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Βιβλιοθήκη ΟΠΑ - Ψηφιακό Αποθετήριο
Τίτλος Αξιολόγηση αμοιβαίων κεφαλαίων
Δημιουργός Καλλιανιώτης, Ιωάννης
Συντελεστής Athens University of Economics and Business, Department of Economics
Τζαβαλής, Ηλίας
Τύπος Text
Φυσική 68σ.
Γλώσσα en
In the present dissertation, mutual funds will be analyzed through theoretical and empirical approach. First, we state some basic information about mutual funds, such as their history
and kinds, whereas after this, we analyze the multiple ways of evaluating a fund.In theoretical approach, we state the risk measurement indicators, such as the Beta coefficient,
R-squared, Alpha, Variance-Cocariance method and more. Then, we move to the analysis of the basic performance measurement indicators, which are the Treynor’s indicator, Jensen’s and
Sharpe’s indicator. Also, having analyzed them,we move to a deeper analysis, adding more indicators, such as the Treynor-Mazuy indicator, the Busse indicator, Fabozzi-Francis model and
much more. Also, in the end of this approach, we state the two market models, the Capital Asset Pricing Model (CAPM) and the Arbitrage theory model.Moving to the empirical approach, we
Περίληψη start by building a sample of 30 Closed-End Equit US funds, observing them in monthly basis, for the time period 01/01/2001-12/31/2014. That means that our sample, consists of 5,040
observations, whereas we have also the monthly returns of the S&P500 index, and the treasury bills rates for the same period, which had been used as the risk free rate.In the begining,
we will evaluate the funds for the time period 01/01/2001-12/31/2011, by regressing the CAPM model Ri,t = ai + biRm,t + εi in order to estimate the risk of each fund. Then, using the
Treynor-Mazuy formula Ri,t − Rf,t = ai + bi(Rm,t − Rf,t) + ci(Rm, t − Rf,t)2 + εi, we will estimate the selectivity (ai) and the market-timing (ci) of each fund. The best funds of this
period, will be evaluated separate out of sample, for the period 01/01/2012-12/31/2014, in order to test their ability to remain the best. Also, we will build a portfolio consisting of
these funds, which will be evaluated for the same period, and for the whole period as well.In the end, we will try to state our conclusions of the results we will get in each level.
Arbitrage theory model
Λέξη κλειδί Mutual funds
Risk measurement indicators
Capital Asset Pricing Model (CAPM)
Ημερομηνία 30-10-2015
Άδεια https://creativecommons.org/licenses/by/4.0/ | {"url":"http://pyxida.aueb.gr/components/objects/print/print_object.php?id=5711","timestamp":"2024-11-06T23:23:10Z","content_type":"application/xhtml+xml","content_length":"6651","record_id":"<urn:uuid:46c36b1c-793f-4a02-b888-1c04e7236833>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00809.warc.gz"} |
Monte Carlo Simulation of Western 649
Someone recently expressed disbelief to me that
Western 649
is actually a smarter gamble than a typical 50/50 draw, so I decided to run a short
Monte Carlo simulation
to prove that
my previous analysis
based on probability theory was correct. I used Excel to generate 500,000 random numbers between 1 and 13,983,816 to simulate one Western 649 draw with 500,000 random combinations played. The
criteria for determining which of my simulated numbers were winners were as follows:
A total of 3,000 draws (each with 500,000 random combinations played) were simulated. The tables below summarize the results of my simulation:
Number of winners in the simulated Western 649 draws
Summary of WCLC's profit and house edge, and Western 649 expected value
In the first table, we see that the expected values generally agree well with the averages from the simulation, apart from there being a few more jackpot winners than expected from probability
theory. At most I had three winners sharing the $50,000 prize, which from my previous analysis I calculated had about 1 in 753 chance of occurrence in a draw with 500,000 random combinations. I also
had two winners sharing the $1,000,000 prize in one of my simulated draws, which I calculated previously to have about 1 in 1,621 chance of occurrence.
The second table shows that the worst individual draw for the WCLC resulted in a net loss of $920,090. But these losses were generally few and far between. The best draw for the WCLC produced
$139,330 in profit. After 3,000 simulated draws, the WCLC earned a total of more than $251 million on the Western 649 game, averaging nearly $84,000 in profits per draw. Note that these numbers
ignore operating costs to run the lottery: in real life, the WCLC has to pay for equipment and personnel to run the lottery, sell tickets, etc. However, it's still illustrative of how lucrative
running this lottery is (WCLC runs two Western 649 draws every week). The simulation results showed an overall house edge of 33.50%. This corresponds to players losing, on average, $0.335 of every $1
spent playing Western 649. This is only about 3.4% off what the analysis based on probability theory indicated. The expected average profit was $86,665.97 per draw, which corresponds to 34.67% house
edge and players losing $0.3467 of every $1 wagered.
Since this is a random number simulation with a finite number of trials, minor differences between the simulation results and the results from probability theory should be expected. The average would
approach the expected value as the number of trials approaches infinity. I've plotted the overall house edge against the number of simulated draws in the graph below to show that my simulation
results do tend to approach the expected value.
Overall house edge versus number of simulated draws
In summary, a simple Monte Carlo simulation of 3,000 Western 649 draws supports the results of my previous analysis based on probability theory. The expected value of Western 649 is better than the
expected value of a typical 50/50 draw. | {"url":"http://alohonyai.blogspot.com/2014/07/monte-carlo-simulation-of-western-649.html","timestamp":"2024-11-05T02:45:34Z","content_type":"text/html","content_length":"54844","record_id":"<urn:uuid:ca05edf9-05bd-4baa-b36f-10448692add1>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00420.warc.gz"} |
Use spectral importance sampling method (mc_spectral_is) to simulate high resolution SCIAMACHY spectra (NO2 channel 425-470 nm).
The directory write_molecular_tau includes a script that generates a molecular_tau_file for 425-470 nm from the uvspec verbose output.
This script has been validated by comparing disort clear sky calculations with and without molecular_tau_file.
(In order to get accurate results the output format of the verbose output needed to be changed. → now committed)
The spectral resolution of the molecular_tau_file and hence of all calculations is 0.1 nm, resulting in 451 spectral points.
For this test the surface albedo was set to a high value (0.5) so that the NO2 absorption features are well visible.
Moved importance sampling of single scattering albedo from escape_probability to scattering. Clearsky still works and the calculation time remains the same.
There is a spectral dependence in the relative difference between DISORT and MYSTIC. The correction of the calculation of the total phase function can be enabled or disabled. Something still wrong
with spectral importance sampling of Rayleigh scattering????
standard deviation at 450 nm: 0.01%, without vroom 2.2% → spikes can follow and increase these values
CPU time with vroom and approx exponential function exp(x) ~ 1+x: 14m 26s, saves less than 10%, so the approximation is not worth implementing!
VROOM does not work with phasecorrection yet. phasecorrection currently in scattering_probability_tot, which is without vroom only used before local estimate, with vroom in many other places, so I
don't know what happens … | {"url":"https://www.en.meteo.physik.uni-muenchen.de/~emde/doku.php?id=studies:resinc:resinc","timestamp":"2024-11-06T21:35:06Z","content_type":"text/html","content_length":"29917","record_id":"<urn:uuid:ea6a852b-2009-4c4b-b533-a3753907cff2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00209.warc.gz"} |
Introduction. Dimensional Analysis - Definition of a turbomachine. Different kinds and applications. - Main defining variables, dimensions and fluid properties. Units. - Dimensional analysis and
performance laws. Compressible flow analysis. Specific speed: machine selection. Model testing. Fluid mechanics and thermodynamics equations - Equations in integral form. - Euler equations for
turbomachines. - Definition of Rothalpy. - Definition of adiabatic / polytropic efficiency. Enthalpy-entropy diagrams. - Equations in differential form. Two-dimensional cascades - Introduction.
Definition of streamsurface, blade-to-blade analysis. - Cascade nomenclature for compressors and turbines. - Cascade kinematics: velocity triangles. Cascade dynamics: forces, momentum. Cascade
entalphy and entropy change: losses. - Compressor cascade performance. Compressor characteristics: enthalpy rise, pressure recovery, deflection, deviation and loss. Blade loading: surface velocity
distribution, diffusion factor. Compressor cascade correlations: optimum solidity, polar curve. Diffusor efficiency. - Turbine cascade performance. Turbine characteristics: turning angle, Zweifel
coefficient. Surface velocity distribution: Back Surface Diffusion parameter. Turbine cascade correlations: loss, optimum pitch-chord ratio. - Cascade wind tunnel testing. Description of tunnels,
measurements. Unsteadiness. Axial flow turbines: two-dimensional stage theory - Dimensional analysis of a single turbine stage. Velocity triangles, loading and flow parameters, reaction. Repeating
stage hypothesis. - Thermodynamics of a turbine stage. Total-to-total stage efficiency. Row loss-stage efficiency relation - Reaction. Effect on efficiency. Optimum reaction - Smith chart. Empirical
versus reversible. - Flow characteristics of a multistage turbine. - Stress/Cooling/Detailed design. Design criteria. Axial flow compressors and fans: two-dimensional stage theory - Dimensional
analysis of a single compressor stage. Velocity triangles, loading and flow parameters, reaction. Repeating stage hypothesis. - Thermodynamics of a compressor stage. Total-to-total stage efficiency.
Row loss-stage efficiency relation. - Loading-Flow coefficient chart. Reaction choice. Lift and Drag in terms of ¿ and ¿. Diffusion Factor and solidity selection. Estimation of compressor efficiency.
Simplify off-design performance. - Blade element theory. - Stall and surge phenomena. Three-dimensional flow in Axial Turbomachines - Theory of radial equilibrium. The indirect problem: free-vortex
flow, forced-vortex flow, general whirl distribution. The direct problem. - Compressible flow through a blade-row. - Constant specific mass flow. - Off-design performance of a stage (free-vortex
turbine). - Actuator disc approach. Blade-row interactions. Computer methods solving through-flow problem. - Secondary flows. Loss, angles and helicity. - Three-dimensional losses. Types and models.
- Three-dimensional design features. Lean, sweep and bow. Centrifugal compressors, fans and pumps - Introduction and definitions. Centrifugal compressor parts. - Theoretical analysis of a centrifugal
compressor. Dimension-less performance parameters. Inlet, impeller and diffuser equations. - Optimum design of a centrifugal compressor inlet. - Radial flow turbo-machine blading design/selection -
Slip factor. Correlations. - Performance of centrifugal compressors. - Diffuser system. Vane and vane-less diffusers. - Chocking in centrifugal compressor stage. Radial turbines - Introduction. Types
of inward flow radial turbine. - Thermodynamics of the 90 degrees IFR turbine - Basic rotor design. Rotor efficiency definition. Mach number relations. Loss coefficients. - Optimum efficiency
considerations. Minimum number of blades. - Design criteria. Pressure ratio limits. | {"url":"https://aplicaciones.uc3m.es/cpa/generaFicha?est=251&plan=421&asig=15362&idioma=2","timestamp":"2024-11-06T23:39:04Z","content_type":"text/html","content_length":"15074","record_id":"<urn:uuid:7d97a0c8-7219-48ee-8c6e-6ec98fefabcc>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00155.warc.gz"} |
CryptoMiniSat assertion error
CryptoMiniSat assertion error
I'm trying to use sage.sat.boolean_polynomials to solve a system of equations for my thesis. I'm ok with the fact it uses CryptoMiniSat as default but every time I run my script I get this error:
Traceback (most recent call last):
File "simon_equations.py", line 104, in <module>
s = alg_attack(p,c)
File "simon_equations.py", line 55, in alg_attack
return solve_sat(F)
File "/home/frollo/sage-6.7/local/lib/python2.7/site-packages/sage/sat/boolean_polynomials.py", line 179, in solve
phi = converter(F)
File "/home/frollo/sage-6.7/local/lib/python2.7/site-packages/sage/sat/converters/polybori.py", line 581, in __call__
File "/home/frollo/sage-6.7/local/lib/python2.7/site-packages/sage/sat/converters/polybori.py", line 538, in clauses
File "/home/frollo/sage-6.7/local/lib/python2.7/site-packages/sage/sat/converters/polybori.py", line 296, in clauses_sparse
File "sage/sat/solvers/cryptominisat/cryptominisat.pyx", line 205, in sage.sat.solvers.cryptominisat.cryptominisat.CryptoMiniSat.add_clause (build/cythonized/sage/sat/solvers/cryptominisat/cryptominisat.cpp:1672)
I got to line 205 of sage/sat/solvers/cryptominisat/cryptominisat.pyx and found this:
I found nothing about this in the documentation, so I decided to try some bad practices and commented away the assertion, restarted sage and tried again. I got the same error, referring to a line
that is now a comment. I tried to restart my shell but, again, I got the same error.
Has anyone a clue about which could be causing this (the same script worked during the tests, the error just started showing up when I tried to run something serious)
changing the code will only be taken into account if you do sage -br to re-build sage | {"url":"https://ask.sagemath.org/question/27039/cryptominisat-assertion-error/","timestamp":"2024-11-15T04:39:59Z","content_type":"application/xhtml+xml","content_length":"50464","record_id":"<urn:uuid:bac1eed2-d699-4a7c-ac31-69d191bc9d8c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00192.warc.gz"} |
Journals starting with emmc
EMMCVPR01 * *EMMCVPR
* 3D Flux Maximizing Flows
* Application of Genetic Algorithms to 3-D Shape Reconstruction in an Active Stereo Vision System
* Articulated Object Tracking via a Genetic Algorithm
* Averaged Template Matching Equations
* Complementary Pivoting Approach to Graph Matching, A
* Continuous Shape Descriptor by Orientation Diffusion, A
* Designing Moiré Patterns
* Designing the Minimal Structure of Hidden Markov Model by Bisimulation
* Discrete/Continuous Minimization Method in Interferometric Image Processing, A
* Double-Loop Algorithm to Minimize the Bethe Free Energy, A
* Edge Based Probabilistic Relaxation for Sub-pixel Contour Extraction
* Efficiently Computing Weighted Tree Edit Distance Using Relaxation Labeling
* Estimation of Distribution Algorithms: A New Evolutionary Computation Approach for Graph Matching Problems
* Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision, An
* Fast MAP Algorithm for 3D Ultrasound, A
* Gabor Feature Space Diffusion via the Minimal Weighted Area Method
* Geodesic Interpolating Splines
* Global Energy Minimization: A Transformation Approach
* Global Feedforward Neural Network Learning for Classification and Regression
* Grouping with Directed Relationships
* Hierarchical Markov Random Field Model for Figure-Ground Segregation, A
* Highlight and Shading Invariant Color Image Segmentation Using Simulated Annealing
* Illumination Invariant Recognition of Color Texture Using Correlation and Covariance Functions
* Image Labeling and Grouping by Minimizing Linear Functionals over Cones
* Learning Matrix Space Image Representations
* Markov Process Using Curvature for Filtering Curve Images, A
* Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics
* Matching Images to Models: Camera Calibration for 3-D Surface Reconstruction
* Maximum Likelihood Estimation of the Template of a Rigid Moving Object
* Maximum Likelihood Framework for Grouping and Segmentation, A
* Metric Similarities Learning through Examples: An Application to Shape Retrieval
* Multiple Contour Finding and Perceptual Grouping as a Set of Energy Minimizing Paths
* Optical Flow and Image Registration: A New Local Rigidity Approach for Global Minimization
* Optimization of Paintbrush Rendering of Images by Dynamic MCMC Methods
* Path Based Pairwise Data Clustering with Application to Texture Segmentation
* Relaxing Symmetric Multiple Windows Stereo Using Markov Random Fields
* Segmentations of Spatio-Temporal Images by Spatio-Temporal Markov Random Field Model
* Shape Tracking Using Centroid-Based Methods
* Spherical Object Reconstruction Using Star-Shaped Simplex Meshes
* Supervised Texture Segmentation by Maximising Conditional Likelihood
* Two Variational Models for Multispectral Image Classification
* Variational Approach to Maximum a Posteriori Estimation for Image Denoising, A
43 for EMMCVPR01
EMMCVPR05 * *EMMCVPR
* Adaptive Simulated Annealing for Energy Minimization Problem in a Marked Point Process Application
* Adaptive Variational Model for Image Decomposition, An
* Bayesian Image Segmentation Using Gaussian Field Priors
* Brain Image Analysis Using Spherical Splines
* Coined Quantum Walks Lift the Cospectrality of Graphs and Trees
* Color Correction of Underwater Images for Aquatic Robot Inspection
* Computational Approach to Fisher Information Geometry with Applications to Image Analysis, A
* Concurrent Stereo Matching: An Image Noise-Driven Model
* Constrained Hybrid Optimization Algorithm for Morphable Appearance Models, A
* Constrained Total Variation Minimization and Application in Computerized Tomography
* Deformable-Model Based Textured Object Segmentation
* Discontinuity Preserving Phase Unwrapping Using Graph Cuts
* Dynamic Shape and Appearance Modeling via Moving and Deforming Layers
* Edge Strength Functions as Shape Priors in Image Segmentation
* Energy Minimization Based Segmentation and Denoising Using a Multilayer Level Set Approach
* Exploiting Inference for Approximate Parameter Learning in Discriminative Fields: An Empirical Study
* Extraction of Layers of Similar Motion Through Combinatorial Techniques
* Geodesic Image Matching: A Wavelet Based Energy Minimization Scheme
* Geodesic Shooting and Diffeomorphic Matching Via Textured Meshes
* Handling Missing Data in the Computation of 3D Affine Transformations
* High-Order Differential Geometry of Curves for Multiview Reconstruction and Matching
* Increasing Efficiency of SVM by Adaptively Penalizing Outliers
* Kernel Methods for Nonlinear Discriminative Data Analysis
* Learning Hierarchical Shape Models from Examples
* Linear Programming Matching and Appearance-Adaptive Object Tracking
* Locally Linear Isometric Parameterization
* Maximum-Likelihood Estimation of Biological Growth Variables
* New Implicit Method for Surface Segmentation by Minimal Paths: Applications in 3D Medical Images, A
* Object Categorization by Compositional Graphical Models
* One-Shot Integral Invariant Shape Priors for Variational Segmentation
* Optimizing the Cauchy-Schwarz PDF Distance for Information Theoretic, Non-parametric Clustering
* Probabilistic Subgraph Matching Based on Convex Relaxation
* Relaxation of Hard Classification Targets for LSE Minimization
* Retrieving Articulated 3-D Models Using Medial Surfaces and Their Graph Spectra
* Reverse-Convex Programming for Sparse Image Codes
* Segmentation Informed by Manifold Learning
* Some New Results on Non-rigid Correspondence and Classification of Curves
* Spatio-temporal Prior Shape Constraint for Level Set Segmentation
* Spatio-temporal Segmentation Using Dominant Sets
* Stable Bounded Canonical Sets and Image Matching
* Stereo for Slanted Surfaces: First Order Disparities and Normal Consistency
* Total Variation Minimization and a Class of Binary MRF Models
43 for EMMCVPR05
EMMCVPR07 * *EMMCVPR
* 3D Computation of Gray Level Co-occurrence in Hyperspectral Image Cubes
* a Contrario Approach for Parameters Estimation of a Motion-Blurred Image, An
* Active Appearance Models Fitting with Occlusion
* Automatic Portrait System Based on And-Or Graph Representation, An
* Bayesian Inference for Layer Representation with Mixed Markov Random Field
* Bayesian Order-Adaptive Clustering for Video Segmentation
* Boosting Discriminative Model for Moving Cast Shadow Detection, A
* Bottom-Up Recognition and Parsing of the Human Body
* CIDER: Corrected Inverse-Denoising Filter for Image Restoration
* Combining Left and Right Irises for Personal Authentication
* Compositional Object Recognition, Segmentation, and Tracking in Video
* Continuous Global Optimization in Multiview 3D Reconstruction
* Decomposing Document Images by Heuristic Search
* Dichromatic Reflection Separation from a Single Image
* Discrete Skeleton Evolution
* Dynamic Feature Cascade for Multiple Object Tracking with Trackability Analysis
* Effective Multi-level Algorithm Based on Simulated Annealing for Bisecting Graph, An
* Efficient Shape Matching Via Graph Cuts
* Energy Minimisation Approach to Attributed Graph Regularisation, An
* Energy-Based Reconstruction of 3D Curves for Quality Control
* Exact Solution of Permuted Submodular MinSum Problems
* Improved Object Tracking Using an Adaptive Colour Model
* Introduction to a Large-Scale General Purpose Ground Truth Database: Methodology, Annotation Tool and Benchmarks
* Marked Point Process for Vascular Tree Extraction on Angiogram
* New Bayesian Method for Range Image Segmentation, A
* Noise Removal and Restoration Using Voting-Based Analysis and Image Segmentation Based on Statistical Models
* Object Category Recognition Using Generative Template Boosting
* Probabilistic Fiber Tracking Using Particle Filtering and Von Mises-Fisher Sampling
* Pupil Localization Algorithm Based on Adaptive Gabor Filtering and Negative Radial Symmetry, A
* Removing Shape-Preserving Transformations in Square-Root Elastic (SRE) Framework for Shape Analysis of Curves
* Shape Analysis of Open Curves in R3 with Applications to Study of Fiber Tracts in DT-MRI Data
* Shape Classification Based on Skeleton Path Similarity
* Simulating Classic Mosaics with Graph Cuts
* Skew Detection Algorithm for Form Document Based on Elongate Feature
* Surface Reconstruction from LiDAR Data with Extended Snake Theory
* Szemerédi's Regularity Lemma and Its Applications to Pairwise Clustering and Segmentation
* Vehicle Tracking Based on Image Alignment in Aerial Videos
38 for EMMCVPR07
EMMCVPR09 * *EMMCVPR
* Bipartite Graph Matching Computation on GPU
* Boundaries as Contours of Optimal Appearance and Area of Support
* Clustering-Based Construction of Hidden Markov Models for Generative Kernels
* Color Image Restoration Using Nonlocal Mumford-Shah Regularizers
* Color Image Segmentation in a Quaternion Framework
* Complementary Optic Flow
* Complex Diffusion on Scalar and Vector Valued Image Graphs
* Computing the Local Continuity Order of Optical Flow Using Fractional Variational Method
* Detection and Segmentation of Independently Moving Objects from Dense Scene Flow
* Efficient Global Minimization for the Multiphase Chan-Vese Model of Image Segmentation
* Exemplar-Based Interpolation of Sparsely Sampled Images
* General Search Algorithms for Energy Minimization Problems
* Geodesics in Shape Space via Variational Time Discretization
* Global Optimal Multiple Object Detection Using the Fusion of Shape and Color Information
* Hierarchical Pairwise Segmentation Using Dominant Sets and Anisotropic Diffusion Kernels
* Hierarchical Vibrations: A Structural Decomposition Approach for Image Analysis
* Human Age Estimation by Metric Learning for Regression Problems
* Image Filtering Driven by Level Curves
* Image Registration under Varying Illumination: Hyper-Demons Algorithm
* Integrating the Normal Field of a Surface in the Presence of Discontinuities
* Intrinsic Second-Order Geometric Optimization for Robust Point Set Registration without Correspondence
* Local Normal-Based Region Term for Active Contours, A
* Locally Parallel Textures Modeling with Adapted Hilbert Spaces
* Multi-label Moves for MRFs with Truncated Convex Priors
* On a Decomposition Model for Optical Flow
* Parallel Hidden Hierarchical Fields for Multi-scale Reconstruction
* Parameter Estimation for Marked Point Processes. Application to Object Extraction from Remote Sensing Images
* PDE Approach to Coupled Super-Resolution with Non-parametric Motion, A
* Pose-Invariant Face Matching Using MRF Energy Minimization Framework
* Quaternion-Based Color Image Smoothing Using a Spatially Varying Kernel
* Reconstructing Optical Flow Fields by Motion Inpainting
* Robust Segmentation by Cutting across a Stack of Gamma Transformed Images
* Schrödinger Wave Equation Approach to the Eikonal Equation: Application to Image Analysis, A
* Three Dimensional Monocular Human Motion Analysis in End-Effector Space
* Tracking as Segmentation of Spatial-Temporal Volumes by Anisotropic Weighted TV
* Variational Framework for Non-local Image Inpainting, A
37 for EMMCVPR09
EMMCVPR11 * *EMMCVPR
* Branch and Bound Strategies for Non-maximal Suppression in Object Detection
* Complex Wave Representation of Distance Transforms, The
* Curvature Regularity for Multi-label Problems: Standard and Customized Linear Programming
* Curvature Regularization for Curves and Surfaces in a Global Optimization Framework
* Data-Driven Importance Distributions for Articulated Tracking
* Detachable Object Detection with Efficient Model Selection
* Discrete Optimization of the Multiphase Piecewise Constant Mumford-Shah Functional
* Distributed Mincut/Maxflow Algorithm Combining Path Augmentation and Push-Relabel, A
* Evaluation of a First-Order Primal-Dual Algorithm for MRF Energy Minimization
* Fast Solver for Truncated-Convex Priors: Quantized-Convex Split Moves, A
* Global Relabeling for Continuous Optimization in Binary Image Segmentation
* Globally Optimal Image Partitioning by Multicuts
* High Resolution Segmentation of Neuronal Tissues from Low Depth-Resolution EM Imagery
* Image Segmentation with a Shape Prior Based on Simplified Skeleton
* Interactive Segmentation with Super-Labels
* Intermediate Flow Field Filtering in Energy Based Optic Flow Computations
* Metrics, Connections, and Correspondence: The Setting for Groupwise Shape Analysis
* Minimizing Count-Based High Order Terms in Markov Random Fields
* Multiple-Instance Learning with Structured Bag Models
* Optical Flow Guided TV-L1 Video Interpolation and Restoration
* Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem
* Optimization of Robust Loss Functions for Weakly-Labeled Image Taxonomies: An ImageNet Case Study
* Robust Trajectory-Space TV-L1 Optical Flow for Non-rigid Sequences
* SlimCuts: GraphCuts for High Resolution Images Using Graph Reduction
* Space-Varying Color Distributions for Interactive Multiregion Segmentation: Discrete versus Continuous Approaches
* Stop Condition for Subgradient Minimization in Dual Relaxed (max,+) Problem
* Temporally Consistent Gradient Domain Video Editing
* Texture Segmentation via Non-local Non-parametric Active Contours
* TV-L1 Optical Flow for Vector Valued Images
* Using the Higher Order Singular Value Decomposition for Video Denoising
31 for EMMCVPR11
EMMCVPR15 * *EMMCVPR
* Adaptive Dictionary-Based Spatio-temporal Flow Estimation for Echo PIV
* Automatic Shape Constraint Selection Based Object Segmentation
* Blind Deconvolution via Lower-Bounded Logarithmic Image Priors
* Coarse-to-Fine Minimization of Some Common Nonconvexities
* Color Image Segmentation by Minimal Surface Smoothing
* Compact Linear Programming Relaxation for Binary Sub-modular MRF, A
* Convex Envelopes for Low Rank Approximation
* Convex Solution to Disparity Estimation from Light Fields via the Primal-Dual Method, A
* Discrete Green's Functions for Harmonic and Biharmonic Inpainting with Sparse Atoms
* Domain Decomposition Methods for Total Variation Minimization
* Efficient Curve Evolution Algorithm for Multiphase Image Segmentation, An
* Expected Patch Log Likelihood with a Sparse Prior
* Fast Projection Method for Connectivity Constraints in Image Segmentation, A
* Hierarchical Planar Correlation Clustering for Cell Segmentation
* How Hard Is the LP Relaxation of the Potts Min-Sum Labeling Problem?
* Inpainting of Cyclic Data Using First and Second Order Differences
* Justifying Tensor-Driven Diffusion from Structure-Adaptive Statistics of Natural Images
* Low Rank Priors for Color Image Regularization
* Mapping the Energy Landscape of Non-convex Optimization Problems
* Marked Point Process Model for Curvilinear Structures Extraction
* Maximizing Flows with Message-Passing: Computing Spatially Continuous Min-Cuts
* Multi-class Graph Mumford-Shah Model for Plume Detection Using the MBO scheme
* Multi-utility Learning: Structured-Output Learning with Multiple Annotation-Specific Loss Functions
* Novel Active Contour Model for Texture Segmentation, A
* Novel Framework for Nonlocal Vectorial Total Variation Based on L p,q,r-norms, A
* On the Link between Gaussian Homotopy Continuation and Convex Envelopes
* Optical Flow with Geometric Occlusion Estimation and Fusion of Multiple Frames
* Point Sets Matching by Feature-Aware Mixture Point Matching Algorithm
* Randomly Walking Can Get You Lost: Graph Segmentation with Unknown Edge Weights
* Segmentation Using SubMarkov Random Walk
* Technique for Lung Nodule Candidate Detection in CT Using Global Minimization Methods, A
* Tensor Variational Formulation of Gradient Energy Total Variation, A
* Training of Templates for Object Recognition in Invertible Orientation Scores: Application to Optic Nerve Head Detection in Retinal Images
* Two-Dimensional Variational Mode Decomposition
* Variational Time-Implicit Multiphase Level-Sets
* Why Does Non-binary Mask Optimisation Work for Diffusion-Based Image Compression?
37 for EMMCVPR15
EMMCVPR17 * *EMMCVPR
* Autonomous Multi-camera Tracking Using Distributed Quadratic Optimization
* Bottom-Up Top-Down Cues for Weakly-Supervised Semantic Segmentation
* Bragg Diffraction Patterns as Graph Characteristics
* Convex Approach to K-Means Clustering and Image Segmentation, A
* Depth-Adaptive Computational Policies for Efficient Visual Tracking
* Discretized Convex Relaxations for the Piecewise Smooth Mumford-Shah Model
* Dominant Set Biclustering
* Euler-Lagrange Network Dynamics
* Fast Asymmetric Fronts Propagation for Voronoi Region Partitioning and Image Segmentation
* Geometric Image Labeling with Global Convex Labeling Constraints
* Graph Theoretic Approach for Shape from Shading, A
* Illumination-Aware Large Displacement Optical Flow
* Inverse Lightfield Rendering for Shape, Reflection and Natural Illumination
* Ising Models for Binary Clustering via Adiabatic Quantum Computing
* Isotropic Minimal Path Based Framework for Segmentation and Quantification of Vascular Networks, An
* Limited-Memory Belief Propagation via Nested Optimization
* Location Uncertainty Principle: Toward the Definition of Parameter-Free Motion Estimators
* Luminance-Guided Chrominance Denoising with Debiased Coupled Total Variation
* Maximum Consensus Parameter Estimation by Reweighted L_1 Methods
* Modelling Stable Backward Diffusion and Repulsive Swarms with Convex Energies and Range Constraints
* Multi-object Convexity Shape Prior for Segmentation
* Multiframe Motion Coupling for Video Super Resolution
* Nonlinear Compressed Sensing for Multi-emitter X-Ray Imaging
* Optimizing Wavelet Bases for Sparser Representations
* PointFlow: A Model for Automatically Tracing Object Boundaries and Inferring Illusory Contours
* Projected Gradient Descent Method for CRF Inference Allowing End-to-End Training of Arbitrary Pairwise Potentials, A
* Quantum Interference and Shape Detection
* Shadow and Specularity Priors for Intrinsic Light Field Decomposition
* Sharpening Hyperspectral Images Using Spatial and Spectral Priors in a Plug-and-Play Algorithm
* Slack and Margin Rescaling as Convex Extensions of Supermodular Functions
* Structured Output Prediction and Learning for Deep Monocular 3D Human Pose Estimation
* Superpixels Optimized by Color and Shape
* Temporal Semantic Motion Segmentation Using Spatio Temporal Optimization
* Unified Functional Framework for Restoration of Image Sequences Degraded by Atmospheric Turbulence
* Variational Approach to Shape-from-Shading Under Natural Illumination, A
* Variational Large Displacement Optical Flow Without Feature Matches
* Vehicle X-Ray Images Registration
38 for EMMCVPR17
Last update: 9-Nov-24 12:05:26
Use price@usc.edu for comments. | {"url":"http://www.visionbib.com/bibliography/journal/emmc.html","timestamp":"2024-11-14T05:25:34Z","content_type":"text/html","content_length":"32595","record_id":"<urn:uuid:373284da-970d-4892-8181-4621cd3a4913>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00543.warc.gz"} |
Background Standard drug development conducts phase I dose finding and phase
Background Standard drug development conducts phase I dose finding and phase
Background Standard drug development conducts phase I dose finding and phase II dose expansion sequentially and Torcetrapib (CP-529414) separately. outcome. Phase I dose escalation dose phase and
graduation II adaptive randomization proceed simultaneously throughout the entire trial. Results Examples are given comparing SEARS with two other designs in which superior performance of SEARS is
demonstrated. An promising and important finding is that SEARS reduces sample sizes without losing power. R program and demo slides of SEARS can be obtained at http://www.northshore.org/research/
investigators/yuan-ji-phd/ Limitation We assume that the binary toxicity and efficacy response can be measured in the same time frame. This is achievable with surrogate efficacy markers in practice
often. (e.g. = 0.3). Let = (= 1is the total number of candidate doses in the trial. The observed data include patients treated at dose the true number of patients among that experienced toxicity. The
likelihood function for data {(= 1and decide future doses that are close to the true MTD. Anticipating that doses might graduate to phase II during the course of phase I dose finding we apply
Torcetrapib (CP-529414) the mTPI design (Ji et al. 2010 to monitor conduct and toxicity dose escalation. The mTPI design is an extension of the toxicity probability interval method (Ji Torcetrapib
(CP-529414) et al. 2007 and employs a simple beta-binomial hierarchical model. Decision rules are based on calculating the unit probability mass (UPM) of three intervals corresponding to ? + 1) and
the proper-dosing interval (? + is used to treat patients. Denote the three dose-finding decisions as escalation (E) de-escalation (D) and stay (S). To apply mTPI we calculate the three UPMs for
under- proper- and over-dosing intervals simply. A dose-assignment rule Bbased on these three UPMs chooses the decision with the largest UPM that is is optimal in that it minimizes a posterior
expected loss in which the loss function is Hhex determined to achieve equal prior expected loss for the three decisions D S and E. The mTPI design assumes vague and independent priors ~ follows
independent 1 + ? = 1denote the number of efficacy responses among patients treated at dose denote the true efficacy probability of dose are independent and follow Jeffreys prior is 0.5 + ? follows a
is due to the setup in the mTPI design mainly. Here we choose Jeffrey’s for due to its invariant and noninformative property prior. The proposed graduation rule is based on posterior probabilities of
and that satisfies and are physician-specified upper toxicity and lower efficacy probability thresholds respectively. For example = could be the historical response rate of the standard treatment. In
words the graduation rule posits that if a dose exhibits low toxicity and reasonable efficacy with high posterior Torcetrapib (CP-529414) probability it will graduate to phase II. An immediate impact
after a dose graduates to phase II is that there will be one fewer dose in phase I dose escalation and one more dose in phase II. Continuing phase I with one fewer dose is unproblematic however.
Consider an arbitrary example in which dose 3 has graduated to phase II just. Remaining Torcetrapib (CP-529414) in phase I are doses 1 2 4 … which can simply be relabeled as doses 1 2 3 … And dose
escalation continues based on mTPI using the decision rule in (1). Since remains the same. Therefore phase I dose escalation proceeds as usual under mTPI with the new dose labels. 2.3 Phase II
Adaptive Randomization For phase II we apply an adaptive randomization scheme similar to that in Huang et al. (2007). To take advantage of the seamless feature we include the efficacy response data
from phase I in computing the adaptive randomization probabilities. Adaptive randomization (AR) procedures aim to Torcetrapib (CP-529414) assign larger numbers of patients to more efficacious dose
arms. Bayesian AR procedures continuously update the randomization probability for arm according to the observed response data. A common approach is to randomize patients to dose arm with a
probability proportional to > |high efficacy rates. We will use the same AR probability to assign patients in the phase II stage of the SEARS design. 2.4 SEARS Design We combine the aforementioned
procedures in phase I dose graduation. | {"url":"https://www.mdm2-inhibitors.com/2016/08/background-standard-drug-development-conducts-phase-i-dose-finding-and-phase/","timestamp":"2024-11-10T01:50:23Z","content_type":"text/html","content_length":"52663","record_id":"<urn:uuid:3444d157-4524-415c-aeef-1f7c7496ab5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00136.warc.gz"} |
Heron's Formula
Heron's formula is a formula that relates the area of any triangle to the lengths of its sides. It is:
A = √s(s − a)(s − b)(s − c)
In the above equation, a, b, and c are the lengths of the sides of the triangle, and s = ½(a + b + c); i.e. half of the perimeter of the triangle (also called the semiperimeter).
Heron's formula is usually proven geometrically or trigonometrically. An algebraic proof is presented below. If you'd like to skip the proof and continue to further discussion, select this link.
Consider triangle ABC, having side lengths a, b, and c. Draw a line through B perpendicular to CA, meeting CA at D. Call its length h, and call the lengths of CD and DA b[1] and b[2], respectively
(note that one of b[1] and b[2] may be negative if the triangle is obtuse). This is all shown in the diagram above.
First, express b[1] and b[2] in terms of a, b, c, and h, then express h in terms of a, b, and c, and then evaluate Area = ½bh:
Using the Pythagorean theorem:
h² + b[2]² = c²
b[2] = √c² − h²
As well:
b[1] = b − b[2]
b[1] = b − √c² − h²
Now, express h in terms of a, b, and c:
h² + b[1]² = a²
h² = a² − (b − √c² − h²)²
h² = a² − (b² − 2b√c² − h² + c² − h²)
h² = a² − b² + 2b√c² − h² − c² + h²
b² + c² − a²2b = √c² − h²
h² = c² − b² + c² − a²2b
h = √(c + b² + c² − a²2b)(c − b² + c² − a²2b)
Having expressed h in terms of the sides of the triangle,
Area = ½bh = ¼√(4b²c² −(b² + c² − a²)²)
= ¼√(2bc + b² + c² − a²)(2bc − b² − c² + a²)
= ¼√((b + c)² − a²)(a² − (b − c)²)
= ¼√(b + c − a)(b + c + a)(a + b − c)(a − b + c) = √½(a + b + c)·½(a + b + c − 2a)·½(a + b + c − 2b)·½(a + b + c − 2c)
Area= √s(s − a)(s − b)(s − c)
It is possible to compute the area of a triangle when given the lengths of its sides. Is it possible to compute the area of a quadrilateral given its side lengths? The answer is no; two
quadrilaterals whose sides are all the same could have different lengths. Unlike triangles, quadrilaterals are not rigid. You can demonstrate this for yourself. Take three straws (or other straight
objects) and use balls of clay or large marshmallows (or something similar) to attach them in the shape of a triangle. Now try moving one of the straws without detaching it from the two balls of clay
it's attached to. You won't be able to do it; the shape is rigid. Now create a square. This time, the square will readily deform into a parallelogram.
However, it can be shown that the maximum area of a quadrilateral with sides a, b, c, and d is:
A = √(s − a)(s − b)(s − c)(s − d) | {"url":"http://mathlair.allfunandgames.ca/heronsformula.php","timestamp":"2024-11-12T17:22:14Z","content_type":"text/html","content_length":"8866","record_id":"<urn:uuid:f331e54b-4fe7-46f7-b9b8-86c332f483ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00107.warc.gz"} |
Nature of Points
Can you write a function whose derivative does NOT exist a some point.🗨 🙋 🕵 🗩 Yes, below is a function fiven by f(x)=|x| whose derivative does NOT exist at x=0 .
Given function is
\(f(x)= |x|\)
In the simplified form, this function can be written as
\( f(x)= \begin{cases} -x &\text { for } x < 0 \\ x &\text { for } x>0 \end{cases} \)
The derivative function is given by
\( f'(x)= \begin{cases} -1 &\text { for } x < 0 \\ 1 &\text { for } x>0 \end{cases} \)
At \( x=0^{-}\), the left hand limit is
\( \displaystyle \lim_{x \to 0^{-}} f'(x)= \lim_{x \to 0^{-} } -1= -1\) LHL
At \( x=0^{+}\), the right hand limit is
\( \displaystyle \lim_{x \to 0^{+}} f'(x)= \lim_{x \to 0^{+} } 1= 1\) RHL
Since, LHL≠RHL, so, the derivative of the function f(x)=|x| does NOT exist at x=0.
Nature of points
There are 6 types of points, they are describes as below.
1. Critical Point
A point where the derivative of a function is either zero or undefined, is called critical points.
In the figure below, A,B,C, and D are critical points in which
1. f'(x)=0 at A
2. f'(x)=0 at B
3. f'(x) is NOT defined at C (f is not differentiable at C)
4. f'(x)=0 at D
Critical points are the points on the graph where the function's rate of change is altered—either a change from increasing to decreasing, in concavity, or in some unpredictable fashion. A
critical point is
1. stationary point if the function changes from increasing/decreasign to decreasing/increasing at that point
2. turning point if the function changes from increasing/decreasign to decreasing/increasing at that point and and is a local minimum/maximum
3. saddle point if the function changes both increasing/decreasign and concavity at that point
NOTE: Maxima and mimina can occurs at critical points where first derivative is undefined.
2. Stationary Point
A point where the derivative of a function (the gradient/slope of a graph) is zero is called a stationary point.
In the figure below, A,B,and D are stationary points in which
1. f'(x)=0 at A
2. f'(x)=0 at B
3. f'(x)=0 at D
Note: all stationary points are critical points, but not all critical points are stationary points.
3. Turning Point
A point where the derivative of a function is (the gradient/slope of a graph) is zero and function has a local maximum or local minimum (extrema point), is called a turning point.There are two
types of turning points: local maximum and local minimum.
1. If f is increasing on the left interval and decreasing on the right interval, then the stationary point is a local maximum
2. If f is decreasing on the left interval and increasing on the right interval, then the stationary point is a local minimum
In the figure below, C is only the turning point[Please note that A and B is not turning point] in which
1. f'(x)=0 at A, but local extrema is NOT defined at A
2. f'(x) is NOT defined at B
3. f'(x)=0 at C, local minima is defined at C
Note: all turning points are stationary points, but not all stationary points are turning points.
4. Straight line/Constant function
Some stationary points are neither turning points nor horizontal points of inflection. For example, every point on the graph of the equation \(y = 1\) (see Figure below), or on any horizontal
line, is a stationary point that is neither a turning point nor point of inflection. It is a straight line.
5. Point of Inflection/ Plateau point
A point where the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa, is called an inflection point. Assuming the
second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.
Mathematically, a point on a function f (x) is said to be a point of inflexion if f''(x) = 0 and f'''(x)≠0.
At this point, the concavity changes from upward to downward or vice-versa.
Therefore the point of inflexion is the transition between concavity of the curves.
6. Saddle point (Horizontal point of inflection)
A saddle point or minimax point is a point on a function where the slopes (derivatives) is zero , but which is not a local extremum of the function. Thus, a saddle point is a point which is both
a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.
Figure below shows the graph of the function \( f(x) = x^3\) .
The derivative of this function is \( f'(x) = 3x^2\), so the 1st derivative f'(x) =0 when x = 0 is
\(f'(0) = 3 \times 0^2 = 0\).
So the function \( f(x) = x^3\) has a stationary point at x = 0.
However, this stationary point isn’t a turning point.
The second derivative of this function \( f''(x) = 6x\), so the second derivative \( f''(x) = 0\) when x = 0 is
\(f''(0) = 6 \times 0 = 0\).
So the function \( f(x) = x^3\) has a inflexional point at x = 0.
In such case, the point is called saddle point
Find the critical points
1. \(f(x)=8x^3+81x^2-42x-8\)
2. \(f(x)=1+80x^3+5x^4-2x^5\)
3. \(f(x)=2x^3-7x^2-3x-2\)
4. \(f(x)=x^6-2x^5+8x^4\)
5. \(f(x)=4x^3-3x^2+9x+12\)
6. \(f(x)=\frac{x+4}{2x^2+x+8}\)
7. \(f(x)=\frac{1-x}{x^2+2x-15}\)
8. \(f(x)=\sqrt[5]{x^2-6x}\)
Find the inflexional point
1. Determine the concavity of \(f(x) = x^ 3 − 6 x^ 2 −12 x + 2\) and identify any points of inflection.
Find the concavity
1. Determine the concavity of \(f(x) = \sin x + \cos x\) on [0,2π] and identify any points of inflection
2. Discuss the curve/function with respect to cancavity, point of inflexion, and local maxima, minima.
a. \(y=x^4-4x^3\)
b. \(y=x^{2/3}(6-x)^{1/3}\)
c. \( f(x) = \frac{1}{6}x^4-2x^3+11x^2-18x\)
d. \( f(x) = \frac{1}{12}x^4-2x^2\)
e. \( f(x) = x^3-3x^2+1\)
f. \( f(x) = \frac{2}{3}x^3-\frac{5}{2}x^2+2x\)
Question for Practice
1. Find the stationary points of the function \( f(x) = 2x^3-3x^2-36x\).
2. Find the approximate values, to two decimal places, of the stationary points of the function \( f(x) = x^3-x^2-2x\).
3. Find the (a) intervals of increasing/decreasing (b) intervals of concave upward/downward (c) point of inflection
a. \(f(x)=2x^3-3x^2-12x\)
b. \( f(x) = \frac{4}{3}x^3 + 5x^2- 6x-2\)
c. \(f(x)=2+3x-x^3\)
d. \( f(x) = 3x^4 -2x^3- 9x^2 + 7\)
e. \(f(x)=x^4-6x^2\)
f. \(f(x)=200+8x^3+x^4\)
g. \(f(x)=3x^5-5x^3+3\)
h. \(f(x)=(x^2-1)^3\)
i. \(f(x)=x \sqrt{x^2+1}\)
j. \(f(x)=x-3x^{1/3}\)
4. Find the (a) vertical and horizental asymptotes (b) intervals of increasing/decreasing (c) intervals of concave upward/downward (d)point of inflection of following functions
a. \(f(x)=\frac{1+x^2}{1-x^2}\)
b. \(f(x)=\frac{x}{(x-1)^2}\)
c. \(f(x)=\sqrt{x^2+1}-x\)
Find the local Extreme
1. Find any turning points of the function \(f(x)=x^3-10x^2+25x+4\) and specify whether each turning point is a global maximum or minimum, or a local maximum or minimum.
Question for Exam
1. Suppose you are given formula for a function f
a. How do you determine where f is increasing/decreasing
b. How do you determine where f is concave upward/downward
c. How do you locate inflexional points
2. Find the critical numbers of \(f(x)=x^4(x-1)^3\).What does the second derivative tells you about the behavior of f at these points. What does the first derivative test tell you?
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G.S. Yablonskii, V.I.Bykov, A.N.Gorban, V.I.Elohin, Kinetic Models of Catalytic Reactions, Elsevier, R.G. Compton (Ed.) Series "Comprehensive Chemical Kinetics", Volume 32, 1991.
The book presents “three kinetics”: (a) detailed, oriented to the elucidation of a detailed reaction mechanism according to its kinetic laws; (b) applied, with the aim of obtaining kinetic
relationships for the further design of chemical reactors; and (c) mathematical kinetics whose purpose is the analysis of mathematical models for heterogeneous catalytic reactions taking place under
steady- or unsteady state conditions
Besides establishing a general theory permitting us to investigate the dependence of kinetic characteristics for complex reactions on the structure of detailed mechanism, the book provides a
comprehensive analysis of some concrete typical mechanisms for catalytic reactions, in particular for the oxidation of carbon monoxide on platinum metals. This reaction is a long-standing traditional
object of catalysis study, “Mona Lisa” of heterogeneous catalysis.
Authors’ preface
Chapter 1 (PDF Preface-Intro-Ch1)
Minimum minumorum
1. Introduction
2. Chemical kinetics and linear algebra
3. Unsteady- and steady-state kinetic models
4 Steady-state reaction theory
5. Elements of qualitative theory of differential equations
6. Relaxation in catalytic reactions
Chapter 2 (PDF Ch2)
The development of basic concepts of chemical kinetics in heterogeneous catalysis
1. Steps in development of general chemical kinetics
2. The development of the kinetics of heterogeneous catalysis
2.1 Ideal adsorbed layer model
2.2 Real adsorbed layer models
2.3 Models accounting for phase and structural transformations of catalysts
2.3.1 Phenomenological model
2.3.2 Lattice gas model
2.3.3 Topochemical models
2.4 Model accounting for diffusional mass transfer
2.5 Heterogeneous-homogeneous catalytic reaction models
2.6 Phenomenological model of branched-chain reactions on a catalyst surface
3. Conclusion
Chapter 3 (PDF Ch3, part 1)
Formalism of chemical kinetics
1. Main concepts of chemical kinetics
1.1 Linear laws of conservation
1.2 Stoichiometry of complex reactions
1.3 Graphical representations of reaction mechanisms
1.4 Chemical kinetics equations
1.5 Reaction polyhedron
1.6 Reaction rate
1.7 Concentration equations
1.8 Non-ideal systems
2. Principle of detailed equilibrium and its consequences
2.1 Principle of detailed equilibrium
2.2 The uniqueness and stability of equilibrium in closed systems
2.3 Thermodynamic limitations on non-steady state kinetic behaviour
2.4 Limitations on non-steady state kinetic behaviour imposed by reaction mechanism
3. Formalism of chemical kinetics for open systems (PDF, Ch2, part 2)
3.1 Kinetic equations for open systems
3.2 “Weakly open” systems
3.3 Stabilization at high flow rates
4. Quasi-stationarity
5. Uniqueness, multiplicity and stability of steady states
5.1 Linear mechanisms
5.2 Mechanisms without intermediate interactions
5.3 Quasi-thermodynamic Horn and Jackson systems
5.4 Criterion for uniqueness and multiplicity associated with the mechanism structure
5.5 Some conclusions
Chapter 4 (PDF, Ch.4)
Graphs in chemical kinetics
1. General description and main concepts
1.1 Simple example
1.2 Two formalisms. Formalism of enzyme kinetics and of steady-state reaction theory
1.3 Non-linear mechanisms on graphs
2. Graphs for steady-state kinetic equations
2.1 Substantiation of the “Mason rule”
2.2 General form of steady-state kinetic equation for complex catalytic reactions with multi-route linear mechanisms
2.3 Analysis of properties for the general steady-state kinetic equation of complex catalytic reactions
2.4 How to find the kinetic equation for reverse reactions
2.5 Matching of reactions and the representation of the kinetic equation of complex catalytic reactions in the Horiuti-Boreskov form
2.6 Observed kinetic regularities and characteristics of detailed mechanisms
2.6.1 Observed reaction order
2.6.2 Observed activation energy
3. Graphs for the analysis of the number of independent parameters
3.1 Simple example
3.2 Reasons for dependence and the impossibility of determining parameters
3.3 Indeterminacy of parameters and graph structure
3.4 The number of determinable parameters and graph colour
3.5 Brutto-reaction, detailed mechanism and the number of parameters under determination
3.5.1 Brutto-equation and the number of steps
3.5.2 Graph colours and kinetic equation structure
4. Graphs to analyze relaxations. General form of characteristic polynomial
5. Conclusion
Chapter 5 (PDF Ch5)
Simplest non-linear mechanisms of catalytic reactions producing critical phenomena
1. Critical phenomena in heterogeneous catalytic reactions
2. The “parallel” adsorption mechanism
3. Steady-state characteristics of the simplest mechanism permitting multiplicity of catalyst steady states
4. Relaxation characteristics of the “parallel” adsorption mechanism
5. Analysis of “consecutive” adsorption mechanisms
6. Models of kinetic self-oscillations in heterogeneous catalytic reactions
Chapter 6 (PDF, Ch6)
Studies of kinetic models for oxidation reactions over metals (exemplified by CO oxidation)
1. Mechanism and model
2. Modelling of kinetic dependences
3. Dynamic studies of CO oxidation
4. “General” kinetic model and prediction of critical effects
Chapter 7 (PDF, Ch7)
Critical retardation effects and slow relaxations
1. The problem of slow relaxations
2. The limit behaviour of dynamic systems
3. Relaxation times. Definition of slow relaxations
4. Bifurcations (explosions) of limit sets
5. Dynamic factors for slow relaxations
6. Taking into account small perturbations and errors of models
7. Conclusion
Chapter 8 (PDF, Ch8)
1. Forecast for tomorrow
2. Afterthoughts to the conclusion
Comment: This book was prepared for publication in late 1980s. At that time we lived deeply in Siberia. This distance caused some problems. The translation is not perfect (translation is never
perfect J): sometimes we can find in the text determination instead of definition, “strong” instead of “exact”, “detailed equilibrium” instead of “detailed balance”, and so on. We can also mention
one global formatting mistake: in almost all equations there is a vector sign (an arrow above a letter) instead of bold font. Sometimes this is fine (for vectors), and sometimes this is funny (for
matrices, spaces and sets). Nevertheless, the book seems to be readable, and all this noise could be filtered. At least one reader was impressed enough to write a review: W. Henry Weinberg, Review on
the book "Comprehensive Chemical Kinetics", Volume 32, Kinetic Models of Catalytic Reactions, Elsevier, 1991). Journal of American Chemical Society (JAChS), V.114, n 13, 1992, 5484-5485. | {"url":"https://thermotree.narod.ru/YBGYcontents.htm","timestamp":"2024-11-03T06:29:04Z","content_type":"text/html","content_length":"26397","record_id":"<urn:uuid:0dcde21d-db46-4ba4-9186-57d13fc87ed7>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00497.warc.gz"} |
GRIN - An Overview of Spiking Neural Networks
This work gives an introduction to SNNs and the underlying biological concepts. It gives an overview and comparison of some of the more commonly used SNN models. It discusses the scope of SNNs and
some of the areas where they have been applied so far. Spiking neural networks or SNNs are inspired by the biological neuron. They are the next step towards the goal of replicating the mammalian
brain in computational speed, efficiency and energy consumption.
First generation artificial neural networks (ANNs) or Perceptron use a [0,1] binary threshold function to approximate digital input and allow for linear classification. Second generation ANNs like
multi-layer perceptron, feed-forward and recurrent neural networks use continuous activation functions like sigmoid which can approximate analog functions. Spiking neural networks, introduced by
Hopfield in 1995, are third generation ANNs and aim at higher biological plausibilty than the first and second generations by including time intrinsically. They use the precise firing times of
neurons to code information. SNNs are modelled on the biological neuron. It is therefore important to understand the basic biological concepts underlying SNNs.
An Overview of Spiking Neural Networks
Garima Mittal
Abstract— Spiking neural networks or SNNs are inspired by the biological neuron. They are the next step towards the goal of replicating the mammalian brain in computational speed, efficiency and
energy consumption. This work gives an introduction to SNNs and the underlying biological concepts. It gives an overview and comparison of some of the more commonly used SNN models. It discusses the
scope of SNNs and some of the areas where they have been applied so far.
Index Terms— Spiking neural network, biological neuron, temporal coding
First generation artificial neural networks (ANNs) or Perceptron use a [0,1] binary threshold function to approximate digital input and allow for linear classification. Second generation ANNs like
multi-layer perceptron, feed-forward and recurrent neural networks use continuous activation functions like sigmoid which can approximate analog functions. Spiking neural networks, introduced by J.
Hopfield in 1995, are third generation ANNs and aim at higher biological plausibilty than the first and second generations by including time intrinsically. They use the precise firing times of
neurons to code information. SNNs are modelled on the biological neuron. It is therefore important to understand the basic biological concepts underlying SNNs.
A. Membrane Potential
The neural cell consists of ions such as sodium(Na+), pottasium(K+ ), calcium(Ca+), chlorine(Cl-). Membrane potential (MP) is based on the balance of ions in and outside the cell membrane. It gets
influenced by the change in the membrane permeability towards specific ion(s) in response to some stimulus. This allows certain ions to flow in or out of the membrane leading to a change in the
overall potential. In the resting state the intracellular space has a negative potential while the extracellular is positively charged. The resting membrane potential (RMP) lies at around -70mV.^1
B. Action Potential
An action potential or a spike allows information to be transferred from one neuron to the next. A stimulus changes the membrane permeability causing MP to become progressively higher. If the MP is
high enough to cross the threshold, usually at around -50mV, depolarization occurs where the MP shoots and peaks to around 30mV. This is when a spike or action potential is said to occur. From here
the MP decays rapidly, called the repolarization stage, and eventually falls below the RMP. This short period is the hyperpolarization stage during which it is not possible for a neuron to spike
again. Eventually the potential gets restored to the RMP. Thus it is important to note that a neuron spikes only when MP exceeds the threshold; and it has a refractory period during which it cannot
fire again.
Abbildung in dieser Leseprobe nicht enthalten
Fig. 1: Generation of action potential1
C. Time as the basis of information coding
The first and second generation ANNs use rate coding where the average number of spikes over time is used to code information. This however is biologically not too realistic. Normally the spike
frequency differs with type of stimulus. Spike trains encoding different information may have the same spike rate but differ in pattern. Spike rate alone is therefore not an accurate measure and
spike timing needs to be taken into account 1.
Abbildung in dieser Leseprobe nicht enthalten
Fig. 2: Spike rate for n spikes over time t 2
Abbildung in dieser Leseprobe nicht enthalten
Fig. 3: Same spike rate for different stimuli generating different responses 2
SNNs use temporal coding to incorporate time intrinsically. Weights in SNN are based on proximity of spikes. They are set higher for closely-timed spikes and lower for spikes which occur further
II. Spiking Neural Network Models
SNN models can be classified in various ways. The two classifications introduced here are:
Abbildung in dieser Leseprobe nicht enthalten
A. Threshold-Fire Models
These models are based on the fundamental principle of biological neurons where a spike is generated when the membrane potential of the neuron crosses a certain threshold value from below.
1) Integrate-and-Fire (I&F) Model: This is the simplest SNN model and describes action potential as an event. This means that only the timing of spike is considered while the form of spike is
ignored. Membrane potential is assumed as the integration of input spikes. These could either be multiple spikes of the same neuron; or spikes from multiple neurons in response to some stimulus; or
Abbildung in dieser Leseprobe nicht enthalten
Fig. 4: Integration of sub-threshold potentials generates a spike2
The following equation of the integrate-and-fire model is simply the derivative of the law of capacitance: C = Q/V.
Abbildung in dieser Leseprobe nicht enthalten
When current I(t) causes summed potential V (t) to increase over time and cross threshold 0, a spike occurs. V (t) is then immediately reset to RMP and the process starts again.  summed potential
lower than 0 does not cause a spike and does not get reset. It is consequently retained till the next spike and does not decay. This is in contrast to the biological neuron, where a sub-threshold
potential eventually decays to RMP. This lack of time-dependent memory^2 ^3 is thus a limitation of this model, reducing its biological plausibility.
2) Leaky-Integrate-and-Fire (LIF) Model: This variant of the I&F model overcomes the memory limitation by using a leak term. This term represents the leakage or decay of subthreshold potential to
RMP before the occurrence of the next spike. The equation of LIF model represents the current I(t) as a combination of capacitor C and resistor R terms. 2
Abbildung in dieser Leseprobe nicht enthalten
The capacitor causes a spike if the integrated potential crosses the threshold. The resistor allows the sub-threshold potential to leak out and decay to the RMP. The next spike then starts building
up from the RMP. This makes the LIF model biologically more realistic than the I&F model.
3) Spike Response Model (SRM): This model can be seen as a generalization of the LIF model. The SRM equation: takes into account parameters such as the time since the last spike t; form of the action
potential y; and the linear membrane response to incoming spikes k. The form of the spike is used to model refractoriness. For example, a hyperpolarizing potential implies that the neuron is in
refractory state. Like previous threshold-based models, the next spike occurs if membrane potential crosses 0, which in case of SRM depends on time since the last spike t. This means that 0 is high
immediately after a spike but decays gradually to RMP as t increases.
Abbildung in dieser Leseprobe nicht enthalten
These additional parameters make SRM more complex but biologically more realistic than the previous models. SRM can be fitted to experimental data where a neuron is stimulated by rapidly varying
time-dependent current. It can predict large fraction of spikes with a precision of +/ — 2ms.
B. Conductance-based Models
These models describe the effect of conductance of individual ions on the membrane potential.
Hodgkin-Huxley (HH) Model: This model is the closest approximation of the biological neuron. It comprises of a set of non-linear differential equations that describe the effect of the conductances of
individual ions on membrane potential over time. Current I(t) is the sum of j ionic current channels.
Abbildung in dieser Leseprobe nicht enthalten
This sum can be decomposed, representing the contribution of individual channels Na +, K + and the leak channel Cl -. The equation thus expands to:
Abbildung in dieser Leseprobe nicht enthalten
where g represents the conductances of individual ion channels; m, n, h represent ion gates which control the flow of ions in and out of membrane; and E denotes the reverse potentials at which the
direction of corresponding current changes. These terms are further decomposed into sub-parameters to enable accurate modelling of the dynamic behaviour of biological neurons including leakage and
refractoriness. This however, requires a minimum of 20 parameters, making the HH model extremely complex. Âlso, because of the differential equations, the implementation of the model requires
numerical approximation techniques such as the Runge-Kutta method.4
III. Comparing SNN Models
Âs apparent from the previous section, complex SNN models like Hodgkin-Huxely are able to capture the dynamic feature of biological neurons much better than simpler models like I&F due to a large
number of parameters. This however, 4https://en.wikipedia.org/wiki/Runge-Kutta_methods makes their simulation computationally much more expensive and their mathematical analysis very difficult.
Simpler models on the other hand, are much more computationally efficient, which comes at the cost of biological plausibility. This plausibility-efficiency tradeoff needs a hybrid approach such as
the SNN model proposed by Izhikevich 3.
Abbildung in dieser Leseprobe nicht enthalten
Defined by the coupled equations (1) and (2), the model is able to capture enough elements of real neurons for a good biological approximation, while still being mathematically tractable. It thus
offers a good compromise between biological plausibility and computational efficiency.
Abbildung in dieser Leseprobe nicht enthalten
IV. Applications of SNNs
Being in the nascent stages of development, SNNs have a tremendous scope in a multitude of applications. Some of the areas explored so far are discussed below.
A. Cognitive Hardware
In contrast with the first and second generation ANNs, SNNs can be modelled in hardware. Neuromorphic chips, neural processing units (NPUs) etc. are designed based on the asynchronous, event-based
information processing of SNNs. This allows for parallel computation and therefore extremely efficient computational speed and energy consumption. An example is the IBM Synapse project^4 for
development of neurosynaptic chips.
B. Vision-based applications
These include pattern recognition, image recognition 5 etc. SNNs when applied to the MNIST6 handwritten digits dataset for the task of handwriting recognition 6, produced interesting results.
Compared to conventional convolution neural network (CNN), evaluation using SNN was much faster. However, the prediction error rose to 0.9% for SNN as compared to 0.21% for CNN. Using SNNs may thus
not necessarily improve accuracy and may even adversely affect it. This needs to be worked upon to allow a more beneficial and widespread application of SNNs.
C. Analysis of spatio-temporal data
The fact that SNNs include time intrinsically, makes them suited for applications involving analysis of spatio-temporal data such as speech recognition 7, autonomous robot navigation 8 etc. Also
being modelled after biological neurons, SNNs are intuitively suited for analysis and understanding of brain data 9.
D. Other areas
Some other areas of SNN research include novel applications such as developing biologically plausible electronic nose for tea odour classification 10 etc.
V. Summary
Spiking neural networks are third generation ANNs that include time intrinsically. Modelled after the biological neuron, SNNs are biologically more plausible, computationally more powerful and
considerably faster than their first and second generation counterparts. There are many SNN models representing the simplest to the most complex features of the biological neuron. The increase in
model complexity is directly proportional to biological plausibility but inversely proportional to computational efficiency. Hybrid models offer a good solution to this plausibility-efficiency
tradeoff. SNNs can also be modelled in hardware as neuromorphic chips and NPUs. They have great scope in analysis of spatio-temporal data; computer vision applications such as image and pattern
recognition; robotics etc. and can be extended to many hitherto unexplored areas in future.
^1 1Image from: Human physiology: from cells to systems, Sherwood, L., St Paul, 1989, Wadsworth
^2 Image from: http://droualb.faculty.mjc.edu/Course%20Materials/Physiology% 20101/Chapter%20Notes/Fall%202011/chapter_8%20Fall%202011.htm
^3 https://en.wikipedia.org/wiki/Biological_neuron_model
^4 http://www.research.ibm.com/cognitive-computing/neurosynaptic-chips.shtml6http://yann.lecun.com/exdb/mnist/
Excerpt out of 3 pages -
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Quote paper
Garima Mittal (Author), 2018, An Overview of Spiking Neural Networks, Munich, GRIN Verlag, https://www.grin.com/document/921629
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School of Electrical and Computer Engineering
Microelectronic Circuits
CMPE Degree: This course is Elective for the CMPE degree.
EE Degree: This course is Required for the EE degree.
Lab Hours: 0 supervised lab hours and 0 unsupervised lab hours.
Course Coordinator: Stephen E Ralph
Prerequisites: See topical outline
Catalog Description
Basic concepts of microelectronic materials, devices and circuits.
Course Outcomes
1. Compute carrier concentrations for semiconductor materials under a variety of conditions.
2. Compute conductivity and resistivity of semiconductor materials under a variety of conditions.
3. Compute terminal voltage and current characteristics for pn junction diodes under a variety of conditions.
4. Compute terminal voltage and current characteristics for bipolar transistors under a variety of conditions.
5. Compute terminal voltage and current characteristics for MOS transistors under a variety of conditions.
6. Compute terminal voltage and current characteristics for ideal operational amplifiers under a variety of conditions.
7. Analyze the DC performance of single-stage analog amplifiers containing these circuit elements.
8. Analyze the AC performance of single-stage analog amplifiers containing these circuit elements.
9. Analyze the DC performance of simple digital circuits (e.g., inverters and logic gates) containing these circuit elements.
Student Outcomes
In the parentheses for each Student Outcome:
"P" for primary indicates the outcome is a major focus of the entire course.
“M” for moderate indicates the outcome is the focus of at least one component of the course, but not majority of course material.
“LN” for “little to none” indicates that the course does not contribute significantly to this outcome.
1. ( P ) An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics
2. ( LN ) An ability to apply engineering design to produce solutions that meet specified needs with consideration of public health, safety, and welfare, as well as global, cultural, social,
environmental, and economic factors
3. ( LN ) An ability to communicate effectively with a range of audiences
4. ( LN ) An ability to recognize ethical and professional responsibilities in engineering situations and make informed judgments, which must consider the impact of engineering solutions in global,
economic, environmental, and societal contexts
5. ( LN ) An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives
6. ( P ) An ability to develop and conduct appropriate experimentation, analyze and interpret data, and use engineering judgment to draw conclusions
7. ( M ) An ability to acquire and apply new knowledge as needed, using appropriate learning strategies.
Strategic Performance Indicators (SPIs)
Not Applicable
Course Objectives
1. will understand the physical, electrical, and optical properties of semiconductor materials and their use in microelectronic circits. [6]
2. relate the atomic and physical properties of semiconductor materials to device and circuit performance issues. [8]
3. develop an understanding of the connection between device-level and circuit-level performance of microelectronic systems. [6,8]
Topical Outline
ECE 3043* and ECE 2031/20X2 and (ECE 2035 or ECE 2036) and ECE 2040 and CHEM 1310/1211K/12X1 and MATH 2401/2411/24X1 and MATH 2403/2413/24X3 [all ECE and MATH courses min C] * ECE 3040 and ECE 3043
normally must be taken
Introduction: Course mechanics, Silicon, Example of silicon devices, Conductivity
Basic Semiconductor Physics: Hydrogen Atom (briefly), Periodic potentials, Band structure, Effective mass, Mobility
Lattices Crystals and Dopants: Metals, Semiconductors and Insulators, Generation/Recombination, Crystal structure, Intrinsic and extrinsic and Doping, Carrier concentrations, electrons and holes,
Donor and acceptor states
Fabrication, DOS, Fermi Statistics: Semiconductor Alloys, Carrier density and bandstructure, Fermi Statistics and Fermi level
Carrier Statistics: Temperature and doping effects, Extrinsic semiconductors, Donor/acceptor occupancy, Determination of Fermi Energy, Recombination and Generation
Carrier Transport: Drift velocity, Effective mass, Mobility and Saturation, Current density, Doping and temperature effects, Energy bands and electrostatic potential
Carrier Transport, Diffusion Fick’s Law, Total current, Einstein Relation, Equilibrium
Optical Properties: Absorption, Recombination and Generation
Return to Equilibrium: Low level injection, Quasi Fermi Levels, Direct recombination, Trap assisted
Equations of State: Continuity equation, Minority carrier diffusion equation (MCDE), Special cases of MCDE, Quasi Fermi levels and current
PN Junctions: Current Flow in PN junctions, Diffusion w forward/reverse bias, Junction electrostatics, Depletion region and bias, Quantitative solution, Carrier density and potential, Minority
injection and Diffusion, Boundary conditions, Total current, Quasi Fermi Levels, Series resistance, High injection, Examples
Real PN Junctions: Capacitance, Recombination/generation, Avalanche/Zener
Circuit Models: Large signal models, Small Signal Models, Small signal model of PN diodes, Diffusion and Junction capacitance, Simple diode circuits
Photonic devices: Absorption, Photodiodes, Solar Cells, LEDs, Lasers
Intro to Transistors: Structure and nomenclature, Currents/band diagram, Biasing modes, Configurations, Alpha, beta (circuit level)
BJT quantitative derivation: Terminal currents, Ebers Moll model, Active mode currents, Simplified Ebers Moll: ideal current results (use to get output resistance in small sig model), Base width
Small Signal Circuit Model: Small Signal analysis, General 2-port models, admittance parameters, DC analysis; Q point, bias stability, Hybrid pi model, Common Emitter examples, Source and Load
MOS Capacitors: Energy levels and flatband, Static and Biased band shapes, Accumulation, depletion and inversion, NMOS and PMOS, Quantitative solution, Fields and Potentials
MOS Transistor: Qualitative description, Triode regime, Pinch-off and saturations regime, Quantitative derivation, Threshold voltage, Square Law
MOS Transistors: Deviations from ideal, Enhancement and depletion modes, MOSFETs small Signal, Admittance parameters, Terminal gain
DC Aspects of Amplifiers: Bias networks for MOSFETs, Current mirrors
Single Transistor Amplifiers: Inverting amplifiers, CS and CE, Follower Amplifiers CD and CC, Non-inverting Amplifiers CG and CB, Amplifier input and output resistance, Voltage and current amplifiers
Multi-stage Amplifiers: Configurations, Cascaded stages, DC equivalent, AC and small signal, Gain and I/O resistance, Op Amps | {"url":"https://ece.gatech.edu/courses/ece3040","timestamp":"2024-11-02T07:39:36Z","content_type":"text/html","content_length":"50380","record_id":"<urn:uuid:60d08192-e5f4-4277-a01a-87611897a2fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00321.warc.gz"} |
NetworkX 2.2
NetworkX 2.2#
Release date: 19 September 2018
Supports Python 2.7, 3.5, 3.6 and 3.7. This is the last release to support Python 2.
NetworkX is a Python package for the creation, manipulation, and study of the structure, dynamics, and functions of complex networks.
For more information, please visit our website and our gallery of examples. Please send comments and questions to the networkx-discuss mailing list.
This release is the result of 8 months of work with over 149 commits by 58 contributors. Highlights include:
• Add support for Python 3.7. This is the last release to support Python 2.
• Uniform random number generator (RNG) handling which defaults to global RNGs but allows specification of a single RNG for all random numbers in NX.
• Improved GraphViews to ease subclassing and remove cyclic references which caused trouble with deepcopy and pickle.
• New Graph method G.update(H)
Each function that uses random numbers now uses a seed argument to control the random number generation (RNG). By default the global default RNG is used. More precisely, the random package’s default
RNG or the numpy.random default RNG. You can also create your own RNG and pass it into the seed argument. Finally, you can use an integer to indicate the state to set for the RNG. In this case a
local RNG is created leaving the global RNG untouched. Some functions use random and some use numpy.random, but we have written a translator so that all functions CAN take a numpy.random.RandomState
object. So a single RNG can be used for the entire package.
Cyclic references between graph classes and views have been removed to ease subclassing without memory leaks. Graphs no longer hold references to views.
Cyclic references between a graph and itself have been removed by eliminating G.root_graph. It turns out this was an avoidable construct anyway.
GraphViews have been reformulated as functions removing much of the subclass trouble with the copy/to_directed/subgraph methods. It also simplifies the graph view code base and API. There are now
three function that create graph views: generic_graph_view(graph, create_using), reverse_view(digraph) and subgraph_view(graph, node_filter, edge_filter).
GraphML can now be written with attributes using numpy numeric types. In particular, np.float64 and np.int64 no longer need to convert to Python float and int to be written. They are still written as
generic floats so reading them back in will not make the numpy values.
A generator following the Stochastic Block Model is now available.
New function all_topological_sort to generate all possible top_sorts.
New functions for tree width and tree decompositions.
Functions for Clauset-Newman-Moore modularity-max community detection.
Functions for small world analysis, directed clustering and perfect matchings, eulerizing a graph, depth-limited BFS, percolation centrality, planarity checking.
The shortest_path generic and convenience functions now have a method parameter to choose between dijkstra and bellmon-ford in the weighted case. Default is dijkstra (which was the only option
API Changes#
empty_graph has taken over the functionality from nx.convert._prep_create_using which was removed.
The create_using argument (used in many functions) should now be a Graph Constructor like nx.Graph or nx.DiGraph. It can still be a graph instance which will be cleared before use, but the preferred
use is a constructor.
New Base Class Method: update H.update(G) adds the nodes, edges and graph attributes of G to H. H.update(edges=e, nodes=n) add the edges and nodes from containers e and n. H.update(e), and H.update
(nodes=n) are also allowed. First argument is a graph if it has edges and nodes attributes. Otherwise the first argument is treated as a list of edges.
The bellman_ford predecessor dicts had sentinel value [None] for source nodes. That has been changed so source nodes have pred value ‘[]’
Graph class method fresh_copy - simply use __class__. The GraphView classes are deprecated in preference to the function interface. Specifically, ReverseView and ReverseMultiView are replaced by
reverse_view. SubGraph, SubDiGraph, SubMultiGraph and SubMultiDiGraph are replaced by subgraph_view. And GraphView, DiGraphView, MultiGraphView, MultiDiGraphView are deprecated in favor of
generic_graph_view(graph, create_using).
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Multiplication Table 1 12 Worksheet
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Encouraging constant practice, supplying support, and creating a positive learning setting are helpful steps. | {"url":"https://crown-darts.com/en/multiplication-table-1-12-worksheet.html","timestamp":"2024-11-04T09:03:08Z","content_type":"text/html","content_length":"29153","record_id":"<urn:uuid:e7983fd0-3ec4-474a-b74d-824cf7e34412>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00113.warc.gz"} |
What is middle?
Middles are the most popular type of bet in sport betting, with more than 80% of all bets placed being middles. This article will explain what a middle is and how to identify them; we'll also look at
the different types of middles available and some common props that you might find when betting on sports.
What is middle?
Middles in sports betting is a situation where you need to make two bets. The result can be one of two:
• One of the two bets will win, which will lead to a small but controlled loss, and in some cases even to a minimal win (such middles are called arb-middles).
• Both bets will win at once, which will lead to an increased profit.
Middle example
Football match between teams Petrolul Ploiesti and Voluntari in Romania. league 1. Fouls bet:
1. Individual Total Petrolul Ploiesti Over 12.5 fouls.
2. Individual Total Petrolul Ploiesti Under 14.5 fouls.
In a situation where there are exactly 13 or 14 fouls, both bets will win at once, which will lead to a profit of as much as 422 euros with a total bet of 500 euros. In all other cases, one of the
two bets will win: if only the first bet wins, there will be a controlled loss of 46 euros, if only the second wins - 32 euros.
Thus, we see that a situation is possible when both bets will play. This situation is called "hitting the middle". If we don't hit the middle, we don't lose the whole bank, but only a small part of
How to find a middles?
We looked at the example of a fouls middle in football, but it is also very common to find middles in basketball and middles in tennis due to the features of these sports. All this does not mean that
there are no middles in other sports, but it is often more difficult to find them yourself. Middles can be searched manually by checking the lines, or you can use a middles scanner, such as
breaking-bet.com, which allows you to find middles according to the given settings of many parameters, such as: Mathematical expectation, Probability of hitting the middle, Profit when hit, Profit/
Loss miss.
How to identify goods middles?
When analyzing middles, it is important to rely on the following indicators:
Mathematical expectation - a value that shows how profitable the middle will be at a distance, based on its probability.
The probability of hitting the middle - the probability that both bets will win (there will be a hit in the middle).
Profit when entering the middle - the percentage of profit when there will be a hit in the middle (both bets will win).
Profit/Loss on Miss - The percentage of win/loss on a miss (only one of the two bets will win).
In the example discussed above, we see that the profit when hitting the middle is quite high - 84.92%, and the loss when missing is 7.54%. At first glance, these are good indicators, but there is a
caveat - the probability of hitting a middle is 19.12%, which is a small indicator (1 out of 5 hits), it is better to consider middles with a higher probability of hitting. Profit from a hit can be
as big as you like, but what's the point if the probability of hitting is small?
The props and cons of middles
Middles are a great way to make money. By finding middles, you can often find arbitrage opportunities that allow you to profit from both sides of the bet. However, this only works if both sides are
available for betting at the same time.
• If the middles are evaluated incorrectly, you can start losing the pot at a distance
• Often searching a middles is easier than searching arbs
• More profit with less losses
• Less attention from bookmakers
In this article, we have covered all you need to know about middles in sport betting. We started with an explanation of what it is and how it works before going on to look at the searching of middle
available on the market today. We also looked at some tips for finding good middles so that you can maximise your chances of making profit from them when placing bets with bookmakers. | {"url":"https://breaking-bet.com/es/help/what-is-middle","timestamp":"2024-11-14T01:15:08Z","content_type":"text/html","content_length":"45984","record_id":"<urn:uuid:b2ff22d5-7b0b-49b7-8c6b-731fc4c232f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00449.warc.gz"} |
How to Use Smart Tables
Smart Tables are used to conduct tests of statistical significance for the dependent question compared with each independent question. The results of these tests are added to the Report tree and
ranked according to statistical significance.
Use Smart Tables, select Create > Tables > Smart Tables.
How Smart Tables work
1. A significance test is conducted on each table formed by crosstabbing the Dependent Question with each Independent Question (see Planned Tests Of Statistical Significance). The function does a
test on all the table cells rather than look at or compare individual cells on the table. It's somewhat like asking, "Is there a significant relationship between these questions?" rather than
asking if there's a significant relationship between any of the table's cells. A p-Value is calculated for each of the question combinations/tables such that each table gets a single p-Value.
2. This p-value collection then has Multiple Comparisons (Post Hoc Testing) applied. The correction depends on the total number of tests conducted (i.e., the number of tables) and on the
distribution of the p-values of those tables.
3. These significance test results are shown in the table name (see the example below).
4. The tables are ordered according to their p-Value, with the most significant results shown at the top.
5. The tables are grouped into the following categories (where applicable):
□ Significant. These tables have a p-Value that is less than or equal to the Overall significance level (considering any Multiple Comparison Corrections that have been specified for the table's
cells). The p-value that equates to a corrected p-value at the specified Overall significance level is shown in brackets in the name of the first Folder (e.g., Significant p <= 0.071). For
more information about corrected p-value, see Multiple Comparisons (Post Hoc Testing).
□ Insignificant.
□ p-Value could not be computed. For example, tables where there is no variation in one of the questions.
Some general comments about Smart Tables
• Smart Tables orders results according to p-Values. While p-Values are a useful measure of the strength of association between different questions, there are sometimes better measures. Also,
sometimes there will be ties, so the actual order in which things are shown should not generally be treated as a result.
• It is possible for there to be a significant relationship between two questions without any specific cell being significant when compared to other cells on the table. For example, comparing age
to preferred brands, you can say that the younger someone is the more they tend to like a certain brand. But, at the same time, this table may not show that any one specific age group likes a
brand so much that they get picked out as being significantly higher than the rest of the sample.
• Statistical significance is not the same as causation.
• It ignores the likelihood that many of the questions are correlated. An alternative approach that resolves this problem but introduces others is using Trees.
Buttons, options, and fields
Dependent question The question that will be used in comparison against each independent question. This comes from the question currently in the blue drop-down.
Available questions The list of all questions available to use as independent questions.
Independent questions The list of questions that will be tested for statistical significance against the dependent question.
Move the selected questions in the Available questions list to the Independent questions list.
Move the selected questions in the Independent questions list to the Available questions list.
Filter drop-downThe filter variable to apply during the significance testing.
Weight drop-down The weight variable to apply during the significance testing.
The Smart Tables output is a set of tables, appended to the Report. The tables form a part of a folder named after the dependent question (with Smart Tables affixed at the end, e.g., Q7. Company
currently with: Smart Tables. Within this folder, there are two sub-folders – one called Significant (p <= 0.05), which contains questions that were significantly related to the dependent question,
and another showing the insignificant questions. The cut-off p-value is determined by the settings of the Overall significance level and Multiple comparisons method. Within each folder, the tables
are ordered according to their p-values, with the most significant results shown at the top. | {"url":"https://help.qresearchsoftware.com/hc/en-us/articles/4425083418895-How-to-Use-Smart-Tables","timestamp":"2024-11-05T07:01:13Z","content_type":"text/html","content_length":"34507","record_id":"<urn:uuid:ebda457b-da2e-4999-912e-9f282b0327ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00505.warc.gz"} |
An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions
An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions ()
1. Introduction
The boundary value problems for isotropic homogeneous performed infinite plates have been discussed by several authors: see Colton and Kress [1] , Popov [2] , Noda et al. [3] and Schinzinger and
Laura [4] . Some authors used Laurent’s theorem to express the solution in the series form, see England [5] , Parkus [6] and Kalandiya [7] . Others used complex variables method of Cauchy integrals
to express the solution of the boundary value problems in the form of two complex potential functions, Goursat functions, by using many rational mappings, see Muskhelishvili [8] , El-Sirafy and Abdou
[9] , Abdou and Khar-Eldin [10] , Abdou and Khamis [11] , Abdou [12] and Abdou et al. [13] . In all previous works, the coefficients of the rational mappings were real.
It is worth mentioning that Exadaktylos and Stavropoulou [14] and Exadaktylos et al. [15] considered rational mapping functions with complex constants that conformally maps the holes inside a unit
circle, using Laurent’s method. Also Abdou and Asseri [16] [17] considered more general rational mapping functions with complex constants that conformally maps the holes outside and inside a unit
circle, using Cauchy singular method. All the previous four works will be considered as special cases of this work.
It is known that, see Muskhelishvili [8] , the first and second fundamental problems in the plane theory of elasticity are equivalent to finding two analytic functions
In terms of the rational mapping function
where X, Y are the components of the resultant vector of all external forces acting on the boundary and
In the absence of body forces, Muskhelishvili [8] has considered the stress components in the plane theory of elasticity in the form
In this work, the complex variables method will be applied to solve the first and second fundamental problems for an infinite plate with a generalized curvilinear hole C conformally mapped on the
domain outside a unit circle
Also, many applications for the first and second fundamental problems are considered and the components of stress and strain have been obtained and plotted to investigate their physical meaning.
Moreover, computer work using maple 9.5 has been used in applications to give the shapes of holes and curves of stresses with some calculations of stresses at their important points.
2. The Rational Mapping
The physical interest of the mapping (6) comes from its special cases and its different shapes of holes that can be obtained, see Figures 1-6.
From the rational mapping we can discuss the following:
1) The number of the holes corners is subjected to
2) The shape of the hole depending on the values of n’s and m’s.
Figure 1. ℓ = 2, n = (0, 0), m = (1, 2), d = (2, 6).
Figure 2. ℓ = 2, n = (0.1, 0.1), m = (0.2, −0.2), d = (1, 1).
Figure 3. ℓ = 3, n = (0.001, 0.001), m = (0.1, −0.05), d = (1, 1).
Figure 4. ℓ = 5, n = (0.001, 0.001), m = (0.1, −0.05), d = (1, 1).
Figure 5. ℓ = 7, n = (0.001, 0.001), m = (0.1, −0.05), d = (1, 1).
Figure 6. ℓ = 10, n = (0.009, 0.007), m = (0.1, −0.05), d = (2, 2).
3) Entering none zero values of the complex constants m and d never gives symmetric graphs. While, entering zero values for all imaginary parts of both m and d, we get symmetric shapes around the
x-axis. On the other hand, entering zero values for all real parts of both m and d, we get symmetric shape around the y-axis.
4) The complex constant m works on circling the shape from the symmetry situation and the circling angle is given by
5) Using the rational mapping function
6) The complex constant d works on expanding the corners of the hole shape.
3. Goursat Functions
In this section, we use the transformation mapping (6) in the boundary conditions (1), and complex variables method, Cauchy method, to obtain a closed form expression for the Goursat functions
Using (7) in the boundary conditions (1) and on
The function
and the complex constant b, will be determined, is given by
Differentiating (15) with respect to z, then using the result in (17), the complex constant b takes the form
Also, the function
The two formulas (15) and (19) are representing the Goursat functions for the first and second fundamental problems for an infinite elastic plate weakened by The two formulas (15) and (19) are
representing the Goursat functions for the first and second fundamental problems for an infinite elastic plate weakened by a generalized curvilinear hole C, that can be transformed outside a unit
circle g by the rational mapping (6).
An important new case for discussion is using the transformation mapping
This mapping function, when
4. Special Cases
Here, we discuss the following:
1) By considering the reality of the constants of the mapping (1.6), the Goursat functions, in this case, are agree with work of Abdou and Khar-Eldin [10] of Equations (15) and (19), on notation the
difference in notation.
2) When
[J] are complex constants (23)
The Goursat functions, in this case, become
The results of the two formulas (24) and (25) are in agreement with the work of Abdou and Asseri [16] , on noting the difference in notation.
3) When
The Goursat functions, in this case, of the two formulas (15) and (19) agree with the all results of Abdou and Asseri [17] .
4) In the mapping function (20) if we let m = 0, then for finite expansion, we will have the following mapping function
with the corresponding Goursat functions
The three Formulas (27)-(29) are equivalent to those derived by Exadaktylos and Stavropoulou [14] , where they used Laurent's theorem, after considering in (27)-(29) the following special cases:
5) Also, in (27), if we allow the index inside the summation sign to take the form
5. Applications
1) For
The complex constant b has been determined by Equation (18) and its value was calculated by using Maple 9.5. Here, we have the Goursat functions for an infinite plate weakened by a curvilinear hole C
which is free from stresses. The plate stretched at infinity by the application of a uniform tensile stress of intensity P, making an angle q with the x-axis.
For[xx], s[yy], s[xy] and the angle q are considered in Figures 7-9.
2) For
Thus, (32) and (33) give the solution of the first fundamental problem for an isotropic infinite plate with a curvilinear hole, when there are no external forces and the edge of the hole is subject
to a uniform pressure P.
If in application (2) we write
For [xx] ,s[yy] ,s[xy] and the angle q, using Maple 9.5 are considered in Figures 10-12.
Figure 7. Max. and Min. values of σ[xx] are [2.09232, [θ = 3.7365]], [−0.49508, [θ = 1.82742]].
Figure 8. Max. and Min. values of σ[yy] are [1.24503, [θ = 1.84222]], [−2.63598, [θ = 0.73521]].
Figure 9. Maximum value of σ[xy] is [0.25045, [θ = 2.75306]], Minimum value of σ[xy] is [−7.61349, [θ = 3.65448]].
Figure 10. Max. and Min. values of σ[xx] are [7.26497, [θ = 3.66684]], [−0.74520, [θ = 2.95389]].
Figure 11. Max. and Min. values of σ[yy] are [0.45410, [θ = 1.87226]], [−6.94161, [θ = 3.66649]].
Figure 12. Maximum value of σ[xy] is [1.99460, [θ = 3.77003]], Minimum value of σ[xy] is [−0.63053, [θ = 0.63788]].
3) For
Here, we have the case of uni-directional tension of an infinite plate with a rigid curvilinear centre. The constant e, which represents the angle of rotation, can be determined from the condition
that the resultant moment of the forces, acting on the curvilinear centre from the surrounding material, must vanish i.e.
Hence, we have
For θ[xx], s[yy], s[xy] and the angle q are considered in Figures 13-15.
From the previous results, we can establish the following Case (1): In the case of Bi-axial tension, we have
Hence, we get
Figure 13. Max. and Min. values of σ[xx] are [19.94724, [θ = 3.72527]], [−31.09023, [θ = 1.48899]].
Figure 14. Max. and Min. values of σ[yy] are [43.21619, [θ = 1.52536]], [−28.86760, [θ = 3.68310]].
Figure 15. Maximum value of σ[xy] is [27.14849, [θ = 4.28087]], Minimum value of σ[xy] is [−18.54277, [θ = 3.15247]].
The complex constant b has been determined by Equation (18) and its value was calculated by using Maple 12. For n = 0.1 + 0.1i, m = 0.2 − 0.2i, d = 1 + i, P = 1/4 and c = 2. The relation θ between
the stress components s[xx], s[yy], s[xy] and the angle q are considered in Figures 16-18.
Case (2): When the curvilinear centre not allowed to rotate, i.e. when
For [xx] ,s[yy] , s[xy] and the angle q , using Maple 12 are considered in Figures 19-21.
4) When the force acts on the centre of the curvilinear kernel and the stresses vanish at infinity. In this case the kernel can not be rotate and it remains in its original position. Hence, we get
Therefore, we have the solution of the second fundamental problem in the case when (X, Y) acts on the centre of curvilinear hole.
For[xx],s[yy], s[xy] and the angle q, using Maple 9.5 are considered in Figures 22-24.
6. Conclusion and Discussion
From the previous work the following discussion and results can be concluded 1) In the theory of two-dimensional linear elasticity one of the most useful techniques for the solution of the boundary
value problem for a region weakened by a curvilinear hole is to transform the region into a simpler shape to get the solution directly without difficulties.
2) The transformation mapping
3) The physical interest of the using mapping transform comes from its different shapes of holes it treats and different directions it takes. This mapping function deals with famous shapes of
tunnels, thereon it is useful in studying the stresses around tunnels. In underground engineering the tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed
geometrical situation. Also, the tunnel is considered to be deep enough such that the stress distribution before excavation is homogeneous. Excavating underground openings in soils and rocks is done
for several purposes and in multi-sizes. At least, excavation of the opening will cause the soil or rock to deform elastically. The excavation in soil or rock is a complicated, dangerous and
expensive process. The mechanics of this can be very complex. However, the use of conformal mapping that allows us to study stresses and strains around a unit circle makes it useful for engineers and
easier for mathematicians.
Figure 16. Max. and Min. values of σ[xx] are [0.19217, [θ = 2.88015]], [−0.21424, [θ = 4.21641]].
Figure 17. Max. and Min. values of σ[yy] are [0.20395, [θ = 4.29250]], [−0.21998, [θ = 2.93249]].
Figure 18. Maximum value of σ[xy] is [0.21042, [θ = 3.54667]], Minimum value of σ[xy] is [−0.13798, [θ = 2.47646]].
Figure 19. Max. and Min. values of σ[xx] are [0.42251, [θ = 3.40048]], [−0.32725, [θ = 2.60422]].
Figure 20. Max. and Min. values of σ[yy] are [0.29017, [θ = 4.09478]], [−0.47644, [θ = 2.04499]].
Figure 21. Maximum value of σ[xy] is [0.24900, [θ = 3.77559]], Minimum value of σ[xy] is [−0.22325, [θ = 1.68785]].
Figure 22. Max. and Min. values of σ[xx] are [0.06313, [θ = 6.28319]], [−0.02863, [θ = 2.37434]].
Figure 23. Max. and Min. values of σ[yy] are [0.04161, [θ = 2.45943]], [−0.02730, [θ = 5.01912]].
Figure 24. Maximum value of σ[xy] is [−0.00237, [θ = 3.58676]], Minimum value of σ[xy] is [−0.04160, [θ = 1.06868]].
4) The complex variables method (Cauchy method) is considered one of the best methods for solving the integro differential equation, boundary value problem, of Equation (1) and obtaining the two
complex potential functions, Goursat functions,
5) The stress is an internal force whereas positive values of it mean that stress is in the positive direction, i.e. stress acts as a tension force. On the other side, negative values of stress mean
that the stress is in the negative direction, i.e. stress acts as a press force.
6) The most important issue deduced from mapping the stress components is that
7) When | {"url":"https://www.scirp.org/journal/PaperInformation?paperID=43794&","timestamp":"2024-11-03T03:18:03Z","content_type":"application/xhtml+xml","content_length":"141167","record_id":"<urn:uuid:5f34e884-25d5-49f8-b2f8-93d187c2c30a>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00031.warc.gz"} |
Dynamics Flashcards
To learn the content of the dynamics section.
Describe how to measure average speed.
• Mark a start line and a finish line.
• Measure the distance between the start and finish line with a ruler/metre stick.
• Start a stopwatch when the object crosses the start line and stop it when the object crosses the finish line.
• Calculate the average speed = distance between start and finish / time on stopwatch
Describe how to measure average speed using the equipment below.
• Two light gates are connected to a timer.
• The car rolls down the slope.
• When the cardboard breaks the first lightgate beam it starts the timer. When the cardboard breaks the second lightgate beam it stops the timer.
• The timer records the time taken to go from the first to the second light gate.
• Measure the distance between the two light gates using a metre stick.
• Calculate the speed = distance between light gates / time on timer
Explain how to use the equipment below to calculate the instantaneous speed of the car.
• Measure the length of the cardboard using a ruler.
• Roll car down the slope.
• When the cardboard breaks the beam it starts the timer connected to the light gate. When the cardboard has passed through the beam is remade and the timer stops.
• Speed = length of cardboard / time on timer
Explain the difference between average and instantaneous speed.
Average speed is over a long period of time, Instantaneous speed is the speed over a very short period of time.
What is the difference between a vector and a scalar quantity?
A scalar needs size/magnitude to be described correctly. A vector needs size/ magnitude and direction to be described correctly.
Distance travelled per second.
change in velocity per second.
What is meant by an acceleration of 15ms^-2 ?
The velocity increases by 15ms^-1 every second.
Explain how to measure the acceleration of the car using the equipment shown below and a stop watch.
• Car starts from rest so u = 0ms^-1.
• When the car is released start the stop watch when it reaches the light gate stop the stop watch.
• This is t, the time for the change in velocity.
• When the cardboard passes through the light gate the timer attached to it records this time.
• Measure the length of the card with a ruler.
• The final velocity, v = length of card / time on timer
• Then calculate acceleration, a = (v-u)/t
Explain how to measure acceleration using the equipment shown below and a stop watch.
• When the cardboard passes through the first light gate the timer attached to it records this time.
• Measure the length of the card with a ruler.
• The initial velocity, u = length of card / time on timer
• As the car rolls down the ramp, start the stop watch when it reaches the first light gate and when it reaches the second light gate stop the stop watch.
• This is t, the time for the change in velocity.
• When the cardboard passes through the second light gate the timer attached to it records this time.
• The final velocity, v = length of card / time on timer
• Then calculate acceleration, a = (v-u)/t
What does this speed-time graph show?
What does this speed-time graph show?
What does this speed - time graph show?
Describe how to calculate acceleration from a veocity - time graph.
• Pick two pints on the slope.
• work out the change in speed,
• t = time to go from the initial to the final speed.
• Then use a = ∆v/t
What are the three effects a force can have?
• Change the speed of the object
• Change the direction of travel of the object
• Change the shape of the object | {"url":"https://www.brainscape.com/flashcards/dynamics-11260856/packs/19948239","timestamp":"2024-11-07T14:03:04Z","content_type":"text/html","content_length":"121275","record_id":"<urn:uuid:a545b491-0154-496d-a7b3-6d66a96ac181>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00636.warc.gz"} |
Introduction to Machine Learning with Ethem Alpaydin - reason.townIntroduction to Machine Learning with Ethem Alpaydin
Introduction to Machine Learning with Ethem Alpaydin
Ethem Alpaydin’s book provides an accessible introduction to the exciting world of machine learning. This blog post covers some key concepts from the book.
Checkout this video:
Introduction to Machine Learning
Ethem Alpaydin is one of the world’s foremost authorities on machine learning. In this book, he provides a comprehensive introduction to the subject, covering both the theoretical foundations and the
practical applications. He also includesworked examples and End-of-Chapter exercises to help readers gain a thorough understanding of the material.
What is Machine Learning?
Machine learning is a branch of artificial intelligence that deals with the design and development of algorithms that can learn from and make predictions on data. These algorithms are used to build
models that can be used to make predictions on new data.
Machine learning is a very broad field and there are many different types of machine learning algorithms. Some of these include supervised learning, unsupervised learning, semi-supervised learning,
reinforcement learning, and deep learning.
Supervised learning is the most common type of machine learning. In supervised learning, the goal is to build a model that can be used to make predictions on new data. This model is built by using a
training dataset, which is a dataset that contains both the input data and the correct output labels. The model is then tested on a test dataset, which is a dataset that contains only the input data.
Unsupervised learning is another type of machine learning. In unsupervised learning, the goal is to build a model that can be used to cluster data points into groups. This model is built by using a
dataset that contains only the input data. The model is then tested on a new dataset to see how well it can cluster the data points into groups.
Semi-supervised learning is a type of machine learning that lies somewhere between supervised and unsupervised learning. In semi-supervised Learning, the goal is to build a model that can be used to
make predictions on new data even though only some of the input data has correct output labels. This type of machine learning often uses both labeled and unlabeled data to train the model.
Reinforcement learning is a type of machine Learning in which an agent interacts with an environment in order to learn how to maximize its reward. In reinforcement Learning, The goal is not to
predict the future but rather to learn how to take actions in an environment so as to maximize some notion of cumulative reward.
Deep Learning is a type of machine Learning that uses artificial neural networks with multiple layers (known as deep neural networks) to learn from data. Deep Learning allows machines to learn from
data in ways that are similar to how humans learn from data.
The Three Types of Machine Learning
There are three primary types of machine learning algorithms: supervised learning, unsupervised learning, and reinforcement learning. Supervised learning algorithms learn from labeled training data.
Unsupervised learning algorithms learn from unlabeled training data. Reinforcement learning algorithms learn from interaction with an environment.
The Five Components of a Machine Learning System
Machine learning is a field of artificial intelligence that deals with the design and development of algorithms that can learn from and make predictions on data. These algorithms are able to
automatically improve given more data.
There are five key components to a machine learning system:
-Data: This is the set of examples that will be used to train the machine learning algorithm. This data can be in the form of text, images, or other forms of structured or unstructured data.
-Features: These are the attributes or characteristics of the data that will be used by the machine learning algorithm to make predictions. For example, in a dataset of images, the features could be
the pixel values.
-Labels: This is the set of correct answers for the training data. In supervised machine learning, the labels are used to train the algorithm so that it can make predictions on new data. In
unsupervised machine learning, there is nolabeled training data, and so the algorithm must learn from the data itself.
-Algorithm: This is the method or technique used to learn from and make predictions ondata. There are many different types of machine learning algorithms, such as linear regression, support vector
machines, and decision trees.
-Evaluation metric: This is a measure used to assess how well the machine learning algorithm performs on a given dataset. Common evaluation metrics include accuracy, precision, recall, and f1 score.
The Seven Steps of a Machine Learning Project
A machine learning project usually follows a similar pipeline, regardless of the problem you are trying to solve or the data you are using. In this article, we will go through the seven steps of a
typical machine learning project so that you can get a better understanding of what goes into building a machine learning system.
1. Preprocessing: The first step in any machine learning project is preprocessing the data. This step is important because it cleans and prepares the data for modeling.
2. Modeling: The next step is to choose and fit a model to the data. This step is where the machine learning algorithms are applied to the data in order to learn from it.
3. Evaluation: After the model has been fit, it must be evaluated on unseen data in order to assess its performance. This step is important in order to identify any areas where the model can be
4. Deployment: If the model is successful, it can be deployed into production so that it can be used by others. This step usually involves putting the model into a service or creating an interface
for it so that it can be used easily by others.
5. Maintenance: Once the model is deployed, it will require regular maintenance in order to keep it running smoothly and to improve its performance over time. This step usually involves monitoring
the data and making changes to the model as new data becomes available.
The Five Types of Data in Machine Learning
There are five types of data that are commonly used in machine learning: numerical, categorical, temporal, text, and image data. Each type of data has its own characteristics and requires different
methods to be effectively used in machine learning algorithms.
Numerical data is the most common type of data used in machine learning. It is data that can be represented by a number, such as the length of a person’s hair or the width of a person’s nose.
Numerical data can be further divided into two subtypes: continuous and discrete. Continuous numerical data can take on any value within a range, such as the height of a person or the temperature of
a room. Discrete numerical data can only take on specific values within a range, such as the number of children in a family or the number of days in a week.
Categorical data is data that can be divided into groups or categories. For example, categorical data can be divided into gender (male or female), hair color (blond, brown, red, etc.), or eye color
(blue, green, brown, etc.). Categorical data is often represented by integers, with each integer representing a different category. For example, in gender classification problems, male may be
represented by 0 and female by 1.
Temporal data is data that represents events that occur over time. Temporal data can be represented as dates (e.g., 5/10/2015) or times (e.g., 10:30am). Temporal data often has to be converted into a
numerical representation before it can be used in machine learning algorithms; for example, dates can be converted into the number of days since some starting point (e.g., 1/1/1970).
Text data is data that consists of natural language text. Text data poses special challenges for machine learning algorithms because it is unstructured and often contains a lot of noise (e.g.,
misspellings, typos). Text pre-processing techniques such as tokenization and stemming are often used to clean up text data before it is used in machine learning algorithms.
Image data isdata that consists of imagesfixel-by-fixel mappingsthat can represent anything from faces to objects to scenes. Image classification is one popular taskin machine learning wherean
algorithm learns to labelimages accordingto their content(eThe taskof image segmentationis another popular image processing taskin machine learning wherethe goalis to partitionan imageinto multiple
regionseach correspondingtoa different objector scene
The Three Types of Machine Learning Algorithms
There are three types of machine learning algorithms: supervised learning, unsupervised learning, and reinforcement learning.
Supervised learning algorithms are used to train models that can predict a target variable, based on a set of input variables. The training data used to train the model includes both the input
variables and the target variable.
Unsupervised learning algorithms are used to find patterns in data. Unlike supervised learning algorithms, unsupervised learning algorithms do not use a target variable.
Reinforcement learning algorithms are used to train agents to take actions that maximizes a reward. Reinforcement learning is different from supervised and unsupervised learning because the data used
to train the agents is based on their actions and not on pre-labeled data.
The Five Types of Neural Networks
Machine learning is a branch of artificial intelligence that deals with the construction and study of systems that can learn from data. Neural networks are a type of machine learning algorithm that
are used to model complex patterns in data. Neural networks are similar to other supervised learning algorithms, but they are composed of a large number of interconnected processing nodes, or
neurons, that can learn to recognize patterns of input data.
Neural networks are classified into five different types:
-Feedforward neural networks: Feedforward neural networks are the simplest type of neural network. They are composed of an input layer, one or more hidden layers, and an output layer. The nodes in
the input layer receive input data, which is then passed through the hidden layers. The nodes in the output layer produce the output of the neural network.
-Recurrent neural networks: Recurrent neural networks are similar to feedforward neural networks, but they also have connections between neurons in adjacent layers. These connections allow
information to be passed backwards through the network, which allows the network to learn from sequences of data.
-Convolutional neural networks: Convolutional neural networks are designed to learn from image data. They are composed of an input layer, a series of hidden layers, and an output layer. The hidden
layers learn to recognize patterns in images by convolving the input image with a set of filters.
-Autoencoders: Autoencoders are a type of neural network that is used for unsupervised learning. They are composed of an input layer and an output layer, but they also have one or more hidden layers
in between. The hidden layers learn to compress the input data into a smaller representation, which is then decompressed by the output layer back into the original input data.
-Generative adversarial networks: Generative adversarial networks (GANs) are a type of unsupervised learning algorithm. They consist of two components: a generator and a discriminator. The generator
learns to generate new data samples that resemble the training data, while the discriminator learns to discriminate between real and generated data samples
The Seven Types of Regression Analysis
Linear regression is the simplest and most widely used form of regression analysis. In linear regression, we are interested in predicting a continuous outcome variable (y) based on one or more
predictor variables (x). The simplest form of linear regression, called simple linear regression, involves only one predictor variable. Multiple linear regression involves multiple predictor
Nonlinear regression is an extension of linear regression that can be used when the relationship between the dependent and independent variables is nonlinear. Nonlinear models can be more difficult
to interpret than linear models, but they can provide a better fit to the data.
Logistic regression is a type of regression analysis that is used when the dependent variable is binary (i.e., takes on only two values). Logistic regression can be used to predict whether a person
will experience a particular outcome (e.g., whether a person will contract a disease) based on one or more predictor variables (e.g., age, sex, smoking status).
Poisson regression is a type of regression analysis that is used when the dependent variable is count data (i.e., data that can take on only whole-number values). Poisson regression can be used to
predict the number of events (e.g., accidents, deaths) that will occur in a given period of time based on one or more predictor variables (e.g., time of day, day of week).
Stepwise regressions are a type of multiple regressions in which the aim is to find the best subset of predictor variables for predicting the dependent variable. In stepwise regressions, predictor
variables are added to or removed from the model in a stepwise fashion until only those predictor variables that have a significant association with the dependent variable remain in the model.
Ridge regressions are another type of multiple regressions that are used when there are many predictor variables and some of these predictor variables are highly correlated with each other. Ridge
regressions help to avoid overfitting byshrinkingthe coefficients of correlated predictors towards each other and reducing their variance.
Lasso regressions are another type of multiple regressions that are similar to ridge regressions but with one important difference: instead of shrinking all the coefficients towards each other, lasso
regressions set some coefficients equal to zero if they are not associated with the dependent variable
The Seven Types of Classification Analysis
There are seven different types of classification analysis: linear discriminant analysis, logistic regression, decision trees, rule-based methods, nearest neighbors, support vector machines, and
neural networks. Each type of classification has its own advantages and disadvantages. The choice of which type ofclassifier to use depends on the nature of the data and the desired properties of the | {"url":"https://reason.town/introduction-to-machine-learning-ethem/","timestamp":"2024-11-15T01:30:42Z","content_type":"text/html","content_length":"107492","record_id":"<urn:uuid:cb6299da-f299-4239-b145-a92cfdf78529>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00601.warc.gz"} |
Concept information
beta-phellandrene mass concentration
• Mass concentration means mass per unit volume and is used in the construction mass_concentration_of_X_in_Y, where X is a material constituent of Y. A chemical species denoted by X may be
described by a single term such as 'nitrogen' or a phrase such as 'nox_expressed_as_nitrogen'. The chemical formula for beta-phellandrene is C10H16. It is a member of the group of organics. It's
IUPAC name is 3-methylene-6-(1-methylethyl)-cyclohexene.
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World Scientific Publ Co Pte Ltd
Let y(x) be a smooth sigmoidal curve, y((n)) be its nth derivative and {x(m,i)} and {x(a,i)}, i = 1, 2, ... , be the set of points where respectively the derivatives of odd and even order reach their
extreme values. We argue that if the sigmoidal curve y(x) represents a phase transition, then the sequences {x(m,i)} and {x(a,i)} are both convergent and they have a common limit x(c) that we
characterize as the critical point of the phase transition. In this study, we examine the logistic growth curve and the Susceptible-Infected-Removed (SIR) epidemic model as typical examples of
symmetrical and asymmetrical transition curves. Numerical computations indicate that the critical point of the logistic growth curve that is symmetrical about the point (x(0), y(0)) is always the
point (x(0), y(0)) but the critical point of the asymmetrical SIR model depends on the system parameters. We use the description of the sol-gel phase transition of polyacrylamide-sodium alginate (SA)
composite (with low SA concentrations) in terms of the SIR epidemic model, to compare the location of the critical point as described above with the "gel point" determined by independent experiments.
We show that the critical point t(c) is located in between the zero of the third derivative t(a) and the inflection point t(m) of the transition curve and as the strength of activation (measured by
the parameter k/eta of the SIR model) increases, the phase transition occurs earlier in time and the critical point, t(c), moves toward t(a).
Gelation, Phase Transition, Epidemic Models
Turkish CoHE Thesis Center URL | {"url":"https://gcris.khas.edu.tr/entities/publication/fa78e584-a9ec-40d3-a6a5-8f1ba7ccdc69","timestamp":"2024-11-10T05:18:37Z","content_type":"text/html","content_length":"410750","record_id":"<urn:uuid:5e79eab4-a0cd-40b7-a145-cabb0a7f6789>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00616.warc.gz"} |
Quantization #
Quantisation is in this context refers to reducing the memory footprint of vectors by converting them to a lower precision format. This is useful when you have a large dataset and want to reduce the
memory footprint of the vectors.
By default, vectors are stored as 32-bit floating point numbers. This means each dimension of the vector is 4 bytes. For example, a 512-dimensional vector would be 2KB in size. This can quickly add
up for large datasets. Another use case is if you already have vectors in a lower precision format and want to use them with SemaDB such as binary vectors.
Quantising floating point vectors to lower precision is always a trade-off between memory footprint and accuracy. Information is inevitably lost when converting to lower precision.
For full details on how to use quantisation, please refer to the API reference.
Binary Quantisation #
type: binary
Binary quantisation is a special case of quantisation where the vectors are converted to binary format using only 1 bit per dimension. Considering again a 512-dimensional vector, the binary version
would be 64 bytes in size. This is a 32x reduction in memory footprint compared to the 32-bit floating point format.
The following parameters are relevant:
• threshold (optional): The threshold to use when converting the vectors to binary format. It determines whether a value is set to 1 or 0 by checking if it is greater than the threshold. For
normally distributed embedding models, a threshold of 0.0 is a good starting point. For other models, you may need to experiment with different thresholds.
• triggerThreshold (optional): It is the number of points after which threshold should be automatically computed. The threshold is set to the mean of all the values in all the vectors. This is
useful when you’re not sure what threshold to use.
• distanceMetric: The distance metric to use when searching. You can use the hamming or jaccard distance metric.
Binary Vectors #
If you have binary vectors to start with and set a binary distance metric at the index schema level, then SemaDB automatically enables binary quantisation with threshold set to 0.5 and the
corresponding distance metric.
When inserting or searching, you need to ensure that the vectors are still in floating point format, e.g. [0.0, 1.0, 0.0, 1.0]. The server will automatically convert the vectors to binary format when
storing them using the 0.5 threshold.
Product Quantisation #
type: product
Product Quantization is a technique to quantise high-dimensional vectors into a low-memory usage representation. The original vector is divided up to m sub-vectors and each sub-vector is quantized to
k centroids. The final quantized vector is the concatenation of the centroid ids of each sub-vector. When, the centroid ids are uint8 values each sub-vector is 1 byte in size and the final quantized
vector is m bytes in size saving a lot of memory. For example, a 512-dimensional vector with m=8 and k=256 would be 8 bytes in size instead of 2KB.
title: Product Quantisation
graph TD
OriginalVector["High-Dimensional Vector"]
OriginalVector --> Subdivide[/"Subdivide"\]
Subdivide --> SV1["Sub-vector 1"]
Subdivide --> SV2["Sub-vector 2"]
Subdivide --> SVM["Sub-vector m"]
SV1 --> Quant1["Quantizer 1 (k centroids)"]
SV2 --> Quant2["Quantizer 2 (k centroids)"]
SVM --> QuantM["Quantizer m (k centroids)"]
Quant1 --> ID1["Centroid ID 1 (uint8)"]
Quant2 --> ID2["Centroid ID 2 (uint8)"]
QuantM --> IDM["Centroid ID m (uint8)"]
ID1 & ID2 & IDM --> Combine[\"Concatenate"/]
Combine --> QuantizedVector["Quantized Vector (m bytes)"]
SemaDB uses k-means clustering to find the centroids for each sub-vector. This is performed only once when the product quantisation is triggered:
• numCentroids (recommended 256): The number of centroids to use for each sub-vector. This is the k parameter in the product quantisation algorithm. It is strongly recommended to use 256 as the
maximum number of possible values of an unsigned 8-bit integer. You may experiment with other values but it will not lower the memory footprint of the quantised vectors.
• numSubVectors (recommended 8): The number of sub-vectors to divide the original vector into. This is the m parameter in the product quantisation algorithm.
• triggerThreshold (recommended 5000): The number of points after which the centroids should be automatically computed and the vectors are quantised. It may be tempting to increase this to get more
vectors, but the centroids are computed in memory and can be quite large. It is recommended to keep this value low to avoid running out of memory.
During search, a pre-computed lookup table is used to find the nearest centroid for each sub-vector. The distance between the original vector and the quantized vector is the sum of the distances
between the original vector and the centroids of each sub-vector. Due to this sum, the distance metric must satisfy the property that the sum of distances is a valid distance metric. For this reason,
euclidean is used even if cosine is given as the distance metric. This is not an issue since squared euclidean distance is proportional to cosine distance, i.e. d = 2(1-cosine(x,y)) for normalised | {"url":"https://semadb.com/docs/concepts/quantization/","timestamp":"2024-11-05T07:04:00Z","content_type":"text/html","content_length":"17532","record_id":"<urn:uuid:22f1664e-92f9-4c23-922d-6ee2a1f4548c>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00710.warc.gz"} |
Newton’s method and Fisher scoring for fitting GLMs
Newton's method and Fisher scoring for fitting GLMs
Generalized linear models are flexible tools for modeling various response disributions. This post covers one common way of fitting them.
Suppose we have a statistical model with log-likelihood $\ell(\theta)$, where $\theta$ is the parameter (or parameter vector) of interest. In maximum likelihood estimation, we seek to find the value
of $\theta$ that maximizes the log-likelihood:
\[\ell_n(\theta) = \sum\limits_{i=1}^n \log f(x_i; \theta)\] \[\hat{\theta}_{\text{MLE}} = \text{arg}\max_{\theta} \ell_n(\theta)\]
There are many ways to solve this optimization problem.
Newton’s Method
One simple numerical method for finding the maximizer is called Newton’s Method. This method essentially uses the local curvature of the log-likelihood function to iteratively find a maximum.
The derivation of Newton’s method only requires a simple Taylor expansion. Below, we focus on the univeriate case (i.e., $\theta \in \mathbb{R}$), but all results can be easily extended to the
multivariate case.
Recall that the first derivative of the log-likelihood function, $\ell’(\theta)$, is called the score function. For any given initial guess of the value of $\theta$, call it $\theta_0$, we can
perform a second-order Taylor expansion around this value:
\[\ell'(\theta) \approx \ell'(\theta_0) + \ell''(\theta_0) (\theta - \theta_0).\]
At the value of $\theta$ that maximizes the loglikelihood, $\theta^*$, we know that the derivative of the log-likelihood is zero, $\ell’(\theta^*) = 0$ (this is usually true under some mild
regularity conditions, such as the maximizer not being at the edge of the support). Thus, if we plug in $\theta = \theta^*$ to our expansion, we have
&\ell’(\theta^*) = 0 \approx \ell’(\theta_0) + \ell’’(\theta_0) (\theta^* - \theta_0) \\ \implies & \theta^* \approx \theta_0 - \frac{\ell’(\theta_0)}{\ell’’(\theta_0)}.
This is known as Newton’s method. Specifically, the algorithm proceeds as follows:
1. Initialize $\theta_0$ to a random value.
2. Until converged, repeat: update $\theta_t = \theta_{t-1} - \frac{\ell’(\theta_{t-1})}{\ell’’(\theta_{t-1})}$
As we can see, Newton’s Method is essentially fitting a parabola to the current location of the log-likelihood function, then taking the minmium of that quadratic to be the next value of $\theta$.
One downside of this method is that it assumes that $\ell’’(\theta)$ is invertible, which may not always be the case. In the next section, we’ll see a method that remedies this issue.
Fisher scoring
Fisher scoring is has the same form as Newton’s Method, but instead of the observed second derivative, it uses the expectation of this second derivative, a quantity that is also known as the Fisher
Information. The update then looks like:
\[\theta_t = \theta_{t-1} - \frac{\ell'(\theta_0)}{\mathbb{E}[\ell''(\theta_0)]}.\]
The benefit of this method is that $\mathbb{E}[\ell’’(\theta_0)]$ is guaranteed to be positive (or a positive definite matrix in the multivariate case).
Relating Newton’s method to Fisher scoring
A key insight is that Newton’s Method and the Fisher Scoring method are identical when the data come from a distribution in canonical exponential form. Recall that $f$ is in the exponential family
form if it has the form
\[f(x) = \exp\left\{ \frac{\eta(\theta(x))x - b(\theta(x))}{a(\phi)} + c(x, \phi) \right\}.\]
The “canonical” form occurs when $\eta(\theta) = \theta$, and so
\[f(x) = \exp\left\{ \frac{\theta(x)x - b(\theta(x))}{a(\phi)} + c(x, \phi) \right\}.\]
The $\log$ density is
\[\log f(x) = \frac{\theta(x)x - b(\theta(x))}{a(\phi)} + c(x, \phi)\]
and the first and second derivatives with respect to $\theta$ are then
\frac{\partial \log f}{\partial \theta} &= \frac{x - b’(\theta(x))}{a(\phi)} \\ \frac{\partial^2 \log f}{\partial \theta^2} &= \frac{-b’’(\theta(x))}{a(\phi)}. \\
In canonical exponential distributions, the second derivative of $b(\theta)$ wrt $\theta$ is also the Fisher information. This can be seen by inspecting the definition of the Fisher information: it
is defined as $\mathcal{I} = -\mathbb{E}[\nabla^2 \log f(x)]$. We saw above that this evaluates to $\mathbb{E}[- A’’(\theta)]$. The expression inside the expectation is constant with respect to $f(x)
$, so we have $\mathcal{I} = - b’’(\theta)$, and this implies that the observed second derivative is identical to the expected second derivative. Thus, in the case of canonical exponential forms,
Newton’s Method and Fisher Scoring are the same.
Generalized linear models
An important application in which these methods play a big role is in fitting generalized linear models (GLMs). As the name suggests, GLMs are a generalization of traditional linear models where the
responses are allowed to come from different types of distributions. GLMs are typically written as
\[g(\mu(x)) = X^\top \beta\]
where $\mu(x) = \mathbb{E}[Y | X = x]$ is called the “regression function”, and $g$ is called the “link function”. In GLMs, we assume the conditional density $Y | X = x$ belongs to the exponential
\[f(y|x) = \exp\left\{ \frac{\theta(x)y + b(\theta(x))}{a(\phi)} + c(y, \phi) \right\}.\]
If we have a data sample ${(X_i, Y_i)}$ for $i = 1, \dots, n$, we can estimate $\theta$ using maximum likelihood estimation (MLE). Notice that $\mu_i = b’(\theta_i)$, which implies that $\theta_i =
(b’)^{-1}(\mu_i)$. Furthermore, since $\mu_i = g^{-1}(X_i^\top \beta$, we have that $\theta_i = (b’ \circ g)^{-1}(X_i^\top \beta)$. For ease of notation, we can define this as a new function $h$: $h
(X_i^\top \beta) \equiv (b’ \circ g)^{-1}(X_i^\top \beta)$.
Then, writing out the log-likelihood function, we have
\[\ell_n(\beta, \phi) = \sum\limits_{i=1}^n \left[\frac{h(X_i^\top \beta)y_i - b(h(X_i^\top \beta))}{a_i(\phi)} + c(y_i, \phi)\right].\]
It’s quite common to take $a(\phi) = \frac{\phi}{w_i}$, which simplifies our expression to
\[\ell_n(\beta, \phi) = \sum\limits_{i=1}^n \left[\frac{w_i h(X_i^\top \beta)y_i - b(h(X_i^\top \beta))}{\phi} + c(y_i, \phi)\right].\]
Ignoring terms that don’t depend on $\beta$, and noticing that $\phi$ is a positive constant, the expression we’d like to maximize is
\[\ell_n(\beta, \phi) = \sum\limits_{i=1}^n w_i\left[h(X_i^\top \beta)y_i - b(h(X_i^\top \beta))\right].\]
After some expert-mode chain rule, we obtain that the first derivative with respect to $\beta$ is
\[\ell'(\beta) = \mathbf{X}^\top \mathbf{W} \mathbf{G} (\mathbf{Y} - \boldsymbol{\mu})\]
$\mathbf{W}$ is a diagonal matrix with $w_i (b’’(\theta)g’(\mu_i)^2)^{-1}$ as the $i$’th diagonal, and $\mathbf{G}$ is a diagonal matrix with $g’(\mu_i)$ as the $i$’th diagonal.
The expectation of the second derivative is then
\[\mathbb{E}[\ell''(\beta)] = -\mathbf{X}^\top \mathbf{W} \mathbf{X}\]
Plugging these results into the Fisher scoring algorithm, we have that the update at time $t+1$ will be
\beta_{t+1} &= \beta_t + (\mathbf{X}^\top \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{W} \mathbf{G} (\mathbf{Y} - \mu) \\ &= (\mathbf{X}^\top \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^\top \
mathbf{W} (\mathbf{G} (\mathbf{Y} - \mu) + \mathbf{X}\beta_t) \\
Notice that this is similar to the estimating equation for weighted least squares
\[\hat{\beta} = (\mathbf{X}^\top \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{W}\mathbf{Y}.\]
In our case, we essentially have $\mathbf{Y} = \mathbf{G} (\mathbf{Y} - \mu) + \mathbf{X}\beta_t.$
One can interpret this as the estimated response $\mathbf{X}\beta_t$, plus the current residuals, $\mathbf{G} (\mathbf{Y} - \mu)$. Thus, we’re basically using a version of the response that has been
“corrected” for the errors in the current estimate for $\beta$.
It turns out that this algorithm is equivalent to iteratively reweighted least squares (IRLS) for maximum likelihood.
• Fan, J., Li, R., Zhang, C.-H., and Zou, H. (2020). Statistical Foundations of Data Science. CRC Press, forthcoming.
• Prof. Steffen Lauritzen’s lecture notes.
• Hua Zhou’s blog post. | {"url":"https://andrewcharlesjones.github.io/journal/fisher-scoring.html","timestamp":"2024-11-13T05:14:05Z","content_type":"text/html","content_length":"15934","record_id":"<urn:uuid:425b4544-24a8-47ac-8669-ebe6e120b745>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00822.warc.gz"} |
R commands: overview of the main commands
R commands are the foundation of data analysis and statistical modeling in the R environment. They provide the tools and flexibility to understand data, detect patterns, and make informed decisions.
R commands: what are they?¶
R commands (or R commands in English) are instructions used in R programming to execute specific tasks or initiate tasks in the R environment. These commands allow you toanalyze data, perform
statistical calculations or create visualizations. R commands can be entered and processed in the R command line or in R scripts. It is important to distinguish commands from R functions.
R functions are code blocks defined and designated under R, which perform specific tasks. They can include the use of R operators and R data to accept arguments or display return values. This means
that functions can save, process, and return data that is associated with different R data types.
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R commands: list of different commands¶
The following list of R commands gives you an overview of the different application areas in R programming. Depending on your specific projects and requirements, you can select and combine the
appropriate R commands.
Data handling and processing¶
• read.csv() : reading data from a CSV file
• data.frame() : creation of a data frame
• subset() : filtering data based on specific conditions
• merge() : merging data from different data frames
• aggregate() : data aggregation based on specific criteria
• transform() : creation of new variables in a data frame
• sort() : sorting vectors or data frames
• unique() : identifying unique values in a vector or column
Data visualization¶
• plot() : creating scatterplots and other basic types of diagrams
• hist() : creation of histograms
• barplot() : creating bar charts
• boxplot() : creating boxplots
• ggplot2::ggplot() : for more demanding and customizable visualizations with the ggplot2 package
Statistical analyzes¶
• summary() : preparation of a collection of data, including key statistical figures
• lm() : running linear regressions
• t.test() : running T tests to test hypotheses
• cor() : calculation of correlation coefficients between variables
• anova() : performing analyzes of variance (ANOVA)
• chi-sq.test() : for chi-square tests
Data processing¶
• ifelse() : for condition evaluations and conditional expressions
• apply() : application of a function to matrices or data frames
• dplyr::filter() : filtering data in a data frame with the dplyr package
• dplyr::mutate() : creation of new variables in data frames with the dplyr package
• lapply(), sapply(), mapply() : for applying functions to lists or vectors
Importing and exporting data¶
• readRDS(), saveRDS() : reading and saving R data objects
• write.csv(), read.table() : export and import of data in different formats
Statistical charts and graphs¶
• qqnorm(), qqline() : for creating quantile-quantile diagrams
• plot(), acf() : representation of autocorrelation diagrams
• density() : representation of density functions and histograms
• heatmap() : creation of density maps
R commands: usage examples¶
The following code examples demonstrate the use of major R commands in various application domains. According to your data and analytics requirementsyou can adapt and expand these commands.
Reading data from a CSV file¶
data <- read.csv("donnees.csv")
Read.csv() is a command allowing you to read the data present in a CSV file, in R. In our example, the data read is saved in the variable data. This command is useful for importing external data into
R and for making analyzes available.
Creating a Scatter Plot¶
plot(data$X, data$Y, main="DiagrammeDispersion")
Plot() is an R command for creating charts and graphs in R. In our example, a scatter plot is created to represent the relationship between variables X And Y of the data frame data. The argument main
sets the title of the diagram.
Running a Linear Regression¶
regression_model <- lm(Y ~ X, data=data)
In this example, we run a linear regression in order to model the relationship between variables X And Y in the data frame data. The command lm() is used to calculate a linear regression in R. The
result of the regression is saved in the variable regression_model and can be used for other analyses.
Filtering data with the dplyr package¶
filtered_data <- dplyr::filter(data, column > 10)
The command dplyr::filter() comes from dplyr package and will be used for data manipulation. The dplyr package provides powerful functions for data filtering. We obtain the variable filtered_data by
selecting the lines of the data frame data for which the value of the column column is greater than 10.
Creating quantile-quantile plots¶
You can use qqnorm() to represent a quantile-quantile diagram in R. In this example, a quantile-quantile diagram is represented for the variable Variable of data. qqline() adds a reference line to
compare the distribution with a normal distribution.
We recommend that all beginners check out the R programming overview tutorial. There you will find plenty of tips and the basic knowledge needed to progress with the R programming language. More tips
and basics are available in our article “Learn programming: basic principles” from the Digital Guide. | {"url":"https://amzdigicom.com/r-commands-overview-of-the-main-commands/","timestamp":"2024-11-07T06:23:30Z","content_type":"text/html","content_length":"103323","record_id":"<urn:uuid:5a745bc1-f0b1-4140-9d04-7417fefd3ddc>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00151.warc.gz"} |
Quantifying the Uncertainty in Deep Bayesian Q-Networks for Robust Decision Making
Deep Q-Networks (DQN) are a powerful class of reinforcement learning algorithms that have been successfully used in various applications, such as robotics, game playing, and finance. However, one
challenge with DQNs is their lack of robustness to uncertainties in the environment, which can result in suboptimal or unsafe decisions. In this blog post, we will discuss how to quantify the
uncertainty in DQNs using Bayesian deep learning, and how to use this uncertainty to make more robust decisions.
Bayesian Deep Learning
Bayesian deep learning is a framework that combines deep learning with Bayesian inference to quantify the uncertainty in neural network models. In Bayesian deep learning, we treat the weights of the
neural network as random variables, and we define a prior distribution over these weights. We then use Bayes' rule to update the prior distribution to a posterior distribution, given the observed
data. The posterior distribution represents our updated belief about the weights, given the data.
Uncertainty in DQNs
In DQNs, the uncertainty arises from two sources: the stochasticity of the environment, and the uncertainty in the neural network model. The stochasticity of the environment refers to the randomness
in the outcomes of the actions taken by the agent, due to the inherent randomness in the environment. The uncertainty in the neural network model refers to our uncertainty about the optimal actions
given the current state of the environment, which is represented by the Q-values predicted by the neural network.
Bayesian DQN
To quantify the uncertainty in DQNs, we can use Bayesian deep learning to model the uncertainty in the neural network weights. Specifically, we can use a Bayesian neural network (BNN), which is a
neural network with weights treated as random variables. We can then use Monte Carlo dropout (MC dropout) to approximate the Bayesian inference process. MC dropout involves adding dropout at test
time and sampling multiple predictions from the network to estimate the distribution of the predictions.
The loss function for training the Bayesian DQN is the negative log-likelihood of the observed data, which includes the rewards received and the transitions between states. The loss function is
modified to include a penalty term for the entropy of the distribution over the Q-values, which encourages exploration and reduces overconfidence in the predictions.
Python Implementation
To implement a Bayesian DQN in Python using TensorFlow, we can start with the standard DQN implementation and modify it to use a BNN and MC dropout. The following code shows an example of how to
modify the Q-network in a DQN to use a BNN and MC dropout:
class BayesianQNetwork(tf.keras.Model):
def __init__(self, num_actions, num_hidden_units):
super(BayesianQNetwork, self).__init__()
self.num_actions = num_actions
self.dense1 = tf.keras.layers.Dense(num_hidden_units, activation='relu')
self.dense2 = tf.keras.layers.Dense(num_hidden_units, activation='relu')
self.logits = tf.keras.layers.Dense(num_actions)
def call(self, inputs):
x = self.dense1(inputs)
x = self.dense2(x)
logits = self.logits(x)
return logits
def sample_predictions(self, inputs, num_samples=10):
outputs = []
for _ in range(num_samples):
return tf.stack(outputs)
In this code, the Q-network is defined as a BNN with two hidden layers and a softmax output layer. The sample_predictions function is used to sample predictions from the network using MC dropout.
To modify the loss function to include the penalty term for entropy, we can use the following code:
def bayesian_loss(model, states, targets, num_samples=10):
Computes the Bayesian loss of a model given the states and targets.
model -- the deep Q-network model
states -- a batch of input states (numpy array of shape (batch_size, state_size))
targets -- a batch of target Q-values (numpy array of shape (batch_size, num_actions))
num_samples -- the number of samples to draw from the posterior distribution (default 10)
The Bayesian loss (scalar).
# Compute the predicted Q-values and log variance for each state-action pair
q_values = []
log_variances = []
for i in range(num_samples):
q_values_i, log_variances_i = model(states, sample=True)
q_values = tf.stack(q_values) # shape: (num_samples, batch_size, num_actions)
log_variances = tf.stack(log_variances) # shape: (num_samples, batch_size, num_actions)
# Compute the mean and variance of the predicted Q-values and log variances
q_mean = tf.reduce_mean(q_values, axis=0) # shape: (batch_size, num_actions)
q_var = tf.math.reduce_variance(q_values, axis=0) # shape: (batch_size, num_actions)
log_var_mean = tf.reduce_mean(log_variances, axis=0) # shape: (batch_size, num_actions)
# Compute the Bayesian loss
precision = tf.exp(-log_var_mean)
loss = 0.5 * precision * tf.reduce_sum(tf.square(targets - q_mean), axis=-1) + \
0.5 * tf.math.log(1 + q_var * precision) # shape: (batch_size,)
loss = tf.reduce_mean(loss) # take the mean over the batch
return loss | {"url":"https://www.gofar.ai/p/quantifying-the-uncertainty-in-deep","timestamp":"2024-11-08T03:14:07Z","content_type":"text/html","content_length":"126778","record_id":"<urn:uuid:6947b815-070f-41e8-aa13-147939d9ce37>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00625.warc.gz"} |
3 | Bridging Practices Among Connecticut Mathematics Educators
I used this task with my 3rd grade students. The purpose of this task was to notice how to use regrouping when subtracting. I wanted students to recognize and understand when it is appropriate to
regroup when subtracting. The students were given 2 different answers for the same subtraction problem. The student had to decide which answer was correct. The student had a place to include their
claim and argument, as well as have time to discuss with a partner and record their partner’s thoughts.
This task is a set of addition and subtraction word problems with three follow up questions created for third graders. Double-digits are used in the word problems and students must decide for
themselves when to use addition or subtraction through the construction of the responses. The follow up questions contain argumentative language and ask students to describe how they solved the
problem, the warrants behind it, and a claim and evidence pertaining to a partner’s strategy.
Microsoft Word version: 3_OA_SubtractionAddition_WordProblem_Construct
PDF version: 3_OA_SubtractionAddition_WordProblem_Construct
This task is geared towards third graders. The task states a problem asking for the difference between two quantities and asks students to crititque two statements about the problem: Is finding the
difference an addition or subtraction problem? Students are provided space to think about the problem, make a claim, and provide evidence. This task could start a class discussion about how students
might think differently about subtraction problems. Some students may state that this is subtraction because one can subtract the two quantities to find a difference while other students may look at
this as an addition problem by looking at how much must be added to one quantity to get to the next.
Microsoft Word version: 3_OA_SubtractionAddition_TalkFrame_Critique
PDF version: 3_OA_SubtractionAddition_TalkFrame_Critique
This think-pair-share task is provided for third-grade students to understand the commutative property for addition. Using a statement with single-digit numbers, students must construct an argument
on whether the true statement is correct and share their ideas with a partner. A graphic organizer is provided to help students create their claim and evidence, as well as record their partner’s
ideas and any similarities/differences.
Microsoft Word version: 3_OA_PropertyCommutativeAddition_ThinkPairShare_Construct
PDF version: 3_OA_PropertyCommutativeAddition_ThinkPairShare_Construct
This task is designed for third grade studenrs learning applications of single digit multiplication. Students are given an application along with a student response and asked to critique the student
response. The task uses argumentation language by first asking students to solve the problem, then having students state a claim and provide evidence to support the claim. The task provides space for
each piece of the argument.
Microsoft Word version: 3_OA_MultiplicationSingleDigit_Problem_Critique
PDF version: 3_OA_MultiplicationSingleDigit_Problem_Critique
In Art Supplies, third graders are given a single-digit multiplication word problem. Students must critique the two given responses, which address the misconception that the number of boxes and the
amount in each box should be added together and not multiplied. A graphic organizer is provided and contains argumentative language to help students create their claim and provide evidence and
Microsoft Word version: 3_OA_MultiplicationSingleDigit_Problem_Critique_ArtSupplies
PDF version: 3_OA_MultiplicationSingleDigit_Problem_Critique_ArtSupplies
Ice Cream Sundaes is geared towards third graders developing multiplication skills. This task asks students to find the total number of possible combinations of two separate entities. Students must
construct an argument and are given space to provide work and space to explain thinking. This problem can open the class to conversations about methods to solve the problem because students can do
the math (single digit multiplication), draw pictures, or create a diagram.
Microsoft Word version: 3_OA_MultiplicationSingleDigit_Problem_Construct_IceCreamSundaes
PDF version: 3_OA_MultiplicationSingleDigit_Problem_Construct_IceCreamSundaes
Third graders are developing an understanding of the commutative property through single-digit multiplication in Same of Different?. Students are asked if two rectangles are the same given reverse
length and width measurements. Argumentation language is present when asking students to explain their thinking using claim, evidence, and warrants.
Microsoft Word version: 3_OA_MultiplicationPropertyCommutative_Problem_Construct_SameOrDifferent
PDF version: 3_OA_MultiplicationPropertyCommutative_Problem_Construct_SameOrDifferent
Pizza Party is a task designed for third grade students working on multiplication. Students must use multiplication to determine how many pizzas to order, knowing how many slices are needed.
Ultimately, a comparison must be made between the operations of 4×8, 3×10, and 2×16. The task is a multiple step problem in which students must critique the answers of two people and decide with whom
to agree.
Microsoft Word version: 3_OA_Multiplication_Problem_Critique_PizzaParty
PDF version: 3_OA_Multiplication_Problem_Critique_PizzaParty | {"url":"https://bridges.education.uconn.edu/tag/3/","timestamp":"2024-11-13T02:43:45Z","content_type":"text/html","content_length":"81427","record_id":"<urn:uuid:a3f8e0f9-ff71-48b0-bc2c-3f6014073cc2>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00285.warc.gz"} |
Multiplication Split Strategy Worksheet
Math, specifically multiplication, creates the keystone of numerous scholastic disciplines and real-world applications. Yet, for lots of learners, mastering multiplication can pose a challenge. To
resolve this obstacle, teachers and moms and dads have actually embraced an effective tool: Multiplication Split Strategy Worksheet.
Introduction to Multiplication Split Strategy Worksheet
Multiplication Split Strategy Worksheet
Multiplication Split Strategy Worksheet -
Multiplication Strategies brettw29 Member for 2 years 7 months Age 9 12 Level 5 Language English en ID 766369 28 02 2021 Country code AU Country Australia School subject Math 1061955 Main content
Multiplication Strategies 1998201 multiplication facts build ups split strategy Other contents Split Strategy and Division
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From standard pen-and-paper exercises to digitized interactive layouts, Multiplication Split Strategy Worksheet have evolved, catering to varied learning designs and preferences.
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This set of worksheet compliments my multiplication strategy posters and was made to practise the Split Strategy for mental multiplication skills I have made 2 different kinds of worksheets to
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Teach Your Students How To Use The Split Strategy For multiplication And Division With This
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Encouraging consistent method, providing aid, and creating a favorable understanding atmosphere are beneficial steps. | {"url":"https://crown-darts.com/en/multiplication-split-strategy-worksheet.html","timestamp":"2024-11-12T05:18:33Z","content_type":"text/html","content_length":"28511","record_id":"<urn:uuid:0f3c883c-bd20-46b1-9e96-315fa39b721e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00835.warc.gz"} |
The Power of Trigonometry: 2 Sin A Cos B - justpersonalcare.com
Trigonometry is an essential branch of mathematics that deals with the relationships between the angles and sides of triangles. It has wide applications in various fields such as engineering,
physics, astronomy, and many others. One of the fundamental trigonometric identities is 2 Sin A Cos B. In this blog post, we will delve into the power and significance of this identity in
Understanding 2 Sin A Cos B
The identity 2 Sin A Cos B stems from the basic trigonometric functions – sine and cosine. In a right-angled triangle with an angle A, sin A is defined as the ratio of the length of the side opposite
the angle A to the length of the hypotenuse, while cos B is defined as the ratio of the length of the side adjacent to the angle B to the length of the hypotenuse. When we multiply sin A by cos B and
then double it, we get 2 Sin A Cos B.
Deriving the Identity
To derive the identity 2 Sin A Cos B, we can use the trigonometric angle addition formula. Let’s consider the formula for sin(A + B):
sin(A + B) = sin A cos B + cos A sin B
Now, if we let A = B, we get:
sin(2A) = sin A cos A + cos A sin A
sin(2A) = 2 sin A cos A
Therefore, 2 Sin A Cos A is the double angle formula for the sine function. By extension, 2 Sin A Cos B is a scaled version of this identity, where A and B are different angles.
Applications of 2 Sin A Cos B
The identity 2 Sin A Cos B finds applications in various areas of mathematics, science, and engineering. Some of the key applications include:
1. Vector Analysis: In vector calculus, 2 Sin A Cos B is used to calculate the cross product of two vectors. The cross product of vectors A and B is given by A x B = |A||B| sin θ n, where n is the
unit vector perpendicular to both A and B, and θ is the angle between A and B. This formula closely resembles the identity 2 Sin A Cos B.
2. Wave Dynamics: In physics, trigonometric identities like 2 Sin A Cos B are applied in wave dynamics to analyze the amplitude and phase of wave functions. Waves can often be described using sine
and cosine functions, making trigonometric identities crucial in wave equations.
3. Signal Processing: In signal processing, trigonometric identities play a vital role in analyzing and manipulating signals. The identity 2 Sin A Cos B can be used to simplify complex signal
functions and facilitate signal transformations.
Further Identities Related to 2 Sin A Cos B
The identity 2 Sin A Cos B is just one of the many trigonometric identities that play a significant role in mathematical calculations. Some other identities related to 2 Sin A Cos B include:
1. Sum-to-Product Identities:
2. Sin A + Sin B = 2 Sin[(A + B)/2] Cos[(A – B)/2]
3. Sin A – Sin B = 2 Cos[(A + B)/2] Sin[(A – B)/2]
4. Cos A + Cos B = 2 Cos[(A + B)/2] Cos[(A – B)/2]
5. Cos A – Cos B = -2 Sin[(A + B)/2] Sin[(A – B)/2]
6. Product-to-Sum Identities:
7. 2 Sin A Sin B = Cos(A – B) – Cos(A + B)
8. 2 Cos A Cos B = Cos(A – B) + Cos(A + B)
These identities can be derived using the sum and difference formulas for sine and cosine functions and are instrumental in simplifying trigonometric expressions.
FAQs (Frequently Asked Questions)
1. What is the geometric interpretation of 2 Sin A Cos B?
The identity 2 Sin A Cos B can be interpreted geometrically as the area of a parallelogram with sides of lengths Sin A and Cos B.
2. How is the double angle formula related to 2 Sin A Cos B?
The double angle formula sin(2A) = 2 sin A cos A is closely related to 2 Sin A Cos B, as it is a special case where A = B.
3. Can 2 Sin A Cos B be expressed in terms of other trigonometric functions?
Yes, 2 Sin A Cos B can be expressed in terms of the sine function as sin(2A + B) – sin(B), using the sum-to-product identity for sine functions.
4. In what kind of problems is the identity 2 Sin A Cos B commonly used?
The identity 2 Sin A Cos B is often used in trigonometric equations involving multiple angles, especially in calculus and physics problems.
5. How does the identity 2 Sin A Cos B contribute to the understanding of periodic functions?
The identity 2 Sin A Cos B helps in analyzing the periodic nature of trigonometric functions and understanding the relationship between different angles in periodic functions.
In conclusion, the identity 2 Sin A Cos B is a powerful tool in trigonometry with diverse applications across various disciplines. Understanding and utilizing this identity not only enhance
mathematical calculations but also shed light on the beauty and elegance of trigonometric functions. | {"url":"https://justpersonalcare.com/the-power-of-trigonometry-2-sin-a-cos-b/","timestamp":"2024-11-06T16:44:46Z","content_type":"text/html","content_length":"352930","record_id":"<urn:uuid:1e5d3e70-5be6-4967-9e4f-6fe68e967ca3>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00498.warc.gz"} |
Question about inverse kinematic processing
Dear All,
I have a basic question about inverse kinematic processing. I previously used V3D software for calculating Inverse Kinematics (mainly joint angles), in this software it was possible to filter the
marker trajectories (low pass) before calculating the kinematics, so the final results was smooth enough and there was no need to filter the inverse kinematic results.
Now I am working with Opensim software, so I am not able to filter the trajectories before calculating the inverse kinematics and the results are noisy, I should apply the lowpass filter on the
From a biomechanical point of view, does the results differ when i filter trajectories at first, in compare with the situation that the filtering is applied on angle data after calculating inverse
kinematics ?
...Any help would be very appreciative.
Farzaneh Yazdani
PhD Candidate, SUMS School of Rehabilitation Sciences
Tags: None
Re: Question about inverse kinematic processing
I will start by saying this is not a basic question and my response may open up a whole debate on the topic. Unfortunately the area around filtering, model/marker design, and error in joint angle
data is poorly understood.
The reason for not filtering 3D marker data to me is relatively straight forward. If you have impacts such as foot strikes then filtering over these with a low pass filter will introduce errors
(oscillations) in the marker coordinate data around the point of impact. This will introduce oscillations into the reconstructed foot segment. With large de-acceleration the foot (running and side
stepping) the foot can be seen to move below and above ground level as well as backwards and forwards prior to reaching a steady position after impact. This mismatch between the foot segment axes and
GRF centre of pressure also produces errors in ankle moments, which propagated through the calculation of knee and hip joint moments of the rigid segment system.
Another source error in the foot segment location is the global optimization method (3 degrees of freedom at each joint) used in the Inverse Kinematics approach. In a traditional six degree of
freedom model the lower leg has the smallest RMS error between static and dynamic segment marker locations (this segment acts most like a rigid body during movement) followed by the foot, pelvis,
thigh and worst for markers on the trunk and shoulder girdle. Global optimization does not allow for errors in joint centre location or skin movement artefact. The result is that larger errors in
shoulder, trunk and thigh segment locations are imposed on the shank and foot. The ankle joint centre being at the end of the linked chain is subject to random and unpredictable errors in the ankle
joint location and consequently foot segment position and orientation. It is for this reason global optimization should never be used in calculating joint angles or joint moments of the legs or arms,
especially when the location of the feet or hands is critical.
The reason for not filtering joint angle data is not so straight forward and requires a greater understanding of axes misalignment and non-linear errors than presently presented in the literature.
Errors in thigh axes alignment produce non-linear errors in calculated knee joint abd/add and ext/int rotations; commonly and I think misleadingly referred to as cross-talk. Consequently, correcting
thigh axes alignment will correct non-linear errors seen in both calculated knee joint abd/add and ext/int rotations. Axes mis-alignment has two components, a static offset produced in the subject
calibration procedure and a dynamic component due to skin movement during movement. When correcting axes misalignment they cannot be considered in isolation, with the static component being the
larger of the two in gait. Axes misalignment can be corrected prior to calculating joint rotations in the static calibration procedure and post-hoc via nonlinear correction of joint angle data
through a mathematical correction to the proximal or distal segment axes alignment. The trick is being able to identify the static and dynamic components. The key to this is recognising that between
sessions the static component is random in the magnitude of the offset but produces characteristic patterns in gait related to the magnitude of the offset; while the dynamic component is varied
during the gait cycle but is consistent across trials and session which use the same markers. Global optimization introduces a third unpredictable source of error in axes misalignment for the lower
leg and foot segments (as mentioned previously) and therefore makes it unsuited to attempt to assess, identify and correct for axes misalignment. Filtering or removing offsets from joint angle data
to correct errors ignores axes misalignment as the source of errors in joint angle data, the interdependency of the calculated joint rotations and the nonlinear nature of the errors, as well as the
opportunity to correct for static offsets and dynamic skin movement artifact.
To answer your question, No - it is not equivalent to smooth 3D marker data or joint angle data. Smoothing 3D marker data effects segment location so it effects everything that follows (predictions
of centre of mass, joint centres, joint moments … ), whereas smoothing joint angle data only effects joint angle data.
So what should I do? My advice: if you are investigating lower limb kinematics or kinetic do not smooth 3D marker data if there are impacts, do not use global optimization, and I would suggest as a
minimum to assess and attempt to correct for thigh axes mis-alignment about the thigh longitudinal axes (error in the direction of the knee flex/ext axis) within the static calibration procedure by
comparing calculated knee abd/add joint rotations during gait to normal gait patterns (no more than 4 degree adduction in stance and 8 degree abduction in swing) to adjust thigh axes alignment
I hope this helps
Allan Carman
Re: Question about inverse kinematic processing
Dear Allan,
Thank you for taking the time to answer me, I really do appreciate it.
According to your advice, I think in my project is more preferred to filter joint kinematics cause I want to accurately detect stance phase and filtering the trajectories may lead to move the events
from their actual points.
Best regards
Farzaneh Yazdani
PhD Candidate, SUMS School of Rehabilitation Sciences
Re: Question about inverse kinematic processing
Like Allan, I usually do not filter 3D marker data.
Filtering of inverse kinematics results is probably needed to avoid noisy joint moments in the inverse dynamic analysis, but you should always filter as little as possible.
For a given filter (e.g. second order, 6 Hz) it will not matter much whether you apply it before or after the inverse kinematic analysis. The final result will be almost the same. However, the
inverse kinematic analysis will already cancel out some of the noise (by combining multiple markers), so you may be doing more filtering than necessary if you aim for smooth 3D marker trajectories.
So it is definitely better to not filter until after inverse kinematics, exactly as Opensim does.
The debate of global optimization vs. 6-DOF joints is a separate issue, and there is no definitive answer. Opensim does inverse kinematics (global optimization) only and in my opinion that is usually
best for analysis of motion and control. Global optimization rejects noise and soft tissue artifact much better than 6-DOF. However, Allan is right that the joint positions (and joint moments) may
not be as good. It depends on the quality of the model and the quality of the data. If the data is bad (or incomplete), global optimization can help a lot. But if the kinematic model introduces more
error than the error in the 3D marker data, you are better off with 6-DOF analysis.
Recently a few good papers have been published comparing IK to 6-DOF.
Ton van den Bogert
Re: Question about inverse kinematic processing
Dear Ton van den Bogert,
thank you for taking the time to write this helpful answer.
could you please provide me your mentioned papers’ references?
Farzaneh Yazdani
PhD Candidate, SUMS School of Rehabilitation Sciences
Re: Question about inverse kinematic processing
Here are a few references.
[1] compares IK to a "direct" method (Vicon's Plug-in Gait) which is not 6-DOF but an older method.
[2] does a comparison between IK (here known as "global optimization") and 6-DOF for a particular application.
It will be hard to give a general conclusion on the performance of IK vs. 6-DOF. It depends on the quality of the model. IK will only improve the kinematic analysis if the model is good enough. It is
hard to define what is good enough.
Also, strengths of both approaches can be combined, an IK model can include 6-DOF joints where needed. In my full-body IK model [3], I used 6-DOF for the "joint" between humerus and thorax, because I
wanted to avoid modeling the shoulder mechanism. A bad shoulder model would be worse than assuming 6 DOF.
1. Kainz H, Modenese L, Lloyd DG, Maine S, Walsh HP, Carty CP (2016) Joint kinematic calculation based on clinical direct kinematic versus inverse kinematic gait models. J Biomech 49(9):1658-69.
doi: 10.1016/j.jbiomech.2016.03.052.
2. Moniz-Pereira V, Cabral S, Carnide F, Veloso AP (2014) Sensitivity of joint kinematics and kinetics to different pose estimation algorithms and joint constraints in the elderly. J Appl Biomech 30
(3):446-60. doi: 10.1123/jab.2013-0105.
3. van den Bogert AJ, Geijtenbeek T, Even-Zohar O, Steenbrink F, Hardin EC (2013) A real-time system for biomechanical analysis of human movement and muscle function. Medical and Biological
Engineering and Computing 51:1069-1077.
Re: Question about inverse kinematic processing
When comparing methods Kainz et al (2016), Moniz-Pereira et al (2014), Duprey et al. (2010) and Duffell etal (2014) have shown that varying methods give different joint kinematics:
un-constrained (6df) != constrained (3df)
constrained (3df) != constrained OS-IK (3-1-2 df’s at the hip-knee-ankle);
Cluster != PiG;
PiG != OS-IK.
So which one should be used?
Increasing the constraints placed on joints has produced greater variations in non-sagital joint kinematics (Kainz et al. 2016; Moniz-Pereira et al. 2014; Duprey et al. (2010) and kinetics
(Moniz-Pereira et al. 2014). The inclusion of planar knee joint (OS-IK model 3-1-2 df’s) has consistently produced the largest variations and erroneous patterns of hip and ankle non-sagital rotations
when compared to 3df and 6df models (Kainz et al. 2016; Moniz-Pereira et al. 2014; Duprey et al. 2010). With Kainz et al. (2016) reporting approximate increases of about 35%, 50% and 20% in hip abd/
add, hip int/ext and ankle flex/ext range of motion (ROM) respectively for the OS-IK model relative to 3df and 6df models in child CP gait. Kainz et al. (2016) did not report the knee non-sagital
rotations. While Moniz-Pereira et al. (2014) found that the OS-IK model could not reproduce any hip knee or ankle non-sagital rotation or hip and knee flexion in stance relative to 3df and 6df
models. With a 1df knee joint the linked segment model of the leg still has to best match the pelvis, thigh, shank and foot segment locations but without the natural knee abd/add or int/ext rotations
or small amounts of translation or errors introduced by skin movement artefact or errors in defining joint centre positions. To do so introduces large errors in hip and ankle joint rotations and
moments (Moniz-Pereira et al. 2014; Duprey et al. 2010) to compensate for a 1 df knee joint. This is in addition to the design not being able to describe the non-sagital rotations of the knee. The
OS-IK model of 3-1-2 df’s has been consistently shown in these studies to be inadequate to describe hip, knee or ankle kinematic or kinetics in gait and should not be used.
The results for PiG model in Duffell et al. (2014) and Kainz et al. (2016) are not much better than the OS-IK (3-1-2) model for reproducing gait kinematics. This is in agreement with reliability
studies that have also found the PiG method unreliable when describing inter-session knee non-sagittal joint rotations in gait (Desloovere 2010, Scheys 2013). As with the similar VCM model (Schwartz
et al. 2004, Charlton et al. 2004, Tsushima et al. 2003, Schache et al. 2006) and Mod-HH (Kadaba et al. 1989, Miller et al. 1996, Gorton et al. 1997, Grownley et al. 1997, Collins et al. 2009,
Kaufman et al. 2016) either with or without the KAD. Results presented for normative gait joint angle data using the VCM with KAD (Baker, Cho, Kirtley www.clinicalgaitanalysis.com/data/ ) and Mod-HH
with wands (Kadaba et al. 1989, Collins et al. 2009, McMulkin & Gorton 2009, Nester et al. 2003) vary widely. Ranging from good (Baker, Collins et al. 2009) to no resemblance (Cho, Kirtley, Kadaba et
al. 1989) to non-sagital joint rotations of the knee.
The use of 3df joints in the legs compared to 6 df joints is less clear through a lack of direct comparisons. Moniz-Pereira et al. (2014) generally found good agreement between the methods, except
for knee abd/add pattern during stance and larger ankle abd/add ROM when using a 3df model. When including the trunk in a 3df model, Bogert et al. (2013) were unable to describe normal hip and ankle
ext/int rotations with large inter-subject avg StdDev (around 6-7 degs). Although the knee non sagittal rotations were not reported it can be expected that due to the large variations in hip ext/int
rotations (and therefore knee flex/ext axis mis-alignment) that the knee non-sagital rotations would suffer large non-linear errors (commonly called cross-talk). It is unclear in Bogert et al. (2013)
how much of the errors seen in ankle non-sagital rotations are a factor of the static calibration procedure and dynamic skin movement artefact producing axes-misalignment or in this case with 3df
joints, the combined effects of errors in the locations of all proximal segments affecting the location of the ankle joint centre and therefore foot segment location and derived ankle joint
If 3df joints are to be used, then I would say use them with caution. Do not use less than 3df at the hips or knee joints; do not include trunk (lumbar, lower thoracic, upper thoracic or shoulder
girdle) segments within a 3df per joint lower body model; do not use with large joint ranges of motion or high skin movement artefact or with joints were the joint centre location are not well
defined, and; still be wary of the non-sagital knee and ankle joint rotations. A mixed constrained leg model of 3 df’s at the hips, 6 df’s at the knees and 3 df’s at the ankles (3-6-3) might be a
viable constrained model option? But certainly not the OS-IK (3-1-2) model.
Does this mean a 6 df model? Unfortunately no, as 6 df models are not all created equally. They vary in the placement and number of markers used or whether a rigid fixation device was used or whether
or not they attempted to correct for axes misalignment and if so what methods were used. The results of these studies do not mean that the unconstrained cluster design is valid or reliable; they just
have a smaller ROM and have produced more consistent gait joint angle data than those which have placed limitations on joint degrees of freedom.
The cluster models presented in Duffell et al. (2014) and Kainz et al. (2016) were quite similar in design. However both studies reported very large inter-subject StdDev in the order of 6 deg for
both knee and ankle abd/adds and around 10 degs for hip, knee and ankle ext/int rotations. Meaning the designs were poor and results unreliable. Limitations include; not assessing knee abd/add or
attempting to align knee medio-lateral axes to correct axes misalignment and non-linear error, use of three markers instead of four or more (three markers is just least squares and not true cluster
with 4 or more markers), poor placement of markers to achieve a distribution along at least two axes, using a rigid fixation device on the thighs, and not using virtual hip, knee and ankle joint
centres within the respective segment’s marker clusters.
As mentioned by Ton in his previous post, 3df vs 6df does very much depend on how the model is constructed and implemented. If done poorly neither will produce usable non-sagital hip, knee and ankle
joint rotations. For comparison to the studies mentioned, using a 6df cluster design (6+ markers per segment, including virtual joint centres with least squares), with no filtering of either raw 3D
coordinate data or calculated joint angle data, and which assesses and attempts to reduce nonlinear error in joint angle data, the inter-subject pooled StdDev in degrees for normal gait of 30
subjects were:
Hips, flex/ext = 3.54, abd/add = 1.65, ext/int = 2.95
Knee flex/ext = 4.11, abd/add = 1.29, ext/int = 1.67
Ankle flex/ext = 3.58, abd/add = 2.01, ext/int = 2.24
Something to think about, cheers
Allan Carman
Re: Question about inverse kinematic processing
There is a lot to digest in Allan's post, so I will just make a few comments.
Starting with the last piece of information. An inter-subject SD of 1-2 degrees for non-sagittal rotations is indeed impressive, but probably more a reflection of the study design than the methods
that were used for kinematic analysis. As Allen mentioned, in my 2013 paper, the SD values were much larger, but subjects walked at their own preferred speed which introduces variations. Also, most
of the inter-subject differences were static offsets between the angle curves. ROM differences were much smaller. In the Kainz paper, CP patients were used, and they had high inter-subject SD values
of around 15 degrees which is not unexpected. Interestingly, the OS-IK results had lower inter-subject SD values than the 6 DOF results. This makes sense, because soft tissue artifact contributes to
the SD, and OS-IK with the 3-1-2 leg model is more robust against soft tissue artifact. However, I don't think these SD values can help us decide which method or model is better. I expect the method
and model choice to produce mostly systematic errors, and not affect the inter-subject SD (other than through robustness, i.e. a lower ratio between number of DOF and the number of markers).
Allen points out one potential problem in the OS-IK method with the 3-1-2 leg degrees of freedom. The knee has 1 dof (flexion only), and therefore, all internal tibia rotation has to be fully
transmitted to hip internal rotation (if knee angle is close to zero). Nothing is absorbed at the knee, while clearly, a real knee has considerable rotational laxity. This explains why Kainz found a
few degrees more hip internal rotation with OS-IK than with 6-DOF, indeed an increase of 50%. But it remains to be seen which one is closer to the truth. Due to soft tissue artifact, some thigh
markers do not follow internal rotation of the femur very well. So a 6 DOF analysis may underestimate hip internal rotation, because it is only based on pelvis and thigh markers, nothing from below
the knee is used. Bone pin studies showed that the soft tissue artifact for internal rotation at the knee is larger than the actual internal rotation that takes place during walking [1]!. So assuming
zero internal rotation in the knee (as in the OS-IK 3-1-2 model) is not a bad assumption at all, and may actually get you closer to the true internal rotation in the hip joint. We don't know. This
question can be answered by labs that can do optical motion capture and stereo-fluoroscopy at the same time. Please do!
If you were looking at a sports motion, or abnormal gait, where the knee internal rotation is much larger than the soft tissue artifact, the 3-1-2 model would certainly be worse than a model with
extra DOFs at the knee. So it really depends on the motion being studied.
It also depends on how good the model is. You can make a good 3-1-2 model or a bad 3-1-2 model. OS-IK is performed after model scaling and this may not produce the best possible subject-specific
skeleton model. Kainz reports 9 mm RMS IK tracking error (with max of 21 mm!) and this is much larger than what I had in my 2013 paper. I don't think I reported it, but I typically get about 3 mm. I
don't scale an existing model, but build a subject model from scratch, using the subject's marker data during standing. So the OS-IK results of Kainz may not be representative of IK and the 3-1-2 leg
model in general.
[1] Reinschmidt et al. (1997) Knee and ankle joint complex motion during walking: skin vs. bone markers. Gait and Posture 6: 98-109. | {"url":"https://biomch-l.isbweb.org/forum/biomch-l-forums/general-discussion/29900-question-about-inverse-kinematic-processing#post37594","timestamp":"2024-11-08T15:03:06Z","content_type":"application/xhtml+xml","content_length":"148094","record_id":"<urn:uuid:d0528c71-42fe-4923-9358-9591c4b99894>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00034.warc.gz"} |
[OS X TeX] Editor with real-time preview?
Ionita, Costel ionita at dixie.edu
Thu Jul 17 21:24:50 CEST 2014
Take a look at http://www.texpadapp.com
Costel Ionita
Department of Mathematics
Dixie State University
Saint George, UT
> On Jul 17, 2014, at 12:39 PM, "Jung-Tsung Shen" <jushen at gmail.com> wrote:
> I was wondering if there is a TeX editor that provides real-time preview? I used to use paper to jot down my calculations and then transcribed to LaTeX documents -- a two-step process (the first step requires some concentration so I do not want to get interrupted by the extra compiling process). Now with the computer faster enough, I was hoping a real-time preview could help simplify the process.
> Thanks,
> JT
> ----------- Please Consult the Following Before Posting -----------
> TeX FAQ: http://www.tex.ac.uk/faq
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More information about the macostex-archives mailing list | {"url":"https://tug.org/pipermail/macostex-archives/2014-July/052711.html","timestamp":"2024-11-03T10:23:52Z","content_type":"text/html","content_length":"4298","record_id":"<urn:uuid:2146d30e-48c3-4774-997e-8478f9d41da2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00108.warc.gz"} |
Percentage based grids and gaps - MrFent.com
Percentage based grids and gaps
A grid with a width of 90%. Six column tracks of 10% each, 5 gutter tracks of 2% each.
The grid-gap property controls columns and rows but as the grid has no height, the row gap resolves to 0.
If we give the grid a height, there is something for 2% to be a percentage of. So we get a gap.
Credit to gridbyexample.com for inspiring this example.
[fl_row] {
grid-gap: 2%;
grid-template-columns: repeat(6, 10%);
height: 500px; /*second example only*/
The grid-gap property controls columns and rows but as the grid has no height, the row gap resolves to 0.
If we give the grid a height, there is something for 2% to be a percentage of. So we get a gap. | {"url":"https://mrfent.com/beaver-builder-css-grid-example/36/","timestamp":"2024-11-04T05:48:55Z","content_type":"text/html","content_length":"60836","record_id":"<urn:uuid:893ab97b-a679-4fd9-9640-ff20f5f6e97c>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00530.warc.gz"} |
Search-Based Synthesis of Probabilistic Models for Quality-of-Service Software Engineering
The formal verification of finite-state probabilistic systems supports the engineering of software with strict quality-of-service (QoS) requirements. However, its use in software design is currently
a tedious process of manual multiobjective optimisation. Software designers must build and verify probabilistic models for numerous alternative architectures and instantiations of the system
parameters. When successful, they end up with feasible but often suboptimal models. The
EvoChecker search-based software engineering approach
introduced in our paper employ multiobjective optimisation genetic algorithms to automate this process and considerably improve its outcome. We evaluate
six variants
of two software systems from the domains of
dynamic power management
foreign exchange trading
. These systems are characterised by different types of design parameters and QoS requirements, and their design spaces comprise between 2E+14 and 7.22E+86 relevant alternative designs. Our results
provide strong evidence that
significantly outperforms the current practice and yields actionable insights for software designers.
The main contributions of this work are:
Our paper has been accepted at the 30th Intl. Conference on Automated Software Engineering (ASE 2015) We have also made available the camera-ready version of our paper. | {"url":"https://www-users.york.ac.uk/~sg778/EvoChecker/","timestamp":"2024-11-06T15:20:45Z","content_type":"application/xhtml+xml","content_length":"51513","record_id":"<urn:uuid:1ce09216-f305-4c8d-af8e-f639d1a2458b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00795.warc.gz"} |
bogo-sort /boh`goh-sort'/ n.
(var. `stupid-sort') The archetypical perversely awful algorithm (as opposed to bubble sort, which is merely the generic bad algorithm). Bogo-sort is equivalent to repeatedly throwing a deck of cards
in the air, picking them up at random, and then testing whether they are in order. It serves as a sort of canonical example of awfulness. Looking at a program and seeing a dumb algorithm, one might
say "Oh, I see, this program uses bogo-sort." Esp. appropriate for algorithms with factorial or super-exponential running time in the average case and probabilistically infinite worst-case running
time. Compare bogus, brute force, lasherism.
A spectacular variant of bogo-sort has been proposed which has the interesting property that, if the Many Worlds interpretation of quantum mechanics is true, it can sort an arbitrarily large array in
linear time. (In the Many-Worlds model, the result of any quantum action is to split the universe-before into a sheaf of universes-after, one for each possible way the state vector can collapse; in
any one of the universes-after the result appears random.) The steps are: 1. Permute the array randomly using a quantum process, 2. If the array is not sorted, destroy the universe (checking that it
is sorted requires O(n) time). Implementation of step 2 is left as an exercise for the reader. | {"url":"http://hackersdictionary.com/html/entry/bogo-sort.html","timestamp":"2024-11-12T16:09:49Z","content_type":"text/html","content_length":"3099","record_id":"<urn:uuid:12994868-dff5-4671-a004-5c66580204bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00418.warc.gz"} |
Local outlier factor model for anomaly detection
Since R2022b
• Outlier detection (detecting anomalies in training data) — Detect anomalies in training data by using the lof function. The lof function creates a LocalOutlierFactor object and returns anomaly
indicators and scores (local outlier factor values) for the training data.
• Novelty detection (detecting anomalies in new data with uncontaminated training data) — Create a LocalOutlierFactor object by passing uncontaminated training data (data with no outliers) to lof,
and detect anomalies in new data by passing the object and the new data to the object function isanomaly. The isanomaly function returns anomaly indicators and scores for the new data.
Create a LocalOutlierFactor object by using the lof function.
X — Predictors
numeric matrix | table
This property is read-only.
Predictors used to train the local outlier factor model, specified as a numeric matrix or a table. Each row of X corresponds to one observation, and each column corresponds to one variable.
BucketSize — Maximum number of data points in each leaf node
positive integer | []
This property is read-only.
Maximum number of data points in each leaf node of the Kd-tree, specified as a positive integer.
This property is valid when SearchMethod is 'kdtree'. If SearchMethod is 'exhaustive', the BucketSize value is empty ([]).
CategoricalPredictors — Categorical predictor indices
vector of positive integers | []
This property is read-only.
Categorical predictor indices, specified as a vector of positive integers. CategoricalPredictors contains index values indicating that the corresponding predictors are categorical. The index values
are between 1 and p, where p is the number of predictors used to train the model. If none of the predictors are categorical, then this property is empty ([]).
ContaminationFraction — Fraction of anomalies in training data
numeric scalar in the range [0,1]
This property is read-only.
Fraction of anomalies in the training data, specified as a numeric scalar in the range [0,1].
• If the ContaminationFraction value is 0, then lof treats all training observations as normal observations, and sets the score threshold (ScoreThreshold property value) to the maximum anomaly
score value of the training data.
• If the ContaminationFraction value is in the range (0,1], then lof determines the threshold value (ScoreThreshold property value) so that the function detects the specified fraction of training
observations as anomalies.
Distance — Distance metric
character vector
This property is read-only.
Distance metric, specified as a character vector.
• If all the predictor variables are continuous (numeric) variables, then the Distance value can be one of these distance metrics.
Value Description
'euclidean' Euclidean distance
Euclidean distance using an algorithm that usually saves time when the number of elements in a data point exceeds 10. See Algorithms. "fasteuclidean" applies only to the
"fasteuclidean" "exhaustive" SearchMethod.
'mahalanobis' Mahalanobis distance — The distance uses the covariance matrix stored in the DistanceParameter property.
'minkowski' Minkowski distance — The distance uses the exponent value stored in the DistanceParameter property.
'chebychev' Chebychev distance (maximum coordinate difference)
'cityblock' City block distance
'correlation' One minus the sample correlation between observations (treated as sequences of values)
'cosine' One minus the cosine of the included angle between observations (treated as vectors)
'spearman' One minus the sample Spearman's rank correlation between observations (treated as sequences of values)
• If all the predictor variables are categorical variables, then the Distance value can be one of these distance metrics.
Value Description
'hamming' Hamming distance, which is the percentage of coordinates that differ
'jaccard' One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ
For more information on the various distance metrics, see Distance Metrics.
DistanceParameter — Distance metric parameter value
positive scalar | []
This property is read-only.
Distance metric parameter value for the Mahalanobis or Minkowski distance, specified as a positive scalar. The DistanceParameter value is empty ([]) for the other distances, indicating that the
specified distance metric formula has no parameters.
• If Distance is 'mahalanobis', then DistanceParameter is the covariance matrix in the Mahalanobis distance formula. The Cov name-value argument of lof sets this property.
• If Distance is 'minkowski', then DistanceParameter is the exponent in the Minkowski distance formula. The Exponent name-value argument of lof sets this property.
IncludeTies — Tie inclusion flag
false or 0 | true or 1
This property is read-only.
Tie inclusion flag indicating whether LocalOutlierFactor includes all the neighbors whose distance values are equal to the kth smallest distance, specified as logical 0 (false) or 1 (true). If
IncludeTies is true, LocalOutlierFactor includes all of these neighbors. Otherwise, LocalOutlierFactor includes exactly k neighbors.
NumNeighbors — Number of nearest neighbors
positive integer value
This property is read-only.
Number of nearest neighbors in X used to compute local outlier factor values, specified as a positive integer value.
PredictorNames — Predictor variable names
cell array of character vectors
This property is read-only.
Predictor variable names, specified as a cell array of character vectors. The order of the elements in PredictorNames corresponds to the order in which the predictor names appear in the training
ScoreThreshold — Threshold for anomaly score
nonnegative scalar
This property is read-only.
Threshold for the anomaly score used to identify anomalies in the training data, specified as a nonnegative scalar.
The software identifies observations with anomaly scores above the threshold as anomalies.
• The lof function determines the threshold value to detect the specified fraction (ContaminationFraction property) of training observations as anomalies.
• The isanomaly object function uses the ScoreThreshold property value as the default value of the ScoreThreshold name-value argument.
SearchMethod — Nearest neighbor search method
'kdtree' | 'exhaustive'
This property is read-only.
Nearest neighbor search method, specified as 'kdtree' or 'exhaustive'.
• 'kdtree' — This method uses a Kd-tree algorithm to find nearest neighbors. This option is valid when the distance metric (Distance) is one of the following:
□ 'euclidean' — Euclidean distance
□ 'cityblock' — City block distance
□ 'minkowski' — Minkowski distance
□ 'chebychev' — Chebychev distance
• 'exhaustive' — This method uses the exhaustive search algorithm to find nearest neighbors.
□ When you compute local outlier factor values for X using the lof function, the function finds nearest neighbors by computing the distance values from all points in X to each point in X.
□ When you compute local outlier factor values for new data Xnew using the isanomaly function, the function finds nearest neighbors by computing the distance values from all points in X to each
point in Xnew.
Object Functions
isanomaly Find anomalies in data using local outlier factor
Detect Outliers
Detect outliers (anomalies in training data) by using the lof function.
Load the sample data set NYCHousing2015.
The data set includes 10 variables with information on the sales of properties in New York City in 2015. Display a summary of the data set.
NYCHousing2015: 91446x10 table
BOROUGH: double
NEIGHBORHOOD: cell array of character vectors
BUILDINGCLASSCATEGORY: cell array of character vectors
RESIDENTIALUNITS: double
COMMERCIALUNITS: double
LANDSQUAREFEET: double
GROSSSQUAREFEET: double
YEARBUILT: double
SALEPRICE: double
SALEDATE: datetime
Statistics for applicable variables:
NumMissing Min Median Max Mean Std
BOROUGH 0 1 3 5 2.8431 1.3343
NEIGHBORHOOD 0
BUILDINGCLASSCATEGORY 0
RESIDENTIALUNITS 0 0 1 8759 2.1789 32.2738
COMMERCIALUNITS 0 0 0 612 0.2201 3.2991
LANDSQUAREFEET 0 0 1700 29305534 2.8752e+03 1.0118e+05
GROSSSQUAREFEET 0 0 1056 8942176 4.6598e+03 4.3098e+04
YEARBUILT 0 0 1939 2016 1.7951e+03 526.9998
SALEPRICE 0 0 333333 4.1111e+09 1.2364e+06 2.0130e+07
SALEDATE 0 01-Jan-2015 09-Jul-2015 31-Dec-2015 07-Jul-2015 2470:47:17
Remove nonnumeric variables from NYCHousing2015. The data type of the BOROUGH variable is double, but it is a categorical variable indicating the borough in which the property is located. Remove the
BOROUGH variable as well.
NYCHousing2015 = NYCHousing2015(:,vartype("numeric"));
NYCHousing2015.BOROUGH = [];
Train a local outlier factor model for NYCHousing2015. Specify the fraction of anomalies in the training observations as 0.01.
[Mdl,tf,scores] = lof(NYCHousing2015,ContaminationFraction=0.01);
Mdl is a LocalOutlierFactor object. lof also returns the anomaly indicators (tf) and anomaly scores (scores) for the training data NYCHousing2015.
Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction.
h = histogram(scores,NumBins=50);
h.Parent.YScale = 'log';
xline(Mdl.ScoreThreshold,"r-",["Threshold" Mdl.ScoreThreshold])
If you want to identify anomalies with a different contamination fraction (for example, 0.05), you can train a new local outlier factor model.
[newMdl,newtf,scores] = lof(NYCHousing2015,ContaminationFraction=0.05);
Note that changing the contamination fraction changes the anomaly indicators only, and does not affect the anomaly scores. Therefore, if you do not want to compute the anomaly scores again by using
lof, you can obtain a new anomaly indicator with the existing score values.
Change the fraction of anomalies in the training data to 0.05.
newContaminationFraction = 0.05;
Find a new score threshold by using the quantile function.
newScoreThreshold = quantile(scores,1-newContaminationFraction)
newScoreThreshold =
Obtain a new anomaly indicator.
newtf = scores > newScoreThreshold;
Detect Novelties
Create a LocalOutlierFactor object for uncontaminated training observations by using the lof function. Then detect novelties (anomalies in new data) by passing the object and the new data to the
object function isanomaly.
Load the 1994 census data stored in census1994.mat. The data set consists of demographic data from the US Census Bureau to predict whether an individual makes over $50,000 per year.
census1994 contains the training data set adultdata and the test data set adulttest. The predictor data must be either all continuous or all categorical to train a LocalOutlierFactor object. Remove
nonnumeric variables from adultdata and adulttest.
adultdata = adultdata(:,vartype("numeric"));
adulttest = adulttest(:,vartype("numeric"));
Train a local outlier factor model for adultdata. Assume that adultdata does not contain outliers.
[Mdl,tf,s] = lof(adultdata);
Mdl is a LocalOutlierFactor object. lof also returns the anomaly indicators tf and anomaly scores s for the training data adultdata. If you do not specify the ContaminationFraction name-value
argument as a value greater than 0, then lof treats all training observations as normal observations, meaning all the values in tf are logical 0 (false). The function sets the score threshold to the
maximum score value. Display the threshold value.
Find anomalies in adulttest by using the trained local outlier factor model.
[tf_test,s_test] = isanomaly(Mdl,adulttest);
The isanomaly function returns the anomaly indicators tf_test and scores s_test for adulttest. By default, isanomaly identifies observations with scores above the threshold (Mdl.ScoreThreshold) as
Create histograms for the anomaly scores s and s_test. Create a vertical line at the threshold of the anomaly scores.
h1 = histogram(s,NumBins=50,Normalization="probability");
hold on
h2 = histogram(s_test,h1.BinEdges,Normalization="probability");
xline(Mdl.ScoreThreshold,"r-",join(["Threshold" Mdl.ScoreThreshold]))
h1.Parent.YScale = 'log';
h2.Parent.YScale = 'log';
legend("Training Data","Test Data",Location="north")
hold off
Display the observation index of the anomalies in the test data.
ans =
0x1 empty double column vector
The anomaly score distribution of the test data is similar to that of the training data, so isanomaly does not detect any anomalies in the test data with the default threshold value. You can specify
a different threshold value by using the ScoreThreshold name-value argument. For an example, see Specify Anomaly Score Threshold.
More About
Local Outlier Factor
Distance Metrics
• You can use interpretability features, such as lime, shapley, partialDependence, and plotPartialDependence, to interpret how predictors contribute to anomaly scores. Define a custom function that
returns anomaly scores, and then pass the custom function to the interpretability functions. For an example, see Specify Model Using Function Handle.
[1] Breunig, Markus M., et al. “LOF: Identifying Density-Based Local Outliers.” Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data, 2000, pp. 93–104.
Version History
Introduced in R2022b
R2023b: "fasteuclidean" Distance Support
The lof function gains support for the "fasteuclidean" Distance algorithm. This algorithm usually computes distances faster than the default "euclidean" algorithm when the number of variables in a
data point exceeds 10. The algorithm, described in Algorithms, uses extra memory to store an intermediate Gram matrix. Set the size of this memory allocation using the CacheSize argument. | {"url":"https://ch.mathworks.com/help/stats/localoutlierfactor.html","timestamp":"2024-11-04T01:53:47Z","content_type":"text/html","content_length":"153487","record_id":"<urn:uuid:43be2754-1918-4940-8f5e-b88bc622a3e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00118.warc.gz"} |
Entropy and Gravity
Faculty of Technology, Art and Design, Oslo and Akershus University College of Applied Sciences, P. O. Box 4, St. Olavs Plass, N-0130 Oslo, Norway
Submission received: 26 October 2012 / Revised: 22 November 2012 / Accepted: 23 November 2012 / Published: 4 December 2012
The effect of gravity upon changes of the entropy of a gravity-dominated system is discussed. In a universe dominated by vacuum energy, gravity is repulsive, and there is accelerated expansion.
Furthermore, inhomogeneities are inflated and the universe approaches a state of thermal equilibrium. The difference between the evolution of the cosmic entropy in a co-moving volume in an
inflationary era with repulsive gravity and a matter-dominated era with attractive gravity is discussed. The significance of conversion of gravitational energy to thermal energy in a process with
gravitational clumping, in order that the entropy of the universe shall increase, is made clear. Entropy of black holes and cosmic horizons are considered. The contribution to the gravitational
entropy according to the Weyl curvature hypothesis is discussed. The entropy history of the Universe is reviewed.
1. Introduction
The arrow of time arises from the universe being far from equilibrium in a state of low entropy. The Second Law of Thermodynamics requires that the entropy of the universe does not decrease. Hence
the universe must initially have been in a state of very low entropy. Callender [
] has discussed this “Past Hypothesis”, writing: “The Boltzmann entropy of the entire universe was very low (compared to now) roughly 15 billion years ago. In particular, the entropy of this state
was low enough to make subsequent entropy increase likely for many billions of years.” Then he notes: “When we look to cosmology for information about the actual Past State, we find early
cosmological states that appear to be states of very high entropy, not very low entropy. Cosmology tells us that the early universe is an almost homogeneous isotropic state of approximately uniform
a very high entropy state.”
Observations favor that the spatial geometry of the universe is Euclidean. This means that with the topology R^3 the universe has infinitely large spatial extension. Then the evolution of “the
entropy within a surface co-moving with the Hubble flow” is representative of the evolution of “the entropy of the universe”.
A homogeneous universe with a perfect fluid expands adiabatically with constant entropy. Measurements of temperature variations in the cosmic microwave background has shown that 400,000 years after
the Big Bang the universe was in a state very close to thermal equilibrium. Later attractive gravity made the original small mass concentrations larger, and created hot regions like stars. The
temperature differences became much larger, and one might wonder if the thermodynamic entropy of the universe had become smaller in conflict with the Second Law of Thermodynamics. Maybe gravity saves
the Past Hypothesis. Callender [
] argues that this is far from obvious. In the present review we shall consider different aspects of this question.
2. Entropy Change during Gravitational Contraction
Let us first calculate the change of entropy of a gas in which gravity may be neglected, evolving away from equilibrium due to some external agent acting on it. The change of thermal entropy of a gas
molecules originally in thermal equilibrium at a temperature
, then separated into two parts with temperatures
$T 1$
$T 2$
with constant thermal energy so that
$T = ( 1 / 2 ) ( T 1 + T 2 )$
, is:
$Δ S = 3 N k 4 ln [ 1 − ( 1 2 Δ T T ) 2 ] ≈ − 3 N k 4 ( 1 2 Δ T T ) 2$
$k = 1 . 38 ⋅ 1 0 − 23 J / K$
is Boltzmann’s constant and
$Δ T = T 1 − T 2$
. This decrease of entropy due to evolution away from thermal equilibrium is a second order effect in
$Δ T / T$
Next we shall make a Newtonian calculation of the effect of gravity upon the change of entropy of an expanding or contracting distribution of ideal gas, giving a further development of a simple model
introduced by Wallace [
]. The thermal entropy of an ideal gas with temperature
and volume
, consisting of
molecules, is:
$S = N k ( 3 2 ln T + ln V ) + C$
where C is a constant. The internal energy of the gas is equal to the kinetic energy
$E K$
of the gas molecules. In the case of monoatomic molecules
$E K = ( 3 / 2 ) N k T$
. We consider a spherically symmetric distribution of
molecules each with mass
inside a radius
. The volume inside
$V = ( 4 π / 3 ) R 3$
, and mass inside
$M = N μ$
. Hence, the thermal entropy of the gas is:
$S = 3 2 N k ln ( R 2 E K ) + C ¯$
$C ¯$
is another constant. The surface with radius
is comoving with the matter. Hence, when the cloud expands or contracts so that the value of
changes, the mass
inside this surface is constant. The change of the thermal entropy when the radius
changes, is:
$d S = 3 2 N k ( d E K E K + 2 d R R )$
The total mechanical energy of the gas molecules is equal to the sum of the kinetic and potential energy in the field of gravity of the cloud,
$E = E K + E P$
. Since the total energy is constant, the kinetic energy changes as
$d E K = − d E P$
. In the case of contraction this is
a conversion of gravitational energy to thermal energy of the gas
. Assuming that the gas is homogeneous the potential energy of the gas inside the radius
where G is Newton’s constant of gravity. Hence the change of kinetic energy of the gas molecules is
$d E K = ( E P / R ) d R$
. The rate of change of the thermal entropy of the gas is:
$( d S d t ) M = 3 2 N K ( 2 + E P M E K ) 1 R d R d t$
which may be written explicitly as a function of
$( d S d t ) M = 3 2 N k 2 R − R 0 R ( R − R 0 ) d R d t , R 0 = − 3 G M 2 5 E$
The thermal entropy is:
$S M = 3 2 N k ln [ E R ( R − R 0 ) ] + C$
$E > 0$
$R > R 0$
$E < 0$
$R < R 0$
. The kinetic energy is:
$E K > 0$
we have the conditions: if
$E > 0$
$R > R 0$
, and there is expansion
motion upwards along the curve (b) in
Figure 1
, and the entropy increases. If
$E < 0$
$R < R 0$
, and there is contraction say from
$R = R 0$
along the right hand part of the curve (a). Again the entropy increases until it reaches a maximum at
$R = R 0 / 2$
. When the cloud contracts further its entropy decreases.
In terms of the energies the entropy has a maximum if $E K = − E P M / 2$. According to the virial theorem the gas is at dynamical equilibrium when this condition is fulfilled. One can show that if
the kinetic energy of the gas is less than $− E P M / 2$, then the cloud will collapse. In this case $R < R 0 / 2 , E < 0$ so that $2 E K + E P M = ( 2 − R 0 / R ) E > 0$, and during the collapse $d
R / d t < 0$ giving $d S / d t < 0$. Hence the thermal entropy of the gas decreases due to the increasing temperature gradient in the gas although there is a conversion of gravitational energy to
thermal energy.
The change of thermal entropy of matter that collapses to a star can be estimated in the following way [
]. About 300,000 years after the Big Bang there was a nearly homogeneous cosmic plasma with density
$ρ 1 ∼ 10 14$
and temperature
$T 1 ∼ 10 4$
K. There are about
$N ∼ 10 57$
baryons in the Sun. The average density and temperature of the Sun are, respectively
$ρ 2 ∼ 10 30$
$T 2 ∼ 10 7$
. The entropy change
$Δ S M$
when the matter of the Sun changed from its state as a nearly homogeneous cosmic plasma 300,000 years after the Big Bang to a star, can be calculated from Equation (2). Using that
$M = ρ 1 V 1 = ρ 2 V 2$
leads to:
$Δ S M = N 2 k ln ( T 2 3 ρ 1 2 T 1 3 ρ 2 2 ) ∼ − 10 35 J K$
Figure 1. Thermal entropy (modulus an undetermined constant and in units of (3/2)Nk) as a function of R for a gas cloud contracting or expanding under the action of its own gravity. (a) Contraction.
Here R < R[0]. Then the entropy first increases towards a maximum at R = R[0]/2, and then decreases. (b) Expansion. Here R > R[0]. The entropy of the gas increases.
This is roughly how much the thermal entropy of the matter decreases when matter that existed in the cosmic plasma 300,000 years after the Big Bang later is formed to a star due to gravity. However,
in order to calculate the entropy change of the universe one also has to find the entropy change of the environment during the collapse. Due to the conversion of gravitational energy to thermal
energy the temperature of the gas increases, leading to an increased rate of electromagnetic radiation from the gas. Following Wallace [
] I will make a rough estimate of the accumulated entropy of the electromagnetic radiation emitted by the gas. The entropy of electromagnetic radiation with energy
and temperature
Assuming that the collapsing mass has lost a tenth of its mass as radiation, the energy of the radiation is
$E = 10 46 J$
. Wallace [
] has calculated the entropy of this radiation for the case that it has the temperature
$T = 100 K$
. The result is
$Δ S R = 10 45 J / K$
. This is a factor
$∼ 10 10$
larger than the decrease in entropy of the star’s matter. Hence the entropy of the universe has increased during the formation of the star due to gravitational collapse.
If $E K > − E P M / 2$ so that $2 E K + E P M > 0$ , then the kinetic energy of the molecules is sufficiently great that the cloud will expand, $d R / d t > 0$, and $d S / d t > 0$. In spite of the
fact that in this case there is a conversion of thermal energy to gravitational energy, the thermal entropy of the gas will increase. This is due to the approach towards thermal equilibrium in this
We shall now see how Lorentz invariant vacuum energy (LIVE) which may be represented by a cosmological constant,
, in the gravitational field equations, modifies these results. In this case the Poisson equation for the gravitational potential
in a distribution of matter with density
takes the form [
$∇ 2 ϕ = 4 π G ρ − Λ c 2$
Assuming a spherical matter distribution with constant density out to a boundary with radius
, the potential inside the matter is:
$ϕ = − 2 π 3 G ρ ( 3 R 2 − r 2 ) − Λ 6 c 2 r 2$
with zero potential at:
$r 0 = 3 1 − 2 R 3 R S R Λ 2 R , R S = 2 G M c 2 , R Λ = 3 Λ$
which is outside the comoving volume with mass
. The potential energy of the mass distribution is:
$E P = E P M + E P Λ = − 3 5 G M 2 R − Λ M c 2 10 R 2$
The rate of change of the entropy of the gas is:
$( d S d t ) G = ( d S d t ) M + ( d S d t ) Λ ,$
where the contribution of the vacuum energy to the rate of change of the entropy is:
$( d S d t ) Λ = − 3 N k E P Λ E K 1 R d R d t .$
$− E P Λ / E K > 0$
this shows that the contribution of LIVE to the rate of change of the entropy is to increase the entropy during expansion and decrease the entropy during contraction. Amarzguioui and Grøn [
] have presented a relativistic calculation of the effect of gravitational contraction upon the cosmic entropy to first order in
$Δ T / T$
. They perturbed a homogeneous FRW universe model and calculated the increase of temperature,
$Δ T$
, due to conversion of gravitational energy to thermal energy. In this way they showed that during this process there would be a corresponding increase of thermal entropy:
$Δ S = 3 N k 4 ln [ 1 + Δ T T ] ≈ 3 N k 4 Δ T T$
Hence, due to the release of gravitational energy there is an increase of thermal entropy to first order in
$Δ T / T$
. Comparing with eq.(1) it is clear that the evolution of a gas in which gravity may be neglected, and the evolution of a self gravitating gas, is very different as illustrated in
Figure 2
Figure 2. Evolution of a system (a) in which gravity may be neglected, and (b) in a self gravitating system where the box is much larger than the Jean’s length of the gas it contains.
The importance of gravity in connection with the Second Law has been discussed by several authors. One of the first of those discussions appeared in the remarkable book
The Physics of Time Asymmetry
by P. C. W. Davies [
]. He clearly expected the result of Amarzuioui and Grøn [
] because already in 1974 he wrote: “gravitational condensation and collapse is itself a means of bringing about an increase of entropy.” And: “At first sight it appears paradoxical that an element
of the cosmological fluid can start out in a quasi-equilibrium condition, and yet still increase in entropy at later epoch.” He further notes that this paradox is resolved because a self-gravitating
system has no equilibrium configuration. Hence the origin of all thermodynamic irreversibility in the real universe depends ultimately on gravitation. In this connection Leubner [
] writes: “In contrast to thermodynamic systems driven to a uniform distribution, the components of gravitating systems tend to clump, thus implying a gravitational arrow of time, which points in the
direction of growing inhomogeneity”, and further: “Increasing inhomogeneity due to gravitational clumping reflects increasing gravitational entropy in a time evolving universe”.
3. Entropy of Black Holes and Cosmic Horizons
In 1974 Bekenstein [
] conjectured that the black hole entropy is proportional to the area
$A B = 4 π R S 2$
of its event horizon divided by the square of the Planck length,
$l P = G ℏ / c 3 = 1.6 ⋅ 10 − 35 m$
, and in the case of a Schwarzschild black hole Hawking [
] deduced the proportionality constant:
$S B = k A B 4 l P 2 = π k ( R S l P ) 2 = π k c 4 4 l P 2 1 κ 2$
$R S = 2 G M / c 2$
is the Schwarzschild radius of the black hole, and
$κ = c 2 / 2 R S$
is the surface gravity at the horizon. This formula shows that the Bekenstein-Hawking entropy of a black hole is equal to the Boltzmann constant times one fourth of the spatial area of its event
horizon in Planck units. A black hole with the mass of the Sun, for example has a Schwarzschild radius
$R S ⊙ = 2 , 95 ⋅ 10 3 m$
and hence a Bekenstein-Hawking entropy
$S ⊙ = 1.5 ⋅ 10 54 J / K$
Damour [
] has given a thorough discussion of the entropy of black holes, and I refer to this article for more references on this topic. Bekenstein’s black-hole entropy was viewed by him as a measure of the
information about the interior of a black hole which is inaccessible to an external observer. Furthermore Bekenstein conjectured the validity of a
generalized version of the second law of thermodynamics
, stating that the sum of the black-hole entropy (19) and of the ordinary entropy in the exterior of the black hole never decreases. However, even after Hawking’s deduction the challenge of
$S B H$
, in the same way as Boltzmann’s interpretation of thermal entropy, as the logarithm of the number of quantum micro-states of a macroscopic black hole, remained. The most striking “explanation” of
black-hole entropy was obtained within the framework of string theory and was reviewed by Damour [
] with many further references. A Boltzmann interpretation of the entropy of a black hole, related to general expected properties of quantum gravity, has recently been deduced in a model independent
way by Saida [
For a star with 3 Sun masses Equation (19) gives
$S B H ∼ 10 55 J / K$
. Wallace [
] calculated that the entropy of a neutron star with this mass is
$S M ∼ 10 29 J / K$
. So when a star collapses to a black hole in a hypernova explosion producing a gamma ray burst, the entropy increases by more than twenty five orders of magnitude.
Vaas [
] noted that “The strongest ‘concentrations’ of gravity, black holes, are also the biggest accumulations of entropy. Physically speaking, gravitational collapse leads to the greatest possible amount
of disorder. The entropy of a single black hole with the mass of a million suns (such as the one at the galactic centre, for example) is a hundred times higher than the entropy of all ordinary
particles in the entire observable universe.”
We have seen that the entropy of black holes is extremely large. This may be an expression of the fact that classically we have lost all information of the matter that collapsed to a black hole since
a black holes have at most three properties: mass, charge and angular momentum.
In spite of the quantum mechanical string theory interpretation of the entropy of black holes, one may wonder whether the large entropy of a black hole is a measure of the inhomogeneity of a
gravitational field on a classical level. Then the entropy of a black hole might be related to the Weyl curvature of its gravitational field.
In 1977 Gibbons and Hawking [
] suggested that one should associate an entropy with an event horizon proportional to its area, interpreted as an expression of the lack of information of an observer about the region which he
cannot see. Similarly to the expression of the entropy associated with the event horizon of a black hole, the entropy associated with a cosmic event horizon with proper area
$A C$
$S C ( t ) = k A C ( t ) 4 l P 2$
This will be called the cosmic event horizon entropy, CEHE. In the case of a flat universe model the proper area of the event horizon is
$A C ( t ) = 2 π a 2 ( t ) χ C 2 ( t )$
. Here
$a ( t )$
is the scale factor is normalized so that its present value is 1, meaning that
represents the distances of objects participating in the Hubble flow at an arbitrary time relative to their present distances. Furthermore
is the cosmic radial standard coordinate comoving with the Hubble flow, and:
$χ C ( t ) = ∫ t ∞ d t ′ a ( t ′ )$
its value at the event horizon, which exists only if this integral is finite. The definition (21) implies that the value of
$χ C$
decreases with time for all universe models. For universe models with an event horizon the CEHE is:
$S C = π k 2 l P 2 a 2 χ C 2$
This shall now be applied to some important and simple classes of universe models in order to find the time evolution of CEHE in these models. A flat universe model dominated by a perfect fluid with
equation of state
$p = w ρ$
with constant value of
has scale factor:
$a ( t ) = ( t t 0 ) 2 3 ( 1 + w )$
$t 0$
is the point of time for which
$a ( t 0 ) = 1$
, which may be taken to be the present time. Inserting the expression (24) into the integral (21) shows that there exists an event horizon only if
$− 1 < w < − 1 / 3$
. This is just the condition that the universe model has accelerated expansion. In this case the coordinate distance to the event horizon is:
$χ C = − 3 ( 1 + w ) 1 + 3 w t 0 ( t t 0 ) 1 + 3 w 3 ( 1 + w )$
The CEHE is:
$S C = 9 π k 2 l P 2 ( 1 + w 1 + 3 w ) t 2$
Hence the CEHE increases with time as $t 2$ for all flat, accelerating universe models with an event horizon dominated by a single cosmic fluid.
A flat LIVE-dominated universe model has scale factor:
$a ( t ) = e Λ / 3 ( t − t 0 )$
In this universe model the coordinate distance to the event horizon is:
$χ C = 3 / Λ e − Λ / 3 ( t − t 0 )$
Hence, the CEHE is:
which is constant.
Davis, Davies and Lineweaver [
] have investigated the change of CEHE with time in a spacetime with LIVE and radiation. Considering a universe model with flat 3-space the scale factor is:
$a ( t ) = A sinh 1 / 2 ( t / t Λ ) , A = Ω r 0 / Ω Λ 0 , t Λ = ( 1 / 2 ) 3 / Λ$
$Ω r 0$
$Ω Λ 0$
are the present values of the density parameters of the radiation and LIVE, respectively. Furthermore
$Λ = κ ρ Λ$
where the density of LIVE,
$ρ Λ$
, is constant. With this scale factor the expression (21) for
$χ C$
is an elliptical integral. However Davis
et al
. [
] have investigated the evolution of CEHE in such universe models with different values of the spatial curvature and of
$Ω r 0$
$Ω Λ 0$
by performing numerical integrations of Friedmann’s equation for such a universe model. They found that the CEHE increases with time in a flat FRW-universe model with LIVE and radiation.
However, the entropy of the radiation inside the horizon decreases with time. The reason is that the entropy is proportional to the number of photons inside the horizon, and inside a surface that is
comoving with the Hubble flow the number of photons and thus the entropy of the radiation is constant. This is in agreement with the fact that there is no heat in a homogeneous universe since there
are no large scale temperature differences. Thus the universe expands adiabatically. But a comoving surface has a constant value of $χ$, and from the definition (21) follows that the radius of the
event horizon, $χ C$, decreases with time. There is a leakage of photons out of the event horizon.
For simplicity the mathematical expression of this will here be deduced only for a flat universe model. Then the volume inside the event horizon is
$V C = ( 4 π / 3 ) a 3 χ C 3$
. As noted by Davis, Davies and Lineweaver [
] the entropy of the radiation inside the event horizon is:
$S r = ( 4 / 3 ) σ 1 / 4 ρ r 3 / 4 V C = ( 16 π / 9 ) σ 1 / 4 ρ r 3 / 4 a 3 χ C 3$
$σ = π 2 k 4 / 15 c 3 ℏ 3$
is the radiation constant. Since the radiation density fulfills
$ρ r a 4 = ρ r 0$
, it follows that:
$S r = ( 16 π / 9 ) σ 1 / 4 ρ r 0 3 / 4 χ C 3$
which decreases with time. However, Davis, Davies and Lineweaver [
] found that in all cases the sum of the CEHE and the radiation entropy increases with time.
4. The Weyl Curvature Hypothesis
A universe which obeys the Second Law of Thermodynamics must come from an initial state with very low entropy. In this context Wald [
] has written that the claim that the entropy of the very early universe must have been extremely low appears to blatantly contradict the “standard model” of cosmology. From observations of the very
small temperature fluctuations of the cosmic microwave radiation we see that the universe was very close to thermal equilibrium 400,000 years after the Big Bang, which seems to correspond to a state
of nearly maximum entropy.
However that would be the case only in the absence of gravity. The situation changes dramatically when gravity is present. Then, for a sufficiently large system, the entropy can always be increased
by clumping the system and using the binding energy that is thereby released to heat up the system. The state of maximum entropy will not correspond to a homogeneous distribution of matter, but
rather to that of a black hole.
In an effort to incorporate the tendency of (attractive) gravity to produce inhomogeneities–and in the most extreme cases black holes–into a generalized Second Law of Thermodynamics, Penrose [
] made some suggestions about how to define a quantity representing the entropy of a gravitational field. Such a quantity should vanish in the case of a homogeneous field and obtain a maximal value
given by the Bekenstein-Hawking entropy (19) for the field of a black hole. In this connection Penrose formulated what is called the
Weyl curvature hypothesis
, saying that the Weyl curvature should be small near the initial singularity of the universe. Wainwright and Anderson [
] interpreted this hypothesis in terms of the ratio of the Weyl and Ricci curvature invariants:
$P W A 2 = W α β γ δ W α β γ δ R μ ν R μ ν$
According to their formulation of the Weyl curvature hypothesis
$P W A$
should vanish at the initial singularity of the universe. The physical content of the hypothesis is that the initial state of the universe is homogeneous and isotropic. Hence, the hypothesis need not
be interpreted as a hypothesis about gravitational entropy. Neither need it refer to an unphysical initial singularity, but should instead be concerned with an initial state, say at the Planck time.
A related attempt to implement Penrose’s suggestions was made in a paper by Mena and Tod [
Wald [
] asked: “What caused the very early universe to be in a very low entropy state?” and noted that there are basically two types of answers: (i) The initial state of the universe was random, but
dynamical evolution caused at least the observable part of the universe to have very low entropy initially. (ii) The universe came into existence in a very special initial state. He then argued that
the latter possibility is the more plausible one, and concluded that it will be more fruitful to seek an answer of the second type than to attempt to pursue dynamical explanations. In this connection
Lebowitz [
] wrote: ”It would certainly be nice to have a theory which would explain the “cosmological initial state”. Grøn and Hervik [
] tried to follow the path that Wald considered the most fruitful one, by investigating whether a quantum calculation of initial conditions for the universe based upon the Wheeler-DeWitt equation
supports Penrose’s Weyl curvature hypothesis, according to which the Ricci part of the curvature dominates over the Weyl part at the initial singularity of the universe.
The vanishing of the Weyl curvature tensor at the initial singularity is a very special initial condition. Due to the evolution during the inflationary era, however, this condition may be relaxed. A
much larger variety of initial conditions are consistent with the observed approximate homogeneity of the universe, as seen in the cosmic background radiation, in inflationary universe models than in
models without inflation. A relevant question in this connection is: if a larger class of initial conditions leads to the same observable universe after a period of inflation, do inflationary
processes lead to the production of entropy? After all, if entropy counts the number of microstates underlying a macrostate, and many more microstates lead to the same macrostate with inflation, then
that should be the case. This topic will be further discussed in section 7 below.
It was shown by Grøn and Hervik [
] that according to the Einstein’s classical field equations
$P W A 2$
diverges at the initial singularity both in the case of the homogeneous, but anisotropic Bianchi type I universe models and the isotropic, but inhomogeneous Tolman-Lemaitre Bondi (LTB) universe
models. This means that there are large anisotropies and inhomogeneities near the initial singularities in these universe models. Hence the classical behavior of
$P W A 2$
is not in agreement with the Weyl curvature hypothesis.
$P W A$
is not a suitable quantity to represent the entropy of a gravitational field. The reason for this failure may be that
$P W A$
is a local quantity. In the case of the LTB universe models Grøn and Hervik showed that:
$P W A = 2 3 ( ρ ¯ − ρ ρ )$
$ρ ¯$
is the average density and
the actual density. Near the initial singularity
$P W A$
will be rapidly fluctuating in space between negative and positive values.
If the entropy represents a large number of gravitational microstates corresponding to a certain gravitational field, it may more properly be represented by a non-local quantity proportional to
$P W A$
. Such a quantity may be finite at the initial singularity even if
$P W A$
diverges. Grøn and Hervik therefore considered the entity:
$S G 1 = k S P W A V , V = ( g 11 g 22 g 33 ) 1 / 2$
$k S$
is a constant which is determined below, and
is the invariant volume corresponding to a unit coordinate volume in coordinates co-moving with the cosmic fluid. The behavior of this quantity in LTB universe models with cold matter and LIVE
represented by a cosmological constant
in Einstein’s field equations, is as follows [
• Expanding universe with large $Λ$: In the initial epoch the dust dominates and $S G 1$ is increasing linearly with $t$. The universe is becoming more and more inhomogeneous. At late times $Λ$
dominates, and $S G 1$ stops growing, evolving asymptotically towards a constant value. The universe inflates.
• Expanding universe with small $Λ$: Again the dust dominates initially. In this case $S G 1$ increases forever, but with a decreasing rate and again approaches a constant value asymptotically.
• Expanding universe with vanishing
: The quantity
$S G 1$
again increases forever, this time approaching asymptotically a function
$f ( t ) = c + b t p$
are constants and
$p = 3$
$F 2 > 1$
$p = 1$
$F 2 = 1$
. (The function
$F ( r )$
is defined in refs. [
] and [
• Recollapsing universe: Due to the dust term the final singularity will be more inhomogeneous than the initial singularity.
This shows that in the LTB universe the quantity $S G 1$ behaves in accordance with the Weyl curvature hypothesis. It should be noted that the Schwarzschild spacetime is a special case of the general
LTB models. It has $m ( t ) = constant$ and a vanishing Ricci tensor. Hence, the quantity $S G 1$ diverges in the Schwarzschild spacetime. This is in some sense the maximal possible value of $S G 1$
, the Weyl tensor is as large as possible, and the Ricci tensor is as small as possible. Thus if one wants to associate entropy with a gravitational field, then at the classical level it seems that
the Schwarzschild spacetime has the largest possible value of such a gravitational entropy.
The unphysical divergence of $S G 1$ is probably not a physical reality, but due to the lack of a quantum gravity theory. Even though the classical vacuum has vanishing Ricci tensor, a quantum field
will fluctuate and cause the expectation value of the square of the Ricci tensor to be non-zero. Hence, there will probably be an upper bound on how large $S G 1$ can be, even in a vacuum.
As to the Bianchi type I universe models Grøn and Hervik [
] showed that magnitude of
$S G 1$
at the initial singularity is proportional to the inverse of the dust density. Furthermore in universe models with LIVE a large value of Λ means that
$S G$
has a small value and a small value of Λ means that
$S G 1$
has a large value. For a more general class of universe models dominated by a fluid with equation of state
$p = w ρ$
it was shown that
$S G 1$
increases as long as
$− 1 / 3 < w < 1 / 3$
5. The Weyl Curvature Hypothesis and Black Hole Entropy
Rudjord, Grøn and Hervik [
] have investigated how far the black hole entropy can be accounted for by an entropy,
$S G 2$
, due to the inhomogeneity of a gravitational field as suggested in the Weyl curvature hypothesis. This work was followed up by Romero, Thomas and Pérez [
Considering the spacetime of a black hole we define a gravitational entropy current vector by the expression:
is a quantity proportional to the Weyl curvature invariant. However, it cannot be given by eq.(32) since the Ricci curvature invariant vanishes in the Schwarzschild spacetime. So Rudjord
et al
. [
] replaced the Wainwright, Anderson expression (32) by:
$P 2 = W α β γ δ W α β γ δ R α β γ δ R α β γ δ$
where the denominator is the Kretschmann curvature scalar. They proved that
$P 2 < 1$
in all spacetimes with vanishing energy flux, which encompasses the spacetime outside the most general Kerr-Newman black hole and the isotropic and homogeneous Friedman universe models.
According to eq. (19) the entropy of a black hole is proportional to the area of the black hole horizon. Hence, the entropy can be described as a surface integral over the horizon, σ:
$S G 2 = k S ∫ σ Ψ ⋅ d σ , k S = k 4 l P 2 = k c 3 4 G ℏ$
$k S$
has been determined by demanding that the condition
$S G 2 = S B H$
as given in eq. (19) shall be valid for a Schwarzschild black hole. This means that according to the present theory all of the entropy of a Schwarzschild black hole is due to the inhomogeneity of the
gravitational field. Writing eq. (37) as a volume integral by means of Gauss’ divergence theorem, the entropy density is:
We now consider static spherically symmetric spaces where the space time line element takes the form:
$d s 2 = − e ν ( r ) c 2 d t 2 + e λ ( r ) d r 2 + r 2 d Ω 2 , d Ω 2 = d θ 2 + sin 2 θ d ϕ 2$
in curvature coordinates defined by the condition that the invariant area of a spherical surface with radius
$4 π r 2$
. In the Schwarzschild spacetime:
$e ν ( r ) = e λ ( r ) = 1 − R S r$
$R S = 2 G M / c 2$
is the Schwarzschild radius, and:
$R α β γ δ R α β γ δ = W α β γ δ W α β γ δ = 12 ( R S r 3 ) 2$
$P 2 = 1$
. Romero, Thomas and Pérez [
] showed that
$P 2 = 1$
also for the Kerr spacetime. This is of course a rather obvious result, since the Weyl curvature scalar is equal to the Kretschmann curvature scalar in empty space in general. Hence,
$P 2 = 1$
in every empty region of an arbitrary spacetime. However in light of both the expressions (32) and (36) something more can be said about the Weyl entropy of a black hole. Considering a black hole as
a limit in which space is empty, and the density profile has diverged to a Dirac delta function, the numerator of eq. (32) diverges, while at the same time, the denominator tends to zero. This shows
the very strong way that black holes can be considered as the most inhomogeneous possible spacetimes. In this sense one may expect that there is maximal gravitational entropy in the Schwarzschild and
Kerr spacetimes, and they can thus be considered to be the most inhomogeneous possible spacetimes.
The covariant expression for the entropy density is:
$s = k S | h | ∂ ∂ x μ ( | h | Ψ μ ) , h i j = g i j − g i 0 g j 0 g 00$
$| h |$
is the determinant of the matrix made of the spatial components
$h i j$
of the spatial metrical tensor. In the Schwarzschild spacetime the gravitational entropy density (33) is:
Rudjord, Grøn and Hervik [
] then calculated the entropy of the horizons due to the inhomogeneity in the gravitational field of a Reissner-Nordström black hole based upon the expression (38). It turned out that it was less
than the Bekenstein-Hawking entropy of a charged black hole. Hence either the gravitational field accounts for only a part of the entropy of a charged black hole, or the expression (38) does not
catch up all of the gravitational entropy.
Further they calculated the gravitational contribution to the entropy of the black hole horizon of the Schwarzschild-de Sitter spacetime. This spacetime has two horizons, a black hole horizon and a
cosmological horizon. Their radii are [
$R B H = 2 Λ cos π + φ 3 , R C H = 2 Λ cos π − φ 3 , φ = arccos ( 3 3 2 R S R Λ )$
$R S$
$R Λ$
are defined in eq. (14). In the limit that the vacuum energy,
Λ vanishes, the black hole horizon has a radius
$R S$
, and in the limit that the mass,
the Schwarzschild radius, vanishes there is de Sitter spacetime and the cosmological horizon has a radius
$R Λ$
. The surface gravities of the horizons in the Schwarzschild-de Sitter spacetime are:
$κ B H = α | R S 2 R B H 2 − R B H R Λ 2 | , κ C H = α | R S 2 R C H 2 − R C H R Λ 2 | , α = 1 1 − 3 ( R S / 2 R Λ ) 2 / 3$
The corresponding entropies are:
$S B H = π κ B H 2 , S C H = π κ C H 2$
Shankaranarayanan [
] suggested that the entropy of the Schwarzschild-de Sitter spacetime is:
$S S d S = S B H + S C H + 2 S B H S C H$
Rudjord, Grøn and Hervik [
] found that the gravitational contribution according to the expression (38) to the black hole horizon in the Schwarzschild-de Sitter spacetime is:
$S G H = S B H 1 + Λ 2 c 4 18 G 2 M 2 R B H 6$
This expression shows that the gravitational contribution to the black hole horizon entropy decreases with increasing value of Λ. The entropy of the cosmological horizon, $S G C$, is given by eq.(48)
with $S B H$ and $R B H$ replaced by $S C H$ and $R C H$.
In the limit that
$M → 0$
there is de Sitter spacetime with no black hole horizon, only a cosmological horizon with radius
$R Λ$
and entropy:
In this limit
$S C G → 0$
. There is no gravitational contribution to the entropy of the de Sitter horizon. This is the “opposite” limit to that of the Schwarzschild black hole limit representing a spacetime in which the
gravitational field is homogeneous. This will be the case in all conformally flat spacetimes, for example in all the Friedmann-Robertson-Walker universe models. Hence the entropy of horizons in
conformally flat spacetimes must be of nongravitational origin according to the theory based upon the Weyl curvature hypothesis. It should also be noted that Penrose has expressed a doubt as to
whether the entropy
$S Λ$
associated with the de Sitter horizon has any physical significance. He writes [
]: “I am inclined to be sceptical about
$S Λ$
representing a true entropy in any case, for at least two further reasons. In the first place if
really is a constant, so
$S Λ$
is just a fixed number, then
does not give rise to any actually discernible physical degrees of freedom. Moreover, I am not aware of any clear physical argument to justify the entropy
$S Λ$
, like the Bekenstein argument for black hole entropy.”
6. Is There a Maximal Entropy for the Universe?
Processes supporting the existence of life in the universe depend upon the possibility that the entropy has not yet reached a maximal value where all organized activity cease. But there still remains
some ambiguity about how to best define the maximum entropy. This question was recently discussed by Egan [
]. There are several conjectures as to what the maximal entropy may be [
]. One is the so-called
Bekenstein bound
]. The maximum entropy of a system with radius
and non-gravitational energy
$S M A X − B e k = 2 π k ℏ c R E$
The physical meaning is that the entropy lost into a black hole cannot be larger than the increase in the horizon entropy of the black hole. In the context of a flat FRW-universe the Bekenstein bound
says that the maximal entropy inside a co-moving surface with standard coordinate radius
$S M A X − B e k = 8 π 2 3 k ℏ c χ 4 ρ a 4$
is the density of the cosmic fluid and
the scale factor. For a universe dominated by a cosmic fluid with equation of state
$p = w ρ$
, the density changes during the expansion according to
$ρ a 3 ( 1 + w ) = ρ 0$
$ρ 0$
is the present density. Hence the Bekenstein bound of the cosmic entropy is proportional to
$a 1 − 3 w$
. In the radiation dominated that lasts for about fifty thousand years after the Big Bang, the dominating fluid,
the radiation, had
$w = 1 / 3$
. Hence
the Bekenstein maximum of entropy was constant during the radiation dominated era
. For cosmic matter with
$w < 1 / 3$
, for example cold dark matter with
$w ≈ 0$
, the Bekenstein bound increases with time, and for cosmic dark energy with
$w < − 1 / 3$
the Bekenstein bound of maximal entropy increases faster that in the matter dominated era.
Secondly the
Holographic bound
] may be formulated by asserting that for a given volume
, the state of maximal entropy is the one containing the largest black hole that fits inside
, and this maximum is given by the finite area that encloses this volume. The maximum entropy is not proportional to the volume as one might have expected. As applied to the space inside the
cosmological horizon in a FRW universe this gives the entropy bound:
$S M A X − H o l = π k ( χ a l P ) 2$
In the present era of our universe which is dominated by dark energy, possibly in the form of LIVE, the holographic bound (52) increases exponentially fast in the future. Several versions of this
bound have been discussed by Custadio and Horvath [
]. However, as noted by Veneziano [
], entropy in cosmology is extensive,
it grows like
$R 3$
, but the holographic entropy boundary grows like
$R 2$
. Therefore at sufficiently large
the holographic entropy boundary must be violated.
Frautschi [
] identified the maximum entropy inside the particle horizon of a universe as the entropy that is produced if all matter inside the horizon collapses to a single black hole. Again as applied to a
flat FRW universe this leads to:
$S M A X − F r a = 64 π 3 k G 9 ℏ c ( χ P H 3 ρ a 3 ) 2$
For a universe dominated by a cosmic fluid with equation of state $p = w ρ$, this implies that $S M A X − F r a ∝ a − 3 w$. Hence, an era dominated by cold matter has approximately constant value for
this upper bound on the entropy. The bound decreased in the radiation dominated era and increases in the present and future LIVE-dominated era.
Pavon and Radicella [
] have recently discussed the question whether the entropy of the universe will tend to a maximum. They took as a point of departure that the dominating contribution to the entropy of the universe is
that associated with a cosmological horizon, and represented for simplicity the horizon by the surface around an observer where the velocity of the Hubble flow is equal to the velocity of light.
Hence the physical radius of the surface is equal to the Hubble radius,
$c / H$
. The area of this surface is
$A = 4 π c 2 / H 2$
, and the entropy is:
$S H = k 4 l P 2 A = π c 2 k l P 2 1 H 2$
The Hubble factor and hence the entropy is here assumed to be given as a function of the scale factor, $S H = S H ( a )$, and differentiation of $S H$ with respect to a is denoted by $S H ′$.
According to the second law of thermodynamics the entropy of the universe does not decrease, $S H ′ > 0$. Then, if the entropy shall reach a maximum the condition $S H ″ < 0$ must be obeyed for large
values of a.
Let us look at the time evolution of
$S H$
in the flat ΛCDM universe model which has:
$H ( a ) = H 0 Ω Λ 0 + Ω M 0 a − 3$
$Ω Λ 0$
$Ω M 0 = 1 − Ω Λ 0$
are the present values of the density parameter of LIVE and dark matter, respectively. Hence
$S H = S ∞ 1 + N a − 3 , S ∞ = π Ω Λ 0 ( c l P H 0 ) 2 k , N = Ω M 0 Ω Λ 0$
In this universe model, which is compatible with all cosmological observations so far, the entropy $S H$ approaches the finite maximal value $S ∞$.
7. Entropy Change During the Inflationary Era
The importance of gravity for the establishment of an arrow of time in the universe was emphasized by Davies [
] in an article titled
Inflation and the time asymmetry in the universe
. He noted that when gravity is attractive, gravity tends to contract matter and, if possible, make black holes. In this case a smooth distribution of matter represents a small gravitational entropy.
However if gravity is repulsive so that gravity tends to smoothen the distribution of matter and energy, it seems natural to say that a state with an irregular mass distribution would be a low
entropy state and a smooth state would be one of maximal entropy. Under attractive gravity an inhomogeneous clumped field has high entropy, but with the switch to repellent gravity it is redefined as
low entropy. Concentrations of mass, i.e. curvature, try to smooth themselves out and the gravitational field tends towards uniformity.
This way of talking about gravitational entropy is in accordance with the Second Law, securing that gravitational entropy increases during a period dominated by repulsive gravity. However, it is in
conflict with the definition of gravitational entropy defined as an expression of inhomogeneity of a gravitational field. A general agreement as to how one should define and describe mathematically
the gravitational contribution to the cosmic entropy during a period dominated by repulsive gravity has not yet been obtained.
Before the Planck time, $10 − 43$ s, the universe was probably in a state of quantum gravitational fluctuations. Then it entered an inflationary era lasting for about $10 − 33$ s dominated by dark
energy with a huge density, possibly in the form of LIVE. The universe then evolved exponentially fast toward a smooth and maximally symmetric de Sitter space which represents the equilibrium end
state with maximal entropy when the evolution is dominated by a non-vanishing cosmological constant. This was an essentially adiabatic expansion with small changes of entropy.
At the end of the inflationary era the entropic situation changed abruptly. The vacuum energy was transformed to radiation and matter, gravity became attractive, and it became favorable for the
Universe to grow clumpy. As pointed out by Veneziano [
] in any inflationary scenario, most of the present entropy is the result of these dissipative processes. But not only did the entropy of the matter increase. Also the value of the maximum possible
entropy increased–and much more than the actual entropy. Hence a large gap opened up between the actual entropy of the universe and the maximum possible entropy. According to Davies this accounts for
all the observed macroscopic time asymmetry in the physical world and imprints an arrow of time on it.
Page [
] has disputed this conclusion. He argues that because the de Sitter spacetime, the perturbed form of which is equal to the spacetime during most of the inflationary era, is time symmetric, then for
every solution of Einstein’s equations which corresponds to decaying perturbations, there will be a time-reversed solution which describes growing perturbations. Page also noted that a sufficient,
though not necessary condition for decaying perturbations is the absence of correlations in the perturbations in the region. Davies answered [
]: The perturbations will only grow if they conspire to organize themselves over a large spatial region in a cooperative fashion. Hence, it is necessary to explain why it is reasonable for the
universe to have been in a state with no correlations initially.
Davies then went on to give the following explanation. Due to repulsive gravitation de Sitter spacetime may be considered to be a state of equilibrium with maximal entropy. But quantum effects will
cause fluctuations about the de Sitter background. Large fluctuations are much rarer than small fluctuations. At the minimum point of such a fluctuation the perturbations will be uncorrelated. A
randomly chosen perturbed state will almost certainly be such a state of no correlations at the minimum of a fluctuation curve. This state is thus one in which the perturbations will decay rather
than grow whichever direction of time is chosen as forward.
Hence inflation lowers the “Weyl-entropy” in a co-moving volume and expansion raises it, and it is not obvious which should dominate. We shall therefore calculate the “Weyl-entropy” change during the
inflationary era in a plane-symmetric Bianchi type I universe dominated by LIVE, using $S G 1$ given in eq. (34) and $S G 2 = P V$ with $P$ given in eq. (38), as measures of gravitational entropy.
The line element of the LIVE-dominated, plane symmetric Bianchi type I universe is [
$d s 2 = − c 2 d t 2 + R i 2 ( d x i ) 2 , i = 1 , 2 , 3$
$R 1 = R 2 = 2 2 / 3 sinh 2 / 3 τ , R 3 = 2 2 / 3 coth 1 / 3 τ cosh 2 / 3 τ , τ = ( 1 / 2 ) ( 3 Λ ) 1 / 2 t$
The co-moving volume is:
For this spacetime the Ricci scalar, the Weyl scalar and the Kretschmann curvature scalar are, respectively:
$R μ ν R μ ν = 36 H 0 2 , W μ ν α β W μ ν α β = 12 H 0 2 sinh 4 τ , R μ ν α β R μ ν α β = 24 H 0 2 + 12 H 0 2 sinh 4 τ ,$
Hence, the quantities
$P W A$
given in eq.(32) and
given in eq.(38) are, respectively:
$P W A = 1 3 sinh 2 τ , P = 1 1 + 2 sinh 4 τ$
This gives:
$S G 1 = k S 3 coth τ , S G 2 = k S sinh τ cosh τ 1 + 2 sinh 4 τ$
The time variations of these quantities at the beginning of the inflationary era are shown in
Figure 3
Figure 3. Time-variation of the candidate gravitational entropies $S G 1$ (upper curve) and $S G 2$ in a comoving volume during the beginning of the inflationary era.
We see that if $S G 1$ or $S G 2$ were a dominating form of entropy, then the entropy of the universe decreased during most part of the inflationary era except possibly during a transient initial
Davies [
] further mentioned the problem of deciding whether the cosmic initial conditions are very inhomogeneous resembling a universe with primordial black holes or more homogeneous. If it was too
inhomogeneous when it entered the inflationary era the inhomogeneities might not have been sufficiently inflated during the inflationary era to let the universe evolve towards the homogeneous state
it was in 400,000 years after the Big Bang when the relative temperature deviations,
$Δ T / T$
from homogeneity was of the order of magnitude 10
More research should be performed on the evolution of the cosmic entropy during the inflationary era. Even during an inflationary era, particularly a late phase lambda dominated one, even though the
lambda term dominates the overall expansion, locally, gravity remains attractive, and so one should consider scenarios which are a mixture of repulsive gravity at large scales, and attractive at
small ones.
8. The Entropy of the Universe
Frampton, Hsu, Kephart and Reeb [
] have recently calculated the entropy within the observable part of the universe. This has been followed up by Egan and Lineweaver [
]. They have presented still more accurate results and also include the entropy of the cosmological horizon.
The observable part of the universe is that part which is inside the event horizon, and has at the present time the magnitude
$V O B S = ( 3 , 65 ± 0 , 10 ) ⋅ 10 80 m 3$
. Egan and Lineweaver found that the baryonic matter in the universe in the form of stars and interstellar gas and dust contributes with
$S B = ( 8 ± 6 ) ⋅ 10 81 k$
. The microwave background radiation has an entropy
$S γ = ( 4 / 3 ) σ V O B S T 3$
$σ = 5 , 67 ⋅ 10 − 8 ( W / m 2 K 4 )$
is the
Stefan-Boltzmann constant
, and the present temperature of the radiation is
$T = 2.73 K$
. This gives
$S γ = ( 2.03 ± 0.15 ) ⋅ 10 89 k$
. The entropy of the relic neutrinos is a factor 11/4 larger,
i.e. $S ν = ( 5.15 ± 0.14 ) ⋅ 10 89 k$
. Egan and Lineweaver [
] have also estimated the entropy of relic gravitons,
$S g r a v = 6.2 ⋅ 10 87 − 2.5 + 0.2 k$
. The entropy of the dark matter was found to be
$S D M = 2 ⋅ 10 88 ± 1 k$
. All of these entropy contributions are thermodynamic. Then come two contributions that may be due to inhomogeneous gravitational fields. The entropy of the stellar black holes with masses
$M < 15 M ⊙$
$M ⊙$
is the mass of the Sun, is calculated to be
$S S B H = 5.9 ⋅ 10 97 − 1.2 + 0.6 k$
. The corresponding entropy of the supermassive black holes in the centers of the galaxies was found to be
$S S M B H = 3.1 − 1.7 + 3.0 ⋅ 10 104 k$
. Hence these gravitational contributions to the entropy of the Universe are much larger than the thermodynamic contributions. Frampton [
] has suggested that intermediate mass black holes in galactic haloes may contribute with even more entropy, up to
$10 106 k$
. The entropy of the cosmic event horizon is vast,
$S C E H = π k ( R C E H / l P ) 2$
$R C E H = ( 15.7 ± 0.4 ) G l y r$
$S C E H = ( 2.6 ± 0.3 ) ⋅ 10 122 k$
. Since the spacetime of the universe model is conformally flat, all of this entropy is of non-gravitational origin according to the theory based upon the Weyl curvature hypothesis. The physical
significance and origin of this dominating entropy is still not known.
The time evolution of the entropy of the universe was summarized briefly by Egan and Lineweaver [
]. I will here neglect the evolution of the entropy associated with the cosmological horizon, as the physical significance of this is rather speculative and not well understood.
At the Planck time,
$t P = 5.4 ⋅ 10 − 44 s$
the universe may have fluctuated into an inflationary era dominated by vacuum energy and lasting for about
$10 − 33 s$
. During this era the universe became increasingly homogeneous. Then the thermodynamic entropy increased, but as shown above (see
Figure 3
) the gravitational entropy based upon the Weyl curvature hypothesis decreased, and the sum of the two can be a decreasing function of time.
If this theory is correct, we have to modify the second law of thermodynamics. The usual formulation is then generally valid only when the dominating cosmic gravity is attractive. In that case the
entropy of the universe cannot decrease, as we are used to. But the entropy of the universe may decrease if the processes determining the change of the entropy in the universe are dominated by
repulsive gravity like in the inflationary era. Hence, if the arrow of time is due to gravitational entropy, then the arrow of time may have been directed along decreasing entropy in the inflationary
era. As mentioned above there is no general agreement as to how the gravitational contribution to the cosmic entropy shall be defined.
At the end of the inflationary era the vacuum energy was transformed to radiation and matter with an enormous heating involved, and gravity became attractive. This produced an increase of
thermodynamic entropy by many orders of magnitude, and the maximal value of the cosmic entropy increased so much that a vast entropy gap opened up, but the Weyl gravitational entropy remained low.
After about a hundred million years the first stars formed from collapsing clouds of hydrogen and helium, and shortly thereafter the first black holes formed. The entropy in stellar black holes
increased rapidly during the galactic evolution. Subsequently the gravitational entropy increased further due to the formation of supermassive black holes in the centers of the galaxies. In a far
away future the black holes may disappear due to Hawking radiation, and the asymptotic future of the entropy budget will then be radiation dominated. One may wonder whether the disappearance of black
holes is an entropy increasing process according to the conventional description of the entropy of black holes and of the form of energy which is Hawking radiated from a black hole. Presumably it is
so, and the entropy increase in Hawking evaporation has been estimated by Zurek [
9. Conclusions
In this article we have considered four types of entropy: the usual thermal entropy as interpreted by Boltzmann, entropy of black holes, entropy of cosmological event horizons and entropy associated
with the inhomogeneity of gravitational fields, as suggested by the Weyl curvature hypothesis. The generalized Second Law of Thermodynamics says that the sum of all these entropies in the universe
cannot decrease.
The gravitational entropy given by the volume integral of the divergence of the square root of the ratio between the Weyl- and Kretschmann curvature scalars is in general different from the
Bekenstein-Hawking entropy of the horizons in an arbitrary spacetime. However the Schwarzschild spacetime seems to be an exceptional case in which all of the horizon-entropy comes from the
gravitational entropy due to the inhomogeneity of the gravitational field. On the other hand all conformally flat spacetimes have vanishing gravitational entropy. This is the case for all of the
Friedmann universe models if local inhomogeneities are neglected. In particular there is no gravitational contribution to the entropy of the cosmological horizon of the de Sitter universe.
If the universe has evolved from an initial rather homogeneous state, the gravitational entropy was low at the beginning. Also a vast reduction of the gravitational entropy may have taken place in
the inflationary era while the thermodynamical entropy increased. At the end of the inflationary era the universe was in a state with a high degree of homogeneity and nearly perfect thermal
equilibrium. During the later condensation of matter and increment of local inhomogeneities the temperature differences increased, but due to release of gravitational energy which was transformed to
thermal energy, the thermodynamical entropy increased also during these processes. Furthermore there was a great increase of gravitational entropy due to gravitational condensation with production of
a large number of black holes in our universe.
Quantum calculations show that an inhomogeneous universe is more likely to be spontaneously created than a homogeneous one, and that a universe with a large cosmological constant is more likely to be
created than a universe with a small cosmological constant.
As to the evolution of entropy in our expanding universe, it was noted by Wallace [
] that although the early universe was at local thermal equilibrium, it was not at global thermal equilibrium because of the expansion of the universe and because it was highly uniform, and the
process of becoming non-uniform is entropy-increasing when conversion of gravitational energy to thermal energy and emission of electromagnetic radiation is taken into account. However, the
dominating entropy-increasing processes are the formation of black holes.
I would like to thank the referees for several useful comments.
© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://
Share and Cite
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Grøn, Ø. Entropy and Gravity. Entropy 2012, 14, 2456-2477. https://doi.org/10.3390/e14122456
AMA Style
Grøn Ø. Entropy and Gravity. Entropy. 2012; 14(12):2456-2477. https://doi.org/10.3390/e14122456
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Grøn, Øyvind. 2012. "Entropy and Gravity" Entropy 14, no. 12: 2456-2477. https://doi.org/10.3390/e14122456
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Graph Metanetworks
Abstract: Neural networks efficiently encode learned information within their parameters. Consequently, many tasks can be unified by treating neural networks themselves as input data. When doing so,
recent studies demonstrated the importance of accounting for the symmetries and geometry of parameter spaces. However, those works developed architectures tailored to specific networks such as MLPs
and CNNs without normalization layers, and generalizing such architectures to other types of networks can be challenging. In this work, we overcome these challenges by building new metanetworks —
neural networks that take weights from other neural networks as input. Put simply, we carefully build graphs representing the input neural networks and process the graphs using graph neural networks.
Our approach, Graph Metanetworks (GMNs), generalizes to neural architectures where competing methods struggle, such as multi-head attention layers, normalization layers, convolutional layers, ResNet
blocks, and group-equivariant linear layers. We prove that GMNs are expressive and equivariant to parameter permutation symmetries that leave the input neural network functions unchanged. We validate
the effectiveness of our method on several metanetwork tasks over diverse neural network architectures. | {"url":"https://research.nvidia.com/labs/toronto-ai/GMN/","timestamp":"2024-11-04T12:01:36Z","content_type":"text/html","content_length":"21589","record_id":"<urn:uuid:9d3f4a62-c778-45dd-b1bf-635071c59ea7>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00339.warc.gz"} |
IF, THEN formula that references a checkbox field and a date field
I'm trying to write an IF, THEN formula that references a checkbox field and a date field, but I keep getting errors. Here's the logic: If the 'EMDS' column is checked, then calculate the 'Date
Needed' column minus five days. If seems simple, but it may actually be too complicated for the system since it essentially involves nesting formulas. Here's what I have so far, which is returning an
#INVALID COLUMN error: =IF(EMDS@row = 1, [Date Needed]@row - 5,). Any advice is appreciated.
Best Answer
• =IF([EMDS]@row, [Date Needed]@row - 5)
The column that contains the formula must be a "Date" column.
• =IF([EMDS]@row, [Date Needed]@row - 5)
The column that contains the formula must be a "Date" column.
• Fantastic. It originally was a Date column, but I kept getting the 'Date Expected' error message so I changed it to a Text column. I changed it back to a Date column and now it works exactly as
intended. Thanks!
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“So what will you do if string theory is wrong?”
In a new preprint of an article entitled “So what will you do if string theory is wrong?”, to appear in the American Journal of Physics, string theorist Moataz Emam gives a striking answer to the
question of the title. He envisions a future in which it has been shown that the string theory landscape can’t describe the universe, but string theorists continue to explore it anyway, breaking off
from physics departments to found new string theory departments:
So even if someone shows that the universe cannot be based on string theory, I suspect that people will continue to work on it. It might no longer be considered physics, nor will mathematicians
consider it to be pure mathematics. I can imagine that string theory in that case may become its own new discipline; that is, a mathematical science that is devoted to the study of the structure
of physical theory and the development of computational tools to be used in the real world. The theory would be studied by physicists and mathematicians who might no longer consider themselves
either. They will continue to derive beautiful mathematical formulas and feed them to the mathematicians next door. They also might, every once in a while, point out interesting and important
properties concerning the nature of a physical theory which might guide the physicists exploring the actual theory of everything over in the next building.
Whether or not string theory describes nature, there is no doubt that we have stumbled upon an exceptionally huge and elegant structure which might be very difficult to abandon. The formation of
a new science or discipline is something that happens continually. For example, most statisticians do not consider themselves mathematicians. In many academic institutions departments of
mathematics now call themselves “mathematics and statistics.” Some have already detached into separate departments of statistics. Perhaps the future holds a similar fate for the unphysical as
well as not-so-purely-mathematical new science of string theory.
This kind of argument may convince physics departments that string theorists don’t belong there, while at the same time not convincing university administrations to start a separate string theory
department. Already this spring the news from the Theoretical Particle Physics Rumor Mill is pretty grim for string theorists, with virtually all tenure-track positions going to phenomenologists.
I have some sympathy for the argument that there are mathematically interesting aspects of string theory (these don’t include the string theory landscape), but the way for people to pursue such
topics is to get some serious mathematical training and go to work in a math department.
The argument Emam is making reflects in somewhat extreme form a prevalent opinion among string theorists, that the failure of hopes for the theory, even if real, is not something that requires them
to change what they are doing. This attitude is all too likely to lead to disaster.
Update: A colleague pointed out this graphic from Wired magazine. Note the lower right-hand corner…
Update: Over at Dmitry Podolsky’s blog, in the context of a discussion of how Lubos’s blog makes much more sense than this one, Jacques Distler explains what it’s like for string theorists these days
trying to recruit students:
Unfortunately, I’ve seen a number of prospective graduate students, who spent their undergraduate days as avid readers of Woit’s blog, and whose perspective on high energy physics is now so
hopelessly divorced from reality that the best one can do is smile and nod one’s head pleasantly and say, “I hear the condensed matter group has openings.”
87 Responses to “So what will you do if string theory is wrong?”
1. Not sure I agree this is the only interest in these systems, but in any event there is currently a new set of ideas of dealing with strongly interacting systems. It may be of some interest for
people with experience dealing with such systems to explore these ideas and discover what they are good for.
2. Eric,
The problem with the argument that physics departments have only stopped hiring string theorists this spring is because of the LHC is that the LHC turn-on has been imminent for quite a few years,
and departments that wanted to move in that direction have been doing so for quite a while.
However, it is only this spring that string theorists have started posting articles on the arXiv about what they plan to do once it is shown that string theory is wrong…
3. Sure, new things need to be explored. The ideas you mentioned aren’t being neglected, while there are other approaches to strongly-interacting problems which have excited less interest, possibly
because they don’t start from string theory (using exact form factors is one of those of which I am familiar).
4. Moshe – If you are interested, the form factor program is reviewed in
an excellent article by Essler and Konik.
5. Thanks Peter O., I’ll take a look.
6. Peter,
I don’t think you can claim that physics departments have stopped hiring string theorists. For example, Hawaii has just made an offer to a string theorist, and I’m sure others will be
7. Again, what about Lehman College, CUNY? Can we not consider Dan Kabat a string theorist?
8. Eric,
I’m not claiming that no physics department is hiring any string theorists, just that dramatically fewer are getting tenure-track jobs this spring than was typical in recent hiring seasons. I’ve
given you my source for this information: the rumor-mill web-site. Go check it for yourself.
9. Peter,
Perhaps you should have a closer look at the rumour mill site yourself, as the information I was just giving you came directly from there. Even if there are fewer string theorists hired this
spring, I don’t think this would be that unusual, as these things are cyclic.
10. Sorry if this is kind of a dumb question, but this is something I’ve been a bit confused about for a while: What exactly is the difference between a “Phenomenologist” and an “Experimentalist”?
11. Coin,
Experimentalists design, build, operate experiments and analyze the data from them. Deciding which theorists are “phenomenologists” is not so clear, the term covers a wide range of activity. But
in general the idea is that they are theorists working not on theoretical or mathematical foundations, not on models far from reality, but on better understanding and extracting predictions from
testable models that can be compared with experiment. At one end of the subject, their activities probably become indistinguishable from some of the activities engaged in by experimentalists as
part of their data analysis.
12. Peter Woit is right; I certainly wasn’t calling for all string theorists to be fired. I was trying to point out that Moataz Emam’s call for string theorists to form their own departments (and I
don’t quite know how seriously to take him on this) was not only completely implausible, but it also wasn’t even in the best interest of the string theorists. If anybody tries to form a separate
Department of String Theory, I would expect and hope that most of the physicists (string theorists and others alike) in the affected Department of Physics would resist.
13. Dimitry and Peter O.
I am well aware that the 3D Ising model is Kramers-Wannier dual to the 3D Ising gauge model and admits a random surface representation. Even better, one can write down a funny O(N) or U(N)
lattice gauge model on a 3D brick lattice, with a log action, that can be exactly mapped onto a model of self-avoiding random surfaces; the weight of each graph equals N^chi u^A, where chi is the
Euler characteristic, u is related to the coupling constant, and A the surface area.
Does this construction solve U(N) gauge theory, or make me into a string theorist? Hardly. Just because I have mapped one untractable model into another does not mean that I have solved it. If
anything, this shows that perturbative string theory is inadequate for this problem because steric repulsion is crucial here (self-avoiding surfaces do not self-intersect!).
A method has only really contributed to our understanding of a model if it helps to extract some kind of quantitative information about it, not necessarily critical exponents and not necessarily
exactly. The methods that Peter O. mentioned do that, as do high- and low-temperature expansions, real-space RG and computer simulations. However, AFAIK no quantitative results about 3D Ising
have come out of string theory.
14. Thomas,
As you can see from my earlier comment, I largely agree with your sentiment that string/random surface approaches to field theory/stat mech problems have not been successful. In fact, as I said,
it has been much easier to obtain string excitations from field theory than the other way around (except in AdS/CFT approaches, which have not yet made a connection with real physics problems.
Maybe the work Moshe referred to will change the situation). I must admit, however, that I still have some fondness for this approach, in spite of its lack of success, since it captured my
imagination in grad school. It was a very daring suggestion, which, unfortunately, may not lead anywhere.
15. As someone with credentials as a scientist and as a writer, I see the argument for splitting String Theory Departments away from Physics Department, to allow Physics Departments to return to the
“Real World” and to allow the String Theorists to pursue their quest; as analogous to an argument that Critical Theory Departments split away from English Literature Departments, to allow Eng Lit
to focus again on poems, stories, and novels, and be free from Deconstructionism, Feminism, Pan-Africanism, Asian Pacific Islandism, Gay Theory, Postcolonialism, and the like, while the Critical
Theory professors and grad students and postdocs explore a rich sociocultural and semiotic landscape unfettered by having to read and explain prose and poetry that naive millions of readers
actually enjoy buying and reading.
16. Is that quote on Podolsky’s blog really by Jacques Distler himself? Or is that quote just somebody else posting under Distler’s name, trying to make him look bad?
(Offhand, I can’t really tell either way).
17. JC,
If it’s not Jacques, it’s someone doing a pitch-perfect impersonation….
18. Peter, this thread shows the difficulty in discussing hypothetical scenarios. Emam referred to a hypothetical situation that “someone shows that the universe cannot be based on string theory” so
he was certainly not calling to form separate departments now.
Overall, Enam’s analysis is not convincing. I think he is simply wrong in his assumption that if someone shows that the universe cannot be based on string theory many string theorist will
continue to do what they do. Showing that the universe cannot be based on string theory (if this will be the way things will go) will represent a major breakthrough which is one of the potential
fruits of string theory research. Probably such a (hypothetical) development will cause most string theory to work on something else (that can still be called “string theory”). Those who will
continue to work on string theory (or “old-fashioned string theory”) will have a place in physics department just like many other theoretical physicists who work on models which while not
describing the universe, still give important physics insights and develop important mathematical physics. Also Emam’s comparison with statistics does not make sense.
Here is an analogy from mathematics. For many centuries algebraists tried to find formulas for solving polynomial equations. If somebody had come say in 1600 and said: “guys in view of the many
years of failure, try to think about the possibility that such formulas are impossibles” this could be a little useful. If the same guy had said “algebraists have failed and they should be fired”
(opening the door to more astrologists and alchemists), then this could be a little harmful. But what was really needed is an understanding why there are no formulas to solve polynomial equations
and not just vague speculations and meditations.
Once this was understood nobody continued the old algebra endeavor but there was plenty of work left for algebraists, in fact, this was the beginning of modern algebra.
The turning point and the beginning of modern algebra was not just the “gut feeling” that the old endeavor might have failed but the detailed technical understanding why it cannot succeed.
19. Gil,
String theory, as an idea about unification, is not in danger of failing because of mathematical contradiction. It is failing because it is becoming clear that, as an idea about unification, it
is essentially vacuous. The reasons for this are well understood (e.g. the landscape). I don’t think the author of this article or anyone else envisions string theory being shown to be
mathematically wrong, or being falsified by showing that it predicts something that disagrees with experiment. What is happening is that conviction within the physics community is growing that
there is no hope to get a real prediction out of string theory, thus the failure. Die-hards will keep claiming that hope remains, but there will be fewer and fewer of them, and at some point the
consensus of the community will be that the cause is hopeless.
Right now, I think many of the leaders of the subject are well aware of how bad the problems are, but are holding out from giving up on the theory, waiting to see what the LHC says, hoping that
LHC results will change the current picture. If the LHC doesn’t come up with something that somehow gives new hope to the string theory unification program, I think it will be conclusively dead,
although some people will insist on pursuing it. I would guess that it is exactly this all too possible situation that Emam is thinking about when he brings up the possibility of string theory
being shown to be “wrong”.
20. Dear Peter,
It is entirely rational to wait to the LHC outcomes and I think most people interested in the scientific endeavor are hoping that it will give new data and lead to important new insights. I have
noticed that your opinion on string theory is negative, but your description of the convictions within the physics and string theory communities may very well be biased by your own personal
position. At least your description differs from the convictions I sense when talking to many colleagues in these areas. (My little comment to Peter (Shor) was that Enam does not call string
theorists to form independent departments but rather discusses a scenario which is hypothetical from his point of view.)
21. Gil
It is not at all reasonable to wait and see what happens at the LHC. Any responsible tax funded theoretician should be working their ass off to come up with quantitative predictions before 2009.
22. Peter O,
I am sorry if my post came across as more negative than it was meant to be. The random surface approach also captured my attention a few years after it captured yours (when I was in grad school),
and I find it frustrating that it has not panned out. When I now think about the problem again after many years, it seems like the obstacle is the self-avoidance constraint, which induces an
interaction which is very non-local on the worldsheet, although it is local in the ambient space.
23. Peter, of course the nature of scientific proofs is different in different disciplines so mathematical proofs just appeared in my analogy since the analogy was drawn from the area of mathematics.
Ahh, I think I see a serious difficulty with your approach. You cannot imagine a way that string theory will be definitely proved to be wrong by physicists and therefore you assume that its
weaknesses already suffice. (But just like the case of solving polynomial equations, in view of the prominence and successes of string theory much stronger scientific arguments will be needed.)
Similarly, you cannot imagine how string theory can prevail as a viable physics theory and therefore you conclude that it failed. But string theory can win in ways we cannot imagine at present
and can also fail in ways we cannot imagine at present.
24. ‘But string theory can win in ways we cannot imagine at present and can also fail in ways we cannot imagine at present.’
This kind of statement is what you expect to find in a religious tract. It’s not really scientific, but is more in the category of belief systems/cults/wishful thinking/crackpotism. If string
theory ‘wins’ or is falsified, then it will be deserving of a media mention. Until then, it’s not a scientific theory.
Feynman told a story of ‘Cargo Cult Science’. During WWII, South Sea islands were used as military airports, and the islanders had business. After the war ended, some islanders tried to attract
business by encouraging passing planes to come down to land by building improvised runways with control towers.
Their efforts looked professional from a distance, but up close they were fake. It’s the same with string theory. The mathematics and alleged physical successes make it superficially look like a
scientific discipline, but it isn’t because it can’t predict anything checkable.
25. I’ve seen a number of prospective graduate students, who spent their undergraduate days as avid readers of Woit’s blog, and whose perspective on high energy physics is now so hopelessly divorced
from reality that the best one can do is smile and nod one’s head pleasantly and say, “I hear the condensed matter group has openings.”
That this blog has steered young students away from string theory is the best endorsement of any blog ever. Sounds like a clear charge of you’re corrupting the youth.
26. Peter Woit Wrote:
“Right now, I think many of the leaders of the subject are well aware of how bad the problems are, but are holding out from giving up on the theory, waiting to see what the LHC says, hoping that
LHC results will change the current picture. If the LHC doesn’t come up with something that somehow gives new hope to the string theory unification program, I think it will be conclusively dead,
although some people will insist on pursuing it.”
Wow, that is not the impression I get at all. Everybody seems to already know what the LHC will find (Higgs for sure, maybe Susy and small chance of extra dimensions and even smaller chance of
something crazy and non standard model). i don’t see how anything bad for string theory can come out of the LHC. If extra dimensions are found, it will be great, if not its no big deal as the
lower bound for these effects (or anything else related to string theory) to be measureable is not even close to being reached by the LHC. If something truely unexpected is found, everybody will
be scrambling to explain it, not just string theorists. As a young grad student that is what I’m routing for, unlikely as it is 🙂
27. Dear Anon, please note that the statement you quote and criticize
“String theory can win in ways we cannot imagine at present and can also fail in ways we cannot imagine at present.”
is not a statement within a scientific theory, but a statement about scientific theories. As such, I think it is perfectly fine and can be supported by examples.
28. Gil, string ‘theory’ is a misnomer. Stringy ideas are an empty framework which fits 10^500 theories. To get a specific string theory, you need to specify all of the moduli of the Calabi-Yau
manifold of 6 compactified dimensions. If there was a string theory then your position would be sensible (and this blog wouldn’t exist).
29. Gil,
It’s true something unimaginable may save string theory. Would you like to invest in Company XYZ? The stock is performing terribly, but something unimaginable may save it.
30. If this looks more accurate to you, anon, replace “string theory’ by ‘the string framework’.
It is possibe that the ‘string framework’ will lead to a definite win (and thus perhaps to a ‘string theory’) in ways we cannot expect, and it is also possible that it will be definitely shown,
in unexpected manners, that the ‘string framework’ is inedequate for its (ultimate) purposes.
There are also various possible forms of partial victory for ‘the string framework’, in terms of new insights and horizons in mathematics and physics which come short of a definite ‘theory’ to
your taste. Some of these partial victories are already in place.
31. The debate about string theory on this thread has become more or less completely content-free. Enough.
My apologies for not providing fresh material for discussion. I’m busy with other things, and there has been remarkably little news on the math-physics front.
32. “I’ve seen a number of prospective graduate students, who spent their undergraduate days as avid readers of Woit’s blog, and whose perspective on high energy physics is now so hopelessly divorced
from reality that the best one can do is smile and nod one’s head pleasantly and say, “I hear the condensed matter group has openings.”
Thank God (anyway, not a believer) !!!! A wise step for the students. Both monetary and scientifically.
33. Why the disdain for condensed matter from the first poster at the other site? Look at what Feynman did with helium. What Einstein did with thermoelectric effect. BCS theory. Fermi-Dirac.
Bose-Einstein. How about figuring out cuprate superconductors, studs? Come on smarties. Direct your theory to something physically tangible.
-solid state chemist
34. A deep statement by Atiyah, discussed by Gowers, recently in turn referenced by Terry Tao, explains why it might be a bad idea for String Theory (if “proven wrong”) reconsolidate itself as
distinct departments away from Physics.
Tao: “In fields such as nonlinear PDE (which Perelman’s result can broadly be included in), individual theorems tend not to be directly applicable much beyond their original intended use, but
general ideas, strategies, tricks, and paradigms are often far broader. A similar point (using combinatorics as the primary example) is discussed in Gowers’ ‘two cultures’ paper
Gowers citing Atiyah:
“The ultimate justification for doing Mathematics is intimately related to its overall unity. If we grant that, on purely utilitarian grounds, mathematics justifies itself by some of its
applications, then the whole of mathematics acquires a rationale provided it remains a connected whole. Any part that drifts away from the main body of the field has then to justify itself in a
more direct fashion.”
If String Theory has drifted away from the main body of Physics, or Mathematics, then (however beautiful it may be) it is sociologically subject to a demand for direct justification.
35. For what it’s worth, this paper seems to highlight the possibility of experimental verifiability at a scale of around 20 orders of magnitude larger than the Planck scale:
36. The paper mentioned above has been criticized to death by a trusted string researcher. My apologies for jumping the gun.
This entry was posted in Uncategorized. Bookmark the permalink. | {"url":"https://www.math.columbia.edu/~woit/wordpress/?p=684","timestamp":"2024-11-11T04:52:54Z","content_type":"text/html","content_length":"115755","record_id":"<urn:uuid:437f2f52-723e-4bd9-8e9f-f3d2ecdfc53b>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00796.warc.gz"} |
Pseudo Random Numbers without Side Effects
Show code cell source Hide code cell source
import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib_inline
import seaborn as sns
Pseudo Random Numbers without Side Effects#
Recall that Jax only uses pure functions. Pseudo random number generators are typically implemented as stateful objects:
• you initialize the generator with a seed
• you call the generator to get a random number
• the generator updates its internal state
This won’t cut it in Jax since it would violate the purity of the function. To deal with this we need to explicitly pass the state around. This is done using the PRNGKey object. Please go through
Pseudo Random Numbers in Jax and Stateful Computations in Jax for more details.
The state object is:
import jax.numpy as jnp
import jax.random as random
key = random.PRNGKey(0)
It is a tuple of two 32-bit unsigned integers:
Array([0, 0], dtype=uint32)
When sampling from a distribution, we explicitly pass the key. Here is a sample from a standard normal:
random.normal(key, shape=(2, 2))
Array([[ 1.8160863 , -0.75488514],
[ 0.33988908, -0.53483534]], dtype=float32)
Now if you pass the same key, you get the same sample:
random.normal(key, shape=(2, 2))
Array([[ 1.8160863 , -0.75488514],
[ 0.33988908, -0.53483534]], dtype=float32)
The key has not been updated:
Array([0, 0], dtype=uint32)
To get a different sample you need to split the key:
key, subkey = random.split(key)
Array([4146024105, 967050713], dtype=uint32)
Array([2718843009, 1272950319], dtype=uint32)
You are kind of branching the key to start two new generators. You can use either one to get a sample:
random.normal(subkey, shape=(2, 2))
Array([[ 1.1378784 , -0.14331433],
[-0.59153634, 0.79466224]], dtype=float32)
So, this is it. You must thread the key through your code. You get used to it when you do it a few times.
Let’s look at an example. We will generate a sample from a random walk using only functional programming. The random walk is starting at \(x_0\):
\[ x_{t+1} = x_t + \sigma z_t, \]
where \(\sigma > 0\) and
\[ z_t \sim N(0, 1). \]
def rw_step(x, sigma, key):
"""A single step of the random walk."""
key, subkey = random.split(key)
z = random.normal(subkey, shape=x.shape)
return key, x + sigma * z
Now we can put it in a loop that takes multiple steps and jit it:
from functools import partial
from jax import jit
from jax import lax
@partial(jit, static_argnums=(3,))
def sample_rw(x0, sigma, key, n_steps):
"""Sample a random walk."""
x = x0
xs = [x0]
for _ in range(n_steps):
key, x = rw_step(x, sigma, key)
xs = jnp.stack(xs)
return key, xs
This works:
sample_rw(jnp.zeros(2), 1.0, key, 10)
(Array([[ 0. , 0. ],
[ 0.00870701, -0.04888523],
[-0.8823462 , -0.71072996],
[ 0.3267806 , 1.6009982 ],
[ 1.2447658 , 2.0843194 ],
[ 1.0294966 , 1.6681931 ],
[ 4.0256443 , 0.41421673],
[ 3.6146142 , 0.53783053],
[ 3.1284342 , -0.39070803],
[ 3.636192 , -1.0362725 ],
[ 4.7213855 , -0.90524477]], dtype=float32),
Array([2172655199, 567882137], dtype=uint32))
But it is not good because the loop is unrolled and the compilation is triggered every time we try a new n_steps. We can use scan to avoid this:
def sample_rw(x0, sigma, keys):
"""Sample a random walk."""
def step_rw(prev_x, key):
"""A single step of the random walk."""
z = random.normal(key, shape=prev_x.shape)
new_x = prev_x + sigma * z
return new_x, prev_x
return lax.scan(step_rw, x0, keys)[1]
n_steps = 100_000
keys = random.split(key, n_steps)
walk = sample_rw(jnp.zeros(2), 0.1, keys)
Let’s plot it:
fig, ax = plt.subplots()
ax.plot(walk[:, 0], walk[:, 1], lw=0.5)
ax.set(xlabel="x", ylabel="y", title="Random Walk")
In case you did not notice, we did 100,000 steps in no time. | {"url":"https://predictivesciencelab.github.io/advanced-scientific-machine-learning/functional_programming/04_random.html","timestamp":"2024-11-13T01:19:54Z","content_type":"text/html","content_length":"44178","record_id":"<urn:uuid:e6156330-0215-47ad-b00c-3a4752ece8c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00839.warc.gz"} |
Linearly Independent Vectors
What is Linear Independence?
Linear independence is an important property of a set of vectors. A set of vectors is called linearly independent if no vector in the set can be expressed as a linear combination of the other vectors
in the set. If any of the vectors can be expressed as a linear combination of the others, then the set is said to be linearly dependent.
Linearly independent sets are vital in linear algebra because a set of n linearly independent vectors defines an n-dimensional space -- these vectors are said to span the space. Any point in the
space can be described as some linear combination of those n vectors.
An example should clarify the definition. Consider the two-dimensional Cartesian plane. We can define a certain point on the plane as (x, y). We could also write this as xî + yĵ, where î = (1, 0) and
ĵ = (0, 1). î and ĵ are linearly independent. î and ĵ also happen to be orthonormal, but this isn't necessarily the case with all linearly independent sets of vectors; if we define k̂ = (2, 1), then
{î, k̂} is a linearly independent set, even though î and k̂ aren't orthogonal and k̂ isn't normalized. We can still define any point on the plane in terms of î and k̂ -- for example, the point (3, 2) can
be expressed as 2î - k̂.
A Mathematical Example:
Chart from TheCleverMachine
The bases b(x) and b(y) are linearly independent. There is no possible way to compose the basis vector b(x) as a linear combination of the other basis vector b(y), nor the other way around. When we
look at this chart in 2D vector space, the blue and red lines are perpendicular to each other (orthogonal). However, linear independence can’t always be represented in 2D space. If we want to
officially determine if two column vectors are linearly independent, we do so by calculating the column rank of a matrix A. We compose this by concatenating the two vectors:
The rank of a matrix is the number of linearly independent columns in the matrix. If the rank of A has the same value as the number of columns in the matrix, then the columns of A forms a linearly
independent set of vectors. In our example, the rank of A is higher than two, and the number of columns is higher than two as well. This means these basis vectors are linearly independent. The same
matrix rank-based test can also verify if vectors of a higher dimension are linearly independent. If we want to be able to define a unique model, then we will care about the linear independence of
the basis set. | {"url":"https://deepai.org/machine-learning-glossary-and-terms/linearly-independent-vectors","timestamp":"2024-11-15T04:48:30Z","content_type":"text/html","content_length":"161240","record_id":"<urn:uuid:b04ef034-db4d-44ff-8a0e-d88fb3014b1a>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00619.warc.gz"} |
Math of ECGs: Fourier Series - Interactive MathematicsMath of ECGs: Fourier Series
Math of ECGs: Fourier Series
By Murray Bourne, 30 Mar 2010
I recently had a medical checkup that included an ECG (electro-cardiograph).
This is what my ECG looked like:
How an ECG is done
The electrodes are connected to various parts of your anatomy (chest, legs, arms, feet) and voltage differences over time are measured to give the ECG readout.
The horizontal axis of the ECG printout represents time and the vertical axis is the amplitude of the voltage.
Amplitude units are millivolts (mV) and on the graph, 1 mV = 10 mm high.
The time scale is 25 mm = 1 second (or 1 mm per 0.04 seconds on the graph).
So here's my readout for Lead II, representing the voltage between the positive electrode on my left leg and the electrode on my right arm. Each thicker red vertical line represents a time of 1
Apparently (according to the doctor), this indicates my heart is quite healthy.
In more detail, the features of the repeated pulse we are looking at are as follows.
[Image source: T. Burke]
The P wave is caused by contraction of the right atrium followed by the left atrium (the chambers at the top of the heart).
The QRS complex represent the point in time when most of the heart muscles are in action, so has highest amplitude.
The T wave represents the polarization of the ventricles (the chambers at the bottom of the heart).
Human heart showing atria and ventricles.
[Image by UCSD, source page no longer available]
Modeling the Heartbeat Using Fourier Series
A heartbeat is roughly regular (if it isn't, it indicates something is wrong). Mathematically, we say something that repeats regularly is periodic.
Such waves can be represented using a Fourier Series.
In my case, my heart rate was about 70 beats per minute. For the sake of simplicity, I'll assume 60 beats per minute or 1 per second. So the period = 1 second = 1000 milliseconds.
Also for simplicity, I will only model the R wave for this article. To get a more accurate model for the heartbeat, I would just need to do a similar process for the P, Q, S and T waves and add them
to my model.
I observed that my R wave was about 2.5 mV high and lasted for a total of 40 ms. The shape of the R wave is almost triangular and so I could have used straight lines for my model, but these don't
give us a smooth curve (especially at the top - it must be continuously differentiable).
A better approach is to use a polynomial (where the ascending and descending lines are close enough to being straight), so my model is as follows (the time units are milliseconds):
f(t) = -0.0000156(t − 20)⁴ + 2.5
f(t) = f(t + 1000)
Explanation of the Model
The model is based on a quartic (power 4) since this will give me close to the shape I need (a parabola would be too broad). I'm using similar thinking to what I was doing in this article, where I
move a curve around to where I want it.
The (t − 20) term comes from deciding the curve should start at (0,0), (which makes our lives easier), it will pass through (40,0) since the pulse is 40 ms long, and be centered on t = 20.
The "+2.5" comes from the fact the amplitude of the pulse is 2.5 mV.
The -0.0000156 comes from solving the following for a when t = 0.
a(t − 20)⁴ + 2.5 = 0.
The "f(t) = f(t + 1000)" part means the function (pulse in this case) is repeated every 1000 ms.
Graph of the Model
This is the graph of part of one period (the part above the t-axis from t = 0 to t = 40):
Of course. this is just one pulse. How do we produce a graph that repeats this pulse at regular intervals?
This is where we use Fourier Series.
I'll spare you all the details, but essentially the Fourier Series is an infinite series involving trigonometric terms. When all the terms are added, you get a mathematical model of the original
periodic function.
To obtain the Fourier Series, wee need to find the mean value, a[0], and 2 coefficient expressions involving n, a[n] and b[n] which are multiplied by trigonometric terms and summed for n = 1 to
Mean Value Term
a[0] is obtained by integration as follows (L is half of the period):
(The section of the curve we need for this part of the problem is from t = 0 to t = 40, so that's why we chose those values for the limits of integration in the second last line.)
First Coefficient Term, a[n]
Next, we compute a[n]:
The answer for this integral is pretty ugly. I've included it in the PDF solution.
Second Coefficient Term, b[n]
Now for b[n]:
Once again, I have spared you from the full details.
Finally, we put it all together and obtain the Fourier Series for our simple model of a heart beat:
When we graph this for just the first 5 terms (n = 1 to 5), we can see the beginnings of a regular 1-second heart beat.
The above graph shows the "noise" you get in a Fourier Series expansion, especially if you haven't taken enough terms.
Taking more terms (this time, adding the first 100 terms) gives us the following, and we see we get a reasonable approximation for a regular R wave with period 1 second.
I added the T wave for this next model (in blue).
I used a parabola for the T wave because the shape of the T wave is broader than the shape of the R wave.
We could keep going, adding the P, Q and S waves to get an even better model.
See the complete solution (up to the T wave, created using Scientific notebook) here:
Model of Human Heartbeat (PDF)
What have we done?
We have taken a single spike representing one R wave of my heartbeat. We then found a formula that repeats our spike at regular time intervals. The Fourier Series (an infinite sum of trigonometric
terms) gave us that formula.
Finally, we added the T wave, using the same theory as before.
Fourier Series is very useful in electronics and acoustics, where waveforms are periodic.
For more on Fourier Series go to:
Don't miss the section on how Fourier is used to create Digital Audio, in
Applications - the Fast Fourier Transform
See the 55 Comments below.
vonjd says:
30 Mar 2010 at 3:21 pm [Comment permalink]
Just to let u know: "The answer for this integral is pretty ugly. I’ve included it in the PDF solution." - brings up a 404
Murray says:
30 Mar 2010 at 3:34 pm [Comment permalink]
Oops - the second link was fine but I forgot to fix the first. Thanks for your feedback.
nahum says:
31 Mar 2010 at 9:46 am [Comment permalink]
i liked the easy-to-understand English.It made it not boring.
sharma k k says:
31 Mar 2010 at 5:29 pm [Comment permalink]
it is quite easy to under stand a mathematical approach.
gagangc says:
11 Mar 2011 at 12:55 am [Comment permalink]
i have modelled R curve in different manner
and fourier series up to two terms is
when i plotted it i got satisfactory answer as i considered only two terms.
please check my answer and let me know if i am correct.
it took lot of effort and time from me to calculate all coefficients by hand.
Murray says:
12 Mar 2011 at 5:15 pm [Comment permalink]
Hi gagangc. This looks good to me! Good on you for churning away with this.
biofredo says:
9 May 2011 at 1:03 am [Comment permalink]
did you calculate de PRS equations?
gagangc says:
11 May 2011 at 5:42 pm [Comment permalink]
i have a doubt regarding fourier transform of rectangular function.If FT indicates frequency contents of time domain signal,then FT of rect function is sinc function which have infinite
frequencies.Does this mean a simple rect function has infinite frequencies??
Murray says:
14 May 2011 at 12:29 pm [Comment permalink]
Hi gagangc. The rectangular function is a single pulse, so it's not relevant to talk about infinite frequencies. These 2 sources may help:
Wikipedia's article
Wolfram Demonstrations (requires a plugin)
Dan Adam says:
5 Feb 2013 at 10:04 pm [Comment permalink]
Hi Murray,
I'm interested to know how you have generated your last 3 graphs. If possible could you please email me your matlab codes. My email address is [hidden for protection from spam]
I'm waiting for your reply.
Murray says:
8 Feb 2013 at 12:14 pm [Comment permalink]
@Dan: Two of the last 3 graphs are based on the formulas I developed in the article (I did 5 terms for the first, then added 100 terms for the next.)
The last one adds the T-wave using the same process (I estimated the "blip" above the horizontal axis, then built it into the Fourier series. I suspect the details are hidden in some archive by now.
I'm using Scientific Notebook for these graphs. It allows for graphing of a summation notation expression.
Sorry it wasn't so helpful!
Kate says:
24 Mar 2014 at 4:37 am [Comment permalink]
Hello! I was curious if you could post your findings for the P, Q, and S waves? I have done it myself, but would love to check if I did it correctly. Many thanks!!
Murray says:
24 Mar 2014 at 8:18 pm [Comment permalink]
@Kate: As I mentioned in the article, I stopped after the T wave.
Basically, in this work, if it looks right, it is (very likely) right! Have you graphed your solution? That usually tells you right away whether you are on the right track. Use any of the online
graphers (like Desmos) to graph the first few terms for you.
Kate says:
26 Mar 2014 at 12:58 am [Comment permalink]
No, I havent graphed it. I was planning on using MatLab to get a graph.
Quick question -- when solving a(t-20)^4+2.5=0 where you got -0.0000156 -- what is a? Or, could you explain this process in more detail? And, why use a quartic for the R wave and a power of 2 when
you solved for the T wave? Thank you!!
Murray says:
27 Mar 2014 at 7:46 pm [Comment permalink]
@Kate: I've written some additions to the article which explain your questions. (You may need to refresh the page to see the changes.)
hari says:
18 Apr 2014 at 8:38 pm [Comment permalink]
Can u clearly explain me how did u get this function f(t)
as shown below. it wil be very helpful for me if u clarify my doubt.
f(t) = -0.0000156(t − 20)⁴ + 2.5
f(t) = f(t + 1000)
Murray says:
19 Apr 2014 at 11:07 am [Comment permalink]
@Hari: Did you go to the article I linked to, How to draw y^2 = x - 2?
I'm using similar thinking to arrive at the function you are asking about.
The minus at front is there to turn the curve "upside down" (so it is n-shaped and not u-shaped).
The "-20" is there because I need to move it over to the right by 20 milliseconds.
It's power 4 because it's more narrow than a parabola (which would be power 2).
The "+2.5" is to move the curve up by 2.5.
I just experimented using this graph facility until I was satisfied.
The f(t) = f(t + 1000) just means "repeat the curve every 1000 milliseconds".
Hope that helps.
Successful Android App Development at a Small Scale says:
13 Feb 2015 at 2:29 pm [Comment permalink]
[…] timings for each phase of the wave (for a mathematical breakdown of the ECG Fourier series, this is a good start). We popped these timings into our code and the result did not disappoint. It’s
Daniel Manogaran says:
14 Mar 2015 at 2:11 pm [Comment permalink]
Hi Murray,
Can you please explain why the limits of integration change from [-500, 500] to [0,40]? Also what software did you use to compute those complex integrals, Mathematica doesn't seem to work when I
tried, I even entered your expression and got a different value.
Murray says:
15 Mar 2015 at 3:54 pm [Comment permalink]
@Daniel: I added an explanation in the post, just under the first time I changed those limits. In the PDF of the complete solution, you'll see I've done a similar thing for the T wave portion, t=200
to t=360.
I was using Scientific Notebook, which at the time used Maple engine. With such examples, the proof is in the pudding - if it looks right, it probably is right! (That is, I wanted a "blip" from 0 to
40 that went up to around 2.5 on the V axis, and that's what I got. You know immediately if your calculations are wrong because the graph will look all wrong.)
Like many trigonometric calculations, the form of the answer could be quite different, but has equivalent value.
All the best with it!
hajar says:
8 Dec 2015 at 12:16 am [Comment permalink]
what is the partial differential equation for this wave ?
is there?
and The initial value problem marginal?
I want search about partial differential equation and solve it by Fourier Series
Phillip says:
7 Jan 2016 at 10:02 am [Comment permalink]
I'm attempting to graph the P wave, and I have a model equation for it, however I cannot find any website or calculator than can integrate the complicated term for your an coefficient.. what did you
use to calculate it?
Murray says:
7 Jan 2016 at 11:24 am [Comment permalink]
@Phillip We actually need to integrate for each different value of n = 1, 2, 3, ...
So a[1] will use n = 1 and so we need to find
We can use Wolfram|alpha for this:
Hope it helps!
Phillip says:
7 Jan 2016 at 11:30 am [Comment permalink]
it very much helped - i made the mistake of using 1/x as a function as part of my piece wise function when i could have just used a positive quadratic function to model the same basic curve. Turns
out if you go along with 1/x, the math needed to calculate the "an" coefficient is too much for wolfram alpha to compute ...LOL
James Jones says:
9 Feb 2016 at 7:33 pm [Comment permalink]
Hi, could you please explain why we need to use the value 20 in the function expression of the f(t) in f(t) = -0.0000156(t-20)^4 + 2.5??
Murray says:
9 Feb 2016 at 7:51 pm [Comment permalink]
@James: In the "Explanation of the model" section, it says:
The (t − 20) term comes from deciding the curve should start at (0,0), (which makes our lives easier), it will pass through (40,0) since the pulse is 40 ms long, and be centered on t = 20.
That's where the 20 comes from - it's half of the width of the "blip" which goes from 0 to 40.
James Jones says:
9 Feb 2016 at 8:35 pm [Comment permalink]
thank you for the quick reply in the previous comment but can you also explain where you got the period value of 1000ms too? Because I thought the period for the R wave region was 40...
Murray says:
9 Feb 2016 at 8:49 pm [Comment permalink]
The period is 1000 ms because I'm assuming my heart rate is 60 beats per minute, or 1 beat per second.
The R wave "blip" lasts for 40 ms, representing the biggest pumping motion, then has a rest for the next 960 ms (while the heart is doing something else). That pulse repeats every 1000 ms.
Dequan says:
12 Feb 2016 at 12:50 am [Comment permalink]
Hi Mr. Murray
Could you explain how you did the integration when solving for an and bn because looking at the attached wolframalpha links, I see that the value for the integration when n is 1,2,3,4,... is all
Regards, DeQuan
Murray says:
12 Feb 2016 at 10:40 am [Comment permalink]
@DeQuan: Thanks for alerting me - I realised those Wolfram|Alpha links had a bracket in the wrong place. I've updated them now.
Dequan says:
12 Feb 2016 at 10:42 am [Comment permalink]
Hmmm I'm sorry but could you please explain how you were able to deduce the integration of an using the integration of each of the n=1,2,3,...?
Murray says:
12 Feb 2016 at 3:05 pm [Comment permalink]
@Dequan: I'm not deducing the integration. I'm just applying the formulas that can be found here: https://www.intmath.com/fourier-series/2-full-range-fourier-series.php
The PDF (linked to in the article) may give you a better idea of what is going on.
Dequan says:
21 Feb 2016 at 2:43 am [Comment permalink]
Is it possible for you to give a clearer explanation in the way that you were able to get the values for the coefficient an? did you generate an arithmetic sequence using the idea that an is simply
the sum function for the terms of an? If not, how were u able to get the "n" in the an function?
Murray says:
21 Feb 2016 at 12:57 pm [Comment permalink]
@Dequan: As I mention in the article, even the answer is quite ugly, let alone all the integration steps involved to get to that answer.
Did you have a look at the PDF solution yet? It shows the general terms for the an and bn coefficients. I haven't gone into the algebra involved because this is certainly a place where we should use
a computer algebra system and not spend significant chunks of our lives doing it on paper. The integral will involve several integration by parts steps. (see https://www.intmath.com/
methods-integration/7-integration-by-parts.php )
The "n" in the Fourier Series expansion just means we need to use integers.
So a0 means we replace all n's with 0,
a1 means we replace all n's with 1,
a2 means replace n's with 2, etc.
You can see how this works in the table in the answer for Example 1 (where I'm finding a0, a1, a2, etc) on this page:
Hope it helps.
Ria T says:
18 Dec 2017 at 8:00 pm [Comment permalink]
I wanted to ask if you could possibly share a short tutorial or give an explanation as to how you managed to graph the equation. I am currently using scientific notebook version 6, but whenever I
type in the equation obtained and click on graph, the graph simply appears blank. I'm not sure if the equation is too complex for the software to process? I typed in the same equation that you have
used but the same keeps happening. Kindly let me know if you know of a possible solution to this issue- if not, it would be great if you could let me know which version of the software you had used
in this article.
Thank you,
Murray says:
19 Dec 2017 at 10:10 am [Comment permalink]
@Ria: I think I was using a version of Scientific Notebook that had Maple as its processing engine when I wrote that article (and produced the graph outputs). When they changed to MuPad, I was never
as happy with the results (both computationally and aesthetically).
One thing I found is that SNB doesn't like square brackets [ ] in expressions (both for computations and graphing). If you have any, try changing them to ordinary parentheses ( ).
Another thing I've tried when facing the "blank graph with no error message to tell you why" problem, is to build up the pieces one at a time until it breaks, then try to rewrite it in some way so it
So for example, try to graph
Then try to graph
Another thing I've resorted to is to expand out inner items (like that integral) first, then substitute that into the expression. So like with humans, if we break it down into its component parts and
do those separately, it (might) work!
I tried it again just now in v5.5 using MuPad and it worked fine.
One final thought - it won't handle infinity as the upper limit for the sum! Try n=1 to 5 as a starter.
Good luck.
Murray says:
21 Dec 2017 at 9:27 am [Comment permalink]
@Ria: I downloaded a trial version of SNB6 for interest.
I haven't managed to graph the heart beat successfully yet, but found it was happier with x as the variable, rather than t.
Singh says:
5 Feb 2018 at 3:23 am [Comment permalink]
Hi, could you please show all the steps involved when you plotted the graph on the scientific notebook because I am unable to plot the graph.
Please reply ASAP.
Murray says:
5 Feb 2018 at 7:30 am [Comment permalink]
@Singh: There was nothing special about it - I just clicked the "2D" plot button.
What version of Scientific Notebook are you using? It works fine for me in v5.5, but I also couldn't plot it using v6.
Amarnath says:
5 Nov 2018 at 9:55 am [Comment permalink]
P wave is caused by the atrial repolarization which is relative to the first half by atrial contraction of right atrium and the second half by contraction of left atrium.
Qrs complex is due to ventricular repolarization
And t wave is due to ventricular repolarization.
Thatyougoon says:
12 Dec 2018 at 6:19 am [Comment permalink]
Why do you approximate your ECG? You know you can just perform a Fourier transform in discrete time right? Can't you just directly transfer the data collected to a matrix/array and performing Fourier
transform on it computationally? I just... I don't know why you went through all this trouble to approximate your data (which resulted in a grave loss of detail), so you could approximate it
Murray says:
12 Dec 2018 at 8:50 am [Comment permalink]
@Thatyougoon: The aim (as per the sub-heading in the article) was to model heartbeats in general, producing a periodic function that could be graphed. Doing a Fourier analysis of the real data was
not the aim - producing a generalised heartbeat-looking function was.
v26 says:
30 Jun 2019 at 10:28 pm [Comment permalink]
hi MR, I have a question, it is possible to use two signal at the same time? and why?
Murray says:
1 Jul 2019 at 8:06 am [Comment permalink]
@v26: Fourier series is, in fact, the combination of 2 or more signals at the same time. So it appears the answer to your question would be yes.
Julia says:
25 Jul 2019 at 8:40 am [Comment permalink]
Hello Sir,
I have been attempting to graph my own heartbeat with Scientific Notebook. However, I am only using the Free Trial (which should be enough). For some reason, I only obtain a straight line when I
graph the series for one wave. The function I am working with is f(x)=-(-0.008x+0.324)^2+0.105, the integral boundaries are from 0 to 80. Could you give me instructions as what to do on Scientific
Notebook to obtain a satisfactory graph? I would be so thankful.
Murray says:
25 Jul 2019 at 11:13 am [Comment permalink]
@Julia: Please see my response to Ria above, as she was facing similar problems.
Could you graph your base function successfully? (I mean -(-0.008x+0.324)^2+0.105) As suggested to Ria, start from something that does work, then add complications until it breaks.
Two other thoughts:
(1) I was not impressed with the latest version of Scientific Notebook and did not upgrade my version. (That doesn't help you, I'm sorry, but it may be a version thing - it did work on earlier
(2) You may also like to try other computer algebra solutions, like SageMath or similar.
SM says:
11 Aug 2019 at 3:59 pm [Comment permalink]
Is it possible to model the individual parts of the ECG (P,Q,R,S and T) in order to determine their functions?
Murray says:
12 Aug 2019 at 2:08 pm [Comment permalink]
@SM: Yes indeed, and that's what I'm doing in this article. The PDF has more details about the process.
Sarthak says:
11 Sep 2019 at 4:54 pm [Comment permalink]
Can we use linear piecewise functions in order to model the QRS complex?
Murray says:
11 Sep 2019 at 8:12 pm [Comment permalink]
@Sarthak: I believe you could, but I think the Fourier Series approach would be more appropriate.
Sarthak says:
15 Sep 2019 at 2:06 pm [Comment permalink]
How did you change the x-axis scale in Scientific Notebook (v5.5)?
Murray says:
15 Sep 2019 at 2:59 pm [Comment permalink]
@Sarthak: From the Doing Math with Scientific Notebook PDF:
"From the Axes page of the Plot Properties dialog you can choose the type of axes displayed and the scaling for the axes."
Hope it helps.
Jim says:
3 Dec 2020 at 1:29 am [Comment permalink]
Interesting article. Have you ever seen a model oscillator for the heart's electrical system?
J says:
16 Feb 2021 at 5:10 pm [Comment permalink]
Thank you so much for the above. I just have a quick question. In many other sources the equation for a0 starts with 1/2L and when I tried to derive/prove the a0 equation I also got 1/2L. So I was
wondering where the 1/L comes from.
Thank you.
Fathima says:
2 Jun 2022 at 11:51 pm [Comment permalink]
Could anyone share P,R,S, waves like in graph for the values used according to the article?!! Actually i tried but graphing uis like a problem for me! | {"url":"https://www.intmath.com/blog/mathematics/math-of-ecgs-fourier-series-4281","timestamp":"2024-11-08T15:37:01Z","content_type":"text/html","content_length":"175556","record_id":"<urn:uuid:726ddb2e-4779-4593-b4f6-ac50f4d63ede>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00184.warc.gz"} |
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1 Ton/Cubic Kilometer (Metric) = 1e-9 Gram/Cubic Centimeter 1 Ton/Cubic Kilometer (Metric) in Gram/Cubic Centimeter is equal to 1e-9
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1 Ton/Cubic Kilometer (Metric) = 1.461601912275e-7 Troy Ounce/Gallon [UK] 1 Ton/Cubic Kilometer (Metric) in Troy Ounce/Gallon [UK] is equal to 1.461601912275e-7
1 Ton/Cubic Kilometer (Metric) = 1.2170381805558e-7 Troy Ounce/Gallon [US] 1 Ton/Cubic Kilometer (Metric) in Troy Ounce/Gallon [US] is equal to 1.2170381805558e-7
1 Ton/Cubic Kilometer (Metric) = 1.6035860567937e-7 Ounce/Gallon [UK] 1 Ton/Cubic Kilometer (Metric) in Ounce/Gallon [UK] is equal to 1.6035860567937e-7
1 Ton/Cubic Kilometer (Metric) = 1.3352647123231e-7 Ounce/Gallon [US] 1 Ton/Cubic Kilometer (Metric) in Ounce/Gallon [US] is equal to 1.3352647123231e-7
1 Ton/Cubic Kilometer (Metric) = 6.2427960576145e-8 Pound/Cubic Feet 1 Ton/Cubic Kilometer (Metric) in Pound/Cubic Feet is equal to 6.2427960576145e-8
1 Ton/Cubic Kilometer (Metric) = 3.6127292000084e-11 Pound/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Pound/Cubic Inch is equal to 3.6127292000084e-11
1 Ton/Cubic Kilometer (Metric) = 0.0000016855549355559 Pound/Cubic Yard 1 Ton/Cubic Kilometer (Metric) in Pound/Cubic Yard is equal to 0.0000016855549355559
1 Ton/Cubic Kilometer (Metric) = 1.0022412854961e-8 Pound/Gallon [UK] 1 Ton/Cubic Kilometer (Metric) in Pound/Gallon [UK] is equal to 1.0022412854961e-8
1 Ton/Cubic Kilometer (Metric) = 8.3454044520193e-9 Pound/Gallon [US] 1 Ton/Cubic Kilometer (Metric) in Pound/Gallon [US] is equal to 8.3454044520193e-9
1 Ton/Cubic Kilometer (Metric) = 1.9403207224936e-9 Slug/Cubic Feet 1 Ton/Cubic Kilometer (Metric) in Slug/Cubic Feet is equal to 1.9403207224936e-9
1 Ton/Cubic Kilometer (Metric) = 1.1228707884801e-12 Slug/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Slug/Cubic Inch is equal to 1.1228707884801e-12
1 Ton/Cubic Kilometer (Metric) = 5.2388659507328e-8 Slug/Cubic Yard 1 Ton/Cubic Kilometer (Metric) in Slug/Cubic Yard is equal to 5.2388659507328e-8
1 Ton/Cubic Kilometer (Metric) = 3.1150617723844e-10 Slug/Gallon [UK] 1 Ton/Cubic Kilometer (Metric) in Slug/Gallon [UK] is equal to 3.1150617723844e-10
1 Ton/Cubic Kilometer (Metric) = 2.5938315213891e-10 Slug/Gallon [US] 1 Ton/Cubic Kilometer (Metric) in Slug/Gallon [US] is equal to 2.5938315213891e-10
1 Ton/Cubic Kilometer (Metric) = 2.7869625257207e-11 Ton/Cubic Feet [Long] 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Feet [Long] is equal to 2.7869625257207e-11
1 Ton/Cubic Kilometer (Metric) = 3.1213980288072e-11 Ton/Cubic Feet [Short] 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Feet [Short] is equal to 3.1213980288072e-11
1 Ton/Cubic Kilometer (Metric) = 1.612825535718e-14 Ton/Cubic Inch [Long] 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Inch [Long] is equal to 1.612825535718e-14
1 Ton/Cubic Kilometer (Metric) = 1.8063646000042e-14 Ton/Cubic Inch [Short] 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Inch [Short] is equal to 1.8063646000042e-14
1 Ton/Cubic Kilometer (Metric) = 7.524798819446e-10 Ton/Cubic Yard [Long] 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Yard [Long] is equal to 7.524798819446e-10
1 Ton/Cubic Kilometer (Metric) = 8.4277746777795e-10 Ton/Cubic Yard [Short] 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Yard [Short] is equal to 8.4277746777795e-10
1 Ton/Cubic Kilometer (Metric) = 4.4742914531074e-12 Ton/Gallon [Long, UK] 1 Ton/Cubic Kilometer (Metric) in Ton/Gallon [Long, UK] is equal to 4.4742914531074e-12
1 Ton/Cubic Kilometer (Metric) = 3.7256269875086e-12 Ton/Gallon [Long, US] 1 Ton/Cubic Kilometer (Metric) in Ton/Gallon [Long, US] is equal to 3.7256269875086e-12
1 Ton/Cubic Kilometer (Metric) = 5.0119597890466e-12 Ton/Gallon [Short, UK] 1 Ton/Cubic Kilometer (Metric) in Ton/Gallon [Short, UK] is equal to 5.0119597890466e-12
1 Ton/Cubic Kilometer (Metric) = 4.1727022260097e-12 Ton/Gallon [Short, US] 1 Ton/Cubic Kilometer (Metric) in Ton/Gallon [Short, US] is equal to 4.1727022260097e-12
1 Ton/Cubic Kilometer (Metric) = 1e-15 Tonne/Cubic Centimeter 1 Ton/Cubic Kilometer (Metric) in Tonne/Cubic Centimeter is equal to 1e-15
1 Ton/Cubic Kilometer (Metric) = 1e-12 Tonne/Cubic Decimeter 1 Ton/Cubic Kilometer (Metric) in Tonne/Cubic Decimeter is equal to 1e-12
1 Ton/Cubic Kilometer (Metric) = 1e-9 Tonne/Cubic Meter 1 Ton/Cubic Kilometer (Metric) in Tonne/Cubic Meter is equal to 1e-9
1 Ton/Cubic Kilometer (Metric) = 1e-12 Tonne/Liter 1 Ton/Cubic Kilometer (Metric) in Tonne/Liter is equal to 1e-12
1 Ton/Cubic Kilometer (Metric) = 1e-15 Tonne/Milliliter 1 Ton/Cubic Kilometer (Metric) in Tonne/Milliliter is equal to 1e-15
1 Ton/Cubic Kilometer (Metric) = 1e-9 Water Density 1 Ton/Cubic Kilometer (Metric) in Water Density is equal to 1e-9
1 Ton/Cubic Kilometer (Metric) = 1000 Kilogram/Cubic Kilometer 1 Ton/Cubic Kilometer (Metric) in Kilogram/Cubic Kilometer is equal to 1000
1 Ton/Cubic Kilometer (Metric) = 1000000 Gram/Cubic Kilometer 1 Ton/Cubic Kilometer (Metric) in Gram/Cubic Kilometer is equal to 1000000
1 Ton/Cubic Kilometer (Metric) = 1e-12 Gram/Cubic Millimeter 1 Ton/Cubic Kilometer (Metric) in Gram/Cubic Millimeter is equal to 1e-12
1 Ton/Cubic Kilometer (Metric) = 1e-12 Gram/Microliter 1 Ton/Cubic Kilometer (Metric) in Gram/Microliter is equal to 1e-12
1 Ton/Cubic Kilometer (Metric) = 5e-11 Gram/Drop 1 Ton/Cubic Kilometer (Metric) in Gram/Drop is equal to 5e-11
1 Ton/Cubic Kilometer (Metric) = 1e-9 Milligram/Cubic Millimeter 1 Ton/Cubic Kilometer (Metric) in Milligram/Cubic Millimeter is equal to 1e-9
1 Ton/Cubic Kilometer (Metric) = 1e-9 Milligram/Microliter 1 Ton/Cubic Kilometer (Metric) in Milligram/Microliter is equal to 1e-9
1 Ton/Cubic Kilometer (Metric) = 0.000001 Microgram/Millimeter 1 Ton/Cubic Kilometer (Metric) in Microgram/Millimeter is equal to 0.000001
1 Ton/Cubic Kilometer (Metric) = 0.000001 Microgram/Microliter 1 Ton/Cubic Kilometer (Metric) in Microgram/Microliter is equal to 0.000001
1 Ton/Cubic Kilometer (Metric) = 1 Nanogram/Millimeter 1 Ton/Cubic Kilometer (Metric) in Nanogram/Millimeter is equal to 1
1 Ton/Cubic Kilometer (Metric) = 0.001 Nanogram/Microliter 1 Ton/Cubic Kilometer (Metric) in Nanogram/Microliter is equal to 0.001
1 Ton/Cubic Kilometer (Metric) = 1000 Picogram/Millimeter 1 Ton/Cubic Kilometer (Metric) in Picogram/Millimeter is equal to 1000
1 Ton/Cubic Kilometer (Metric) = 1 Picogram/Microliter 1 Ton/Cubic Kilometer (Metric) in Picogram/Microliter is equal to 1
1 Ton/Cubic Kilometer (Metric) = 1e-9 Ton/Cubic Meter (Metric) 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Meter (Metric) is equal to 1e-9
1 Ton/Cubic Kilometer (Metric) = 5e-9 Carat/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Carat/Cubic Inch is equal to 5e-9
1 Ton/Cubic Kilometer (Metric) = 0.000001 Gamma/Cubic Millimeter 1 Ton/Cubic Kilometer (Metric) in Gamma/Cubic Millimeter is equal to 0.000001
1 Ton/Cubic Kilometer (Metric) = 0.001 Gamma/Milliliter 1 Ton/Cubic Kilometer (Metric) in Gamma/Milliliter is equal to 0.001
1 Ton/Cubic Kilometer (Metric) = 0.000001 Gamma/Microliter 1 Ton/Cubic Kilometer (Metric) in Gamma/Microliter is equal to 0.000001
1 Ton/Cubic Kilometer (Metric) = 6.8521779647661e-8 Slug/Cubic Meter 1 Ton/Cubic Kilometer (Metric) in Slug/Cubic Meter is equal to 6.8521779647661e-8
1 Ton/Cubic Kilometer (Metric) = 2.2046226218488e-9 Pound/Liter 1 Ton/Cubic Kilometer (Metric) in Pound/Liter is equal to 2.2046226218488e-9
1 Ton/Cubic Kilometer (Metric) = 3.527396194958e-8 Ounce/Liter 1 Ton/Cubic Kilometer (Metric) in Ounce/Liter is equal to 3.527396194958e-8
1 Ton/Cubic Kilometer (Metric) = 0.000015432358352941 Grain/Liter 1 Ton/Cubic Kilometer (Metric) in Grain/Liter is equal to 0.000015432358352941
1 Ton/Cubic Kilometer (Metric) = 0.0001 Milligram/Deciliter 1 Ton/Cubic Kilometer (Metric) in Milligram/Deciliter is equal to 0.0001
1 Ton/Cubic Kilometer (Metric) = 0.1 Microgram/Deciliter 1 Ton/Cubic Kilometer (Metric) in Microgram/Deciliter is equal to 0.1
1 Ton/Cubic Kilometer (Metric) = 0.000016387064 Milligram/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Milligram/Cubic Inch is equal to 0.000016387064
1 Ton/Cubic Kilometer (Metric) = 1.6387064e-8 Gram/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Gram/Cubic Inch is equal to 1.6387064e-8
1 Ton/Cubic Kilometer (Metric) = 0.016387064 Microgram/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Microgram/Cubic Inch is equal to 0.016387064
1 Ton/Cubic Kilometer (Metric) = 7.64554857984e-7 Kilogram/Cubic Yard 1 Ton/Cubic Kilometer (Metric) in Kilogram/Cubic Yard is equal to 7.64554857984e-7
1 Ton/Cubic Kilometer (Metric) = 2.8316846592e-8 Kilogram/Cubic Feet 1 Ton/Cubic Kilometer (Metric) in Kilogram/Cubic Feet is equal to 2.8316846592e-8
1 Ton/Cubic Kilometer (Metric) = 7.64554857984e-10 Ton/Cubic Yard (Metric) 1 Ton/Cubic Kilometer (Metric) in Ton/Cubic Yard (Metric) is equal to 7.64554857984e-10
1 Ton/Cubic Kilometer (Metric) = 0.016387064 Gamma/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Gamma/Cubic Inch is equal to 0.016387064
1 Ton/Cubic Kilometer (Metric) = 8.193532e-8 Carat/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Carat/Cubic Inch is equal to 8.193532e-8
1 Ton/Cubic Kilometer (Metric) = 1 Part/Billion 1 Ton/Cubic Kilometer (Metric) in Part/Billion is equal to 1
1 Ton/Cubic Kilometer (Metric) = 1e-7 Percent 1 Ton/Cubic Kilometer (Metric) in Percent is equal to 1e-7
1 Ton/Cubic Kilometer (Metric) = 0.001 Part/Million 1 Ton/Cubic Kilometer (Metric) in Part/Million is equal to 0.001
1 Ton/Cubic Kilometer (Metric) = 2.2046226218488e-12 Kilopound/Liter 1 Ton/Cubic Kilometer (Metric) in Kilopound/Liter is equal to 2.2046226218488e-12
1 Ton/Cubic Kilometer (Metric) = 1.6855549355559e-9 Kilopound/Cubic Yard 1 Ton/Cubic Kilometer (Metric) in Kilopound/Cubic Yard is equal to 1.6855549355559e-9
1 Ton/Cubic Kilometer (Metric) = 6.2427960576145e-11 Kilopound/Cubic Feet 1 Ton/Cubic Kilometer (Metric) in Kilopound/Cubic Feet is equal to 6.2427960576145e-11
1 Ton/Cubic Kilometer (Metric) = 3.6127292000084e-14 Kilopound/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Kilopound/Cubic Inch is equal to 3.6127292000084e-14
1 Ton/Cubic Kilometer (Metric) = 7.0988848424626e-8 Poundal/Liter 1 Ton/Cubic Kilometer (Metric) in Poundal/Liter is equal to 7.0988848424626e-8
1 Ton/Cubic Kilometer (Metric) = 0.000054274868925738 Poundal/Cubic Yard 1 Ton/Cubic Kilometer (Metric) in Poundal/Cubic Yard is equal to 0.000054274868925738
1 Ton/Cubic Kilometer (Metric) = 0.0000020101803305829 Poundal/Cubic Feet 1 Ton/Cubic Kilometer (Metric) in Poundal/Cubic Feet is equal to 0.0000020101803305829
1 Ton/Cubic Kilometer (Metric) = 1.1632988024206e-9 Poundal/Cubic Inch 1 Ton/Cubic Kilometer (Metric) in Poundal/Cubic Inch is equal to 1.1632988024206e-9
1 Ton/Cubic Kilometer (Metric) = 1e-24 Exagram/Liter 1 Ton/Cubic Kilometer (Metric) in Exagram/Liter is equal to 1e-24
1 Ton/Cubic Kilometer (Metric) = 1e-21 Petagram/Liter 1 Ton/Cubic Kilometer (Metric) in Petagram/Liter is equal to 1e-21
1 Ton/Cubic Kilometer (Metric) = 1e-18 Teragram/Liter 1 Ton/Cubic Kilometer (Metric) in Teragram/Liter is equal to 1e-18
1 Ton/Cubic Kilometer (Metric) = 1e-15 Gigagram/Liter 1 Ton/Cubic Kilometer (Metric) in Gigagram/Liter is equal to 1e-15
1 Ton/Cubic Kilometer (Metric) = 1e-8 Hectogram/Liter 1 Ton/Cubic Kilometer (Metric) in Hectogram/Liter is equal to 1e-8
1 Ton/Cubic Kilometer (Metric) = 1e-7 Dekagram/Liter 1 Ton/Cubic Kilometer (Metric) in Dekagram/Liter is equal to 1e-7
1 Ton/Cubic Kilometer (Metric) = 0.00001 Decigram/Liter 1 Ton/Cubic Kilometer (Metric) in Decigram/Liter is equal to 0.00001
1 Ton/Cubic Kilometer (Metric) = 0.0001 Centigram/Liter 1 Ton/Cubic Kilometer (Metric) in Centigram/Liter is equal to 0.0001
1 Ton/Cubic Kilometer (Metric) = 1000000000000 Attogram/Liter 1 Ton/Cubic Kilometer (Metric) in Attogram/Liter is equal to 1000000000000
1 Ton/Cubic Kilometer (Metric) = 4.33527504001e-7 PSI/1000 feet 1 Ton/Cubic Kilometer (Metric) in PSI/1000 feet is equal to 4.33527504001e-7
1 Ton/Cubic Kilometer (Metric) = 1.8122508155129e-10 Earth Density 1 Ton/Cubic Kilometer (Metric) in Earth Density is equal to 1.8122508155129e-10
1 Ton/Cubic Kilometer (Metric) = 1e-7 Kilogram/Hectoliter 1 Ton/Cubic Kilometer (Metric) in Kilogram/Hectoliter is equal to 1e-7
1 Ton/Cubic Kilometer (Metric) = 0.0001 Gram/Hectoliter 1 Ton/Cubic Kilometer (Metric) in Gram/Hectoliter is equal to 0.0001
1 Ton/Cubic Kilometer (Metric) = 7.7688850894913e-8 Pound/Bushel [US] 1 Ton/Cubic Kilometer (Metric) in Pound/Bushel [US] is equal to 7.7688850894913e-8
1 Ton/Cubic Kilometer (Metric) = 8.0179302839684e-8 Pound/Bushel [UK] 1 Ton/Cubic Kilometer (Metric) in Pound/Bushel [UK] is equal to 8.0179302839684e-8
1 Ton/Cubic Kilometer (Metric) = 0.0000012430216143186 Ounce/Bushel [US] 1 Ton/Cubic Kilometer (Metric) in Ounce/Bushel [US] is equal to 0.0000012430216143186
1 Ton/Cubic Kilometer (Metric) = 0.0000012828688454349 Ounce/Bushel [UK] 1 Ton/Cubic Kilometer (Metric) in Ounce/Bushel [UK] is equal to 0.0000012828688454349 | {"url":"https://www.kodytools.com/units/density/from/tnmetricpkm3","timestamp":"2024-11-06T08:57:25Z","content_type":"text/html","content_length":"162132","record_id":"<urn:uuid:e1276f0a-ab39-4d62-b3bd-f77c2f4560af>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00388.warc.gz"} |
Functional programming with lambda.r | R-bloggersFunctional programming with lambda.r
Functional programming with lambda.r
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After a four month simmer on various back burners and package conflicts, I’m pleased to announce that the successor to futile.paradigm is officially available on CRAN. The new package is lambda.r
(source on github), which hopefully conveys the purpose of the library better than its whimsical predecessor. In some ways this new version deserves a more serious name as the package has matured
quite a bit not to mention is part and parcel of a book I’m writing on computational systems and functional programming.
So what exactly is lambda.r? Put simply, lambda.r gives you functional programming in the R language. While R has many functional features built into the language, application development in R is a
decidedly object-oriented affair. I won’t go into all the reasons why it’s better to write computational systems in a functional paradigm since that is covered in depth in my forthcoming book
“Computational Finance and the Lambda Calculus”. However, here are the salient points:
• Conceptual consistency with mathematics resulting in less translation error between model and system (see my slides from R/Finance 2011)
• Modular and encapsulated architecture that makes growth of a system easier to manage (not to mention easier to accommodate disparate computing needs — think parallel alternatives of the same
processing pipeline)
• Efficiency in application development since wiring is trivial
The fundamental goal of lambda.r is to provide a solid architectural foundation that remains intact through the prototyping and development phases of a model or application. One half is accomplished
with a functional syntax that builds in modularity and encapsulation into every function. The other half is through the incremental adoption of constraints in the system. This article will focus
primarily on the features, and in a separate article I will outline how to best leverage these features as a system matures.
Pattern Matching
Functional languages often use pattern matching to define functions in multiple parts. This syntax is reminiscent of sequences or functions with initial values in addition to multi-part definitions.
Removing control flow from function definitions makes functions easier to understand and reduces the translation error from math to code.
Fibonacci Sequence
For example, the ubiquitous Fibonacci sequence is defined as
$F_{n} = F_{n-1} + F_{n-2}$, where $F_{1} = F_{2} = 1$
In standard R, one way to define this is with an if/else control block or function [1].
fib <- function(n)
ifelse(n < 2, 1, fib(n - 1) + fib(n - 2))
Using lambda.r, pattern matching defines the function in three clauses. The behavior of the function is free of clutter as each clause is self-contained and self-explanatory.
fib(0) %as% 1
fib(1) %as% 1
fib(n) %as% { fib(n-1) + fib(n-2) }
Heaviside Step Function
When represented as a piecewise constant function, the Heaviside step function is defined in three parts. [2]
Using pattern matching in lambda-r, the function can be defined almost verbatim.
h.step(n) %when% { n < 0 } %as% 0
h.step(0) %as% 0.5
h.step(n) %as% 1
In languages that don’t support pattern matching, again if/else control structures are used to implement these sorts of functions, which can get complicated as more cases are added to a function. A
good example of this is the ‘optim’ function in R, which embeds a number of cases within the function definition.
Guard Statements
The last example sneaks in a guard statement along with pattern matching. Guards provide a rich vocabulary to control when a specific function clause is executed. Each guard statement is a logical
expression. Multiple expressions can be present in a guard block, so that the function clause only executes when all the expressions evaluate to TRUE. Using the Fibonacci example above, we can add an
argument check to only allow integers.
fib(0) %as% 1
fib(1) %as% 1
fib(n) %when% {
n > 1
} %as% { fib(n-1) + fib(n-2) }
If none of the clauses are satisfied, lambda.r will complain telling you that it couldn’t find a matching function clause.
> fib(2)
Error in UseFunction("fib", ...) : No valid function for 'fib(2)'
> fib(as.integer(2))
[1] 2
Note: If you are running the examples as you are reading along, then you need to either seal() the functions or rm() the current definition prior to redefining the function. The reason is that
function clauses are additive. You can add as many clauses as you want, and they will be evaluated in the order they were declared. Since lambda.r has no way of knowing when you are done defining
your function you must explicitly tell it via the seal() function.
Custom types can be defined in lambda.r. These types can be used in type constraints to provide type safety and distinguish one function clause from another. All types must be defined using
Type Constructors
A type constructor is simply a function that creates a type. The name of the function is the name of the type. The return value will automatically be typed while also preserving existing type
information. This means that you can create type hierarchies as needed.
Point(x,y) %as% list(x=x,y=y)
Polar(r,theta) %as% list(r=r,theta=theta)
In this example we use a list as the underlying data structure. To create an instance of this type simply call the constructor.
point.1 <- Point(2,3)
point.2 <- Point(5,7)
Under the hood lambda.r leverages the S3 class mechanism, which means that lambda.r types are compatible with S3 classes.
Type Constraints
Types by themselves aren’t all that interesting. Once we define the types, they can be used as constraints on a function.
distance(a,b) %::% Point : Point : numeric
distance(a,b) %as% { ((a$x - b$x)^2 + (a$y - b$y)^2)^.5 }
distance(a,b) %::% Polar : Polar : numeric
distance(a,b) %as%
(a$r^2 + b$r^2 - 2 * a$r * b$r * cos(a$theta - b$theta))^.5
As shown above each function clause can have its own constraint. Since type constraints are greedy, a declared constraint will apply to every successive function clause until a new type constraint is
> distance(point.1, point.2)
[1] 5
Types are great for adding structure and safety to an application. However types can have diminishing returns as more types are introduced. In general lambda.r advocates using existing data
structures where possible to minimize type clutter. Of course if data.frames and matrices are used for most operations, how do you differentiate function clauses? The answer of course are attributes,
which come standard with R. Attributes can be considered meta-data that is orthogonal to the core data structure. They are preserved during operations, so can be carried through a process. Lambda.r
makes working with attributes so easy that they should become second nature fairly quickly.
With lambda.r you can access attributes via the ‘@’ symbol. Define them in a type constructor as shown below.
Temperature(x, system='metric', units='celsius') %as%
x@system <- system
x@units <- units
Function clauses can now be defined based on the value of an attribute.
freezing(x) %::% Temperature : logical
freezing(x) %when% {
x@system == 'metric'
x@units == 'celsius'
} %as% {
if (x < 0) { TRUE }
else { FALSE }
freezing(x) %when% {
x@system == 'metric'
x@units == 'kelvin'
} %as% {
if (x < 273) { TRUE }
else { FALSE }
It is trivial then to check whether a given temperature is freezing, based on the units. This approach can be extended to objects like covariance matrices to preserve information that is normally
lost in the creation of the matrix (e.g. number of observations).
ctemp <- Temperature(20)
Note that the Temperature type extends the type of ‘x’, so it is also a numeric value. This means that you can add a scalar to a Temperature object, and everything behaves as you would expect.
> ctemp1 <- ctemp - 21
> freezing(ctemp1)
[1] TRUE
The combination of types and attributes are two essential tools in the lambda.r toolkit. In this section I’ve also illustrated how S3 classes can naturally be mixed and matched with lambda.r classes.
The goal of lambda.r is to provide rich functionality with a simple and intuitive syntax. To accomplish this there is a lot of wiring behind the scenes. While most of the implementation can safely be
ignored, there are times when it is necessary to look under the hood for troubleshooting purposes. Lambda.r provides a number of tools to make debugging and introspection as simple as possible.
The default output of a lambda.r function or type gives a summary view of the function clauses associated with this function. This is an abridged view to prevent long code listings from obscuring the
high level summary. Any type constraints and guard statements are included in this display as well as default values.
> freezing
freezing(x) %::% Temperature:logical
freezing(x) %when% {
x@system == "metric"
x@units == "celsius"
} %as% ...
freezing(x) %::% Temperature:logical
freezing(x) %when% {
x@system == "metric"
x@units == "kelvin"
} %as% ...
Index values prefix each function clause. Use this key when looking up the definition of an explicit function clause with the ‘describe’ function.
> describe(freezing,2)
function(x) { if ( x < 273 ) {
else {
} }
<environment: 0x7f8cfcd7de60>
Since lambda.r implements its own dispatching function (UseFunction), you cannot use the standard ‘debug’ function to debug a function clause. Instead use the supplied ‘debug.lr’ and ‘undebug.lr’.
These functions will allow you to step through any of the function clauses within a lambda.r function.
All examples are in the source package as unit tests. Below are some highlights to give you an idea of how to use the package.
Prices and Returns
This example shows how to use attributes to limit the scope of a function for specific types of data. Note that the definition of Prices makes no restriction on series, so this definition is
compatible with a vector or data.frame as the underlying data structure.
Prices(series, asset.class='equity', periodicity='daily') %as%
[email protected] <- asset.class
series@periodicity <- periodicity
returns(x) %when% {
[email protected] == "equity"
x@periodicity == "daily"
} %as% {
x[2:length(x)] / x[1:(length(x) - 1)] - 1
Taylor Approximation
This is a simple numerical implementation of a Taylor approximation.
fac(1) %as% 1
fac(n) %when% { n > 0 } %as% { n * fac(n - 1) }
d(f, 1, h=10^-9) %as% function(x) { (f(x + h) - f(x - h)) / (2*h) }
d(f, 2, h=10^-9) %as% function(x) { (f(x + h) - 2*f(x) + f(x - h)) / h^2 }
taylor(f, a, step=2) %as% taylor(f, a, step, 1, function(x) f(a))
taylor(f, a, 0, k, g) %as% g
taylor(f, a, step, k, g) %as% {
df <- d(f,k)
g1 <- function(x) { g(x) + df(a) * (x - a)^k / fac(k) }
taylor(f, a, step-1, k+1, g1)
Use the following definitions like so:
> f <- taylor(sin, pi)
> v <- f(3.1) | {"url":"https://www.r-bloggers.com/2012/11/functional-programming-with-lambda-r/","timestamp":"2024-11-10T21:48:49Z","content_type":"text/html","content_length":"108140","record_id":"<urn:uuid:4ac6a87b-cea6-4e9f-8ce0-1a6999efb8ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00698.warc.gz"} |
Analytica User FAQs/Expressions - How to
(In progress -- two things need to occur. Common questions need to be collected here and organized, and then answers need to be added. Feel free to contribute to either aspect)
Array Abstraction
How do I access a single row of an array?
See Subscript/Slice Operator. The syntax is A[I=x], where A is the array, I is the index that identifies the row dimension, and x identifies the row using the value or label from I.
How do I represent a square matrix?
The rule in Analytica is that every dimension must correspond to a different index. So when you have two dimensions, you need two separate indexes. In the case of a square matrix, the second index is
often a copy of the first.
Try this. Create an index node, name it State, and define it as 1..5. How create a second index, name it State2, and define it as CopyIndex(State). Now you have two indexes with identical value. You
have what you need to represent a square matrix. Finally, create a variable, name it State_Identity, define it as State=State2, and show the result table. You just created a square identity matrix.
How do I re-index an array, exchanging one index, I, for another of the same length, J?
Again, this is accomplished using the Subscript/Slice Operator. Two methods:
The first re-indexes associatively, the second re-indexes by position. You can use the first when I and J have the same labels or values with no duplicates (in general, it is bad practice to have
duplicate values in indexes). The second can be used if they are the same length, but have different values.
How do I aggregate an array from a fine grain index (e.g., days) to a coarser index (e.g., months)?
Generally, to aggregate, you will need a mapping (many-to-one) from the fine grain to the coarse grain. This mapping will take the form of an array, indexed by the fine grain (e.g., days), where each
value in the array is an element in the coarser index (e.g., months). For illustration, let's call this Day2Month.
Once you have this map, use the Aggregate function. Here is an example:
Index theDate
Variable Revenue_by_date { indexed by theDate }
Variable Date_to_year_map := DatePart( theDate, "Y" )
Variable Year := 2000..2020
Variable Revenue_by_year := Aggregate(Revenue_by_date, Date_to_year_map, theDate, Year )
See Aggregate for more details.
• How do I generate independent distributions across an index, for example, so that Noise := Normal(0,1) is independent for each time point t?
• How do I define a chance variable so that its uncertainty is correlated with an existing chance variable?
User-Defined Functions
• How do I create my own User-Defined function?
• How do I create a custom distribution function?
How do I model XXX?
How can I represent missing values?
The constant Null is useful to represent a missing value.
Library functions will similarly ignore NULL values if they rely on built-in functions that do that. If you want your user-defined functions to treat Null differently, you can insert appropriate
conditional logic to deal with the missing values in the appropriate fashion. For example:
Variance( MaybeMissing(x), I )
Cumulate( MaybeMissing(x,1), I )
Function MaybeMissing( x ; defX : optional )
Definition: if x=Null then if IsNotSpecified(defX) then 0 else defX else X
How do I go about debugging a model?
See Debugging Hints | {"url":"https://docs.analytica.com/index.php/Analytica_User_FAQs/Expressions_-_How_to","timestamp":"2024-11-09T17:41:24Z","content_type":"text/html","content_length":"27489","record_id":"<urn:uuid:461fb691-c83e-47b4-ae0e-7b74651c7b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00203.warc.gz"} |
Essbase - Functions
Functions are predefined routines that perform specialized calculations and return :
• sets of members
• or sets of data values.
Type of functions
The Essbase functions include more than 100 predefined routines to extend the calculation capabilities of Essbase. Essbase supports the following functions:
• Boolean functions, which provide a conditional test by returning a TRUE or FALSE value
• Mathematical functions, which perform specialized mathematical calculations
• Relationship functions, which look up data values within a database during a calculation based on the position of the current member
• Range functions, which declare a range of members as an argument to another function or to a command
• Financial functions, which perform specialized financial calculations
• Member set functions, which are based on a specified member and which generate lists of members
• Allocation functions, which allocate values that are input at a parent level across child members
• Forecasting functions, which manipulate data for the purpose of smoothing data, interpolating data, or calculating future values
• Statistical functions, which calculate advanced statistics
• Date and time functions, which use date and time characteristics in calculation formulas
• Calculation mode functions, which specify the calculation mode that Essbase uses to calculate a formula
Documentation / Reference | {"url":"https://datacadamia.com/db/essbase/functions","timestamp":"2024-11-08T21:03:08Z","content_type":"text/html","content_length":"192541","record_id":"<urn:uuid:a3644716-67d3-4b86-9c8f-d8f1d3762363>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00737.warc.gz"} |
Epicyclic Train Example
We use the method introduced in Epicyclic Ratio Calculation for determining the final gear ratio of an epicyclic gear train. This method is extremely methodical, which is appropriate since use of
intuition is quite futile with an epicyclic gear train such as the following example.
A plan view of the epicyclic gear train arrangement
A 3D view of the epicyclic gear train arrangement
Unlike the previous section, however, we begin by giving the ARM one positive turn (instead of the entire assembly one positive turn). The technique is to choose an arbitrary rotation in which gear
movements are intuitively obvious. Some iteration may be required to determine an appropriate set of gear movements to obtain the overall gear ratio.
STEP 1 Set up the following table after having thought about a set of movements which are obvious to visualize:
Action Arm A B C D E
Gears locked, arm given one positive turn.
Arm locked in vertical position, C turns CW (negatively) once.
Add the above two columns to obtain the RESULTANT TURNS
STEP 2 Mentally perform the first row action on the gear train and list the turns, which are obviously all +1 (CCW).
Action Arm A B C D E
Gears locked, arm given one positive turn. +1 +1 +1 +1 +1 +1
Arm locked in vertical position, C turns CW (negatively) once.
Add the above two columns to obtain the RESULTANT TURNS
We now must calculate the effects of the second row action. The rotation response of the gears to giving C one negative (CW) turn is determined as follows:
The ARM is stationary:
Gear A turns are calculated as follows. The negative sign is a result of A and B being external gears.
Gear B turns are calculated as follows.
Gear D turns are the same as those for Gear B.
Gear E turns are calculated as follows.
STEP 3 Now, inputting into the table:
Action Arm A B C D E
Gears locked, arm given one positive turn. +1 +1 +1 +1 +1 +1
Arm locked in vertical position, C turns CW (negatively) once. 0 6.7 -2.3 -1 -2.3 -.78
Add the above two columns to obtain the RESULTANT TURNS
STEP 4 Now, utilizing superposition to add the two rows:
Action Arm A B C D E
Gears locked, arm given one positive turn. +1 +1 +1 +1 +1 +1
Arm locked in vertical position, C turns CW (negatively) once. 0 6.7 -2.3 -1 -2.3 -.78
Add the above two columns to obtain the RESULTANT TURNS 1 7.7 -1.3 0 -1.3 .22
The overall ratio of the epicyclic gear train can be derived from the last row, as with the previous example. | {"url":"https://www.efunda.com/designstandards/gears/gears_epicyclic_train.cfm","timestamp":"2024-11-10T23:51:58Z","content_type":"text/html","content_length":"31692","record_id":"<urn:uuid:3acbc97d-8b47-48d8-9d78-efbd903047e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00433.warc.gz"} |
Math Techniques and Strategies
Zombies and mathematics looks like it would be two things that didn't quite go together. Andrew Miller had a project-based learning project about Zombie-based Learning.
With math and zombies most of the material has to do with diseases that increase at an exponential rate. Students could analyze different population centers and predict its spread using exponential
functions. They could determine when everyone is infected and map the spread using the math data they calculate, or even explore the rate of decay. Students could also investigate what happens when a
certain number of people are vaccinated to help prevent the spread.
These are some ideas that have been implemented as part of a PBL project or would be a good entry point for zombie-based learning across the curriculum.
Zombie-based Learning
Featured in today's post is an elementary school level mathematics glossary for English to Spanish glossary. This works great for many teachers who teach in a ELL math classroom or in a high
population of Hispanic students. Some of these glossary terms work great in middle and high school math classrooms.
algebraic expression expresión algebraica
algebraic patterns patrones algebraicos
algebraic relationship relación algebraica
algebraic relationships relaciones algebraicas
algebraically algebraicamente
algorithm algoritmo
distributive property propiedad distributiva
divide dividir
dividend dividendo
divisibility test prueba de divisibilidad
geometric fact hecho geométrico
geometric figure figura geométrica
geometric pattern patrón geométrico
geometric solid sólido geométrico
geometry geometría
line línea
line graph gráfico lineal
line of symmetry línea de simetría
line plot diagrama lineal
line segment segmento lineal
See more glossary terms in spanish here: Glossary
I have recently blogged about Ada Lovelace and her paper doll which could be used as a history piece in a history of mathematics center that students can learn about mathematics and how math was
developed back in the day. Today, are some activities that you can use in your classroom that revolve around paper dolls and mathematics.
Patterns: Figures alternate, for example: right arm up, left arm up, right arm, left arm.
Reflections: Students love that every doll is a flipped copy of the one next to it. Technically, that's called a reflection, one of three kinds of geometry transformation students study in elementary
school, the other two being rotation and translation.
Powers of Two: Fold the paper twice, you get four figures. Fold the paper three times, you get 8 figures. Fold four times, you get 16 figures. Every fold is a power of two of the figures.
Multiplying fractions: Every time you fold, the dolls become half as wide. That's a visual illustration of what it means to multiply a fraction, in this case x 1/2. Or, if the first fold divides the
paper in half, and the second fold divides it in quarters.
Learn more here: Paper Doll Math
View other activities and information here:
Math Manipulatives
Geometry of Folding
I'm trying to piece together a beginning of the year student survey. What are some questions you ask your students?
Please Call Me:
Home Phone:
Parent's Email:
Activities and Hobbies:
Favorite subject in school:
Why do you like math?
Why do you dislike math?
I obviously need more questions, comments are welcome below! | {"url":"https://new-to-teaching.blogspot.com/2013/06/","timestamp":"2024-11-11T16:13:14Z","content_type":"text/html","content_length":"84391","record_id":"<urn:uuid:66d9784c-89c7-4620-a543-a34796a458d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00250.warc.gz"} |
Left and Right, Above and Below Worksheets
Worksheets and No Prep Teaching Resources
Math Worksheets
Left and Right, Above and Below Worksheets
Understanding left and right shouldn't be left to chance, and these exciting worksheets will give students the right ideas to thoroughly ground them in this important topic. These sensational
worksheets cover upper and lower-case letters and numbers in a variety of attractive formats that will be both fun and appealing for students. Other creative worksheets include fill-ins and multiple
choice, so students are never bored as they learn this necessary concept that carries over into so many subjects. | {"url":"https://www.edhelper.com/left_right_above_below.htm","timestamp":"2024-11-06T04:59:32Z","content_type":"text/html","content_length":"27983","record_id":"<urn:uuid:c6c0b5e3-ac4c-4516-9b17-dd85a999f38a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00587.warc.gz"} |
Does Matplotlib have heatmap?
Does Matplotlib have heatmap?
As already mentioned heatmap in matplotlib can be build using imshow function. You can either use random data or a specific dataset. After this imshow function is called where we pass the data,
colormap value and interpolation method (this method basically helps in improving the image quality if used).
How do I create a heat map in Matplotlib?
1. Syntax: matplotlib.pyplot.imshow(X, cmap=None, norm=None, aspect=None, interpolation=None, alpha=None, vmin=None,
2. Syntax: seaborn.heatmap(data, *, vmin=None, vmax=None, cmap=None, center=None, robust=False,annot=None,
3. Syntax: matplotlib.pyplot.pcolormesh(*args, alpha=None, norm=None, cmap=None, vmin=None, vmax=None,
How do you plot a 3D surface plot in Python?
Creating 3D surface Plot The axes3d present in Matplotlib’s mpl_toolkits. mplot3d toolkit provides the necessary functions used to create 3D surface plots. Surface plots are created by using ax.
plot_surface() function.
How do you visualize a 3D matrix in python?
Creating a 3D plot in Matplotlib from a 3D numpy array
1. Create a new figure or activate an existing figure using figure() method.
2. Add an ‘~. axes.
3. Create a random data of size=(3, 3, 3).
4. Extract x, y, and z data from the 3D array.
5. Plot 3D scattered points on the created axis.
6. To display the figure, use show() method.
How do you plot heatmap in pandas?
How to Display Pandas DataFrame As a Heatmap
1. import numpy as np import pandas as pd from pandas. util. testing import makePeriodSeries s = makeTimeSeries(10) cols = [‘col_1’, ‘col_2’] df = pd.
2. df. style.
3. import seaborn as sns sns. heatmap(df[[‘col_1’, ‘col_2’]])
4. sns. heatmap(df.
5. import plotly. express as px fig = px.
How do you draw a heatmap?
1. Step 1: Enter data. Enter the necessary data in a new sheet.
2. Step 2: Select the data. Select the dataset for which you want to generate a heatmap.
3. Step 3: Use conditional formatting.
4. Step 4: Select the color scale.
How do you create a geographic heatmap in python?
Create a geographic heat map of the City of Toronto in Python
1. Download the dataset.
2. Download shape files of Toronto.
3. Download The GeoPandas library to read the shape files.
4. Read The Shape Files using GeoPandas.
5. Create the group-by Count on the dataset.
6. Join the Dataset and Shape file Dataframe.
7. Plot the data.
8. Add Labels.
Why heatmap is used in python?
A heatmap contains values representing various shades of the same colour for each value to be plotted. Usually the darker shades of the chart represent higher values than the lighter shade. For a
very different value a completely different colour can also be used.
Can we plot 3D using matplotlib?
In order to plot 3D figures use matplotlib, we need to import the mplot3d toolkit, which adds the simple 3D plotting capabilities to matplotlib. Once we imported the mplot3d toolkit, we could create
3D axes and add data to the axes.
How do I use matplotlib 3D?
Plot a single point in a 3D space
1. Step 1: Import the libraries. import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D.
2. Step 2: Create figure and axes. fig = plt.figure(figsize=(4,4)) ax = fig.add_subplot(111, projection=’3d’)
3. Step 3: Plot the point.
How do you show 3D images in Python?
In this example, we use numpy. linspace() that creates an array of 10 linearly placed elements between -1 and 5, both inclusive after that the mesh grid function returns two 2-dimensional arrays,
After that in order to visualize an image of 3D wireframe we require passing coordinates of X, Y, Z, color(optional).
How do you visualize a 3D array?
Visualizing 3D array:
1. int shows that the 3D array is an array of type integer.
2. arr is the name of array.
3. first dimension represents the block size(total number of 2D arrays).
4. second dimension represents the rows of 2D arrays.
5. third dimension represents the columns of 2D arrays.
How do you draw a heat map?
What is the use of heatmap in Python?
Why heatmap is used in Python?
What is geo heatmap?
Geographic Heat Map (Geo Heatmap for short) is an interactive visualization that displays your data points on a real map and signifies areas of low and high density.
How do you plot a heat map?
Heat maps are a standard way to plot grouped data. The basic idea of a heat map is that the graph is divided into rectangles or squares, each representing one cell on the data table, one row and one
data set. The rectangle or square is color coded according to the value of that cell in the table.
What type of data is best visualized with a heat map?
The primary purpose of Heat Maps is to better visualize the volume of locations/events within a dataset and assist in directing viewers towards areas on data visualizations that matter most.
How do I use matplotlib 3d?
How to create a heatmap with Matplotlib?
X : Array-like or PIL Image – Here the input data is provided in the form of arrays or images.
cmap : str or Colormap,default: ‘viridis’ – This parameter takes the colormap instance or registered colormap name.
norm : Normalize,optional – This parameter helps in data normalization.
How to create heatmap calendar using NumPy and Matplotlib?
Calendar Example using HeatMap import datetime as dt import matplotlib.pyplot as plt import numpy as np def main(): dates, data = generate_data() fig, ax = plt
How to create a heat map in Excel?
First,select the column of the data on which we want to apply the Heat map.
Now,Got to Home tab,then go to Styles&Click on conditional formatting,then you will get a list.
After selecting conditional formatting,click on Color Scales from the list.
How to plot with Matplotlib?
Matplotlib plot a linear function. A linear function represents a straight line on the graph. You can use the slope-intercept form of the line that is y = m * x + c; Here, x and y are the X-axis and
Y-axis variables respectively, m is the slope of the line, and c is the x-intercept of the line. | {"url":"https://darkskiesfilm.com/does-matplotlib-have-heatmap/","timestamp":"2024-11-10T01:40:24Z","content_type":"text/html","content_length":"51944","record_id":"<urn:uuid:110d981d-7538-4e06-87fd-edae91d245da>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00000.warc.gz"} |
MCQ: Line Charts - 1 Free MCQ Practice Test with Solutions - SSC CGL
MCQ: Line Charts - 1 - Question 3
MCQ: Line Charts - 1 - Question 2
Direction: Two different finance companies declare fixed annual rate of interest on the amounts invested with them by investors. The rate of interest offered by these companies may differ from year
to year depending on the variation in the economy of the country and the banks rate of interest. The annual rate of interest offered by the two Companies P and Q over the years are shown by the line
graph provided below.
Q. If two different amounts in the ratio 8:9 are invested in Companies P and Q respectively in 2002, then the amounts received after one year as interests from Companies P and Q are respectively in
the ratio?
MCQ: Line Charts - 1 - Question 1
Direction: Two different finance companies declare fixed annual rate of interest on the amounts invested with them by investors. The rate of interest offered by these companies may differ from year
to year depending on the variation in the economy of the country and the banks rate of interest. The annual rate of interest offered by the two Companies P and Q over the years are shown by the line
graph provided below.
Q. In 2000, a part of ₹ 30 lakhs was invested in Company P and the rest was invested in Company Q for one year. The total interest received was ₹ 2.43 lakhs. What was the amount invested in Company
Direction: Two different finance companies declare fixed annual rate of interest on the amounts invested with them by investors. The rate of interest offered by these companies may differ from year
to year depending on the variation in the economy of the country and the banks rate of interest. The annual rate of interest offered by the two Companies P and Q over the years are shown by the line
graph provided below.
Q. An investor invested ₹ 5 lakhs in Company Q in 1996. After one year, the entire amount along with the interest was transferred as investment to Company P in 1997 for one year. What amount will be
received from Company P, by the investor?
MCQ: Line Charts - 1 - Question 4
Direction: Two different finance companies declare fixed annual rate of interest on the amounts invested with them by investors. The rate of interest offered by these companies may differ from year
to year depending on the variation in the economy of the country and the banks rate of interest. The annual rate of interest offered by the two Companies P and Q over the years are shown by the line
graph provided below.
Q. An investor invested a sum of ₹ 12 lakhs in Company P in 1998. The total amount received after one year was re-invested in the same Company for one more year. The total appreciation received by
the investor on his investment was?
MCQ: Line Charts - 1 - Question 5
Direction: Two different finance companies declare fixed annual rate of interest on the amounts invested with them by investors. The rate of interest offered by these companies may differ from year
to year depending on the variation in the economy of the country and the banks rate of interest. The annual rate of interest offered by the two Companies P and Q over the years are shown by the line
graph provided below.
Q. A sum of ₹ 4.75 lakhs was invested in Company Q in 1999 for one year. How much more interest would have been earned if the sum was invested in Company P?
MCQ: Line Charts - 1 - Question 6
Study the following line graph and answer the questions.
Q. Average annual exports during the given period for Company Y is approximately what percent of the average annual exports for Company Z?
MCQ: Line Charts - 1 - Question 7
Study the following line graph and answer the questions.
Q. For which of the following pairs of years the total exports from the three Companies together are equal?
MCQ: Line Charts - 1 - Question 8
Study the following line graph and answer the questions.
Q. What was the difference between the average exports of the three Companies in 1993 and the average exports in 1998?
MCQ: Line Charts - 1 - Question 9
Study the following line graph and answer the questions.
Q. In how many of the given years, were the exports from Company Z more than the average annual exports over the given years?
MCQ: Line Charts - 1 - Question 10
Study the following line graph and answer the questions.
Q. In which year was the difference between the exports from Companies X and Y the minimum?
MCQ: Line Charts - 1 - Question 11
Direction: Study the following line graph which gives the number of students who joined and left the school in the beginning of year for six years, from 1996 to 2001.
Q. For which year, the percentage rise/fall in the number of students who left the school compared to the previous year is maximum?
MCQ: Line Charts - 1 - Question 12
Direction: Study the following line graph which gives the number of students who joined and left the school in the beginning of year for six years, from 1996 to 2001.
Q. The number of students studying in the school in 1998 was what percent of the number of students studying in the school in 2001?
MCQ: Line Charts - 1 - Question 13
Direction: Study the following line graph which gives the number of students who joined and left the school in the beginning of year for six years, from 1996 to 2001.
Q. The strength of school increased/decreased from 1997 to 1998 by approximately what percent?
MCQ: Line Charts - 1 - Question 14
Direction: Study the following line graph which gives the number of students who joined and left the school in the beginning of year for six years, from 1996 to 2001.
Q. The number of students studying in the school during 1999 was?
MCQ: Line Charts - 1 - Question 15
Direction: Study the following line graph which gives the number of students who joined and left the school in the beginning of year for six years, from 1996 to 2001.
Q. The ratio of the least number of students who joined the school to the maximum number of students who left the school in any of the years during the given period is? | {"url":"https://edurev.in/test/41639/MCQ-Line-Charts-1","timestamp":"2024-11-06T07:44:21Z","content_type":"text/html","content_length":"375523","record_id":"<urn:uuid:c8e46219-df19-4b92-bf50-1e534476bc3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00552.warc.gz"} |
Translate inequalities involving absolute values to inequalities without absolute values - Stumbling Robot
Translate inequalities involving absolute values to inequalities without absolute values
Write the following inequalities in equivalent forms without the absolute values.
a1 is equivalent to b2:
a2 is equivalent to b5:
a3 is equivalent to b7:
a4 is equivalent to b10:
a5 is equivalent to b3:
a6 is equivalent to b8:
a7 is equivalent to b9:
a8 is equivalent to b4:
a9 is equivalent to b6:
a10 is equivalent to b1:
2 comments
1. Commenting to point out a minor typo. For a5, the solution number is b3, not b5.
Thanks a lot for these solutions. They are a big help for me to go through this book.
□ Thanks! Fixed. No problem on making the solutions.
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): | {"url":"https://www.stumblingrobot.com/2015/07/07/translate-inequalities-involving-absolute-values-to-inequalities-without-absolute-values/","timestamp":"2024-11-10T10:00:18Z","content_type":"text/html","content_length":"70008","record_id":"<urn:uuid:63e31c4d-253e-4540-b34b-fff1e8b2b7f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00456.warc.gz"} |
[Solved] Include another public method in the Empl | SolutionInn
Include another public method in the Employee class from Pencil and Paper Exercise 3. The method should
Include another public method in the Employee class from Pencil and Paper Exercise 3. The method should calculate an Employee object’s new salary, which is based on the raise percentage provided by
the program using the object. Before making the calculation, the method should verify that the raise percentage is greater than or equal to 0.0. If the raise percentage is less than 0.0, the method
should assign the number 0.0 as the new salary.
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
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required to heat
Informational content provided by Jamie at Immersion Heaters UK Ltd.
Click for our online water heat up time calculations page HERE.
One question which comes up time and again is “How many kW do I need to heat up my tank?”
Or phrased a different way, “How long is it going to take my ? litres of solution to raise ? °C using my ? kW heater?”
If we can calculate the volume of water and the required temperature rise, we can answer these questions using the following formula.
It is used to calculate the power of heating element needed to heat a specific volume of water by a given temperature rise in 1 hour.
volume in litres x 4 x temperature rise in degrees centigrade / 3412
(4 being a factor and 3412 being a given constant)
for example 100 litres of water, to be heated from 20ºC to 50ºC, giving a temperature rise of 30ºC would give –
100 x 4 x 30 / 3412 = 3.52
meaning that the water would be heated in 1 hour by 3.5kW of applied heat.
Also we can use this information to extrapolate both ways. To heat the same water volume in half the time (30 minutes) would need twice the heating power, ie, 7kW.
Converesely, if we only use half the heating power, 1.75kW, it will take twice as long to heat up to desired temperature, ie, 2 hours.
If we only have a 1kW element available, we will expect a heat up time circa 3.5 hours.
Also we can use this formula as the basis of similar calculations for heating oil. Generally speaking, oil heats up in about half the time of water, due to its viscosity & density. However, oil
requires a much lower watts density element than water, as described here in the “How to choose an oil heater” article.
Another variant of this formula, given here at the excellent website Sciencing.com gives the following varaint of the formula & subsequent explanation-
Pt = (4.2 × L × T ) ÷ 3600
Calculate Kilowatt-Hours
Calculate the kilowatt-hours (kWh) required to heat the water using the following formula: Pt = (4.2 × L × T ) ÷ 3600. Pt is the power used to heat the water, in kWh. L is the number of liters of
water that is being heated and T is the difference in temperature from what you started with, listed in degrees Celsius.
Solve for Thermal Power
Substitute in the appropriate numbers into the equation. So imagine you are heating 20 liters of water from 20 degrees to 100 degrees. Your formula would then look like this: Pt = (4.2 × 20 ×
(100-20)) ÷ 3600, or Pt = 1.867
Divide by Heater Element Rating
Calculate the amount of time it takes to heat the water by dividing the power used to heat the water, which was determined to be 1.867 with the heater element rating, listed in kW. So if your heater
element rating was 3.6 kW, your equation would look like this: heating time = 1.867 ÷ 3.6, or heating time =0.52 hours. Therefore, it would take 0.52 hours to heat 20 liters of water, with an element
with a rating of 3.6 kW.
Which made better sense in my little brain when I put a multiplication sign between P and t, allowing 30+ year old math class memories to clarify that if you move the Power (P) or the Hour (t) to the
other side of the equals symbol, we gotta divide by that number also. “Change the side, change the sign” Thanks Mr Phipps, some of it actually stuck, hope you are still above ground, happy & healthy.
P x t = (4.2 × L × T ) ÷ 3600
…which doesn’t usually “show” as t = 1 hour, as in kW(1)h.
Hope you found this useful.
Any feedback, suggestions, improvements, etc, PLEASE COMMENT, I promise to read ’em.
If you now want to buy something to actually do some heating, you have options.
All content provided by Jamie, who is Immersion Heaters UK Ltd, BreweryHeaters.co.uk, Heating Elements.co.uk & FlangedImmersionHeaters.co.uk. And we do Vat Heaters as well, if you wanna go “Over the
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Traversals - 5
Arrays Hands-On: Warming Up
Traversals - 5
Max Consecutive Ones
In this problem, we'll have an array of integers only, but it'll contain just 0's and 1's. Our target is to find out the maximum number of consecutive 1's
In other words, our target is to find the length of the largest subarray, that contains no 0's at all.
Subarray is a contiguous sub-part of an array. E.g. [2, 3] , [3, 4, 5], [1] are subarrays of [1, 2, 3, 4, 5], whereas [2, 4, 5], [1, 3] are not.
How to solve it?
If I try to think like a beginner, the first approach that comes to my mind is:
The Approach Let's keep a variable final_ans that is initialised with 0 in the beginning, and then do the following:
1. Let's iterate over all the subarrays
2. Somehow, check if the condition of all ones is satisfied.
□ If yes, update the final_ans variable if length of this subarray is greater than final_ans
□ Else, keep going.
int findMaxConsecutiveOnes(vector<int>& arr) {
int final_ans = 0, n = arr.size();
// iterate over all possible
// start values for a subarray
for(int l = 0; l < n; ++l) {
int num_of_zeroes = 0;
// iterate over all possible
// end values for a subarray
// for this particular l value.
for(int r = l; r < n; ++r) {
if(arr[r] == 0)
if(num_of_zeroes == 0)
final_ans = max(final_ans, r - l + 1); // because current_length = (r - l + 1)
return final_ans;
Explaining some elements of the code
I hope the overal idea is clear, as it has already been explained in The Approach section.
I'll just cover the inner loop part here, which is to explain why are we checking if arr[r] is equal to 0 or not. Some of you may think, don't we need to check all the elements? why are we just
checking for arr[r]?
The answer is that if you check for all the elements b/w the indices l and r, that'll be absolutely right, but that'll make our code even slower. Care to guess the time complexity in that case?
Click to reveal. The time complexity in that case will be O(N^3). I urge you to think about the reason yourself.
Okay, coming back to the question that is checking for just arr[r] enough? The answer is yes; take an example array and try to dry-run the code line-by-line on that, you'll see that although we're
only checking for arr[r], yet it's correct because we've already stored the number of 0's in the num_of_zeroes variable for the range [l, r-1] during the previous iterations.
Time and Space Complexity
• Time Complexity: O(N^2)
• Space Complexity: O(1)
A more efficient solution
The above discussed solution is kinda brute force, as we're going to every subarray and checking if it meets the condition.
Can we do it by traversing through the array just once? Let's give it a try.
Traversing is like a box of chocolates, you never know what you'll get.
Well, in this case, it'll be either 1 or 0. If it's one, then the window of 1's is basically expanding as you've gotten one more 1. If you get 0, then the window of 1's has to be reset, right?
Complete Explanation
The idea is similar to what's already told in the hint. We'll keep track of the number of 1's as we get 1's while iterating, and we'll reset the count to 0 if a 0 is enocountered in between.
We'll keep 2 variables, final_ans and cur_count, both initialised with 0. Then, we'll iterate over the array and do the following in each iteration:
1. If arr[i] == 1, then increment the value of cur_count by 1.
2. Otherwise if arr[i] == 0, then reset the value of cur_count equal to 0.
3. Keep updating the value of final_ans appropriately in each iteration so that we have the maximum number of consecutive 1's so far saved safely.
int findMaxConsecutiveOnes(vector<int>& arr) {
// final_ans: stores the maximum consecutive ones.
// cur_count: stores the number of consecutive ones in the current window.
int final_ans = 0, cur_count = 0;
for(int num : arr) {
if(num == 1)
cur_count = 0;
final_ans = max(final_ans, cur_count);
return final_ans;
Time and Space Complexity If we look at the above code carefully, all we've done is traverse tha array once and a couple of if-else operations in each iteration. Also, for space, we've just used a
couple of int variables, nothing more. Therefore:
• Time Complexity: O(N)
• Space Complexity: O(1)
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Safe Haskell Safe
Language Haskell2010
Abstract binding trees with a nameless internal representation.
Abstract binding tree terms
Abstract binding trees take the form Term v f a, or, more commonly, Flat v f. These types are abstract—you will never construct or analyze them directly.
data Term v f Source #
An abstract Term v f is an abstract binding tree of the shape described by the pattern functor f augmented with variables named by v. Equality is alpha-equivalence. In particular, Term v f is
(morally) equivalent to the fixed-point of the pattern-algebra View respecting the binding properties of VAbs and VVar.
(Eq v, Eq (f (Term v f))) => Eq (Term v f) Source #
(Ord v, Ord (f (Term v f))) => Ord (Term v f) Source #
(Show v, Show (Nameless v f (Term v f))) => Show (Term v f) Source #
Constructing terms with View patterns
To construct or analyze a Term, the Var, Abs, and Pat pattern synonyms are useful. These synonyms let you essentially treat Term as if it weren't abstract and both construct new terms and case
analyze them.
pattern Var :: forall f v. (Freshen v, Foldable f, Ord v, Functor f) => v -> Term v f Source #
Var v creates and matches a Term value corresponding to a free variable.
pattern Abs :: forall f v. (Freshen v, Foldable f, Ord v, Functor f) => v -> Term v f -> Term v f Source #
Abs v t creates and matches a Term value where the free variable v has been abstracted over, becoming bound.
pattern Pat :: forall f v. (Freshen v, Foldable f, Ord v, Functor f) => f (Term v f) -> Term v f Source #
Pat f creates and matches a Term value built from a layer of the pattern functor f.
Working with free variables
Abstract binding trees take the form Term v f a, or, more commonly, Flat v f. These types are abstract---you will never construct or analyze them directly.
freeVars :: Term v f -> Set v Source #
Returns the free variables used within a given Term.
NOTE: We have to define a new function in order to not accidentally break encapsulation. Just exporting free direction would allow uses to manipulate the Term value and break invariants (!).
Freshen class
class Eq v => Freshen v where Source #
A type which can be freshened has an operation which attempts to find a unique version of its input. The principal thing that must hold is that `freshen n /= n`. It's not necessary that `freshen n`
be totally fresh with respect to a context---that's too much to ask of a value---but it is necessary that freshen *eventually* produces a fresh value.
Variable identifier types must be instances of Freshen.
Freshen Int Source #
Freshen Index Source #
Freshen Name Source #
View in detail
data View v f x Source #
VVar !v
VPat (f x)
VAbs !v x
(Eq v, Eq x, Eq (f x)) => Eq (View v f x) Source #
(Ord v, Ord x, Ord (f x)) => Ord (View v f x) Source #
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Input Data Sets
You can read group summary statistics and outlier information from a BOX= data set specified in the PROC BOXPLOT statement. This enables you to reuse OUTBOX= data sets that have been created in
previous runs of the BOXPLOT procedure to reproduce schematic box plots.
A BOX= data set must contain the following variables:
• the group variable
• _VAR_, containing the analysis variable name
• _TYPE_, identifying features of box-and-whiskers plots
• _VALUE_, containing values of those features
Each observation in a BOX= data set records the value of a single feature of one group’s box-and-whiskers plot, such as its mean. Consequently, a BOX= data set contains multiple observations per
group. These must appear consecutively in the BOX= data set.
The _TYPE_ variable identifies the feature whose value is recorded in a given observation. The following table lists valid _TYPE_ variable values.
Table 26.9: Valid _TYPE_ Values in a BOX= Data Set
_TYPE_ Description
N group size
MIN group minimum value
Q1 group first quartile
MEDIAN group median
MEAN group mean
Q3 group third quartile
MAX group maximum value
STDDEV group standard deviation
LOW low outlier value
HIGH high outlier value
LOWHISKR low whisker value, if different from MIN
HIWHISKR high whisker value, if different from MAX
FARLOW low far outlier value
FARHIGH high far outlier value
The features identified by _TYPE_ values N, MIN, Q1, MEDIAN, MEAN, Q3, and MAX are required for each group.
Other variables that can be read from a BOX= data set include the following:
• the variable _ID_, containing labels for outliers
• the variable _HTML_, containing URLs to be associated with features on box plots
• block variables
• symbol variable
• BY variables
• ID variables
When you specify the keyword SCHEMATICID or SCHEMATICIDFAR with the BOXSTYLE= option, values of _ID_ are used as outlier labels. If _ID_ does not exist in the BOX= data set, the values of the first
variable listed in the ID statement are used.
You can read group summary statistics from a HISTORY= data set specified in the PROC BOXPLOT statement. This enables you to reuse OUTHISTORY= data sets that have been created in previous runs of the
BOXPLOT procedure or to read output data sets created with SAS summarization procedures, such as PROC UNIVARIATE.
Note that a HISTORY= data set does not contain outlier information. Therefore, in general you cannot reproduce a schematic box plot from summary statistics saved in an OUTHISTORY= data set. To save
and reproduce schematic box plots, use OUTBOX= and BOX= data sets.
A HISTORY= data set must contain the following:
• the group variable
• a group minimum variable for each analysis variable
• a group first-quartile variable for each analysis variable
• a group median variable for each analysis variable
• a group mean variable for each analysis variable
• a group third-quartile variable for each analysis variable
• a group maximum variable for each analysis variable
• a group standard deviation variable for each analysis variable
• a group size variable for each analysis variable
The names of the group summary statistics variables must be the analysis variable name concatenated with the following special suffix characters.
Group Summary Statistic Suffix Character
group minimum L
group first quartile 1
group median M
group mean X
group third quartile 3
group maximum H
group standard deviation S
group size N
For example, consider the following statements:
proc boxplot history=Summary;
plot (Weight Yieldstrength) * Batch;
The data set Summary must include the variables Batch, WeightL, Weight1, WeightM, WeightX, Weight3, WeightH, WeightS, WeightN, YieldstrengthL, Yieldstrength1, YieldstrengthM, YieldstrengthX,
Yieldstrength3, YieldstrengthH, YieldstrengthS, and YieldstrengthN.
Note that if you specify an analysis variable whose name contains the maximum of 32 characters, the summary variable names must be formed from the first 16 characters and the last 15 characters of
the analysis variable name, suffixed with the appropriate character.
These other variables can be read from a HISTORY= data set:
• block variables
• symbol variable
• BY variables
• ID variables | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_boxplot_details05.htm","timestamp":"2024-11-02T10:53:01Z","content_type":"application/xhtml+xml","content_length":"35109","record_id":"<urn:uuid:b976a8aa-3966-4358-ac95-a3edc276f79a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00684.warc.gz"} |
Bronze coins - Dismounting Rider
Coin catalogue sections: Nagidos, Bronze coins;
Nagidos, Uncertain issues
Coin corpus datasets: Nagidos, Bronze coins
Box plots1 of individual coin types and basic descriptive statistics are presented in Figure 1 and Table 1 (Std. Dev. denotes the standard deviation and IQR the interquartile range), respectively.
Figure 1: Box plots
Type Count Mean Median Std. Dev. IQR
8.1 36 1.07 1.04 0.22 0.32
8.2 18 3.11 3.12 0.57 0.54
8.3 5 1.49 1.48 0.08 0.10
X.7 8 1.24 1.21 0.19 0.18
Table 1: Basic descriptive statistics of coin types
Figure 2 presents relative frequency histograms of individual types, i.e. the bars represent the relative frequencies of observations ranging from 0.60 to 3.60 g in increments of 0.20 g. Cumulative
distributions are shown in Figure 3.
Figure 2: Relative frequency histograms
Figure 3: Cumulative distributions
1The bottom and top of each box are the 25th and 75th percentiles of the dataset, respectively (the lower and upper quartiles). Thus, the height of the box corresponds to the interquartile range
(IQR). The red line inside the box indicates the median. Whiskers (the dashed lines extending above and below the box) indicate variability outside the upper and lower quartiles. From above the upper
quartile, a distance of 1.5 times the IQR is measured out and a whisker is drawn up to the largest observed data point from the dataset that falls within this distance. Similarly, a distance of 1.5
times the IQR is measured out below the lower quartile and a whisker is drawn down to the lowest observed data point from the dataset that falls within this distance. Observations beyond the whisker
length are marked as outliers and are represented by small red circles.
8 October 2023 – 31 August 2024 | {"url":"https://www.dismountingrider.info/weight-analyses/nagidos-weight-analyses/nagidos-bronze-coins-weight-analysis/","timestamp":"2024-11-05T02:54:58Z","content_type":"text/html","content_length":"35416","record_id":"<urn:uuid:96f5945d-cf04-4048-9f7c-b0652ded455c>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00027.warc.gz"} |
Parallel comparison algorithms for approximation problems
Suppose we have n elements from a totally ordered domain, and we are allowed to perform p parallel comparisons in each time unit (=round). In this paper we determine, up to a constant factor, the
time complexity of several approximation problems in the common parallel comparison tree model of Valiant, for all admissible values of n, p and e{open}, where e{open} is an accuracy parameter
determining the quality of the required approximation. The problems considered include the approximate maximum problem, approximate sorting and approximate merging. Our results imply as special
cases, all the known results about the time complexity for parallel sorting, parallel merging and parallel selection of the maximum (in the comparison model), up to a constant factor. We mention one
very special but representative result concerning the approximate maximum problem; suppose we wish to find, among the given n elements, one which belongs to the biggest n/2, where in each round we
are allowed to ask n binary comparisons. We show that log^*n+O(1) rounds are both necessary and sufficient in the best algorithm for this problem.
All Science Journal Classification (ASJC) codes
• Discrete Mathematics and Combinatorics
• Computational Mathematics
• AMS subject classification (1980): 68E05
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On Two Segmentation Problems
The hypercube segmentation problem is the following: Given a set S of m vertices of the discrete d-dimensional cube {0,1}^d, find k vertices P[1],..., P[k], P[i] ∈ {0, 1}^d and a partitions of S into
k segments S[1],..., S[k] so as to maximize the sum ∑^k[i]=[1] ∑[c∈Si] P[i]⊙c, where ⊙ is the overlap operator between two vertices of the d-dimensional cube, defined to be the number of positions
they have in common. This problem (among other ones) was considered by Kleinberg, Papadimitriou, and Raghavan, where the authors designed a deterministic approximation algorithm that runs in
polynomial time for every fixed k and produces a solution whose value is within 0.828 of the optimum, as well as a randomized algorithm that runs in linear time for every fixed k and produces a
solution whose expected value is within 0.7 of the optimum. Here we design an improved approximation algorithm; for every fixed ε > 0 and every fixed k our algorithm produces in linear time a
solution whose value is within (1 - ε) of the optimum. Therefore, for every fixed k, this is a polynomial approximation scheme for the problem. The algorithm is deterministic, but it is convenient to
first describe it as a randomized algorithm and then to derandomize it using some properties of expander graphs. We also consider a segmented version of the minimum spanning tree problem, where we
show that no approximation can be achieved in polynomial time, unless P = NP.
All Science Journal Classification (ASJC) codes
• Control and Optimization
• Computational Mathematics
• Computational Theory and Mathematics
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Two-Photonsorption Calculator
Physical Chemistry
Two-Photonsorption Calculator
Two-Photon Absorption Calculator
What is the Two-Photon Absorption Calculator?
The Two-Photon Absorption Calculator is a handy tool designed for calculating the rate of two-photon absorption in a given medium. This process is integral in fields such as photochemistry and
material science where understanding light-matter interactions is crucial.
Applications of Two-Photon Absorption
Two-photon absorption has several practical applications, particularly in areas like microscopy, imaging, and photodynamic therapy. With this calculator, scientists and researchers can gain insights
into the efficiency and dynamics of two-photon processes, facilitating advancements in these fields.
In microscopy, two-photon absorption allows for high-resolution imaging of biological tissues with deep penetration and minimal photodamage. This technique enhances the visualization of cellular
structures in greater detail.
Two-photon imaging techniques are employed in various scientific and medical fields. By using this calculator, researchers can optimize parameters for better image quality and more accurate data.
Photodynamic Therapy
In photodynamic therapy, two-photon absorption is used to activate photosensitive drugs at targeted sites within the body, minimizing side effects and increasing treatment efficacy. The calculator
assists in determining optimal dosages and exposure times.
How Does the Calculator Work?
The Two-Photon Absorption Calculator operates based on several input parameters:
• Photon Energy: The energy of each photon involved in the process. Higher energies can lead to more efficient two-photon absorption.
• Absorption Cross Section: This measures the probability of the absorption process per molecule. A larger cross section indicates a higher likelihood of absorption.
• Photon Flux: The number of photons passing through a unit area per second. Higher photon flux increases the chances of two-photon absorption events.
• Concentration of Absorbing Species: The concentration of molecules that can absorb photons. Higher concentrations generally result in more absorption events.
• Path Length: The distance the photons travel through the medium. A longer path length provides more opportunities for photon absorption.
Benefits of Using the Two-Photon Absorption Calculator
This calculator simplifies complex calculations, enabling users to rapidly determine the rate of two-photon absorption. Researchers can focus more on analyzing results instead of performing
meticulous calculations.
By inputting relevant parameters, users can streamline their experiments and research, leading to better resource management and more accurate experimental setups.
Understanding the Calculation
To derive the rate of two-photon absorption, the calculator multiplies the absorption cross-section by the square of the photon flux, concentration of the absorbing species, and the path length. This
product gives an insightful measure of how the variables interact.
The Two-Photon Absorption Calculator is an essential tool for anyone involved in photochemistry and material science. By providing precise calculations, it aids researchers in optimizing their
experiments and gaining deeper insights into the two-photon absorption process.
What is two-photon absorption?
Two-photon absorption is a nonlinear optical process where two photons are simultaneously absorbed by a molecule, resulting in an electronic transition to a higher energy state. It requires high
photon flux and is commonly encountered in processes like microscopic imaging and photodynamic therapy.
Why should I use the Two-Photon Absorption Calculator?
This calculator provides a straightforward method to compute the rate of two-photon absorption by inputting relevant parameters. It eliminates the need for manual calculations, saving time and
reducing errors.
What units should the input parameters be in?
Photon Energy: Electron volts (eV)
Absorption Cross Section: Square centimeters (cm²)
Photon Flux: Photons per square centimeter per second (photons/cm²/s)
Concentration of Absorbing Species: Molecules per cubic centimeter (molecules/cm³)
Path Length: Centimeters (cm)
How does the photon flux influence the rate of two-photon absorption?
The photon flux significantly affects the rate of two-photon absorption. As photon flux increases, the number of photon pairs available for simultaneous absorption also increases, thereby enhancing
the two-photon absorption rate. This relationship is quadratic, meaning the rate is proportional to the square of the photon flux.
Can this calculator be used for any type of medium?
Yes, the Two-Photon Absorption Calculator can be used for various mediums, provided you have the necessary input parameters (photon energy, absorption cross section, photon flux, concentration of
absorbing species, and path length). However, different mediums may have varying absorption cross sections and other properties.
What role does the absorption cross section play in the calculations?
The absorption cross section measures the likelihood of a molecule absorbing photons. A larger cross section indicates a higher probability of photon absorption. It directly influences the rate of
two-photon absorption; the larger the cross section, the higher the absorption rate.
How can I interpret the results from the calculator?
The calculator provides the rate of two-photon absorption, which indicates how many two-photon absorption events occur per unit volume per unit time. Higher rates suggest more efficient absorption,
which can be used to optimize experimental conditions and applications in microscopy, imaging, or photodynamic therapy.
What kind of limitations should I be aware of?
The accuracy of the calculator depends on the precision of the input parameters. Incorrect or estimated values can lead to inaccurate results. Additionally, the calculator assumes that the input
parameters are within a valid range for two-photon absorption to occur.
Who can benefit from using this calculator?
This calculator is intended for researchers, scientists, and students working in fields such as photochemistry, material science, and biomedical imaging. It helps streamline experimental setups and
provides a clearer understanding of light-matter interactions.
How can I improve the accuracy of my calculations?
To improve accuracy, ensure that you use precise and accurate input values, including experimentally determined absorption cross sections and photon flux measurements. Regularly calibrating your
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A cord is tied to a pail of water and the pail is swung in a vertical circle of 1.2 m. What is the minimum velocity the pail must have at the top of the circle if no water spills? | Socratic
A cord is tied to a pail of water and the pail is swung in a vertical circle of 1.2 m. What is the minimum velocity the pail must have at the top of the circle if no water spills?
2 Answers
At the top of the circle the weight is balanced by the centripetal force so:
$\cancel{m} g = \frac{\cancel{m} {v}^{2}}{r}$
$\therefore v = \sqrt{g r}$
$v = \sqrt{9.8 \times 1.2} = 3.43 \text{m/s}$
Let us examine the forces acting not on the pail, but on the water inside the pail at the top of the circular path.
Imagine the pail being swung at a fairly high speed. At the top of the path, the pail exerts a normal force on the water, directed downwards. This combined with the force of gravity gives us a
centripetal force that keeps the entire pail-water system moving in a circle.
Here's an appropriate free-body diagram I found online: (ignore the "bottom of circle" portion)
Now, if we lower the speed enough there is a certain value for which the normal force becomes zero at the exact instant in which the pail is at the top of the path. And, any speed lower than this
value would result in the water falling out of the pail - the water would simply not have enough momentum to resist the force of gravity.
Recall that for an object undergoing circular motion at a constant speed we have
$a = {v}^{2} / r$
Since ${F}_{\text{net" = F_"grav" + F_"norm}}$, If we let ${F}_{\text{norm}} = 0$ then we have ${F}_{\text{net" = F_"grav}}$.
So we can apply Newton's second law and obtain
${F}_{\text{net}} = m a = m {v}^{2} / r$
Since we know that ${F}_{\text{net" = F_"grav}}$, and ${F}_{\text{grav}} = m g$, we can substitute this into the above equation to find
$m g = m {v}^{2} / r$
Cancelling $m$'s and solving for $v$ yields
$v = \sqrt{r g}$
Assuming that $g = 9.81 \text{m"/"s}$ we have
$v = 3.4 \text{m"/"s}$
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Quantum Error Correction
July 24th, 2009
Given a completely positive linear map E: M[n] → M[n], its multiplicative domain, denoted MD(E), is an algebra defined as follows:
Roughly speaking, MD(E) is the largest subalgebra of M[n] on which E behaves multiplicatively. It will be useful to make this notion precise:
Definition. Let A be a subalgebra of M[n] and let π : A → M[n]. Then π is said to be a *-homomorphism if π(ab) = π(a)π(b) and π(a^*) = π(a)^* for all a,b ∈ A.
Thus, MD(E) is roughly the largest subalgebra of M[n] such that, when E is restricted to it, E is a *-homomorphism (I keep saying “roughly speaking” because of the “∀b ∈ M[n]” in the definition of MD
(E) — the definition of a *-homomorphism only requires that the multiplicativity hold ∀b ∈ A). Probably the most well-known result about the multiplicative domain is the following theorem of Choi
[1,2], which shows how the multiplicative domain simplifies when E is such that E(I) = I (i.e., when E is unital):
Theorem [Choi]. Let E: M[n] → M[n] be a completely positive map such that E(I) = I. Then
Let $\phi : \cl{L}(\cl{H}) \rightarrow \cl{L}(\cl{H})$ be a completely positive, unital map. Then
MD(\phi) = & \big\{ a \in \cl{L}(\cl{H}) : \phi(a)^{*}\phi(a) =
\phi(a^*a)\text{ and } \phi(a)\phi(a)^{*} =
One thing in particular that this theorem shows is that, when E(I) = I, the multiplicative domain of E only needs to be multiplicative within MD(E) (i.e., we can remove the “roughly speaking” that I
spoke of earlier).
MD(E) in Quantum Error Correction
Before moving onto how MD(E) plays a role in quantum error correction, let’s consider some examples to get a better feeling for what the multiplicative domain looks like.
• If E is the identity map (that is, it is the map that takes a matrix to itself) then MD(E) = M[n], the entire matrix algebra.
• If E(a) = Diag(a) (i.e., E simply erases all of the off-diagonal entries of the matrix a), then MD(E) = {Diag(a)}, the set of diagonal matrices.
Notice that in the first example, the map E is very well-behaved (as well-behaved as a map ever could be); it preserves all of the information that is put into it. We also see that MD(E) is as large
as possible. In the second example, the map E does not preserve information put into it (indeed, one nice way to think about matrices in the quantum information setting is that the diagonal matrices
are “classical” and rest of the matrices are “quantum” — thus the map E(a) = Diag(a) is effectively removing all of the “quantumness” of the input data). We also see that MD(E) is tiny in this case
(too small to put any quantum data into).
The above examples then hint that if the map E preserves quantum data, then MD(E) should be large enough to store some quantum information safely. This isn’t quite true, but the intuition is right,
and we get the following result, which was published as Theorem 11 in this paper:
Theorem. Let E: M[n] → M[n] be a quantum channel (i.e., a completely positive map such that Tr(E(a)) = Tr(a) for all a ∈ M[n]) such that E(I) = I. Then MD(E) = UCC(E), the algebra of
unitarily-correctable codes for E.
What this means is that, when E is unital, its multiplicative domain encodes exactly the matrices that we can correct via a unitary operation. This doesn’t tell us anything about correctable codes
that are not unitarily-correctable, though (i.e., matrices that can only be corrected by a more complicated correction operation). To capture these codes, we have to generalize a bit.
Generalized Multiplicative Domains
In order to generalize the multiplicative domain, we can require that the map E be multiplicative with another map π that is already a *-homomorphism, rather than require that it be multiplicative
with itself. This is the main theme of this paper, which was submitted for publication this week. We define generalized multiplicative domains as follows:
Definition. Let A be a subalgebra of M[n], let E : M[n] → M[n] be completely positive, and let π : A → M[n] be a *-homomorphism. Then the multiplicative domain of E with respect to π, denoted MD[π]
(E), is the algebra given by
It turns out that these generalized multiplicative domains are reasonably well-behaved and generalize the standard multiplicative domain in exactly the way that we wanted: they capture all
correctable codes for arbitrary quantum channels (see Theorem 11 of the last paper I mentioned). Furthermore, there are even some characterizations of MD[π](E) analogous to the theorem of Choi above
(see Theorems 5 and 7, as well as Corollary 12).
1. M.-D. Choi, A Schwarz inequality for positive linear maps on C*-algebras. Illinois Journal of Mathematics, 18 (1974), 565-574.
2. V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2003.
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The Montana Mathematics Enthusiast Monograph in Mathematics Education Monograph 12 <i>Crossroads in the History of Mathematics and Mathematics Education</i> PREFACE
Original language American English
Title of host publication Crossroads In The History Of Mathematics And Mathematics Education
Editors B Sriraman
Pages IX-IX
Number of pages 1
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2.3 Time, Velocity, and Speed
9 2.3 Time, Velocity, and Speed
• Explain the relationship between instantaneous velocity, average velocity, instantaneous speed, average speed, displacement, and time.
• Calculate velocity and speed given initial position, initial position, final position, and final time.
• Derive a graph of velocity vs. time given a graph of position vs. time.
• Interpret a graph of velocity vs. time.
Figure 1. The motion of these racing snails can be described by their speeds and their velocities. (credit: tobitasflickr, Flickr).
There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other
concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion.
As discussed in Chapter 1.2 Physical Quantities and Units, the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time
involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple—time is
change, or the interval over which change occurs. It is impossible to know that time has passed unless something changes.
The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full
swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us
to not only measure the amount of time, but also to determine a sequence of events.
How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To
find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would
be 50 min. Elapsed time
Life is simpler if the beginning time
In this text, for simplicity’s sake,
• motion starts at time equal to zero
• the symbol t is used for elapsed time unless otherwise specified
Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has
units of distance divided by time, such as miles per hour or kilometers per hour.
Average velocity is displacement (change in position) divided by the time of travel,
where average (indicated by the bar over the v) velocity,
Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many
other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use. Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the negative sign indicates that
displacement is toward the back of the plane). His average velocity would be
The minus sign indicates the average velocity is also toward the rear of the plane.
The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether
the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals.
Figure 2. A more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip.
The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval.
Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car’s speedometer, for example, shows the magnitude (but not the direction) of
the instantaneous velocity of the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to
use average velocity.) Instantaneous velocity is the average velocity at a specific instant in time (or over an infinitesimally small time interval).
Mathematically, finding instantaneous velocity,
In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is
that speed has no direction. Thus speed is a scalar. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and
average speed.
Instantaneous speed is the magnitude of instantaneous velocity. For example, suppose the airplane passenger at one instant had an instantaneous velocity of −3.0 m/s (the minus meaning toward the rear
of the plane). At that same time his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at
that instant would be 40 km/h—the same magnitude but without a direction. Average speed, however, is very different from average velocity. Average speed is the distance traveled divided by elapsed
We have noted that distance traveled can be greater than displacement. So average speed can be greater than average velocity, which is displacement divided by time. For example, if you drive to a
store and return home in half an hour, and your car’s odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero, because your
displacement for the round trip is zero. (Displacement is change in position and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity.
Figure 3. During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in
position. Thus the average velocity is zero.
Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the
position, velocity, and speed-vs.-time graphs are displayed in Figure 4. (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip,
which is unrealistic given that we’ll probably stop at the store. But for simplicity’s sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store
and the house is a perfectly straight line.)
Figure 4. Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative.
If you have spent much time driving, you probably have a good sense of speeds between about 10 and 70 miles per hour. But what are these in meters per second? What do we mean when we say that
something is moving at 10 m/s? To get a better sense of what these values really mean, do some observations and calculations on your own:
• calculate typical car speeds in meters per second
• estimate jogging and walking speed by timing yourself; convert the measurements into both m/s and mi/h
• determine the speed of an ant, snail, or falling leaf
Check Your Understanding
1: A commuter train travels from Baltimore to Washington, DC, and back in 1 hour and 45 minutes. The distance between the two stations is approximately 40 miles. What is (a) the average velocity of
the train, and (b) the average speed of the train in m/s?
Section Summary
• Time is measured in terms of change, and its SI unit is the second (s). Elapsed time for an event is
• Average velocity
• The SI unit for velocity is m/s.
• Velocity is a vector and thus has a direction.
• Instantaneous velocity
• Instantaneous speed is the magnitude of the instantaneous velocity.
• Instantaneous speed is a scalar quantity, as it has no direction specified.
• Average speed is the total distance traveled divided by the elapsed time. (Average speed is not the magnitude of the average velocity.) Speed is a scalar quantity; it has no direction associated
with it.
Conceptual Questions
1: Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
2: There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.
3: Does a car’s odometer measure position or displacement? Does its speedometer measure speed or velocity?
4: If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity?
Under what circumstances are these two quantities the same?
5: How are instantaneous velocity and instantaneous speed related to one another? How do they differ?
Problems & Exercises
1: (a) Calculate Earth’s average speed relative to the Sun. (b) What is its average velocity over a period of one year?
2: A helicopter blade spins at exactly 100 revolutions per minute. Its tip is 5.00 m from the center of rotation. (a) Calculate the average speed of the blade tip in the helicopter’s frame of
reference. (b) What is its average velocity over one revolution?
3: The North American and European continents are moving apart at a rate of about 3 cm/y. At this rate how long will it take them to drift 500 km farther apart than they are at present?
4: Land west of the San Andreas fault in southern California is moving at an average velocity of about 6 cm/y northwest relative to land east of the fault. Los Angeles is west of the fault and may
thus someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is 590 km northwest, assuming the motion
remains constant?
5: On May 26, 1934, a streamlined, stainless steel diesel train called the Zephyr set the world’s nonstop long-distance speed record for trains. Its run from Denver to Chicago took 13 hours, 4
minutes, 58 seconds, and was witnessed by more than a million people along the route. The total distance traveled was 1633.8 km. What was its average speed in km/h and m/s?
6: Tidal friction is slowing the rotation of the Earth. As a result, the orbit of the Moon is increasing in radius at a rate of approximately 4 cm/year. Assuming this to be a constant rate, how many
years will pass before the radius of the Moon’s orbit increases by
7: A student drove to the university from her home and noted that the odometer reading of her car increased by 12.0 km. The trip took 18.0 min. (a) What was her average speed? (b) If the
straight-line distance from her home to the university is 10.3 km in a direction
8: The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord
to your feet is 1.1 m long, and the nerve impulse speed is 18 m/s, how long does it take for the nerve signal to travel this distance?
9: Conversations with astronauts on the lunar surface were characterized by a kind of echo in which the earthbound person’s voice was so loud in the astronaut’s space helmet that it was picked up by
the astronaut’s microphone and transmitted back to Earth. It is reasonable to assume that the echo time equals the time necessary for the radio wave to travel from the Earth to the Moon and back
(that is, neglecting any time delays in the electronic equipment). Calculate the distance from Earth to the Moon given that the echo time was 2.56 s and that radio waves travel at the speed of light
10: A football quarterback runs 15.0 m straight down the playing field in 2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He breaks the tackle and runs straight forward another
21.0 m in 5.20 s. Calculate his average velocity (a) for each of the three intervals and (b) for the entire motion.
11: The planetary model of the atom pictures electrons orbiting the atomic nucleus much as planets orbit the Sun. In this model you can view hydrogen, the simplest atom, as having a single electron
in a circular orbit
average speed
distance traveled divided by time during which motion occurs
average velocity
displacement divided by time over which displacement occurs
instantaneous velocity
velocity at a specific instant, or the average velocity over an infinitesimal time interval
instantaneous speed
magnitude of the instantaneous velocity
change, or the interval over which change occurs
simplified description that contains only those elements necessary to describe the physics of a physical situation
elapsed time
the difference between the ending time and beginning time
Check Your Understanding
1: (a) The average velocity of the train is zero because
(b) The average speed of the train is calculated below. Note that the train travels 40 miles one way and 40 miles back, for a total distance of 80 miles.
Problems & Exercises
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Deconvolution Helps Break Down Imaging Barriers
Researchers can use this method to improve clarity — without additional optical hardware — and to obtain access to structures previously hidden above and below the focal plane.
Deconvolution is a computationally intensive image processing technique used to improve the contrast and sharpness of images captured with a light microscope. Light microscopes are diffraction
limited, which means they are unable to resolve individual structures unless they are more than half the wavelength of light away from one another. Each point source below this diffraction limit is
blurred by the microscope into what is known as a point spread function (PSF). With traditional wide-field fluorescence microscopy, out-of-focus light from areas above or below the focal plane causes
additional blurring in the captured image. This becomes a problem when trying to identify or measure structures inside thicker samples.
Deconvolution reverses this degradation by using the optical system’s PSF and reconstructing an ideal image made from a collection of smaller point sources. This is especially helpful for samples
such as spheroids or tissue sections, where areas of interest are often hidden within the sample rather than at the top or bottom, where they can be viewed easily.
Maximum intensity projections of plant seeds, acquired with a wide-field microscope (top). Deconvolution reverses the degradation with the aid of an advanced maximum likelihood constrained iterative
algorithm (bottom). Courtesy of Joe Dragavon.
A light microscope’s PSF varies based on the optical properties of both the microscope and the sample, making it difficult to experimentally determine the exact optical transformation of the complete
system. For this reason, mathematical algorithms have been developed to determine the PSF and to make the best possible reconstruction of the ideal image using deconvolution. Nearly any image
acquired with a fluorescence microscope can be deconvolved, including those that are not three-dimensional, meaning that even a single image can benefit from these image processing techniques.
Commercially available software brings these algorithms together into cost-effective, user-friendly packages. Each deconvolution algorithm differs in how the point spread and noise functions of the
convolution operations are determined. The basic imaging formula is:
g(x) = f(x) * h(x) + n(x)
where x: spatial coordinate; g(x): observed image; f(x): object; h(x): PSF; n(x): noise function; *: convolution.
Deblurring algorithms
Deblurring algorithms apply an operation to each two-dimensional plane of a three-dimensional image stack. A common deblurring technique, “nearest neighbor,” operates on each z-plane by blurring the
neighboring planes (z + 1 and z − 1, using a digital blurring filter), then subtracting the blurred planes from the z-plane. Multineighbor techniques extend this concept to a selectable number of
planes, allowing users to reduce processing time by eliminating z-planes that are too far out of focus to provide usable data. A three-dimensional stack is processed by applying the algorithm to
every selected plane in the stack (figure below).
This class of deblurring algorithms is workable because it involves calculations performed on a small number of image planes. However, there are several disadvantages to these approaches. Structures
whose PSFs overlap each other in nearby z-planes may be localized in planes where they do not belong, altering the apparent position of the object when multiple areas of interest are close in
proximity to one another. This problem is particularly severe when deblurring a single two-dimensional image because it often contains diffraction rings or light from out-of-focus structures that
will then be sharpened as if they were in the correct focal plane.
Inverse filter algorithms
An inverse filter functions by taking the Fourier transform of an image and dividing it by the Fourier transform of the PSF. Division in Fourier space is equivalent to deconvolution in real space,
making inverse filtering the simplest method to reverse the convolution in the image and providing the most faithful representation of the image. The calculation is fast, with a similar speed as
two-dimensional deblurring methods. However, the method’s utility is limited by noise amplification. During division in Fourier space, small noise variations in the Fourier transform are amplified by
the division operation. The result is that blur removal is compromised as a trade-off against a gain in noise. This technique can also introduce an artifact known as ringing, where ghost structures
appear surrounding the original object.
Additional noise and ringing can be reduced by plugging in some assumptions about the structure of the object that gave rise to the image. For instance, if the object is assumed to be relatively
smooth, noisy solutions with rough edges can be eliminated. Regularization can be applied in one step within an inverse filter, or it can be applied iteratively. The result is an image stripped of
high Fourier frequencies, with a smoother appearance. Much of the “roughness” removed in the image resides at Fourier frequencies well beyond the resolution limit and, therefore, the process does not
eliminate structures recorded by the microscope. However, because there is a potential for loss of detail, software implementations of inverse filters typically include an adjustable parameter that
enables the user to control the trade-off between smoothing and noise amplification for the truest representation of data. In most image-processing software programs, these algorithms have a variety
of names, including Wiener deconvolution, regularized least squares, linear least squares, and Tikhonov-Miller regularization.
Constrained iterative algorithms
A typical constrained iterative algorithm improves the performance of inverse filters by applying additional algorithms to restore photons to the correct position in the image. These methods operate
in successive cycles based on results from previous cycles, hence the term “iterative.” An initial estimate of the object is performed and convolved with the PSF. The resulting “blurred estimate” is
compared with the raw image to compute an error criterion that represents how similar the blurred estimate is to the raw image. Using the information contained in the error criterion, a new iteration
takes place. The new estimate is convolved with the PSF, a new error criterion is computed, and so on (figure on previous page). The best estimate is the one that minimizes the error criterion. As
the algorithm progresses, each time the software determines that the error criterion has not been minimized, a new estimate is blurred again, and the error criterion is recomputed. The cycle is
repeated until the error criterion is minimized or reaches a defined threshold. The final restored image is the object estimate at the last iteration.
A z-stack for deconvolution, before processing (top) and after (bottom). Courtesy of Gavin Ryan/ Olympus Sales.
The constrained iterative algorithms offer good results, but they are not suitable for all imaging setups. They require long calculation times and place a high demand on computer processors, reducing
available processing power for image acquisition or simple data reporting. This can be overcome with modern technol- ogies, such as GPU-based processing, which significantly improves speed. To take
full advantage of the algorithms, three-dimensional images are required, though two-dimensional images can be used with limited performance.
Clarifying microscopic images
Some recommend deconvolution as an alternate technique to using a confocal microscope^1. This recommendation is not strictly true because deconvolution techniques can also be applied to the images
acquired using the pinhole aperture in a confocal microscope. In fact, it is possible to restore images acquired with a confocal, multiphoton, or superresolution light microscope through convolution.
The combination of optical image improvement through confocal or superresolution microscopy and deconvolution techniques improves sharpness beyond what is generally attainable with either technique
alone. However, the major ben- efit of deconvolving images from these specialized microscopes is decreased noise in the final image. This is particularly helpful for low-light applications such as
live cell superresolution or confocal imaging. Deconvolution of multiphoton images has also been successfully used to remove noise and improve contrast for particularly convoluted images. In all
cases, care must be taken to apply an appropriate PSF, especially if the confocal pinhole aperture is adjustable.
Deconvolution in practice
Processing speed and quality are dramatically affected by how software and user controls implement the deconvolution algorithm. The algorithm can be optimized to reduce the number of iterations and
accelerate convergence to produce a stable estimate. For example, an unoptimized Jansson-Van Cittert algorithm usually requires between 50 and 100 iterations to converge to an optimal estimate. By
prefiltering the raw image to suppress noise and correcting with an additional error criterion on the first two iterations, the algorithm converges in only 5 to 10 iterations.
When using an empirical PSF, it is critical to use a high-quality PSF with minimal noise. To achieve this, commercial software packages contain preprocessing routines that reduce noise and enforce
radial symmetry by averaging the Fourier transform of the PSF. Many software packages also enforce axial symmetry in the PSF and assume the absence of spherical aberration. These steps reduce the
empirical PSF’s noise and aberrations and make a significant difference in the restoration’s quality.
Preprocessing can also be applied to raw images using routines such as background subtraction and flat-field correction. These operations can improve the signal-to-noise ratio and remove certain
artifacts that are detrimental to the final image.
In general, the more faithful the data representation, the more computer memory and processor time are required to deconvolve an image. Previously, images would be divided into subvolumes to
accommodate processing power, but modern technologies have reduced this barrier and expanded into larger data sets.
Meet the author
Lauren Alvarenga is a product manager for life science research imaging at Olympus Corp. of the Americas. She is currently responsible for inverted microscopes, imaging software, and superresolution
microscopy. Alvarenga received a B.S. in biomedical photographic communications from the Rochester Institute of Technology in 2010.
1. P.J. Shaw and D.J. Rawlins (1991). The point-spread function of a confocal microscope: its measurement and use in deconvolution of 3-D data. J Microsc, Vol. 163, Issue 2, pp. 151-165. | {"url":"https://www.photonics.com/Articles/Deconvolution_Helps_Break_Down_Imaging_Barriers/a65198","timestamp":"2024-11-15T02:46:49Z","content_type":"application/xhtml+xml","content_length":"65279","record_id":"<urn:uuid:66ec0fc5-0eb5-48b3-9948-25aaa00c5c75>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00372.warc.gz"} |
Conditional formatting to find 3 closest values
Im currently trying to use conditional formating to highlight a col of numbers (A1:A99) to find the closest 3 values to the value entered into a cell (B2), including a exact match and ties.
Any help would be great.
Excel Facts
How to calculate loan payments in Excel?
Use the PMT function: =PMT(5%/12,60,-25000) is for a $25,000 loan, 5% annual interest, 60 month loan.
Jul 13, 2011
Hi confusedjo,
you could insert two auxiliary columns:
column C is the absolute delta of column A to B2, column D is the RANK of each element in column C.
Now you can add three conditional formats for column with the following formula (with cell A1 being the active cell): =$D1=1
If there are 5 closest values all the same you'll want those 5 cells highlighted, but will you want to highlight any more cells?
What version of Excel?
Additionally to the two questions in my last message, say you had to highlight the closest
values to the value
from this list (I realise the list might not be sorted in reality):
you might highlight the yellow cells (they're all 1 away from the target
). Now you want the closest
values; would you stick with the yellow cells or include the orange cells (which are all 2 away from the target) too?
Re first reply
Excel 2010 version
If there are 5 values (ties) then yes i would wont all highlighted.
Re second reply
I would want the list to look as it does - I only need the 3 closest values but if there are ties like you have in your example then it would be the 2 closest values.
scores:- 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,34 and the value i want to find the 3 closest is 27 - then 27, 28, 26 would highlight.
Scores:- 23, 24, 25, 26, 27, 27, 28 , 29, 30, 31, 32, 33,34 and the value i want to find the 3 cloest is 27 - then 27, 27, 28 (or 26) would highlight.
Thank you for you time in responding and hope this makes sense.
Hi confusedjo,
you could insert two auxiliary columns:
column C is the absolute delta of column A to B2, column D is the RANK of each element in column C.
Now you can add three conditional formats for column with the following formula (with cell A1 being the active cell): =$D1=1
Sorry i dont really understand this (rather new to excel)
Right, this is going to take some study!
Excel Workbook
A B C D E
Cells with Conditional Formatting
Cell Condition Cell Format Stop If True
A1 1. / Formula is =ISNUMBER(MATCH(ABS($B$2-A1),SMALL(ABS($B$2-$A$1:$A$13),diddy),0)) Abc
C1 1. / Formula is =ABS($B$2-C1)<=SMALL(IF(FREQUENCY(ABS($B$2-$C$1:$C$13),ABS($B$2-$C$1:$C$13))>0,ABS($B$2-$C$1:$C$13)),$B$1) Abc
E1 1. / Formula is =ABS($B$2-E1)<=SMALL(ABS($B$2-$E$1:$E$13),$B$1) Abc
There are three type of conditional formatting above, 1 each in columns A, C & E. They behave quite differently from one another.
I've used B2 as the target value for all three columns.
B1 is referred to in the conditional formatting for columns C and E only, and is the
in 'highlight the nearest
Column E first:
This is the simplest, but has limitations when there are lots of ties, because only the nearest 1 value will be highligted if there are more than 3 ties nearest. Play with the values in B1 and B2.
Column A:
B1 isn't used in this formula. It uses a defined Name (
) which is defined as:
because conditional formatting didn't like using hard-coded arrays.
Actually, this one might be quite similar to column E.
Column C:
This one's my favourite; it first strips out duplicates from the list to get a unique list, then takes the three closest
and highlights them, regardless of how many there might be. It will include values which are equidistant from the target, above or below, so may seem to highlight quite a lot when there are such
equidistant values. It does use both B1 and B2 - so play with these values.
I couldn't easily find a way to get the results exactly as you want in your last message.
WOW! (is my first comment) thank you so much for your work.
I will try all the suggested forumla's to see which works best, will leave message when i have find which appears to be the one that suits my list of scores.
Thanks heaps!
Column C worked great thanks - I played with the values as you suggested and it appears having a 2 in B1 gives a closer result to what i was wanting.
Thank you so much, this is going to help me out so much!! | {"url":"https://www.mrexcel.com/board/threads/conditional-formatting-to-find-3-closest-values.586998/","timestamp":"2024-11-03T04:12:20Z","content_type":"text/html","content_length":"143824","record_id":"<urn:uuid:e60a89d9-932b-4e71-8f25-ce7a2f92c206>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00139.warc.gz"} |
Teaching the least common multiple using the lattice method
Algebra Tutorials! Saturday 2nd of November
teaching the least common multiple using the lattice method
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Miscellaneous Equations
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Two Variables BillgVabBongo Posted: Thursday 28th of Dec 09:51
Literal Numbers Will somebody be able to crack my difficulty with my math? I have attempted to find myself a tutor who can assist me . But until now I have
Adding and Subtracting not succeeded . Its not easy to locate somebody easy to get to and inexpensive . But then I must get over my problem with teaching the
Polynomials least common multiple using the lattice method as my exams are coming up just now. It will be a huge help for me if somebody can guide me.
Subtracting Integers
Simplifying Complex Registered:
Fractions 16.02.2005
Decimals and Fractions From: On The Interweb
Multiplying Integers Thingy
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial Jahm Xjardx Posted: Saturday 30th of Dec 07:24
Factoring Trinomials Algebrator is the latest hot favourite of teaching the least common multiple using the lattice method students. I know a couple of teachers
The Pythagorean Theorem who actually ask their students to have a copy of this program at their residence .
Solving Radical
Equations in One
Variable Registered:
Multiplying Binomials 07.08.2005
Using the FOIL Method From: Odense,
Imaginary Numbers Denmark, EU
Solving Quadratic
Equations Using the
Quadratic Formula
Solving Quadratic DoniilT Posted: Saturday 30th of Dec 08:49
Equations Hello ! Algebrator is a very useful piece of software. I use it daily to solve equations . You must try it. It may solve your issue .
Order of Operations
Dividing Complex Numbers
Polynomials Registered:
The Appearance of a 27.08.2002
Polynomial Equation From:
Standard Form of a Line
Positive Integral
Dividing Fractions Vnode Posted: Saturday 30th of Dec 11:05
Solving Linear Systems Algebrator is a incredible software and is surely worth a try. You will find quite a few exciting stuff there. I use it as reference
of Equations by software for my math problems and can say that it has made learning math much more enjoyable.
Multiplying and Dividing Registered:
Square Roots 27.09.2001
Functions and Graphs From: Germany
Dividing Polynomials
Solving Rational
Use of Parentheses or
Brackets (The
Distributive Law)
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1
Adding Fractions
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
teaching the least common multiple using the lattice method
Related topics:
Home indefinite integral problem solver | 3 variable two equation solver | simplifying fractions algebra activity | year 11 maths algebra | factoring polynomials
Rotating a Parabola calculator | nonhomogeneous differential equation | solving polynomial functions on ti-83 | convert .62 into fraction | subtracting equations calculator | solve
Multiplying Fractions the system with fractions | online polar graphing calculator and table | solutions to fundamental of physics sixth edition problem supplement #1 | algebraic
Finding Factors subtraction help
Miscellaneous Equations
Mixed Numbers and
Improper Fractions Author Message
Systems of Equations in
Two Variables BillgVabBongo Posted: Thursday 28th of Dec 09:51
Literal Numbers Will somebody be able to crack my difficulty with my math? I have attempted to find myself a tutor who can assist me . But until now I have
Adding and Subtracting not succeeded . Its not easy to locate somebody easy to get to and inexpensive . But then I must get over my problem with teaching the
Polynomials least common multiple using the lattice method as my exams are coming up just now. It will be a huge help for me if somebody can guide me.
Subtracting Integers
Simplifying Complex Registered:
Fractions 16.02.2005
Decimals and Fractions From: On The Interweb
Multiplying Integers Thingy
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial Jahm Xjardx Posted: Saturday 30th of Dec 07:24
Factoring Trinomials Algebrator is the latest hot favourite of teaching the least common multiple using the lattice method students. I know a couple of teachers
The Pythagorean Theorem who actually ask their students to have a copy of this program at their residence .
Solving Radical
Equations in One
Variable Registered:
Multiplying Binomials 07.08.2005
Using the FOIL Method From: Odense,
Imaginary Numbers Denmark, EU
Solving Quadratic
Equations Using the
Quadratic Formula
Solving Quadratic DoniilT Posted: Saturday 30th of Dec 08:49
Equations Hello ! Algebrator is a very useful piece of software. I use it daily to solve equations . You must try it. It may solve your issue .
Order of Operations
Dividing Complex Numbers
Polynomials Registered:
The Appearance of a 27.08.2002
Polynomial Equation From:
Standard Form of a Line
Positive Integral
Dividing Fractions Vnode Posted: Saturday 30th of Dec 11:05
Solving Linear Systems Algebrator is a incredible software and is surely worth a try. You will find quite a few exciting stuff there. I use it as reference
of Equations by software for my math problems and can say that it has made learning math much more enjoyable.
Multiplying and Dividing Registered:
Square Roots 27.09.2001
Functions and Graphs From: Germany
Dividing Polynomials
Solving Rational
Use of Parentheses or
Brackets (The
Distributive Law)
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1
Adding Fractions
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and
Improper Fractions
Systems of Equations in
Two Variables
Literal Numbers
Adding and Subtracting
Subtracting Integers
Simplifying Complex
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical
Equations in One
Multiplying Binomials
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the
Quadratic Formula
Solving Quadratic
Order of Operations
Dividing Complex Numbers
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral
Dividing Fractions
Solving Linear Systems
of Equations by
Multiplying and Dividing
Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational
Use of Parentheses or
Brackets (The
Distributive Law)
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1
Adding Fractions
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
teaching the least common multiple using the lattice method
Related topics:
indefinite integral problem solver | 3 variable two equation solver | simplifying fractions algebra activity | year 11 maths algebra | factoring polynomials calculator | nonhomogeneous
differential equation | solving polynomial functions on ti-83 | convert .62 into fraction | subtracting equations calculator | solve the system with fractions | online polar graphing
calculator and table | solutions to fundamental of physics sixth edition problem supplement #1 | algebraic subtraction help
Author Message
BillgVabBongo Posted: Thursday 28th of Dec 09:51
Will somebody be able to crack my difficulty with my math? I have attempted to find myself a tutor who can assist me . But until now I have not succeeded . Its not
easy to locate somebody easy to get to and inexpensive . But then I must get over my problem with teaching the least common multiple using the lattice method as my
exams are coming up just now. It will be a huge help for me if somebody can guide me.
From: On The Interweb
Jahm Xjardx Posted: Saturday 30th of Dec 07:24
Algebrator is the latest hot favourite of teaching the least common multiple using the lattice method students. I know a couple of teachers who actually ask their
students to have a copy of this program at their residence .
From: Odense,
Denmark, EU
DoniilT Posted: Saturday 30th of Dec 08:49
Hello ! Algebrator is a very useful piece of software. I use it daily to solve equations . You must try it. It may solve your issue .
Vnode Posted: Saturday 30th of Dec 11:05
Algebrator is a incredible software and is surely worth a try. You will find quite a few exciting stuff there. I use it as reference software for my math problems and
can say that it has made learning math much more enjoyable.
From: Germany
Author Message
BillgVabBongo Posted: Thursday 28th of Dec 09:51
Will somebody be able to crack my difficulty with my math? I have attempted to find myself a tutor who can assist me . But until now I have not succeeded . Its not easy to
locate somebody easy to get to and inexpensive . But then I must get over my problem with teaching the least common multiple using the lattice method as my exams are coming up
just now. It will be a huge help for me if somebody can guide me.
From: On The Interweb
Jahm Xjardx Posted: Saturday 30th of Dec 07:24
Algebrator is the latest hot favourite of teaching the least common multiple using the lattice method students. I know a couple of teachers who actually ask their students to
have a copy of this program at their residence .
From: Odense,
Denmark, EU
DoniilT Posted: Saturday 30th of Dec 08:49
Hello ! Algebrator is a very useful piece of software. I use it daily to solve equations . You must try it. It may solve your issue .
Vnode Posted: Saturday 30th of Dec 11:05
Algebrator is a incredible software and is surely worth a try. You will find quite a few exciting stuff there. I use it as reference software for my math problems and can say
that it has made learning math much more enjoyable.
From: Germany
Posted: Thursday 28th of Dec 09:51
Will somebody be able to crack my difficulty with my math? I have attempted to find myself a tutor who can assist me . But until now I have not succeeded . Its not easy to locate somebody easy to get
to and inexpensive . But then I must get over my problem with teaching the least common multiple using the lattice method as my exams are coming up just now. It will be a huge help for me if somebody
can guide me.
Posted: Saturday 30th of Dec 07:24
Algebrator is the latest hot favourite of teaching the least common multiple using the lattice method students. I know a couple of teachers who actually ask their students to have a copy of this
program at their residence .
Posted: Saturday 30th of Dec 08:49
Hello ! Algebrator is a very useful piece of software. I use it daily to solve equations . You must try it. It may solve your issue .
Posted: Saturday 30th of Dec 11:05
Algebrator is a incredible software and is surely worth a try. You will find quite a few exciting stuff there. I use it as reference software for my math problems and can say that it has made
learning math much more enjoyable. | {"url":"https://gre-test-prep.com/algebra-1-practice-test/exponent-rules/teaching-the-least-common.html","timestamp":"2024-11-02T20:25:02Z","content_type":"text/html","content_length":"113242","record_id":"<urn:uuid:b794a7f3-cc19-496c-a753-1cc8793ac701>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00285.warc.gz"} |
Acceleration over Distance and Time
Question Video: Acceleration over Distance and Time Physics • First Year of Secondary School
A bicycle has an initial velocity of 12 m/s and it accelerates for 15 s in the direction of its velocity, moving 220 m during that time. What is the bicycle’s rate of acceleration? Answer to two
decimal places.
Video Transcript
A bicycle has an initial velocity of 12 meters per second. And it accelerates for 15 seconds in the direction of its velocity, moving 220 meters during that time. What is the bicycle’s rate of
acceleration? Answer to two decimal places.
Alright, so in this question, we’ve got a bicycle. So let’s start by saying that it’s moving in this direction. We choose this arbitrarily. And we’ve been told that this bicycle has an initial
velocity of 12 meters per second. Now, we’ve been told that this bicycle accelerates for 15 seconds in the direction of its velocity. In other words, it accelerates for 15 seconds in this direction,
the same direction that it was initially travelling in. So its velocity is increasing. It’s getting faster and faster. So here’s our bicycle after accelerating. And we’ve been told that the
acceleration phase takes a time of 15 seconds. And in that time, we’ve been told that the bicycle moves a distance of 220 meters.
Now, based on all of this information, we need to find the rate of the bicycle’s acceleration. So if we say that what we’re trying to find, the acceleration, is 𝑎. And if we further say that the
initial velocity of the bicycle was 𝑢, the time taken to accelerate was 𝑡. And the distance moved by the bicycle was 𝑠. Then we need to find an equation that links together the quantities 𝑢, 𝑡, 𝑠,
and 𝑎.
Now, the equation that we’re looking for is one of the kinematic equations. Specifically, the equation that tells us that the distance moved by the bicycle in a straight line or, in other words, the
displacement of the bicycle. Is equal to the initial velocity of the bicycle multiplied by the time over which the bicycle accelerates plus half multiplied by the acceleration multiplied by the time
taken for the acceleration squared. In other words, 𝑠 is equal to 𝑢𝑡 plus half 𝑎𝑡 squared.
Now, in this equation, we know the value of 𝑠, the value of 𝑢, and the value of 𝑡. And the only thing we don’t know is the value of 𝑎. So we need to rearrange and solve for 𝑎. To do this, we can
start by subtracting 𝑢𝑡 from both sides of the equation. This way, on the right-hand side, 𝑢𝑡 cancels with minus 𝑢𝑡. And so what we’re left with is 𝑠 minus 𝑢𝑡 is equal to half 𝑎𝑡 squared. Then, we
can multiply both sides of the equation by two divided by 𝑡 squared. And this way, the factor of half on the right-hand side cancels with the factor of two in the numerator. And we have 𝑡 squared
divided by 𝑡 squared. Those cancel too, leaving us with just 𝑎 on the right-hand side.
And so when we finally rearrange the equation, we find that the acceleration of the bicycle is equal to two multiplied by 𝑠 minus 𝑢𝑡, all divided by 𝑡 squared. Then, we just need to plug in some
values. We could say that the acceleration of the bicycle 𝑎 is equal to two multiplied by 𝑠, which is 220 meters, minus 𝑢𝑡. So we plug in 𝑢, which is 12 meters per second, and 𝑡, which is 15 seconds.
And then we divide this whole thing by 𝑡 squared, which happens to be 15 seconds whole squared. Now when it comes to units, everything might look a little bit messy. We’ve got metres here. We’ve got
meters per second here, seconds here, and second squared in the denominator. However, things are gonna cancel out very nicely. And to see that, let’s start by simplifying this pair of parentheses
What we’ve got is 12 meters per second multiplied by 15 seconds. So to calculate that, we first need to calculate 12 multiplied by 15 to give us the numerical value. And then we need to calculate
meters divided by seconds multiplied by seconds. And so when we put the whole thing together, the numerical value, 12 times 15, ends up being 180. And then we see that this factor of seconds in the
numerator cancels with this factor of seconds in the denominator. And what we’re left with is simply meters.
Hence, we can replace the power of parentheses with 180 meters. Then we can see that all we have in the numerator of this large fraction now is the unit of meters in both case. We’re subtracting 180
meters from 220 meters. And then we multiply all of that simply by a number, which is two in this case. So simplifying the whole of the numerator, we have 220 meters minus 180 meters, which is 40
meters. And then we multiply 40 meters by two, which is 80 meters. And now our fraction is looking a lot nicer.
To make it look even more clean, let’s simplify this parenthesis. We’ve got 15 seconds whole squared. So to work that out, we need to square the 15. And we need to square the seconds. So 15 squared
is 225. And second squared is well second squared. Then we can see that our fraction is going to have a numerical value of 80 divided by 225. And a unit of meters divided by second squared or meters
per second squared. Which is perfect because we are trying to calculate an acceleration here. And the base unit of acceleration is meters per second squared.
So all that’s left to do now is to calculate the numerical value 80 divided by 225 to two decimal places. When we do this, we find that the acceleration of the bicycle is 0.36 meters per second
squared. And so at this point, we found the answer to our question. | {"url":"https://www.nagwa.com/en/videos/965184240265/","timestamp":"2024-11-05T02:34:15Z","content_type":"text/html","content_length":"251724","record_id":"<urn:uuid:15b9b4db-a8f5-45a7-adc9-d91725d7e71a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00001.warc.gz"} |
How to Convert 9mm Bullet Speed from Miles per Hour to Meters per Second
What is the process to convert the speed of a 9mm bullet from miles per hour to meters per second? To convert the speed of a 9mm bullet from miles per hour to meters per second, you need to use
conversion factors for miles to kilometers and kilometers to meters per second. By utilizing dimensional analysis, you can calculate the speed in meters per second.
When dealing with units of measurement, it's essential to convert between different scales to ensure accuracy. In this case, we are converting the speed of a 9mm bullet from miles per hour to meters
per second.
Conversion Factors:
1 mile = 1.60934 kilometers
1 kilometer = 0.621371 miles
1 hour = 3600 seconds
Dimensional Analysis:
First, we convert miles to kilometers by multiplying the speed of the bullet in miles per hour by 1.60934 km/mi. This step gives us the speed of the bullet in kilometers per hour.
Next, we convert kilometers per hour to meters per second by multiplying the speed in kilometers per hour by 0.277778 m/s/km. This final step gives us the speed of the bullet in meters per second.
Let's calculate the speed of a 9mm bullet that travels at 853 miles per hour in meters per second:
853 mph * 1.60934 km/mi * 0.277778 m/s/km = 385.698 m/s
Final Answer:
The speed of a 9mm bullet traveling at 853 miles per hour is equivalent to 385.698 meters per second. | {"url":"https://www.ledgemulepress.com/chemistry/how-to-convert-9mm-bullet-speed-from-miles-per-hour-to-meters-per-second.html","timestamp":"2024-11-10T05:18:50Z","content_type":"text/html","content_length":"21314","record_id":"<urn:uuid:68be1293-3d0c-498a-ae9b-61886df3bd93>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00496.warc.gz"} |
The Linear Coefficient of Expansion of Titanium?
by Jim
(United Kingdom)
Q : I am currently working on an A-Level Physics research project on Titanium, it's properties and uses. I read about the fact that Titanium has a low Linear Coefficient of Expansion, but am unable
to find an explanation of the reason why Titanium's is low. I have only found a generic piece of information saying the L-C of E is related to bond energies, not information specific to Titanium.
A :
A great question and a not so simple answer. I’ll try to take it gradually, so that everybody can understand what we’re talking about, Jim.
So, the coefficient linear of expansion is about the thermal expansion of different materials, when exposed to an increased temperature. When a substance is heated, its particles begin moving more
and thus usually maintain a greater average separation. The degree of expansion divided by the change in temperature is called the material's coefficient of thermal expansion and generally varies
with temperature. The linear thermal expansion coefficient relates the change in a material's linear dimensions to a change in temperature. It is the fractional change in length per degree of
temperature change. There is also possible to calculate the area and volume coefficients of thermal expansion.
The difference linear coefficient of expansion of materials is determined by the different thermal conductivity of each material, which is subsequently affected by the different specific heats of
To clarify the definition – the thermal conductivity the thermal conductivity can be thought of as the container for the medium-dependent properties which relate the rate of heat loss per unit area
to the rate of change of temperature. The specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius.
It is generally known that metals are better heat conductors than non-metallic solids and gases, and the physics fundaments for this property is the actual heat-transfer in mechanism which involves
electrons, similar as in electrical conductivity. . You may have already guessed that this two are closely related. At a given temperature, the thermal and electrical conductivities of metals are
proportional. It should also be noted that raising the temperature increases the thermal conductivity while decreasing the electrical conductivity
And now, we return to the matter at hand. Titanium is documented to display both low thermal conductivity and low electrical conductivity properties. The answer to your questions lies in the way
electrons are distributed in titanium atoms, as compared to other metals that have better conductivity properties.
More precisely, in physics, the notion is known as Free Electron Density, which should be pretty self – explanatory. This coefficient differs from metals to metal based on atomic mass, density of the
material and some other more coefficients related to advanced physics theories (Fermi energies)
In titanium metal, both the atom mass and density of materials are lower than of other, better conductive metals, say, copper.
I think this should be enough for a start in the right direction. You may find further references about titanium’s electrical and thermal conductivity in E.W. Collings books: “A sourcebook of
titanium alloy superconductivity” and “Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys”
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Keep it simple
Can all unit fractions be written as the sum of two unit fractions?
Keep it Simple printable sheet
Unit fractions (fractions which have numerators of 1) can be written as the sum of two different unit fractions.
For example
$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$
Charlie thought he'd spotted a rule and made up some more examples.
$\frac{1}{2} = \frac{1}{10} + \frac{1}{20}$
$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$
$\frac{1}{3} = \frac{1}{7} + \frac{1}{21}$
$\frac{1}{4} = \frac{1}{5} + \frac{1}{20}$
Can you describe Charlie's rule?
The denominator of the last fraction is the product of the denominators of the first two fractions.
Are all his examples correct?
What do you notice about the sums that are correct?
Find some other correct examples..
How would you explain to Charlie how to generate lots of correct examples?
Alison started playing around with $\frac{1}{6}$ and was surprised to find that there wasn't just one way of doing this.
She found:
$\frac{1}{6} = \frac{1}{7} + \frac{1}{42}$
$\frac{1}{6} = \frac{1}{8} + \frac{1}{24}$
$\frac{1}{6} = \frac{1}{9} + \frac{1}{18}$
$\frac{1}{6} = \frac{1}{10} + \frac{1}{15}$
$\frac{1}{6} = \frac{1}{12} + \frac{1}{12}$ (BUT she realised this one didn't count because they were not different.)
Charlie tried to do the same with $\frac{1}{8}$. Can you finish Charlie's calculations to see which ones work?
$\frac{1}{8} = \frac{1}{9} + ?$
$\frac{1}{8} = \frac{1}{10} + ?$
$\frac{1}{8} = \frac{1}{11} + ?$
Can all unit fractions be made in more than one way like this?
Choose different unit fractions of your own to test out your theories.
Getting Started
Try working systematically through all the possibilities.
$\frac{1}{8} = \frac{1}{9} + ?$
$\frac{1}{8} = \frac{1}{10} + ?$
$\frac{1}{8} = \frac{1}{11} + ?$
$\frac{1}{8} = \frac{1}{12} + ?$
$\frac{1}{8} = \frac{1}{13} + ?$
$\frac{1}{8} = \frac{1}{14} + ?$
$\frac{1}{8} = \frac{1}{15} + ?$
$\frac{1}{8} = \frac{1}{16} + ?$ (but this won't count)
Why is $\frac{1}{9}$ the first one you can use?
Why don't you need to go further than $\frac{1}{16}$?
Student Solutions
Catherine and Poppy from Stoke by Nayland Middle School made a good start on this problem, and Kijung from Wind Point Elementary School found that:
Not all of Charlie's examples were right.
To be correct, one of the unit fractions must have a denominator which is 1 more than the denominator of the original unit fraction, and the other unit fraction must have a denominator which is the
product of the other two denominators:
$$ \frac{1}{n} = \frac{1}{n+1}+\frac{1}{n(n+1)}$$
Here are some other examples that work:
$ \frac{1}{5} = \frac{1}{6}+\frac{1}{30}$
$ \frac{1}{6} = \frac{1}{7}+\frac{1}{42}$
$ \frac{1}{105} = \frac{1}{106}+\frac{1}{11130}$
$\frac{1}{8}$ can also be expressed as the sum of two unit fractions in several ways:
$\frac{1}{8} = \frac{1}{9} +\frac{1}{72}$
$\frac{1}{8} = \frac{1}{10} +\frac{1}{40}$
$\frac{1}{8} = \frac{1}{11} + \frac{1}{n}$ is not possible
$\frac{1}{8} = \frac{1}{12} +\frac{1}{24}$
Felix from Condover Primary acutely observed that unit fractions with denominators which are prime numbers can only be written in one way as the sum of two distinct unit fractions.
Rose, from Claremont Primary School in Tunbridge Wells, Kent worked out a general formula:
$ \frac{1}{z} = \frac{1}{y}+\frac{1}{x}$ (where $z$, $y$ and $x$ are positive integers and $y < x$)
Using $\frac{1}{10}$ as an example:
$ \frac{1}{10} = \frac{1}{11}+\frac{1}{110}$
$ \frac{1}{10} = \frac{1}{12}+\frac{1}{60}$
$ \frac{1}{10} = \frac{1}{14}+\frac{1}{35}$
$ \frac{1}{10} = \frac{1}{15}+\frac{1}{30}$
I listed the values of $y-z$ that provide solutions:
$1$, $2$, $4$ and $5$
These are also the factors of $z^ 2$ (i.e. $100$) that are smaller than its square root: $1\times100$
This pattern also occurred for $\frac{1}{12}$:
$ \frac{1}{12} = \frac{1}{13}+\frac{1}{156}$
$ \frac{1}{12} = \frac{1}{14}+\frac{1}{84}$
$ \frac{1}{12} = \frac{1}{15}+\frac{1}{60}$
$ \frac{1}{12} = \frac{1}{16}+\frac{1}{48}$
$ \frac{1}{12} = \frac{1}{18}+\frac{1}{36}$
$ \frac{1}{12} = \frac{1}{20}+\frac{1}{30}$
$ \frac{1}{12} = \frac{1}{21}+\frac{1}{28}$
Here $y - z = 1, 2, 3, 4, 6, 8, 9$
and the factors of $z ^ 2$ (i.e.$144$) are:
$\frac{1}{10}$ can be written as the sum of two different unit fractions in $4$ ways.
In this case $z ^ 2$ has $9$ factors and $y-z = 4$
$\frac{1}{12}$ can be written as the sum of two different unit fractions in $7$ ways.
In this case $z ^ 2$ has $15$ factors and $y-z = 7$
If $n$ is the number of factors of $z ^ 2$,
$\frac{1}{z}$ can be written as the sum of two different unit fractions in $\frac{n -1}{2}$ ways.
Rose's conclusion draws on her two examples, but when we generalise in mathematics, we need to be sure that what we have noticed will be true in all other cases.
Can anyone provide a convincing explanation for why Rose's conclusion is, or is not, correct?
Teachers' Resources
Why do this problem?
This is the first problem in a set of three linked activities. Egyptian Fractions and The Greedy Algorithm follow on.
It's often difficult to find interesting contexts to consolidate addition and subtraction of fractions. This problem offers that, whilst also requiring students to develop and analyse different
strategies and explain their findings.
Possible approach
Pose the initial part of the problem as it is set and ask the students to suggest what Charlie's rule might be. Allow some time for them to work out which sums are correct and ask them to modify
Charlie's rule so that it always generates correct solutions. Working in pairs, invite students to generate some more examples that confirm their new rule. Collect some of these on the board for a
general discusion. (With some classes this could lead to an algebraic explanation/proof.)
Alison's question offers an opportunity to involve the whole class in a collaborative activity. Talk through what Alison might have been thinking as she generated different pairs which worked. This
might be an opportunity to talk to the class about the value of working systematically. How can Alison be sure that she has found all the possible pairs?
In pairs, ask students to choose their own unit fraction and find all the correct pairs.
Collect all results on the board and encourage students to share their strategies for finding all possible combinations.
Key questions
Can a unit fraction always be written as the sum of two different unit fractions?
Which unit fractions can only be written in one way?
What is the strategy for finding all the combinations of two unit fractions that add up to a third unit fraction?
Possible support
Some students may find it easier to contribute to the class discussion by working systematically to generate lots of unit sum calculations and highlighting any that result in a unit fraction as an
For example
$\frac{1}{6} + \frac{1}{7} = \frac{13}{42}$ Image
$\frac{1}{6} + \frac{1}{8} = \frac{7}{24}$ Image
$\frac{1}{6} + \frac{1}{12} = \frac{1}{4}$ Image
Possible extension
Ask students to produce an algebraic or visual proof of Charlie's revised rule.
Can they predict how many different pairs of unit fractions will add up to any given unit fraction?
You may wish to move students on to Egyptian Fractions. | {"url":"https://nrich.maths.org/problems/keep-it-simple","timestamp":"2024-11-14T23:30:53Z","content_type":"text/html","content_length":"48380","record_id":"<urn:uuid:1efe594b-3336-4975-9e9b-23e6edf20e11>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00689.warc.gz"} |
1.1 Overview of Time Series Characteristics | STAT 510
In this lesson, we’ll describe some important features that we must consider when describing and modeling a time series. This is meant to be an introductory overview, illustrated by example, and not
a complete look at how we model a univariate time series. Here, we’ll only consider univariate time series. We’ll examine relationships between two or more time series later on.
Univariate Time Series
A univariate time series is a sequence of measurements of the same variable collected over time. Most often, the measurements are made at regular time intervals.
One difference from standard linear regression is that the data are not necessarily independent and not necessarily identically distributed. One defining characteristic of a time series is that it is
a list of observations where the ordering matters. Ordering is very important because there is dependency and changing the order could change the meaning of the data.
Basic Objectives of the Analysis
The basic objective usually is to determine a model that describes the pattern of the time series. Uses for such a model are:
1. To describe the important features of the time series pattern.
2. To explain how the past affects the future or how two time series can “interact”.
3. To forecast future values of the series.
4. To possibly serve as a control standard for a variable that measures the quality of product in some manufacturing situations.
Types of Models
There are two basic types of “time domain” models.
1. Models that relate the present value of a series to past values and past prediction errors - these are called ARIMA models (for Autoregressive Integrated Moving Average). We’ll spend substantial
time on these.
2. Ordinary regression models that use time indices as x-variables. These can be helpful for an initial description of the data and form the basis of several simple forecasting methods.
Important Characteristics to Consider First
Some important questions to first consider when first looking at a time series are:
• Is there a trend, meaning that, on average, the measurements tend to increase (or decrease) over time?
• Is there seasonality, meaning that there is a regularly repeating pattern of highs and lows related to calendar time such as seasons, quarters, months, days of the week, and so on?
• Are there outliers? In regression, outliers are far away from your line. With time series data, your outliers are far away from your other data.
• Is there a long-run cycle or period unrelated to seasonality factors?
• Is there constant variance over time, or is the variance non-constant?
• Are there any abrupt changes to either the level of the series or the variance?
The following plot is a time series plot of the annual number of earthquakes in the world with seismic magnitude over 7.0, for 99 consecutive years. By a time series plot, we simply mean that the
variable is plotted against time.
Some features of the plot:
• There is no consistent trend (upward or downward) over the entire time span. The series appears to slowly wander up and down. The horizontal line drawn at quakes = 20.2 indicates the mean of the
series. Notice that the series tends to stay on the same side of the mean (above or below) for a while and then wanders to the other side.
• Almost by definition, there is no seasonality as the data are annual data.
• There are no obvious outliers.
• It’s difficult to judge whether the variance is constant or not.
One of the simplest ARIMA type models is a model in which we use a linear model to predict the value at the present time using the value at the previous time. This is called an AR(1) model, standing
for autoregressive model of order 1. The order of the model indicates how many previous times we use to predict the present time.
A start in evaluating whether an AR(1) might work is to plot values of the series against lag 1 values of the series. Let \(x_t\) denote the value of the series at any particular time \(t\), so \(x_
{t-1}\) denotes the value of the series one time before time \(t\). That is, \(x_{t-1}\) is the lag 1 value of \(x_t\). As a short example, here are the first five values in the earthquake series
along with their lag 1 values:
t \(x_t\) \(x_{t-1}\) (lag 1 value)
1 13 *
For the complete earthquake dataset, here’s a plot of \(x_t\) versus \(x_{t-1}\) :
Although it’s only a moderately strong relationship, there is a positive linear association so an AR(1) model might be a useful model.
The AR(1) model
Theoretically, the AR(1) model is written
\(x_t = \delta + \phi_1 x_{t-1} + w_t\)
• \(w_t \overset{iid}{\sim} N(0, \sigma^2_w)\), meaning that the errors are independently distributed with a normal distribution that has mean 0 and constant variance.
• Properties of the errors \(w_t\) are independent of \(x\).
This is essentially the ordinary simple linear regression equation, but there is one difference. Although it’s not usually true, in ordinary least squares regression we assume that the x-variable is
not random but instead is something we can control. That’s not the case here, but in our first encounter with time series we’ll overlook that and use ordinary regression methods. We’ll do things the
“right” way later in the course.
Following is Minitab output for the AR(1) regression in this example:
quakes = 9.19 + 0.543 lag1
98 cases used, 1 cases contain missing values
Predictor Coef SE Coef T P
Constant 9.191 1.819 5.05 0.000
lag1 0.54339 0.08528 6.37 0.000
S = 6.12239 R-Sq = 29.7% R-Sq(adj) = 29.0%
We see that the slope coefficient is significantly different from 0, so the lag 1 variable is a helpful predictor. The \(R^2\) value is relatively weak at 29.7%, though, so the model won’t give us
great predictions.
Residual Analysis
In traditional regression, a plot of residuals versus fits is a useful diagnostic tool. The ideal for this plot is a horizontal band of points. Following is a plot of residuals versus predicted
values for our estimated model. It doesn’t show any serious problems. There might be one possible outlier at a fitted value of about 28.
1The following plot shows a time series of quarterly production of beer in Australia for 18 years.
Some important features are:
• There is an upward trend, possibly a curved one.
• There is seasonality – a regularly repeating pattern of highs and lows related to quarters of the year.
• There are no obvious outliers.
• There might be increasing variation as we move across time, although that’s uncertain.
There are ARIMA methods for dealing with series that exhibit both trend and seasonality, but for this example, we’ll use ordinary regression methods.
Classical regression methods for trend and seasonal effects
To use traditional regression methods, we might model the pattern in the beer production data as a combination of the trend over time and quarterly effect variables.
Suppose that the observed series is \(x_t\), for \(t = 1,2, \dots, n\).
• For a linear trend, use \(t\) (the time index) as a predictor variable in a regression.
• For a quadratic trend, we might consider using both \(t\) and \(t^2\).
• For quarterly data, with possible seasonal (quarterly) effects, we can define indicator variables such as \(S_j=1\) if the observation is in quarter \(j\) of a year and 0 otherwise. There are 4
such indicators.
Let \(\epsilon_t \overset{iid}{\sim} N(0, \sigma^2)\). A model with additive components for linear trend and seasonal (quarterly) effects might be written
\(x_t = \beta_1t+\alpha_1S_1+\alpha_2S_2 + \alpha_3S_3 +\alpha_4S_4 + \epsilon_t\)
To add a quadratic trend, which may be the case in our example, the model is
\(x_t = \beta_1t + \beta_2t^2 +\alpha_1S_1 + \alpha_2S_2 + \alpha_3S_3 +\alpha_4S_4 + \epsilon_t\)
We’ve deleted the “intercept” from the model. This isn’t necessary, but if we include it we’ll have to drop one of the seasonal effect variables from the model to avoid collinearity issues.
Back to Example 2: Following is the Minitab output for a model with a quadratic trend and seasonal effects. All factors are statistically significant.
Predictor Coef SE Coef T P
Time 0.5881 0.2193 2.68 0.009
tsqrd 0.031214 0.002911 10.72 0.000
quarter_1 261.930 3.937 66.52 0.000
quarter_2 212.165 3.968 53.48 0.000
quarter_3 228.415 3.994 57.18 0.000
quarter_4 310.880 4.018 77.37 0.000
Residual Analysis
For this example, the plot of residuals versus fits doesn’t look too bad, although we might be concerned by the string of positive residuals at the far right.
When data are gathered over time, we typically are concerned with whether a value at the present time can be predicted from values at past times. We saw this in the earthquake data of example 1 when
we used an AR(1) structure to model the data. For residuals, however, the desirable result is that the correlation is 0 between residuals separated by any given time span. In other words, residuals
should be unrelated to each other.
Sample Autocorrelation Function (ACF) Section
The sample autocorrelation function (ACF) for a series gives correlations between the series \(x_t\) and lagged values of the series for lags of 1, 2, 3, and so on. The lagged values can be written
as \(x_{t-1}, x_{t-2}, x_{t-3}\), and so on. The ACF gives correlations between \(x_t\) and \(x_{t-1}\), \(x_t\) and \(x_{t-2}\), and so on.
The ACF can be used to identify the possible structure of time series data. That can be tricky going as there often isn’t a single clear-cut interpretation of a sample autocorrelation function. We’ll
get started on that in Lesson 1.2 this week. The ACF of the residuals for a model is also useful. The ideal for a sample ACF of residuals is that there aren’t any significant correlations for any
Following is the ACF of the residuals for Example 1, the earthquake example, where we used an AR(1) model. The "lag" (time span between observations) is shown along the horizontal, and the
autocorrelation is on the vertical. The red lines indicated bounds for statistical significance. This is a good ACF for residuals. Nothing is significant; that’s what we want for residuals.
The ACF of the residuals for the quadratic trend plus seasonality model we used for Example 2 looks good too. Again, there appears to be no significant autocorrelation in the residuals. The ACF of
the residual follows:
Lesson 1.2 will give more details about the ACF. Lesson 1.3 will give some R code for examples in Lesson 1.1 and Lesson 1.2. | {"url":"https://online.stat.psu.edu/stat510/lesson/1/1.1","timestamp":"2024-11-12T23:54:57Z","content_type":"text/html","content_length":"83958","record_id":"<urn:uuid:5d22f831-ea23-4fdf-a91c-ff69b6a92691>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00410.warc.gz"} |
Bezier curves
Bill Casselman's home page
Bezier curves
Bezier curves are used in computer graphics to produce curves which appear reasonably smooth at all scales (as opposed to polygonal lines, which will not scale nicely). Mathematically, they are a
special case of cubic Hermite interpolation (whereas polygonal lines use linear interpolation). What this means is that curves are constructed as a sequence of cubic segments, rather than linear
ones. But whereas Hermite interpolating polynomials are constructed in terms of derivatives at endpoints, Bezier curves use a construction due to Sergei Bernstein, in which the interpolating
polynomials depend on certain control points. The mathematics of these curves is classical, but it was a French automobile engineer Pierre Bezier who introduced their use in computer graphics.
Roughly speaking, to each set of four points P[0], P[1], P[2], P[3] we associate a curve with the following properties:
* It starts at p0 and ends at p3.
* When it starts from p0 it heads directly towards p1, and when it arrives at p3 it is coming from the direction of p2.
* The entire curve is contained in the quadrilateral whose corners are the four given points (their convex hull).
One reason these curves are used so much in computer graphics is that they are very efficient to construct, since a simple recursion process means that the basic arithmetic operation needed to build
the points along one is just division by two. For this reason, also, the most efficient implementations use scaled integers instead of floating point numbers as basic numerical data.
Bezier curves are discussed in a lot of places in the computer graphics literature, but as far as I know the most detailed discussion of practical implementation is in D. E. Knuth's book Metafont:
the Program, pp. 123-131. My own program follows Knuth's quite closely, except that it doesn't include refinements to get that extra smoothness that makes Knuth's own curves so beautiful. Knuth's
entire book is available as a WEB file in the standard TEX distribution (it is in fact the source code for METAFONT.
I make available here my own Java source code for Bezier curves, and notes on Bezier curves from my UBC geometry course. And in addition some personal notes. Finally, a complete package can be found
in the file bezier.zip. Note that this package can be compiled with version 1.0.2 of Java (or later).
The main difference between Knuth's program and mine is that Knuth is concerned with writing all the pixels, whereas I am concerned with building up a polygon of straight line segments. For him it is
necessary to smooth out the pixelization as much as possible, and this causes him a lot of work. Most of his code is concerned with the octant subdivision, necessary in order to maintain the symmetry
of square pixelization.
A good reference on cubic interpolation is Henrici's book Essentials of numerical analysis (with pocket calculator demonstrations). Bernstein polynomials of all degrees are discussed in the book
Infinitesimal Calculus by Jean Dieudonne.
The program for building Bezier curves needs also two classes for dealing with real numbers and real points. | {"url":"https://personal.math.ubc.ca/~cass/gfx/bezier.html","timestamp":"2024-11-07T10:14:54Z","content_type":"text/html","content_length":"4062","record_id":"<urn:uuid:b4ff4755-e138-4feb-b115-9300636e5bf6>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00340.warc.gz"} |
Lesson: Representing 2-digit numbers (Part 1) | Oak National Academy
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Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom.
Lesson details
Key learning points
1. In this lesson, we will represent how many tens and ones there are in numbers within 100 in different ways and then representing these numbers on a number line.
This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.
5 Questions
What multiple of ten do these Dienes represent?
Which part-whole model matches the Dienes representation below?
Here is a representation of the number 90. How many more tens are needed to make 100?
Here is a representation of the number 40. How many more tens are needed to make 100?
5 Questions
What number do these strawberries represent?
Which bead string image represents the number line below?
Which bead string image represents the number line below?
The value of the digit 3 in my number is 3 ones. My number is less than 45. The digit in the tens place is one less than 5. What is my number?
The value of the digit 4 in my number is 4 tens. My number is less than 45. The digit in the ones place is one less than 2. What is my number? | {"url":"https://www.thenational.academy/teachers/lessons/representing-2-digit-numbers-part-1-6dhkgd","timestamp":"2024-11-15T04:25:17Z","content_type":"text/html","content_length":"291691","record_id":"<urn:uuid:6fa8ffa4-af8b-4344-8961-9108669777c5>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00172.warc.gz"} |
What does similarity mean in geometry?
What does similarity mean in geometry?
the same shape
Two figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent , and the ratios of the lengths of
their corresponding sides are equal.
How do you find structural similarity?
Structural similarity between two proteins is generally given as the root mean square deviation (RMSD) of their atomic positions after they have been structurally aligned. More on RMSD can be found
on wikipedia: https://en.wikipedia.org/wiki/Root-mean-square_deviation_of_atomic_positions.
What is the similarities of shape?
Lesson Summary Two shapes are similar when all corresponding sides have the same ratio of proportionality. This means that when you take the ratio of one pair of corresponding sides, this ratio must
be the same as the ratios of the other corresponding sides of the shape.
What are the similarity between the two triangles?
Two triangles are similar if they meet one of the following criteria. : Two pairs of corresponding angles are equal. : Three pairs of corresponding sides are proportional. : Two pairs of
corresponding sides are proportional and the corresponding angles between them are equal.
What is similarity transformation in mathematics?
▫ A similarity transformation is a composition of a finite number of dilations or rigid motions. Similarity transformations precisely determine whether two figures have the same shape (i.e., two
figures are similar).
What is the similarity?
A similarity is a sameness or alikeness. When you are comparing two things — physical objects, ideas, or experiences — you often look at their similarities and their differences. Difference is the
opposite of similarity. Both squares and rectangles have four sides, that is a similarity between them.
What tool is used to find structural similarities proteins?
MinRMS – A Tool for Determining Protein Structure Similarity.
How do you determine the similarity of protein structure?
Currently, the structural similarity between two proteins is measured by the root-mean-square-deviation (RMSD) in their best-superimposed atomic coordinates.
What shapes are always similar in geometry?
Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar. If two polygons are similar, we know
the lengths of corresponding sides are proportional.
How do you find the similarity of a triangle?
If all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar. Thus, if AB/XY = BC/YZ = AC/XZ then ΔABC ~ΔXYZ.
What is the definition of similar triangles?
geometry. triangles that are similar due to the equality of corresponding angles and the proportional similarity of the corresponding sides. Similar triangles have an identical shape.
How can you use similarity transformations to demonstrate that two figures are similar?
In general, similarity transformations preserve angles. Side lengths are enlarged or reduced according to the scale factor of the dilation. This means that similar figures will have corresponding
angles that are the same measure and corresponding sides that are proportional.
What is structural similarity index?
Structural similarity. Jump to navigation Jump to search. The structural similarity (SSIM) index is a method for predicting the perceived quality of digital television and cinematic pictures, as well
as other kinds of digital images and videos.
What is structural similarity in image processing?
Structural similarity. SSIM is used for measuring the similarity between two images. The SSIM index is a full reference metric; in other words, the measurement or prediction of image quality is based
on an initial uncompressed or distortion-free image as reference. SSIM is designed to improve on traditional methods such as peak signal-to-noise…
What is similarity in geometry?
This rings a bell, and you remember that last week in math class, your teacher was talking about something called similarity in geometry. You try to remember what it was that he said about this
subject. Oh yeah! You remember that, in geometry, similar objects, shapes, or figures have the same shape but different sizes.
What are similar shapes in geometry?
“Similar” shapes in geometry must have equal corresponding angles and equal ratios of corresponding sides. Just because two shapes are rectangles does not mean they are similar. | {"url":"https://federalprism.com/what-does-similarity-mean-in-geometry/","timestamp":"2024-11-10T03:47:16Z","content_type":"text/html","content_length":"58112","record_id":"<urn:uuid:d26d671e-d1d2-4547-8b3b-d80dc6b7c5d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00878.warc.gz"} |
Introduction to broom.mixed
Ben Bolker
broom.mixed is a spinoff of the broom package. The goal of broom is to bring the modeling process into a “tidy”(TM) workflow, in particular by providing standardized verbs that provide information on
• tidy: estimates, standard errors, confidence intervals, etc.
• augment: residuals, fitted values, influence measures, etc.
• glance: whole-model summaries: AIC, R-squared, etc.
broom.mixed aims to provide these methods for as many packages and model types in the R ecosystem as possible. These methods have been separated from those in the main broom package because there are
issues that need to be dealt with for these models (e.g. different types of parameters: fixed, random-effects parameters, conditional modes/BLUPs of random effects, etc.) that are not especially
relevant to the broader universe of models that broom deals with.
Mixed-model-specific issues
• the upper-level parameters that describe the distribution of random variables (variance, covariance, precision, standard deviation, or correlation) are called random-effect parameters (ran_pars
in the effects argument when tidying)
• the values that describe the deviation of the observations in a group level from the population-level effect (which could be posterior means or medians, conditional modes, or BLUPs depending on
the model) are called random-effect values (ran_vals in the effects argument when tidying)
• the parameters that describe the population-level effects of (categorical and continuous) predictor variables are called fixed effects (fixed in the effects argument when tidying)
• the categorical variable (factor) that identifies which group or cluster an observation belongs to is called a grouping variable (group column in tidy() output)
• the particular level of a factor that specifies which level of the grouping variable an observation belongs to is called a group level (level column in tidy() output)
• the categorical or continuous predictor variables that control the expected value (i.e., enter into the linear predictor for some part of the model) are called terms (term column in tidy()
output); note that unlike in base broom, the term column may have duplicated values, because the same term may enter multiple model components (e.g. zero-inflated and conditional models; models
for more than one response; fixed effects and random effects)
Time-consuming computations
Some kinds of computations needed for mixed model summaries are computationally expensive, e.g. likelihood profiles or parametric bootstrapping. In this case broom.mixed may offer an option for
passing a pre-computed object to tidy(), eg. the profile argument in the tidy.merMod (lmer/glmer) method.
Automatically retrieve table of available methods:
## # A tibble: 26 × 4
## class tidy glance augment
## <chr> <lgl> <lgl> <lgl>
## 1 MCMCglmm TRUE FALSE FALSE
## 2 MixMod TRUE FALSE FALSE
## 3 TMB TRUE FALSE FALSE
## 4 allFit TRUE TRUE FALSE
## 5 brmsfit TRUE TRUE TRUE
## 6 clmm FALSE FALSE TRUE
## 7 gamlss TRUE TRUE FALSE
## 8 gamm4 TRUE TRUE TRUE
## 9 glmm TRUE FALSE FALSE
## 10 glmmTMB TRUE TRUE TRUE
## # ℹ 16 more rows
Manually compiled list of capabilities (possibly out of date):
lme4 glmer y y y y y y y NA y NA NA NA ?
lme4 lmer y y y y y y y NA y NA NA NA ?
nlme lme y y y y y y y NA n NA NA ? ?
nlme gls y y y y NA NA NA NA n NA NA ? ?
nlme nlme y y y y y n y NA n NA NA ? ?
glmmTMB glmmTMB y y y y y y n NA NA y y ?
glmmADMB glmmadmb y y y y y y n NA NA y ? ?
brms brmsfit y y y y y y n NA NA y ? ?
rstanarm stanreg y y y y y y n NA NA NA NA ?
MCMCglmm MCMCglmm y y y y y y n NA NA ? ? ?
TMB TMB y n n n n n n NA NA NA NA ?
INLA n n n n n n n NA NA ? ? ? | {"url":"http://cran.rediris.es/web/packages/broom.mixed/vignettes/broom_mixed_intro.html","timestamp":"2024-11-01T23:58:10Z","content_type":"text/html","content_length":"24985","record_id":"<urn:uuid:a7a693d0-f885-48d2-8b30-d235402d9345>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00623.warc.gz"} |
266 drygallon to usquart - How much is 266 dry gallons in US quarts? [CONVERT] ✔
266 dry gallons in US quarts
Conversion formula
How to convert 266 dry gallons to US quarts?
We know (by definition) that: $1⁢drygallon ≈ 4.65458874458874⁢usquart$
We can set up a proportion to solve for the number of US quarts.
$1 ⁢ drygallon 266 ⁢ drygallon ≈ 4.65458874458874 ⁢ usquart x ⁢ usquart$
Now, we cross multiply to solve for our unknown $x$:
$x ⁢ usquart ≈ 266 ⁢ drygallon 1 ⁢ drygallon * 4.65458874458874 ⁢ usquart → x ⁢ usquart ≈ 1238.120606060605 ⁢
Conclusion: $266 ⁢ drygallon ≈ 1238.120606060605 ⁢ usquart$
Conversion in the opposite direction
The inverse of the conversion factor is that 1 US quart is equal to 0.000807675758811375 times 266 dry gallons.
It can also be expressed as: 266 dry gallons is equal to $1 0.000807675758811375$ US quarts.
An approximate numerical result would be: two hundred and sixty-six dry gallons is about one thousand, two hundred and thirty-eight point one one US quarts, or alternatively, a US quart is about zero
times two hundred and sixty-six dry gallons.
[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).
Results may contain small errors due to the use of floating point arithmetic. | {"url":"https://converter.ninja/volume/dry-gallons-to-us-quarts/266-drygallon-to-usquart/","timestamp":"2024-11-09T00:44:40Z","content_type":"text/html","content_length":"20379","record_id":"<urn:uuid:0c952139-8613-4918-bdf5-bf679cd9f681>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00117.warc.gz"} |
RS Aggarwal Class 8 Solutions Chapter 15 Quadrilaterals Latest PDF | Utopper
RS Aggarwal Class 8 Solutions Chapter 15
RS Aggarwal Solutions for Class 8 Chapter 15 – Quadrilaterals PDF
RS Aggarwal Class 8 Maths Solutions Chapter-wise – Free PDF Download
The correct answers to all of the exercise questions can be found in the RS Aggarwal Class 8 Solutions Chapter 15 “Quadrilaterals.” There is only one exercise with 9 questions in this section of RS
Aggarwal Class 8 Solutions. But all of these questions will give you a solid foundation for geometry topics like area, constructions, quadrilaterals, and lines and angles.
In this chapter, you’ll find out what quadrilaterals are and how they work. You will learn about a quadrilateral’s sides that are next to each other, sides that are opposite each other, angles,
vertices, and sides. The angle sum property is a very important idea in this chapter that will help you in future classes as well. To remember the formulas and properties talked about in the chapter,
it is very important to do the chapter’s exercises.
We have experts on Class 8 Maths here at Utopper who have solved all of the problems in this chapter’s exercises for you. You will understand the topics better if they are explained well and have the
right diagrams. We follow the CBSE Class 8 Maths syllabus to the letter and make sure that all of the answers match the exam pattern.
Utopper is a website where students can get free Reference Book Solutions and other study materials like Revision notes, Sample papers, and Important Question class 8. Science and Maths will be
easier to learn if you have access to RS Aggarwal Solutions for Class 8 and solutions for other courses.
Click Here – To Buy/Purchase RS Aggarwal Class 8 Solutions Online
RS Aggarwal Class 8 Solutions Chapter 15 – Quadrilaterals
RS Aggarwal Class 8 Solutions
Download a free PDF of RS Aggarwal Class 8 Solutions Chapter 15
Students can’t solve problems from RS Aggarwal Class 8 Solutions Chapter 15 if they don’t have the right skills and a strong grasp of math and geometry. The formulas for RS Aggarwal Class 8 Solutions
Quadrilaterals are thought to be one of the most difficult parts of the chapter. The best way to learn this chapter is to regularly work on a lot of questions from different exercises.
For help with these questions, look at the free PDF version of RS Aggarwal Class 8 Solutions Chapter 15 from Utopper, which you can find online. This PDF has the answers to all of RS Aggarwal’s
Chapter 15 Class 8’s questions and problems. The PDF gives clear, complete explanations of how to solve the problems. With this PDF, students can improve their skills.
How to Start with Quadrilaterals
Chapter 15 of RS Aggarwal Solutions for Class 8 is about quadrilaterals. The quadrilateral is a hard chapter in the subject of geometry. Let’s talk about a few things that students will learn while
studying this chapter:
• A quadrilateral is a polygon with four sides and four corners.
• Quadrilaterals come in six different shapes: square, rectangle, parallelogram, rhombus, kite, and trapezium. All six of these shapes have four corners, four sides, and four angles.
• When you add up a quadrilateral’s four angles, you get a total of 360 degrees. This number comes from the equation for the angles inside the polygon, which is (n – 2) x 180.
A quadrilateral’s sides and angles
Most of the time, the sides and angles of quadrilaterals are different lengths and sizes. In rare situations, however, some of the sides and angles are the same. For example, in a square, all of the
sides and angles are the same. In a rectangle, only the opposite sides are the same, but all of the angles are the same. Because different quadrilaterals have different parts, the area of a
quadrilateral is different for each quadrilateral.
Special Types of Quadrilaterals
• Rectangle: It is a four-sided shape where all of the angles on the inside are 90°. In a rectangle, the sides on either end are always the same length.
• Square: A square is a four-sided shape with all angles on the inside equal to 90°. It has four equal sides.
• Rhombus: A rhombus is a square with a small change. It has four sides that are all the same length, but the sides that are opposite each other are straight.
• Parallelogram: A parallelogram is a rectangle that has been changed in a small way. All its opposite sides are equal and parallel.
• Trapezium: It is a four-sided shape with any two of its sides going in the same direction.
• Kite: It is a four-sided shape with equal sides next to each other.
How to calculate the area of some quadrilaterals
• Area of a Square – (side)²
• Area of a Rectangle – Length x Breadth
• Parallelogram Area = Base x Height
• Area of a Trapezium – { (Sum of two opposite sides / 2) x Height }
• Area of a Rhombus – Product of two diagonals / 2
• Area of a Kite – Product of two diagonals / 2
Some quadrilaterals’ perimeter formulas
Every quadrilateral has the same perimeter formula, which is the sum of all four sides. When you add up all four sides of a quadrilateral, whether they are all the same length or not, you get the
quadrilateral’s perimeter.
All of these ideas seem hard to understand, but for students to become experts in this area, they need to work on problems from RS Aggarwal Class 8 Chapter 15. Remember to use Utopper’s RS Aggarwal
Solutions for Class 8 Chapter 15 as a guide when you work on these problems.
Students’ skills will get better and their core knowledge will be stronger if they work on and solve different problems often. With the help of RS Aggarwal Class 8 Solutions Chapter 15, this practice
is easier to do.Â
FAQ ( Frequently Asked Questions )
1. What is a Parallelogram?
Ans – A parallelogram is a four-sided shape with two sets of sides that are parallel to each other. Both sides that are parallel to each other are the same length. In this Chapter, both angles are
measured the same way. The two diagonals that make up the shape meet in the middle. This means that they both cut each other in the same way. The rectangle is a parallelogram that forms 90-degree
angles at each corner of a quadrilateral.
2. List all of the important topics that are talked about in the quadrilaterals chapter.
Ans – Definition of –
• a) Quadrilaterals
• b) Rectangle
• c) Square
• d) Rhombus
• e) Parallelogram
• f) Trapezium
• g) Kite
• Properties of Quadrilaterals, parallelograms, and Rhombus
• Rectangle, square, and parallelogram diagonals.
• How to find the area of certain quadrilaterals with formulas
• To figure out the perimeter of any quadrilateral, you use the same formula, which is the sum of all four sides.
3. What is a quadrilateral, and what are some of the different kinds?
Ans – A quadrilateral is a closed two-dimensional shape with four sides and four points. In every quadrilateral, students can find four angles. The sum of these four angles always adds up to 360°.
Quadrilaterals can be put into too many different groups, so there are many different kinds of quadrilaterals in math. Some of the most common are the square and rectangle, the rhombus, the
parallelogram, the trapezium, and the kite. There may be more types of these special types within themselves.
4. What are parallelogram properties and how does it work?
Ans – In a parallelogram, the angles next to each other are complementary. This means that the sum of the angles next to each other is 180 degrees. Each diagonal of a parallelogram cuts the shape
into two triangles that are the same. A parallelogram can also be a square or a rectangle. If one angle is right, then all of the angles must also be right. Opposite sides and angles are congruent.
The RS Aggarwal class 8 solutions can tell the students more about them. | {"url":"https://utopper.com/rs-aggarwal-solutions/class-8-maths-chapter-15-solutions/","timestamp":"2024-11-04T23:50:42Z","content_type":"text/html","content_length":"374422","record_id":"<urn:uuid:c815f98e-5f1e-4e9c-9f37-d620e335195c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00313.warc.gz"} |
Hi all,
The bold statement in the below code is working, at least I did not get any errors. Based on this code I'm supposed to have 4 variables: p_m1, pm_2, pm_3, and pm_4. The values for pm_2 are correct
however others are wrong. Also all values are the same for each of those variables. Could you help me to find what I'm missing?
data all_pars1;
set all_pars;
array t {3} theta1-theta3 ;
array ex {3,4} &vlist;
array s {4};
array in {4};
array sumex{3} 8.;
array p_m{4} ;
do i=1 to 3;
do j=1 to 4;
01-27-2019 01:24 AM | {"url":"https://communities.sas.com/t5/SAS-Procedures/arrays/td-p/530427","timestamp":"2024-11-07T03:33:19Z","content_type":"text/html","content_length":"207141","record_id":"<urn:uuid:a17b39f2-db1e-409e-bfe2-628a0b3dd041>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00865.warc.gz"} |
quicksort linked list python
You can write merge sort , partition sort , tree sort and compare results Quicksort is a popular sorting algorithm and is often used, right alongside Merge Sort. Recursion To learn about Quick Sort,
you must know: 1. To understand Quick Sort letâ s take an example:-Example. How does Hoare's quicksort work, even if the final position of the pivot after partition() is not what its position is in
the sorted array? singly linked list The general idea of the quicksort algorithm over a linked list is: 1 - To choice a pivot key, in your case you are choosing the last key in the list. Data
Structures and Algorithms from Zero to Hero and Crack Top Companies Interview questions (supported by Python Code) Rating: 4.6 out of 5 4.6 (228 ratings) 11,292 students We have an array
[48,44,19,59,72,80,42,65,82,8,95,68] First of all we take first element and place it at its proper place. Like Merge Sort, QuickSort is a Divide and Conquer algorithm. Firstly we should limit extra
storage to the recursion needs Python data structures - Lists 3. Once we have pointer to last node, we can recursively sort the linked list using pointers to first and last nodes of linked list. It's
a good example of an efficient sorting algorithm, with an average complexity of O(nlogn). The slow random-access performance of a linked list makes some other algorithms (such as quicksort) perform
poorly, and others (such as heapsort) completely impossible. 4. Inserting a new node in a linked list in C. So in total it takes $\mathcal O(n)$ time for $n$ elements. This algorithm is a sorting
algorithm which follows the divide and conquer algorithm. edit You can quick sort linked lists however you will be very limited in terms of pivot selection, restricting you to pivots near the front
of the list which is bad for nearly sorted inputs, unless you want to loop over each segment twice (once for pivot and once for partition). Can I print in Haskell the type of a polymorphic function
as it would become if I passed to it an entity of a concrete type? We call this element Pivot element. Following is C++ implementation for same. Our list have to have poiner to the node if we want
our pivot Divide and Conquer:- Yes, quicksort on a linked list and mergesort on a linked list are the same complexity class, but those constants you're not seeing make mergesort uniformly faster.
Implement quicksort in C, Java and Python with illustrated graphics and examples. When could 256 bit encryption be brute forced? In the average case both algorithms are in $\mathcal O(n\log n)$ so
the asymptotic case is the same, but preference is strictly due to hidden constant and sometimes the stability is the issue (quicksort is inherently unstable, mergsort is stable). Though with some
extra optimizations like ignoring the previous pointer during all but the last merge operation. Each data element contains a connection to another data element in form of a pointer. Last link carries
a link as null to mark the end of the list. The access is more sequential in nature. Once we have a pointer to the last node, we can recursively sort the linked list using pointers to first and last
nodes of a linked list, similar to the above recursive function where we pass indexes of first and last array elements. Now, if I always choose last element as pivot, then identifying the pivot
element (1st step) takes $\mathcal O(n)$ time. How to prevent guerrilla warfare from existing. We also add a few convenience methods: one that returns the stored data, another that returns the next
node (thâ ¦ Singly Linked-List. Following is C++ implementation for same. However adding a single 64 element array of pointers you can avoid that extra iteration and sort lists of up to $2^{64}$
elements in $O(1)$ additional extra space. Use MathJax to format equations. We will use simple integers in the first part of this article, but we'll give an example of how to change this algorithm to
sort objects of a custom class. A linked list is a linear collection of data elements, called nodes, each pointing to the next node by means of a pointer. Essential Concepts - Big O Notation - Memory
- Logarithms - Recursion 2. Using quick sort as well as merge sort for linked list . QuickSort on Doubly Linked List - Searching and sorting -. Linked List before sorting 23 ->1 ->50 ->15 ->16 ->6.
Because of this, the quicksort is not natural choice for linked list while merge sort takes great advantage. We use cookies to ensure you have the best browsing experience on our website. QuickSort
on Singly linked list was given as an exercise. The idea is simple, we first find out pointer to the last node. Contribute to TheAlgorithms/Python development by creating an account on GitHub. What
is the origin of Faerûn's languages? TSLint extension throwing errors in my Angular application running in Visual Studio Code, Cryptic Family Reunion: Watching Your Belt (Fan-Made). ... C++ : Linked
lists in C++ (Singly linked list) May 30, 2017. @Zephyr, you need to remember that complexity notation drops constant factors. brightness_4 Algorithm for Quicksort. For tree sort each node of linked
list must have two pointers even if we sort $T(n) = 2T(n/2) + n$ which is $\mathcal O(n \log n)$ which is the same as in merge sort with a linked list. It is a data structure consisting of a group of
nodes which together represent a sequence. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Nodes are connected (or
organized) in a specific way to make data structures like linked lists, trees, etc. Please write to us at contribute@geeksforgeeks.org to report any issue with the above content. Python Server Side
Programming Programming. We know that a node for a single linked list contains the item and the reference to the next node. QuickSort on Doubly Linked List Code definitions. In Python, thereâ s a
specific object in the collections module that you can use for linked lists called deque (pronounced â deckâ ), which stands for double-ended queue. Random QuickSort cannot be efficiently
implemented for Linked Lists by picking random pivot. To learn more, see our tips on writing great answers. Attention reader! A linked list can be implemented using an array or using a class. Asking
for help, clarification, or responding to other answers. The partition function for a linked list is also similar to partition for arrays. Write a Python program to delete a specific item from a
given doubly linked list. Making statements based on opinion; back them up with references or personal experience. Hoare partition can be done only for doubly linked list, This code needs some
improvements I am afraid I could not see how this answers the question. May 30, 2017. The unordered list is a collection of items where new items are inserted at the end of the list regardless of
order or value. All Algorithms implemented in Python. Written below is the algorithm for quicksort for linked list: After selecting an element as pivot, which is the last element of the linked list
in our case, we divide the array for â ¦ Writing code in comment? Here, we have taken the But the average time complexity is same right ? Each comparison will take $\mathcal O(1)$ time as we store
the pivot data and each swap takes $\mathcal O(1)$ time. You can choose any element from the array as the pviot element. A pivot element is chosen from the array. To subscribe to this RSS feed, copy
and paste this URL into your RSS reader. First, we will learn what is divide and conquer algorithm. Write a Python program to search a specific item in a singly linked list and return true if the
item is â ¦ 3. Unlike array, in linked list, we can insert items in the middle in O(1) extra space and O(1) time. The Python list â ¦ Divide â ¦ Example. Quick sort algorithm can be divided into
following steps. Don’t stop learning now. Welcome to the Complete Data Structures and Algorithms in Python Bootcamp, the most modern, and the most complete Data Structures and Algorithms in Python
course on the internet. We can use queue operations for this step By clicking â Post Your Answerâ , you agree to our terms of service, privacy policy and cookie policy. Unlike arrays, linked list
nodes may not be adjacent in memory. rev 2020.12.10.38158, The best answers are voted up and rise to the top, Computer Science Stack Exchange works best with JavaScript enabled, Start here for a
quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring
developers or posting ads with us, There is no need to pick the last element as the pivot instead of the first. Each link is linked with its next link using its next link. It is quite easy to write
tree sort if you allow some extra space Maybe I will show steps for quicksort, Ad 1. It takes O(n^2) time in the worst case and O(nLogn) in average and best cases. Divide the linked list recursively
into 2 parts. 200 lines of code doesn't do anything to explain why merge sort is preferred over quick sort for linked lists. and concatenation run faster with presence of this pointer. Each link
carries a data field(s) and a link field called next. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. ... Python /
data_structures / linked_list / singly_linked_list.py / Jump to. Why MergeSort is preferred over QuickSort for Linked Lists? Python Quick Sort. Linked List after sorting 1 ->6 ->15 ->16 ->23 ->50.
Belgian formats when choosing US language - regional & language settings issue. As per the above illustration, following are the important points to be considered. This tutorial shows how to sort a
Linked List, putting the nodes in sorted order. The node initializes with a single datum and its pointer is set to None by default (this is because the first node inserted into the list will have
nothing to point at!). In case of linked lists the case is different mainly due to difference in memory allocation of arrays and linked lists. Python 3 2. A linked list is a sequence of data
elements, which are connected together via links. First, we need to create a class for the nodes(s) that will constitute the linked list. The node is where data is stored in the linked list (they
remind me of those plastic Easter eggs that hold treats). The above implementation is for a doubly linked list. Modify it for a singly linked list. In the following implementation, quickSort() is
just a wrapper function, the main recursive function is _quickSort() which is similar to quickSort() for array implementation. And you will need to keep a stack of the partition boundaries for the
lists you still need to sort. Linked list can be visualized as a chain of nodes, where every node points to the next node. Because of this, the quicksort is not natural choice for linked list while
merge sort takes great advantage. Is there a difference between a tie-breaker and a regular vote? compare its key with pivot key and enqueue node to the correct sublist The important things about
implementation are, it changes pointers rather swapping data and time complexity is same as the implementation for Doubly Linked List. Thanks for your detailed contribution but this isn't a coding
site. Following is a typical recursive implementation of QuickSort for arrays. Data structures: - Arrays - Linked Lists (Singly Linked List, Doubly Linked List, Circular Linked List) - Stacks -
Queues - â ¦ In a linked list, a node is connected to a different node forming a chain of nodes. Experience. Algorithm Partition Algorithm. Time Complexity: Time complexity of the above
implementation is same as time complexity of QuickSort() for arrays. With a linked list, you could sort items as you insert them more efficiently, and that might be one of the few advantages with a
linked list over the vector implementation in Pythons builtin list, but since you mention quicksort, that's obviously not your plan. Merge sort on linked lists can be executed using only $O(1)$ extra
space if you take a bottom-up approach by counting where the boundaries of the partitions are and merging accordingly. The access is more sequential in nature. Computer Science Stack Exchange is a
question and answer site for students, researchers and practitioners of computer science. Read the clear explanation and codes. In this article, we will learn about the solution to the problem
statement given below. Random QuickSort cannot be efficiently implemented for Linked Lists by picking random pivot. 1. In the above picture, each node has two parts, one stores the data and another
is connected to a different node. Empirically, quicksort is quicker. The sort phase simply sorts the two smaller problems that are generated in â ¦ @Zephyr Basically it's the difference of
theoretical and empirical results. If we want to choose pivot fast the choice is limited MathJax reference. 2 - To split the list of elements in 2 sublist: one with elements < pivot (or <=) and the
other with elements >= pivot (or >). 2. Time Complexity : O() In this method the main idea is to swap pointers rather than swaping data. Can we implement random quicksort for a linked list? Linked
List contains a link element called first. You are calling this list left and right. This Python tutorial helps you to understand what is Quicksort algorithm and how Python implements this algorithm.
Inserting a new node to a linked list in C++. 1. How do I convert Arduino to an ATmega328P-based project? Fist we dequeue node from original linked list So why is merge sort preferred over quick sort
for linked lists? up to date? Please use ide.geeksforgeeks.org, generate link and share the link here. Contribute to TheAlgorithms/Python development by creating an account on GitHub. Quicksort.
Thanks for contributing an answer to Computer Science Stack Exchange! Instead of returning index of the pivot element, it returns a pointer to the pivot element. close, link You have to start
somewhere, so we give the address of the first node a special name called HEAD.Also, the last node in the linkedlist can be identified because its next portion points to NULL. Recursive calls for
sublists which contain nodes not equal to pivot node, Concatenate sorted sublists into one sorted list. That stack can grow to $O(n)$ when pivot selection is bad along with the time complexity
growing to $O(n^2)$. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Throughout the course, we will explore the most important Data Structures and Algorithms
topics step-by-step: 1. Quicksort is a representative of three types of sorting algorithms: divide and conquer, in-place, and unstable. Python Algorithms Data Structures Linked Lists A linked list is
a data structure that uses pointers to point to the next item in the list. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry
ready. acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Stack
Data Structure (Introduction and Program), Doubly Linked List | Set 1 (Introduction and Insertion), Find the middle of a given linked list in C and Java, Function to check if a singly linked list is
palindrome, Delete a Linked List node at a given position, Reverse a Linked List in groups of given size | Set 1, Program for n'th node from the end of a Linked List, Find Length of a Linked List
(Iterative and Recursive), Write a function to get the intersection point of two Linked Lists, Implement a stack using singly linked list, Circular Linked List | Set 1 (Introduction and
Applications), Implementing a Linked List in Java using Class, Remove duplicates from a sorted linked list, Add two numbers represented by linked lists | Set 1, Remove duplicates from an unsorted
linked list, Search an element in a Linked List (Iterative and Recursive), Write a function to get Nth node in a Linked List, Clone a linked list with next and random pointer | Set 1, XOR Linked List
- A Memory Efficient Doubly Linked List | Set 1, XOR Linked List â A Memory Efficient Doubly Linked List | Set 2, Difference between Singly linked list and Doubly linked list, Construct a Doubly
linked linked list from 2D Matrix, Convert a given Binary Tree to Doubly Linked List | Set 1, Convert a given Binary Tree to Doubly Linked List | Set 2, Extract Leaves of a Binary Tree in a Doubly
Linked List, Convert a given Binary Tree to Doubly Linked List | Set 3, Convert a given Binary Tree to Doubly Linked List | Set 4, Convert a Binary Tree into Doubly Linked List in spiral fashion,
Create a Doubly Linked List from a Ternary Tree, Check if a doubly linked list of characters is palindrome or not, Doubly Circular Linked List | Set 1 (Introduction and Insertion), Doubly Circular
Linked List | Set 2 (Deletion), Convert a given Binary Tree to Circular Doubly Linked List | Set 2, Given a linked list which is sorted, how will you insert in sorted way, Insert a node at a specific
position in a linked list, Write Interview The idea is simple, we first find out pointer to last node. The worst case occurs when the linked list is already sorted. Linked Lists. The memory access
pattern in Quicksort is random, also the out-of-the-box implementation is in-place, so it uses many swaps if cells to achieve ordered result. QuickSort on Singly linked list was given as an exercise.
We can choose head node or tail node ... QuickSort on Doubly Linked List; QuickSort on Singly Linked List; Can QuickSort be implemented in O(nLogn) worst case time complexity? Why don't we use quick
sort on a linked list? be accessible fast otherwise we have to search for node, Ad 2. Are the vertical sections of the Ackermann function primitive recursive? Why is it impossible to measure position
and momentum at the same time with arbitrary precision? Partition the linked list based on pivot. I was bitten by a kitten not even a month old, what should I do? The important things about
implementation are, it changes pointers rather swapping data and time complexity is same as the implementation for Doubly Linked List. Hoare. When average , worst and best case time complexity
happens in quick sort? Why don’t you capture more territory in Go? Merge sort is often preferred for sorting a linked list. Let head be the first node of the linked list to be sorted and headRef be
the pointer to head. There are two types of linked-list we would go over: Unordered Linked-List; Ordered Linked-List; Nodes Unordered List. In partition Sort choosing pivot is limited to the first or
last element (last if we keep pointer to the tail node) otherwise choosing pivot is slow Hoare partition is possible only for doubly linked lists Swapping should be replaced with inserting and
deleting Tree sort with unbalanced tree has the same compexity as quicksort if we ignore constant factor but it is easier to avoid worst case in tree sort For merge sort there is to few characters in
the comment. Algorithms: Coding a linked list in python. How to purge a linked list in $\mathcal{O}(n\log n)$ time? Below is a simple implementation. It only takes a minute to sign up. How to best
use my hypothetical “Heavenium” for airship propulsion? Part of its popularity also derives from the ease of implementation. If you take the last element, like OP suggests, that means that the worst
case ($O(n^2)$) is an already sorted list or sublist. Note that we don’t have prev pointer in a singly linked list. allocate memory for new nodes, Pointer to the tail node will be useful because
queue operations 1. The value of the itemwill be set by the value passed through the constructor, while the reference will be initially set to null. Quicksort can be implemented for Linked List only
when we can pick a fixed point as the pivot (like the last element in the above implementation). Quicksort can be implemented for Linked List only when we can pick a fixed point as the pivot (like
the last element in the above implementation). Article, we will learn about the topic discussed above data structures and algorithms step-by-step! All the important points to be considered Your Belt
( Fan-Made ) time for n... Average and best cases as pivot this, the quicksort is a typical recursive implementation of quicksort a... Our tips on writing great answers RSS reader the above
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Conversion Rates: How Many Grams Are In An Ounce, Pound, Or Milliliter? | Food Readme
Conversion Rates: How Many Grams Are In An Ounce, Pound, Or Milliliter?
If you’re like most people, you probably don’t know how many grams are in an ounce, pound, or milliliter. But did you know that this information is important when it comes to conversion rates? By
understanding the relationship between these units of measurement, you can make sure that your conversions are accurate.
How many grams is 113 ounces
There are 28.35 grams in an ounce, so 113 ounces is equal to 3,185.05 grams.
How many pounds is 113 ounces
There are a lot of ways to answer this question, but the most straightforward answer is that there are approximately 7.1 pounds in 113 ounces. This is because there are 16 ounces in a pound, and when
you divide 113 by 16, you get approximately 7.1. Of course, this is not an exact answer, but it should give you a general idea of how many pounds are in 113 ounces.
How many milliliters is 113 grams
How many milliliters is 113 grams?
This is a question that often comes up, especially when baking. But the answer is actually quite simple.
There are 1000 milliliters in a liter, and a liter is about 4 and a half cups. So, to answer the question, there are 453.592 milliliters in 113 grams.
Now, this isn’t an exact answer, because the density of different substances can vary slightly. For example, water has a density of 1 gram per milliliter, while honey has a density of about 1.4 grams
per milliliter. So, depending on what you’re measuring, the answer could be slightly different.
But in general, if you need to know how many milliliters are in 113 grams, just remember that there are 1000 milliliters in a liter, and use that as a starting point.
How many teaspoons is 113 grams
This is a question that often plagues those who are new to the world of baking. How many teaspoons is 113 grams? This conversion can be tricky, but we have a few tips to help make things a bit
First, it’s important to note that there are different types of teaspoons. The standard teaspoon is 4.93 mL, while the metric teaspoon is 5 mL. For the purposes of this conversion, we will use the
standard teaspoon.
Now that we have that out of the way, let’s get down to business. There are 28.35 grams in an ounce, so 113 grams is equal to 4 ounces. There are 3 teaspoons in an ounce, so 4 ounces is equal to 12
teaspoons. Therefore, 113 grams is equal to 12 teaspoons.
We hope this has helped clear things up! Now you can rest easy knowing exactly how many teaspoons are in 113 grams.
How many tablespoons is 113 grams
When it comes to baking, precision is key. Baking is a science, after all, and even a small mistake can result in an unpalatable (or even inedible) final product. This is why many bakers prefer to
measure their ingredients by weight rather than by volume.
While teaspoons, tablespoons, and other measurement units are handy for measuring smaller quantities of liquid ingredients like water or oil, they’re not as accurate when it comes to dry ingredients
like flour or sugar. That’s because these ingredients can vary greatly in density. For example, one cup of all-purpose flour weighs 4.5 ounces, while one cup of cake flour weighs 4 ounces. So if
you’re following a recipe that calls for “1 cup of flour” and you use a standard tablespoon to measure it out, you could be adding as much as 30% more flour than the recipe intends!
But how do you convert grams into tablespoons? The answer depends on the ingredient in question. Here are some common conversions:
All-purpose flour: 1 gram = 0.035 ounces = 0.076 tablespoons
Sugar: 1 gram = 0.035 ounces = 0.065 tablespoons
Butter: 1 gram = 0.065 ounces = 0.13 tablespoons
As you can see, there is no definitive answer when it comes to converting grams into tablespoons. The best way to ensure accuracy is to use a digital kitchen scale.
How many kilograms is 113 grams
How many kilograms is 113 grams?
This is a question that we get asked a lot here at the office. And it’s one that we have to answer a lot, too – because it’s not always as simple as it seems.
First of all, let’s just get something straight: a gram is a unit of mass, not weight. So, when you’re asking how many kilograms there are in 113 grams, what you’re really asking is how many times
heavier than a gram is a kilogram.
The answer to that question is 1,000. In other words, a kilogram is 1,000 times heavier than a gram.
Now, let’s put that into perspective. A standard sheet of paper weighs about 5 grams. So, if you took 113 sheets of paper and stacked them on top of each other, they would weigh about the same as a
Here’s another way to think about it: a liter of water weighs 1 kilogram. So, if you had 113 grams of water – which is just over 1/8th of a liter – it would weigh the same as a kilogram.
So there you have it! Now you know how many kilograms are in 113 grams.
What is the conversion rate for ounces to grams
If you’re a Cooking aficionado, you might be well aware that recipes often call for either grams or ounces, and converting between the two is often necessary. But what is the conversion rate for
ounces to grams?
One ounce (oz) of flour is equal to 28.3495231 grams (g). This means that if a recipe calls for 1 oz of flour, you can use 28.3495231 g, or about 28 grams.
To convert ounces to grams, simply multiply the number of ounces by 28. For example, 2 oz of flour would be equal to 2 x 28, or 56 grams.
On the other hand, if you need to convert from grams to ounces, divide the number of grams by 28. So, if a recipe calls for 60g of flour, you can use 60 / 28, or about 2.1 ounces.
Keep in mind that these conversions are for general guidance only – depending on the type of flour (and other ingredients), the exact conversion rate may vary slightly.
What is the conversion rate for grams to ounces
When it comes to measuring food ingredients, both grams and ounces are commonly used units. But what is the conversion rate for grams to ounces?
In general, there are 28.35 grams in one ounce. This means that when you’re converting from grams to ounces, you can multiply the number of grams by 0.035. For example, 10 grams would be equal to
0.35 ounces.
Of course, this is just a general guide – the exact conversion rate will depend on the specific ingredient you’re measuring. For example, 1 cup of flour weighs around 125 grams, while 1 cup of sugar
weighs around 200 grams.
So if you’re ever in a pinch and need to convert grams to ounces (or vice versa), remember that there are 28.35 grams in one ounce. This simple conversion will help you get accurate measurements for
all your baking needs!
What is the conversion rate for pounds to ounces
A pound is a unit of mass in the imperial system, and an ounce is a unit of weight in the same system. The conversion rate between the two is therefore dependent on the specific gravity of the
material being measured. For example, water has a specific gravity of 1.0, meaning that 1 pound of water weighs 16 ounces. This means that the conversion rate for pounds to ounces is 16:1.
There are many online calculators that can help you convert between pounds and ounces, or you can use this simple formula:
(pounds x 16) + (ounces ÷ 16) = ounces
For example, to convert 2 pounds and 3 ounces to ounces, you would calculate it as follows:
(2 x 16) + (3 ÷ 16) = 32.1875 ounces
To convert from ounces to pounds, you can use the inverse of this formula:
(ounces ÷ 16) – (pounds x 16) = pounds
For example, to convert 32.1875 ounces to pounds, you would calculate it as follows:
(32.1875 ÷ 16) – (2 x 16) = 2.015625 pounds
As you can see, the conversion rate between pounds and ounces is not a simple 1:1 ratio. It is important to keep this in mind when working with both units of measurement.
What is the conversion rate for milliliters to grams
There are many factors that can affect the conversion rate for milliliters to grams. The type of material being converted, the density of the material, and the accuracy of the measurement, can all
play a role in the conversion rate.
In general, one milliliter of water is equal to one gram. However, this is not always the case for other materials. For example, one milliliter of milk is equal to 1.03 grams, and one milliliter of
honey is equal to 1.19 grams. The conversion rate will also vary depending on the density of the material being converted. For instance, one milliliter of lead is equal to 11.34 grams, while one
milliliter of gold is only equal to 0.032 grams.
The most accurate way to convert milliliters to grams is to use a scale. This will ensure that the conversion rate is as accurate as possible. When using a scale, it is important to weigh the object
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TopKCategoricalAccuracy — PyTorch-Ignite v0.4.6 Documentation
class ignite.metrics.TopKCategoricalAccuracy(k=5, output_transform=<function TopKCategoricalAccuracy.<lambda>>, device=device(type='cpu'))[source]#
Calculates the top-k categorical accuracy.
□ update must receive output of the form (y_pred, y) or {'y_pred': y_pred, 'y': y}.
☆ k (int) – the k in “top-k”.
☆ output_transform (Callable) – a callable that is used to transform the Engine’s process_function’s output into the form expected by the metric. This can be useful if, for example, you
have a multi-output model and you want to compute the metric with respect to one of the outputs. By default, metrics require the output as (y_pred, y) or {'y_pred': y_pred, 'y': y}.
☆ device (Union[str, device]) – specifies which device updates are accumulated on. Setting the metric’s device to be the same as your update arguments ensures the update method is
non-blocking. By default, CPU.
compute Computes the metric based on it's accumulated state.
reset Resets the metric to it's initial state.
update Updates the metric's state using the passed batch output.
Computes the metric based on it’s accumulated state.
By default, this is called at the end of each epoch.
the actual quantity of interest. However, if a
is returned, it will be (shallow) flattened into engine.state.metrics when
is called.
Return type
NotComputableError – raised when the metric cannot be computed.
Resets the metric to it’s initial state.
By default, this is called at the start of each epoch.
Return type
Updates the metric’s state using the passed batch output.
By default, this is called once for each batch.
output (Sequence[Tensor]) – the is the output from the engine’s process function.
Return type | {"url":"https://pytorch.org/ignite/v0.4.6/generated/ignite.metrics.TopKCategoricalAccuracy.html","timestamp":"2024-11-05T06:27:49Z","content_type":"text/html","content_length":"37132","record_id":"<urn:uuid:ab1d1c2d-77e1-4381-8ae0-00d533102308>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00021.warc.gz"} |
Profile Matching in Electronic Social Networks Using a Matching Measure for Fuzzy Numerical Attributes and Fields of Interests
Profile Matching in Electronic Social Networks Using a Matching Measure for Fuzzy Numerical Attributes and Fields of Interests ()
1. Introduction
Social activities in electronic networks play an increasingly important role in our every-day lives. We are exchanging important information via electronic mails, wikis, web-based forums, or blogs,
and meet new friends or business contacts in Internet communities and via social network services. Parallel to this growing sociali- zation of the World Wide Web, the requirements on the electronic
services become more ambitious. Huge data quantities have to be processed, user-friendly interfaces are to be designed, and more and more sophisticated computations must be implemented to offer
complex solutions.
One of the most important subjects in complex networks is the search for specific items or objects in it. For instance this comprises the search for web sites which are relevant with respect to a
specific term, or the request for specific resources as consumers do for electricity in smart grids. Therefore there is, and has always been, a great interest in enabling and maintaining efficient
special network services which perform precisely these tasks.
This paper studies a special aspect of general electronic networks, but in particular of social network services, the profile matching problem. In essence it asks, given a search profile, for
offering profiles matching it best. This problem is in principle well-known in Grid computing, where computational tasks are seeking for appro- priate resources such as CPU time and memory space on
different computers. In electronic social networks, however, the problem is more general because not only specified attribute ranges such as resource sizes are to be compared but more or less vaguely
describable interests. Questions of this kind have recently been under consideration, for instance to implement multiple interest matching in personel business networks [1] , to study automated user
interest matching in online classified offering systems [2] , or to apply semantic fuzzy search techniques to web service matchmaking [3] .
Here, however, a different approach is presented, leaving the important but difficult problem of semantic search aside and enabling an automated matching of general types of attributes. The aim of
this paper is to formulate a mathematical model for the problem of matching attribute ranges and fields of interests in electronic social networks. It tackles the following fundamental questions. How
can an appropriate system and its data structures be designed? How is the mathematical formulation of a matching problem as an optimization problem? In particular, what is its search space? What is
its objective function? It is obvious to use a fuzzy function to calculate the matching degree of two numerical ranges or to use the characteristic function to check for the equa- lity of discrete
attribute sets, but how could a function calculating a matching degree of two fields of interest look like? One of the central results of this paper is the proposal of a precise definition of such a
function computing the matching degree of a pair of profiles and the presentation of a concrete example. The applica- bility of the profile matching model is not confined to electronic social
networks but can also operate to enable and control electricity resource allocation in electrical power systems or in smart grids. By this approach technological challenges of energy transition
programs such as the Energiewende to integrate renewable energy into electrical power systems can be tackled.
The paper is organized as follows. After a definition of electronic social networks is given in the next section, a mathematical model of the matching problem as an optimization problem is proposed,
especially the data structure of search and offering profiles, the search space, and the matching degree as the objective function. A short discussion concludes the paper.
2. Electronic social networks
A social network consists of a finite set of actors and the direct relations defined between them. An actor here may be an individual, a group, or an organization, and the direct relation between two
actors may indicate that they directly interact with each other, have immediate contact, or are connected through social familiarities such as casual acquaintance, friendship, or familial bonds [4] -
[6] , see also ( [7] , ğ3.6). Thus a social network is naturally represented by a graph in which each node represents an actor and each edge a direct relation. Empirically, the mean number of direct
relationships of an individual in a biological social network depends on the size of the neocortex of its individuals; the maximum size of such relationships in human social networks tends to be
around 150 people (“Dunbar’s number”) and the average size around 125 people [8] .
Since the popularization of the World Wide Web in the middle of the 1990’s and in particular around 2005 after the introduction of the Web 2.0 paradigm [9] , there have rapidly emerged several
Internet based social networking services, maintaining “circle of friends” networks, business platforms, knowledge networks, or virtual world online games ( [5] , ğ13.5). Examples of each of these
types of networks services are as follows: Orkut (www.orkut.com), Facebook (facebook.com) or Google+ (plus.google.com); LinkedIn (linkedin.com) or XING (xing.com); Wikipedia (wikipedia.org); World of
Warcraft, Star Wars: The Old Republic (swtor.com), or Second Life (secondlife.com).
In this paper, an electronic social network is defined as a network of at least three human individuals or organizations which use essentially, albeit not exclusively, electronic devices and media to
get in contact and acquaintance to each other, to meet new partners, to communicate, and to exchange information. Examples of electronic social networks are Internet social networks, as well as
videoconference sessions and conference calls, especially if they serve to meet new people as in party lines, or as long as they admit spontaneous communi- cation between each network actor [10] -
[12] .
3. The matching problem
In computer science, the term matching or sometimes matchmaking in general refers to the process of evaluating the degree of similarity or agreement of two objects. Each object is characterized by a
set of properties or attributes, which in many systems are given by name-value pairs [13] . Matching plays a vital role in many areas of computer science and communication systems. For instance, it
is studied for resource discovery and resource allocation in grid computing where matching services are needed to intermediate between resource requesters and resource providers [14] . Other examples
are given by the problem of matching demands and supply of business or personal profiles in online auctions, e-commerce, recruitment agencies, or dating services.
3.1. Basic definitions
In most matching problems, the objects under consideration take asymmetric roles, viz., some search for information or request for a service, others offer information or provide a service. A single
object may naturally do both activities at a time, in electronic social networks this even is the usual case. In the sequel we will therefore more accurately consider the matching of a search
profile, containing information for a request, and an offering profile presenting information for a supply or a provided service.
Given a specific search profile, the matching problem then is to find those offering profiles which match it best, in a sense to be specified in the sequel. Generalizations of this problem ask for
best global matchings, given a whole set of search profiles and a set of offering profiles. For instance, the global pairwise matching problem seeks pairs of search/offering-profiles such that the
entity of the pairs matches the best under the constraint that any profile is member of at most one pair. The global multiple matching problem searches for possibly multiple combinations of search
and offering profiles which as a whole match the best. The pairwise version of the problem typically occurs for dating services or classical marriage matching tasks, whereas the multiple version
appears in grid computing or in brokering interest groups.
In this paper we will focus on the local version of the matching problem, i.e., finding an optimum offering profile to a specified search profile. Thus the matching problem is an optimization
problem, and to formulate it precisely we have to specify the search space and the objective function. The search space will turn out to be the set of pairs of the fixed search profile and the
offering profiles, and the objective function will be a function measuring the “matching degree”. We will work out these notions in the next sections.
3.2. Profile data structures
A profile consists of its owner corresponding to an actor of the electronic social network, a list of attributes of a given set Figure 1).
In principle, there are two different types of attributes, subsumed in the two disjoint sets
The set
Correspondingly, the stencil of an attribute is determined by the attribute’s name and its range, being of a
Figure 1. UML diagram of the data structure of a profile and its relationship to the owner’s attributes, the attribute stencils and the fields of interest. An attribute stencil consists of an owner’s
attribute name and its (searched or offered) range.
certain set called type
denotes the set of ranges for the numerical attributes, and
denotes the set of possible value sets for discrete non-numerical attributes. That is,
On the other hand, a field of interest is a name-value pair specifying the field itself as well as its level ranging on a scale from ‒1 to 1, coded by the interpolation of the following table,
The set of fields of interests is denoted by
Example 1. In grid computing, a main matching problem is resource discovery and resource allocation [15] [16] . Assume a toy grid consisting of two resource providers Haegar and Bond, and two
resource requests by some computational process. In our terminology, Bond and Haegar each provide an offering profile, whereas the requests are represented by search profiles. Moreover, in the widely
used matchmaking framework Condor-G [17] - [21] , the profiles are called ClassAds (classified advertisements). Let us assume a system of four objects with search and offering profiles according to
the following tables.
In each column of a profile there is listed its owner and some attributes and their values. Then a solution to the global matching problem is obviously given by the matchings (194.94.2.21, Haegar)
and (194.1.1.3, Bond). ,
3.3. The search space
Let be a set
problem is given by all pairs of search and offering profiles, i.e.,
considering the local matching problem, given a single search profile
denote the set of searched numerical attributes, the set of discrete non-numerical attributes, and the set of fields of interests, respectively, specified by the search profile. Then a search profile
is the set of attribute-range pairs, with the given mapping
numerical attributes to their associated desired ranges (
is the set of attribute-set pairs, with the mapping
is the set of searched fields of interest with their desired levels, with the given mapping
that for a usual software system each of the pairs
table or a hash map. Analogously, an offering profile is given by
where the three sets are defined the same way as in the search case, but with the index “s” (for “search”) replaced by “o” (for “offering”).
Example 2. Assume a small social network for pooling interest groups, consisting of three persons, Alice, Bob, and Carl, who provide search and offering profiles according to the following tables.
In each column of a profile there is listed its owner, some attributes and their values, and the fields of interests with their levels. For instance, Alice looks for someone between 20 and 40 years
of age being enthusiastic in tennis and having some penchant to chess, whereas Carl seeks a tall person in the 20’s with highest preference for basketball. Looking at the offering profiles in this
social network, one sees that Alice may contact Bob, but Carl cannot find an ideal partner in this community. On the other hand, Alice would be a “better” partner for Carl than Bob, since she is
partly interested in basketball. Formally Alice’s search profile, for instance, is given as follows. The sets for the searched attributes and fields of interest are
the mapping
and the mapping
The mapping
Note in particular that
With the definitions
the search space
3.4. Matching degree as objective function
The matching degree of a search profile and an offering profile is a real number
3.4.1. Matching degree for a numerical attribute
To determine the matching degree of a searched value range
The parameter
For instance, if the searched attribute is “height > 180” and an offered attribute is “height = 165” then for a fuzzy level of
i.e., the matching degree is 16.7%. Then the matching degree of two numerical ranges
3.4.2. Matching degree for a non-numerical attribute
If the values of specific attribute are constrained to be of a finite set, or an enum, say
If the searched attribute, for instance, is “name Î {‘Smith’, ‘Taylor’}” and the offered attribute is “name = ‘Tailor’” then “E = {‘Smith’, ‘Taylor’}” and “
3.4.3. Matching degree of a field of interest
First we notice that the matching degree as a function of the levels of interest
greater than 0, but if the search requires
case, the searcher is indifferent about the field of interest, in the second case he demands high interest.
Definition 3. An interest matching degree function is a function
conditions are satisfied.
The first condition expresses the perfect matching of the diagonal, the second the search indifference, and the last the search necessity. ,
A possible matching degree function is given by
By construction,
It is asymmetric with respect to its arguments, since we have
the other hand, it is an even function, i.e.,
3.4.4. The objective function
Putting together all partial matching degrees considered above, i.e., the matching degree (22) for numerical attributes, the matching degree (24) for numerical attributes, the matching degree (26) of
fields of interest, we construct a function
Thus for the computation of the matching degree, the attributes and fields of interest of the search profile
Example 2 (revisited). For Alice’s search space we have the two solutions (19), i.e.,
Hence Bob’s offering profile has a matching degree of 73.15% with Alice’s search profile, whereas Carl’s matches it only by 54.36%. ,
Notice that the objective function (29) is constructed in such a way that each searched item
of all weights
3.5. Mathematical formulation
A simple algorithm to solve this maximum problem is to exhaustively compute the matching degrees of all possible profile pairs
4. Discussion
In this paper, a mathematical model of the matching of search and offering profiles in electronic social networks is proposed. Basing on the data structure described by Figure 1 and distinguishing
between matching of attri- bute ranges via stencils and matching of fields of interests via comparison, the matching problem is formulated as an optimization problem, with the search space consisting
of a fixed search profile and several offering profiles as in (8), and the matching degree as its objective function in (29). The main difficulty is to define a function measuring adequately the
matching degree of two fields of interest and obeying the necessary condi- tions listed in Definition 3. A proposed solution is the function given in (26) and depicted in Figure 2. The implementation
of a matching service in an electronic social network basing on this matching optimization is straightforward.
The applicability of a profile matching algorithm is not restricted to electronic social networks but could be adapted for resource discovery in grid computation or in matchmaking energy resources in
grids. In particular energy transition projects aiming to integrate renewable energy into electrical power systems have to solve the problem of matching energy supply and demand, caused by the high
variability of renewable energy supply such as wind or solar power ( [22] , ğğ7.5, 8.2.1). This way profile matching might become one of the relevant technologies to support eager energy transition
projects like the Energiewende towards a sustainable economy.
I am indebted to Thomas Kowalski for valuable discussions. | {"url":"https://scirp.org/journal/paperinformation?paperid=49622","timestamp":"2024-11-08T10:58:28Z","content_type":"application/xhtml+xml","content_length":"141417","record_id":"<urn:uuid:c8d24f25-baa7-4fb4-b608-8827e0fe4db5>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00890.warc.gz"} |
Chapter 5, Exercise 3 – CSS Tranformations To Matrices
Common 2D CSS transformations include translate, rotate, scale, and skew. Which of these can you reproduce using a matrix transformation, and which can’t you reproduce?
This is a bit of a trick question, because the answer is that all of the regular transformations can be reproduced by a matrix. In fact, if you tried to query the transform property of an element
then you’ll get a matrix back, as that’s the way the browser keeps track of transformations.
So, for example, if you added the following to the start_game element:
Then tried to query its transform property with JavaScript with $('#start_game').css(Modernizr.prefixed('transform')) you’d get something like matrix(0, 1, -1, 0, 0, 0) back.
You probably won’t need to know matrices in-depth at the moment, but it’s worth reading about the principals involved so you know what to look for if you need to do some more advanced transformations
in the future. This article on matrices https://dev.opera.com/articles/understanding-the-css-transforms-matrix/ is a good place to start. | {"url":"http://buildanhtml5game.com/?page_id=92","timestamp":"2024-11-14T04:31:00Z","content_type":"text/html","content_length":"15471","record_id":"<urn:uuid:8493efc3-c8c3-4e37-9bef-e7fcd695c0c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00772.warc.gz"} |
nonabelian cocycle
nLab nonabelian cocycle
A nonabelian cocycle is, generally, a cocycle in nonabelian cohomology.
This means usually that it is modeled in either of the following two ways:
• a descent datum, i.e. an object in the category of descent data (sometimes briefly called a descent category);
• an $\omega$-anafunctor.
Notice that for $n\geq 2$, descent data in a setup of strict $n$-categories still form a strict $(n+1)$-category, while $n$-anafunctors form weak $(n+1)$-category.
Last revised on June 9, 2009 at 02:20:52. See the history of this page for a list of all contributions to it. | {"url":"https://ncatlab.org/nlab/show/nonabelian+cocycle","timestamp":"2024-11-02T14:31:04Z","content_type":"application/xhtml+xml","content_length":"14851","record_id":"<urn:uuid:acb79bba-fc1f-4b89-befe-4580deaa03c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00586.warc.gz"} |
sptotal implements finite population block kriging (Ver Hoef (2008)), a geostatistical approach to predicting means and totals of count data for finite populations.
See sptotal’s Website for more information.
sptotal can be installed from CRAN
or using devtools
Simple Example
The sptotal package can be used for spatial prediction in settings where there are a finite number of sites and some of these sites were not sampled. Note that, to keep this example simple, we are
simulating response values that are spatially independent. In a real example, we assume that there is some spatial dependence in the response.
spatial_coords <- expand.grid(1:10, 1:10)
toy_df <- data.frame(xco = spatial_coords[ ,1],
yco = spatial_coords[ ,2], counts = sample(c(rpois(50, 15),
rep(NA, 50)), size = 100, replace = TRUE))
mod <- slmfit(formula = counts ~ 1, xcoordcol = "xco",
ycoordcol = "yco", data = toy_df)
pred <- predict(mod)
We can look at the predictions with
pred$Pred_df[1:6, c("xco", "yco", "counts", "counts_pred_count")]
Methods and Basic Functions
sptotal Main Functions:
slmfit() fits a spatial linear model to the response on the observed/sampled sites. can be used to construct an empirical variogram of the residuals of the spatial linear model.
predict.slmfit() uses the spatial linear model fitted with slmfit() and finite population block kriging to predict counts/densities at unobserved locations. A prediction for the total count as well
as a prediction variance are given by default.
For more details on how to use these functions, please see the Vignette by running
and clicking HTML.
The methods in this package are based on the following reference:
Ver Hoef, Jay M. “Spatial methods for plot-based sampling of wildlife populations.” 15, no. 1 (2008): 3-13.
To cite this package in the literature, run the following line: | {"url":"https://cran.ms.unimelb.edu.au/web/packages/sptotal/readme/README.html","timestamp":"2024-11-12T02:30:27Z","content_type":"application/xhtml+xml","content_length":"3838","record_id":"<urn:uuid:3ea76b83-1f51-4af8-821a-f8387b91eecf>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00308.warc.gz"} |
Building software
This page has a software dedicated to the building. Its purpose is to enable building professionals to hobbyists who like to do construction or renovation to understand how this software can help
them in their work.
Bricocalculette is a downloadable shareware that allows for 40 types of calculations and conversions useful in many different areas of habitat. It can also easily make complex calculations through a
system of original colaboration between these modules.
The calculation function: electrical conductor max allows you to calculate the maximum length that can have an electrical conductor so that the voltage drop does not exceed the percentage you set. A
voltage drop of 3% is usually considered acceptable. The maximum length of the conductor is calculated based on the voltage, amperage, type of circuit and the section of wire.
You can calculate the length of wire using the standard calculator or the calculator electrical circuit that can be accessed from the dropdown list in the formula box.
To calculate the maximum length of an electrical conductor:
1) Type the value of the voltage in volts.
2) Click "&".
3) Type the value of the electrical current in amperes.
4) Enter the value of the section of wire in mm2.
5) Click "&".
6) Enter the percentage of voltage drop.
7) Click "f=" to perform the calculation. The calculation result is displayed in the input box and the formula box in response to the entered parameters. Using the result of the calculation:
The result of calculating the maximum length of the driver is automatically saved and you can reuse it as the parameter value of another calculation by clicking on the "MR" button.
You can also copy this result into the clipboard of Windows by running the key combination "Ctrl + C". This allows you to retrieve the value in another program or a Word document, Excel, or email.
Download and try for free HERE BricoCalculette The calculation function: electric circuits allows you to calculate the cross wire size, the amperage of the fuse and the circuit breaker of an
electrical circuit. These characteristics of the circuit are calculated based on the type and number of shots. wire size is given in mm2 (square millimeters). The intensities electric fuse and
circuit breaker are given in amperes. To calculate the wire size and amperage fuse and breaker of an electrical circuit 1) Select the type of electrical circuit in the list. 2) Click the "&". 3)
Select the number of shots that should have the electrical circuit. 4) Click "f=" to perform the calculation. By default this is the section of the wire that appears.
5) Click fuse to view the value of the amperage of the fuse. If the type of circuit there is no fuse is the "#NV" (no value see Error Handling) that appears. 6) Click for Circuit Breaker display the
value of the amperage of the circuit breaker. Using the result of the calculation Calculation result of electrical circuit is automatically saved and you can reuse it as the parameter value to
another calculation by clicking on the button MR. You can also copy the result into the clipboard of Windows by running the key combination "Ctrl C". This allows you to retrieve the value in another
program or a Word document, Excel, or email. Download and try for free HERE BricoCalculette The calculation function: calculation for flooring allows you to calculate the amount of flooring and
joists to cover a rectangular area. These quantities of flooring and joists are calculated according to the dimensions of the surface to be covered and the type of installing the flooring. To
calculate the amount of flooring and joists 1) Select the unit of measure you want to work. 2) Type the value of the length of the surface. 3) Type the value of the width of the surface. 4) Click "f=
". The default is the number of square meters of flooring that appears.
5) Click "Joists" to display the amount of joists. Using the result of the calculation The result of the calculation of quantities of flooring joists and is automatically saved and you can reuse it
as the parameter value to another calculation by clicking on the "MR" button. You can also copy the result into the clipboard of Windows by running the key combination "Ctrl + C". This allows you to
retrieve the value in another program or a Word document, Excel, or email. Download and try for free HERE BricoCalculette The calculation function: stairs allows you to calculate the height, the
horizontal stair run and number of steps of a straight staircase.
The calculation is performed so that the height of steps is between 16 and 19 so as to ensure the comfort of walking.
To calculate the number and height of the steps of a stair 1) Select the unit of measurement with which you wish to work. 2) Enter the value of the height to ground floor. This height is the vertical
distance between the starting point of the stairway are ending point on the upper floor. 3) Click "f=". By default it is the step height that appears. 4) Click Run view the run of the stairs. 5)
Click Number of steps to view the number of steps of the stair. Using the result of the calculation The calculation result of the staircase is automatically saved and you can reuse it as the
parameter value to another calculation by clicking on the "MR" button. You can also copy the result into the clipboard of Windows by running the key combination "Ctrl C". This allows you to retrieve
the value in another program or a Word document, Excel, or email. Download and try for free HERE BricoCalculette The calculation function: quantities of materials for concrete allows you to calculate
the quantities of the various components of a concrete or mortar in order to obtain the desired volume. To calculate the quantities for concrete 1) Select the unit of measure you want to work. 2)
Type the value of the volume of concrete or mortar as you want. 3) Click "&". The input field is replaced by a list. 4) Select the type of concrete or mortar you want to do.
5) Click "f=" to perform the calculation. The default is the amount of cement needed to obtain the volume of concrete or mortar that appears. 6) Click "Water" to view the volume of water. 7) Click
"Sand" to view the amount of sand. 8) Click "Gravel" to view the amount of gravel. Using the result of the calculation The result of the calculation of quantities for concrete is automatically saved
and you can reuse it as the parameter value to another calculation by clicking on the "MR" button. You can also copy the result into the clipboard of Windows by running the key combination "Ctrl C".
This allows you to retrieve the value in another program or a Word document, Excel, or email. Download and try for free HERE BricoCalculette For more information, see the online guide for using | {"url":"http://woodsoft.blogspot.com/p/building-software.html","timestamp":"2024-11-14T11:42:50Z","content_type":"text/html","content_length":"50510","record_id":"<urn:uuid:bd29173b-a7dc-4b2d-b9d3-af1e575f5e03>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00410.warc.gz"} |
N5 Physics – N5 (National 5)
This helpsheet is on the front page in the general resources but I forgot, so I’ll repost it here so that those who are looking for it get a double chance. ENJOY
Your calculator can go in to all Physics exams unless it is one that isn’t allowed. Here are the two most common calculators and how to set them up. Learning these tips will give you the edge in the
No programmable calculators are allowed in the Physics Exam!
November 2023
November 2023
Past Papers for National 5 Physics
Please read the course report after completing the paper and marking it according to the General Marking Principles.
N5 Papers Year Marking Exam Digital QP
tagging xls 2014- tagging pdf Skills Tagging
2024 S1QP 2024 2024 MI 2024Report 2024 S1DQP
2024 S2QP 2024 S2 DQP
2023S1QP 2023 MI 2023S1DQP
2023S2QP 2023 Relations 2023 Report 2023S2 DQP
2022S1QP 2022 2022 MI 2022Report 2022S1DQP
2022S2QP 2022S2DQP
2021S1QP 2021 2021 MI 2021 QP evidence S1DQP
2021S2QP S2DQP
2019QP 2019 2019 MI 2019Report 2019DQP
2018QP 2018 2018 MI 2018Report 2018 DQP
Specimen New Specimen Assignment
QP & MI Model QP & MI Assessment
2017QP 2017 2017 MI 2017Report 2017 DQP
2016QP 2016 2016 MI 2016Report 2016 DQP
2015QP 2015 2015 MI 2015Report 2015 S1 DQP
2015 S2 DQP
2014QP 2014 2014 MI 2014Report 2014 DQP
QP & MI Specimen QP & MI
READ THIS FIRST MARK GUIDE
This is the legendary file from Mr Davie, with all the past paper questions matched to the topics.
I’ve just found this file, to give you some additional access to other questions to practice. Here some of the topics have got questions from SG, Int 2 and H questions.
Wordwall Revision Games
Practice your Physics using these Wordwalls, don’t forget this forms only PART of your revision.
Sorry I don’t know why some of these wont embed, I’ve had to post them as links. I hope you can still get to play.
Continue reading “Wordwall Revision Games”
This is the main Radiation post. Start here!
Here’s the video
Thanks to Miss Horn for the Radiation Notes. Worked Answers to follow.
Thanks to Miss Horn who started these off
click on the image for the pdf Summary Sheet for Radiation
Radiation Mind Map- only print page 1 and 2. If anyone knows how to delete p3 I’d be grateful for a helping hand.
From Helpmyphysics
Fusion is the process when two SMALL NUCLEI join to form a LARGER NUCLEI with the production of ENERGY
Fission is the process when two large nuclei split to form two smaller nuclei with the production of energy. This can occur spontaneously or due to a collision with a neutron. Often extra neutrons
are produced.
Chain Reaction
When neutrons split nuclei by fission and extra neutrons are produced which can split further nuclei. Large quantities of energy are produced.
Reducing exposure to ionising radiation.
There are 3 groups of category to reduce harm caused by radiation:
1. MONITOR
2. SHIELD
3. DISTANCE
Monitor includes things like wearing radiation badges or EPUs, timing how long you are exposed to radiation, checking with radiation counters any contamination on clothes.
Shielding is placing layers of absorbers between you and the source, BEWARE, goggles and a lab coat are great at protecting against alpha but have no effect on gamma. Only thick layers of lead would
offer protection against gamma.
Distance. Radiation obeys the inverse square law, as you double the distance from a source the level you are exposed to decreases by ÂĽ . Using tongs is an effective method of keeping your distance
from a source.
To give you an idea of the radiation dose that would occur with radiotherapy, here is my mum’s dose. I know that she’d have been happy to share this with you as a learning experience. I really miss
you mum x
When it goes wrong
Chernobyl Nuclear Disaster 1986- Effects and Summary
Chernobyl Surviving Disaster (BBC Drama Documentary)
Chernobyl Questions
1. What date was the Chernobyl Disaster?
2. What was the name of the man who hanged himself at the start, who was narrating the story?
3. Which reactor blew?
4. What was the cause of the accident?
5. How many people went to see what had happened?
6. What happened to the people who saw the hole in the reactor?
7. What day of the week was the disaster?
8. What town was evacuated?
9. How did they drain the water from the reactor?
10. How did they put out the fire?
11. What was the reading on the counter when they measured the radiation levels?
12. Why was this reading misleading and wrong?
13. What was the real count when it was measured correctly?
14. What were some of the symptoms of radiation poisoning?
15. Who was sent to prison for crimes to do with the disaster? (or record how many people went to jail)
16. Who was president of the USSR when the disaster occurred?
17. What was the trigger that caused the man to hang himself?
18. What is the “elephant’s foot?” in the reactor?
19. Have there been any other nuclear disasters? Can you find out about them and name them?
20. What other things did you learn about nuclear power stations and radioactivity?
updated October 2020
Space Learning Outcome Questions
Final Space Learning Outcomes.
Sept 2020
Dynamics 2021
A couple of songs to start this unit. I think we should start all topics with a song.
I can’t condone where this guy is putting his hands!
Most up-to-date notes,
covering all outcomes
This is the updated version of the Dynamics booklet, updated to match the 2017 SQA changes.
The latest version February 2020
Dynamics Summary Notes
Now these appear to be called “Knowledge Organisers!” Who thinks up these fancy names?
This one is a joint effort by Miss Horn and Mrs Physics with formatting help from Mr Risbridger.
These are perfect Mind Maps by Ms Milner, thanks these are the best out!
From A Milner, these are so comprehensive
Other mind maps by Melanie Ehsan, with thanks to eSgoil (who provide lots of online materials), the first of a collection of mindmaps.
Scalars and Vectors
Speed, Velocity, Acceleration
Velocity time graphs
Newton’s Laws of Motion
With a little help from the IoP here is my updated Newton’s 3 Laws. I hope you can understand it. I’m quite scared to share it with you! The pdf will miss some detail as I’ve overlapped some images.
A mixture of some notes not yet tidied up!
Here are some practice questions with worked answers and 6 to a page diagram of the sky diving graph
Mrs Physics
29th December 2021
Waves Resources
Let’s start with a song!
and I’ve downloaded the lyrics and made them into a songsheet for you. Hope Jonny doesn’t charge me for copyright!
and if you like that one, then this is Physics legend
this has got lots more information on the EM Spectrum
2018 Wave Notes as produced by Miss Horn
Wave notes pdf
Wave notes word
Waves Summary Notes
These are waves summary notes I’ve produced. Hope you like them. I’d appreciate someone telling me if a photodiode can detect gamma radiation!
Revision Mind Map
This is part of a series of brilliant Mind Maps made by Miss Milner for the N5 Physics Course. I’ve broken it up into sections so here are the waves mind maps!
Here are a list of current wave resources. I will add more as I go through them. Thanks to other schools if you have kindly supplied material. I really appreciate it as do my students.
Reflection is not in the N5 Course, but it is good to know about reflection!
This is a pdf of the power point that I a using
waves-summary-notes-gairloch1 Some of these notes are for National 4, use with the content statements so you don’t spend too long learning the National 4 work.
vflambda-vdt This starts with a practical model that you can complete in class using the Virtual Physics/ Flash Learning. It then shows how v=fλ is equivalent to v=d/t. Finally some questions will
let you practise what you know.
WAVES questions word WAVES questions pdf
Januarty 2021
Virtual Flash Video
The audio can be turned off it is annoys. Here is the Virtual converted to an mp4 if I can get it to work. If people comment and find them useful I can do the rest.
PLEASE NOTE: I KNOW I HAVE A FEW BLOOPERS IN HERE. I’VE GOT TO FIND AN EDITING PACKAGE AND FIND TIME TO USE IT.
HSDU powerpoint questions
These questions will be great for student self study. Beware I will need to edit some of them later as there are some things that are out of date.
eg Q= quality factor, now called Radiation weighting factor
H = dose equivalent now called equivalent dose.
Week 2: The Maths Bit
Week 2: Significant Figures
You will need to be able to use and understand significant figures in N5 Physics. Don’t worry if you don’t get it straight away, we’ve almost a year to get it right. The video I’ve found is clearer
than I could do and sorry it is a bit long, but well worth getting to grips with. What I will add today is a document explaining the importance of significant figures to a physicist, which I will
post on here and in the class Notebook section. I wouldn’t watch the hour long video as we need to move on.
• Watch it here on Youtube :Â Significant Figures Video
• Read and make notes on significant figures: It is in Class Notebook, and on Mrsphysics
• Read and make notes on Rounding (Sheet to follow)
• Make sure you’ve checked the answers to the Compendium Questions on Significant Figures. (section 0)
• I’ll add to the calculator work this week, and you can work through that as soon as you can.
Week 2, part 2. Rounding
You will need to correctly round to the correct number of significant figures in N5 Physics. Again you might not get it straight away, but you’ll get plenty of practice. I’ll do another helpsheet for
the Class Notebook.
• Watch the video on Youtube: Rounding in more detail it explains the reason for rounding and how it does it
• For an additional help try this one Rounding Videos This is by the same guy who did the sig fig video.
• Make notes on rounding: it will eventually be in the class notebook and on MrsPhysics in the N5 maths section.
• Complete the Sig fig and Rounding Quiz (10 questions). You ought to be able to get at least 7/10. Review the work if you get less than this.
Scientific Notation Week 2 extension
…..but you will need to be able to do this. You will need to know how to do Scientific Notation. I will not test you in this just now, but you should be confident about it by August. Watch this video
on YouTube:Â Scientific Notation
Make a note on Scientific Notation in your Class Notebook
There will be a sheet this week to help you with this, which will be in the class materials here and in your note book as well, and on this site in the Maths bit.
Significant Figures
Watch the video below on significant figures.
Figure 1: The red and brown is called a counting stick and can only measure to 10 cm.
Figure 2: The top part of this metre stick can read to the nearest 1 cm, the bottom to the nearest mm.
When Physicist use numbers it is usually because they have measured something. Significant figures tell us how precise our measurement.
For example a student uses a metre stick to measure the length of a jotter.
If the student measures a jotter with the “counting stick” (in the top picture in the red and brown) which is marked in 10 cm graduations they will not be able to get a very good value. You would
get that the jotter was just under 30 cm long but you wouldn’t be able to say much more.
If the student uses a ruler marked in centimetre marks they could say that the jotter was over 29 cm but less than 30 cm and closer to 30 cm than 29 cm, you’d say it was about 30 cm long.
If the jotter was measured with a metre stick marked in millimetres the jotter could be measured as 29.7 cm long
Figure 3 Here is a diagram of the jotter measured with different metre stick.
30 cm is one significant figure and means a number between 25 cm and 34 cm which would be rounded to 30 cm. This is how you could record the number if you used the counting stick.
29 cm is two significant figures and means a number between 29.5 cm and 30.4 cm, which would be rounded to 29 cm. This is how you could record the number if you used the metre stick marked in cm only
29.7 cm is three significant figures and means a number between 29.65 cm and 29.74 cm, which would be rounded to 29.7 cm. This is probably the best measurement we should aim to make and to do this we
would need a metre stick with millimetre graduations.
29.76 cm is four significant figures and means a number between 29.755 cm and 29.764 cm, it is unlikely that you could measure a jotter to that level of precision as the pages would vary by more than
this. You would need a better piece of apparatus than a metre stick to measure this.
How many Significant Figures?
The simple rule is this: Your answer should have no more than the number of significant figures given in the question.
If different numbers in the question are given to a different number of significant figure you should use the number of significant figures in the value given to the smallest number of significant
Question: A rocket motor produces 4,570 N (3 sig fig) of thrust to a rocket with a mass of 7.0 kg (2 sig fig). What is the acceleration of the rocket?
The calculated answer to this question would be 652.8571429 ms^-2 . However the least accurate value we are given in the question is the value of the mass. This is only given to two significant
figures. Therefore our answer should also be to two significant figures: 650 ms^–2 .
You might not think that this makes a difference, but during the SQA Intermediate 2 paper in 2006 Q25 was written to test significant figures. | {"url":"https://mrsphysics.co.uk/n5/tag/n5-physics/","timestamp":"2024-11-15T02:38:54Z","content_type":"text/html","content_length":"238832","record_id":"<urn:uuid:dd04356a-9e51-4489-b8ba-dadb9526a050>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00073.warc.gz"} |
[ANSWERED] Let f t be the temperature of a cup of coffee t minutes - Kunduz
Let f t be the temperature of a cup of coffee t minutes
Last updated: 2/2/2024
Let f t be the temperature of a cup of coffee t minutes after it has been poured Interpret f 5 180 and f 5 8 Estimate the temperature of the coffee after 5 minutes and 12 seconds that is after 5 2
minutes What does f 5 180 imply OA 5 minutes after the coffee has been poured the temperature of the cup of coffee is rising at a rate of 180 degrees per minute OB 180 minutes after the coffee has
been poured the temperature of the cup of coffee is 5 degrees OC 5 minutes after the coffee has been poured the temperature of the cup of coffee is 180 degrees OD 180 minutes after the coffee has
been poured the temperature of the cup of coffee is rising at a rate of 5 degrees per minute | {"url":"https://kunduz.com/questions-and-answers/let-f-t-be-the-temperature-of-a-cup-of-coffee-t-minutes-after-it-has-been-poured-interpret-f-5-180-and-f-5-8-estimate-the-temperature-of-the-coffee-after-5-minutes-and-12-seconds-that-is-after-5-2-333571/","timestamp":"2024-11-10T18:05:27Z","content_type":"text/html","content_length":"210444","record_id":"<urn:uuid:6914504c-5e0c-4243-b4fc-164e77781490>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00245.warc.gz"} |
[QSMS Geometry Seminar 2022-05-17, 19] Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications
• Date : 2022-05-17, 05-19 14:00 ~ 15:30
• Place : 129-406 (SNU)
• Speaker : Dahye Cho (Stony Brook University)
• TiTle : Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications
Abstract : We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology,
which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance
properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of
log pairs in the direction of the Minimal Model Program.
Part 1. After briefly reviewing the Stein/Weinstein domain, Liouville domain, and Weinstein handle-decomposition, we will learn the definition of symplectic cohomology and some vanishing, invariance
properties of symplectic cohomology. We will learn how symplectic cohomology changes under algebro-geometric operations,
Part 2, We will learn how symplectic cohomology changes under Kishimoto's half-point attachment. We define a criteria for affine varieties to admit uniruled subvarieties of certain dimensions from
the long exact sequence of symplectic cohomology. We will talk about the key idea proving the main theorem and provide some applications. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&document_srl=2292&listStyle=viewer&page=3","timestamp":"2024-11-09T04:24:47Z","content_type":"text/html","content_length":"21357","record_id":"<urn:uuid:a842d1fa-a8c1-41e5-a204-17f069b04ece>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00441.warc.gz"} |
Charge in a Magnetic Field - Important Concepts and Tips for JEE
What is a Magnetic Field?
A magnetic field is defined as a field in which moving electric charges, electric currents, and other magnetic materials can experience the magnetic influence. A moving charge present in the magnetic
field experiences a force perpendicular to its own velocity and to the magnetic field. In this article, we will study the concept of magnetic field due to moving charges and magnetism in detail.
What are Moving Charges?
A moving charge is an electric current that is circulating in any electric or magnetic substance, whether or not the charge is flowing through it. These moving charges or currents generate a magnetic
field, which is denoted by $B$.
Force on a Moving Charge in a Magnetic Field
When a charge particle is moving, the magnetic field exerts the force on electric charge same as moving charge producing a magnetic field. With the increase in the magnetic field strength, the force
will also increase. So, we can say that if charge has higher velocities, then the force will be larger.
A magnetic field experiences a force that is perpendicular to both its own velocity and to the magnetic field.
Let an electric charge $+Q$ move with velocity v through the magnetic field $B$, then a force Fm is experienced by the electric charge. Now, consider a magnetic field that is acting along the
$y-$axis, the charge moves along the $XY$ making angle $\theta$ with the magnetic field. Now, we know that
$ {{F}_{m}}\propto Q~v~B~\sin \theta $
$ {{F}_{m}}=k~Q~v~B~\sin \theta $
This force is also called the Lorentz magnetic force.
The cross product suggests that the force is always perpendicular to both the magnetic field and the velocity. As a result, it always acts out of the plane and will not contribute in charge's work.
It can merely change the velocity's direction but will not change its magnitude. Fleming's Right-hand Rule makes it simple to establish the force's direction.
Fleming’s Right-hand Rule
According to Fleming's Right-Hand Rule, if we arrange our right hand's thumb, forefinger, and middle finger perpendicular to one another, the thumb will point in the direction of the magnetic force,
the forefinger will point in the direction of the magnetic field, and the middle finger will point in the direction of the current.
A Brief on Lorentz Force
The force exerted on the charged particle $q$ moves with a velocity $v$ through an electric and magnetic field. This entire electromagnetic force $F$ on the charged particle is known as the Lorentz
Force. This is given by the equation:
$\vec F=q~\vec E+q\left(~\vec{v}\times \vec B\right)$
Here, the first term is found by the electric field and the second term is the magnetic force which has a direction perpendicular to both the velocity and the magnetic field. The magnetic force
exerted on a moving charge reveals the sign of the charge carriers present in the conductor. If the current is flowing from right to left in a conductor, the sign of the charge carriers is positive,
and if the current is flowing from left to right, the sign of the charge carriers is negative.
How to Determine the Direction of Magnetic Force?
To determine the direction of the magnetic force on a moving charge, use the right-hand rule. Right Hand Rule state that to determine the direction of the magnetic force on a positive moving charge,
point the thumb of your right hand in the direction of the velocity of the moving particle, the fingers in the direction of the magnetic field, and a perpendicular to the face of the palm points in
the direction of Magnetic Force. The Magnetic Force is perpendicular to the plane formed by the velocity and magnetic field.
Right Hand Rule
Magnetism is defined as the force which is exerted by the magnets when they attract or repel each other. Magnetism in a substance is generally caused by the motion of electric charges. This affected
region around the moving charge consists of both an electric and magnetic field. A bar magnet is the most familiar example of magnetism which clearly gives us an overview of the concept, the bar
magnet is attracted to a magnetic field and can attract or repel other magnets.
Magnetism can be classified into the following:
It is the tendency to be repelled by a magnetic field. Diamagnetism can be only observed in materials that contain no unpaired electrons. Examples of diamagnetic materials are gold, quartz, copper,
The magnetic moments align and thus are magnetised in the direction of the applied field. Examples of paramagnetic materials include lithium, molybdenum, etc.
They can form permanent magnets and are also attracted to magnets. They have unpaired electrons and magnetic properties are experienced in a substance even after it is removed from the magnetic
field. Examples of ferromagnetic materials include iron, cobalt, nickel, etc.
Causes of Magnetism
Magnetism is produced by the spin magnetic moment of elementary particles, such as electrons and the current travelling through the wire. Therefore, we can also say that magnetism is caused by the
electromagnetic force present inside a substance.
Important Facts on Magnetism in Everyday Life
• When a bar magnet is cut into two pieces along its length, the two pieces will behave like an independent magnet altogether, but the pole strength of the magnet will get reduced.
• A plane which passes through the geographic axis and is perpendicular to the surface of Earth (vertical plane) is called a geographic meridian.
• The earth’s magnetic field varies from point to point in space. Does it change according to time? The Earth’s magnetic field does change with time. Although it takes a few hundred years to change
by an appreciable amount, the variation in earth’s magnetic field with time cannot be neglected.
When a charge is moving in a magnetic field, it experiences a force which is perpendicular to both the velocity of the moving charge and the magnetic field. This force is given by the formula ${{F}_
{m}}=QvB\sin \theta $. This force is also known as the Lorentz Magnetic Force. The direction of the force on a moving charge in the magnetic field is perpendicular to the plane formed by the velocity
and magnetic field of the moving charge.
FAQs on Charge in a Magnetic Field - Definitions, Concepts, and Formulas for JEE
1. Is work done by magnetic field zero?
Yes, the direction of the magnetic force and velocity are always perpendicular to one another due to the way the magnetic force interacts with matter. The amount of work done will be zero if $\theta
= 90$ because if force and velocity are perpendicular, then force and displacement are also perpendicular. Thus, work done by the magnetic force is always zero and the kinetic energy of the charge
remains unchanged.
2. What is a magnetic field?
A magnetic field is defined as a field in which moving electric charges, electric currents, and other magnetic materials can experience the magnetic influence. The magnetic fields are produced by the
moving electric charges and the intrinsic magnetic moments of elementary particles associated with their spin.
3. What is meant by Lorentz force?
When a moving charge passes through both the electric and magnetic fields, the net force experienced by the moving charge is known as Lorentz force. It is basically the sum of the magnetic and
electric forces. | {"url":"https://www.vedantu.com/jee-main/physics-charge-in-a-magnetic-field","timestamp":"2024-11-09T10:30:12Z","content_type":"text/html","content_length":"202018","record_id":"<urn:uuid:d60975a1-efd1-482f-a7db-637689fb636d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00353.warc.gz"} |
Why does a stone released from circular motion move along a straight line tangent to the circle?
You must login to ask question.
When a stone is released from uniform circular motion, it continues to move along a straight line tangent to the circular path because it retains the direction of motion it had at that instant. This
is due to the principle of inertia, which states that an object in motion will continue to move in a straight line at a constant velocity unless acted upon by an external force. | {"url":"https://discussion.tiwariacademy.com/question/why-does-a-stone-released-from-circular-motion-move-along-a-straight-line-tangent-to-the-circle/","timestamp":"2024-11-11T01:44:56Z","content_type":"text/html","content_length":"149307","record_id":"<urn:uuid:679d7b9a-b183-4427-9d0c-e4e681aac96f>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00669.warc.gz"} |
A New Approach to Data Storage - Research & Development World
Researchers at Johannes Gutenberg University Mainz (JGU) have succeeded in developing a key constituent of a novel unconventional computing concept. This constituent employs the same magnetic
structures that are being researched in connection with storing electronic data on shift registers known as racetracks.
In this, researchers investigate so-called skyrmions, which are magnetic vortex-like structures, as potential bit units for data storage. However, the recently announced new approach has a particular
relevance to probabilistic computing.
This is an alternative concept for electronic data processing where information is transferred in the form of probabilities rather than in the conventional binary form of 1 and 0. The number 2/3, for
instance, could be expressed as a long sequence of 1 and 0 digits, with 2/3 being ones and 1/3 being zeros.
The key element lacking in this approach was a functioning bit reshuffler, i.e., a device that randomly rearranges a sequence of digits without changing the total number of 1s and 0s in the sequence.
That is exactly what the skyrmions are intended to achieve.
The results of this research have been published in the journal Nature Nanotechnology (“Thermal skyrmion diffusion used in a reshuffler device”).
The researchers used thin magnetic metallic films for their investigations. These were examined in Mainz under a special microscope that made the magnetic alignments in the metallic films visible.
The films have the special characteristic of being magnetized in vertical alignment to the film plane, which makes stabilization of the magnetic skyrmions possible in the first place.
Skyrmions can basically be imagined as small magnetic vortices, similar to hair whorls. These structures exhibit a so-called topological stabilization that protects them from collapsing too easily—as
a hair whorl resists being easily straightened. It is precisely this characteristic that makes skyrmions very promising when it comes to use in technical applications such as, in this particular
case, information storage.
The advantage is that the increased stability reduces the probability of unintentional data loss and ensures the overall quantity of bits is maintained.
The reshuffler receives a fixed number of input signals such as 1s and 0s and mixes these to create a sequence with the same total number of 1 and 0 digits, but in a randomly rearranged order. It is
relatively easy to achieve the first objective of transferring the skyrmion data sequence to the device, because skyrmions can be moved easily with the help of an electric current.
However, the researchers working on the project now have for the first time managed to achieve thermal skyrmion diffusion in the reshuffler, thus making their exact movements completely
unpredictable. It is this unpredictability, in turn, which made it possible to randomly rearrange the sequence of bits while not losing any of them.
This newly developed constituent is the previously missing piece of the puzzle that now makes probabilistic computing a viable option.
“There were three aspects that contributed to our success. Firstly, we were able to produce a material in which skyrmions can move in response to thermal stimuli only. Secondly, we discovered that we
can envisage skyrmions as particles that move in a fashion similar to pollen in a liquid. And ultimately, we were able to demonstrate that the reshuffler principle can be applied in experimental
systems and used for probability calculations.
The research was undertaken in collaboration between various institutes and I am pleased I was able to contribute to the project,” emphasized Dr. Jakub Zázvorka, lead author of the publication.
Zázvorka conducted his research into skyrmion diffusion as a research associate in the team headed by Professor Mathias Kläui and is meanwhile working at Prague University.
“It is very interesting that our experiments were able to demonstrate that topological skyrmions are a suitable system for investigating not only problems relating to spintronics, but also to
statistical physics. Thanks to the MAINZ Graduate School of Excellence, we were able to bring together different fields of physics here that so far usually work on their own, but that could clearly
benefit from working together. I am particularly looking forward to future collaboration in the field of spin structures with the Theoretical Physics teams at Mainz University that will feature our
new TopDyn—Dynamics and Topology Center,” emphasized Mathias Kläui, Professor at the Institute of Physics at JGU and Director of the Graduate School of Excellence Materials Science in Mainz (MAINZ).
“We can see from this work that the field of spintronics offers interesting new hardware possibilities with regard to algorithmic intelligence, an emerging phenomenon also being investigated at the
recently founded JGU Emergent Algorithmic Intelligence Center,” added Dr. Karin Everschor-Sitte, a member of the research center’s steering committee and head of the Emmy Noether research group TWIST
at the JGU Institute of Physics. | {"url":"https://www.rdworldonline.com/a-new-approach-to-data-storage/","timestamp":"2024-11-02T08:01:22Z","content_type":"text/html","content_length":"62471","record_id":"<urn:uuid:9a9d550f-1ce6-470e-8ef3-0c4f4b9176bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00789.warc.gz"} |
Event-Based Proof of the Mutual Exclusion Property of Peterson's Algorithm
Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and
event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based
approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible. Analysis of the literature [1], [7], [12],
[9], [13], [14], [15] shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are
often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile.
Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time
case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time/hybrid models of distributed systems. In the paper [2]
we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is
determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson's algorithm [11]. We have
also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in [8]. Using Mizar [3], in this paper
we give a formal proof of the mutual exclusion property of the Peterson's algorithm in Mizar on the basis of the event-based approach proposed in [2]. Firstly, we define an event-based model of a
shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of
this structure include a binary precedence relation ≤ on the set of events which turns it into a linear preorder (events are considered simultaneous, if e[1] ≤ e[2] and e[2] ≤ e[1]), special
predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial
function "val of" on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) [2]. Then we define several natural consistency
requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in [2]). After this we formulate and prove the
main theorem about the mutual exclusion property of the Peterson's algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of
two processes, and in some trace there occur two events e[1] and e[2] in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed
by execution of the Peterson's algorithm (setting the flag of the current process, writing the identifier of the opposite process to the "turn" shared variable, and reading zero from the flag of the
opposite process or reading the identifier of the current process from the "turn" variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier
to the "turn" variable, then either the events e[1] and e[2] coincide, or they are not simultaneous (mutual exclusion property).
• algorithm
• distributed system
• mathematical model
• parallel computing
• verification
ASJC Scopus subject areas
• Computational Mathematics
• Applied Mathematics
Dive into the research topics of 'Event-Based Proof of the Mutual Exclusion Property of Peterson's Algorithm'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/event-based-proof-of-the-mutual-exclusion-property-of-petersons-a","timestamp":"2024-11-11T04:10:44Z","content_type":"text/html","content_length":"69741","record_id":"<urn:uuid:f55ea0cc-8225-4999-8530-de1a89044101>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00094.warc.gz"} |
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Casey, J.
A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl.
Dublin: Hodges, Figgis, & Co., p. 66, 1893.Eddy, R. H. and Fritsch, R. "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle."
Math. Mag. 67
, 188-205, 1994.Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer. 129
, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(76)=3rd Brocard Point."
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Direct Assimilation of Chinese FY-3E Microwave Temperature Sounder-3 Radiances in the CMA-GFS: An Initial Study
Earth System Modeling and Prediction Centre, China Meteorological Administration, Beijing 100081, China
State Key Laboratory of Severe Weather, Institute of Meteorological, China Meteorological Administration, Beijing 100081, China
Joint Center for Data Assimilation for Research and Application, Nanjing University of Information Science and Technology, Nanjing 210044, China
Author to whom correspondence should be addressed.
Submission received: 15 October 2022 / Revised: 19 November 2022 / Accepted: 21 November 2022 / Published: 24 November 2022
FengYun-3E (FY-3E), the fifth satellite in China’s second-generation polar-orbiting satellite FY-3 series, was launched on 5 July 2021. FY-3E carries a third-generation microwave temperature sounder
(MWTS-3). For the first time, this study demonstrates that MWTS-3 radiances data assimilation can improve the China Meteorological Administration Global Forecast System (CMA-GFS). By establishing a
cloud detection module based on the retrieval results of the new channels of MWTS-3, a quality control module according to the error characteristics of MWTS-3 data, and a bias correction module
considering the scanning position of satellite and weather systems, the effective assimilation of MWTS-3 data in the CMA-GFS has been realized. Through one-month cycling experiments of assimilation
and forecasts, the error characteristics and assimilation effects of MWTS-3 data are carefully evaluated. The results show that the observation errors in MWTS-3 data are similar to those in advanced
technology microwave sounder (ATMS) data within the same frequency channel, are slightly larger than those in the advanced microwave-sounding unit-A (AMSU-A) data, and are much better than those in
the MWTS-2 data. The validation of the assimilation and prediction results demonstrate the positive contribution of MWTS-3 data assimilation, which can remarkably reduce the analysis errors in the
Northern and Southern Hemispheres. Specifically, the error growth on the upper layer of the model is obviously suppressed. When all other operational satellite observations are included, the
assimilation of MWTS-3 data has a neutral or slightly positive contribution to the analysis and forecast results, and the improvement is mainly found in the Southern Hemisphere. The relevant
evaluation results indicate that the MWTS-3 data assimilation has good application prospects for operation.
1. Introduction
In recent years, satellite observations have become a key component of the global operational numerical weather prediction (NWP) system due to their high spatial-temporal resolution and wide spatial
coverage. Many studies have shown that direct assimilation of microwave-sounding data can remarkably improve the initial conditions of numerical models so as to improve the prediction levels of
global and regional models [
]. Most NWP centers have reported a substantial reduction in the root mean square error (RMSE) in forecasts by effectively assimilating the data from the Advanced Television and Infrared Observation
Satellite (TIROS) operational vertical sounder (ATOVS) onboard the National Oceanic and Atmospheric Administration satellites (NOAA-15, -16, -17, -18 -19, and -20), the Meteorological Operational
satellite-A/B (MetOp-A/B) and the Aqua earth observing system. Adjoint sensitivity experiments [
] have proven that microwave temperature-sounding data has become the most influential observation in almost all operational forecasting systems [
Recently, China’s polar-orbiting meteorological satellites have become an important part of the global polar-orbiting satellite observing system. Since the successful launch of China’s new generation
polar-orbiting satellite Fengyun-3A (FY-3A) on 26 May 2008 [
], Fengyun-3B/C/D (FY-3B/C/D) satellites have been launched successively. The performance of microwave sounders onboard these satellites is similar to those of the advanced microwave-sounding unit-A
(AMSU-A) onboard the NOAA and MetOp satellites [
]. FY-3A/B are equipped with the first-generation microwave temperature sounder (MWTS-1), which has four channels with frequencies comparable to channels 3, 5, 7, and 9 of the AMSU-A [
]. FY-3C/D are equipped with the second-generation microwave temperature sounder (MWTS-2). MWTS-2 has 13 channels, and the channels located in the oxygen absorption band (50–60 GHz) are identical to
those of the AMSU-A. Various studies on data evaluation [
] and assimilation have been carried out for the MWTS, and many of them have indicated that the assimilation of MWTS-1 and MWTS-2 data has positive impacts on NWP results [
FY-3E satellite was successfully launched on 5 July 2021, which is the world’s first meteorological satellite sent into the early-morning orbit for civil use [
]. It has a local equatorial crossing time of about 5:40 am. This satellite carries a third-generation microwave temperature sounder (MWTS-3). A systematic evaluation study [
] has demonstrated that the performance of MWTS-3 is remarkably better than the previous two generations of instruments, with more observational information and well-suppressed observational noises.
The purpose of this study is to evaluate the impacts of the direct assimilation of the MWTS-3 radiance data on the China Meteorological Administration global forecast system (CMA-GFS) for the first
time. By establishing the quality control (QC) and bias correction modules suitable for the MWTS-3 data, the effective assimilation of MWTS-3 data in the CMA-GFS is realized. The influence of MWTS-3
data assimilation on the CMA-GFS is evaluated based on the results of one-month assimilation and forecasting. It should be noted that the original name of the operational numerical prediction system
in China was the Global and Regional Estimation and PrEdiction System (GRAPES) [
]. After September 2021, it was renamed CMA-GFS.
The remainder of this paper is organized as follows.
Section 2
introduces the CMA-GFS four-dimensional variational assimilation (4D-Var) system. The general details of the FY-3E MWTS-3 radiance data is described here.
Section 2
also provides the QC and bias correction scheme for the MWTS-3 radiance data, and the initial assessments of MWTS-3 data.
Section 3
presents the analysis of the numerical results of the FY-3E MWTS-3 radiance data assimilation experiments. The discussion and conclusion are given in
Section 4
Section 5
2. Materials and Methods
2.1. CMA-GFS 4D-Var System
The main components of CMA-GFS include: four-dimensional variational (4D-Var) data assimilation; fully compressible non-hydrostatical model core with semi-implicit and semi-Lagrangian discretization
scheme; modularized model physics package, and global and regional assimilation and prediction systems [
The CMA-GFS 4D-Var system is an analysis system designed for operational application [
]. This assimilation system adopts an incremental analysis method, and the assimilation process is divided into outer circulation and inner circulation. In order to reduce the amount of computation,
the horizontal resolution of the nonlinear model in the outer circulation of the assimilation is 0.25 degrees, the horizontal resolution of the tangent linear model and the adjoint model in the inner
circulation is 1.0 degrees, and only the simplified physical process is applied. The model has 87 vertical layers, with the top being approximately 0.1 hPa. The 4D-Var data assimilation system
applies the incremental analysis scheme proposed by Courtier et al. (1998) [
]. By using the observations distributed within a time interval (t0, tn) in the assimilation, the cost function can be defined as follows:
$J ( x ( t 0 ) ) = 1 2 ( x ( t 0 ) − x b ( t 0 ) ) T B − 1 ( x ( t 0 ) − x b ( t 0 ) ) + 1 2 ∑ i = 0 N ( H ( x i ) − y i o ) T R i − 1 ( H ( x i ) − y i o ) + J c$
) is a state vector composed of atmospheric and surface variables;
) is a background estimate of the state vector provided by a 6 h forecast, and
$y i o$
is a vector of all the observations;
is the observation operator that transforms the state vector
into observation space;
is the estimated error covariance of the observations at time
is a constraint term added to control various noises and errors generated in variational analysis. For the CMA-GFS data assimilation system,
is the weak constraints of the digital filtering.
is the error covariance matrix of
. In order to solve the problem that the inverse of the background error covariance matrix (
) is too large to be computed, the background term is preconditioned, which improves the convergence in the minimization process and avoids calculating
directly. In the CMA-GFS 4D-Var system, the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm [
] is used to perform the minimization.
Currently, the CMA-GFS can directly assimilate radiosonde data, surface synoptic observations (SYNOPs), ship reports, aircraft reports (Airep), atmospheric motion vectors (AMVs), the AMSU-A and the
microwave humidity sounder (MHS) data of NOAA-15/18/19, the AMSU-A, MHS and infrared atmospheric sounding interferometer (IASI) data of MetOp-A/B, Suomi National Polar-orbiting Partnership (NPP) ATMS
data, the MWHS-2, micro-wave radiation imager (MWRI) and Global Navigation Satellite System (GNSS) radio occultation sounder (GNOS) data of FY-3C/D, the MWTS-2 and hyperspectral infrared atmospheric
sounder-2 (HIRAS) radiance data of FY-3D, the Constellation Observing System for Meteorology, Ionosphere and Climate radio occultation (COSMIC RO) data, etc.
The radiative transfer for TIROS operational vertical sounder-12 (RTTOV-12) [
] is used as the observation operator for the direct assimilation of satellite radiance data in the CMA-GFS 4D-Var system. The transmittance coefficients applicable to the RTTOV-12 for FY-3E MWTS-3
simulation are provided by the National Satellite Meteorological Center of CMA.
2.2. FY-3E MWTS-3 Observations
The MWTS-3 radiance data in L1b format from September to October 2021 were used in this study. Channel characteristics of FY-3E MWTS-3 are shown in
Table 1
. Compared with the MWTS-2, MWTS-3 has improved its detection capability and performance indicators. The number of detection channels of MWTS-3 is 17, which is 4 more than that of MWTS-2. Channels 1
and 2 are horizontally polarized, while the other channels are vertically polarized. The noise equivalent differential temperature (NEDT) of channels 1–11 is about 0.3–0.35 K. The NEDT of channels
12–17 is slightly larger, around 0.6–2.1 K. The swath width of MWTS-3 is 2700 km, which is much larger than that of the MWTS-2 (2250 km) and is also wider than that of similar instruments in the
world, such as the AMSU-A (2300 km) and ATMS (2500 km). For MWTS-3, the number of fields of view (FOV) in a single scan line also increases to 98 from 90 for the MWTS-2, which is larger than that of
AMSU-A (30) and ATMS (96).
In terms of channel settings, two channels that can detect cloud water content were added into MWTS-3 for the first time, with the detection frequencies being 23.8 GHz and 31.4 GHz, respectively. As
a result, the existing mature scheme can be used to identify the microwave data in cloudy areas [
] based on cloud liquid water path (CLWP) retrieval [
]. In addition, MWTS-3 also has two detection channels at the oxygen absorption band 50–60 GHz, which can be used to detect the atmospheric temperature information at central altitudes of about 500
and 700 hPa.
The weighting function of MWTS-3 is shown in
Figure 1
, which is calculated using the RTTOV-12 based on the American standard atmosphere profile. The MWTS-3 can detect atmospheric temperature information from the troposphere to the stratosphere. The
peaks of the weighting function of channels 1–4 are mainly located on the ground, and those of channels 5–17 are uniformly distributed in the vertical direction, which allows the MWTS-3 to detect the
atmospheric temperature information at different heights. The weighting function of channel 17 has the highest peak at about 2 hPa.
2.3. Cloud Detection
The MWTS-3 instrument observes the Earth from outer space, which is inevitably affected by clouds. Although the long wavelength allows microwave radiation to penetrate most nonprecipitation clouds,
it is inevitably influenced by cloud absorption, large-particle scattering, etc. At present, the assimilation of radiance data in cloudy areas is very challenging due to the lack of reliable
information about clouds in the input atmospheric profiles and the inability to accurately involve the cloud impact in the fast radiative transfer model. Many schemes have been developed to
assimilate the cloud-influenced observations of microwave-sounding data [
]. However, in order to ensure the stability of the operational NWP system, the CMA-GFS is still assimilating the clear sky data of microwave temperature-sounding. Hence, it is necessary to perform
cloud detection on the MWTS-3 data in this study.
The microwave sounders onboard the satellites (from FY-3A to FY-3D) lack channels that are sensitive to cloud absorption and scattering, which makes it difficult to perform cloud detection in MWTS-1/
2 data assimilation. In the early stage, cloud products of the visible and infrared radiometer (VIRR) mounted on the same platform were used to assist in cloud detection [
]. In order to meet the needs of cloud detection, the MWTS-3 onboard FY-3E has included the channels of 23.8 GHz and 31.4 GHz for the first time. Previous studies have developed a mature CLWP
retrieval method over the ocean area based on the brightness temperatures observed at these two frequencies [
], which provides an effective way for cloud detection in MWTS-3.
Figure 2
shows the distribution of FY-3E MWTS-3 observed brightness temperatures at channels 1–2 and the retrieved CLWP during 0300–1500 Universal Time (UTC) on 1 July 2014. Note that only the CLWP over the
ocean area is retrieved (areas covered by sea ice are also excluded), which ranges from 0.01 to 2.0 g kg
The accuracy of the retrieval product is assessed by comparing it with the brightness temperature of a 12 μm-channel (channel 7) in the medium resolution spectral imager with a low light level
(MERSI-ll) [
] onboard the same platform.
Figure 3
shows the distribution of the retrieved CLWP and MERSI channel 7 brightness temperature during 0300–1500 UTC on 24 September 2021. As shown in
Figure 3
, there is a tropical cyclone over the north Pacific with an obvious high brightness temperature center, which has a good spatial correspondence with the large-value area of the retrieved CLWP. A
larger CLWP indicates thicker clouds.
For the land area, the differences between the observed and simulated brightness temperature (O-B) on window channel 3 of MWTS-3 is used for cloudy data identification. When the O-B exceeds 1.5 K,
this FOV is determined to be the data over cloud and will be rejected.
2.4. The Initial Evaluation of Observation Bias and Error
The accurate estimation of observation bias and error is an important prerequisite for the effective assimilation of satellite data. Observation from 10–23 September 2021 was selected for the
evaluation of MWTS-3 data before assimilation. The RTTOV-12 was used to simulate the brightness temperature during the same period based on the ERA-5 reanalysis data released by the European Centre
for Medium-Range Weather Forecasts (ECMWF). On this basis, the observation bias and error of MWTS-3 were estimated by analyzing the difference between the observed (O) and simulated (B) brightness
temperature. In order to avoid the influence of the uncertainty of land surface emissivity, only clear-sky observations over the ocean were selected for the estimation. The means and standard
deviations (STDs) of the calculated O-B of the MWTS-3 data are shown in
Figure 4
It can be seen that the biases and errors have great channel differences. For all channels, the biases are basically between ±2.0 K. Specifically, the biases of channels 1, 4, 7, 8, 10, and 11 are
negative, whereas the biases of channels 4, 8, and 10 reach about −2.0 K. Channels 2, 3, 5, and 9 and the four channels in the upper stratosphere, i.e., channels 14–17, all have positive biases,
which are basically around 1.5 K. While the biases of rest channels are close to 0.
The O-B STD (
Figure 4
b) gradually decreases and then increases with the increasing height from the ground to higher altitudes. The O-B STDs of channels 1–5 are sensitive to clouds and are also greatly affected by weather
systems. Influenced by the relatively larger error of the lower-layer background, the O-B STDs of these channels are the largest. The peak values of the weighting function of channels 6–14 are mainly
between 20–700 hPa, and the overall STD is within 1 K (except channel 8). The STD of channel 8 is about 1.2 K, obviously higher than those of adjacent channels. The peak values of the weighting
function of channels 15–17 appear in the upper stratosphere, where remarkable increases in the observation errors are found in these channels, which may be due to the large error of the upper-level
temperature profile in the background field and large NEDTs of these channels.
In general, the observation errors of MWTS-3 are within the normal range; only the noise of channel 8 is greater than expected. In addition, the biases of channels 6–7 also exceed those of similar
channels of the same-type instruments, such as the ATMS (personal communication with Prof. Wen F. Z.).
2.5. Channel Selection
As shown in
Figure 1
, it is found that the maximums of the weighting function of MWTS-3 channels 1–5 are close to the ground and are sensitive to the underlying surface. Due to those inaccurate surface physical
variables, such as the surface temperature and surface emissivity, these near-ground channels were not included in the data assimilation. When considering the bias problems of channels 6–8 in the
preliminary evaluation in
Section 2.4
, these three channels were also excluded. For the upper tropospheric or stratospheric channels 11–17, since the error of the CMA-GFS is relatively larger near the model top (10 hPa to 0.1 hPa), the
two high-level channels of 16 and 17 were excluded. As a preliminary study, MWTS-3 channels 9–15 were directly assimilated in the CMA-GFS.
2.6. Quality Control Based on Scan and Surface Characteristics
In addition to cloud detection, some extra QC procedures were applied to eliminate the observation data with abnormal O-B values caused by complex underlying surfaces and large terrain height.
Extra QC procedures were carried out in the following order. (I) The observations of channels 9 and 10 in the cloudy area are removed. (II) All FOVs covering the coastline are removed. A land mask
database with longitudinal and latitudinal resolutions of 0.1° is used for land/ocean/coast identification. (III) The 10 outermost FOVs on each side of a scan line are not used; (IV) The observations
of channel 9 over the sea ice or the land are not used. Sea ice surface is identified by the criteria that the sea surface temperature is lower than 271.45 K. (V) If the terrain height is greater
than 500 m, the data of channel 10 is rejected. This threshold is based on previous experience. In the CMA-GFS, the QC of AMSU-A, ATMS, MWTS-1/2 data all adopt this threshold. Lastly, the MWTS-3
data, which passes all the above QC procedures, is thinned to a spatial resolution of 120 km according to the distance between the observations and the nearest model grid.
2.7. Bias Correction
The CMA-GFS 4D-VAR system was used in this study. The basic theory of variational data assimilation is the Bayesian conditional probability theorem [
]. This theory assumes that the background error and observation error satisfy the Gaussian distribution, and that there is no systematic bias. However, in practical application, systematic biases
generally exist in the background and are mainly caused by the continuous forward integration of the numerical model. Meanwhile, there are inevitably systematic errors in the radiation transfer model
simulations. These also lead to a certain degree of systematic biases in the O-B, meaning effective bias correction is necessary.
The importance of bias correction for satellite radiances in data assimilation has been realized by many meteorologists, and a lot of studies have been conducted to develop effective bias correction
methods. It is found that systematic bias mainly consists of the bias caused by the scanning position difference and the bias depending on the air-mass property. Harris and Kelly (2001) developed a
static bias correction scheme [
]. After a lot of practical applications, it has proven to be an effective correction scheme and is widely used in operational NWP centers around the world [
]. In 2007, Liu et al. (2007) added this scheme to the CMA-MESO model and also achieved good results [
]. However, considering the limitation of the static bias correction scheme in estimating the bias caused by the change in the weather system, an air-mass bias correction method has been proposed to
take into account the impact of weather systems on systematic biases [
]. In addition, a variational bias correction scheme has also been established, which considers the variation of biases in combination with the minimization process of the assimilation system [
]. At present, this scheme has been applied in many national operational forecast centers, such as the National Centers for Environmental Prediction (NCEP) and the ECMWF [
]. After selecting the appropriate forecast factors, the variational bias correction scheme statistically updates the correction coefficients in the minimization process of the cost function. This
scheme has also been tested in the CMA-GFS, and it is expected to achieve operational application in 2023. However, only the scan bias correction and the air-mass bias correction are involved in this
2.7.1. Scan Bias Correction
Since the scan angle bias obviously changes with the latitude, the statistics of scan bias also need to be conducted in different latitude bands. The whole hemisphere was divided into 18 latitudinal
bands using 10° intervals. For each latitude band, the O-B difference between each scanning position and the nadir point in each scan line was calculated, and then the average value of all the O-B
differences in the same latitude band was obtained as the systematic bias in this latitude band. A linear smoothing method was also applied to avoid discontinuous correction between the two adjacent
latitudinal bands.
2.7.2. Air-Mass Bias Correction
In this study, two predictors were selected for the air-mass bias correction, namely, the thicknesses between 300–1000 hPa and 50–200 hPa of the background. Using the two-week thickness data, a
linear regression equation was established for each channel, and the coefficients,
, in the regression equation were obtained for the channel
data with a scan angle of
. The regression equation is as follows:
$Bias j ( θ ) = a j 0 + ∑ i = 1 2 a j i ( θ ) X j i ( θ )$
Here $Bias j$ is the O-B bias, and X[ji] is for the thickness. a[jo] and a[ji] represent the linear relationship between the O-B bias and the two thickness data. Using these coefficients, the O-B
bias was calculated and subtracted from each observation in the assimilation process.
After the bias correction, the QC module also removes the observation data with large O-B values, and the pixels with O-B values greater than two times that of the observation error are rejected.
According to the analysis results in
Figure 4
, the observation errors were set to 0.55 K for MWTS-3 channels 9, 10, and 12–14, 0.4 K for channel 11, and 1.1 K for channel 15 in this study.
3. Results
3.1. Experimental Design
Four experiments were conducted to demonstrate the impact of MWTS-3 data on the CMA-GFS during the period from 24 September to 25 October 2021.
Table 2
shows the specific experimental designs. Experiment 1 assimilated only the conventional observations, called CTRL1. The conventional observations contain a global set of surface and upper-air
reports, including radiosondes, SYNOP, ship reports, Aireps, and AMVs from the Global Telecommunications System (GTS). Experiment 2 assimilated the conventional observations: NOAA-15/18/19 AMSU-A,
NOAA-18/19 MHS, MetOp-A/B AMSU-A, MHS and IASI, NPP ATMS, FY-3C/D MWHS-2 and MWRI, FY-3D MWTS-2 and HIRAS radiance data, FY-3C/D GNOS, COSMIC RO data, etc., called CTRL2. The setup of the two
sensitive experiments (TEST1 and TEST2) is identical to the control experiments (CTRL1 and CTRL2), except that the FY-3E MWTS-3 radiance data were added in TEST1 and TEST2.
3.2. Analysis and Forecast of the Cycling Experiments
3.2.1. Characteristics of Data after Quality Control and Bias Correction
Figure 5
shows scatter plots of the observed and simulated brightness temperature of MWTS-3 channels 11 and 14 before and after QC during September 24–30, 2021. It can be seen that the differences between the
O and B of channel 11 are larger before QC, which are scatter distributed, especially in the range of 210–220 K. Besides, the scatter plots obviously deviate from the diagonal. After QC, only the
clear-sky observations over the ocean are retrained, which makes the distributions of O and B closer to each other, and the differences between them are from −3 K to −5 K.
Figure 5
b is for channel 14, where the scatter plots are already close to the diagonal before QC, only the plots with a brightness temperature higher than 230 K slightly deviate from the diagonal. The QC
removes those abnormal observations effectively and makes the plots closer to the diagonal after QC.
Figure 6
shows the probability density functions of O-B for channels 11 and 14 before and after bias correction. Before bias correction, the biases of channels 11 and 14 are about −0.8 K and 0.8 K, and the
STDs are 0.31 K and 0.55 K, respectively. The biases after correction are within ±0.1 K, and the STDs are slightly reduced to 0.29 K and 0.48 K, respectively. This indicates that the systematic
biases of O-B have been corrected.
3.2.2. Comparisons of Observation Biases and Errors between MWTS-3 and Other Microwave Temperature Sounders
In order to further clarify the performance of MWTS-3, the bias and error characteristics of various microwave temperature-sounding data assimilated in TEST2 are given in this subsection, where the
microwave temperature sounders include FY-3E MWTS-3, FY-3D MWTS-2, AMSU-A onboard NOAA-15/18/19, MetOp-A/B, and NPP ATMs.
Figure 7
shows the biases and STDs of O-B from MWTS-3 and AMSU-A before and after bias correction in TEST2 during the period from 24 September to 25 October 2021. Among them, the frequencies of AMSU-A
channels 5 and 6–14 are the same as those of MWTS-3 channels 7 and 9–17.
As shown in
Figure 7
, the overall bias of AMSU-A mid- and low-level channels is generally small before bias correction, where most channels show negative biases. Among them, the negative bias of AMSU-A onboard NOAA-18
is the most obvious. The bias of MWTS-3 is slightly larger than that of the same frequency channel of other instruments, where channel 9 shows a positive bias, while channel 6 of AMSU-A with the same
frequency shows a slightly smaller negative bias. The biases of MWTS-3 channels 10–11 are twice those of the AMSU-A channels with the same frequency. Channels 12–17 (of MWTS-3) show positive biases
that are opposite to those of AMSU-A. The upper-level channels of MWTS-3 and AMSU-A both exhibit large biases, which may be related to the large temperature errors in the upper-level of the
Figure 7
c shows that the O-B STD of MWTS-3 is also larger than that of AMSU-A before the bias correction, which may be related to the fact that the MWTS-3 has more pixels per scan line and a shorter sampling
residence time. After the bias correction, the biases of all instruments are close to 0 (
Figure 7
b), indicating that the bias correction method for the CMA-GFS data assimilation system has a good correction effect. Besides, the STDs of all instruments also obviously decrease after the correction
Figure 7
Figure 8
shows the biases and STDs of FY-3D MWTS-2 and NPP ATMS for the same period. It can be found that, before bias correction, the bias of ATMS is also smaller than that of MWTS-3, which is comparable to
that of AMSU-A, but the STD is larger than that of AMSU-A and is only slightly smaller than that of MWTS-3. The magnitudes of the bias and STD of MWTS-2 are comparable to those of MWTS-3. After the
bias correction, the biases of all instruments are close to 0, and the STDs are also remarkably reduced. However, the STD of MWTS-3 is smaller than that of FY-3D MWTS-2 but is more similar to the STD
features of the ATMS channels with the same frequency.
As indicated above, the comparisons among the observation errors of these microwave-sounding data before and after bias correction reveal that the error of AMSU-A is the smallest, followed by that of
MWTS-3 and ATMS, and the observation error of MWTS-2 is the largest.
3.2.3. Analysis and Forecast
After investigating the characteristics of the MWTS-3 data, the assimilation effect of MWTS-3 data was further evaluated. The effect of adding the MWTS-3 data to the conventional data assimilation
was explored first.
Figure 9
shows that the RMSE of the geopotential height and the potential temperature differences between the analysis field and ERA-5 reanalysis data in the southern and Northern Hemispheres are reduced
remarkably during the period from 24 September to 25 October 2021. Due to the lack of conventional observations in the Southern Hemisphere, the RMSE reduction in the Southern Hemisphere is most
pronounced by adding the MWTS-3 data. Since only channels 9–15 of MWTS-3 are assimilated, and the peak heights of the weighting functions are located in the range of 10–400 hPa, the variables in the
middle and high layers of the model are improved the most. Because there are a large number of conventional observations in the middle and lower layers of the Northern Hemisphere, the influence of
MWTS-3 data assimilation over these regions is very small. However, as there are few conventional data above the height of 10 hPa, the improvement of adding MWTS-3 data on the geopotential height and
potential temperature above 10 hPa is more obvious.
Figure 10
shows the daily RMSE of the geopotential height differences between the analysis results and the ERA-5 reanalysis data for CTRL1 and TEST1 in the Southern Hemisphere at 10 hPa during the period from
24 September to 25 October 2021. It can be seen that, in the CTRL1, due to the lack of observation data above the altitude of 10 hPa, the model error increases rapidly with time. On the other hand,
although only the MWTS-3 data is added in TEST1, the growth of the model error above 10 hPa is obviously suppressed, and the RMSE is greatly reduced in the first 15 days and then stably maintained
within 60 gpm.
For operational assimilation applications, the impact of assimilating the FY-3E MWTS-3 data on the operational NWP system using all observation data needs to be paid more attention. CTRL2 assimilates
all observation data used in the operations, including conventional and various satellite data, while TEST2 assimilates the MWTS-3 data additionally. The comparison shows that, after adding the
MWTS-3 data, the errors of geopotential height, potential temperature, the U and V wind exhibit little change compared with the CTRL2 results at almost all altitudes. As shown in
Figure 11
, below an altitude of 2 hPa, the errors of the TEST2 results are slightly lower than those of the CTRL2 results. However, near the model top of above 2 hPa, the analysis results are slightly worse.
This may be related to the imperfection of the bias correction scheme for the upper-level satellite data.
Figure 12
shows that the error of wind below 5 hPa is slightly reduced. Overall, the effects of MWTS-3 data assimilation are neutral or slightly positive.
Using the analysis results of CTRL2 and TEST2 (at 1200 UTC of each day) as the initial conditions, a 10-day prediction was achieved. The comprehensive scorecard for the evaluation of the forecast
results shows the abnormal correlation coefficients (ACCs) and RMSEs of various variables at different levels and in different regions (
Figure 13
). It can be seen that the assimilation of MWTS-3 data has a positive contribution to the 10-day forecasts in the Northern and Southern Hemispheres, especially to the first two-day forecasts of the
Southern Hemisphere. The overall impact in East Asia is neutral. In tropical areas, the impact on the ACCs is also generally neutral, but the RMSEs have increased, especially for the errors of
geopotential height and potential temperature, which need to be further investigated in the future.
4. Discussion
This study demonstrates the impact of the FY-3E MWTS-3 radiances data assimilation for the first time. In the follow-up study, a further comparison of the impacts of data assimilation between MWTS-3
and MWTS-2 will be conducted to provide a reference for the improvement of the Fengyun satellite microwave temperature sounder. For the two newly added channels (6 (53.246 ± 0.08 GHz) and 8 (53.948 ±
0.081 GHz)) of MWTS-3, further assessment and application research for their assimilation is needed. In addition, as the new generation of early-morning orbiting satellites, the FY-3E, NOAA series,
and the MetOp series polar-orbiting satellites have formed a complete three-orbit observation system. The supplementary effect of MWTS-3 is worth further exploration.
5. Conclusions
FY-3E is the fifth polar-orbiting satellite in the FY-3 series launched in July of 2021 in China, which carries the new-generation microwave temperature-sounding instruments. Compared with its
predecessor satellites, two channels capable of retrieving the CLWP have been designed for the first time, which is of great help for cloud detection in data assimilation. In addition, two detection
channels have been added with their peak weighting functions near 700 hPa and 500 hPa, and the ability to detect atmospheric temperature has been improved compared with the previous generation
After the effective QC, bias correction, and accurate error specification of the MWTS-3 data, the direct assimilation of MWTS-3 radiance data has been realized in the CMA-GFS. The near one-month
cycling experiments have indicated that the errors of analysis results can be remarkably reduced by adding the MWTS-3 data to the conventional data, especially for the variables on the upper layer of
the model, where there is a lack of sufficient conventional observations. When all the observations in operation are included, the MWTS-3 data assimilation has a neutral contribution to the forecasts
in the Northern Hemisphere and a slightly positive contribution in the Southern Hemisphere. However, in the tropics, the forecast errors of geopotential height and potential temperature have
increased after adding the MWTS-3 data, which needs further investigation.
Author Contributions
Conceptualization, J.L.; methodology, J.L.; software, J.L., X.Q. and G.L.; validation, J.L., X.Q. and G.L.; formal analysis, J.L. and X.Q.; investigation, J.L. and G.L.; data curation, G.L.;
writing—original draft preparation, J.L. and G.L.; writing—review and editing, J.L. and Z.Q. All authors have read and agreed to the published version of the manuscript.
This research was jointly funded by the Fengyun Application Pioneering Project (FY-APP-2021.0201) and FY-3 Meteorological Satellite Ground Application System Project (FY-3 (03)-AS-11.08).
Data Availability Statement
All data used in this paper are available from the authors upon request (
[email protected]
The authors would like to acknowledge the National Satellite Meteorological Centre of CMA for providing the satellite data.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Weighting Functions of FY-3E MWTS-3 calculated by RTTOV based on US standard atmosphere profile.
Figure 2. Spatial distribution of observed brightness temperature of FY-3E MWTS-3 channel 1 (a), channel 2 (b) and retrieved cloud LWP (c) for descending orbit data on 24 September 2021.
Figure 3. Spatial distribution of retrieved cloud LWP from FY-3E MWTS-3 channel 1 and 2, and brightness temperature of MERSI channel 7 during 0300–1500 UTC 24 September 2021.
Figure 4. Bias (a) and standard deviation (STD) (b) of the differences between the brightness temperature observations and ERA simulations for FY-3E MWTS-3 channels during 10–23 September 2021.
Figure 5. Scatterplots of observed (y-axis) and simulated (x-axis) brightness temperature for MWTS-3 channels 11 (a) and 14 (b) before (black dots) and after (green dots) quality control during 24–30
September 2021.
Figure 6. Frequency distributions of O-B differences for channels 11 (top) and 14 (bottom) before (hatched bars) and after (solid bars) bias correction for MWTS-3 channels 11 (upper panels) and 14
(down panels) during 24 September–3 October 2021.
Figure 7. Bias (upper panels) and STD (lower panels) of the O-B for FY-3E MWTS-3, NOAA-15/18/19, and MetOp-A/B AMSU-A channels before (a,c) and after (b,d) bias correction calculated from the
analysis results of the TEST2 experiment during 24 September–25 October 2021.
Figure 8. Bias (upper panels) and STD (lower panels) of the O-B for FY-3E MWTS-3, FY-3D MWTS-2 and NPP ATMS channels before (a,c) and after (b,d) bias correction calculated from the analysis results
of the TEST2 experiment during 24 September–25 October 2021.
Figure 9. RMS of geopotential height from the analysis field difference between CTL1 and ERA (black) and TEST1 and ERA (red) in the (a) Southern Hemisphere and (b) Northern Hemisphere from 24
September–25 October 2021. (c,d) are similar to (a,b) but for the potential temperature.
Figure 10. The daily RMS of geopotential height for the analysis field difference between CTL1 and ERA (black) and TEST1 and ERA (red) at 10 hPa in the Southern Hemisphere from 24 September–25
October 2021.
Figure 11. RMS of geopotential height for the analysis field difference between CTL2 and ERA (black) and TEST2 and ERA (red) in the (a) Southern Hemisphere and (b) Northern Hemisphere from 24
September–25 October 2021. (c,d) are similar to (a,b) but for the potential temperature.
Figure 12. RMS of U wind for the analysis field difference between CTL2 and ERA (black), TEST2 and ERA (red) in the (a,c) Southern Hemisphere and (b,d) Northern Hemisphere.
Channel Number Center Frequency (GHz) Bandwidth (MHz) Polarization NEΔT (K)
1 23.8 270 QH 0.3
2 31.4 180 QH 0.35
3 50.3 180 QV 0.35
4 51.76 400 QV 0.3
5 52.8 400 QV 0.3
6 53.246 ± 0.08 2 × 140 QV 0.35
7 53.596 ± 0.115 2 × 170 QV 0.3
8 53.948 ± 0.081 2 × 142 QV 0.35
9 54.40 400 QV 0.3
10 54.94 400 QV 0.3
11 55.50 330 QV 0.3
12 57.290 330 QV 0.6
13 57.290 ± 0.217 2 × 78 QV 0.7
14 57.290 ± 0.3222 ± 0.048 4 × 36 QV 0.8
15 57.290 ± 0.3222 ± 0.022 4 × 16 QV 1.0
16 57.290 ± 0.3222 ± 0.010 4 × 8 QV 1.2
17 57.290 ± 0.3222 ± 0.0045 4 × 3 QV 2.1
EXP Observation Data
CTL1 Conventional data
CTL2 Conventional data+ NOAA-15/18/19 AMSU-A+ NOAA-18/19 MHS+ MetOp-A/B AMSU-A/MHS/IASI+ NPP ATMS + FY-3C/D MWHS-2/MWRI + FY-3D MWTS-2/HIRAS + FY-3C/D GNOS + COSMIC RO, etc
TEST1 CTL1+FY-3E MWTS-3
TEST2 CTL2+FY-3E MWTS-3
Notes: conventional data consists of radiosondes, SYNOP, ship, Airep, and AMVs.
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Li, J.; Qian, X.; Qin, Z.; Liu, G. Direct Assimilation of Chinese FY-3E Microwave Temperature Sounder-3 Radiances in the CMA-GFS: An Initial Study. Remote Sens. 2022, 14, 5943. https://doi.org/
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Li J, Qian X, Qin Z, Liu G. Direct Assimilation of Chinese FY-3E Microwave Temperature Sounder-3 Radiances in the CMA-GFS: An Initial Study. Remote Sensing. 2022; 14(23):5943. https://doi.org/10.3390
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What is matrix triangulation?
What is matrix triangulation?
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are
zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.
What is the best way to multiply a chain of matrices?
Take the sequence of matrices and separate it into two subsequences. Find the minimum cost of multiplying out each subsequence. Add these costs together, and add in the cost of multiplying the two
result matrices.
How do you solve matrix chain multiplication problems?
For example, suppose A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × 60 matrix. Then, (AB)C = (10×30×5) + (10×5×60) = 1500 + 3000 = 4500 operations A(BC) = (30×5×60) + (10×30×60) = 9000 +
18000 = 27000 operations. Clearly the first parenthesization requires less number of operations.
What is matrix chain multiplication example?
Example of Matrix Chain Multiplication. Example: We are given the sequence {4, 10, 3, 12, 20, and 7}. The matrices have size 4 x 10, 10 x 3, 3 x 12, 12 x 20, 20 x 7. We need to compute M [i,j], 0 ≤
i, j≤ 5. We know M [i, i] = 0 for all i.
Where is matrix chain multiplication used?
Matrix Chain Multiplication is one of the optimization problem which is widely used in graph algorithms, signal processing and network industry [1–4]. We can have several ways to multiply the given
number of matrices because the matrix multiplication is associative.
Why Parenthesization is important in matrix multiply?
Matrix Chain Multiplication Problem can be stated as “find the optimal parenthesization of a chain of matrices to be multiplied such that the number of scalar multiplication is minimized”. Number of
ways for parenthesizing the matrices: There are very large numbers of ways of parenthesizing these matrices.
What is time complexity of matrix chain multiplication?
As before, if we have n matrices to multiply, it will take O(n) time to generate each of the O(n2) costs and entries in the best matrix for an overall complexity of O(n3) time at a cost of O(n2)
What is the triangular formula?
The basic formula for the area of a triangle is equal to half the product of its base and height, i.e., A = 1/2 × b × h. This formula is applicable to all types of triangles, whether it is a scalene
triangle, an isosceles triangle, or an equilateral triangle.
How do you find the triangulation of matrices?
Triangulation of matrices. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their union. If the union does not form a basis
of , then we would add some extra elements from the canonical basis of to that union to form a basis of .
Is Matrix Triangulation Gaussian elimination?
Matrix triangulation (Bareiss method) It’s also Gaussian elimination as it’s a method of successive elimination of variables, when with the help of elementary transformations the equation systems is
reduced to a row echelon (or triangular) form, in which all other variables are placed (starting from the last).
How to find a triangular matrix similar to a triangularizable matrix?
Suppose that A is some triangularizable matrix in M n ( R). The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their union.
What are the applications of matrix multiplication?
Although there are many applications of matrices, essentially, multiplication of matrices is an operation in linear algebra. The linear mapping which includes scalar addition and multiplication is
represented by matrix multiplication. One can also find a wide range of algorithms on meshes. | {"url":"https://thecrucibleonscreen.com/what-is-matrix-triangulation/","timestamp":"2024-11-03T06:37:57Z","content_type":"text/html","content_length":"54985","record_id":"<urn:uuid:5b0ee1cc-c289-4de0-bee4-03e01210430a>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00741.warc.gz"} |
Use multiplication to scale up or down speeds and sizes
Learn creative coding writing simple programs
Multiplication is something we often use in our programs, so it's important to feel comfortable with it. In programming languages, the sign used for multiplication is the asterisk: *. When we
multiply any number by 1, we get the same number: 77 * 1 = 77. When we multiply by a number larger than 1, we get a number larger than the original number: 77 * 1.2 = 92.4. When we multiply by a
positive number smaller than 1, we make it smaller: 77 * 0.8 = 61.6. Knowing these simple rules we create a program where objects move a different speeds. We also update the program we created with 5
rotating rectangles to give each rectangle a different rotation speed.
Code editor
You can make changes to the code below. Then
Questions and comments
Try to stay close to the topic of this episode. Use the Processing forums for help with unrelated Processing projects (or hire me for help ;-)
To indicate that a word in your comment is code, use the `backtick`. Example
Do `float` and `int` smell similar?
To highlight code blocks, surround it with ``` code-fences ``` like this:
``` void setup() { size(600, 600); } ``` | {"url":"https://funprogramming.org/33-Use-multiplication-to-scale-up-or-down-speeds-and-sizes.html","timestamp":"2024-11-13T18:53:42Z","content_type":"text/html","content_length":"8497","record_id":"<urn:uuid:e97d63a4-850c-41ad-9635-1166390f233a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00857.warc.gz"} |
算法代写 | COMP2550/COMP4450/COMP6445 – Advanced Computing R&D Methods - VIPDUE
算法代写 | COMP2550/COMP4450/COMP6445 – Advanced Computing R&D Methods
COMP2550/COMP4450/COMP6445 – Advanced Computing R&D Methods
Assignment 3: Theoretical Methods
Maximum marks 100 points + 5 bonus points
Submission deadline May 12, 11:59pm
No late submission unless permission is obtained from the convener.
Submission mode One PDF file via Wattle
This assignment contains questions to both Theoretical Methods I and Theoretical Methods II. The first part
gets 65 points the second gets 40 points. The maximum will be 105 of 100, meaning that you can earn up to 5
bonus points (and you can lose 5 points without actually losing them).
Theoretical Methods I
Although the main purpose on this lecture was to study and prove properties of algorithms (illustrated with
the classical planning search algorithm and heuristics), we start with some general planning and search exercises
before focusing on analyzing and proving theoretical properties of algorithms and, most importantly, heuristics.
Thus, by first doing the exercises within the block 1. Foundational Planning and Search Exercises you can
familiarize yourself with the topic and get a more intuitive understanding, before you move on to the more
challenging – and more theoretical – questions.
Do no leave out any exercises! When you need to prove something, check the required definition and try prove
or argue (as formal as possible) why it is fulfilled or why not. — Pascal
1. Foundational Planning and Search Exercises
A. Checking (Proving) Plan Executability (5 points)
a) Let P1 = (V, A, sI , g1) and P2 = (V, A, sI , g2) be two classical planning problems, which base upon
the same state variables, actions, and initial state. Only their goal descriptions are different.
Let ¯a
1 = a
, a1
, . . . , a1
n be a plan (i.e., a solution action sequence) for P1 and ¯a
2 = a
, a2
, . . . , a2
n be a
plan for P2.
Answer the following questions (1.5+1pts). If a statement is wrong provide a counter example and
briefly explain why it is a counter-example. If a statement is correct provide a proof sketch or make a
reasonable argument that shows that you understood why the respective property is true (we do not
demand anything overly formal here).
• If ¯a
1 ◦ a¯
is executable in sI , then it is also a solution for P2. (Like in the lecture, the sign ◦
represents the concatenation of actions.)
• If P1 (and thus P2) is delete-free, then a
, a2
, a1
, a2
, . . . , a1
, a2
n a solution for P3 = (V, A, sI , g1∪g2).
b) Answer the following questions (1+1.5pts). If a statement is wrong provide a counter example and
briefly explain why it is a counter-example. If a statement is correct provide a proof sketch or make a
reasonable argument that shows that you understood why the respective property is true (we do not
demand anything overly formal here).
• Let a be an action and s be a state. If a is applicable in s once, then it is also applicable twice.
Formally: If τ (a, s) = true, then τ (a, s0
) = true for s
0 = γ(a, s).
• Let a be an action and s be a state. If a is applicable in s twice in a row, then it is also applicable
thrice. Formally: If τ (aa, s) = true, then τ (aaa, s) = true.
B. Modeling a Planning Problem (5 points)
(Re-)Consider the sliding tile puzzle from the lecture:
Initial State
Goal State
As the lecture explained, the left side shows the initial configuration of the puzzle (meant to represent a
shuffled picture), and the right side shows the goal configuration. The numbers are provided so we can give
individual tiles individual names – i.e., their number. The position without a number does not have a tile
in it, this is the only empty position into which neighboring tiles could be moved in.
We also assume that every “grid position” has a name that starts from 1 and ends at 16, in the same way
as the goal configuration is shown. Thus, for example:
• In the initial state, the tile with name 9 is located at position 5.
• In both the initial state and the goal state, the empty position without any tile in it is number 16.
• In the goal state, any tile with name i is also at position i.
In this exercise you should model that problem as a classical planning problem (V, A, sI , g). To make things
easier for you, we already provide you with a partial domain model.
You can (i.e., have to!) assume that we are given the following problem components:
• V = {posi-is-free|1 ≤ i ≤ 16} ∪ {tilei-at-posj |1 ≤ i ≤ 15 and 1 ≤ j ≤ 16}
• sI = {tile2-at-pos1, tile1-at-pos2, tile4-at-pos3, tile8-at-pos4
tile9-at-pos5, tile7-at-pos6, tile11-at-pos7, tile10-at-pos8
tile6-at-pos9, tile5-at-pos10, tile15-at-pos11, tile3-at-pos12
tile13-at-pos13, tile14-at-pos14, tile12-at-pos15, pos16-is-free}
Exercises (1.5+3.5pts):
a) Specify the goal description.
b) We are also still missing the action set A. However, because there are too many special cases to
consider, we do not demand that you model the entire set. Instead, we only ask you to:
• Model all (four) actions for moving tile number 6 from position 11 to all possible, directly neighboring positions.
→ Please provide names for your actions, e.g., move-tilei-from-posj -to-posk = (pre, add, del) with …
Make sure that you covered all required preconditions and effects.
C. Proving Simple Heuristic Properties (8 points)
We will again take a look at the Sliding Tile puzzle. This time, however, we do not assume it to be modeled
as a planning problem. It is simply the sliding tile puzzle! Its formal representation is not important for
this exercise.
Instead, we only look at the three heuristics that we already took a look at in the lecture:
• The “number of misplaced tiles” heuristic.
This heuristic counts the number of misplaced tiles. Please note that it counts tiles! Not positions!
In other words, if the position (i, j) is empty in the current puzzle state, but it’s supposed to be
occupied by some tile in the goal state, then the “wrong position” (i, j) does not contribute towards
the heuristic value (as there is no tile on it).
• The “Manhatten distance” heuristic.
For each tile (again, tile!, not positions), add the horizontal plus vertical distance to its goal position.
• The “ignore some tiles” heuristic.
This heuristic ignores a certain number of tiles, both in the current puzzle and the goal configuration.
The resulting problem is solved optimally and the resulting solution cost is used as heuristic. For a
fully defined heuristic, we had of course to define which tiles had to be ignored given some puzzle.
This, however, is not important for the sake of this exercise. Just assume that for each puzzle state
you are provided with a set of tiles that’s being ignored.
Prove or disprove the following propositions. We do not expect anything formal here, but your arguments
should be convincing enough to allow for your conclusion (i.e., using only natural language is sufficient,
but don’t be too abstract). If a proposition is wrong, a proof would consist of a counter-example and an
explanation why it is a counter-example.
a) (2pts) The number of misplaced tiles heuristic h#MT is:
• goal-aware.
• safe.
• admissible.
• perfect.
b) (3 pts) The Manhatten distance heuristic hMD is:
• goal-aware.
• safe.
• admissible.
• perfect.
c) (3 pts) The ignore some tiles heuristic hIT is:
• goal-aware.
• safe.
• admissible.
• perfect. It is meant that the heuristic is perfect for any pattern. So if it is indeed perfect, show
this for any possible pattern. It is not perfect if there exists a pattern for which it is not perfect.
(So you can pick a pattern to construct a counter-example.)
D. Executing A∗
(7 points)
In this exercise you are supposed to experience the influence of heuristics’ guiding power. For this, you will
execute the classical planning progression algorithm on the simple Cranes in the Harbor domain. You will
not have to compute heuristics. Since the entire reachable state space consist of only six states we provide
the heuristic values for you. (They are not computed by an actual heuristic but just chosen more-or-less
This is the transition system:
put take put take
move move
Title: Lecture Slides for Automated Planning
Source: http://www.cs.umd.edu/ nau/planning/slides/chapter01.pdf
Licence: Attribution-NonCommercial-ShareAlike 2.0 Generic
Copyright: Dana S. Nau
Note that the transition system only mentions the action name “move”, but we differentiate between
moveLeft and moveRight to differentiate between moving to the left location and the right location. Please
be meticulous when writing down the move actions, because it is tempting to write down the “wrong
direction” (e.g., when moving from s2 to s0, the truck moves to the right, although the node s2 is on the
left of s0, so don’t get that wrong). Assume that every action has a cost of 1. Assume that the initial state
is s2 and the goal state is s5.
Below, given the heuristic provided, you are supposed to execute the progression planning algorithm from
the lecture. You do not have to show action applicability, instead you can simply use the transition system
to identify which actions that are applicable in the respective states. You also do not have to write down
the encoding of the planning states, it suffices to use the states’ names s0 to s5 (given in the graphic).
You have to follow the annotation of the lecture! I.e., you have to draw one single search tree, just like in
slide 19, which shows the A* search example of section AI search. Write to each node of the tree:
• The content of the respective search node n. For example, the root node of the next exercise a) has
to be labeled with “n = (s2, ε)”.
• The f equation for n. E.g., you have to write “f(n) = g(n) + h(n) = 0 + 2 = 2” for the root note as
mentioned above.
• A number to indicate when you expanded the search node n (“expansion” meaning when you selected
n from the search fringe to create its successors or to return the solution). The root note would
obviously be annotated by “(1)”. Then, its successor with the smallest f value would get the “(2)”,
and so on, until the solution node gets the highest number.
a) (4 pts) Assume our heuristic, h1, has the following heuristic values: h1(s0) = 1, h1(s2) = 2, h1(s3) = 2,
h1(s1) = 0, h1(s4) = 1, h1(s5) = 0. Execute the algorithm as described above.
• The algorithm from the lecture does not check for duplicates. Since the transition system contains
cycles it is likely that you will explore some states multiple times.
• There is only one correct solution. The respective search tree will have 14 search nodes.
• Again: Do not confuse moveLeft and moveRight!
b) (1.5 pts) Assume that instead of using h1 we would have used the heuristic h2, which has the same
heuristic values for s3, s4, s5 as h1, but for the others it uses h2(s0) = 4, h2(s1) = 3, h2(s2) = 3.
Can you make at least two observations about the resulting search tree (and, ideally, how it relates
to the one from the previous exercise)? (Just one sentence each, this exercise is meant to make you
think about the impact of using different heuristics, rather than testing your capabilities of proving
c) (1.5 pts) Answer and prove (do not just say “yes” or “no”) the following questions:
• Is h1 admissible?
• Is h1 goal-aware?
• Is h1 perfect?
• (Remark: We do not ask whether h1 is safe because that would not make much sense as there are
no states for which we defined h1 to be ∞.)
2. Relaxed Planning Heuristics.
A. Algorithm to solve Delete-Relaxed Planning Problems (10 points)
In the lecture we introduced delete-relaxation as a basis for heuristics. They were motivated as a domainindependent problem relaxation that makes a problem “easier” because all facts made true once
will always
remain true. It was claimed that solving a delete-relaxed problem can be done in polynomial time (which is
a significant improvement because solving “normal” planning problems (where delete lists are not empty)
takes exponential time).
You have to prove that deciding whether a delete-relaxed problem has a solution can be done in polynomial
time. To do so,
• (6 pts) provide a decision procedure (i.e. an algorithm in pseudo code similar to the classical planning
algorithm) that takes a delete-relaxed planning problem P
+ = (V, A, sI , g) as input and that returns
true if there is a solution and f alse if there is none.
• (2 pts) Prove that your algorithm runs in polynomial time.
• (2 pts) Prove (or explain) why the returned result (i.e., yes or no) is correct.
• Exploit the fact that executing an action (to progress some state to another) can never be a disadvantage for the purpose of solving a (delete-free) planning problem,
• exploit that there is no reason to ever execute an action more than once (i.e., once an action is executed
it can be ignored). (By exploiting this the right way you can show that the algorithm runs in quadratic
time (which is polynomial).)
In the lecture (recordings) we saw the proofs for:
• h
max is not perfect.
• h
max is safe.
• h
max is goal-aware.
• h
+ dominates h
Two of these proofs are based on the following fact: Let ¯a a solution plan for a planning problem P. Then,
the delete-relaxed variant of ¯a, ¯a
is a solution for the delete-relaxed variant of P, P
You can assume the correctness of this property if you need it in the following.
We will introduce two further heuristics, h
add (the add heuristic), and h
PDB (the Pattern Database Heuristic). For these, you are supposed to prove these properties as well.
B. Add Heuristic h
add (15 points)
The h
max heuristic from the lecture can be interpreted as making the hardest fact in a goal condition true
– the same applies to all the preconditions of all actions. Thus the preconditions of actions are taken into
account by only a very limited extent, as only their hardest precondition contributes to the heuristic value,
which is the reason why this heuristic is not very well informed.
The add heuristic h
add tries to compensate that. Just like h
max we define h
add based on the relaxed planning
graph. In contrast to h
max, rather than estimating an action’s cost based on it’s hardest precondition, we
do this by adding over the cost of all preconditions.
Based on the relaxed planning graph, h
add can be calculated as follows:
• Action vertex: The cost of an action vertex a ∈ Ai
is c(a) plus the sum of the predecessor vertex costs.
Mathematically, this can be expressed as follows:
add(a; s) = c(a) + h
add(pre(a); s)
• Variable vertex:
– The cost of a variable vertex v is 0 if v ∈ V
– For all v ∈ V
, i > 0, the cost of v equals the minimum cost of all predecessor vertices (these
might be either action or variable vertices).
Mathematically, this can be expressed as follows:
add(v; s) =
0 if v ∈ V
add(a; s) otherwise
Here, pred(v) stands for “predecessors”, which is the set of all actions that occur in an action layer
before the vertex layer of v and add v.
• Vertex set: For a set of state variables ¯v ⊆ V , the cost equals the sum of costs of the variables in ¯v.
Mathematically, this can be expressed as follows:
add(¯v; s) = X
add(v; s)
For a state s ∈ S, h
add(s) equals the cost of g in the rPG that we start constructing in s. Just like h
add returns ∞ if and only if there does not exist a delete-relaxed solution for g (which is equivalent to
stating that the final vertex layer does not contain all goals).
Hint: Before you answer the following questions We recommend to compute the heuristic at least once in
a small example, for instance in the Cranes in the Harbor Domain (you do already have a correct planning
graph available, so you can easily and quickly annotate the respective values).
Answer and prove the following questions. As always: If some proposition does not hold, you can prove it
by providing a counter-example (and showing/explaining that/why it is a counter-example).
a) (2 pts) Is h
add perfect?
b) (3 pts) Is h
add safe?
c) (2 pts) Is h
add goal-aware?
d) (4 pts) Is h
add admissible?
e) (4 pts) Does h
add dominate h
C. Pattern Database Heuristics h
PDB (15 points)
Patter database heuristics (PDB heuristics) are a type of heuristic obtained by dropping a a subset of
variables from the planning problem. The main idea is closely related to ignoring tiles in the sliding tile
puzzle: Just make a world easier by completely ignoring certain parts of it!
Given a planning problem P = (V, A, sI , g) and a set of variables – called a pattern – X ⊆ V , we restrict
the planning problem to the pattern, thereby effectively ignoring all other state variables.
Formally, we can define the resulting planning problem as P
X = (X, AX, sI
X, gX) with:
• AX = {a
X|a ∈ A} with
– a
X = (pre(a) ∩ X, add(a) ∩ X, del(a) ∩ X, c(a)) for a ∈ A
• sI
X = sI ∩ X
• g
X = g ∩ X
It is easy to see that the resulting planning problem P
X is just a normal planning problem again, without
any further special properties, it’s just smaller! (This is in contrast to, for example, delete-relaxation
heuristics, because there the resulting planning problem has no delete-effects; here, however, depending on
how we choose the pattern, we might still have delete effects.) The resulting problem P
X is also referred
to as “abstraction of P”.
We will ignore many details about this heuristic as they will not be important for the purpose of this
exercise. You only have to know that each “original” state s from the original problem’s search corresponds
to exactly one “relaxed” state in the abstraction. Provided the heuristic uses a pattern X ⊆ V , we obtain
the abstract state s
X that corresponds to s by simply restricting s to X, i.e., s
X := s ∩ X. So if, during
search, we encounter a state s, say {a, c, e, f}, and our heuristic uses the pattern X = {d, e, f}, then the
PDB heuristic uses the state {e, f} for its computation.
Now, completely similar to “ignoring tiles heuristic” in the Sliding Tile Puzzle, the PDB heuristic simply
uses the cost of a perfect solution to the abstract problem as heuristic for the original state. Formally:
Given the heuristic uses the pattern X, h
PDB(s) (for P) is defined as h
X) = h
(s ∩ X) in the problem
X. So, back in our example, h
PDB({a, c, e, f}) for the original problem P would be defined as h
({e, f})
in the abstracted problem P
Answer (try to prove/disprove) the following questions for PDB heuristics.
a) (2 pts) Is it perfect? Hint: It is completely obvious that this is not the case, so there is no harm in
revealing that it’s not. For proving that it is not you are allowed to choose any pattern you want (for
the counter example that you construct).
b) (2 pts) If P = (V, A, sI , g) is the planning problem you are facing, can you provide a pattern X ⊆ V
for which h
PDB will be perfect?
c) (3 pts) Is it safe?
d) (2 pts) Is it goal-aware?
e) (3 pts) Is it admissible?
f) (3 pts) Let P = (V, A, sI , g) be a planning problem and X ⊆ V and Y ⊆ V patterns. Let h
P BD
and h
P BD
Y be the respective PDB heuristics that use X and Y as patterns, respectively. Assume
X ⊇ Y . Does h
P BD
X dominate h
P BD
or the other way round? Or is there no dominance among these
heuristics? Explain (prove) your answer.
Theoretical Methods II
(see next pages)
3 Classical Propositional Logic (8pts.).
Answer questions 1.1 to 1.3 below using only rules of propositional logic below
and the lecture content. Clearly formulate and justify every argument in your
proof using proper English. If you use a property listed as one of the questions on
this assignment or listed in the lecture slides, please clearly state which question
is it and how do you use it in your argument. Let A and B be logical statements.
Propositional Logic Rules (there are more, but these set of rules are sufficient for the purpose
of writing proofs for problems in this assignment):
1. To show that A ⇒ B holds, we assume that A holds, then we deduce B.
2. If we know that both A and A ⇒ B hold, then we can deduce B.
3. If we know that A holds and ¬A holds, then we can deduce F.
4. If we assume A holds, then deduced F, therefore ¬A must hold.
5. (Proof by Contradiction) To show that A ⇒ B holds, we assume that both A and ¬B
hold and deduce F.
6. If we know that A ∧ B holds, then we can deduce A and B (or equivalently in English,
B and A).
7. If we know that A holds and B holds, we can deduce A ∧ B
8. If we know that at least one of the following:
(1) A holds,
(2) B holds,
then, we can deduce A ∨ B.
Note: Logical operators precedence order is the following ¬, ∧, ∨, and ⇒.
Question A
Show that (A ∧ B) ⇒ (B ∧ A). (2pts.)
Question B
Show that (A ⇒ B) ⇒ (¬B ⇒ ¬A). Deduce ¬A from ¬B using proof by contradiction.
Question C
Show that ¬¬A ⇒ A using proof by contradiction. (2pts.)
4 Natural Numbers (32pts.).
Using only given definitions below and the lecture content, answer the questions
below. Clearly formulate and justify every argument in your proof using proper
English. If you use a property listed as one of the questions on this assignment
or listed in the lecture slides, please clearly state which question is it and how
do you use it in your argument.
Definition 1 (Natural Numbers N). Let 0 be a term. For n = (S x), n is a term if and only
if x is a term, and we say that n is a successor of x. Let the set of Natural Numbers N and
for its element n, we write n ∈ N and we say that n is a natural number. We have n ∈ N
if and only if either n = 0 or n = (S x) if x ∈ N, and nothing else is a natural number.
Thus, the natural numbers N is the set N = {0,(S 0),(S (S 0)), . . . }, which we write as
{0, 1, 2, . . . }. J
As mentioned in the slides, we will also assume arithmetical operations ”+” and ”×”
over n ∈ N which for natural numbers a, b ∈ N, have the properties:
• closed: a + b and a × b are also natural numbers, and
• commutative: a + b = b + a and a × b = b × a.
• ”×” is distributive: (a + b) × c = (a × c) + (b × c)
Definition 2 (odd and even). For unary functions Odd and Even that accepts a natural
number n, we have:
• Even(n) = T if and only if ∃m ∈ N such that n = m × 2, and
• Odd(n) = T if and only if ∃m ∈ N such that n = (S (m × 2)).
Question A
Show by a direct proof that Odd(n) ∧ ¬Even(S n) ⇒ F. (2pts.)
For questions 2.2 to 2.7, consider the definition below.
Definition 3 (ordering). For functions Geq and Leq that take two natural numbers, we
• Geq(n, m) = T if and only if ∃k ∈ N such that n = m + k, and
• Leq(n, m) = T if and only if ∃k ∈ N such that n + k = m.
Question B
Show that for n, m ∈ N, Leq(n, m) if and only if Geq(m, n) (i.e: this is equivalent to showing
Leq(n, m) ⇒ Geq(m, n) and Leq(n, m) ⇐ Geq(m, n). (2pts.)
Question C
Show that for n, m, j ∈ N such that n 6= m 6= j, if Leq(n, m) ∧ Leq(m, p), then Leq(n, p).
Question D
Show that Geq(n, m) ∧ Leq(n, m) ⇒ n = m. (2pts.)
Question E
Show by induction that ∀n ∈ N, Geq(n, 0). (4pts.)
Question F
Show by a direct proof that ∀n ∈ N, ∃m ∈ N where m 6= n such that Leq(n, m). (2pts.)
Question G
Show that for all n ∈ N, there exist m ∈ N such that ((Odd(n) ⇒ Even(m)) ∨ (Even(n) ⇒
Odd(m)))∧Geq(m, n). Use a method of proof that you think is appropriate (i.e: induction,
direct, by contradiction, etc.). Justify your choice in a few sentences. (6pts.)
Hint: Consider the definition of ’+’, which is: a + b = c iff c = (S (S . . . S (a). . .) (i.e:
we apply S b-times to a).
Question H
Show that for all natural numbers n and m, we have Geq(n, m) ∨ Leq(n, m). (4pts.)
Hint: Again, consider the definition of ’+’, which is: a+b = c iff c = (S (S . . . S (a). . .)
(i.e: we apply S b-times to a). Also recall the definition of a natural number.
Question I
Show that ∀a, b ∈ N such that b 6= 0, there exist q, r ∈ N such that (a = (b × q) + r) ∧
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