what is wrong??
0
Why burst mode is used here.Anyone please tell
+6
@rahul
Here burst mode is used because of this line " constantly transferring data to memory using DMA " .
Because in cycle stealing mode , instead of the data being transferred all at once, it is transferred one byte at a time. so we can easily guess here Burst mode can be used .
+1
+2
I think the following wording is making it Burst mode
If the size of the transfer is 20 Kbytes
Because in cycle stealing transfer is one byte.
@Bikram Sir:- And question may say like:- "Hard disk is actively transferring data in cycle stealing mode", ?
+3
@rahul
They say size of data which is transfer is 20 KB , it does not say 20 KB at one time they can send or divided by multiple times transfer so by seeing the word "constantly " means without a break , i am sure it is burst mode used here ..
+3
@Bikram sir,in DMA while Disk is reading/writing memory, during this time CPU may perform some internal work but not the BUS related activity unlike the interupt driven IO where CPU does not perform any internal work too.In DMA,CPU only intiate the transfer and control it.But here explicitly given that CPU takes 900 clock cycles to complete the transfer.Does it mean here that CPU is ideal during whole transfer like Interrupt driven.
+7
In this question, it is explicitly mention that " The processor takes 900 clock cycles to complete DMA transfer " .
CPU is usually the master and in this DMA case, instead of this master, another master DMA takes its place. So you can see that CPU is also a processor, DMA controller is also a processor but then DMA controller is a special purpose processor meant only for data transfer, nothing else.
CPU is kept in suspended animation and it is going to wait until the DMA controller releases , until the block of data transfer is over, the CPU will be kept in suspended animation and at the end of the block transfer the controller will release the bus to the CPU.
So it means CPU is idle theoritacally in this time period when DMA transfer the data but practically CPU is not idle , it is doing some other task through other bus .. but for our convenience we just think that CPU is siting idle in this time period .
+2
Clock cycle time = 1/600×106
Although it doesn't affect the answer, shouldn't it be 1/(600 x 106) sec or 1/600 x 10-6 sec?
Must be a typo.
+1
@Bikram sir
So it means CPU is idle theoritacally in this time period when DMA transfer the data but practically CPU is not idle , it is doing some other task through other bus .. but for our convenience we just think that CPU is siting idle in this time period .
But, isn't we have just one bus in the system? So,how cpu is using some other bus?
0
I think, There will be only one bus, but in case of DMA, the bus is connected from cpu to memory and DMA controller to memory. so while DMA transfering data through bus, the CPU can perform computations other the accessing the bus. so cpu is not idle
0
How 20kbytes is 2000 microseconds
Clock cycle time = (1/600) x 10 ^6
For DMA initiation and competion = (900+300)/(600 x 10 ^6) = 2 microsec .
Disk Transfer rate = 10 M bytes. 1 byte = 1/10^7 sec
20 K bytes = 2000 microsec Percentage = (2/2000)*100 =0.1%
so, ans D
.
0
How is it 2/2000 ??
It is simple logical question. No need of so many calculations.
(20KB/10MB)x100
= (20KB/10x1024KB)x100
= 0.1%
The reason for this is whatever may be the processor speed and clock cycle . While taking percentage ratio they will get cancel led out.
Processor Speed = 600MHZ = $6*10^{8}$ cycles/second
Transfer time = $\frac{20KB}{10MB}$ = 2ms = $12*10^{5}$ cycles
Total cycles consumed = 1200,000 + 300 + 900 = 120,1200 cycles
% of processor time consumed = $\frac{120,1200*100}{6*10^{8}} = \frac{12012}{60000}$= 0.2%
So, (D) can be chosen as option!
edited
+1
@manu00x
how you are getting 0.2?
i am getting 0.1022 %(approx)
0
how?? what will be 12012/60000 = ?
Transfer rate=10 MB per second
Data=20 KB=20* 2 10
So Time=(20 * 2 10)/(10 * 2 20)= 2* 10-3 =2 ms
Processor speed= 600 MHz=600 Cycles/sec
Cycles required by CPU=300+900 =1200
For DMA=1200
So time=1200/(600 *10 6)=.002 ms
In %=.002/2*100=.1%
So (D) is correct option
0
what kind of logic is this, first converted into millisecond and converted it into %??
Transfer rate=10 MB per second
Data=20 KB=20* 2 10
So Time=(20 * 2 10)/(10 * 2 20)= 2* 10-3 =2 ms
Processor speed= 600 MHz=600 Cycles/sec
Cycles required by CPU=300+900 =1200
For DMA=1200
So time=1200/(600 *10 6)=.002 ms
In %=.002/2*100=.1%
So (D) is correct option
Data transfer time= initialization time + DMA transfer time =300+900=1200 cycles
Data preparation time = time disk is taking to transfer 20KB to DMA
= (20KB)/(10MBps) = 2msec or 1200000 cycles
thus CPU time consumed = (transfer time)/(prepartion time + transfer time)
= [1200/(1200000+1200)] * 100
=0.0999 = 0.1%
Option d is correct
1
2
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|
https://vismor.com/documents/network_analysis/graph_algorithms/S1.SS6.php
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1.6 Acyclic Graph
A directed graph with no cycles is called directed acyclic graph or a DAG for short. An acyclic digraph has at least one vertex with an out-degree of zero.
Figure 4 shows a variant of the directed graph of Figure 1 that contains no cycles. You will observe that vertex 4 has an out-degree of zero.
A directed tree is a connected DAG with the following properties:
• There is one vertex, called the root, which no edges enter.
• All vertices except the root have one entering edge.
• There is a unique path from each vertex to the root.
A DAG consisting of one or more trees is called a forest. If the graph $F=(V,E)$ is a forest and the edge $(v,w)$ is in $E(F)$, vertex $v$ is the parent of $w$ and vertex $w$ is the child of $v$. If there is a path from $v$ to $w$, then vertex $v$ is an ancestor of $w$ and vertex $w$ is a descendent of $v$. A vertex with no proper descendants is a leaf. A vertex $v$ and its descendants form a subtree of $F$. The vertex $v$ is the root of this subtree. The depth of vertex $v$ is the length of the path from the root to $v$. The height of vertex $v$ is the length of the longest path from $v$ to a leaf. The height of a tree is the height of its root. The level of vertex $v$ is its depth subtracted from the height of the tree.
Figure 5 depicts a directed tree. Its root is vertex 1. Its leaves are the set of vertices $L(G)=\{ 3,4,6,8,9\}$.
An undirected, connected, acyclic graph is called a free tree or an undirected tree. A rooted free tree is a free tree in which one vertex has been designated as the root. A directed tree is converted into a rooted free tree by discarding the orientation of the edges. A rooted free tree is converted into a directed tree by orienting each edge away from the root. The terminology which applies to directed trees also applies to rooted free trees.
Figure 6 depicts the directed tree of Figure 5 converted into a rooted free tree.
|
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|
https://www.r-bloggers.com/2012/11/matching-clustering-solutions-using-the-hungarian-method/
|
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
Some time ago I stumbled upon a problem connected with the labels of a clustering. The partition an instance belongs to, is mostly labeled through an integer ranging from 1 to K, where k is the number of clusters. The task at that time was to plot a map of the results from the clustering of spatial polygons where every cluster is represented by some color. Like in most projects the analysis was performed multiple times and we used plotting to monitor the changes resulting from the iterations. But after rerunning the clustering algorithm (k-means in this case) the assignment between the clusters and the labeling completely changed, even when using the same parameters. This is because there is no unique connection between a partition (a group of elements) and a specific label (eg. “1”). So even when two solutions match perfectly the assigned labels changed completely. So the graphical representations of two clusterings (which only have some slight differences) look like they are completely different. This is because the coloring relates to the labels. The following R code depicts a simple example for this matter:
require(spdep)
require(rgdal)
require(maptools)
require(ggplot2)
require(plyr)
library(grid)
# panel function for the plots
vplayout <- function(x, y) viewport(layout.pos.row = x, layout.pos.col = y)
gpclibPermit() # required for fortify method
bh <- readShapePoly(system.file("etc/shapes/bhicv.shp", package = "spdep")[1])
# prepare input data
[email protected]$id = rownames([email protected]) bh.points = fortify(bh, region = "id") bh.df = join(bh.points, [email protected], by = "id") # clustering after standardization of data dpad <- data.frame(scale([email protected][, 5:8])) set.seed(1234) res1 <- kmeans(dpad, centers = 10) set.seed(9999) res2 <- kmeans(dpad, centers = 10) # add cluster id to polygon layer bh.df.cl = merge(bh.df, data.frame(id = [email protected]$id, CL1 = res1$cluster, CL2 = res2$cluster), by = "id")
# plot
p1 <- ggplot(bh.df.cl) + aes(long, lat, group = group, fill = as.factor(CL1)) +
geom_polygon() + geom_path(color = "white") + coord_equal() +
scale_fill_brewer(palette = "Set3") +
theme(plot.margin = unit(c(0.2, 0.2, 0.2, 0.2), "cm"), legend.position = "none")
p2 <- ggplot(bh.df.cl) + aes(long, lat, group = group, fill = as.factor(CL2)) +
geom_polygon() + geom_path(color = "white") + coord_equal() +
scale_fill_brewer(palette = "Set3") +
theme(plot.margin = unit(c(0.2, 0.2, 0.2, 0.2), "cm"), legend.position = "none")
grid.newpage()
pushViewport(viewport(layout = grid.layout(1, 2)))
print(p1, vp = vplayout(1, 1))
print(p2, vp = vplayout(1, 2))
After searching the internet for a possible approach my first results point to the direction of methods for cluster validation (subsequently I found out that this problem is also evident when it comes to consensus clustering). In a research paper from Lange et. al. “Stability-based validation of clustering solutions” the authors describe a sampling based approach for evaluating the stability of clustering solutions. Therefore they hade to compare partitions from different runs over the data. This matches exactly the same question I described above. Here a method from Kuhn called the ‘Hungarian method’ for minimum weighted bipartite matching is mentioned which should solve the assignment of two different clustering solutions onto each other. As a result we could rearrange the labels from one clustering.
But what is the idea of formulating this correspondence problem as an optimization exercise? You can relate this type of question to weighted bipartite graphs and subsets of them. In a bipartite graph a matching is a subset of the edges so that no two edges in the subset share a common vertex. It is called a minimum weighted bipartite matching when the graph is a weighted bipartite graph and the sum of all edges in the subset is minimal. This could be represented as a distance matrix having the dimension of the number of clusters where the value between two instances depicts the agreement between these two partitions (one constraint for this approach is that there is the same number of partitions in both clusterings). So, one clustering is represented by columns and the other one by row or vice versa. The agreement can be calculated as follows: Calculate the number of elements in the intersection of the two partitions and subtract it twice from the sum of the number of elements in both clusters. The notion behind this computation is that if all elements are in the intersection, the value is zero and hence it is very likely that these two partitions are mapped on each other. The higher the value the more different are the two partitions. One approach for calculating this distance matrix in R looks like the following (herby we us the method solve_LSAP from the package clue, where some additional explanations could also be found inside the associated paper “A CLUE for CLUster Ensembles”):
# labels from cluster A will be matched on the labels from cluster B
minWeightBipartiteMatching <- function(clusteringA, clusteringB) {
require(clue)
idsA <- unique(clusteringA) # distinct cluster ids in a
idsB <- unique(clusteringB) # distinct cluster ids in b
nA <- length(clusteringA) # number of instances in a
nB <- length(clusteringB) # number of instances in b
if (length(idsA) != length(idsB) || nA != nB) {
stop("number of cluster or number of instances do not match")
}
nC <- length(idsA)
tupel <- c(1:nA)
# computing the distance matrix
assignmentMatrix <- matrix(rep(-1, nC * nC), nrow = nC)
for (i in 1:nC) {
tupelClusterI <- tupel[clusteringA == i]
solRowI <- sapply(1:nC, function(i, clusterIDsB, tupelA_I) {
nA_I <- length(tupelA_I) # number of elements in cluster I
tupelB_I <- tupel[clusterIDsB == i]
nB_I <- length(tupelB_I)
nTupelIntersect <- length(intersect(tupelA_I, tupelB_I))
return((nA_I - nTupelIntersect) + (nB_I - nTupelIntersect))
}, clusteringB, tupelClusterI)
assignmentMatrix[i, ] <- solRowI
}
# optimization
result <- solve_LSAP(assignmentMatrix, maximum = FALSE)
attr(result, "assignmentMatrix") <- assignmentMatrix
return(result)
}
A simple example will illustrate the matching:
minWeightBipartiteMatching(
c(1, 1, 2, 3, 3, 4, 4, 4, 2),
c(2, 2, 3, 1, 1, 4, 4, 4, 3)
)
## Optimal assignment:
## 1 => 2, 2 => 3, 3 => 1, 4 => 4
The mapping resulting for the example should be done in the following way: rename cluster 1 from the first partition as 2, cluster 2 as 3, cluster 3 as 1 and cluster 4 keeps its name.
The last step is to write some code to carry out the mapping automatically.
matching <- minWeightBipartiteMatching(res1$cluster, res2$cluster)
clusterA <- res1$cluster # map the labels from cluster A tmp <- sapply(1:length(matching), function(i) { clusterA[which(res1$cluster == i)] <<- matching[i]
})
clusterB <- res2$cluster bh.df.cl.mwbm = merge(bh.df, data.frame(id = [email protected]$id, CL1 = clusterA, CL2 = clusterB),
by = "id")
# plot
p1 <- ggplot(bh.df.cl.mwbm) + aes(long, lat, group = group, fill = as.factor(CL1)) +
geom_polygon() + geom_path(color = "white") + coord_equal() +
scale_fill_brewer(palette = "Set3") +
theme(plot.margin = unit(c(0.2, 0.2, 0.2, 0.2), "cm"), legend.position = "none")
p2 <- ggplot(bh.df.cl.mwbm) + aes(long, lat, group = group, fill = as.factor(CL2)) +
geom_polygon() + geom_path(color = "white") + coord_equal() +
scale_fill_brewer(palette = "Set3") +
theme(plot.margin = unit(c(0.2, 0.2, 0.2, 0.2), "cm"), legend.position = "none")
grid.newpage()
pushViewport(viewport(layout = grid.layout(1, 2)))
print(p1, vp = vplayout(1, 1))
print(p2, vp = vplayout(1, 2))
Looking at the final plot reveals, that besides the inherent instability from the k-mean method the clustering looks approximately identical and thus the mapping was done successfully. Hoping that this code will also help others, I’m looking forward for any helpful comment.
|
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|
http://semantic-domain.blogspot.ca/2011/
|
## Monday, December 19, 2011
### Adding Equations to System F, at ESOP 2012
Our paper on adding equations to system F was accepted to ESOP 2012!
The reviewers suggested rather a lot of improvements to the paper, so I'm taking the draft down while we incorporate their suggestions. It will go up again in a few weeks, after the revisions are done, and undoubtedly in a much-improved state.
## Tuesday, November 15, 2011
### Updata
The final version of our POPL paper on space-bounded FRP is available now -- we bought two extra pages from the ACM, and used them to add a lot more detail to the examples. With luck the paper will be a lot clearer now.
Also, Bob Atkey has a new blog post on using delay operators to model guarded definitions in type theory. He takes advantage of the fact that delay is a strong lax monoidal functor (with respect to the monoidal structure of products) to use the syntax of applicative functors.
This is a very elegant way of embedding these things into Haskell. Unfortunately, I have never liked the typing rules for the idiom syntax, since they don't work solely on the outermost type constructor. This means type-theoretic properties like normalization are messier to prove. (Bob evades this problem with semantic techniques, though.)
All his other posts are very good, too.
## Monday, October 17, 2011
### Adding Equations to System F
Nick and I have a new draft out, on adding types for term-level equations to System F. Contrary to the experience of dependent types, this is not a very hairy extension -- in fact, I would not even hesitate to call it simple.
However, it does open the door to all sorts of exciting things, such as many peoples' long-standing goal of putting semantic properties of modules into the module interfaces. This is good for documentation, and also (I would hope) good for compilers --- imagine Haskell, if the Monad typeclass definition also told you (and ghc!) all the equational rewriting that it was supposed to do.
## Tuesday, October 4, 2011
### Higher-Order Functional Reactive Programming in Bounded Space
I just learned that our (with Nick Benton and Jan Hoffmann) paper, Higher-Order Functional Reactive Programming in Bounded Space, was accepted for publication at POPL 2012!
I am very happy with this work, since it resolves some of the thorniest problems in FRP (memory usage and space leaks) without giving up the highly expressive higher-order abstractions that makes FRP attractive in the first place.
A technical surprise in this work is that this stuff is the first place I've seen where the magic wand of separation logic is actually essential. The denotational model in this paper is a variation on Martin Hofmann's length space model, which forms a doubly-closed category modelling bunched implications. In this paper, we needed both function spaces: we needed the linear function space to control allocation, and we needed the ordinary function space to make use of sharing.
It would be interesting to see if there are ways to transport some of these ideas back to Hoare-style separation logic to find some interesting new invariants.
## Thursday, September 29, 2011
This isn't really on-topic for this blog, but I can't resist posting this: the Princeton mathematician Edward Nelson claims to have found an inconsistency in arithmetic! On the FOM mailing list, he posted:
I am writing up a proof that Peano arithmetic (P), and even a small fragment of primitive-recursive arithmetic (PRA), are inconsistent. This is posted as a Work in Progress at http://www.math.princeton.edu/~nelson/books.html
A short outline of the book is at:
http://www.math.princeton.edu/~nelson/papers/outline.pdf
The outline begins with a formalist critique of finitism, making the case that there are tacit infinitary assumptions underlying finitism. Then the outline describes how inconsistency will be proved. It concludes with remarks on how to do modern mathematics within a consistent theory.
There's some discussion of this at The n-Category Cafe, including a brief (so far) discussion between Terry Tao and Nelson himself. Obviously, I expect a flaw will be found in his proof -- but I sure hope he's right! That would mean all of mathematics will be in need of revision.
Update: Nelson says that Tao has indeed found a hole in the proof. Exponentiation remains total, for now.
## Wednesday, September 28, 2011
### The most surprising paper at ICFP
I was just at ICFP, which was very nice -- it was my first trip ever to Japan, and I found the people very friendly. (The cuisine, alas, is not so vegetarian-friendly, if you do not regard fish as a vegetable. But the people made up for it!) There were many excellent talks, but only one result which really shocked me:
Linearity and PCF: a semantic insight!, by Marco Gaboardi, Luca Paolini, and Mauro Piccolo.
Linearity is a multi-faceted and ubiquitous notion in the analysis and the development of programming language concepts. We study linearity in a denotational perspective by picking out programs that correspond to linear functions between coherence spaces.
We introduce a language, named SlPCF*, that increases the higher-order expressivity of a linear core of PCF by means of new operators related to exception handling and parallel evaluation. SlPCF* allows us to program all the finite elements of the model and, consequently, it entails a full abstraction result that makes the reasoning on the equivalence between programs simpler.
Denotational linearity provides also crucial information for the operational evaluation of programs. We formalize two evaluation machineries for the language. The first one is an abstract and concise operational semantics designed with the aim of explaining the new operators, and is based on an infinite-branching search of the evaluation space. The second one is more concrete and it prunes such a space, by exploiting the linear assumptions. This can also be regarded as a base for an implementation.
In this paper, the authors considered a really simple language based on coherence spaces, consisting of the flat coherence space of natural numbers and the linear function space. The nice thing about this model is that tokens of function spaces are just trees with natural numbers at the leaves, with branching determined by the parenthesization of the function type. This really shows off how concrete, simple, and easy-to-use coherence spaces are. Then they found the extension to PCF for which this model was fully abstract, as semanticists are prone to doing.
But: the additional operator and its semantics are really bizarre and ugly. I don't mean this as a criticism -- in fact it is exactly why I liked their paper so much! It implies that there are fundamental facts about linear types which we don't understand. I asked Marco Gaboardi about it after his talk, and he told me that he thinks the issue is that the properties of flat domains in coherence spaces are not well understood.
Anyway, this was a great paper, in a "heightening the contradictions" sort of way.
## Monday, August 8, 2011
### Functional Programming, Program Transformations, and Compiler Construction
I just discovered that Lex Augusteijn's 1993 PhD thesis, Functional Programming, Program Transformations, and Compiler Construction, is available online. It was from reading his papers with Renee Leermakers and Frans Kruseman-Aretz that I finally understood how LR parsing worked, so I'm looking forward to reading the extended exposition in this thesis.
## Wednesday, July 27, 2011
### A new lambda calculus for bunched implications
I am going to put off syntactic extensions for a bit, and talk about an entirely different bit of proof theory instead. I am going to talk about a new lambda-calculus for the logic of bunched implications that I have recently been working on. $\newcommand{\lolli}{\multimap} \newcommand{\tensor}{\otimes}$
First, the logic of bunched implications (henceforth BI) is the substructural logic associated with things like separation logic. Now, linear logic basically works by replacing the single context of intuitionistic logic with two contexts, one for unrestricted hypotheses (which behaves the same as the intuitionistic one, and access to which is controlled by a modality $!A$), and one which is linear --- the rules of contraction and weakening are no longer allowed. The intuitionistic connectives are then encoded, so that $A \to B \equiv !A \lolli B$. BI also puts contraction and weakening under control, but does not do so by simply creating two zones. Instead, contexts now become trees, which lets BI simply add the substructural connectives $A \tensor B$ and $A \lolli B$ to intuitionistic logic. Here's what things look like in the implicational fragment:
$$\begin{array}{lcl} A & ::= & P \bnfalt A \lolli B \bnfalt A \to B \\ \Gamma & ::= & A \bnfalt \cdot_a \bnfalt \Gamma; \Gamma \bnfalt \cdot_m \bnfalt \Gamma, \Gamma \\ \end{array}$$
Note that contexts are trees, since there are two context concatenation operators $\Gamma;\Gamma'$ and $\Gamma,\Gamma'$ (with units $\cdot_a$ and $\cdot_m$) which can be freely nested. They don't distribute over each other in any way, but they do satisfy the commutative monoid properties. The natural deduction rules look like this (implicitly assuming the commutative monoid properties):
$$\begin{array}{ll} \frac{}{A \vdash A} & \\ \frac{\Gamma; A \vdash B} {\Gamma \vdash A \to B} & \frac{\Gamma \vdash A \to B \qquad \Gamma' \vdash A} {\Gamma;\Gamma' \vdash B} \\ \frac{\Gamma, A \vdash B} {\Gamma \vdash A \lolli B} & \frac{\Gamma \vdash A \lolli B \qquad \Gamma' \vdash A} {\Gamma,\Gamma' \vdash B} \\ \frac{\Gamma(\Delta) \vdash A} {\Gamma(\Delta;\Delta') \vdash A} & \frac{\Gamma(\Delta;\Delta) \vdash A} {\Gamma(\Delta) \vdash A} \end{array}$$
Note that the substructural implication adds hypotheses with a comma, and the intuitionistic implication uses a semicolon. I've given the weakening and contraction explicitly in the final two rules, so we can reuse the hypothesis rule.
Adding lambda-terms to this calculus is pretty straightforward, too:
$$\begin{array}{ll} \frac{}{x:A \vdash x:A} & \\ \frac{\Gamma; x:A \vdash e:B} {\Gamma \vdash \fun{x}{e} : A \to B} & \frac{\Gamma \vdash e : A \to B \qquad \Gamma' \vdash e' : A} {\Gamma;\Gamma' \vdash e\;e' : B} \\ \frac{\Gamma, x:A \vdash e : B} {\Gamma \vdash A \lolli \hat{\lambda}x.e : B} & \frac{\Gamma \vdash e : A \lolli B \qquad \Gamma' e': \vdash A} {\Gamma,\Gamma' \vdash e\;e' : B} \\ \frac{\Gamma(\Delta) \vdash e : A} {\Gamma(\Delta;\Delta') \vdash e : A} & \frac{\Gamma(\Delta;\Delta') \vdash \rho(e) : A \qquad \Delta \equiv \rho \circ \Delta'} {\Gamma(\Delta) \vdash e : A} \end{array}$$
Unfortunately, typechecking these terms is a knotty little problem. The reason is basically that we want lambda terms to tell us what the derivation should be, without requiring us to do much search. But contraction can rename a host of variables at once, which means that there is a lot of search involved in typechecking these terms. (In fact, I personally don't know how to do it, though I expect that it's decidable, so somebody probably does.) What would be really nice is a calculus which is saturated with respect to weakening, so that the computational content of the weakening lemma is just the identity on derivation trees, and for which the computational content of contraction is an \emph{easy} renaming of a single obvious variable.
Applying the usual PL methodology of redefining the problem to one that can be solved, we can give an alternate type theory for BI in the following way. First, we'll define nested contexts so that they make the alternation of spatial and intuitionistic parts explicit:
$$\begin{array}{lcl} \Gamma & ::= & \cdot \bnfalt \Gamma; x:A \bnfalt r[\Delta] \\ \Delta & ::= & \cdot \bnfalt \Delta, x:A \bnfalt r[\Gamma] \\ & & \\ e & ::= & x \bnfalt \fun{x}{e} \bnfalt \hat{\lambda}x.\;e \bnfalt e\;e' \bnfalt r[e] \bnfalt \rho r.\;e \\ \end{array}$$
The key idea is to make the alternation of the spatial parts syntactic, and then to name each level shift with a variable $r$. So $x$ are ordinary variables, and $r$ are variables naming nested contexts. Then we'll add syntax $r[e]$ and $\rho r.e$ to explicitly annotate the level shifts:
$$\begin{array}{ll} \frac{}{\Gamma; x:A \vdash x:A} & \frac{}{\cdot,x:A \vdash x:A} \\ \frac{\Gamma; x:A \vdash e:B} {\Gamma \vdash \fun{x}{e} : A \to B} & \frac{\Gamma \vdash e : A \to B \qquad \Gamma \vdash e' : A} {\Gamma \vdash e\;e' : B} \\ \frac{\Delta, x:A \vdash e : B} {\Delta \vdash A \lolli \hat{\lambda}x.e : B} & \frac{\Delta \vdash e : A \lolli B \qquad \Delta' e': \vdash A} {\Delta,\Delta' \vdash e\;e' : B} \\ \frac{r[\Gamma] \vdash e : A} {\Gamma \vdash \rho r.\;e : A} & \frac{r[\Delta] \vdash e : A} {\Delta \vdash \rho r.\;e : A} \\ \frac{\Gamma \vdash e : A} {\cdot, r[\Gamma] \vdash r[e] : A} & \frac{\Delta \vdash e : A} {\Gamma; r[\Delta] \vdash r[e] : A} \end{array}$$
It's fairly straightforward to prove that contraction and weakening are admissible, and doing a contraction is now pretty easy, since you just have to rename some occurences of $r$ to two different variables, but may otherwise leave the branching contexts to be the same.
It's obvious that you can take any derivation in this calculus and erase the $r$-variables to get a well-typed term of the $\alpha\lambda$-calculus. It's only a bit more work to go the other way: given an $\alpha\lambda$-derivation, you prove that given any new context which erases to the $\alpha\lambda$-context, you can find a new derivation proving the same thign. Then there's an obvious proof that you can indeed find a new-context which erases properly.
One interesting feature of this calculus is that the $r$-variables resemble regions in region calculi. The connection is not entirely clear to me, since my variables don't show up in the types. This reminds me a little of the contextual modal type theory of Nanevski, Pfenning and Pientka. It reminds me even more of Bob Atkey's PhD thesis on generalizations of BI to allow arbitrary graph-structured sharing. But all of these connections are still speculative.
There is one remaining feature of these rules which is still non-algorithmic: as in linear logic, the context splitting in the spatial rules with multiple premises (for example, the spatial application rule) just assumes that you know how to divide the context into two pieces. I think the standard state-passing trick should still work, and it may even be nicer than the additives of linear logic. But I haven't worked out the details yet.
## Tuesday, July 26, 2011
### Functional Programming as a Particular Use of Modules
Bob Harper sometimes gets grumbly when people say that ML is an impure language, even though he knows exactly what they mean (and, indeed, agrees with it), because this way of phrasing things does not pay data abstraction its full due.
Data abstraction lets us write verifiably pure programs in ML. By "verifiably pure", I mean that we can use the type system to guarantee that our functions are pure and total, even though ML's native function space contains all sorts of crazy effects involving higher-order state, control, IO, and concurrency. (Indeed, ML functions can even spend your money to rent servers: now that's a side effect!) Given that the ML function space contains these astonishing prodigies and freaks of nature, how can we control them? The answer is data abstraction: we can define new types of functions which only contains well-behaved functions, and ensure through type abstraction that the only ways to form elements of this new type preserve well-behavedness.
Indeed, we will not just define a type of pure functions, but give an interface containing all the type constructors of the guarded recursion calculus I have described in the last few posts. The basic idea is to give an ML module signature containing:
• One ML type constructor for each type constructor of the guarded recursion calculus.
• A type constructor for the hom-sets of the categorical interpretation of this calculus
• One function in the interface for each categorical combinator, such as identity, composition, and each of the natural transformations corresponding to the universal properties of the functors interpreting the type constructors.
That's a mouthful, but it is much easier to understand by looking at the following (slightly pretty-printed) Ocaml module signature:
module type GUARDED =sig type one type α × β type α ⇒ β type α stream type num type •α type (α, β) hom val id : (α, α) hom val compose : (α, β) hom -> (β, γ) hom -> (α, γ) hom val one : (α, one) hom val fst : (α × β, α) hom val snd : (α × β, β) hom val pair : (α, β) hom -> (α, γ) hom -> (α, β × γ) hom val curry : (α × β,γ) hom -> (α,(β,γ) exp) hom val eval : ((α ⇒ β, α) times, β) hom val head : (α stream, α) hom val tail : (α stream, •(α stream)) hom val cons : (α × •(α stream), α stream) hom val zero : (one,num) hom val succ : (num, num) hom val plus : (num × num, num) hom val prod : (num × num, num) hom val delay : (α, •α) hom val next : (α,β) hom -> (•α, •β) hom val zip : (•α × •β, •(α × β)) hom val unzip : (•(α × β), •α × •β) hom val fix : (•α ⇒ α, α) hom val run : (one, num stream) hom -> (unit -> int)end
As you can see, we introduce abstract types corresponding to each of our calculus's type constructors. So α × β and α ⇒ β are not ML pairs and functions, but rather is the type of products and functions of our calculus. This is really the key idea -- since ML functions have too much stuff in them, we'll define a new type of pure functions. I replaced the "natural" numbers of the previous posts with a stream type, corresponding to our LICS 2011 paper, since they are really a kind of lazy conatural, and not the true inductive type of natural numbers. The calculus guarantees definitions are productive, but it's kind of weird in ML to see something called nat which isn't. So I replaced it with streams, which are supposed to yield an unbounded number of elements. (For true naturals, you'll have to wait for my post on Mendler iteration, which is a delightful application of parametricity.)
The run function takes a num stream, and gives you back an imperative function that successively enumerates the elements of the stream. This is the basic trick for making streams fit nicely into an event loop a la FRP.
However, we can implement these functions and types using traditional ML types:
module Guarded : GUARDED =struct type one = unit type α × β = α * β type α ⇒ β = α -> β type •α = unit -> α type α stream = Stream of α * α stream delay type num = int type (α, β) hom = α -> β let id x = x let compose f g a = g (f a) let one a = () let fst (a,b) = a let snd (a,b) = b let pair f g a = (f a, g a) let curry f a = fun b -> f(a,b) let eval (f,a) = f a let head (Stream(a, as')) = a let tail (Stream(a, as')) = as' let cons (a,as') = Stream(a,as') let zero () = 0 let succ n = n + 1 let plus (n,m) = n + m let prod (n,m) = n * m let delay v = fun () -> v let next f a' = fun () -> f(a'()) let zip (a',b') = fun () -> (a'(), b'()) let unzip ab' = (fun () -> fst(ab'())), (fun () -> snd(ab'())) let rec fix f = f (fun () -> fix f) let run h = let r = ref (h()) in fun () -> let Stream(x, xs') = !r in let () = r := xs'() in xend
Here, we're basically just implementing the operations of the guarded recursion calculus using the facilities offered by ML. So our guarded functions are just plain old ML functions, which happen to live at a type in which they cannot be used to misbehave!
This is the sense in which data abstraction lets us have our cake (effects!) and eat it too (purity!).
Note that when we want to turn a guarded lambda-term into an ML term, we can basically follow the categorical semantics to tell us what to write. Even though typechecking will catch all misuses of this DSL, actually using it is honestly not that much fun (unless you're a Forth programmer), since even even small terms turn into horrendous combinator expressions -- but in another post I'll show how we can write a CamlP4 macro/mini-compiler to embed this language into OCaml. This macro will turn out to involve some nice proof theory there, just as this ML implementation shows off how to use the denotational semantics in our ML programming.
## Wednesday, July 20, 2011
### Termination of guarded recursion
I will now give a termination proof for the guarded recursion calculus I sketched in the last two posts. This post got delayed because I tried to oversimplify the proof, and that didn't work -- I actually had to go back and look at Nakano's original proof to figure out where I was going wrong. It turns out the proof is still quite simple, but there's one really devious subtlety in it.
First, we recall the types, syntax and values.
A ::= N | A → B | •Ae ::= z | s(e) | case(e, z → e₀, s(x) → e₁) | λx.e | e e | •e | let •x = e in e | μx.e | xv ::= z | s(e) | λx.e | •e
The typing rules are in an earlier post, and I give some big-step evaluation rules at the end of this post. Now, the question is, given · ⊢ e : A[n], can we show that e ↝ v?
To do this, we'll turn to our old friend, Mrs. step-indexed logical relation. This is a Kripke logical relation in which the Kripke worlds are given by the natural numbers and the accessibility relation is given by ≤. So, we define a family of predicates on closed values indexed by type, and by a Kripke world (i.e., a natural number n).
V(•A)ⁿ = {•e | ∀j<n. e ∈ E(A)ʲ}V(A ⇒ B)ⁿ = {λx.e | ∀j≤n, v ∈ V(A)ʲ. [v/x]e ∈ E(B)ʲ}V(N)ⁿ = {z} ∪ { s(e) | ∀j<n. e ∈ E(N)ʲ}E(A)ʲ = {e | ∃v. e ↝ v and v ∈ V(A)ʲ}
This follows the usual pattern of logical relations, where we give a relation defining values mutually recursively with a relation defining well-behaved expressions (i.e., expressions are ok if they terminate and reduce to a value in the relation at that type).
Note that as we expect, j ≤ n implies V(A)ⁿ ⊆ V(A)ʲ. (The apparent antitonicity comes from the fact that if v is in the n-relation, it's also in the j relation.) One critical feature of this definition is that at n = 0, the condition on V(•A)⁰ always holds, because of the strict less-than in the definition.
The fun happens in the interpretation of contexts:
Ctxⁿ(· :: j) = ()Ctxⁿ(Γ,x:A[i] :: j) = {(γ,[v/x]) | γ ∈ Ctxⁿ(Γ) and v ∈ Vⁿ(A)} when i ≤ jCtxⁿ(Γ,x:A[j+l] :: j) = {(γ,[e/x]) | γ ∈ Ctxⁿ(Γ) and •ˡe ∈ Vⁿ(•ˡA)} when l > 0
The context interpretation has a strange dual nature. At times less than or equal to j, it is a familiar context of values. But at future times, it is a context of expressions. This is because the evaluation rules substitute values for variables at the current time, and expressions for variables at future times. We abuse the bullet value relation in the third clause, to more closely follow Nakano's proof.
On the one hand, the fixed point operator is μx.e at any type A, and unfolding this fixed point has to substitute an expression (the mu-term itself) for the variable x. So the fixed point rule tells us that there is something necessarily lazy going on.
One the other hand, the focusing behavior of this connective is quite bizarre. It is not apparently positive or negative, since it distribute neither through all positives (eg, •(A + B) ≄ •A + •B) nor is it the case that it distributes through all negatives (eg, •(A → B) ≄ •A → •B). (See Noam Zeilberger, The Logical Basis of Evaluation Order and Pattern Matching.)
I take this to mean that •A should probably be decomposed further. I have no present idea of how to do it, though.
Anyway, this is enough to let you prove the fundamental property of logical relations:
Theorem (Fundamental Property). If Γ ⊢ e : A[j], then for all n and γ ∈ Ctxⁿ(Γ :: j), we have that γ(e) ∈ Eⁿ(A).
The proof of this is a straightforward induction on typing derivations, with one nested induction at the fixed point rule. I'll sketch that case of the proof here, assuming an empty context Γ just to reduce the notation:
Case: · ⊢ μx.e : A[j]By inversion: x:A[j+1] ⊢ e : A[j]By induction, for all n,e'. if •e' ∈ Vⁿ(•A) then [e'/x]e ∈ Eⁿ(A)By nested induction on n, we'll show that [μx.e/x]e ∈ Eⁿ(A)Subcase n = 0: We know if •μx.e ∈ V⁰(•A) then [μx.e/x]e ∈ E⁰(A) We know •μx.e ∈ V⁰(•A) is true, since · ⊢ μx.e : A[j] Hence [μx.e/x]e ∈ E⁰(A) Hence μx.e ∈ E⁰(A)Subcase n = x+1: We know if •μx.e ∈ Vˣ⁺¹(•A) then [μx.e/x]e ∈ Eˣ⁺¹(A) By induction, we know [μx.e/x]e ∈ Eˣ(A) Hence μx.e ∈ Eˣ(A) Hence •μx.e ∈ Vˣ⁺¹(A) So [μx.e/x]e ∈ Eˣ⁺¹(A) Hence μx.e ∈ Eˣ⁺¹(A)
Once we have the fundamental property of logical relations, the normalization theorem follows immediately.
Corollary (Termination). If · ⊢ e : A[n], then ∃v. e ↝ v.
Evaluation rules:
e₁ ↝ (λx.e) e₂ ↝ v [v/x]e ↝ v' —————— ———————————————————————————————— v ↝ v e₁ e₂ ↝ v' e ↝ z e₀ ↝ v e ↝ s(e) [e/x]e₁ ↝ v —————————————————————————————— —————————————————————————————————case(e, z → e₀, s(x) → e₁) ↝ v case(s(e), z → e₀, s(x) → e₁) ↝ ve₁ ↝ •e [e/x]e' ↝ v [μx.e/x]e ↝ v ————————————————————— ——————————————let •x = e₁ in e₂ ↝ v μx.e ↝ v
## Friday, July 15, 2011
### Semantics of a weak delay modality
In my previous post, I sketched some typing rules for a guarded recursion calculus. Now I'll give its categorical semantics. So, suppose we have a Cartesian closed category with a delay functor and the functorial action and natural transformations:
•(f : A → B) : •A → •Bδ : A → •Aι : •A × •B → •(A × B)ι⁻¹ : •(A × B) → •A × •Bfix : (•A ⇒ A) → A
I intend that the next modality is a Cartesian functor (ie, distributes through products) and furthermore we have a delay operator δ. We also have a fixed point operator for the language. However, I don't assume that the delay distributes through the exponential. Now, we can follow the usual pattern of categorical logic, and interpret contexts and types as objects, and terms as morphisms. So types are interpreted as follows:
〚A → B〛 = 〚A〛 ⇒ 〚B〛〚•A〛 = •〚A〛
Note that we haven't done anything with time indices yet. They will start to appear with the interpretation of contexts, which is relativized by time:
〚·〛ⁿ = 1〚Γ, x:A[j]〛ⁿ = 〚Γ〛 × •⁽ʲ⁻ⁿ⁾〚A〛 if j > n〚Γ, x:A[j]〛ⁿ = 〚Γ〛 × 〚A〛 if j ≤ n
The idea is that we interpret a context at time n, and so all the indices are interpreted relative to that. If the index j is bigger than n, then we delay the hypothesis, and otherwise we don't. Then we can interpret morphisms at time n as 〚Γ ⊢ e : A[n]〛 ∈ 〚Γ〛ⁿ → 〚A〛, which we give below:
〚Γ ⊢ e : A[n]〛 ∈ 〚Γ〛ⁿ → 〚A〛〚Γ ⊢ μx.e : A[n]〛 = fix ○ λ(〚Γ, x:A[n+1] ⊢ e : A[n]〛)〚Γ ⊢ x : A[n]〛 = π(x) (where x:A[j] ∈ Γ)〚Γ ⊢ λx.e : A → B[n]〛 = λ(〚Γ, x:A[n] ⊢ e : B[n]〛)〚Γ ⊢ e e' : B[n]〛 = eval ○ ⟨〚Γ ⊢ e : A → B[n]〛, 〚Γ ⊢ e' : B[n]〛⟩〚Γ ⊢ •e : •A[n]〛 = •〚Γ ⊢ e : A[n+1]〛 ○ Nextⁿ(Γ)〚Γ ⊢ let •x = e in e' : •B[n]〛 = 〚Γ, x:A[n+1] ⊢ e' : B[n]〛 ○ ⟨id(Γ), 〚Γ ⊢ e : •A[n]〛⟩
Most of these rules are standard, with the exception of the introduction rule for delays. We interpret the body •e at time n+1, and then use the functorial action to get an element of type •A. This means we need to take a context at time n and produce a delayed one interpreted at time n+1.
Nextⁿ(Γ) ∈ 〚Γ〛ⁿ → •〚Γ〛ⁿ⁺¹Nextⁿ(·) = δ₁Nextⁿ(Γ, x:A[j]) = ι ○ (Nextⁿ(Γ) × δʲ⁻ⁿ) if j > nNextⁿ(Γ, x:A[j]) = ι ○ (Nextⁿ(Γ) × δ) if j ≤ n
I think this is a pretty slick way of interpreting hybrid annotations, and a trick worth remembering for other type constructors that don't necessarily play nicely with implications.
Next up, if I find a proof small enough to blog, is a cut-elimination/normalization proof for this calculus.
## Wednesday, July 13, 2011
### Guarded recursion with a weaker-than-Nakano guard modality
We have a new draft paper up, on controlling the memory usage of FRP. I have to say that I really enjoy this line of work: there's a very strong interplay between theory and practice. For example, this paper --- which is chock full of type theory and denotational semantics --- is strongly motivated by questions that arose from thinking about how to make our implementation efficient.
In this post, I'm going to start spinning out some consequences of one minor point of our current draft, which we do not draw much attention to. (It's not really the point of the paper, and isn't really enough to be a paper on its own -- which makes it perfect for research blogging.) Namely, we have a new delay modality, which substantially differs from the original Nakano proposed in his LICS 2000 paper.
Recall that the delay modality $\bullet A$ is a type constructor for guarded recursion. I'll start by giving a small type theory for guarded recursion below.
$$\begin{array}{lcl} A & ::= & A \to B \;\;|\;\; \bullet A \;\;|\;\; \mathbb{N} \\ \Gamma & ::= & \cdot \;\;|\;\; \Gamma, x:A[i] \end{array}$$
As can be seen above, the types are delay types, functions, and natural numbers, and contexts come indexed with time indices $i$. So is the typing judgement $\Gamma \vdash e : A[i]$.
$$\begin{array}{ll} \frac{x:A[i] \in \Gamma \qquad i \leq j} {\Gamma \vdash x : A[j]} & \frac{\begin{array}{l} \Gamma \vdash e:A[i] & \Gamma,x:A[i] \vdash e' : B[i] \end{array}} {\Gamma \vdash \mathsf{let}\; x = e \;\mathsf{in}\; e' : B[i]} \\ & \\ \frac{\Gamma, x:A[i] \vdash e : B[i]} {\Gamma \vdash \lambda x.\;e : A \to B} & \frac{\begin{array}{l} \Gamma \vdash e : A \to B [i] & \Gamma \vdash e' : A[i] \end{array}} {\Gamma \vdash e \; e' : B[i]} \\ & \\ \frac{\Gamma \vdash e : A[i+1]} {\Gamma \vdash \bullet e : A[i]} & \frac{\begin{array}{ll} \Gamma \vdash e : \bullet A[i] & \Gamma, x:A[i+1] \vdash e' : B[i] \end{array}} {\Gamma \vdash \mathsf{let}\; \bullet x = e \;\mathsf{in}\; e' : B[i]} \\ & \\ \frac{} {\Gamma \vdash \mathsf{z} : \mathbb{N}[i]} & \frac{\Gamma \vdash e : \mathbb{N}[i+1]} {\Gamma \vdash \mathsf{s}(e) : \mathbb{N}[i]} \\ & \\ \frac{\Gamma, x:A[i+1] \vdash e : A[i]} {\Gamma \vdash \mu x.\;e : A[i]} & \frac{\begin{array}{l} \Gamma \vdash e : \mathbb{N}[i] \\ \Gamma \vdash e_1 : A[i] \\ \Gamma, x:\mathbb{N}[i+1] \vdash e_2 : A[i] \end{array}} {\Gamma \vdash \mathsf{case}(e, \mathsf{z} \to e_1, \mathsf{s}(x) \to e_2) : A[i]} \end{array}$$
The $i+1$ in the successor rule for natural numbers pairs with the rule for case statements, and this is what allows the fixed point rule to do its magic. Fixed points are only well-typed if the recursion variable occurs at a later time, and the case statement for numbers gives the variable one step later. So by typing we guarantee well-founded recursion!
The intro and elim rules for delays internalize increasing the time index, so that an intro for $\bullet A$ at time $i$ takes an expression of type $A$ at time $i+1$. We have a let-binding elimination form for delays, which differs from our earlier LICS paper, which had a direct-style elimination for delays. The new elimination is weaker than the old one, in that it cannot prove the isomorphism of $\bullet(A \to B) \simeq \bullet A \to \bullet B$.
This is really quite great, since that isomorphism was really hard to implement! (It was maybe half the complexity of the logical relation.) The only question is whether or not we can still give a categorical semantics to this language. In fact, we can, and I'll
describe it in my next post.
## Friday, June 17, 2011
The constructive lift monad is the coinductive type $T(A) \equiv \nu\alpha.\; A + \alpha.$
The intuition is that an element of this type either tells you a value of type $A$, or tells you to compute some more and try again. Nontermination is modeled by the element which never returns a value, and always keeps telling you to compute some more.
Our goal is to construct a general fixed-point combinator $\mu(f : TA \to TA) : 1 \to TA$, which takes an $f$ and then produces a computation corresponding to the fixed point of $f$. To fix notation, we'll take the constructors to be:
$$\begin{array}{lcl} \mathsf{roll} & : & A + TA \to TA \\ \mathsf{unroll} & : & TA \to A + TA \end{array}$$
Since this is a coinductive type, we also have an unfold satisfying the following equation:
$$\mathsf{unfold}(f : X \to A + X) : X \to TA \equiv \mathsf{roll} \circ (\mathit{id} + (\mathsf{unfold}\; f)) \circ f$$
First, we will explicitly construct the bottom element, corresponding to the computation that runs forever, with the following definition:
$$\bot : 1 \to TA \equiv \mathsf{unfold}(\mathsf{inr})$$
This definition just keeps telling us to wait, over and over again. Now, we can define the fixed point operator:
$$\begin{array}{l} \mu(f : TA \to TA) : 1 \to TA \\ \mu(f) \equiv (\mathsf{unfold} (\mathsf{unroll} \circ f)) \circ \bot \end{array}$$
What this does is to pass bottom to $f$. If $f$ returns a value, then we're done. Otherwise, $f$ returns us another thunk, which we can pass back to $f$ again, and repeat.
Of course, this is exactly the intuition behind fixed points in domain theory. Lifting in domains is usually defined directly, and I don't know who invented the idea of defining it as a coinductive type. I do recall a 1999 paper by Martin Escardo which uses (a slight variant of) it, and he refers to it as a well-known construction in metric spaces, so probably the papers from the Dutch school of semantics are a good place to start the search. This construction has seen a renewed burst of interest in the last few years, since it offers a relatively convenient way to represent nonterminating functions in dependent type theory. It's also closely related to step-indexed models in operational semantics, since, given a terminating element of the lift monad, you can compute how many steps it takes to return a value.
It's stuff like this that makes me say I can't tell the difference between operational and denotational semantics any more.
## Tuesday, May 31, 2011
### ICFP 2011
I was just notified that the paper I wrote with Nick Benton, A Semantic Model of Graphical User Interfaces, was accepted to ICFP 2011!
This is my first paper at ICFP, so I'm quite excited. Also, I accidentally misread the submission guidelines and wrote a 10-page draft, instead of a 12-page draft. As a result, I have two whole pages to add examples, intuition, and motivation. This feels positively decadent. :)
## Thursday, May 5, 2011
### GUI Programming and MVC
This post was inspired by this post on William Cook's blog.
I've been thinking a lot about GUIs lately, and I think model-view-controller is a very good idea -- but one that existing languages do a poor job supporting. Smalltalk deserves credit for making these thoughts possible for Trygve Reenskaug to express, but it's been over 30 years; we really ought to have made it easy to express by now.
Now, the point of a controller is to take low-level events and turn them into high-level events. Note that "low-level" and "high-level" ought to be relative notions -- the toolkit might give us mouse and keyboard events, from which we write button widgets to turn them into click events, from which we write calculator button turns into application events (numbers and operations).
However, in reality new widgets tend to be extremely painful to write, since it usually involves mucking around with the guts of the event loop, and so people tend to stick with what the toolkit gives them. This way, they don't need to understand the deep implementation invariants of the GUI toolkit. As a result, they don't really build GUI abstractions, and so they don't need separate controllers. (A good example would be to think about how hard it would be to write a new text entry widget in your toolkit of choice.)
So I see the move from MVC to MV as a symptom of a problem. The C is there to let you build abstractions, but since it's really hard we don't. As a result new frameworks drop the C, and just grow by accretion -- the toolkit designers add some new widgets with each release, and everybody just uses them. I find this a little bit sad, honestly.
As you might guess from this blog, I find this quite an interesting problem. I think functional reactive programming offers a relatively convenient model (stream transducers) to write event processors with, but we have the problem that FRP systems tend towards the unusably inefficient. In some sense this is the opposite problem of MVC, which can be rather efficient, but can require very involved reasoning to get correct.
This basically leads to my current research program: can we compile functional reactive programming into model-view-controller code? Then you can combine the ease-of-use of FRP, with the relative efficiency[*] of the MVC design.
IMO, the system in our LICS paper is a good first step towards fixing this problem, but only a first step. It's quite efficient for many programs, but it's a bit too expressive: it's possible to write programs which leak rather a lot of memory without realizing it. Basically, the problem is that promoting streams across time requires buffering them, and it's possible to accidentally write programs which repeatedly buffer a stream at each tick, leading to unbounded memory use.
[*] As usual, "efficiency" is relative to the program architecture. MVC is a retained mode design, and if the UI is constantly changing a lot, you lose. For things like games, immediate mode GUIs seem like a better design to me. In the longer term, I'd like to try synthesizing these two designs, For example, even a live-document app like a spreadsheet or web page (the ideal cases for MVC) may want to embed video or 3D animations (which work better in immediate mode).
## Tuesday, April 19, 2011
### Models of Linear Logic: Length Spaces
One of the things I've wondered about is what the folk model of linear logic is. Basically, I've noticed that people use linear logic an awful lot, and they have very clear intuitions about the operational behavior they expect, but they are often not clear about what equational properties they expect to hold are.
Since asking that question, I've learned about a few models of linear logic, which are probably worth blogging about. Morally, I ought to start by showing how linear algebra is a model of linear logic, but I'll set that aside for now, and start with some examples that will be more familiar to computer scientists.
First, recall that the general model of simply-typed functional languages are cartesian closed categories, and that the naive model of functional programming is "sets and functions". That is, types are interpreted as sets, and terms are functions between sets, with the cartesian closed structure corresponding to function spaces on sets.
Next, recall that the general model of intuitionistic linear logic are symmetric monoidal closed categories. So the question to ask is, what are some natural symmetric monoidal closed category given by sets-with-structure and functions preserving that structure? There are actually many answers to this question, but one I like a lot I learned from a 1999 LICS paper by Martin Hofmann, and is called "length spaces". It is very simple, and I find it quite beautiful.
To understand the construction, let's first set out the problem: how can we write programs that respect some resource bounds? For example, we may wish to bound the memory usage of a language, so that every definable program is size-bounded: the output of a computation is no bigger than its input.
Now, let's proceed in a really simple-minded way. A length space is a pair $(A, \sigma_A)$, where $A$ is a set and $\sigma_A : A \to \N$ is a size function. Think of $A$ as the set of values in a type, and $\sigma_A$ as a function which tells you how big each element is. These pairs will be the objects of our category of length spaces.
Now, define a morphism $f : (A, \sigma) \to (B, \tau)$ to be a function on sets $A \to B$, with the property that $\forall a \in A.\; \tau(f\;a) \leq \sigma(a)$. That is, the size of the output is always less then or equal to the size of the input.
Here's the punchline: length spaces are a symmetric monoidal closed category.
Given two length spaces $(A, \sigma)$ and $(B, \tau)$, they have a monoidal product $(A, \sigma) \otimes (B, \tau) = (A \times B, \fun{(a,b)}{\sigma(a) + \tau(b)})$. The unit is $I = (\{*\}, \fun{*}{0})$, and the associators and unitors are inherited from the cartesian product. The intuition here is that strict pair, requires resources equal to the sum of the resources of the components.
The function space construction is pretty clever, too. The exponential $(B, \sigma) \multimap (C, \tau) = (B \to C, \fun{f}{\min\comprehend{a \in \N}{\forall b \in B.\;\tau(f\;b) \leq \sigma(b) + a}})$. The intuition here is that if you have a morphism $A \otimes B \to C$, when we curry it we want a morphism $A \to (B \multimap C)$. However, the returned function may have free variables in $A$, so the size of the value it can return should be bounded by the size of its input $b \in B$ plus the size of its environment.
We also have a cartesian product, corresponding to lazy pairs. We define $(A, \sigma) \times (B, \tau) = (A \times B, \fun{(a,b)}{\max(\sigma(a), \tau(b)})$. The intuition is that with a lazy pair, where we have the (linear) choice of either the first or second component, we can bound the space usage by maximum of the two alternatives.
Note how pretty this model is -- you can think about functions in a totally naive way, and that's fine. We just carry around a little extra data to model size information, and now we have a very pretty extensional model of an intensional property -- space usage. This also illustrates the general principle that increasing the observations you can do creates finer type distinctions.
## Friday, March 25, 2011
### A Semantic Model for GUI Programs
Our paper on ultrametric semantics will appear at LICS 2011!
We've got a new draft, too, on applying this to GUI programming. This turns out to be not entirely straightforward, since GUI widgets accept user input, and so the semantics of a GUI has to model that input. The usual way of modelling input is via a reader monad -- that is, we model computations as functions which accept input as an argument -- but in a GUI you can (for instance) write programs which dynamically create new buttons, and so create new input channels.
The thing we did -- which seems obvious to me, but which Nick says is unusual, and for which I would like references -- was to view things like buttons as giving you a set of values. Then you can say that creating a button is an operation in the nondeterminism monad (basically). Then you don't need to model state at all.
Since this is notionally a research blog, let me point out a bit of mathematics I really liked in this paper (not actually due to me). Since we were working in ultrametric spaces, we couldn't use powersets, obviously -- we needed power spaces. It turns out that the right notion of powerspace for a space $X$ is the collection of closed subsets of $X$, under the Hausdorff metric. Then when $X$ is a complete 1-bounded ultrametric space, then $\mathcal{P}(X)$ will be too.
This is all totally standard, but one thing that wasn't obvious to me is whether powerspace actually forms a monad! The multiplication for powerset (in sets) is $\mu : \mathcal{P}^2(A) \to \mathcal{P}(A) = \fun{U}{\{a \in A \;|\; \exists X \in U.\; a \in X\}}$ -- that is, $\mu(U) = \cup U$.
Since only finite unions of closed sets are closed, for powerspace the corresponding multiplication ought to be $\mu(U) = \mathrm{cl}(\cup U)$. However, it wasn't obvious to me whether this was a nonexpansive map or not. I learned the following super-slick proof from a paper of Steve Vickers, "Localic Completion of Generalized Metric Spaces II: Powerlocales". Despite the formidable title, it is a very readable paper.
Below is just the fragment of the proof for the lower half of the Hausdorff metric.
$$\begin{array}{lcl} d(U, V) \leq q & \iff & \forall X \in U, \exists Y \in V.\; d(X, Y) < q \\ & \iff & \forall X \in U, \exists Y \in V.\; \forall x \in X, \exists y \in Y.\; d(x,y) \leq q \\ & \Longrightarrow & \forall X \in U, \forall x \in X.\; \exists Y \in V, \exists y \in Y.\; d(x,y) \leq q \\ & \iff & \forall x \in \cup X, \exists y \in \cup V, d(x,y) \leq q \\ & \iff & d(\cup U, \cup V) \leq q \\ & \iff & d(\mathrm{cl}(\cup U), \mathrm{cl}(\cup V)) \leq q \\ & \iff & d(\mu(U), \mu(V)) \leq q \\ \end{array}$$
That quantifier flip is clever, but what makes this proof beautiful to me is introducing the auxilliary $q$. My own attempts just drowned in quantifier alternations, but this splits things up at just the right point. I'm reminded of Dijkstra's observation that most mathematicians do not make enough use of logical equivalence in their proofs, and clearly I fall into this category.
## Friday, January 14, 2011
### Ultrametric Semantics of Reactive Programs
I've got a new draft paper out with Nick Benton, Ultrametric Semantics of Reactive Programs.
We answer some longstanding open questions in FRP, namely:
1. What are higher-order reactive functions at all? How do you interpret them, and what are their semantics?
2. How can we implement FRP in a reasonable way, without compromising on either the equational theory or the ability to fit into the mutable callback event loop world of MVC?
It's got a little something for everyone -- some very pretty proof theory, some nice denotational semantics, and a hardcore separation logic proof with a step-indexed logical relation.
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http://bdlisle.com/wp5okk/water-wave-symbol-text-9b0fdb
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Download this Premium Vector about Water wave symbol and icon logo template, and discover more than 10 Million Professional Graphic Resources on Freepik
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https://en.wikipedia.org/wiki/Enzyme_inhibitor
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# Enzyme inhibitor
A figure comparing the three types of enzyme inhibitors and how they work in regards to substrate binding sites and inhibitors binding sites.
An enzyme binding site that would normally bind substrate can alternatively bind a competitive inhibitor, preventing substrate access. Dihydrofolate reductase is inhibited by methotrexate which prevents binding of its substrate, folic acid. Binding site in blue, inhibitor in green, and substrate in black. ()
An enzyme inhibitor is a molecule that binds to an enzyme and decreases its activity. By binding to enzymes' active sites, inhibitors reduce the compatibility of substrate and enzyme and this leads to the inhibition of Enzyme-Substrate complexes' formation, preventing the catalysis of reactions and decreasing (at times to zero) the amount of product produced by a reaction. It can be said that as the concentration of enzyme inhibitors increases, the rate of enzyme activity decreases, and thus, the amount of product produced is inversely proportional to the concentration of inhibitor molecules. Since blocking an enzyme's activity can kill a pathogen or correct a metabolic imbalance, many drugs are enzyme inhibitors. They are also used in pesticides. Not all molecules that bind to enzymes are inhibitors; enzyme activators bind to enzymes and increase their enzymatic activity, while enzyme substrates bind and are converted to products in the normal catalytic cycle of the enzyme.
The binding of an inhibitor can stop a substrate from entering the enzyme's active site and/or hinder the enzyme from catalyzing its reaction. Inhibitor binding is either reversible or irreversible. Irreversible inhibitors usually react with the enzyme and change it chemically (e.g. via covalent bond formation). These inhibitors modify key amino acid residues needed for enzymatic activity. In contrast, reversible inhibitors bind non-covalently and different types of inhibition are produced depending on whether these inhibitors bind to the enzyme, the enzyme-substrate complex, or both.
Many drug molecules are enzyme inhibitors, so their discovery and improvement is an active area of research in biochemistry and pharmacology.[1] A medicinal enzyme inhibitor is often judged by its specificity (its lack of binding to other proteins) and its potency (its dissociation constant, which indicates the concentration needed to inhibit the enzyme). A high specificity and potency ensure that a drug will have few side effects and thus low toxicity.
Enzyme inhibitors also occur naturally and are involved in the regulation of metabolism. For example, enzymes in a metabolic pathway can be inhibited by downstream products. This type of negative feedback slows the production line when products begin to build up and is an important way to maintain homeostasis in a cell. Other cellular enzyme inhibitors are proteins that specifically bind to and inhibit an enzyme target. This can help control enzymes that may be damaging to a cell, like proteases or nucleases. A well-characterised example of this is the ribonuclease inhibitor, which binds to ribonucleases in one of the tightest known protein–protein interactions.[2] Natural enzyme inhibitors can also be poisons and are used as defenses against predators or as ways of killing prey.
## Reversible inhibitors
### Types of reversible inhibitors
Reversible inhibitors attach to enzymes with non-covalent interactions such as hydrogen bonds, hydrophobic interactions and ionic bonds. Multiple weak bonds between the inhibitor and the active site combine to produce strong and specific binding. In contrast to substrates and irreversible inhibitors, reversible inhibitors generally do not undergo chemical reactions when bound to the enzyme and can be easily removed by dilution or dialysis.
There are four kinds of reversible enzyme inhibitors. They are classified according to the effect of varying the concentration of the enzyme's substrate on the inhibitor.[3][4][1]
Types of inhibition. This classification was introduced by W.W. Cleland.[5]
• In competitive inhibition, the substrate and inhibitor cannot bind to the enzyme at the same time, as shown in the figure on the right. This usually results from the inhibitor having an affinity for the active site of an enzyme where the substrate also binds; the substrate and inhibitor compete for access to the enzyme's active site. This type of inhibition can be overcome by sufficiently high concentrations of substrate (Vmax remains constant), i.e., by out-competing the inhibitor. However, the apparent Km will increase as it takes a higher concentration of the substrate to reach the Km point, or half the Vmax. Competitive inhibitors are often similar in structure to the real substrate (see examples below).
• In uncompetitive inhibition, the inhibitor binds only to the substrate-enzyme complex. This type of inhibition causes Vmax to decrease (maximum velocity decreases as a result of removing activated complex) and Km to decrease (due to better binding efficiency as a result of Le Chatelier's principle and the effective elimination of the ES complex thus decreasing the Km which indicates a higher binding affinity).
• In non-competitive inhibition, the binding of the inhibitor to the enzyme reduces its activity but does not affect the binding of substrate. As a result, the extent of inhibition depends only on the concentration of the inhibitor. Vmax will decrease due to the inability for the reaction to proceed as efficiently, but Km will remain the same as the actual binding of the substrate, by definition, will still function properly.
• In mixed inhibition, the inhibitor can bind to the enzyme at the same time as the enzyme's substrate. However, the binding of the inhibitor affects the binding of the substrate, and vice versa. This type of inhibition can be reduced, but not overcome by increasing concentrations of substrate. Although it is possible for mixed-type inhibitors to bind in the active site, this type of inhibition generally results from an allosteric effect where the inhibitor binds to a different site on an enzyme. Inhibitor binding to this allosteric site changes the conformation (i.e., tertiary structure or three-dimensional shape) of the enzyme so that the affinity of the substrate for the active site is reduced.
These types can also be distinguished by the effect of increasing the substrate concentration [S] on the degree of inhibition caused by a given amount of inhibitor. For competitive inhibition the degree of inhibition is reduced by increasing [S], for noncompetitive inhibition the degree of inhibition is unchanged, and for uncompetitive (also called anticompetitive) inhibition the degree of inhibition increases with [S].[6]
### Quantitative description of reversible inhibition
Reversible inhibition can be described quantitatively in terms of the inhibitor's binding to the enzyme and to the enzyme-substrate complex, and its effects on the kinetic constants of the enzyme. In the classic Michaelis-Menten scheme below, an enzyme (E) binds to its substrate (S) to form the enzyme–substrate complex ES. Upon catalysis, this complex breaks down to release product P and free enzyme. The inhibitor (I) can bind to either E or ES with the dissociation constants Ki or Ki', respectively.
Competitive inhibitors can bind to E, but not to ES. Competitive inhibition increases Km (i.e., the inhibitor interferes with substrate binding), but does not affect Vmax (the inhibitor does not hamper catalysis in ES because it cannot bind to ES). Uncompetitive inhibitors bind to ES. Uncompetitive inhibition decreases both Km' and 'Vmax. The inhibitor affects substrate binding by increasing the enzyme's affinity for the substrate (decreasing Km) as well as hampering catalysis (decreases Vmax). Non-competitive inhibitors have identical affinities for E and ES (Ki = Ki'). Non-competitive inhibition does not change Km (i.e., it does not affect substrate binding) but decreases Vmax (i.e., inhibitor binding hampers catalysis). Mixed-type inhibitors bind to both E and ES, but their affinities for these two forms of the enzyme are different (Ki ≠ Ki'). Thus, mixed-type inhibitors affect substrate binding (increase or decrease Km) and hamper catalysis in the ES complex (decrease Vmax). Kinetic scheme for reversible enzyme inhibitors
When an enzyme has multiple substrates, inhibitors can show different types of inhibition depending on which substrate is considered. This results from the active site containing two different binding sites within the active site, one for each substrate. For example, an inhibitor might compete with substrate A for the first binding site, but be a non-competitive inhibitor with respect to substrate B in the second binding site.[7]
### Measuring the dissociation constants of a reversible inhibitor
As noted above, an enzyme inhibitor is characterised by its two dissociation constants, Ki and Ki', to the enzyme and to the enzyme-substrate complex, respectively. The enzyme-inhibitor constant Ki can be measured directly by various methods; one extremely accurate method is isothermal titration calorimetry, in which the inhibitor is titrated into a solution of enzyme and the heat released or absorbed is measured.[8] However, the other dissociation constant Ki' is difficult to measure directly, since the enzyme-substrate complex is short-lived and undergoing a chemical reaction to form the product. Hence, Ki' is usually measured indirectly, by observing the enzyme activity under various substrate and inhibitor concentrations, and fitting the data[9] to a modified Michaelis–Menten equation
${\displaystyle V={\frac {V_{max}[S]}{\alpha K_{m}+\alpha ^{\prime }[S]}}={\frac {(1/\alpha ^{\prime })V_{max}[S]}{(\alpha /\alpha ^{\prime })K_{m}+[S]}}}$
where the modifying factors α and α' are defined by the inhibitor concentration and its two dissociation constants
${\displaystyle \alpha =1+{\frac {[I]}{K_{i}}}}$
${\displaystyle \alpha ^{\prime }=1+{\frac {[I]}{K_{i}^{\prime }}}.}$
Thus, in the presence of the inhibitor, the enzyme's effective Km and Vmax become (α/α')Km and (1/α')Vmax, respectively. However, the modified Michaelis-Menten equation assumes that binding of the inhibitor to the enzyme has reached equilibrium, which may be a very slow process for inhibitors with sub-nanomolar dissociation constants. In these cases, it is usually more practical to treat the tight-binding inhibitor as an irreversible inhibitor (see below); however, it can still be possible to estimate Ki' kinetically if Ki is measured independently.
The effects of different types of reversible enzyme inhibitors on enzymatic activity can be visualized using graphical representations of the Michaelis–Menten equation, such as Lineweaver–Burk plots, Eadie-Hofstee plots or Hanes-Woolf plots. For example, in the Lineweaver–Burk plots at the right, the competitive inhibition lines intersect on the y-axis, illustrating that such inhibitors do not affect Vmax. Similarly, the non-competitive inhibition lines intersect on the x-axis, showing these inhibitors do not affect Km. However, it can be difficult to estimate Ki and Ki' accurately from such plots,[10] so it is advisable to estimate these constants using more reliable nonlinear regression methods, as described above.
### Reversible inhibitors
Traditionally reversible enzyme inhibitors have been classified as competitive, uncompetitive, or non-competitive, according to their effects on Km and Vmax. These different effects result from the inhibitor binding to the enzyme E, to the enzyme–substrate complex ES, or to both, respectively. The division of these classes arises from a problem in their derivation and results in the need to use two different binding constants for one binding event. The binding of an inhibitor and its effect on the enzymatic activity are two distinctly different things, another problem the traditional equations fail to acknowledge. In noncompetitive inhibition the binding of the inhibitor results in 100% inhibition of the enzyme only, and fails to consider the possibility of anything in between.[11] The common form of the inhibitory term also obscures the relationship between the inhibitor binding to the enzyme and its relationship to any other binding term be it the Michaelis–Menten equation or a dose response curve associated with ligand receptor binding. To demonstrate the relationship the following rearrangement can be made:
{\displaystyle {\begin{aligned}{\cfrac {V_{\max }}{1+{\cfrac {\ce {[I]}}{K_{i}}}}}&={V_{\max }}\left({\cfrac {K_{i}}{K_{i}+[{\ce {I}}]}}\right)&&{\text{multiply by }}{\cfrac {K_{i}}{K_{i}}}=1\\&={V_{\max }}\left({\cfrac {K_{i}+[{\ce {I}}]-[{\ce {I}}]}{K_{i}+[{\ce {I}}]}}\right)&&{\text{add }}[{\ce {I}}]-[{\ce {I}}]=0{\text{ to numerator}}\\&={V_{\max }}\left(1-{\cfrac {[{\ce {I}}]}{K_{i}+[{\ce {I}}]}}\right)&&{\text{simplify }}{\cfrac {K_{i}+[{\ce {I}}]}{K_{i}+[{\ce {I}}]}}=1\\&=V_{\max }-V_{\max }{\cfrac {\ce {[I]}}{K_{i}+[{\ce {I}}]}}&&{\text{multiply out by }}V_{\max }\end{aligned}}}
This rearrangement demonstrates that similar to the Michaelis–Menten equation, the maximal rate of reaction depends on the proportion of the enzyme population interacting with its substrate.
fraction of the enzyme population bound by substrate
${\displaystyle {\cfrac {\ce {[S]}}{[{\ce {S}}]+K_{m}}}}$
fraction of the enzyme population bound by inhibitor
${\displaystyle {\cfrac {\ce {[I]}}{[{\ce {I}}]+K_{i}}}}$
the effect of the inhibitor is a result of the percent of the enzyme population interacting with inhibitor. The only problem with this equation in its present form is that it assumes absolute inhibition of the enzyme with inhibitor binding, when in fact there can be a wide range of effects anywhere from 100% inhibition of substrate turn over to just >0%. To account for this the equation can be easily modified to allow for different degrees of inhibition by including a delta Vmax term.
${\displaystyle V_{\max }-\Delta V_{\max }{\cfrac {\ce {[I]}}{[{\ce {I}}]+K_{i}}}}$
or
${\displaystyle V_{\max 1}-(V_{\max 1}-V_{\max 2}){\cfrac {\ce {[I]}}{[{\ce {I}}]+K_{i}}}}$
This term can then define the residual enzymatic activity present when the inhibitor is interacting with individual enzymes in the population. However the inclusion of this term has the added value of allowing for the possibility of activation if the secondary Vmax term turns out to be higher than the initial term. To account for the possibly of activation as well the notation can then be rewritten replacing the inhibitor "I" with a modifier term denoted here as "X".
${\displaystyle V_{\max 1}-(V_{\max 1}-V_{\max 2}){\cfrac {\ce {[X]}}{[{\ce {X}}]+K_{x}}}}$
While this terminology results in a simplified way of dealing with kinetic effects relating to the maximum velocity of the Michaelis–Menten equation, it highlights potential problems with the term used to describe effects relating to the Km. The Km relating to the affinity of the enzyme for the substrate should in most cases relate to potential changes in the binding site of the enzyme which would directly result from enzyme inhibitor interactions. As such a term similar to the one proposed above to modulate Vmax should be appropriate in most situations:[12]
${\displaystyle K_{m1}-(K_{m1}-K_{m2}){\cfrac {\ce {[X]}}{[{\ce {X}}]+K_{x}}}}$
### Special cases
• The mechanism of partially competitive inhibition is similar to that of non-competitive, except that the EIS complex has catalytic activity, which may be lower or even higher (partially competitive activation) than that of the enzyme–substrate (ES) complex. This inhibition typically displays a lower Vmax, but an unaffected Km value.[13]
• Uncompetitive inhibition occurs when the inhibitor binds only to the enzyme–substrate complex, not to the free enzyme; the EIS complex is catalytically inactive. This mode of inhibition is rare and causes a decrease in both Vmax and the Km value.[13]
• Substrate and product inhibition is where either the substrate or product of an enzyme reaction inhibit the enzyme's activity. This inhibition may follow the competitive, uncompetitive or mixed patterns. In substrate inhibition there is a progressive decrease in activity at high substrate concentrations. This may indicate the existence of two substrate-binding sites in the enzyme.[1] At low substrate, the high-affinity site is occupied and normal kinetics are followed. However, at higher concentrations, the second inhibitory site becomes occupied, inhibiting the enzyme.[14] Product inhibition is often a regulatory feature in metabolism and can be a form of negative feedback.
• Slow-tight inhibition occurs when the initial enzyme–inhibitor complex EI undergoes isomerisation to a second more tightly held complex, EI*, but the overall inhibition process is reversible. This manifests itself as slowly increasing enzyme inhibition. Under these conditions, traditional Michaelis–Menten kinetics give a false value for Ki, which is time–dependent.[1] The true value of Ki can be obtained through more complex analysis of the on (kon) and off (koff) rate constants for inhibitor association. See irreversible inhibition below for more information.
TGDDF / GDDF MAIs where blue depicts the tetrahydrofolate cofactor analog, black GAR or thioGAR and red, the connecting atoms.
Peptide-based HIV-1 protease inhibitor ritonavir
Nonpeptidic HIV-1 protease inhibitor tipranavir
• Bi-substrate analog inhibitors are high affinity and selectivity inhibitors that can be prepared for enzymes that catalyze bi-molecular reactions by capturing the binding energy of each substrate into one molecule.[15][16] For example, in the formyl transfer reactions of purine biosynthesis, a potent multi-substrate adduct inhibitor (MAI) to GAR TFase was prepared synthetically by linking analogs of the glycinamide ribonucleotide (GAR) substrate and the N-10-formyl tetrahydrofolate cofactor together to produce thioglycinamide ribonucleotide dideazafolate (TGDDF),[17] or enzymatically from the natural GAR substrate to yield GDDF.[18] Here the subnanomolar dissociation constant (KD) of TGDDF was greater than predicted presumably due to entropic advantages gained and/or positive interactions acquired through the atoms linking the components. MAIs have also been observed to be produced in cells by reactions of pro-drugs such as isoniazid[19] or enzyme inhibitor ligands (e.g., PTC124) [20] with cellular cofactors such as NADH and ATP respectively.
### Examples of reversible inhibitors
As enzymes have evolved to bind their substrates tightly, and most reversible inhibitors bind in the active site of enzymes, it is unsurprising that some of these inhibitors are strikingly similar in structure to the substrates of their targets. Inhibitors of DHFR are prominent examples. Other example of these substrate mimics are the protease inhibitors, a very successful class of antiretroviral drugs used to treat HIV.[21] The structure of ritonavir, a protease inhibitor based on a peptide and containing three peptide bonds, is shown on the right. As this drug resembles the protein that is the substrate of the HIV protease, it competes with this substrate in the enzyme's active site.
Enzyme inhibitors are often designed to mimic the transition state or intermediate of an enzyme-catalyzed reaction. This ensures that the inhibitor exploits the transition state stabilising effect of the enzyme, resulting in a better binding affinity (lower Ki) than substrate-based designs. An example of such a transition state inhibitor is the antiviral drug oseltamivir; this drug mimics the planar nature of the ring oxonium ion in the reaction of the viral enzyme neuraminidase.[22]
However, not all inhibitors are based on the structures of substrates. For example, the structure of another HIV protease inhibitor tipranavir is shown on the left. This molecule is not based on a peptide and has no obvious structural similarity to a protein substrate. These non-peptide inhibitors can be more stable than inhibitors containing peptide bonds, because they will not be substrates for peptidases and are less likely to be degraded.[23]
In drug design it is important to consider the concentrations of substrates to which the target enzymes are exposed. For example, some protein kinase inhibitors have chemical structures that are similar to adenosine triphosphate, one of the substrates of these enzymes. However, drugs that are simple competitive inhibitors will have to compete with the high concentrations of ATP in the cell. Protein kinases can also be inhibited by competition at the binding sites where the kinases interact with their substrate proteins, and most proteins are present inside cells at concentrations much lower than the concentration of ATP. As a consequence, if two protein kinase inhibitors both bind in the active site with similar affinity, but only one has to compete with ATP, then the competitive inhibitor at the protein-binding site will inhibit the enzyme more effectively.[24]
## Irreversible inhibitors
### Types of irreversible inhibition (covalent inactivation)
Reaction of the irreversible inhibitor diisopropylfluorophosphate (DFP) with a serine protease
Irreversible inhibitors usually covalently modify an enzyme, and inhibition can therefore not be reversed. Irreversible inhibitors often contain reactive functional groups such as nitrogen mustards, aldehydes, haloalkanes, alkenes, Michael acceptors, phenyl sulfonates, or fluorophosphonates. These nucleophilic groups react with amino acid side chains to form covalent adducts. The residues modified are those with side chains containing nucleophiles such as hydroxyl or sulfhydryl groups; these include the amino acids serine (as in DFP, right), cysteine, threonine, or tyrosine.[25]
Irreversible inhibition is different from irreversible enzyme inactivation. Irreversible inhibitors are generally specific for one class of enzyme and do not inactivate all proteins; they do not function by destroying protein structure but by specifically altering the active site of their target. For example, extremes of pH or temperature usually cause denaturation of all protein structure, but this is a non-specific effect. Similarly, some non-specific chemical treatments destroy protein structure: for example, heating in concentrated hydrochloric acid will hydrolyse the peptide bonds holding proteins together, releasing free amino acids.[26]
Irreversible inhibitors display time-dependent inhibition and their potency therefore cannot be characterised by an IC50 value.[1][27] This is because the amount of active enzyme at a given concentration of irreversible inhibitor will be different depending on how long the inhibitor is pre-incubated with the enzyme. Instead, kobs/[I] values are used,[28] where kobs is the observed pseudo-first order rate of inactivation (obtained by plotting the log of % activity vs. time) and [I] is the concentration of inhibitor. The kobs/[I] parameter is valid as long as the inhibitor does not saturate binding with the enzyme (in which case kobs = kinact).
### Analysis of irreversible inhibition
Kinetic scheme for irreversible inhibitors
As shown in the figure to the right, irreversible inhibitors have a short instance where they form a reversible non-covalent complex with the enzyme (EI or ESI) and this then reacts to produce the covalently modified "dead-end complex" EI* (an irreversible covalent complex). The rate at which EI* is formed is called the inactivation rate or kinact. Since formation of EI may compete with ES, binding of irreversible inhibitors can be prevented by competition either with substrate or with a second, reversible inhibitor. This protection effect is good evidence of a specific reaction of the irreversible inhibitor with the active site.
The binding and inactivation steps of this reaction are investigated by incubating the enzyme with inhibitor and assaying the amount of activity remaining over time. The activity will be decreased in a time-dependent manner, usually following exponential decay. Fitting these data to a rate equation gives the rate of inactivation at this concentration of inhibitor. This is done at several different concentrations of inhibitor. If a reversible EI complex is involved the inactivation rate will be saturable and fitting this curve will give kinact and Ki.[29]
Another method that is widely used in these analyses is mass spectrometry. Here, accurate measurement of the mass of the unmodified native enzyme and the inactivated enzyme gives the increase in mass caused by reaction with the inhibitor and shows the stoichiometry of the reaction.[30] This is usually done using a MALDI-TOF mass spectrometer. In a complementary technique, peptide mass fingerprinting involves digestion of the native and modified protein with a protease such as trypsin. This will produce a set of peptides that can be analysed using a mass spectrometer. The peptide that changes in mass after reaction with the inhibitor will be the one that contains the site of modification.
### Special cases
Chemical mechanism for irreversible inhibition of ornithine decarboxylase by DFMO. Pyridoxal 5'-phosphate (Py) and enzyme (E) are not shown. Adapted from[31]
Not all irreversible inhibitors form covalent adducts with their enzyme targets. Some reversible inhibitors bind so tightly to their target enzyme that they are essentially irreversible. These tight-binding inhibitors may show kinetics similar to covalent irreversible inhibitors. In these cases, some of these inhibitors rapidly bind to the enzyme in a low-affinity EI complex and this then undergoes a slower rearrangement to a very tightly bound EI* complex (see figure above). This kinetic behaviour is called slow-binding.[32] This slow rearrangement after binding often involves a conformational change as the enzyme "clamps down" around the inhibitor molecule. Examples of slow-binding inhibitors include some important drugs, such methotrexate,[33] allopurinol,[34] and the activated form of acyclovir.[35]
### Examples of irreversible inhibitors
Trypanothione reductase with the lower molecule of an inhibitor bound irreversibly and the upper one reversibly. Created from PDB 1GXF.
Diisopropylfluorophosphate (DFP) is shown as an example of an irreversible protease inhibitor in the figure above right. The enzyme hydrolyses the phosphorus–fluorine bond, but the phosphate residue remains bound to the serine in the active site, deactivating it.[36] Similarly, DFP also reacts with the active site of acetylcholine esterase in the synapses of neurons, and consequently is a potent neurotoxin, with a lethal dose of less than 100 mg.[37]
Suicide inhibition is an unusual type of irreversible inhibition where the enzyme converts the inhibitor into a reactive form in its active site. An example is the inhibitor of polyamine biosynthesis, α-difluoromethylornithine or DFMO, which is an analogue of the amino acid ornithine, and is used to treat African trypanosomiasis (sleeping sickness). Ornithine decarboxylase can catalyse the decarboxylation of DFMO instead of ornithine, as shown above. However, this decarboxylation reaction is followed by the elimination of a fluorine atom, which converts this catalytic intermediate into a conjugated imine, a highly electrophilic species. This reactive form of DFMO then reacts with either a cysteine or lysine residue in the active site to irreversibly inactivate the enzyme.[31]
Since irreversible inhibition often involves the initial formation of a non-covalent EI complex, it is sometimes possible for an inhibitor to bind to an enzyme in more than one way. For example, in the figure showing trypanothione reductase from the human protozoan parasite Trypanosoma cruzi, two molecules of an inhibitor called quinacrine mustard are bound in its active site. The top molecule is bound reversibly, but the lower one is bound covalently as it has reacted with an amino acid residue through its nitrogen mustard group.[38]
## Discovery and design of inhibitors
Robots used for the high-throughput screening of chemical libraries to discover new enzyme inhibitors
New drugs are the products of a long drug development process, the first step of which is often the discovery of a new enzyme inhibitor. In the past the only way to discover these new inhibitors was by trial and error: screening huge libraries of compounds against a target enzyme and hoping that some useful leads would emerge. This brute force approach is still successful and has even been extended by combinatorial chemistry approaches that quickly produce large numbers of novel compounds and high-throughput screening technology to rapidly screen these huge chemical libraries for useful inhibitors.[39]
More recently, an alternative approach has been applied: rational drug design uses the three-dimensional structure of an enzyme's active site to predict which molecules might be inhibitors.[40] These predictions are then tested and one of these tested compounds may be a novel inhibitor. This new inhibitor is then used to try to obtain a structure of the enzyme in an inhibitor/enzyme complex to show how the molecule is binding to the active site, allowing changes to be made to the inhibitor to try to optimise binding. This test and improve cycle is then repeated until a sufficiently potent inhibitor is produced.[41] Computer-based methods of predicting the affinity of an inhibitor for an enzyme are also being developed, such as molecular docking[42] and molecular mechanics.
## Uses of inhibitors
Enzyme inhibitors are found in nature and are also designed and produced as part of pharmacology and biochemistry. Natural poisons are often enzyme inhibitors that have evolved to defend a plant or animal against predators. These natural toxins include some of the most poisonous compounds known. Artificial inhibitors are often used as drugs, but can also be insecticides such as malathion, herbicides such as glyphosate, or disinfectants such as triclosan. Other artificial enzyme inhibitors block acetylcholinesterase, an enzyme which breaks down acetylcholine, and are used as nerve agents in chemical warfare.
### Chemotherapy
The structure of sildenafil (Viagra) The coenzyme folic acid (left) compared to the anti-cancer drug methotrexate (right) The structure of a complex between penicillin G and the Streptomyces transpeptidase. Generated from PDB 1PWC.
The most common uses for enzyme inhibitors are as drugs to treat disease. Many of these inhibitors target a human enzyme and aim to correct a pathological condition. However, not all drugs are enzyme inhibitors. Some, such as anti-epileptic drugs, alter enzyme activity by causing more or less of the enzyme to be produced. These effects are called enzyme induction and inhibition and are alterations in gene expression, which is unrelated to the type of enzyme inhibition discussed here. Other drugs interact with cellular targets that are not enzymes, such as ion channels or membrane receptors.
An example of a medicinal enzyme inhibitor is sildenafil (Viagra), a common treatment for male erectile dysfunction. This compound is a potent inhibitor of cGMP specific phosphodiesterase type 5, the enzyme that degrades the signalling molecule cyclic guanosine monophosphate.[43] This signalling molecule triggers smooth muscle relaxation and allows blood flow into the corpus cavernosum, which causes an erection. Since the drug decreases the activity of the enzyme that halts the signal, it makes this signal last for a longer period of time.
Another example of the structural similarity of some inhibitors to the substrates of the enzymes they target is seen in the figure comparing the drug methotrexate to folic acid. Folic acid is a substrate of dihydrofolate reductase, an enzyme involved in making nucleotides that is potently inhibited by methotrexate. Methotrexate blocks the action of dihydrofolate reductase and thereby halts the production of nucleotides. This block of nucleotide biosynthesis is more toxic to rapidly growing cells than non-dividing cells, since a rapidly growing cell has to carry out DNA replication, therefore methotrexate is often used in cancer chemotherapy.[44]
### Antibiotics
Drugs also are used to inhibit enzymes needed for the survival of pathogens. For example, bacteria are surrounded by a thick cell wall made of a net-like polymer called peptidoglycan. Many antibiotics such as penicillin and vancomycin inhibit the enzymes that produce and then cross-link the strands of this polymer together.[45] This causes the cell wall to lose strength and the bacteria to burst. In the figure, a molecule of penicillin (shown in a ball-and-stick form) is shown bound to its target, the transpeptidase from the bacteria Streptomyces R61 (the protein is shown as a ribbon-diagram).
Antibiotic drug design is facilitated when an enzyme that is essential to the pathogen's survival is absent or very different in humans. In the example above, humans do not make peptidoglycan, therefore inhibitors of this process are selectively toxic to bacteria. Selective toxicity is also produced in antibiotics by exploiting differences in the structure of the ribosomes in bacteria, or how they make fatty acids.
### Metabolic control
Enzyme inhibitors are also important in metabolic control. Many metabolic pathways in the cell are inhibited by metabolites that control enzyme activity through allosteric regulation or substrate inhibition. A good example is the allosteric regulation of the glycolytic pathway. This catabolic pathway consumes glucose and produces ATP, NADH and pyruvate. A key step for the regulation of glycolysis is an early reaction in the pathway catalysed by phosphofructokinase-1 (PFK1). When ATP levels rise, ATP binds an allosteric site in PFK1 to decrease the rate of the enzyme reaction; glycolysis is inhibited and ATP production falls. This negative feedback control helps maintain a steady concentration of ATP in the cell. However, metabolic pathways are not just regulated through inhibition since enzyme activation is equally important. With respect to PFK1, fructose 2,6-bisphosphate and ADP are examples of metabolites that are allosteric activators.[46]
Physiological enzyme inhibition can also be produced by specific protein inhibitors. This mechanism occurs in the pancreas, which synthesises many digestive precursor enzymes known as zymogens. Many of these are activated by the trypsin protease, so it is important to inhibit the activity of trypsin in the pancreas to prevent the organ from digesting itself. One way in which the activity of trypsin is controlled is the production of a specific and potent trypsin inhibitor protein in the pancreas. This inhibitor binds tightly to trypsin, preventing the trypsin activity that would otherwise be detrimental to the organ.[47] Although the trypsin inhibitor is a protein, it avoids being hydrolysed as a substrate by the protease by excluding water from trypsin's active site and destabilising the transition state.[48] Other examples of physiological enzyme inhibitor proteins include the barstar inhibitor of the bacterial ribonuclease barnase.[49]
### Pesticides
Many pesticides are enzyme inhibitors. Acetylcholinesterase (AChE) is an enzyme found in animals, from insects to humans. It is essential to nerve cell function through its mechanism of breaking down the neurotransmitter acetylcholine into its constituents, acetate and choline. This is somewhat unusual among neurotransmitters as most, including serotonin, dopamine, and norepinephrine, are absorbed from the synaptic cleft rather than cleaved. A large number of AChE inhibitors are used in both medicine and agriculture. Reversible competitive inhibitors, such as edrophonium, physostigmine, and neostigmine, are used in the treatment of myasthenia gravis and in anaesthesia. The carbamate pesticides are also examples of reversible AChE inhibitors. The organophosphate pesticides such as malathion, parathion, and chlorpyrifos irreversibly inhibit acetylcholinesterase.
The herbicide glyphosate is an inhibitor of 3-phosphoshikimate 1-carboxyvinyltransferase,[50] other herbicides, such as the sulfonylureas inhibit the enzyme acetolactate synthase. Both these enzymes are needed for plants to make branched-chain amino acids. Many other enzymes are inhibited by herbicides, including enzymes needed for the biosynthesis of lipids and carotenoids and the processes of photosynthesis and oxidative phosphorylation.[51]
### Natural poisons
To discourage seed predators, pulses contain trypsin inhibitors that interfere with digestion.
Animals and plants have evolved to synthesise a vast array of poisonous products including secondary metabolites, peptides and proteins that can act as inhibitors. Natural toxins are usually small organic molecules and are so diverse that there are probably natural inhibitors for most metabolic processes.[52] The metabolic processes targeted by natural poisons encompass more than enzymes in metabolic pathways and can also include the inhibition of receptor, channel and structural protein functions in a cell. For example, paclitaxel (taxol), an organic molecule found in the Pacific yew tree, binds tightly to tubulin dimers and inhibits their assembly into microtubules in the cytoskeleton.[53]
Many natural poisons act as neurotoxins that can cause paralysis leading to death and have functions for defence against predators or in hunting and capturing prey. Some of these natural inhibitors, despite their toxic attributes, are valuable for therapeutic uses at lower doses.[54] An example of a neurotoxin are the glycoalkaloids, from the plant species in the family Solanaceae (includes potato, tomato and eggplant), that are acetylcholinesterase inhibitors. Inhibition of this enzyme causes an uncontrolled increase in the acetylcholine neurotransmitter, muscular paralysis and then death. Neurotoxicity can also result from the inhibition of receptors; for example, atropine from deadly nightshade (Atropa belladonna) that functions as a competitive antagonist of the muscarinic acetylcholine receptors.[55]
Although many natural toxins are secondary metabolites, these poisons also include peptides and proteins. An example of a toxic peptide is alpha-amanitin, which is found in relatives of the death cap mushroom. This is a potent enzyme inhibitor, in this case preventing the RNA polymerase II enzyme from transcribing DNA.[56] The algal toxin microcystin is also a peptide and is an inhibitor of protein phosphatases.[57] This toxin can contaminate water supplies after algal blooms and is a known carcinogen that can also cause acute liver hemorrhage and death at higher doses.[58]
Proteins can also be natural poisons or antinutrients, such as the trypsin inhibitors (discussed above) that are found in some legumes, as shown in the figure above. A less common class of toxins are toxic enzymes: these act as irreversible inhibitors of their target enzymes and work by chemically modifying their substrate enzymes. An example is ricin, an extremely potent protein toxin found in castor oil beans. This enzyme is a glycosidase that inactivates ribosomes. Since ricin is a catalytic irreversible inhibitor, this allows just a single molecule of ricin to kill a cell.[59]
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https://telescopeauthority.com/brightness-and-magnitude-in-astronomy/
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# Brightness and Magnitude in Astronomy
3 years ago
Magnitude is an astronomical term that is used to describe precisely how bright a stellar object is. It can be done with both objective scientific measurements and a more qualitative classification of how bright the object is in the sky. Magnitude is measured on an inverse scale where lower numbers equal a brighter object. The traditional magnitude scale used in astronomy runs from 1 to 6.
The two major scales of magnitude are visual magnitude and absolute magnitude. Absolute magnitude is a scientific scale of how much light an object would shed if the observer was precisely 10 parsecs away from it. This distance equals about 33 light years, or 200 trillion miles, and the exact absolute brightness value of a heavenly body is determined by complex calculations of the elemental composition of the object, the amount of light it sheds measured in lumens and other scientific considerations. Absolute magnitude is rarely used outside of the scientific context.
Apparent magnitude, also known as visual magnitude, is a more intuitive form of classification. It measures how bright the object is by the time that its light reaches the earth. This value can be measured precisely through the use of light meters and telescopes, or it can be approximated simply by looking at the stellar objects with binoculars or the naked eye.
In fact, the magnitude scale of stellar brightness was invented thousands of years before telescopes or binoculars. The scale of apparent magnitude that we use today was laid down by the ancient Greek astronomers Hipparchus and Ptolemy, who classified the objects in the sky according to six categories of brightness. They chose about twenty of the stars that looked brightest to them and assigned them to the category of the first magnitude. The next brightest set of stellar objects was assigned to the second magnitude, all the way to the ones which could only barely be seen, which were all grouped in the sixth magnitude.
When telescopes, prisms and other optical devices were invented, they brought with them two important discoveries related to magnitude. The first was that there were a whole lot more stars below the sixth category than anyone had ever expected. The telescope revealed uncountable numbers of tiny lights that were far too dim to be seen with the unassisted eye. What is more, the telescope made it possible to measure the brightness coming from a particular star with ever-increasing specificity. Since the first magnitude was such a well-defined category and the sixth magnitude was used to apply to a vast number of objects at the very edge of human decision, the arbitrary decision was made to set the brightness of the sixth magnitude at 100 times less than the 1st magnitude. This transformed the magnitude scale into a logarithmic scale, which means that the quantities measured increased exponentially as the measurements increased arithmetically.
This historical tradition of astronomy explains why the stellar magnitude chart appears almost upside-down. Since the scale has been extended to cover things far outside the original scope, and since the measured brightness falls as the magnitude climbs, the chart can sometime seem to operate in a counter-intuitive manner. However, so long as one remembers that a bright star has a magnitude of about one, it is easy to reason the rest through.
The magnitude chart was expanded in the other direction as well. It became possible to measure exactly how much brighter planets such as Venus were than the heavenly bodies around them. In other words, they recorded precisely how much light there was coming from Venus when it was at its fullest brightness. It was found to be about a hundred times brighter than the objects that made up the traditional first magnitude, so the scale was simply extended into the negative numbers to accommodate. Venus has a visual magnitude of about -4. The Moon, by far the brightest of the planets we can see in the sky, has a magnitude of about -12 when it is at its fullest. The sun puts out about 400,000 times as much light as the full moon. This means that its visual magnitude is about -26.
As can be seen, this scale represents an excellent way to deal with the widely divergent lights seen in the sky. Although local atmospheric conditions are always paramount, on a clear night it should be possible to see stellar bodies all the way up to magnitude 6. A typical pair of field binoculars should be sufficient to see objects as dim as magnitude 9.5, which is more than a hundred times darker than the eyes can see alone. A telescope will naturally improve this ability to detect dimmer objects to an almost unimaginable degree. The wider the diameter of the telescope, the more light it will capture and the higher magnitude objects that can be seen. The only limits to this are found in the diameter of lens that can be procured and the atmospheric conditions prevailing at the viewing location. The Hubble Space Telescope has been able to resolve objects as dim as 31.5 magnitude in its nearly perfect viewing conditions outside the Earth’s atmosphere.
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http://math.stackexchange.com/questions/264250/notation-in-set-theory-applied-to-counters
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# Notation in set theory applied to counters
I have written a notation representing a counter for a condition:
$M \leftarrow \displaystyle \sum_{i =1}^{|X|} [B_j = X_i]$
So far this gives me a number for a specific j (the counter), but I want to turn this into a set for all values of j in such a way M is representing a multiset. M would be something like this:
$M = \{1,1,2,4,5\}$
How can I fix my notation to represent what I want?
thanks
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Assuming your indexing starts at $0$ for the first element, then you want to sum to the order of $X - 1$. For example, the set
I think this might work:
$$M = \left\{M_j \mid M_j \leftarrow \displaystyle \sum_{i =0}^{|X|-1} [B_j = X_i]; 0\le j < |M|\right\}.$$
Of course, determining $|M|$ requires knowing in advance the number of counters = $C$, so the condition $0 \le j < C$ should probably replace the condition $0 \le j < |M|$:
$$M = \left\{M_j \mid M_j \leftarrow \displaystyle \sum_{i =0}^{|X|-1} [B_j = X_i]; 0\le j < C\right\}.$$
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Hi, thanks for your solution, it looks good. I just would like to know if this is true for a multiset as well, because a set does not allow repetition of the elements and considering I have a counter, I need definitely a multiset. If I get no better answer, I accept yours. – sfelixjr Dec 23 '12 at 20:21
I will use your solution with double brackets. According to this, that's how we should represent a multiset: fr.m.wikipedia.org/wiki/Multiensemble – sfelixjr Dec 23 '12 at 20:26
This would be a multiset: it is the set of all $M_j$ each depending on/associated with $B_j$, where $j$ ranges through the number of counters. – amWhy Dec 23 '12 at 20:27
Good enough!... – amWhy Dec 23 '12 at 20:27
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https://geo.libretexts.org/Bookshelves/Meteorology/Book%3A_Practical_Meteorology_(Stull)/22%3A_Atmospheric_Optics/22.00%3A_New_Page
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# 22.1: Ray Geometry
When a monochromatic (single color) light ray reaches an interface between two media such as air and water, a portion of the incident light from the air can be reflected back into the air, some can be refracted as it enters the water (Fig. 22.1), and some can be absorbed and changed into heat (not sketched). Similar processes occur across an air-ice interface.
# 22.1.1. Reflection
The angle θ3 of the reflected ray always equals the angle θ1 of the incident ray, measured with respect to a line normal (perpendicular) to the interface:
\begin{align}\theta_{1}=\theta_{3}\tag{22.1}\end{align}
The reflected angle does not depend on color (i.e., is not a function of the wavelength of light).
# 22.1.2. Refraction
## 22.1.2.1. Refractive index
The refractive index ni for medium i relative to a vacuum is defined as:
\begin{align}n_{i}=\frac{c_{o}}{c_{i}}\tag{22.2}\end{align}
where ci is the speed of light through medium i and co = 3 x 108 m s–1 is the speed of light in a vacuum.
The ratio of refractive indices is sometimes defined as:
\begin{align}\mu_{12}=\frac{n_{1}}{n_{2}}\tag{22.3}\end{align}
where subscripts 1 and 2 refer to the media containing the incident ray and refracted rays, respectively. Different colors and different media have different refractive indices, as indicated in Table 22-1 and Fig. 22.2.
## 22.1.2.2. Snell’s Law
The relationship between the incident angle θ1 and refracted angle θ2 (Fig. 22.1) is called Snell’s Law:
\begin{align}\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{c_{1}}{c_{2}}=\frac{n_{2}}{n_{1}}=\frac{1}{\mu_{12}}\tag{22.4}\end{align}
Incident and refracted rays are always in the same plane. This plane includes the line that is normal to the surface. It is the plane that gives the smallest angle between the incident ray and the surface.
Solving Snell’s Law for the refracted ray angle gives:
\begin{align}\theta_{2}=\arcsin \left[\mu_{12} \cdot \sin \left(\theta_{1}\right)\right]\tag{22.5}\end{align}
Red light is bent less than violet light as it passes through an interface. Thus, refraction causes white light to be spread into a spectrum of colors. This phenomenon is called dispersion
A viewer looking toward the incoming light ray (Fig. 22.3) would see a light point in the celestial sphere overhead at angle θ2 from normal.
Sample Application
Rays of red and violet light in air strike a water surface, both with incident angle of 60°. Find the angle of refraction for each color, given T = 20°C, P = 101 kPa?
Given: θ1 = 60° , T = 20°C , P = 101 kPa
Find: θ2 = ?° for red (λ=0.7 µm) and violet (λ=0.4 µm).
Use eq. (22.5): θ2 = arcsin[(n1/n2)·sin(θ1)] = arcsin[ µ12 ·sin(θ1)]
with µ12 from eq. (22.3) and refractive indices from Table 22-1.
For red: µ12 = n1/n2 = 1.0002704/1.3305 = 0.7518
θ2 = arcsin[0.7518·sin(60°)]
θ2 = 40.62°
Similarly for violet: θ2 = 40.15°
Check: Units OK. Physics OK.
Exposition: Had there been no bending, then both answers would have been 60°. Angles closer to 60° for this example correspond to less bending. The answers above confirm the statement that red light is bent less than violet. The amount of bending (difference between incident and refracted angles) is large: 60° – 40° = 20°.
## 22.1.2.3. Snell’s Law in 3 Dimensions
λ (µm) Temperature (°C) –40 –20 0 20 Table 22-1a. Refractive index (nair) for air with pressure P = 101.325 kPa and relative humidity RH = 75%. From NIST. 0.4 violet 0.45 1.0003496 1.0003469 1.0003219 1.0003194 1.0002982 1.0002958 1.0002773 1.0002751 1.0002583 1.0002562 0.5 0.55 0.6 1.0003450 1.0003436 1.0003425 1.0003176 1.0003163 1.0003154 1.0002942 1.0002930 1.0002921 1.0002736 1.0002725 1.0002716 1.0002548 1.0002537 1.0002529 0.65 0.7 red 1.0003417 1.0003410 1.0003146 1.0003140 1.0002914 1.0002908 1.0002710 1.0002704 1.0002523 1.0002523
λ (µm) Temperature (°C) –40 –20 0 20 Table 22-1b. Refractive index (nair) for air with P = 80 kPa and RH = 75%. Based on data from NIST modified Edlen Eq. calculator. 0.4 violet 0.45 1.0002760 1.0002738 1.0002541 1.0002521 1.0002353 1.0002335 1.0002188 1.0002171 1.0002036 1.0002019 0.5 0.55 0.6 1.0002723 1.0002712 1.0002704 1.0002507 1.0002497 1.0002489 1.0002322 1.0002312 1.0002305 1.0002158 1.0002150 1.0002143 1.0002008 1.0001999 1.0001993 0.65 0.7 red 1.0002697 1.0002692 1.0002483 1.0002479 1.0002300 1.0002295 1.0002138 1.0002134 1.0001988 1.0001984
λ (µm) Temperature (°C) –40 –20 0 20 Table 22-1c. Refractive index (nair) for air with P = 40 kPa and RH = 75%. Based on data from NIST modified Edlen Eq. calculator. 0.4 violet 0.45 1.0001379 1.0001369 1.0001270 1.0001260 1.0001176 1.0001166 1.0001091 1.0001082 1.0001009 1.0001000 0.5 0.55 0.6 1.0001361 1.0001355 1.0001351 1.0001253 1.0001248 1.0001244 1.0001160 1.0001155 1.0001151 1.0001076 1.0001071 1.0001068 1.0000994 1.0000990 1.0000987 0.65 0.7 red 1.0001348 1.0001346 1.0001241 1.0001239 1.0001149 1.0001147 1.0001066 1.0001064 1.0000985 1.0000983
λ (µm) Temperature (°C) –40 –20 0 20 Table 22-1d. Refractive index (nwater) for liquid water. Based on data from IAPWS 1997 release and CRC Handbook. 0.4 violet 0.45 (values below 0°C are for supercooled water) 1.3429 1.3390 1.3446 1.3406 1.3436 1.3396 1.3411 1.3371 0.5 0.55 0.6 1.3362 1.3341 1.3324 1.3378 1.3357 1.3340 1.3368 1.3347 1.3330 1.3344 1.3323 1.3306 0.65 0.7 red 1.3310 1.3299 1.3326 1.3315 1.3317 1.3305 1.3293 1.3281 density (kg m–3): 993.547 999.840 998.207 992.200
λ (µm) Temperature (°C) –40 –20 0 Table 22-1e. Refractive index (nice) for ice (Ih). Based on data from S.G. Warren (1984) and V.F. Petrenko & R.W. Whitworth (1999). 0.4 violet 0.45 1.3206 1.3169 1.3199 1.3162 1.3192 1.3155 (values at 0°C were extrapolated) (See INFO Box in Precipitation chapter for info on ice phases such as Ih.) 0.5 0.55 0.6 1.3142 1.3122 1.3106 1.3135 1.3115 1.3099 1.3128 1.3108 1.3092 0.65 0.7 red 1.3092 1.3081 1.3085 1.3074 1.3078 1.3067
Sometimes it is easier to work with x and y components of the incident ray, where the x-axis might be aligned with the axis of a columnar ice crystal, for example, and the y-axis might be on the crystal prism surface (Fig. 22.3). The relationship between the component angles and the incident angle is:
\begin{align}\tan ^{2} \theta_{1}=\tan ^{2} \alpha_{1}+\tan ^{2} \beta_{1}\tag{22.6}\end{align}
where θ, α and β are as illustrated in Fig. 22.3. This relationship also applies to refracted angles (θ2, α2, β2), and will be used extensively later in this chapter to discuss ice-crystal optics.
Sample Application
Find the speed of red light through liquid water and through air at T = 20°C and P = 101.325 kPa.
Given: λ = 0.7 µm, T = 20°C and P = 101.325 kPa.
Find: cair = ? m s–1 , cwater = ? m s–1.
Use eq. (22.2) with refractive indices from Table 22-1:
cair = co/nair =(3x108m s–1)/1.0002704 =2.999x108 m s–1
cwater = co/nwater =(3x108m s–1)/1.3305 =2.255x108 m s–1
Check: Speeds reasonable. Units OK.
Exposition: The difference in speeds of light is useful for understanding Huygens’ principle.
Component angles α and β of the refracted and incident rays do NOT individually obey Snell’s law (eq. 22.4). Nevertheless, Snell’s law can be reformulated in terms of components as follows:
\begin{align}\tan ^{2} \alpha_{2}=b_{\alpha \beta} \cdot \tan ^{2} \alpha_{1}\tag{22.7a}\end{align}
\begin{align}\tan ^{2} \beta_{2}=b_{\alpha \beta} \cdot \tan ^{2} \beta_{1}\tag{22.7b}\end{align}
where α2 and β2 are the components of the refracted ray (analogous to α1 and β1), and
\begin{align}b_{\alpha \beta}=\frac{\mu_{12}^{2}}{1+\left(1-\mu_{12}^{2}\right) \cdot\left\{\tan ^{2} \alpha_{1}+\tan ^{2} \beta_{1}\right\}}\tag{22.8}\end{align}
These equations are abbreviated as
\begin{align}\alpha_{2}=S_{\alpha}\left(\alpha_{1}, \beta_{1,}, \mu_{12}\right)=\arctan \left[\left(b_{\alpha \beta} \cdot \tan ^{2} \alpha_{1}\right)^{1 / 2}\right]\tag{22.9b}\end{align}
\begin{align}\beta_{2}=S_{\beta}\left(\alpha_{1}, \beta_{1,}, \mu_{12}\right)=\arctan \left[\left(b_{\alpha \beta} \cdot \tan ^{2} \beta_{1}\right)^{1 / 2}\right]\tag{22.9b}\end{align}
where S represents Snell’s law for components.
Sample Application
A ray of red light in air strikes a water surface, with incidence angle components of α1 = 45° and β1 = 54.74°. What are the corresponding component angles of the refracted ray? Assume T = 20°C and P = 101.325 kPa.
Given: α1 = 45°, β1 = 54.74°.
Find: α2 = ?° , β2 = ?°
From Table 22-1 for red light:
µ122 = (1.0002704/1.3305)2 = (0.7518)2 = 0.5652
Also: tan(45°) = 1, and tan(54.74°) = 1.414
Next, solve eq. (22.8):
$$\ b_{\alpha \beta}=\frac{0.5652}{1+(1-0.5652) \cdot(1+2)}=0.2453$$
Then use eqs. (22.7):
tan22) = 0.2453·[tan2(45°)] = 0.2453
tan22) = 0.2453·[tan2(54.74°)] = 0.4906
Thus: α2 = arctan[(0.2453)0.5] = 26.35°
β2 = arctan[(0.4906)0.5] = 35.01°
Check: Units OK. Physics OK.
Exposition: Using eq. (22.6) with the incident angle components of 45° and 54.74°, we find that the incident ray angle is θ1=arctan[(1+2)0.5] = 60°. This is the same as a previous Sample Application. Using eq. (22.6) on the answers above, we find θ2 = arctan[ (0.2453+0.4906)0.5] = 40.62°. This is also the same as a previous Sample Application, which verifies eqs. (22.6 - 22.9).
# 22.1.3. Huygens’ Principle
Huygens suggested that every point along a wave front acts like a generator of new spherical secondary wavelets. Wave-front position after some time interval is located at the tangent to all of the new wavelets. Thus, when a portion of a wave front encounters a medium with a slower light velocity, then that portion of the wave slows, causing the whole wave front to turn into the medium (Fig. 22.4).
# 22.1.4. Critical Angle
When Snell’s law is applied to light rays moving from a denser medium (having slower light velocity) to a less-dense medium (having faster light velocity), there is a critical angle at which light is bent so much that it follows the interface. At angles greater than this critical incidence angle, light cannot refract out of the dense medium at all. Instead, all of the light reflects (Fig. 22.5).
The critical angle θc is found from:
\begin{align}\sin \left(\theta_{c}\right)=\frac{n_{2}}{n_{1}}\tag{22.10}\end{align}
where n1 is the refractive index for the incident ray (i.e., in the denser medium with slower light velocity). An example is a light ray moving out of water and into air (i.e., n2 < n1). For red light the critical angle is about 48.7°, while for violet light the it is about 48.1°. There is no critical angle for light moving into a denser medium (i.e., n2 > n1).
Sample Application
Find the critical angle for red light going from a cirrus-cloud ice crystal to air, where the air state is T = –20°C & P = 40 kPa.
Given: red light λ = 0.7 µm, T = –20°C & P = 40 kPa.
Find: θc = ? °
From Table 22-1: nice = 1.3074 and nair = 1.0001239.
Use eq. (22.10):
θc = arcsin(1.0001239/1.3074) = 49.9°.
Check: Magnitude OK. Units OK.
Exposition: This critical angle is close to that for liquid water, because the refractive indices for liquid and solid water are nearly the same.
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https://mersenneforum.org/showthread.php?s=068ac4b4a2a4412184ffec9d46066970&p=587872
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2021-08-31, 16:51 #3092 bur Aug 2020 5*10398e-4;3*2539e-3 5·7·11 Posts If it makes sense I could also test-sieve, though only with 10 cores. If so, which would be a good set of parameters?
2021-08-31, 18:52 #3093 charybdis Apr 2020 33×17 Posts There's nothing stopping you from test-sieving, but it wouldn't be useful for me because I'm comparing speeds as well as yields for lots of different parameter choices, and for speeds to be consistent I have to run everything on the same machine.
2021-08-31, 20:56 #3094 bur Aug 2020 5*10398e-4;3*2539e-3 5×7×11 Posts Ok, then I won't get into it.
2021-08-31, 23:33 #3095 charybdis Apr 2020 33×17 Posts Test-sieving done, at least for today. I only used I=16, lpb 34/35, and given the yields I don't expect anything smaller than this to be better. Even if 35/35 does sieve faster, it will make postprocessing more of a pain. The machine I used only has 16GB RAM so I can't test A=32 which requires ~26GB, but I don't think we're at a size where that would be clearly better. Raising ncurves0 from 25 doesn't find any more relations. It could possibly be dropped a bit, but 2LP cofactorization is a tiny contribution to the total time so the effect would be negligible. ncurves1=21 does miss a few relations, but it's just a fraction of a percent, so with the extra 3LP cofactorization time I don't think there's any need to change this from the default. There is a clear benefit of several percent from using -adjust-strategy 2, as seems to be standard with large jobs. The price paid for this is that memory use with I=16 is 13GB rather than 10GB, but I assume this won't be a problem. Higher lims = more yield and slower sieving, and with the added complication of the duplication rate depending on the lim choice, it's not easy to measure the effects with test-sieving. 500M/800M probably isn't optimal but it'll be close enough. mfb0 choice is always complicated by the fact that higher mfb0 leads to a small increase in the number of relations required, and we don't have a good handle on this. This makes 68 less attractive, but in the interests of keeping yield high I stuck with 67 for the main test. mfb1 can go quite high and still show a decent improvement in sec/rel. There doesn't seem to be any problem with going as high as 102. I test-sieved the 500M/800M, 67/102 combination at various intervals between 200M and 2500M. Sec/rel worsened from ~0.7 to ~1.5 over this range. If the aim is 3G raw relations, the range 250M-2000M looks about right, and I estimate this will take only(!) ~105 thread-years on the machine I used. This figure seems suspiciously low even given that the threads in question are fast, but I can't find a mistake anywhere, so I have to believe it. Hopefully this doesn't mean Ryan's in for a nasty surprise with the duplication rate! Last fiddled with by charybdis on 2021-08-31 at 23:37
2021-09-01, 02:45 #3096 VBCurtis "Curtis" Feb 2005 Riverside, CA 115128 Posts I appreciate the detail in the writeup- for e.g. Bur who sees a future in test-sieving, that post is a nice place to start. Here's the conclusion for Ryan: Code: tasks.lim0 = 500000000 tasks.lim1 = 800000000 tasks.lpb0 = 34 tasks.lpb1 = 35 tasks.sieve.mfb0 = 67 tasks.sieve.mfb1 = 102 tasks.sieve.ncurves0 = 25 tasks.sieve.ncurves1 = 21 tasks.I = 16 tasks.qmin = 250000000 tasks.sieve.qrange = 5000 tasks.sieve.adjust_strategy = 2 tasks.sieve.rels_wanted = 3000000000 If memory is plentiful, consider running A=32 for the first ~500M relations for the extra yield at small Q, and then changing down to I=16. That should help with the duplicate-relations rate, too- making it more likely to get 2G unique relations before Q=2000M.
2021-09-01, 03:02 #3097 charybdis Apr 2020 33·17 Posts I'd make one small change for practical purposes, which is to set rels_wanted artificially high (say 4G rather than 3G) as the intention is presumably to run msieve filtering. We don't want CADO filtering to be triggered, for a couple of reasons: 1. There is a bug whereby sieving with adjust_strategy = 2 can produce a relation with a (square) composite factor. This is extremely rare, but in 3G relations there's a decent chance it will happen, and unlike msieve, I don't think CADO filtering handles it well. 2. Even if this doesn't happen, CADO filtering on a job this big likely requires hundreds of GB of memory. Setting a large value of rels_wanted has the added benefit that ETA values are more accurate early in the sieving while the yield is high
2021-09-01, 17:19 #3098
bur
Aug 2020
5*10398e-4;3*2539e-3
5×7×11 Posts
Quote:
for e.g. Bur who sees a future in test-sieving, that post is a nice place to start.
That sound like test-sieving is a career-option. Professional test-siever.
But seriously, thanks, I always enjoy these posts.
2021-09-12, 18:56 #3099
ryanp
Jun 2012
Boulder, CO
24×19 Posts
Quote:
Originally Posted by VBCurtis I appreciate the detail in the writeup- for e.g. Bur who sees a future in test-sieving, that post is a nice place to start. Here's the conclusion for Ryan:
I'm sieving now. Thanks to those who found the poly and params!
Will give an update in a few days.
2021-09-14, 13:58 #3100 ryanp Jun 2012 Boulder, CO 24×19 Posts Update: I'm at 555M uniques now. I currently seem to be pulling about 250M unique (not total) relations per day, so hopefully another 6-10 days of sieving should be enough, if the rate keeps up.
2021-09-14, 16:56 #3101 bur Aug 2020 5*10398e-4;3*2539e-3 38510 Posts How can the percentage of uniques be found during sieving? By doing a manual filtering step on the relations? Impressive progress anyway, about the same timeframe I need for a C167...
2021-09-14, 20:21 #3102
EdH
"Ed Hall"
Dec 2009
1111010010002 Posts
Quote:
Originally Posted by bur How can the percentage of uniques be found during sieving? By doing a manual filtering step on the relations? Impressive progress anyway, about the same timeframe I need for a C167...
remdups4 is what I use. It can be found in the ggnfs package. (ggnfs/contrib/remdups/remdups4.c)
Edit: Depending on which sieving package you are using, you will need to gather all the relations into a single file and:
Code:
[z]cat <file> | ./remdups4 1000 ><uniquerels>
Last fiddled with by EdH on 2021-09-14 at 20:24
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https://brilliant.org/discussions/thread/fr-rating-historyplots/
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# FR: rating history/plots
After the new rating system was introduced, I found myself wondering if my rating was actually increasing or decreasing over time, yet there was no obvious way to find this out (except for writing all of my ratings down in the beginning of a week). Plots of participant's ratings in each topic over time would let both the participants and other people (probably the ones the participant is bragging to about how awesome he is, as well as prospective employers looking for students who adapt and learn quickly) see the participant's tendency to improve their knowledge and problem-solving skills.
Codeforces, a website with a mission similar to Brilliant's yet focused on Computer Science, has already implemented such plots and the users like them a lot.
Note by Aleksejs Popovs
6 years ago
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Rather than use this for bragging rights, I'd use this for a different reason. As it stands, It's pretty difficult to see how you are progressing. In the Brilliant Ted Talk at UChicago, I noticed that they gave the questions tags like modulo arithmetic. I would want to see my progress over time as a graph, and in between each point it would say something like
"+50 points, answered Happy Hour tagged: Derangements, PIE"
That way you could see if there are any questions based on techniques which you routinely get wrong.
- 6 years ago
I am always in favor of more graphs. We will keep a ratings plot in mind as we prioritize the improvements we need to make.
If you you guy's could have a plot of your ratings over time, what would you want to have publicly visible vs. only visible to you?
In the past we have brain-stormed having a plot of your levels through time public to everyone, and have a more specific ratings through time plot visible to only you.
Staff - 6 years ago
Well, since ratings are private at the moment, I suppose what you suggest is the best thing that can be implemented. Though you could also add a switch to make precise ratings public, which would also make rating plots public.
- 6 years ago
Really don't mind/care whether people could see it or not so I'll agree with whatever the consensus may be. As long as it can be used to find out finer details about what you are good at so you can better target certain concepts. Know Thyself.
- 6 years ago
how i can answer the key question in my level to increase my rating.. I tried to answer some questions in the news feed and got correct i some questions but it seem no increase in my rating ... this new brilliant page makes me hard to find out what I'm going to do ,,, ??????????????????
- 5 years, 11 months ago
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https://socratic.org/questions/how-do-you-differentiate-g-y-x-2-2x-1-4-4x-6-5-using-the-product-rule
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Calculus
Topics
# How do you differentiate g(y) =(x^2 - 2x + 1)^4 (4x^6 + 5) using the product rule?
Jun 13, 2018
Treat the two terms as a normal product rule, but then use the chain rule on the first function when differentiating to get the full solution: $g ' \left(x\right) = 8 {\left(x - 1\right)}^{7} \left(7 {x}^{6} - 3 {x}^{5} + 5\right)$
#### Explanation:
First, let's designate the two pieces of the function as ${f}_{1} \left(x\right)$ and ${f}_{2} \left(x\right)$:
${f}_{1} \left(x\right) = {\left({x}^{2} - 2 x + 1\right)}^{4}$
${f}_{2} \left(x\right) = 4 {x}^{6} + 5$
For a given function, the derivative using the product rule is:
$\frac{\mathrm{dy}}{\mathrm{dx}} \left({f}_{1} \left(x\right) {f}_{2} \left(x\right)\right) = {f}_{1} ' \left(x\right) \cdot {f}_{2} \left(x\right) + {f}_{1} \left(x\right) \cdot {f}_{2} ' \left(x\right)$
This means we'll need to know the derivatives of each $f$ function. The first function requires a chain rule expansion:
$\frac{\mathrm{dy}}{\mathrm{dx}} f \left(g \left(x\right)\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$
Deriving ${f}_{1} ' \left(x\right)$:
$\frac{\mathrm{dy}}{\mathrm{dx}} {\left({x}^{2} - 2 x + 1\right)}^{4} = 4 {\left({x}^{2} - 2 x + 1\right)}^{3} \cdot \left(2 x - 2\right)$
${f}_{1} ' \left(x\right) = {\left({x}^{2} - 2 x + 1\right)}^{3} \left(8 x - 8\right)$
Deriving ${f}_{2} ' \left(x\right)$:
$\frac{\mathrm{dy}}{\mathrm{dx}} \left(4 {x}^{6} + 5\right) = 24 {x}^{5}$
Now, we reassemble (and simplify):
$g ' \left(x\right) = {\left({x}^{2} - 2 x + 1\right)}^{3} \left(8 x - 8\right) \left(4 {x}^{6} + 5\right) + 24 {x}^{5} {\left({x}^{2} - 2 x + 1\right)}^{4}$
$g ' \left(x\right) = 8 \left({\left({x}^{2} - 2 x + 1\right)}^{3} \left(x - 1\right) \left(4 {x}^{6} + 5\right) + 3 {x}^{5} {\left({x}^{2} - 2 x + 1\right)}^{4}\right)$
Note that the factored form of ${x}^{2} - 2 x + 1$ is ${\left(x - 1\right)}^{2}$
$g ' \left(x\right) = 8 \left({\left({\left(x - 1\right)}^{2}\right)}^{3} \left(x - 1\right) \left(4 {x}^{6} + 5\right) + 3 {x}^{5} {\left({\left(x - 1\right)}^{2}\right)}^{4}\right)$
$g ' \left(x\right) = 8 \left({\left(x - 1\right)}^{6} \left(x - 1\right) \left(4 {x}^{6} + 5\right) + 3 {x}^{5} {\left(x - 1\right)}^{8}\right)$
$g ' \left(x\right) = 8 \left({\left(x - 1\right)}^{7} \left(4 {x}^{6} + 5\right) + 3 {x}^{5} {\left(x - 1\right)}^{8}\right)$
$g ' \left(x\right) = 8 {\left(x - 1\right)}^{7} \left(\left(4 {x}^{6} + 5\right) + 3 {x}^{5} \left(x - 1\right)\right)$
$g ' \left(x\right) = 8 {\left(x - 1\right)}^{7} \left(4 {x}^{6} + 5 + 3 {x}^{6} - 3 {x}^{5}\right)$
color(green)(g'(x)=8(x-1)^7(7x^6-3x^5+5)
##### Impact of this question
66 views around the world
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http://mathhelpforum.com/calculus/79297-help-natural-logs.html
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# Math Help - help with natural logs
1. ## help with natural logs
I am trying to brush up on logs and I am having a hard time finding some more challenging problems. Here is an example of one.
$ln\frac{{(2x+3)^5}{\sqrt{4-sinx}}}{{(9+e^{3x})}{(9-2x)^4}}$
I am looking to practice more problems like this one. For example, what would I do if I had a third root along with the 4-sinx?
2. $\ln{\frac{{(2x+3)^5}{\sqrt[3]{4-\sin{x}}}}{{(9+e^{3x})}{(9-2x)^4}}}$
$5\ln(2x+3) + \frac{1}{3}\ln(4-\sin{x}) - \ln(9+e^{3x}) - 4\ln(9-2x)$
3. Thanks, but since you took 4-sinx out of the square root, don't you normally have to multiply that to the 1/2, so why would you then not have (1/3)(1/2)?
4. Originally Posted by gammaman
I am trying to brush up on logs and I am having a hard time finding some more challenging problems. Here is an example of one.
$ln\frac{{(2x+3)^5}{\sqrt{4-sinx}}}{{(9+e^{3x})}{(9-2x)^4}}$
Please pardon my confusion, but what are you supposed to do with this? You posted this to calculus; are you supposed to differentiate the associated function?
Thank you!
5. Oh yes, i am supposed to differentiate. However as said before I need some of the more complicated ones to practice, both diff and integration. I don't mean to beg but the ones in the text book and the ones I am finding on-line are far less complicated then the ones that will be on my test.
6. Originally Posted by gammaman
Thanks, but since you took 4-sinx out of the square root, don't you normally have to multiply that to the 1/2, so why would you then not have (1/3)(1/2)?
$\sqrt[3]{a} = a^{\frac{1}{3}}$
$\sqrt{a} = a^{\frac{1}{2}}$
see the difference?
a radical with no index is presumed to be a square root.
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http://sourceforge.net/p/hugin/hugin/ci/8f0f49ab22cc45a0541e5da0ca6402b1861f31a0/tree/src/foreign/vigra/stdimagefunctions.hxx
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Work at SourceForge, help us to make it a better place! We have an immediate need for a Support Technician in our San Francisco or Denver office.
## [8f0f49]: src / foreign / vigra / stdimagefunctions.hxx Maximize Restore History
Download this file
### stdimagefunctions.hxx 97 lines (88 with data), 4.8 kB
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 /************************************************************************/ /* */ /* Copyright 1998-2002 by Ullrich Koethe */ /* Cognitive Systems Group, University of Hamburg, Germany */ /* */ /* This file is part of the VIGRA computer vision library. */ /* ( Version 1.4.0, Dec 21 2005 ) */ /* The VIGRA Website is */ /* http://kogs-www.informatik.uni-hamburg.de/~koethe/vigra/ */ /* Please direct questions, bug reports, and contributions to */ /* [email protected] or */ /* [email protected] */ /* */ /* Permission is hereby granted, free of charge, to any person */ /* obtaining a copy of this software and associated documentation */ /* files (the "Software"), to deal in the Software without */ /* restriction, including without limitation the rights to use, */ /* copy, modify, merge, publish, distribute, sublicense, and/or */ /* sell copies of the Software, and to permit persons to whom the */ /* Software is furnished to do so, subject to the following */ /* conditions: */ /* */ /* The above copyright notice and this permission notice shall be */ /* included in all copies or substantial portions of the */ /* Software. */ /* */ /* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND */ /* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES */ /* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND */ /* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT */ /* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, */ /* WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING */ /* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR */ /* OTHER DEALINGS IN THE SOFTWARE. */ /* */ /************************************************************************/ #ifndef VIGRA_STDIMAGEFUNCTIONS_HXX #define VIGRA_STDIMAGEFUNCTIONS_HXX /** \page PointOperators Point Operators
\ref InitAlgo
init images or image borders
\ref InspectAlgo
Apply read-only functor to every pixel
\ref InspectFunctor
Functors which report image statistics
\ref CopyAlgo
Copy images or regions
\ref TransformAlgo
apply functor to calculate a pixelwise transformation of one image
\ref TransformFunctor
frequently used unary transformation functors
\ref CombineAlgo
apply functor to calculate a pixelwise transformation from several image
\ref CombineFunctor
frequently used binary transformations functors
\ref MultiPointoperators
Point operators on multi-dimensional arrays
\#include "vigra/stdimagefunctions.hxx"
Namespace: vigra see also: \ref FunctorExpressions "Automatic Functor Creation" */ #include "vigra/initimage.hxx" #include "vigra/inspectimage.hxx" #include "vigra/copyimage.hxx" #include "vigra/transformimage.hxx" #include "vigra/combineimages.hxx" #include "vigra/resizeimage.hxx" #endif // VIGRA_STDIMAGEFUNCTIONS_HXX
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https://link.springer.com/chapter/10.1007%2F978-3-642-10217-2_15
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IWOCA 2009: Combinatorial Algorithms pp 125-133
# Note on Decomposition of Kn,n into (0,j)-prisms
• Sylwia Cichacz
• Dalibor Fronček
• Petr Kovář
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5874)
## Abstract
R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K 6n,6n . In [2] the first two authors established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph K n,n into certain families of 3-regular graphs of order 2n. In this paper we tackle the problem of decompositions of K n,n into 3-regular graphs some more. We will show that certain families of 3-regular graphs of order 2n decompose the complete bipartite graph $$K_{\frac{3n}{2},\frac{3n}{2}}$$.
## References
1. 1.
Bermond, J.C., Huang, C., Sotteau, D.: Balanced cycle and circuit designs: Even case. Ars. Combinat. 5, 293–318 (1978)
2. 2.
Cichacz, S., Fronček, D.: Factorization of K n,n into (0,j)-prisms. Information Processing Letters 109, 932–934 (2009)
3. 3.
El-Zanati, S.I., Vanden Eynden, C.: Decompositions of K m,n into cubes. J. Comb. Designs 4, 51–57 (1996)
4. 4.
Fronček, D.: Cyclic type decompositions of complete bipartite graphs into hypercubes. Australasian Journal of Combinatorics 25, 201–209 (2002)
5. 5.
Fronček, D.: Oberwolfach rectangular table negotiation problem. Discr. Math. 309, 501–504 (2009)
6. 6.
Fronček, D., Kovář, P., Kubesa, M.: Decomposition of complete graphs into C m[2] (submitted)Google Scholar
7. 7.
Frucht, R.W., Gallian, J.A.: Labelling Prisms. Ars. Combin. 26, 69–82 (1988)
8. 8.
Häggkvist, R.: Decompositions of complete bipartite graphs. In: Surveys in combinatorics, Norwich. London Math. Soc. Lecture Note Ser., vol. 141, pp. 115–147. Cambridge Univ. Press, Cambridge (1989)Google Scholar
9. 9.
Piotrowski, W.-L.: The solution of the bipartite analogue of the Oberwolfach problem. Discr. Math. 97, 339–356 (1991)
10. 10.
Sotteau, D.: Decomposition of K m,n (K m,n*) into cycles (circuits) of length 2k. J. Comb. Theory, Ser. B 30, 75–81 (1981)
11. 11.
Rosa, A.: On certain valuations of the vertices of a graph. In: Theory of Graphs (Internat. Symposium), Rome, July 1966, pp. 349–355. Gordon and Breach, N. Y. and Dunod Paris (1967)Google Scholar
12. 12.
Ringel, G., Lladó, A.S., Serra, O.: Decomposition of complete bipartite graphs into trees. DMAT Research Report 11/96, Univ. Politecnica de CatalunyaGoogle Scholar
## Authors and Affiliations
• Sylwia Cichacz
• 1
• Dalibor Fronček
• 2
• Petr Kovář
• 3
1. 1.AGH University of Science and Technology
2. 2.University of Minnesota Duluth
3. 3.VŠB – Technical University of Ostrava
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https://mattbaker.blog/2014/04/
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# Effective Chabauty
One of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if $X / {\mathbf Q}$ is an algebraic curve of genus at least 2, then the set $X({\mathbf Q})$ of rational points on $X$ is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set $X({\mathbf Q})$. The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on $X$, but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of $X({\mathbf Q})$ provided that the Mordell-Weil rank of the Jacobian of $X$ is not too large. (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)
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https://deepai.org/publication/linear-tsne-optimization-for-the-web
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DeepAI
# Linear tSNE optimization for the Web
The t-distributed Stochastic Neighbor Embedding (tSNE) algorithm has become in recent years one of the most used and insightful techniques for the exploratory data analysis of high-dimensional data. tSNE reveals clusters of high-dimensional data points at different scales while it requires only minimal tuning of its parameters. Despite these advantages, the computational complexity of the algorithm limits its application to relatively small datasets. To address this problem, several evolutions of tSNE have been developed in recent years, mainly focusing on the scalability of the similarity computations between data points. However, these contributions are insufficient to achieve interactive rates when visualizing the evolution of the tSNE embedding for large datasets. In this work, we present a novel approach to the minimization of the tSNE objective function that heavily relies on modern graphics hardware and has linear computational complexity. Our technique does not only beat the state of the art, but can even be executed on the client side in a browser. We propose to approximate the repulsion forces between data points using adaptive-resolution textures that are drawn at every iteration with WebGL. This approximation allows us to reformulate the tSNE minimization problem as a series of tensor operation that are computed with TensorFlow.js, a JavaScript library for scalable tensor computations.
• 7 publications
• 10 publications
• 5 publications
• 7 publications
• 9 publications
• 6 publications
02/19/2017
### Compressive Embedding and Visualization using Graphs
Visualizing high-dimensional data has been a focus in data analysis comm...
09/05/2022
### Opening the black-box of Neighbor Embedding with Hotelling's T2 statistic and Q-residuals
In contrast to classical techniques for exploratory analysis of high-dim...
02/13/2019
### Do Subsampled Newton Methods Work for High-Dimensional Data?
Subsampled Newton methods approximate Hessian matrices through subsampli...
12/25/2017
### Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding
t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for ...
10/24/2022
### A fast multilevel dimension iteration algorithm for high dimensional numerical integration
In this paper, we propose and study a fast multilevel dimension iteratio...
05/22/2019
### Learning Networked Exponential Families with Network Lasso
The data arising in many important big-data applications, ranging from s...
03/15/2022
### Scalable Bigraphical Lasso: Two-way Sparse Network Inference for Count Data
Classically, statistical datasets have a larger number of data points th...
## 1 Introduction
Understanding how data points are arranged in a high-dimensional space plays a crucial role in exploratory data analysis [25]
. In recent years, non-linear dimensionality reduction techniques, also known as manifold learning algorithms, became a powerful tools for mining knowledge from data, e.g., the presence of clusters at different scales. Manifold learning algorithms for data visualization reduce the dimensionality of the points to 2 or 3 dimensions while preserving some characteristic of the data such as the preservation of the local neighborhoods. The success of this approach is motivated by the fact that most of the real-world data satisfy the “manifold hypothesis”, i.e., it lies on relatively low-dimensional manifolds embedded in a high-dimensional space. The manifolds are typically mapped to a lower dimensional space and visualized and analyzed, for example, in a scatterplot.
The t-distributed Stochastic Neighbor Embedding (tSNE) algorithm [27] has been accepted as the state of the art for nonlinear dimensionality reduction applied to visual analysis of high-dimensional space in several application areas, such as life sciences [2, 4, 15]
and machine learning model understanding and human-driven supervision
[17, 21, 9]
. tSNE can be logically separated in two computation modules; first it computes the similarities of the high-dimensional points as a joint probability distribution and, second, it minimizes the Kullback–Leibler divergence
[12] of a similarly computed joint probability distribution that measures the closeness of the points in the low dimensional space. The memory and computational complexity of the algorithm is , where is the number of data points.
Given its popularity, research efforts have been spent on improving the computational and memory complexity of the algorithm. While many works focused on the improvement of the similarity computation [26, 20, 24, 19, 16], only limited effort have been spent in improving the minimization algorithm employed for the creation of the embedding [26, 16, 11]. The most notable of these improvements is the Barnes-Hut-SNE that makes use of an -body simulation approach [1] to approximate the repulsive forces between the data points. Despite the improvements, the minimization requires many minutes using a highly-optimized C++ implementation.
In this work we focus on the minimization of the objective function for the creation of the embedding. We observe that the heavy tail of the t-Student distribution used by tSNE makes the application of the -body simulation not particularly effective. To address this problem we propose a novel minimization approach that embraces this characteristic and we reformulate the gradient of the objective function as a function of scalar fields and tensor operations. Our technique has linear computational and memory complexity and, more importantly, is implemented in a GPGPU fashion. The latter allowed us to implement a version for the browser that minimizes the objective function for standard datasets in a matter of seconds.
The contribution of this work is twofold:
• A linear complexity minimization of the tSNE objective function that makes use of the modern WebGL rendering pipeline. Specifically, we
• approximate the repulsive forces between data points by drawing low-resolution textures and
• An efficient implementation of our result that is released as part of Google’s TensorFlow.js library
The rest of the paper is structured as follows. In the next section, we provide a theoretical primer on the tSNE algorithm that is needed to understand the related work (Section 3) and our contributions (Section 4). In Section 5, we provide the details regarding our implementation, released within Google’s TensorFlow.js library.
## 2 tSNE
In this section, we provide a short introduction to tSNE [27], which is needed to understand the related work and our contribution. tSNE interprets the overall distances between data points in the high-dimensional space as a symmetric joint probability distribution that encodes their similarities. Likewise a joint probability distribution is computed that describes the similarity in the low-dimensional space. The goal is to achieve a representation, referred to as embedding, in the low dimensional space, in which faithfully represents . This is achieved by optimizing the positions in the low-dimensional space to minimize the cost function given by the Kullback-Leibler () divergence between the joint-probability distributions and :
C(P,Q)=KL(P||Q)=N∑i=1N∑j=1,j≠ipijln(pijqij) (1)
Given two data points and in the dataset , the probability models the similarity of these points in the high-dimensional space. To this extent, for each point, a Gaussian kernel
is chosen, whose variance
is defined according to the local density in the high-dimensional space and then is described as follows:
pij=pi|j+pj|i2N, (2)
wherepj|i=exp(−(||xi−xj||2)/(2σ2i))∑Nk≠iexp(−(||xi−xk||2)/(2σ2i)) (3)
can be seen as a relative measure of similarity based on the local neighborhood of a data point . The perplexity value is a user-defined parameter that describes the effective number of neighbors considered for each data point. The value of is chosen such that for fixed and each :
μ=2−∑Njpj|ilog2pj|i (4)
A Student’s t-Distribution
with one degree of freedom is used to compute the joint probability distribution in the low-dimensional space
, where the positions of the data points should be optimized. Given two low-dimensional points and , the probability that describes their similarity is given by:
qij=((1+||yi−yj||2)Z)−1 (5)
(6)
The gradient of the Kullback-Leibler divergence between and is used to minimize (see Eq. 1). It indicates the change in position of the low-dimensional points for each step of the gradient descent and is given by:
δCδyi =4(Fattri−Frepi) (7) =4(N∑j≠ipijqijZ(yi−yj)−N∑j≠iq2ijZ(yi−yj)) (8)
The gradient descent can be seen as an -body simulation [1], where each data-point exerts an attractive and a repulsive force on all other points ( and ).
## 3 Related Work
We now present the work that has been done to improve the tSNE computation of tSNE embeddings in term of quality and scalability. Van der Maaten proposed the Barnes-Hut-SNE (BH-SNE) [26], which reduces the complexity of the algorithm to for both the similarity computations and the objective function minimization. More specifically, in the BH-SNE approach the similarity computations are seen as a -nearest neighborhood graph computation problem, which is obtained using a Vantage-Point Tree [29]. The minimization of the objective function is then seen as an -body simulation, which is solved by applying the Barnes-Hut algorithm [3].
Pezzotti et al. [20] observed that the computation of the
-nearest neighborhood graph for high-dimensional spaces using the Vantage-Point Tree is affected by the curse of dimensionality, limiting the efficiency of the computation. To overcome this limitation, they proposed the Approximated-tSNE (A-tSNE) algorithm
[20], where approximated -nearest neighborhood graphs are computed using a forest of randomized KD-trees [18]. Moreover, A-tSNE adopts the novel Progressive Visual Analytics paradigm [23, 7], allowing the user to observe the evolution of the embedding during the minimization of the objective function. This solution does not only enable a user-driven early termination of the algorithm but also led to novel discoveries in cell differentiation pathways [15].
A similar observation was later made by Tang et al. that led to the development of the LargeVis technique [24]. LargeVis uses random projection trees [5]
followed by a kNN descent procedure
[6]
for the computation of the similarities and a different objective function that is minimized using a Stochastic Gradient Descent approach
[10]. Despite the improvements, both the A-tSNE and LargeVis tools require 15 to 20 minutes to compute the embedding of the MNIST dataset [14], a 784-dimensional dataset of 60k images of handwritten digits, that is often used as benchmark for manifold-learning algorithms. Better performance is achieved by the UMAP algorithm [16], which provides a different formulation of the dimensionality-reduction problem as a cross-entropy minimization between topological representations. Computationally, UMAP follows very closely LargeVis and adopts a kNN descent procedure [6] and Stochastic Gradient Descent minimization of the objective function.
A completely different approach is taken in the Hierarchical Stochastic Neighbor Embedding algorithm (HSNE) [19]. HSNE efficiently builds a hierarchical representation of the manifolds and embeds only a subset of the initial data that represent an overview of the available manifolds. This approach can embed the MNIST dataset in less than 2 minutes. The user can “drill-in” the hierarchy by requesting more detailed embeddings that reveal smaller clusters of data points. HSNE is implemented in the Cytosplore [8] tool and led to the discovery of new cell populations [28] in large samples, i.e., containing more that 5 million cells. While HSNE produces better embeddings due to an easier minimization process, it has the downside that it does not produce a single embedding that depicts the complete dataset but it requires the user to actively explore the data and it generates embeddings on request.
The techniques presented so far do not take advantage of the target domain, in which the data is embedded. As a matter of fact, tSNE is mostly used for data visualization in 2-dimensional scatterplots, while the previously introduced techniques are general and can be used for higher dimensional spaces. Based on this observation, Kim et al. introduced the PixelSNE technique [11] that employs a -body simulation approach similar to the BH-SNE, but quantizes the embedding space to the pixels used for visualizing the embedding. However, PixelSNE requires to scale the number of used pixels with respect to the size of the dataset in order to achieve a good embedding quality due to the quantization of the embedding space.
In this work, we take advantage of the 2-dimensional domain in which the embedding resides and we propose a more efficient way to minimize the tSNE objective function. Contrary to PixelSNE we observe that, by quantizing only the 2-dimensional space for the computation of the repulsive forces presented in Equation 8, embeddings that are hardly distinguishable from those generated by the BH-SNE implementation are computed in a fraction of the time. Moreover, we this approach allows to develop a linear complexity GPGPU implementation that runs in the client side of the browser.
## 4 Linear complexity tSNE minimization
In this section, we present our approach to minimize the objective function, presented in Equation 1, by rewriting its gradient, presented in Equation 7. The computation of the gradient relies on a scalar field and a vector field that are computed in linear time on the GPU.
### 4.1 Gradient of the objective function
The gradient of the objective function has the same form as the previously introduced one:
δCδyi =4(^Fattri−^Frepi), (9)
with attractive and repulsive forces acting on every point . We denote the forces with a to distinguish them from their original counterparts. Our main contribution is to rewrite the computation of the gradient as a form of a scalar field and a vector field .
S(p)=N∑i(1+||yi−p||2)−1,S:R2⇒R (10) V(p)=N∑i(1+||yi−p||2)−2(yi−p),V:R2⇒R2 (11)
Intuitively, represents the density of the points in the embedding space, according to the t-Student distribution, and it is used to compute the normalization of the joint probability distribution . An example of the field is shown in Figure 1b. The vector field represents the directional repulsive force applied to the entire embedding space. An example of is presented in Figure 1c-d, where the horizontal and vertical components are visualized separately. In the next section, we will present how both and are computed with a complexity of and sampled in constant time. For now, we assume these fields given and we present how the gradient of the objective function are derived from these two fields, accelerating hereby their calculation drastically.
For the attractive forces, we adopt the restricted neighborhood contribution as presented in the Barnes-Hut-SNE technique [26]. The rationale of this approach is that, by imposing a fixed perplexity to the Gaussian kernel, only a limited number of neighbors effectively apply an attractive force on any given point (see Equation 3 and 4). Therefore we limit the number of contributing points to a multiple of the value of perplexity, equal to three times the value of the chosen perplexity, effectively reducing the computational and memory complexity to , since where is the size of the neighborhood.
^Fattri=^Z∑l∈kNN(i)pilqil(yi−yl) (12)
The normalization factor , as it was presented in Equation 6, has complexity . In our approach we compute in linear time by sampling the scalar field .
^Z=N∑l=1(S(yl)−1) (13)
Note that and formulation is identical but, since we assume that is computed in linear time, computing has linear complexity. Moreover, since does not depend on the point , for which we are computing the gradient, it needs to be computed only once for all the points.
The repulsive force assumes even a simpler form
^Frepi=V(yi)/^Z, (14)
being the value of the vector field in the location identified by the coordinates normalized by . Similarly as for , has an equivalent formulation as but with computational and memory complexity equal to . So far, we assumed that and are computed in linear time and queried in constant time. In the next section we present how we achieve this result by using the WebGL rendering pipeline to compute an approximation of these fields.
### 4.2 Computation of the fields
In the previous section, we formulated the gradient of the objective function as dependent from a scalar field and a vector field . If the fields are evaluated independently, the complexity of the approach is due to the summation in Equations 10 and 11. We achieve a linear complexity by precomputing and approximating the fields on the GPU using textures of appropriate resolution. An example of the fields for the MNIST dataset [14] is given in Figure 1b-d.
A similar approach is used for Kernel Density Estimation
[22] that has applications in visualization [13] and non-parametric clustering [8]
. In this setting, given a number of points, the goal is to estimate a 2-dimensional probability density function, from which the points were sampled. This is usually achieved by overlaying a Gaussian kernel, whose
has to be estimated, on top of every data point.
Lampe et al. [13] were the first to propose a computation of the kernel density on the GPU for a visualization purpose. They observed that the Gaussian kernel used for estimating the density has a limited support, i.e., having value almost equal to zero if they are sufficiently far away from the origin. A good approximation of the density function is then achieved by drawing, instead of the points, little quads that are textured with a precomputed Gaussian kernel. By using additive blending available in OpenGL, i.e., by summing the values in every pixel, the resulting drawing corresponds to the desired density function.
If we analyze Equations 10 and 11, we can observe that every element in the summations for both and have a limited support, making it indeed very similar to the Kernel Density Estimation case discussed before. The drawn functions, however, are different and Figure 2 shows them for and . Therefore, we can compute the fields by drawing over a texture with a single additive drawing operation. Each point is drawn as a quad and colored with a floating-point RGB texture where each channel encodes one of the functions shown in Figure 2.
Contrary to the Kernel Density Estimation case, where the size of the quads changes according to the chosen for the Gaussian kernel, our functions have a fixed support in the embedding space. Therefore, given a certain embedding , the resolution of the texture influences the quality of the approximation but not the overall shape of the fields. To achieve linear complexity, we define the resolution of the target texture according to the size of the embedding. In this way, every data point updates the value of a constant number of pixels in the target texture, effectively leading to complexity for the computation of the fields.
Computing the value of and for a point
corresponds to extracting the interpolated value in the textures that represents the fields. This operation is extremely fast on the GPU, as WebGL natively supports the bilinear interpolation of texture values. In the next section, we provide a more detailed overview of the computational pipeline as a number of tensor operations and custom drawing operations.
## 5 Implementation
In this section, we present how the ideas presented in the previous section are concretely implemented in a JavaScript library that can be used to execute an efficient tSNE computation directly in the user’s browser. Figure 3 shows an overview of the overall approach. We rely on TensorFlow.js, a WebGL accelerated, browser-based JavaScript library for training and deploying machine-learning models. TensorFlow.js has extensive support for tensor operations that we integrate with custom shader computations to derive the tSNE embeddings.
The randomly initialized tSNE embedding is stored in a 2-dimensional tensor. We then proceed to compute the repulsive forces and attractive forces , shown respectively in the lower and upper side of Figure 3. The attractive forces are computed in a custom shader that measures the sum of the contribution of every neighboring point in the high-dimensional space. The neighborhoods are encoded in the joint probability distribution that is stored in a WebGL texture. can be computed server-side, for example using an approximated -nearest-neighborhood algorithm [18, 6, 5] or by the Hierarchical-SNE technique [19]. However, we provide a WebGL implementation of the kNN-Descent algorithm [6] and the computation of directly in the browser to enable a client-side only computational workflow.
The repulsive forces are computed using the approach presented in previous sections. In a custom shader, we draw for each point, whose location is defined by the value in the embedding tensor, a quad that is textured with the functions presented in Figure 2. The resulting 3-channel texture, an example of which is presented in Figure 1b-d, represents the scalar field and the vector field . For each embedding point , the values of and are stored in tensors and are computed by a custom WebGL shader that interpolates the value of the texture in the corresponding channel. The normalization factor is then obtained by summing all the elements in the tensor with the interpolated values of , an operation that is efficiently performed on the GPU by TensorFlow.js.
The remaining computational steps are computed as tensor operations. is obtained by dividing the interpolated values of by , and, by adding the attractive forces , the gradient of the objective function is obtained. The gradient is then added to the embedding, hence, modifying the position of the points according to their similarities. Our work is released as part of the TensorFlow.js library and can be found on GitHub at the following address: https://github.com/tensorflow/tfjs-tsne
## 6 Conclusion
In this work we presented a novel approach for the optimization of the objective function of the t-Distributed Stochastic Neighbor Embedding algorithm (tSNE) that scales to large datasets in the client side of the browser. Our approach relies on modern graphics hardware to efficiently compute the gradient of the objective function from a scalar field that represents the point density and and the directional repulsive forces in the embedding space. The implementation of the technique is based on the TensorFlow.js library and can be found on GitHub at the following address: https://github.com/tensorflow/tfjs-tsne. Examples that validate our approach can also be found on GitHub https://github.com/tensorflow/tfjs-tsne-examples.
As future work, we want to perform a systematic analysis of the results of our technique, both in terms of speed and quality of the results. More specifically we plan to perform a comparison with single-embedding techniques such as LargeVis [24], UMAP [16] and PixelSNE [11]. Moreover, we want to integrate our technique in the Hierarchical-SNE technique [19]
and in tools for the analysis of Deep Neural Networks such as the Embedding Projector
, TensorBoard and DeepEyes [21]. To conclude, we believe that having a scalable tSNE implementation that runs in the browsers open exciting possibilities for the development of new analytical systems.
## References
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• [2] E.-a. D. Amir, K. L. Davis, M. D. Tadmor, E. F. Simonds, J. H. Levine, S. C. Bendall, D. K. Shenfeld, S. Krishnaswamy, G. P. Nolan, and D. Pe’er, “viSNE enables visualization of high dimensional single-cell data and reveals phenotypic heterogeneity of leukemia,” Nature biotechnology, vol. 31, no. 6, pp. 545–552, 2013.
• [3] J. Barnes and P. Hut, “A hierarchical o (n log n) force-calculation algorithm,” nature, vol. 324, p. 446, 1986.
• [4] B. Becher, A. Schlitzer, J. Chen, F. Mair, H. R. Sumatoh, K. W. W. Teng, D. Low, C. Ruedl, P. Riccardi-Castagnoli, and M. Poidinger, “High-dimensional analysis of the murine myeloid cell system,” Nature immunology, vol. 15, no. 12, pp. 1181–1189, 2014.
• [5] S. Dasgupta and Y. Freund, “Random projection trees and low dimensional manifolds,” in
Proceedings of the fortieth annual ACM symposium on Theory of computing
, 2008, pp. 537–546.
• [6] W. Dong, C. Moses, and K. Li, “Efficient k-nearest neighbor graph construction for generic similarity measures,” in Proceedings of the 20th international conference on World wide web. ACM, 2011, pp. 577–586.
• [7] J.-D. Fekete and R. Primet, “Progressive analytics: A computation paradigm for exploratory data analysis,” arXiv preprint arXiv:1607.05162, 2016.
• [8] T. Höllt, N. Pezzotti, V. van Unen, F. Koning, E. Eisemann, B. Lelieveldt, and A. Vilanova, “Cytosplore: Interactive immune cell phenotyping for large single-cell datasets,” in Computer Graphics Forum, vol. 35, 2016, pp. 171–180.
• [9] M. Kahng, P. Y. Andrews, A. Kalro, and D. H. P. Chau, “A cti v is: Visual exploration of industry-scale deep neural network models,” IEEE transactions on visualization and computer graphics, vol. 24, no. 1, pp. 88–97, 2018.
• [10] J. Kiefer and J. Wolfowitz, “Stochastic estimation of the maximum of a regression function,” The Annals of Mathematical Statistics, pp. 462–466, 1952.
• [11] M. Kim, M. Choi, S. Lee, J. Tang, H. Park, and J. Choo, “Pixelsne: Visualizing fast with just enough precision via pixel-aligned stochastic neighbor embedding,” arXiv preprint arXiv:1611.02568, 2016.
• [12] S. Kullback, Information theory and statistics. Courier Corporation, 1997.
• [13] O. D. Lampe and H. Hauser, “Interactive visualization of streaming data with kernel density estimation,” in Visualization Symposium (PacificVis), 2011 IEEE Pacific, 2011, pp. 171–178.
• [14] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, 1998.
• [15] N. Li, V. van Unen, T. Höllt, A. Thompson, J. van Bergen, N. Pezzotti, E. Eisemann, A. Vilanova, S. M. Chuva de Sousa Lopes, B. P. Lelieveldt, and F. Koning, “Mass cytometry reveals innate lymphoid cell differentiation pathways in the human fetal intestine,” Journal of Experimental Medicine, 2018.
• [16] L. McInnes and J. Healy, “Umap: Uniform manifold approximation and projection for dimension reduction,” arXiv preprint arXiv:1802.03426, 2018.
• [17] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski et al.
, “Human-level control through deep reinforcement learning,”
Nature, vol. 518, no. 7540, pp. 529–533, 2015.
• [18] M. Muja and D. Lowe, “Scalable nearest neighbor algorithms for high dimensional data,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 36, no. 11, pp. 2227–2240, 2014.
• [19] N. Pezzotti, T. Höllt, B. Lelieveldt, E. Eisemann, and A. Vilanova, “Hierarchical stochastic neighbor embedding,” in Computer Graphics Forum, vol. 35, 2016, pp. 21–30.
• [20] N. Pezzotti, B. Lelieveldt, L. van der Maaten, T. Hollt, E. Eisemann, and A. Vilanova, “Approximated and user steerable tsne for progressive visual analytics,” IEEE Transactions on Visualization and Computer Graphics, vol. PP, no. 99, pp. 1–1, 2016.
• [21] N. Pezzotti, T. Höllt, J. Van Gemert, B. P. Lelieveldt, E. Eisemann, and A. Vilanova, “Deepeyes: Progressive visual analytics for designing deep neural networks,” IEEE transactions on visualization and computer graphics, vol. 24, no. 1, pp. 98–108, 2018.
• [22] M. Rosenblatt, “Remarks on some nonparametric estimates of a density function,” The Annals of Mathematical Statistics, pp. 832–837, 1956.
• [23] C. Stolper, A. Perer, and D. Gotz, “Progressive visual analytics: User-driven visual exploration of in-progress analytics,” IEEE Transactions on Visualization and Computer Graphics, vol. 20, no. 12, pp. 1653–1662, 2014.
• [24] J. Tang, J. Liu, M. Zhang, and Q. Mei, “Visualizing large-scale and high-dimensional data,” in Proceedings of the 25th International Conference on World Wide Web, 2016, pp. 287–297.
• [25] J. W. Tukey, “The future of data analysis,” The Annals of Mathematical Statistics, pp. 1–67, 1962.
• [26] L. Van Der Maaten, “Accelerating t-sne using tree-based algorithms,” The Journal of Machine Learning Research, vol. 15, no. 1, pp. 3221–3245, 2014.
• [27] L. van der Maaten and G. Hinton, “Visualizing data using t-SNE,” Journal of Machine Learning Research, vol. 9, no. 2579-2605, p. 85, 2008.
• [28] V. van Unen, T. Hollt, N. Pezzotti, N. Li, M. J. T. Reinders, E. Eisemann, A. Vilanova, F. Koning, and B. P. F. Lelieveldt, “Interactive visual analysis of mass cytometry data by hierarchical stochastic neighbor embedding reveals rare cell types,” Nature Communications, vol. 8, 2017.
• [29] P. N. Yianilos, “Data structures and algorithms for nearest neighbor search in general metric spaces,” in Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, 1993, pp. 311–321.
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https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-7-exponents-and-exponential-functions-7-8-geometric-sequences-practice-and-problem-solving-exercises-page-470/29
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Algebra 1: Common Core (15th Edition)
The recursive formula is $a_{1}$=1 and $a_{n}$=$a_{n-1}$ $\times$ 5.
Use the recursive formula $a_{1}$=1, $a_{n}$=$a_{n-1}$$\times$r. The starting value is 1 so it is the $a_{1}$ You have the sequence 1,5,25 so use the common ratio formula(r=$\frac{{a_{2}}}{{a_{1}}}$) r=$\frac{5}{1}$=5 and r=$\frac{25}{5}$=5.So the common ratio is 5.Substitute the value of a1 and R into the recursive formula: The recursive formula is $a_{1}$=1 and $a_{n}$=$a_{n-1}$ $\times$ 5.
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http://openstudy.com/updates/5168962be4b05937219b6224
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## yashar806 2 years ago need help !!! Delete Cancel Submit
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1. yashar806
• 2 years ago
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We have four boxes, each of which contains ten coloured balls: Box 1 contains 4 red balls, 5 green balls and 1 yellow ball Box 2 contains 3 red balls, 5 green balls and 2 yellow balls Box 3 contains 2 red balls, 5 green balls and 3 yellow balls Box 4 contains 1 red ball, 5 green balls and 4 yellow balls If we randomly select one ball from each box, what is the probability of selcecting exactly one red ball? If we randomly select one ball from each box, what is the probability of selcecting at one red ball?
2. honourznr
• 2 years ago
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ı think (4/10)*(3/10)*(2/10)*(1/10)
3. yashar806
• 2 years ago
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4. yashar806
• 2 years ago
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but im not sure how to get that answer
5. honourznr
• 2 years ago
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6. honourznr
• 2 years ago
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we use all box isnt it?
7. yashar806
• 2 years ago
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i guess so
8. yashar806
• 2 years ago
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any ideA?
9. honourznr
• 2 years ago
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ı dont understand question very well 4 balls will red?
10. honourznr
• 2 years ago
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all the balls must be red ? 4 balls ? like red red red red ?
11. electrokid
• 2 years ago
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${4\over4+5+1}\times{3\over3+5+2}\times{2\over2+5+3}\times{1\over1+5+4}=?$
12. yashar806
• 2 years ago
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13. yashar806
• 2 years ago
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of course its add up to 1
14. electrokid
• 2 years ago
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no wait got
15. electrokid
• 2 years ago
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1 ball from EACH box
16. electrokid
• 2 years ago
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so, P = 1red from box1 and 3no-red + 1 red from box2 and 3 no red +...
17. yashar806
• 2 years ago
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still dont get it
18. PeterPan
• 2 years ago
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Okay... what's the probability of getting a red ball from ONLY the first box? That's the product of the probability of getting a red ball from the first box and NOT getting a red ball from the other three boxes...
19. yashar806
• 2 years ago
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should I use bionomial probability?
20. PeterPan
• 2 years ago
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I don't know what that is ^ (c'mon go easy on me :> ) But it seems the odds of getting a red ball from the first box is 2 out of 5 odds of NOT getting a red ball from the second box is 7 out of 10 not getting a red ball from the third : 4 out of 5 not getting a red ball from the fourth : 9 out of 10 Now just multiply these, and this'll be one of the things you add...
21. honourznr
• 2 years ago
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ı did the questions
22. PeterPan
• 2 years ago
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merhaba, @honourznr What did you get? ^.^
23. yashar806
• 2 years ago
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merhaba?
24. honourznr
• 2 years ago
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(4/10*7/10*8/10*9/10)+ (3/10*6/10*8/10*9/10)+(2/10*9/10*7/10*6/10)+(1/10*8/10*7/10*6/10)= 0.436=0.44
25. PeterPan
• 2 years ago
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It's like "hello" only more exotic
26. honourznr
• 2 years ago
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:)
27. honourznr
• 2 years ago
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merhaba
28. PeterPan
• 2 years ago
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Well, as long as @honourznr did the arithmetic correctly... there's no reason for it to be wrong :)
29. PeterPan
• 2 years ago
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On second thought, maybe you need to redo that, @honourznr
30. honourznr
• 2 years ago
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why?
31. PeterPan
• 2 years ago
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Okay, it's correct.. just jumbled up in the products, but correct :D
32. yashar806
• 2 years ago
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woww asweome what about the second one ?
33. PeterPan
• 2 years ago
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At least one red ball?
34. honourznr
• 2 years ago
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for example red green green yellow
35. yashar806
• 2 years ago
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ben turk
36. PeterPan
• 2 years ago
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a little confusing...
37. PeterPan
• 2 years ago
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also, @honourznr I did not understand that :D
38. honourznr
• 2 years ago
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so yashar806 do you understand the answer
39. yashar806
• 2 years ago
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evet
40. honourznr
• 2 years ago
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bileydim en baştan türkçe konuşıurdum:D
41. yashar806
• 2 years ago
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Nerelisiniz?
42. honourznr
• 2 years ago
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sne össye falanmı hazırlnısn
43. yashar806
• 2 years ago
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DogoTürkistan'yi biliyor musun?
44. honourznr
• 2 years ago
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evet biliyorum bnde türkiyedeyim
45. honourznr
• 2 years ago
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46. yashar806
• 2 years ago
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bilmiyorm zatan
47. yashar806
• 2 years ago
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48. honourznr
• 2 years ago
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onur... senin
49. yashar806
• 2 years ago
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50. honourznr
• 2 years ago
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memnun oldum yaşar :) iyiakşamlar ben gidiyorum
51. yashar806
• 2 years ago
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Doğu türkistan'da
52. yashar806
• 2 years ago
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benda qok memnun oldum onur
53. honourznr
• 2 years ago
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bana mesaj yolla konuşuruz sonra ....
54. yashar806
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tamam , iyiakshamlar
55. honourznr
• 2 years ago
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https://chem.libretexts.org/Courses/Lumen_Learning/Book%3A_Statistics_for_the_Social_Sciences_(Lumen)/02%3A_1-_Types_of_Statistical_Studies_and_Producing_Data/2.12%3A_Sampling_(1_of_2)
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# 2.12: Sampling (1 of 2)
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### Learning Objectives
• For an observational study, critique the sampling plan. Recognize implications and limitations of the plan.
We now focus on observational studies and how to collect reliable and accurate data for an observational study.
We know that an observational study can answer questions about a population. But populations are generally large groups, so we cannot gather data from every individual in the population. Instead, we select a sample and gather data from the sample. We use the data from the sample to make statements about the population.
Here are two examples:
• A political scientist wants to know what percentage of college students consider themselves conservatives. The population is college students. It would be too time consuming and expensive to poll every college student, so the political scientist selects a sample of college students. Of course, the sample must be carefully selected to represent the political perspectives that are present in the population.
• A government agency plans to test airbags from Honda to determine if the airbags work properly. Testing an airbag means it has to be inflated and punctured, which ruins the airbag, so the researchers certainly cannot test every airbag. Instead, they test a sample of airbags and draw a conclusion about the quality of airbags from Honda.
### Important Point
Our goal is to use a sample to make valid conclusions about a population. Therefore, the sample must be representative of the population. A representative sample is a subset of the population that reflects the characteristics of the population.
A sampling plan describes exactly how we will choose the sample. A sampling plan is biased if it systematically favors certain outcomes.
In our discussion of sampling plans, we focus on surveys. The next example is a famous one that illustrates how biased sampling in a survey leads to misleading conclusions about the population.
## The 1936 Presidential Election
In 1936, Democrat Franklin Roosevelt and Republican Alf Landon were running for president. Before the election, the magazine Literary Digest sent a survey to 10 million Americans to determine how they would vote. More than 2 million people responded to the poll; 60% supported Landon. The magazine published the findings and predicted that Landon would win the election. However, Roosevelt defeated Landon in one of the largest landslide presidential elections ever.
What happened?
The magazine used a biased sampling plan. They selected the sample using magazine subscriptions, lists of registered car owners, and telephone directories. The sample was not representative of the American public. In the 1930s, Democrats were much less likely to own a car or have a telephone. The sample therefore systematically underrepresented Democrats. The poll results did not represent the way people in the general population voted.
Before we discuss a method for avoiding bias, let’s look at some examples of common survey plans that produce unreliable and potentially biased results.
### Example
Online polls: The American Family Association (AFA) is a conservative Christian group that opposes same-sex marriage. In 2004, the AFA began a campaign in support of a constitutional amendment to define marriage as strictly between a man and a woman. The group posted a poll on its website asking AFA members to voice their opinion about same-sex marriage. The AFA planned to forward the results to Congress as evidence of America’s opposition to same-sex marriage. Almost 850,000 people responded to the poll. In the poll, 60% favored legalizing same-sex marriage.
What happened? Against the wishes of the AFA, the link to the poll appeared in blogs, social-networking sites, and a variety of email lists connected to gay/lesbian/bisexual groups. The AFA claimed that gay rights groups had skewed its poll. Of course, the results of the poll would have been skewed in the other direction had only AFA members been allowed to participate.
This is an example of a voluntary response sample. The people in a voluntary response sample are self-selected, not chosen. For this reason, a voluntary response sample is biased because only people with strong opinions make the effort to participate.
Mall surveys: Have you ever noticed someone surveying people at a mall? People shopping at a mall are more likely to be teenagers, retired people, or people who have more money than the typical American. In addition, unless interviewers are carefully trained, they tend to interview people with whom they are comfortable talking. For these reasons, mall surveys frequently overrepresent the opinions of white middle-class or retired people. Mall surveys are an example of a convenience sample.
## How to Eliminate Bias in Sampling
In a voluntary response sample, people choose whether to respond. In a convenience sample, the interviewer chooses who will be part of the sample. In both cases, personal choice produces a biased sample. Random sampling is the best way to eliminate bias. Collecting a random sample is like pulling names from a hat (assuming every individual in the population has a name in the hat!). In a simple random sample everyone in the population has an equal chance of being chosen.
Reputable polling firms use techniques that are more complicated than pulling names out of a hat. But the goal is the same: eliminate bias by using random chance to decide who is in the sample.
Random samples will eliminate bias, even bias that may be hidden or unknown. The next three activities will reveal a bias that most of us have but don’t know that we have! We will see how random sampling avoids this bias.
### Random Samples
Instructions: Use the simulation below for this activity. You will see 60 circles. This is the “population.” Our goal is to estimate the average diameter of these 60 circles by choosing a sample.
1. Choose a sample of five circles that look representative of the population of all 60 circles. Mark your five circles by clicking on each of them. They will turn orange. Record the average diameter for the five circles. (Make sure you have five orange circles before you record the average diameter.)
2. Reset the simulation.
3. Choose another five circles and record the average diameter for this sample of circles. You can reuse a circle, but the sample should not have all the same circles. You now have the averages for two samples.
4. Reset and repeat for a total of 10 samples. Record the average diameter for each sample.
An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/sss/?p=36
Now we estimate the average diameter of the 60 circles using random samples.
Instructions: Use the simulation below for this activity. You will again see the same 60 circles. As before, this is the “population.” Our goal is to estimate the average diameter of these 25 circles by choosing a random sample.
1. Click on the “Generate sample” button to get a random sample of five circles by clicking on the random sample button. The simulation randomly chooses five circles. Record the average diameter for the random sample.
2. Reset the simulation using the reset button.
3. Click on the “Generate sample” button to get another random sample. Record the average diameter for this random sample. You now have the averages for two samples.
4. Reset and repeat for a total of 10 samples. Record the average diameter for each sample.
An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/sss/?p=36
### Learn By Doing
https://assessments.lumenlearning.co...sessments/3827
### Comment
Random selection also guarantees that the sample results do not change haphazardly from sample to sample. When we use random selection, the variability we see in sample results is due to chance. The results obey the mathematical laws of probability. We looked at this idea briefly in the Big Picture of Statistics. Probability is the machinery for drawing conclusions about a population on the basis of samples. To use this machinery, the sample must be chosen by random chance.
### Learn By Doing
https://assessments.lumenlearning.co...sessments/3406
### Learn By Doing
https://assessments.lumenlearning.co...sessments/3407
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https://www.varsitytutors.com/hotmath/hotmath_help/topics/converting-fractions-to-decimals.html
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# Converting Fractions to Decimals
To convert a fraction to a decimal, just divide the numerator by the denominator .
Example 1:
Write $\frac{3}{15}$ as a decimal.
Since $15$ is larger than $3$ , in order to divide, we must add a decimal point and some zeroes after the $3$ . We may not know how many zeroes to add but it doesn’t matter. If we add too many we can erase the extras; if we don’t add enough, we can add more.
$\begin{array}{l}\begin{array}{c}\\ 15\end{array}\begin{array}{c}\hfill 0.2\\ \hfill \overline{)3.0}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{-3\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}$
So $\frac{3}{15}=0.2$ .
Remember that a decimal is really just a special way of writing a fraction that has a power of ten as a denominator. What we're doing here is rewriting $\frac{3}{15}$ as $\frac{2}{10}$ .
Sometimes, you may get a repeating decimal .
Example 2:
Write $\frac{5}{6}$ as a decimal.
$\begin{array}{l}\begin{array}{c}\\ 6\end{array}\begin{array}{c}\hfill 0.8333\\ \hfill \overline{)5.0000}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{-4\text{\hspace{0.17em}}8\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}20\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{-18}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}20\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{-18}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\end{array}$
We can write the result using a bar over the repeating digit (or digits):
$\frac{5}{6}=0.8\stackrel{¯}{3}$ .
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http://www.leunkim.com/tag/hyperbolic/
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# [M2 Seminar I] Week 1~4 : $\sqrt{\log t}$를 달기 위한 여정의 시작
[latexpage] 블로그 활동이 너무 뜸해서 연구일지나 쓰려고 합니다. 구체적인 연구내용이나, 세미나 발표자료들은 연구가 끝나는 대로 공개하도록 하겠습니다! 2013년 4월 12일 (금) 본격적인 M2 세미나가 시작되었다. 지도교수로부터 1-dimensional Klein-Gordon equation에 대한 여러 논문들을 추천받아 왔다. 방콕모드를 가동시켜서 일단 받아 온 논문들을 … Continue reading
# [M1 Seminar II] Week 5 : High Energy Estimates
We prove the high energy estimates for the nonlinear wave equation on the nonlinearity $\alpha =1$. In fact, this problem is a special case of the quasi-linear symmetric hyperbolic system with some assumptions. By the help of the previous existence … Continue reading
# [M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)
This week, we will prove the local existence of quasi-linear symmetric hyperbolic systems by using $u^{k}$ which is iteratively defined by the solution of the linear symmetric hyperbolic system. For this purpose we first proved the boundedness of $u^k$ in … Continue reading
# [M1 Seminar II] Week 1 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems
This week, we prove the uniqueness and local existence for quasi-linear symmetric hyperbolic systems using general energy method. M1_Semi2_Week1
# [M1 Seminar] Week 6,7 : A Global Existence Theorem to the Linear Symmetric Hyperbolic Systems
This week, we prove the global existence theorem to the linear symmetric hyperbolic system by using the standard energy inequality. M1_Semi_Week6-7
# [M1 Seminar] Week 5 : Energy Estimates for the Linear Symmetric Hyperbolic System
We derive the corresponding estimates for higher derivatives from the basic energy estimates which was given in the last week. We also prove a simple corollary using this result. And we discuss about the finite propagation speed of the hyperbolic … Continue reading
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https://bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/Book%3A_General_Biology_(OpenStax)/5%3A_Biological_Diversity/29%3A_Vertebrates/29.5%3A_Birds
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# 29.5: Birds
Skills to Develop
• Describe the evolutionary history of birds
• Describe the derived characteristics in birds that facilitate flight
The most obvious characteristic that sets birds apart from other modern vertebrates is the presence of feathers, which are modified scales. While vertebrates like bats fly without feathers, birds rely on feathers and wings, along with other modifications of body structure and physiology, for flight.
## Characteristics of Birds
Birds are endothermic, and because they fly, they require large amounts of energy, necessitating a high metabolic rate. Like mammals, which are also endothermic, birds have an insulating covering that keeps heat in the body: feathers. Specialized feathers called down feathers are especially insulating, trapping air in spaces between each feather to decrease the rate of heat loss. Certain parts of a bird’s body are covered in down feathers, and the base of other feathers have a downy portion, whereas newly hatched birds are covered in down.
Feathers not only act as insulation but also allow for flight, enabling the lift and thrust necessary to become airborne. The feathers on a wing are flexible, so the collective feathers move and separate as air moves through them, reducing the drag on the wing. Flight feathers are asymmetrical, which affects airflow over them and provides some of the lifting and thrusting force required for flight (Figure $$\PageIndex{1}$$). Two types of flight feathers are found on the wings, primary feathers and secondary feathers. Primary feathers are located at the tip of the wing and provide thrust. Secondary feathers are located closer to the body, attach to the forearm portion of the wing and provide lift. Contour feathers are the feathers found on the body, and they help reduce drag produced by wind resistance during flight. They create a smooth, aerodynamic surface so that air moves smoothly over the bird’s body, allowing for efficient flight.
Figure $$\PageIndex{1}$$: Primary feathers are located at the wing tip and provide thrust; secondary feathers are located close to the body and provide lift.
Flapping of the entire wing occurs primarily through the actions of the chest muscles, the pectoralis and the supracoracoideus. These muscles are highly developed in birds and account for a higher percentage of body mass than in most mammals. These attach to a blade-shaped keel, like that of a boat, located on the sternum. The sternum of birds is larger than that of other vertebrates, which accommodates the large muscles required to generate enough upward force to generate lift with the flapping of the wings. Another skeletal modification found in most birds is the fusion of the two clavicles (collarbones), forming the furcula or wishbone. The furcula is flexible enough to bend and provide support to the shoulder girdle during flapping.
An important requirement of flight is a low body weight. As body weight increases, the muscle output required for flying increases. The largest living bird is the ostrich, and while it is much smaller than the largest mammals, it is flightless. For birds that do fly, reduction in body weight makes flight easier. Several modifications are found in birds to reduce body weight, including pneumatization of bones. Pneumatic bones are bones that are hollow, rather than filled with tissue (Figure $$\PageIndex{2}$$). They contain air spaces that are sometimes connected to air sacs, and they have struts of bone to provide structural reinforcement. Pneumatic bones are not found in all birds, and they are more extensive in large birds than in small birds. Not all bones of the skeleton are pneumatic, although the skulls of almost all birds are.
Figure $$\PageIndex{2}$$: Many birds have hollow, pneumatic bones, which make flight easier.
Other modifications that reduce weight include the lack of a urinary bladder. Birds possess a cloaca, a structure that allows water to be reabsorbed from waste back into the bloodstream. Uric acid is not expelled as a liquid but is concentrated into urate salts, which are expelled along with fecal matter. In this way, water is not held in the urinary bladder, which would increase body weight. Most bird species only possess one ovary rather than two, further reducing body mass.
The air sacs that extend into bones to form pneumatic bones also join with the lungs and function in respiration. Unlike mammalian lungs in which air flows in two directions, as it is breathed in and out, airflow through bird lungs travels in one direction (Figure $$\PageIndex{3}$$). Air sacs allow for this unidirectional airflow, which also creates a cross-current exchange system with the blood. In a cross-current or counter-current system, the air flows in one direction and the blood flows in the opposite direction, creating a very efficient means of gas exchange.
Figure $$\PageIndex{3}$$: Avian respiration is an efficient system of gas exchange with air flowing unidirectionally. During inhalation, air passes from the trachea into posterior air sacs, then through the lungs to anterior air sacs. The air sacs are connected to the hollow interior of bones. During exhalation, air from air sacs passes into the lungs and out the trachea. (credit: modification of work by L. Shyamal)
## Evolution of Birds
The evolutionary history of birds is still somewhat unclear. Due to the fragility of bird bones, they do not fossilize as well as other vertebrates. Birds are diapsids, meaning they have two fenestrations or openings in their skulls. Birds belong to a group of diapsids called the archosaurs, which also includes crocodiles and dinosaurs. It is commonly accepted that birds evolved from dinosaurs.
Dinosaurs (including birds) are further subdivided into two groups, the Saurischia (“lizard like”) and the Ornithischia (“bird like”). Despite the names of these groups, it was not the bird-like dinosaurs that gave rise to modern birds. Rather, Saurischia diverged into two groups: One included the long-necked herbivorous dinosaurs, such as Apatosaurus. The second group, bipedal predators called theropods, includes birds. This course of evolution is suggested by similarities between theropod fossils and birds, specifically in the structure of the hip and wrist bones, as well as the presence of the wishbone, formed by the fusing of the clavicles.
One important fossil of an animal intermediate to dinosaurs and birds is Archaeopteryx, which is from the Jurassic period (Figure $$\PageIndex{4}$$). Archaeopteryx is important in establishing the relationship between birds and dinosaurs, because it is an intermediate fossil, meaning it has characteristics of both dinosaurs and birds. Some scientists propose classifying it as a bird, but others prefer to classify it as a dinosaur. The fossilized skeleton of Archaeopteryx looks like that of a dinosaur, and it had teeth whereas birds do not, but it also had feathers modified for flight, a trait associated only with birds among modern animals. Fossils of older feathered dinosaurs exist, but the feathers do not have the characteristics of flight feathers.
Figure $$\PageIndex{4}$$: (a) Archaeopteryx lived in the late Jurassic Period around 150 million years ago. It had teeth like a dinosaur, but had (b) flight feathers like modern birds, which can be seen in this fossil.
It is still unclear exactly how flight evolved in birds. Two main theories exist, the arboreal (“tree”) hypothesis and the terrestrial (“land”) hypothesis. The arboreal hypothesis posits that tree-dwelling precursors to modern birds jumped from branch to branch using their feathers for gliding before becoming fully capable of flapping flight. In contrast to this, the terrestrial hypothesis holds that running was the stimulus for flight, as wings could be used to improve running and then became used for flapping flight. Like the question of how flight evolved, the question of how endothermy evolved in birds still is unanswered. Feathers provide insulation, but this is only beneficial if body heat is being produced internally. Similarly, internal heat production is only viable if insulation is present to retain that heat. It has been suggested that one or the other—feathers or endothermy—evolved in response to some other selective pressure.
During the Cretaceous period, a group known as the Enantiornithes was the dominant bird type (Figure $$\PageIndex{5}$$). Enantiornithes means “opposite birds,” which refers to the fact that certain bones of the feet are joined differently than the way the bones are joined in modern birds. These birds formed an evolutionary line separate from modern birds, and they did not survive past the Cretaceous. Along with the Enantiornithes, Ornithurae birds (the evolutionary line that includes modern birds) were also present in the Cretaceous. After the extinction of Enantiornithes, modern birds became the dominant bird, with a large radiation occurring during the Cenozoic Era. Referred to as Neornithes (“new birds”), modern birds are now classified into two groups, the Paleognathae (“old jaw”) or ratites, a group of flightless birds including ostriches, emus, rheas, and kiwis, and the Neognathae (“new jaw”), which includes all other birds.
Figure $$\PageIndex{5}$$: Shanweiniao cooperorum was a species of Enantiornithes that did not survive past the Cretaceous period. (credit: Nobu Tamura)
Career Connection: Veterinarian
Veterinarians treat diseases, disorders, and injuries in animals, primarily vertebrates. They treat pets, livestock, and animals in zoos and laboratories. Veterinarians usually treat dogs and cats, but also treat birds, reptiles, rabbits, and other animals that are kept as pets. Veterinarians that work with farms and ranches treat pigs, goats, cows, sheep, and horses.
Veterinarians are required to complete a degree in veterinary medicine, which includes taking courses in animal physiology, anatomy, microbiology, and pathology, among many other courses. The physiology and biochemistry of different vertebrate species differ greatly.
Veterinarians are also trained to perform surgery on many different vertebrate species, which requires an understanding of the vastly different anatomies of various species. For example, the stomach of ruminants like cows has four compartments versus one compartment for non-ruminants. Birds also have unique anatomical adaptations that allow for flight.
Some veterinarians conduct research in academic settings, broadening our knowledge of animals and medical science. One area of research involves understanding the transmission of animal diseases to humans, called zoonotic diseases. For example, one area of great concern is the transmission of the avian flu virus to humans. One type of avian flu virus, H5N1, is a highly pathogenic strain that has been spreading in birds in Asia, Europe, Africa, and the Middle East. Although the virus does not cross over easily to humans, there have been cases of bird-to-human transmission. More research is needed to understand how this virus can cross the species barrier and how its spread can be prevented.
## Summary
Birds are endothermic, meaning they produce their own body heat and regulate their internal temperature independently of the external temperature. Feathers not only act as insulation but also allow for flight, providing lift with secondary feathers and thrust with primary feathers. Pneumatic bones are bones that are hollow rather than filled with tissue, containing air spaces that are sometimes connected to air sacs. Airflow through bird lungs travels in one direction, creating a cross-current exchange with the blood. Birds are diapsids and belong to a group called the archosaurs. Birds are thought to have evolved from theropod dinosaurs. The oldest known fossil of a bird is that of Archaeopteryx, which is from the Jurassic period. Modern birds are now classified into two groups, Paleognathae and Neognathae.
## Review Questions
A bird or feathered dinosaur is ________.
1. Neornithes
2. Archaeopteryx
3. Enantiornithes
4. Paleognathae
B
Which of the following feather types helps to reduce drag produced by wind resistance during flight?
1. flight feathers
2. primary feathers
3. secondary feathers
4. contour feathers
D
## Free Response
Explain why birds are thought to have evolved from theropod dinosaurs.
This is suggested by similarities observed between theropod fossils and birds, specifically in the design of the hip and wrist bones, as well as the presence of a furcula, or wishbone, formed by the fusing of the clavicles.
Describe three skeletal adaptations that allow for flight in birds.
The sternum of birds is larger than that of other vertebrates, which accommodates the force required for flapping. Another skeletal modification is the fusion of the clavicles, forming the furcula or wishbone. The furcula is flexible enough to bend during flapping and provides support to the shoulder girdle during flapping. Birds also have pneumatic bones that are hollow rather than filled with tissue.
## Glossary
Archaeopteryx
transition species from dinosaur to bird from the Jurassic period
contour feather
feather that creates an aerodynamic surface for efficient flight
down feather
feather specialized for insulation
Enantiornithes
dominant bird group during the Cretaceous period
flight feather
feather specialized for flight
furcula
wishbone formed by the fusing of the clavicles
Neognathae
birds other than the Paleognathae
Neornithes
modern birds
Paleognathae
ratites; flightless birds, including ostriches and emus
pneumatic bone
air-filled bone
primary feather
feather located at the tip of the wing that provides thrust
secondary feather
feather located at the base of the wing that provides lift
theropod
dinosaur group ancestral to birds
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http://mathhelpforum.com/trigonometry/206479-require-help-projectiles-trigonometry-question-print.html
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require help on projectiles and trigonometry question
• October 31st 2012, 08:12 AM
aidanbk
1 Attachment(s)
require help on projectiles and trigonometry question
Hi all, I'd be extremely grateful if anyone could point me in the right direction on this question as I've been looking at it for half an hour now and nothing seems to be working. Thanks in advance!
Attachment 25490
• October 31st 2012, 10:45 AM
skeeter
Re: require help on projectiles and trigonometry question
$\Delta y = v_0\sin{\theta} \cdot t - \frac{g}{2} t^2$
$t = \frac{\Delta x}{v_0 \cos{\theta}}$
sub for $t$ ...
$\Delta y = \Delta x \tan{\theta} - \frac{g}{2} \cdot \frac{(\Delta x)^2 \sec^2{\theta}}{v_0^2} > h$
where $h$ is the height of the crossbar.
• October 31st 2012, 11:31 AM
aidanbk
Re: require help on projectiles and trigonometry question
Thank you very much :)
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https://motls.blogspot.com/2014/02/universe-is-maths-but-only-some-maths.html
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## Thursday, February 13, 2014 ... /////
### Universe is maths, but only some maths is relevant or true
Max Tegmark of MIT has released his book, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, a month ago and it just appeared in the Kindle edition, too.
I won't write a full-fledged review, especially because the book looks way too similar to Brian Greene's The Hidden Reality. Greene's latest major book talks about "parallel universes of all types" we find in physics and the most speculative ones, simulated universes and the universe of the mathematical totality as promoted by Tegmark, were described in the last chapters of Greene's book.
Off-topic: Dominique Gisin (Switzerland), a particle physics fan, visited CERN to see how the protons are accelerated to 99.9999% of the speed of light. This CMS enthusiast who thinks that ATLAS sucks in comparison has used the know-how to accelerate herself while downhill skiing in Sochi. Congratulations to the gold medal! She shared the medal with Tina Maze, Slovenian skier and piano player who must have been trained by ATLAS. Click the photo for more info.
Tegmark introduces the reader to some basic physics – including quantum mechanics and cosmology – and then begins to discuss inflation. He is led to define "Level N parallel universes" with different values of the integer label N.
Ultimately, he claims that the world isn't just described by mathematics. It is mathematics. And he tries to deduce some far-reaching consequences of this claim.
Tegmark is clearly a mathematical Platonist, and so am I. Mathematics exists independently of the humans. Humans are actually discovering (pre-existing) mathematics rather than inventing it. Mathematicians – and physicists – are therefore more analogous to Christopher Columbus than to Thomas Alva Edison. The newly found mathematics may appear in front of the human eyes for the first time but at least if it is important enough, Nature or the "mathematical wisdom of the world" knew about it before the discovery.
I actually think that most mathematicians consider themselves Platonists like Tegmark does (and I do). There are surely exceptions, e.g. Brian Rotman who reviewed Tegmark's book for The Guardian. I just think it is silly to make a big deal out of it because the Platonism doesn't really mean anything tangible. It is a philosophical way to think about the "place where ideas live". Moreover, neither Tegmark nor your humble correspondent "invented" this philosophy. Perhaps Plato did. ;-)
In 1998, a senior top physicist at Princeton (whose name I will keep confidential) informed his younger colleague, an otherwise mainstream-oriented cosmologist Max Tegmark, about the following insight (via e-mail):
Your crackpot papers are not helping you.
He or she clearly meant all these "Level N bullshit universes" that Tegmark had already been emitting for quite some time. The times have changed and it's plausible that the senior physicist is no longer courageous to publicly point out that Tegmark's papers are crackpot papers. Well, TRF isn't under this pressure by the crackpot movement so it is still perfectly OK – even now, in 2014 – for you to point out that Tegmark's papers are crackpot papers. ;-) And if you look at the most recent one, Consciousness as the state of matter called perceptronium, you will both laugh and agree that Tegmark as an individual writer of papers is a kook.
Is Tegmark right? Well, his propositions primarily lack any beef.
Is reality just "described" by mathematics or "is it" mathematics itself? I don't mind if you say the latter; I have said it many times myself. The objects we actually perceive obey some rules that are mathematical, so in this sense, any question about the real objects' behavior is equivalent to a question about their mathematical description. So the real objects are isomorphic to the mathematical concepts or structures, so it doesn't really hurt if you identify them. As long as you care about the derivation of the true statements, you may say that the real objects are the same things as the mathematical structures that represent them.
Fine. But that doesn't mean anything.
Even if you identify the world with the "totality of mathematical objects and structures", you won't be able to answer particular physical questions, like whether mammograms save at least a thousand of American women's lives a year. Or any other question, for that matter. Both answers, Yes and No in this case, may be given a mathematical representation and may be connected with additional mathematical structures and their networks and relationships. However, only one answer – in this case No – is the right one.
Off-topic: my ex-colleague Subir Sachdev talks about AdS/CFT applied to condensed-matter physics, his primary expertise.
So one may imagine a "totality of mathematical laws, objects, structures, axioms, lemmas, and proofs" but it's still true that only a part of this megaverse is relevant for Nature around us. Mathematics is really the "glue" that connects some mathematically represented "objects" with others – by links that are of mathematical character, too. Much of the power of mathematics is really hiding in the "patterns of the links", in the glue and its detailed distribution. But when we say statements about physics or the real world, we need some particular mathematical links – and mathematical objects connected by these links – that may be glued to the objects in the real world. They're very special animals, relatively to the unselective "total world of all mathematical concepts".
So only some candidate physical theories may be relevant for the Universe around us. They must be quantum theories which are relativistic at the same moment and which contain some spectrum of particles and forces whose existence is demonstrable. There may be other conditions, too. This is an extremely different set from the universal "set of all mathematical objects".
Off-topic: a crow solves an 8-step puzzle I still have trouble to comprehend. ;-) Hat tip: Peter F.
In physics – and in all of science – we are looking for the truth by falsifying i.e. eliminating conjectures that disagree with the empirical evidence. So in physics, a conjecture really has to be kind of mathematical to be considered in the first place. But it's not enough for it to survive. It must also survive tests – comparisons of its predictions with some empirical data, with some observed properties of Nature around us.
This process may be shown to prove that creationism, the phlogiston, loop quantum gravity, geocentrism, causal dynamical triangulation, local hidden-variable theories, spin foams, the luminiferous aether, Alain Connes' noncommutative theory of the particle spectrum, de Broglie-Bohm pilot wave theory, Ghirardi-Rimini-Weber collapse models of quantum phenomena, and millions of other potentially mathematical thoughts that people have invented over centuries are simply wrong. It doesn't matter that they belong to some abstract, all-encompassing world of mathematical objects or structures or propositions. The claims about the real world based on them are wrong. In a similar way, whole portions of otherwise totally legitimate mathematics are wrong "even as a starting point" to do physics – Cantor's diagonal tricks, Gödel's theorems, denial of the existence of Feynman's path integral, and so on, and so on.
The reason why Tegmark tries to "radically enhance" the mathematical Lebensraum that is identified with the physical world look utterly incomprehensible. But this enhancement is anti-falsification and therefore anti-science in effect if not in intent (thanks for the linguistic construct, Larry). I think that there can only be one "big reason" why one would want to "resuscitate" all the parts of maths that don't seem relevant or right for physics – that have been eliminated as a description of physics. The "big reason" is the desire to deny that they have been falsified; to deny that it's right for physics and science to discriminate empirically valid or viable mathematical structures from the... challenged ones that are still waiting for their success, if I am allowed to copy a politically correct description of the word "losers" from one commercial promoting expensive enough cars. :-)
And if you want to suppress this discrimination of right and wrong theories, Max, then it's too bad because this is really the defining property of the scientific method. Science doesn't necessarily say that certain mathematical concepts and structures "don't exist anywhere"; it just says that they should be used (at least) less frequently than others whenever we talk about Nature unless we want to be wrong!
Because Tegmark effectively throws away the scientific method – the falsification of the wrong ideas – it is not surprising that most of his comments about the impact of his largely fairy-tale "parallel universes" on observable questions are just wrong, and demonstrably so. I have written too many blog posts that rather surgically isolated the flaws in the reasoning based on the multiverse or anthropic paradigm and I feel it was a waste of time because the human stupidity clearly cannot be eliminated, anyway, so I decided not to write anything like that again.
Your musings may be as good as important and correct physical theories from the viewpoint of the indiscriminate realm of all mathematical structures, Max; but they're not equally good from the viewpoint of science. Your crackpot musings may be helping you to sell books but they are not helping you you to improve your credibility as a scientist among those who actually have a clue about the field, Max.
#### snail feedback (129) :
Physics contains no errors of derivation. It has forever suffered errors of assumption. That is why we perform heretical experiments - and have trash barrels
Given Ptolemy, curve fit astronomy. No – Copernicus. Given cartography, curve fit Euclid. No – Bolyai. Given cyclotrons, curve fit Newton. No – SR. Given Mercury’s orbit, curve fit Newton. No – GR. Given helium’s emission spectrum, curve fit Bohr. No – QM. Given the Dirac equation, curve fit proton magnetic moment. No – quarks. Given spiral galaxies, curve fit the Tully-Fisher relation. No dark matter – Milgrom acceleration. Given Yang and Lee, curve fit vacuum mirror symmetry toward hadrons. No – trace vacuum chiral anisotropy toward matter measurable with geometric Eötvös experiments.
Reality is not administrative triumph of Aristotle over Galileo. Reality is theory free of empirical failure. When theory fails, physics must look orthogonally elsewhere, not curve fit. A black swan in Australia is not a parameterized Northern Hemisphere white swan.
Hear hear! But things are about to change ;-)
LOL, I like the combination of adjectives "stupid, man-made" as if they were nearly synonyma.
Flight is great but I would like to disappoint you by pointing out that the spaceships that flew to Mars couldn't have used wings because wings depend on the air. Instead, they used utterly unnatural, stupid, man-made devices such as rockets.
Sorry for having crippled your belief system so much.
Well, if you define "beauty" as the ultimate ugliness that is ready to be mindlessly adopted in the name of the defense of the old order against new heretical revolutions – in your examples, quantum theory and relativity – and if you define "reasonable things" as the contrived models stubbbornly proposed to fight against all principles that Nature is rather directly showing us, then they were beautiful and reasonable, indeed!
Dear Luboš,
very interesting opinion! Thanks again. For sure I hate bureaucratic tricks :-) and I don't want to question your "moral assessment"... but at least some physics believe in the *physical* Church-Turing thesis
( http://arxiv.org/pdf/1102.1612v1.pdf )
and other ones think that there could be a *Gödel undecidable* problem in real physics like the answer to a simple question: ‘will the orbit of a particle become chaotic ?’ (... from page 13 of http://arxiv.org/pdf/physics/0612253.pdf )
Dear Giulio, thanks for your interesting thoughts.
To be sure, most problems - differential equations or analogous problems - cannot be solved analytically in terms of elementary functions, and not even "not so elementary" functions.
However, if you phrase a completely well-defined physical question like "will the evolution be chaotic", I am sure that none of them may be undecidable. After all, they may really be solved by catching some qualitative properties in an accurate enough numerical simulation.
It was my fear of over diagnosis that stopped me going for mammogram and breast exams too ! Where is Gordon? I chose homeopathy instead :).
When I was nearing menopause I wanted to look into hormone replacement therapy, and for that you need a doctor. I used to go to a good gynecologist, but he was so good that his time was taken over by women trying to conceive and I believed his time would be better spent there. I tried a friend's gynecologist, and he examined me and found by palpitation the same remnants of infections during breast feeding that the previous doctor had seen. He said : to be sure, go and have a mammogram at XXX medical center.
So I went. It was the first and last one.. a) it hurt very much because my breasts were small and b) the specialist doctor there looked at the mammogram and said "hmm, I see this and that shadow but it is better that you discuss it with your surgeon".
The minute I heard "surgeon" I took it for a one way road to over diagnosis ( I had great trust in my first doctor ).
So I went to my sister's homeopathic MD and now take a little pill every morning and that's it. I am now 74, if I reach 80 that will be fine. We will all die one day and something will be written on the death certificate.
Dear Lubos,
I still don't get. You have a question, let's call it Q. The question can be checked by experiment. In the case of SETI the question is whether we are able to catch a message from aliens. Now you say, well, is Q true or false? Then you say, nobody knows either way. let's check. That for me is the very definition of science. Choosing a good question Q and deciding whether you have the right tools to answer it can make the difference between good and bad science. But that's it.
I fully agree that making physical contact with those aliens may end up very badly. I was just thinking about point in time when we receive the first signal and said that it would be important. But yes I would link some positive excitement to this event. I would agree that the risk of answering that message should be carefully evaluated depending on the content of the message, from where it comes, whether they can detect our TV signals anyway etc. Probably it would be too tempting to answer though.
"Tegmark is clearly a mathematical Platonist, and so am I. Mathematics
exists independently of the humans. Humans are actually discovering
(pre-existing) mathematics rather than inventing it."
Yes! Me too. I annoys the crap out of me when people try to say mathematics is just a human construct.
On the other hand, I also agree with you that Tegmark draws from this a total non sequitur.
http://space.mit.edu/home/tegmark/mathematical.html
Dear Lubos, probably no ET has been detected because they don't use EM waves to communicate ... most likely, advanced civilizations use teleportation of signals ( http://pra.aps.org/abstract/PRA/v89/i1/e012311 ) ;-)
In the most ideal society there is a device that recognizes the distinction that the DNA has markers that can be treated and until that time is reached, even one life is worth saving. Anybody who sacrifices a life, because consensus can't be reached about recognizing the architecture of the individual cancer diagnosis, not one of those treatments or advice to not have a mammogram can be fended off with statistics.
Horgan failed here and does not know his science. He does not say what needs to be told about those markers. Emotion has nothing to do with it and neither does politics.
Rockets are ok for transporting objects through space, but not as good as planets (or perhaps comets as you alluded to a few days ago)... nature wins again... :-)
Enough of this... I think we get each others points and I have other things to do. Have a nice weekend!
Lubos boldly asserts: “Mathematics exists independently of the humans. Humans are actually discovering (pre-existing) mathematics rather than inventing it.” One easily visualizes him in climbing harness scaling icy walls on the wind swept lofty peaks of ST, here and there flipping a prettier piece of shale to find his math inscribed by Nature. He states: “I just think it is silly to make a big deal out of it because the Platonism doesn't really mean anything tangible.” I agree heartily, but perhaps it does matter in some small way in that it could be a pointer into the nature of consciousness.
Just as the realization that atomic particles and the devices used in there investigation were inextricably linked was absolutely central to the formulation of QM, it may be that our mathematics is inextricably linked with our stream of consciousness viewed as a mapping from or, pace Thomas Metzinger, a representation of the neurons constituting our minds. No sane individual doubts reality’s causality, but the “logic” represented in our minds by this causality in no way exists in the real world as anything other than an ephemeral: a 3D array of dynamic binary registers (our brain) creating the representation in our minds that we call “logic” is what actually exists in cold, hard, absolute empirical reality.
Sartre, that most incoherent of philosophers, actually said something meaningful (at least to me) along these lines: “the consciousness saying “I think, therefore I am” is not the same as the consciousness doing the thinking”. Such temperance leads one to take applied mathematics as a supplier of quantitative models, where accord with empirical data becomes the final, and only, arbitrator of truth, and one must be happy with this. To expect more as in “the precise form of physics” may be a bridge too far (think of technology’s advance yielding data from which Le Verrier extracted Mercury’s perihelion precession, leading to GR, leading to ST, leading to …).
Haha ;). This is from Quine's 48 'on what there is'. To cite a pre-'two dogmas' Quine on the relationship between mathematics and physics is not quite unlike citing a 10 month old Quine on potty training. 'Quine was against defecating on the toilet'; you read it here first, people.
That said, the TL;DR of this piece is that neither formalists nor Platonists have got it quite figured out. He has yet to draw his more radical conclusions at this point which defined him as a thinker, but nonetheless: amen to that.
Want to try again?
Planets are not able to move anything through space at all. Planets, like any objects under no forces except for gravity, including excrements and pieces of industrial waste, are just being mindlessly moved along geodesics.
Have a nice Friday.
A friend of mine was diagnosed with breast cancer two years ago. The few mammograms she did never showed any tumor. She never felt any pain in her breast. It was only by visual that her doctor decided that there was something wrong. I'm not even sure the further analysis she did ever confirmed the supposed tumor. Her doctor never sounded 100% sure. However she went through the whole chemotherapy, lost her hair, had her left breast removed, tough times... She is recovering now, but I do wonder if she ever had a malignant tumor at the end of the day.
Do you think I have nothing better to do than engage in never ending arguments with ignoramus like you? Arrogance is irritating even when there is a justification for it but in your case it is nothing but stupidity. You can ponder on this while trying to digest:
"From a mathematical point of view, indeed, the important opposition of doctrines here is precisely the opposition between un-willingness and willingness to posit, out of hand, an infinite universe. This is a clearer division than that between nominalists and others as ordinarily conceived, for the latter division depends on a none too clear distinction between what qualifies as particular and what counts as universal. In the opposition between conceptualists an.d platonists, in turn, we have an opposition between those who admit just one degree of infinity and those who admit a Cantorian hierarchy of infinities.
The nominalist, or he who preserves an agnosticism about the infinitude of entities, can still accommodate in a certain indirect way the mathematics of the infinitist-the conceptualist or platonist. Thought he cannot believe such mathematics, he can formulate the rules of its prosecution.” But he would like to show also that whatever service classical mathematics performs for science can in theory be performed equally, if less simply, by really nominalistic methods-unaided by a meaning- less mathematics whose mere syntax is nominalistically de- scribed. And here he has his work cut out for him."
The only relevant maths in the Universe is the maths I learned at school to get my points for my Baccalaureat and that has got me by in life so far. ;-)
"some spectrum of particles"
Could you elaborate on that sometime? Is it anything like the spectrum of light through a prism?
I think Tegmark's idea has genuine merit to some degree, but "show me the money": derive mathematical consequences out of it, make actual predictions.
One thing I do not agree with this post is placing non-commutative geometry in the crackpot bucket. Yes, the theory had its low points, like the incorrect prediction of Higgs mass, but all prior crises are now solved (I mean in the last 12 months). Non-commutative geometry is nothing but a mathematical duality at core (the Hilbert space formulation of QM vs. the phase space formulation of QM is a manifestation of this duality), and the standard model happens to be described in this formalism in a very compact way. The strange notion of distance in NCG using sup versus inf (like in Euclidean space) arise out of the operator norm on C* algebras. Is it a ToE? Not at all. Solves quantum gravity? Nope. It is an equivalent mathematical description of the SM Lagrangian weekly coupled with gravity.
Hi Lubos, what do you think about paraconsistent mathematics? It has some applications in quantum mechanics and could deal with things like Gödel's incompleteness theorem or the halting problem.
Greets
The design of the world can be 'mathematical' without having to admit that 'incompressible fluids' actually exist. I suspect that the hallmark of whatever that design is entails that one can no longer draw a nice boundary between that math and the physics - it would not be merely a tool
Popper (and no doubt many others) makes the distinction between the world of apriori truths (which he calls world 1), empirical truths based on the evidence (ie, science) which he calls world 2, and the truths of human experience (feelings of beauty, patterns of history, etc.) which he calls world 3.
To me this seems like a pretty good way to divide up the world of truths. Apriori truths are true by definition. Every mathematical true statement turns out to be a (sometimes deeply hidden) tautology.
About the nature of empirical truth I would not dare breathe a word on this blog. Why else do I come here except to learn.
Now, about that world no. 3: it may (no doubt is) a subset of world 2, yet trying to appreciate it in terms of abstract mathematical propositions seems (to me at least) inappropriate because unfruitful. I sometimes think of it as a kind of supra-natural realm: completely within the laws of nature but best appreciated by other means, namely, our own experience of ourselves and other people.
Anyway, I'm just an old man babbling and having fun. Cheers everyone! I love Lubot Motl!
This nice article neatly underlines why I always get annoyed when on Physics SE questions involving mathematical concepts / ideas get way too aggressively migrated away to Math SE ...
http://meta.physics.stackexchange.com/q/5473/2751
And it is a nice introduction to some set theory issues ... :-)
Hi Lubos,
Thanks for this spirited critique, which I think actually wins the prize as the most explective-filled one I've seen to date! You're obviously free to call me a "kook" etc., but if you in addition have some sober scientific critique of something I've claimed in the book, please let me know. You raise the important question of whether parallel universes are science or mere speculation. I'm not sure whether you had a chance to read the book before writing this post, but I discuss this question extensively in chapter 6 before starting to explore parallel universes.
First of all, please note that my book does *not* claim that parallel universes exist. Instead, all my arguments involve what logicians know as “modus ponens”: that if X implies Y and X is true, then Y must also be true. Specifically, I argue that if some scientific theory X has enough experimental support for us to take it seriously, then we must take seriously also all its predictions Y, even if these predictions are themselves untestable (involving parallel universes, for example). In other words, I argue that parallel universes are not a scientific theory, but prediction of certain scientific theories. Specifically, I claim that there are four implications:
1) Cosmological inflation generically implies Level I multiverse
2) Inflation + string landscape generically implies Level II multiverse
3) Unitary quantum mechanics implies Level III multiverse
4) The Mathematical Universe Hypothesis implies Level IV multiverse
I'd be interested to hear if you have objections to any specific claims that I make in the book. Would you object to the claims 1), 2), 3) or 4)? Or are you arguing that one of the theories (inflation, say) makes no testable predictions and is therefore unscientific?
Early detection increases the years between detection and death. Good luck explaining to people that this statement, while true, does not necessarily mean there is a benefit to early detection. This study will be dismissed as anti-woman and pinned on greedy insurance companies who don't want to pay for mammograms.
Well, you almost had me convinced of your position by the sheer volume of your humorless name-calling; but im sorry to say, your claim that you don't have anything better to do is at odds with the observable facts.
What these rather neutral observations you are quoting are supposed to prove eludes and ignoramus such as myself; but ill give you the benefit of the doubt, and will assume you have more important things to do than making a coherent argument.
One nice aspect of Tegmark's line of thinking is that it provides a possible solution to the age-old conundrum of why there is something rather than nothing. If the mathematical Platonists are correct, there is always something.
Ralph K
As a math prof, I really despise the use of 'maths' in place of math or mathematics etc... Your title would be more eloquent with two less 's''es.
Lukelea,
Nicely put discretization of truth. But why stop at 3? If you think of all human activities disciplined enough to be seen as methodologically seeking truth, one can discern equivalence relations leading to classes having as kernel certain models - physics, sculpture, music, theology, cinema, economics, … . It becomes unclear just how many axes are needed to describe all truths’ subtleties. And, jee, is empiricism orthogonal to fideism? Hmm, before you know it you are in Hilbert space and need to throw back another single malt. Have a good one.
This is somewhat misleading, mesocyclone. High efficiency (high bps/Hz) transmissions look like noise unless the receiver knows something in advance about the character of the transmitter. It is true that low bps/Hz are very inefficient but they are easy to understand.
Given the problems of interpretation that would exist even if the alien sender had a direct, noise-free, fiber-optic link to our planet, the sender is not likely to use high bps/Hz methods if he is trying to initiate contact.
Of course if we accidentally pick up alien communications not intended for initial contact what you say is correct but, then, not much has really changed since SETI’s early days.
There was some pretty good science done at the beginning of SETI by ex-Hewlett Packard engineers. Of course it was understood that a very large transmitting antenna would be needed to overcome diffraction limitations along with immense transmitted power unless the sender happened to be in our immediate cosmic neighborhood. Most thoughtful observers considered success to be very unlikely and they still do.
If we could accurately identify the senders direction and if the sender were to beam a powerful signal exactly in our direction it would be fairly easy but those are two huge “ifs”. We have no idea where to look and they likely would not know where to send even if they wanted to and why, exactly, should they?
The idea that mathematics consists essentially of “tautologies” is, I think, a common misconception. It occurs to people who think of mathematics as consisting of proofs and of proofs as being “tautologies”. But firstly, mathematics does not consists only of proofs, it also consists of “definitions” and even before one makes a formal definition there are mathematical objects that are at first understood intuitively. Secondly, the only proofs that can be considered “tautologies” are purely formalised ones, but “real life” proof are rarely completely formalised and are almost never discovered in this form.
Take for example the idea of a knot: this is a mathematical idea which formalizes a concept that is quite familiar from our experience. In real life we meet “knots” in various manifestations: they can be made of strings, rope or of DNA. We soon notice that these have something in common and can ask the same questions about them (like can they unknotted). From these considerations we move to an informal mathematical definition: a knot is a closed non-intersecting curve in three space. This definition is sufficient for certain “informal proofs”, but since these rely on geometric intuition it’s hard to justify calling them “tautologies”. One can (and does) go further and give a precise mathematical definition: an unoriented knot is the image of an embedding of the unit circle into the 3 dimensional euclidean space. One can now define certain computable invariants (Alexander-Conway polynomials, Jones polynomial, Vassiliev invariants) which enable one to answer various questions about unknoting of knots etc.
Even if the proofs of the properties of these invariants are, in some sense “tautologies” (and they can only be considered as such when they have been completely formalised since no argument that relies on geometric intuition, however seemingly obvious, can be called a “tautology”), surely the invariants themselves are not “tautologies”, and their properties are every bit as “real” as the properties of physical objects.
And this sort of stuff is what, I think, Popper’s “World 3” consists of.
I'm sorry if I didn't clarify. I was talking in the context of "intercepting their WiFi" - the article headline - i.e. capturing signals not intended for us.
Modern signals, very unlike those when SETI was created, are noise-like - effectively noise if one doesn't have the information to cohere it.
Lubos,
" Humans are actually discovering (pre-existing) mathematics rather than inventing it.'
This isn't exactly the platonic ideal forms, existing independent of their physical manifestation. Because physical reality expresses order, doesn't mean that order exists independent of its expression. There isn't much order in the void.
Women in general are confused about the value of mammography because they've been told disinformation about the medical procedure instead of the real truth (read "The Mammogram Myth" by Rolf Hefti).
The mammogram industry and their adherents, do what they've always done: either ignore or downplay the significance of the data of this new study or partly concede to, in a face-saving manner, the inconvenient facts by claiming now that the choice to have a mammogram is a "an individual patient decision," as if these tactics of damage-control would change an iota on the true facts about the exceedingly harmful procedure.
Dear Max, thanks for your visit. The word is written as "expletive", not "explective". I sometimes use them but it wasn't the case of this blog post. In particular, "kook" isn't an expletive. I think it's dishonest for you to pretend that you might get offended by this label. You know that all good theoretical physicists consider your papers on levels of multiverses to be crackpottery,don't you?
First of all, I find it very likely that *none* of the multiverses exists in the physical sense - so none of the claims 1,2,3,4 is "implied" by physics as we know it with the meaning of the verb "imply" as physicists are used to.
MWI i.e. what you rebrand as Level III Universe is a misunderstanding of quantum mechanics. If you study this blog really carefully, you may be able to learn these basics of QM and why "Level III Universe" is completely missing what QM is actually about. It is no accident that Niels Bohr recommended Hugh Everett to get out of physics. Everett's musings were crackpottery, at least a bit more originally than yours at that time. So your item 3) is fundamentally flawed.
The item 4) is containing no argument whatsoever, it is just a circular reasoning without any scientific backing. The Level IV multiverse and your MHU are two words for the very same thing, so saying that one implies another only shows that you're not capable of thinking logically.
Concerning 1), Level 1 multiverse - just regions behind the cosmic horizon - may look innocent and harmless classically, and some people wouldn't even call them a multiverse. However, it seems likely that quantum gravity implies that the regions behind the cosmic horizon don't have a separate existence due to the horizon complementarity - in analogy with the black hole complementarity. So the physics of the "better physicists" today will surely favor the answer that even the Level 1 multiverse doesn't exist in any physical sense.
So the only multiverse that is motivated enough according to modern science is the Level II multiverse envisioned by 2) - meaning the set of bubbles in the eternal inflation. However, a survey of all attempts to incorporate this meme into physics implies that there exists no credible, convincing, or motivated argument that would suggest that this Level II multiverse may be used to deduce anything about physics in any particular bubble. Like Level I universe, the other bubbles may very well be unphysical from a given observer's viewpoint.
Even if Level I and/or II multiverses exist, you didn't discover them. You just tried to give them new, stupid names - names that fallaciously encourage people to "accept" the other "multiverses" as well although they have nothing do with the evidence for a particular multiverse. And everything "new" you are actually saying about these "multiverses" is either vacuous or downright wrong.
SETI search to science is like lottery to hard work. It makes sense to not confuse them and that is the point.
You say no one knows for sure if we can detect them or not. Well guess what you never know anything for sure in modern science, that does not justify looking for anything you can come up with.
Hi Lucretius!
I am mostly impressed by your contributions. In this instance I am mainly impressed by that you don't seem to be humbled and scared by the fact that both your (mathematicians') and physicists' concepts are built from (!!approximately!!) 1st to 2nd to 3rd level pattern-detecting neurons' activity that forms an aha type 'intuitive insight'-type pattern in 'stringy extra-dimensional brainspacetime' -- and that then this specific pattern of 'pseudo platonic' output is and must be passed to and between Wernicke's and Broca's brain-areas before written representations can finally be put onto electronic paper to be largely and usually correctly critiqued by our humble but hard-nosed correspondent (Lumo). :-)
I like your comment and I refuse to think that age has anything to do with it! ;-)
If need be, I'll help you to defend it best I (someone as young as I) can! ;-)
@uncle al: beautiful argument by orthogonal reasoning. How many degrees of freedom do you posit - 11 maybe? I like your black swan analogy BTW.
This is not a topic that it takes many sentences to effectively philosophically terminate or finalize!
What gave rise to this our universe (definitely no thing, or not suitably characterized as 'thingy' and as surely a timelessly eternal potential or "pseudo process") did so only as a once off event (which both intuitively and by string/M-theoretical indications seem utterly unlikely) or it is a cosmological big bang producing process that does its 'mathematically kaleidoscopic energy producing and patterning thing' consecutively or in parallel or both sequentially and in parallel.
Next time I make a comment here (or directly to Lubos) I aim it to be on a topic most relevant to us humans, and in respect of which I think I just might have something 'atheistic enlightenment' promoting (or of complementary significance) to say about mainly (but not only) from a vantage point of evolutionary psycho[physio]logy type'.
This has been argued before in the skeptical community.
A Skeptical Look at Screening Tests
http://www.sciencebasedmedicine.org/a-skeptical-look-at-screening-tests/
What tests can explicitly prove/disprove multi-verse interpretation? It really seems MWI is an interpretation of QM/cosmology rather than a physical theory; it seems to provide a philosophical explanation, trying to make personal sense of nature rather than providing general rules about nature that can be tested and make realizable predictions.
Scrupulous avoidance of the substance of a fundamental critique of the premises of a scholastic theory is an essential feature of any scholastic theory. You insist in effect that what can be discussed here is the genealogy and fine anatomical structure of centaurs, but that asking whether centaurs can exist in nature is naive and evidence of the doubter's ignorance. What you call questions, or what should be formulated as questions in your view, are in fact legitimate criticisms of the fundamentals of the black hole hypothesis. I made two points:
1. There is no experimental evidence that warrants the assumption that is fundamental to the black hole hypothesis, namely that large masses of ordinary matter are compressible to arbitrarily small volumes. In fact such an assumption is patently absurd, invalidating the entire black hole hypothesis.
2. The notion that stars can avoid gravitational collapse by means of thermonuclear reactions violates Newton’s Third Law. In fact, if matter is ejected from the star during its active phase, this means that the pressure on the core due to gravitation is augmented by the inward directed counter-pressure generated by the ejection, and is thus episodically stronger than in the subsequent phase when the thermonuclear reactions are dying down. If the star did not collapse due to such extra pressures, why should it collapse after those pressures have ceased?
You remain true to the scholastic tradition by dismissing fundamental objections to the theory’s premises as evidence of ignorance, or as being somehow inappropriate. Contrary to what you state, I did not present a theory of my own, I only presented two fundamental objections to the strange hypothesis that you choose to believe. In the space you took to argue (twice) that I ought to ask someone else at another forum to answer my naive questions, you could have addressed the two fundamental objections I stated in my first comment. But as said, the fundamental assumptions of a scholastic hypothesis are not subject to scrutiny.
Instead of another patronizing reply, perhaps you could divulge to the patient readers of this blog the experimental or observational evidence that has led you to believe the patently absurd notion that the entire mass of a star substantially larger than the sun can be compressed to an arbitrarily small volume, i.e., a volume with a radius approaching zero.
According to Libet, a famous neuroscientist : Regarding the unity of conscious experience, it was increasingly evident that many functions of the cortex are localized, even to a microscopic level in a region of the brain, and yet the conscious experiences related to these areas are integrated and unified. We do not experience an infinite array of individual events but rather a unitary integrated consciousness, for example, with no gaps in spatial and colored images. Some unifying process or phenomenon likely mediates the transformation of localized, particularized neuronal representations into our unified conscious experience.
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It should appear obvious that any particular brain representation and doing mathematics is just a particular case of generating brain representations, is a property of the material device, e.g the brain, generating them.
This is even a general property of all brains, so not only human, and that is certainly the reason why evolution has selected and developped this particular device as the most efficient way to adapt and control the environment.
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So a cat jumping on a fly will aim at a point in space where the fly is NOT. This is only possible because its brain has extrapolated the fly's movement and considering the relative velocities has estimated the point of intersection.
Even if a cat has arguably no consciousness of itself, its brain has abilities which can be interpreted as generating mathematical representations of the reality and these representations guide actions (here signals to muscles orienting the jump).
Even more obvious is observing a cheetah attacking an antelope.
In the first phase the cheetah approaches slowly and low to stay hidden.
Then at a particular distance it decides to start the attack.
It knows that if it starts the attack earlier, the distance will be too large for the velocity differential and if it starts too late the antelope will see/smell/hear it and initiate evasive manoeuvers before the cheetah started accelerating.
Clearly the cheetah brain is able to estimate distances and this metrical representation of space can also be interpreted as a mathematical activity.
There are millions of similar examples and experiments (some concerning even more "sophisticated" activities like adition and order relations) that show that different "mathematical" brain representations of the reality are an universal property of brains.
I would agree with Lubos that the Platonic or non Platonic philosophy of mathematics is probably irrelevant to the real process which is neural activity generating brain states.
However as this specific kind of neural activity is extremely efficient in dealing with/predicting the real world, I am convinced that what the Nature wanted to do was to realise a duality between the real world governed by her "laws" and the brain representations of this world.
What is called mathematics would then be just the brain side of this duality.
Even if we are still very far from it, understanding the detailed brain functions that generate the mathematical representations and understanding what we (e.g science) can say about the real world would then be equivalent.
How the former is studied can be for example seen here : http://sulcus.berkeley.edu
Sorry, but you mistake me for someone else. I am very far from being an expert, as you would have realized had you read my own question further upthread (which, btw, has not been answered to date but you don't see me complaining).
I merely offered you some friendly advice on maximizing your chances of getting a useful reply, nothing more.
I am slowly getting the impression that the purpose of your "questions" is not to improve your knowledge by learning from what experts could tell you in answers, but to push your personal opinion about certain topics through and start unscientific not constructive discussions.
However, there seem to be rather interesting (astro)physics questions hidden in your comments. So if you are able to reformulate them in a more neutral less subjective opinion based way, you may indeed be able to get some good answers on the websites Eugene suggested to you ...
Cheers
Ahoy Lubos!
I'm not offended by your aggressive tone, merely amused. You make many strong claims about the views of the physics community that are at odds with my impressions from recent physics conferences, but you're obviously entitled to your sociological interpretations. If you have any concrete critique of my publications ("equation 115 in my consciousness paper http://arxiv.org/pdf/1401.1219v1.pdf is incorrect", say), just let me know.
Wow, now I remember why I like reading your stuff. If anything is beyond the empirical veil it is this multiverse BS, right up there with Russell’s teacup.
Mammography machines feel like instruments of torture. :-) They have a pair of glass plates between which the tissue is pressed quite firmly to get the image, and the plates are rotated 90 degrees for a second set of images and one's tissue pressed firmly in that orientation. Ouch! One really big improvement some years ago was to design automatic release of the plates once image was taken. First time I had one, the technician had to come over and manually release the plates - those additional seconds of discomfort were huge. Thank you to the engineer who tweaked this design! I dislike the nagging from the medical community to get these images so regularly. There is a sense of anxiety and fear in these mammography locales, overlaid with a veneer of childish cheer; pink teddy bears and pink hearts and pink ribbons. As others here have pointed out, let's face our mortality with some style and courage.
It's the other way around. Many well defined questions that are formally decidable (there is an algorithm), can *not* be solved by any physical process. They have resource requirements much bigger than the universe (in Plank units). You might call those problems "unphysical", and you would be right; they are not about the result of any physical experiment.
The "bureaucratic trick" is mostly physically irrelevant not because it labels too *many* problems "undecidable", but too *few*.
It is possible in principle that the reverse (mathematically undecidable problems with physical solutions) could happen, but all evidence so far is against that. So far as I know, no physical theories (including string theory) permit it.
Yep, that noncommutative geometry should belong to the no good things surprised me a bit too, maybe because I know not enough about it ...
So maybe somebody could explain to me what its problematic features are (just curious now)?
Are you joking ...?
Wow thanks Peter F. & Lubos for the video of the crow. These birds are really clever. It is said that most animals can become clever when they want food.
Common sense is neither common nor sense. So patently absurd is the well established theory fitting the observational evidence.
If you want an interesting read get "The Perfect Theory" by Pedro Ferreira. It answers your questions in nontechnical way and tells you where you will turn if you wish to learn. See chapter on collapsing stars especially.
Here I am, Anna.
"I chose homeopathy instead..."---in other words you chose a totally crackpot delusional system of belief in placebo and anecdotal-based drops of water. Gad. And you are proud to trumpet that here, on a science blog! If homeopathy worked, you would be cured of virtually everything by drinking tap water (geez, it must have one or two molecules of nearly everything in it, "succussed" by earth tremors.
This reminds me of the anti-vaccination stuff where people don't get say, polio shots, because they never get polio---that is, until they do. Do you have any other bat-shit crazy beliefs, like Lemurians in Mount Shasta?
BTW, I also think that, particularly in the USA, tests are grossly overordered.
The controversy (?) over mammograms saving lives is similar to that over ordering PSAs for men. For individual cases, of course they can save lives. The papers saying that they don't are churned out by bean counters and epidemiologists. In any case, ultrasounds should be used more.
Finding an isolated reasonably well differentiated breast CA earlier before it metastasizes is life saving. If it is anaplastic, probably not.
All those high tech tests in the US are often driven by profit motives of HPOs and hospitals, and by the lawsuit-crazy situation there.
Stop speculating about what my purpose might be, Dilaton, and try to deal with the substance of my arguments, which are not in any way hidden in my comments, as you claim, but on the contrary are quite clear and explicit. I am also entirely on topic, so your trying to shoo me off to another forum won't work either. It is your dodging and refusal to address the points I have made that is unscientific. Equally unscientific is the notion of infinite density, to which you seem to subscribe. I have also pointed to a fallacy in the notion that gravitational collapse can be triggered by the cessation of thermonuclear activity in the core of a star, since thermonuclear activity at the core of a star cannot alleviate the gravitational pressure of the outer layers without violating Newton's Third Law. The core experiences the entire pressure of the outer layers regardless of any nuclear activity taking place at the core and may occasionally experience additional pressure as a result of mass ejections. Hence if the core did not collapse when it was experiencing such extra pressures, why should it collapse upon the cessation of the activity giving rise to such pressures, simply due to gravitational pressure that was active all along?
Thank you for your suggestion, Rehbock, I will try to find the book. However the fact that a theory fits the observational evidence is not sufficient to establish its veracity. The notion that earthquakes are caused by movements of the turtle on whose back the world rests also fits the observational evidence, but it is patently absurd, though perhaps less so than the notion that stars can collapse to a singularity. Stars can obviously collapse, as they have been observed to do, but the notion that there is no countervailing force and that the collapse continues indefinitely, reducing the volume of an entire star to a size smaller than an atomic nucleus makes the turtle theory of earthquakes sound reasonable by comparison.
@ Florin Moldoveanu What is really non commutative geometry? What are its successes ? Can you give a simple minded description or point to a web reference?
It is really hard to have a simple description. But here are some references: http://arxiv.org/pdf/math/0408416v1.pdf and http://www.alainconnes.org/en/ For a popular introduction (if you manage to suffer through the very frequent commercials) http://www.dailymotion.com/video/xuiyfo_the-music-of-shapes_tech#from=embediframe
A nice explanation by Connes can be found here: http://www.alainconnes.org/docs/maths.pdf
The major application for the Standard Model is here: http://fr.arxiv.org/abs/0706.3690v3 and here: http://fr.arxiv.org/pdf/1304.8050v3
Connes is an expert in QM and von Neumann algebras, won the Fields medal and started a very important and active mathematical branch of non-commutative geometry. At core, NCG is about mathematical dualities (see the table on page 6 on http://arxiv.org/pdf/math/0408416v1.pdf) For a deeper dive in the subject I recommend the 2 books listed under: http://www.alainconnes.org/en/downloads.php
@Timo: You must be in U.S. When I was in India, under British educational system we used to call Mathematics as maths. This must be true for Europe also. They learnt English from Britishers! But you are absolutely right . In U.S. it is called math without the last s!!
Another flawed study. Once again, early diagnosis may not make a noticeable difference if the tumor is undifferentiated and highly anaplastic because metastasis may have already occurred with small tumors. Also, with highly differentiated tumors, it may take 15-20 years or so to kill someone...Really, if you find a solitary tumor in your prostate, and have screening to see if it has metastasized and biopsy to get the cell type and aggressiveness of the tumor, would you really not want to get treatment?
Rehbock, I just read a few reviews of the book you suggested and see no point in acquiring it or reading it. I have read dozens of such hagiographies of Albert the Stoned, basically fawning glorifications of his failed attempt at formulating a general theory of gravitation. Normally a theory fails because of some error in its fundamental premises. However the spectacular failure of general relativity is an exception to this general rule, as it started out with a correct fundamental premise (the equivalence of mass and acceleration) and still managed to screw up physics for at least a century. As I see it, it was Al's uncritical adoption of the fatally flawed Riemannian geometry that led him astray. In reality, Euclidean geometry is basically sound since the fifth postulate is rigorously provable on the basis of prior postulates, though Euclid, having failed to demonstrate the proof, treated what should have been a theorem as an additional postulate. This gave Riemann what he thought was a license to build geometries based on a rejection of the fifth postulate. Given that the fifth postulate is actually a provable theorem, Al's attempt at building his general relativity theory on the Riemannian rejection of the fifth postulate was doomed from the start. Al grasped at this straw since he needed added degrees of freedom to account for gravitation. Yet he proceeded to try to fit his entire structure within the constraints of the Cartesian reference system, an impossible task. Rather than rejecting Euclid and embracing Riemann, Al should have adhered to Euclid, or an updated version of Euclid, while recognizing and surpassing the constraints of the Cartesian reference system. As a bonus, such an approach would have given him a handle on quantum phenomena, which he was never able to properly account for.
In any event, I base my critique of the current astrophysics on standard texts such as Stars and Stellar Evolution by de Boer and Seggweis, or Evolution of Stars and Stellar Populations by Salaris and Cassini, not on The Complete Idiot's Guide to Stellar Evolution.
I had to look up hagiography. I did not mean to insult you. But you make no sense to ask a question for which you insist you have all the answers.
Given that treatment is continuing to improve (for example taxol) a 25 year study may not be appropriate. Also remember the shit storm when the US Preventative Services Task Force recommended screening start at 50 and not 40.
Needless to say, I do not claim to have all the answers; all I did was to venture an educated guess as to where Al took the wrong turn with GR. Had he accepted that the principle of equivalence is in fact the principle of identity, i.e., that acceleration is not just equivalent to gravitation, but is the same as gravitation and indistinguishable from it, i.e., gravitation is acceleration,though not in a single dimension as in his elevator thought experiment, but in three dimensions simultaneously, as experienced on the surface of any spherical body with mass, he might not have been so tempted to reach for Riemannian geometry, his fatal error in my opinion. He would have had to question the general validity of the Cartesian reference system, which cannot represent the true three-dimensional acceleration of a body such as the earth, Einstein did not succeed in generalizing his elevator thought experiment in that he failed to extend it to three dimensions, and tried to fudge the issue by bending space in the spirit of Riemann. Space cannot be bent, but acceleration can act in three dimensions simultaneously,not just in one dimension, as in his elevator example. What we perceive as gravitation is actually three-dimensional acceleration, which the Cartesian reference system cannot represent correctly, In the Cartesian reference system, only one-dimensional acceleration can be fully represented and all of its three axes, x, y and z are employed for the purpose. Acceleration in the other two dimensions is perceived as a force, the gravitational force in this case.
Just to clarify,I am engaging in a critique and not begging to be brought up to speed on current astrophysics, which is in a dismal state in my view and almost entirely absorbed in what amounts to scholastic debates about the precise temperature of the fires of hell.
Agree with you in this instance lucretius.
Unfortunately many Americans have adopted the unfortunate habit of believing the rhetoric of their own cultural ' exceptionalism' and 'selfieism' consistently downplaying or exploiting the intellectual and social achievements of other nations.
In the process they have ' dumbed down' trashed or distorted their educational system, language, metric system, environmental heritage, political system and ethical values in general to the point where they may be classified as a dysfunctional society.
Timos 'maths' rhetoric as a supposed intellectal member of the US respected academic community only confirms this worrying trend
Dear Gordon, how did you get the interval 15-20 years? Is there any science behind it or are we supposed to trust you just because your MD degree gave you extraordinary powers to determine such numbers without any science? It may also give you 80 years, right? This is really the point and meaning of the term "overdiagnosis".
I would surely not okay the treatment mindlessly. It would depend on what the treatment means and what is the prognosis with and without the treatment.
No smiley? you really take yourself seriously :).
The following is not for you, you have made up your mind and will not be bothered with the facts. It is for people, if they exist, wondering about my choices and in order to set strait the "theory" behind homeopathy, which will not give medicinal properties to the tap water.
First of all I am an experimentalist by training and an experiential sponge by construction of my DNA.
Homeopathy has worked for me, my sister, my deceased mother and father , my children and a wider variety of people. . This is the fact, and I consider it an experimental fact. It works.
The "theory" behind it depends on the old etheric model . I will state it though, I am as skeptical/dismissive as the next physicist about it, the theory I mean.
In this model, the physical body is the end product of materializing in these three dimensions. There exists the etheric body, whose total attributes can be called "soul" as the western culture defines it.
This etheric body is composed of seven consecutive layers and is the matrix/mold ( master plan) on which the physical body fits. Diseases appear first on the etheric body and it is the etheric body that has to be healed for the body to heal.
There are people who claim to see the etheric body of people. ( see Barbara Brennan physicist turned healer !) .
All atoms, molecules, etc have an etheric body too. That is what homeopathy is based on. Supposedly the appropriate medicine, usually a poison which would kill a person , has an etheric body. If these molecules are shaken and diluted many times , the water's etheric is imprinted with the etheric properties of the poison but will be absolutely not dangerous because of the great dilution in the three dimensions. It is the shaking that does the imprinting.
The idea is that the etheric medicine will resonate with the etheric problem and neutralize it ( that is why it is good if the symptoms increase the first day, the resonance shows that it is the correct medicine). Once the etheric is cured the body is cured.
Again, this is a model not based in physics as we know it and there have been no physics experiments to verify it. A colleague of mine, during the time of polywater fuss( extra properties in water that finally were not found) tried to make up a theory at the time.
And again, homeopathy works. I am sure that a physical theory will be found as physics advances towards a theory of everything. (think of all those extra dimensions in strings ;) ). History of physics shows that what was metaphysics before became physics for the future generations of physicists.
@dhtow01: Although some of your criticisms of American systems are worth considering, in this particular case of 'maths' I do not agree. May be I have lived in U.S. for too long! But if you want to have 4 lettered short names for the subjects, math is more reasonable, just as Physics is called phys rather than physs , Chemistry is called chem rather than chemy, Economics is called econ etc!! Of course this is just a little remark. Such differences in languages should not be taken seriously. Actually much more serious may be the way Britishers distorted names of Indian cities so much (because they could not pronounce them) that the old names are stuck in my head for 76 years!!!
Is any of this even worth discussing, especially on this post, which has nothing to do with whether "commonwealth conventions" or "U.S. conventions" are better?, but instead to do with Tegmark's papers, etc. ? You further deviate the conversation by going into things like criticising anglicisation of city names, etc. ?! I What about calling the British as "Britishers" , as you do above, then? ... And by the way I really doubt that the names are changed because they "Can't pronounce" anything. I am an Indian and I can pronounce most Indian city names, but I choose to use the anglicised names when speaking in English.
You state yourself, that "Such differences in languages should not be taken seriously", and defy it in the very next sentence!
Interesting post, and I agree with you. Tegmark's Level N multiverses are surely not physics, and the paper on consciousness in indeed laughable. But the Level N universes are fine as philosophy, and should keep them selfves in the pop-sci section of the ArXiV. The paper on consciousness, unfortunately, has gotten into the quant-ph section : (
When I first heard someone refer to mathematics as "math" it sounded to me like they had a cleft palate or some other sneech imfedinen — that kind of thing.
There's a lot of it about. Another fave is "sixth". Loads of people can't do that one. They say "sikth". Obviously the sibilant in the middle is too difficult for them. Jeremy Paxman has this diforder. But then he's an al beeboid so there's bound to be something wrong with him.
Murcans have no excuse. :)
Come on now, Luboš, don't hold back — tell us what you really think. :)
Tom, me too. Ace stuff!
@CentralCharge15: Yes. I agree we should not clutter up Lubos' blog with trivial things. The only reason I responded was that a math professor did not like the usage 'maths' although I had noticed it long time back.If you prefer anglicized names, that is your prerogative. As for me I firmly believe Britishers were using their colonial power in a completely unjustifiable manner to distort officially the city names : Vadodara became Baroda, Kolkata became Calcutta etc. I do not know how Chennai became Madras! This is not just a question of different nationalities pronouncing the names in a different manner. Anyway let me stop before other readers get irritated!!
There's a lo' of i' abou'.
Wow John, your glottal stops could halt the eurotrain in its tracks... 'Estuary English', I believe it's called?
Yes, I agree. Repulsive aren't they. I fucking hate them.
It's true: an Englishman only has to open his mouth to have another Englishman immediately detest him. But that's fine — it's healthy to maintain a certain level of native aggression in the population. After all, the world full of aliens, and you never know when you might need draw on the resource in earnest.
It's one reason I admire the efforts of our football hooligans. They keep the spirit alive.
P.S. Well done your hockey team! No mention of any crippling though. Well, never mind. Maybe next time. :)
Lubos--next time read my post more carefully.
1. 15-20 years was the time for the well-differentiated tumors to cause significant problems or to kill and those numbers were approx and plucked from the ether of experience.
That is why (gasp) MDs often DO NOT TREAT those and just suggest periodic psa blood tests to see if the growth is linear or exponential in the psa-exponential means the tumor has turned aggressive or metastasized.
The "anaplastic" tumors kill quickly. That is why they are operated on or treated.
Note: there are many ways of treating prostate tumors including focused infrared.
2. Yes, my MD does give me special insight and knowledge, just like your Phd gives you insight into string theory. Not all doctors are procedure crazy (just most US ones).
3. To my mind, chemotherapy and radiation should be used as a last resort, not a first---new treatments involve monoclonal antibodies, and stimulating the immune system.
Also, when a small sample (biopsy) of the tumor is available, it can be used to produce personal targeted therapy.
3. Tumors ARE omnipresent in the body...mutant cells crop up continuously but are mopped up by our immune systems, intrinsic and adaptive. It is when they escape the surveillance and reach a critical mass, that our evolutionary systems are overwhelmed.
The intrinsic system was fairly recently discovered (ie how it works) and there are innovative ways of bootstrapping it to kill tumors without nuking normal cells.
4. Just like physicists, there are smart doctors and dumb ones. Also, there are those who are tied to mainstream protocols and those who are not. Some of us have other advanced degrees and read books.
I do agree that often by the time treatments etc have reached the populace, that, for many reasons, the options have tunnel vision (legal reasons, regulatory reasons, in the US, financial reasons, etc) and that there are many routes abandoned in the past that may be resurrected (Coley's toxins, phage therapy for infections) as, say, antibiotic resistance accelerates.
As Leslie Orgel said "Evolution is smarter than we are"---
No one in Canada is treating every patient as a cancer patient. It is interesting how many people are conspiracy theorists when it comes to MDs, but flock to homeopaths, naturopaths and other ridiculous Smoits and Elmer Gantrys of biological sciences.
Dear Gordon, could you please at least show me a paper where this 15-20-year timescale is defended by some evidence?
My PhD isn't giving me any supernatural skills to determine the right values of things in the Universe without either measurements or arguments. But that's probably a difference between physics and medicine. Physics is a science rooted in the evidence, medicine is a form of shamanism that sometimes uses science but sometimes uses the old shaman authority-based methods - whatever is more convenient for a physician.
Homeopathy works because credulous people do not know what the placebo effect is. Their "experiments" are anecdotal in the extreme (my auntie had this xxxx, stood on her head and said drat three times and it went away, so do this :) (there is a smiley).
As interesting as your etheric theory sounds, it is really high on the crackpot scale....way, way to the right of the Gaussian, but I am sure if you wrote a book about it, it would be a best seller. Try "Extraordinary Popular Delusions and the Madness of Crowds"--a wonderful book that "cures" crackpottery...the cure is miraculous (sort of like homeopathic cures).
Good for you, and for it. ;-)
The literature is somewhat inconsistent but there are lots of powerful papers, like this paper with 200+ cits,
http://jnci.oxfordjournals.org/content/96/14/1094.long
that concluded that the reduction of the risk is huge, indeed.
Just google survival times for well-differentiated prostate tumors yourself Lubos. Stop fixating on the numbers I pulled out of the air--they are approx right. You don't understand that some tumors are indolent and others rapidly evolving and aggressive----prostate tumors is a set with many subsets and treatment depends on that. I absolutely makes a difference to survival in certain individual cases, the same as breast tumors. It absolutely makes a difference http://www.cancer.org/cancer/breastcancer/detailedguide/breast-cancer-breast-cancer-types
There is little point in arguing with you about this stuff. It is sort of like me arguing with you about papers on heterotic string theory.
Sorry, Gordon, prostate cancer and breast cancer just don't necessarily follow the same laws, timing, classification.
The paper that shows that regular mammography is useless in practice is a paper about breast cancer only and I won't immediately deduce any direct conclusions about prostate cancer because such a reasoning would be totally sloppy.
The only timing mentioned on the page you linked to is one that says that a form of cancer appears because of a previous treatment 5-10 years later. This isn't even remotely the evidence I was asking you about.
If we had a disagreement about heterotic string theory, I would lead you to the very focus of the disagreement and analyzed the finest detail you would ask for. You don't do anything like that - you are just referring to some supernatural skills to guess the right answer and "debunk" papers without anhy counter-evidence, skillls that you must have because of your occupation or degree. I was joking but it seems to me that this is really how you're thinking. Sorry, I don't respect any authority based on a similarly extremely shaky ground.
Lubos, there is loads of evidence. Things are not so simple or black and white as you seem to think. Einstein's warning not to make things simpler than they are is certainly true. Because i don't want to waste my time scouring for good papers amongst the thousands does not mean there is no evidence.Obviously I am feeling very frustrated by this blog post and by what I see as simplistic thinking prompted by media reports in people with no expertise in immunology and oncology.
In short, as you like saying when you are frustrated, "pearls before swine", even stubborn genius swine :)
I am much too busy to waste more time on this.
You are putting way too much faith in one (large) study. Things are not so simple as you think. You also seem to look on this as an intellectual exercise. I have operated on and treated people with these tumors for over 30 years, and have seen some live and some die. The sort of studies that "show" that mammograms and PSAs make no difference to population survival figures are imo always flawed because individual patients are saved in the right cases.
Tumors are not homogeneous. I think if you were immersed in the medical culture, you would understand and also understand the motivations of various authors of studies. Often a medical study that makes some Delphic pronouncement definitively, a year later is definitively negated. I don't really care if you don't respect my opinion on this--you should, but wtf. I never set out in these posts to make a case based on a paper fight, but, to put it mildly, the study's conclusions are controversial in the medical community. But you wouldn't know about that.
Some years ago, studies "showed" that hormone replacement therapy with estrogen increased risks of breast cancer. Then it was shown that more cancers were being diagnosed (estrogen receptor positive ones); now it seems that the culprit is progesterone, not estrogen. It is a prudent idea to place a mind holding pattern on medical studies for at least 6 months.
too narrow column
Wow!, Lubos. You said, “-medicine is a form of modernized shamanism-“.
Perhaps your doctors in Czechia are total crackpots but you would be very, very foolish, indeed, to so dismiss the doctors that my wife and I rely on in California. Our doctors are professionals in every sense of the word and, believe it or not, many of them are your intellectual equals. They save lives and prevent human suffering every day, including ours. I have enormous respect for the medical profession and I think you should too.
Gordon is not my doctor but I have every reason to believe that he is a credit to his profession and I would be very fortunate to have him as my primary care provider.
I would humbly suggest that you take him seriously just as he should take you seriously as a physicist.
Thanks, Gordon, for the reference but i have just made my position clear in my above response to our host.
I love you, Anna, but I have not read such bullshit in a very long time. Holy cow!
Universe has a tendency towards Development,
not Death by Entropy.
Well said, Ann. Facing our mortality is a necessary key to successful living, isn’t it?
Dear Gordon, I am no swine and I am working hard to learn about cancer from all sources that have something meaningful to say - which unfortunately didn't include you in this case, sorry.
This is about the quantification of benefits of a particular procedure and public health policy based on it and you have shown vastly more reasons to think that you are simply biased and financially vested than evidence. Check what you have shown and you will surely agree.
Dear Gene, there was no fear in the original paper I promoted.
The radiation from Fukushima is small and the radiation from X-rays methods in medicine is arguably "small", too. But it's clearly not negligible and in many cases, it may exceed any benefits.
It's not a fantasy that women have been killed by the treatment by getting a kind of cancer that only arises due to the radiation treatment. See in particular the paper about the cases ironically mentioned by Gordon:
http://www.ncbi.nlm.nih.gov/pubmed/12963926
55 women got cutaneous angiosarcoma which is only acquired as a result of radiation treatment. The survival rate for this condition is just 20%. I think that to remain silent about similar negatives of similar treatments is fraudulent - in fact, it is mass manslaughter.
I am just saying what I was always saying. Medicine as a practical occupation isn't science according to the normal hard scientific definitions of the word. It's reflected at all places, starting from the ludicrous low-confidence-level standards "sufficient" in medical papers to the actual practice that is only loosely connected to the research findings.
And I know that Gordon was eager to "cure" conditions he had absolutely no clue about, without ever seeing the potential patient, so I just don't trust this kind of doing things. It's shamanism and group think.
It is or should be a science-inspired profession but the work in it isn't really doing science itself, and the people aren't really validating their opinions. they're mostly parroting what they've heard from others who were also not validating it, with some extra bias for the doctors to help themselves financially. And indeed, papers that could imply negative changes to the funding are immediately - and without evidence - being attacked.
This applies to Czech medicine as well as Canadian or U.S. medicine. I have some experience with both Czech and U.S. medicine. It is silly for you to try to lower my credibility by the nationality-based attack: in the overall, our healthcare system is as good as the U.S. one.
Dear Gene, I think I have put enough disclaimers to show that it is not "my theory". It is a "theory" with ancient roots, proposed by people who do not know physics and have just a terminology of "energy and "vibrations". Humans need explanations and causes and that is the way they explain it to themselves.
On the other hand I have my experience with the method and it is positive in its effects. Thus I expect that at some point a physical theory will be found , of which the above will be a distorted myth.
Dear Anna, excuse me for butting in. I don't have a dog in this fight, but I can't help wondering why you do not even consider the possibility that no present or future physical theory explaining the efficacy of homeopathy exists and that instead a "double placebo" effect explains everything.
The first part of this effect is that physicians offering homeopathy (and probably even so-called naturopaths without a medical degree) are a self-selected group marked by idealism and a people-centric approach. They spend more time on patient interviews and they listen more intently to what the patients tell them. Most of them also genuinely, sincerely believe in the efficacy of homeopathic treatments. They exude a glow of positivity, which some see as fanaticism, to others it is inspiring and empowering.
The second part that completes the picture is patients' self-suggestion: encouraged by anecdotal "success stories" and by the belief of their doctors (or naturopaths), the body's considerable powers of self-healing are activated. More than once I have been told by a believer in homeopathy that they used to get frequent colds (sniffles, coughing, elevated temperature) but since taking homeopathic globuli they are free of recurrences. I have no reason to disbelieve them. The psyche is known to be intertwined with the body's immune system. Autosuggestion, coupled with an increase in "taking care of oneself" spurred by the attention received from a physician, is sufficient to bring about such a change.
Sadly, I cannot say the same for the cancer patient I knew who placed all her hopes in homeopathy. There is, after all, a limit to what autosuggestion can accomplish.
But in addition to the psychological devastation that comes from disappointed expectations for a cure from a terminal disease, there is also the "nocebo" effect to consider. The mirror image of the placebo effect, this denotes an adverse effect in the absence of a physical mechanism that could explain it. Since I am not in the medical field, I won't presume to write more about the nocebo effect, but you can easily find more information on the Web.
In previous exchanges with Gordon I had accepted the possibility that what I have seen in myself and my near and dear ones comes from the placebo effect, and stated that in that case it is good that we have collectively as a family found something that triggers the placebo effect. After all it has maybe 30% cure rate and is very much cheaper and no side effects .. If we manage to get that to 90% ( since there are a number of us) great.
The MD we go to always asks for diagnosis by specialists if something serious is happening before prescribing the homeopathy and would never dream of treating anybody with cancer just by pills. Rather he prescribes homeopathy that alleviates the side effects of surgery . After all a broken arm is a broken arm :).
The reason I expect a physical explanation of this "dilution and shakings" is because physics has explained a lot of things unexplainable in the 19th century when this therapy started. After all the placebo effect is still unexplainable except with handwaving like "psyche" ( soul) etc, not too far away from the etheric mold after all.
As for one molecule in a zillion, theY have found that butterflies can detect 1 molecule in a cubic meter of air of the pheromone and find a mate kilometers away. The physics explanation I have read is that that single molecule during its brownian motion at temperature T, radiates specific electromagnetic photons that fit the antennae in the insect's reception center for pheromones.
This is not too different from the model my colleague was trying to set up in water, that the shakings would generate "holes" with the signature of the seed chemical, which could have a similar effect on the field of the cells which balance the immune system, i.e. alert them to react. Unfortunately polywater was shown to be due to contaminations.
I am NOT CLAIMING this is the mechanism, I am just offering a potential model within the existing physics framework that might be tested experimentally sometime.
If a system is decidable it must be complete.
For instance, full arithmetic is not decidable, neither complete.
Stephen Wolfram, mentioned by Luboš below, speculates that
"undecidability is common in all but the most trivial physical theories".
I'm not sure why Luboš is so confident in accurate enough numerical simulation,
but I think that his point is that physical world only makes use of the decidable part of mathematics.
In fact, many mathematical systems used in physics, like lattice theory,
projective geometry, and Abelian group theory are decidable, while others, notably non-
Abelian group theory are not (A. Tarski, Undecidable theories)
"And I know that Gordon was eager to "cure" conditions he had absolutely
no clue about, without ever seeing the potential patient"
That is so wrong in in so many ways, as was your comment about my being biased by financial concerns. We are not paid to do tests here in Canada. We have no need to see more patients---we are totally swamped.
You are also confusing medicine at the patient level with the sciences of oncological research, immunology,etc...but there is no telling you this because you are making yourself an expert at these things through self study....read some Alexander Pope ("a little knowledge...).
Trying to pigeon-hole me as a biased non-scientific doctor tied to med-biz is simply stupid. You sort of know me. I have a broad scientific background including a stint at a cancer research institute that was treating melanoma patients with adjuvant-enhanced inactivated tumor cells to stimulate their intrinsic immune system.
I will not add to anything on this particular blog post because your mind is slammed shut... ( you are not a swine--I was pointing out my frustration and quoting you when you were irritated that people were commenting without reading your blog post.
As a second further reply, it has been found a strong connection between the halting problem and quantum physics, leading to undecidable measurement, according to http://phys.org/news/2012-07-classical-problem-undecidable-quantum.html (pdf: http://arxiv.org/abs/1111.3965 )
I welcome studies like this one, Lubos, if it can de-emphasize the importance of preventive medicine, and especially so-called "wellness programs." I can't think of a more wasteful use of our healthcare dollars than for employers to use their earnings to pay for wellness programs for their employees. Wellness programs are based on good ol’ common sense, so these programs should cost hardly anything to run. But the truth of the matter is that those in the wellness industry will find a way to overcharge you for giving out common-sense information. I already see this happening at the hospital where I work. The hospital has recently hired a gold medal winner at the 1999 Pan American games with a PhD in “Human Studies” ( I suppose that’s the opposite of “non-Human Studies”) and is paying her six figures to tell employees to do such common sense things like eat right, don’t smoke, look both ways before crossing the street, and always wear a seat belt when your behind the wheel of a car. Sorry, but this is something that a minimum-wage daycare worker is more than qualified to do. What an absolute waste of healthcare dollars!
Don't get me wrong, but I believe that preventative medicine is a bit overrated. Think about it, obesity has reached epidemic proportions, despite there being a whole slew of diet and exercise programs made available free of charge to the general public. (Michelle Obama and her "let's move and eat healthy" campaign deserves some of the most recent credit for this wasteful use of taxpayers' money.) This is why I believe ObamaCare will failed to achieve its primary goal, which is prevent many types of illnesses, chief among them is breast cancer. Keep in mind, ObamaCare mandates that patients pay nothing out of pocket for mammograms and other preventive screening tests. This will result in higher healthcare costs with very little to show for it terms of preventing illnesses.
Even though preventive medicine, aka primary care, has been short changed for years and deserves more funded, this shouldn’t be done by underfunding emergency and rehab medicine, aka acute and long-term care, respectively. Needless to say, most preventative medicine requires self-motivation. This means that if patients aren’t motivated to get a flu shot every year, eat right and exercise regularly, or stop smoking and wear a seat belt every time they get behind the wheel, no amount of prevention can change that fact.
Tegmark is correct on reality being nothing but a mathematical structure (literally), however, he is most likely wrong about the multiverse.
.http://www.qsa.netne.net/a.htm
If you look at the population density of stars (and by logical relationship, planets) in areas of the galaxy, then the central "bulge" the Milky Way galaxy is where all the action is, by several orders of magnitude. Therefore any intelligent life is by the same margin more likely to exist in this very high star/planet density part of the galaxy. Now where in this galaxy would such a species look for other intelligent life? The same high density area of the galaxy. There may be many intelligent species interacting with each other as we speak and we will never be a part of it because no one would bother to look out at the "boonies" part of the galaxy that we inhabit.
Those of us who step into the realm of physics from outside are at once met by an atmosphere of acrimony and insults -- crackpot being a favorite term of abuse. I was once tempted to take Jack Sarfatti seriously until I discovered that he at one time took Uri Gelller's spoon-bending seriously, having studied it and being fooled by it. The insults are somewhat amusing and make this most beautiful subject even more entertaining.
Dear Lupos. As a Platonist, you say that we discover pre-existing patterns by doing mathematics. What if the discrete disconnectedness of objects we perceive is merely an illusion? Then exact isomorphisms are impossible and the things we perceive can only be approximated by mathematics. Structures like objects, biochemistry, minds and computers are only seemingly discrete because they emerge in a metastable way from a possibly non-mathematical structure, a structure that cannot be described by our mathematical means. Where do mathematical patterns preexist then?
Or do you mean that discrete structures such as our brains yield the potential to produce all mathematical theorems (to a certain finite length) and therefore, when doing mathematics, we explore and therefore discover the theorems that are producible within this system?
Have not had time to respond until 'now. Your reference specifically points out the difference between US/Canada and almost the rest of the worlds written and spoken English. Well, dear Lucretius, Mr. Motl has an interesting diction to say the least, but his use of color, center, theater etc... lends one to infer his use of American/Canadian english is dominant. The world 'maths' is crossing over, albeit slowly into the US/Canadian system and to our ear it sound 'low' and uneducated. Your aligning Luboš' writing style to British/Australian I find surprising, I feel it is pronounced Americanized. That said, amazed how quickly and fun you choose to covert my "I despise the use" to "I despise the people". You are welcome to assume some American and Canadian are like that, but "I despise the use of" maths and feel math or mathematics is far more eloquent. I guess I should rush to judgement and determine that from whereever you come, are largely impudent, arrogant, miscreant's who choose to label people by their passport rather than their believe. But i will just keep judging England and Australia on their failed political analysis of climate change and cheerfully leave them with their own problems and keep mine any day.
Wow, my recommendation to Lubos was not rhetorical while your acerbic attack against a nation based on a twitter sized comment clearly shows some animosity. I will put my children, my students and most of my country up against most of Australia, England or any "fill-in your choice of English dialect" school system any day. My students, on average, are well prepared and reasonably versed. I will put our 'dysfunctional' education system up against any European 'tracking' system as well. Bash Americans all you like, continue to show us the depth of you modern cultural anthropology and how accurately it reflects who I am. Lastly, trashing Americans is just so much fun between you and yours, but I smile as I flip you off and am glad you represent the pinnacle of whatever education system wrought you. They deserve you.
Sounding "Low and uneducated" ? Sure people like Penrose, Dirac, Turing all must sound pretty dumb to an American math prof. I for one think maths is more accurate given that there are so many branches and parts.
I am an American and have seen five children, mine and others through our schools.
Fortunately none of their professors was you. Unfortunately none was the professor you think you are. Given your attitude it is you who need teaching.
Thank you. I just read an account of the "perceptonium" paper, realized that whenever contemporary cosmology (or whatever the "perceptonium" paper is) sounds like BS, it's likely to have Tegmark's name attached to it, felt compelled to Google "Max Tegmark is full of shit," and the first interesting-looking search result was this article. Thanks again.
@Rehbock, I comment that I feel math for Lubos to use is more eloquent. Get "mis place patriotism or ignorace" crack. I comment that the animosity towards american bubbles up quickly from only my opinion on Lubos' diction. They then continue to state we are dumbing down and are being classified as dysfunctional.
I then state our educated are well educated and could stand up to anyothers. And semi-tactfully them to bugger off.
Your response is: You don't like, me and my attitude. I will listen and kindly respond if you like. Curious what indiscretion I passed over?
Well, I have not spoke with Penrose, but I have heard Dirac and Turning and in their writings they don't use Maths but math or mathematics So these great men sound just fine to most american ears. As to low sounding,among many intercity youth and extreme rural young expression that are conjugated poorly or out right miss used, they sound 'uneducated'. Lastly, you may like the variation on math to maths as better, few if any educators use it in American Schools. And having searched the Turning archive I have not found one reference to maths, but many to the mathematics only. So perhaps you can just enlighten me as to how my comments enlighted you as to how I must surely do a disservice to my students?
Hi, just a few words so that you know where I am on this issue. I don't really care much.
Normally, I am conservative, including the usage of words such as "mathematics" (good) or "maths" (bad). Still, I find the efficiency of formulations important and "maths" is shorter. And because "maths" is used by some native speakers whose English still looks vastly superior over mine, I think that "maths" must be OK enough for me, too.
It is incorrect to consider others are inferior and ignorant by whether math is it's own plural.
The America is better than anyone attitude is also not agreeable with me. I think Lucretius and dhtow1 said it well.
That the Universe is completely described by mathematics is indeed an old idea, however Pythagoras and Galileo did not provide enough arguments why it is so! It is more like a postulate in their philosophy. It is easy to say that the Universe is mathematical but we need epistemological and ontological basis for such claims. I saw in your website the link to ontic structural realism (OSR) so I guess you are familiar with it. OSR indeed provides good arguments about the underlying invariant structure of all our theories.
I think Immanuel Kant is the first philosopher to provide the strong arguments why the Universe is described by mathematics. If you are familiar with history of philosophy, Kant reacted to the famous debate between Rationalists (Leibniz, Descartes, Spinoza) and Empiricists (Locke and especially David Hume). Rationalists claim that the source of knowledge is reason and innate ideas, while empiricist claim that the source of knowledge is experience through the senses. Both are right from their perspective. Kant said that to speak about innate ideas in our mind which ground mathematics, metaphysics (a priori knowledge) as rationalists did is lazy business. David Hume has shown that everything comes from experience but he had problems with establishing mathematics on firm ground because maths speak of experience a priori. He could not explain how mathematics is possible! Kant tried to defend this a priori knowledge (mathematics, theoretical physics) and so-called synthetic a priori judgments. That's why I have used Kant to model our cognitive framework (and the Universe as it appears to us) as a quantum computer defined on a grid of cells. I claim that this grid is invariant structure within which all our thoughts, knowledge and theories originate. The structure OSR seeks.
Kant had influenced such mathematicians as Henri Poincaré and David Hilbert. In philosophy of mathematics Kant belongs to intuitionist school. It is also interesting to study the logicist school, that is Frege, Russell. I know that you are involved with FQXi. I claim that we will not understand ultimate reality unless we view everything as a system of mathematics, theoretical physics, philosophy of science and cognitive science. Cognitive science is of absolute importance in understanding ultimate reality because all our thoughts about the world originate in our brain. I know that you come from strictly scientific background but philosophy of science, philosophy of mind cannot be left out if you want to understand the ultimate reality.
It seems that you have buried the philosophy of corporeal nature. This is the true purpose of proper metaphysics of corporeal nature - to assist mathematics and physics. They should go together. It does not matter that people did not know about the Higgs boson or the mathematical description of general relativity 200 years ago. What Kant and Hegel knew is fundamentals - how our knowledge about the world in general is possible. If you know the roots of your knowledge, the epistemological basis of mathematics and physics, everything else is just details. To understand ultimate reality we must understand how we understand things in the first place! That is, we must have the picture of our cognitive faculties in general. This yields the big picture of the Universe how it appears to us.
That's why I took Kant who asked and provided answers in his work to the questions: ''how is mathematics possible?''. ''How is physics possible?''. ''How is metaphysics as science possible?''.
Max, RE modus ponendo ponens , ("P implies Q; P is asserted to be true, so therefore Q must be true." )
I think it's a mistake to place such confidence in modus ponendo ponens. E.g. B. Russell remarks in an essay (Chap. 16, Non-demonstrative inference) in My Philosophical Development (Routledge, London)
"I came to the conclusion that inductive arguments, unless they are confined within the limits of common sense, will lead to false conclusions much more often than to true ones."
I think that this is too little appreciated by all kinds of scientists today. Perhaps you should consider Russell's arguments in that chapter.
Max, RE modus ponendo ponens , ("P implies Q; P is asserted to be true, so therefore Q must be true." )
I think it's a mistake to place such confidence in modus ponendo ponens. E.g. B. Russell remarks in an essay (Chap. 16, Non-demonstrative inference) in My Philosophical Development (Routledge, London)
"I came to the conclusion that inductive arguments, unless they are confined within the limits of common sense, will lead to false conclusions much more often than to true ones."
I think that this is too little appreciated by all kinds of scientists today. Perhaps you should consider Russell's arguments in that chapter.
I'm at a slight disadvantage here in that I can't lay my hands on a copy of the book right now but on the basis simply of what you have written above it sounds to me as if you have completely misunderstood Russell.
First of all, judging from your quotation, by "non-demonstrative inference" it seems that Russell here simply means what others might call inductive reasoning i.e. "induction". (Maybe he talks of other things too, such as heuristics, but I think most of us would assume he's primarily talking about induction when he uses such a description as "non-demonstrative inference". At least that would be default assumption in the absence of a specific classification.)
That being so, I have to say that modus ponens is NOT "non-demonstrative inference". Quite the opposite — modus ponens forms part of deductive reasoning, and only deductive reasoning. Indeed, it's the ONLY part. Deductive reasoning could equally be called "demonstrative inference", which is what I presume Russell is doing here.
As I said, I don't have access to the book, so perhaps my guess that Russell simply means induction (inductive reasoning) when he talks of "non-demonstrative inference" is wrong. But I very much doubt it. And if it is wrong then, again, my guess is that it is wrong only in so far as it doesn't also include possibly other types of reasoning (e.g. heuristic, which types I wouldn't expect Russell would spend any time on) since the expression "non-demonstrative inference" has a rather general ring to it.
Most of the relevant parts of chapter may be read via googlebooks.com (Key-terms: Chapter 16, Russell, My Philosophical Development). His arugument applies equally to "induction" and "deduction" since the point is that the quality of each turns on the validity of the logical inferences employed, not whether they are operate via a deductive or inductive use of empirical data.
Heuristics are merely more or less useful 'reasoning techniques'--aids to logical thought--and they also apply to both inductive and deductive processes. An analogy is a type of heuristic device--it may, according to the particular instance, be valid or invalid, just as both poor deduction and poor induction can produce erroneous conclusions. The essence is not in one or the other but in the virtue of the lack of it in the soundness of the logical relations of the premises and the conclusions on which the inferences rely.
John - You are,of course, correct regarding the quintesentially deductive nature of modus ponens.
While I have absolutely no idea what *anonymous* was attempting to convey I think that Russell was writing about the inclusion of uncertain or probabilistic premises within valid deductive arguments.
Many philosophers of that time thought that only axiomatic certainty was scientific.
If such things interest you, the Australian philosopher David Stove wrote in defense of induction (which is really just Bayesian inference). Much of his writing is available online, including some cogent criticism of Karl Popper and a wonderful essay titled 'What is wrong with our thoughts?'.
RE: "While I have absolutely no idea what *anonymous* was attempting to convey I think that Russell was writing about the inclusion of uncertain or probabilistic premises within valid deductive arguments."
To find out what I meant--and, more importantly, what Russell meant--read the relevant chapter I cited. Few writers are as clear as Russell and you are clearly confused about (or simply ignorant of) what are Russell's points concerning these matters.
Another cite from the article:
" I realised that all the inferences used in both common sense and in science are of a different sort than deductive logic, [ here, Russell is referring to formal logic as a branch of mathematics ] and are such that, when the premises are true and the reasoning correct, the conclusion is only probable."
This passage, by the way, is the one I'd intended to cite in the first post of mine, above.
I have just read the relevant chapter (again), and have just replaced the book in it's spot on the two shelves devoted to Russell. John's reply to you was correct and so was my reply to John.
Yes, Russell was always clear, but he was often wrong. His writings on these matters are no longer taken seriously, and with good reason. His Human Knowledge is also, in hindsight, almost risible.
You, it seems, are always wrong and often unclear (admittedly a non demonstrative inference; a phrase which goes back to Hume) - e.g. -
1) The passage you intended to cite makes a clear distinction between deductive logic and what he thought of as scientific inference, and yet you would cite it to cast doubt on modus ponens - wrong
2) your reply to John (just below) is almost entirely gibberish - unclear.
I don't think you understand Russell or any of the issues he was addressing.
RAF III,
Thank you very much. :) You saved me a lot of hassle.
Two whole shelves, eh? Now that's what I call keen! :)
I like Russell too and have read a few of his, but much as (some) philosophers are fun to read, one always finds 'flies in their ointment', somewhat like discovering a turd with a hideous brown miasmic aura at the bottom of a nice swimming pool after you've spent an afternoon diving in and splashing around, but just not quite as unsettling. The discovery doesn't stop me from doing it again though — reading philosophers, that is. The turd on the other hand ... well, it's a turd! And you don't want one of those on your hand, or anywhere else. Ever. :)
And thanks for the heads-up on Stove. I had heard about him around 20 years ago but never followed it up as I was philosophied out at the time. I'll put him back on my to-do list.
If you're a woman ... mwha! XXX. If not, I'll leave it with a thumbs-up.
However, I strongly suspect it'll have to be a thumbs-up, with probability approaching 1. :)
___
Anon,
I skimmed that chapter with its missing pages. I read nothing to change my mind. My comment stands.
Just a thought, but have you heard the one about the two builders and the professor of probability logic? If not, I'd like to ask you a few questions. :)
On second thoughts, I'll cut to the chase. I can't tell if you have a goldfish or not but I'd put my shirt on you not being overly familiar with the statement/propositional calculus.
If so, I'd recommend any book on it (usually covering the predicate calculus too) that's ordinarily recommended to undergraduate mathematicians, as opposed to books for philosophers — I never liked those books; there was always something odd about them; but maybe they've gotten better; dunno.
John - Thanks for the thumbs-up.
I've been a voracious reader since childhood and have kept every book (bought, inherited, or 'other') I've ever had. It makes moving house a nightmare. What you call keen others might call a psychological disorder.
I wouldn't go so far as your swimming pool example, however.
In many cases the conversation has simply moved on and the arguments have been rendered moot. Does anyone today,concerned with probability theory, care that Russell found Keyne's theory of probability defective because Keynes viewed probability as a relation between propositions rather than between propositional functions?
I also think it's important to understand the intellectual and cultural context in which such things were written in order to understand and appreciate what was meant - something that young and callow people seem unable to do.
Speaking of which, I expect anon to come back at me with some damn fool nonsense, and while I'm happy to have saved you some hassle, I hope you realize that someday you may have to take a bullet for me.
Cheers
RAF III,
Haha!
I found the essay you mentioned by Stove (What is Wrong with Our Thoughts?).
What a fun read. Great stuff.
Yes, that's much more to my liking. And it's funny too. I think Stove would make an excellent hooligan — he does seem to enjoy ripping each of those fakers and windbags a new arsehole. Some nice doses of ridicule there.
I'll find some more of his stuff and read that too.
But one thing about him puzzles me. Given his views and his experience (for example, he alludes to the hell of marking a myriad of bozo undergraduate essays) I wonder why he persisted with philosophy at all. I don't understand that. Once he worked out that it was a load of crap why didn't he jack it all in and turn his mind to something more rewarding? That was a rhetorical question by the way. I don't expect an answer.
Another thing I don't understand. The French are nothing if not proud of their nation. So why then do they appear put up with the likes of Foucault and the rest, let alone pander to them? Can't they see they're a national disgrace? Jean Bricmont (with Alan Sokal) had a go at them but I understand he's a Belgian, not French. How come we don't hear home grown criticism? Maybe there is such and it's just that it doesn't travel much? That was also largely rhetorical, but if Shannon is reading this then maybe she will shed some some light for us? :)
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http://en.wikipedia.org/wiki/Six_factor_formula
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# Six factor formula
The six-factor formula is used in nuclear engineering to determine the multiplication of a nuclear chain reaction in a non-infinite medium. The formula is[1]
$k = \eta f p \varepsilon P_{FNL} P_{TNL}$
Symbol Name Meaning Formula Typical Thermal Reactor Value
$\eta$ Thermal Fission Factor (Eta) The number of fission neutrons produced per absorption in the fuel. $\eta = \frac{\nu \sigma_f^F}{\sigma_a^F}$ 1.65
$f$ The thermal utilization factor Probability that a neutron that gets absorbed does so in the fuel material. $f = \frac{\Sigma_a^F}{\Sigma_a}$ 0.71
$p$ The resonance escape probability Fraction of fission neutrons that manage to slow down from fission to thermal energies without being absorbed. $p \approx \mathrm{exp} \left( -\frac{\sum\limits_{i=1}^{N} N_i I_{r,A,i}}{\left( \overline{\xi} \Sigma_p \right)_{mod}} \right)$ 0.87
$\varepsilon$ The fast fission factor (Epsilon)
$\tfrac{\mbox{total number of fission neutrons}}{\mbox{number of fission neutrons from just thermal fissions}}$
$\varepsilon \approx 1 + \frac{1-p}{p}\frac{u_f \nu_f P_{FAF}}{f \nu_t P_{TAF} P_{TNL}}$ 1.02
$P_{FNL}$ The fast non-leakage probability The probability that a fast neutron will not leak out of the system. $P_{FNL} \approx \mathrm{exp} \left( -{B_g}^2 \tau_{th} \right)$ 0.97
$P_{TNL}$ The thermal non-leakage probability The probability that a thermal neutron will not leak out of the system. $P_{TNL} \approx \frac{1}{1+{L_{th}}^2 {B_g}^2}$ 0.99
The symbols are defined as:[2]
• $\nu$, $\nu_f$ and $\nu_t$ are the average number of neutrons produced per fission in the medium (2.43 for Uranium-235).
• $\sigma_f^F$ and $\sigma_a^F$ are the microscopic fission and absorption cross sections for fuel, respectively.
• $\Sigma_a^F$ and $\Sigma_a$ are the macroscopic absorption cross sections in fuel and in total, respectively.
• $N_i$ is the number density of atoms of a specific nuclide.
• $I_{r,A,i}$ is the resonance integral for absorption of a specific nuclide.
• $I_{r,A,i} = \int_{E_{th}}^{E_0} dE' \frac{\Sigma_p^{mod}}{\Sigma_t(E')} \frac{\sigma_a^i(E')}{E'}$.
• $\overline{\xi}$ (often referred to as worm-bar or squigma-bar) is the average lethargy gain per scattering event.
• Lethargy is defined as decrease in neutron energy.
• $u_f$ (fast utilization) is the probability that a fast neutron is absorbed in fuel.
• $P_{FAF}$ is the probability that a fast neutron absorption in fuel causes fission.
• $P_{TAF}$ is the probability that a thermal neutron absorption in fuel causes fission.
• ${B_g}^2$ is the geometric buckling.
• ${L_{th}}^2$ is the diffusion length of thermal neutrons.
• ${L_{th}}^2 = \frac{D}{\Sigma_{a,th}}$.
• $\tau_{th}$ is the age to thermal.
• $\tau = \int_{E_{th}}^{E'} dE'' \frac{1}{E''} \frac{D(E'')}{\overline{\xi} \left[ D(E'') {B_g}^2 + \Sigma_t(E') \right]}$.
• $\tau_{th}$ is the evaluation of $\tau$ where $E'$ is the energy of the neutron at birth.
## Multiplication
The multiplication factor, k, is defined as (see Nuclear chain reaction):
$k = \frac{\mbox{number of neutrons in one generation}}{\mbox{number of neutrons in preceding generation}}$
If k is greater than 1, the chain reaction is supercritical, and the neutron population will grow exponentially.
If k is less than 1, the chain reaction is subcritical, and the neutron population will exponentially decay.
If k = 1, the chain reaction is critical and the neutron population will remain constant.
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http://love2d.org/forums/viewtopic.php?f=4&t=2093&start=50&sid=973625a2afda428995211ade00a3df65
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## Camera Shake? / Collisions / Enemies...
Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
Before you make a thread asking for help, read this.
Ryne
Party member
Posts: 444
Joined: Fri Jan 29, 2010 11:10 am
### Re: Camera Shake? / Collisions / Enemies...
Robin wrote:Three things:
1. Now I'm not sure what the issue was. Wasn't it the initial position of the enemy weird?
2. You could change the if-statement to if not CheckCollision(player, enemy) then. It works the same, but you're less likely to get odd looks from others. Perhaps if you rename the function to hasCollision or isCollision or so it will make more sense?
3. The enemy speed doesn't have an upper limit: the further they are away, the faster they will go. Off the top of my head, that will probably mean that:
1. You can't outrun the zombies. Ever.
2. Any horde will end up in one jumbling mess, since the ones at the back move faster than the ones in front.
Right now you can outrun the single zombie no problem since he slows down as he gets closer. I think the way it is right now (it will need tweaking when there is more than 1 enemy) is fine, you can outrun the zombies but they always look like they will catch you.
The issue is this:
When dragging my "game folder" (the one containing main.lua etc.) onto the love shortcut it works fine, the exact way it was coded.
If I package the game as a ".love" the enemy oddly approaches from the left instead of the right? Though after some testing this doesn't happen every time, I think only times where the .love file has a name like "zombie_test_long_name.love" instead of just something like "zombie.love".
Solution?:
I was thinking that giving the player an ACTUAL starting value might help. Basically a freeze time since right now start position could be different every time (depending on the load time)
Another thing I found was that if I copy and paste the game folder (to create "game - copy"" and drag the COPY onto the love shortcut the same issue occurs, though after changing the name of the folder from "game - copy" to something smaller like "random" it works fine again. It's so weird lol
On an unrelated side-node: I almost lost this whole post since my internet "cut-out" luckily the back button let me copy it...
@rynesaur
Robin
The Omniscient
Posts: 6506
Joined: Fri Feb 20, 2009 4:29 pm
Location: The Netherlands
Contact:
### Re: Camera Shake? / Collisions / Enemies...
I think I know how to solve the problem (also, it seems independent of the things you mentioned, in my testing).
Try adding this at the beginning of love.update:
Code: Select all
dt = math.min(dt, 0.1) -- or some other not too large value
Help us help you: attach a .love.
Ryne
Party member
Posts: 444
Joined: Fri Jan 29, 2010 11:10 am
### Re: Camera Shake? / Collisions / Enemies...
Robin wrote:I think I know how to solve the problem (also, it seems independent of the things you mentioned, in my testing).
Try adding this at the beginning of love.update:
Code: Select all
dt = math.min(dt, 0.1) -- or some other not too large value
That fixed it. How exactly did you come to this conclusion?
@rynesaur
Robin
The Omniscient
Posts: 6506
Joined: Fri Feb 20, 2009 4:29 pm
Location: The Netherlands
Contact:
### Re: Camera Shake? / Collisions / Enemies...
Because of this:
Robin wrote:The enemy speed doesn't have an upper limit: the further they are away, the faster they will go. Off the top of my head, that will probably mean that:
1. You can't outrun the zombies. Ever.
2. Any horde will end up in one jumbling mess, since the ones at the back move faster than the ones in front.
This also means that if the dt is very large (as is the case at the beginning of the game), their speed is very large as well, causing the zombies to be "warped" to the other side of the player.
Help us help you: attach a .love.
Ryne
Party member
Posts: 444
Joined: Fri Jan 29, 2010 11:10 am
### Re: Camera Shake? / Collisions / Enemies...
Robin wrote:Because of this:
Robin wrote:The enemy speed doesn't have an upper limit: the further they are away, the faster they will go. Off the top of my head, that will probably mean that:
1. You can't outrun the zombies. Ever.
2. Any horde will end up in one jumbling mess, since the ones at the back move faster than the ones in front.
This also means that if the dt is very large (as is the case at the beginning of the game), their speed is very large as well, causing the zombies to be "warped" to the other side of the player.
I assumed it was because of the function, I just didn't know why, thanks again!
@rynesaur
Robin
The Omniscient
Posts: 6506
Joined: Fri Feb 20, 2009 4:29 pm
Location: The Netherlands
Contact:
### Re: Camera Shake? / Collisions / Enemies...
Help us help you: attach a .love.
zac352
Party member
Posts: 496
Joined: Sat Aug 28, 2010 8:13 pm
Contact:
### Re: Camera Shake? / Collisions / Enemies...
What did I miss? My brother has been hogging the computer for 4 hours.
Here's some generic code to make a zombie move towards you, seeing as I noticed some posts about it.
Code: Select all
local zombieX,zombieY=-100,-100
local playerX,playerY=0,0
function magnitude(x,y)
return math.sqrt(x*x + y*y)
end
function norm(x,y)
local dist=magnitude(x,y)
return x/dist,y/dist
end
while true do --main loop
local dirx,diry=norm(playerX-zombieX,playerY-zombieY)
zombieX=zombieX+dirx*speed*dt
zombieY=zombieY+diry*speed*dt
end
When I get some time, I'll look over your code and make some awesome zombie hording stuff.
Hello, I am not dead.
Ryne
Party member
Posts: 444
Joined: Fri Jan 29, 2010 11:10 am
### Re: Camera Shake? / Collisions / Enemies...
zac352 wrote:What did I miss? My brother has been hogging the computer for 4 hours.
Here's some generic code to make a zombie move towards you, seeing as I noticed some posts about it.
Code: Select all
local zombieX,zombieY=-100,-100
local playerX,playerY=0,0
function magnitude(x,y)
return math.sqrt(x*x + y*y)
end
function norm(x,y)
local dist=magnitude(x,y)
return x/dist,y/dist
end
while true do --main loop
local dirx,diry=norm(playerX-zombieX,playerY-zombieY)
zombieX=zombieX+dirx*speed*dt
zombieY=zombieY+diry*speed*dt
end
When I get some time, I'll look over your code and make some awesome zombie hording stuff.
Thanks for the post, but the zombie already walks toward the player. I like how it's set up now, the zombie is faster if it's further away and slows as it gets closer, it gives the illusion that the zombie will catch you when you can slightly outrun him. The challenge comes with hordes, even though you can outrun one, you wont be able to outrun a mob.
@rynesaur
### Who is online
Users browsing this forum: Google [Bot] and 34 guests
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https://email.esm.psu.edu/pipermail/macosx-tex/2008-July/035922.html
|
# texwork (was Re: [OS X TeX] synctex)
Mon Jul 14 08:00:29 EDT 2008
On Jul 11, 2008, at 12:50 PM, Richard Koch wrote:
> On Jul 11, 2008, at 8:28 AM, William Adams wrote:
>
>> \begin{historicnitpicking}
>>
>> To be fair, I think it'd be nice to mention that TeXshop in turn
>> was inspired by TeXview.app on NeXTstep....
>>
>> \end{historicnitpicking}
>
> Thanks for adding this historical note, which is correct. It may
> matter only to those few of us who had NeXT machines.
I still think it's important from an historical accuracy / perspective
angle.
>> Actually, have you considered just taking TeXshop and making it
>> compile w/ GNUstep? That'd get one running on Windows and Linux and
>> Mac OS X would just work''.
>
> I think that would be a nightmare; you'd get a program which "sorta,
> kinda worked." When you see what Jonathan has actually produced,
> you'll be amazed.
I dunno, NovaMind is actually using GNUstep to distribute their
commercial mind-mapping application on Windows --- I know this mostly
reflect my irritation at not getting Yellow Box for Windows'' and
losing PasteUp.app, and FreeHand not switching back to being a Cocoa
app &c., but it seems a lot of redundant work to code a new
application which mostly re-creates what one would get for free'' w/
the *step frameworks.
n Jul 11, 2008, at 3:02 PM, Bruno Voisin wrote:
> But TeXview drew inspiration from Textures on the Mac, isn't it?
> Textures, if I'm not mistaken, was a project launched by Addison-
> Wesley which finally was released independently, by Blue Sky
> Research (then Barry Smith and Doug Henderson) in 1986.
>
> It was made easier by the fact the Mac was programmed in Pascal at
> the time, and Don Knuth's web, the language TeX was originally
> programmed in, was a dialect of Pascal.
>
> I think I read a text mentioning this history somewhere, maybe by
> Nelson Beebe.
This is a good example of why it's important to get the history correct.
There was a lot of cross-pollination in this timeframe --- sorting it
out would require more than just cursory searching (I'm seeing a
usenet post on Textures 1.2 from 23 Mar 1990, but a post on TeXview
1.0 from 5 Oct 1989) --- I'm almost certain Textures was available
first though. Searching on Beebe Textures Teview'' yielded TUGboat
Volume 14, Number 2, July 1993 which mentioned both in different
articles (mental note, must dig out archive of TUGboat and transfer
over to pen slate).
On Jul 13, 2008, at 8:28 PM, Joseph C. Slater PE, PhD wrote:
> On Jul 13, 2008, at 4:13 PM, Louis Talman wrote:
>
>>
>> On Jul 11, 2008, at 10:50 AM, Richard Koch wrote:
>>
>>> Thanks for adding this historical note, which is correct. It may
>>> matter only to those few of us who had NeXT machines.
>>
>> ...and still mourn their departure.
>
>
> Next isn't completely gone:
Yeah, but they keep taking stuff away, like the input hack that let
one have a top-level Services menu in Cocoa apps, and of course
commands (punch in Altsys Virtuoso) to be gestural.
William
--
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http://mathhelpforum.com/statistics/25288-binomial-probability-graphing-calculator.html
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# Thread: Binomial Probability on graphing calculator
1. ## Binomial Probability on graphing calculator
These questions are to be done on a graphing calculator. The questions give n, p, x but I am having difficulty figuring out what is what.
1. In a history class, Colin and Diana both write a multiple choice quiz. There are 10 questions. Each question has five possible answers. What is the probability that:
a) Colin will pass the test if he guesses an answer to each question.
b) Diana will pass the test if she studies so that she has a 75% chance of answering each question correctly.
I really don't have a clue on how to approach these types of questions, any help would be really appreciated
thanks
~G
2. Originally Posted by the-G
These questions are to be done on a graphing calculator. The questions give n, p, x but I am having difficulty figuring out what is what.
1. In a history class, Colin and Diana both write a multiple choice quiz. There are 10 questions. Each question has five possible answers. What is the probability that:
a) Colin will pass the test if he guesses an answer to each question.
b) Diana will pass the test if she studies so that she has a 75% chance of answering each question correctly.
I really don't have a clue on how to approach these types of questions, any help would be really appreciated
thanks
~G
You must use the three-prong approach:
1. Define the random variable.
2. State the distribution the random variable follows.
3. Write down a probability statement of the problem.
a)
1. Let C be the random variable number of answers Colin gets correct.
2. C ~ Bin(n = 10, p = 1/5)
3. $Pr(C \geq 5) = ?$
Depending on the type of graphics calculator you have, you might need to re-write $Pr(C \geq 5)$ as $1 - Pr(C \leq 4)$
b)
is left for you.
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https://asmedigitalcollection.asme.org/PVP/proceedings-abstract/PVP2018/51623/V03AT03A036/277567
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Some fixed tubesheet heat exchangers suffer amount of system startup and shutdown causing large thermal stress for tubesheets. It will make an impact on the fatigue damage of the equipment. Transient thermal and stress analysis were carried out in many papers, but the geometry models were simplified, the non-expanded part of the tube and tubesheet were not modeled in detail at the end of tube. In fact, there is a tiny gap enclosed by tube, tubesheet and weld. So in this paper, a simplified model and an accurate model are established, the transient thermal and stress analysis were performed in ANSYS Workbench. The difference between two models was compared and the influence of temperature change rate on thermal stress was discussed.
This content is only available via PDF.
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http://mathhelpforum.com/calculus/92603-expansion-problem-print.html
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# Expansion problem
• Jun 11th 2009, 09:36 PM
jimmyp
Expansion problem
All edges of a cube are expanding at a rate of 3cm per second. How fast is the surface area changing when each edge is (a) 1cm and (b) 10 cm?
Not sure how to approach the probme. (Wait)(Crying)
• Jun 11th 2009, 10:06 PM
alexmahone
$S=6a^2$
$\frac{dS}{dt}=6*2a\frac{da}{dt}$
$\frac{da}{dt}=3 cm/sec$
$\frac{dS}{dt}=6*2a*3$
$\frac{dS}{dt}=36a$
a) $\frac{dS}{dt}=36*1=36 cm^2/sec$
b) $\frac{dS}{dt}=36*10=360 cm^2/sec$
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https://curriculum.illustrativemathematics.org/MS/students/3/2/5/index.html
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Lesson 5
More Dilations
Let’s look at dilations in the coordinate plane.
5.1: Many Dilations of a Triangle
Explore the applet and observe the dilation of triangle $$ABC$$. The dilation always uses center $$P$$, but you can change the scale factor. What connections can you make between the scale factor and the dilated triangle?
5.2: Info Gap: Dilations
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
3. Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.
4. Share the problem card and solve the problem independently.
If your teacher gives you the data card:
2. Ask your partner “What specific information do you need?” and wait for them to ask for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.
4. Read the problem card and solve the problem independently.
5. Share the data card and discuss your reasoning.
Triangle $$EFG$$ was created by dilating triangle $$ABC$$ using a scale factor of 2 and center $$D$$. Triangle $$HIJ$$ was created by dilating triangle $$ABC$$ using a scale factor of $$\frac12$$ and center $$D$$.
1. What would the image of triangle $$ABC$$ look like under a dilation with scale factor 0?
2. What would the image of the triangle look like under dilation with a scale factor of -1? If possible, draw it and label the vertices $$A’$$, $$B’$$, and $$C’$$. If it’s not possible, explain why not.
3. If possible, describe what happens to a shape if it is dilated with a negative scale factor. If dilating with a negative scale factor is not possible, explain why not.
Summary
One important use of coordinates is to communicate geometric information precisely. Let’s consider a quadrilateral $$ABCD$$ in the coordinate plane. Performing a dilation of $$ABCD$$ requires three vital pieces of information:
1. The coordinates of $$A$$, $$B$$, $$C$$, and $$D$$
2. The coordinates of the center of dilation, $$P$$
3. The scale factor of the dilation
With this information, we can dilate the vertices $$A$$, $$B$$, $$C$$, and $$D$$ and then draw the corresponding segments to find the dilation of $$ABCD$$. Without coordinates, describing the location of the new points would likely require sharing a picture of the polygon and the center of dilation.
Glossary Entries
• center of a dilation
The center of a dilation is a fixed point on a plane. It is the starting point from which we measure distances in a dilation.
In this diagram, point $$P$$ is the center of the dilation.
• dilation
A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point. The fixed point is the center of the dilation. All of the original distances are multiplied by the same scale factor.
For example, triangle $$DEF$$ is a dilation of triangle $$ABC$$. The center of dilation is $$O$$ and the scale factor is 3.
This means that every point of triangle $$DEF$$ is 3 times as far from $$O$$ as every corresponding point of triangle $$ABC$$.
• scale factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because $$4 \boldcdot (1.5) = 6$$, $$5 \boldcdot (1.5)=7.5$$, and $$6 \boldcdot (1.5)=9$$.
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https://mathtuition88.com/tag/calculator/
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Best Calculator for Linear Algebra and Calculus
Many universities and colleges allow students to use advanced graphical calculators for exams in math courses, including linear algebra, calculus and engineering. Having the appropriate calculator can prove to be a huge advantage in terms of getting the correct answer and saving time. Even if the handwritten manual working is required, a good calculator can help to check the correctness of the answer.
Best Calculator for Linear Algebra
The key calculations in linear algebra include calculating the inverse and determinant of matrices. Additional features that are useful in calculators include the ability to calculate eigenvalues, eigenvectors, and RREF (reduced row echelon form). These features can be great time savers (or at the minimum, useful tools for checking your answer), as matrix operations are often tedious and prone to human error.
Our first choice would be Texas Instruments TI-84 Plus CE Color Graphing Calculator, Black. This is one of the flagship calculators by Texas Instruments. This type of calculator is something like a smart phone, you would need to download free apps that will greatly magnify its power in solving Linear Algebra equations. Some of the linear algebra capabilities of the TI-84 Plus are:
• Calculate adjoint of a matrix
• Solve Simultaneous Equations
• Calculate dot product, cross product of vectors
• Calculate inverse of matrix (when matrix is invertible)
• Calculate LDU decomposition, Cholesky factorization
• Calculate Eigenvalues and plot the characteristic polynomial
• Compute the Frobenius norm of a matrix
• Perform Gram Schmidt orthonormalization
• Calculate null space (kernel) if a matrix
• and more!
This calculator is good enough up till senior undergraduate level or even graduate level, for computational intensive modules, such as computational physics or any module that requires linear algebra calculations.
For a more budget-friendly version, you may check out the Texas Instruments TI-84 Plus Graphing Calculator, Black which is essentially the black and white version of the TI-84 Plus CE Color.
Or for an even cheaper option, you could opt for the older version Texas Instruments TI-83 Plus Graphing Calculator which still supports the linear algebra apps.
Often, linear algebra courses include applications to differential equations as well, which overlaps with the next topic on Calculus.
Update: For the latest TI-85/86 models, there is a new “Matrix Mode” inbuilt into the calculator that can perform basic matrix operations (including inverse) and other advanced operations such as LU Decompositions, and finding eigenvalues/eigenvectors.
Note that paradoxically, the newer TI calculator models may be cheaper than the older ones.
Best Calculator for Calculus
For Calculus, the important features that a calculator should have are the ability to perform numerical differentiation and integration. There are also some amazing apps for TI83/84 that can perform symbolic differentiation, that is, find the derivative of a function in symbols (not just the numerical value at certain values of x).
So once again, the TI-84 Plus CE Color Graphing Calculator, White is a good suggested calculator for Calculus modules.
If your budget (and course requirements) permit, you should definitely check out the Texas Instruments TI-Nspire CX II CAS Color Graphing Calculator with Student Software (PC/Mac). CAS stands for “computer algebra system”, which means that the calculator is capable of producing symbolic results rather than just numerical results.
The TI-Nspire™ CX II features a deSolve wizard function for reducing syntax errors in solving differential equations. The TI-Nspire CX can even do vector calculus. It is a very powerful smart calculator that is essentially a mini-computer.
TI Calculator Alternatives
In my experience as a student and educator, TI (Texas Instruments) calculators dominate the market and are generally considered a good choice. But there are other viable alternatives that can also be considered good for linear algebra, calculus or engineering math.
A popular alternative is Casio calculators, for instance Casio fx-9750GII Graphing Calculator with icon based menu. Color white. An immediate advantage of Casio calculators is their low price.
The Casio fx-9750 (and related series) calculator is able to calculate determinant of matrix as well as basic matrix operations such as multiplying matrices, finding the inverse of matrix, etc.
As mentioned above, the main advantage of Casio graphical calculators is their low price. For advanced features, the TI calculators definitely trump over the Casio calculators.
Another alternative to TI calculators are the HP graphical calculators. Let us check out the most powerful HP calculator, the HP 50g Graphing Calculator.
The HP 50g Graphing Calculator has a “MatrixWriter form” to facilitate the entry of matrices. It is essentially like an Excel spreadsheet. Once the matrix is entered, there is a myriad of functions that can be applied, such as finding the determinant, trace, transpose, and rank of a matrix.
In terms of Calculus, the HP 50g can perform a wide range of calculus functions such as:
• Limits and derivatives
• Anti-derivatives and Integrals
• Calculate the Taylor/Maclaurin series of a function symbolically (up to 4-th order relative power, i.e., the difference between the highest and lowest power in the expansion is 4)
• Partial Derivatives (Multivariate Calculus)
• Multiple Integrals such as double integral
• Vector Calculus such as del, gradient, divergence, curl
• Differential Equations
• Fourier series
The list above is quite impressive! Overall, the HP 50g Graphical Calculator is a very strong competitor to the TI series graphical calculators.
Best Calculator for Engineering
Engineers often need to use Linear Algebra, as well as Calculus. Hence, the best calculator for engineering often overlaps with the best calculator for linear algebra or calculus. Do check out our above reviews and pick the calculator that best suits your needs.
In general, we have the below summary.
Best Budget Calculator: Casio graphical calculators, such as Casio fx-9750GII Graphing Calculator with icon based menu. Color white.
Safe choice, all-round best calculator: TI Texas Instruments Calculator, for instance the iconic Texas Instruments TI-84 Plus CE Color Graphing Calculator, Black
Super Powerful Calculator (very advanced and comprehensive features): HP 50g Graphing Calculator
Indian bride walks out of wedding when groom fails math test
Here’s one more reason to learn math!
NEW DELHI (AP) — An Indian bride walked out of her wedding ceremony after the groom failed to solve a simple math problem, police said Friday.
The bride tested the groom on his math skills and when he got the sum wrong, she walked out.
The question she asked: How much is 15 plus six?
The news may seem like a joke, but on a more serious note, the usage of calculators in Primary 5 & Primary 6 has led to a drop in the mental arithmetic standards of children in Singapore! As a tutor, I have clearly observed the difference before and after the calculator usage was introduced. Many a times, students need a calculator to calculate what is 18+15, for example. Also, 8 times 8 (mental calculation) would pose a challenge nowadays to some students ages 9-12, whereas in the past students would know it is 64 in less than a second.
What is happening is that students are getting a bit too reliant on the calculator, and hence their human calculator (a.k.a brain) has lack of training in this aspect. How to remedy it is to practice and train in more multiplication questions.
How to Remember Trigonometric Special Angles easily using Calculator
How to Remember Special Angles easily using Calculator
Step 1: Type the expression into calculator, eg. $\sin (\frac{\pi}{3})$ (in radian mode, for this case)
Step 2: You will get 0.8660254038. Square the Answer. (Ans^2)
Step 3: You will get 3/4. That means $\sin (\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
How to use tables in CASIO FX-9860G Slim (H2 Maths Tuition)
Tables in CASIO FX-9860G Slim
The most popular Graphical Calculator for H2 Maths is currently the TI-84 PLUS series, but some students do use Casio Graphical Calculators.
The manual for CASIO FX-9860G Slim is can be found here:
http://edu.casio.com/products/fx9860g2/data/fx-9860GII_Soft_E.pdf
The information about Tables and how to generate a table is on page 121.
Generating tables is useful to solve some questions in sequences and series, and also probability. It makes guess and check questions much faster to solve.
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https://mathoverflow.net/questions/236151/conjugacy-classes-of-mathrmsl-2-mathbbz/236162
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# Conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$
I was wondering if there is some description known for the conjugacy classes of $$\mathrm{SL}_2(\mathbb{Z})=\{A\in \mathrm{GL}_2(\mathbb{Z})|\;\;|\det(A)|=1\}.$$ I was not able to find anything about this. Most references only give solutions for $$\mathrm{SL}_2(\mathbb{R})$$.
• There is plenty of literature about conjugacy within ${\rm GL}(2,\mathbb{Q}),$ which is essentially the theory of the rational canonical form (in a special case). – Geoff Robinson Apr 13 '16 at 21:40
• – Steve Huntsman Apr 14 '16 at 12:20
One can proceed as follows for $$SL_2(\mathbb{Z})$$.
1. First, the trace is a conjugacy invariant.
2. For trace $$0$$ there are two conjugacy classes represented by $$\pmatrix{0 & 1 \\ -1 & 0}$$ and $$\pmatrix{0 & -1 \\ 1 & 0}$$. These representatives can be thought of as $$90^\circ$$ and $$270^{\circ}$$ degree rotations of a lattice generated by the corners of a square centered on the origin.
3. For trace $$1$$ and $$-1$$ there are two conjugacy classes each, represented by the matrices $$M=\pmatrix{1 & -1 \\ 1 & 0}, M^2=\pmatrix{0 & 1 \\ -1 & -1}, M^4=\pmatrix{-1 & 1 \\ -1 & 0}, M^5 = \pmatrix{0 & -1 \\ 1 & 1}$$ These representatives can be thought of as $$60^\circ$$, $$120^\circ$$, $$240^\circ$$, and $$300^\circ$$ degree rotations of a lattice generated by the vertices of a regular hexagon centered at the origin.
4. For trace $$2$$ there is a $$\mathbb{Z}$$-indexed family of conjugacy classes, represented by $$\pmatrix{1 & n \\ 0 & 1}$$; these are all "shear" transformations except for the identity. For trace $$-2$$ there is a similar $$\mathbb{Z}$$-indexed family of conjugacy classes represented by $$\pmatrix{-1 & n \\ 0 & -1}$$.
5. In general, for nonzero trace the conjugacy classes come in opposite pairs, represented by a matrix $$M$$ with trace $$t>0$$ and an opposite representative $$-M$$ with trace $$-t<0$$.
6. For trace of absolute value $$> 2$$, there is one conjugacy class for each word of the form $$R^{j_1} L^{k_1} R^{j_2} L^{k_2} \cdots R^{j_I} L^{k_I}$$ up to cyclic conjugacy, where $$I \ge 1$$ and all the exponents are positive integers. A matrix representing this form is obtained from the above word by making the replacements $$R=\pmatrix{1 & 1 \\ 0 & 1}, \quad L=\pmatrix{1 & 0 \\ 1 & 1}$$ These are all "hyperbolic" transformations, having an independent pair of real eigenvectors. The slope of the expanding eigenvector is a quadratic irrational, and hence has eventually repeating continued fraction expansion. The cyclic sequence $$(j_1,k_1,j_2,k_2,\ldots,j_I,k_I)$$ can be thought of as the fundamental repeating portion of the continued fraction expansion of the slope of the expanding eigenvector, or, better, as an appropriate power of the fundamental repeating portion where the power is equal to the exponent of the given matrix.
Number theorists will tell you that the number of conjugacy classes of each trace $$t>2$$ is closely related to the class number of the number field generated by $$\sqrt{t^2-4}$$.
• Not the square root of the trace, but rather $\sqrt{t^2-4}$, where $t$ is the trace. – David E Speyer Apr 14 '16 at 2:52
• @LeeMosher Nice answer. Do you have reference for this? For the case 6, you probably have to say that the conjugacy class is of the form $\pm R^{j_1}L^{k_1}\cdots R^{j_I}L^{k_I}$. Indeed, if the trace is negative, you cannot be conjugate to a product of $R,L$. – Jérémy Blanc Mar 11 '17 at 6:28
• I don't have a specific reference. I learned it over a long process of ingestion, maybe starting with my first book on continued fractions as a kid, and continuing with my education in mapping class groups ($PSL_2(\mathbb{Z})$ is the mapping class group of the torus). The answer of @IgorRivin answer gives some references. – Lee Mosher Mar 11 '17 at 13:57
This is the subject of Gauss' reduction theory, as discussed in Karpenkov's book (among many other places). In this 2007 paper, Karpenkov also extends the method to $SL(n, \mathbb{Z}).$
The conjugacy classes of elements of ${\rm SL}(2,\mathbb{Z})$ with given trace are counted in:
S. Chowla, J. Cowles and M. Cowles: On the number of conjugacy classes in SL(2,Z). Journal of Number Theory 12(1980), Issue 3, Pages 372-377.
• Extraordinary: the paper which has Chowla as an author, was also communicated by Chowla! – Lucia Apr 13 '16 at 22:29
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http://www.jiskha.com/members/profile/posts.cgi?name=Ali&page=6
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# Posts by Ali
Total # Posts: 834
Science
thank you sooooo much!
Science
Was Daniel Bernoulli for the atomic theory or against it and explain why?
english
can anyone help me with explicationg act 3 scene 1 (setting) (charcater) (plot) (theme) of macbeth?
math
The force F (in Newtons) required to move a box of mass m kg in motion by pulling on an attached rope is F(x) = f m g/(cos(x))+ f (sin(x)) where x is the angle between the rope and the horizontal, f is the coefficient of static friction and g = 9.8 m/s^2. Find the angle x that...
Calculus
The answer might be easy if you told us the minimum height requirement. Since the ceiling is cheaper, the lower the height, the cheaper it will be. Once we know that, the squarer the building, As equal as possible in length & width, the smaller the perimeter and therefore the ...
Finance
Trying to figure out how to do problems like these. So confused! 1. The receivables turnover for 2009 is 10 times. 2. All sales are on account. 3. The profit margin for 2009 is 14.5%. 4. Return on assets is 22% for 2009. 5. The current ratio on December 31, 2009, is 3.0. 6. ...
calculus
Differentiate the function. h(x)= ln(x+ sqrt(x^2 -4))
chemistry
A student measures 0.2452 g of a monoprotic solid acid, and dissolves the solid in water to make 100.0 mL of solution. the entire solution is titrated to a phenolphthalein endpoint with 45.06mL of 0.1725M KOH. A. what is the molarity of the acid solution? B. what is the molar ...
Calculator Help
1. The csc^-1(1.23) I got 1.061 in radian mode 2. Cot^-1(-2.26) I got 0.8240 in radians 3. Cos^-1(-0.84) I got 2.568 Are these correct?
Physics
Sam, whose mass is 72.0 kg, takes off down a 51.0 m high, 11.0^\circ slope on his jet-powered skis. The skis have a thrust of 180 N. Sam's speed at the bottom is 43.0 {\rm m/s}. What is the coefficient of kinetic friction of his skis on snow?
algebra
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physics
Batman, whose mass is 119 kg, is holding on to the free end of a 14.6 m rope, the other end of which is fixed to a tree limb above. He is able to get the rope in motion as only Batman knows how, eventually getting it to swing enough that he can reach a ledge when the rope ...
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Josh hit a baseball straight up into the air. The equation below describes the height of the ball (h), in meters, as a function of time (t) in seconds. h(t) = -9.8t² + 30t + 1.5
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In the diagram shown the pulley at frictionless and the cylindrical load weighs 800. A. Determine the tension in the cable. B. Determine the horizontal component of the reaction of the pin A. C. Determine the vertical component of the reaction of the pin A.
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a hockey puck rebounds from a board. the puck is in contact with the board for 2.5 ms. dertermine the acceleration of the puck over the time interval. V(i) = 26 m/s...at angle of 22 degrees. V(f) = 21 m/s...at angle of 22 degrees. a=?
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I don't know
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1356 83 degrees
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if the length of RS is 12 and the length of ST is 13 , which of the following could be the length of RT ? A> 24 , B> 26 , C> 28 , D> 29 E> 30
Law and ethics in allied health
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thank you. this was very helpful.
physics
thank you for responding: i did: (9.0*10^9)cos30((4.0*10^-6)^2)/(0.2^2) using this i got 3.1. so then i need to double it because it's two spheres. right? and sorry about the formula in the question. i confused my self with the "times" and "exponent".
physics
Three spheres, each with a negative chrge of 4.0x10^6C, are fixed at the vertices of an equilateral triangle whose sides are 0.20 m long. Calculate the magnitude and direction of the net electric force on each spehere. i used this: (K)Cos30((4.0*10-6)*2)/(0.2*2) and i get 3.1 ...
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Three spheres, each with a negative chrge of 4.0x10^6C, are fixed at the vertices of an equilateral triangle whose sides are 0.20 m long. Calculate the magnitude and direction of the net electric force on each spehere. i used this: (K)Cos30((4.0*10-6)*2)/(0.2*2) and i get 3.1 ...
physics
Three spheres, each with a negative chrge of 4.0x10^6C, are fixed at the vertices of an equilateral triangle whose sides are 0.20 m long. Calculate the magnitude and direction of the net electric force on each spehere. i used this: (K)Cos30((4.0*10-6)*2)/(0.2*2) and i get 3.1 ...
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Assume a population of any 4 numbers(suppose: 4,8,15,19). Select a random sample of size 2 with replacement from the population. List all the possible samples and their corresponding values of x̄ “x-bar” and then find the probability distribution of “X-bar...
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Graph the results of a coin toss: Toss Coins 0 20 1 12 2 5 3 1 4 0 State an equation that best fits the results (use n to represent the number of the toss). What physical phenomenon does this model? I was able to graph this, I just do not understand how to write the equation, ...
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http://cms.math.ca/cmb/kw/Toeplitz%20operators
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Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle. Keywords:Toeplitz operators, irrational rotation algebras, index theoryCategories:47B35, 46L80
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https://en.wikipedia.org/wiki/Subtractor
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# Subtractor
Jump to: navigation, search
In electronics, a subtractor can be designed using the same approach as that of an adder. The binary subtraction process is summarized below. As with an adder, in the general case of calculations on multi-bit numbers, three bits are involved in performing the subtraction for each bit of the difference: the minuend (${\displaystyle X_{i}}$), subtrahend (${\displaystyle Y_{i}}$), and a borrow in from the previous (less significant) bit order position (${\displaystyle B_{i}}$). The outputs are the difference bit (${\displaystyle D_{i}}$) and borrow bit ${\displaystyle B_{i+1}}$. The subtractor is best understood by considering that the subtrahend and both borrow bits have negative weights, whereas the X and D bits are positive. The operation performed by the subtractor is to rewrite ${\displaystyle X_{i}-Y_{i}-B_{i}}$ (which can take the values -2, -1, 0, or 1) as the sum ${\displaystyle -2B_{i+1}+D_{i}}$.
${\displaystyle D_{i}=X_{i}\oplus Y_{i}\oplus B_{i}}$
${\displaystyle B_{i+1}=X_{i}<(Y_{i}+B_{i})}$
Subtractors are usually implemented within a binary adder for only a small cost when using the standard two's complement notation, by providing an addition/subtraction selector to the carry-in and to invert the second operand.
${\displaystyle -B={\bar {B}}+1}$ (definition of two's complement negation)
{\displaystyle {\begin{alignedat}{2}A-B&=A+(-B)\\&=A+{\bar {B}}+1\\\end{alignedat}}}
## Half subtractor
Logic diagram for a half subtractor
The half subtractor is a combinational circuit which is used to perform subtraction of two bits. It has two inputs, the minuend ${\displaystyle X}$ and subtrahend ${\displaystyle Y}$ and two outputs the difference ${\displaystyle D}$ and borrow out ${\displaystyle B_{\text{out}}}$. The borrow out signal is set when the subtractor needs to borrow from the next digit in a multi-digit subtraction. That is, ${\displaystyle B_{\text{out}}=1}$ when ${\displaystyle X. Since ${\displaystyle X}$ and ${\displaystyle Y}$ are bits, ${\displaystyle B_{\text{out}}=1}$ if and only if ${\displaystyle X=0}$ and ${\displaystyle Y=1}$. An important point worth mentioning is that the half subtractor diagram aside implements ${\displaystyle X-Y}$ and not ${\displaystyle Y-X}$ since ${\displaystyle B_{\text{out}}}$ on the diagram is given by
${\displaystyle B_{\text{out}}={\overline {X}}\cdot Y}$.
This is an important distinction to make since subtraction itself is not commutative, but the difference bit ${\displaystyle D}$ is calculated using an XOR gate which is commutative.
The truth table for the half subtractor is:
Inputs Outputs
X Y D Bout
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0
Using the table above and a Karnaugh map, we find the following logic equations for ${\displaystyle D}$ and ${\displaystyle B_{\text{out}}}$:
${\displaystyle D=X\oplus Y}$
${\displaystyle B_{\text{out}}={\overline {X}}\cdot Y}$.
## Full subtractor
The full subtractor is a combinational circuit which is used to perform subtraction of three input bits: the minuend ${\displaystyle X}$, subtrahend ${\displaystyle Y}$, and borrow in ${\displaystyle B_{\text{in}}}$. The full subtractor generates two output bits: the difference ${\displaystyle D}$ and borrow out ${\displaystyle B_{\text{out}}}$. ${\displaystyle B_{\text{in}}}$ is set when the previous digit borrowed from ${\displaystyle X}$. Thus, ${\displaystyle B_{\text{in}}}$ is also subtracted from ${\displaystyle X}$ as well as the subtrahend ${\displaystyle Y}$. Or in symbols: ${\displaystyle X-Y-B_{\text{in}}}$. Like the half subtractor, the full subtractor generates a borrow out when it needs to borrow from the next digit. Since we are subtracting ${\displaystyle X}$ by ${\displaystyle Y}$ and ${\displaystyle B_{\text{in}}}$, a borrow out needs to be generated when ${\displaystyle X. When a borrow out is generated, 2 is added in the current digit. (This is similar to the subtraction algorithm in decimal. Instead of adding 2, we add 10 when we borrow.) Therefore, ${\displaystyle D=X-Y-B_{\text{in}}+2B_{\text{out}}}$.
The truth table for the full subtractor is:
Inputs Outputs
X Y Bin D Bout
0 0 0 0 0
0 0 1 1 1
0 1 0 1 1
0 1 1 0 1
1 0 0 1 0
1 0 1 0 0
1 1 0 0 0
1 1 1 1 1
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https://chat.stackexchange.com/transcript/41?m=47862121
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2:42 AM
@marmot -- (coming late to the discussion.) i think the "do-it-for-me" questions that have raised the most annoyance are ones that were clearly homework questions from a student. in the past there have been times when multiple instances of exactly the same image with the question "how to get this result?" have appeared within a few hours of one another. usually these have been challenged, but since the challenge is in a comment, they aren't easy to find. but the annoyance lingers on.
3:28 AM
@barbarabeeton I fully understand that you find them annoying. After all they dilute the real questions. I do, however, not see why any one needs to justify oneself when answering them. (For a while I thought that answering them will encourage more users asking such questions. However, I do not think any more that is a real issue. Rather, it seems to me that these are mostly independent individuals. Of course, I do not have the possibility to check their IP addresses.)
4:10 AM
^^^^ finally I can cast proper shadows! I'm really relieved. No more cold and long winters! ;-)
3
2 hours later…
5:52 AM
@marmot I don't want you to justify yourself for answering, I would like to make the OP know that, even if s/he got an answer, it is not fair to post a just-do-it-for-me question, s/he got an answer only because is a new user, etc., not because the question is perfect and s/he can continue to ask such question in the future. Of course, this is my opinion, everyone is free to think differently, I don't want to establish new rules I only think it should be better doing it, also for the OPs.
6:19 AM
@marmot Further clarification added to my Meta post
2 hours later…
8:13 AM
8:33 AM
Morning all
@JosephWright Morning quack :)
9:09 AM
@ShreevatsaR Hi mr. velociraptor!
hello!
9:56 AM
Morning all
20
There have been a few questions here on MSE along with some on various child metas about how fixed/permanent the new site redesigns are. I'm posting a new question rather than answering so that we can have more real estate for dialogue (in answers) and because I was missing all of your pings from...
10:16 AM
@PauloCereda Probably not worth it
@JosephWright thought so
10:39 AM
@JosephWright But maybe we can lobby for getting back the fonts and some colors.
@egreg Fonts are explicitly listed as network-wide
@JosephWright Fonts are set network-wide with either a serif or sans-serif stack. Which fonts are in the stack may change as we're realizing the current stack isn't optimal for all of our sites, both technically and for design reasons.
@JosephWright We could see what serif font they're proposing.
11:09 AM
@egreg I guess
@egreg Hope you like the latest PRs :)
@DavidCarlisle Seen meta.stackexchange.com/questions/318910/…? Submit xii?
@UlrikeFischer, @DavidCarlisle Hope you are both enjoying my email rants :)
Waiting for my domain to move ... so I can finish of the move of texdev.net from WordPress to Jekyll ...
.... so while I wait, I'll do a beamer release :)
12:12 PM
@egreg comic sans would be good, cheer the site up a bit.
2
@CarLaTeX There is no reason for you to justify yourself either. ;-) I understand that this is only your opinion, agree with you that it is worthwhile to inform users about the best practice (and it is mostly the same users who ask newcomers for an MWE). However, I just don't want to have to justify not adding such a comment when I do not feel like adding one.
12:39 PM
@marmot What I'm telling is that I don't want to oblige anybody, it's only a suggestion, you can follow it or not as you like
3 hours later…
3:33 PM
@TeXnician As for your comment here: I fully agree. But doesn't that also apply to AboAmmar's "answer"? (... which got 3 upvotes even...)
@marmot it does but one answer was posted after the question was clarified and one before
@CarLaTeX Yes, I guess we more or less agree. In the end it is just a wording issue. I guess most users will agree that explaining newcomers why an MWE can be useful is a good idea. However, several users read in your post a suggested that everyone should do that. You got two answers that suggest that their authors read it this way, as well as a lot of comments.
@marmot and one other answer should be downvoted for gratuitous use of tikz
@DavidCarlisle Yes, but at this time the "answer" had one upvote, not three. (Please feel free to downvote... but if you at the same time argue that the answer that is not to the point deserves upvotes, I find that a bit hard to understand.)
@marmot oh I gave up trying to understand the voting logic ages ago, I can spend two days on a tex answer and get a couple of votes or 30 seconds stealing an image from wikipedia and get several hundred. people vote for what they vote for, Some people even voted for brexit apparently
6
3:46 PM
@DavidCarlisle Yes, sure, I know all that. I am just thinking that the user whose answer got deleted (and does not have the opportunity to see earlier versions of the question) may feel that there are double standards. IMHO the other incorrect answer should not get upvoted either.
@marmot Well, one problem with review is that you see the question and the answer and based on the question it did not answer the question. But I would agree with @DavidCarlisle that sometimes timing is everything.
@TeXnician So far I agree. But I do not see why the wrong answer should get upvoted after the OP clarified that this is not what they asked for. There are tons of questions which only get clear after edits.
@marmot Well, I do not understand that either, but adding another answer that basically even duplicates the content of another answer is definitely not quite useful. Nobody knows why some people vote on answers that do not answer the question, sometimes it's just the interesting approach (without wanting to be disrespectful it's probably not the case in that answer) and sometimes it's because it has a colorful picture (well, it has).
@marmot There are double standards. There are actually thousands of standards - every user has his/her own. And quite a lot probably upvote only because of a nice picture. It doesn't make much sense to care about it (my current goal is to get 200.000 points with less than 10 gold badges ...)
@TeXnician Well, probably I should not say this in public, but the OP of the "not-to-the-point" answer is IMHO the last person who should complain if somebody just copies their answer and posts it, because this is the main skill of this special OP.
3:56 PM
@DavidCarlisle I didn't really could look yet (just came home from a chess match). Is the pdfsavepos problem only with xetex?
@UlrikeFischer I could offer some bounties on questions you have answered. AFAIK you don't get gold badges from harvesting bounties, do you. (I am only concerned that this could be regarded as voter fraud.)
@UlrikeFischer I haven't checked but it will be pdftex as well, usual tex--xet issue that the box gets written backwards so non-text related whatsits get handled in the wrong order
@marmot no need. I think I will manage - I only will have to answer miktex questions only in comments and leave biblatex to moewe ;-).
@DavidCarlisle but luatex should be ok I guess?
@marmot I think I badly wrote that post, that's why I added the clarifications. As you know, I'm not native English speaker.
@UlrikeFischer it has a better chance of being correct:-)
4:01 PM
@CarLaTeX Which is good, so you do not put pineapple on a pizza. ;-)
@marmot Indeed I am always right only when I talk about pineapple pizza
@UlrikeFischer You can afford two more gold badges. (As long as they do not change the rules and give you 2 gold badges if your score reaches 2000 and so on.)
@CarLaTeX I agree. And I guess a meta post along the lines: "Why should one never put pineapple on a pizza?" will be very well received, at least by those who know what pizza is. ;-)
@marmot Of course, but it would be closed as off topic. Btw, I'm thinking of a seasonal challenge for Xmas...
@DavidCarlisle ;-) Do you have an idea how to get the commented line working (that's for github.com/ho-tex/hyperref/issues/63):
\documentclass{article}
\begin{document}
\let \mychar !
\ifnum ! <100 \number! \fi %fine
%\ifnum \mychar <100 \number\mychar \fi %error
\end{document}
@UlrikeFischer steal code from bm (uses \meaning) and dissects the result
4:17 PM
@DavidCarlisle I feared it would boil down to this. Injecting this in the middle of the hyperref processing could get challenging ;-(. With luatex I could use the token library (if I dared to use a function described as "experimental" in the manual) but this doesn't help with xetex.
\documentclass{article}
\begin{document}
\let \mychar !
\ifnum ! <100 \number! \fi %fine
\def\zzz#1 #2 #3#4\relax{#3}
\edef\zz{\expandafter\zzz\meaning\mychar {} {} {} \relax}
\ifnum \zz <100 \number\zz \fi %error
\end{document}
@UlrikeFischer ^^
1 hour later…
5:49 PM
@DavidCarlisle I think it doesn't work ;-(. Basically the whole title is fed to the converter with an \xdef and as the code is not expandable ...
\def\hy@tempa{some title with \mychar zzz}
\xdef\hy@tempa{\expandafter\HyPsd@DoConvert\hy@tempa\@empty\@empty\@empty}
@DavidCarlisle more or less like this ^^
6:36 PM
@UlrikeFischer a better solution really is not to do \let\mychar! and do \edef\mychar{\string!} instead. Are there lots of csnames \let to character tokens around in the wild?
@DavidCarlisle tons:
\documentclass{article}
\usepackage{unicode-math}
\begin{document}
\show\mitOmega
\show\mitnabla
\show\mbfitChi
\show\euro
\end{document}
@UlrikeFischer those ones don't count (as we know who to blame)
I’m looking for some big/complex LaTeX documents using fontspec that I can use for testing/benchmarking my font loader and shaper.
@DavidCarlisle Imho it is the largest batch (difficult to search for this time of command). I will ask Will if he can change it.
6:51 PM
It is funny how many years I spent hacking various TeX engines and packages and I never got to use it seriously, that I don’t even have documents to test with.
"time of command" ? I seem to be a bit tired ...
7:12 PM
@UlrikeFischer either change the default or if that's tricky hyperref could input unicode-math-table.tex while setting up the bookmark mapping and locally define them all (perhaps:-)
@UlrikeFischer I didn't even notice:-)
@DavidCarlisle I added an issue to unicode-math.
@DavidCarlisle ;-)
@KhaledHosny Will may have something in his fontspec tests, the main latex2e test suite is rather lacking tests in that area. However since the tex macro side of things isn't really affected, you might be better just to take "simple" text paragraphs as borrowed from web pages in whatever scripts to see if harfbuzz in luatex does as well as a browser?
7:39 PM
@KhaledHosny I have a number of tests in the luaotfload github (a lot stolen from fontspec/unicode math), but they are normally rather short as this is more suitable for l3build tests.
@DavidCarlisle I do that already, I’m trying to test real world files, you never no what you are missing.
know*
@KhaledHosny Oh OK (I don't use tex at all apart from small examples copied from this site:-)
@ShreevatsaR may have some examples
8:00 PM
@DavidCarlisle I just saw that ifluatex is in texlive in its own package. Did Karl do this?
@UlrikeFischer issue number 1 :-)
8:16 PM
@UlrikeFischer yes Karl had that set up before ho-tex was set up, and added that issue on day one, if I finally get round to splitting this up I'll finally get tp that issue
8:28 PM
@KhaledHosny As far as real-world examples go, here is something that someone used to typeset a 400+ page book with xelatex: github.com/shreevatsa/nmisra-arapv
(Note I'm just mirroring someone else's LaTeX sources; I don't know anything more about it, but I just now tried compiling it and it seems to work (with copious warnings), after fixing the font name.)
8:52 PM
@DavidCarlisle I’m in good company then :)
@ShreevatsaR Thanks!
@UlrikeFischer It doesn't work yet but... github.com/ho-tex/oberdiek/tree/split
@DavidCarlisle wow. I better not interfere. How do you plan to handle the hobsub problem? Will they pull from the parallel folders?
@DavidCarlisle I found at least a luatex solution which seems to work for the \mychar problem:
\documentclass{article}
\usepackage{luacode}
\usepackage[pdfencoding=unicode]{hyperref}
\begin{luacode}
function getcharactercode ()
local t=token.get_next()
tex.print(t.mode)
end
\end{luacode}
\makeatletter
\def\HyPsd@Char#1{%
\ifnum\directlua{getcharactercode ()}#1<128 %
\@backslashchar 000\Uchar\directlua{getcharactercode ()}#1%
\else
\ifnum\directlua{getcharactercode ()}#1<65536 %
\expandafter\HyPsd@CharTwoByte\directlua{getcharactercode ()}#1!%
\else`
@UlrikeFischer something like that, it's on a branch so I can easily junk it all if it doesn't work, I just decided to do it in last few minutes so it's not exactly a fully worked plan yet:-)
@PauloCereda ^^^^ choko ducks.
9:07 PM
@UlrikeFischer oh more edible ducks?
@DavidCarlisle a moral conflict for Paulo ;-).
5
1 hour later…
10:38 PM
@UlrikeFischer yay oh no but it's chocolate but it's a duck but it's chocolate but it's a duck oh no quacks in despair
@PauloCereda dinner and pudding
@DavidCarlisle oh no
|
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|
https://arxiv.org/abs/1804.06884
|
math.CV
# Title:On the normal bundle of Levi-flat real hypersurfaces
Abstract: Let $X$ be a connected complex manifold of dimension $\geq 3$ and $M$ a smooth compact Levi-flat real hypersurface in $X$. We show that the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This generalizes a result obtained by Brunella.
Comments: minor changes, final version, in Math. Ann. (online first) Subjects: Complex Variables (math.CV); Differential Geometry (math.DG); Dynamical Systems (math.DS) MSC classes: 32V15, 32V40 Journal reference: Math. Ann. 375, 343--359 (2019) DOI: 10.1007/s00208-018-1723-7 Cite as: arXiv:1804.06884 [math.CV] (or arXiv:1804.06884v2 [math.CV] for this version)
## Submission history
From: Judith Brinkschulte [view email]
[v1] Wed, 18 Apr 2018 19:28:14 UTC (17 KB)
[v2] Mon, 23 Jul 2018 17:06:51 UTC (17 KB)
|
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|
http://lavalle.pl/planning/node204.html
|
#### Making good grids
Optimizing dispersion forces the points to be distributed more uniformly over . This causes them to fail statistical tests, but the point distribution is often better for motion planning purposes. Consider the best way to reduce dispersion if is the metric and . Suppose that the number of samples, , is given. Optimal dispersion is obtained by partitioning into a grid of cubes and placing a point at the center of each cube, as shown for and in Figure 5.5a. The number of cubes per axis must be , in which denotes the floor. If is not an integer, then there are leftover points that may be placed anywhere without affecting the dispersion. Notice that just gives the number of points per axis for a grid of points in dimensions. The resulting grid will be referred to as a Sukharev grid [922].
The dispersion obtained by the Sukharev grid is the best possible. Therefore, a useful lower bound can be given for any set of samples [922]:
(5.20)
This implies that keeping the dispersion fixed requires exponentially many points in the dimension, .
At this point you might wonder why was used instead of , which seems more natural. This is because the case is extremely difficult to optimize (except in , where a tiling of equilateral triangles can be made, with a point in the center of each one). Even the simple problem of determining the best way to distribute a fixed number of points in is unsolved for most values of . See [241] for extensive treatment of this problem.
Suppose now that other topologies are considered instead of . Let , in which the identification produces a torus. The situation is quite different because no longer has a boundary. The Sukharev grid still produces optimal dispersion, but it can also be shifted without increasing the dispersion. In this case, a standard grid may also be used, which has the same number of points as the Sukharev grid but is translated to the origin. Thus, the first grid point is , which is actually the same as other points by identification. If represents a cylinder and the number of points, , is given, then it is best to just use the Sukharev grid. It is possible, however, to shift each coordinate that behaves like . If is rectangular but not a square, a good grid can still be made by tiling the space with cubes. In some cases this will produce optimal dispersion. For complicated spaces such as , no grid exists in the sense defined so far. It is possible, however, to generate grids on the faces of an inscribed Platonic solid [251] and lift the samples to with relatively little distortion [987]. For example, to sample , Sukharev grids can be placed on each face of a cube. These are lifted to obtain the warped grid shown in Figure 5.6a.
Example 5..15 (Sukharev Grid) Suppose that and . If , then the Sukharev grid yields points for the nine cases in which either coordinate may be , , or . The dispersion is . The spacing between the points along each axis is , which is twice the dispersion. If instead , which represents a torus, then the nine points may be shifted to yield the standard grid. In this case each coordinate may be 0, , or . The dispersion and spacing between the points remain unchanged.
One nice property of grids is that they have a lattice structure. This means that neighboring points can be obtained very easily be adding or subtracting vectors. Let be an -dimensional vector called a generator. A point on a lattice can be expressed as
(5.21)
for independent generators, as depicted in Figure 5.6b. In a 2D grid, the generators represent up'' and right.'' If and a standard grid with integer spacing is used, then the neighbors of the point are obtained by adding , , , or . In a general lattice, the generators need not be orthogonal. An example is shown in Figure 5.5b. In Section 5.4.2, lattice structure will become important and convenient for defining the search graph.
Steven M LaValle 2012-04-20
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https://runescape.wiki/w/Archmage_Sedridor
|
# Archmage Sedridor
Sedridor is the current archmage of the second Wizards' Tower, which trains wizards in the magical arts. He can be found in his office on the 2nd floor[UK]3rd floor[US] of the tower.
He is an avid drinker of Wizard's mind bombs, with Wizard Shug stating that he could rival a Fremennik in a drinking contest.
## History
### Legacy of Blood
Sedridor played a minor role in the RuneScape novel Legacy of Blood. In 164, he was one of the three judges, alongside Wizard Grayzag and Aubury who decided the fate of the wizard Castimir for the crime of stealing knowledge. They voted to expel Castimir from the Tower permanently but spared his life. Afterwards, he had Aubury equip Castimir to become a rogue wizard, revealing that the Tower occasionally hired rogues to perform tasks that Tower wizards could not be seen doing.
Aubury and he later fought the army of Tenebra at the Second Battle of the Salve but the two wizards were quickly incapacitated by a darkness attack.
### Rebuilding the Mage Training Arena
After one of the wizards lost an apprentice after a spell misfired and exploded on them, during his free time drinking, Sedridor decided to rebuild the Mage Training Arena so that budding mages could safely train while starting with beginner-level spells. With the help of Wizards Shug, Dougal and Edvin, the Mage Training Arena was successfully rebuilt, with the original rune guardians and charmed warriors helping to train and protect the staff and students.
### Archmage of the Tower
The following takes place during Rune Mysteries.
In the year 169 of the Fifth Age, Ariane began having troubling visions regarding the future of the tower. Despite her warnings, Sedridor soon discovered her "demon-worshipping" (she had made a deal with Azacorax, the lesser demon imprisoned in the tower) and banned her from setting foot in the tower, even erecting a magical ward to ensure she could not enter. Although he trusted her knowledge, Sedridor stated that he had to uphold the rules of the tower, and that he intended to use an Icyenic Purge to cleanse the power beam of whatever creature was lurking inside after it attacked Wizard Borann.
Sedridor's stubbornness worried Ariane, as she knew the Icyenic Purge would not dispel the vortices; in fact, they were attracted to magic, and the Icyenic Purge would prompt the vortices to roam the tower and cause further troubles. Ariane tasked the adventurer with stopping Sedridor so that she could cast the proper spell to keep the vortices in the ruins. The adventurer did so by playing the grand organ in the tower, which angered Sedridor, and he even threatened them, but regardless, Ariane was able to cast her spell before the wizards could. The wizards mistakenly believed they had cast Icyenic Purge, and Sedridor told the adventurer to let Ariane know that all was good and well in the tower.
The following takes place during Rune Memories.
After the quest Rune Memories, Sedridor will be saddened by the betrayal of Wizard Ellaron and offer the adventurer to take from his personal rune chest after the quest which yields pure essence equal ${\displaystyle 2.5\times level\times (level+1)}$ - this is updated each time a level is gained, for a total of 24,750 essence granted once level 99 is reached. He will also offer the Archmage title to players who have 99 Magic and Runecrafting.
### Attack on the Tower
The following takes place during Love Story.
After Dionysius reconciled with Zenevivia, the two old mages decided to raid the Wizards' Tower to replenish their stockpile of runes, as they were running out after their recent fight with each other. They left their house, prompting the guard staking out at the tree to state that the subjects were on the move towards Wizards' Tower.
Sedridor organized the Wizards Grayzag, Traiborn and Mizgog to defend the tower while Wizard Elriss sealed up the Runecrafting Guild. Much to Sedridor's dismay, Traiborn had his usual delusion, so Sedridor reluctantly told the others to prepare for battle as they would be one wizard short. Wizards Grayzag and Mizgog were forced to retreat, leaving Sedridor alone, but Traiborn quickly came back with a thingummywut. Sedridor was shocked by the thingummywut as it rapid-casted Earth Surge on Dionysius and Zenevivia, forcing the two mages to retreat. Following the incident, Sedridor warned the two not to try anything funny again or they would put a stop to it.
### Summer Beach Party
Archmage Sedridor is one of the four NPCs who can teleport to the sandcastle building area in the Summer Beach Party. While he is present, building the Wizards' Sandtower (a replica of the Wizards' Tower) will grant double experience.
Sedridor speaks
## Historical
In the old Rune Mysteries quest, Sedridor, having struggled to rediscover the knowledge of runestone creation, which was lost after the destruction of the Tower by Zamorakian mages decades ago, recently succeeded in this task largely due to the efforts of an adventurer who brought him an unidentified air talisman from Lumbridge's Duke Horacio, who knows of Sedridor and his vast lore well enough to send it to him for examination. He is located in the tower's basement. You are required to talk to him twice during the Rune Mysteries quest. After the said quest, you can use him to teleport to the rune essence mine, where you can mine rune or pure essence. He is seldom visited for this, because he is so far away from a bank. Sedridor has conducted extensive research on runes and the secrets of crafting runestones, eventually combining his research with that of Aubury's rune knowledge to unlock the secrets of the air talisman, thus considerably increasing the Wizards' supply of runes and leading to the opening of a Runecrafting Guild, which is run by a friend of Sedridor's, Wizard Elriss.
## Dialogue
Archmage Sedridor speaks in:
Predecessor Title Successor
Unnamed female Archmage of the Wizards' Tower Incumbent
## Update history
This information has been compiled as part of the update history project. Some updates may not be included - see here for how to help out!
## Trivia
Concept art of Sedridor and Traiborn.
• Before the graphical update of the wizards tower, Sedridor could be found in the tower basement.
• Archmage Sedridor will not allow players to bank essence at the deposit box, which can be found at the very top floor of the Wizards' Tower. This may be a conscious decision by Jagex to prevent an easier way of power-mining rune/pure essence. Upon attempting to do so, a message will appear stating: "Sedridor does not want you banking rune essence in the tower."
|
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|
https://docs.scipy.org/doc/scipy-1.3.3/reference/generated/scipy.sparse.linalg.minres.html
|
# scipy.sparse.linalg.minres¶
scipy.sparse.linalg.minres(A, b, x0=None, shift=0.0, tol=1e-05, maxiter=None, M=None, callback=None, show=False, check=False)[source]
Use MINimum RESidual iteration to solve Ax=b
MINRES minimizes norm(A*x - b) for a real symmetric matrix A. Unlike the Conjugate Gradient method, A can be indefinite or singular.
If shift != 0 then the method solves (A - shift*I)x = b
Parameters
A{sparse matrix, dense matrix, LinearOperator}
The real symmetric N-by-N matrix of the linear system Alternatively, A can be a linear operator which can produce Ax using, e.g., scipy.sparse.linalg.LinearOperator.
b{array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
Returns
x{array, matrix}
The converged solution.
infointeger
Provides convergence information:
0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown
Other Parameters
x0{array, matrix}
Starting guess for the solution.
tolfloat
Tolerance to achieve. The algorithm terminates when the relative residual is below tol.
maxiterinteger
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
M{sparse matrix, dense matrix, LinearOperator}
Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.
callbackfunction
User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
References
Solution of sparse indefinite systems of linear equations,
C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629. https://web.stanford.edu/group/SOL/software/minres/
This file is a translation of the following MATLAB implementation:
https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip
#### Previous topic
scipy.sparse.linalg.lgmres
#### Next topic
scipy.sparse.linalg.qmr
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|
https://thespectrumofriemannium.wordpress.com/2012/09/01/log026-boosts-rapidity-hep/
|
# LOG#026. Boosts, rapidity, HEP.
In euclidean two dimensional space, rotations are easy to understand in terms of matrices and trigonometric functions. A plane rotation is given by:
$\boxed{\begin{pmatrix}x'\\ y'\end{pmatrix}=\begin{pmatrix}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}}\leftrightarrow \boxed{\mathbb{X}'=\mathbb{R}(\theta)\mathbb{X}}$
where the rotation angle is $\theta$, and it is parametrized by $0\leq \theta \leq 2\pi$.
Interestingly, in minkovskian two dimensional spacetime, the analogue does exist and it is written in terms of matrices and hyperbolic trigonometric functions. A “plane” rotation in spacetime is given by:
$\boxed{\begin{pmatrix}ct'\\ x'\end{pmatrix}=\begin{pmatrix}\cosh \varphi & -\sinh \varphi \\ -\sinh \varphi & \cosh \varphi \end{pmatrix}\begin{pmatrix}ct\\ x\end{pmatrix}}\leftrightarrow \boxed{\mathbb{X}'=\mathbb{L}(\varphi)\mathbb{X}}$
Here, $\varphi = i\psi$ is the so-called hiperbolic rotation angle, pseudorotation, or more commonly, the rapidity of the Lorentz boost in 2d spacetime. It shows that rapidity are a very useful parameter for calculations in Special Relativity. Indeed, it is easy to check that
$\mathbb{L}(\varphi_1+\varphi_2)=\mathbb{L}(\varphi_1)\mathbb{L}\mathbb(\varphi_2)$
So, at least in the 2d spacetime case, rapidities are “additive” in the written sense.
Firstly, we are going to guess the relationship between rapidity and velocity in a single lorentzian spacetime boost. From the above equation we get:
$ct'=ct\cosh \varphi -x\sinh \varphi$
$x'=-ct\sinh \varphi +x\cosh \varphi$
Multiplying the first equation by $\cosh \varphi$ and the second one by $\sinh \varphi$, we add the resulting equation to obtain:
$ct'\cosh\varphi+x'\sinh \varphi =ct\cosh^2 \varphi -ct\sinh^2 \varphi =ct$
that is
$ct'\cosh\varphi+x'\sinh \varphi =ct$
From this equation (or the boxed equations), we see that $\varphi=0$ corresponds to $x'=x$ and $t'=t$. Setting $x'=0$, we deduce that
$x'=0=-ct\sinh \varphi +x\cosh \varphi$
and thus
$ct\tanh \varphi =x$ or $x=ct\tanh\varphi$.
Since $t\neq 0$, and the pseudorotation seems to have a “pseudovelocity” equals to $V=x/t$, the rapidity it is then defined through the equation:
$\boxed{\tanh \varphi=\dfrac{V}{c}=\beta}\leftrightarrow\mbox{RAPIDITY}\leftrightarrow\boxed{\varphi=\tanh^{-1}\beta}$
If we remember what we have learned in our previous mathematical survey, that is,
$\tanh^{-1}z=\dfrac{1}{2}\ln \dfrac{1+z}{1-z}=\sqrt{\dfrac{1+z}{1-z}}$
We set $z=\beta$ in order to get the next alternative expression for the rapidity:
$\varphi=\ln \sqrt{\dfrac{1+\beta}{1-\beta}}=\dfrac{1}{2}\ln \dfrac{1+\beta}{1-\beta}\leftrightarrow \exp \varphi=\sqrt{\dfrac{1+\beta}{1-\beta}}$
In experimental particle physics, in general 3+1 spacetime, the rapidity definition is extended as follows. Writing, from the previous equations above,
$\sinh \varphi=\dfrac{\beta}{\sqrt{1-\beta^2}}$
$\cosh \varphi=\dfrac{1}{\sqrt{1-\beta^2}}$
and using these two last equations, we can also write momenergy components using rapidity in the same fashion. Suppose that for some particle(objetc), its mass is $m$, its energy is $E$, and its (relativistic) momentum is $\mathbf{P}$. Then:
$E=mc^2\cosh \varphi$
$\lvert \mathbf{P} \lvert =mc\sinh \varphi$
From these equations, it is trivial to guess:
$\varphi=\tanh^{-1}\dfrac{\lvert \mathbf{P} \lvert c}{E}=\dfrac{1}{2}\ln \dfrac{E+\lvert \mathbf{P} \lvert c}{E-\lvert \mathbf{P} \lvert c}$
This is the completely general definition of rapidity used in High Energy Physics (HEP), with a further detail. In HEP, physicists used to select the direction of momentum in the same direction that the collision beam particles! Suppose we select some orientation, e.g.the z-axis. Then, $\lvert \mathbf{P} \lvert =p_z$ and rapidity is defined in that beam direction as:
$\boxed{\varphi_{hep}=\tanh^{-1}\dfrac{\lvert \mathbf{P}_{beam} \lvert c}{E}=\dfrac{1}{2}\ln \dfrac{E+p_z c}{E-p_z c}}$
In 2d spacetime, rapidities add nonlinearly according to the celebrated relativistic addition rule:
$\beta_{1+2}=\dfrac{\beta_1+\beta_2}{1+\frac{\beta_1\beta_2}{c^2}}$
Indeed, Lorentz transformations do commute in 2d spacetime since we boost in a same direction x, we get:
$L_1^xL_2^x-L_2^xL_1^x=0$
with
$L_1^x=\begin{pmatrix}\gamma_1 & -\gamma_1\beta_1\\ -\gamma_1\beta_1 &\gamma_1 \end{pmatrix}$
$L_2^x=\begin{pmatrix}\gamma_2 & -\gamma_2\beta_2\\ -\gamma_2\beta_2 &\gamma_2 \end{pmatrix}$
This commutativity is lost when we go to higher dimensions. Indeed, in spacetime with more than one spatial direction that result is not true in general. If we build a Lorentz transformation with two boosts in different directions $V_1=(v_1,0,0)$ and $V_2=(0,v_2,0)$, the Lorentz matrices are ( remark for experts: we leave one direction in space untouched, so we get 3×3 matrices):
$L_1^x=\begin{pmatrix}\gamma_1 & -\gamma_1\beta_1 &0\\ -\gamma_1\beta_1 &\gamma_1 &0\\ 0& 0& 1\end{pmatrix}$
$L_2^y=\begin{pmatrix}\gamma_2 & 0&-\gamma_2\beta_2\\ 0& 1& 0\\ -\gamma_2\beta_2 & 0&\gamma_2 \end{pmatrix}$
and it is easily checked that
$L_1^xL_2^y-L_2^yL_1^x\neq 0$
Finally, there is other related quantity to rapidity that even experimentalists do prefer to rapidity. It is called: PSEUDORAPIDITY!
Pseudorapidity, often denoted by $\eta$ describes the angle of a particle relative to the beam axis. Mathematically speaking is:
$\boxed{\eta=-\ln \tan \dfrac{\theta}{2}}\leftrightarrow \mbox{PSEUDORAPIDITY}\leftrightarrow \boxed{\exp (\eta)=\dfrac{1}{\tan\dfrac{\theta}{2}}}$
where $\theta$ is the angle between the particle momentum $\mathbf{P}$ and the beam axis. The above relation can be inverted to provide:
$\boxed{\theta=2\tan^{-1}(e^{-\eta})}$
The pseudorapidity in terms of the momentum is given by:
$\boxed{\eta=\dfrac{1}{2}\ln \dfrac{\vert \mathbf{P}\vert +P_L}{\vert \mathbf{P}\vert -P_L}}$
Note that, unlike rapidity, pseudorapidity depends only on the polar angle of its trajectory, and not on the energy of the particle.
In hadron collider physics, and other colliders as well, the rapidity (or pseudorapidity) is preferred over the polar angle because, loosely speaking, particle production is constant as a function of rapidity. One speaks of the “forward” direction in a collider experiment, which refers to regions of the detector that are close to the beam axis, at high pseudorapidity $\eta$.
The rapidity as a function of pseudorapidity is provided by the following formula:
$\boxed{\varphi=\ln\dfrac{\sqrt{m^2+p_T^2\cosh^2\eta}+p_T\sinh \eta}{\sqrt{m^2+p_T^2}}}$
where $p_T$ is the momentum transverse to the direction of motion and m is the invariant mass of the particle.
Remark: The difference in the rapidity of two particles is independent of the Lorentz boosts along the beam axis.
Colliders measure physical momenta in terms of transverse momentum $p_T$ instead of the momentum in the direction of the beam axis (longitudinal momentum) $P_L=p_z$, the polar angle in the transverse plane (genarally denoted by $\phi$) and pseudorapidity $\eta$. To obtain cartesian momenta $(p_x,p_y,p_z)$ (with the z-axis defined as the beam axis), the following transformations are used:
$p_x=P_T\cos\phi$
$p_y=P_T\sin\phi$
$p_z=P_T\sinh\eta$
Thus, we get the also useful relationship
$\vert P \vert=P_T\cosh\eta$
This quantity is an observable in the collision of particles, and it can be measured as the main image of this post shows.
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https://openstax.org/books/introductory-business-statistics/pages/10-4-comparing-two-independent-population-proportions
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# 10.4Comparing Two Independent Population Proportions
Introductory Business Statistics10.4 Comparing Two Independent Population Proportions
When conducting a hypothesis test that compares two independent population proportions, the following characteristics should be present:
1. The two independent samples are random samples that are independent.
2. The number of successes is at least five, and the number of failures is at least five, for each of the samples.
3. Growing literature states that the population must be at least ten or even perhaps 20 times the size of the sample. This keeps each population from being over-sampled and causing biased results.
Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance in the sampling. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the two population proportions.
Like the case of differences in sample means, we construct a sampling distribution for differences in sample proportions: $(pA'-pB')(pA'-pB')$where $p A ' = X A n A p A ' = X A n A$ and $p B ' = X B n B p B ' = X B n B$ are the sample proportions for the two sets of data in question. XA and XB are the number of successes in each sample group respectively, and nA and nB are the respective sample sizes from the two groups. Again we go the Central Limit theorem to find the distribution of this sampling distribution for the differences in sample proportions. And again we find that this sampling distribution, like the ones past, are normally distributed as proved by the Central Limit Theorem, as seen in Figure 10.5 .
Figure 10.5
Generally, the null hypothesis allows for the test of a difference of a particular value, 𝛿0, just as we did for the case of differences in means.
$H0 : p1 − p2 = 𝛿0 H0:p1−p2=𝛿0$
$H1 : p1 − p2 ≠ 𝛿0 H1:p1−p2≠𝛿0$
Most common, however, is the test that the two proportions are the same. That is,
$H 0 : p A = p B H 0 : p A = p B$
$H a : p A ≠ p B H a : p A ≠ p B$
To conduct the test, we use a pooled proportion, pc.
The pooled proportion is calculated as follows:
$p c = x A + x B n A + n B p c = x A + x B n A + n B$
The test statistic (z-score) is:
$Zc = ( p A ′ − p B ′ ) − δ0 p c (1− p c )( 1 n A + 1 n B ) Zc= ( p A ′ − p B ′ ) − δ0 p c (1− p c )( 1 n A + 1 n B )$
where δ0 is the hypothesized differences between the two proportions and pc is the pooled variance from the formula above.
### Example 10.6
A bank has recently acquired a new branch and thus has customers in this new territory. They are interested in the default rate in their new territory. They wish to test the hypothesis that the default rate is different from their current customer base. They sample 200 files in area A, their current customers, and find that 20 have defaulted. In area B, the new customers, another sample of 200 files shows 12 have defaulted on their loans. At a 10% level of significance can we say that the default rates are the same or different?
Solution 10.6
This is a test of proportions. We know this because the underlying random variable is binary, default or not default. Further, we know it is a test of differences in proportions because we have two sample groups, the current customer base and the newly acquired customer base. Let A and B be the subscripts for the two customer groups. Then pA and pB are the two population proportions we wish to test.
Random Variable:P′AP′B = difference in the proportions of customers who defaulted in the two groups.
$H0:pA=pBH0:pA=pB$
$Ha:pA≠pBHa:pA≠pB$
The words "is a difference" tell you the test is two-tailed.
Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal:
$p c = x A + x B n A + n B = 20+12 200+200 =0.08 1– p c =0.92 p c = x A + x B n A + n B = 20+12 200+200 =0.08 1– p c =0.92$
(p′Ap′B) = 0.04 follows an approximate normal distribution.
Estimated proportion for group A: $p ′ A = x A n A = 20 200 =0.1 p ′ A = x A n A = 20 200 =0.1$
Estimated proportion for group B: $p ′ B = x B n B = 12 200 =0.06 p ′ B = x B n B = 12 200 =0.06$
The estimated difference between the two groups is : p′Ap′B = 0.1 – 0.06 = 0.04.
Figure 10.6
$Zc = (P′A−P′B)−δ0 Pc(1−Pc) (1nA+1nB) = 0.54 Zc= (P′A−P′B)−δ0 Pc(1−Pc) (1nA+1nB) =0.54$
The calculated test statistic is .54 and is not in the tail of the distribution.
Make a decision: Since the calculate test statistic is not in the tail of the distribution we cannot reject H0.
Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference between the proportions of customers who defaulted in the two groups.
Try It 10.6
Two types of valves are being tested to determine if there is a difference in pressure tolerances. Fifteen out of a random sample of 100 of Valve A cracked under 4,500 psi. Six out of a random sample of 100 of Valve B cracked under 4,500 psi. Test at a 5% level of significance.
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http://www.transtutors.com/questions/a-uniform-thin-rod-of-length-0-500-m-and-mass-4-00-kg-can-431277.htm
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+1.617.933.5480
# Q: A uniform thin rod of length 0.500 m and mass 4.00 kg can
A uniform thin rod of length 0.60 m and mass 3.5 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling in the rotation plane is fired into one end of the rod. As viewed from above, the bullet's path makes angle θ = 60° with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 14 rad/s immediately after the collision, what is the bullet's speed just before impact? (Answer is in m/s)
We must conservation of angular velocity. Initially: (3/1000)v 0.6 sin 60°+(3/1000)(0
Related Questions in General Physics
Question Status: Solved
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http://link.springer.com/chapter/10.1007/978-1-4419-6594-3_30
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Springer Optimization and Its Applications Volume 42, 2011, pp 447-460
Date: 02 Sep 2010
# Context Hidden Markov Model for Named Entity Recognition
* Final gross prices may vary according to local VAT.
## Abstract
Named entity (NE) recognition is a core technology for understanding low-level semantics of texts. In this paper we consider the combination of two classifiers: our version of probabilistic supervised machine learning classifier, which we named the Context Hidden Markov Model, and grammar rule-based system in named entity recognition. In order to deal with the problem of estimating the probabilities of unseen events, we have applied the probability mixture models which were estimated using another machine learning algorithm: Expectation Maximization. We have tested our Named Entity Recognition system on MUC 7 corpus.
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https://www.arxiv-vanity.com/papers/1002.0145/
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# From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-box Identity Test for Depth-3 Circuits
###### Abstract.
We study the problem of identity testing for depth- circuits of top fanin and degree (called identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic -time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes -time. Our structure theorem essentially says that the number of independent variables in a real depth- identity is very small. This theorem settles affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field) for black-box identity tests.
Our main theorem (almost optimally) pins down the relation between higher dimensional Sylvester-Gallai theorems and the rank of depth- identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of Sylvester-Gallai rank bounds for any field, and show the intimate connection between this and depth- identity rank bounds. We also prove the first ever theorem about high dimensional Sylvester-Gallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth- identity : there is a nucleus of that forms a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.
Hausdorff Center for Mathematics, Bonn - 53115, Germany.
IBM Almaden, San Jose - 95126, USA.
## 1. Introduction
Polynomial identity testing (PIT) ranks as one of the most important open problems in the intersection of algebra and computer science. We are provided an arithmetic circuit that computes a polynomial over a field , and we wish to test if is identically zero (in other words, if is the zero polynomial). In the black-box setting, the circuit is provided as a black-box and we are only allowed to evaluate the polynomial at various domain points. The main goal is to devise a deterministic polynomial time algorithm for PIT. Kabanets & Impagliazzo [KI04] and Agrawal [Agr05, Agr06] have shown connections between deterministic algorithms for identity testing and circuit lower bounds, emphasizing the importance of this problem. To know more about the current state of the general identity testing problem see the surveys [Sax09, AS09].
The first randomized polynomial time PIT algorithm, which was a black-box algorithm, was given (independently) by Schwartz [Sch80] and Zippel [Zip79]. Randomized algorithms that use less randomness were given by Chen & Kao [CK00], Lewin & Vadhan [LV98], and Agrawal & Biswas [AB03]. Klivans & Spielman [KS01] observed that even for depth- circuits for bounded top fanin, deterministic identity testing was open. Progress towards this was first made by Dvir & Shpilka [DS06], who gave a quasi-polynomial time algorithm, although with a doubly-exponential dependence on the top fanin. The problem was resolved by a polynomial time algorithm given by Kayal and Saxena [KS07], with a running time exponential in the top fanin. As expected, the current understanding of depth- circuits is even more sparse. Identity tests are known only for rather special depth- circuits [AM07, Sax08, SV09, KMSV09]. Why is progress restricted to such small depth circuits? Agrawal and Vinay [AV08] showed that an efficient black-box identity test for depth- circuits will actually give a quasi-polynomial black-box test, and subexponential lower bounds, for circuits of all depths (that compute low degree polynomials). Thus, understanding depth- identities seems to be a natural first step towards the goal of proving more general lower bounds.
For deterministic black-box testing, the first results were given by Karnin & Shpilka [KS08]. Based on results in [DS06], they gave an algorithm for bounded top fanin depth- circuits having a quasi-polynomial running time (with a doubly-exponential dependence on the top fanin). The dependence on the top fanin was later improved (to singly-exponential) by the rank bound results of Saxena & Seshadhri [SS09] (for any ). But the time complexity also had a quasi-polynomial dependence on the degree of the circuit. This dependence is inevitable in rank-based methods over finite fields (as shown by [KS07]). However, over the field of rationals, Kayal & Saraf [KS09b] showed how to remove this quasi-polynomial dependence on the degree at the cost of doubly-exponential dependence on the top fanin, thus giving a polynomial time complexity for bounded top fanin. In this work we achieve the best of the two works [SS09] and [KS09b], i.e. we prove (for rationals) a time complexity that depends only polynomially on the degree and “only” singly-exponentially on the fanin.
In a quite striking result, Kayal & Saraf [KS09b] proved how Sylvester-Gallai theorems can get better rank bounds over the reals. We introduce the concept of Sylvester-Gallai rank bounds that deals with the rank of vectors (over some given field) that have some special incidence properties. This is a very convenient way to express known Sylvester-Gallai results. These are inspired by the famous Sylvester-Gallai theorem about point-line incidences. We show how this very interesting quantity is tightly connected to depth- identities. Sylvester-Gallai rank bounds over high dimensions were known over the reals, and are used to prove depth- rank bounds over reals. We prove the first ever theorem for high dimensional Sylvester-Gallai configurations over any field.
### 1.1. Definitions and Previous Work
This work focuses on depth- circuits. A structural study of depth- identities was initiated in [DS06] by defining a notion of rank of simple and minimal identities. A depth- circuit over a field is:
C(x1,…,xn)=k∑i=1Ti
where, (a multiplication term) is a product of linear polynomials over . Note that for the purposes of studying identities we can assume wlog (by homogenization) that ’s are linear forms (i.e. linear polynomials with a zero constant coefficient) and that . Such a circuit is referred to as a circuit (or depending on the context), where is the top fanin of and is the degree of . We give a few definitions from [DS06].
###### Definition 1.
[Simple Circuit] is a simple circuit if there is no nonzero linear form dividing all the ’s.
[Minimal Circuit] is a minimal circuit if for every proper subset , is nonzero.
[Rank of a circuit] Every can be seen as an -dimensional vector over . The rank of the circuit, , is defined as the rank of the set of all linear forms ’s viewed as -dimensional vectors.
Can all the forms be independent, or must there be relations between them? The rank can be interpreted as the minimum number of variables that are required to express . There exists a linear transformation converting the variables of the circuit into independent variables. A trivial upper bound on the rank (for any -circuit) is , since that is the total number of linear forms involved in . The rank is a fundamental property of a circuit and it is crucial to understand how large this can be for identities. A substantially smaller rank bound than shows that identities do not have as many “degrees of freedom” as general circuits, and leads to deterministic identity tests. Furthermore, the techniques used to prove rank bounds show us structural properties of identities that may suggest directions to resolve PIT for circuits.
The rank bounds, in addition to being a natural property of identities, have found applications in black-box identity testing [KS08] and learning circuits [Shp09, KS09a]. The result of [KS08] showed rank bounds imply black-box testers: if is a rank bound for simple minimal identities over field , then there is a deterministic black-box identity tester for such circuits, that runs in -operations. (For the time complexity over , we actually count the bit operations.)
Dvir & Shpilka [DS06] proved that the rank of a simple, minimal identity is bounded by . This rank bound was improved to by Saxena & Seshadhri [SS09]. Fairly basic identity constructions show that the rank is over the reals and for finite fields [DS06, KS07, SS09]. Dvir & Shpilka [DS06] conjectured that should be some over the reals. Through a very insightful use of Sylvester-Gallai theorems, Kayal & Saraf [KS09b] subsequently bounded the rank of identities, over reals, by . This means that for a constant top fanin circuit, the rank of identities is constant, independent of the degree. This also leads to the first truly polynomial-time deterministic black-box identity testers for this case.
Unfortunately, as soon as becomes even , this bound becomes trivial. We improve this rank bound exponentially, to , which is almost optimal. This gives a major improvement in the running time of the black-box testers. We also improve the rank bounds for general fields from to . We emphasize that we give a unified framework to prove all these results. Table 1 should make it easier to compare the various bounds.
Kayal & Saraf [KS09b] connect Sylvester-Gallai theorems to rank bounds. They need advanced versions of these theorems that deal with colored points and have to prove certain hyperplane decomposition theorems. We make the connection much more transparent (at the loss of some color from the theorems). We reiterate that our techniques are completely different, and employ a very powerful algebraic framework to dissect identities. This allows us to use as a “black-box” the most basic form of the higher dimensional Sylvester-Gallai theorems.
### 1.2. Our Results
Before we state our results, it will be helpful to understand Sylvester-Gallai configurations. A set of points with the property that every line through two points of passes through a third point in is called a Sylvester-Gallai configuration. The famous Sylvester-Gallai theorem states: for a set of points in , not all collinear, there exists a line passing through exactly two points of . In other words, the only Sylvester-Gallai configuration in is a set of collinear points. This basic theorem about point-line incidences was extended to higher dimensions [Han65, BE67]. We introduce the notion of Sylvester-Gallai rank bounds. This is a clean and convenient way of expressing these theorems.
###### Definition 2.
Let be a finite subset of the projective space . Alternately, is a subset of vectors in without multiples: no two vectors in are scalar multiples of each other111When , such an is, wlog, a subset of distinct vectors with first coordinate .. Suppose, for every set of linearly independent vectors, the linear span of contains at least vectors of . Then, the set is said to be -closed.
The largest possible rank of an -closed set of at most vectors in (for any ) is denoted by .
The classic Sylvester-Gallai theorem essentially states222To see this, take an -closed set of vectors. Think of each vector being represented by an infinite line through the origin, hence giving a set in the projective space. Take a -dimensional plane not passing through the origin and take the set of intersection points of the lines in with . Observe that the coplanar points have the property that a line passing through two points of passes through a third point of . that for all , . Higher dimensional analogues [Han65, BE67] prove that . One of our auxiliary theorems is such a statement for all fields.
###### Theorem 3 (SGk for all fields).
For any field and , .
Our main theorem is a simple, clean expression of how Sylvester-Gallai influences identities.
###### Theorem 4 (From SGk to Rank).
Let . The rank of a simple and minimal identity over is at most .
Remark. If is small, then we choose an extension of size and get a rank bound with .
Plugging in -rank bounds gives us the desired theorem for depth- identities. We have a slightly stronger version of the above theorem that we use to get better constants (refer to Theorem 18).
###### Theorem 5 (Depth-3 Rank Bounds).
Let be a circuit, over field , that is simple, minimal and zero. Then,
• For , .
• For any , .
As discussed before, a direct application of this result to Lemma 4.10 of [KS08] gives a deterministic black-box identity test for circuits (we will only discuss here as the other statement is analogous). Formally, we get the following hitting set generator for circuits with real coefficients.
###### Corollary 6 (Black-box PIT over Q).
There is a deterministic algorithm that takes as input a triple of natural numbers and in time , outputs a hitting set with the following properties:
• Any circuit over computes the zero polynomial iff , .
• has at most points.
• The total bit-length of each point in is .
Remark.
• Our black-box test has quasi-polynomial in time complexity (with polynomial-dependence on ) for top fanin as large as , and sub-exponential in time complexity (with polynomial-dependence on ) even for top fanin as large as . This is the first tester to achieve such bounds.
• The fact that the points in are integral and have “small” bit-length is important to estimate the time complexity of our algorithm in terms of bit operations. Thus, the hitting set generator takes at most bit operations to compute .
## 2. Proof Outline, Ideas, and Organization
Our proof of the rank bound comprises of several new ideas, both at the conceptual and the technical levels. In this section we will give the basic intuition of the proof. The three notions that are crucially used (or developed) in the proof are: ideal Chinese remaindering, matchings and Sylvester-Gallai rank bounds. These have appeared (in some form) before in the works of Kayal & Saxena [KS07], Saxena & Seshadhri [SS09] and Kayal & Saraf [KS09b] respectively, to prove different kinds of results. Here we use all three of them together to show quite a strong structure in identities. We will talk about them one by one in the following three subsections outlining the three steps of the proof. Each step proves a new property of identities which is interesting in its own right. The first two steps set up the algebraic framework and prove theorems that hold for all fields. The third step is where the Sylvester-Gallai theorems are brought in. Some (new and crucial) algebraic lemmas and their proofs have been moved to the Appendix. The flow of the actual proof will be identical to the overview that we now provide.
### 2.1. Step 1: Matching the Gates in an Identity
We will denote the set by . We fix the base field to be , so the circuits compute multivariate polynomials in the polynomial ring .
A linear form is a linear polynomial in with zero constant term. We will denote the set of all linear forms by . Clearly, is a vector (or linear) space over and that will be quite useful. Much of what we do shall deal with multi-sets of linear forms (sometimes polynomials in too), equivalence classes inside them, and various maps across them. A list of linear forms is a multi-set of forms with an arbitrary order associated with them. The actual ordering is unimportant: we will heavily use maps between lists, and the ordering allows us to define these maps unambiguously. The object, list, comes with all the usual set operations naturally defined.
###### Definition 7.
We collect some important definitions from [SS09]:
[Multiplication term, & ] A multiplication term is an expression in given as (the product may have repeated ’s), , where and is a list of nonzero linear forms. The list of linear forms in , , is just the list of forms occurring in the product above. For a list of linear forms we define the multiplication term of , , as or if .
[Forms in a Circuit] We will represent a circuit as a sum of multiplication terms of degree , . The list of linear forms occurring in is . Note that is a list of size exactly . The rank of , , is just the number of linearly independent linear forms in . (Remark: for the purposes of this paper ’s are given in circuit representation and thus the list is unambiguously defined from )
[Similar forms] For any two polynomials we call similar to if there exists such that . We say is similar to mod , for some ideal of , if there is some such that . Note that “similarity mod ” is an equivalence relation (reflexive, symmetric and transitive) and partitions any list of polynomials into equivalence classes.
[Span ] For any we let be the linear span of the linear forms in over the field . (Conventionally, .)
[Matchings] Let be lists of linear forms and be a subspace of . An -matching between is a bijection between lists such that: for all , .
When are multiplication terms, an -matching between would mean an -matching between .
We will show that all the multiplication terms of a minimal identity can be matched by a “low” rank space.
###### Theorem 8 (Matching-Nucleus).
Let be a circuit that is minimal and zero. Then there exists a linear subspace of such that:
1) .
2) , there is a -matching between .
The idea of matchings within identities was first introduced in [SS09], but nothing as powerful as this theorem has been proven. This theorem gives us a space of small rank, independent of , that contains most of the “complexity” of . All forms in outside are just mirrored in the various terms. This starts connecting the algebra of depth- identities to a combinatorial structure. Indeed, the graphical picture (explained in detail below) that this theorem provides, really gives an intuitive grasp on these identities. The proof of this involves some interesting generalizations of the Chinese Remainder Theorem to some special ideals.
###### Definition 9 (mat-nucleus).
Let be a minimal identity. The linear subspace given by Theorem 8 is called mat-nucleus of .
The notion of mat-nucleus is easier to see in the following unusual representation of the circuit . The four bubbles refer to the four multiplication terms of and the points inside the bubbles refer to the linear forms in the terms. The proof of Theorem 8 gives mat-nucleus as the space generated by the linear forms in the dotted box. The linear forms that are not in mat-nucleus lie “above” the mat-nucleus and are all (mat-nucleus)-matched, i.e. , there is a form similar to modulo mat-nucleus in each . Thus the essence of Theorem 8 is: the mat-nucleus part of the terms of has low rank , while the part of the terms above mat-nucleus all look “similar”.
#### Proof Idea for Theorem 8
The key insight in the construction of mat-nucleus is a reinterpretation of the identity test of Kayal & Saxena [KS07] as a structural result for identities. Again, refer to the following figure depicting a circuit and think of each bubble having linear forms. Roughly, [KS07] showed that iff for every path (where ): or in ideal terms, . Thus, roughly, it is enough to go through all the paths to certify the zeroness of . This is why the time complexity of the identity test of [KS07] is dominated by .
Now if we are given a identity which is minimal, then we know that . Thus, by applying the above interpretation of [KS07] to we will get a path such that . Since this means that but (if is in then so will be ). Thus, is a nontrivial congruence and it immediately gives us a -matching between (see Lemma 44). By repeating this argument with a different permutation of the terms we could match different terms (by a different ideal), and finally we expect to match all the terms (by the union of the various ideals).
This fantastic argument has numerous technical problems, but they can all be taken care of by suitable algebraic generalizations. The main stumbling block is the presence of repeating forms. It could happen that , occurs in many terms, or in the same term with a higher power. The most important tool developed is an ideal version of Chinese remaindering that forces us to consider not just linear forms , but multiplication terms dividing respectively. We give the full proof in Section 3. (Interestingly, the non-blackbox identity test of [KS07] guides in devising a blackbox test of “similar” complexity over rationals.)
### 2.2. Step 2: Certificate for Linear Independence of Gates
Theorem 8 gives us a space , of rank , that matches to each term . In particular, this means that the list has the same cardinality for each . In fact, if we look at the corresponding multiplication terms , , then they again form a identity! Precisely, for some ’s in (see Lemma 46) is an identity. We would like to somehow mimic the structure of . Of course is simple but is it again minimal? Unfortunately, it may not be. For reasons that will be clear later, minimality of would have allowed us to go directly to Step 3. Now step 2 will involve increasing the space (but not by too much) that gives us a that “behaves” like . Specifically, if are linearly independent (i.e. s.t. ), then so are .
###### Theorem 10 (Nucleus).
Let be a minimal identity and let be a maximal set of linearly independent terms (). Then there exists a linear subspace of such that:
• .
• , there is a -matching between .
• (Define , .) The terms are linearly independent.
###### Definition 11 (nucleus).
Let be a minimal identity. The linear subspace given by Theorem 10 is called the nucleus of . By Lemma 46, the subspace induces an identity which we call the nucleus identity.
The notion of the nucleus is easier to grasp when is a identity that is strongly minimal, i.e. are linearly independent. Clearly, such a is also minimal333If for some proper , then linear independence of is violated.. For such a , Theorem 10 gives a nucleus such that the corresponding nucleus identity is strongly minimal. The structure of is very strongly represented by . As a bonus, we actually end up greatly simplifying the polynomial-time PIT algorithm of Kayal & Saxena [KS07] (although we will not discuss this point in detail in this paper).
#### Proof Idea for Theorem 10
The first two properties in the theorem statement are already satisfied by mat-nucleus of . So we incrementally add linear forms to the space mat-nucleus till it satisfies property (3) and becomes the nucleus. The addition of linear forms is guided by the ideal version of Chinese remaindering. For convenience assume to be linearly independent. Then, by homogeneity and equal degree, we have an equivalent ideal statement: and (see Lemma 42). Even in this general setting the path analogy (used in the last subsection) works and we essentially get linear forms and such that: and . We now add these forms to the space mat-nucleus, and call the new space . It is expected that the new are now linearly independent.
Not surprisingly, the above argument has numerous technical problems. But it can be made to work by careful applications of the ideal version of Chinese remaindering. We give the full proof in Section 4.
### 2.3. Step 3: Invoking Sylvester-Gallai Theorems
We make a slight, but hopefully interesting, detour and leave depth- circuits behind. We rephrase the standard Sylvester-Gallai theorems in terms of Sylvester-Gallai closure (or configuration) and rank bounds. This is far more appropriate for our application, and seems to be very natural in itself.
###### Definition 12 (SGk-closed).
Let . Let be a subset of non-zero vectors in without multiples: no two vectors in are scalar multiples of each other444This is just a set of elements in the projective space , but this formulation in terms of vectors is more convenient for our applications.. Suppose that for every set of linearly independent vectors in , the linear span of contains at least vectors of . Then, the set is said to be -closed.
We would expect that if is finite then it will get harder to keep -closed as is gradually increased. This intuition holds up when . As we mentioned earlier, the famous Sylvester-Gallai Theorem states: if a finite is -closed, then . It is optimal as the line has rank and is -closed.
In fact, there is also a generalization of the Sylvester-Gallai theorem known (as stated in Theorem 2.1 of [BE67]) : Let be a finite set in spanning that projective space. Then, there exists a -flat such that , and is spanned by those points .
Let be a finite set of points with first coordinate being and let . We claim that if is -closed, then . Otherwise the above theorem guarantees vectors in whose -flat has only points of . If has a point then as has first coordinates , it would mean that a convex linear combination of (i.e. sum of coefficients in the combination is ) is . In other words, , which contradicts having only points of . Thus, also has no point in , but this contradicts -closure of . This shows that higher dimensional Sylvester-Gallai theorem implies that if is -closed then . We prefer using this rephrasal of the higher dimensional Sylvester-Gallai Theorem. This motivates the following definitions.
###### Definition 13 (SG operator).
Let .
[] The largest possible rank of an -closed set of at most points in is denoted by . For example, the above discussion entails which is, interestingly, independent of . (Also verify that for , and .)
[] Suppose a set has rank greater than (where ). Then, by definition, is not -closed. In this situation we say the -dimensional Sylvester-Gallai operator (applied on ) returns a set of linearly independent vectors in whose span has no point in .
The Sylvester-Gallai theorem in higher dimensions can now be expressed succintly.
###### Theorem 14 (High dimension Sylvester-Gallai for R).
[Han65, BE67] .
Remark. This theorem is also optimal, for if we set to be a union of “skew lines” then has rank and is -closed. For example, when define . It is easy to verify that and the span of every three linearly independent vectors in contains a fourth vector!
Using some linear algebra and combinatorial tricks, we prove the first ever Sylvester-Gallai bound for all fields. The proof is in Section 6, where there is a more detailed discussion of this (and the connection with LDCs).
Theorem 3 ( for all fields). For any field and , .
#### 2.3.1. Back to identities
Let be a simple and strongly minimal identity. Theorem 10 gives us a nucleus , of rank , that matches to each term . As seen in Step 2, if we look at the corresponding multiplication terms , , then they again form a “nucleus identity” , for some ’s in , which is simple and strongly minimal. Define the non-nucleus part of as , for all ( in the exponent annotates “complement”, since ). What can we say about the rank of ?
Define the non-nucleus part of as . Our goal in Step 3 is to bound by when the field is . This will give us a rank bound of for simple and strongly minimal identities over . The proof is mainly combinatorial, based on higher dimensional Sylvester-Gallai theorems and a property of set partitions, with a sprinkling of algebra.
We will finally apply operator not directly on the forms in but on a suitable truncation of those forms. So we need another definition.
###### Definition 15 (Non-K rank).
Let be a linear subspace of . Then is again a linear space (the quotient space). Let be a list of forms in . The non- rank of is defined to be (i.e. the rank of when viewed as a subset of ).
Let be a identity with nucleus . The non- rank of the non-nucleus part is called the non-nucleus rank of . Similarly, the non- rank of the non-nucleus part is called the non-nucleus rank of .
We give an example to explain the non- rank. Let . Suppose and . We can take any element in and simply drop all the terms, i.e. ‘truncate’ -part of . This gives a set of linear forms over the variables. The rank of these is the non- rank of .
We are now ready to state the theorem that is proved in Step 3. It basically shows a neat relationship between the non-nucleus part and Sylvester-Gallai.
###### Theorem 16 (Bound for simple, strongly minimal identities).
Let . The non-nucleus rank of a simple and strongly minimal identity over is at most .
Given a simple, minimal identity that is not strongly minimal. Let be linearly independent and form a basis of . Then it is clear that such that is a strongly minimal identity (for some ). Hence, we could apply the above theorem on this identity and get a rank bound for the non-nucleus part. The only problem is this fanin- identity may not be simple. Our solution for this is to replace by the suitable linear combination of in and repeat the above argument on the new identity. In Section 5.2 we show this takes care of the whole non-nucleus part and bounds its rank by . To state the theorem formally, we need a more refined notion than the fanin of a circuit.
###### Definition 17 (Independent-fanin).
Let be a circuit. The independent-fanin of , , is defined to be the size of the maximal such that are linearly independent polynomials. (Remark: If then . Also, for an identity , is strongly minimal iff .)
We now state the following stronger version of the main theorem.
###### Theorem 18 (Final bound).
Let . The rank of a simple, minimal , independent-fanin , identity is at most .
Remark: In particular, the rank of a simple, minimal identity over reals is at most , proving the main theorem over reals. Likewise, for any , we get the rank bound of , proving the main theorem.
#### Proof Idea for Theorem 16
Basically, we apply the operator on the non-nucleus part of the term , i.e. we treat a linear form as the point for the purposes of Sylvester-Gallai and then we consider assuming that the non-nucleus rank of is more than . This application of Sylvester-Gallai is much more direct compared to the methods used in [KS09b]. There, they needed versions of Sylvester-Gallai that dealt with colored points and had to prove a hyperplane decomposition property after applying essentially a operator on . Since, modulo the nucleus, all multiplication terms look essentially the same, it suffices to focus attention on just one of them. Hence, we apply the -operator on a single multiplication term.
To continue with the proof idea, assume is a simple, strongly minimal identity with terms and let be its nucleus given by Step 2. It will be convenient for us to fix a linear form and a subspace of such that we have the following orthogonal vector space decomposition (i.e. implies and implies ). This means for any form , there is a unique way to express , where , and . Furthermore, we will assume wlog that for every form the corresponding is nonzero, i.e. each form in is monic wrt (see Lemma 40).
###### Definition 19 (trun(⋅)).
Fix a decomposition . For any form , there is a unique way to express , where , and .
The truncated form is the linear form obtained by dropping the part and normalizing, i.e. .
Given a list of forms we define to be the corresponding set (thus no repetitions) of truncated forms.
To be precise, we fix a basis of so that each form in has representation (). We view each such form as the point while applying Sylvester-Gallai on . Assume, for the sake of contradiction, that the non-nucleus rank of , then (by definition) gives linearly independent forms whose span contains no other linear form of .
For simplicity of exposition, let us fix , spanned by ’s, spanned by ’s and . Note that (by definition) . We want to derive a contradiction by using the -tuple and the fact that is a simple, strongly minimal identity. The contradiction is easy to see in the following configuration: Suppose the linear forms in that are similar to a form in are exactly those depicted in the figure. Let us consider modulo the ideal . It is easy to see that these forms (call them ) “kill” the first three gates, leaving . As is an identity this means , thus there is a form such that . Now none of the forms divide . Also, their non-trivial combination, say for , cannot occur in . Otherwise, by the matching property will be in . This contradicts the ’s being a -tuple. Thus, cannot be in , a contradiction. This means that the non-nucleus rank of is , which by matching properties implies the non-nucleus rank of is .
We were able to force a contradiction because we used a set of forms in an SG-tuple that killed three terms and “preserved” the last term. Can we always do this? This is not at all obvious, and that is because of repeating forms. Suppose, after going modulo form , the circuit looks like . This is not simple, but it does not have to be. We are only guaranteed that the original circuit is simple. Once we go modulo , that property is lost. Now, the choice of any form kills all terms. In the figure above, , does not yield a contradiction. We will use our more powerful Chinese remaindering tools and the nucleus properties to deal with this. We have to prove a special theorem about partitions of and use strong minimality (which we did not use in the above sketch). The full proof is given in Section 5.1.
## 3. Matching the Terms in an Identity: Construction of mat-nucleus
### 3.1. Chinese Remaindering for Multiplication Terms
Traditionally, Chinese remaindering is the fact: if two coprime polynomials (resp. integers) divide a polynomial (resp. integer) then divides . The key tool in constructing mat-nucleus is a version of Chinese remaindering specialized for multiplication terms but generalized to ideals. Similar methods appeared first in [KS07] but we turn those on their head and give a “simpler” proof. In particular, we avoid the use of local rings and Hensel lifting.
Let be multiplication terms generating an ideal . Define linear space .
When the set of generators are clear from the context we will also use the notation . Similarly, would be a shorthand for .
Remark. Radical-span is motivated by the radical of an ideal but it is not quite that, for example, but . It is easy to see that the ideal generated by radsp always contains the radical ideal.
Now we can neatly state Chinese remaindering as an ideal decomposition statement.
###### Theorem 21 (Ideal Chinese remaindering).
Let be multiplication terms. Define the ideal . Assume while, and . Then, .
###### Proof.
If is a polynomial in then clearly it is in each of the ideals , and .
Suppose is a polynomial in . Then by definition there exist and such that,
h=i1+az=i2+bf=i3+cg.
The second equation gives . Since , repeated applications of Lemma 41 give us, . Implying , hence . This ensures the existence of and a polynomial such that,
h=i′2+b′zf=i3+cg.
Again this system says that . Since , repeated applications of Lemma 41 give us . Implying , hence . This finishes the proof. ∎
The conditions in this theorem suggest that factoring a multiplication term into parts corresponding to the equivalence classes of “similarity mod ” would be useful.
###### Definition 22 (Nodes).
Let be a multiplication term and let be an ideal generated by some multiplication terms. As the relation “similarity mod ” is an equivalence relation on , it partitions, in particular, the list into equivalence classes.
[] For each such class pick a representative and define . (Note that form can also appear in this set, it represents the class .)
[] For each , we multiply the forms in that are similar to mod . We define nodes of mod as the set of polynomials . (Remark: When , nodes of are just the coprime powers-of-forms dividing .)
[…wrt a subspace] Let be a linear subspace of . Clearly, the relation “similarity mod ” is an equivalence relation on . It will be convenient for us to also use notations and . They are defined by replacing in the above definitions by .
Observe that the product of polynomials in just gives . Also, modulo , each node is just a form-power . In other words, modulo , a node is rank-one term. The choice of the word “node” might seem a bit mysterious, but we will eventually construct paths through these. To pictorially see what is going on, think of each term as a set of its constituent nodes.
We prove a corollary of the ideal Chinese remaindering theorem that will be very helpful in both Steps 1 and 2.
###### Corollary 23.
Let , be a multiplication term, and let be an ideal generated by some multiplication terms. Then, iff such that .
###### Proof.
If , for some , then clearly .
Conversely, assume . Let and correspondingly, . If then is similar to , hence and we are done. So assume . Also, in case has a form in , assume wlog is the representative of the class . Define , for all .
We claim that for all , . Otherwise such that either or . Former case contradicts being the representative of the class , while the latter case contradicts
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http://petegustafson.com/CalculiX/ccx_2.15/doc/ccx/node181.html
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Next: Green functions Up: Types of analysis Previous: Electromagnetism Contents
### Sensitivity
A sensitivity analysis calculates how a variable G (called the objective function) changes with some other variables s (called the design variables), i.e. DG/Ds.
If s are the coordinates of some nodes, then the objective function usually takes the form , i.e. it is a direct function of the coordinates and it is a direct function of the displacements, which are again a function of the coordinates. One can write (vector- and matrix-denoting parentheses have been omitted; it is assumed that the reader knows that and are vectors, and are matrices and that and are potentially vectors):
(430)
The governing equation for static (linear and nonlinear) calculations is , which leads to
(431)
or
(432)
where
(433)
Since for linear applications and , the above equations reduce in that case to
(434)
or
(435)
Consequently one arives at the equation:
(436)
For the speed-up of the calculations it is important to perform the calculation of the term on element level and to calculate the term before multiplying with the last term in brackets. Furhermore, should be calculated by solving an equation system and not by inverting .
For special objective functions this relationship is further simplified:
• if is the mass .
• if is the shape energy .
• if are the displacements Equation (435) applies directly
For eigenfrequencies as objective function one starts from the eigenvalue equation:
(437)
from which one gets:
(438)
Premultiplying with and taking the eigenvalue equation and the normalization of the eigenvectors w.r.t. into account leads to
(439)
Notice that this is the sensitivity of the eigenvalues, not of the eigenfrequencies (which are the square roots of the eigenvalues). This is exactly how it is implemented in CalculiX: you get in the output the sensitivity of the eigenvalues.
Subsequently, one can derive the eigenvalue equation to obtain the derivatives of the eigenvectors:
(440)
If s is the orientation in some or all of the elements, the term is in addition zero in the above equations.
In CalculiX, G is defined with the keyword *OBJECTIVE, s is defined with the keyword DESIGNVARIABLES and a sensitivity analysis is introduced with the procedure card *SENSITIVITY.
If the parameter NLGEOM is not used on the *SENSITIVITY card, the calculation of does not contain the large deformation and stress stiffness, else it does. Similarly, without NLGEOM is calculated based on the linear strains, else the quadratic terms are taken into account.
If the objective function is the mass, the shapeenergy or the displacements a *STATIC step must have been performed. The displacements and the stiffness matrix from this step are taken for and in Equation (436) (in the presence of a subsequent sensitivity step is stored automatically in a file with the name jobname.stm). If the static step was calculated with NLGEOM, so should the sensitivity step in order to be consistent. So the procedure cards should run like:
*STEP
*STATIC
...
*STEP
*SENSITIVITY
...
or
*STEP,NLGEOM
*STATIC
...
*STEP,NLGEOM
*SENSITIVITY
...
If the objective functions are the eigenfrequencies (which include the eigenmodes), a *FREQUENCY step must have been performed with STORAGE=YES. This frequency step may be a perturbation step, in which case it is preceded by a static step. The displacements , the stiffness matrix and the mass matrix for equations (439) and (440) are taken from the frequency step. If the frequency step is performed as a perturbation step, the sensitivity step should be performed with NLGEOM, else it is not necessary. So the procedure cards should run like:
*STEP
*FREQUENCY,STORAGE=YES
...
*STEP
*SENSITIVITY
...
or
*STEP
*STATIC
...
*STEP,PERTURBATION
*FREQUENCY,STORAGE=YES
...
*STEP,NLGEOM
*SENSITIVITY
...
or
*STEP,NLGEOM
*STATIC
...
*STEP,PERTURBATION
*FREQUENCY,STORAGE=YES
...
*STEP,NLGEOM
*SENSITIVITY
...
(a perturbation frequency step only makes sense with a preceding static step).
The output of a sensitivity calculation is stored as follows (frd-output only if the SEN output request was specified underneath a *NODE FILE card):
For TYPE=COORDINATE design variables the results of the target functions MASS, SHAPE ENERGY, EIGENFREQUENCY and DISPLACEMENT (i.e. the sum of the squares of the displacements in all objective nodes) are stored in the .frd-file and can be visualized using CalculiX GraphiX.
For TYPE=ORIENTATION design variables the eigenfrequency sensitivity is stored in the .dat file whereas the displacement sensitivity (i.e. the derivative of the displacements in all nodes w.r.t. the orientation) is stored in the .frd-file. The order of the design variables is listed in the .dat-file. All orientations defined by *ORIENTATION cards are varied, each orientation is defined by 3 independent variables. So for n *ORIENTATION cards there are 3n design variables. The sensitivity of the mass w.r.t. the orientation is zero.
Finally, it is important to know that a sensitivity analysis in CalculiX only works for true 3D-elements (no shells, beams, plane stress, etc...).
Next: Green functions Up: Types of analysis Previous: Electromagnetism Contents
guido dhondt 2018-12-15
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kb1408 Jan20-13 02:05 AM
Electric field at a distance from a charged disk
A disk of radius 2.4 cm carries a uniform surface charge density of 3.1 μ C/m2. Using reasonable approximations, find the electric field on the axis at the following distances.
I have used the equation E=(Q/ε0)(1/(4*pi*r2))
I also tried the equation E=(Q/2(ε0))(1-(z/(√(z2)+(r2)))
Thanks in advance for the help. Both equations have not led me to the correct answer.
*note, there is not a figure provided for this question*
Doc Al Jan20-13 07:43 AM
Re: Electric field at a distance from a charged disk
Quote:
Quote by kb1408 (Post 4235461) I have used the equation E=(Q/ε0)(1/(4*pi*r2))
This looks like the field from an infinite sheet of charge. You should write it as σ/2ε, where σ is the surface charge density. Not what you want.
Quote:
I also tried the equation E=(Q/2(ε0))(1-(z/(√(z2)+(r2)))
That's the one you want, but you need to replace Q with σ.
andrien Jan20-13 07:58 AM
Re: Electric field at a distance from a charged disk
http://www.physics.udel.edu/~watson/...n/efield1.html
kb1408 Jan20-13 11:37 PM
Re: Electric field at a distance from a charged disk
Doc Al, thanks for your quick reply. Unfortunately I am still doing something wrong. I am using 3.1E-6 C/m2 for σ. Is that wrong?
haruspex Jan21-13 12:05 AM
Re: Electric field at a distance from a charged disk
kb1408 Jan21-13 12:14 AM
Re: Electric field at a distance from a charged disk
E= (3.1E-6/(2*8.85E-12))(1-((.0001/(√(.00012)+(.0242))
so E= 1.79E5 N/C
where:
σ=3.1E-6 C/m2
ε0=8.85E-12 C2/N*m2
z=.01E-2 m
r= 2.4E-2 m
haruspex Jan21-13 12:34 AM
Re: Electric field at a distance from a charged disk
Did you simply replace Q by σ? What about the disc area?
kb1408 Jan21-13 12:45 AM
Re: Electric field at a distance from a charged disk
Yes, that's what I did. Is it σ=Q/A then?
haruspex Jan21-13 12:58 AM
Re: Electric field at a distance from a charged disk
Yes, as in the link andrien posted.
kb1408 Jan21-13 01:09 AM
Re: Electric field at a distance from a charged disk
cheers!
kb1408 Jan21-13 01:13 AM
Re: Electric field at a distance from a charged disk
And thank you andrien for the link!
jtbell Jan21-13 01:25 AM
Re: Electric field at a distance from a charged disk
Note: This thread had already developed quite a bit before I noticed that it really should have been in one of the homework help forums. Therefore I've simply moved it instead of deleting it and asking the original poster to start over, which is the normal practice.
In the future, please post requests for help on specific exercises like this in one of the homework help forums, even if they're not actually assignments for a class. The "normal" forums are more for conceptual questions and general discussion of their topics.
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https://peeterjoot.wordpress.com/tag/pressure/
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• 328,167
# Posts Tagged ‘pressure’
## A final pre-exam update of my notes compilation for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti
Posted by peeterjoot on April 22, 2013
Here’s my third update of my notes compilation for this course, including all of the following:
April 21, 2013 Fermi function expansion for thermodynamic quantities
April 20, 2013 Relativistic Fermi Gas
April 10, 2013 Non integral binomial coefficient
April 10, 2013 energy distribution around mean energy
April 09, 2013 Velocity volume element to momentum volume element
April 04, 2013 Phonon modes
April 03, 2013 BEC and phonons
April 03, 2013 Max entropy, fugacity, and Fermi gas
April 02, 2013 Bosons
April 02, 2013 Relativisitic density of states
March 28, 2013 Bosons
plus everything detailed in the description of my previous update and before.
## Fermi-Dirac function expansion for thermodynamic quantities
Posted by peeterjoot on April 21, 2013
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
In section 8.1 of [1] are some Fermi-Dirac \index{Fermi-Dirac function} expansions for $P$, $N$, and $U$. Let’s work through these in detail.
Our starting point is the relations
\begin{aligned}P V \beta = \ln Z_{\mathrm{G}} = \sum \ln \left( 1 + z e^{-\beta \epsilon} \right)\end{aligned} \hspace{\stretch{1}}(1.0.1a)
\begin{aligned}N = \sum \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }}.\end{aligned} \hspace{\stretch{1}}(1.0.1b)
Recap. Density of states
We’ll employ the 3D non-relativisitic density of states
\begin{aligned}\mathcal{D}(\epsilon) &= \sum_\mathbf{k} \delta(\epsilon - \epsilon_\mathbf{k}) \\ &\sim V \int \frac{d^3 \mathbf{k}}{(2 \pi)^3}\delta(\epsilon - \epsilon_\mathbf{k}) \\ &= \frac{4 \pi V}{(2 \pi)^3}\int dk k^2 \delta\left( \epsilon - \frac{\hbar^2 k^2}{2 m} \right) \\ &= \frac{4 \pi V}{(2 \pi)^3}\int dk k^2 \frac{ \delta\left( k - \sqrt{2 m \epsilon}/\hbar \right)}{ \frac{\hbar^2}{m} \frac{\sqrt{2 m \epsilon}}{\hbar}} \\ &= \frac{2 V}{(2 \pi)^2 }\frac{m}{\hbar^2}\sqrt{\frac{2 m \epsilon}{\hbar^2}},\end{aligned} \hspace{\stretch{1}}(1.0.1b)
or
\begin{aligned}\boxed{\mathcal{D}(\epsilon)=\frac{V}{(2 \pi)^2 }\left( \frac{2 m}{\hbar^2} \right)^{3/2}\epsilon^{1/2}.}\end{aligned} \hspace{\stretch{1}}(1.0.1b)
Density
Now let’s make our integral approximation of the sum for $N$. That is
\begin{aligned}N &= g \int d\epsilon \mathcal{D}(\epsilon) \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }} \\ &= g \frac{V}{(2 \pi)^2 }\left( \frac{2 m}{\hbar^2} \right)^{3/2}\int_0^\infty d\epsilon \frac{\epsilon^{1/2}}{ z^{-1} e^{\beta \epsilon} + 1 } \\ &= g \frac{V}{(2 \pi)^2 \beta^{3/2}}\left( \frac{2 m}{\hbar^2} \right)^{3/2}\int_0^\infty du \frac{u^{1/2}}{ z^{-1} e^{u} + 1 } \\ &= g \frac{V}{(2 \pi)^2 \beta^{3/2}}\left( \frac{2 m}{\hbar^2} \right)^{3/2}\Gamma(3/2) f_{3/2}(z) \\ &= g \frac{V}{(2 \pi)^2 \beta^{3/2}}\frac{\left( 2 m k_{\mathrm{B}} T \right)^{3/2}}{\hbar^3}\frac{1}{{2}} \sqrt{\pi}f_{3/2}(z)\\ &= g V \not{{2}} \pi\frac{\left( 2 m k_{\mathrm{B}} T \right)^{3/2}}{h^3}\frac{1}{{\not{{2}}}} \sqrt{\pi}f_{3/2}(z),\end{aligned} \hspace{\stretch{1}}(1.0.1b)
or
\begin{aligned}\frac{N}{V} = g \frac{\left( 2 \pi m k_{\mathrm{B}} T \right)^{3/2}}{h^3}f_{3/2}(z).\end{aligned} \hspace{\stretch{1}}(1.0.5)
With
\begin{aligned}\lambda = \frac{h}{\sqrt{ 2 \pi m k_{\mathrm{B}} T }},\end{aligned} \hspace{\stretch{1}}(1.0.6)
this gives us the desired density result from the text
\begin{aligned}\boxed{\frac{N}{V}=\frac{g}{\lambda^3} f_{3/2}(z).}\end{aligned} \hspace{\stretch{1}}(1.0.7)
Pressure
For the pressure, we can do the same, but have to integrate by parts
\begin{aligned}P V \beta &= g \sum \ln \left( 1 + z e^{-\beta \epsilon} \right) \\ &\sim g \frac{V}{(2 \pi)^2 }\left( \frac{2 m}{\hbar^2} \right)^{3/2}\int_0^\infty d\epsilon \epsilon^{1/2} \ln \left( 1 + z e^{-\beta \epsilon} \right) \\ &= - g \frac{V}{(2 \pi)^2 }\left( \frac{2 m}{\hbar^2} \right)^{3/2}\int_0^\infty d\epsilon \frac{2}{3} \epsilon^{3/2} \frac{-\beta z e^{-\beta \epsilon} }{ 1 + z e^{-\beta \epsilon} } \\ &= g\frac{V}{(2 \pi)^2 }\left( \frac{2 m}{\hbar^2} \right)^{3/2}\frac{2}{3} \frac{1}{{\beta^{3/2}}}\int_0^\infty dx\frac{x^{3/2}}{z^{-1} e^{x} + 1 } \\ &= g\frac{2}{3} 2 \pi V\frac{\left( 2 m k_{\mathrm{B}} T \right)^{3/2}}{h^3 }\Gamma(5/2)f_{5/2}(z) \\ &= g\frac{2}{3} 2 \pi V\frac{\left( 2 m k_{\mathrm{B}} T \right)^{3/2}}{h^3 }\frac{3}{2} \frac{1}{2} \sqrt{\pi}f_{5/2}(z) \\ &= g V\frac{\left( 2 \pi m k_{\mathrm{B}} T \right)^{3/2}}{h^3 }f_{5/2}(z),\end{aligned} \hspace{\stretch{1}}(1.0.7)
or
\begin{aligned}\boxed{P \beta = \frac{g}{\lambda^3} f_{5/2}(z).}\end{aligned} \hspace{\stretch{1}}(1.0.9)
Energy
The average energy is the last thermodynamic quantity to come very easily. We have
\begin{aligned}U &= - \frac{\partial {}}{\partial {\beta}} \ln Z_{\mathrm{G}} \\ &= - \frac{\partial {T}}{\partial {\beta}} \frac{\partial {}}{\partial {T}} \ln Z_{\mathrm{G}} \\ &= - \frac{\partial {(1/k_{\mathrm{B}} T)}}{\partial {\beta}} \frac{\partial {}}{\partial {T}} P V \beta \\ &= \frac{1}{{k_{\mathrm{B}} \beta^2}}\frac{\partial {}}{\partial {T}} \frac{g V}{\lambda^3} f_{5/2}(z) \\ &= g V k_{\mathrm{B}} T^2f_{5/2}(z)\frac{\partial {}}{\partial {T}} \frac{\left( 2 \pi m k_{\mathrm{B}} T \right)^{3/2}}{h^3} \\ &= \frac{3}{2} \frac{g V k_{\mathrm{B}} T}{\lambda^3}f_{5/2}(z).\end{aligned} \hspace{\stretch{1}}(1.0.9)
From eq. 1.0.7, we have
\begin{aligned}\frac{g V}{\lambda^3} = \frac{N}{f_{3/2}(z) },\end{aligned} \hspace{\stretch{1}}(1.0.11)
so the energy takes the form
\begin{aligned}\boxed{U = \frac{3}{2} N k_{\mathrm{B}} T \frac{f_{5/2}(z)}{f_{3/2}(z) }.}\end{aligned} \hspace{\stretch{1}}(1.0.11)
We can compare this to the ratio of pressure to density
\begin{aligned}\frac{P \beta}{n} = \frac{f_{5/2}(z)}{f_{3/2}(z) },\end{aligned} \hspace{\stretch{1}}(1.0.11)
to find
\begin{aligned}U= \frac{3}{2} N k_{\mathrm{B}} T \frac{P V \beta}{N}= \frac{3}{2} P V,\end{aligned} \hspace{\stretch{1}}(1.0.11)
or
\begin{aligned}\boxed{P V = \frac{2}{3} U.}\end{aligned} \hspace{\stretch{1}}(1.0.11)
# References
[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.
## Relativisitic Fermi gas
Posted by peeterjoot on April 20, 2013
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
## Question: Relativisitic Fermi gas ([1], pr 9.3)
Consider a relativisitic gas of $N$ particles of spin $1/2$ obeying Fermi statistics, enclosed in volume $V$, at absolute zero. The energy-momentum relation is
\begin{aligned}\epsilon = \sqrt{(p c)^2 + \epsilon_0^2},\end{aligned} \hspace{\stretch{1}}(1.1)
where $\epsilon_0 = m c^2$, and $m$ is the rest mass.
Find the Fermi energy at density $n$.
With the pressure $P$ defined as the average force per unit area exerted on a perfectly-reflecting wall of the container.
Set up expressions for this in the form of an integral.
Define the internal energy $U$ as the average $\epsilon - \epsilon_0$.
Set up expressions for this in the form of an integral.
Show that $P V = 2 U/3$ at low densities, and $P V = U/3$ at high densities. State the criteria for low and high densities.
There may exist a gas of neutrinos (and/or antineutrinos) in the cosmos. (Neutrinos are massless Fermions of spin $1/2$.) Calculate the Fermi energy (in eV) of such a gas, assuming a density of one particle per $\text{cm}^3$.
Attempt exact evaluation of the various integrals.
We’ve found [3] that the density of states associated with a 3D relativisitic system is
\begin{aligned}\mathcal{D}(\epsilon) = \frac{4 \pi V}{(c h)^3} \epsilon \sqrt{\epsilon^2 -\epsilon_0^2},\end{aligned} \hspace{\stretch{1}}(1.0.2)
For a given density $n$, we can find the Fermi energy in the same way as we did for the non-relativisitic energies, with the exception that we have to integrate from a lowest energy of $\epsilon_0$ instead of $0$ (the energy at $\mathbf{p} = 0$). That is
\begin{aligned}n &= \frac{N}{V} \\ &= \left( 2 \frac{1}{{2}} + 1 \right)\frac{4 \pi}{(c h)^3} \int_{\epsilon_0}^{\epsilon_{\mathrm{F}}}d\epsilon \epsilon \sqrt{ \epsilon^2 -\epsilon_0^2} \\ &= \frac{8 \pi}{(c h)^3}\frac{1}{{3}} {\left.{{\left( x^2 - \epsilon_0^2 \right)^{3/2}}}\right\vert}_{{\epsilon_0}}^{{\epsilon_{\mathrm{F}}}} \\ &= \frac{8 \pi}{3 (c h)^3}\left( \epsilon_{\mathrm{F}}^2 - \epsilon_0^2 \right)^{3/2}.\end{aligned} \hspace{\stretch{1}}(1.0.2)
Solving for $\epsilon_{\mathrm{F}}/\epsilon_0$ we have
\begin{aligned}\frac{\epsilon_{\mathrm{F}}}{\epsilon_0} =\sqrt{\left( \frac{3 (c h)^3 n}{8 \pi \epsilon_0^3} \right)^{2/3}+ 1}.\end{aligned} \hspace{\stretch{1}}(1.0.2)
We’ll see the constant factor above a number of times below and designate it
\begin{aligned}n_0 = \frac{8 \pi}{3} \left( \frac{\epsilon_0}{c h} \right)^3,\end{aligned} \hspace{\stretch{1}}(1.0.2)
so that the Fermi energy is
\begin{aligned}\frac{\epsilon_{\mathrm{F}}}{\epsilon_0} =\sqrt{\left( \frac{n}{n_0} \right)^{2/3}+ 1}.\end{aligned} \hspace{\stretch{1}}(1.0.2)
For the pressure calculation, let’s suppose that we have a configuration with a plane in the $x,y$ orientation as in fig. 1.1.
Fig 1.1: Pressure against x,y oriented plane
It’s argued in [4] section 6.4 that the pressure for such a configuration is
\begin{aligned}P = n \int p_z u_z f(\mathbf{u}) d^3 \mathbf{u},\end{aligned} \hspace{\stretch{1}}(1.7)
where $n$ is the number density and $f(\mathbf{u})$ is a normalized distribution function for the velocities. The velocity and momentum components are related by the Hamiltonian equations. From the Hamiltonian eq. 1.1 we find \footnote{ Observe that by squaring and summing one can show that this is equivalent to the standard relativisitic momentum $p_x = \frac{m v_x}{\sqrt{ 1 - \mathbf{u}^2/c^2}}$.} (for the x-component which is representative)
\begin{aligned}u_x \\ &= \frac{\partial {\epsilon}}{\partial {p_x}} \\ &= \frac{\partial {}}{\partial {p_x}}\sqrt{(p c)^2 +\epsilon_0^2} \\ &= \frac{ p_x c^2 }{\sqrt{(p c)^2 +\epsilon_0^2}}.\end{aligned} \hspace{\stretch{1}}(1.8)
For $\alpha \in \{1, 2, 3\}$ we can summarize these velocity-momentum relationships as
\begin{aligned}\frac{u_\alpha}{c} = \frac{ c p_\alpha }{ \epsilon }.\end{aligned} \hspace{\stretch{1}}(1.9)
Should we attempt to calculate the pressure with this parameterization of the velocity space we end up with convergence problems, and can’t express the results in terms of $f^+_\nu(z)$. Let’s try instead with a distribution over momentum space
\begin{aligned}P=n \int \frac{(c p_z)^2}{\epsilon} f(c \mathbf{p}) d^3 (c \mathbf{p}).\end{aligned} \hspace{\stretch{1}}(1.10)
Here the momenta have been scaled to have units of energy since we want to express this integral in terms of energy in the end. Our normalized distribution function is
\begin{aligned}f(c \mathbf{p})\propto \frac{\frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }}}{\int \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }} d^3 (c \mathbf{p})},\end{aligned} \hspace{\stretch{1}}(1.11)
but before evaluating anything, we first want to change our integration variable from momentum to energy. In spherical coordinates our volume element takes the form
\begin{aligned}d^3 (c \mathbf{p}) &= 2 \pi (c p)^2 d (c p) \sin\theta d\theta \\ &= 2 \pi (c p)^2 \frac{d (c p)}{d \epsilon} d \epsilon \sin\theta d\theta.\end{aligned} \hspace{\stretch{1}}(1.12)
Implicit derivatives of
\begin{aligned}c^2 p^2 = \epsilon^2 - \epsilon_0^2,\end{aligned} \hspace{\stretch{1}}(1.13)
gives us
\begin{aligned}\frac{d (c p)}{d\epsilon}= \frac{\epsilon}{c p}=\frac{\epsilon}{\sqrt{\epsilon^2 -\epsilon_0^2}}.\end{aligned} \hspace{\stretch{1}}(1.0.14)
Our momentum volume element becomes
\begin{aligned}d^3 (c \mathbf{p}) \\ &= 2 \pi (c p)^2 \frac{\epsilon}{\sqrt{\epsilon^2 - \epsilon_0^2 }}d \epsilon \sin\theta d\theta \\ &= 2 \pi \left( \epsilon^2 - \epsilon_0^2 \right)\frac{\epsilon}{\sqrt{\epsilon^2 - \epsilon_0^2 }}d \epsilon \sin\theta d\theta \\ &= 2 \pi \epsilon \sqrt{ \epsilon^2 - \epsilon_0^2} d \epsilon \sin\theta d\theta.\end{aligned} \hspace{\stretch{1}}(1.0.14)
For our distribution function, we can now write
\begin{aligned}f(c \mathbf{p}) d^3 (c \mathbf{p})= C \frac{\epsilon \sqrt{ \epsilon^2 - \epsilon_0^2} d \epsilon }{ z^{-1} e^{\beta \epsilon} + 1 }\frac{ 2 \pi \sin\theta d\theta }{ 4 \pi \epsilon_0^3 },\end{aligned} \hspace{\stretch{1}}(1.0.14)
where $C$ is determined by the requirement $\int f(c \mathbf{p}) d^3 (c \mathbf{p}) = 1$
\begin{aligned}C^{-1} = \int_{0}^\infty \frac{(y + 1)\sqrt{ (y + 1)^2 - 1} dy }{ z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }.\end{aligned} \hspace{\stretch{1}}(1.0.14)
The z component of our momentum can be written in spherical coordinates as
\begin{aligned}(c p_z)^2= (c p)^2 \cos^2\theta= \left( \epsilon^2 - \epsilon_0^2 \right)\cos^2\theta,\end{aligned} \hspace{\stretch{1}}(1.0.18)
Noting that
\begin{aligned}\int_0^\pi \cos^2\theta \sin\theta d\theta =-\int_0^\pi \cos^2\theta d(\cos\theta)= \frac{2}{3},\end{aligned} \hspace{\stretch{1}}(1.0.19)
all the bits come together as
\begin{aligned}P &= \frac{C n}{3 \epsilon_0^3 } \int_{\epsilon_0}^\infty\left( \epsilon^2 - \epsilon_0^2 \right)^{3/2} \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1 }} d \epsilon \\ &= \frac{n \epsilon_0}{3} \int_{1}^\infty\left( x^2 - 1 \right)^{3/2} \frac{1}{{ z^{-1} e^{\beta \epsilon_0 x} + 1 }} dx.\end{aligned} \hspace{\stretch{1}}(1.0.19)
Letting $y = x - 1$, this is
\begin{aligned}P= \frac{C n \epsilon_0}{3} \int_{0}^\infty \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } dy.\end{aligned} \hspace{\stretch{1}}(1.0.19)
We could conceivable expand the numerators of each of these integrals in power series, which could then be evaluated as a sum of $f^+_\nu(z e^{-\beta \epsilon_0})$ terms.
Note that above the Fermi energy $n$ also has an integral representation
\begin{aligned}n &= \left(2\left( \frac{1}{{2}} \right) + 1\right)\int_{\epsilon_0}^\infty d\epsilon \mathcal{D}(\epsilon) \frac{1}{{ z^{-1} e^{\beta \epsilon} + 1}} \\ &= \frac{8 \pi}{(c h)^3} \int_{\epsilon_0}^\infty d\epsilon\frac{\epsilon \sqrt{\epsilon^2 - \epsilon_0^2} }{ z^{-1} e^{\beta \epsilon} + 1} \\ &= \frac{8 \pi \epsilon_0^3}{(c h)^3} \int_{0}^\infty dy\frac{(y + 1)\sqrt{(y + 1)^2 - 1} }{ z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1},\end{aligned} \hspace{\stretch{1}}(1.0.19)
or
\begin{aligned}\boxed{n = \frac{3 n_0}{C}.}\end{aligned} \hspace{\stretch{1}}(1.0.23)
Observe that we can use this result to remove the dependence of pressure on this constant $C$
\begin{aligned}\boxed{\frac{P}{n_0 \epsilon_0}= \int_{0}^\infty dy \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }.}\end{aligned} \hspace{\stretch{1}}(1.0.24)
Now for the average energy difference from the rest energy $\epsilon_0$
\begin{aligned}U &= \left\langle{{\epsilon - \epsilon_0}}\right\rangle \\ &= \int_{\epsilon_0}^\infty d\epsilon \mathcal{D}(\epsilon) f(\epsilon) (\epsilon - \epsilon_0) \\ &= \frac{8 \pi V}{(c h)^3}\int_{\epsilon_0}^\infty d\epsilon \frac{ \epsilon(\epsilon - \epsilon_0) \sqrt{ \epsilon^2 - \epsilon_0 } }{ z^{-1} e^{\beta \epsilon} + 1} \\ &= \frac{8 \pi V \epsilon_0^4}{(c h)^3}\int_{0}^\infty dy\frac{ y ( y - 1 ) \sqrt{ (y + 1)^2 - 1 }}{ z^{-1} e^{\beta \epsilon} + 1}.\end{aligned} \hspace{\stretch{1}}(1.0.24)
So the average energy density difference from the rest energy, relative to the rest energy, is
\begin{aligned}\boxed{\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0} =3 n_0 \int_{0}^\infty dy \frac { y (y + 1)\sqrt{(y + 1)^2 - 1} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }.}\end{aligned} \hspace{\stretch{1}}(1.0.26)
From eq. 1.0.24 and eq. 1.0.26 we have
\begin{aligned}\frac{1}{{n_0}} &= 3 \frac{V \epsilon_0} {\left\langle{{\epsilon - \epsilon_0}}\right\rangle} \int_{0}^\infty \frac { y (y + 1)\sqrt{(y + 1)^2 - 1} dy } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } \\ &= \frac{\epsilon_0}{P} \int_{0}^\infty \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } dy,\end{aligned} \hspace{\stretch{1}}(1.0.26)
or
\begin{aligned}P V =\frac{U}{3}\frac{ \int_{0}^\infty \frac{ \left( (y + 1)^2 - 1 \right)^{3/2} } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 } dy}{ \int_{0}^\infty \frac { y (y + 1)\sqrt{(y + 1)^2 - 1} dy } { z^{-1} e^{\beta \epsilon_0 (y + 1)} + 1 }}.\end{aligned} \hspace{\stretch{1}}(1.0.26)
This ratio of integrals is supposed to resolve to 1 and 2 in the low and high density limits. To consider this let’s perform one final non-dimensionalization, writing
\begin{aligned}\begin{aligned} \\ x &= \beta \epsilon_0 y \\ \theta &= \frac{1}{{\beta \epsilon_0}} = \frac{k_{\mathrm{B}} T}{\epsilon_0} \\ \bar{\mu} &= \mu - \epsilon_0 \\ \bar{z} &= e^{\beta \bar{\mu}}.\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.29)
The density, pressure, and energy take the form
\begin{aligned}\frac{n}{n_0}= 3 \theta\int_{0}^\infty dx\frac{(\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} }{ \bar{z}^{-1} e^{x} + 1}\end{aligned} \hspace{\stretch{1}}(1.0.30a)
\begin{aligned}\frac{P}{n_0 \epsilon_0}= \theta \int_{0}^\infty dx \frac{ \left( (\theta x + 1)^2 - 1 \right)^{3/2} } { \bar{z}^{-1} e^{x} + 1 }\end{aligned} \hspace{\stretch{1}}(1.0.30b)
\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} =3 \theta^2 \int_{0}^\infty dx \frac { x (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} } { \bar{z}^{-1} e^{x} + 1 }.\end{aligned} \hspace{\stretch{1}}(1.0.30c)
We can rewrite the square roots in the number density and energy density expressions by expanding out the completion of the square
\begin{aligned}(1 + \theta x) \sqrt{ (1 + \theta x)^2 - 1}=(1 + \theta x) \sqrt{ 1 + \theta x + 1 }\sqrt{ 1 + \theta x - 1 }= \sqrt{2 \theta} x^{1/2} (1 + \theta x) \sqrt{ 1 + \frac{\theta x}{2}},\end{aligned} \hspace{\stretch{1}}(1.0.30c)
Expanding the distribution about $\bar{z} e^{-x} = 0$, we have
\begin{aligned}\frac{1}{ \bar{z}^{-1} e^{x} + 1}=\frac{\bar{z} e^{-x}}{ 1 + \bar{z} e^{-x}}=z e^{-x} \sum_{s = 0}^\infty (-1)^s \left( \bar{z} e^{-x} \right)^s,\end{aligned} \hspace{\stretch{1}}(1.0.32)
allowing us to write, in the low density limit with respect to $\bar{z}$
\begin{aligned}\frac{n}{n_0}= 3 \sqrt{2}\theta^{3/2} \sum_{s=0}^\infty(-1)^s\bar{z}^{s + 1}\int_{0}^\infty dx x^{1/2}(1 + \theta x) \sqrt{ 1 + \frac{\theta x}{2}} e^{-x(1 + s)} \end{aligned} \hspace{\stretch{1}}(1.0.33a)
\begin{aligned}\frac{P}{n_0 \epsilon_0}= \theta\sum_{s=0}^\infty(-1)^s\bar{z}^{s + 1} \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} e^{-x(1 + s)} \end{aligned} \hspace{\stretch{1}}(1.0.33b)
\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} =3 \sqrt{2} \theta^{5/2} \sum_{s=0}^\infty(-1)^s\bar{z}^{s + 1} \int_{0}^\infty dx x^{3/2} (1 + \theta x) \sqrt{ 1 + \frac{\theta x}{2}} e^{-x(1 + s)} .\end{aligned} \hspace{\stretch{1}}(1.0.33c)
Low density result
An exact integration of the various integrals above is possible in terms of special functions. However, that attempt (included below) introduced an erroneous extra factor of $\theta$. Given that this end result was obtained by tossing all but the lowest order terms in $\theta$ and $\bar{z}$, let’s try that right from the get go.
For the pressure we have an integrand containing a factor
\begin{aligned}\left( (\theta x + 1)^2 -1 \right)^{3/2}&= \left( \theta x + 1 - 1 \right)^{3/2}\left( \theta x + 1 + 1 \right)^{3/2} \\ &= \theta^{3/2} x^{3/2} 2^{3/2} \left( 1 + \frac{\theta x}{2} \right)^{3/2} \\ &= 2 \sqrt{2} \theta^{3/2} x^{3/2} \left( 1 + \frac{\theta x}{2} \right)^{3/2}\approx2 \sqrt{2} \theta^{3/2} x^{3/2} \end{aligned} \hspace{\stretch{1}}(1.0.33c)
Our pressure, to lowest order in $\theta$ and $\bar{z}$ is then
\begin{aligned}\frac{P}{\epsilon_0 n_0} = 2 \sqrt{2} \theta^{5/2} \bar{z} \int_0^\infty x^{3/2} e^{-x} dx= 2 \sqrt{2} \theta^{5/2} \bar{z} \Gamma(5/2).\end{aligned} \hspace{\stretch{1}}(1.0.33c)
Our energy density to lowest order in $\theta$ and $\bar{z}$ from eq. 1.0.33c is
\begin{aligned}\frac{U}{V \epsilon_0 n_0} &= 3 \sqrt{2} \theta^{5/2} \bar{z} \int_{0}^\infty dx x^{3/2} e^{-x} \\ &= 3 \sqrt{2} \theta^{5/2} \bar{z} \Gamma(5/2).\end{aligned} \hspace{\stretch{1}}(1.0.33c)
Comparing these, we have
\begin{aligned}\frac{1}{{\epsilon_0 n_0\sqrt{2} \theta^{5/2} \bar{z} \Gamma(5/2)}} &= 3 \frac{V}{U} \\ &= \frac{2}{P},\end{aligned} \hspace{\stretch{1}}(1.0.37)
or in this low density limit
\begin{aligned}\boxed{P V = \frac{2}{3} U.}\end{aligned} \hspace{\stretch{1}}(1.0.38)
High density limit
For the high density limit write $\bar{z} = e^y$, so that the distribution takes the form
\begin{aligned}f(\bar{z}) &= \frac{1}{ \bar{z}^{-1} e^{x} + 1} \\ &= \frac{1}{ e^{x - y} + 1}.\end{aligned} \hspace{\stretch{1}}(1.0.39)
This can be approximated by a step function, so that
\begin{aligned}\frac{P}{n_0 \epsilon_0}\approx \int_{0}^y \theta dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} \end{aligned} \hspace{\stretch{1}}(1.0.40a)
\begin{aligned}\frac{U}{V \epsilon_0 n_0} \approx3 \int_{0}^\infty \theta dx \theta x (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} \end{aligned} \hspace{\stretch{1}}(1.0.40b)
With a change of variables $u = \theta x + 1$, we have
\begin{aligned}\begin{aligned}\frac{P}{n_0 \epsilon_0} &\approx \int_{1}^{\theta y + 1x} du\left( u^2 - 1 \right)^{3/2} \\ &=\frac{1}{8} \left((2 \theta y (\theta y+2)-3) \sqrt{\theta y (\theta y+2)} (\theta y+1)+3 \ln \left(\theta y+\sqrt{\theta y (\theta y+2)}+1\right)\right) \\ &\approx\frac{1}{4} \left( \theta \ln \bar{z} \right)^4\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.41a)
\begin{aligned}\begin{aligned}\frac{U}{V \epsilon_0 n_0} &\approx3 \int_{1}^{\theta y + 1x} (u^2 - u)\sqrt{u^2 - 1} \\ &=\frac{3}{24} \left(\sqrt{\theta y (\theta y+2)} (\theta y (2 \theta y (3 \theta y+5)-1)+3)-3 \left(\ln \left(\theta y+\sqrt{\theta y (\theta y+2)}+1\right)\right)\right) \\ &\approx\frac{3}{4} \left( \theta \ln \bar{z} \right)^4\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.41b)
Comparing both we have
\begin{aligned}\frac{4}{\epsilon_0 n_0 \left( \theta \ln \bar{z} \right) } = \frac{1}{{P}} = \frac{3 V}{U},\end{aligned} \hspace{\stretch{1}}(1.0.42)
or
\begin{aligned}\boxed{P V = \frac{1}{{3}} U.}\end{aligned} \hspace{\stretch{1}}(1.0.43)
\begin{aligned}{\left.{{\epsilon_{\mathrm{F}}}}\right\vert}_{{n = 1/(0.01)^3}} = 6.12402 \times 10^{-35} \text{J} \times 6.24150934 \times 10^{18} \frac{\text{eV}}{\text{J}} = 3.82231 \times 10^{-16} \text{eV}\end{aligned} \hspace{\stretch{1}}(1.0.43)
Wow. That’s pretty low!
Pressure integral
Of these the pressure integral is yields directly to Mathematica
\begin{aligned}\begin{aligned} \int_{0}^\infty & dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} e^{-x(1 + s)} \\ &=\frac{3 \theta e^{(s+1)/\theta}}{(s + 1)^2} K_2\left( \frac{s+1}{\theta } \right) \\ &=\frac{3 \sqrt{\frac{\pi }{2}} \theta ^{3/2}}{(s+1)^{5/2}}+\frac{45 \sqrt{\frac{\pi }{2}} \theta ^{5/2}}{8 (s+1)^{7/2}}+\frac{315 \sqrt{\frac{\pi }{2}} \theta ^{7/2}}{128 (s+1)^{9/2}}-\frac{945 \sqrt{\frac{\pi }{2}} \theta ^{9/2}}{1024 (s+1)^{11/2}}+\frac{31185 \sqrt{\frac{\pi }{2}} \theta ^{11/2}}{32768 (s+1)^{13/2}} + \cdots\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.45)
where $K_2(z)$ is a modified Bessel function [5] of the second kind as plotted in fig. 1.2.
Fig 1.2: Modified Bessel function of the second kind
Plugging this into the series for the pressure, we have
\begin{aligned}\frac{P}{n_0 \epsilon_0}= 3 \left( \frac{k_{\mathrm{B}} T}{\epsilon_0} \right)^2\sum_{s=0}^\infty(-1)^s\frac{\left( \bar{z} e^{\epsilon_0/k_{\mathrm{B}} T} \right)^{s + 1}}{(s + 1)^2}K_2\left( (s+1) \epsilon_0/k_{\mathrm{B}} T \right).\end{aligned} \hspace{\stretch{1}}(1.0.46)
Plotting the summands $3 (-1)^s \frac{\theta^2}{(s + 1)^2} \left( \bar{z} e^{ 1/\theta} \right)^{s + 1} K_2\left((s+1)/\theta\right)$ for $\bar{z} = 1$ in fig. 1.4 shows that this mix of exponential Bessel and quadratic terms decreases with $s$.
Plotting this sum in fig. 1.3 numerically to 10 terms, shows that we have a function that appears roughly polynomial in $\bar{z}$ and $\theta$.
Fig 1.3: Pressure to ten terms in z and theta
Fig 1.4: Pressure summands
For small $\bar{z}$ it can be seen graphically that there is very little contribution from anything but the $s = 0$ term of this sum. An expansion in series for a few terms in $\bar{z}$ and $\theta$ gives us
\begin{aligned}\begin{aligned}\frac{P}{\epsilon_0 n_0}&=\sqrt{\pi} \theta^{5/2} \left(\frac{3 \bar{z}}{\sqrt{2}}-\frac{3 \bar{z}^2}{8}+\frac{\bar{z}^3}{3 \sqrt{6}}-\frac{3 \bar{z}^4}{32 \sqrt{2}}+\frac{3 \bar{z}^5}{25 \sqrt{10}}\right) \\ &+\sqrt{\pi} \theta^{7/2} \left(\frac{45 \bar{z}}{8 \sqrt{2}}\right) -\frac{45 \bar{z}^2}{128}+\frac{5 \bar{z}^3}{24 \sqrt{6}}-\frac{45 \bar{z}^4}{1024 \sqrt{2}}+\frac{9 \bar{z}^5}{200 \sqrt{10}}\\ &+\sqrt{\pi} \theta^{9/2} \left(\frac{315 \bar{z}}{128 \sqrt{2}}-\frac{315 \bar{z}^2}{4096}+\frac{35 \bar{z}^3}{1152 \sqrt{6}}-\frac{315 \bar{z}^4}{65536 \sqrt{2}}+\frac{63 \bar{z}^5}{16000 \sqrt{10}}\right).\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.47)
This allows a $k_{\mathrm{B}} T \ll m c^2$ and $\bar{z} \ll 1$ approximation of the pressure
\begin{aligned}\frac{P}{\epsilon_0 n_0} = \frac{3}{2} \sqrt{2 \pi} \bar{z} \theta^{5/2}.\end{aligned} \hspace{\stretch{1}}(1.0.48)
Number density integral
For the number density, it appears that we can evaluate the integral using integration from parts applied to eq. 1.0.30.30
\begin{aligned}\frac{n}{n_0}= \theta\int_{0}^\infty dx\frac{3 (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} }{ \bar{z}^{-1} e^{x} + 1}=\theta\int_{0}^\infty dx\left( \frac{d}{dx} \left( (\theta x + 1)^2 - 1 \right) ^{3/2} \right)\frac{1}{ \bar{z}^{-1} e^{x} + 1}={\left.{{\theta\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{1}{ \bar{z}^{-1} e^{x} + 1}}}\right\vert}_{{0}}^{{\infty}}-\theta\int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{ -\bar{z}^{-1} e^{x} }{ \left( \bar{z}^{-1} e^{x} + 1 \right)^2}=\theta\int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{ \bar{z} e^{-x} }{ \left( 1 + \bar{z} e^{-x} \right)^2}.\end{aligned} \hspace{\stretch{1}}(1.0.48)
Expanding in series, gives us
\begin{aligned}\frac{n}{n_0}=\theta\sum_{s = 0}^\infty\binom{-2}{s}\bar{z}^{s + 1} \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2} e^{-x(s + 1)}=3 \theta^2\sum_{s = 0}^\infty\binom{-2}{s}\frac{\left( \bar{z} e^{1/\theta} \right)^{s + 1}}{(s + 1)^2}K_2\left( \frac{s+1}{\theta } \right).\end{aligned} \hspace{\stretch{1}}(1.0.48)
Here the binomial coefficient has the meaning given in the definitions of \statmechchapcite{nonIntegralBinomialSeries}, where for negative integral values of $b$ we have
\begin{aligned}\binom{b}{s}\equiv(-1)^s \frac{-b}{-b + s} \binom{-b+s}{-b}.\end{aligned} \hspace{\stretch{1}}(1.0.51)
Expanding in series to a couple of orders in $\theta$ and $\bar{z}$ we have
\begin{aligned}\frac{n}{n_0} = \frac{\sqrt{2 \pi}}{36} \theta^{1/2} \left(\left(2 \sqrt{3} \bar{z} - 9/\sqrt{2} \right) \bar{z} +18 \right) \bar{z}+\frac{5 \sqrt{ 2 \pi}}{576} \theta^{3/2} \left(\left(4 \sqrt{3} \bar{z} - 27/\sqrt{2}\right) \bar{z} +108 \right) \bar{z}+ \cdots\end{aligned} \hspace{\stretch{1}}(1.0.52)
To first order in $\theta$ and $\bar{z}$ this is
\begin{aligned}\frac{n}{n_0} = \frac{1}{{2}} \sqrt{ 2 \pi } \bar{z} \theta^{1/2},\end{aligned} \hspace{\stretch{1}}(1.0.53)
which allows a relation to pressure
\begin{aligned}P V = 3 N (k_{\mathrm{B}} T)^2 /\epsilon_0.\end{aligned} \hspace{\stretch{1}}(1.0.54)
It’s kind of odd seeming that this is quadratic in temperature. Is there an error?
Energy integral
Starting from eq. 1.0.30c and integrating by parts we have
\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} &= 3 \theta^2 \int_{0}^\infty dx \frac { x (\theta x + 1)\sqrt{(\theta x + 1)^2 - 1} } { \bar{z}^{-1} e^{x} + 1 } \\ &= -\theta^2 \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\frac{d}{dx} \left( \frac{x} { \bar{z}^{-1} e^{x} + 1 } \right) \\ &= -\theta^2 \int_{0}^\infty dx\left( (\theta x + 1)^2 - 1 \right)^{3/2}\left( \frac{1} { \bar{z}^{-1} e^{x} + 1 } - \frac{x \bar{z}^{-1} e^{x} } { \left( \bar{z}^{-1} e^{x} + 1 \right)^2 } \right) \\ &= \theta^2 \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \frac{ (x - 1)\bar{z}^{-1} e^{x} - 1} { \left( \bar{z}^{-1} e^{x} + 1 \right)^2 } \\ &= \theta^2 \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \frac{ (x - 1)\bar{z} e^{-x} - \bar{z}^2 e^{-2 x}} { \left( 1 + \bar{z} e^{-x} \right)^2 } \\ &= \theta^2\sum_{s=0}^\infty \binom{-2}{s} \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \left( (x - 1)\bar{z} e^{-x} - \bar{z}^2 e^{-2 x} \right) (\bar{z} e^{-x})^s \\ &= \theta^2\sum_{s=0}^\infty \binom{-2}{s} \bar{z}^{s + 1} \int_{0}^\infty dx \left( (\theta x + 1)^2 - 1 \right)^{3/2} \left( (x - 1) e^{-x(s + 1)} - \bar{z} e^{-x(s + 2)} \right).\end{aligned} \hspace{\stretch{1}}(1.0.54)
The integral with the factor of $x$ doesn’t have a nice closed form as before (if you consider the $K_2$ a nice closed form), but instead evaluates to a confluent hypergeometric function [6]. That integral is
\begin{aligned}\int_0^{\infty } x \left((\theta x+1)^2-1\right)^{3/2} e^{-x (1+s)} dx = \frac{15 \sqrt{\pi } \theta^3 U\left(-\frac{3}{2},-4,\frac{2 (s+1)}{\theta }\right)}{8 (s+1)^5},\end{aligned} \hspace{\stretch{1}}(1.0.54)
and looks like fig. 1.5. Series expansion shows that this hypergeometricU function has a $\theta^{3/2}$ singularity at the origin
Fig 1.5: Plot of HypergeometricU, and with theta^5 scaling
\begin{aligned}U\left(-\frac{3}{2},-4,\frac{2 (s+1)}{\theta }\right)=\frac{2 \sqrt{2} \sqrt{s+1} s+2 \sqrt{2} \sqrt{s+1}}{\theta^{3/2}}+\frac{21 \sqrt{s+1}}{2 \sqrt{2} \sqrt{\theta }}+ \cdots\end{aligned} \hspace{\stretch{1}}(1.57)
so our multiplication by $\theta^5$ brings us to zero as seen in the plot. Evaluating the complete integral yields the unholy mess
\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} &= \sum_{s=0}^\infty \theta^2 (-1)^s (s+1) \bar{z}^{s+1} \Bigl( \\ &\frac{105 \sqrt{\pi } \theta^3 U\left(-\frac{1}{2},-4,\frac{2 (s+1)}{\theta }\right)}{16 (s+1)^5} \\ &- \frac{3 \sqrt{\pi } \theta^2 U\left(-\frac{1}{2},-2,\frac{2 (s+1)}{\theta }\right)}{2 (s+1)^3} \\ &- \frac{3 \sqrt{\pi } \theta^2 \bar{z} U\left(-\frac{1}{2},-2,\frac{2 (s+2)}{\theta }\right)}{2 (s+2)^3} \\ &+\frac{(\theta -2) (-3 \theta +2 s+2) e^{\frac{s+1}{\theta }} K_2\left(\frac{s+1}{\theta }\right)}{\theta (s+1)^2} \\ &-\frac{2 (\theta -2) e^{\frac{s+1}{\theta }} K_1\left(\frac{s+1}{\theta }\right)}{\theta (s+1)} \\ &+\frac{\bar{z} (-3 \theta +2 s+4) e^{\frac{s+2}{\theta }} K_2\left(\frac{s+2}{\theta }\right)}{(s+2)^2} \\ &-\frac{2 \bar{z} e^{\frac{s+2}{\theta }} K_1\left(\frac{s+2}{\theta }\right)}{s+2} \Bigr),\end{aligned} \hspace{\stretch{1}}(1.58)
to first order in $\bar{z}$ and $\theta$ this is
\begin{aligned}\frac{\left\langle{{\epsilon - \epsilon_0}}\right\rangle}{V \epsilon_0 n_0} =\frac{9}{4} \sqrt{2 \pi} \bar{z} \theta^{7/2}.\end{aligned} \hspace{\stretch{1}}(1.59)
Comparing pressure and energy we have for low densities (where $\bar{z} \approx 0$)
\begin{aligned}\frac{1}{{\epsilon_0 n_0 \sqrt{2 \pi} \bar{z} \theta^{5/2}}} = \frac{3}{2} \frac{1}{{P}} = \frac{9}{4} \theta \frac{V}{U},\end{aligned} \hspace{\stretch{1}}(1.0.60)
or
\begin{aligned}\theta P V = \frac{2}{3} U.\end{aligned} \hspace{\stretch{1}}(1.0.61)
It appears that I’ve picked up an extra factor of $\theta$ somewhere, but at least I’ve got the $2/3$ low density expression. Given that I’ve Taylor expanded everything anyways around $\bar{z}$ and $\theta$ this could likely have been done right from the get go, instead of dragging along the messy geometric integrals. Reworking this part of this problem like that was done above.
# References
[1] Kerson Huang. Introduction to statistical physics. CRC Press, 2001.
[2] Peeter Joot. Basic statistical mechanics., chapter {Non integral binomial coefficient}. \natexlab{a}. URL http://sites.google.com/site/peeterjoot2/math2013/phy452.pdf.
[3] Peeter Joot. Basic statistical mechanics., chapter {Relativisitic density of states}. \natexlab{b}. URL http://sites.google.com/site/peeterjoot2/math2013/phy452.pdf.
[4] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.
[5] Wolfram. BesselK, \natexlab{a}. URL http://reference.wolfram.com/mathematica/ref/BesselK.html. [Online; accessed 11-April-2013].
[6] Wolfram. HyperGeometricU, \natexlab{b}. URL http://reference.wolfram.com/mathematica/ref/HypergeometricU.html. [Online; accessed 17-April-2013].
## An updated compilation of notes, for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti
Posted by peeterjoot on March 27, 2013
Here’s my second update of my notes compilation for this course, including all of the following:
March 27, 2013 Fermi gas
March 26, 2013 Fermi gas thermodynamics
March 26, 2013 Fermi gas thermodynamics
March 23, 2013 Relativisitic generalization of statistical mechanics
March 21, 2013 Kittel Zipper problem
March 18, 2013 Pathria chapter 4 diatomic molecule problem
March 17, 2013 Gibbs sum for a two level system
March 16, 2013 open system variance of N
March 16, 2013 probability forms of entropy
March 14, 2013 Grand Canonical/Fermion-Bosons
March 13, 2013 Quantum anharmonic oscillator
March 12, 2013 Grand canonical ensemble
March 11, 2013 Heat capacity of perturbed harmonic oscillator
March 10, 2013 Langevin small approximation
March 10, 2013 Addition of two one half spins
March 10, 2013 Midterm II reflection
March 07, 2013 Thermodynamic identities
March 06, 2013 Temperature
March 05, 2013 Interacting spin
plus everything detailed in the description of my first update and before.
## PHY452H1S Basic Statistical Mechanics. Lecture 16: Fermi gas. Taught by Prof. Arun Paramekanti
Posted by peeterjoot on March 27, 2013
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer
Peeter’s lecture notes from class. May not be entirely coherent.
# Fermi gas
Review
Continuing a discussion of [1] section 8.1 content.
We found
\begin{aligned}n_{\mathbf{k}} = \frac{1}{{e^{\beta(\epsilon_k - \mu)} + 1}}\end{aligned} \hspace{\stretch{1}}(1.2.1)
With no spin
\begin{aligned}\int n_\mathbf{k} \times \frac{d^3 k}{(2\pi)^3} = \rho\end{aligned} \hspace{\stretch{1}}(1.2.2)
Fig 1.1: Occupancy at low temperature limit
Fig 1.2: Volume integral over momentum up to Fermi energy limit
\begin{aligned}\epsilon_{\mathrm{F}} = \frac{\hbar^2 k_{\mathrm{F}}^2}{2m}\end{aligned} \hspace{\stretch{1}}(1.2.3)
gives
\begin{aligned}k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3}\end{aligned} \hspace{\stretch{1}}(1.2.4)
\begin{aligned}\sum_\mathbf{k} n_\mathbf{k} = N\end{aligned} \hspace{\stretch{1}}(1.2.5)
\begin{aligned}\mathbf{k} = \frac{2\pi}{L}(n_x, n_y, n_z)\end{aligned} \hspace{\stretch{1}}(1.2.6)
This is for periodic boundary conditions \footnote{I filled in details in the last lecture using a particle in a box, whereas this periodic condition was intended. We see that both achieve the same result}, where
\begin{aligned}\Psi(x + L) = \Psi(x)\end{aligned} \hspace{\stretch{1}}(1.2.7)
Moving on
\begin{aligned}\sum_{k_x} n(\mathbf{k}) = \sum_{p_x} \Delta p_x n(\mathbf{k})\end{aligned} \hspace{\stretch{1}}(1.2.8)
with
\begin{aligned}\Delta k_x = \frac{2 \pi}{L} \Delta p_x\end{aligned} \hspace{\stretch{1}}(1.2.9)
this gives
\begin{aligned}\sum_{k_x} n(\mathbf{k}) = \sum_{n_x} \frac{L}{2\pi} \Delta k_x \rightarrow \frac{L}{2\pi} \int d k_x\end{aligned} \hspace{\stretch{1}}(1.2.10)
Over all dimensions
\begin{aligned}\sum_{\mathbf{k}} n_\mathbf{k} = \left( \frac{L}{2\pi} \right)^3 \left( \int d^3 \mathbf{k} \right)n(\mathbf{k})=N\end{aligned} \hspace{\stretch{1}}(1.2.11)
so that
\begin{aligned}\rho = \int \frac{d^3 \mathbf{k}}{(2 \pi)^3}\end{aligned} \hspace{\stretch{1}}(1.2.12)
Again
\begin{aligned}k_{\mathrm{F}} = (6 \pi^2 \rho)^{1/3}\end{aligned} \hspace{\stretch{1}}(1.2.13)
## Example: Spin considerations
{example:basicStatMechLecture16:1}{
\begin{aligned}\sum_{\mathbf{k}, m_s} = N\end{aligned} \hspace{\stretch{1}}(1.2.14)
\begin{aligned}\sum_{\mathbf{k}, m_s} \frac{1}{{e^{\beta(\epsilon_k - \mu)} + 1}} = (2 S + 1)\left( \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} n(\mathbf{k}) \right)L^3\end{aligned} \hspace{\stretch{1}}(1.2.15)
This gives us
\begin{aligned}k_{\mathrm{F}} = \left( \frac{ 6 \pi^2 \rho }{2 S + 1} \right)^{1/3}\end{aligned} \hspace{\stretch{1}}(1.2.16)
and again
\begin{aligned}\epsilon_{\mathrm{F}} = \frac{\hbar^2 k_{\mathrm{F}}^2}{2m}\end{aligned} \hspace{\stretch{1}}(1.2.17)
}
High Temperatures
Now we want to look at the at higher temperature range, where the occupancy may look like fig. 1.3
Fig 1.3: Occupancy at higher temperatures
\begin{aligned}\mu(T = 0) = \epsilon_{\mathrm{F}}\end{aligned} \hspace{\stretch{1}}(1.2.18)
\begin{aligned}\mu(T \rightarrow \infty) \rightarrow - \infty\end{aligned} \hspace{\stretch{1}}(1.2.19)
so that for large $T$ we have
\begin{aligned}\frac{1}{{e^{\beta(\epsilon_k - \mu)} + 1}} \rightarrow e^{-\beta(\epsilon_k - \mu)}\end{aligned} \hspace{\stretch{1}}(1.2.20)
\begin{aligned}\rho &= \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} e^{\beta \mu} e^{-\beta \epsilon_k} \\ &= e^{\beta \mu} \int \frac{d^3 \mathbf{k}}{(2 \pi)^3} e^{-\beta \epsilon_k} \\ &= e^{\beta \mu} \int dk \frac{4 \pi k^2}{(2 \pi)^3} e^{-\beta \hbar^2 k^2/2m}.\end{aligned} \hspace{\stretch{1}}(1.2.21)
Mathematica (or integration by parts) tells us that
\begin{aligned}\frac{1}{{(2 \pi)^3}} \int 4 \pi^2 k^2 dk e^{-a k^2} = \frac{1}{{(4 \pi a )^{3/2}}},\end{aligned} \hspace{\stretch{1}}(1.2.22)
so we have
\begin{aligned}\rho &= e^{\beta \mu} \left( \frac{2m}{ 4 \pi \beta \hbar^2} \right)^{3/2} \\ &= e^{\beta \mu} \left( \frac{2 m k_{\mathrm{B}} T 4 \pi^2 }{ 4 \pi h^2} \right)^{3/2} \\ &= e^{\beta \mu} \left( \frac{2 m k_{\mathrm{B}} T \pi }{ h^2} \right)^{3/2}\end{aligned} \hspace{\stretch{1}}(1.2.23)
Introducing $\lambda$ for the thermal de Broglie wavelength, $\lambda^3 \sim T^{-3/2}$
\begin{aligned}\lambda \equiv \frac{h}{\sqrt{2 \pi m k_{\mathrm{B}} T}},\end{aligned} \hspace{\stretch{1}}(1.2.24)
we have
\begin{aligned}\rho = e^{\beta \mu} \frac{1}{{\lambda^3}}.\end{aligned} \hspace{\stretch{1}}(1.2.25)
Does it make any sense to have density as a function of temperature? An inappropriately extended to low temperatures plot of the density is found in fig. 1.4 for a few arbitrarily chosen numerical values of the chemical potential $\mu$, where we see that it drops to zero with temperature. I suppose that makes sense if we are not holding volume constant.
Fig 1.4: Density as a function of temperature
We can write
\begin{aligned}\boxed{e^{\beta \mu} = \left( \rho \lambda^3 \right)}\end{aligned} \hspace{\stretch{1}}(1.2.26)
\begin{aligned}\frac{\mu}{k_{\mathrm{B}} T} = \ln \left( \rho \lambda^3 \right)\sim -\frac{3}{2} \ln T\end{aligned} \hspace{\stretch{1}}(1.2.27)
or (taking $\rho$ (and/or volume?) as a constant) we have for large temperatures
\begin{aligned}\mu \propto -T \ln T\end{aligned} \hspace{\stretch{1}}(1.2.28)
The chemical potential is plotted in fig. 1.5, whereas this $- k_{\mathrm{B}} T \ln k_{\mathrm{B}} T$ function is plotted in fig. 1.6. The contributions to $\mu$ from the $k_{\mathrm{B}} T \ln (\rho h^3 (2 \pi m)^{-3/2})$ term are dropped for the high temperature approximation.
Fig 1.5: Chemical potential over degenerate to classical range
Fig 1.6: High temp approximation of chemical potential, extended back to T = 0
Pressure
\begin{aligned}P = - \frac{\partial {E}}{\partial {V}}\end{aligned} \hspace{\stretch{1}}(1.2.29)
For a classical ideal gas as in fig. 1.7 we have
Fig 1.7: Ideal gas pressure vs volume
\begin{aligned}P = \rho k_{\mathrm{B}} T\end{aligned} \hspace{\stretch{1}}(1.2.30)
For a Fermi gas at $T = 0$ we have
\begin{aligned}E &= \sum_\mathbf{k} \epsilon_k n_k \\ &= \sum_\mathbf{k} \epsilon_k \Theta(\mu_0 - \epsilon_k) \\ &= \frac{V}{(2\pi)^3} \int_{\epsilon_k < \mu_0} \frac{\hbar^2 \mathbf{k}^2}{2 m} d^3 \mathbf{k} \\ &= \frac{V}{(2\pi)^3} \int_0^{k_{\mathrm{F}}} \frac{\hbar^2 \mathbf{k}^2}{2 m} d^3 \mathbf{k} \\ &= \frac{V}{(2\pi)^3} \frac{\hbar^2}{2 m} \int_0^{k_{\mathrm{F}}} k^2 4 \pi k^2 d k\propto k_{\mathrm{F}}^5\end{aligned} \hspace{\stretch{1}}(1.2.31)
Specifically,
\begin{aligned}E(T = 0) = V \times \frac{3}{5} \underbrace{\epsilon_{\mathrm{F}}}_{\sim k_{\mathrm{F}}^2}\underbrace{\rho}_{\sim k_{\mathrm{F}}^3}\end{aligned} \hspace{\stretch{1}}(1.2.32)
or
\begin{aligned}\frac{E}{N} = \frac{3}{5} \epsilon_{\mathrm{F}}\end{aligned} \hspace{\stretch{1}}(1.2.33)
\begin{aligned}E = \frac{3}{5} N \frac{\hbar^2}{2 m} \left( 6 \pi^2 \frac{N}{V} \right)^{2/3} = a V^{-2/3},\end{aligned} \hspace{\stretch{1}}(1.2.34)
so that
\begin{aligned}\frac{\partial {E}}{\partial {V}} = -\frac{2}{3} a V^{-5/3}.\end{aligned} \hspace{\stretch{1}}(1.2.35)
\begin{aligned}P &= -\frac{\partial {E}}{\partial {V}} \\ &= \frac{2}{3} \left( a V^{-2/3} \right)V^{-1} \\ &= \frac{2}{3} \frac{E}{V} \\ &= \frac{2}{3} \left( \frac{3}{5} \epsilon_{\mathrm{F}} \rho \right) \\ &= \frac{2}{5} \epsilon_{\mathrm{F}} \rho.\end{aligned} \hspace{\stretch{1}}(1.2.36)
We see that the pressure ends up deviating from the classical result at low temperatures, as sketched in fig. 1.8. This low temperature limit for the pressure $2 \epsilon_{\mathrm{F}} \rho/5$ is called the degeneracy pressure.
Fig 1.8: Fermi degeneracy pressure
# References
[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.
## Pathria chapter 4 diatomic molecule problem
Posted by peeterjoot on March 18, 2013
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
## Question: Diatomic molecule ([1] pr 4.7)
Consider a classical system of non-interacting, diatomic molecules enclosed in a box of volume $V$ at temperature $T$. The Hamiltonian of a single molecule is given by
\begin{aligned}H(\mathbf{r}_1, \mathbf{r}_2, \mathbf{p}_1, \mathbf{p}_2) = \frac{1}{{2m}} \left( \mathbf{p}_1^2 + \mathbf{p}_2^2 \right)+\frac{1}{{2}} K \left\lvert {\mathbf{r}_1 - \mathbf{r}_2} \right\rvert^2.\end{aligned} \hspace{\stretch{1}}(1.0.1)
Study the thermodynamics of this system, including the dependence of the quantity $\left\langle{{r_{12}^2}}\right\rangle$ on $T$.
Partition function
First consider the partition function for a single diatomic pair
\begin{aligned}Z_1 &= \frac{1}{{h^6}} \int d^6 \mathbf{p} d^6 \mathbf{r} e^{-\beta \frac{ \mathbf{p}_1^2 + \mathbf{p}_2^2 }{2m}} e^{-\beta K\frac{ \left\lvert {\mathbf{r}_1 - \mathbf{r}_2} \right\rvert^2 }{2}} \\ &= \frac{1}{{h^6}} \left( \frac{2 \pi m}{\beta} \right)^{6/2}\int d^3 \mathbf{r}_1 d^3 \mathbf{r}_2 e^{-\beta K\frac{ \left\lvert {\mathbf{r}_1 - \mathbf{r}_2} \right\rvert^2 }{2}}\end{aligned} \hspace{\stretch{1}}(1.0.2)
Now we can make a change of variables to simplify the exponential. Let’s write
\begin{aligned}\mathbf{u} = \mathbf{r}_1 - \mathbf{r}_2\end{aligned} \hspace{\stretch{1}}(1.0.3a)
\begin{aligned}\mathbf{v} = \mathbf{r}_2,\end{aligned} \hspace{\stretch{1}}(1.0.3b)
or
\begin{aligned}\mathbf{r}_2 = \mathbf{v}\end{aligned} \hspace{\stretch{1}}(1.0.4a)
\begin{aligned}\mathbf{r}_1=\mathbf{u} + \mathbf{v}.\end{aligned} \hspace{\stretch{1}}(1.0.4b)
Our volume element is
\begin{aligned}d^3 \mathbf{r}_1 d^3 \mathbf{r}_2 = d^3 \mathbf{u} d^3 \mathbf{v} \frac{\partial(\mathbf{r}_1, \mathbf{r}_2)}{\partial(\mathbf{u}, \mathbf{v})}.\end{aligned} \hspace{\stretch{1}}(1.0.5)
It wasn’t obvious to me that this change of variables preserves the volume element, but a quick Jacobian calculation shows this to be the case
\begin{aligned}\frac{\partial(\mathbf{r}_1, \mathbf{r}_2)}{\partial(\mathbf{u}, \mathbf{v})} &= \begin{vmatrix}\partial r_{11}/\partial u_1 & \partial r_{11}/\partial u_2 &\partial r_{11}/\partial u_3 &\partial r_{11}/\partial v_1 &\partial r_{11}/\partial v_2 &\partial r_{11}/\partial v_3 \\ \partial r_{12}/\partial u_1 & \partial r_{12}/\partial u_2 &\partial r_{12}/\partial u_3 &\partial r_{12}/\partial v_1 &\partial r_{12}/\partial v_2 &\partial r_{12}/\partial v_3 \\ \partial r_{13}/\partial u_1 & \partial r_{13}/\partial u_2 &\partial r_{13}/\partial u_3 &\partial r_{13}/\partial v_1 &\partial r_{13}/\partial v_2 &\partial r_{13}/\partial v_3 \\ \partial r_{21}/\partial u_1 & \partial r_{21}/\partial u_2 &\partial r_{21}/\partial u_3 &\partial r_{21}/\partial v_1 &\partial r_{21}/\partial v_2 &\partial r_{21}/\partial v_3 \\ \partial r_{22}/\partial u_1 & \partial r_{22}/\partial u_2 &\partial r_{22}/\partial u_3 &\partial r_{22}/\partial v_1 &\partial r_{22}/\partial v_2 &\partial r_{22}/\partial v_3 \\ \partial r_{23}/\partial u_1 & \partial r_{23}/\partial u_2 &\partial r_{23}/\partial u_3 &\partial r_{23}/\partial v_1 &\partial r_{23}/\partial v_2 &\partial r_{23}/\partial v_3 \end{vmatrix} \\ &= \begin{vmatrix}\partial r_{11}/\partial u_1 & \partial r_{11}/\partial u_2 &\partial r_{11}/\partial u_3 &\partial r_{11}/\partial v_1 &\partial r_{11}/\partial v_2 &\partial r_{11}/\partial v_3 \\ \partial r_{12}/\partial u_1 & \partial r_{12}/\partial u_2 &\partial r_{12}/\partial u_3 &\partial r_{12}/\partial v_1 &\partial r_{12}/\partial v_2 &\partial r_{12}/\partial v_3 \\ \partial r_{13}/\partial u_1 & \partial r_{13}/\partial u_2 &\partial r_{13}/\partial u_3 &\partial r_{13}/\partial v_1 &\partial r_{13}/\partial v_2 &\partial r_{13}/\partial v_3 \\ 0 & 0 & 0 &\partial r_{21}/\partial v_1 &\partial r_{21}/\partial v_2 &\partial r_{21}/\partial v_3 \\ 0 & 0 & 0 &\partial r_{22}/\partial v_1 &\partial r_{22}/\partial v_2 &\partial r_{22}/\partial v_3 \\ 0 & 0 & 0 &\partial r_{23}/\partial v_1 &\partial r_{23}/\partial v_2 &\partial r_{23}/\partial v_3 \end{vmatrix} \\ &= 1.\end{aligned} \hspace{\stretch{1}}(1.0.6)
Our remaining integral can now be evaluated
\begin{aligned}\int d^3 \mathbf{r}_1 d^3 \mathbf{r}_2 e^{-\beta K\frac{ \left\lvert {\mathbf{r}_1 - \mathbf{r}_2} \right\rvert^2 }{2}} &= \int d^3 \mathbf{u} d^3 \mathbf{v} e^{-\beta K \left\lvert {\mathbf{u}} \right\rvert^2 /2 } \\ &= V \int d^3 \mathbf{u} e^{-\beta K \left\lvert {\mathbf{u}} \right\rvert^2 /2 } \\ &= V \int d^3 \mathbf{u} e^{-\beta K \left\lvert {\mathbf{u}} \right\rvert^2 /2 } \\ &= V \left( \frac{ 2 \pi }{ K \beta } \right)^{3/2}.\end{aligned} \hspace{\stretch{1}}(1.0.7)
Our partition function is now completely evaluated
\begin{aligned}Z_1 = V\frac{1}{{h^6}} \left( \frac{2 \pi m}{\beta} \right)^{3}\left( \frac{ 2 \pi }{ K \beta } \right)^{3/2}.\end{aligned} \hspace{\stretch{1}}(1.0.8)
As a function of $V$ and $T$ as in the text, we write
\begin{aligned}Z_1 = V f(T)\end{aligned} \hspace{\stretch{1}}(1.0.9a)
\begin{aligned}f(T) = \left( \frac{m }{h^2 } \sqrt{\frac{(2\pi)^3}{K}} \right)^3\left( k_{\mathrm{B}} T \right)^{9/2}.\end{aligned} \hspace{\stretch{1}}(1.0.9b)
Gibbs sum
Our Gibbs sum, summing over the number of molecules (not atoms), is
\begin{aligned}Z_{\mathrm{G}} &= \sum_{N_r = 0}^\infty \frac{z^{N_r}}{N_r!} Z_1^{N_r} \\ &= e^{ z V f(T) },\end{aligned} \hspace{\stretch{1}}(1.0.10)
or
\begin{aligned}q &= \ln Z_{\mathrm{G}} \\ &= z V f(T) \\ &= P V \beta.\end{aligned} \hspace{\stretch{1}}(1.0.11)
The fact that we can sum this as an exponential series so nicely looks like it’s one of the main advantages to this grand partition function (Gibbs sum). We can avoid any of the large $N!$ approximations that we have to use when the number of particles is explicitly fixed.
Pressure
The pressure follows
\begin{aligned}P &= z f(T) k_{\mathrm{B}} T \\ &= e^{\mu/k_{\mathrm{B}} T}\left( \frac{m }{h^2 } \sqrt{\frac{(2\pi)^3}{K}} \right)^3\left( k_{\mathrm{B}} T \right)^{11/2}.\end{aligned} \hspace{\stretch{1}}(1.0.12)
Average energy
\begin{aligned}\left\langle{{H}}\right\rangle &= -\frac{\partial {q}}{\partial {\beta}} \\ &= - z V \frac{9}{2} \frac{f(T)}{T} \frac{\partial {T}}{\partial {\beta}} \\ &= z V \frac{9}{2} \frac{f(T)}{T^3} \frac{1}{{k_{\mathrm{B}}}},\end{aligned} \hspace{\stretch{1}}(1.0.13)
or
\begin{aligned}\left\langle{{H}}\right\rangle = e^{\mu/k_{\mathrm{B}} T} V \frac{9}{2} k_{\mathrm{B}}^2 \left( \frac{m }{h^2 } \sqrt{\frac{(2\pi)^3}{K}} \right)^3\left( k_{\mathrm{B}} T \right)^{3/2}.\end{aligned} \hspace{\stretch{1}}(1.0.14)
Average occupancy
\begin{aligned}\left\langle{{N}}\right\rangle &= z \frac{\partial {}}{\partial {z}} \ln Z_{\mathrm{G}} \\ &= z \frac{\partial {}}{\partial {z}} \left( z V f(T) \right) \\ &= z V f(T)\end{aligned} \hspace{\stretch{1}}(1.0.15)
but this is just $q$, or
\begin{aligned}\left\langle{{N}}\right\rangle &= e^{\mu/k_{\mathrm{B}} T} V\left( \frac{m }{h^2 } \sqrt{\frac{(2\pi)^3}{K}} \right)^3\left( k_{\mathrm{B}} T \right)^{9/2}.\end{aligned} \hspace{\stretch{1}}(1.0.16)
Free energy
\begin{aligned}F &= - k_{\mathrm{B}} T \ln \frac{ Z_{\mathrm{G}} }{z^N} \\ &= - k_{\mathrm{B}} T \left( q - N \ln z \right) \\ &= N k_{\mathrm{B}} T \beta \mu - k_{\mathrm{B}} T q \\ &= z V f(T) \mu - k_{\mathrm{B}} T z V f(T) \\ &= z V f(T) \left( \mu - k_{\mathrm{B}} T \right)\end{aligned} \hspace{\stretch{1}}(1.0.17)
\begin{aligned}F = e^{\mu/k_{\mathrm{B}} T} V \left( \mu - k_{\mathrm{B}} T \right)\left( \frac{m }{h^2 } \sqrt{\frac{(2\pi)^3}{K}} \right)^3\left( k_{\mathrm{B}} T \right)^{9/2}.\end{aligned} \hspace{\stretch{1}}(1.0.18)
Entropy
\begin{aligned}S &= \frac{U - F}{T} \\ &= \frac{V}{T} e^{\mu/k_{\mathrm{B}} T} \left( k_{\mathrm{B}} T \right)^{3/2}\left( \frac{m }{h^2 } \sqrt{\frac{(2\pi)^3}{K}} \right)^3\left( \frac{9}{2} k_{\mathrm{B}}^2 - \left( \mu - k_{\mathrm{B}} T \right) \left( k_{\mathrm{B}} T \right)^3 \right).\end{aligned} \hspace{\stretch{1}}(1.0.19)
Expectation of atomic separation
The momentum portions of the average will just cancel out, leaving just
\begin{aligned}\left\langle{r_{12}^2}\right\rangle &= \frac{\int d^3 \mathbf{r}_1 d^3 \mathbf{r}_2 \left( \mathbf{r}_1 - \mathbf{r}_2 \right)^2 e^{-\beta K \left( \mathbf{r}_1 - \mathbf{r}_2 \right)^2 /2 }}{\int d^3 \mathbf{r}_1 d^3 \mathbf{r}_2 e^{-\beta K \left( \mathbf{r}_1 - \mathbf{r}_2 \right)^2 /2 }} \\ &= \frac{ \int d^3 \mathbf{u} \mathbf{u}^2 e^{-\beta K \mathbf{u}^2 /2 }}{\int d^3 \mathbf{u} e^{-\beta K \mathbf{u}^2 /2 }} \\ &= \frac{\int da db dc \left( a^2 + b^2 + c^2 \right) e^{-\beta K \left( a^2 + b^2 + c^2 \right) /2}}{\int e^{-\beta K \left( a^2 + b^2 + c^2 \right)/2}} \\ &= 3 \frac{\int da a^2 e^{-\beta K a^2/2}\int db dc e^{-\beta K \left( b^2 + c^2 \right) /2}}{\int e^{-\beta K \left( a^2 + b^2 + c^2 \right)/2 }} \\ &= 3 \frac{\int da a^2 e^{-\beta K a^2/2}}{\int e^{-\beta K a^2/2}}\end{aligned} \hspace{\stretch{1}}(1.0.20)
Expanding the numerator by parts we have
\begin{aligned}\int da a^2 e^{-\beta K a^2/2} \\ &= \int a d\frac{ e^{-\beta K a^2/2}}{- 2 \beta K/2} \\ &= \frac{1}{\beta K}\int e^{-\beta K a^2/2}.\end{aligned} \hspace{\stretch{1}}(1.0.21)
This gives us
\begin{aligned}\boxed{\left\langle r_{12}^2 \right\rangle = \frac{3}{\beta K} = \frac{3 k_{\mathrm{B}} T}{K}.}\end{aligned} \hspace{\stretch{1}}(1.0.22)
# References
[1] RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.
## Thermodynamic identities
Posted by peeterjoot on March 7, 2013
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
Impressed with the clarity of Baez’s entropic force discussion on differential forms [1], let’s use that methodology to find all the possible identities that we can get from the thermodynamic identity (for now assuming $N$ is fixed, ignoring the chemical potential.)
This isn’t actually that much work to do, since a bit of editor regular expression magic can do most of the work.
Our starting point is the thermodynamic identity
\begin{aligned}dU = d Q + d W = T dS - P dV,\end{aligned} \hspace{\stretch{1}}(1.0.1)
or
\begin{aligned}0 = dU - T dS + P dV.\end{aligned} \hspace{\stretch{1}}(1.0.2)
It’s quite likely that many of the identities that can be obtained will be useful, but this should at least provide a handy reference of possible conversions.
Differentials in $P, V$
This first case illustrates the method.
\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {P}} \right)_{V} dP +\left( \frac{\partial {U}}{\partial {V}} \right)_{P} dV- T\left( \left( \frac{\partial {S}}{\partial {P}} \right)_{V} dP + \left( \frac{\partial {S}}{\partial {V}} \right)_{P} dV \right)+ P dV \\ &= dP \left( \left( \frac{\partial {U}}{\partial {P}} \right)_{V} - T \left( \frac{\partial {S}}{\partial {P}} \right)_{V} \right)+dV \left( \left( \frac{\partial {U}}{\partial {V}} \right)_{P} - T \left( \frac{\partial {S}}{\partial {V}} \right)_{P} + P \right).\end{aligned} \hspace{\stretch{1}}(1.0.3)
Taking wedge products with $dV$ and $dP$ respectively, we form two two forms
\begin{aligned}0 = dP \wedge dV \left( \left( \frac{\partial {U}}{\partial {P}} \right)_{V} - T \left( \frac{\partial {S}}{\partial {P}} \right)_{V} \right)\end{aligned} \hspace{\stretch{1}}(1.0.4a)
\begin{aligned}0 = dV \wedge dP \left( \left( \frac{\partial {U}}{\partial {V}} \right)_{P} - T \left( \frac{\partial {S}}{\partial {V}} \right)_{P} + P \right).\end{aligned} \hspace{\stretch{1}}(1.0.4b)
Since these must both be zero we find
\begin{aligned}\left( \frac{\partial {U}}{\partial {P}} \right)_{V} = T \left( \frac{\partial {S}}{\partial {P}} \right)_{V}\end{aligned} \hspace{\stretch{1}}(1.0.5a)
\begin{aligned}P =-\left( \frac{\partial {U}}{\partial {V}} \right)_{P}- T \left( \frac{\partial {S}}{\partial {V}} \right)_{P}.\end{aligned} \hspace{\stretch{1}}(1.0.5b)
Differentials in $P, T$
\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {P}} \right)_{T} dP + \left( \frac{\partial {U}}{\partial {T}} \right)_{P} dT-T \left( \left( \frac{\partial {S}}{\partial {P}} \right)_{T} dP + \left( \frac{\partial {S}}{\partial {T}} \right)_{P} dT \right)+\left( \frac{\partial {V}}{\partial {P}} \right)_{T} dP + \left( \frac{\partial {V}}{\partial {T}} \right)_{P} dT,\end{aligned} \hspace{\stretch{1}}(1.0.6)
or
\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {P}} \right)_{T} -T \left( \frac{\partial {S}}{\partial {P}} \right)_{T} + \left( \frac{\partial {V}}{\partial {P}} \right)_{T}\end{aligned} \hspace{\stretch{1}}(1.0.7a)
\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {T}} \right)_{P} -T \left( \frac{\partial {S}}{\partial {T}} \right)_{P} + \left( \frac{\partial {V}}{\partial {T}} \right)_{P}.\end{aligned} \hspace{\stretch{1}}(1.0.7b)
Differentials in $P, S$
\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {P}} \right)_{S} dP + \left( \frac{\partial {U}}{\partial {S}} \right)_{P} dS- T dS+ P \left( \left( \frac{\partial {V}}{\partial {P}} \right)_{S} dP + \left( \frac{\partial {V}}{\partial {S}} \right)_{P} dS \right),\end{aligned} \hspace{\stretch{1}}(1.0.8)
or
\begin{aligned}\left( \frac{\partial {U}}{\partial {P}} \right)_{S} = -P \left( \frac{\partial {V}}{\partial {P}} \right)_{S}\end{aligned} \hspace{\stretch{1}}(1.0.9a)
\begin{aligned}T = \left( \frac{\partial {U}}{\partial {S}} \right)_{P} + P \left( \frac{\partial {V}}{\partial {S}} \right)_{P}.\end{aligned} \hspace{\stretch{1}}(1.0.9b)
Differentials in $P, U$
\begin{aligned}0 &= dU - T dS + P dV \\ &= dU - T \left( \left( \frac{\partial {S}}{\partial {P}} \right)_{U} dP + \left( \frac{\partial {S}}{\partial {U}} \right)_{P} dU \right)+ P\left( \left( \frac{\partial {V}}{\partial {P}} \right)_{U} dP + \left( \frac{\partial {V}}{\partial {U}} \right)_{P} dU \right),\end{aligned} \hspace{\stretch{1}}(1.0.10)
or
\begin{aligned}0 = 1 - T \left( \frac{\partial {S}}{\partial {U}} \right)_{P} + P \left( \frac{\partial {V}}{\partial {U}} \right)_{P} \end{aligned} \hspace{\stretch{1}}(1.0.11a)
\begin{aligned}T \left( \frac{\partial {S}}{\partial {P}} \right)_{U} = P \left( \frac{\partial {V}}{\partial {P}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.11b)
Differentials in $V, T$
\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {V}} \right)_{T} dV + \left( \frac{\partial {U}}{\partial {T}} \right)_{V} dT - T \left( \left( \frac{\partial {S}}{\partial {V}} \right)_{T} dV + \left( \frac{\partial {S}}{\partial {T}} \right)_{V} dT \right)+ P dV,\end{aligned} \hspace{\stretch{1}}(1.0.12)
or
\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {V}} \right)_{T} - T \left( \frac{\partial {S}}{\partial {V}} \right)_{T} + P \end{aligned} \hspace{\stretch{1}}(1.0.13a)
\begin{aligned}\left( \frac{\partial {U}}{\partial {T}} \right)_{V} = T \left( \frac{\partial {S}}{\partial {T}} \right)_{V}.\end{aligned} \hspace{\stretch{1}}(1.0.13b)
Differentials in $V, S$
\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {V}} \right)_{S} dV + \left( \frac{\partial {U}}{\partial {S}} \right)_{V} dS - T dS+ P dV,\end{aligned} \hspace{\stretch{1}}(1.0.14)
or
\begin{aligned}P = -\left( \frac{\partial {U}}{\partial {V}} \right)_{S}\end{aligned} \hspace{\stretch{1}}(1.0.15a)
\begin{aligned}T = \left( \frac{\partial {U}}{\partial {S}} \right)_{V} .\end{aligned} \hspace{\stretch{1}}(1.0.15b)
Differentials in $V, U$
\begin{aligned}0 &= dU - T dS + P dV \\ &= dU- T \left( \left( \frac{\partial {S}}{\partial {V}} \right)_{U} dV + \left( \frac{\partial {S}}{\partial {U}} \right)_{V} dU \right)+ P \left( \left( \frac{\partial {V}}{\partial {V}} \right)_{U} dV + \left( \frac{\partial {V}}{\partial {U}} \right)_{V} dU \right)\end{aligned} \hspace{\stretch{1}}(1.0.16)
or
\begin{aligned}0 = 1 - T \left( \frac{\partial {S}}{\partial {U}} \right)_{V} + P \left( \frac{\partial {V}}{\partial {U}} \right)_{V} \end{aligned} \hspace{\stretch{1}}(1.0.17a)
\begin{aligned}T \left( \frac{\partial {S}}{\partial {V}} \right)_{U} = P \left( \frac{\partial {V}}{\partial {V}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.17b)
Differentials in $S, T$
\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \left( \frac{\partial {U}}{\partial {S}} \right)_{T} dS + \left( \frac{\partial {U}}{\partial {T}} \right)_{S} dT \right)- T dS+ P \left( \left( \frac{\partial {V}}{\partial {S}} \right)_{T} dS + \left( \frac{\partial {V}}{\partial {T}} \right)_{S} dT \right),\end{aligned} \hspace{\stretch{1}}(1.0.18)
or
\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {S}} \right)_{T} - T + P \left( \frac{\partial {V}}{\partial {S}} \right)_{T} \end{aligned} \hspace{\stretch{1}}(1.0.19a)
\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {T}} \right)_{S} + P \left( \frac{\partial {V}}{\partial {T}} \right)_{S}.\end{aligned} \hspace{\stretch{1}}(1.0.19b)
Differentials in $S, U$
\begin{aligned}0 &= dU - T dS + P dV \\ &= dU - T dS+ P \left( \left( \frac{\partial {V}}{\partial {S}} \right)_{U} dS + \left( \frac{\partial {V}}{\partial {U}} \right)_{S} dU \right)\end{aligned} \hspace{\stretch{1}}(1.0.20)
or
\begin{aligned}\frac{1}{{P}} = - \left( \frac{\partial {V}}{\partial {U}} \right)_{S} \end{aligned} \hspace{\stretch{1}}(1.0.21a)
\begin{aligned}T = P \left( \frac{\partial {V}}{\partial {S}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.21b)
Differentials in $T, U$
\begin{aligned}0 &= dU - T dS + P dV \\ &= dU - T \left( \left( \frac{\partial {S}}{\partial {T}} \right)_{U} dT + \left( \frac{\partial {S}}{\partial {U}} \right)_{T} dU \right)+ P\left( \left( \frac{\partial {V}}{\partial {T}} \right)_{U} dT + \left( \frac{\partial {V}}{\partial {U}} \right)_{T} dU \right),\end{aligned} \hspace{\stretch{1}}(1.0.22)
or
\begin{aligned}0 = 1 - T \left( \frac{\partial {S}}{\partial {U}} \right)_{T} + P \left( \frac{\partial {V}}{\partial {U}} \right)_{T} \end{aligned} \hspace{\stretch{1}}(1.0.23a)
\begin{aligned}T \left( \frac{\partial {S}}{\partial {T}} \right)_{U} = P \left( \frac{\partial {V}}{\partial {T}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.23b)
# References
[1] John Baez. Entropic forces, 2012. URL http://johncarlosbaez.wordpress.com/2012/02/01/entropic-forces/. [Online; accessed 07-March-2013].
## PHY454H1S Continuum Mechanics. Lecture 15: More on surface tension and Reynold’s number. Taught by Prof. K. Das.
Posted by peeterjoot on March 10, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Disclaimer.
Peeter’s lecture notes from class. May not be entirely coherent.
# Review. Surface tension clarifications
For a surface like figure (\ref{fig:continuumL15:continuumL15Fig1})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Fig1}
\caption{Vapor liquid interface.}
\end{figure}
we have a discontinuous jump in density. We will have to consider three boundary value constraints
\begin{enumerate}
\item Mass balance. This is the continuity equation.
\item Momentum balance. This is the Navier-Stokes equation.
\item Energy balance. This is the heat equation.
\end{enumerate}
We have not yet discussed the heat equation, but this is required for non-isothermal problems.
The boundary condition at the interface is given by the stress balance. Denoting the difference in the traction vector at the interface by
\begin{aligned}[\mathbf{t}]_1^2 = \mathbf{t}_2 - \mathbf{t}_1 = -\frac{\sigma}{2 R} \hat{\mathbf{n}} - \boldsymbol{\nabla}_I \sigma\end{aligned} \hspace{\stretch{1}}(2.1)
Here the gradient is in the tangential direction of the surface as in figure (\ref{fig:continuumL15:continuumL15Fig2})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Fig2}
\caption{normal and tangent vectors on a curve.}
\end{figure}
In the normal direction
\begin{aligned}[\mathbf{t}]_1^2 \cdot \hat{\mathbf{n}}&= (\mathbf{t}_2 - \mathbf{t}_1) \cdot \hat{\mathbf{n}} \\ &= -\frac{\sigma}{2 R} \end{aligned}
With the traction vector having the value
\begin{aligned}\mathbf{t} &= \mathbf{e}_i T_{ij} n_j \\ &= \mathbf{e}_i \left( -p \delta_{ij} + \mu \left( \frac{\partial {u_i}}{\partial {x_j}}+\frac{\partial {u_j}}{\partial {x_i}}\right)\right)n_j\end{aligned}
We have in the normal direction
\begin{aligned}\mathbf{t} \cdot \mathbf{n} =n_i \left( -p \delta_{ij} + \mu \left( \frac{\partial {u_i}}{\partial {x_j}}+\frac{\partial {u_j}}{\partial {x_i}}\right)\right) n_j\end{aligned} \hspace{\stretch{1}}(2.2)
With $\mathbf{u} = 0$ on the surface, and $n_i \delta_{ij} n_j = n_j n_j = 1$ we have
\begin{aligned}\mathbf{t} \cdot \mathbf{n} = -p\end{aligned} \hspace{\stretch{1}}(2.3)
Returning to $(\mathbf{t}_2 - \mathbf{t}_1) \cdot \hat{\mathbf{n}}$ we have
\begin{aligned}\boxed{-p_2 + p_1 = -\frac{\sigma}{2 R} }\end{aligned} \hspace{\stretch{1}}(2.4)
This is the Laplace pressure. Note that the sign of the difference is significant, since it effects the direction of the curvature. This is depicted pictorially in figure (\ref{fig:continuumL15:continuumL15Fig3})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Fig3}
\caption{pressure and curvature relationships}
\end{figure}
Question In [1] the curvature term is written
\begin{aligned}\frac{1}{{2 R}} \rightarrow \frac{1}{{R_1}} + \frac{1}{{R_2}}.\end{aligned} \hspace{\stretch{1}}(2.5)
Why the difference? Answer: This is to account for non-spherical surfaces, with curvature in two directions. Illustrating by example, imagine a surface like as in figure (\ref{fig:continuumL15:continuumL15Figq})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Figq}
\caption{Example of non-spherical curvature.}
\end{figure}
## Followup required to truly understand things
While, this review clarifies things, we still don’t really know how the surface tension term $\sigma$ is defined. Nor have we been given any sort of derivation of 2.1, from which the end result follows.
I’m assuming that $\sigma$ is a property of the two fluids at the interface, so if you have, for example, oil and vinegar in a bottle, we have surface tension and curvature that’s probably related to how the two of these interact. If there is still a mixing or settling process occurring, I’d imagine that this could even vary from point to point on the surface (imagine adding soap to a surface where stuff can float until the soap mixes in enough that things start sinking in the radius of influence of the soap).
Now consider the tangential component of the traction vector
\begin{aligned}\mathbf{t}_2 \cdot \hat{\boldsymbol{\tau}} - \mathbf{t}_1 \cdot \hat{\boldsymbol{\tau}} = - \not{{ \frac{\sigma}{2 R} \hat{\mathbf{n}} \cdot \hat{\boldsymbol{\tau}}}} - \hat{\boldsymbol{\tau}} \cdot \boldsymbol{\nabla}_I \sigma\end{aligned} \hspace{\stretch{1}}(2.6)
So we see that for a static fluid, we must have
\begin{aligned}\boldsymbol{\nabla}_I \sigma = 0\end{aligned} \hspace{\stretch{1}}(2.7)
For a static interface there cannot be any surface tension gradient. This becomes very important when considering stability issues. We can have surface tension induced flow called capillary, or mandarin (?) flow.
# Reynold’s number.
In Navier-Stokes after making non-dimensionalization changes of the form
\begin{aligned}x \rightarrow L x'\end{aligned} \hspace{\stretch{1}}(3.8)
the control parameter is like Reynold’s number.
In NS
\begin{aligned}\rho \frac{\partial {\mathbf{u}}}{\partial {t}} + \rho ( \mathbf{u} \cdot \boldsymbol{\nabla} ) \mathbf{u} = - \boldsymbol{\nabla} p + \mu \boldsymbol{\nabla}^2 \mathbf{u}\end{aligned} \hspace{\stretch{1}}(3.9)
We call the term
\begin{aligned}\rho ( \mathbf{u} \cdot \boldsymbol{\nabla} ) \mathbf{u}\end{aligned} \hspace{\stretch{1}}(3.10)
the inertial term. It is non-zero only when something is being “carried along with the velocity”. Consider a volume fixed in space and one that is moving along with the fluid as in figure (\ref{fig:continuumL15:continuumL15Fig4})
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Fig4}
\caption{Moving and fixed frame control volumes in a fluid.}
\end{figure}
All of our viscosity dependence shows up in the Laplacian term, so we can roughly characterize the Reynold’s number as the ratio
\begin{aligned}\text{Reynold's number}&\rightarrow\frac{{\left\lvert{\text{effect of inertia}}\right\rvert}}{{\left\lvert{\text{effect of viscosity}}\right\rvert}} \\ &=\frac{ {\left\lvert{\rho ( \mathbf{u} \cdot \boldsymbol{\nabla} ) \mathbf{u} }\right\rvert} }{{\left\lvert{ \mu \boldsymbol{\nabla}^2 \mathbf{u}}\right\rvert}} \\ &\sim\frac{ \rho U^2/L }{\mu U/L^2} \\ &\sim\frac{ \rho U L }{\mu }\end{aligned}
In figures (\ref{fig:continuumL15:continuumL15Fig5}), (\ref{fig:continuumL15:continuumL15Fig6}) we have two illustrations of viscous and non-viscous regions the first with a moving probe pushing its way through a surface, and the second with a wing set at an angle of attack that generates some turbulence. Both are illustrations of the viscous and inviscous regions for the two flows. Both of these are characterized by the Reynold’s number in some way not really specified in class. One of the points of mentioning this is that when we are in an essentially inviscous region, we can neglect the viscosity ($\mu \boldsymbol{\nabla}^2 \mathbf{u}$) term of the flow.
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Fig5}
\caption{continuumL15Fig5}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.2\textheight]{continuumL15Fig6}
\caption{continuumL15Fig6}
\end{figure}
# References
[1] L.D. Landau and E.M. Lifshitz. A Course in Theoretical Physics-Fluid Mechanics. Pergamon Press Ltd., 1987.
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|
http://math.stackexchange.com/questions/188195/inverse-of-binary-entropy-function-for-0-le-x-le-frac12?answertab=votes
|
# Inverse of binary entropy function for $0 \le x \le \frac{1}{2}$
I'm trying to find the inverse of $H_2(x) = -x \log_2 x - (1-x) \log_2 (1-x)$[1] subject to $0 \le x \le \frac{1}{2}$. This is for a computation, so an approximation is good enough.
My approach was to take the Taylor series at $x=\frac{1}{4}$, cut it off as a quadratic, then find the inverse of that. That yields
$$H_2^{-1}(x) \approx -\frac{1}{16} \, \sqrt{-96 \, x \log\left(2\right) + 9 \, \log\left(3\right)^{2} - 72 \, \log\left(3\right) + 96 \, \log\left(4\right)} + \frac{3}{16} \, \log\left(3\right) + \frac{1}{4}$$
Unfortunately, that's a pretty bad approximation and it's complex at $H_2^{-1}(1)$. What other approaches can I take?
[1] I originally forgot to write the base 2 subscript (I added that in a later edit)
-
A rather coarse approximation of $H_2(x)$ on the said interval is $4 \log(2) x(1-x)$. Hence a crude approximation for the inverse is: $$H_2^{-1}(y) \approx \frac{1}{2} \left(1- \sqrt{1-\frac{y}{\log(2)}}\right) = \frac12\frac{y}{\log(2) + \sqrt{\log(2)\left(\log(2)-y\right)}}$$ This initial approximation should be refined with Newton-Raphson method.
-
Thanks! How did you come up with $H_2(x) \approx 4 \log(2) x (1-x)$? – Red Aug 29 '12 at 3:17
The entropy is symmetric function of $x$, suggesting to approximate it by a polynomial in $x(1-x)$. – Sasha Aug 29 '12 at 3:19
@Red I meant symmetric around $x=1/2$. – Sasha Aug 29 '12 at 3:27
+1. A slightly better approximation would be to choose $a$ such that $$\int_0^1 \left(H_2(x) - ax(1-x) \right)^2 dx$$ is minimized. This gives us $a= \dfrac{35}{12} \approx 2.91667$ as opposed to $a = 4 \log(2) \approx 2.7726$. Sasha's $4 \log 2$ ensures that the approximation is always $\leq H_2(x)$, whereas the above approximation optimizes the choice over the entire domain. – user17762 Nov 28 '12 at 23:53
A nice approximation I found in this thesis:
$\frac{x}{2 \log_2 (6/x)} \leq H_2^{-1}(x) \leq \frac{x}{ \log_2 (1/x)}$
-
|
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|
https://www.physicsforums.com/threads/check-the-working-set.9088/
|
# Check the working ( set )
1. Nov 17, 2003
### denian
using algebraic laws of sets, show that, for any set A and set B that
( A U B ) [inter] ( A U B' ) = A
this is my working
( A U B ) [inter] ( A U B' )
= A U ( B [inter] A ) U ( B [inter] B' )
= ( A U B ) [inter] ( A U A )
= ( A U B ) [inter] A
= A U ( B [inter] A )
= A ( shown )
i think this working is wrong. is there other way better than this?
2. Nov 17, 2003
### arcnets
denian,
I think the 1st step should be
( A U B ) [inter] ( A U B' )
= A[inter]A U A[inter]B' U B[inter]A U B[inter]B'
3. Nov 17, 2003
### denian
ok. i dont even know we can expand it that way.
thank you.
|
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|
https://brilliant.org/problems/lucas-numbers/
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# Luca's numbers....
Just like Fibonacci numbers , there are $$\color{Blue}{Luca's}$$ numbers... They are defined by following recurrence relation. $L_0=2$ $L_1 =1$ $\text{and}$ $L_n = L_{n-1} + L_{n-2}$
Find the value of $$\displaystyle \sum_{n=0} ^{10} L_n$$
Note :- I read about Luca's numbers at this problem in the tag recurrence relations, and i think the same thing as why the maker had chosen $$L_{10}$$ as asked answer, the same is why I also have chosen it to be sum till 10. After you get the answer, you'll come to know it, $$\color{Red}{\text{the answer number looks very good.}}$$
×
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http://mathematica.stackexchange.com/questions/11352/efficient-dynamic-drawing/11354
|
# Efficient dynamic drawing
So for example I have this code:
points = {}; k = 0;
Dynamic@k
Dynamic@Graphics[Point@points]
x = .1; y = .3; K = .9;
While[True,
k++;
{x, y} = {FractionalPart[x + K y], FractionalPart[x]};
AppendTo[points, {x, y}];
]
The problem is that the list of points is growing and it slows down with time. What I want is to be able to add a point (or another primitive) directly on the dynamic graph without storing unnecessary chunks of data.
-
AppendTo[] is notoriously slow... – Guess who it is. Oct 1 '12 at 9:08
@J.M. And increasing the number of points to plot ad infinitum doesn't help – belisarius Oct 1 '12 at 9:18
For systems like Matlab this works because it uses a more PostScript-like approach to rendering (just adding on top)... – Yves Klett Oct 1 '12 at 12:55
As it was mentioned in the comments: your loop is infinite, which eventually will cause the slowdown of any computation that accumulates data in the memory. The important rule of thumb for Dynamic updating is: only update when necessary and only update what is necessary. Accordingly, you can speed up the performance of the dynamic drawing by wrapping only points and k in Dynamic. By this way, only the list of points is updated (and the label) and Mathematica does not have to redraw the whole Graphics object again and again (which involves a lot of extra computation).
points = {};
Graphics[Point@Dynamic@points, PlotLabel -> Dynamic@k]
x = .1; y = .3; K = .9;
Do[
{x, y} = {FractionalPart[x + K y], FractionalPart[x]};
points = Append[points, {x, y}];
, {k, 20000}]
Starting from @belisarius' comment, I came up with a more economic version (time scales linearly with k). If one does not have to keep all the points we can apply a reasonable resolution to bin the ranges and saving new datapoints in a matrix, overwriting previous data.
resolution = 256; (* divide the (0,1) range into 256 bins *)
array = Array[0 &, {resolution, resolution}];
Dynamic@ArrayPlot[array, PlotLabel -> Dynamic@k]
x = .1; y = .3; K = .9;
Do[
{x, y} = {FractionalPart[x + K y], FractionalPart[x]};
array = ReplacePart[array, (Min[#, resolution] & /@ (Round[{x, y}*resolution] + 1)) -> 1],
{k, 1000000}]
-
Yes it's much faster but execution still slows down with time. The goal is to get rid off list and just add primitives directly to the existing graph. – swish Oct 1 '12 at 10:14
@swish: I think that in any case a list must be maintained to store points if you want to build up your graphics by successively introducing points. – István Zachar Oct 1 '12 at 10:32
How about keeping a rasterized pre-image, and adding a point at a time. It's slow, but O(1) – belisarius Oct 1 '12 at 12:33
Thanks for the tip @belisarius. See edit. – István Zachar Oct 1 '12 at 13:39
Thank you all, it is really fast now. I was playing with ArrayPlot too. – swish Oct 1 '12 at 14:25
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http://wealoneonearth.blogspot.de/2012/01/
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## 20120129
### Review: The Art of the Long View
I've been calling myself a futurist for the past five years, and for five years, I've been lying. But no longer, because I've read this book, which is every bit as a thought-provoking as Science Fiction for Prototyping proved disappointing. Peter Schwartz is one of the founders of the Global Business Network consulting firm, and honed his skills designing scenarios for Shell Oil in the 1980s. In The Art of the Long View, he makes a strong case for the utility of scenario planning, explains how to develop a proper futurist mindset, and how to create your own scenarios.
Scenario planning is not predicting the future. Rather, it is about challenging the official future, and the assumptions that underlie it. Scenarios force you to examine your unspoken beliefs and values, the evidence supporting them, and how you might react in the future. An organization that includes scenario planning in its process is better able to react to rapidly changing conditions, and less likely to be rendered slowly obsolete through technological change.
Scenario planning is inherently interdisciplinary. A scenario plan has to include technological, economic, cultural, and political factors, as well as individual psychology. Broad areas of knowledge rather than deep and narrow research is better suited at picking up on trends. The ideas and forces that most powerfully influence the future originate on the margins of society, among the dispossessed, the utopian, or the just plain weird. Finally, Schwartz includes a detailed, 8 stage guide to using scenarios in your own organization, with a good balance of theories and examples. Perhaps the ultimate success of scenario planning is that it creates a shared language to talk about the future.
Scenario planning might not be about predicting the future, but a futurist who makes no predictions isn't very useful. The book was published in 1991, and some parts feel oddly anachronistic, like the Japanophilia, the groping towards a 'digital global teenager', and the absence of the War on Terror. On the other hand, he offers three scenarios for the world in 2005: New Empires focused on regional militarism, Market World with multicultural entrepreneurialism, and Change Without Progress, where the wealthy hollow out states, and fear of losing what little remains prevents successful action. Change Without Progress is strikingly similar to the world today, with our 1%ers and 99%ers, paralyzed multinational bodies, and collapsing infrastructure.
Scenario planning is not a strict methodology that automatically produces valid results, it's an attitude towards the future that is based on broad understandings of historical forces and skepticism about the status quo. The results will vary on the quality of the questions you can ask, the data available, and the conversation you foster. But as far as crystal balls go, scenario planning is one of the best.
## 20120117
### Why Andrew Sullivan is Wrong About Obama
Andrew Sullivan has been blowing up the internet with an article about how Obama has outsmarted his critics on the Left and on the Right, by playing a long game that has allowed him to achieve meaningful policy advances without grandstanding or drama. Yet while Obama has achieved policy successes, he has failed to establish the government as a credible force for good. Andrew Sullivan misses the cultural forest for the policy trees. I didn’t want Obama to be FDR; I just wanted him to reverse the worst parts of the Reagan revolution. Instead, at this rate Obama is going to wind up looking more like Richard Nixon than Ronald Reagan.
In Sullivan’s words:
Obama was not elected, despite liberal fantasies, to be a left-wing crusader. He was elected as a pragmatic, unifying reformist who would be more responsible than Bush.
And what have we seen? A recurring pattern. To use the terms Obama first employed in his inaugural address: the president begins by extending a hand to his opponents; when they respond by raising a fist, he demonstrates that they are the source of the problem; then, finally, he moves to his preferred position of moderate liberalism and fights for it without being effectively tarred as an ideologue or a divider.
This is essentially correct. Obama has achieved some notable policy successes, and I for one have greatly enjoyed the frothy fury of the Republican primary, but come November, the election will be over, and somebody will have to govern. And the fact is they’ll do so with a population that trusts the government less than ever before. It’s what David Brooks calls the instrument problem; 10% of Americans trust the government to do the right thing, even as they rely on the government to secure the borders, ensure the safety of food and drugs, and provide healthcare, social security, and unemployment insurance. Suzanne Mettler identifies the problem as the submerged state. Most of the government programs that benefit the middle class are either invisible or run so smoothly that more than half the people who use them don’t think they’re using a government program. If you don’t think the government supports you, why would you support the government?
I’d like to compare two Republican presidents, not on their conservative credentials, but on their legacy. By conventional measures, Nixon is a far better president. He founded the EPA, opened relations with China, unilaterally renounced the development of biological weapons, negotiated the first arms control treaty with the USSR, ended the Vietnam War (eventually), got American off the gold standard, reduced inflation, launched the War on Cancer, and saw American land on the moon. Reagan presided over ballooning budget deficits, used government power to crush the unions, cut taxes only to raise them, ignored AIDS for several years, supervised the Iran-Contra affair, presided over a massive arms race, slashed anti-poverty programs, pushed the war on drugs, and saw the Challenger explode.
Yet for all this, Nixon’s legacy is “I am not a crook,” and Reagan’s legacy is “The nine most terrifying words in the English language are: ‘I'm from the government and I'm here to help.’” Sure, Nixon was a paranoid lowlife with the moral instincts of a hammerhead shark, but he led many initiatives which America is rightfully proud of. Reagan’s accomplishments are far thinner, but he was the Great Communicator, and he established a political dialogue that is with us today, an undead ideology that flows through the Tea Party and cripples the ability to govern.
Sullivan thinks that Obama’s opponents will be punished for their carelessness with the truth, but I’m not so sure. Paul Krugman believes that Mitt Romney is running a post-truth campaign, and he’s the most reasonable of the Republican candidates, or at least the least insane. Being elected today requires that you believe a dozen contradictory things before breakfast (to link to Krugman again). I’ve not seen any backlash towards public figures for spouting obvious falsehoods, and even the New York Times is wondering if it should challenge people who lie in its articles.
I supported Obama because I believed that he could articulate a vision for American democracy in the 21st century. I thought that the author of Dreams from my Father, the 2004 Democratic Convention Keynote, and the speech on Reverend Wright, would be somebody who could inspire America in the same way that Kennedy and Reagan did. We needed, and still need, inspiration more than any specific policy solution. I believed that roused to action, the American people would find their own solutions to major problems, like healthcare, energy, education, and the war.
Instead, Barack Obama has presided over an ugly and secretive government. It is a government that uses drones to kill terrorists on the other side of the world, while making the absurd claim that “There hasn’t been a single collateral death because of the exceptional proficiency, precision of the capabilities we’ve been able to develop,” (according to senior counter-terrorism official John O. Brennan) despite ample evidence to the contrary. It is a government that has failed to address basic concerns about hidden risks and ‘shadow banks’ in the financial system. And while the rancor and insanity of the 112th Congress is not Obama’s fault, the White House is little better. On the Keystone XL pipeline, and Plan B birth control pill, the Obama administration has given the impression that it does not make decisions based on evidence, or what he believes would be right for the country, but what is most politically expedient. It is a short-sighted tactic that reduces his own credibility.
David Brooks, at the end of his editorial on the instrument problem, says:
“If Democrats can’t restore Americans’ trust in government, it really doesn’t matter what problems they identify and what plans they propose. No one will believe in the instrument they rely on for solutions.”
I do not want people to uncritically trust Big Government, but American has passed the point of reasonable skepticism to the point of political solipsism. Congress is less popular than polygamy, the BP oil spill and Maoism. If Obama cannot restore some basic faith in government, then he will be a failure, no matter how many policy successes he manages.
## 20120116
### Equivalence of Statistics on a Pair of Gaussian Channels
This post is a departure from our usual discussions. It relates to statistics and information theory, as applied to a somewhat limited model of communication. In this model, we have two variables with normally distributed amplitudes. One variable is the "true" signal, and the other contains the "true" signal mixed in with some noise. This is a model of a noisy Guassian communication channel. The main purpose is to show that, under these conditions, correlation, squared error, mutual information, and signal-to-noise ratio all become equally good and interconvertible measurements of how related the two signals are. This topic was of interest because many people apply all of these statistics separately to the same data. These notes are not written for a general audience, I'm just putting this out here in case someone, somewhere, finds it interesting.
For a pair of Gaussian channels ( continuous random variables who's values follow a normal distribution ), the mutual information, correlation, root mean squared error, correlation, and signal to noise ratio, are all equivalent and can be computed from each-other. Without loss of generality we restrict this discussion to zero-mean unit variance channels. This discussion elaborates on the discussion of mutual information between Gaussian channels presented in the third chapter of Spikes.
Correlation & Mutual Information
Consider a single gaussian channel $y = g x + n$, where $x$ is the input, $y$ is the output, $g$ is the gain, and $n$ is addative gaussian noise. Without loss of generality, assume that $x$, $n$ and $y$ have been converted to z-scores. Reconstructed z-scores can always be mapped back to the original gaussian variable by multiplying by the original standard deviations and adding in the original means. This means that all random variables have zero mean and unit variance. If we do this, we will need a separate gain for the signal and noise, say, $a$ and $b$.
$y = a x + b n$
Since ths signal and noise are independent, their variances add:
$\sigma^2_{y} = \sigma^2_{a x} + \sigma^2_{b n}$ and the gain parameters can be factored out
$\sigma^2_{y} = a^2 \sigma^2_{x} + b^2 \sigma^2_{n}.$
Since $\sigma^2_{y}=\sigma^2_{x}=\sigma^2_{n}=1$,
$a^2+b^2=1$
This can be parameterized as
$\sigma^2_{y} = \alpha \sigma^2_{x} + (1-\alpha) \sigma^2_{n},\,\,\alpha=a^2\in[0,1]$
and
$y = x\sqrt{\alpha} + n\sqrt{1-\alpha}$
The relationships between mutual information $I$ and signal-to-noise ration $SNR$ come from Spikes, chapter 3.
$I=\frac{1}{2}lg(1+\frac{\sigma^2_{a x}}{\sigma^2_{b n}})=\frac{1}{2}lg(1+SNR)$
Where $lg(\dots)$ is the base-2 logarithm.
The $SNR$ simplifies as :
$SNR=\frac{\sigma^2_{a x}}{\sigma^2_{b n}}=\frac{\alpha \sigma^2_x}{(1-\alpha) \sigma^2_n}=\frac{\alpha}{1-\alpha}$
Mutual information simplifies as :
$I=\frac{1}{2}lg(1+SNR)=\frac{1}{2}lg{\frac{\sigma^2_y}{\sigma^2_{b n}}}=\frac{1}{2}lg{\frac{\sigma^2_y}{(1-\alpha)\sigma^2_n}}=\frac{1}{2}lg{\frac{1}{1-\alpha}}$
The correlation $\rho$ is the standard definition of Pearson's product-moment correlation coefficient, which can be viewed as the angle $\theta$ between vectors defined by the samples of random variables $x$ and $y$.
$\rho=cos(\theta)=\frac{x y}{|x||y|}$
Since $x$ and $n$ are independent, the samples of $x$ and $n$ can be viewed as an orthonormal basis for the samples of $y$, where the weights of the components are just previously defined $a$ and $b$, respectively. This relates our gain parameters to the correlation coefficient: the tangent of the angle between $y$ and $x$ is just the ratio of the noise gain to the signal gain
$tan(\theta)=\frac{b}{a}=\frac{\sqrt{1-\alpha}}{\sqrt{\alpha}}$
Then $tan(\theta)$ can be expressed in terms of the correlation coefficient $\rho$ :
$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}=\frac{\sqrt{1-cos(\theta)^2}}{cos(\theta)}=\frac{\sqrt{1-\rho^2}}{\rho}$
This gives the relationship $\sqrt{1-\alpha}/\sqrt{\alpha}=\sqrt{1-\rho^2}/\rho$, which implies that that $\alpha=\rho^2$, or $a=\rho$. (There is a slight problem here in that correlation can be negative, but it is the magnitude of the correlation that really matters. As a temporary fix, correlation now means "absolute value of the correlation".) This can be used to relate $\rho$ to $SNR$ and mutual informtaion:
$SNR=\frac{\rho^2}{1-\rho^2}$
$I=\frac{1}{2}lg{\frac{1}{1-\rho^2}}=-\frac{1}{2}lg(1-\rho^2)$
As a corollary, if $\phi=\sqrt{1-\rho^2}$ is the correlation of $y$ and the noise $n$, then information is simply $I=-lg(\phi)$. Mean squared error ($MSE$) is also related :
$MSE=(1-\rho)^2+(1-\rho^2)=1-2\rho+1=2(1-\rho)$
which implies that
$\rho=1-\frac{1}{2}MSE$
and gives a relationship between mutual information and mean squared error:
$I=-\frac{1}{2}lg(1-\rho^2)=-\frac{1}{2}lg(1-(1-MSE/2)^2)$
The relationships between correlation $\rho$, root mean squared error $RMSE$, information $I$, and signal to noise ratio $SNR$, all increase monotonically, implying that correlation, SNR, and mutual information, all give the same quality ranking for a collection of channels.
Further Speculation
This can be generalized (as in chapter 3 of Spikes) to vector-valued Gaussian variables by transforming into a space where $Y=AX+BN$ is diagonal, treating each component independently, and then transforming back into the original space.
Similarly to how chapter 3 of Spikes generalizes mutual information of a Gaussian channel into a bound on mutual information for possibly non-gaussian, vector valued, channels, these relationships can be generalized to inequalities for non-Gaussian channels :
$I\geq-lg(\Phi)=-\frac{1}{2}lg(1-\Sigma^2)=-\frac{1}{2}lg(1-(1-MSE/2)^2)$
Where, for vector valued variables, $\phi$, $\rho$, and $MSE$ become matrices $\Phi$, $\Sigma$, and $MSE$.
## 20120110
### D&D: Working the Rules
Last time, I wrote about how gameplay is enacted as a negotiation at the table, with the rules serving as one component of the common grounding for negotiation, a means of abstracting tedious or low-player knowledge negotiations, and furthermore as a wagering game that is fun in and of itself. I was planning to write a follow-up about that last part, but before I could, Wizards of the Coast announced the next Dungeons and Dragons, with generally favorable articles in The New York Times and Forbes. Most interestingly, they’re inviting participation from the community to design and playtest the new edition. While I haven’t yet decided if I want to formally be part of the process, it seems like a good time to throw in my two cents.
The last edition change, from 3.5 to 4e, was a little bit of a disaster. While I thought it was a major step forward, many players didn’t like the how all the classes in the new edition had the same core structure (at-will, encounter, daily, and utility powers) compared to simple fighters and complex spellcasters, the mandatory use of a battlemat, the length of combat, or just the aesthetics of the books and character sheets. While some of the complaints were fundamentally off-base, D&D4e is not an MMO, it is not ‘dumbed down’, and it is just as viable to roleplay in as any other edition of D&D, and other complaints were corrected by Essentials classes and new math for monsters, the damage had been done. The D&D community split into 4e players, a solid core of 3.5/Pathfinder fans, and the new Old School Rules movement, which leveraged digital distribution and a vibrant community to create cleaned up versions of the first editions of D&D.
Wizards claims that 5e will reunite the scattered fans and become the One Edition to Rule Them All. Personally, I’m doubtful. People have diverse tastes, and the only common ground is that they like what they already know. I’m also not sure how much participation the community will have, or even if the community actually knows what it wants (A common joke in the RPG industry is that to make a successful game, you should look at what RPG.net says and then do the exact opposite). All I know is that right now the geekier corners of the internet are exploding with speculation, demands, and manifestos. So without further delay, this is what I think about rules, and how to write good ones for D&D.
Pretty much every RPG relies on some random element, usually dice, for a core task resolution mechanic. The exact details of this resolution mechanic aren’t particularly important, you can use XdY+modifier vs a target number, percentile roll under, dice pools, or pretty much anything that can be imagined, but at the end of the day the player will declare some action, with probability P between 0 and 1 of success.
Finding the right P is an important part of game design. Too low, and plays will miss frequently, leading to a high whiff factor. Too high, and players will be rolling simply to see if they fail, which is irritating. Casinos are the experts in balancing probabilities and payoffs, ensuring enough winning to keep players hooked, while the law of averages drains their pockets. Personally, I find a chance of success at p=0.6 for an average task is most satisfying (coincidentally, this is what D&D 4e uses). Easy tasks can go up to about p=0.85, and hard task should be p=0.4, with extremely difficult or desperate tasks at p=0.2.
But rolling and trying to get above an 8 isn’t a game that’s likely to hold a person’s attention for long. So what else is there? I’m going to refer to Greg Costikyan classic essay, “I have no words and I must design”, which you really should read in full.
The thing that makes a game a game is the need to make decisions. Consider Chess: it has few of the aspects that make games appealing -- no simulation elements, no roleplaying, and damn little color. What it's got is the need to make decisions. The rules are tightly constrained, the objectives clear, and victory requires you to think several moves ahead. Excellence in decision making is what brings success.
Negotiation is fun, but decision making is also fun. The rules tell us what kind of decisions can be made, what the option space is, and what outcomes are. And to describe what makes rules fun, or good, I’m going to have to pull in a couple of other theories.
The first one is the Paradox of Choice, which I’ll use to talk about building a character. Having too few options makes us unhappy for obvious reasons-we are constrained from doing what we want to do. Having too many options also makes us unhappy, as the energy involved in coming to a decision outweighs the benefits gain by making the best decision. We become obsessed with alternatives and missed opportunities. In a competitive environment, like a game, the feeling like you’re not making good choices is equivalent to feeling like you’re lost, like you’re losing. It’s not fun.
With that in mind, D&D4e (and 3.5 before it) simply has too many options. At first level, the player has 44 classes, 50 races, hundreds of backgrounds, themes, feats, and powers. In play, every turn a player has a choice of the two At-Will powers, one encounter, one daily, and perhaps a power or two gained from class or background. This is about the limit of most people’s working memories (conventional wisdom is that most people can remember about seven things, so that’s your powers, your HP, and the general state of the game).
At 30th level, this elegant structure has totally imploded. A character has 20 feats, selected out of a list of over 2300, 12 core attack powers select from a list of hundreds, and perhaps a couple dozen more miscellaneous class abilities and magic items chosen from a list of thousands. If you’ve grown organically into that 30th level character, you might understand its full capabilities, but most players are overwhelmed.
And worst of all, there’s not much too clearly evaluate many of these choices. Powers can play very differently, but there are lots of powers of the form “deal 2[weapon]+stat damage and push 1 square”, or the like. Now, I like the power system introduced in 4e, I think it goes a long way towards distinguishing different characters of the same class, and one class from another, but the power system needs to be toned down so that even a high level character doesn’t have more than 12 total. As for feats and magic items, they need to be seriously curtailed. Group similar feats into Themes, or Packages, or Kits, and give each character a choice of one. Magic items that simply fix the game’s math should be combined with the character’s intrinsic skill. Instead, magic items should be singular, rare, and synergize with the characters. A warrior will have a weapon of legend, a mage might have a staff that defends the user on its own, a rogue might have a Ring of Invisibility. Either way, I’d like it if even top level characters have no more than three magic items, and each one was a significant part of the character.
The second theory is the OODA loop, developed by fighter pilot John Boyd. OODA (observe, orient, decide, act) is the process by which people make decisions in a strategic or tactical game. You observe the enemy and try and figure out what their capabilities are, using your past experience you orient yourself, make a decision, and execute. In warfare, the trick is to get inside the enemy’s OODA loop, to move with the chaos of combat and direct it at your enemy until they make a mistake that can be capitalized on.
What a player observes is what the DM shows and tells them: A description of the monsters, where they are on the battlefield, how they move and act. But in D&D 4e, and other grid based systems, this description has two parts, the verbal side of what the GM says, and the visual side of what the map looks like. And if the rules and GM are not well coordinate, the descriptions will begin to diverge. Players will be constantly switching from a verbal-negotiation cognitive frame to a visual-puzzle solving cognitive frame.
There are good reasons to represent combat on a grid with some kind of figurines. People are usually fairly skilled in visual puzzle solving. It allows decisions to be very finely grained-precisely how far to move, where to drop that fireball, which rewards players that are using tactical maneuvers to their advantage. Without a map, players have to hold the entire encounter in their heads, and with about 10 combatants on the board (the player, 4 allies, 5 enemies) that can get tricky, especially when people begin to disagree about who was where.
But the downside of using a map is that it focuses attention on this visual puzzle, diminishing attention on the verbal aspect of the game, or to use the phrase of gamers, “immersive roleplaying”. I used to think that this was a silly complaint, but thinking about players having to switch between cognitive systems, it makes sense. Most of us are lousy multitaskers. WotC design Mike Mearls has indicated the next edition will be grid-optional. “The new edition is being conceived of as a modular, flexible system, easily customized to individual preferences. Just like a player makes his character, the Dungeon Master can make his ruleset. He might say ‘I’m going to run a military campaign, it’s going to be a lot of fighting’… so he’d use the combat chapter, drop in miniatures rules, and include the martial arts optional rules.” It’ll be interesting to see how this modularity works out in practice.
Finally, there’s one other issue I want to bring up: pacing and flow. Flow, in games studies, is the sensation of full immersion in the activity, and of playful challenge. One of the major complaints against D&D 4e is that the fights take too long, a critique which I agree with. The length of a fight depends on the total number of rounds, and the time it takes each player to take their action. In my experience, a fight in 4e lasts about 5 rounds. On the one hand, this seems like a good number—it prevents fights from being decided in the opening action, and gives the players time to recover from bad luck.
The problem is that it’s slow. I haven’t taken a stopwatch to my group, but it seems to take about two minutes for a person to take their turn. Multiply by six people (five players + GM) and five rounds, and you have an hour long fight. I think turns could be speeded up by reducing the complexity of the decisions that people have to make, even in 4e there’s too much referring to the character sheet for rarely used powers, and reducing the number of dice rolls. It takes about 10 seconds to find, roll, and do the math on a single die. Powers with multiple attacks and multiple damage dice, which are pretty common, can easily take two minutes just to crank through. Thinking and remembering also add time, every situational modifier bestowed by a power or position adds another few seconds to the turn, and few minutes to the fight. Interrupt actions are even worse. I like big set-piece battles, and the interactivity of 4e turns, but they’re not suited for every situation. It’ll be interesting to see what the designers come up with to speed up play. This is really my biggest open question.
I’ll close with a quote from John Boyd, “The second O, orientation –the repository of our genetic heritage, cultural tradition, and previous experiences –is the most important part of the O-O-D-A loop since it shapes the way we observe, the way we decide, the way we act.” D&D has nearly 40 years of history. Pretending to be elves and casting magic missile on the darkness and unearthing horrors and then running away are part of our gaming DNA. The biggest reason why 4e failed was not that it was poorly marketed, but that it required a different orientation from gamers, one that they didn’t want to learn. Hopefully, people wiser than me are figuring out what the D&D player’s orientation is, and designing rules that work with it, instead of fighting against it. If they can do that, everything else from the modular rules, to tactical depth, to combat speed, will fall into place.
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https://zh-cn.unz.com/pgiraldi/john-brennans-cia-trump-task-force/?showcomments
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博客浏览菲利普·吉拉迪（Philip Giraldi）档案
记住我的信息
=>
• B
6年2017月XNUMX日，就在特朗普就职前不久，俄罗斯的“选举干扰”叙事一直是类固醇，当时由克拉珀（Clapper）和布伦南（Brennan）策划的“情报界评估”（ICA）得以发表。 次日本身就是“深州”的组成部分的纽约头上的大标题，为接下来的事情定下了基调：“普京领导向特朗普提供援助的计划，报道说。”
[]
1. The magic words are “FISA warrants”.
The entire FISA court process has been exposed as an insane sham.
“The Secret Team” just took the absurdity of the process and raised it to the next level–injecting it into a political campaign.
It would be wonderful if they could fill a jail with every empty suit who touched those warrants–but I would be stunned if even one of them gets paraded around in the orange jump-suit they so richly deserve.
• 回复： ,
2. NPleeze 说：
好像特朗普不是深州的一部分。
他在叙利亚、玻利维亚、香港、伊拉克、阿富汗、以色列等地的行动都无可争议地证明，他是（并且一直是）深州中地位良好的成员。 如果他是满洲候选人，他就是深州的真正傀儡，而不是人民或俄罗斯的傀儡。
One can argue that his actions “against” China are the one thing that proves the contrary, but that omits that the entire point of the TPP, and the “Pivot to Asia”, by Obama was to contain China. Trump is doing nothing very different. The TPP has been replaced, incidentally, by the “Blue Dot” program.
Globalism and globalists are doing fine. Trump will continue to be labeled by his patrons as a “nationalist” because he is the “controlled opposition”.
• 巨魔： Sbaker
3. AWM 说：
Somebody needs to question Brennan’s “source” in the Kremlin who was spirited out after Trump’s election and hopefully we may find out something about this in the IG report.
If this “source” in the Kremlin was the basis for the 17 (actually 3) Intelligence Agencies deciding Putin wanted Trump to win well that’s the rest of the story.
Also, Putin arrested several FSB members for illegally and without his knowledge for co-operating with the CIA, Brennan’s CIA.
4. 不见得。 如果他是（（（他们的）））柏忌人，他们就不会如此致力于全力以赴至死。 （（（他们）））必须害怕他这种程度的歇斯底里。
• 回复： ,
5. Democracy died a long time ago in America. We just didn’t notice. So, who should be killed? Everyone in the American government, from the top down. You can skip Trump, since he is a powerless outsider, but everyone else? They are all compromised.
坦率地说，我们所能希望的最好的结果是一个仁慈的独裁者摧毁现有结构。
• 回复： ,
6. Alfred 说：
Looking on at this affair from outside the USA, it is clear that the power and influence of the USA is waning a lot faster than most people expected.
The replacement of the US military by mercenaries who are called other names was a first step. The sanctioning and punishing of allies for stepping out of line is the second step. BTW, it is notable how Japan and Australia are very keen to stay in line but the Europeans less so.
I suspect the third step will be to encourage a collapse of the Euro – so as to make wealthy Europeans shift their money to the USA in a panic.
It seems to me that the US public will be the last to learn of what is really happening. Even on this website there are sometimes letters or articles that mention 9/11 as a “terrorist” or “Saudi” act. How can one take anything such a person writes seriously?
The control of media and the internet seems to be the last part of the collapse. They will hang on to that to the very last moment.
• 回复：
7. The emperor is naked, and even though many see it and say it, the charade goes on.
8. Crazy Horse [又名“ Gall”] 说：
In the old days they used to give traitors the option of being hung or shot. Now they work for the CIA and become TV celebrities.
• 回复：
9. I campaigned for Trump but was tricked. The die is caste now. There must be civil war and secession.
• 同意： Realist
10. Realist 说：
约翰·布伦南（John Brennan）的中央情报局特朗普特遣部队
可以成为奥巴马门吗？
Perhaps, but what is the point? All this bullshit is engineered to make dumbass Americans think justice is being served. Nothing will come of it…no one will go to prison.
11. Realist 说：
好像特朗普不是深州的一部分。
他在叙利亚、玻利维亚、香港、伊拉克、阿富汗、以色列等地的行动都无可争议地证明，他是（并且一直是）深州中地位良好的成员。 如果他是满洲候选人，他就是深州的真正傀儡，而不是人民或俄罗斯的傀儡。
Exactly. I voted for Trump, but, as long ago as mid April 2017, I determined that he was a Deep Stater…his actions are just too obvious to ignore.
12. The Magic Negro cannot be touched and that is why nothing will be done about the biggest crime in the history of our nation……Obama orchestrated the destruction of a political rival and he will get away scott free….because he’s an oppressed and downtrodden dindu.
13. Realist 说：
有相当多的证据表明，美国政府系统可能已成为美国组织和执行的非法秘密行动的受害者
你指的是哪一个，1953 年的伊朗，1963 年的肯尼迪遇刺事件，1964 年的北部湾，还是其他几十个例子？
• 回复：
14. Sbaker 说：
The truth of your comment is way understated. The Semite from Chitcago is protected from all the evidence by the old media empire. If the Semite goes down, the end for the rat swamp and the old media empire follow. As always, a corrupt media has the upper hand.
15. Sbaker 说：
Exactly. I voted for Trump, but, as long ago as mid April 2017, I determined that he was a Deep Stater…his actions are just too obvious to ignore.
This certainly explains the incessant attacks on Trump by the deep state.
16. “And this was not a CIA-only operation. Personnel from the FBI also were assigned to the Task Force “
This just in: both the CIA and FBI are unconstitutional, agencies.Get rid of them -and all of the other unconstitutional alphabet-soup agencies[FDA,EPA,SEC etc.etc.etc.]. No downsizing-trash them all- NOW!
This also just in: “Because they are all ultimately funded via both direct and indirect theft [taxes], and counterfeiting [via central bank monopolies], all governments are essentially, at their very cores, 100% corrupt criminal scams which cannot be “reformed”,”improved”, simply because of their innate, unchangeble criminal nature.” onebornfree
“把国家带到任何地方，随时进入其历史，看不出有什么办法可以将其创建者，管理者和受益者的活动与专业犯罪分子的活动区分开。” 阿尔伯特·诺克
Regards, onebornfree
i
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17. Biff 说：
The Magic Negro cannot be touched
That’s pretty much it in a nutshell – put him on a shelf with the Clintons, and the Bush’s.
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18. Moi 说：
Unfortunately, an overwhelming majority of ignorant Americans think they have a democracy and a representative government. The country is going down the proverbial tube, and pronto!
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19. John Brennan just looks sinister. It would not surprise me if he were in the top 1% of most evil persons alive.
20. Anonymous[381]• 免责声明 说：
Your point about 9/11 can’t be made forcefully enough. We’re going straight to hell unless Israel and its American confederates are brought to justice, these wars ended, and order restored. Clapper, Comey, Brennan, Mueller, Chertoff and the whole traitorous bunch are probably guilty as principals but almost certainly they’re at least complicit as accessories before and after the fact. So naturally all we hear about is Russiagate.
The evidence overwhelmingly implicates Israel and not Saudi Arabia as you point out. That Building 7 was brought down by explosives has been proved beyond doubt by Architects & Engineers for 911 Truth, and as Dr. Alan Sobrosky put it, if Building 7 was brought down by explosives, so too were the Twin Towers. The official NIST reports and all related government narratives are preposterous. They’re fairytales for fools inasmuch as the official mechanisms rely on a suspension of the laws of physics more fanciful than Jack and the Beanstalk. The story of the nineteen Arabs who couldn’t handle Cessna 150s magically flying jetliners into precise targets is more absurd than fairytale tropes about flying carpets.
Yet for Conservatism Inc and Fox News, which both claim to oppose the Deep State and its narratives, there’s no standard of evidence so low or preposterous that these cucks won’t cling to it to cover up what they must now know is Israel’s guilt. We can assume it’s precisely because they’re aware of Israel’s guilt that they rule out the overwhelmingly conclusive circumstantial evidence pointing to Israel on the grounds such evidence is “anti-Semitic” and consequently false on apriori grounds. Moreover, any expert investigator qualified in the relevant field who uncovers and presents evidence implicating Israel is cast as the actual terrorist. It should go without saying they’ve reversed a millennium in the development of Western thought regarding the connection between evidence and conclusion, and they’ve done so for the basest of reasons. At least Conservatism Inc is being daily exposed for the controlled opposition and worthless club of preppy snots it’s always been.
• 同意： Robjil
21. Brilliant Article.
The question is when deep state will finally admit that Globalism after all that sacrifice and evildoing are just sour grapes. As fox said in Ezops tale.
22. Trump isn’t so much a Manchurian candidate as a Khazarian candidate, dancing to the merry tune of his Zionist handlers.
23. Agree. Only proviso, let’s stop calling it the “Deep State”. I prefer Kiriakou’s term, the Federal Bureaucracy.
24. Indeed tRump is just the NY street corner mobster being told what to do by the”godfather”.
• 同意： Realist
25. “the historical practices of the Russians, who typically, are almost genetically driven to co-opt, penetrate, gain favor, whatever, which is a typical Russian technique,” adding later that “It’s in their DNA.”
Right, along with drinking vodka and eating borscht.
The one nation that did interfere in the 2016 election, and has inserted themselves into other elections to get their candidate elected, Israel remains untouched by this (((Deep State))).
There’s plenty of evidence for the Zionists and Israeli-Firsters corrupting the election process for their fav nation, Israel, but the Operation Mockingbird asshats in the MSM won’t go near that, not if they want to keep their cushy job, 5th Avenue penthouse and that chauffeured limo.
Anytime AIPAC comes to town, Congress gets into a fight with each other, trying to be the one that shows the most slavish loyalty to the nation that has attacked the USA numerous times, spies constantly on us, stealing our military, business and industrial secrets, had a hand in both murdering JFK, RFK and masterminded the 9/11 FF, and has an overwheling presence on the FED, yet most Americans don’t know that, because the MSM keeps reporting lies, distortions and half-truths, and always presenting a boogeyman to hate, sometimes Russia, most times Muslims.
But fear not, that will soon come to an end, for when those TBTF Wall Street banks–in collusion with the FED–again crash the stock market and drag the economy down with their greed, that coming crash will make the one of 1929 seem like a picnic.
When that happens, what’s left won’t be of any interest to Israel to steal or manipulate.
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26. The Deep State murdered Kennedy.
He planned on destroying the Fed and the CIA.
The Deep State required a president that COLLABORATED, like LBJ.
Then they figured, “why not put our guy in?’
Thus Bush 1.
Then Clinton (Bush 1 was his ‘mentor’) a pervert stooge.
Then Bush 2, a gaymail stooge.
Then Obomber, the gay Kenyan C_A stooge.
Then America says ‘Enough’, rejects the Witch and elects Trump.
Now the Deep State wants to kill America.
I think it’s time for America to kill its’ Deep State.
It is, after all, self-defense. Besides, hasn’t this gone on long enough?
The alternative is to end up as the modern parallel of Rome.
27. Brennan is an admitted communist and so is undermining the government for his zionist bolshevik masters, in fact most of the creatures in the DC swamp are communists aka bolsheviks.
28. Clapper, Brennan and Comey ” may not have operated on their own.” Duh!
You just remember, a donkey won’t carry a heavy burden unless it’s fed regularly. Find out who owns the beast and you will have the culprit!
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29. It goes without saying to the more informed, but the Deep State is of course yid to the core. They direct it all.
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30. Z-man 说：
Phil, you should offer your services to the Trump defense/attack team*. Just stay away from Giuliani (grin). Good article and salvo against Brennan and the rest who deserve all the pain thay can get.
*Hey money is good especially around Christmas time. (Grin)
31. Realist 说：
This certainly explains the incessant attacks on Trump by the deep state.
You have no concept of a charade being perpetrated on the American people. You don’t find it a little strange that Trump keeps hiring the Deep State denizens he purports to be fighting? You are incapable of detecting the friend/foe, psychological tactic used to deceive?
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32. Sbaker 说：
That’s it. Your hearsay trumps thousands of eyewitnesses and conversations of the victims on flight 93 and in the pentagon.. You must be a first responder too. Talk about easily influenced–you are why the old media gets away with their corruption. Some anonymous source writes, says it on TV, and the lemmings follow.
33. As we write, Obama probably is banging away some groupie in a DC mansion basement while the gorrila is frying chicken upstairs for Oprah.
And Bubba, most likely, is watching porn in the garage in Westchester and the wicked witch is massaging mrs. Wiener.
But it’s Dubya worth looking into because he is out in the cowshed, buck naked save the cowboy boots and the ten-gallon hat, whipping himself silly for the “mission accomplished!”
34. Or, Obama and the police agencies investigated a known organized crime stooge when it became apparent the GOP could offer no other candidate. Ironically, this was the original intent of the creation of the FBI.
Don’t play into the “victim trump” brand, he needs no help with it.
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35. Thanks much for the most comprehensive précis yet of this bungled CIA putsch. The articles in sequence teasingly open Gina’s kimono, giving us horripilating glimpses of her bushy penetralia. The question of Obama’s involvement is the next step. CIA bots have been pushing a partisan perspective for some time. Those darn Democrats!
But in fact Obama too is CIA nomenklatura, with the same depth of dynastic ties as GW Bush, albeit not at the same lofty level. Just look at the oppo research, the best of which comes from sanitized glimpses of the errands candidates run for CIA. Obama’s other passport is not Kenyan but Indonesian. It facilitated the youngster’s schooling during Mom’s year of living dangerously in Indonesia. Obama’s dad and stepdad were CIA skins on the wall. Grandma was not in fact a drunk – she laundered the money for forcible overthrow and genocide in Indonesia at her bank job in Hawaii. Grandpa was a “furniture salesman,” like Bibi, travelling around Asia under the hoariest old chestnut of NOC cover.
Young Barack was groomed as carefully as Bush minor. His only real job was BIC, a sheepish front perennially stuffed to bursting with NOCs. While he was still wet behind the ears he sported at falconry with a future head of state of Pakistan, for chrissakes, at a time when nobody could get in there. And he got out without getting his head sawed off. How? The youthful promise of this sullen stoner was somewhat obscure at that time. His GF was the Aussie daughter of Mike Barry’s opposite number. And the Mockingbird unison of ecstatic acclaim when he rose to public prominence out of nowhere is the proof. His empty suit belings to CIA.
This past election was CIA office politics, nothing more. Russigate is simply CIA eminence Hillary, the Queen of Mena, ratfucking a bumptious queue-jumper. She outranks Trump, who was merely a junior money-launderer for the CIA agents who looted the Soviet Union. It was her turn to take the figurehead head-of-state sinecure. She and Bill earned it with their lucrative Clinton Foundation covert-ops slush fund.
Don’t overthink it. Your government is CIA.
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36. They sense the rumblings of White solidarity among “the forgotten men and women” of Trump’s base and they do not cotton to this one little bit. Solidarity is forbidden to Whites. It is only for the coalition of the fringes, all of those groups whose alienation can be stoked to weaponize them against the descendants of those who founded and built the United States.
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37. The FISA warrants had nothing to do with organized crime.
On second thought, that is not correct.
They _were_ organized crime.
(It is not necessary to defend Trump to understand this. FISA warrants based on known fake “evidence” are a stunning abuse of power–even within the slime-pit of DC.)
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38. BL 说：
Anyone at all familiar with Brennan knows that he was and remains the driver of the conspiracy to destroy first candidate, then President -Elect, and then POTUS Trump.
The same cannot be said of Comey and Clapper (especially).
It literally makes me sick to my stomach whenever I think about what it says about this great republic that seditious filth like him rose to such a powerful position. It’s rather obvious he was willing to do anything including, I would submit, gift Russia and God Knows Who Else anything they wanted in return for helping him destroy our constitutional republic.
As I’m sure most here know, long before his 2016 election malefactions he had brazenly engaged in spying on Congress and, most despicably, had debased President Obama and the Office of the President through NYT revelations that every week Obama picked from his list of drone assassination targets.
. . . and it inevitably leads to the question “What did Obama know?”
Yes, though more important than that was what Obama was told (by Brennan) and what real options did he have as president given that Brennan had him by the short hairs.
I’ve long considered anyone’s efforts to prematurely direct liability to President Obama as a bald attempt to protect Brennan. That worked for the purposes of a general, earlier on, cover up. It won’t at this stage because it isn’t even a close call when it comes to Democrats, elected and rank and file, choosing between the first black president and Brennan, the American Beria.
39. Herald 说：
This certainly explains the incessant attacks on Trump by the deep state.
Oh dear me, how little it takes to bullshit some people!
• 同意： Realist
40. anonymous[307]• 免责声明 说：
[Israel Kills Senior Islamic Jihad Commander in Gaza
In a surprise airstrike before dawn on Tuesday, Israeli forces killed a senior commander of the Palestinian Islamic Jihad militant group in the Gaza Strip, setting off waves of retaliatory rocket attacks that immediately raised fears of an escalating new conflict.
The timing of the attack, after a period of relative calm along the border and amid a protracted, high-stakes negotiation over who will lead Israel’s next government, led some critics of Prime Minister Benjamin Netanyahu to charge that it was politically motivated.]
This is UNACCEPTABLE. The UN petty chairman, even does not care to condemn these assassination by the scums like Netanyahu and its Jewish mafia pimps in Washington. They deserve the right to kill and bomb whoever they want and where ever they like for political gain.
For the past few years Netanyahu and the Jewish mafia pimp, trump to assassinate General Suleimani where has been failed. Trump the terrorist and thief tried to assassinate Nicolás Maduro Moros the president of Venezuela and recently US have staged a coup in Bolivia toppled the elected president Evo Morales to steal other countries resources.
If their economy is so good, why these mass murderers, racists and thieves stage coup, invade and occupy so many countries. This show their economy is based on killing other groups and stealing their resources.
Sine ICC and UN are the evil empire tools to kill and robe other nations and these scums use assassination regularly to eliminate leaders, then it is the right of others to assassinate these thieves and terrorists and remove these scum from the earth.
Why ICC does not do its job. The mass murderer who invaded Iraq and looted Iraqi libraries, museum and resources, George Bush, is receiving his salary and working on his latest hobby, ‘painting’ where are at the Southwest ‘museum’.
Now, a mass murderer who have committed many CRIMES against humanity is a painter according to Evil Empire. Shame on this culture.
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41. Hey iSmellBagels!
Question. Do you think Israel has a Deep State? Please let me know & thanks.
42. anonymous[307]• 免责声明 说：
Assassinate Netanyhu the terrorist, baby killer and thief
[Israeli forces kill another young Palestinian man ‘for no reason’ in Hebron refugee camp
Israeli forces shot and killed a young Palestinian man outside his home today in the al-Arroub refugee camp in the southern occupied West Bank district of Hebron.
Omar al-Badawi, reported to be 22-years-old, was shot in the abdomen near the front steps of his home in the camp during protests commemorating the death of revered Palestinian leader Yasser Arafat.]
CONDEMN ZIONIST KILLING and its supporters who are silent at the WH. Expose them all over the world.
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43. Anonymous[381]• 免责声明 说：
There are dozens of different lines of proof developed by acknowledged experts in the relevant fields proving beyond a shadow of a doubt that the official narrative is bullshit. You’re resorting to ad hominem and mischaracterizing what I said proves you’re a bluff.
Try Yahoo’s hasbara stooge comment board where stuff like “thousands of eyewitnesses” gets a free pass. In fact, hundreds of firefighters and cops have reported, as a matter of public record, that sequential explosions were going off in Building 7 right before it came down. You couldn’t last two minutes in a debate with anyone even vaguely familiar with the facts of 9/11.
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44. The emperor is naked
That is the best argument for voting for young beautiful women as POTUS.
America needs its own Yulia Tymoshenko
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45. “What did Obama know?”
Is Giraldi turning on his hero, President Zero?
And has been since the early 1950s when the Allen Dulles-Frank Wisner-James Jesus Angleton crew got rolling. They managed to thoroughly penetrate federal and state bureaucracies, the court systems, major corporations, the financial sector, and the mockingbird media.
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47. kikz 说：
“What did Obama know?”
whatever his handlers like gollum jarrett fed him.
• 同意： Desert Fox
48. anon[113]• 免责声明 说：
So Phil, was there any cooperation/communication between the Trmp Task Force and the DNC dirt-diggers in Ukraine (Ali Chalupa et al), or were they completely independent actions?
Erdogan is now the object of general invective. Is this spite, for decline in US power? Yesterday at the Press Club, the Investigative Journal and friends convened to bash him –
Erdogan’s End Game: Turkey’s Long Arm in Syria and America
https://investigativejournal.org/live/
49. Unfortunately you are correct. There is a racial angle to this situation. The mockingbird media has been weaponized against whitey — both people and culture. The New York Times being the most glaring example. Eight years of Obama-Holder-Jarrett and their progressive white minions in the Executive Branch really did a number on the USA.
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51. … it was plausible to maintain that [the Russians] would have hoped that a weakened Clinton would be less able to implement the anti-Russian agenda that she had been promoting.
Not sure I follow this line of reasoning. If the Russians had tried unsuccessfully to throw the election to Trump and Hellary won anyway, how exactly would that leave her “weakened”? And wouldn’t she have that much more reason to go after Russia?
My theory is that Hellary and her deep-swamp creatures only messed with Trump because they were 一定 he was going lose. And if they could then plausibly claim after the election that the Russians had interfered (albeit unsuccessfull) in the election, Hell-bitch could’ve used that as a pretext for … well, I don’t know. War? More sanctions? Inducting Ukraine into NATO? Invading Syria?
Clapper revealed that the preparers of the ICA were “handpicked analysts” from only the FBI, CIA, and NSA. He explained rather unconvincingly during an interview on May 28, 2017, that “the historical practices of the Russians, who typically, are almost genetically driven to co-opt, penetrate, gain favor, whatever, which is a typical Russian technique,” adding later that “It’s in their DNA.”
I had no idea Clapper was into HBD. Damn, he’s based!
52. anonymous[307]• 免责声明 说：
[Even though none of the alleged Kremlin plotters would have expected Trump to actually beat Hillary, it was plausible to maintain that they would have hoped that a weakened Clinton would be less able to implement the anti-Russian agenda that she had been promoting. Many observers in both Russia and the U.S. believed that if she had been elected armed conflict with Moscow would have been inevitable, particularly if she moved to follow her husband’s example and push to have both Georgia and Ukraine join NATO, which Russia would have regarded as an existential threat.]
This is a propaganda piece for Trump re-election using SCARE tactic to fool the dummies. Don’t trust Trump apologists, an occupier and thief. He has not honor single promise he made during selection process last time, only delivered tax cut for the thieves and stolen land for the zionist Maifa that he and his family are members.
Trump needs the cooperation of another zionist Mafia like PUTIN to run its ‘foreign policy’ smoothly to keep the dummies on board. Have you seen him someone who is against invasion, looting, assassination and terrorism. Only fools answer positively.
Trump is supported by the zionist bankers, who are part of the Deep State, who are keeping the ‘economy’ based on gambling, in ‘shape’ and very profitable for the thieves who run the show. If there is someone who is not as cooperative as the MAFIA Trump, then the economy will goes down, as Trump warned the dummies so many times in order to threaten the public.
the Old Michael Rubens Bloomberg, another zionist Mafia, has been entered the SELECTION process to boost another Mafia’s chance of selection although he is a ‘democrat’. The pathological liar and jewish mafia Trump has been so good for both thieves, the zionist bankers, and Israel especially that Russia and Putin are protecting the same scum. Putin and Russia were helping the criminal West, even Obama regime and Hillary Clinton. Hillary Clinton many times said that Putin and Russia have been very cooperative on illegal sanctions against Iranian people, but Putin as a Jewish mafia servant is helping the criminal Mafia gang even MORE.
Vote for Elizabeth WARREN to kill the reelection of the zionist mafia in Washington and Tel Aviv.
If you were deceived by a con man once, then shame on him, if twice then shame on you.
Trump cannot be trusted for any job except at the brothel houses and Escorts.
53. There is a theater play going on, unending series, each episode catching some other superficial drift. A-l-l actors in the public view, their dialogues and declamations are scripted. Trump´s also. He is not the major character, just a single, temporary one. All media opinion pieces, what is news, are prompt readings. Rectal extraction is close.
Why is this possible? The public is beyond understanding. The ones who do, at least part of what is going on, being closer to some sectors of society where a whiff of the smell of power is perceived at clouded times, are interested. The middle classes are scraping and grabbing and bickering for the scraps of the table of the powerful. It takes them most of their career to even get to under the table. They are happy dogs, and scraps comparing to scraps makes them a diverse world of nothings.
It is hard work to come up with alternative policies, not rail into historical models proven wrong as to long term interests and goals of society. A path not to venture into, against instinct.
That makes for a fine world, while it lasts, and is upended by another cycle. The empty drum feeling in the head of most is stuffed with images and sound-bites that makes for a life behind a velvet curtain(Apple´s i-phone).
There is very little cognitive difference between the individuals at the top and the glorious bottom undesirables, they both like the sniff of the glue.
54. Donald Trump’s election (which was not supposed to be allowed to happen) forced into public view, the existence of a Deep State that’s been in existence for more than 75 years. Although not widely recognized as such, JFK’S election accomplished the same thing, but to an even greater extent. Leaving me puzzled as to why Trump has been allowed to remain in office as long as he has without the Deep State subjecting him to a similar fate. With one logical explanation being that, at this point in time, it would become obvious, even to the brain dead, who’s actually in control of the US government.
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55. “It is now known that President Barack Obama’s CIA Director John Brennan created a Trump Task Force in early 2016,” says Mr. Giraldi, and his assertion is supported by a link to a website called Sic Semper Tyrannis, where someone writing under the byline of Larry C. Johnson says, “I am told by an knowledgeable source that Brennan created a Trump Task Force in early 2016.” Perhaps Mr. Giraldi has good reasons for his implicit trust in the statement of Mr. Johnson’s unnamed “knowledgeable source.” But some of the rest of us might be inclined to think that Mr. Giraldi is not waiting for a real starting gun to be off to the races.
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56. Herald 说：
Trump cannot be trusted for any job except at the brothel houses and Escorts.
He’s likely too old to do a proper job at the whore-house.
57. Best argument I have ever heard for the Burka.
58. The zionists control the CIA and the CIA is in bed with the Mossad and MI6.
59. anonymous # 307 declared: “CONDEMN ZIONIST KILLING and its supporters who are silent at the WH. Expose them all over the world.”
Above, agreed, #307.🇺🇸
Nontheless, there’s a huge practical problem in doing so. Subsequently, following is a reasonable😏working question.
Will the awesome ZUS Deep Staters, who are relentlessly condemning the president & salivate for his impeachment,🇮🇱have enough chutzpah to overcome Contra-Deep Staters who persist to rule out of D.C.’s Blue & White House? 🇮🇱
P.S.: Am rather disappointed Philip Giraldi has left behind the deeper (double edge) good/bad stuff.🙁
‘Cuz the devil lurks in the fine print, and thanks a lot, #307!
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60. With one logical explanation being that, at this point in time, it would become obvious, even to the brain dead, who’s actually in control of the US government.
Another explanation is that his VP does not have the stomach for such messy stuff–because remember that the next President is in charge of the cover-up.
LBJ 没有这样的顾虑。
61. At Sunbaked suburb 48, yeah, and the remarkable thing is how our so-called history fixates on the the constitutional convention, the revolutionary war, the civil war, and how desperately it ignores that uniquely drastic discontinuity in US history. All that schoolbook stuff is the ancien régime. The Central Intelligence Agency Act set the constitution and the laws aside except as weapons of the state against its enemies. And everybody’s brainwashed to act like nothing changed. That induced selective blindness is amazing. Americans are way more brainwashed than North Koreans. You can see that when they interact in treaty and charter bodies.
62. schrub 说：
Trump’s biggest weakness is that he appears incapable of friendships with other adult males because he trusts no one. This is probably partially the result of his dealing in the absolutely cut throat New York real estate industry along with his own relentless and long time need for publicity no matter how outrageous this publicity is? (Remember Trump’s forays into professional wrestling?)
Trump decided to hang out with the dogs and, no surprise, ended up getting fleas.
His continual purging of his cabinet members and his bad mouthing of them afterwards has probably made his White House staff paranoid about challenging anything that comes out of his mouth no matter how outrageous it is.
Along with all this self promotion has come an increasing inability to accept any sort of criticism whatsoever. To claim he is slightly “prickly” is a gross understatement.
Where is Trump’s James Baker, or better yet, his Sergey Lavrov. to moderate and control his goofier instincts
.
63. If Trump is smart enough and wants History to write his name with Golden Letters, he has to order a new and true investigation on 9/11 in his second term.
• 同意： Carroll Price
64. geokat62 说：
Brother Nathanael’s take on the so-called Deep State.
Excerpts from, Deep State For Jews:
… ALEPH~BET (ALPHABET) are HEBREW letters signaturing Mossad ownership.
Don’t be fooled, the Deep State doesn’t flow from DC but from Tel Aviv…
The deeper you look the creepier it gets. The Deep State is crawling with Jews.
If we don’t start naming the Jew, we’re all going to be in deep s##t.
The entire article is worth a read.
[更多]
Deep State For Jews
HOW DEEP DOES A PERSON have to dive before finding the deep state?
Some say not far since agencies like the CIA, FBI, and the DOJ have wrapped their roots so tightly around the political mass that no president or senator could ever uproot the tangle.
It’s a good start in finding the buried treasure, or rather, shrouded criminality, but it fetches only a few paltry sea worms—no real fish to fry.
For is Pompeo, Wray, Haspel, and Bolton the Deep State?
If so, then Tillerson, Comey, Brennan, and McMaster were also ‘deep staters’ of essentially a shallow marsh easy to weed out.
But we’re diving into a deep ocean, not a mud puddle, and the first place to look is Jewish Wall Street—financial pillar of the political-military-intelligence-security-corporate industrial apparatus before it was called by the hazy name “Deep State.”
The Jew boys of Goldman Sachs led by Gary Cohn, Lloyd Blankfein, and David Solomon, cut the checks of the Generals via the Fed and underwrite the stocks of defense contractors.
WAR IS A RACKET and bloodshed abroad fills the coffins of goys dying young and the coffers of Jew boys dying old.
This is why Trump’s tariffs won’t bring manufacturing back home.
It’s a financially-enabled game out of Jewish Wall Street that keeps defense contractors home and production factories abroad.
Aluminum and steel will not return as an American commodity–despite tariffs, but to other countries who will buy it sans tariffs with all the Jew-financing they need.
ALEPH~BET (ALPHABET) are HEBREW letters signaturing Mossad ownership.
Don’t be fooled, the Deep State doesn’t flow from DC but from Tel Aviv.
“Alphabet” is the parent company of Google acting as a cover for International Jewry.
Evidently, it was a Mossad handler, Professor Jeffrey Ullman, acting in tandem with a CIA/Mossad agent, Rick Steinheiser, that funded Sergei Brin and Larry Page, Google’s ‘founders,’ when they were students at Stanford in the 1990’s.
Stanford has been tied to secret military research projects for decades and was the steam propelling Silicon Valley’s nexus with the Pentagon and the CIA.
Creating the Massive Digital Data Systems Project (MDDS) to fund computer scientists at Stanford, the Pentagon and Intelligence Community (CIA/Mossad) tapped into a ready-made way to track individuals and groups online…which led up to Google.
Strategic partner Mossad was eager to increasingly connect with monied Jews of the Silicon Valley–(fourth largest Jewish community in the US)–who dominate the high-tech world, and grab some technology of their own while expanding its global network via cyber-science.
Attached to multiple projects sponsored by the Pentagon, Professor Jeffrey Ullman and CIA’s Rick Steinheiser, recognized Brin and Page’s pioneering search-engine technology and pursued a long term goal: Control of the information highway.
NONE SHOULD BE SURPRISED, given Mossad’s veiled ownership of Google, that this cyber cyclops has recently entered into a direct partnership with the Pentagon.
The 18 month Google-Pentagon Project, ending March 2019, is called by the Pentagon, “Project Maven,” a Yiddishee moniker that means “know-it-all.”
And only Jews—as far as they’re concerned—are the genuine “mavens,” while the Goys who work for them, (like Eric Schmidt, who with his nice goyisha look announced Google’s de-ranking program), are useful shills providing Gentile window dressing.
Jews are getting nervous. More and more are bypassing deep-state ambiguity and looking for specific actors and entities.
THE JEWISH MIST thrown over the goy’s face of ‘innocent victimhood’ is being dissipated by Jewish criminality observed all over the world.
Thus Jews legislate mandatory Holocaust education…again made into law in Connecticut; and criminalize criticism of Israel with a Jew-sponsored bill dubbed the “Anti-Semitism Awareness Act of 2018″ now moving from the House into the Senate.
The deeper you look the creepier it gets. The Deep State is crawling with Jews.
If we don’t start naming the Jew we’re all going to be in deep s##t.
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(occupation: Investor, hedge fund manager, and philanthropist-Early life and education: Cooperman was born to a Jewish family in the South Bronx, New York City)
plans to support billionaire Michael Bloomberg’s potential run for president.
到目前为止，一切都不错。
Now Ilhan Omar comes and uncovers the reason with a tweet’ asking her self: “I wonder WHY”?
And the Jews attacking her for …anti-semitism.
等一下。
Is it against the law NOT TO LIKE JEWS?
And then Dr. Giraldi is talking about the deep state.
Deep State in USA, is FULL of Jews, Phil.
PS
Now old Martina Navratilova jump in, to save the Benjamins, like a swollen cl1t.
• 回复：
66. anonymous[222]• 免责声明 说：
It is, after all, self-defense. Besides, hasn’t this gone on long enough?
Beat you to it, steinbergfeldwitzcohen
Classrooms without Borders, a Jewish Federation cutout, is rolling out
Lesson plans to teach 9 – 12th graders how to curb rising auntiesemitism and stop all that Hate Hate Hate!
Jews have suffered enough! It has to stop!
The following lesson plan was written as part of Classrooms Without Borders’ Call for Lesson Plans about antisemitism and hate, in commemoration of the shooting at the Tree of Life synagogue in Pittsburgh.
On the morning of October 27, at Sabbath services—the holy day of the week for the Jewish community—Robert Bowers entered the Tree of Life Synagogue in Pittsburgh, PA and yelled “All Jews must die,” then opened fire upon the congregants. He was armed with an assault rifle and several handguns and killed eleven congregants and wounded six others, four of whom are police officers. When surrendering to law enforcement, Bowers told an officer that he “wanted all Jews to die” and that Jews “were committing genocide against his people.” This shooting is the deadliest attack on the Jewish community in the U.S.
Our world is changing, our understanding of truths are being challenged, and that makes our job as educators more important than ever before. Fostering an understanding of what it means to be a global citizen, engage in civic discussions, and understand other cultures is of the utmost importance. As a history teachers we are acutely aware that human memory is fragile, and the records of the past can be destroyed or distorted. Without an understanding of the past, students have nothing with which to judge what they are told in the present. The truth then becomes the narrative of the ruler or government often distorted and anchoring a single agenda. Our increasingly pluralistic society has ushered in the urgent need for more focus in our classrooms that support and foster an understanding of cultural and racial differences. The unpleasant and divisive language used by many to vilify and denounce various groups within society suggests that the development of this important capacity is urgently required.
***
Suggested Technology: Projector, Laptops for Students, Internet Connection.
Instructional Time: 5 hours.
A free account is required to access the files for this lesson plan.
Just 5 little hours.
300 minutes — 20,000 Jews a minute
Is that too much to ask to save Jews and prevent another holocaust?
67. anonymous[307]• 免责声明 说：
They impeached Bill Clinton, were you defending him because ‘deep state’ was against him?
Trump is embedded in the deep state.
They do their criminal act. This is another SHOW to entertain the ignorant. Name one or two good piece of work he has done?
He is a show man and a clown who is determined to protect the interest of the Evil Empire. He is coming from a Mafia circle. Even Adam Schiff who is the symbol of anti Trump calls him MAFIA but leaving the zionist (jewish) behind to protect the Tribe.
Both are part of the zionist mafia, he is part of George Soros gang who is against Trump, they act differently for the same end game. Soros is pushing for ‘world government’ where ‘greater Israel’ is part of it, Trump is working for the same thing, ‘greater Israel’.
Even Bloomberg, a ‘democrat’, is trying to make division among his own gang, so Trump who has been been so good for the pockets of these thieves stays in power. He enters to benefit Trump, both are working for the same goal, their pockets and the Tribe. The zionist bankers love him.
• 回复：
68. anonymous[222]• 免责声明 说：
Is it against the law NOT TO LIKE JEWS?
是的。
Mr. Elan S. Carr, US State Department Special Envoy for Monitoring and Combatting anti-Semitism. February, 2019.
69. Thank you for correction.
回报：
Tale : (fairy tale)
Tail : Dogs and cats have a tail (in the back.)
70. Trump knows the zionists did 911, and he is part of the coverup!
71. Lloyd 说： • 您的网站
Trump has not been president as long as JFK. But he will be soon. Maybe I am a bit out with the dates. Trump has his own private security and is well aware what happened to JFK. Augustus likewise appointed his Praetorian Guards after assassinations of his uncle and the Gracchi.
• 回复：
72. renfro 说：
You don’t find it a little strange that Trump keeps hiring the Deep State denizens he purports to be fighting?
LOL…Trump doesnt even know anything about most people he appoints…his ‘handlers” and ‘advisors” select them.
• 同意： Ilyana_Rozumova
73. geokat cc’d Brother Nathanael’s deep view into the Deep State and said: “The deeper you look the creepier it gets. The Deep State is crawling with Jews.”
谢谢你，小鬼geo，👍！
And Brother Nathanel continued to go down & dirty into The Land of Bilk & Money, warned: “If we don’t start naming the Jew, we’re all going to be in deep s##t.”
向拿但业弟兄提出一个问题。
(Zigh) You do realize my fellow “Homelanders” have been in “deep shit” ever since the killing of the Roman Catholic President JFK who resisted the Israeli faction of the deep state? Others assembled here would shake me by recalling the Federal Reserve inauguration & persuant deep shit. 🥴
• 同意： Desert Fox
• 回复：
74. That the investigation into trumps organized crimes ties may have been corrupted, doesn’t change the fact of those ties. You put a known stooge up for president, he is gonna be anally inspected. Would expect no less from the GOP. Nice FISA distraction tho.
• 回复：
75. Ahoy 说：
Today another big white hater bit the dust.
https://www.latimes.com/obituaries/story/2019-11-11/noel-ignatiev-dies-race-whiteness
It looks like NEMESIS has come down on earth and is thinning out the ranks of those that have succumbed to hubris, like H.E.W. Bush, McCain, Brzezinski, et. al.
It is the ultimate hubris, sin, to hate the White Race.
• 回复：
76. renfro 说：
Phil, you need to get on the State Department and NSC re the coup against Trump by the Ukraine cabal . The State Department has been stuffed with people like the below who try to set US policy according their personal loyalties and /or hatreds or love for any foreign country. And as we all know the State Department lost all objectivity when the Jews infiltrated it decades ago to run out the ‘Arbarist”.
Currently staring in Congress Impeachment Ukraine testimony against Trump
Lt. Vindman————Ukraine Jewish refugee NSC
Amb Gordon Sondland—-Russian Jewish refugee
Amb Marie Yovanovich- Russian Jewish refugee
Fiona Hill ————Dual US-UK citizen. Studied under Richard Pipes, in 1998 at Harvard, Russian expert.
I have read the testimonies and several things jump out. All these people are outspoken anti Russia activist and pro Ukraine. According to their statements Russia is the ultimate evil. Vindman, Yovanovitch and Hill all use the same description…”Ukraine needs US aid because it is fighting for US interest and against Russian aggression’. ….same spin Jews put on ”Israel fighting for US and world interest against Iran’.
Their testimonies were as much or more about why we should support Ukraine then about what Trump said or didn’t say.
It is clear…and was even said by Hill in her testimony that they…..”should formulate foreign policy, not they president’. And in several cases that is what they have done…going even further with sanctions on countries then what was called for and the unattentive Trump just accepts it .
This Trump coup is coming from the Deep State of the NSC and the State Department, not the CIA this time.
77. Pancho 说：
Democracy never died in America, because the Founding Fathers, knowing the fact that democracies always en up in becoming tyrannies, created a representative Republic, not a Democracy.
78. Herr Trump has declared bankruptcy at least 4 times, more if you count his cut-outs. One doesn’t get away with those kind of financial shenanigans in NYC unless one is in debt to the ones who really run this nation.
Now that he accidentally fell into the WH, the bill has come due and the Deep State will use Tubby the Grifter as a carny barker, raising all sorts of hell on Twitter, to give the appearance of something being done–in the morning–until the adults find out and tell him to backtrack. While Trump is causing another tweet storm, Jared the Snake and the rest of the Kosher Nostra will go about BAU.
Trump reminds me of a mentally challenged kid one of my neighbor’s had. Cute little guy, but absolutely helpless without adults.
But they kept him occupied for hours on end by letting him play video games, it was the only way to keep him out of trouble while they took a break from caring for him 24/7.
• 回复： ,
79. Wally 说：
– Given the obstacles, Trump has done rather well.
– Considering Hillary and the current Communist alternatives, I’ll take Trump any day.
事实：
特朗普的60分成就列表 : https://www.washingtonexaminer.com/washington-secrets/media-blackout-trumps-60-point-accomplishment-list-of-american-greatness
给予特朗普应有的成就 : https://www.washingtontimes.com/news/2018/may/6/giving-trumps-accomplishments-their-due/
特朗普总统取消了860项奥巴马时代的联邦法规: http://www.breitbart.com/big-government/2017/07/21/great-again-trump-eliminates-860-obama-era-federal-regulations/
特朗普总统现在制定行动议程，已签署40项立法: http://dailycaller.com/2017/06/25/president-trump-has-now-signed-40-pieces-of-legislation-as-he-moves-to-enact-his-agenda/
特朗普宣布“共产主义受害者国庆日”。 : https://www.whitehouse.gov/the-press-office/2017/11/07/national-day-victims-communism
特朗普宣布“共产主义受害者国庆日”。 “Holocau\t Industry”的骗子疯了: https://www.salon.com/2017/11/07/trumps-national-day-for-the-victims-of-communism-is-opposite-of-holocaust-statement/ 特朗普的移民改革到 1 年将节省 2027 万亿美元: http://www.breitbart.com/big-government/2017/08/14/donald-trumps-merit-immigration-reform-saves-1-trillion-2027-says-study/ NASA数据证明特朗普有权退出《巴黎气候协定》: https://www.prisonplanet.com/nasa-data-proves-trump-right-to-exit-paris-climate-accord.html 特朗普是对的：新研究显示多达 5.7 万非法选民在选举中投票: https://www.infowars.com/trump-was-right-new-study-reveals-up-to-5-7-million-illegals-voted-in-election/ • 回复： , 80. CIA, as in “Central Idiot Agency”. Langley should have been shut down years ago. The once-proud OSS of Wild Bill Donovan has morphed into a clown show featuring the likes of “Slam Dunk” Tenet! 81. Art 说： Clapper, Brennan and Comey ” may not have operated on their own.” Duh! Please do not let these three bastards off the hook. Remain focused on THEM – not Obama. This “task force” thing is a pure bureaucratic mindset operation. It did not start in the White House. Obama did not do this – he is not the mastermind. His nature is to be cautious. He would not risk his historical future doing this. What Obama should be condemned for is putting these three haughty feminists in charge of foreign policy — Susan Rice, Samantha Powers, and Hillary Clinton. Rice & Powers had no real experience in world affairs, and Hillary was a proven arrogant bloody bitch. (Serbia – the only bombing of Europe since WWII.) The ongoing war they put on Libya and Syria is as bloody, as what Bush did to Iraq after 9/11. p.s. After these guys are taken down – it’s on to 9/11. 82. Realist 说： – Given the obstacles, Trump has done rather well. Obstacles of Trump’s own making. • 同意： lavoisier • 回复： 83. JFK served 2 years and 2 days. At present, Trump has served a few days longer. • 回复： 84. Then they figured, “why not put our guy in?’ Thus Bush 1. Are you including the assassination attempt on Reagan? 85. Concerning CIA agent whistle blower Eric Ciaramella. Can anybody help me here please!!!!!! There is the name and under is a picture of woman. Eric is northern Germanic surname of a male. Erica is a female surname, (My sister in low carries a surname Erica.} 86. 我喜欢这里的深刻思考者，他们居高临下地告诉你，联邦调查局、中央情报局、国家安全局和军事情报局对特朗普的明确而明显的攻击都是虚假的骗局，他们实际上都在一个相互钦佩的社会中。 愚蠢的人往往太聪明了，不利于他们自己的利益。 Trump is a member of the ruling class, but not of the deep state. Thus, he cannot stop what the deep state is doing, fomenting and continuing forever wars in all these countries. Thus, he gets blamed for wars he would really rather stop. Trump is the first anti-war president perhaps in American history. No wonder he can’t get anything done. The unelected legislators, and I don’t mean the poets, don’t know what to do with him. So they just undermine him and then he rolls with the punches and acts like it was all his idea all the time. Trump, being a member of the ruling class, can’t be threatened by the deep state the way non-members can be. But his policies can be ignored. And they can try to impeach him. Although they can’t impeach him for actual criminal act, because they all commit the criminal acts he does, and so revealing his crimes would mean revealing those of the rest of the ruling class. Impossible. So they attack him on trumped-up nonsense–comparable to Watergate–the type that has usually worked to scandolize the masses in the past, but that is not working very well now, perhaps because the trumpers-up are so incompetent, perhaps because enough of the general public is so cynical now, they don’t swallow stuff as easy, or perhaps because the general public is just so stolid and numbed and bovine and uninterested that they just don’t care either way, not to mention it is almost impossible to get them up in arms about almost anything, much less some half-ass incomprehensible scandal about requests and e-mails and feelings, or whatever it is, who can tell. • 回复： 87. 特朗普先生至少四次宣布破产， What’s your point? The moneyed class uses the law to its own benefit. Sucking out the assets through fronts is pretty ordinary these days. It’s a different form of “Jewish lightning”. There is no debt to anyone in NYC, it’s all allowed by the IRS. Trump is unlikable, to be sure. But let’s put something into context, does a 68 year old billionaire really “need” to be POTUS? Did he really need to spend millions self funding his campaign for the Republican nomination? Does he really need all of the horseshit heaped on his family? My take is that he understood what was wrong and knew the FBI, CIA, and NSA were scum, but was politically naive, thinking that the military types he hired, would actually follow the chain of command. I have no skin in the game, but people laughed at me when I said he would win the primary, and laughed harder when I said he would beat Clinton. To me, it was obvious. He was palying massive venues, sometimes 2 a day, where there would be 15-20k with thousands outside unable to get in. Meanwhile, the opposition, including Clinton, were playing occasional half empty school gymnasiums. Yet all of the (rigged) polls showed him massively behind. The coup started with his announcing that he was running. Trump is similar to a local politician I know. He keeps getting re-elected, but is a party outsider. He seldom has party support, because he’s not afraid to tell the party fixers just how dumb his constituents think the grand plan is. The party, whether Republocrat or Demican, will do anything to stop the outsider. 88. You put a known stooge up for president, he is gonna be anally inspected. Would expect no less from the GOP. Nice FISA distraction tho. If you are hanging out here you know we have had nothing but stooges as President for _decades_. Most of them were not investigated by _anyone_ in the bureaucracy. Nice FISA distraction tho. Nice distraction tho. 89. Curmudgeon, is you want to see who really controls DC, watch this nauseating video of Israeli Crime Minister Nuttyahoo getting verbal hand jobs from our corrupt Congress, setting a record that surpassed any ovation a US president received. As for the rest of your thoughts, riddle me this: Why did Trump pardon five of those TBTF banks for fraud, over the 2017 Christmas holidays? Including one he does business with? 没有人看到，特朗普赦免了5家大型银行的腐败指控 在美国人庆祝假期的同时，特朗普总统跟随他的前任们的脚步，为华尔街的利益行事，并分心做一些不符合美国人民最大利益的事情。 他赦免了五家大型银行的猖corruption欺诈和腐败，由于他欠他们很多钱，这一点尤其值得注意。 90. MarkinLA 说： Bankruptcy today isn’t what it used to be. Trump “buys” a casino but uses mostly OPM by floating bonds. The casinos don’t make enough money to pay off the bonds. He files chapter 11 and does a “reorganization”. He threatens to walk if they take the casino away from him. He issues more debt and pays out a few bucks to the bondholders. The casino can’t pay off the new debt. Rinse and repeat until there is nothing left for the bondholders. I saw this scam firsthand on California development properties. He had other indebted holdings to pull this on as well. 91. Mr. Dolan, Exactly, just remember that Harry was there first…and there is a very long book, or at least article, inside your comment but I run out of juice at the end of short comment. Several old college buddies have rewritten the same bad novel a couple of times over the last 40 years, never published…I got nuthin’… 问候， 哈利 92. Lloyd 说： • 您的网站 Actually JFK served two years ten months and two days. Trump on twenty second November will have served same time. So that makes Trump nine days behind Kennedy’s service. 93. “This is UNACCEPTABLE” Nope. You’ll accept it and forget about it by next week. 94. A question to anyone who is not a shallow thinker.🤗 Re, the antiwar article, posted below; Is this apparent Trump administration action permitted by his opposition, Deep State? • 回复： 95. Rurik 说： Some might reasonably regard the whole Brennan affair, to include its spear carriers among the current and retired national security state leadership, as a case of institutionalized treason, and it inevitably leads to the question “What did Obama know?” If a video of Obama talking to Brennan and telling him to spy on, and find dirt on Trump, and then ordering him to sabotage Trump’s election victory, ‘by hook or by crook’, and to use all the power of the state to destroy the democratically elected president of the US.. .. then what difference would that make? The media would simply ignore it, thereby making it all go away, and the people who worship Obama, would worship him even more. And the people who hate Trump, would continue to hate him, only perhaps a little more. The whores in the GOP, would do what their ((donors)) tell them to do, as always, and the outrage would fester only in the ranks of the ignored Deplorables. We’re living in a real-life Idiocracy. Treason and sedition are in-your-face blatant, and there’s nothing anyone can do about it. Just consider the hysterically idiotic impeachment farce. They’re saying Trump was trying to do what Joe Biden (as Obama’s emissary) bragged about doing. Only in Trump’s case, it is a high crime to do what Biden bragged about doing. 专制。 There’s nothing that Obama could have done that would make the ladies of the view stop worshiping him. And there’s nothing that Trump could do that would make them stop hating him. This is the nation we live in now. A society of ‘identity politics’, self-absorbed imbeciles, as far as the eye can see. From the former heads of the CIA and FBI, all the way to the Senate and House. to the entire media, and academia. ‘Respectability’ in our society is based on your respective level of butt-hurt. The more butt-hurt you are, the more respectable you are, regardless of truth or even treason. Just consider the case of Adam Schiff. A liar and a scumbag without a shred of character, but he’s arguably the most butt-hurt disgrace of a human being in the entire congress, (and that’s saying a lot!) and so he’s appointed as the Chairman of the House Permanent Select Committee on Intelligence. An unhinged, utterly dishonest and unscrupulous asshole. Given an enormously powerful position, based solely on his rabid and pathological hatred for Trump (and the Deplorables). We have arrived, America! • 回复： 96. Rurik 说： Noel Ignatiev, dies at 78 yee haa! Now I wonder if they’ll tell us where his carcass will be planted. It might become the world’s biggest urinal. His corpse will be so besotted in piss, that it could help preserve him as a curiousity to science, as the only carcass so saturated with urine, that the marrow of his very bones will reek for eons. If the lucky cemetery is smart, they could make a fortune charging a fee to relieve oneself. I imagine a line where they sell good German beer, and the pilgrims could grab a pint on the way to the back of the line, and by the time they get to the front, perhaps might have tossed back two or three pints, and would be ready to pay their ‘respects’ over and over… A good time to have by all! • 哈哈： NoseytheDuke 97. 令人担忧的是，尽管缺乏证据，但似乎有一半的人实际上愿意挑剔。 我对这对我们的司法系统意味着什么深感不安。 上周我正在观看一个名为“十三岁恐惧症”的节目，其中一个陪审团在一个案件中被判无罪，另一个陪审团在没有任何支持指控的情况下被定罪。 这对于我们系统的一个重要基石来说意味着什么，这简直是一件不小的事情。 尽管对整个系统的信任受到侵蚀，但该国仍然相信他们声称不信任的机构的话，并且这样做没有任何审查。 98. Ive thought about it many times. With Trump repeatedly proving himself a wild card, and thus unreliable, 9/11 could be one of the things they’re most worried about. Helping to explain why George Bush, Dick Cheney and the rest of that crowd, hate his guts. 99. Sbaker 说： You have no concept of a charade being perpetrated on the American people. You don’t find it a little strange that Trump keeps hiring the Deep State denizens he purports to be fighting? You are inca pable of detecting the friend/foe, psychological tactic used to deceive? Gee, that must be why he has fired so many of them, or have you avoided that observation. • 回复： 100. beau 说： re: brennan – the only thing i want to read about that muslim communist is his jail sentence and his obit. 101. Carroll Price testified: “Helping to explain why George Bush, Dick Cheney and the rest of that crowd, hate his (Trumpstein’s) guts.” Oh, so you know such is the case, Carroll? Hm.🤔 Maybe Team Trumpstein can use evidence of that “crowd’s” hatred to help him during this difficult All Zio impeachment process. Excuse me, but haha. Thanks. 102. Sbaker 说： Really, how many of those first responders have you talked to? People who train first responders know many of them and not one from NYC has ever expressed what you claim. Anonymous sources remain anonymous for good reason. Anonymous proof is not proof and is not evidence in support of preposterous conspiracies. Provide some names of those people mr. Hearsay. 103. A123 说： There are so many investigations going, hopefully some of the authoritarian left will have to stand public trial. https://www.zerohedge.com/political/horowitz-report-will-be-damning-criminal-referrals-likely Given how ‘loyal’ anti-Christian Globalists are, this could be an entertaining span of years as they flip on each other to save themselves. The biggest risk to Christian Populism is the 2024 election after Trump term limits out. Having anti-Christian Globalism on trial will pay huge dividends. Keeping the submissive GOP establishment on the defensive will protect the movement. They will not be able to force thru a Globalist compliant candidate to follow in the footsteps of Romney & McCain. 和平😇 104. Jeez, funny you mention Flight 93. I wonder whether we will ever find out what really happened to Flight 93, and its passengers. 105. Let’s try it again! Aesop’s tale (fable). 106. geokat62 说： 107. I don’t find Kariakou’s term useful because a lot of people in the federal bureaucracy are not in the Deep State. In fact, I’d say that the Deep State is actually a very few people, whereas the federal bureaucracy encompasses millions. So the term is very imprecise, which creates confusion. 108. geokat62 说： ADL stands with 3rd world Dreamers in America, but opposes sub-Saharan African Dreamers in the Apartheid State! • 回复： 109. MarkinLA 说： The judicial system is already a stinking pile of crap and it has more to do with Roberts than Ginzberg. When clowns like Roberts base their decisions on their personal feelings about a law and not what the statutes say then the judiciary is more powerful that the legislature or the executive. 110. Apeon 说： Your comment reveals that you do not understand the subleties, and twistings of international power struggles, which is why you are not Pr. Pr. Trump learned it in NYC doing his business, dealing with the NYC politicians, the party bosses, the local bosses, the unions, the PD, FD, and the mob. Very few can navigate that jungle without getting knocked off. DJT did so successfully, and is fully qualified to be President, and DTS 111. Any military action by president Trump is warming up the hearts of deep state. Even also military build up of forces, But this is cautionarymove, and it is only saving the face at best, US did stop in Syria being commanding entity, and now it did become only opportunity waiting entity. Eventually the case of Syria can be resolved only by UN. • 不同意： ChuckOrloski • 回复： 112. 拉屎！ You are right. This damn T9 113. anon[117]• 免责声明 说： Former national security adviser and noted war hawk John Bolton told a private gathering of Morgan Stanley’s largest hedge fund clients that President Trump’s approach is not in alignment with any of his key advisers, according to NBC News, citing six attendees. Bolton, who is currently working on a book, suggested that Trump’s “win-or-lose” negotiation style that works in real estate is bad for foreign policy, and that the president doesn’t understand the interconnected nature of geopolitical relationships. Where Bolton does agree with Trump is their mutual stance against China on trade, according to the report, however the two have had far more disagreements over Iran, North Korea, Syria and Ukraine. https://www.zerohedge.com/political/john-bolton-trashes-trump-during-private-speech-says-real-estate-tactics-hurting-foreign Deep state allows Bolton to bring his agenda and discuss his frustration aired on NBC and expressed to Morgan Stanley What is the difference between him and well known dictators from Sudan Nigeria NKoera or Liberia or Pakistan ? . • 回复： , 114. 嗯嗯！ Then, we run to some serious discrimination issues. Why it is OK for some-one to be anti-American, anti-Canadian, anti-German and so on, but it is NOT OK to be anti-jew. Cause Semites are millions of Arabs as well. Including the occupied Palestinians. 115. Realist 说： Gee, that must be why he has fired so many of them, or have you avoided that observation. How god damn stupid would have to be, to not know the reputation of the assholes he hired. • 回复： 116. MarkinLA 说： Isn’t it Bolton’s inw-or-lose (my way of the highway) stance on Iran that was part of derailing anything related to Iran or his and Pompeo’s similar stance on North Korea that shitcanned that effort? 117. “The judicial system is already a stinking pile of crap and it has more to do with Roberts than Ginzberg.” I am going to hold fire here a tad. But I will say, as far as I am concerned, Justice Ginzberg’s comments regarding the passing of Justice Scalia has a left a very very sour taste in my being. her attitudes are reflective of Republicans and conservative in general in my view but that she would air such a comment and is now being celebrated — it left me heartsick for what is happening on that bench. The history of personal views of justices is not new. 118. anon[113]• 免责声明 说： Who’s your “well known dictator” in Pakistan? 119. If Trump has to stand “trial” before the Senate, it’ll be interesting to see if his lawyers subpoena Biden and his former boss, Obongo, to testify. Watch the fur fly if/when that happens. 120. Excellent analysis, Phil. May it be spread far and wide. 121. Wally 说： Yeah boy, those Russians are everywhere. Watch out! The Steele Dossier is all real. LOL And of course you ignored his accomplishments. Hurts don’t it? • 回复： 122. oops onebornfree, you forgot to include the US Military – all of them funded by direct theft as you put it. 军队 the airforce the navy all private military contractors close them down Build new bridges and rail and hospitals and schools and universities. End every tax concession to every industry as they just suck off the teat of the workers taxes. 修复了它！ 123. 多么冗长的一篇文章，典型的右翼福克斯新闻在这里编排了低智商旅鼠。 PCR在他过去的几篇文章中也暴露了他的真面目。 If Trump is in a fight with the CIA he should probably stop stuffing their pockets with cash. He should probably stop doing their bidding by overthrowing yet another govt. in Bolivia. Jesus fucking Christ how stupid do people have to be to believe this ridiculous shit. If he were a threat to the CIA or any “deep state” his brains would have been blown out by now, or imprisoned. This Q-anon BS is truly so fucking stupid that only boomers who get their news fro TV could believe it. Don’t want to hear nothing else about “invasions” white genocide or replacement now that Trump regime has allowed CIA and Wall St to overthrow another country in Latin America. Don’t expect your border to be respected when you respect no ones borders. Don’t expect anyone to give a shit about your culture or heritage when your hero Trump doesn’t give a shit about the Bolivians. No excuses for Democrats either, they’ve been pretty much silent on the coup as well, just as they were when obama regime allowed it so many times. In fact at this point we may as well stop with the democracy charade and admit that this country is ruled by the CIA, a military dictatorship. Who runs the CIA? Fascists, Wall St oligarchs, the big investment banks. They own the fucking media, they own both fucking parties (uniparty), they own the fucking military, they own the fucking police, they fucking own you and me, they own GOD, or what most brainwashed idiots believe to be. Zionists fascists, CIA, the Mafia, the banksters, whatever the fuck you wantt to call them, all the same demons. Satanic terrorists one and all… Its all a distraction folks, make you believe you have a choice, you DON’T. Haven’t since 1963 when they blew JFKs brains out, certainly haven’t since 9/11, that was the fucking end. Carlin was right, Germany lost WW2, the fascists won. So sick of it, everyone should really say a prayer for the Bolivians, theres some really evil people being put in charge down there. The kind of demons trained here at the School of the Americas who have run the death squads so many times before down there. 124. ADL stands with 3rd world Dreamers in America, but opposes sub-Saharan African Dreamers in the Apartheid State! So, Webster’s new spelling of hypocrite is ADL? • 同意： geokat62 125. I agree, except Trump is not controlled opposition. The war between Trump and his enemies is very real, but it is just a gang war between two different factions of the Deep State. They war with each other, but at the end of the day they are all still mafia, and that is why no one will be prosecuted for trying to overthrow Trump. However the end result of this situation is that Trump is effectively the same thing as controlled opposition, even though he is technically not controlled opposition. I am also not sure that Trump has always been Deep State. He might have actually intended to drain the swamp and MAGA early on in his candidacy, but somewhere along the way he became controlled by the Deep State, either willingly or as a result of threats or blackmail. But at any rate he is definitely Deep State now. 126. bjondo 说： Israeli forces kill another young Palestinian Rites of passage: circumcision, bar mitzvah, murder. Jew becomes Israeli. 5ds • 回复： 127. getaclue 说： Your statements fly in the face of the evidence– the “Establishment” “Elites” “Deep State” or whatever term to describe these haters of our country/traitors– Globalist NWO agenda proponents– have been trying to destroy Trump from day one, to say he is “one of them” is just stupid– or you’re a disinformationist/propagandist for pay– which is also quite likely these days sadly…. • 回复： 128. getaclue 说： Your statements in total are bs but as to first responders, they’ve been threatened against telling the truth and the Mainslime Media has enforced silence but the truth is coming out now– : “The Franklin Square and Munson Fire District 9/11 Resolution Below is the full resolution from the district. I’d like to highlight in bold this statement: “Whereas, the overwhelming evidence presented in said petition demonstrates beyond any doubt that pre-planted explosives and/or incendiaries — not just airplanes and the ensuing fires — caused the destruction of the three World Trade Center buildings, killing the vast majority of the victims who perished that day.” Whereas, the attacks of September 11, 2001, are inextricably and forever tied to the Franklin Square and Munson Fire Department; 鉴于 11 年 2001 月 290 日，在纽约市的世界贸易中心工作时，纽约富兰克林广场和芒森消防局 Hook and Ladder Company #1 徽章 #2,976 的消防员 Thomas J. Hetzel 在与其他 XNUMX 名应急响应人员和平民一起履行职责； 鉴于富兰克林广场和芒森消防局的成员被要求协助随后的救援和恢复行动以及对世界贸易中心场地的清理，他们中的许多人由于吸入了致命的毒素而患上了危及生命的疾病在现场; 鉴于，富兰克林广场和芒森消防区的消防专员委员会认识到，11 月 2001 日纽约南区美国检察官在世界贸易中心报告未起诉的联邦罪行的请愿书具有重大和令人信服的性质, 18 年，并呼吁美国检察官根据美国宪法和 3332 USC SS XNUMX(A) 将该请愿书提交给特别大陪审团； 鉴于上述请愿书中提供的压倒性证据毫无疑问地表明，预先埋设的爆炸物和/或燃烧弹——不仅仅是飞机和随后发生的火灾——造成了世贸中心三座建筑的毁坏，造成了绝大多数遇难者死亡。那天死了； 鉴于 9/11 的受害者、他们的家人、纽约市人民和我们的国家应该对与 11 年 2001 月 XNUMX 日袭击有关的每起罪行进行最充分的调查，并且每个应负责任的人都面临正义; 因此，现在决定，富兰克林广场和芒森消防区的消防专员委员会完全支持联邦大陪审团对与 11 年 2001 月 XNUMX 日袭击有关的所有罪行进行全面调查和起诉，以及任何和所有努力由其他政府实体调查并揭露围绕那个可怕日子的事件的全部真相。 https://www.lewrockwell.com/2019/07/no_author/new-york-fire-commissioners-call-for-new-9-11-investigation-about-pre-planted-explosives/ • 回复： 129. The country is going down the proverbial tube, and pronto! Yep! And there will be no break in the fall. 130. The mockingbird media has been weaponized against whitey — both people and culture. 那就对了。 And the problem is made worse by the fact that, when it comes to racial self-preservation, Whites organize like blacks swim. In both cases they couldn’t do it if their lives depended on it. 131. anon[113]• 免责声明 说： Video: Israel attacks a CNN reporter and his crew at Gaza fence and prevented him from reporting on the latest Israeli attacks on Gaza. https://daysofpalestine.com/post/13087/video-israel-attacks-a-cnn-reporter-and-his-crew-at-gaza-fence YouTube – Israel attacks a CNN reporter and his crew at Gaza fence • 回复： 132. I agree, except I don’t think it was primarily Israel that pulled off 9/11. It was a combined effort of many factions of the Deep State, which included Israel and many neocons in this country, as well as many others. I have been paying attention to the recent situation with Giuliani and his Ukrainegate tribulations, because he was obviously involved up to his eyeballs in the planning and cover-up of 9/11. In their desperation to find evidence to impeach Trump, Trump’s enemies are furiously digging up everything they can on Giuliani. There is a small chance that they will dig up something on Giuliani’s involvement with 9/11 that will get past the gatekeepers into the mainstream news cycle, which would be interesting indeed. 133. NPleeze 说： What fight to the death? You mean the public theater? The Orange Man got his start in WWE, where he got into some colossal “fights” with Vince McMahon, and then in the Deep State program The Apprentice, where he got into colossal “fights” with his apprentices. Just like his colossal “fights” with his buddies now. Remember when the Orange Man berated the Clintons during the campaign? Pure theater, he even admitted it after the election, calling them “good people”. He’s been friends with them for decades. He’s been a NY liberal, in cahoots with the Deep State (from investment banking to the most giant media arms) for decades. He’s been surrounded by Jews for decades – not just his daughter and grandkids but his personal doctor, personal lawyer, foundation chair, finance chair, etc., etc., etc. You are willfully blind – that is the legal term for someone who knows the truth, as it is self-evident, but pretends not to see it. • 回复： , 134. NPleeze 说： to say he is “one of them” is just stupid What is stupid is to believe you are witnessing reality when you are watching a Hollywood movie. 135. new 说： this article was really helpful. makes sense now. if this guy would tie this all in with the chicago gang would be super helpful. Its been a long time since ive read anything i felt wasn’t designed to screw with me. this feels like real journalism. never known it in my lifetime. thank god. there is hope. 136. Realist 说： Yeah boy, those Russians are everywhere. Watch out! The Steele Dossier is all real. LOL And of course you ignored his accomplishments. Hurts don’t it? You seem to have a hard time understanding simple logic. I voted for Trump. If you had read 1/10 of my comments you would know I am not a believer in the Russia gate bullshit. Trump has had no accomplishments for 90% of Americans. The US is a hegemonic behemoth, I had hoped Trump would not be a part of that. If you contemplate replying to me again…at least know what you’re talking about. 137. Ahoy 说： @ChuckOrloski #97 The US world policeman took a sabbatical after Syria and Ucraine. Don’t think will be coming back. The last moves in Syria are dictated by the need to save some grace. General Milley told us that in the ME we did a lot of things that we regret. Hallooo!!! In Ucraine Putin told them the SE Ucraine must be autonomous, US, Merkel, UN said OK. For Crimea he told them will talk in the far out future. This is translated in plain english, case is closed shut up and keep digging. Johnson is taking England out of UE, Macron says NATO is “brain dead” and the Italians are itching to tell Brussels fuck you. What a nice picture of coherent Ziowest. The Rothschild/Rockeffeller oil empire had a serious car accident in Syria. The insurence company said the car can’t be fixed. It is a total loss. As always, all the best to an upright and far looking American. May be is not by accident that you are giving your share of the fight for the rebirth of America from the State of the Bell. God moves in mysterious ways. 138. Realist 说： How about an article on Julian Assange? 139. There is no doubt that Alexander Downer, former Australian Foreign Minister and High Commissioner [= Ambassador] to the UK passed on information from George Panopoulos about dirt the Russians might have on Hillary Clinton to Australia’s Department of Foreign Affairs and Trade as one would expect him to do, and that the information was then passed to an appropriate US body. Nothing surprising or inappropriate in that. But I am unaware of Australian cooperation wiyh the US Deep State going any further wrt to Trump and the election campaign. Can you, PG, enlighten your readers by reference to any evidence there may be about “collegiality” which may have led in effect to Australian assistance to Deep State impropriety? 140. geokat62 说： He’s been surrounded by Jews for decades – not just his daughter and grandkids but his personal doctor, personal lawyer, foundation chair, finance chair, etc., etc., etc. Here’s the proof to back it up: Trump’s Jewish Donors 视频链接 141. I disagree with your statement; Trump has not started any wars and is keeping the Feds in check. That does not sound so deep state friendly, plus he is taking down some of their operatives. You need to listen to the X22 report… 142. Sparkon 说： So in your mind is Vladimir Putin (b. 1952) a “boomer”? Anatoly Karlin has claimed the right to define “boomerism” any old way he chooses, i.e. making it up as he goes along. Let me announce here that I have for some time now considered it an “own goal,” and a proclamation of stupidity by anyone using “boomer” as a pejorative or epithet. It seems TPTB aka Deep State or Establishment, can always rely on ranks of useful idiots to help them divide and conquer, and poison the well in the process. We don’t need people like you in the 9/11 truth movement. JKF’s assassination was in 1963. All the Baby Boomers were in school that day when they killed our President. 143. getaclue 说： In your world Hillary Clinton threw the election to Trump? OK then…. And he is on board with 24/7 defamation by the Mainslime DNC Media of both him and his family? I never liked Trump before he got elected, nada, until I saw who his enemies are and he has all the right ones basically in Washington as he has slowly done what he can in a very, very bad environment where they have done all they can to hurt him short of direct violence (he isn’t Emperor and can’t just make Proclamations by the way, and the Admin State is very hard to overcome at this point, you can’t even fire most of them even for egregious behavior under the current system….) but your position is ludicrous. You’re delusional or a disinformationist. Who do you think would be doing things better than him at this point? Given everybody at issue he is as good as it gets — at least more people are waking up to the fact of the incredible corruption of the entire “system” — that alone is a very big thing at this point given the dire future at issue if the “Elites” get to pull off what they want with the NWO One World Government. • 同意： Sbaker • 回复： 144. (Zigh) My “Homeland” tax dollars at work. Please refer to report below? https://www.presstv.com/Detail/2019/11/12/611012/EU-court-Israel-settlements-Palestine Americans lost their homes in the Goldman Sachs housing scandal, and at the same time, Israel builds homes for Israelis with “Homelander” tax dollars. 😞 How asleep can Americans get while narcotized with the televised Zio impeachment “hearings”🙉 including a percentage who are resolute in thinking Trump wants to rule with America First principles? Thank you, (argh). • 回复： 145. 嗯，你说的有道理。 但我们必须等待他的第二个任期再看。 鼓励我以积极方式思考的是特朗普的性格。 他不仅是一张外卡，而且是一个自私的 SOB，他绝对希望自己的名字能够超越所有其他美国总统，比如将美国从犹太人和以色列手中解放出来的那一位。 可能是我太乐观了，但我们会拭目以待。 • 回复： 146. Sbaker 说： Your assessment is yours. You are not a DC insider and many of Trump’s firings were of obama holdovers. It’s a good thing we are not in a high school gym with you swearing at me like some teenager. I bet you are a real tough guy/gal. • 回复： 147. Sbaker 说： Nice, I get the lawyer speak and the incentive to make money from the massive murderous tragedy. Speculations from a writer at lewrockwell, are not necessarily consistent with the whole truth and nothing but the truth. For the record, speculation does not equal truth even if it comes from LEW’s ad selling site. Tell me about the personal contact you have had with those first responders. Tell me about the people you knew that went down on that day. 148. I will admit to some level of hat off to Ambassador Haley who spilled the beans about the intentions of her fellow co-workers in the administration — and disappointingly, that Sec Tillerson is among them is just sad. ------------ I echo the support for this article in doting the eyes of what seems to have been in my view a very obvious gambit. And I suspect its much worse. Though, I am not sure how deep the previous executive is in this. This really sounds like Sec Clinton and her supporters. 149. Sbaker 说： One Semite killing another–very unique story. • 回复： 150. Realist 说： You are not a DC insider and many of Trump’s firings were of obama holdovers. Like Bolton??? It’s a good thing we are not in a high school gym with you swearing at me like some teenager. I bet you are a real tough guy/gal. I did not swear at you…if you hadn’t spent so much time in gym you wight know how to read Is that a threat??/ Probably a good thing for you. • 回复： 151. Rurik 说： 93 号航班和五角大楼内的数千名目击者和遇难者的谈话。 ‘Hi, Mom, this is Mark Bingham!’ Have you seen the site where Flight 93 ‘crashed’ into the field at Shanksville? that long scar you see, was already there. They simply shot a missile into the middle of it as an impromptu ‘crash site’. Only there’s no plane, and no passengers, or wreckage of any kind. Just a crater from a missile, shot into an existing scar on the field. Apparently we’re supposed to believe that the jet and all its contents plunged into the ground so deep that there’s nothing left to see. Just as we’re supposed to believe so many other patent absurdities. Like all the ‘black boxes’ are all missing or the recorders are blank. And there’s no video of the jet hitting the Pentagon, even tho that’s likely the most surveilled real estate on the planet. They even sent FBI agents to confiscate all the surrounding gas station and convenience store and hotel video recordings. Why? One wonders… 事实上，大量证据表明整个官方叙述是一堆腐烂的谎言，可以填满沃伦委员会的大部头。 从七号楼开始，看看那栋楼是如何倒塌的，以及记者如何报道它的倒塌情况， before 它发生了。 To this day, the only explanation for building seven’s collapse is controlled demolition. If forensic scientists and structural engineers agree that it could not have collapsed due to office fires and ‘vibrations’, then how could 记者 知道它会跌倒之前吗？ 告诉你所有你需要知道的。 152. Show me 说： I have to agree. Just one person brought to justice would be a huge victory for justice and the rule of law. People are committing immense crimes, but nothing is being done. There is an IG report said to be coming out, and they have said it was going to come out in the next few days for months now, but it has never come out publicly. It would be better if they didn’t mention it at all. If there are crimes, it would just be better for law enforcement, DOJ and FBI to just go after them as they’re supposed to do, not wait indefinitely for some IG report. IG reports are important, but they are intended for internal use in the government, to see if they’re operating as intended, so that corrective action can be put in place. I don’t think the IG was ever intended to be a law enforcement mechanism and they don’t have prosecutor powers. So I’m really not expecting anything to come of all this. In fact, it seems to be going the other way, rather than the government actually prosecuting crimes, now they’re impeaching the President for what amounts to non-criminal acts, not crimes or even misdemeanors, but just being a populist rather than a functionary of the Democrat and/or Republican parties. 153. “联邦官僚机构”一词甚至与“深层国家”都不接近。 深层国家本质上是新世界秩序的较低层级，其中包括政府内外各种实体的精选成员，包括以下内容： 1. 许多政府 3 字母机构和内阁——中央情报局、联邦调查局、国家安全局、国务院、财政部等。 2. 一些国会议员和一些总统。 3. 大到不能倒的银行和美联储（它是私人的，不是政府的一部分）。 4、主流媒体的高层职位，尤其是纽约时报、华盛顿邮报、CNN等。 5. 以色列游说团体、摩萨德、许多新保守主义者等。 6. 军工安全综合体，包括国防部和各种国防/安全承包商。 7. 其他各种企业领导人，尤其包括一些硅谷领导人。 8. 各类智库、说客、总统顾问。 • 回复： , 154. 9/11 is THE KEY to Trump’s success for his presidency. 他进行调查，尝试绞死美国人的叛徒，不管他是不是，他都将成为世界英雄。 人类将是自由的。 155. Israel builds homes for Israelis with “Homelander” tax dollars. Those are shelters for the mentally circumcised…. you… you… anti-semantic, you!!! 156. Hi non-King David!🤗 Interesting findings, shall mull, thanks. A question. Do you contemplate a US Deep State “Lowerarchy”* which could plausibly resemble the o.o.b., Soviet Central Committee? I do. * C.S. Lewis, “Lowerarchy” from his book “The Screwtape Letters.” He passed away, same day as JFK. 157. Sorry Curmudgeon, 你是对的。 Bush 1 was such an arrogant [电子邮件保护] he wanted the power right away but failed. Having failed he had to bide his time. 158. I don’t think the UN will be able to “solve” Syria. And why should it? Syria is a sovereign nation and doesn’t need to have the UN tell it what to do. It is obvious that international law is being violated in Syria. Any resolutions with teeth will be neutralized by the USA and its allies. Look at the decades that Israel has gotten away with being an international rogue and scofflaw. Failure to make its resolutions stick just weakens the UN further. The UN needs a militia that can enforce its resolutions and also international law. But you and I know that this is not going to happen. First, the UN would have to be moved off US territory. To a state/spot with enough land and infrastructure so that the UN can train up and base its own professional militias. Even then the USA and the JSP (Jewish State in Palestine) will not accede to the UN. So, the situation in Syria will grind on until domestic politics in the states that are currently there illegally shift enough to mandate a pullout, or Syria-Russia-Iran-Hezbollah figure out how to winkl them out of there via a combo of military and nonmilitary carrots and sticks. 159. You believe Warren? 你完全疯了。 160. LBJ 没有这样的顾虑。 --------- 起初，LBJ 有着巨大的野心。 几年后，他很高兴离开了WH。 他在震惊和难以置信的状态下度过了他的退休生活，抽着大麻，努力不完全失去理智。 哭泣的破坏和释放战争之犬是有代价的。 161. Trump knows that Israel and traitors in the ZUS gov did 911 and Trump is part of the coverup. • 同意： ChuckOrloski 162. 随着特朗普（到目前为止）通过避免另一场战争同时似乎想要一场战争来战胜战争党，我非常希望他认识到世界的问题（犹太复国主义），并且为了保持活力，可能会玩一场“长期游戏”冒充内塔尼亚胡的傀儡。 与此同时，以色列更容易受到伊朗和叙利亚等新兴军事大国的影响，他们最终将变得足够强大，以唯一能够解决的方式解决世界问题。 163. LBJ 在目睹了他们对肯尼迪所做的事情后，他可能会害怕深州，因为他们拒绝接受命令，退出攻击古巴，并且未能扩大越南战争。 164. Rurik 说： 9/11 是关键…… 他进行调查，试图绞死叛徒 我羡慕你的乐观，Patrikios。 对特朗普来说，实现这一目标类似于调查和逮捕暗杀肯尼迪的其余叛徒。 特朗普甚至无法调查和逮捕联邦调查局中那些监视他的竞选活动并试图颠覆他的选举的叛国败类。 或者只是谁杀死了赛斯·里奇。 找出谁暗杀了里奇，会导致很多其他有趣的信息，但他甚至无法调查。 从我坐的地方看，布伦南、科米、斯特佐克、克拉珀、奥尔、麦凯布和其他许多人应该在一根漂亮的绳子末端跳舞。 如果， 他们实际上为党派政治议程滥用了各自职位的权力，同时试图掩护（如果不直接参与）针对这个国家的民选总统的政变。 那是叛国罪，为此，他们肯定应该被绞死。 如果， 正如我所说，他们有罪。 然而，他们在 MSNBC 和 CNN 上写书并被（（媒体））视为英雄受害者。 试想一下，如果特朗普说他想将下令袭击自由号的凶残懦夫从以色列引渡到美国接受审判。 我们整个社会都会变得松懈。 在距离地球六千光年的蟹状星云上，媒体和政界妓女的尖叫声将被听到。 耳塞值得用黄金来衡量。 它们将成为地球上最受追捧的商品。 这仅仅是因为想要将一些恶毒、凶残的罪犯绳之以法，他们对我们的国家犯下了臭名昭著的懦弱战争行为。 米特和马可为了抗议会自焚 将是，这只是对以色列的任何指责今天在 A 的好 ol'Z-US 中引起的歇斯底里的一小部分。 暗杀我们的总统，对我们的水手犯下战争罪行和暴行，在 9/11 上对成千上万的美国人进行可怕的屠杀…… ..都只是课程的标准，对于我们（（（最好的小伙伴）））在整个广阔的世界中。 任何关于实际调查的谈话？！？！！！ 会导致人类从未想象过的尖叫声，更不用说幸存下来了。 我想特朗普能胜任这项任务，但我怀疑任何凡人都能在伴随 9/11 实际调查的那种尖叫中幸存下来。 不仅如此，像朱利安尼这样与特朗普非常亲近的人也参与了当天的进攻。 多么令人作呕的讽刺，整个人格都基于 9/11 事件的男人，作为那天的某种英雄，正是帮助实现这一目标的凶残罪犯之一。 我想知道当朱利安尼靠近一些纽约消防员时，他需要什么样的安全保障。 • 同意： Carroll Price 165. anon[113]• 免责声明 说： 考虑到每个有争议的人，他都很好——至少有更多的人意识到整个“系统”令人难以置信的腐败这一事实——在这一点上，仅这一点就是一件非常大的事情...... 敏锐的观察力。 此外，“考虑到每个有问题的人”，考虑到每个有问题的问题，图尔西·加巴德 2020 是最好的。 Kim Iversen：为什么我支持 Tulsi Gabbard 166. anon[201]• 免责声明 说： 你和你的“小子”geokat，完全沉浸在物质中，无论如何都不知道有什么区别，是吗？ 从摇篮到坟墓。 167. anon[113]• 免责声明 说： “联邦官僚机构”一词甚至与“深层国家”都不接近。 正确的。 （这是典型的约翰·基里亚库，尽管他经历了一切，但略带怀疑，不诚实。） 您的 8 点定义与凯文·希普 (Kevin Shipp) 的解释一致，他非常详细地区分了深州政府和影子政府，以及两者之间的关系。 他的演讲可以在网上找到，通常长达一个多小时。 这是一个 2 分钟的剪辑，它给出了味道，并提供了完整演讲的链接。 我已经有一段时间没有听希普了。 一个警告：他谈到中央情报局，好像它是同质的。 对于 2003 年的伊拉克战争，有 CIA 官员感到愤怒（实际上是愤怒和痛苦），CIA 局长没有正确表达他们发现伊拉克没有大规模杀伤性武器的证据，因此没有战争的理由。 中央情报局举报人凯文希普揭露影子政府的黑人行动 • 回复： , 168. renfro 说： 如果有人正在观看民主党的特朗普-乌克兰弹劾听证会和证词，那么他们就会知道“袋鼠法庭”的含义。 我不是特朗普的粉丝，但这是民主党发起的明显政变。 所有的证词都是基于 有人从某人那里听到了什么...... 民主党不会允许发起这一切的“告密者”作证或被点名。 这就像一个人被指控并因殴打而受审，但声称被殴打的人是保密的，不会出庭。 无论特朗普做什么或不做什么，都违反了总统行为的完整性，损害了美国的信誉，这不会比在美国国会进行的让全世界看到的这个字面上的袋鼠法庭更糟糕。 那些作证反对特朗普的人一遍又一遍地说......“乌克兰对美国的国家安全很重要”。 这与我们对以色列的看法相同。 和a 额外问题 问他们是......“乌克兰干预美国大选了吗？” 他们都回答......'不' 在他们回答“不”之后，民主党律师问他们......“谁干涉了美国选举？” 他们都回答……“俄罗斯”。 把他们都杀了，让上帝把他们解决……这是我的政治立场。 • 回复： 169. 除了它的纠结和时间，现在粗糙的痕迹之外，还有什么可以在戏剧之上开始的地方吗？ 意识本身是我们不能不……通过它以及超越它的所有生命和物理系统感知我们人类形象的镜头。 我在这个论坛上用这些神秘的句子自我介绍，部分是为了问一个问题，但我的目的是插入另一个观点和核心思想，许多人在他们的额叶舌尖上有这样的想法：我们害怕失去什么，失去什么我们可以保留还是应该保留？ 如果我能以高尚的借口下降片刻……并建议一系列不同的事件。 美国宪法有很多东西，但从广义上看，创始人们同意以下两点： 1. 自由是从宇宙事实和人类意识事实的奥秘开始的更深层次过程的一部分， 2. 人类合作，比其他人类更亲近他们的朋友、部落和家庭，这会导致政府腐败，任何希望通过实现创造性忍耐而长期坚持下去的政府的轮廓必须释放个人创造性反应。 因此，为了实现这些目标，该政府的结构必须是建立在一定程度上相互冲突的机构，并以此来制衡彼此的权力，然后它们在理论上可能是可渗透的，但至少是一个循环的大坝政府以外的腐败圈子。 但是：这一切都是在基督教道德基础的背景下进行的，我在神学之外将其解释为：说实话，所有人都是我们的兄弟姐妹。 后者，道德一致性，是至关重要的，一些创始人直接评论了这句话，如果我们的道德假设不成立，我们的共和国就会分裂和垮台。 这是对功能的观察，而不是对任何可能对神明的判断的恐惧：随着时间的推移，基督教的道德框架可以为任何事情带来真相，而有道德的人将通过选举成为最终的法官。 但遗憾的是，每个想法都被物理现实、权力的变化潮流、偏见的揭示以及所有这些重要的技术变革所修改。 对基督教道德的无情无情攻击始于 20 世纪，正如许多人所观察到的长期以来令人恐惧的联邦储备系统叛国行为，在圣诞节前夕几乎秘密投票，当时大多数参议院和国会都在家中。在很短的时间内，美国国税局的成立是为了通过联邦税收为私人银行卡特尔系统提供资金。 这一切都是由基于情感的宣传新技术发挥作用的，即精神控制……以推动美国公众舆论，或者至少，对这种观点的看法……进入第一次世界大战，而这种力量的第一个行动是资助第一次世界大战，双方。 （请阅读道格拉斯·里德（Douglas Reed）的书，这是 20 世纪最重要和最勇敢的书之一，“锡安之争”。） 经济周期与腐败周期步调一致，美国势不可挡的经济和体系发展极其迅速，人们认为最终可能会吞并世界，因此是一个巨大的权力杠杆，可以想象相对地决定世界的命运。至少在一段时间内隔离其控制器选择。 在我们剩下的基督教道德结构和可悲的轻信中，美国人和美国是一个软目标，因为西方的旅行癖和财富获取让位于财富保护，阶级意识成为一种有害的习惯，因为革命精神和移民国家来之不易的自由和财富消退和取而代之的是历史上普遍的社会竞争。 我们今天当然看到这一点，因为现在第 3 代和第 4 代幽灵通过原始国家安全法案的许多附录传递了美国宪法中排除的精英品味和特权。 但是，经济周期本身就是......复苏和创造性反应的面孔，这些最终无法利用，在狭窄的历史时刻短时间受到干扰，甚至在疯狂的蒸发行动中保持一致，正如我们今天看到的那样互联网的审查和可怕的社会信用评分，但最终这个创意流是宇宙本身……与人类力量的虚荣相比，是浅薄的废话。 试图将手指伸入创造力堤坝的项目（我们在这里微笑）是如此愚蠢，以至于只有古代患有严重精神病的自恋者和他们轻量级的 Goyim 情报前线才能认为这是可能的。 在一些核心假设中密切相关的两个来源是美国和西方道德观念衰落的工具： 1. 塔木德和锡安作为一个世界网络，以宗派形式“马克思主义”和相关团体的部落种族为基础的反应。 2. 更深刻、更“外邦人”的进化论选择性达尔文批判……今天被称为“进化生物学”，它淹没了每一个高等机构的假设，在每一个逻辑序列中都可以听到其微妙而明显的嘲讽语气……每个情报机构及其战线的官僚主义，每个技术官僚的波形，今天是每个专业协会的舞台声音，当然，科学本身的声音是无可指责的。 可悲和最关键的是，所有这些都是美国课程的源泉，我怀疑，所有欧洲学校“系统”，除其他外，今天将变装皇后的变态带入学前班及以后，使无辜的成长中的大脑/思想敏感一个扭曲的议程，今天以欢乐和完全开放的态度攻击所有以生物学的基本事实为基础的道德结构。 当然，后现代思想将所有道德贬低为只是伪装的权力，并说权力是包括道德在内的所有文化的内在存在理由，这当然是进化生物学的确切形象。 取代道德的是彻底的平等，因为按照这种逻辑，所有系统都是权力和选择性力量的面孔。 所以我们要生活在一个没有道德的世界。 亲眼目睹索罗斯资助的马克思主义对城市的攻击实际上是一个更深的浪潮，DAs 拒绝资助和起诉这些政策的受害者的“小罪”，即盗窃、吸毒和公共空间占用，以迫使大规模的社会程式。 一个巧妙的狡猾伎俩，但是…… 如果一个社会允许它的孩子被宣传为原始的生物学变态，目的是破坏家庭第一个政治单位……这是通过 100 多年字面上的“狂热者”的行为，一小部分人以及这种努力的精神后代那么这次毫无造作地投票给原始的马克思主义积极分子并为这一革命行动欢呼，那么人们至少必须接受一个优越的意志的想法，这也是进化的产物，最后的警告和武装的呼吁？ Moloch 有一个新的烤箱口，它是当地公立学校的前门，可悲的是，许多私立学校和几乎所有“高等教育”席位。 人类不可能存在……没有道德。 道德实际上是作为工具的物理系统，将人类提升到至少一个全意识阶段。 它不能被抛在后面甚至变得迟钝，只能暂时从我们的视线中消失。 亚里士多德处理了 Telos 或宇宙的目的驱动循环的想法，这些循环包含创造性意识，可以在人类及其社会系统中观察到。 每一个谎言我们都在某处深深地知道......揭示了黑暗事物的面貌，由此，光流出来。 对我们自己社会的成就赋予独特而深刻的重要性是如此容易，甚至几乎不可抗拒，但这是否是一种近视的错觉？ 在美国大众思想宣传及其技术的泡沫中，现在变化如此迅速，包括今天的深层神经学方面，我们失去了方向和这个事实： 美国只是这个星球上人类行动创造力的一股。 我像这里的大多数人一样爱她，即使她扭曲成丑陋的东西，但最终保留的东西将远远超过……而不是琐碎的种族仇恨。 情报机构的面貌可能只不过是一种症状，但它作为一个隐藏的丑陋怪物的启示......是关于民主的弱点和技术官僚主义作为旧时代转移腐败背信弃义的一个深刻的历史教训，因此集体面前是一个悬崖： 我们要么背弃马克思主义的塔木德旧约计划及其 19 世纪英语人民统治世界的梦想，要么我们作为一个国家死去，共和国的分裂可能比悲观主义者想象的更近。 自欺欺人的重量压垮了我们的勇气。 二战的宣传现在几乎是宗教教条，因此没有任何东西可以生存，所以危机是一种福气？ 这有利于美国再次看清，最后一次机会？ 这是我们都参与的伟大创意流以及人类如何学习 看看，对真理的半意识感如何总是带来全意识。 创造本身将吞没并淹没浅滩。 有信仰。 • 回复： , 170. 特朗普宣布破产还是你在撒谎？ 我知道他控制的许多公司都有，但这不是一回事，是吗？ 171. renfro 说： 那么除了国家安全委员会和国务院的乌克兰犹太裔和俄罗斯难民的乌克兰阴谋……以及国会和希拉里的反俄罗斯民主党……还有什么在推动美国提升乌克兰对亲乌克兰 - 反俄罗斯阴谋集团的重要性？ 乌克兰的新犹太喜剧演员普雷兹泽连斯基被政界人士誉为反腐败救星，不过……看着犹太人和全球秃鹰吞噬乌克兰的农田。 这等于中国买下了美国所有的农田，然后运到中国去多少？... 乌克兰的农产品有多少留在乌克兰，又有多少出口到乌克兰？ 乌克兰人可能会陷入另一场高价半饥饿的境地，或者成为新外国土地所有者政治的人质。 泽伦斯基要求迅速结束乌克兰土地销售禁令 https://www.ft.com > 内容 2 年 2019 月 XNUMX 日 - 乌克兰总统已命令其新政府在 XNUMX 月之前起草立法，推翻禁止出售该国农田的禁令…… 乌克兰出售农田计划引发外国人担忧 ... https://www.nytimes.com › 世界 › 欧洲 › bc-eu-ukraine-land-for-sale 31 年 2019 月 XNUMX 日 – 乌克兰大部分肥沃的农田被分割成大约...... • 同意： Robjil • 回复： , 172. LBJ 是犹太复国主义者的傀儡，并参与了对自由号航空母舰的袭击，为了证明这一点，请阅读琼梅伦的《水中之血》和菲利普·纳尔逊的《记住自由》。 LBJ 应该为他为以色列所做的一切而下地狱。 173. anon[113]• 免责声明 说： 我所指的分析师 [疯狂而痛苦地] 是保罗·皮拉尔 (Paul Pillar)，他甚至写了一本关于它的书。 我相信这是一个： “情报与美国外交政策：伊拉克、9/11 和被误导的改革” 174. “谁干涉了美国大选？” 他们都回答……“俄罗斯”。 _那_ 正是我声称 LSD 在 DC 的供水中的原因。 智商高于室温的人都不会认真对待这种说法。 175. 糟糕，我的前任应该是针对培根的。 176. geokat62 说： 我想特朗普能胜任这项任务...... “我是铁杆犹太复国主义者，特朗普总统也是！” — 罗杰·斯通 • 同意： ChuckOrloski • 回复： , 177. 在肯尼亚人巴里取代小布什之后不久，我四处寻找如果/当孟山都到期时会受益的公司。 它的杀气腐败的深度越来越明显，它的时代似乎越来越近了。 看着 Global Ag 中的大玩家，不可避免地会导致乌克兰（当时被称为），我有点惊讶地发现美国/乌克兰商会的成员之一是 Dick Cheyney。 我想那个池子里的鱼太大了。 然而，我越看越明显，这个国家是一个巨大的腐败洗钱坑。 Little Vicky's Fuck the EU 政变并不令人意外，目前的这批爆料也不令人意外。 某处有一本 Saker 的书。 178. anon[113]• 免责声明 说： 乌克兰出售农田的计划引发了外国人的恐惧…… https://www.nytimes.com › 世界 › 欧洲 › bc-eu-ukraine-land-for-sale 31 年 2019 月 XNUMX 日 - 乌克兰大部分肥沃的农田被划分为大约……也反对允许一般土地出售，这种观点广泛 重要的分享，TY。 她是谁？ （纽约时报现在在付费墙后面。） 今天及时发布： 伊戈尔·洛帕托诺克/奥利弗·斯通电影， http://www.revealingukraine.com/ 今天采访导演_ “揭示乌克兰”在亚马逊 Prime 上发布，@Lopatonok • 回复： 179. chris 说： 这是一篇文章的重磅炸弹，它解释了深层政府和民主党急于弹劾作为防止他们的叛国行为暴露的预防策略。 我不知道，但这些笨拙的举动似乎对他们来说有点反应过度； 在完全控制媒体的情况下，即使完全曝光也不会对美国人口产生重大影响（……当然，直到有一天它会发生）。 但是，深州的害羞者也不会轻易倒下，再一次，轮到我们找出我们一直赋予他人的这些颜色革命看起来像什么近距离和个人。 然而，帝国也即将得到自己的一剂良药，因为无论弹劾审判如何进行，通过颜色革命传播“民主”意识形态的时代即将结束。 面具终于掉了下来，露出了一个毒害整个政府的蛇坑。 而在这次曝光之后，帝国工具箱中唯一剩下的工具将是武力，它的销量远不及被意识形态迷惑的受害者的善意。 180. geokat62 说： “我是铁杆犹太复国主义者，特朗普总统也是！” — 罗杰·斯通 支持罗杰披露的 20 秒剪辑： 特朗普在纽约的犹太筹款活动中获得全场起立鼓掌 描述： 美国优先？ 几乎不。 • 同意： Desert Fox • 回复： 181. Rurik 说： 我想特朗普能胜任这项任务...... “我是铁杆犹太复国主义者，特朗普总统也是！” — 罗杰·斯通 Geo，我确实说过'我会 喜欢 认为'......这与'我认为'不同。 他当然是犹太复国主义者。 任何人都不得在塔木德一英里范围内使用权力杠杆，除非他们是犹太洁食者。 我们都知道。 所以问题不在于他是不是犹太复国主义者，问题在于他愿意在多大程度上出卖他领导的国家，以取悦超级犹太复国主义者。 到目前为止，没有与伊朗发生战争。 或委内瑞拉或黎巴嫩或任何其他超级犹太复国主义败类希望看到我们摧毁的无辜国家。 在某些方面，我也是一名犹太复国主义者。 我不认为应该驱逐以色列/巴勒斯坦的所有犹太人。 我认为有些现实无法完全消除。 所以如果你问我，‘以色列应该不复存在吗？ 我不得不说，恕我直言.. '不'。 但我会坚持，（如果我可以的话），所有巴勒斯坦人的损失都得到补偿，他们的回归权得到承认，他们至少有权获得与谋杀他们并继续下去的犹太至上主义者相同的保护和合法权利。窃取他们的土地。 我会让以色列回到 1948 年的边界，并要求他们平等对待自己土地上的每个人。 但是通过允许他们甚至保留 1948 年的以色列边界，我仍然会被认为是犹太复国主义者，如果你对此嗤之以鼻。 • 回复： 182. Cleburne 说： 我以为他指的是林肯对南方的战争。 • 同意： Carroll Price • 回复： 183. geokat62 说： 所以如果你问我，‘以色列应该不复存在吗？ 我不得不说，恕我直言.. '不'。 嘿，留里克。 如果你问我，我不得不说，鉴于巴勒斯坦被占领土上的所有实地事实，两国解决方案实际上已经死了。 鉴于这一现实，我支持一国解决方案，即犹太人和阿拉伯人并肩生活在圣地。 没有犹太国家，没有穆斯林国家，只有所有公民的国家。 184. 这里没有人在乎玻利维亚，当然也不希望美国政府卷入其中。 您是否只是在您能找到的每个“右翼”网站上复制粘贴相同的毫无意义的咆哮？ lt • 回复： 185. 人们需要对 9/11 事件的真实真相有很大程度的认识和理解，远远超出临界点，也需要有凝聚力的反对。 国家的深层反应将包括“恐怖袭击”、停电和停水、火灾、食品和燃料分配瓶颈、街头大规模抗议以及警察和军队的严厉镇压。 186. 实际上，评论是特朗普并不关心玻利维亚人，就像人民一样。 许多人对玻利维亚本身感兴趣，因为它是一个非常丰富的矿产资源来源，其中许多矿产具有重要的战略意义。 187. 如果我能从高处下来片刻…… 我想你的意思是“......从我的崇高借口......看来你不能。 188. renfro 说： 太棒了......我必须看到......谢谢anon。 189. Paul C. 说： 让我们不要忘记他正在推出以反犹太主义为幌子剥夺我们言论自由的法律。 非常MAGA。 除了谈话之外，他在限制非法移民方面做了零。 我以为他要资助庇护城市？ 我猜他忘记了。 他让美国卷入了叙利亚的非法战争，就像其他所有傀儡总统所做的那样。 与他作为候选人的谈话相反，他没有触及 9/11 的真相。 每个城市、县、城镇和州都有联邦资助的“气候行动计划”，因此离开巴黎协议只是烟幕弹。 Here’s who Trump serves, raising \40M: https://www.breakingisraelnews.com/139915/trump-anything-happens-here-im-taking-trip-over-israel-prime-minister/
特朗普是公认的共济会成员。 共济会是一个秘密社团（他们撒谎），他们在那里宣誓。 他们崇拜路西法，因此不是基督徒（尤其是在学位链上的更高）。
他是游戏中的棋子。 我不乐意这么说，我希望这不是真的。 我们都需要了解真相才能恢复国家。 随着美国越来越多地被洗劫一空，每天发生的事情是烟雾缭绕。
我们正在进行天气战，因为政府正在从事地球工程，喷洒天空来决定事件的结果。 特朗普并没有阻止它。 他也没有解决 2012 年允许政府宣传美国公众的《史密斯-蒙特法案》的变化，这在之前是非法的。 谎言，谎言和更多的谎言每天。 习惯它。 现在是弹劾大戏。 This will make his supporters forget all about his broken promises and will be gleeful when he's re-elected to continue doing the bidding of our Central Bank masters.
我是否忘记提到我们仍在推翻世界各地的政府，我想到了南美洲。 特朗普领导下的中央情报局没有任何改变。
• 同意： ChuckOrloski
• 回复： ,
190. renfro 说：
然而，我越看越明显，这个国家是一个巨大的腐败洗钱坑。
同意。 它被腐败破坏得如此之大，它现在唯一的好处就是作为美国雇来的枪来捅俄罗斯。
191. 谢谢（你的）信息。 我以前听说过凯文·希普，但对他一无所知。 我刚刚看了他的一个视频（下面的链接），非常棒，所以我订阅了他的 YouTube 频道。
• 回复：
192. 从 Pollyanna 土地上权衡：就调查 Seth Rich 谋杀案而言，这应该是阿桑奇被带到美国的目的。 在发现之下，他放弃了作为线人的里奇； 美国人民将点点滴滴联系起来，并初步了解深州的罪恶。 阿桑奇让美国成为人类著名的英雄。
无论如何，这就是理论。 诚然，矛盾没有比“美国人”和“连接点”更明显。 Zio 媒体不知疲倦地工作以确保他们不会这样做。
• 回复：
193. chris 说：
是的，是特朗普第一，以色列第二，美国第三； 这就是为什么他们讨厌他并且不信任他。
他应该是第二，而不是第一，就像所有其他工具一样。
194. Ahoy 说：
@沙漠之狐#177
“LBJ 应该为他为以色列所做的一切而在地狱中煎熬”
我还要补充一些美国和欧洲的其他知名人士。 他们都玩了一个险恶的游戏，即除了犹太人之外，其他人都是压迫者。 为自己保留了未来伟大解放者的角色。
不过美国人正在觉醒。 这可能看起来是一个非常缓慢的过程，但在历史的进程中，你所需要的只是改变方向的火花。
这家伙说了一些话。 水已经以小溪的形式流动。 它会变成一条河流。
• 同意： Desert Fox
195. JimDandy 说：
是的，比尔·克里斯托、马克斯·布特、詹妮弗·鲁宾等人一直在假装特朗普引发的神经衰弱。 这都是他们精心设计的 12 维国际象棋游戏的一部分，目的是欺骗我们并掩饰他们对特朗普的支持。
哥奇亚。
196. JimDandy 说：
闭嘴，布伦南。 我们都知道是你。
197. ” 也许吉拉尔迪先生有充分的理由对约翰逊先生的未透露姓名的“知识渊博的消息来源”的声明表示信任。 但我们其他人可能会倾向于认为吉拉尔迪先生并没有等待真正的发令枪开始比赛。”
来吧，如果有组织的尝试或官方所说的尝试在“x”会议或“x”时间发生或发生的方式，任何稍加注意的人都不需要坚持下去。 事实是，在竞选期间的选举之前，民主党人都在谈论弹劾。 当总统宣誓就职时，您开始使用枪支，而民主党失去了炮弹震惊。
稍有注意的人都不会错过 Dir 的信号。 辞职后的布伦南走上新闻圈，对俄罗斯人和总统提出指控。 在揭露出联邦雇员的电子邮件和对话之后，没有人关注俄罗斯调查期间发生的事情，这些电子邮件和对话是他们既定的罢免总统的目标。 即使现在乌克兰的调查显示，如果有任何外国政治程序的干预，包括刑事正当程序，民主人士也一直深陷于交换行为，其中一些是为了个人利益。
我不需要秘密的深喉咙来获得显而易见的东西。 唯一比努力更令人惊讶的是它曾经和仍然是完全非隐蔽的。
所有这些都减去一点证据表明曾经存在过犯罪。
• 同意： SBaker
198. geokat62 说：
那并没有持续多久！ 正在敦促groypers停止谈论房间里的800磅大猩猩。
哦，好吧……它持续的时候很有趣！
尼克富恩特斯“不再谈论以色列
探索者”文斯·詹姆斯“USS LIBERTY TALK IS CRINGE”
美国优先的 Nick Fuentes，Telegram 帖子：
https://t.me/nickjfuentes1/1871
来自 Red Elephants 的 Vince James，Telegram 发帖：
https://t.me/RealVincentJames/527
199. SBaker 说：
显然，您错过了与住在附近的人的对话以及他们所看到的内容。 有的甚至还拍了照。 再多的第一手证据或目击者都无法说服住在一千英里外的阴谋小丑。 我建议你联系那些在谋杀穆斯林团伙摧毁他们生命之前与亲人取得联系的乘客的亲属。 如果我冒犯了真主，请告诉我。
• 回复：
200. geokat62 说：
ADL 首席执行官乔纳森·格林布拉特 (Jonathan Greenblatt) 的评论（注意：我们 = 犹太至上主义者；我们的 = goyim）：
如果我们可以为我们的教室接种疫苗，如果我们可以接触到我们的孩子，我们就有机会一劳永逸地在打击偏见方面取得进展。
201. GetReal 说：
那么，你会和沃伦一起去吗？ 二元决定。
• 回复：
202. 巴拉克奥巴马是一个愚蠢的人。 他只是，他说话，他走路，但他是个白痴。 让他摆脱准备好的文本，他是业内最糟糕的广告。 我可以很容易地看到这些年来他们带着这个或那个计划去找他的场景，从伊朗到利比亚到奥萨马·本·拉登，再到特朗普在这里。 他参与假期，他的运动伙伴和高尔夫，他的括号，他的 ESPN，那个婊子的妻子读他关于他的兴奋剂和吸烟的骚乱行为以及其他任何事情以及他周围的 Valerie Jarrett 盾牌让他保持又瞎又蠢，他能知道什么？ 除了福利之外，他对所有事情都充满了兴趣并且不感兴趣。 所以他们带着当天的计划去找他，你知道他的反应可能是什么吗？
“你们都做你想做的事”。 我真的能看到那个场景。 他经常表现出对这些人的存在以及他自己的白宫和机构的漠不关心，他对他愚蠢的鼻子下发生的事情完全漠不关心，这表明他的愚蠢和无法辨别那些把文书工作的人的动机在他面前签名。 他会声称自己无知。
事实上，你必须在 Valerie Jarrett 的指甲下画竹子。 她曾经沉浸在作为守门人、终极内幕者的光辉中，没有她的话，任何东西都不会进入巴拉克奥巴马的耳朵、办公室或电话。 曾经。 时至今日，她仍与奥巴马一家住在华盛顿。 WashPost 过去常常庆祝这些事实。 她不能否认她什么都不知道，她什么都知道。 从一开始，就连他自己在参众两院的党员也打不通奥巴马的电话，也打不过贾勒特。 事实是，克拉珀和布伦南以及其他人掌握着王国的钥匙，所有玩具都在他们手中，而 POTUS 在开关处完全睡着了。 这两个人和瓦莱丽·贾勒特是这一切的关键。 科米、麦凯布、佩奇、罗森斯坦，有几十个，他们是特工，应该失去工作、养老金和职业生涯。
布伦南和克拉珀应该被处决。
• 回复：
203. The ZUS gov has been spraying the skies over Montana for over 30 years by my personal observation, and to get more information, go to 地球工程观察网.
这是联合国 21 和 2030 年议程的一部分，这是一个邪恶的 NWO 计划！
204. SBaker 说：
你说的有些是真的，但 BHO 完全是他支持穆斯林兄弟会的同谋，以及他对白人和美国的恶毒仇恨。 他躲在瓦莱丽的裤子后面保护自己——他一直否认对文明的白人世界犯下的罪行一无所知，这是他从一开始就精心策划的策略。 他完全意识到并且完全有罪，应该和 Brennan 和 Clap 在一起。
205. Rurik 说：
显然，您错过了与住在附近的人的对话以及他们所看到的内容。
你的意思是像宾夕法尼亚州萨默塞特县的验尸官华莱士米勒一样，他是第一批到达现场的人吗？
“大约 20 分钟后，我不再担任验尸官，因为那里没有尸体。 “
事实上，在 12/9 之后的 11 个月里，米勒反复明确地描述了 93 号航班坠机现场令人惊讶的缺乏人类遗骸的情况：他告诉作者大卫麦考尔：
“我到了实际的坠机现场，简直不敢相信我所看到的。 … 通常你会看到很多碎片、残骸，以及很多噪音和骚动。 这次崩溃是不同的。 没有残骸，没有尸体，也没有噪音。 ……这架飞机上似乎没有乘客或机组人员。” （大卫·麦考尔，《从悲剧到胜利》，2002 年，第 86-87 页）
http://humansarefree.com/2015/03/shanksville-coroner-no-bodies-found-at.html
有的甚至还拍了照。
是的我知道。 竟然还有视频！
再多的第一手证据或证人都无法说服一直存在的阴谋小丑，
谁是小丑？
如果我冒犯了真主，请告诉我。
你没有冒犯真主，但你是对故意无知的事实和证据不可渗透的性质的冒犯性提醒。
至于
杀害穆斯林的团伙毁了他们的生活。
.. 自从 11 年 2001 月 XNUMX 日犹太复国主义/犹太至上主义对我心爱的国家进行前所未有的、背信弃义的袭击以来，谋杀和破坏数百万无辜穆斯林生命的正是像你这样的混蛋。
这里唯一的问题是你是否只是另一个假装是美国牛白痴的狡猾恶棍，（最有可能的），或者你是否真的是 非常愚蠢。
206. anonymous[307]• 免责声明 说：
[特朗普是统治阶级的成员，但不是深层国家的成员。 因此，他无法阻止深层国家正在做的事情，在所有这些国家煽动和继续永远的战争。]
因此，“深刻的思想家”必须表明犹太黑手党仆人结束了哪场战争。 叙利亚？，伊拉克？，黎巴嫩，伊朗？，委内瑞拉？，玻利维亚？，巴勒斯坦？，阿富汗？，也门？等等？
在你还没有这样做之前，他们应该在评论之前闭嘴并思考一下。
这个无知的教父，有这么多犹太复国主义犹太人在执行其政策，包括为以色列造福的犹太复国主义外交政策，其种族主义移民政策由一个名叫斯蒂芬米勒的犯罪犹太复国主义男孩执行，许多有关人士正试图将其踢出去，因为这个犹太复国主义男孩像内塔尼亚胡一样思考、说话和走路，成为特朗普“移民政策”的面孔。 特朗普政权接受内塔尼亚胡的指示，因为教父无知，其外交政策是由犹太复国主义者皮条客推动的，名叫兰迪·朱利安尼（Randy Juliani），他与所有黑手党和恐怖分子同床共枕，并证明他像特朗普一样为以色列的利益服务，而不是美国。
Everone 曾代表以色列和特朗普看到 Randy Juliani 与恐怖组织 MEK 合作，他的服务由另一个恐怖主义国家沙特阿拉伯支付。 为你们所有人感到羞耻
甚至其“国家议程右翼”的面孔是犹太复国主义男孩斯蒂芬米勒，人们正试图踢出去，然后它将成为这个叛徒和第五纵队必须去的地方。
特朗普是第五纵队，也是深层国家的一部分。 犹太复国主义宣传者正在攻击使用“深层国家”的人，因为他们认为“深层国家”是犹太人的词，现在特朗普的宣传者通过散布“深层国家”是针对一名犹太黑手党成员的谎言来误导公众。
这太可笑了。
你知道金融就像“深层国家”是一个“反犹太”词，因为根据犹太复国主义宣传者的说法，金融已经成为犹太人的面孔。
https://www.tabletmag.com/jewish-news-and-politics/293389/finance-codeword-for-jewish
207. Rurik 说：
据说阿桑奇被带到美国是为了什么。 在发现之下，他放弃了作为线人的里奇； 美国人民将点点滴滴联系起来，并初步了解深州的罪恶。 阿桑奇让美国成为人类著名的英雄。
诚然，矛盾没有比“美国人”和“连接点”更明显。
是的，我刚刚给其中一头美国奶牛写了一条评论，（或者至少假装是；）
尽管如此，了解管理我们联邦机构的精神病患者实际上是多么令人难以理解的邪恶，这是一个很大的飞跃。
毫无疑问，赛斯·里奇在他的灵魂中拥有一个体面的地方，这与希拉里·克林顿军政府的背叛格格不入。 他顽固的人性和正派是他的毁灭。
对于很多人来说，布什总统和副总统切尼（等人）升任美国最高职位的想法会背叛美国人民，对他们来说，这简直是难以理解的。
那种冷酷的邪恶会让像切尼这样的精神病患者与这个国家的凶残敌人一起策划，大规模屠杀美国人，这样切尼在飞往他的 CFR 晚会时可以享受更豪华的私人飞机，真的是太远了，对他们中的许多人来说。
正常人根本不具备理解临床精神病患者的心理指标。
Blake Liebel 漂亮的乌克兰女友认为她和他在一起很安全……
直到那天她发现自己错了。
208. Realist 说：
那么，你会和沃伦一起去吗？ 二元决定。
你怎么知道沃伦会被提名？ 这不是二元选择。 深州控制着我们的政府。
只要“两党”的重要议题得以维持，“深国”就不会在乎“两党”的无关紧要的内部争执。 事实上，它加剧了人们对投票时有选择权的错误认识。
209. Realist 说：
以下是我两年半前给 UNZ 的回复。
将伊万卡带出白宫
Ilana Mercer•13年2017月55日•XNUMX条评论•回复
“唐纳德·特朗普必须把那些孩子赶出白宫，”几周前，南非一位直言不讳的南非政治观察家对我咆哮。 “您看起来越来越像我们。” 她指的是特朗普白宫展示的裙带关系。 自总统开始袭击叙利亚以来，很明显……
现实主义者说：
15年2017月9日，格林尼治标准时间上午51:XNUMX
深州在不到 100 天的时间里就给特朗普绝育了。国家领导人的布鲁斯詹纳。
他现在将被称为The Caitlyn。
[更多]
现实主义者说：
7年2017月6日，格林尼治标准时间上午19:XNUMX
特朗普真是个白痴。 他是深层国家好战霸权混蛋的一部分。 我不应该投票给他。 他肯定是一个术语。
唐纳德·特朗普的怪异先知
以色列Shamir•3年2017月152日•XNUMX条评论•回复
如果特朗普总统说“太阳从东方升起，在西方落下”，无疑，《纽约时报》会讽刺地大笑，并声称由于杀手普京的干涉而成为总统的愚蠢的乡下人不知道没有-天文学的最新进展使他的说法真正荒谬。 那里…
现实主义者说：
3年2017月9日，格林尼治标准时间上午26:100•XNUMX字
特朗普要么愚蠢要么腐败，我怀疑是前者。 他选择了迈克·彭斯 (Mike Pence) 担任副总裁……如果有的话，他就可以和阴户相处。 他选择 Reince Priebus 作为他的参谋长，Reince 是 Paul Ryan 的一个非常亲密的朋友，他讨厌特朗普，是权力精英的一部分。 特朗普早就应该在该机构打扫房子。 他雇佣了他的敌人，并允许其他敌人留在政府中。 他将有一个不可能的四年。
• 回复：@Wally
• 回复：
210. 嘿现实主义者。
一个基本问题。
Deep Stater 的工作人员如何与 Deep Stater 同事绝育？ 🥴
• 回复：
211. Realist 说：
嘿现实主义者。
一个基本问题。
Deep Stater 的工作人员如何与 Deep Stater 同事绝育？ 🥴
正如明确指出的那样，这些评论是在两年半以前写的。 在我发表上述言论后的相对较短的时间内，特朗普的持续口是心非迫使我相信他是深州的成员或推动者。 正如你应该能够从我 3 年 2017 月 XNUMX 日的评论中注意到的那样，我说过
特朗普要么愚蠢要么腐败，我怀疑是前者。 一段时间后，我确定他两者兼而有之。 但我不像你那样生活在妄想中，你是个巨魔。
• 回复：
212. 唤醒超现实主义现实主义者的缓慢，宣布我有罪，并判我：“......成为巨魔。”
哈哈。 在成为“愚蠢或腐败”方面，将有很多轻松的持续艰苦的劳动。 🤗
后记：对我来说，奉承会让你到处都是幻觉。 😏 所以非常感谢！
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213. Vetran 说：
FISA 调查已演变为刑事调查。
让我们看看约翰达勒姆会起诉谁......沼泽中的一些大鱼最近可能睡不好。
等等，看看他们是否会穿橙色西装……或者不！
214. 是的，我认为在这些方面做了一些事情。
像中央情报局局长这样的人从来没有像布伦南那样自己公开露面。
当然，就像总统对国外政变等事情点头同意一样，奥巴马组织这样的行动不会有书面记录。
这种针对特朗普的行动对于奥巴马总统来说将是一个合适的墓志铭，他在任时表现出的主要品质是保密、对举报者的强烈厌恶以及对海外大量暴力的认可。
但话又说回来，真的有美国总统再决定这些事情了吗？ 自从肯尼迪半个头被炸到达拉斯街头以来？
总统在国家和国际事务的重大问题上实际决定什么，以及他作为黑暗国家的建议微笑签署的内容根本不清楚。
是的，允许在语言、重点和气质方面存在差异，但有一条连接脐带。 布什和奥巴马的实际工作与现在特朗普在国外的实际工作差异并不像人们通常想象的那么大。
The days of believing that a person's being merely elected decides the direction of things for a gigantic global empire do seem distant and rather naive, almost like thinking in terms of men in frock coats and wigs signing off on matters with quill pens.
但即使是安全服务人员，尽管他们的资源无限，而且冷酷无情，但有时仍然会出错。 我们在国外看到了拙劣的政变和战争，在国内看到了拙劣的阴谋。
215. Realist 说：
唤醒超现实主义现实主义者的缓慢，宣布我有罪，并判我：“......成为巨魔。”
在特朗普担任总统的三个月里，这对我来说显然是……不是你，特朗普在解决这个国家的问题方面将毫无用处。 吉拉尔迪的一个抱怨的小丑打电话给我 慢慢醒来 很有钱。 与其抱怨问题，不如提出解决方案。
• 回复：
216. 白人并不是在缓慢自杀； 他们被多元化的 AA 深州老鼠杀害。 和他们自己令人作呕的放纵肥胖猪，比如 Swine-Scot！
217. 哈哈。 现实主义者深深地撅起嘴：“。 与其抱怨问题，不如提出解决方案。
由于另一个 Zio 经典弹劾剧院正在进行中，我确实为您提供了绞刑架解决方案，请参阅下面链接的复古 SNL 视频。
嘲讽与欢笑，让美国ZOG的生活变得宜居。 所以放松点，现实主义者？ （Zigh）'因为要及时交易，😏也许被猎头的特朗普斯坦会打电话给齐奥猎杀的鲁哈尼？
218. 考虑到美国公众的批判性思维水平已经下降，我最害怕的是被诬告并任由 12 个白痴摆布。
219. 锡安门的深处，再； 下面的文章，想知道基辛格博士是否透露了有关美国吸血乌贼（深州🇮🇱）在防止与中国发生全球灾难性战争方面的作用的任何信息？
220. 尽管宫廷历史学家刻意忽视和避免，1861 年的政变是第一次也是最重要的政变。 In that it enabled all others that have followed, including current attempts to remove a duly elected US president.
221. 然而，我越看越明显，这个国家是一个巨大的腐败洗钱坑。
我们今天看到的华盛顿内部人士（如拜登家族）和腐败的乌克兰寡头之间的腐败，是普京下令停摆之前整个俄罗斯国家的缩影。 难怪普京是 Zio 机构的#1 敌人。
• 同意： Rurik
• 回复：
222. 机敏的卡罗尔·普莱斯提醒说：“……1861 年的政变是第一次也是最重要的政变。 In that it enabled all others that have followed, including current attempts to remove a duly elected US president.”
尊敬的卡罗尔，一个重要的问题
What exactly do you mean by “duly” elected?
顺便说一句，1868 年，汤姆·杜利 (Tom Dooley) 被选为不当绞刑。
就其价值而言，😏 我认为平均 Joe Amerikan unDoolie 在每个总统“选举”日都会挂起。
1.
Jackson Stephens and other members of the Stephens family bankrolled Bill Clinton’s rise to political prominence … Worthen Bank, which was majority-owned by Stephens, provided Clinton’s first presidential campaign a \$3.5 million line of credit. In addition, Stephens’ many businesses were frequently represented by the Rose Law Firm, where Hillary Clinton was a partner. ……中央情报局官员似乎参与了银行的成立……中央情报局文件后来在关于银行活动和相关丑闻的国会听证会上浮出水面，称BCCI直接参与了“洗钱、贩毒、走私和持有大笔资金”对于恐怖组织。” 2. Rose 律师事务所在协助 BCCI 进入美国金融体系的同时，还代表 Stephens 拥有的金融服务公司 Stephens Inc. 以及 Stephens 在 1960 年代后期收购的数据处理公司 Systematics Inc.。 … 已故记者迈克尔·鲁珀特断言，这个“被窃听的软件”正是 Promis 软件，美国和以色列的情报机构都曾窃听过该软件以监视情报，并且部分由杰弗里·爱泼斯坦夫人的父亲罗伯特·麦克斯韦销售, 吉斯莱恩麦克斯韦。 ... Promis 最初由比尔·汉密尔顿 (Bill Hamilton) 创立的一家小型软件公司 Inslaw Inc. 出租给司法部——后来司法部从 Inslaw 那里偷走了它，迫使其宣布破产。 … Systematics “秘密地将 [软件] 植入其银行客户的计算机”，允许“盟军情报机构暗中跟踪和监控通过银行系统的资金流向”，并“应美国国家安全局的要求这样做” (NSA) 及其在以色列情报方面的合作伙伴。” Inslaw 还表示，该软件被这些相同的情报机构用于“洗钱，尤其是毒品利润”。 224. 看… 美国的深层国家是由犹太人在 150 多年的时间里建立起来的。 所以美国总统很难在4年甚至8年内把它打倒。 但我们必须从某个地方开始。 我认为特朗普为此奠定了基础。 225. 这是我唯一留下的东西。 乐观 但是，永远不要说永远。 我把整件事看成是光明与黑暗之间的斗争。 你认为谁会赢？ • 回复： 226. Art 说： 这段视频很好地解释了影子政府 vs. 深州。 影子政府是未经选举的常任官僚——而深州则是控制他们的私人机构。 中央情报局和五角大楼是影子政府——MIC 和 AIPAC 是深州。 • 回复： 227. “考虑到美国公众的批判性思维水平已经下降，我最大的恐惧是被诬告并任由 12 个白痴摆布。” 两个回应： 1. 这就是人们恳求的原因——它更简单，感知风险更低 但有这个 2. 将一个人的命运交由有政治动机的法官掌握 https://www.unz.com/ishamir/the-deep-state-vs-netanyahu-lawyers-and-statemen/ 站出来冒着被嘲笑的风险是人们的一个严肃的开局。 这里在线是一回事，但站在流行的观点上，说 “否” 搞不的花言巧语。 . . 连法官都畏缩了。 228. 不会很快想到现任或前任“仁慈的独裁者”。 229. 艺术指示：“中央情报局和五角大楼是影子政府——MIC 和 AIPAC 是深州。” 很抱歉，艺术，但你上面的话，激发了廉价和住宅区骗子的欢乐。 在 SolontoCroesus 曾经反思他高尚的 UR 评论辛劳并写道：“这已经变得无聊了。” 太糟糕了，“这个”是唯一可以从 Zio 阴影中搜索并找到光线的地方。 谢谢，最好不要冒犯，艺术？ 我想我最好前往雅克希特高地。 230. 早在 98 年，The Nation 就发表了这篇不言自明的标题作品。 https://www.thenation.com/article/harvard-boys-do-russia/ 在小布什政权的早期，我突然想到哈佛男孩开始在美国做事，事实证明确实如此。 231. 比尔·琼斯 (Bill Jones) 说，👍：“在小布什执政初期，我想到哈佛男孩开始在美国做事，事实证明确实如此。” 哈哈。 在您对 Carroll Price 发表深刻而有价值的评论后，我从这篇文章线程中撤出并没有持续多久。 好吧，Bill，今天，感谢您的深刻洞察，Naomi 的“冲击学说”一书又翻开了 ZUS 的一页，我正在看。 谢谢和我的感激！ 232. bjondo 说： Yid/假闪米特人谋杀闪米特人。 哑巴哈欠。 233. anon[113]• 免责声明 说： “埃里克·恰拉梅拉 (Eric Ciaramella) 是一名职业中情局分析师，在奥巴马政府结束期间担任国家安全委员会 (NSC) 乌克兰主任。 据 Real Clear Investigations 报道，恰拉梅拉在 2015 年和 2016 年为前总统乔拜登处理乌克兰问题，当时拜登担任乌克兰问题的“关键人物”。 ” Ciaramella 是一个重要的证人，无论他是不是举报人。 234. geokat62 说： 摘自《打破以色列新闻》文章， 拉比：祝福特朗普，因为国王表明他是大卫王朝回归前的最后一位总统: President Trump attended an Orthodox Jewish fundraising event at the Intercontinental Hotel in Manhattan on Tuesday evening. Four hundred Orthodox Jews participated in the event, contributing an estimated \$100,000 each for the honor of paying a personal tribute to the Commander-in-Chief.
拉比 Yosef Yitzchak Jacobson 在见到一位非犹太统治者时背诵了祝福语。
ברוך אתה ה' אלוקינו מלך העולם שחלק מכבודו לבשר ודם
祝福你，我们的上帝，宇宙之王，你已经与一个永远维护每个无辜者和每个犹太人的荣誉的人类分享了你的部分爱、荣耀和同情心。”
演讲全文录像：
• 回复：
235. 嘿，地理！
当然，我不指望你回心转意，因为我没有必要让你卷入我对深奥深州存在的异议。 我坚持认为 Phil Giraldi 有点不情愿地为 Zio 追击的 Trumpstein 骑猎枪。 😟告诉我，geo？ 美国被令人毛骨悚然的弹劾听证会着迷，这是自疯狂的 OJ 审判以来前所未有的。 这对美国有什么好处？ 齐尔奇！
你的评论 #242 提供了更多的证据表明深州不是 anno domini，但正如 Robjil 一直指出的那样，是 BC 强大的“深州”已经走向国际，所有被认定的深州特工都受制于吸血乌贼银行家。
• 回复：
236. geokat62 说：
我不希望你回心转意，因为我没有必要让你卷入我对深奥深奥存在的异议。
嘿，罗德诺小子，查克。
辛西娅·麦金尼 (Cynthia McKinney) 在安吉洛·约翰·盖奇 (Angelo John Gage) 采访她的精彩视频中深入探讨了 DS， Cynthia McKinney 博士解释政府腐败
她基本上将深层国家定义为两个犹太至上主义者派系，为了完全控制而相互对抗。
最喜欢的报价从 @ 37:14 开始：
......这些控制权的人然后他们在你面前游行，然后他们只是在测试他们成为你的领导......这就是为什么我没有非常关注民主党总统初选，因为它是一样的他们只是在我们面前游行的现象，看看他们已经选择了我们将与谁一起去……
从@ 46:06开始：
所以，现在你所拥有的，在我看来，美国基本上被那些爱以色列的人所控制，而不是他们爱美国的人......描述我实际发生的事情的最好方式是有一群人在我能够爱乔治亚州的石山之前，让我爱以色列……
……如果你不为以色列做事，他们就会把你赶走！
• 回复：
237. geokat62 说：
有人告诉我科赫兄弟不是犹太人。 E. Michael Jones 在 Groypers、Coomers 和 Kochsuckers——Groypcast 上的 E Michael Jones
从@ 21:56开始：
Charlie Kirk 就是我们所说的 Kochsucker。 好的，我指的是科赫家族基金会。 他们创建这样的团体，然后这些团体代表他们的利益。 所以，是科赫兄弟， 这是一个犹太基金会. 因此，查理柯克提倡自由市场经济和对以色列的支持。
• 回复：
238. 哈哈，对我来说，geokat 违背了传统智慧，并说：“嘿，Rodnoy 小子，查克。”
哈哈。 👍！ 什么是小子之间的“rodnoy”？
很高兴你把 Cynthia McKinney 引出来，我相信 Phil Giraldi 想成为大元帅，但他做出了选择，犹豫不决。
尽管如此，geo，“不要为以色列做事”，上帝保佑菲尔的溃烂！
239. 令人着迷的是，geokat 分享了这一点：“所以，这是科赫兄弟，这是一个犹太基金会。 因此，查理柯克提倡自由市场经济和对以色列的支持。”
我想知道我在这里做什么。
嗯。 在海军上将斯托克代尔令人难忘的总统辩论中，他在塔木德电视上光荣地面对美国人，哈哈，并反问：“我在这里做什么？”
可惜，他成了美国家庭的笑柄，geo。
现在我在 Unz Review 上做了什么，并抱怨 Ron Unz、Phil Giraldi 和评论者使用深奥的名字 Deep State。 🤔 我认为这样的行动让 8100 部队有了一些有趣的事情要做。 *
谢谢geo，（Zigh），不是可口可乐，而是kochsucker和微笑。🇮🇱
* 我认为 8100 单元是那个深沉而奇怪的名字。
• 哈哈： geokat62
• 回复：
240. 基于我 1950 年代田园诗般的童年，以及当时正常的父母和亲密的家庭成员希望我保持谨慎和安全的时候，我被警告要担心如果我不听，不正常的布吉人会抓住我。😕
现在我正在学习可怕的深州让我掌握了它🇮🇱，不幸的是，即使是其他可靠的布吉人也没有提供庇护所。 😕
• 回复：
241. 我把整件事看成是光明与黑暗之间的斗争。 你认为谁会赢？
可以使用电灯开关的那个？
• 同意： ChuckOrloski
242. 嘿查克，
希望你和你的校车一切顺利。
是否有力量在那里看到你和我想要的大部分东西 [政治/政府/等] 顺其自然，而不是你和/或我的？
有人将肯尼迪的一半大脑射到榆树街的停机坪上。 我们无法说出扣动那个特定扳机或在阵容中指出他的人的名字，这并不意味着肯尼迪没有那么死。 下令打击，设置打击，掩盖打击的女性/男性也是如此。 在某些情况下，我们可以命名名称，但大多数情况下不能或无法建立足够的确定性以使其坚持下去。 我将执行那些马基雅维利阴谋的能力扩展到“TPTB”或“深州”或任何贴在上面的标签。
它就像宇宙中的一个黑洞； 我们看到周围发生的事情，却无法说出内部发生的事情。 我们看到了“结果”，但 [还] 无法获得答案。
瓦本，compagno！
243. 票价 = 公平
谢谢，自动更正！
[被困在学校的“公共汽车”位？]
244. geokat62 说：
有人将肯尼迪的一半大脑射到榆树街的停机坪上。 我们无法说出扣动特定扳机或在阵容中指出他的人的名字，这一事实并不意味着肯尼迪没那么死。
就像任何称职的刑警一样，他到达犯罪现场后问自己的第一个问题是 谁是bono，谁受益？
就肯尼迪·肯尼迪遇刺案而言，显然有一个非常大的受益人在其余的人中脱颖而出，以色列。
到1963年XNUMX月，肯尼迪对他进行了两次大罢工：
1. 1963年，肯尼迪总统要求本古里安结束以色列的核威慑计划。 肯尼迪在18年1963月8日的一封信中警告说，除非美国检查员被允许进入以色列的Dimona设施，否则以色列将发现自己完全孤立。 “我们对以色列发展核武器能力对世界稳定造成的令人不安的影响感到关切……正是由于我们对这一问题的关注，我国政府寻求与您安排定期访问迪莫纳。” 事实证明，他随后的保证置若de闻：“正如我在XNUMX月XNUMX日的新闻发布会上明确指出的那样，我们对以色列的安全有着坚定的承诺。”
2. 1963年，肯尼迪总统要求美国犹太复国主义委员会根据FARA登记为外国代理人。 在17年1963月XNUMX日（肯尼迪被暗杀前一个月）的信中，里夫金德法官（AZC的代表）回应了肯尼迪政府的要求：“……大多数与安理会有联系的人认为， 这样的注册 将由美国犹太教委员会如此宣传，以至于 最终将摧毁犹太复国主义运动。=
以典型的犹太复国主义风格，他们决定在三击之前炸掉他的半个头骨！
• 同意： ChuckOrloski, Robjil
245. 根据您所在的时区，早上好，
丹尼尔·里奇！
丰富而肆无忌惮的诗意是你的深层状态冥想，例如，“它就像宇宙中的一个黑洞； 我们看到周围发生的事情，却无法说出内部发生的事情。 我们看到了‘结果’，但[还]无法获得答案。”
一种有关“黑洞”的相关但可能不相关的想法。 人们可以尝试窥探黑立方的邪恶活动，但充其量会被吐出来，最坏的情况是死。😏
关于肯尼迪总统在 22 年 1963 月 XNUMX 日对选举的打击和不负责任的推翻这一话题，和许多人一样，我一直在抱怨。
1962 年，肯尼迪总统和他的小子 AG Bobby 出于爱国的目的想要迫使强大的美国犹太复国主义委员会 (AZC) 注册为外国代理人，这是黑洞掩盖的事实。 （1951-1953，Isaiah Kenen 是 AZC 的华盛顿代表😈）
年轻、雄心勃勃、聪明伶俐的伊利诺伊州众议员唐纳德·拉姆斯菲尔德 (Donald Rumsfeld) 致信 RFK，前者的 Zio 股票开始开花。 下面链接是 Lucky Rummy 有“合理否认”的信件。😟
https://www.israellobby.org/AZCDOJ/congress/default.asp
仅供参考，丹尼尔·里奇 (Daniel Rich) 现在一切正常，但事实上我的腰部手术并没有像我希望的那样好。 哦，人类的苦难是人类潜在的救赎之恩。 当一个人拒绝 Zio Dystopia 时，他们的肉体就会遇到麻烦。 哦，好吧，我们只是借来的。
最后，1981 年 XNUMX 月，我从一次令人大跌眼镜的中东冒险，包括（受感染的）西岸返回家中，阅读了大卫·利夫顿 (David Lifton) 的非凡著作《最佳证据》。
我不是为即将到来的新年分配阅读清单的人，而是在你的句子“有人将肯尼迪的一半大脑射到榆树街的停机坪上”之后，我将分享一个事实：穿着粉红色连衣裙，杰基追求了一部分她丈夫的大脑在死亡豪华轿车的后备箱上飞来飞去。 😟
因此，在这个 Zio 改造的圣诞节期间，请考虑仔细阅读科学的“最佳证据”，其中大卫利夫顿推出了不可否认的证据，证明特勤局是榆树街谋杀案的主要帮凶。 非常感谢，丹尼尔，嗯，你是出现在这里的更富有、更深刻的思想家之一！
246. geokat62 说：
Groyper Wars 的最新更新
权利正在崛起：关于 Don Jr 事件和 Groyper 战争的真相
从15:47开始：
我和很多人合影……他们不是白人，就这么说吧。 但他们明白如果允许这种人口变化继续下去会发生什么。 他们完全理解这一点。 他们明白，如果对[大规模移民]不采取措施，美国将变成第三世界。
伙计们，每个关心自己国家未来的美国人都需要宣传人口变化正在摧毁这个国家。 每个人都应该发推文、写信、打电话，并携带一个标有简单信息的标志：
多样性不是我们的强项。 我们要求立即暂停大规模移民！
247. geokat62 说：
摘录自 人口变化：想象的还是不可避免的？
从7:42开始：
贾里德泰勒在解释说 LBJ 是最后一位大多数白人投票并获胜的总统之后，
“这意味着白人没有得到他们想要的总统。 为什么？ 因为“伟大的……”不，不……因为白人至上主义的阴谋论。 现在，Farhad Manjoo [NYT 专栏作家] 和他的朋友认为他们在愚弄谁？ 他们告诉我们，我们注意到那些不仅客观真实而且每个人都知道是真实的事情在道德上是错误的。 很难把这看作是想要你的国家的人的蓄意欺骗！”
• 同意： ChuckOrloski
• 回复：
248. + 查克·奥尔洛夫斯基，
谢谢各位，你们的回复。
当我看到 这部电影/视频剪辑 肯尼迪车队和奇怪的特勤局在他们袭击达拉斯街道之前的行为，很明显[对我来说]有人知道“孤独的肯尼迪杀手，李哈维奥斯瓦尔德”的角度必须得到保护，并且两个特勤局特工实际上会站在道路。
看看 [如果你愿意] 在 0:30 标记附近发生的事情以及 SS 特工的后续反应。
被占领的巴勒斯坦是文明社会结构中的一根刺，正因为如此，自然机制被激活以将其从时间的页面中移除。 这就像一群智力低下的孩子在经营一家枪支商店。 最终，真正的正义将占上风，即使女士本人暂时被闪亮的谢克尔的光芒蒙蔽了双眼。
至于 David S.Lefton 的“最佳证据”，肯定会从亚马逊订购这本书。 这是 一个链接，以防其他人也想订购/阅读。
感谢您对这个无尽且极其复杂的难题的另一部分的指导。
我欠你和geokat62很多啤酒！
干杯，希望你背部的疼痛 {Chuck] 改善到可以忍受的水平 [或者，最好完全消失]。
249. geokat62 说：
最终，真正的正义将占上风，即使女士本人暂时被闪亮的谢克尔的光芒蒙蔽了双眼。
伟大的线路，丹尼尔！
干杯，希望你背部的疼痛 {Chuck] 改善到可以忍受的水平 [或者，最好完全消失]。
我的心情，没错！
250. 嗨，丹尼尔·里奇！
我的大卫利夫顿“最佳证据”的旧硬拷贝有关键的草图和照片，它们带回了使沃伦委员会报告蒙羞的证据，特别是雄心勃勃的犹太律师阿伦斯佩克特，他炮制了谎言的魔法子弹理论。
仅供参考，除了“最佳证据”中的内容，在委员会的报告发布后不久，Spectre 的伟人和法律同事文森特萨兰德里亚先生与他交谈，并试图恢复阿伦的正直，并直言不讳地告诉他他“不诚实和腐败。”
生气，当然，Arlen Specter 不喜欢 Salandria 先生的形象，但后者也公开谈论杰克 'Ruby' Rubinstein 在沉重的达拉斯 PD 中轻松杀死 Lee Harvey Oswald
监护权。😟
萨兰德里亚先生担心鲁比是犹太人，一旦他得知已故的 LHO 不会接受审判，他就开始担心对美国犹太人的反击，并担心城市爆发的大屠杀震惊了美国。
哦，好吧，（叹气）已经偏离了菲利普的话题，深（州）尽头，😏，我感谢你和 geokat 对我的下背部的关注，它保持完整，我继续直立行走。
251. 任何想要和平的总统都会被深层国家深深打倒。 肯尼迪、尼克松和现在的特朗普。
252. 又是我，丹尼尔，哈哈，而不是阿黛尔！
在利夫顿先生的“最佳证据”中，你会欣赏另一件事（哥伦比亚侦探☺️）。
这本书展示了贝塞斯达海军医院的尸检医生对肯尼迪头部的草图，这与帕克兰医院急诊室医生报告的发现完全矛盾。😟 是的，“草图”，没有验证图片。
太糟糕了，“最佳证据”因为其他优秀作品而被降级，例如，詹姆斯·道格拉斯广受好评的书“肯尼迪与无法言说”，遗憾地忽略了特勤局在促进“打击”和掩盖总统的罪行方面的作用榆树街谋杀案。
253. “那是叛国罪，为此，他们肯定应该被绞死。”
从技术上讲，叛国罪是在战争时期向敌人提供帮助（不确定我们甚至不再有“战争”，如果宪法定义为我们无所作为的国会所宣布的。）在美国只有十几个人因叛国罪被定罪，主要是二战时期的广播公司。
人们应该使用的术语是煽动叛乱。
254. Rurik 说：
叛国在技术上是在战时帮助敌人
无论它在法律术语中的技术用途如何，该词都有其含义。
我最近阅读了 PCR 的一篇文章，这个词在其完全正确的上下文中被使用了两次。
“我不是那个时代英国法律的专家，但民选官员在战争与和平事务上为外国人竞标以换取巨额秘密支付似乎几乎是教科书式的例子 叛逆 对我来说，我认为丘吉尔的及时处决肯定会挽救数以千万计的生命。”
...
“1944 年，英国军事情报部门的成员认为战后亚瑟·哈里斯元帅爵士会被绞死或被枪决是理所当然的。 叛逆 反对英国人民……”
如果你'Bing'这个词，定义如下：
叛国罪。
[ˈtrēzən]
名词
背叛国家的罪行，尤其是企图杀死主权者或 推翻政府.
维基：
在法律上，叛国罪是对国家不忠的犯罪行为。 这是一种犯罪，涵盖了一些针对一个国家或主权的更极端的行为。 这通常包括诸如参加对自己祖国的战争， 企图推翻其政府， 监视它的军队，它的外交官， 或其为敌对和外国势力提供的秘密服务， 或试图杀死其国家元首。 犯有叛国罪的人在法律上被称为叛徒。
虽然在美国法律中，为了审判、定罪和处决叛徒，这个词的实际含义确实令人费解，以至于毫无意义（正如您正确指出的，因为 ZUS 参与了无休止的战争） all over the globe, but none of them are “declared” – by our treasonous congress' ; ), so the word becomes mute as a legal term, nevertheless, when people commit acts of espionage and treachery against our elected president, with the intention颠覆和推翻民选的合法政府（为了他们自己邪恶的目标），那么他们仍然犯有叛国罪，如果这个词有任何意义，超出了非常狭窄的范围（到了荒谬的无意义）宪法。
我最近听到 NPR 上的人说乔和亨特拜登在乌克兰所做的事情完全合法，因此不是腐败的例子。
这可能是合法的，但如果这个词 腐败 有任何意义，那么拜登在乌克兰所做的就是一个教科书式的例子。
• 回复：
255. Dannyboy 说：
约翰·布伦南应该被关押在潮湿、黑暗、虫害横行的地牢中大约三年，然后在美国白人工人阶级喧闹的欢呼人群面前被拉出来，隆重地折磨和勒死。
• 回复：
256. 民主党和自由党在电视上无休止地运行 Schiff For Brains，希望说服某人他有头脑，特朗普总统应该被弹劾。
美国现在已经充分展示了特朗普精神错乱综合症，他们不再被逗乐了。
是时候重新选举特朗普总统并加强他在众议院和参议院的影响力，这样民主党人和自由党人就可以被追杀至灭绝。
弹劾妄想政变实际上比俄罗斯勾结妄想政变更难看
是时候将布伦南、科米、克拉珀、希夫、佩洛西和其他一些人召集起来进行处决了。
257. 煽动叛乱是一种可执行的罪行，我会放弃对他的处决，以确保他可以在一般监狱人口中每天被强奸 50 次，持续 XNUMX 年。
Personal 古典文学
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http://models.street-artists.org/2017/03/08/bet-proofness-as-a-property-of-confidence-interval-construction-procedures-not-realized-intervals/comment-page-1/
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There was a bunch of discussion over at Andrew Gelman's blog about "bet proof" interpretations of confidence intervals. The relevant paper is here.
The basic principal of bet-proofness was essentially that if a sample of data X comes from a RNG with known distribution $D(\Theta)$ that has some parameter $\Theta$, then even if you know $\Theta$ exactly, so long as you don't know what the X values will be, you can't make money betting on whether the constructed CI will contain the $\Theta$ (the paper writes this in terms of $f(\Theta)$ but the principal is the same since f is a deterministic function).
The part that confused me, was that this was then taken to be a property of the individual realized interval... "Because an interval came from a bet-proof procedure it is a bet-proof realized interval" in essence. But, this defines a new term "bet-proof realized interval" which is meaningless when it comes to actual betting. The definition of "bet-proof procedure" explicitly uses averaging over the possible outcomes of the data collection procedure $X$ but after you've collected $X$ and told everyone what it is, if someone knows $\Theta$ and knows $X$ they can calculate exactly whether the confidence interval does or does not contain $\Theta$ and so they win every bet they make.
So "bet-proof realized confidence interval" is really just a technical term meaning "a realized confidence interval that came from a bet proof procedure" however it doesn't have any content for prediction of bets about that realized interval. The Bayesian with perfect knowledge of $\Theta$ and $X$ and the confidence construction procedure wins every bet! (there's nothing uncertain about these bets).
2 Responses leave one →
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https://weberblog.net/considerations-about-ipsec-pre-shared-keys-psks/?replytocom=47231
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# Considerations about IPsec Pre-Shared Keys
Pre-shared keys (PSK) are the most common authentication method for site-to-site IPsec VPN tunnels. So what’s to say about the security of PSKs? What is its role for the network security? How complex should PSKs be? Should they be stored additionally? What happens if an attacker catches my PSKs?
I am listing my best practice steps for generating PSKs.
This is one of many VPN tutorials on my blog. –> Have a look at this full list. <–
## Pre-Shared Keys in IPsec
The following section is related to site-to-site VPNs only and NOT to remote access VPNs.
1. The pre-shared key is merely used for authentication, not for encryption! IPsec tunnels rely on the ISAKMP/IKE protocols to exchange the keys for encryption, etc. But before IKE can work, both peers need to authenticate each other (mutual authentication). This is the only part in which the PSKs are used (RFC 2409).
2. If static IP addresses are used on both sides (= main mode can be used), an attacker who has the PSK must also spoof/redirect these public addresses over himself in order to establish a VPN connection. That is: Even if an attacker has a PSK, he must spoof a public IP address to use it to authenticate against the other side. This is quite unrealistic for normal persons with common ISP connections. Even skilled hackers must be able to inject falsified BGP routes or to sit nearby the customers default gateway/router.
3. But: If one remote side has only a dynamic IP address, IKE must use the aggressive mode for its authentication. In this scenario, a hash from the PSK traverses the Internet. An attacker can do an offline brute-force attack against this hash. That is: If the PSK is not complex enough, the attacker could succeed and would be able to establish a VPN connection to the network (if he furthermore knows the IDs of the site-to-site VPN peers which is no problem since they traverse through the Internet in plaintext, too).
## Best Practice for PSKs
Since the PSKs must be configured on each side only once, it should be no problem to write 20-40 letters on the firewall. Thereby, a really complex key can be generated and used for the authentication of the VPN peer. Here are my tips:
1. Generate a new/different PSK for every VPN tunnel.
2. Use a password/passphrase generator for the creation of the PSK.
3. Generate a long PSK with at least 30 chars, to resist a brute-force attack. (See my article about password complexity.) To avoid problems, use only alphanumeric chars. Since the PSK with 30 chars is really long, the “small” character set of only 62 alphabets and numerals is no problem. The security level in this example would be round about 178 bit (since $log_{2}(62^{30})=178$).
4. Do NOT send the PSK to your peer over the Internet, but via phone, fax, or SMS.
5. There is no need to store the PSK anywhere else. If it is configured on both sides, you can discard it. In the worst case, you need to generate and transfer a new one.
## 10 thoughts on “Considerations about IPsec Pre-Shared Keys”
1. aaa says:
Is there a way to require the PSK to expire?
1. What exactly do you mean? A kind of expiration timer that automatically blocks the VPN if the same PSK is used for x days? This must be a firewall feature, but I have not heard of a feature like that.
Or do you mean whether it is a security issue if the PSK is never changed? Well, as long as both sites use static IPs, and as long as the PSK is complex enough, there is no reason to change the PSK. However, if it is never changed, this is not “good” either. Maybe it is exposed through another way (social engineering, etc.). So, in my opinion, a PSK change every 3-5 years is a good choice. But even more it is relevant to check every 3-5 years if appropriate security algorithms (ciphers) are used for phase 1 and phase 2. If you are still using “no-DH” or “DH-2”, this is NOT secure anymore.
That is: Change your PSK every 3-5 years AND review your P1 and P2 proposals. ;)
2. Saghar says:
Hello,
What do you mean by “Generate a new PSK for every VPN tunnel”? Do the communicating parties need to exchange a nonce/random everytime (with which they generate a new PSK)?
Imagine that we have several embedded devices that they need to authenticate whenever they want to communicate with each other. Do you think that authentication with PSK is a good idea ? Do you know of any mechanism with which we can securely distribute the PSK to all these devices? Or should we configure the PSK seperately on each device?
Thanks
1. Well, it depends. If you have multiple embedded devices, you should consider using authentication via certificates. There are options to distribute certificates automatically.
Concerning “Generate a new PSK for every VPN tunnel”: If you are a company that has 10-50 static VPN tunnels that do not change that often (i.e., the IP addresses of the partners do not change that ofen), you can use PSKs for authentication. The PSK must be configured only once (!) during the setup of the VPN. It must not be changed later on.
But if you have a few VPNs coming from dynamic IP addresses, I do not recommend to use the same PSK for these VPNs, but to use a different/new PSK for each of these. If one PSK is exposed, it can be deleted without the need for a change of the PSKs from the other VPNs.
1. LOL says:
So instead of worrying if a third party might know your PSK, you definately know a third party knows it [you]?
:P
3. Karlfife says:
Can you speak to the plausibility of an attack on a main-mode Phase 1 Internet Key Exchange IF the attacker can snoop on the traffic and if s/he ALSO knows the pre-shared key?
I’m thinking that in thinking that in main-mode, If one were 100% certain that your IP address were not being spoofed by an attacker (impossible, I know, but this is a thought experiment), that absolutely NO phase 1 key would be needed at all, is that right? The Diffie Hellman key exchange itself guarantees that the key is exchanged privately. The key, as you say, is only used for validating the identity of the remote parties.
1. Hey Karlfife.
I am not sure whether I am understanding your question correctly. You’re asking about a passive attacker that 1) has the PSK and 2) is only listening but NO man-in-the-middle, correct?
If so, he is NOT able to read anything in plain text because of Diffie-Hellman. You’re assumption is correct. DH guarantees a secure key exchange. (To be correct, it is a “key agreement” rather than a “key exchange” since both parties have their input rather than one party transmitting the key). However, DH does NOT authenticate the peers. That’s what the PSK is used for.
Hence in theory, if no one spoofes your IP connection you can simply trust in your connection (as authentication) and must not use any PSK. However, I would definitely NOT suggest that! ;)
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http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=1436023
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0
Research Papers: Fundamental Issues and Canonical Flows
# Patterns of Airflow in Circular Tubes Caused by a Corona Jet With Concentric and Eccentric Wire Electrodes
[+] Author and Article Information
Reza Baghaei Lakeh
Department of Mechanical Engineering, Southern Illinois University Edwardsville, Edwardsville, IL [email protected]
Majid Molki
Department of Mechanical Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1805
J. Fluids Eng 132(8), 081201 (Aug 16, 2010) (10 pages) doi:10.1115/1.4002008 History: Received December 22, 2009; Revised June 16, 2010; Published August 16, 2010; Online August 16, 2010
## Abstract
A computational investigation is conducted to study the patterns of airflow induced by corona discharge in the cross section of a circular tube. The secondary flow induced by corona wind in various flow passages has been the subject of numerous investigations. The flow patterns are often identified by multiple recirculation bubbles. Such flow patterns have also been anticipated for circular cross sections where the corona discharge is activated by an electrode situated at the center of the cross section. In this investigation, it is shown that, contrary to public perception, a symmetric corona discharge does not generate a secondary flow in circular cross sections. This investigation then proceeds to demonstrate that the flow responsible for thermal enhancements in circular tubes often reported in the published literature is induced only when there is a slight asymmetry in the position of the electrode. The present computations are performed in two parts. In part one, the electric field equations are solved using the method of characteristics. In part two, the flow equations are solved using a finite-volume method. It is shown that the method of characteristics effectively eliminates the dispersion errors observed in other numerical solutions. The present computations show that the flow in the eccentric configuration is characterized by a corona jet that is oriented along the eccentricity direction and two recirculation zones situated on either sides of the jet. In addition to the computational approach, a number of analytical solutions are presented and compared with the computational results.
<>
## Figures
Figure 1
Geometry of the problem and characteristic lines
Figure 2
The influence of grid refinement on the computational results of electric field for Vo=7.5 kV, ReEHD=2085, and ε=0% (concentric configuration)
Figure 3
Potential distribution along the characteristic lines, shown for Vo=7.5 kV, ReEHD=2085, and comparison with analytical solution (concentric configuration)
Figure 4
Electric field and charge density distributions for Vo=7.5 kV, ReEHD=2085, and ε=0 (concentric configuration)—the inset shows the smooth distribution of charge density in the vicinity of the corona electrode
Figure 5
Grid refinement results of corona jet centerline velocity −Vo=7.5 kV, ReEHD=2085, and ε=1%
Figure 6
Static pressure gradient and electric body force for Vo=7.5 kV, ReEHD=2085, and ε=0% (concentric configuration)
Figure 7
Contour plots of velocity magnitude, kinetic energy and streamfunction in eccentric configuration for different applied potentials and ε=1%. The time-averaged corona currents and ReEHD are (0.66 mA, 2085), (2.4 mA, 5305), and (5.31 mA, 8145), respectively.
Figure 8
The centerline velocity of the corona jet in different potentials and eccentricities
Figure 9
The analytical and numerical results of velocity in rapid acceleration zone for Vo=7.5 kV, ReEHD=2085, and ε=1%
Figure 10
Comparison between von Karman, computational and analytical results of corona jet velocity in impingement zone for Vo=7.5 kV, ReEHD=2085, and ε=1%
Figure 11
Pressure distribution along the corona jet and tube perimeter for Vo=7.5 kV, ReEHD=2085, and ε=1%
Figure 12
Contour plots of vorticity and pressure for different applied potentials and ε=1%. The time-averaged corona currents and ReEHD are (0.66 mA, 2085), (2.4 mA, 5305), and (5.31 mA, 8145), respectively.
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http://mathoverflow.net/questions/87022/kernel-of-the-representation-of-the-mapping-class-group-to-autf-n
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# Kernel of the representation of the mapping class group to $Aut(F_n)$
Let $S_{g,1}$ be a orientable compact surface of genus $g$ with one boundary component and $\Gamma_{g,1}$ the mapping class group. By $F_n$ I denote the free group on $n$ generators.
One obtains a representation $\rho: \Gamma_{g,1} \rightarrow Aut(F_{2g})$.
What is the kernel of $\rho$?
-
To be clear, you are placing a base point on the boundary of $S_{g,1}$. Otherwise, you only get a representation into $\mathrm{Out}(F)$. – HJRW Jan 30 '12 at 13:30
@HW and lsw: Could you clarify what is $\rho$ and why it depends on where the basepoint is (as long as it is fixed)? – Mark Sapir Jan 30 '12 at 15:07
@HW: Thank you for making this precise. @Mark Sapir: You have to consider the induced action on the fundamental group of $S_{g,1}$. By fixing a base-point there is no $Inn(\pi_1)$-action. – lsw Jan 30 '12 at 15:16
@Isw: Why is the kernel non-trivial? – Mark Sapir Jan 30 '12 at 16:24
@Mark Sapir: I don't know. Why is it trivial? – lsw Jan 30 '12 at 16:39
The representation is faithful, since a mapping class is determined by its action on the fundamental group of the surface. A surface is a $K(\pi,1)$, so given any element $Aut(S_{g,1})$, one obtains a (pointed) map $\varphi:S_{g,1}\to S_{g,1}$ which is unique up to homotopy. Now one needs to know that two homotopic homeomorphisms of a surface are isotopic, which is classic (at least one may find this in a paper of Waldhausen). In fact, one may identify the image in $Aut(F_{2g})$ as the subgroup preserving the peripheral element. Also, note that everything should be fixing a basepoint in the boundary, as in HW's comment.
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http://mathhelpforum.com/advanced-algebra/118311-diagonalizable-matrices.html
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## Diagonalizable Matrices
If a matrix has complex eigenvalues, can it be diagonalizable? Furthermore, what ratio of square matrices will be diagonalizable? Does it vary with the size of the matrix?
I'm working on a project and these questions came up and will effect which direction I end up taking it.
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https://codegolf.stackexchange.com/questions/24681/self-containing-logs/24683
|
# The "problem"
Define a function log (or other 3 letter name) which when called will log/print/write (whatever is the default for the language in question) both the instruction (as source) and the first argument. In other words:
i=777
j=333
log(i) //outputs: "log(i) 777"
log(i+j+1) //outputs: "log(i+j+1) 1111"
For all practical purposes the output i: 777 would be enough, but in some languages there are very specific reflection libraries for that, and that would be no challenge, so the entire instruction should be outputted.
# Inspiration
The inspiration for this was me and another programmer discussing how irritating it is that often (with bad debuggers), you write things like console.log("i:", i), next we made a (pretty crazy) javascript (node only) solution (it outputs i: 777 rather than the entire line of source) which was suprisingly long and reminded me of codegolfing and made me wonder how much better other (especially code golfing) languages would fare.
# Bonuses
-10%: No file reading (beyond the compiler)
PS. This is my first 'question' here, so feel free to point out any mistakes I made.
• Welcome to CodeGolf.SE! I personally think your question is pretty decent, but it's usually a better idea to run question ideas through the sandbox to resolve ambiguities etc before people start working on answers. – Martin Ender Mar 21 '14 at 18:25
• Thx and useful@sandbox, might be good to explain it's use on help/on-topic (it's mentioned, but I didn't consider it worth checking the way it was described there). – David Mulder Mar 21 '14 at 18:36
• @WolleVanillebärLutz: Of course it's not, did you see anyone claim that to be true then? – David Mulder Mar 27 '14 at 20:45
• The bounty is for TrungDQ (I think his solution is amazing just from a code perspective (better than our node only solution), regardless of length), have to wait 24 hours though before I can award it. – David Mulder Jun 8 '14 at 15:53
# C (40 -10% = 36) (38 -10% = 34.2)
Note that, in C, a log function can only be defined for a specific type. Therefore, this log "function" takes only int arguments.
#define log(x)printf("log("#x") %d",x)
A more general solution specifies how to print the argument, in addition to the argument itself:
#define lg2(f,x)printf("lg2("#x") "f,x)
which would be used as e.g. lg2("%s", "I'm a string"); or lg2("%f", 3.1415).
• I don't think the final brackets around x are necessary. – user12205 Mar 21 '14 at 19:07
• @ace: I thought they might be needed if the user puts in some odd characters in the argument, but upon reflection I think you're right. I'll remove them. – nneonneo Mar 21 '14 at 19:09
# Python (65 -10% = 58.5)
This assumes your code is in a file (it produces odd output if invoked in the interactive interpreter):
import traceback as t
def log(x):print t.extract_stack()[-2][3],x
It has been tested on Python 2.7.6.
Example:
def foo():
x = 1
log(x)
for i in xrange(10):
log(x+i+1)
return x
log(foo())
outputs
log(x) 1
log(x+i+1) 2
log(x+i+1) 3
log(x+i+1) 4
log(x+i+1) 5
log(x+i+1) 6
log(x+i+1) 7
log(x+i+1) 8
log(x+i+1) 9
log(x+i+1) 10
log(x+i+1) 11
log(foo()) 1
• Nice! Got to say, this is the kind of crazy stuff that interests me as a programmer (negative index on a native function :O ) :P wonders off to find some docs – David Mulder Mar 21 '14 at 19:16
# C++ 12171 67 -10% = 60.3
#include<iostream>
#define log(x)std::cout<<"log("#x") "<<(x)<<"\n"
Used like this:
int main() {
int i = 777;
int j = 333;
log(i);
log(i+j+1);
}
Outputs:
log(i) 777
log(i+j+1) 1111
• You can remove 30 char and make a one-liner if you write it in C instead of C++: #define log(x)printf("log(%s) %d\n",#x,x), but that will only work of integers. – user12205 Mar 21 '14 at 19:02
• @ace: then it only works for one type. (Also, this is the solution I proposed, see below) – nneonneo Mar 21 '14 at 19:02
• @nneonneo I hate it when I forgot to refresh before posting a comment. – user12205 Mar 21 '14 at 19:05
# Rebol3 - 31.5 (35 - 10 %)
Here is a simple implementation shortened from @draegtun that works well for numbers:
log: func[p][print[{log[}p{]}do p]]
Running it outputs:
>> log: func[p][print[{log[}p{]}do p]]
>> i: 777
>> j: 333
>> log [i]
log[ 777 ] 777
>> log[i + j + 1]
log[ i + j + 1 ] 1111
It can be much more flexible (for displaying the form of non-number types) at 42.3 chars (47 - 10%)
log: func[p][print[{log}mold p mold/only do p]]
The output:
>> log: func[p] [print[{log}mold p mold/only do p]]
>> log [join "4" 4]
log [join "4" 4] "44" ;; shows a string
>> log [1 + 2]
log [1 + 2] 3
# Javascript (325)
I think this is the log function you are looking for:
function log(m){L=(new Error()).stack.match(/(at log \([\s\S]+?at .+?:)\d+:\d+/m)[0].split('\n')[1].match(/:\d+:\d+/)[0];N=L.split(':')[1];C=parseInt(L.split(':')[2]);R=new XMLHttpRequest();R.open('GET',location.href,0);R.onload=function(){console.log(R.response.split('\n')[N-1].substr(C-1).split(';')[0]+' = '+m)};R.send()}
# Usage
<script>
function log(m){L=(new Error()).stack.match(/(at log \([\s\S]+?at .+?:)\d+:\d+/m)[0].split('\n')[1].match(/:\d+:\d+/)[0];N=L.split(':')[1];C=parseInt(L.split(':')[2]);R=new XMLHttpRequest();R.open('GET',location.href,0);R.onload=function(){console.log(R.response.split('\n')[N-1].substr(C-1).split(';')[0]+' = '+m)};R.send()}
function doSomething() {
var a = 123; log(a); var b = "Hello, I am TrungDQ!"; log(b);
}
doSomething();
var message = "...or just do it out here";
log(message + "!");
</script>
## Output
log(a) = 123
log(b) = Hello, I am TrungDQ!
log(message + "!") = ...or just do it out here!
## Long code
<script>
function log(msg) {
// Get the line number and offset of the line where is function is called
var lineInfo = (new Error()).stack.match(/(at log \([\s\S]+?at .+?:)\d+:\d+/m)[0].split('\n')[1].match(/:\d+:\d+/)[0];
var lineNum = lineInfo.split(':')[1];
var charOffset = parseInt(lineInfo.split(':')[2]);
// Get the file source
request = new XMLHttpRequest();
request.open('GET', window.location.href, true);
// Get file source code
var response = request.responseText;
// Get the log line
var line = response.split('\n')[lineNum - 1];
// Get the log statement
var logStatement = line.substr(charOffset - 1).split(';')[0];
// Print it
console.log(logStatement + ' = ' + msg);
};
request.send();
}
function doSomething() {
var a = 123; log(a); var b = "Hello, I am TrungDQ!"; log(b);
}
doSomething();
</script>
Only works when the script is put inside <script> tag which is put in .html document because it sends a request to location.href to get the source code. JSfiddle, F12 Dev Tool Console, embbed .js files won't work, I am trying to make it available everywhere...
Anyway, this question is interesting.
• I'm little skeptical this is cross browsers. – Farid Nouri Neshat Mar 23 '14 at 12:27
# Scala - (221 - 10%) = 198.9
Yay macros! This is actually exactly the type of stuff they're for.
import language.experimental.macros
def log(p:Any)=macro l
def l(c:reflect.macros.Context)(p:c.Expr[Any])={import c.universe._;reify{println("log("+(c.Expr[String](Literal(Constant(show(p.tree)))).splice)+") "+p.splice)}}
import language.experimental.macros
def log(p: Any) = macro l
def l(c: reflect.macros.Context)(p: c.Expr[Any]) = {
import c.universe._
val inputString = show(p.tree)
val inputStringExpr = c.Expr[String](Literal(Constant(inputString)))
reify {
println("log(" + (inputStringExpr.splice) + ") " + p.splice)
}
}
Example:
log(1)
val x = 3
log(x)
val y = 4
log(x+y)
Outputs:
log(1) 1
log(x) 3
log(x.+(y)) 7
Since addition is a method call in Scala, it adds that verbose syntax back in, but it's pretty close! It's also a bit more verbose in a couple of other cases.
• Wow, that's quite interesting to see@the addition of the function. So much cool stuff to learn :D – David Mulder Mar 21 '14 at 23:46
# bash (21 - 10% = 18.9)
This:
alias log=echo;set -v
Then use log like you would use echo:
log $((1+1)) or A=2 B=3 log$((A+B))
This method will do all what is required; as a bonus, some extra information will also be printed, but no explicit rule forbid it.
# BASH
Arguments are not passed using "(...)" in BASH, so I let the output of 'log()' fit that style:
$log(){ echo "$FUNCNAME $@:$(($@))"; }$ i=333
$j=777$ log i
log i: 333
$log i+j+1 log i+j+1: 1111 • $((...)) can be $[...] instead but I did not count the chars, so it doesn't matter up to now. – user19214 Jun 8 '14 at 16:28 # Clojure (defmacro log[x] (let [x# ~x] (println "log("'~x")" x#))) Homoiconicity has its benefits! To use: (def i 777) (def j 333) (log i) ;Prints log( i ) 777 (log (+ i j 1)) ;Prints log( (+ i j 1) ) 1111 Let's see what's happening with macroexpand: (macroexpand '(log (+ i j 1))) ;; Prints the following: (let* [x__1__auto__ (+ i j 1)] (clojure.core/println "log(" (quote (+ i j 1)) ")" x__1__auto__)) • If you quote x, do you really need to use an intermediate gensym (ie. x#)? I think you will only evaluate the expression once (btw, I am no Clojure expert) – coredump Nov 2 '14 at 14:40 # Julia, 51*0.9=45.9 julia> x=4 4 julia> macro log(x) println("log($x) $(log(eval(x)))") end julia> @log(x) log(x) 1.3862943611198906 Alternatively, but not meeting the rules julia> @show log(x) log(x) => 1.3862943611198906 ## Tcl, 42.3 (47 - 10%) proc log c {puts [dict g [info fr -1] cmd]\$c}
Usage:
set i 777
set j 333
log $i ;#outputs: "log$i 777"
log [expr {$i+$j+1}] ;#outputs: "log [expr {$i+$j+1}] 1111"
Edit: small improvement
# Common Lisp - 119.7 (133 -10%)
(defmacro @(&whole f &rest r)(let((g(gensym)))(let((,g(multiple-value-list,@r)))(progn(format t"~s~{ ~a~}
"',f,g)(values-list,g)))))
• Named @ because log is the standard logarithm function and locked by default (at least on SBCL). Also, @ is only one character long.
• Acts as a progn, taking a variable number of arguments, but prints to standard output. In real applications, I would probably signal a condition with an S-expression instead of printing space-separated output.
• Contrary to the existing Clojure solution, we ultimately returns the value of the logged expression, so that (@ x) can be used whenever x is used.
• Printing uses prin1, which outputs a read-able string. This is useful when trying to reproduce logged expressions.
• Handles all possible types (see C answer)
• Takes into account mutliple values
• Does not produce different outputs (see Scala answer)
• Works from a file and from REPL (See Pyhton answer)
• Does not require browser/interpreter trick (Python traceback, Javascript request)
# Sample outputs:
CL-USER>(@ (+ 3 2)) ; user input
(@ (+ 3 2)) 5 ; printed output
5 ; result of expression
CL-USER> (@ (values 3 4)) ; input
(@ (VALUES 3 4)) 3 4 ; logging
3 ; first value
4 ; second value
CL-USER>(@ (round 3.4))
(@ (ROUND 3.4)) 3 0.4000001
3 ; rounded value
0.4000001 ; delta
And finally, if I log the above defmacro, I have the ungolfed version:
CL-USER> (@ (defmacro @(&whole f &rest r)(let((g(gensym)))(let((,g(multiple-value-list,@r)))(progn(format t"~s~{ ~a~}
"',f,g)(values-list,g))))))
STYLE-WARNING: redefining COMMON-LISP-USER::@ in DEFMACRO
(@
(DEFMACRO @ (&WHOLE F &REST R)
(LET ((G (GENSYM)))
(LET ((,G (MULTIPLE-VALUE-LIST ,@R)))
(PROGN
(FORMAT T ,"~s~{ ~a~}
"
',F ,G)
(VALUES-LIST ,G)))))) @
@ ; actual result
# PHP 138
You can't redeclare log in PHP without using another module (APD) so I used logg instead, i can resubmit with log example if needed. That's minor, but more sinful i guess is that this assume the log function is on a line by itself. I can update my answer as dictated by comments.
<?php function logg($v){$b=debug_backtrace()[0];$h=fopen($b['file'],"r");for($i=0;$i<$b['line']&&$l=fgets($h);$i++);echo trim($l)."$v";}
example output:
for ($i=1;$i<10; $i++) {$j=$i+1;$k=$j+1; logg($i+$j+$k);
echo "\n";
}
/*
logg($i+$j+$k); 6 logg($i+$j+$k); 9
logg($i+$j+$k); 12 logg($i+$j+$k); 15
logg($i+$j+$k); 18 logg($i+$j+$k); 21
logg($i+$j+$k); 24 logg($i+$j+$k); 27
logg($i+$j+\$k); 30
*/
# JavaScript 55 53
function log(x){console.log('log("'+x+'") '+eval(x))}
Usage:
var i = 777,
j = 333;
log("i")
log("j")
log("12*4")
log("i*j-4")
Output:
log("i") 777
log("j") 333
log("12*4") 48
log("i*j-4") 258737
You MUST use double quotes " otherwise it will not work.
• Regardless of it already bending the rules by not following the example pseudo code I provided, the bigger problem is that it only works if and only if the variables are defined in the global context (I know the eval evaluation context is more complex than that, but point stands) – David Mulder Mar 21 '14 at 19:06
• The point of the challenge was that you don't pass a string... -1 – Doorknob Mar 21 '14 at 20:31
• The point was not doing log("i:", i) ... I don't think it can't be done without ' or " in js ... you can make it smaller using console.log('log('+o+')'+eval(x)) but the output won't match the code line (who cares) – rafaelcastrocouto Mar 21 '14 at 22:34
• You can do it in a single line, I did it in node, how? By throwing an error, getting the stack, reading the file and extracting the line. Yep, kinda crazy :D. Additionally it might be possible using arguments.callee.caller.toString(), but I wasn't able to figure out which line is which when you have two logs. – David Mulder Mar 21 '14 at 23:34
|
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|
https://chemistry.stackexchange.com/questions/49642/is-axis-of-symmetry-considered-a-true-symmetry
|
# Is axis of symmetry considered a true symmetry?
In most of the books when i read about optical activity of a compound this is what they say:
A compound that does not posses any kind of symmetry is optically active. In other words chiral compounds without having any kind of of symmetry are optically active.
However, this holds true when we refer to plane of symmetry or centre of symmetry, because molecules having axis of symmetry or alternate axis of symmetry are still optically active.
For example this compound possesses axis of symmetry but is still optically active:
And therefore i am totally confused about relation of optical activity with symmetry.
• It has to be non trivial. A rotation of $0$ or $2\pi$ about an axis being trivial. – K_P Apr 17 '16 at 16:18
• No, this is not what they say. If you skip a few words from a meaningful sentence, it is quite easy to arrive at nonsense. It is not just "any kind of symmetry"; it is "any kind of inversion symmetry", or "any kind of symmetry with determinant -1". Rotational symmetry does not apply, though it is surely true and real (and non-trivial, for that matter). – Ivan Neretin Apr 17 '16 at 17:24
A molecule is chiral (1, 2) if it does not have an improper axis of rotation. Since your drawing does not seem to show the meso form of 2,3-dichlorobutane, it is optically active.
This is to complement the answer given already and to address @Ivan Neretin's comment under OP's question. It is an example that generally (with lots of asterisks) rotations cannot be excluded. This is a copy from J. March's 6ed (p. 158).
From the text:
Such compounds possess ... an alternating axis of symmetry as in 1
Of course Ivan is correct that you only care for "determinant $-1$" motions or else orientation reversing and in this case we use a rotation by $\pi$ which has determinant 1 (or else belongs to $SO_3$). My group theory is not as good since I am an organic chemist but I think the actual symmetry element is something like $\rho_{\pi/2} \circ r = 1$ whereas the rotation element above is $\rho_\pi=(\rho_{\pi/2} \circ r)\circ (\rho'_{\pi/2} \circ r')=1 \circ 1=1$ and it does have determinant 1 but is not one of the symmetry group elements (in the sense that $x^{34}$ is in $C_4=\{1,x,x^2,x^3 \}$ but only as $x^{34}=x^2$ ).
To make things more complicated March is defining an "alternating axis of symmetry" as:
An alternating axis of symmetry 17 of order n is an axis such that when an object containing such an axis is rotated by 360/n about the axis and then reflection is effected across a plane at right angles to the axis, a new object is obtained that is indistinguishable from the original one
So it appears that in the example of (1) we talk of $Z_4$ symmetry (if we call $Z$ the alternating axis subgroup) and $Z_4 \sim C_2$
Finally, I think that (1) is an extreme example and almost always for organic compounds chirality (or not) is determined by the lack (or not) of a plane of symmetry
• isn't symmetry of axis a disqualifying property for a molecule being an isomer? – bonCodigo Jul 17 '17 at 3:12
|
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|
http://www.cl.cam.ac.uk/~jrh13/hol-light/HTML/term_order.html
|
term_order : term -> term -> bool
SYNOPSIS
Term order for use in AC-rewriting.
DESCRIPTION
This binary predicate implements a crude but fairly efficient ordering on terms that is appropriate for ensuring that ordered rewriting will perform normalization.
FAILURE CONDITIONS
Never fails.
EXAMPLE
This example shows how using ordered rewriting with this term ordering can give normalization under associative and commutative laws given the appropriate rewrites:
# ADD_AC;;
val it : thm =
|- m + n = n + m /\ (m + n) + p = m + n + p /\ m + n + p = n + m + p
\noindent
# TOP_DEPTH_CONV
d + (f + a) + b + (c + e):num;;
|
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|
https://cs.stackexchange.com/questions/87743/weight-constrained-shortest-path-problem-variants
|
# weight constrained shortest path problem variants
Given a graph $G=(V,E)$, metric spaces $\delta:E\rightarrow \mathbb{Z}^{+}$ and $w:E\rightarrow \mathbb{Z}^{+}$, terminal vertices $s,t\in V$, do there exists $s\rightarrow t$ path $P=(V_{p},E_{p})$ such that $\sum_{e\in E_{p}} w(e) \leq W$ and $\sum_{e\in E_{p}} \delta(e) \leq K$, where $W,K\in Z^{+}$.
This is Weight constrained Shortest Path Problem, and known to be NP-complete for undirected as well as directed even acyclic graphs. as long as $\delta$ and $w$ is not equal for all edges. Now if we move weight from edges to vertices and change $\leq$ of total weight constraint to $=$ then the new problem is:
Given a directed acyclic graph $G=(V,E)$, metric spaces $\delta:E\rightarrow \mathbb{Z}^{+}$ and $w:V\rightarrow \mathbb{Z}^{+}$, terminal vertices $s,t\in V$, do there exists $s\rightarrow t$ path $P=(V_{p},E_{p})$ such that $\sum_{v\in V_{p}} w(v) = W$ and $\sum_{e\in E_{p}} \delta(e) \leq K$.
Is this problem known ? is this solvable in P time ? or this is NP-complete too ? I think even if we replace $\mathbb{Z}^{+}$ with $\mathbb{R}^{+}$ it will not change nature of the problem.
If we set $W=\vert V \vert$ then it tries to reach all the vertices which looks like hamiltonian path problem, but it is a DAG.
## Update
I was trying to reduce it from Subset Sum as suggested by @D.W.
Given $X=\{1,2,3\}$ of size $\vert X \vert = n$ I can convert that to a graph of $n^{2}+n+2$ vertices. We can make $n$ layers of $X$ and connect every vertex with all vertices of next layer except itself. Keep a vertex with 0 weight on each layer. As show below.
But this reduction has 2 restrictions.
1. Every Layer is not complete bipartite (e.g. $2_{i}$ does not connect to $2_{i+1}$).
2. Every layer must have a vertex with 0 weight.
Now this does not prove the second problem for General purpose DAG. I can not connect $x_{i}\rightarrow x_{i+1}$ because that will change the original problem. So if there is a DAG that looks very similar to this one but have $n^{2}$ edges between each layer and have no 0 weight vertex this reduction does not apply to that graph.
Also in this formulation I can take the same element twice $1_{1}\rightarrow 2_{2}\rightarrow 1_{3}$ which will add to $4$. Which I should not be allowed
• Have you tried reducing from subset-sum? – D.W. Feb 6 '18 at 8:05
• Yes that was a typo, Thanks. The problem is with the acyclic. But if the problem is known I don't need to reduce. Also I don't think I can reduce the second one from first one. I thought I will check the proof for the first one but that is a private communication. – Neel Basu Feb 6 '18 at 9:37
• I suggest you try reducing from subset-sum. The reduction looks immediate to me, if I'm not missing something, but I'll let you check the details. – D.W. Feb 6 '18 at 13:08
• Yes I understand why it looks immediate each element will be translated to a vertex of a complete graph. The only thing I am afraid of is to prove it for DAG and remove the complete graph restriction. I am not sure how far I can go with vertex splitting to get rid of that. I will try tonight. – Neel Basu Feb 6 '18 at 15:35
• Thanks. Partial Success. But the DAG generated after reduction is not a general DAG. Please check update. – Neel Basu Feb 6 '18 at 19:22
• Definition requires P-time transformation of Input $I_{x}\rightarrow I_{y}$. Now how can I defy this argument ? if $Y$ is chain of complete bipartite graphs (with no infinite edge cost allowed) then how will that transform function generate $I_{y}$ ? – Neel Basu Feb 7 '18 at 8:19
• Also is this even correct ? because I can take the same element twice $1_{1}\rightarrow 2_{2}\rightarrow 1_{3}$ which will add to $4$. Which I should not be allowed. This may lead to a scenario when there is no such subset in the original problem but there is a path in the reduced problem. Also number of solutions not preserved. – Neel Basu Feb 7 '18 at 11:57
|
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|
https://physics.stackexchange.com/questions/154510/wave-particle-duality-as-result-of-taking-different-limits-of-a-qft
|
Wave/particle-duality as result of taking different limits of a QFT
There is an account on dualities in quantum field theories and string theories by Polchinski from last week
http://arxiv.org/abs/1412.5704
At the end of page 4, he writes the wave/particle dichotomy arises from different limits you can take in a quantum field theory.
Which limits are meant here exactly, and can one give a proper example? I assume it might relate to many/few quanta states.
• The particle picture arises from QFT e.g. by looking at the path that gives the larges contribution to the action in a Feynman path integral, which happens to be the classical path. Experimentally one can arrive at this with weak measurements e.g. in a cloud or bubble chamber on a single particle, with each interaction changing the momentum of a particle very little. This works on a "single" high energy particle (although there are still a lot of individual quantum processes!). The wave picture emerges by looking at the collective of many quanta, each of which makes a single interaction. – CuriousOne Dec 22 '14 at 15:06
• I would be interested in seeing a for the interested amateur type answer to this. That is, an answer that expects you to work at it but doesn't assume any specific knowledge of QFT. If the consensus is that such an answer is possible/useful I'd be willing to place a bounty on the question. – John Rennie Dec 23 '14 at 10:54
• @CuriousOne : The least action principle is known from classical mechanics. But I have doubts that picking a path gives a "particle picture". One can pick whatever one wishes. With classical bodies we can deprive the object from the other paths, i.e. limit its evolution to that one path, and it's O.K. But with a quantum object, if we only dare to limit its evolution at one point of that path, to $\Delta r = 0$ the particle subsequently may follow whatever paths in the universe. – Sofia Dec 23 '14 at 11:19
• OP here. The paper suggestion you have two limits, a particle limit and another field limit. I'd like to know/see both different limits, and explicitly with some QFT. – Nikolaj-K Dec 23 '14 at 13:05
• I've placed a (large!) bounty on this because I see it as an important contribution to writing the definitive article on wave particle duality. An answer targeted at the mathematically sophisticated amateur (like me :-) would be ideal. An answer of this type is likely to be long, because I'm guessing lots of side issues will also need to be explained. But then I'm offering the maximum bounty, and you have the Christmas/New Year holiday to write it in :-) – John Rennie Dec 26 '14 at 7:31
There are probably various answers to this question and I will try to provide one that I consider quite interesting. It is a specific realization/example of the fact that the path integral is dominated by estrema of the action.
The wave aspect of a QFT is probably trivial as QFT is dealing with wave equations. This is particularly apparent for massless particles and I will not discuss it any further.
Let me thus focus instead on the opposite limit when the particles are very heavy. I will use the Schwinger proper time and heavily follow Matt Schwartz textbook.
For simplicity, consider the propagator of a scalar particle in an external field source $A_\mu$ that in the Schwinger proper time takes a path-integral form over the particle trajectory $$G_A(x,y)=\langle A|T\phi(x)\phi(y)|A\rangle=\int_0^\infty ds e^{-is m^2}\langle y| e^{-i\hat{H}s}|x\rangle$$ where $$\langle y| e^{-i\hat{H}s}|x\rangle =\int_{z(0)=x}^{z(s)=y} [dz(\tau)] e^{i\mathcal{L}(z,\dot{z})}$$ with $$\mathcal{L}=-\int_0^s d\tau \left(\frac{dz^\mu(\tau)}{2d\tau}\right)^2+e \int A_\mu(z) dz^\mu\,.$$ It is convenient to rescale the variables with the mass, $s\rightarrow s/m^2$ and $\tau\rightarrow/m^2$ so that the path-integral is clearly dominated by the free kinetic energy when the mass is large $$G_A(x,y)=\frac{1}{m^2}\int_0^\infty ds e^{-is}\int_{z(0)=x}^{z(s/m^2)=x} [dz(\tau)]e^{-i\int_0^s d\tau m^2(\frac{dz^\mu}{d\tau})^2+i\int eA_\mu dz^\mu}$$ This is the limit of particle that takes a well definite trajectory since the path-integral is dominated by the point of stationary phase that corresponds to the free particle solution $$z^\mu(\tau)=x^\mu+\tau v^\mu\qquad v^\mu=(y-x)^\mu/s\,.$$ Moreover, on this solution the propagator becomes (after rescaling back to the original variables) $$G_A(x,y)=\int_0^\infty ds e^{-i\left[s m^2+\frac{(y-x)^2}{4s}-ev^\mu\int_0^s d\tau A_\mu z(\tau)\right]}$$ where the last term is the same that one get by adding the source current $$J_\mu=v_\mu \delta(x-v\tau)$$ so that the heavy particle creates the field $A_\mu$ as if moving in a classical trajectory at constant speed. As Schwartz says, when a particle is heavy the QFT can be approximated by treating the particle as a classical source (but treating everything else as quantum, e.g. the particle can possibly generates quantum radiation $A_\mu$ upon which we haven't integrated over yet).
• tl,dr: The stationary phase approximation is better for heavier, i.e. more classical, particles. Therefore you can understand the physics in that limit by considering only the extrema of the action, which is like tracing the path of a particle. – DanielSank Dec 29 '14 at 16:58
• @DanielSank I think you don't realize that extrema of the action can describe a priori also classical wave solutions, not necessarily particle-like solutions. In my answer I show a particular limit where the extremum of the action of a (quantum) field theory gives in fact a (classical) particle-like behavior as opposed to a (classical) wave behavior. – TwoBs Jan 1 '15 at 17:41
• Good point! I did not appreciate that. Thanks. – DanielSank Jan 1 '15 at 17:42
Wave particle duality is not a quantum physical issue! Here is a full description of its simple mechanism, exclusively based on special relativity which is easily understandable by any interested person.
Wave-particle duality is deeply embedded into the foundations of quantum mechanics (Wikipedia).
This statement is entirely disproved in the following by showing one case which may entirely be explained classically: light in vacuum.
The following derivation is based exclusively on the two postulates of special relativity from which is resulting directly and compellingly the entire model for light in vacuum.
There is one unexplored zone in special relativity which seems to yield only meaningless results. When particles are moving not only near speed of light (v < c) but at speed of light (v=c), the Lorentz transforms cease to operate. The proper time is reduced mathematically to zero, but there is no reference system from which this could be observed. Also, lengths would be reduced to zero for such a hypothetical non-existent reference system.
As a consequence, up to now the corresponding equations deriving from special relativity (time dilation and length contraction) were simply confined to massive particles, excluding the case v=c from the domain of definition of these equations. There is no physical legitimation for such a break in their application (implying de facto a limitation of universal validity of special relativity), and Einstein's special relativity does not cease to exist at v=c as it is shown by the means of an example in the following chart:
By consequence, it follows from the equations for proper time and length contraction that a photon which is traveling the distance Sun-Earth according to our observations in t=8 minutes for a distance of s=8 light minutes, has from its (hypothetical) own point of view a proper time t'=0 and travels a distance s'= 0.
If time and traveled distance are both zero that would mean that there was no movement. When I am traveling zero meters in zero seconds, I did not move, and there is no movement which could be subject to a measurement of velocity. My velocity is not defined (0m/ 0 sec.)
The Lorentz factor splits realities
The twin paradox shows with unequaled clarity the effects of the Lorentz factor.
Example: A twin brother undertakes a space travel and returns after 20 years. At his return to Earth the twin brother who remained at home observes that the traveling twin aged only by 5 years.
In this example the observed time on the observer's clock is 20 years. The proper time (and thus the real aging) is only 5 years instead of 20 years. These two realities are linked arithmetically by the proper time equation and by the Lorentz factor.
Moreover we can notice a hierarchical order of realities: We cannot say that the traveling twin has become 20 years older, even if all observers on Earth have measured 20 years. This would be in contradiction with the physical condition of the traveling twin who looks younger than the twin who stayed on Earth. This means with regard to photons that the proper reality of the photon, even if it may not be observed by anyone, reflects its primary reality. All observations are secondary with regard to this primary reality. Even the constant of speed of light c.
By consequence, and in accordance with the wording of the second postulate of special relativity, light velocity c is a secondary observer's reality. We observe a movement of light which according to the primary reality of the photon is a standstill.
The Lorentz factor is assigning to photons two realities, that means, the transmission of the light momentum is double-tracked:
The secondary reality is the (commonly known) observed reality: Maxwell equations are describing a light quantum in the form of an electromagnetic wave moving at speed of light (v = c, t = 8 min, s = 8 light minutes). The transmission of the momentum occurs indirectly from Sun to the wave and then from the wave to Earth.
The primary reality is the unobserved proper reality of the photon: t'=0 and s'=0, proper time and distance are zero, there is no velocity. That means that the momentum is transmitted directly outside of spacetime from Sun to Earth, without intermediate medium.
Result:
1. A classical explanation of Young's double slit: while we are observing nothing but an interfering wave, the particle characteristics of light in vacuum are transmitted directly (path length = 0) and in parallel to the electromagnetic wave.
2. Light in vacuum is a primitive border case of quantum physics which can be explained classically. As a result, the mere wave-particle duality can be described without non-locality issue (see also the open (former bounty) question) as a classical phenomenon.
3. This fact does not change at all quantum physics with all its non locality issues. But it shows that there is one classical case of wave-particle duality, with no need of recourse to quantum mechanics and/ or QFT.
4. A simple answer to the question of NikolajK and John Rennie what the nature of wave-particle duality is.
• I appreciate the effort you've put in, but your answer seems to be unrelated to the original question or to the targets I laid out for the bounty. Just to clarify, while I'm interested in the question because I have a deeper interest in wave-particle duality, to earn this bounty you need to answer Nikolaj's question. – John Rennie Dec 26 '14 at 16:11
• @john rennie: No problem, I understand! Anyway thank you for this very nice Christmas bounty you offered to Stack Exchange users, I find this a very good idea! For your private interests in wave-particle duality, I remain at your disposal with regard to my text. – Moonraker Dec 26 '14 at 16:26
• What you call different "realities" are just different coordinate systems, it's fundamentally no different from the fact that you can describe the same Newtonian scenario with different Galilean coordinate systems that assign different x and y coordinates to a given event. And I don't see how your answer gives a non-quantum version of wave-particle duality, since classical electromagnetic waves aren't measured to set off detectors at highly localized positions like individual quanta (photons, electrons) are. – Hypnosifl Dec 31 '14 at 20:28
• @hypnosifl : Within the first reality there is no coordinate system, instead there is a banal pointlike reality which, however, is real. - Wave-particle duality of photons in vacuum can be explained mathematically and by analogy with the classical twin phenomenon ---- not to be confounded with the fact that photons may be subject to measurements of quantum physics. Photons in vacuum may be considered as primitive border case of quantum physics, characterized by their empty space time interval. – Moonraker Jan 1 '15 at 15:56
• "Wave-particle duality of photons in vacuum can be explained mathematically and by analogy with the classical twin phenomenon" -- Your answer doesn't make clear what precisely this analogy is supposed to consist of, it all seems rather handwavey. What exactly is the SR analogue of the "particle" aspect of the photon and what exactly is the SR analogue of the "wave" aspect, and what is the SR analogue of the "duality" aspect where a given experimental setup will only reveal one or the other? – Hypnosifl Jan 1 '15 at 16:55
These might be two different issues. Wave-particle duality is one issue, different classical limits is another issue.
Wave-particle duality often refers to the fact that when choosing an experiment historically, people sometimes chose options that revealed wave properties and sometimes chose options that revealed particle properties. So the hypothesis was that nature has both qualities waiting to be revealed by different choices of experimental setups that measure the same initial input.
As for classical limits, (assuming you aren't doing MIW or dBB) a classical limit is one where you can ignore (relative) phases (classical fields and particles are entirely real, they have no phase).
For a bosonic field, you can take something like a classical wave limit. You have the option to take a high quantum number limit that is also a coherent state, then there is no relative phase, so the phase can be ignored, and it looks like a classical field. So it's not just a high quanta limit, you also need the coherence. I didn't go into much detail because Motl seems to cover it in detail at the level you are looking for in http://motls.blogspot.com/2011/11/how-classical-fields-particles-emerge.html
You can also take something like a classical particle limit, this is a low quanta limit but also a limit where the energy is kept high. So for the electromagnetic case, this would be single gamma rays, and now the scattering of a single quanta (where QFT reduces to just relativistic quantum mechanics since there is only a single quanta). In this limit the phase doesn't matter for the scattering angle, and you can compute it as compton scattering by a photon of fixed momentum $h\nu$. The details about the QFT to RQM limit (single quanta) is well known and how the high energy RQM scattering reduces to that of compton scattering I think is simply because there are so few options that conserve energy and momentum and scattering states have to be on shell. Again, probably well known.
None of this is as deep as I think you expected, but I wanted to provide the filling in of what I thought the authors meant and it might be things you probably already knew but they just didn't give enough details for you to know it was stuff you already knew.
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http://repository.aust.edu.ng/xmlui/handle/123456789/3902?show=full
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# The monotone wrapped Fukaya category and the open-closed string map
dc.creator Ritter, Alexander F dc.creator Smith, Ivan dc.date.accessioned 2016-07-19 dc.date.accessioned 2018-11-24T23:26:51Z dc.date.available 2016-09-13T09:31:29Z dc.date.available 2018-11-24T23:26:51Z dc.date.issued 2016-08-09 dc.identifier https://www.repository.cam.ac.uk/handle/1810/260145 dc.identifier.uri http://repository.aust.edu.ng/xmlui/handle/123456789/3902 dc.description.abstract We build the wrapped Fukaya category $\textit{W}$($\textit{E}$)for any monotone symplectic manifold $\textit{E}$, convex at infinity. We define the open-closed and closed-open string maps, OC : HH$_{*}$($\textit{W}$($\textit{E}$)) → $\textit{SH}^{*}$($\textit{E}$) and CO : $\textit{SH}^{*}$($\textit{E}$) → HH$^{*}$($\textit{W}$($\textit{E}$)). We study their algebraic properties and prove that the string maps are compatible with the $\textit{c}_1$($\textit{TE}$)-eigenvalue splitting of $\textit{W}$($\textit{E}$). We extend Abouzaid’s generation criterion from the exact to the monotone setting. We construct an acceleration functor $\textit{AF}$ : $\textit{F}$($\textit{E}$) → $\textit{W}$($\textit{E}$) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map $\textit{c}^{*}$ : $\textit{QH}^{*}$($\textit{E}$) → $\textit{SH}^{*}$($\textit{E}$). We define the $\textit{SH}^{*}$($\textit{E}$)-module structure on the Hochschild (co)homology of $\textit{W}$($\textit{E}$) which is compatible with the string maps (this was proved independently for exact convex symplectic manifolds by Ganatra). The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category $\textit{F}$($\textit{E}$), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of $\textit{O}$(−$\textit{k}$) → $\Bbb {CP}^m$ is proper (cohomologically finite) for 1 ≤ $\textit{k}$ ≤ $\textit{m}$. For any monotone negative line bundle $\textit{E}$ over a closed monotone toric manifold $\textit{B}$, we show that $\textit{SH}^{*}$($\textit{E}$) $\neq$ 0, $\textit{W}$($\textit{E}$) is non-trivial and $\textit{E}$ contains a non-displaceable monotone Lagrangian torus $\textit{L}$ on which OC is non-zero. dc.language en dc.publisher Springer dc.publisher Selecta Mathematica dc.title The monotone wrapped Fukaya category and the open-closed string map dc.type Article
## Files in this item
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Ritter_et_al-2016-Selecta_Mathematica-AM.pdf1.052Mbapplication/pdfView/Open
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http://en.wikipedia.org/wiki/Cayley-Dickson_construction
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# Cayley–Dickson construction
(Redirected from Cayley-Dickson construction)
In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras. They are useful composition algebras frequently applied in mathematical physics.
The Cayley–Dickson construction defines a new algebra based on the direct sum of an algebra with itself, with multiplication defined in a specific way and an involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of this) is called the norm.
The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, and next associativity of multiplication.
More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.[1]
## Complex numbers as ordered pairs
Main article: Complex number
The complex numbers can be written as ordered pairs (ab) of real numbers a and b, with the addition operator being component-by-component and with multiplication defined by
$(a, b) (c, d) = (a c - b d, a d + b c).\,$
A complex number whose second component is zero is associated with a real number: the complex number (a, 0) is the real number a.
Another important operation on complex numbers is conjugation. The conjugate (ab)* of (ab) is given by
$(a, b)^* = (a, -b).\,$
The conjugate has the property that
$(a, b)^* (a, b) = (a a + b b, a b - b a) = (a^2 + b^2, 0),\,$
which is a non-negative real number. In this way, conjugation defines a norm, making the complex numbers a normed vector space over the real numbers: the norm of a complex number z is
$|z| = (z^* z)^{1/2}.\,$
Furthermore, for any nonzero complex number z, conjugation gives a multiplicative inverse,
$z^{-1} = {z^* / |z|^2}.\,$
In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional vector space over the real numbers.
Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.
## Quaternions
Main article: Quaternion
The next step in the construction is to generalize the multiplication and conjugation operations.
Form ordered pairs $(a, b)$ of complex numbers $a$ and $b$, with multiplication defined by
$(a, b) (c, d) = (a c - d^* b, d a + b c^*).\,$
Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.
The order of the factors seems odd now, but will be important in the next step. Define the conjugate $(a, b)^*\,$ of $(a, b)$ by
$(a, b)^* = (a^*, -b).\,$
These operators are direct extensions of their complex analogs: if $a$ and $b$ are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.
The product of an element with its conjugate is a non-negative real number:
$(a, b)^* (a, b) = (a^*, -b) (a, b) = (a^* a + b^* b, b a^* - b a^*) = (|a|^2 + |b|^2, 0 ).\,$
As before, the conjugate thus yields a norm and an inverse for any such ordered pair. So in the sense we explained above, these pairs constitute an algebra something like the real numbers. They are the quaternions, named by Hamilton in 1843.
Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional vector space over the real numbers.
The multiplication of quaternions is not quite like the multiplication of real numbers, though. It is not commutative, that is, if $p$ and $q$ are quaternions, it is not generally true that $p q = q p$, but it is true that $p q = (q p)'$, where $(a, b)' = (a, -b)$.
## Octonions
Main article: Octonion
From now on, all the steps will look the same.
This time, form ordered pairs $(p, q)$ of quaternions $p$ and $q$, with multiplication and conjugation defined exactly as for the quaternions:
$(p, q) (r, s) = (p r - s^* q, s p + q r^*).\,$
Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were $r^*q$ rather than $qr^*$, the formula for multiplication of an element by its conjugate would not yield a real number.
For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.
This algebra was discovered by John T. Graves in 1843, and is called the octonions or the "Cayley numbers".
Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space over the real numbers.
The multiplication of octonions is even stranger than that of quaternions. Besides being non-commutative, it is not associative: that is, if $p$, $q$, and $r$ are octonions, it is generally not true that
$(p q) r = p (q r).\$
For the reason of this non-associativity, octonions have no matrix representation.
## Further algebras
The algebra immediately following the octonions is called the sedenions. It retains an algebraic property called power associativity, meaning that if $s$ is a sedenion, $s^n s^m = s^{n + m}$, but loses the property of being an alternative algebra and hence cannot be a composition algebra.
The Cayley–Dickson construction can be carried on ad infinitum, at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. All the algebras generated in this way over a field are quadratic: that is, each element satisfies a quadratic equation with coefficients from the field.[2]
## General Cayley–Dickson construction
Albert (1942, p. 171) gave a slight generalization, defining the product and involution on B=AA for A an algebra with involution (with (xy)* = y*x*) to be
$(p, q) (r, s) = (p r - \gamma s^* q, s p + q r^*)\,$
$(p, q)^* = (p^*, -q)\$
for γ an additive map that commutes with * and left and right multiplication by any element. (Over the reals all choices of γ are equivalent to −1, 0 or 1.) In this construction, A is an algebra with involution, meaning:
• A is an abelian group under +
• A has a product that is left and right distributive over +
• A has an involution *, with x** = x, (x + y)* = x* + y*, (xy)* = y*x*.
The algebra B=AA produced by the Cayley–Dickson construction is also an algebra with involution.
B inherits properties from A unchanged as follows.
• If A has an identity 1A, then B has an identity (1A, 0).
• If A has the property that x + x*, xx* associate and commute with all elements, then so does B. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.
Other properties of A only induce weaker properties of B:
• If A is commutative and has trivial involution, then B is commutative.
• If A is commutative and associative then B is associative.
• If A is associative and x + x*, xx* associate and commute with everything, then B is an alternative algebra.
## Notes
1. ^ Schafer (1995) p.45
2. ^ Schafer (1995) p.50
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https://www.physicsforums.com/threads/how-do-we-formalize-the-following-for-all-natural-nos.521923/
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How do we formalize the following:For all natural Nos :
1. Aug 16, 2011
solakis
How do we formalize the following:
For all natural Nos : $n^2$ is even$\Rightarrow$ n is even
2. Aug 16, 2011
Rasalhague
Re: formalization
Here's one way:
$$(\forall n \in \mathbb{N})[((\exists p \in \mathbb{N})[n^2=2p])\Rightarrow((\exists q\in \mathbb{N})[n=2q])].$$
Here's another:
$$\left \{ n \in \mathbb{N} \; | \; (\exists p \in \mathbb{N})[n^2 = 2p] \right \} \subseteq \left \{ n \in \mathbb{N} \; | \; (\exists q \in \mathbb{N})[n=2q] \right \}.$$
The symbol $\mathbb{N}$, like the name "the natural numbers", is ambiguous, so you might want to specify whether you mean the positive integers, $\mathbb{N}_1=\mathbb{Z}_+ = \left \{ 1,2,3,... \right \}$, or the non-negative integers, $\mathbb{N}_0 =\mathbb{Z}_+ \cup \left \{ 0 \right \}=0,1,2,3,...$.
3. Aug 18, 2011
solakis
Re: formalization
Thank you ,i think the first formula is the more suitable to use in formalized mathematics.
The question now is how do we prove this formula in formalized mathematics
4. Aug 18, 2011
Rasalhague
Re: formalization
$$1. \enspace (\exists p \in \mathbb{N}_0)[n^2 = 2p]$$
$$\Rightarrow ((\exists a \in \mathbb{N}_0)[n=2a]) \vee ((\exists b \in \mathbb{N}_0)[n=2b+1]).$$
$$2. \enspace (\exists b \in \mathbb{N}_0)[n=2b+1])$$
$$\Rightarrow (\exists q \in \mathbb{N}_0)[n^2=(2b+1)^2=4b^2+4b+1=2q+1]$$
$$\Rightarrow \neg (\exists p \in \mathbb{N}_0)[n^2 = 2p].$$
$$3. \enspace\therefore ((\exists p \in \mathbb{N}_0)[n^2 = 2p]) \Rightarrow (\exists a \in \mathbb{N}_0)[n=2a] \enspace\enspace \blacksquare$$
Or, to see the argument clearer, let $A = (\exists a \in \mathbb{N}_0)[n=2a]$ and let $B = (\exists b \in \mathbb{N}_0)[n=2b+1],$ and suppose $(\exists p \in \mathbb{N}_0)[n^2 = 2p]$. Then $A \vee B.$ But $\neg B.$ So $A.$
5. Aug 18, 2011
solakis
Re: formalization
I think to change the variables inside the existential quantifier without dropping it first it is not allowed by logic.
Also it becomes confusing and impossible to follow.
I also wander is necessarily a formalized proof also a formal one??
Or to put in a better way is it possible to have a proof like the one above ,where one uses only formulas , without mentioning the laws of logic and the theorems upon which these laws act to give us the formalized conclusions of the above proof??
6. Aug 19, 2011
Rasalhague
Re: formalization
Simple answer: I don't know. I'm curious. I hope someone better informed can give say more about this. Seems like it would at least be a good idea to make explicit which statements and which rules of inference are used in each line; so, we could append to the final line: / 1,2,DS. (Disjunctive syllogism.) I could be mistaken but here is what looks like an example of someone using formal and formalized as synonyms: http://arxiv.org/abs/math/0410224
A problem with my proof as it stands: a full, formal proof would have to justify the assumption that natural numbers are odd or even but not both.
I don't see what the problem is with substitution. Maybe you could elaborate?
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http://mathematica.stackexchange.com/users/2983/luser-droog?tab=activity&sort=comments
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# luser droog
less info
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bio website code.google.com/p/xpost location St. Louis, MO age 34 member for 7 months seen Apr 20 at 6:00 profile views 1
I've written a postscript interpreter and a postscript debugger as well as some pretty pictures, available at http://code.google.com/p/xpost/downloads/list. Xpost has fewer features than ghostscript (most notably, no definefont, no <</dictionary/syntax>>, no garbage collection, output devices limited to PNG, PDF(rather bloated, by default), and X11 windows); but for certain tasks, this proves to make xpost faster than ghostscript (computation-heavy, memory-light programs mostly).
The profession of scribe was considered to be most honourable, and its rewards were great, for no rank and no dignity were too high for the educated scribe.
E.A. Wallis Budge, The Literature of the Ancient Egyptians
Positive philosophy stands in a very ambiguous relation to Kant's views. It accepts them and does not accept them: it accepts and considers them correct in their relation to the direct experience of the organs of sense -- what we see, hear, touch. That is, positive philosophy recognizes the subjectivity of our receptivity, and recognizes everything that we perceive in objects as imposed upon them by ourselves -- but this in relation to the direct experiences of the senses only.
When it concerns itself with "scientific experience" however, in which precise instruments and calculations are used, positive philosophy evidently considers Kant's view in relation to that invalid, assuming tht "scientific experience" makes known to us the very substance of things, the true causes of our sensations -- or if it does not do so now, it brings us closer to the truth of things, and can inform us later.
... Of course it is possible not to know Kant, but it is impossible to controvert him.
Pyotr Dem'anovich Ouspensky, Tertium Organum
It is impossible to disassociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based, the abstract concepts which call these phenomena to mind, and the words in which the concepts are expressed. To call forth a concept, a word is needed; to portray a phenomenon, a concept is needed. All three mirror one and the same reality.
Antoine Lavoisier, 1789
[L]ogic, in so far as it exhibits the universal and necessary laws of the understanding, must in these very laws present us with criteria of truth. Whatever contradicts these rules is false, because thereby the understanding is made to contradict its own universal laws of thought; that is, to contradict itself.
Kant, Critique of Pure Reason, I, 2nd Part, II. Of Transcendental Logic
He who in reasoning cites authority is making use of his memory rather than of his intellect.
Leonardo Da Vinci, Thoughts on Art and Life
|
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|
https://parasol-lab.gitlab.io/stapl-home/docs/sgl/creation/generators/make_torus_3D.html
|
# make_torus_3D
Defined in <stapl/containers/graph/generators/torus_3D.hpp>
template<typename GraphView>
GraphView make_torus_3D(size_t nx,size_t ny,size_t nz,bool bidirectional)
## Summary
Generates an x-by-y-by-z 3D torus.
The generated torus will have x vertices in the horizontal direction, y vertices in the vertical direction, and z vertices in the in-out direction. This is distributed n/p in the z-direction, where n = xyz.
The returned view owns its underlying container.
#### Parameters
• nx: Size of the x-dimension of the torus.
• ny: Size of the y-dimension of the torus.
• nz: Size of the z-dimension of the torus.
• bidirectional: True to add back-edges in a directed graph, false for forward edges only.
#### Returns
A view over the generated graph.
#### Example
using view_type = stapl::graph_view<stapl::multidigraph<int>>;
auto v = stapl::generators::make_torus_3D<view_type>(16, 8, 16, true);
|
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|
http://math.stackexchange.com/questions/5746/second-order-taylor-method-to-solve-system-of-equations
|
# Second order Taylor method to solve system of equations
How do I use second order Taylor method to solve a system of non-linear equations? Is there a good reference that gives details? I found mentions of it in a dozen of numerical analysis books, but no examples
Specifically, $f:\mathbb{R}^n \to \mathbb{R}^m$, solve $f(\mathbf{x})=\mathbf{0}$ using second order Taylor expansion of $f$ around initial guess $\mathbf{x_0}$
-
Which NA books have you been looking at? Note that the multidimensional version of Newton-Raphson involves expanding $f(\mathbf{x})$ up to the Jacobian-containing term (first-order). – Guess who it is. Sep 30 '10 at 4:07
the ones that come up in google books when I search for "higher order Taylor" – Yaroslav Bulatov Sep 30 '10 at 4:14
I don't see any practical generalizations of Halley's method to multidimensional equations, if that's what you're getting at; the quadratic term involves a rank-3 tensor, and it looks unwieldy to manipulate in manner of how one would derive Halley's method from the Taylor expansion. – Guess who it is. Sep 30 '10 at 4:24
It looks unwieldy, which is why I'm looking for some reference that goes through the details – Yaroslav Bulatov Sep 30 '10 at 4:33
Actually, what I was getting at is that there's one question you have to ask first: "how does one 'invert' a rank-3 tensor?" – Guess who it is. Sep 30 '10 at 23:47
I still consider converting simultaneous nonlinear equations into a nonlinear least squares problem as a bit of a cheat, but... :D Anyway, what Yaroslav has originally is a $m$-dimensional vector function with $n$ independent variables, so the first term of the Taylor expansion has a vector, the second (linear) term has a matrix... you get the idea. Rather unfortunate "order" is a very overloaded mathematical term, I must say. – Guess who it is. Sep 30 '10 at 23:44
|
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|
https://link.springer.com/chapter/10.1007/978-3-642-03869-3_41
|
Euro-Par 2009: Euro-Par 2009 Parallel Processing pp 417-428
# MapReduce Programming Model for .NET-Based Cloud Computing
• Chao Jin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5704)
## Abstract
Recently many large scale computer systems are built in order to meet the high storage and processing demands of compute and data-intensive applications. MapReduce is one of the most popular programming models designed to support the development of such applications. It was initially created by Google for simplifying the development of large scale web search applications in data centers and has been proposed to form the basis of a ‘Data center computer’ This paper presents a realization of MapReduce for .NET-based data centers, including the programming model and the runtime system. The design and implementation of MapReduce.NET are described and its performance evaluation is presented.
## Keywords
Cloud Computing Runtime System Reduce Task MapReduce Program Model Sort Phase
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
## References
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McNabb, A.W., Monson, C.K., Seppi, K.D.: Parallel PSO Using MapReduce. In: Proc. of the Congress on Evolutionary Computation (2007)Google Scholar
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Jin, C., Vecchiola, C., Buyya, R.: MRPGA: An Extension of MapReduce for Parallelizing Genetic Algorithms. In: Proc. of 4th International Conference on e-Science (2008)Google Scholar
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Ranger, C., Raghuraman, R., Penmetsa, A., Bradski, G., Kozyrakis, C.: Evaluating MapReduce for Multi-core and Multiprocessor Systems. In: Proc. of the 13th Intl. Symposium on High-Performance Computer Architecture (2007)Google Scholar
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Patterson, D.A.: Technical perspective: the data center is the computer. Communications of the ACM 51(1), 105 (2008)
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Gregor, D., Lumsdaine, A.: Design and Implementation of a High-Performance MPI for C$$\sharp$$ and the Common Language Infrastructure. In: Proc. of the 13th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (2008)Google Scholar
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Yang, H.C., Dasdan, A., Hsiao, R.L., Stott Parker, D.: Map-Reduce-Merge: simplified relational data processing on large clusters. In: Proc. of SIGMOD (2007)Google Scholar
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Dean, J., Ghemawat, S.: MapReduce: Simplified Data Processing on Large Clusters. In: Proc. of the 6th Symposium on Operating System Design and Implementation (2004)Google Scholar
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Varia, J.: Cloud Architectures. White Paper of Amazon (2008), jineshvaria.s3.amazonaws.com/public/cloudarchitectures-varia.pdf
10. 10.
Isard, M., Budiu, M., Yu, Y., Birrell, A., Fetterly, D.: Dryad: Distributed Data-Parallel Programs from Sequential Building Blocks. In: Proc. of European Conference on Computer Systems, EuroSys (2007)Google Scholar
11. 11.
Kruijf, M., Sankaralingam, K.: MapReduce for the Cell B.E. Architecture. TR1625, Technical Report, The University of Wisconsin-Madison (2007)Google Scholar
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Buyya, R., Yeo, C.S., Venugopal, S., Broberg, J., Brandic, I.: Cloud Computing and Emerging IT Platforms: Vision, Hype, and Reality for Delivering Computing as the 5th Utility. Future Generation Computer Systems 25(6), 599–616 (2009)
13. 13.
Bryant, R.E.: Data-Intensive Supercomputing: The Case for DISC. CMU-CS-07-128, Technical Report, Carnegie Mellon University (2007)Google Scholar
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Chen, S., Schlosser, S.W.: Map-Reduce Meets Wider Varieties of Applications. IRP-TR-08-05, Technical Report, Intel Research Pittsburgh (2008)Google Scholar
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Hey, T., Trefethen, A.: The data deluge: an e-Science perspective. In: Grid Computing: Making the Global Infrastructure a Reality, pp. 809–824 (2003)Google Scholar
|
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|
https://vr-spec.readthedocs.io/en/1.0rc/terms_and_model.html
|
# Terminology & Information Model¶
When biologists define terms in order to describe phenomena and observations, they rely on a background of human experience and intelligence for interpretation. Definitions may be abstract, perhaps correctly reflecting uncertainty of our understanding at the time. Unfortunately, such terms are not readily translatable into an unambiguous representation of knowledge.
For example, “allele” might refer to “an alternative form of a gene or locus” [Wikipedia], “one of two or more forms of the DNA sequence of a particular gene” [ISOGG], or “one of a set of coexisting sequence alleles of a gene” [Sequence Ontology]. Even for human interpretation, these definitions are inconsistent: does the definition precisely describe a specific change on a specific sequence, or, rather, a more general change on an undefined sequence? In addition, all three definitions are inconsistent with the practical need for a way to describe sequence changes outside regions associated with genes.
The computational representation of biological concepts requires translating precise biological definitions into data structures that can be used by implementers. This translation should result in a representation of information that is consistent with conventional biological understanding and, ideally, be able to accommodate future data as well. The resulting computational representation of information should also be cognizant of computational performance, the minimization of opportunities for misunderstanding, and ease of manipulating and transforming data.
Accordingly, for each term we define below, we begin by describing the term as used by biologists (biological definition) as available. When a term has multiple biological definitions, we explicitly choose one of them for the purposes of this specification. We then provide a computer modelling definition (computational definition) that reformulates the biological definition in terms of information content. We then translate each of these computational definitions into precise specifications for the (logical model). Terms are ordered “bottom-up” so that definitions depend only on previously-defined terms.
Note
The keywords “MUST”, “MUST NOT”, “REQUIRED”, “SHALL”, “SHALL NOT”, “SHOULD”, “SHOULD NOT”, “RECOMMENDED”, “MAY”, and “OPTIONAL” in this document are to be interpreted as described in RFC 2119.
## Data Model Notes and Principles¶
• VR uses snake_case to represent compound words. Although the schema is currently JSON-based (which would typically use camelCase), VR itself is intended to be neutral with respect to languages and database.
• VR objects are value objects. Two objects are considered equal if and only if their respective attributes are equal. As value objects, VR objects are used as primitive types and SHOULD NOT be used as containers for related data. Instead, related data should be associated with VR objects through identifiers. See Computed Identifiers.
• Error handling is intentionally unspecified and delegated to implementation. The VR-Spec provides foundational data types that enable significant flexibility. Except where required by this specification, implementations may choose whether and how to validate data. For example, implementations MAY choose to validate that particular combinations of objects are compatible, but such validation is not required.
• We recognize that a common desire may be to have human-readable identifiers associated with VR objects. We recommend using the _id field (see Optional Attributes below) to create a lookup for any such identifiers (see example usage), and provide reference methods for creating VR identifiers from other common variant formats (see the HGVS translation example).
## Optional Attributes¶
• VR attributes use a leading underscore to represent optional attributes that are not part of the value object. Such attributes are not considered when evaluating equality or creating computed identifiers. Currently, the only such attribute in the specification is the _id attribute.
• The _id attribute is available to identifiable objects, and MAY be used by an implementation to store the identifier for a VR object. If used, the stored _id element MUST be a CURIE. If used for creating a Truncated Digest (sha512t24u) for parent objects, the stored element must be a GA4GH Computed Identifier.
## Primitive Concepts¶
### CURIE¶
Biological definition
None.
Computational definition
A CURIE formatted string. A CURIE string has the structure prefix:reference (W3C Terminology).
Implementation guidance
• All identifiers in VR-Spec MUST be a valid Compact URI (CURIE), regardless of whether the identifier refers to GA4GH VR objects or external data.
• For GA4GH VR Objects, this specification RECOMMENDS using globally unique Computed Identifiers for use within and between systems.
• For external data, CURIE-formatted identifiers MUST be used. When an appropriate namespace exists at identifiers.org, that namespace MUST be used. When an appropriate namespace does not exist at identifiers.org, support is implementation-dependent. That is, implementations MAY choose whether and how to support informal or local namespaces.
• Implemantions MUST use CURIE identifiers verbatim and MUST NOT be modified in any way (e.g., case-folding). Implementations MUST NOT expose partial (parsed) identifiers to any client.
Example
Identifiers for GRCh38 chromosome 19:
ga4gh:SQ.IIB53T8CNeJJdUqzn9V_JnRtQadwWCbl
refseq:NC_000019.10
grch38:19
See Identifier Construction for examples of CURIE-based identifiers for VR objects.
### Residue¶
Biological definition
A residue refers to a specific monomer within the polymeric chain of a protein or nucleic acid (Source: Wikipedia Residue page).
Computational definition
A character representing a specific residue (i.e., molecular species) or groupings of these (“ambiguity codes”), using one-letter IUPAC abbreviations for nucleic acids and amino acids.
### Sequence¶
Biological definition
A contiguous, linear polymer of nucleic acid or amino acid residues.
Computational definition
A character string of Residues that represents a biological sequence using the conventional sequence order (5’-to-3’ for nucleic acid sequences, and amino-to-carboxyl for amino acid sequences). IUPAC ambiguity codes are permitted in Sequences.
Information model
A Sequence is a string, constrained to contain only characters representing IUPAC nucleic acid or amino acid codes.
Implementation guidance
• Sequences MAY be empty (zero-length) strings. Empty sequences are used as the replacement Sequence for deletion Alleles.
• Sequences MUST consist of only uppercase IUPAC abbreviations, including ambiguity codes.
• A Sequence provides a stable coordinate system by which an Allele MAY be located and interpreted.
• A Sequence MAY have several roles. A “reference sequence” is any Sequence used to define an Allele. A Sequence that replaces another Sequence is called a “replacement sequence”.
• In some contexts outside the VR specification, “reference sequence” may refer to a member of set of sequences that comprise a genome assembly. In the VR specification, any sequence may be a “reference sequence”, including those in a genome assembly.
• For the purposes of representing sequence variation, it is not necessary that Sequences be explicitly “typed” (i.e., DNA, RNA, or AA).
## Composite Concepts¶
### Interval (Abstract Class)¶
Biological definition
None.
Computational definition
The Interval abstract class defines a range on a Sequence, possibly with length zero, and specified using Interbase Coordinates. An Interval MAY be a SimpleInterval with a single start and end coordinate. Future Location and Interval types will enable other methods for describing where Variation occurs. Any of these MAY be used as the Interval for Location.
#### SimpleInterval¶
Computational definition
An Interval (Abstract Class) with a single start and end coordinate.
Information model
Field Type Limits Description
type string 1..1 Interval type; MUST be set to ‘SimpleInterval
start uint64 1..1 start position
end uint64 1..1 end position
Implementation guidance
• Implementations MUST enforce values 0 ≤ start ≤ end. In the case of double-stranded DNA, this constraint holds even when a feature is on the complementary strand.
• VR uses Interbase coordinates because they provide conceptual consistency that is not possible with residue-based systems (see rationale). Implementations will need to convert between interbase and 1-based inclusive residue coordinates familiar to most human users.
• Interbase coordinates start at 0 (zero).
• The length of an interval is end - start.
• An interval in which start == end is a zero width point between two residues.
• An interval of length == 1 MAY be colloquially referred to as a position.
• Two intervals are equal if the their start and end coordinates are equal.
• Two intervals intersect if the start or end coordinate of one is strictly between the start and end coordinates of the other. That is, if:
• b.start < a.start < b.end OR
• b.start < a.end < b.end OR
• a.start < b.start < a.end OR
• a.start < b.end < a.end
• Two intervals a and b coincide if they intersect or if they are equal (the equality condition is REQUIRED to handle the case of two identical zero-width Intervals).
• <start, end>=<0,0> refers to the point with width zero before the first residue.
• <start, end>=<i,i+1> refers to the i+1th (1-based) residue.
• <start, end>=<N,N> refers to the position after the last residue for Sequence of length N.
• See example notebooks in GA4GH VR Python Implementation.
Example
{
"end": 44908822,
"start": 44908821,
"type": "SimpleInterval"
}
### Location (Abstract Class)¶
Biological definition
As used by biologists, the precision of “location” (or “locus”) varies widely, ranging from precise start and end numerical coordinates defining a Location, to bounded regions of a sequence, to conceptual references to named genomic features (e.g., chromosomal bands, genes, exons) as proxies for the Locations on an implied reference sequence.
Computational definition
The Location abstract class refers to position of a contiguous segment of a biological sequence. The most common and concrete Location is a SequenceLocation, i.e., a Location based on a named sequence and an Interval on that sequence. Additional Intervals and Locations may also be conceptual or symbolic locations, such as a cytoband region or a gene. Any of these may be used as the Location for Variation.
Implementation Guidance
• Location refers to a position. Although it MAY imply a sequence, the two concepts are not interchangable, especially when the location is non-specific (e.g., a range) or symbolic (a gene).
#### SequenceLocation¶
Biological definition
None
Computational definition
A Location subclass for describing a defined Interval (Abstract Class) over a named Sequence.
Information model
Field Type Limits Description
_id CURIE 0..1 Location Id; MUST be unique within document
type string 1..1 Location type; MUST be set to ‘SequenceLocation
sequence_id CURIE 1..1 An id mapping to the Computed Identifiers of the external database Sequence containing the sequence to be located.
interval Interval (Abstract Class) 1..1 Position of feature on reference sequence specified by sequence_id.
Implementation guidance
• For a Sequence of length n:
• 0 ≤ interval.startinterval.endn
• interbase coordinate 0 refers to the point before the start of the Sequence
• interbase coordinate n refers to the point after the end of the Sequence.
• Coordinates MUST refer to a valid Sequence. VR does not support referring to intronic positions within a transcript sequence, extrapolations beyond the ends of sequences, or other implied sequence.
Important
HGVS permits variants that refer to non-existent sequence. Examples include coordinates extrapolated beyond the bounds of a transcript and intronic sequence. Such variants are not representable using VR and MUST be projected to a genomic reference in order to be represented.
Example
{
"interval": {
"end": 44908822,
"start": 44908821,
"type": "SimpleInterval"
},
"type": "SequenceLocation"
}
### State (Abstract Class)¶
Biological definition
None.
Computational definition
State objects are one of two primary components specifying a VR Allele (in addition to Location (Abstract Class)), and the designated components for representing change (or non-change) of the features indicated by the Allele Location. As an abstract class, State may encompass concrete Sequence changes (see SequenceState), complex translocations, copy number changes, expression variation, rule-based variation, and more (see State Classes).
#### SequenceState¶
Biological definition
None.
Computational definition
The SequenceState class specifically captures a Sequence as a State (Abstract Class). This is the State class to use for representing “ref-alt” style variation, including SNVs, MNVs, del, ins, and delins.
Information model
Field Type Limits Description
type string 1..1 State type; MUST be set to ‘SequenceState
sequence string 1..1 The string of sequence residues that is to be used as the state for other types.
Example
{
"sequence": "T",
"type": "SequenceState"
}
### Variation¶
Biological definition
In biology, variation is often used to mean genetic variation, describing the differences observed in DNA among individuals.
Computational definition
The Variation abstract class is the top-level object in the VR Schema Diagram and represents the concept of a molecular state. The representation and types of molecular states are widely varied, and there are several Variation Classes currently under consideration to capture this diversity. The primary Variation subclass defined by the VR 1.0 specification is the Allele, with the Text subclass for capturing other Variations that are not yet covered.
#### Allele¶
Biological definition
One of a number of alternative forms of the same gene or same genetic locus. In the context of biological sequences, “allele” refers to one of a set of specific changes within a Sequence. In the context of VR, Allele refers to a Sequence or Sequence change with respect to a reference sequence, without regard to genes or other features.
Computational definition
An Allele is a specific, single, and contiguous Sequence at a Location (Abstract Class). Each alternative Sequence may be empty, shorter, longer, or the same length as the interval (e.g., due to one or more indels).
Information model
Field Type Limits Description
_id CURIE 0..1 Variation Id; MUST be unique within document
type string 1..1 Variation type; MUST be set to ‘Allele
location Location (Abstract Class) 1..1 Where Allele is located
state State (Abstract Class) 1..1 State at location
Implementation guidance
• Implementations MUST enforce values interval.end ≤ sequence_length when the Sequence length is known.
• Alleles are equal only if the component fields are equal: at the same location and with the same state.
• Alleles MAY have multiple related representations on the same Sequence type due to normalization differences.
• Implementations SHOULD normalize Alleles using “justified” normalization whenever possible to facilitate comparisons of variation in regions of representational ambiguity.
• Implementations MUST normalize Alleles using “justified” normalization when generating a Computed Identifiers.
• When the alternate Sequence is the same length as the interval, the lengths of the reference Sequence and imputed Sequence are the same. (Here, imputed sequence means the sequence derived by applying the Allele to the reference sequence.) When the replacement Sequence is shorter than the length of the interval, the imputed Sequence is shorter than the reference Sequence, and conversely for replacements that are larger than the interval.
• When the replacement is “” (the empty string), the Allele refers to a deletion at this location.
• The Allele entity is based on Sequence and is intended to be used for intragenic and extragenic variation. Alleles are not explicitly associated with genes or other features.
• Biologically, referring to Alleles is typically meaningful only in the context of empirical alternatives. For modelling purposes, Alleles MAY exist as a result of biological observation or computational simulation, i.e., virtual Alleles.
• “Single, contiguous” refers the representation of the Allele, not the biological mechanism by which it was created. For instance, two non-adjacent single residue Alleles could be represented by a single contiguous multi-residue Allele.
• The terms “allele” and “variant” are often used interchangeably, although this use may mask subtle distinctions made by some users.
• In the genetics community, “allele” may also refer to a haplotype.
• “Allele” connotes a state whereas “variant” connotes a change between states. This distinction makes it awkward to use variant to refer to the concept of an unchanged position in a Sequence and was one of the factors that influenced the preference of “Allele” over “Variant” as the primary subject of annotations.
• See Use “Allele” rather than “Variant” for further details.
• When a trait has a known genetic basis, it is typically represented computationally as an association with an Allele.
• This specification’s definition of Allele applies to all Sequence types (DNA, RNA, AA).
Example
{
"location": {
"interval": {
"end": 44908822,
"start": 44908821,
"type": "SimpleInterval"
},
"type": "SequenceLocation"
},
"state": {
"sequence": "T",
"type": "SequenceState"
},
"type": "Allele"
}
#### Text¶
Biological definition
None
Computational definition
The Text subclass of Variation is intended to capture textual descriptions of variation that cannot be parsed by other Variation subclasses, but are still treated as variation.
Information model
Field Type Limits Description
_id CURIE 0..1 Variation Id; MUST be unique within document
type string 1..1 Variation type; MUST be set to ‘Text
definition string 1..1 The textual variation representation not parsable by other subclasses of Variation.
Implementation guidance
• An implementation MUST represent Variation with subclasses other than Text if possible.
• An implementation SHOULD define or adopt conventions for defining the strings stored in Text.definition.
• If a future version of VR-Spec is adopted by an implementation and the new version enables defining existing Text objects under a different Variation subclass, the implementation MUST construct a new object under the other Variation subclass. In such a case, an implementation SHOULD persist the original Text object and respond to queries matching the Text object with the new object.
• Additional Variation subclasses are continually under consideration. Please open a GitHub issue if you would like to propose a Variation subclass to cover a needed variation representation.
Example
{
"definition": "APOE loss",
"type": "Text"
}
|
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https://blogs.mathworks.com/developer/2016/05/31/scaling-fibonacci/
|
# Small Fibs Eventually Become Large Fibs – Another exercise in scaling
OK, I had way too much fun writing that last blog post exploring algorithmic scaling. I want to explore this a bit more with a some "real" code, and what is more real than a function to calculate the lovely fibonacci number, the darling of interview questions, algorithms classes, and our own previous blogs. This will be fun to explore not only to cement the method we are using to explore scalability, but also because it is fun to see how fast and elegantly the fibonacci function can be written in MATLAB.
First, let's look at the most obvious solution based on the definition directly:
function f = fibonacci1(n)
if n < 1
f = 0;
elseif n < 3
f = 1;
else
f = fibonacci1(n-1) + fibonacci1(n-2);
end
end
This is very nice and readable and reads exactly like the definition of fibonacci numbers, but we will see that unfortunately it is not very efficient at all. This is not efficient because the recursive approach needs to continually recalculate lower values of $n$ .
Let's create a parameterized test to invoke this algorithm. As I mentioned, this solution explodes pretty quickly so I am just going to take 10 linearly spaced points from 0 to 25. Also note that the calculation is fast for small n so we need to do this in a loop to get valid measurements.
classdef FirstFibonacciTest < matlab.perftest.TestCase
properties(TestParameter)
size = num2cell(round(linspace(0,25,10)));
end
methods(Test)
function testFibonacci(~, size)
for idx = 1:1000
fibonacci1(size);
end
end
end
end
How does it do?
recursiveResult = runperf('FirstFibonacciTest');
findMins = @(resultArray) arrayfun(@(result) min(result.Samples.MeasuredTime), resultArray);
ax = axes('XScale','log','YScale','log');
ylabel('Execution time (s)');
xlabel('Problem Size');
hold on;
grid on;
problemSize = linspace(0,25,10);
plot(ax, problemSize, findMins(recursiveResult), 'LineWidth', 2)
leg = legend({'Recursive'});
Running FirstFibonacciTest
..........
..........
..........
..........
..........
..........
..........
..........
.
Done FirstFibonacciTest
__________
$\mathcal{O}(2^n)$. Exponential. Problem sizes larger than 25 quickly become unproductive.
Well, what's the next approach? A common solution to calculate fibonacci numbers is to recognize that we don't need to recalculate $n-1$ and $n-2$ for every single $n$. Rather, we can just store these values into an array and calculate the number iteratively. Note this is actually a simple form of dynamic programming as it holds on to previous solutions of the problem in memory and it uses these prior solutions to calculate subsequent values. In essence we get some pretty substantial efficency gains because we leverage memory to our advantage. How does this look? We can just store the $nth$ fibonacci number along with those for $n+1$ and $n+2$ in a three element array and work through each of the fibonacci numbers iteratively continually updating our 3 element vector as we go.
function f = fibonacci2(n)
values = [0 1 1];
for q = 1:n
values(1:2) = values(2:3);
values(3) = sum(values(1:2));
end
f = values(1);
end
To observe how it scales I am going to modify our test to pass the algorithm in as another test parameter so we can reuse the test we have already written and ensure consistency of our measurements. We don't want one algorithm to be measured using a loop of 1000 and another accidently measured using a loop of 2000.
The test looks the same with the addition of the algorithm test parameter which contains function handles to our different fibonacci implementations.
classdef SecondFibonacciTest < matlab.perftest.TestCase
properties(TestParameter)
algorithm = struct(...
'Recursive', @fibonacci1, ...
'Iterative', @fibonacci2);
size = num2cell(round(linspace(0,25,10)));
end
methods(Test)
function testFibonacci(~, algorithm, size)
for idx = 1:1000
algorithm(size);
end
end
end
end
Note that I have used the struct syntax to define the parameter which allows us to give nice names like "Recursive" and "Iterative" to each of the parameter values. This also allows us to select just these parameters and run the exact subset of interest, in this case, the iterative algorithm.
iterativeResult = runperf('SecondFibonacciTest', 'ParameterName', 'Iterative');
plot(ax, problemSize, findMins(iterativeResult), 'LineWidth', 2)
leg = legend([leg.String, 'Iterative']);
Running SecondFibonacciTest
..........
..........
..........
..........
..........
..........
..........
..........
...
Done SecondFibonacciTest
__________
Alright we now see the iterative approaches scales linearly ($\mathcal{O}(n)$)) which us much nicer, but as it turns out we can do better with a closed form solution using Binet's formula:
$$f_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}$$
Here is how we code that up in MATLAB:
function f = fibonacci3(n)
sqrt5 = sqrt(5);
num = (1 + sqrt5)^n - (1 - sqrt5)^n;
denom = (2^n)*sqrt5;
f = num/denom;
end
...and now we can simply add this new algorithm as new parameter to our existing test:
properties(TestParameter)
size = num2cell(round(linspace(0,25,10)));
algorithm = struct(...
'Recursive', @fibonacci1, ...
'Iterative', @fibonacci2, ...
'ClosedForm', @fibonacci3);
end
and runperf it!
closedFormResult = runperf('FibonacciTest', 'ParameterName', 'ClosedForm');
plot(ax, problemSize, findMins(closedFormResult), 'LineWidth', 2)
leg = legend([leg.String, 'Closed Form']);
Running FibonacciTest
..........
..........
..........
..........
..........
..........
..........
..........
.
Done FibonacciTest
__________
That is looking very nice! In fact it demonstrates $\mathcal{O}(1)$ constant behavior. Can't get much better than that from a complexity perspective. However, there is one more solution I'd like to show using matrix exponentials. Hmmmm, I seem to remember hearing somewhere that MATLAB is pretty good with matrices. Does that sound familiar? Well lucky us, because the solution is quite trivial with the power of MATLAB. This is because the $nth$ fibonacci number $f_n$ is governed by the following equation:
$$\left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right]^n = \left[ \begin{array}{cc} f_{n+1} & f_n \\ f_n & f_{n-1} \end{array} \right]$$
Yes, that's right, the solution in MATLAB is just that simple.
function f = fibonacci4(n)
F = [1 1;1 0]^n;
f = F(1,2);
end
How does this scale? Add the test and run it.
properties(TestParameter)
size = num2cell(round(linspace(0,25,10)));
algorithm = struct(...
'Recursive', @fibonacci1, ...
'Iterative', @fibonacci2, ...
'ClosedForm', @fibonacci3, ...
'MatrixExponential', @fibonacci4);
end
matrixResult = runperf('FibonacciTest', 'ParameterName', 'MatrixExponential');
plot(ax, problemSize, findMins(matrixResult), 'LineWidth', 2)
leg = legend([leg.String, 'Matrix Exponential']);
Running FibonacciTest
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
Done FibonacciTest
__________
Look at that! The closed form and the matrix solutions both show very scalable behavior, but the matrix form is not quite as good, showing logarithmic ($\mathcal{O}(\log n)$) scaling instead of constant and the constant factor is greater for the matrix case. However, there is another benefit enjoyed by the matrix exponential solution but not by the closed form solution. To show this, I am going to add a quick spot check to this test to confirm that we are actually getting the right answer. Note for this post I am just going to add another test method which validates a single size, but this could be done in the existing test method across all sizes if desired.
classdef VerifiedFibonacciTest < matlab.perftest.TestCase
properties(TestParameter)
size = num2cell(round(linspace(0,25,10)));
algorithm = struct(...
'Recursive', @fibonacci1, ...
'Iterative', @fibonacci2, ...
'ClosedForm', @fibonacci3, ...
'MatrixExponential', @fibonacci4);
end
methods(Test)
function testFibonacci(~, algorithm, size)
for idx = 1:1000
algorithm(size);
end
end
testCase.verifyEqual(algorithm(16), 987, ...
'The 16th fibonacci number should be 987');
end
end
end
Let's run it with runtests to confirm correctness.
result = runtests('VerifiedFibonacciTest')
Running VerifiedFibonacciTest
..........
..........
..........
..........
..
================================================================================
----------------
Test Diagnostic:
----------------
The 16th fibonacci number should be 987
---------------------
Framework Diagnostic:
---------------------
verifyEqual failed.
--> The values are not equal using "isequaln".
--> Failure table:
Actual Expected Error RelativeError
______ ________ ____________________ ____________________
987 987 4.54747350886464e-13 4.60736930989325e-16
Actual double:
9.870000000000005e+02
Expected double:
987
------------------
Stack Information:
------------------
================================================================================
..
Done VerifiedFibonacciTest
__________
Failure Summary:
Name Failed Incomplete Reason(s)
============================================================================================================
result =
1x44 TestResult array with properties:
Name
Passed
Failed
Incomplete
Duration
Details
Totals:
43 Passed, 1 Failed, 0 Incomplete.
7.3549 seconds testing time.
...and there is the problem with the closed form solution. It is subject to floating point precision errors. Of course we can simply make this test pass by leveraging a tolerance in verifyEqual
testCase.verifyEqual(algorithm(16), 987, 'RelTol', 1e-14...
'The 16th fibonacci number should be 987');
but the fact remains that using the closed form solution we compute something that is close to the $nth$ fibonacci number, but the matrix solution computes it exactly. Which one to use depends on your application, but given the constant factor is so small and logarithmic behavior approaches constant for large problem sizes I think I would tend to choose the exact solution. How about you?
Published with MATLAB® R2016a
|
|
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|
http://www.zora.uzh.ch/27116/
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# Translated Poisson approximation to equilibrium distributions of Markov population processes
Socoll, S N; Barbour, A D (2010). Translated Poisson approximation to equilibrium distributions of Markov population processes. Methodology and Computing in Applied Probability, 12(4):567-586.
## Abstract
The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, with $O( 1 /{\sqrt{n}})$ error as measured in Kolmogorov distance. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our arguments are based on the Stein–Chen method and Dynkin’s formula.
The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, with $O( 1 /{\sqrt{n}})$ error as measured in Kolmogorov distance. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our arguments are based on the Stein–Chen method and Dynkin’s formula.
## Altmetrics
28 downloads since deposited on 04 Feb 2010
Detailed statistics
Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2010 04 Feb 2010 14:54 05 Apr 2016 13:44 Springer 1387-5841 The original publication is available at www.springerlink.com https://doi.org/10.1007/s11009-009-9124-8 http://arxiv.org/abs/0902.0884
Permanent URL: https://doi.org/10.5167/uzh-27116
Preview
Content: Accepted Version
Filetype: PDF
Size: 1MB
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Size: 1MB
## TrendTerms
TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.
You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.
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|
https://www2.ccs.tsukuba.ac.jp/Astro/achievements/ja/2021/11/01/inoue2021a/
|
# 研究成果・発表論文
## Fragmentation of ring galaxies and transformation to clumpy galaxies
### Inoue, Shigeki, Yoshida, Naoki, & Hernquist, Lars
##### 要旨
We study the fragmentation of collisional ring galaxies (CRGs) using a linear perturbation analysis that computes the physical conditions of gravitational instability, as determined by the balance of self-gravity of the ring against pressure and Coriolis forces. We adopt our formalism to simulations of CRGs and show that the analysis can accurately characterize the stability and onset of fragmentation, although the linear theory appears to underpredict the number of fragments of an unstable CRG by a factor of 2. In addition, since the orthodox ’density- wave’ model is inapplicable to such self-gravitating rings, we devise a simple approach that describes the rings propagating as material waves. We find that the toy model can predict whether the simulated CRGs fragment or not using information from their pre-collision states. We also apply our instability analysis to a CRG discovered at a high redshift, z = 2.19. We find that a quite high-velocity dispersion is required for the stability of the ring, and therefore the CRG should be unstable to ring fragmentation. CRGs are rarely observed at high redshifts, and this may be because CRGs are usually too faint. Since the fragmentation can induce active star formation and make the ring bright enough to observe, the instability could explain this rarity. An unstable CRG fragments into massive clumps retaining the initial disc rotation, and thus it would evolve into a clumpy galaxy with a low surface density in an interclump region.
We study the fragmentation of collisional ring galaxies (CRGs) using a linear perturbation analysis that computes the physical conditions of gravitational instability, as determined by the balance of self-gravity of the ring against pressure and Coriolis forces. We adopt our formalism to simulations of CRGs and show that the analysis can accurately characterize the stability and onset of fragmentation, although the linear theory appears to underpredict the number of fragments of an unstable CRG by a factor of 2. In addition, since the orthodox ’density- wave’ model is inapplicable to such self-gravitating rings, we devise a simple approach that describes the rings propagating as material waves. We find that the toy model can predict whether the simulated CRGs fragment or not using information from their pre-collision states. We also apply our instability analysis to a CRG discovered at a high redshift, z = 2.19. We find that a quite high-velocity dispersion is required for the stability of the ring, and therefore the CRG should be unstable to ring fragmentation. CRGs are rarely observed at high redshifts, and this may be because CRGs are usually too faint. Since the fragmentation can induce active star formation and make the ring bright enough to observe, the instability could explain this rarity. An unstable CRG fragments into massive clumps retaining the initial disc rotation, and thus it would evolve into a clumpy galaxy with a low surface density in an interclump region.
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https://www.physicsforums.com/threads/summer-assignment-resulting-slick.128419/
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# Summer Assignment - resulting slick
1. Aug 8, 2006
Ive seen questions like this one done here before but I dont think im understanding this well. The question is:
Suppose 300 cubic meters of oil is spilled into the ocean. Find the are of the resulting slick, assuming that it is one molecule thick, and each moelcule occupies a cube .50 um (micrometers) on a side.
I believe the proper formula for this situation is Area=Volume/Depth. So then I would do Area=300 meters cubed/.5 micrometers. Convert the micrometers to meters and get Area=300 meters cubed/.0000005 meters and have my answer being 600,000,000 meters squared.
Is that right though? For some reason I think its wrong, and I hadn't any idea what to do at all until i found the formula but it just seems like...I dont know, a bit too easy and like maybe im missing something. It also appears that the answer is ridiculously large, but maybe its supposed to be like that? Any help on this would be appreciated, I would be shocked if what I did was correct.
Last edited: Aug 8, 2006
2. Aug 8, 2006
### Staff: Mentor
Maybe an easier way (or just a double-check on your answer) would be to figure out how many molecules there are in 300m^3 of oil and do the area math. Do you know how to figure out how many molecules there are in 300m^3 of oil?
3. Aug 8, 2006
### Office_Shredder
Staff Emeritus
Looks right to me.
berkeman, I don't think that's an easier way ;)
4. Aug 9, 2006
Im not sure what you mean by your way but, you say I actually did it right? Thanks for confirming what I did, I was having a lot of trouble on that problem and didn't even get as far as finding the area formula until I foudn this website, which is when I quit guessing what to do and trying to look it up becuase I was so confused.
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https://arizona.pure.elsevier.com/en/publications/impulse-entrainment-computer-simulations-and-studies-on-the-paral
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# Impulse entrainment: Computer simulations and studies on the parallel fibers of the cerebellum
J. D. Kocsis, K. L. Cummins, S. G. Waxman, R. C. Malenka
Research output: Contribution to journalArticlepeer-review
5 Scopus citations
## Abstract
The interspike interval for a pair of propagating impulses was studied from a simulation model of axons with a triphasic recovery cycle (relative refractory, supernormal, and subnormal periods) and from electrophysiologic results on the parallel fibers of the cerebellar cortex. Impulse entrainment, whereby convergence of the interspike interval to a fixed value occurs, was predicted from the model for impulses initiated at any point during the recovery cycle. Convergence of the interspike interval toward a fixed value was also observed for the parallel fibers when impulses were initiated during the relative refractory or the early supernormal periods. We suggest that convergence of the interspike interval contributes to regularization of constitutent impulses within a spike train.
Original language English (US) 628-637 10 Experimental Neurology 72 3 https://doi.org/10.1016/0014-4886(81)90011-X Published - Jun 1981 Yes
## ASJC Scopus subject areas
• Neurology
• Developmental Neuroscience
## Fingerprint
Dive into the research topics of 'Impulse entrainment: Computer simulations and studies on the parallel fibers of the cerebellum'. Together they form a unique fingerprint.
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http://nrich.maths.org/8059/note?nomenu=1
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## Two Numbers Under the Microscope
Choose any two odd numbers, such as $5$ and $9$. Add them together.
Draw a picture or make a model to show how the numbers add together.
Adam found some dominoes with $5$ and $9$ spots on them:
Abdul drew a picture of $5$ add $9$ like this:
Look closely at the models and pictures.
Can you see anything in any of them that would work in exactly the same way if you used two different odd numbers?
Can you use your one example to prove what will happen every time you add any two odd numbers?
See if you can explain this to someone else. Are they convinced by your argument?
Once you can convince someone else, see if you can find a way to show your argument. You might draw something or take a photo of things you have used to prove that your result is always true from your example.
### Why do this problem?
This problem supports the development of the idea of generic proof with the children. This is a tricky concept to grasp but it draws attention to mathematical structures that are not often addressed at primary school level. It is possible that only very few children in the class may grasp the idea but this is still a worthwhile activity which provides opportunities for children to explore odd and even numbers and the relationship between them. Proof is a fundamental idea in mathematics and in encouraging them to do this problem you will be helping them to behave like mathematicians.
By addressing the case of adding two odd numbers, a generic proof that adding two odd numbers always gives an even answer is developed based on the structure of odd and even numbers. The article entitled Take One Example will help you understand how this problem supports the development of the idea of generic proof with the children. Reading it will help you to see what is involved.
### Possible approach
Ask the children to choose two odd numbers and add them together. It is probably easiest if they choose ones that are easy to model and numbers that they are secure with.
Suggest that they make a model of their numbers using apparatus that is widely available in the classroom. Resist pointing them in specific directions unless they become stuck, but if they are then resources such as Multilink cubes, Numicon or squared paper will be helpful. After some time exploring they may need some prompting to move them towards looking at the pairing of their dots or cubes. These pictures may help.
The idea is that they take a particular example and then see if they can see the general structure within that one example.
### Key questions
How would you like to show these numbers?
Can you see anything in your example that would work in exactly the same way if you used two different odd numbers?
Can you say what will happen every time you add any two odd numbers?
Can you convince your friend that this is true?
### Possible extension
##### Even plus even
See what happens if you add two even numbers such as $4$ and $12$.
Can you 'see' in this example what will happen every time you add two even numbers?
You may find it helpful to work in a similar way to the way you worked for odd numbers.
##### Adding an odd and an even.
See what happens if you add an odd and an even or an even and an odd such as $6$ and $9$.
Can you 'see' in this example what will happen every time you add an odd and an even or an even and an odd number?
You may find it helpful to work in a similar way to the way you worked before.
You could go on to look at Take Three Numbers.
### Possible support
It may be helpful to encourage children to use paper cut outs of the numbers such as these. In these the oddness of the odd numbers is very clear. Laminated sets of these cards could be a very useful addition to your classroom resources. Numicon would be an alternative resource to use.
|
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|
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-9-section-9-3-the-parabola-exercise-set-page-996/15
|
## Precalculus (6th Edition) Blitzer
focus $(0,-\frac{1}{8})$, directrix $y=\frac{1}{8}$ see graph.
Step 1. Rewriting the equation as $x^2=-\frac{1}{2}y$, we have $4p=-\frac{1}{2}$ and $p=-\frac{1}{8}$ with the parabola opening downwards and vertex at $(0,0)$. Step 2. We can find the focus at $(0,-\frac{1}{8})$ and directrix as $y=\frac{1}{8}$ Step 3. We can graph the parabola as shown in the figure.
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http://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/statug_hpfmm_syntax02.htm
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# The HPFMM Procedure
### BAYES Statement
• BAYES bayes-options;
The BAYES statement requests that the parameters of the model be estimated by Markov chain Monte Carlo sampling techniques. The HPFMM procedure can estimate by maximum likelihood the parameters of all models supported by the procedure. Bayes estimation, on the other hand, is available for only a subset of these models.
In Bayesian analysis, it is essential to examine the convergence of the Markov chains before you proceed with posterior inference. With ODS Graphics turned on, the HPFMM procedure produces graphs at the end of the procedure output; these graphs enable you to visually examine the convergence of the chain. Inferences cannot be made if the Markov chain has not converged.
The output produced for a Bayesian analysis is markedly different from that for a frequentist (maximum likelihood) analysis for the following reasons:
• Parameter estimates do not have the same interpretation in the two analyses. Parameters are fixed unknown constants in the frequentist context and random variables in a Bayesian analysis.
• The results of a Bayesian analysis are summarized through chain diagnostics and posterior summary statistics and intervals.
• The HPFMM procedure samples the mixing probabilities in Bayesian models directly, rather than mapping them onto a logistic (or other) scale.
The HPFMM procedure applies highly specialized sampling algorithms in Bayesian models. For single-component models without effects, a conjugate sampling algorithm is used where possible. For models in the exponential family that contain effects, the sampling algorithm is based on Gamerman (1997). For the normal and t distributions, a conjugate sampler is the default sampling algorithm for models with and without effects. In multi-component models, the sampling algorithm is based on latent variable sampling through data augmentation (Frühwirth-Schnatter 2006) and the Gamerman or conjugate sampler. Because of this specialization, the options for controlling the prior distributions of the parameters are limited.
Table 51.3 summarizes the bayes-options available in the BAYES statement. The full assortment of options is then described in alphabetical order.
Table 51.3: BAYES Statement Options
Option
Description
Options Related to Sampling
Specifies how to construct initial values
Specifies the number of burn-in samples
Specifies the number of samples after burn-in
Forces a Metropolis-Hastings sampling algorithm even if conjugate sampling is possible
Generates a data set that contains the posterior estimates
Controls the thinning of the Markov chain
Specification of Prior Information
Specifies the prior parameters for the Dirichlet distribution of the mixing probabilities
Specifies the parameters of the normal prior distribution for individual parameters in the vector
Specifies the parameters of the prior distribution for the means in homogeneous mixtures without effects
Specifies the parameters of the inverse gamma prior distribution for the scale parameters in homogeneous mixtures
Specifies additional options used in the determination of the prior distribution
Posterior Summary Statistics and Convergence Diagnostics
Displays convergence diagnostics for the Markov chain
Displays posterior summary information for the Markov chain
Other Options
Specifies which estimate is used for the computation of OUTPUT statistics and graphics
Specifies the time interval to report on sampling progress (in seconds)
You can specify the following bayes-options in the BAYES statement.
BETAPRIORPARMS=pair-specification
BETAPRIORPARMS(pair-specificationpair-specification)
specifies the parameters for the normal prior distribution of the parameters that are associated with model effects (s). The pair-specification is of the form , and the values a and b are the mean and variance of the normal distribution, respectively. This option overrides the PRIOROPTIONS option.
The form of the BETAPRIORPARMS with an equal sign and a single pair is used to specify one pair of prior parameters that applies to all components in the mixture. In the following example, the two intercepts and the two regression coefficients all have a prior distribution:
proc hpfmm;
model y = x / k=2;
bayes betapriorparms=(0,100);
run;
You can also provide a list of pairs to specify different sets of prior parameters for the various regression parameters and components. For example:
proc hpfmm;
model y = x/ k=2;
bayes betapriorparms( (0,10) (0,20) (.,.) (3,100) );
run;
The simple linear regression in the first component has a prior for the intercept and a prior for the slope. The prior for the intercept in the second component uses the HPFMM default, whereas the prior for the slope is .
DIAGNOSTICS=ALL | NONE | (keyword-list)
DIAG=ALL | NONE | (keyword-list)
controls the computation of diagnostics for the posterior chain. You can request all posterior diagnostics by specifying DIAGNOSTICS=ALL or suppress the computation of posterior diagnostics by specifying DIAGNOSTICS=NONE. The following keywords enable you to select subsets of posterior diagnostics; the default is DIAGNOSTICS=(AUTOCORR).
AUTOCORR <(LAGS= numeric-list)>
computes for each sampled parameter the autocorrelations of lags specified in the LAGS= list. Elements in the list are truncated to integers, and repeated values are removed. If the LAGS= option is not specified, autocorrelations are computed by default for lags 1, 5, 10, and 50. See the section Autocorrelations in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.
ESS
computes an estimate of the effective sample size (Kass et al. 1998), the correlation time, and the efficiency of the chain for each parameter. See the section Effective Sample Size in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.
GEWEKE <(geweke-options)>
computes the Geweke spectral density diagnostics (Geweke 1992), which are essentially a two-sample t test between the first portion and the last portion of the chain. The default is and , but you can choose other fractions by using the following geweke-options:
FRAC1=value
specifies the fraction for the first window.
FRAC2=value
specifies the fraction for the second window.
See the section Geweke Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.
HEIDELBERGER <(Heidel-options)>
HEIDEL <(Heidel-options)>
computes the Heidelberger and Welch diagnostic (which consists of a stationarity test and a half-width test) for each variable. The stationary diagnostic test tests the null hypothesis that the posterior samples are generated from a stationary process. If the stationarity test is passed, a half-width test is then carried out. See the section Heidelberger and Welch Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.
These diagnostics are not performed by default. You can specify the DIAGNOSTICS=HEIDELBERGER option to request these diagnostics, and you can also specify suboptions, such as DIAGNOSTICS=HEIDELBERGER(EPS=0.05), as follows:
SALPHA=value
specifies the level for the stationarity test. By default, SALPHA=0.05.
HALPHA=value
specifies the level for the half-width test. By default, HALPHA=0.05.
EPS=value
specifies a small positive number such that if the half-width is less than times the sample mean of the retaining iterates, the half-width test is passed. By default, EPS=0.1.
MCERROR
MCSE
computes an estimate of the Monte Carlo standard error for each sampled parameter. See the section Standard Error of the Mean Estimate in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.
MAXLAG=n
specifies the largest lag used in computing the effective sample size and the Monte Carlo standard error. Specifying this option implies the ESS and MCERROR options. The default is MAXLAG=250.
RAFTERY <(Raftery-options)>
RL <(Raftery-options)>
computes the Raftery and Lewis diagnostics, which evaluate the accuracy of the estimated quantile ( for a given Q ) of a chain. can achieve any degree of accuracy when the chain is allowed to run for a long time. The algorithm stops when the estimated probability reaches within of the value Q with probability S; that is, . See the section Raftery and Lewis Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The Raftery-options enable you to specify Q, R, S, and a precision level for a stationary test.
These diagnostics are not performed by default. You can specify the DIAGNOSTICS=RAFERTY option to request these diagnostics, and you can also specify suboptions, such as DIAGNOSTICS=RAFERTY(QUANTILE=0.05), as follows:
QUANTILE=value
Q=value
specifies the order (a value between 0 and 1) of the quantile of interest. By default, QUANTILE=0.025.
ACCURACY=value
R=value
specifies a small positive number as the margin of error for measuring the accuracy of estimation of the quantile. By default, ACCURACY=0.005.
PROB=value
S=value
specifies the probability of attaining the accuracy of the estimation of the quantile. By default, PROB=0.95.
EPS=value
specifies the tolerance level (a small positive number between 0 and 1) for the stationary test. By default, EPS=0.001.
MIXPRIORPARMS=K
MIXPRIORPARMS(value-list)
specifies the parameters used in constructing the Dirichlet prior distribution for the mixing parameters. If you specify MIXPRIORPARMS=K, the parameters of the k-dimensional Dirichlet distribution are a vector that contains the number of components in the model (k), whatever that might be. You can specify an explicit list of parameters in value-list. If the MIXPRIORPARMS option is not specified, the default Dirichlet parameter vector is a vector of length k of ones. This results in a uniform prior over the unit simplex; for k=2, this is the uniform distribution. See the section Prior Distributions for the distribution function of the Dirichlet as used by the HPFMM procedure.
ESTIMATE=MEAN | MAP
determines which overall estimate is used, based on the posterior sample, in the computation of OUTPUT statistics and certain ODS graphics. By default, the arithmetic average of the (thinned) posterior sample is used. If you specify ESTIMATE=MAP, the parameter vector is used that corresponds to the maximum log posterior density in the posterior sample. In any event, a message is written to the SAS log if postprocessing results depend on a summary estimate of the posterior sample.
INITIAL=DATA | MLE | MODE | RANDOM
determines how initial values for the Markov chain are obtained. The default when a conjugate sampler is used is INITIAL=DATA, in which case the HPFMM procedure uses the same algorithm to obtain data-dependent starting values as it uses for maximum likelihood estimation. If no conjugate sampler is available or if you use the METROPOLIS option to explicitly request that it not be used, then the default is INITIAL=MLE, in which case the maximum likelihood estimates are used as the initial values. If the maximum likelihood optimization fails, the HPFMM procedure switches to the default INITIAL=DATA.
The options INITIAL=MODE and INITIAL=RANDOM use the mode and random draws from the prior distribution, respectively, to obtain initial values. If the mode does not exist or if it falls on the boundary of the parameter space, the prior mean is used instead.
METROPOLIS
requests that the HPFMM procedure use the Metropolis-Hastings sampling algorithm based on Gamerman (1997), even in situations where a conjugate sampler is available.
MUPRIORPARMS=pair-specification
MUPRIORPARMS(pair-specificationpair-specification)
specifies the parameters for the means in homogeneous mixtures without regression coefficients. The pair-specification is of the form , where a and b are the two parameters of the prior distribution, optionally delimited with a comma. The actual distribution of the parameter is implied by the distribution selected in the MODEL statement. For example, it is a normal distribution for a mixture of normals, a gamma distribution for a mixture of Poisson variables, a beta distribution for a mixture of binary variables, and an inverse gamma distribution for a mixture of exponential variables. This option overrides the PRIOROPTIONS option.
The parameters correspond as follows:
Beta:
The parameters correspond to the and parameters of the beta prior distribution such that its mean is and its variance is .
Normal:
The parameters correspond to the mean and variance of the normal prior distribution.
Gamma:
The parameters correspond to the and parameters of the gamma prior distribution such that its mean is and its variance is .
Inverse gamma:
The parameters correspond to the and parameters of the inverse gamma prior distribution such that its mean is and its variance is .
The two techniques for specifying the prior parameters with the MUPRIORPARMS option are as follows:
• Specify an equal sign and a single pair of values:
proc hpfmm seed=12345;
model y = / k=2;
bayes mupriorparms=(0,50);
run;
• Specify a list of parameter pairs within parentheses:
proc hpfmm seed=12345;
model y = / k=2;
bayes mupriorparms( (.,.) (1.4,10.5));
run;
If you specify an invalid value (outside of the parameter space for the prior distribution), the HPFMM procedure chooses the default value and writes a message to the SAS log. If you want to use the default values for a particular parameter, you can also specify missing values in the pair-specification. For example, the preceding list specification assigns default values for the first component and uses the values 1.4 and 10.5 for the mean and variance of the normal prior distribution in the second component. The first example assigns a prior distribution to the means in both components.
NBI=n
specifies the number of burn-in samples. During the burn-in phase, chains are not saved. The default is NBI=2000.
NMC=n
SAMPLE=n
specifies the number of Monte Carlo samples after the burn-in. Samples after the burn-in phase are saved unless they are thinned with the THIN= option. The default is NMC=10000.
OUTPOST<(outpost-options)>=data-set
requests that the posterior sample be saved to a SAS data set. In addition to variables that contain log likelihood and log posterior values, the OUTPOST data set contains variables for the parameters. The variable names for the parameters are generic (Parm_1, Parm_2, , Parm_p). The labels of the parameters are descriptive and correspond to the "Parameter Mapping" table that is produced when the OUTPOST= option is in effect.
You can specify the following outpost-options in parentheses:
LOGPRIOR
adds the value of the log prior distribution to the data set.
NONSINGULAR | NONSING | COMPRESS
eliminates parameters that correspond to singular columns in the design matrix (and were not sampled) from the posterior data set. This is the default.
SINGULAR | SING
adds columns of zeros to the data set in positions that correspond to singularities in the model or to parameters that were not sampled for other reasons. By default, these columns of zeros are not written to the posterior data set.
PHIPRIORPARMS=pair-specification
PHIPRIORPARMS( pair-specificationpair-specification)
specifies the parameters for the inverse gamma prior distribution of the scale parameters (’s) in the model. The pair-specification is of the form , and the values are chosen such that the prior distribution has mean and variance .
The form of the PHIPRIORPARMS with an equal sign and a single pair is used to specify one pair of prior parameters that applies to all components in the mixture. For example:
proc hpfmm seed=12345;
model y = / k=2;
bayes phipriorparms=(2.001,1.001);
run;
The form with a list of pairs is used to specify different prior parameters for the scale parameters in different components. For example:
proc hpfmm seed=12345;
model y = / k=2;
bayes phipriorparms( (.,1.001) (3.001,2.001) );
run;
If you specify an invalid value (outside of the parameter space for the prior distribution), the HPFMM procedure chooses the default value and writes a message to the SAS log. If you want to use the default values for a particular parameter, you can also specify missing values in the pair-specification. For example, the preceding list specification assigns default values for the first component a prior parameter and uses the value 1.001 for the b prior parameter. The second pair assigns 3.001 and 2.001 for the a and b prior parameters, respectively.
PRIOROPTIONS <=>(prior-options)
PRIOROPTS <=>(prior-options)
specifies options related to the construction of the prior distribution and the choice of their parameters. Some prior-options apply only in particular models. The BETAPRIORPARMS= and MUPRIORPARMS= options override this option.
You can specify the following prior-options:
CONDITIONAL | COND
chooses a conditional prior specification for the homogeneous normal and t distribution response components. The default prior specification in these models is an independence prior where the mean of the hth component has prior . The conditional prior is characterized by .
DEPENDENT | DEP
chooses a data-dependent prior for the homogeneous models without effects. The prior parameters a and b are chosen as follows, based on the distribution in the MODEL statement:
Binary and binomial:
, b=1, and the prior distribution for the success probability is .
Poisson:
, , and the prior distribution for is . See Frühwirth-Schnatter (2006, p. 280) and Viallefont, Richardson, and Greene (2002).
Exponential:
, , and the prior distribution for is inverse gamma with parameters a and b.
Normal and t:
Under the default independence prior, the prior distribution for is where f is the variance factor from the VAR= option and
Under the default conditional prior specification, the prior for is where and . The prior for the scale parameter is inverse gamma with parameters 1.28 and . For further details, see Raftery (1996) and Frühwirth-Schnatter (2006, p. 179).
VAR=f
specifies the variance for normal prior distributions. The default is VAR=1000. This factor is used, for example, in determining the prior variance of regression coefficients or in determining the prior variance of means in homogeneous mixtures of t or normal distributions (unless a data-dependent prior is used).
MLE<=r>
specifies that the prior distribution for regression variables be based on a multivariate normal distribution centered at the MLEs and whose dispersion is a multiple r of the asymptotic MLE covariance matrix. The default is MLE=10. In other words, if you specify PRIOROPTIONS(MLE), the HPFMM procedure chooses the prior distribution for the regression variables as where is the vector of maximum likelihood estimates. The prior for the scale parameter is inverse gamma with parameters 1.28 and where
For further details, see Raftery (1996) and Frühwirth-Schnatter (2006, p. 179). If you specify PRIOROPTIONS(MLE) for the regression parameters, then the data-dependent prior is used for the scale parameter; see the PRIOROPTIONS(DEPENDENT) option above.
The MLE option is not available for mixture models in which the parameters are estimated directly on the data scale, such as homogeneous mixture models or mixtures of distributions without model effects for which a conjugate sampler is available. By using the METROPOLIS option, you can always force the HPFMM procedure to abandon a conjugate sampler in favor of a Metropolis-Hastings sampling algorithm to which the MLE option applies.
STATISTICS <(global-options)> = ALL | NONE | keyword | (keyword-list)
SUMMARIES <(global-options)> = ALL | NONE | keyword | (keyword-list)
controls the number of posterior statistics produced. Specifying STATISTICS=ALL is equivalent to specifying STATISTICS=(SUMMARY INTERVAL). To suppress the computation of posterior statistics, specify STATISTICS=NONE. The default is STATISTICS=(SUMMARY INTERVAL). See the section Summary Statistics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.
The global-options include the following:
ALPHA=numeric-list
controls the coverage levels of the equal-tail credible intervals and the credible intervals of highest posterior density (HPD) credible intervals. The ALPHA= values must be between 0 and 1. Each ALPHA= value produces a pair of % equal-tail and HPD credible intervals for each sampled parameter. The default is ALPHA=0.05, which results in 95% credible intervals for the parameters.
PERCENT=numeric-list
requests the percentile points of the posterior samples. The values in numeric-list must be between 0 and 100. The default is PERCENT=(25 50 75), which yields for each parameter the 25th, 50th, and 75th percentiles, respectively.
The list of keywords includes the following:
SUMMARY
produces the means, standard deviations, and percentile points for the posterior samples. The default is to produce the 25th, 50th, and 75th percentiles; you can modify this list with the global PERCENT= option.
INTERVAL
produces equal-tail and HPD credible intervals. The default is to produce the 95% equal-tail credible intervals and 95% HPD credible intervals, but you can use the ALPHA= global-option to request credible intervals for any probabilities.
THIN=n
THINNING=n
controls the thinning of the Markov chain after the burn-in. Only one in every k samples is used when THIN=k, and if NBI= and NMC=n, the number of samples kept is
where [a] represents the integer part of the number a. The default is THIN=1—that is, all samples are kept after the burn-in phase.
TIMEINC=n
specifies a time interval in seconds to report progress during the burn-in and sampling phase. The time interval is approximate, since the minimum time interval in which the HPFMM procedure can respond depends on the multithreading configuration.
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|
https://tex.stackexchange.com/questions/427394/appendices-in-tables-of-contents-under-appendices-section
|
# Appendices in tables of contents under Appendices section
I have a very simple article which consists of the following:
\documentclass{article}
\usepackage{appendix}
\begin{document}
\tableofcontents
\appendixtitleon
\appendixtitletocon
\section{Writing}
\begin{appendices}
\section{First appendix}
\subsection{First}
\subsection{Second}
\section{Second appendix}
\subsection{First}
\subsection{Second}
\end{appendices}
\end{document}
This produces the following output:
This is almost what I want, except, I actually want the appendices to come under an appendices section in the table of contents (everything indented a level so to speak). The output would be like:
Add the [toc] option:
\documentclass{article}
\usepackage[toc]{appendix}
\begin{document}
\tableofcontents
\appendixtitleon
\appendixtitletocon
\section{Writing}
\begin{appendices}
\section{First appendix}
\subsection{First}
\subsection{Second}
\section{Second appendix}
\subsection{First}
\subsection{Second}
\end{appendices}
\end{document}
• I have tried this, doesn't this add them as sections? They are not under the Appendices section, they are just listed below them? – Warmley Apr 19 '18 at 12:30
• I dont' really understand what you mean. Appendix A, Appendix B are sections. The [toc] just adds an intertitle, just to say: here begins the appendices part. – Bernard Apr 19 '18 at 12:36
• Yeah that wasn't very clear. What I'm looking for is the appendix section to exist in the table of contents, but with each appendix being a subsection (I'm not sure if that is right). So the indent is something like the provided output. – Warmley Apr 19 '18 at 12:39
• I think it should be possible to leave each appendix as a section, but have a subsection layout in the table of contents with titletoc and apptools. – Bernard Apr 19 '18 at 12:46
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https://www.physicsforums.com/threads/rotational-mechanics-and-angular-momentum-help-needed-to-check-over-work.269422/
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# Rotational mechanics and Angular Momentum- help needed to check over work
1. Nov 4, 2008
### ultrapowerpie
Please note, this is a series of problems, with a series of answers. I believe that I understand how to do the problems, but for some reason, my answer is wrong, and I can think of no reason why.
1. The problem statement, all variables and given/known data
Problem 1: http://img355.imageshack.us/my.php?image=physics1jw9.png
i) I need to find the speed of the mass as it passes point B in m/s
ii) I need to find the Tension of the string (Which I am sure I can do, but without a correct answer for part i, I'm not going to attempt yet)
Problem 2: http://img186.imageshack.us/my.php?image=physics2pt6.png
Problem 3: http://img355.imageshack.us/my.php?image=physics3el2.png
(Note, the word that is cut off is longitudinal. I had to copy and paste parts of it into one picture, sorry it's a little shoddy <.<)
2. Relevant equations
All the lovely rotational mechanics equations, and their linear counterparts
3. The attempt at a solution
Problem 1
i) Ok, I did not take into account the spoke on teh wheel whatsoever, since the mass of inertia was given to me, and the spokes of the wheel did not seem to have their own mass. Not sure if this would effect the problem.
T=I(angular a)
Angular acceleration = Torque/I
Torque= 46 KG * 9.8 m/s^2 * 3m = 1352.4
I= 3/4* 23 KG * 3^2 = 155.25
Ang A= 8.7111
a = (Ang A)*r
a= 8.711 * 3 = 26.133
Plugging this acceleration into the classic Xf= Xi + Vi +(at^2)*2 (assuming Vi and Xi are 0)
I get t=2.106
Then, using the Vf= Vi +at
I get Vf= 55.05
This answer is wrong though, not sure where I screwed up
[hr]
Problem 2
This one seemed simple enough, just a basic torque= I * ang a
Ang a = T/I
I=(1/12)*(2.7)*(3^2)= 2.025
T= rxF
F= M*G= 9.8*2.7= 26.46
r= 3cos(43)=2.045
T= 54.137
Ang a= 26.734
It wants it in Radians/sec^2. Is it bad that I was using degrees for the cos?
[hr]
Problem 3
However, upon trying the forumulas given there, I still got it wrong. Using the formula ShawnD had at the end of his post, I got the wrong answer.
m= IW/rV
I= 2950
r=2.9
V=755
I got M to be .282 kg
I then divided by .012 kg, and got that time = 21.87 s
This answer is also wrong, not sure what happened.
Thank you in advance for looking over this work.
2. Nov 5, 2008
### ultrapowerpie
Slight bump, sorry, but homework's due by Thursday Night. >.>
Similar Discussions: Rotational mechanics and Angular Momentum- help needed to check over work
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https://www.physicsforums.com/threads/dimensional-analysis-of-planck-length.827312/
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# Dimensional Analysis of Planck Length
1. Aug 12, 2015
### noahcharris
Hello!
So in looking at black hole thermodynamics, I came across the equation
$l_p = \sqrt{G\hbar}$
But in doing a dimensional analysis of
$\sqrt{G\hbar}$
I get
$[\sqrt{G \hbar}] = \sqrt{ \frac{Nm^2}{kg^2} \frac{m^3}{kgs} }$
This obviously doesn't amount to a length. What am I doing wrong here?
2. Aug 12, 2015
### willem2
The planck length is equal to
$$\sqrt { \frac {G \hbar} { c^3} }$$
You got the dimension of plancks constant wrong. A joule second is not m^3/(kg.s)
You also need to convert newton to kg, meter and second.
Similar Discussions: Dimensional Analysis of Planck Length
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http://mathoverflow.net/questions/24849/path-connected-set-of-matrices/24876
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# (Path) connected set of matrices?
Let $N \in \mathfrak{M}_n(\mathbb{C})$ nilpotent, such that there exists $X \in \mathfrak M_n(\mathbb{C})$ with $X^2=N$ (take for instance $n>2$ and $N(1,n)=1$; $N(i,j)=0$ otherwise).
Denote by $\mathcal{S}_N$ the set of $X \in \mathfrak M_n(\mathbb{C})$ such that $X^2=N$. Is $\mathcal{S}_N$ connected or path-connected ? What happens when we change $2$ by $3,4,\ldots$?
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This is a memorial to an incorrect solution that used to be here. Unfortunately, I can't delete it since it was accepted.
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thanks, that finishes it off! – Xandi Tuni May 17 '10 at 8:22
The first statement about closure and refining partitions is not clear, can you elaborate? What about the following 6x6 matrix: the upper left and the lower right quarters are standard 3x3 Jordan blocks, and the upper right quarter has $\epsilon$ in its center (everything else is 0). I believe it has Jordan type (4,2) but converges to (3,3) as $\epsilon$ goes to 0. – Sergei Ivanov May 17 '10 at 9:28
The relevant order on partitions is not refinement, but dominance (en.wikipedia.org/wiki/Dominance_order). Thus two elements of $S_N$ can be in the same connected component only if the corresponding partitions can be "linked", using dominance, via partitions whose "square" is the partition of N. It is not clear to me that this is also sufficient: you are only allowed to conjugate elements of $S_N$ by matrices commuting with N, so even if the "poset connectedness" condition holds, you may not be able to conclude that the various elements of $S_N$ are in the same connected component. – damiano May 17 '10 at 10:44
Sergei -- I don't know about that particular example, but yes, I was too hasty with the first statement. – algori May 17 '10 at 11:50
Edit: This is just half an answer: I can only show that the sets matrices with $X^2=N$ and fixed Jordan type are path connected.
Every nilpotent matrix is conjugate to a nilpotent matrix in Jordan form, which is unique up to permutation of Jordam blocks. So we have a bijection
$$\mathrm{Nilp}_n(\mathbb C)/\mathrm{conjugation} \quad \cong \quad \mbox{integer partitions of }n$$
associating with a conjugacy class of a nilpotent matrix $X$ the sizes of its Jordan blocks $(a_1, \ldots, a_r)$ which sum up to $n$. The max of the $a_i$'s is the nilpotency-degree of $X$. To the class of $X^2$ is associated the partition $$(\lfloor (a_1 +1)/2 \rfloor, \lfloor a_1/2\rfloor ,\lfloor (a_2 +1)/2 \rfloor ,\lfloor a_2/2\rfloor, \ldots, \lfloor a_r/2 \rfloor)$$
From here we can derive a necessary and sufficient condition on a nilpotent matrix to be a square.
Now fix your preferred nilpotent matrix $N$. Let $X$ be a matrix with $X^2=N$ and Jordan type $(a_1, \ldots, a_r)$. Conjugating the whole setup, we may assume $X$ is in Jordan form.
Let $Y$ be a matrix with $Y^2=X^2 = N$ having the same Jordan type as $X$, and let us construct a path from $X$ to $Y$. Since $X$ and $Y$ have the same Jordan type, there exists an invertible matrix $S$ with $Y=SXS^{-1}$. Because $Y^2=X^2=N$ the matrix $S$ commutes with $N$. It is enough to construct a path from the identity matrix to $S$ in the set $\mathcal C_N$ of invertible matrices that commute with $N$.
I claim $\mathcal C_N$ is path connected (for just any $N$). Indeed, the set $[N]$ of all commutators of $N$ is linear subspace of the vector space $\mathrm M_n(\mathbb C)$. The determinant, as a function on $[N]$ is a polynomial function which is not identically zero since the identity matrix belongs to $[N]$. Thus, the zero set of the determinant is Zariski closed, so $\mathcal C_N$ is Zariski open in $[N]$. Any Zariski-open in a complex vector space is path connected.
What remains is to connect different Jordan types. We certainly can connect $(5,2)$ with $(5,1,1)$ by the 1 in the $2\times 2$--block with $t$ and vary $t$ from $1$ to zero. The problem that remains is to connect, say, type $(4,2)$ with type $(3,3)$ as pointed out in the comments.
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How do you connect Jordan type (3,3) to (4,2)? They have the same square. – Sergei Ivanov May 16 '10 at 11:45
Jordan type $(3,3)$ has square $(2,1,2,1) = (2,2,1,1)$ and Jordan type $(4,2)$ has square $(3,1,1,1)$. Is that not so? – Xandi Tuni May 16 '10 at 11:57
No, the square of Jordan type (4) is (2,2). – Sergei Ivanov May 16 '10 at 12:02
Oops, my fault. I will look if I can fix that. – Xandi Tuni May 16 '10 at 12:11
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