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Maya math node summary - Rodolphe Vaillant's homepage
Cheat sheet of Maya nodes. - 09/2023 - #
Maya math nodes
A small list of useful Maya nodes, I only list some of the most useful math nodes here. You can find a list of Maya nodes in the technical documentation as well as other utility nodes here. Some
nodes are not documented, you will have to go to the node editor press 'tab' and search for yourself... Some of those undocumented nodes are listed here.
If you don't find the native Maya node you are looking for, Maya animation expressions might do the job. Expression nodes are notoriously slow but allows for quick prototyping, also keep in mind that
it is named based and any object renaming will potentially break your expressions.
A list of available functions inside an expression:
- math functions: trunc(), ceil(), floor() clamp(min, max, value), sin(), cos(), tan(), min(), max(), sqrt(), exp(), log(), sign() (and other trig functions ...)
- Random functions: gauss(), noise(), dnoise(), rand(), sphrand(), seed()
- Vector functions: angle(), cross(), dot(), mag(), rot(), unit()
- Conversion functions: deg_to_rad(), rad_to_deg(), hsv_to_rgb(), rgb_to_hsb()
- Array functions: clear(), size(), sort()
- Curve functions: linstep(), smoothstep(), hermite()(vector), hermite()(scalar)
Bifrost is a powerful node system usually known for its ability to simulate fluids, smokes and such, but that can be used to work on geometry as well, and provides all the basic and more advanced
math nodes and help you with rigging. (See also Autodesk Japan bifrost tutorials on skin collision)
Native nodes
plusMinusAverage math node specification. Originally designed with shaders in mind those operations work on vectors, i.e. addition, division etc is usually applied on a per component basis (out.x =
in1.x + in2.x), (out.y = in1.y + in2.y) etc. only supports floats
Related see: "addDoubleLinear", "substract" or "average" nodes for double attributes (works only on scalars though).
Note: when disconnecting make sure to reset the value of the attribute, otherwise it will keep adding, subtracting etc. whatever was the value at disconnection.
Output1D = \( \sum_i \) Input1D[ i ]
Output2Dx = \( \sum_i \) Input2D[ i ]x
Output2Dy = \( \sum_i \) Input2D[ i ]y
Output3Dx = \( \sum_i \) Input3D[ i ]x
Output3Dy = \( \sum_i \) Input3D[ i ]y
Output3Dz = \( \sum_i \) Input3D[ i ]z
Output1D = Input1D[ 0 ] - \( \sum_{i+1} \) Input1D[ i ]
Output2Dx = Input2Dx[ 0 ] - \( \sum_{i+1} \) Input2Dx[ i ]
Output2Dy = Input2Dy[ 0 ] - \( \sum_{i+1} \) Input2Dy[ i ]
Output3Dx = Input2Dx[ 0 ] - \( \sum_{i+1} \) Input2Dx[ i ]
Output3Dy = Input2Dy[ 0 ] - \( \sum_{i+1} \) Input2Dy[ i ]
Output3Dz = Input2Dz[ 0 ] - \( \sum_{i+1} \) Input2Dz[ i ]
Note: when disconnecting make sure to delete the array element otherwise it will average based on the current size.
N = getAttr -size plusMinusAverage.Input1D
Output1D = ( \( \sum_i \) Input1D[ i ] ) / N
N = getAttr -size plusMinusAverage.Input2D
Output2Dx = ( \( \sum_i \) Input2D[ i ]x ) / N
Output2Dy = ( \( \sum_i \) Input2D[ i ]y ) / N
N = getAttr -size plusMinusAverage.Input3D
Output3Dx = ( \( \sum_i \) Input3D[ i ]x ) / N
Output3Dy = ( \( \sum_i \) Input3D[ i ]y ) / N
Output3Dz = ( \( \sum_i \) Input3D[ i ]z ) / N
multiplyDivide math node specification. (for float attributes)
(to work with
see "divide" node or
) (only scalars though)
Output.x = Input1.x
Output.y = Input1.y
Output.z = Input1.z
Input2.xyz is ignored.
Output.x = Input1.x * Input2.x
Output.y = Input1.y * Input2.y
Output.z = Input1.z * Input2.x
Output.x = Input1.x / Input2.x
Output.y = Input1.y / Input2.y
Output.z = Input1.z / Input2.x
Output.x = \( (Input1.x) ^ { Input2.x } \)
Output.y = \( (Input1.y) ^ { Input2.y } \)
Output.z = \( (Input1.z) ^ { Input2.x } \)
condition math node specification. Compare 'firstTerm' and 'secondTerm' with some boolean operator (<,>,== etc.) OutColor is assigned ColorIfTrue or ColorIfFalse according to the result of the
if( firstTerm operation secondTerm ){
OutColor.r = ColorIfTrue.r
OutColor.g = ColorIfTrue.g
OutColor.b = ColorIfTrue.b
} else {
OutColor.r = ColorIfFalse.r
OutColor.g = ColorIfFalse.g
OutColor.b = ColorIfFalse.b
BlendColors math node specification. Linear interpolation between Color1 and Color2 given the parameter \( blender \in [0.0, 1.0]\).
Output = Color1.rgb * (1.0 - blender) + Color2.rgb * blender
Reverse math node specification. Reverse a parametric parameter: Output = 1.0 - Input.
Output.x = 1.0 - Input.x
Output.y = 1.0 - Input.y
Output.z = 1.0 - Input.z
Remaps an input scalar to an output scalar according to a user defined curve. Also outputs a color according to the position of the input scalar into the user defined color gradient.
Note: 'animCurve' node is similar but always prefer 'remapValue' which is really fast to evaluate contrary to the 'animCurve' node.
OutputValue = Curve( InputValue )
Where the function "float Curve(float)" is the curve displayed in the attribute editor and which control points are stored in the "Value[]" attribute;
OutputColor.RGB = Gradient( InputValue )
Where the function "Vec3 Gradient(float)" is the gradient strip displayed in the attribute editor and which interpolated colors are stored in the "Color[]" attribute
Not covered here:
- pointMatrixMult (matrix multiplication against vector or point) (double)
- See also (starting Maya 2024) (also double):
multiplyPointByMatrix, multiplyVectorByMatrix
- vectorProduct (dot product, cross product, vector or point matrix product) (floats)
- For doubles (starting Maya 2024):
crossProduct, dotProduct, vectorMatrix, pointMatrix
- clamp (clamp(vec3, minVec3, maxVec3))
- "min", "max" (out = minimum or maximum of a list of doubles) (undocumented)
- multDoubleLinear (scalar multiplication: out = realValue1 * realValue2) (double)
- addDoubleLinear (scalar addition: out = realValue1 + realValue2) (double)
- pointConstraint can be used to compute the barycenter of several points (outPoint3 = ∑ point_i / n)
Although not natively present you can emulate trigonometry functions (sin, cos etc)
Official list of built in matrix nodes
Multiplication order, from right to left as we increase the index:
Indices: n ... 1 0
Matrix : mat_n * ... * mat_1 * mat_0
Other nodes not covered here (see matrix utility doc): addMatrix, aimMatrix, blendMatrix, composeMatrix, fourByFourMatrix, holdMatrix, passMatrix, pickMatrix, uvPin, proximityPin
Multiplies a list of 4x4 matrices together.
\( MatrixSum = \prod_{i=n}^{0} MatrixIn[i] \)
MatrixSum = MatrixIn[n] * ... * MatrixIn[1] * MatrixIn[0]
Weighted sum of a list of 4x4 matrices.
\( MatrixSum = \sum_{i=n}^{0} WtMatrix[i].MatrixIn * WtMatrix[i].WeightIn \)
MatrixSum =
WtMatrix[n].MatrixIn * WtMatrix[n].WeightIn *
... *
WtMatrix[0].MatrixIn * WtMatrix[0].WeightIn *
WtMatrix[0].MatrixIn * WtMatrix[0].WeightIn
Takes a 4x4
matrices and outputs its rotation, translation, scale, shear components.
(From Maya 2024) For
see also: axisFromMatrix, translationFromMatrix, rotationFromMatrix, columnFromMatrix, rowFromMatrix.
Takes rotation, translation, scale, shear components and builds a 4x4 matrix.
Takes a 4x4 matrix in input and outputs a matrix with only certain components (e.g. translate, scale) of the input matrix. Which component is set to identity and which component's value will be
picked is decided by the attributes useScale, useTranslate etc.
Extra nodes in 'matrixNodes.mll' plugin:
- inverseMatrix node
- transposeMatrix node
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LightGBM in SAS Model Studio
For the 2022.10 release (October 2022) of Model Studio, the very popular LightGBM gradient boosting framework has been added to Model Studio as an available supervised learning algorithm in the
Gradient Boosting node. LightGBM is an open-source gradient boosting package developed by Microsoft, with its first release in 2016.
In Model Studio, because it is a variant of gradient boosting and shares many of its properties, the LightGBM algorithm has been integrated into the Gradient Boosting node. The following image
contains a pipeline with the Gradient Boosting node. The Perform LightGBM checkbox, which is in the node properties, enables the LightGBM algorithm. When you select Perform LightGBM, the node
displays the available LightGBM properties.
Clicking on the Run pipeline button at the top executes the Gradient Boosting node. Under the covers, the node executes the SAS LIGHTGRADBOOST procedure, which calls the lightGradBoost.lgbmTrain CAS
action to run LightGBM. This trains the LightGBM model with the options that you specified, and produces training and assessment reports in the results. When you right-click on the Gradient Boosting
node and select Results, the training reports are displayed on the Node tab. One of those reports is the Iteration History report, a line plot illustrating the change in training and validation
accuracy as the boosting iterations (number of trees) increase. Note that the right-hand pane provides an automated description to help you interpret the plot.
An additional report is the Training Code report, which contains the Proc Lightgradboost training code. You can use this as an example syntax with which to train your own LightGBM models in SAS
Clicking on the Assessment tab in the results brings up a handful of model assessment reports. These reports assess the LightGBM model against all available data partitions, including Train,
Validate, and Test, and are the standard assessment reports generated for any supervised learning node in Model Studio.
If you had selected post-training node properties to produce one or more Model Interpretability reports, these will display when you click on the Model Interpretability tab in the results. Gradient
Boosting models, while very accurate, are not very interpretable, which makes these reports very important in understanding the LightGBM model. The reports displayed here include Surrogate Variable
Importance, PD and ICE Plots (Partial Dependence and Individual Conditional Expectations), LIME Explanations (Local Interpretable Model-agnostic Explanations), and HyperSHAP Values (Shapley).
After exiting the node results, you can view and compare pipeline performance across pipelines by clicking on the Pipeline Comparison tab. Shown here are two LightGBM models that you can compare,
with a flag that identifies the champion model. You can score new data by clicking your model and selecting Score holdout data from the Project pipeline menu (three vertical dots) at the top.
You can also do a side-by-side assessment comparison by selecting both models and clicking the “Compare” at the top, which produces assessment plots that include both models. And then you can
register your model in Model Manager by selecting Register models from the Project pipeline menu. Once registered, you can maintain and track the performance of your model in Model Manager, in
addition to publishing your model for deployment (you can also publish your model from the Project pipeline menu).
Given its popularity and wide usage, providing LightGBM as an available modeling algorithm within Model Studio increases the breadth of modeling options available to Model Studio users. With the
power of Model Studio, LightGBM users will appreciate the ease with which assessment and model interpretability reports can be generated, the ease with which models can be compared, and the ease with
which models can be registered and published for deployment into production.
Below are descriptions of the LightGBM specific properties in the Gradient Boosting node, with corresponding open-source parameters in parentheses.
Basic Options
• Boosting type (boosting) – A selector to choose the type of boosting algorithm to execute.
□ Gradient boosting decision tree (gbdt) – This is the traditional gradient boosting method. Default.
□ Dropouts additive regression trees (dart) – Mutes the effect of, or drops, one or more trees from the ensemble of boosted trees. This is effective in preventing over specialization.
□ Gradient-based one-side sampling (goss) – Retains data instances with large gradients, or large training error, and down-samples instances with small gradients, or small training error.
• Number of trees (num_iterations) – The number of boosting iterations. The default value is 100.
• Learning rate (learning_rate) – The rate at which the gradient descent method converges to the minimum of the loss function. The default value is 0.1.
• Bagging frequency rate (bagging_freq) – The iteration frequency at which the training data is sampled. As an example, for a value of 5, the data is sampled before training begins, and then after
every five iterations. Sampling is enabled for a value greater than 0. The default value is 0.
• Bagging fraction rate (bagging_fraction) – The fraction of the training data that is sampled when sampling is enabled (Bagging frequency rate > 0). This option is hidden until a Bagging frequency
rate greater than 0 is entered. A value less than 1 is required. The default value is 0.5.
• L1 regularization (lambda_l1) – In a regression model, a regularization parameter (lambda) which is applied to the absolute value of the coefficient in the penalty term to the loss function. The
default value is 0.
• L2 regularization (lambda_l2) – In a regression model, a regularization parameter (lambda) which is applied to the squared value of the coefficient in the penalty term to the loss function. The
default value is 1.
• Interval target objective function (objective) – A selector to choose the objective loss function for an interval target.
□ Fair loss (fair)
□ Gamma (gamma)
□ Huber loss (huber)
□ L1 regression (MAE) (regression_l1)
□ L2 regression (MSE) (regression) – Default
□ Mean absolute percentage error (mape)
□ Poisson (poisson)
□ Quantile (quantile)
□ Tweedie (tweedie)
• Nominal target objective function (objective) – A selector to choose the objective loss function for a nominal target. For a binary target, the binary log loss function is used.
□ Multinominal logistic regression (multiclass) – Default
□ One vs. rest classification (multiclassova)
• Ensure deterministic results across job executions (deterministic) – A checkbox to enable deterministic results for the same data and parameters. Not selected by default.
• Seed (seed) – The value used to generate random numbers for data sampling. The default value is 12345.
Tree-splitting Options
• Maximum depth (max_depth) – The maximum number of generation of nodes, where generation 0 is the root node. The default value is 4.
• Minimum leaf size (min_data_in_leaf) – The minimum number of training observations in a leaf. The default value is 5.
• Use missing values (use_missing) – A checkbox to enable the handling of missing values. Selected by default.
• Number of interval bins (max_bin) – The maximum number of bins for an interval input. The default value is 50.
• Proportion of inputs to consider per tree (feature_fraction) – Proportion of inputs randomly sampled for use per tree. The default value is 1.
• Maximum class levels (max_cat_threshold) – The maximum number of levels for a class input. The default value is 128. | {"url":"https://communities.sas.com/t5/SAS-Communities-Library/LightGBM-in-SAS-Model-Studio/ta-p/850376","timestamp":"2024-11-11T16:39:02Z","content_type":"text/html","content_length":"125371","record_id":"<urn:uuid:9e64d2e9-a3cd-4374-9684-42765fbbf086>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00818.warc.gz"} |
Compute the Net present value | Accounting homework help
1. Find the present value of $985, invested for 15 years at 6.125% compound interest, compounded quarterly.
2. What is the present value of $800 to be received 8 years from now discounted back to the present at 11 percent?
3. What is the present value of $1,250 to be received 4 years from now and discounted back to the present at 18%?
4. What is the present value of $200 to be received 7 years from now discounted back to the present at 7 percent?
5. Altima plans to invest money today at an interest rate of 3% compounded annually to have $60,000 available for the purchase of a car 6 years from now. How much does the firm need to invest today?
6. Which of the following is a capital budgeting method? a. net present value b. return on assets c. inventory turnover d. return on equity | {"url":"https://www.essay-writing.com/compute-the-net-present-value-accounting-homework-help-16/","timestamp":"2024-11-14T01:47:18Z","content_type":"text/html","content_length":"63332","record_id":"<urn:uuid:35d3f15f-4cc1-4145-9878-6a982bd307bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00594.warc.gz"} |
RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.1
These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.1
Question 1.
Is zero a rational number? Can you write it P in the form \(\frac { p }{ q }\) , where p and q are integers and q ≠ 0? [NCERT]
Yes, zero is a rational number e.g.
Question 2.
Find five rational numbers between 1 and 2. [NCERT]
We know that one rational number between two numbers a and b = \(\frac { a+b }{ 2 }\)
Therefore one rational number between 1 and 2
Question 3.
Find six rational numbers between 3 and 4. [NCERT]
One rational number between 3 and 4
Question 4.
Find five rational numbers between \(\frac { 3 }{ 5 }\) and \(\frac { 4 }{ 5 }\)
Question 5.
Are the following statements true or false?
Give reason for your answer.
(i) Every whole number is a natural number. [NCERT]
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every natural number is a whole number,
(v) Every integer is a whole number.
(vi) Every rational number is a whole number.
(i) False, as 0 is not a natural number.
(ii) True.
(iii) False, as \(\frac { 1 }{ 2 }\), \(\frac { 1 }{ 3 }\) etc. are not integers.
(iv) True.
(v) False, ∵ negative natural numbers are not whole numbers.
(vi) False, ∵ proper fraction are not whole numbers
Hope given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.1 are helpful to complete your math homework.
If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you. | {"url":"https://www.learninsta.com/rd-sharma-class-9-solutions-chapter-1-number-systems-ex-1-1/","timestamp":"2024-11-08T11:39:40Z","content_type":"text/html","content_length":"57178","record_id":"<urn:uuid:7f7f19f4-847d-41a2-a1d0-34f6608ab653>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00061.warc.gz"} |
Discontinuity, Nonlinearity, and Complexity
Dimitry Volchenkov (editor), Dumitru Baleanu (editor)
Dimitry Volchenkov(editor)
Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA
Email: dr.volchenkov@gmail.com
Dumitru Baleanu (editor)
Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania
Email: dumitru.baleanu@gmail.com
Fluid Flow and Solute Transfer in a Permeable Tube with Influence of Slip Velocity
Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 153--166 | DOI:10.5890/DNC.2020.03.011
M. Varunkumar$^{1}$, P. Muthu$^{2}$
$^{1}$ Department of BS&H, GMRIT Rajam, Srikakulam-532127, India
$^{2}$ Department of Mathematics, National Institute of Technology Warangal-506004, India
Download Full Text PDF
In this paper, the influence of slip velocity on the fluid flow and solute transfer in a tube with permeable boundary is studied as a mathematical model for blood flow in glomerular capillaries. The
viscous incompressible fluid flow across the permeable tube wall, as a result of differences in both hydrostatic and osmotic pressure, is considered. The solutions of differential equations governing
the fluid flow and solute transfer are obtained using analytical and Crank-Nicolson type numerical methods. It is observed that the effect of slip on the hydrostatic and osmotic pressures, velocity
profiles, concentration profile, solute mass flux and total solute clearance is significant and the results are presented graphically.
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8. [8]  Brenner, B.M., Troy, J.L., Daugharty, T.M., and Deen, W.M. (1972), Dynamics of glomerular ultrafiltration in the rat. II. Plasma flow dependence of GFR, Am. J. Physio., 223, 1184-1190.
9. [9]  Deen, W.M., Robertson, C.R., and Brenner, B.M. (1972), A model of glomerular ultrafiltration in the rat, Americal Journal of Physiology , 223, 1178-1183.
10. [10]  Marshall, E.A. and Trowbridge, E.A. (1974), A mathematical model of the ultrafiltration process in a single glomerular capillary, Journal of Theoretical Biology, 48, 389-412.
11. [11]  Papenfuss, H.D. and Gross, J.F. (1978), Analytical study of the influence of capillary pressure drop and permeability on glomerular ultrafiltration, Microvascular Research, 16, 59-72.
12. [12]  Papenfuss, H.D. and Gross, J.F. (1987), Transcapillary exchange of fluid and plasma proteins, Biorheology, 24, 319- 335.
13. [13]  Salathe, E.P. (1988), Mathematical studies of capillary tissue exchange, Bulletin of Mathematical Biology, 50-3, 289- 311.
14. [14]  Deen, W.M., Robertson, C.R., and Brenner, B.M. (1974), Concentration polarization in an ultrafiltering capillary, BioPhysical Journal, 14, 412-431.
15. [15]  Ross, M.S. (1974), A mathematical model of mass transport in a long permeable tube with radial convection, Journal of Fluid Mechanics, 63, 157-175.
16. [16]& Tyagi, V.P. and Abbas, M. (1987), An exact analysis for a solute transport, due to simultaneous dialysis and ultrafiltration, in a hollow-fiber artificial kidney, Bulletin of
nbsp Mathematical Biology, 49-3, 697-717.
17. [17]  Chaturani, P. and Ranganatha, T.R. (1991), Flow of Newtonian fluid in non-uniform tubes with variable wall permeability with application to flow in renal tubules, Acta Mechanica, 88,
18. [18]  Beavers, G.S. and Joshep, D.D. (1967), Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, 197– 207.
19. [19]  Misra, J.C. and Shit, G.C. (2007), Role of slip velocity in blood flow through stenosed arteries: A non-Newtonian model, J. ofMech. Med. Biol., 7, 337-353.
20. [20]  Moustafa, E., (2004), Blood flow in capillary under starling hypothesis, Appl. Math. Comput., 149, 431-439.
21. [21]  Shankararaman, C., Mark, R.W., and Clint, D. (1992 Slip at uniformly porous boundary: effect on fluid flow and mass transfer, J. of Eng. Math., 26, 481-492.
22. [22]  Singh, R. and Laurence, R. L., (1979) Influence of slip velocity at a membrane surface on ultrafiltration performance - I. Channel flow system, I. J. of Heat and Mass Trans., 22,
23. [23]  Singh, R., and Laurence, R.L. (1979), Influence of slip velocity at a membrane surface on ultrafiltration performance - II. Tube flow system, I. J. of Heat and Mass Trans., 22, 731-737.
24. [24]  Apelblat, A., Katchasky, A.K., and Silberberg, A. (1974), A mathematical analysis of capillary tissue fluid exchange, Biorheology, 11, 1-49.
25. [25]  Palatt, J.P., Henry, S., and Roger, I.T. (1974), A hydrodynamical model of a permeable tubule, J. theor. Biol., 44, 287-303.
26. [26]  Pozrikidis, C. (2013), Leakage through a permeable capillary tube into a poroelastic tumor interstitium, Engg. Anal. Boun. Elements., 37, 728-737.
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30. [30]  Brenner, B.M., Baylis, C., and Deen, W.M. (1978), Transport of molecules across renal glomerular capillaries, Physiological Reviews, 56, 502-534. | {"url":"https://www.lhscientificpublishing.com/Journals/articles/DOI-10.5890-DNC.2020.03.011.aspx","timestamp":"2024-11-08T05:01:15Z","content_type":"application/xhtml+xml","content_length":"28896","record_id":"<urn:uuid:914eed12-acd0-4e08-a05a-374ebfa17cae>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00013.warc.gz"} |
Emslander and Scherer (2022)
| Number of studies (k): 47 | Effect size: Correlations | ABSTRACT: Executive functions (EFs) are key skills underlying other cognitive skills that are relevant to learning andeveryday life. Although
a plethora of evidence suggests a positive relation between the three EFsubdimensions, inhibition, shifting, and updating, and math skills for schoolchildren and adults, thefindings on the magnitude
of and possible variations in this relation are inconclusive for preschoolchildren and several narrow math skills (i.e., math intelligence). Therefore, the present meta-analysisaimed to (a)
synthesize the relation between EFs and math intelligence (an aggregate of math skills) inpreschool children; (b) examine which study, sample, and measurement characteristics moderate thisrelation;
and (c) test the joint effects of EFs on math intelligence. Utilizing data extracted from 47 studies (363 effect sizes, 30,481 participants) from 2000 to 2021, we found that, overall, EFs are
significantlyrelatedtomathintelligence (r=.34, 95% CI [.31, .37]), as are inhibition (r=.30, 95% CI [.25, .35]),shifting (r=.32, 95% CI [.25, .38]), and updating (r=.36, 95% CI [.31, .40]). Key
measurementcharacteristics of EFs, but neither children’s age nor gender, moderated this relation. Thesefindingssuggest a positive link between EFs and math intelligence in preschool children and
emphasize theimportance of measurement characteristics. We further examined the joint relations between EFs andmath intelligence via meta-analytic structural equation modeling. Evaluating different
models andrepresentations of EFs, we did notfind support for the expectation that the three EF subdimensionsare differentially related to math intelligence | {"url":"https://matthewbjane.github.io/opensynthesis/emslander-and-scherer-2022/","timestamp":"2024-11-04T11:37:08Z","content_type":"text/html","content_length":"8097","record_id":"<urn:uuid:5ec91d44-14af-4d44-b973-cc3b75887717>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00171.warc.gz"} |
natural language processing blog
When I was working on what turned into an old short paper (Markov Random Topic Fields) I decided it might be pedagogically interesting to keep a journal of what I was doing. This journal ended when I
ran out of steam and I never got back to it. My whole original idea was, after the paper got published, post everything: the journal, the code, the paper, the paper reviews, etc. It's now been 6
years and that's not going to happen, but in case anyone finds it interesting, here in the report.
Anyway, here is the report. I'm posting this so that perhaps new students can see that things don't ever work the first time, faculty still have trouble getting their code to work, etc.
The progress of a research idea
============= DAY 1 =============
* Idea
Want to do a form of topic modeling, but where there is meta
information. There are ways to do this, eg., Supervised LDA or
Dirichlet-Multinomial Regression. These both operate on a *feature*
level. For some tasks, it is more natural to operate over a graph.
Along these lines, there's Pachinko Allocation, but this posits a
graph over the vocabulary, not over the documents. (Plus, it is a
DAG, which doesn't make sense for our application.)
Question: how can we augment a standard topic model (eg., LDA), with
an underlying graph, where we assume topics vary smoothly over the
* Technology
What technology exists for statistical modeling over graphs? Sounds
like a Markov Random Field. So let's marry topic models (LDA) with
MRFs, to give a "Topical Markov Random Field" (TMRF).
We think of LDA a generating documents by first choosing a topic
mixture \theta, and then choosing topics z=k for each word w, where w
is drawn from a multinomial \beta_k.
Where can a graph fit in this? The first idea is to put an MRF over
* MRF over \theta
If we have an MRF over theta, then two issues arise. First, we almost
certainly can't collapse out theta as we might like. Okay, we'll live
with that.
Second, from an MRF perspective, what do the potential functions look
The simplest idea is to use pairwise potentials of the form e^{-dist},
where dist is the distance between two thetas that touch on the
graph. What Distance metric should we use? We could use
Bhattacharyya, Hellinger, Euclidean, LogitEuclidean, etc. Let's start
with Hellinger.
What about a variance? We could have lengths in the graph that are
either latent or known. Let's say they're latent and our potentials
have the form e^{-dist/l}, where l is the length (so that if you're
far away, distance doesn't matter.
** Getting data together
We have about 1000 docs and three graphs over those docs. We get them
in a reasonable format and then subsample about 400 of the docs. We
do this both for speed and to make sure we don't overfit the model on
the data too much.
============= DAY 2 =============
** Implementation
We implement this with an option to use or not use graphs (so we can
tell if they're helping). We collapse out \beta, but not \theta in
both cases, and we compute log posteriors.
We run first on some simple test data (testW) from HBC and find that
it seems to be doing something kind of reasonable. We then run on
some real data and it puts everything in one cluster after about 20-30
Gibbs iterations.
Debugging: First, we turn off all graph stuff (sampling lengths) and
things are still broken. Then we initialize optimally and things are
still broken. Then we turn off resampling \theta and things are still
broken. The problem is that I'm using the collapsed \beta incorrectly
when sampling z. I fix it and things work as expected (i.e., not
everything goes in the same cluster).
** Evaluating
So now the code seems to be working, so we want to evaluate. We run a
model with an without a graph (where the graph is something we expect
will help). The posterior probabilities coming out of the two
different models are all over the place.
So we do the standard trick of holding out 20% of the data as "test"
data and then evaluating log likelihood on the test. Here, we do the
"bad" thing and just use 20% of the words in each document (so that we
already have \theta for all the documents). Not great, but easy to
implement. This time, no bugs.
At this point, it's a pain to recompile for every configuration change
and we'd like to be able to run a bunch of configs simultaneously. So
we add a simple command line interface.
In order to evaluate, we plot either posteriors or held-out
likelihoods (usually the latter) as a function of iteration using
xgraph (interacts nicely with the shell and I'm used to it).
Things now seem mixed. There's very little difference when you're not
using a graph between sampling \theta from the true posterior versus
using an MH proposal (this is good for us, since we have to use MH).
There is also little difference between the baseline LDA model and the
MRF models.
We turn off sampling of the lengths and just fix them at one. For the
three graphs we're trying, only one seems to be doing any better than
the baseline LDA model.
** Optimization
Now that we're running experiments, we find that things are taking way
too long to run. So we do some optimization. First, we cache the sum
of all \beta_k posteriors. This helps. Second, we flip \beta from
\beta_{k,w} to \beta_{w,k}, which we've heard helps. It doesn't. We
put it back the other way.
All the time is being spent in resample_z, so we waste a half day
trying to figure out if there's a way to only resample a subset of the
zs. For instance, track how often they change and only resample those
that change a lot (probabilistically). This hurts. Resampling those
with high entropy hurts. I think the issue is that there are three
types of zs. (1) those that change a lot because they have high
uncertainty but are rare enough that they don't really matter, (2)
those that change a lot and do matter, (3) those that just don't
change very much. Probably could do something intelligent, but have
wasted too much time already.
In order to really evaluate speed, we add some code that prints out
We do one more optimization that's maybe not very common. resample_z
loops over docs, then words, then topics. For each word, the loop
over topics is to compute p(z=k). But if the next word you loop over
is the same word (they are both "the"), then you don't need to
recompute all the p(z=k)s -- you can cache them. We do this, and then
sort the words. This gives about a 20% speedup with no loss in
performance (posterior or held-out likelihood).
** Evaluating again
Since we had some success fixing the lengths at 1, we try fixing them
at 20. Now that one graph is doing noticably better than the baseline
and the other two slightly better. We try 5 and 10 and 50 and nothing
much seems to happen. 20 seems like a bit of a sweet spot.
============= DAY 3 =============
** A more rigorous evaluation
Running with lengths fixed at 20 seems to work, but there's always
variance due to randomness (both in the sampling and in the 80/20
split) that we'd like to account for.
So we run 8 copies of each of the four models (8 just because we
happen to have an 8 core machine, so we can run them simultaneously).
Now, we require more complex graphing technology than just xgraph.
We'll probably eventually want to graph things in matlab, but for now
all we really care about it how things do over time. So we write a
small perl script to extract scores every 50 iterations (we've
switched from 200 to 300 just to be safe) and show means and stddevs
for each of the models.
While we're waiting for this to run, we think about...
* MRF over z?
Our initial model which may or may not be doing so well (we're waiting
on some experiments) assumes an MRF over \th. Maybe this is not the
best place to put it. Can we put it over z instead?
Why would we want to do this? There are some technological reasons:
(1) this makes the MRF discrete and we know much better how to deal
with discrete MRFs. (2) we can get rid of the MH step (though this
doesn't seem to be screwing us up much). (3) we no longer have the
arbitrary choice of which distance function to use. There are also
some technological reasons *not* to do it: it seems like it would be
computationally much more burdensome.
But, let's think if this makes sense in the context of our
application. We have a bunch of research papers and our graphs are
authorship, citations, time or venue. These really do feel like
graphs over *documents* not *words*.
We could turn them in to graphs over words by, say, connecting all
identical terms across documents, encouraging them to share the same
topic. This could probably be done efficiently by storing an inverted
index. On the other hand, would this really capture much? My gut
tells me that for a given word "foo", it's probably pretty rare that
"foo" is assigned to different topics in different documents. (Or at
least just as rare as it being assigned to different topics in the
same document.) Note that we could evaluate this: run simple LDA, and
compute the fraction of times a word is assigned it's non-majority
topic across all the data, versus just across one documents. I
suspect the numbers would be pretty similar.
The extreme alternative would be to link all words, but this is just
going to be computationally infeasible. Moreover, this begins to look
just like tying the \thetas.
So for now, we put this idea on the back burner...
* MRF over \theta, continued...
We're still waiting for these experiments to run (they're about half
of the way there). In thinking about the graph over z, though, it did
occur to me that maybe you have to use far more topics than I've been
using to really reap any benefits here. We begin running with 8, and
then bumped it up to 20. But maybe we really need to run with a lot
So, I log in to two other machines and run just one copy with 50, 100,
150 and 200 topics, just for vanilla LDA. The larger ones will be
slow, but we'll just have to go do something else for a while...
============= DAY 4 =============
* MRF over \theta, re-continued...
Both experiments finish and we see that: (1) with 20 topics and
lengths fixed at 10, there is no difference between raw LDA and TMRF.
(2) More topics is better. Even 200 topics doesn't seem to be
overfitting. Going 20, 50, 100, 150, 200, we get hidden log
likelihoods of -1.43, -1.40, -1.38, -1.36, -1.36 (all e+06). The
significance (from the first experiments) seems to be around .002
(e+06), so these changes (even the last, which differs by 0.005) seem
to be real. Since we weren't able to overfit, we also run with 300
and 400, and wait some more...
...and they finish and still aren't overfitting. We get -1.35 and
-1.35 respectively (differing by about 0.004, again significant!).
This is getting ridiculous -- is there a bug somewhere? Everything
we've seen in LDA-like papers shows that you overfit when you have a
ton of topics. Maybe this is because our documents are really long?
** Model debugging
One thing that could be going wrong is that when we hide 20% of the
words, and evaluate on that 20%, we're skewing the evaluation to favor
long documents. But long documents are probably precisely those that
need/want lots of topics. Our mean document length is 2300, but the
std is over 2500. The shortest document has 349 words, the longest
has 37120. So, instead of hiding 20%, we try hiding a fixed number,
which is 20% of the mean, or 460.
============= DAY 5 =============
** Read the papers, stupid!
At this point, we've done a large number of evals, both with 20% hid,
and 460 words/doc hid (actually, the min of this and doclen/2), and
50-1600 (at *2) topics. We do actually see a tiny bit of overfitting
at 800 or 1600 documents.
Then we do something we should have done a long time ago: go back and
skim through some LDA papers. We look at the BNJ 2003 JMLR paper. We
see that one of the selling points of LDA over LSI is that
it *doesn't* overfit! Aaargh! No wonder we haven't been able to get
substantial overfitting.
However, we also notice something else: aside from dropping 50 stop
words (we've been dropping 100), on one data set they don't prune rare
words at all, and on the other they prune df=1 words only. We've been
pruning df<=5 or <=10 words (can't remember which). Perhaps what's
going on is that the reason the graphs aren't helping is just because
there vocabulary (around 3k) isn't big enough for them to make a
We recreate the text, pruning only the df=1 words. This leads to a
vocab of about 10k (which means inference will be ~3 times slower).
We run at 50, 100, 200 and 400 and we actualy see a tiny bit of
overfitting (maybe) on 400. We accidentally only ran 100 iterations,
so it's a bit hard to tell, but at the very least there's
no *improvement* for going from 200 topics to 400 topics. Strangely
(and I'd have to think about this more before I understand it),
running on the new text is actually about 5-20% *faster*, despite the
larger vocabulary!
** Re-running with graphs
At this point, we're ready to try running with graphs again. Despite
the fact that it's slow, we settle on 200 topics (this took about 3.5
hours without graphs, so we will be waiting a while). We also want to
run for more iterations, just to see what's going to happen: we do 200
And again there's not much difference. One of the graphs seems to be
just barely one std above everyone else, but that's nothing to write
home about.
============= DAY 6 =============
* Abstracts only?
At this point, things are not looking so spectacular. Perhaps the
problem is that the documents themselves are so big that there's
really not much uncertainty. This is reflected, to some degree, by
the lack of variance in the predictive perplexities.
So we rebuild the data on abstracts only. This makes running
significantly faster (yay). We run 5 copies each of 10, 20, 40, 80,
160 and 320 topics. 40 is a clear winner. 80 and above overfit
fairly badly.
Now, we turn on graphs and get the following results (5 runs):
40-top-nog.default -69239.4 (86.2629700392932)
40-top-nog.auth -68920.0 (111.863756418243)
40-top-nog.cite -68976.4 (174.920839238783)
40-top-nog.year -69174.2 (133.769204228776)
If we compare default (not graphs) to auth (best), we see that we get
a 2-3 std separation. This is looking promising, FINALLY! Also, if
we plot the results, it looks like auth and cite really dominate.
Year is fairly useless.
It suggests that, maybe, we just need more data to see more of a
* Getting More Data
There are two ways we could get more data. First, we could crawl
more. Second, we could switch over to, say, PubMed or ACM. This
would work since we only need abstracts, not full texts. These sites
have nice interfaces, so we start downloading from ACM.
============= DAY 7 =============
Okay, ACM is a pain. And it's really not that complete, so we switch
over to CiteSeer (don't know why we didn't think of this before!). We
seed with docs from acl, cl, emnlp, icml, jmlr, nips and uai. We
notice that CiteSeer is apparently using some crappy PDF extractor (it
misses ligatures! ugh!) but it's not worth (yet!) actually
downloading the pdfs to do the extraction ourselves ala Braque. From
these seeds, we run 10 iterations of reference crawling, eventually
ending up with just over 44k documents. We extract a subset
comprising 9277 abstracts, and six graphs: author, booktitle,
citation, url, year and time (where you connect to year-1 and year+1).
The 9k out of 44k are those that (a) have >=100 characters
"reasonable" in the abstract and (b) have connections to at least 5
other docs in *all* the graphs. (We're no longer favoring citations.)
We then begin the runs again....
The question of how "traditional conference publication" should react to arxiv prepublications is raised quite frequently. I'm not particularly shy about the fact that I'm not a fan, but that's not
what this post is about. This post is about data.
In any discussion about the "arxiv question," proponents of the idea typically cite the idea that by posting papers early on arxiv, they are able to get feedback from the community about their work.
(See for example here, which at least tries to be balanced even if the phrasing is totally biased, for instance in the poll at the end :P.)
At any rate, the question I was curious about is: is this actually borne out in practice?
I did the following experiment. Arxiv nicely lets us see revision histories. So we can see, for instance, whether a paper that was placed on arxiv before notifications for the corresponding
conference have gone out, is updated more than a paper that was placed on arxiv after notifications.
For NIPS papers that were first posted to arxiv after camera ready, 75% were never updated and 19% were updated once (on average, they were updated 0.36 times +- 0.713 std).
For papers that were first posted to arxiv before notification, all were updated at least once. The real question is: how many times were they updated between posting to arxiv and acceptance to the
conference. The answer is that 82% were never updated during that period. Of course, all were updated at some point later (after the camera ready deadline), and 55% were updated only once, and
another 18% were updated twice.
[Note: I only count updated that come at least two week after the first posting to arxiv because before is more likely to be typo fixing, rather than real feedback from the community.]
The sample size is small enough that I can actually look at all of the ones that were updated between posting to arxiv and notification. One of these seems was first posted in mid-Feb, updated twice
is late March, and then again in Nov (acceptance) and Dec (camera ready). Another is very similar. Two were most likely posted the previous year when it was submitted to AIStats (the dates match up)
and then updated when submitted to NIPS. Those were the only four, and two of them seem like a legit possible case of update due to community feedback.
As far as the question of "rapid impact on the field" this is harder to answer. I took a random sample of ten papers from each of the groups (prepub versus non-prepub) and got citation counts from
google scholar. The median citation count was 10 for both sets. The average was slightly higher for the prepub set (15 versus 11, but with giant standard deviations of 12 and 16). Considering the
prepub set has been out at least 6 months longer (this is NIPS 2013 and 2014 so this is a sizeable percentage), this is a pretty small difference. And it's a difference that might be attributable to
other factors like "famous people are perhaps more likely to prepub" [actually it's not clear the data play this out; in a totally unscientific study of "does Hal think this person is famous" and a
sample of 20 for each, it's even split 10/10 in both sets].
Anyway, I'm posting this because I've heard this argument many times and I've always felt it's a bit dubious. I've never seen data to back it up. This data suggests it's not true. If someone really
believes this argument, it would be nice to see it backed up with data!
[Notes: I took only papers that were marked on arxiv as having appeared in NIPS, and which were first posted to arxiv in 2013 or 2014; this is 175 papers. I then hand-checked them all to exclude
things like workshops or just submissions, and labeled them as to whether they appeared in 2013 or 2014. That left a sample of papers. The rest of the data was extracted automatically from the arxiv
abstract. The total number that was posted before notification (the prepub cases) is 22 (or 27%) and the total number that were posted after notification is 59 (or 73%). So the sample is indeed
small. Not much I can do about that.] | {"url":"https://nlpers.blogspot.com/2015/10/?m=0","timestamp":"2024-11-09T02:43:04Z","content_type":"application/xhtml+xml","content_length":"121168","record_id":"<urn:uuid:c28ca5cd-987c-4823-9c6e-f2171dd9d8c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00570.warc.gz"} |
Karel Devriendt
About me
Hello! I am an applied mathematician working at the intersection of discrete mathematics, geometry and (linear) algebra with a strong interest in applications. Currently, I am a postdoc with Renaud
Lambiotte at Oxford. Previously, I was a postdoc with Bernd Sturmfels , Jürgen Jost and Raffaella Mulas at the Max Planck Institute in Leipzig and obtained my PhD at Oxford.
My research revolves around graphs and their applications. Over the last few years, I have focused on the concept of effective resistance and how it captures the geometry of graphs. Currently, I am
interested in discrete curvature and discrete geometry and related questions on matroids, tropical geometry and algebraic statistics.
I have worked on applications such as power grid robustness, network epidemics and polarization in social networks.
News & Travels
• [new!] preprint: "Graphs with nonnegative resistance curvature" (link)
• My paper "Kemeny's constant and the Lemoine point of a simplex" (link) was accepted for publication in the Electronic Journal of Linear Algebra
• New preprint with Eric Boniface & Serkan Hoşten: "Tropical toric maximum likelihood estimation" (link) | {"url":"https://sites.google.com/view/kareldevriendt/home","timestamp":"2024-11-08T15:17:56Z","content_type":"text/html","content_length":"82908","record_id":"<urn:uuid:5193ae0f-628d-40cf-970a-83499ce8d8f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00619.warc.gz"} |
IM 3 - M1-T3-L6
Module 1 - Topic 3 - Lesson 5 - Analyzing Polynomial Functions
Essential Question(s):
How does context change your interpretation of a polynomial function?
Follow the steps to complete your notes and review the content.
STEP 1: Preparation
Title your spiral with the heading above, copy the essential question(s), and draw your border line for the Cornell notes.
STEP 2: Textbook
Answer the follow questions by using your workbook. Read more than enough to ensure a complete answer to the question.
Following a Cornell notes format, the questions should be written in the left-hand column and with the answers to the right of the question.
Make sure to write enough to answer the question.
Getting Started/Activity 5.1/Activity 5.2
1. What does multiplicity tell you about the behavior of a polynomial graph?
2. What is the difference between zeros and x-intercepts?
3. What does the a-value of a polynomial function tell you about its graph?
4. What conclusions can you make about a polynomial function if you know the degree of a polynomial function?
5. What is average rate of change?
STEP 3: Self-check
Perform a self-check after the lesson is completed in class by asking yourself the following question:
"Can you answer the essential question(s) completely?"
• Yes? Awesome job! You took effective notes, and paid attention. You are on your way to success! :D
• No? Ok. We all have struggles. Determine why you said no, revise your notes and self-check again. Do not get discouraged. | {"url":"https://www.msstevensonmath.com/im-3---m1-t3-l6.html","timestamp":"2024-11-09T03:53:41Z","content_type":"text/html","content_length":"88100","record_id":"<urn:uuid:6bc6fa7f-798f-4cff-ac29-42604e841c54>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00193.warc.gz"} |
On the stochastic behaviour of the run length of EWMA control schemes for the mean of correlated output in the presence of shifts in sigma
Morais, M. C. ; Okhrin, Y.; Pacheco, António; Schmid, W.
Statistics & Decisions, 24 (2006), 1001-1018
This paper discusses in detail the impact of shifts on the process variance (sigma^2) on the run length (RL) of modified upper one-sided EWMA charts for the process mean (mu) when the output is
Quite apart from the relevance of a process variance change in its own right, a dilation in sigma^2 can cause an undesirable stochastic decrease in the detection speed of some specific shifts in mu.
This and other stochastic results are proved and illustrated with a few examples. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=5&member_id=90&doc_id=1489","timestamp":"2024-11-05T09:11:28Z","content_type":"text/html","content_length":"8655","record_id":"<urn:uuid:1e1ca1f4-28a3-4e4c-8ec1-bb55b8c324cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00539.warc.gz"} |
Orthogonal partitions in designed experiments
A survey is given of the statistical theory of orthogonal partitions on a finite set. Orthogonality, closure under suprema, and one trivial partition give an orthogonal decomposition of the
corresponding vector space into subspaces indexed by the partitions. These conditions plus uniformity, closure under infima and the other trivial partition give association schemes. Examples covered
by the theory include Latin squares, orthogonal arrays, semilattices of subgroups, and partitions defined by the ancestral subsets of a partially ordered set (the poset block structures).
Isomorphism, equivalence and duality are discussed, and the automorphism groups given in some cases. Finally, the ideas are illustrated by some examples of real experiments.
• association scheme
• block structure
• crossing
• infimum
• nesting
• orthogonality
• partition
• poset
• supremum
Dive into the research topics of 'Orthogonal partitions in designed experiments'. Together they form a unique fingerprint. | {"url":"https://research-portal.st-andrews.ac.uk/en/publications/orthogonal-partitions-in-designed-experiments","timestamp":"2024-11-06T08:36:04Z","content_type":"text/html","content_length":"51720","record_id":"<urn:uuid:054d9d89-96d0-4d82-927b-a80045798e25>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00579.warc.gz"} |
CSE 245: Computer Aided Circuit Simulation and Verification
Title: CSE 245: Computer Aided Circuit Simulation and Verification 1
CSE 245 Computer Aided Circuit Simulation and
Fall 2004, Oct 19 Lecture 7 Matrix Solver II
-Iterative Method
• Iterative Method
• Stationary Iterative Method (SOR, GS,Jacob)
• Krylov Method (CG, GMRES)
• Multigrid Method
Iterative Methods
• Stationary
• x(k1) Gx(k)c
• where G and c do not depend on iteration count
• Non Stationary
• x(k1) x(k)akp(k)
• where computation involves information that
change at each iteration
Stationary Jacobi Method
• In the i-th equation solve for the value of xi
while assuming the other entries of x remain
• In matrix terms the method becomes
• where D, -L and -U represent the diagonal, the
strictly lower-trg and strictly upper-trg parts
of M
• MD-L-U
• Like Jacobi, but now assume that previously
computed results are used as soon as they are
• In matrix terms the method becomes
• where D, -L and -U represent the diagonal, the
strictly lower-trg and strictly upper-trg parts
of M
• MD-L-U
Stationary Successive Overrelaxation (SOR)
• Devised by extrapolation applied to Gauss-Seidel
in the form of weighted average
• In matrix terms the method becomes
• where D, -L and -U represent the diagonal, the
strictly lower-trg and strictly upper-trg parts
of M
• MD-L-U
• Choose w to accelerate the convergence
• W 1 Jacobi / Gauss-Seidel
• 2gtWgt1 Over-Relaxation
• W lt 1 Under-Relaxation
Convergence of Stationary Method
• Linear Equation MXb
• A sufficient condition for convergence of the
solution(GS,Jacob) is that the matrix M is
diagonally dominant.
• If M is symmetric positive definite, SOR
converges for any w (0ltwlt2)
• A necessary and sufficient condition for the
convergence is the magnitude of the largest
eigenvalue of the matrix G is smaller than 1
• Jacobi
• Gauss-Seidel
• SOR
• Iterative Method
• Stationary Iterative Method (SOR, GS,Jacob)
• Krylov Method (CG, GMRES)
• Steepest Descent
• Conjugate Gradient
• Preconditioning
• Multigrid Method
Linear Equation an optimization problem
• Quadratic function of vector x
• Matrix A is positive-definite, if
for any nonzero vector x
• If A is symmetric, positive-definite, f(x) is
minimized by the solution
Linear Equation an optimization problem
• Quadratic function
• Derivative
• If A is symmetric
• If A is positive-definite
• is minimized by setting to 0
For symmetric positive definite matrix A
Gradient of quadratic form
The points in the direction of steepest increase
of f(x)
Symmetric Positive-Definite Matrix A
• If A is symmetric positive definite
• P is the arbitrary point
• X is the solution point
We have,
If p ! x
If A is not positive definite
• Positive definite matrix b) negative-definite
• c) Singular matrix d) positive
indefinite matrix
Non-stationary Iterative Method
• State from initial guess x0, adjust it until
close enough to the exact solution
• How to choose direction and step size?
Adjustment Direction
Step Size
Steepest Descent Method (1)
• Choose the direction in which f decrease most
quickly the direction opposite of
• Which is also the direction of residue
Steepest Descent Method (2)
• How to choose step size ?
• Line Search
• should minimize f, along the direction of
, which means
Steepest Descent Algorithm
Given x0, iterate until residue is smaller than
error tolerance
Steepest Descent Method example
• Starting at (-2,-2) take the
• direction of steepest descent of f
• b) Find the point on the intersec-
• tion of these two surfaces that
• minimize f
• c) Intersection of surfaces.
• d) The gradient at the bottommost
• point is orthogonal to the gradient
• of the previous step
Iterations of Steepest Descent Method
Convergence of Steepest Descent-1
Energy norm
Convergence of Steepest Descent-2
Convergence Study (n2)
Spectral condition number
Plot of w
Case Study
Bound of Convergence
It can be proved that it is also valid for ngt2,
Conjugate Gradient Method
• Steepest Descent
• Repeat search direction
• Why take exact one step for each direction?
Search direction of Steepest descent method
Orthogonal Direction
Pick orthogonal search direction
Orthogonal ? A-orthogonal
• Instead of orthogonal search direction, we make
search direction A orthogonal (conjugate)
Search Step Size
Iteration finish in n steps
Initial error
The error component at direction dj is eliminated
at step j. After n steps, all errors are
Conjugate Search Direction
• How to construct A-orthogonal search directions,
given a set of n linear independent vectors.
• Since the residue vector in
steepest descent method is orthogonal, a good
candidate to start with
Construct Search Direction -1
• In Steepest Descent Method
• New residue is just a linear combination of
previous residue and
• Let
We have
Krylov SubSpace repeatedly applying a matrix to
a vector
Construct Search Direction -2
For i gt 0
Construct Search Direction -3
• can get next direction from the previous one,
without saving them all.
Conjugate Gradient Algorithm
Given x0, iterate until residue is smaller than
error tolerance
Conjugate gradient Convergence
• In exact arithmetic, CG converges in n steps
(completely unrealistic!!)
• Accuracy after k steps of CG is related to
• consider polynomials of degree k that are equal
to 1 at 0.
• how small can such a polynomial be at all the
eigenvalues of A?
• Thus, eigenvalues close together are good.
• Condition number ?(A) A2 A-12
?max(A) / ?min(A)
• Residual is reduced by a constant factor by
O(?1/2(A)) iterations of CG.
Other Krylov subspace methods
• Nonsymmetric linear systems
• GMRES for i 1, 2, 3, . . . find xi ? Ki
(A, b) such that ri (Axi b) ? Ki (A,
b)But, no short recurrence gt save old vectors
gt lots more space (Usually restarted every k
iterations to use less space.)
• BiCGStab, QMR, etc.Two spaces Ki (A, b) and Ki
(AT, b) w/ mutually orthogonal bases Short
recurrences gt O(n) space, but less robust
• Convergence and preconditioning more delicate
than CG
• Active area of current research
• Eigenvalues Lanczos (symmetric), Arnoldi
• Suppose you had a matrix B such that
• condition number ?(B-1A) is small
• By z is easy to solve
• Then you could solve (B-1A)x B-1b instead of Ax
• B A is great for (1), not for (2)
• B I is great for (2), not for (1)
• Domain-specific approximations sometimes work
• B diagonal of A sometimes works
• Better blend in some direct-methods ideas. . .
Preconditioned conjugate gradient iteration
x0 0, r0 b, d0 B-1 r0, y0
B-1 r0 for k 1, 2, 3, . . . ak
(yTk-1rk-1) / (dTk-1Adk-1) step length xk
xk-1 ak dk-1 approx
solution rk rk-1 ak Adk-1
residual yk B-1 rk
solve ßk (yTk rk) / (yTk-1rk-1)
improvement dk yk ßk dk-1
search direction
• One matrix-vector multiplication per iteration
• One solve with preconditioner per iteration
• Iterative Method
• Stationary Iterative Method (SOR, GS,Jacob)
• Krylov Method (CG, GMRES)
• Multigrid Method
What is the multigrid
• A multilevel iterative method to solve
• Axb
• Originated in PDEs on geometric grids
• Expend the multigrid idea to unstructured problem
Algebraic MG
• Geometric multigrid for presenting the basic
ideas of the multigrid method.
The model problem
Ax b
Simple iterative method
• x(0) -gt x(1) -gt -gt x(k)
• Jacobi iteration
• Matrix form x(k) Rjx(k-1) Cj
• General form x(k) Rx(k-1) C (1)
• Stationary x Rx C (2)
Error and Convergence
• Definition error e x - x (3)
• residual r b Ax (4)
• e, r relation Ae r (5) ((3)(4))
• e(1) x-x(1) Rx C Rx(0) C Re(0)
• Error equation e(k) Rke(0) (6) ((1)(2)(3))
• Convergence
Error of diffenent frequency
• Wavenumber k and frequency ?
• k?/n
• High frequency error is more oscillatory between
Iteration reduce low frequency error efficiently
• Smoothing iteration reduce high frequency error
efficiently, but not low frequency error
k 1
k 2
k 4
Multigrid a first glance
• Two levels coarse and fine grid
Idea 1 the V-cycle iteration
• Also called the nested iteration
Start with
A2hx2h b2h
A2hx2h b2h
Iterate gt
Prolongation ?
Restriction ?
Ahxh bh
Iterate to get
Question 1 Why we need the coarse grid ?
• Prolongation (interpolation) operator
• xh x2h
• Restriction operator
• xh x2h
• The basic iterations in each level
• In ?ph xphold ? xphnew
• Iteration reduces the error, makes the error
smooth geometrically.
• So the iteration is called smoothing.
Why multilevel ?
• Coarse lever iteration is cheap.
• More than this
• Coarse level smoothing reduces the error more
efficiently than fine level in some way .
• Why ? ( Question 2 )
Error restriction
• Map error to coarse grid will make the error more
K 4, ? ?
K 4, ? ?/2
Idea 2 Residual correction
• Known current solution x
• Solve Axb eq. to
• MG do NOT map x directly between levels
• Map residual equation to coarse level
• Calculate rh
• b2h Ih2h rh ( Restriction )
• eh Ih2h x2h ( Prolongation )
• xh xh eh
Why residual correction ?
• Error is smooth at fine level, but the actual
solution may not be.
• Prolongation results in a smooth error in fine
level, which is suppose to be a good evaluation
of the fine level error.
• If the solution is not smooth in fine level,
prolongation will introduce more high frequency
Revised V-cycle with idea 2
?2h ?h
• Smoothing on xh
• Calculate rh
• b2h Ih2h rh
• Smoothing on x2h
• eh Ih2h x2h
• Correct xh xh eh
What is A2h
Going to multilevels
• V-cycle and W-cycle
• Full Multigrid V-cycle
h 2h 4h
h 2h 4h 8h
Performance of Multigrid
Gaussian elimination O(N2)
Jacobi iteration O(N2log?)
Gauss-Seidel O(N2log?)
SOR O(N3/2log?)
Conjugate gradient O(N3/2log?)
Multigrid ( iterative ) O(Nlog?)
Multigrid ( FMG ) O(N)
Summary of MG ideas
• Three important ideas of MG
• Nested iteration
• Residual correction
• Elimination of error
• high frequency fine grid
• low frequency coarse grid
AMG for unstructured grids
• Axb, no regular grid structure
• Fine grid defined from A
Three questions for AMG
• How to choose coarse grid
• How to define the smoothness of errors
• How are interpolation and prolongation done
How to choose coarse grid
• Idea
• C/F splitting
• As few coarse grid point as possible
• For each F-node, at least one of its neighbor is
a C-node
• Choose node with strong coupling to other nodes
as C-node
How to define the smoothness of error
• AMG fundamental concept
• Smooth error small residuals
• r ltlt e
How are Prolongation and Restriction done
• Prolongation is based on smooth error and strong
• Common practice I
AMG Prolongation (2)
AMG Prolongation (3)
• Multigrid is a multilevel iterative method.
• Advantage scalable
• If no geometrical grid is available, try
Algebraic multigrid method
The landscape of Solvers
More Robust
Less Storage (if sparse)
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What is the Bivariate Analysis? | Data Basecamp
In the realm of data analysis, understanding the relationships between variables is often the key to unraveling valuable insights. Welcome to the world of bivariate analysis, where the focus shifts
from individual variables to the dynamic interplay between pairs of them. This analytical technique is a cornerstone of statistical inquiry, allowing us to dig deeper into data, draw connections, and
make informed decisions.
Imagine you’re investigating the impact of advertising spending on product sales or exploring how student performance relates to study hours. Bivariate analysis is your trusty compass, guiding you
through the intricate terrain of associations, dependencies, and correlations.
In this article, we embark on a journey into the heart of bivariate analysis. We’ll delve into its fundamental concepts, practical methodologies, and real-world applications. Whether you’re a data
enthusiast, a researcher, or a business professional seeking to uncover the secrets hidden within your datasets, join us as we navigate the world of bivariate analysis and harness the power of two
variables working in concert.
What is Bivariate Analysis?
Bivariate analysis is a statistical method used in data analysis and research to examine the relationship between two variables. Unlike univariate analysis, which focuses on a single variable at a
time, bivariate analysis explores how two different variables interact or influence each other.
At its core, bivariate analysis seeks to answer questions such as:
• Does a change in one variable correspond to a change in another variable?
• Are the two variables related in some way?
• Can one variable be predicted or explained based on the values of the other variable?
This analytical approach is particularly valuable in various fields, including economics, social sciences, healthcare, and marketing, where understanding the connections between variables is crucial
for making informed decisions.
Bivariate analysis can take on many forms, depending on the types of variables being studied. Some common scenarios include:
1. Continuous vs. Continuous: When both variables under investigation are continuous (numeric), researchers often use techniques such as correlation analysis or scatterplots to visualize and measure
the strength and direction of the relationship.
2. Categorical vs. Categorical: In cases where both variables are categorical (non-numeric), methods like chi-square tests of independence help determine if there is a significant association
between the two variables.
3. Continuous vs. Categorical: When one variable is continuous and the other is categorical, techniques like t-tests or analysis of variance (ANOVA) are employed to compare means across different
categories or groups.
4. Time-Series Bivariate Analysis: In time-series data, researchers may investigate how changes in one variable impact another variable over time. This can involve techniques like cross-correlation
or Granger causality tests.
Bivariate analysis serves as a fundamental building block for more complex multivariate analyses and modeling. It allows researchers and analysts to gain valuable insights into the dependencies and
interactions between variables, paving the way for better-informed decision-making and a deeper understanding of underlying patterns in data.
What are the different types of Bivariate Analysis?
Bivariate analysis encompasses a variety of techniques, each designed to explore the relationship between two different variables. The choice of which method to use depends on the types of variables
you are working with and the research questions you aim to answer. Here are some common types of bivariate analysis:
1. Correlation Analysis: This is one of the most widely used techniques for examining the relationship between two continuous variables. The Pearson correlation coefficient measures the strength and
direction of a linear relationship. A value close to 1 indicates a strong positive correlation, close to -1 suggests a strong negative correlation, and close to 0 implies little to no
2. Scatterplots: Scatterplots are graphical representations of data points in a two-dimensional space. They are particularly useful for visualizing the relationship between two continuous variables.
The pattern of points on the scatterplot can provide insights into the nature of the relationship.
3. Covariance: Covariance is a statistical measure that assesses how two continuous variables change together. It indicates the direction of the linear relationship (positive or negative) but
doesn’t provide a standardized measure like correlation coefficients.
4. Chi-Square Test: When dealing with two categorical variables, the chi-square test of independence is a common choice. It helps determine whether there is a significant association between the two
variables. For example, it can be used to analyze whether there’s a relationship between gender and voting preferences.
5. T-Test: The t-test is used when you want to compare the means of two groups for a continuous variable. For instance, you might use a t-test to determine if there’s a significant difference in the
test scores of two different teaching methods.
6. Analysis of Variance (ANOVA): ANOVA is an extension of the t-test and is used when there are more than two groups to compare. It assesses whether there are statistically significant differences
among the means of three or more groups.
7. Regression Analysis: Bivariate regression analysis is used to model and predict the relationship between one dependent variable and one independent variable. For example, you might use simple
linear regression to predict how changes in temperature (independent variable) affect ice cream sales (dependent variable).
8. Logistic Regression: This technique is suitable when the outcome variable is binary (e.g., yes/no, success/failure). Logistic regression helps understand the relationship between one or more
predictor variables and the probability of the binary outcome.
9. Spearman’s Rank-Order Correlation: When dealing with ordinal or non-normally distributed continuous variables, Spearman’s correlation provides a non-parametric measure of association. It’s based
on the rank order of data points rather than their actual values.
These are just a few examples of the many bivariate analysis techniques available. The choice of method depends on your data types, research objectives, and the assumptions you are willing to make
about your data. Properly executed bivariate analysis can provide valuable insights into the relationships between variables, informing decision-making and guiding further research.
What are Scatterplots and Scatterdiagrams?
In the realm of bivariate analysis, scatterplots, and scatter diagrams are indispensable tools for visually exploring and understanding the relationship between two continuous variables. These
graphical representations offer valuable insights into patterns, trends, and potential associations within your data.
What is a Scatterplot?
A scatterplot, also known as a scatter diagram or scatter graph, is a two-dimensional graphical representation of data points. In bivariate analysis, it displays each data point as a dot on a
Cartesian plane, with one variable plotted on the x-axis (horizontal) and the other on the y-axis (vertical).
How to Create a Scatterplot:
Creating a scatterplot is straightforward:
1. Data Preparation: Ensure you have a dataset containing two continuous variables that you want to analyze together.
2. Choose Axes: Select which variable will go on the x-axis and which on the y-axis, depending on your research question and the variables’ roles.
3. Plot Data Points: For each data point, find the corresponding values of the two variables and place a dot at the intersection of the corresponding x and y values.
4. Label Axes: Label the x-axis and y-axis to indicate the variables being represented.
Interpreting a Scatterplot:
Interpreting a scatterplot involves examining the pattern of dots and identifying trends:
1. Direction: Look at the general direction of the dots. Are they sloping upwards from left to right, indicating a positive correlation? Or are they sloping downwards, suggesting a negative
2. Strength: Assess the degree of scatter or clustering of the dots. A tight cluster indicates a strong correlation, while a scattered pattern suggests a weak correlation.
3. Outliers: Identify any data points that lie far from the main cluster of dots. Outliers can be influential in the analysis and may require further investigation.
4. Linearity: Consider whether the data points form a linear pattern. If so, a linear relationship may exist between the two variables. If not, the relationship may be nonlinear.
Use Cases of Scatterplots in Bivariate Analysis:
1. Correlation Assessment: Scatterplots are often used to assess the strength and direction of the relationship between two continuous variables. A strong positive correlation appears as a tight
cluster of dots sloping upwards, while a strong negative correlation appears as a tight cluster sloping downwards.
2. Outlier Detection: Outliers, which are data points significantly different from the others, can be easily spotted on a scatterplot. Their presence may indicate data quality issues or unique
observations requiring special attention.
3. Pattern Recognition: Scatterplots help recognize patterns, such as seasonality in time-series data, cyclic behavior, or linear trends. These patterns can inform decision-making and guide further
4. Heteroscedasticity Detection: In regression analysis, scatterplots can reveal heteroscedasticity, which is a situation where the spread of data points varies systematically with the independent
variable. This information is crucial for model selection and interpretation.
In summary, scatterplots and scatter diagrams play a pivotal role in bivariate analysis by providing a visual means to explore and interpret the relationship between two continuous variables. These
visualizations serve as a foundation for more advanced statistical analyses and guide decision-making processes in various fields, from economics and finance to healthcare and environmental science.
What is the Correlation Analysis?
Correlation analysis is a fundamental component of bivariate analysis, aimed at quantifying and understanding the strength and direction of the relationship between two continuous variables. It
provides valuable insights into how changes in one variable may be associated with changes in another.
What is Correlation Analysis?
Correlation analysis involves calculating a correlation coefficient, typically Pearson’s correlation coefficient (often denoted as “r”), to quantify the linear relationship between two variables.
This coefficient provides information about the following aspects of the relationship:
1. Strength: The magnitude of the correlation coefficient indicates how strong the relationship is. A coefficient close to 1 signifies a strong positive correlation, while a coefficient close to -1
indicates a strong negative correlation. A coefficient near 0 suggests a weak or no linear correlation.
2. Direction: The sign of the correlation coefficient (+ or -) reveals the direction of the relationship. A positive coefficient indicates a positive correlation, meaning that as one variable
increases, the other tends to increase as well. Conversely, a negative coefficient signifies a negative correlation, where one variable tends to decrease as the other increases.
Interpreting Correlation Coefficients:
The correlation coefficient ranges from -1 to 1:
• Positive Correlation (r > 0): When “r” is greater than 0, it suggests that as one variable increases, the other tends to increase as well. The closer “r” is to 1, the stronger the positive
• Negative Correlation (r < 0): A correlation coefficient less than 0 implies that as one variable increases, the other tends to decrease. The closer “r” is to -1, the stronger the negative
• No Correlation (r = 0): When the correlation coefficient is close to 0, it indicates little to no linear relationship between the variables.
Different Examples of Correlation Coefficients | Source: Author
Creating a Scatterplot for Correlation Visualization:
Before calculating the correlation coefficient, it’s common to create a scatterplot to visualize the relationship between the two variables. Scatterplots help you assess the linearity of the
relationship, detect outliers, and identify patterns.
Use Cases of Correlation Analysis in Bivariate Analysis:
1. Strength of Association: Correlation analysis is used to determine the strength of the relationship between variables. For example, in finance, it can quantify the relationship between interest
rates and stock market returns.
2. Model Building: In predictive modeling, understanding the correlations between variables helps select relevant predictors for a model. Highly correlated variables may be redundant.
3. Quality Control: In manufacturing, correlation analysis can identify variables that are strongly related, which may indicate a quality control issue that needs attention.
4. Healthcare: In medical research, correlation analysis can be used to assess the relationship between factors like diet and health outcomes.
5. Social Sciences: In sociology or psychology, correlation analysis can explore connections between variables like income and happiness.
6. Environmental Studies: Correlation analysis can reveal links between environmental factors like pollution levels and public health.
It’s important to note that correlation does not imply causation. While a correlation suggests an association between two variables, it does not prove that changes in one variable cause changes in
the other. Establishing causation often requires additional research and experimentation.
In summary, correlation analysis is a powerful tool in bivariate analysis for quantifying and interpreting the relationship between two continuous variables. It provides valuable insights into how
variables are related, which is essential for making informed decisions in various fields of study and industry sectors.
What is Regression Analysis?
Regression analysis is a statistical technique employed in bivariate analysis to examine the relationship between two continuous variables. Unlike correlation analysis, which quantifies the strength
and direction of a relationship, regression analysis goes a step further by modeling the relationship and making predictions based on that model.
What is Regression Analysis?
Regression analysis involves fitting a mathematical model to the data to describe how one variable (the dependent or response variable) changes in relation to changes in another variable (the
independent or predictor variable). The goal is to find the best-fitting model that explains the relationship between the two variables.
Example of a Linear Regression | Source: Author
Key Elements of Regression Analysis:
1. Dependent Variable: The variable that you want to predict or explain is known as the dependent variable (Y). It’s the outcome variable or what you’re trying to understand.
2. Independent Variable: The variable that you believe influences or explains changes in the dependent variable is the independent variable (X). It’s the predictor variable.
3. Regression Model: The regression model is a mathematical equation that represents the relationship between the dependent and independent variables. The simplest form is the linear regression
model, which assumes a linear relationship between the variables:
\(\) \[ y = \beta_{0} + \beta_{1} \cdot X + \epsilon \]
• Y is the dependent variable.
• X is the independent variable.
• β₀ is the intercept (the value of Y when X is 0).
• β₁ is the slope (how much Y changes for a one-unit change in X).
• ε represents the error term (unexplained variability).
β₀ and β₁ are the parameters of the regression model. They are estimated from the data to find the best-fit line that minimizes the sum of squared errors (the differences between the observed and
predicted values).
Interpreting Regression Analysis:
Regression analysis provides insights into the following aspects of the relationship between variables:
• Strength and Significance: The coefficient β₁ indicates the strength and direction of the relationship. A positive β₁ suggests that as X increases, Y tends to increase, while a negative β₁
implies the opposite. The magnitude of β₁ reflects the degree of impact.
• Prediction: Regression analysis allows you to predict the value of the dependent variable for a given value of the independent variable. This predictive capability is valuable in various fields,
such as finance (predicting stock prices), economics (predicting inflation rates), and healthcare (predicting patient outcomes).
Use Cases of Regression Analysis in Bivariate Analysis:
1. Economics: Predicting the relationship between factors like income and spending or employment rates and economic growth.
2. Healthcare: Modeling the impact of variables like diet and exercise on health outcomes, such as weight loss or blood pressure.
3. Marketing: Analyzing the influence of advertising spending on product sales.
4. Environmental Science: Understanding the relationship between pollution levels and biodiversity.
5. Engineering: Predicting the relationship between variables like temperature and material strength.
It’s important to conduct regression analysis with caution and consider potential limitations. Correlation does not imply causation, and regression results should not be interpreted as causal
relationships without further evidence. Additionally, the assumptions of the regression model, such as linearity and homoscedasticity, should be assessed.
In summary, regression analysis is a powerful tool in bivariate analysis that goes beyond correlation by modeling and predicting the relationship between two continuous variables. It is widely used
in various fields to understand, explain, and make predictions based on data.
This is what you should take with you
• Bivariate analysis is a fundamental statistical technique that uncovers relationships and associations between two variables.
• Through correlation analysis, we can quantify the strength and direction of relationships. Positive, negative, or no correlation provides valuable insights.
• Scatterplots are a powerful tool for visualizing bivariate data. They allow us to identify patterns and trends by plotting data points on a graph.
• Regression analysis takes bivariate analysis to the next level by modeling the relationship and making predictions. It’s a versatile technique used in various fields.
• Bivariate analysis has applications in economics, healthcare, marketing, environmental science, engineering, and more.
• While bivariate analysis is valuable, it doesn’t imply causation. It’s crucial to interpret results cautiously and consider underlying assumptions.
• Bivariate analysis empowers data-driven decision-making by providing insights into how two variables are related, which is essential for understanding complex systems and making informed choices.
• Bivariate analysis serves as a foundation for more advanced multivariate analysis, where relationships among multiple variables are explored.
Other Articles on the Topic of Bivariate Analysis
The University of West Georgia has an interesting article on the topic of Bivariate Analysis that you can find here. | {"url":"https://databasecamp.de/en/data/bivariate-analysis","timestamp":"2024-11-08T21:30:24Z","content_type":"text/html","content_length":"291708","record_id":"<urn:uuid:3beddd8d-f50e-4482-8cbe-07095f4e1bcc>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00191.warc.gz"} |
937 research outputs found
We present an efficient and exact Monte Carlo algorithm to simulate reversible aggregation of particles with dedicated binding sites. This method introduces a novel data structure of dynamic bond
tree to record clusters and sequences of bond formations. The algorithm achieves a constant time cost for processing cluster association and a cost between $\mathcal{O}(\log M)$ and $\mathcal{O}(M)$
for processing bond dissociation in clusters with $M$ bonds. The algorithm is statistically exact and can reproduce results obtained by the standard method. We applied the method to simulate a
trivalent ligand and a bivalent receptor clustering system and obtained an average scaling of $\mathcal{O}(M^{0.45})$ for processing bond dissociation in acyclic aggregation, compared to a linear
scaling with the cluster size in standard methods. The algorithm also demands substantially less memory than the conventional method.Comment: 8 pages, 3 figure
A \emph{metric tree embedding} of expected \emph{stretch~$\alpha \geq 1$} maps a weighted $n$-node graph $G = (V, E, \omega)$ to a weighted tree $T = (V_T, E_T, \omega_T)$ with $V \subseteq V_T$ such
that, for all $v,w \in V$, $\operatorname{dist}(v, w, G) \leq \operatorname{dist}(v, w, T)$ and $operatorname{E}[\operatorname{dist}(v, w, T)] \leq \alpha \operatorname{dist}(v, w, G)$. Such
embeddings are highly useful for designing fast approximation algorithms, as many hard problems are easy to solve on tree instances. However, to date the best parallel $(\operatorname{polylog} n)$
-depth algorithm that achieves an asymptotically optimal expected stretch of $\alpha \in \operatorname{O}(\log n)$ requires $\operatorname{\Omega}(n^2)$ work and a metric as input. In this paper, we
show how to achieve the same guarantees using $\operatorname{polylog} n$ depth and $\operatorname{\tilde{O}}(m^{1+\epsilon})$ work, where $m = |E|$ and $\epsilon > 0$ is an arbitrarily small
constant. Moreover, one may further reduce the work to $\operatorname{\tilde{O}}(m + n^{1+\epsilon})$ at the expense of increasing the expected stretch to $\operatorname{O}(\epsilon^{-1} \log n)$.
Our main tool in deriving these parallel algorithms is an algebraic characterization of a generalization of the classic Moore-Bellman-Ford algorithm. We consider this framework, which subsumes a
variety of previous "Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss it in depth. In our tree embedding algorithm, we leverage it for providing efficient query access
to an approximate metric that allows sampling the tree using $\operatorname{polylog} n$ depth and $\operatorname{\tilde{O}}(m)$ work. We illustrate the generality and versatility of our techniques by
various examples and a number of additional results
Bellman's optimality principle has been of enormous importance in the development of whole branches of applied mathematics, computer science, optimal control theory, economics, decision making, and
classical physics. Examples are numerous: dynamic programming, Markov chains, stochastic dynamics, calculus of variations, and the brachistochrone problem. Here we show that Bellman's optimality
principle is violated in a teleportation problem on a quantum network. This implies that finding the optimal fidelity route for teleporting a quantum state between two distant nodes on a quantum
network with bi-partite entanglement will be a tough problem and will require further investigation.Comment: 4 pages, 1 figure, RevTeX
A central endeavor of thermodynamics is the measurement of free energy changes. Regrettably, although we can measure the free energy of a system in thermodynamic equilibrium, typically all we can say
about the free energy of a non-equilibrium ensemble is that it is larger than that of the same system at equilibrium. Herein, we derive a formally exact expression for the probability distribution of
a driven system, which involves path ensemble averages of the work over trajectories of the time-reversed system. From this we find a simple near-equilibrium approximation for the free energy in
terms of an excess mean time-reversed work, which can be experimentally measured on real systems. With analysis and computer simulation, we demonstrate the accuracy of our approximations for several
simple models.Comment: 5 pages, 3 figure
Learning meaningful topic models with massive document collections which contain millions of documents and billions of tokens is challenging because of two reasons: First, one needs to deal with a
large number of topics (typically in the order of thousands). Second, one needs a scalable and efficient way of distributing the computation across multiple machines. In this paper we present a novel
algorithm F+Nomad LDA which simultaneously tackles both these problems. In order to handle large number of topics we use an appropriately modified Fenwick tree. This data structure allows us to
sample from a multinomial distribution over $T$ items in $O(\log T)$ time. Moreover, when topic counts change the data structure can be updated in $O(\log T)$ time. In order to distribute the
computation across multiple processor we present a novel asynchronous framework inspired by the Nomad algorithm of \cite{YunYuHsietal13}. We show that F+Nomad LDA significantly outperform
state-of-the-art on massive problems which involve millions of documents, billions of words, and thousands of topics
Solving mazes is not just a fun pastime. Mazes are prototype models in graph theory, topology, robotics, traffic optimization, psychology, and in many other areas of science and technology. However,
when maze complexity increases their solution becomes cumbersome and very time consuming. Here, we show that a network of memristors - resistors with memory - can solve such a non-trivial problem
quite easily. In particular, maze solving by the network of memristors occurs in a massively parallel fashion since all memristors in the network participate simultaneously in the calculation. The
result of the calculation is then recorded into the memristors’ states, and can be used and/or recovered at a later time. Furthermore, the network of memristors finds all possible solutions in
multiple-solution mazes, and sorts out the solution paths according to their length. Our results demonstrate not only the first application of memristive networks to the field of massively-parallel
computing, but also a novel algorithm to solve mazes which could find applications in different research fields
The one-way measurement model is a framework for universal quantum computation, in which algorithms are partially described by a graph G of entanglement relations on a collection of qubits. A
sufficient condition for an algorithm to perform a unitary embedding between two Hilbert spaces is for the graph G, together with input/output vertices I, O \subset V(G), to have a flow in the sense
introduced by Danos and Kashefi [quant-ph/0506062]. For the special case of |I| = |O|, using a graph-theoretic characterization, I show that such flows are unique when they exist. This leads to an
efficient algorithm for finding flows, by a reduction to solved problems in graph theory.Comment: 8 pages, 3 figures: somewhat condensed and updated version, to appear in PR
A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing
the position of the center of each box, and thereby, vertices in preassigned boxes can divide subsequent boxes into more than one pieces, but divided boxes are counted once. We find that such
box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fractal scaling for fractal networks; however, it is inessential for regular lattice and conventional fractal
objects embedded in the Euclidean space. Next the algorithm is viewed from the cluster-growing perspective that boxes are allowed to overlap and thereby, vertices can belong to more than one box.
Then, the number of distinct boxes a vertex belongs to is distributed in a heterogeneous manner for SF fractal networks, while it is of Poisson-type for the conventional fractal objects.Comment: 12
pages, 11 figures, a proceedings of the conference, "Optimization in complex networks." held in Los Alamo
We present a model supported by simulation to explain the effect of temperature on the conduction threshold in disordered systems. Arrays with randomly distributed local thresholds for conduction
occur in systems ranging from superconductors to metal nanocrystal arrays. Thermal fluctuations provide the energy to overcome some of the local thresholds, effectively erasing them as far as the
global conduction threshold for the array is concerned. We augment this thermal energy reasoning with percolation theory to predict the temperature at which the global threshold reaches zero. We also
study the effect of capacitive nearest-neighbor interactions on the effective charging energy. Finally, we present results from Monte Carlo simulations that find the lowest-cost path across an array
as a function of temperature. The main result of the paper is the linear decrease of conduction threshold with increasing temperature: $V_t(T) = V_t(0) (1 - 4.8 k_BT P(0)/ p_c)$, where $1/P(0)$ is an
effective charging energy that depends on the particle radius and interparticle distance, and $p_c$ is the percolation threshold of the underlying lattice. The predictions of this theory compare well
to experiments in one- and two-dimensional systems.Comment: 14 pages, 10 figures, submitted to PR
This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in CERI '16 Proceedings of the 4th Spanish
Conference on Information Retrieval, http://dx.doi.org/10.1145/2934732.2934747Recommender systems is an active research area where the major focus has been on how to improve the quality of gen-
erated recommendations, but less attention has been paid on how to do it in an e cient way. This aspect is increas- ingly important because the information to be considered by recommender systems is
growing exponentially. In this pa- per we study how di erent data structures a ect the perfor- mance of these systems. Our results with two public datasets provide relevant insights regarding the
optimal data struc- tures in terms of memory and time usages. Speci cally, we show that classical data structures like Binary Search Trees and Red-Black Trees can beat more complex and popular
alternatives like Hash Tables | {"url":"https://core.ac.uk/search/?q=author%3A(Cormen%20T.%20H.)","timestamp":"2024-11-07T03:44:56Z","content_type":"text/html","content_length":"182128","record_id":"<urn:uuid:eef58a0c-159d-4dcf-9c9d-9d04da1b0c03>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00733.warc.gz"} |
Logic for Systems: Lightweight Formal Methods for Everybody
In the following, <x> is a variable, <expr> is an expression of arity 1, and <fmla> is a formula (that can use the variable <x>). You can quantify over a unary set in the following ways:
• some <x>: <expr> | { <fmla> }: true when <fmla> is true for at least one element in <expr>; and
• all <x>: <expr> | { <fmla> }: true when <fmla> is true for all elements in <expr>
If you want to quantify over several variables, you can also do the following:
• some <x>: <expr-a>, <y>: <expr-b> | { <fmla> }; or
• some <x>, <y>: <expr> | { <fmla> }.
The syntax is the same for other quantifiers, such as all.
Forge also provides 3 additional quantifiers, which encode somewhat richer constraints than the above:
• no <x>: <expr> | { <fmla> }: true when <fmla> is false for all elements in <expr>
• lone <x>: <expr> | { <fmla> }: true when <fmla> is true for zero or one elements in <expr>
• one <x>: <expr> | { <fmla> }: true when <fmla> is true for exactly one element in <expr>
The above 3 quantifiers (no, lone, and one) should be used carefully. Because they invisibly encode extra constraints, they do not commute the same way some and all quantifiers do. E.g., some x : A |
some y : A | myPred[x,y] is always equivalent to some y : A | some x : A | myPred[x,y], but one x : A | one y : A | myPred[x,y] is NOT always equivalent to one y : A | one x : A | myPred[x,y]. (Why
not? Try it out in Forge!)
Beware combining the no, one, and lone quantifiers with multiple variables at once; the meaning of, e.g., one x, y: A | ... is "there exists a unique pair <x, y> such that ...". This is different
from the meaning of one x: A | one y: A | ..., which is "there is a unique x such that there is a unique y such that ...".
Sometimes, it might be useful to try to quantify over all pairs of elements in A, where the two in the pair are distinct atoms. You can do that using the disj keyword, e.g.:
• some disj x, y : A | ... (adds an implicit x != y and ...); and
• all disj x, y : A | ... (adds an implicit x != y implies ...)` | {"url":"https://tnelson.github.io/forgebook/docs/building-models/constraints/formulas/quantifiers.html","timestamp":"2024-11-12T13:48:07Z","content_type":"text/html","content_length":"25119","record_id":"<urn:uuid:5773f9ac-8be5-4992-b3d6-031f598e346d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00287.warc.gz"} |
Linear density explained
Linear density is the measure of a quantity of any characteristic value per unit of length. Linear mass density (titer in textile engineering, the amount of mass per unit length) and linear charge
density (the amount of electric charge per unit length) are two common examples used in science and engineering.
The term linear density or linear mass density is most often used when describing the characteristics of one-dimensional objects, although linear density can also be used to describe the density of a
three-dimensional quantity along one particular dimension. Just as density is most often used to mean mass density, the term linear density likewise often refers to linear mass density. However, this
is only one example of a linear density, as any quantity can be measured in terms of its value along one dimension.
Linear mass density
Consider a long, thin rod of mass
and length
. To calculate the average linear mass density,
, of this one dimensional object, we can simply divide the total mass,
, by the total length,
$\bar\lambda_m = \frac$
If we describe the rod as having a varying mass (one that varies as a
of position along the length of the rod,
), we can write:
$m = m(l)$
unit of mass,
, is equal to the product of its linear mass density,
, and the infinitesimal unit of length,
$dm = \lambda_m dl$
The linear mass density can then be understood as the
of the mass function with respect to the one dimension of the rod (the position along its length,
$\lambda_m = \frac$
The SI unit of linear mass density is the kilogram per meter (kg/m).
Linear density of fibers and yarns can be measured by many methods. The simplest one is to measure a length of material and weigh it. However, this requires a large sample and masks the variability
of linear density along the thread, and is difficult to apply if the fibers are crimped or otherwise cannot lay flat relaxed. If the density of the material is known, the fibers are measured
individually and have a simple shape, a more accurate method is direct imaging of the fiber with a scanning electron microscope to measure the diameter and calculation of the linear density. Finally,
linear density is directly measured with a vibroscope. The sample is tensioned between two hard points, mechanical vibration is induced and the fundamental frequency is measured.^[1] ^[2]
Linear charge density
See main article: Linear charge density.
Consider a long, thin wire of charge
and length
. To calculate the average linear charge density,
, of this one dimensional object, we can simply divide the total charge,
, by the total length,
$\bar\lambda_q = \frac$
If we describe the wire as having a varying charge (one that varies as a function of position along the length of the wire,
), we can write:
$q = q(l)$
Each infinitesimal unit of charge,
, is equal to the product of its linear charge density,
, and the infinitesimal unit of length,
$dq = \lambda_q dl$
The linear charge density can then be understood as the derivative of the charge function with respect to the one dimension of the wire (the position along its length,
$\lambda_q = \frac$
Notice that these steps were exactly the same ones we took before to find $\lambda_m = \frac$.
The SI unit of linear charge density is the coulomb per meter (C/m).
Other applications
In drawing or printing, the term linear density also refers to how densely or heavily a line is drawn.
The most famous abstraction of linear density is the probability density function of a single random variable.
See also: Units of textile measurement. Common units include:
• kilogram per meter
• ounce (mass) per foot
• ounce (mass) per inch
• pound (mass) per yard: used in the North American railway industry for the linear density of rails
• pound (mass) per foot
• pound (mass) per inch
• tex, a unit of measure for the linear density of fibers, defined as the mass in grams per 1,000 meters
• denier, a unit of measure for the linear density of fibers, defined as the mass in grams per 9,000 meters
• decitex (dtex), the SI unit for the linear density of fibers, defined as the mass in grams per 10,000 meters
See also
Notes and References
1. 10.1177/004051755802800809. Findings and Recommendations on the Use of the Vibroscope . Textile Research Journal. 28. 8 . 691–700 . 1958 . Patt . D.H.. 137534752 . | {"url":"https://everything.explained.today/Linear_density/","timestamp":"2024-11-14T07:51:37Z","content_type":"text/html","content_length":"16209","record_id":"<urn:uuid:2c7294b1-3b6d-421b-a460-6efd7df455aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00188.warc.gz"} |
Automated feature recognition in CFPD analyses of DMA or supply area flow data
The recently introduced comparison of flow pattern distributions (CFPD) method for the identification, quantification and interpretation of anomalies in district metered areas (DMAs) or supply area
flow time series relies, for practical applications, on visual identification and interpretation of features in CFPD block diagrams. This paper presents an algorithm for automated feature recognition
in CFPD analyses of DMA or supply area flow data, called CuBOid, which is useful for objective selection and analysis of features and automated (pre-)screening of data. As such, it can contribute to
rapid identification of new leakages, unregistered changes in valve status or network configuration, etc., in DMAs and supply areas. The method is tested on synthetic and real flow data. The obtained
results show that the method performs well in synthetic tests and allows an objective identification of most anomalies in flow patterns in a real life dataset.
In recent years, utilities have been moving towards more data based decision making for network operation and management. Flow rate time series for district metered areas (DMAs) and distribution
areas provide meaningful insights into the flow performance of the network, but due to their complexity these are not always fully explored and used. These data contain information about leakage
(which continues to be an issue, with numbers worldwide ranging from 3% to more than 50% (Lambert 2002; Beuken et al. 2006)), unauthorized consumption, customer behavior, network configuration and
isolation (valve statuses), among others. Many methods exist to obtain information out of these data, and most focus on leakage. Classically, the most important are top-down and bottom-up methods (
Farley & Trow 2003; Wu 2011). The top-down method consists of a water balance in which the registered amount of water delivered to a supply area over the period of a year is compared to the billed
amount of water. The bottom-up method essentially compares the minimum flow rate during the quiet night hours into a DMA or demand zone, or the integrated flow of a 24-hour period, to an estimate for
the demand for this DMA or demand zone based on the number of connections (Puust et al. 2010).
Different methods to determine the amount of non-revenue water, leakage, bursts and the location of leakages have been the focus of research. These methods include inverse transient analysis (Liggett
& Chen 1994; Savic et al. 2005; Vítkovský et al. 2007), alternative statistical (e.g. Palau et al. 2012; Romano et al. 2013) and machine learning methods (e.g. Aksela et al. 2009; Mounce et al. 2010;
Mamo et al. 2014), or a combination of both (e.g. Romano et al. 2014), probabilistic leak detection (Poulakis et al. 2003; Puust et al. 2006), pressure dependent leak detection (Wu et al. 2010), and
meta-methods including a comparison of results for neighboring DMAs (Montiel & Nguyen 2011, 2013).
The comparison of flow pattern distributions (CFPD) method was introduced (
Van Thienen 2013
Van Thienen et al. 2013a
) as a new tool to assess flow data of a DMA or supply area in order to pinpoint (in time, not space), identify, and quantify changes in the amount of water supplied (see
Figure 1
). It has since been successfully applied in multiple projects with Dutch drinking water companies to identify, for example, leakages and incorrect valves statuses and network connections (
Van Thienen et al. 2013b
The interpretation of changes in water flow time series through CFPD block diagrams is intuitive in all but the most complex cases. However, it relies on the visual interpretation of these diagrams,
which is still a limitation. This paper is aimed at overcoming this limitation by presenting a support algorithm for automated feature recognition in CFPD block diagrams. Such an algorithm offers
several advantages: automated pre-screening of data to limit manual inspection and interpretation to the most interesting cases; objective rather than (to some degree) subjective selection and
analysis of features.
This paper presents a method for automated feature recognition in CFPD block diagrams, called the CuBOid (CFPD Block Optimization) algorithm. Its principle is presented, and the method is applied to
synthetic and real network data to evaluate its performance.
For a complete description of the CFPD method, the reader is referred to
Van Thienen (2013)
. A concise introduction is provided in Appendix 1 (available with the online version of this paper). An overview of the analysis and interpretation of the method is presented in
Figure 1
. In the matrices resulting from the analysis, each event (change in the flow pattern) is characterized by a typical structure (
Figure 2
). These matrices should be read as follows: going from the left to the right (i.e. time arrow), a change in color or color intensity represents a change of the CFPD parameter. The CFPD parameter
changes as a consequence of a flow pattern alteration: the color intensity is proportional to magnitude of the alteration. The presence of this change in multiple rows of the matrix means that the
anomaly is not caused by an anomaly in the reference signal. Changes in flow patterns which are most interesting are systematic changes, which are indicative of changes in demand, network
configuration, leakage, etc., rather than stochastic variations of short duration. These systematic changes show up in CFPD block diagrams as blocks with a similar color intensity (see
Figure 2(b)
The CuBOid feature recognition algorithm presented in this paper seeks to describe these typical patterns observed in a CFPD block diagram as a summation of permissible block functions. In this way,
it is somewhat similar to, for example, the Discrete Cosine Transform (
Ahmed et al. 1974
) or the Discrete Wavelet Transform (e.g.
Akansu et al. 2009
) which are used, for example, in image compression methods. The typical shape of the permissible blocks representing anomalies in the flow pattern stems from the nature of the CFPD analysis
procedure (see
Figure 2
). This typical shape can be described by the following function:
In this expression, i and j are the row and column number, respectively, and j[1] and j[2] are the first and last column of a perturbation block. Note that consecutive columns in a CFPD block diagram
correspond to consecutive days (or weeks, or some other duration), which are compared to each other in consecutive rows (for more details see Van Thienen (2013)).
In this expression, w[1] and w[2] are two weight factors.
The approach to automatically identify the block functions representing anomalies, whilst ignoring the regular weekday-weekend pattern, is by means of an optimization algorithm, in which only a
limited number of block configurations, which can logically be present in a CFPD block diagram (shown in Figure 2), combined with the typical weekend pattern, are considered. The process has six
steps, which are as follows (for a block (slopes or intercepts) of dimensions mxm).
1. Break detection: in order to quantify changes (breaks) between neighboring columns, firstly the L[x] norm (with x having a value typically around 1.0) is computed for the difference vector of each
pair of consecutive column vectors (skipping the first column of the matrix and the last item of each second vector). The values of these norms are divided by the size of the column vectors – m,
obtaining a measure for the step sizes between consecutive columns, representing analysis periods. The exponent of the norm determines the focus of sensitivity of break detection within the matrix:
smaller values result in more breaks being detected on the left side of the matrix, while larger values result in more breaks being detected on the right side of the matrix.
2. Generation of permissible blocks: using the n biggest changes (with n decided by the user), all permissible block functions are generated for . These permissible block functions correspond to all
possible combinations of two steps (starting and ending) which are taken from the n biggest changes. The number of functions generated is .
3. Combination of block functions: the user choses the number of block functions p, with , which is used to resolve a single CFPD block diagram. All possible combinations of p block functions from
the k functions generated in step 2 are generated. Thus, in total there are combinations.
4. For each combination, an optimization is performed in which the function amplitudes are the decision variables and the objective is to minimize the difference between the summation of this
function combination and the block matrix which is being fitted. The weekday-weekend pattern is included in this computation, so the parameters and are free parameters of the optimization problem as
well. This is described by the following objective function:for combination , with j and k the indices for the matrix rows and columns, the weights or amplitudes for block function l in combination ,
the amplitude of block function of combination i at matrix row and column k (expression (1)), the weekend day factor for weekend day q, b[ij] the amplitude of the weekend block function for day q at
matrix row and column k, and the actual CFPD matrix value at at matrix row j and column k. No constraints were applied to the optimization.
This optimization can be done in parallel. This step results in block amplitudes for each combination generated in step 3. Note that the amplitude is dimensionless for the matrix of slope factors a
and has the same unit of volumetric flow rate as the original input time series for the matrix of intercept factors b.
5. The performance of each combination (blocks and amplitudes) is quantified using the following expression:in which
is the fitness of the solution, is the Euclidian 2-norm of the difference between the original matrix and the reconstructed matrix (Equation (
)), is the penalty factor for the number of block functions, is the number of block functions used (
), is the overlap penalty factor, is the number of overlapping blocks in the set of block functions (
(number of block functions in column - 1)), and
is the sum of the lengths of the block functions. This cost function reflects the fit of the candidate blocks with respect to the actual CFPD matrix. It is designed to penalize both a large number of
block functions and a large degree of overlap. The fitness parameter
is minimized.
6. The best performing combination of block functions and weekend parameters is selected.
Thus, the process combines a combinatorial problem with a parameter fitting problem. The former is addressed in steps 2, 3, 5, 6, the latter in step 4.
Note that the method can be asked to fit a large number of functions simultaneously, but this will most certainly lead to overfitting the data, with noise being described by additional block
functions. Therefore, parsimony is important to obtain meaningful results. This will be illustrated later. The penalty parameters become relevant for larger values of n and . Choosing a larger value
of n and/or p results in a significant increase in computation time, which is the reason why these parameters were introduced. With unlimited computation power, n should be the number of columns – 2,
and p should be chosen to represent the largest number of anomalies expected in a single matrix.
The optimization criterion is formulated in terms of a Euclidean norm of the difference vector of the diagram data and sum of all block and weekday/weekend functions for which the optimization is
being performed. For slope diagrams, the log of the actual values is taken, since multiple anomalies simply add up in log space and values are zero centered. The optimization method used for the
parameter fitting part of the algorithm is the quasi-Newton method of Broyden, Fletcher, Goldfarb, and Shanno, as implemented in scipy (Oliphant 2007).
In order to test the performance of the proposed approach and the influence of the different parameters on the results, a series of tests was performed on synthetic data. Bearing in mind that several
combinations of parameter values are possible, note that only a limited set of tests that focus on the influence of each parameter one at a time (and hence facilitate interpretation) have been
carried out and reported in this paper. Table 1 summarizes the considered tests and corresponding parameter values. In addition to this, the results of a series of tests on real flow data are also
Synthetic data
The synthetic data considered for the tests consist of repetitions of actual measured flow patterns, with in total three sequences of five identical weekdays and two identical weekend days, starting
at day 2 and ending at day 22 of an arbitrary month in an arbitrary year. Different datasets were generated from these original data, by adding anomalies with different amplitude and duration, as
well as different levels of normally distributed noise. The characteristics of the generated datasets are summarized in
Table 2
, and the flow patterns corresponding to the unperturbed signal and datasets 1a, 2a and 3a can be seen in
Figure 3
Table 2
. Anomaly 1 . Anomaly 2 . (%) .
Dataset ID . Start-end day . Amplitude (m^3/h) . Start-end day . Amplitude (m^3/h) . Gaussian noise .
0 None – None – None
1a 05–08 10 15–18 5 None
1b 05–08 10 15–18 5 5
1c 05–08 10 15–18 5 10
1d 05–08 10 15–18 5 20
2a 04–10 10 14–20 5 None
2b 04–10 10 14–20 5 5
2c 04–10 10 14–20 5 10
2d 04–10 10 14–20 5 20
3a 05–15 10 12–20 5 None
3b 05–15 10 12–20 5 5
3c 05–15 10 12–20 5 10
3d 05–15 10 12–20 5 20
. Anomaly 1 . Anomaly 2 . (%) .
Dataset ID . Start-end day . Amplitude (m^3/h) . Start-end day . Amplitude (m^3/h) . Gaussian noise .
0 None – None – None
1a 05–08 10 15–18 5 None
1b 05–08 10 15–18 5 5
1c 05–08 10 15–18 5 10
1d 05–08 10 15–18 5 20
2a 04–10 10 14–20 5 None
2b 04–10 10 14–20 5 5
2c 04–10 10 14–20 5 10
2d 04–10 10 14–20 5 20
3a 05–15 10 12–20 5 None
3b 05–15 10 12–20 5 5
3c 05–15 10 12–20 5 10
3d 05–15 10 12–20 5 20
The difference between week and weekend days is clearly visible in the flow patterns. The added anomalies are also visible, corresponding to upward shifts in the patterns.
Block functions
The CuBOid algorithm identifies block functions representing anomalies in flow patterns of some days with respect to earlier days (or any other time scale – weeks, months, …) in a certain period of
time. Since for the different datasets the anomalies were manually added to the data, it is known beforehand what the block functions should look like. The block functions are described by a start
and an end column in the matrix diagram, and by an amplitude. The start and end columns should correspond to the start and end dates of the anomalies, and the amplitude should be equal to the
amplitude of the actual anomaly (recall Table 2).
For datasets 1a to 1d, two block functions should be identified. The start and end days of the block function describing the first anomaly should be 5 and 8, and the amplitude should be 10 m^3/h. For
the second anomaly the start and end columns of the block function should be 15 and 18, and the amplitude should be 5 m^3/h. Table 3 summarizes the start and end days as well as the amplitudes of the
block functions obtained by the different tests. The last two columns in the table display the actual used steps to form the block functions and the norm (Equation (3)), i.e. the difference (or
distance) between the difference between the original matrix and the reconstructed matrix. This norm can also be used to compare results obtained by different tests in a more straightforward way.
Table 3
. . Block function 1 . Block function 2 . Block function 3 . . .
Dataset . Test . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . C[i] . Number of steps used .
1a All 5 8 10.05 15 18 5.04 . . . 3.5 4
1b All except 5 8 9.84 15 18 4.48 . . . 4.3 4
11 5 8 9.75 15 18 4.39 9 14 –0.21 4.6 4
1c All except 5 8 8.96 15 20 3.75 . . . 22.0 4
7, 8 5 8 9.03 15 18 5.04 . . . 7.6 4
All except 5 8 8.18 15 20 3.68 . . . 19.2 4
1d 6, 10 5 8 8.51 15 22 3.32 . . . 24.1 4
11 5 8 8.82 15 20 4.50 6 13 1.68 17.1 5
15 . . . . . . . . . 50.8 0
. . Block function 1 . Block function 2 . Block function 3 . . .
Dataset . Test . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . C[i] . Number of steps used .
1a All 5 8 10.05 15 18 5.04 . . . 3.5 4
1b All except 5 8 9.84 15 18 4.48 . . . 4.3 4
11 5 8 9.75 15 18 4.39 9 14 –0.21 4.6 4
1c All except 5 8 8.96 15 20 3.75 . . . 22.0 4
7, 8 5 8 9.03 15 18 5.04 . . . 7.6 4
All except 5 8 8.18 15 20 3.68 . . . 19.2 4
1d 6, 10 5 8 8.51 15 22 3.32 . . . 24.1 4
11 5 8 8.82 15 20 4.50 6 13 1.68 17.1 5
15 . . . . . . . . . 50.8 0
First of all, the influence of added noise on the estimated amplitude of the block functions is clearly visible: when adding more noise to the data, the estimated amplitude decreases, and the
estimated end day of the second block also tends to get worse, being shifted forward (this is, ending later than it should). Accordingly, the norm increases with the increase of random Gaussian
When no noise is added to the data (dataset 1a), all performed tests lead to the same resulting block functions, and these are a very close fit to the actual introduced anomalies. The slight
deviation from the actual values is presumably due to numerical issues and/or the stop criterion for the optimization algorithm.
When adding 5% random Gaussian noise, not all tests lead to the same block functions. While most tests perform well in identifying the two anomalies, test 11 (lowest f[1] penalty coefficient), leads
to the identification of three blocks instead. The third block is probably fitting the noise added to the data.
When adding 10% of noise, the tests perform generally worse, overestimating the duration of the second anomaly (by identifying the end column as being 20 instead of 18), and underestimating the
amplitudes of the anomalies. The best results are obtained for test 7 and 8 (with higher number of steps and a lower L
norm, respectively)
Figure 4(a)
illustrate the obtained results when performing test 7. In
Figure 4(a)
, the matrix of
-factors is visible. In
Figure 4(b)
, the estimated block functions are visible. The visual interpretation of
Figure 4(b)
is clearer, since the added noise is not visually represented.
For 20% of random noise, all tests underestimate the amplitudes of both anomalies, and overestimate the duration of the second anomaly. The best performance, in terms of the norm, is obtained for
test 11 (with the lowest f
penalty). However, for this case, the algorithm identifies three block functions, i.e. a false positive which is fitting the noise. For test 15 (highest f
penalty coefficient) the algorithm does not identify block functions, i.e. the results are false negatives.
Figure 5(a)
illustrate the obtained results when performing test 1. In
Figure 5(a)
, the matrix of
-factors is visible. In
Figure 5(b)
, the estimated block functions are visible. With the increased noise, it becomes more difficult to interpret and visually identify anomalies in the matrix of
-factors. The visual interpretation of
Figure 5(b)
is much easier, since the added noise is not visually represented. The longer duration of the second block function and the lower estimated amplitude are also clear in
Figure 5(b)
Regarding datasets 2a–2d, two block functions should be identified. The start and end columns of the block function describing the first anomaly should be 4 and 10, and the amplitude should be 10 m^3
/h. For the second anomaly the start and end columns of the block function should be 14 and 20, and the amplitude should be 5 m^3/h. Table 4 summarizes the obtained results from the different tests
performed to datasets 2a–2d.
Table 4
. . Block function 1 . Block function 2 . Block function 3 . Block function 4 . . .
Dataset Test . Start End . Amplitude (m^3/h) Start End . Amplitude (m^3/h) Start End . Amplitude (m^3/h) Start End . Amplitude (m^3/h) C[i] Number of steps used
. . . . . . . . . . .
All except 4 10 8.80 16 20 4.19 21 22 –3.51 . . . 16.4 4
2a 2, 14, 15 4 10 9.18 16 20 4.62 . . . . . . 19.6 4
6, 9, 10 4 10 10.50 11 20 3.92 . . . . . . 27.2 3
All except 4 10 10.56 16 20 5.86 11 15 2.2 . . . 15.5 4
2, 13, 14, 15 4 10 9.26 16 20 4.61 . . . 17.9 4
2b 3 4 6 10.23 7 10 7.96 16 20 4.42 21 22 −3.56 14.1 5
10 4 10 10.55 11 20 3.78 . . . . . . 26.8 3
16 4 10 12.39 11 20 5.68 7 16 –3.01 . . . 14.9 5
1, 3, 11, 12, 16, 17 4 10 10.61 16 20 7.189 9 16 2.08 . . . 17.6 5
2, 13, 14, 15, 18, 19, 4 10 10.002 16 20 5.815 . . . . . . 21.5 4
2c 20
6, 9, 10 4 10 11.37 11 20 4.41 . . . . . . 34.7 3
7 4 10 12 9 20 7.5 7 16 –5.61 . . . 16.7 6
All except 4 10 9.51 16 20 5.08 . . . . . . 23.9 4
7 4 10 11.74 9 20 6.22 6 15 –4.54 . . . 17.7 6
2d 10 4 10 7.9 . . . . . . . . . 40.9 2
11, 12, 16 4 10 10.17 16 20 6.56 9 15 2.24 . . . 23.7 5
15 . . . . . . . . . . . . 69.6 0
. . Block function 1 . Block function 2 . Block function 3 . Block function 4 . . .
Dataset Test . Start End . Amplitude (m^3/h) Start End . Amplitude (m^3/h) Start End . Amplitude (m^3/h) Start End . Amplitude (m^3/h) C[i] Number of steps used
. . . . . . . . . . .
All except 4 10 8.80 16 20 4.19 21 22 –3.51 . . . 16.4 4
2a 2, 14, 15 4 10 9.18 16 20 4.62 . . . . . . 19.6 4
6, 9, 10 4 10 10.50 11 20 3.92 . . . . . . 27.2 3
All except 4 10 10.56 16 20 5.86 11 15 2.2 . . . 15.5 4
2, 13, 14, 15 4 10 9.26 16 20 4.61 . . . 17.9 4
2b 3 4 6 10.23 7 10 7.96 16 20 4.42 21 22 −3.56 14.1 5
10 4 10 10.55 11 20 3.78 . . . . . . 26.8 3
16 4 10 12.39 11 20 5.68 7 16 –3.01 . . . 14.9 5
1, 3, 11, 12, 16, 17 4 10 10.61 16 20 7.189 9 16 2.08 . . . 17.6 5
2, 13, 14, 15, 18, 19, 4 10 10.002 16 20 5.815 . . . . . . 21.5 4
2c 20
6, 9, 10 4 10 11.37 11 20 4.41 . . . . . . 34.7 3
7 4 10 12 9 20 7.5 7 16 –5.61 . . . 16.7 6
All except 4 10 9.51 16 20 5.08 . . . . . . 23.9 4
7 4 10 11.74 9 20 6.22 6 15 –4.54 . . . 17.7 6
2d 10 4 10 7.9 . . . . . . . . . 40.9 2
11, 12, 16 4 10 10.17 16 20 6.56 9 15 2.24 . . . 23.7 5
15 . . . . . . . . . . . . 69.6 0
For datasets 2a, the majority of the performed tests identify three block functions. The start day of the second anomaly is identified 2 days later than the actual start date of the anomaly.
For dataset 2b, with 5% of Gaussian noise, several tests identify three block functions, overestimate the amplitude of both anomalies, and test three even identifies four blocks. Tests 2, 10, 13, 14,
15 lead to the identification of two block functions, solving the false positive issue. When adding 10% of noise to the data, the average norm increases. The algorithm continues to overestimate the
amplitude of the anomalies. Several tests identify the two anomalies, although the best results in terms of the norm are obtained for test 7.
Figure 6(a)
represents the results for dataset 2c.
Figure 6(a)
represents the matrix of
Figure 6(b)
represents the block functions estimated by test 1, and
Figure 6(c)
represents the block function estimated by test 14. In
Figure 6(a)
, the effect of the added noise is visible. When performing test 1, this noise is approximated by a third block function, visible in
Figure 6(b)
. Test 14 is able to ignore this noise and identifies only two block functions (
Figure 6(c)
For the last dataset, with 20% of added noise, most tests are able to identify the two block functions describing the added anomalies. The best results in terms of the norm are again obtained by test
7. For test 10, affecting the step size, only one block function is identified and the norm is the highest of all tests. For test 15 (highest f[1] penalty coefficient), no blocks are identified.
Datasets 3a–3d consider two anomalies that overlap during a few days. Two block functions should be identified. The start and end columns of the block function describing the first anomaly should be
5 and 15, and the amplitude should be 10 m^3/h. For the second anomaly the start and end columns of the block function should be 12 and 20, and the amplitude should be 5 m^3/h. Table 5 summarizes the
results obtained by performing the different tests.
Table 5
. . Block function 1 . Block function 2 . Block function 3 . . .
Dataset . Test . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . C[i] . Number of steps used .
3a All except 5 15 10.16 12 20 5.01 . . . 4.2 4
1 5 13 10.25 14 16 9.7 12 20 5.16 4 5
3b All except 5 15 10.08 12 20 5.39 . . . 4.4 4
1 12 15 15.46 7 11 10.11 16 20 5.43 4.4 4
3c All except 5 15 9.98 12 20 5.12 . . . 7.1 4
1 5 15 9.69 12 20 4.86 21 22 –1.28 5.9 4
All except 5 15 10.71 14 20 4.49 . . . 20.4 4
3d 1 9 15 12.76 5 9 9.42 16 20 5.59 16.9 4
10 5 15 8.12 9 20 4.45 . . . 18.8 4
. . Block function 1 . Block function 2 . Block function 3 . . .
Dataset . Test . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . Start . End . Amplitude (m^3/h) . C[i] . Number of steps used .
3a All except 5 15 10.16 12 20 5.01 . . . 4.2 4
1 5 13 10.25 14 16 9.7 12 20 5.16 4 5
3b All except 5 15 10.08 12 20 5.39 . . . 4.4 4
1 12 15 15.46 7 11 10.11 16 20 5.43 4.4 4
3c All except 5 15 9.98 12 20 5.12 . . . 7.1 4
1 5 15 9.69 12 20 4.86 21 22 –1.28 5.9 4
All except 5 15 10.71 14 20 4.49 . . . 20.4 4
3d 1 9 15 12.76 5 9 9.42 16 20 5.59 16.9 4
10 5 15 8.12 9 20 4.45 . . . 18.8 4
For dataset 3a, all tests except the default test identify two block functions. The estimated amplitudes are close to real amplitudes of the anomalies. For test 1, where the start and end dates are
less accurate, the algorithm also identifies a third block function with positive amplitude.
When adding 5% Gaussian noise, the results are similar. However, for test 1 the first identified block function actually describes the overlap of both anomalies, by estimating an amplitude equal to
15.46 m^3/h, and estimating accurately the start and end days of the overlap, this is 12–15.
For dataset 3c, most of the tests identify two block functions.
For dataset 3d, with 20% added noise, the becomes significantly higher. Again, test 1 leads to the best fit between the block functions and the anomalies.
Figure 7(a)
illustrate some of the obtained results when considering dataset 3c, namely the block functions obtained by test 1. Between days 12 and 15 the block functions overlap and the color of the block is
darker, illustrating the higher amplitude.
Influence of noise and parameters
Table 6 gives an overview of the influence of the noise, gap between anomalies, and the parameters considered to run the algorithm on the obtained results.
Table 6
of dataset and Effect .
parameter .
Higher noise values lead to a decrease of the estimated amplitude of the block functions – especially visible in dataset 1
Noise Higher noise values make the algorithm more sensitive to the f[1] penalty coefficient: for datasets 1 and 2 the algorithm fails to identify block functions when higher values for the
f[1] penalty coefficient are considered
Overall results for datasets 1 are better than the results for datasets 2. The difference between sets 1 and 2 is the duration of the added anomalies: for sets 2 anomalies last
Gap between longer, and the gap between them is shorter. This makes it harder for the algorithm to clearly identify two separate block functions
anomalies For datasets 2, the algorithm has more difficulties in identifying the four necessary steps to describe the block functions. For several tests, the algorithm uses, or less or more
steps, than the ones required for the block identification. For datasets 1 and 3, and for the majority of the tests, the four necessary steps are well identified
The number of clusters significantly influences the computational time. When three and four clusters are considered the average computational times are respectively 6 to 17 times
Number of longer than when two clusters are considered. Since the generated datasets have only two anomalies, setting the number of clusters equal to two is ideal. However, when performing the
clusters test to real data, from which anomalies are not known beforehand, but instead are desired to be identified, setting the number of clusters to two can entail some risks such as not
identifying more anomalies than two, if they exist. On the other hand increasing the number of clusters can lead to the identification of more blocks than the actual anomalies,
mainly if anomalies occur soon after each other and there is some noise in the data. A suitable value for the f[1] penalty factor should be chosen to prevent this issue
The number of considered steps also influences the computational time. When using five or six steps instead of four, the computational times are five and eight times longer,
Number of steps respectively
Increasing the number of steps can lead to better results, especially when more noise is added to the data. However, it also leads to the identification of extra block functions in
some cases. A suitable value for the f[1] penalty factor should be chosen to prevent this issue
Using the L[2] norm to determine the steps size leads to worse results in terms of the distance between the identified block functions and the matrix of b-factors. This effect
L[x] norm becomes even more evident when the added noise increases. On the other hand, the use of the L[2] norm seems to decrease the risk of identifying a third block
Two intermediate values for the L[x] norm were also considered (0.7 and 1.25). In some tests the lower value lead to better results, while the higher value leads to worse results
For several tests, when using a very small f[1] penalty, (0.01), the algorithm identifies a third block function, located between the anomalies. With this very small penalty, the
Penalty f[1] algorithm is not penalizing the use of more block functions and adds a block which is fitting the added noise. Increasing the f[1] penalty solves this problem. For datasets 1a–1c, it
is sufficient to consider a f[1] penalty of 0.33 However, for datasets 2a–d, the algorithm benefits from higher f[1] penalty values, and in some cases to avoid the identification of
a third block it is necessary to increase the f[1] value to 0.7
Penalty f[2] For most of the performed tests the value of the f[2] penalty has no influence on the results. The exceptions are for datasets 2c where increasing the f[2] penalty avoids identifying
a third block
of dataset and Effect .
parameter .
Higher noise values lead to a decrease of the estimated amplitude of the block functions – especially visible in dataset 1
Noise Higher noise values make the algorithm more sensitive to the f[1] penalty coefficient: for datasets 1 and 2 the algorithm fails to identify block functions when higher values for the
f[1] penalty coefficient are considered
Overall results for datasets 1 are better than the results for datasets 2. The difference between sets 1 and 2 is the duration of the added anomalies: for sets 2 anomalies last
Gap between longer, and the gap between them is shorter. This makes it harder for the algorithm to clearly identify two separate block functions
anomalies For datasets 2, the algorithm has more difficulties in identifying the four necessary steps to describe the block functions. For several tests, the algorithm uses, or less or more
steps, than the ones required for the block identification. For datasets 1 and 3, and for the majority of the tests, the four necessary steps are well identified
The number of clusters significantly influences the computational time. When three and four clusters are considered the average computational times are respectively 6 to 17 times
Number of longer than when two clusters are considered. Since the generated datasets have only two anomalies, setting the number of clusters equal to two is ideal. However, when performing the
clusters test to real data, from which anomalies are not known beforehand, but instead are desired to be identified, setting the number of clusters to two can entail some risks such as not
identifying more anomalies than two, if they exist. On the other hand increasing the number of clusters can lead to the identification of more blocks than the actual anomalies,
mainly if anomalies occur soon after each other and there is some noise in the data. A suitable value for the f[1] penalty factor should be chosen to prevent this issue
The number of considered steps also influences the computational time. When using five or six steps instead of four, the computational times are five and eight times longer,
Number of steps respectively
Increasing the number of steps can lead to better results, especially when more noise is added to the data. However, it also leads to the identification of extra block functions in
some cases. A suitable value for the f[1] penalty factor should be chosen to prevent this issue
Using the L[2] norm to determine the steps size leads to worse results in terms of the distance between the identified block functions and the matrix of b-factors. This effect
L[x] norm becomes even more evident when the added noise increases. On the other hand, the use of the L[2] norm seems to decrease the risk of identifying a third block
Two intermediate values for the L[x] norm were also considered (0.7 and 1.25). In some tests the lower value lead to better results, while the higher value leads to worse results
For several tests, when using a very small f[1] penalty, (0.01), the algorithm identifies a third block function, located between the anomalies. With this very small penalty, the
Penalty f[1] algorithm is not penalizing the use of more block functions and adds a block which is fitting the added noise. Increasing the f[1] penalty solves this problem. For datasets 1a–1c, it
is sufficient to consider a f[1] penalty of 0.33 However, for datasets 2a–d, the algorithm benefits from higher f[1] penalty values, and in some cases to avoid the identification of
a third block it is necessary to increase the f[1] value to 0.7
Penalty f[2] For most of the performed tests the value of the f[2] penalty has no influence on the results. The exceptions are for datasets 2c where increasing the f[2] penalty avoids identifying
a third block
Real data
The aforementioned synthetic data share a common characteristic: the introduced anomalies are relatively abrupt. In real data anomalies can occur either in a progressive or in an abrupt manner and
the signal may be noisier than that in the considered synthetic tests. Therefore, anomalies can be harder to detect. Thus, to assess the performance and capability of the CuBOid algorithm on the
detection of natural anomalies in real data with varying noise conditions, flow measurements series from the municipal drinking water company of the city of Paris, Eau de Paris, were considered. The
water company serves 4 million consumers during the day, and 2 million during night time, and has an average water consumption of 550,000 m³/day. For a detailed description of the Parisian drinking
water distribution system the reader is referred to Montiel & Nguyen (2011, 2013).
The flow data considered in this paper are an extract from Paris real-time SCADA system historical records. The quality of the registered data is varying, with data gaps and periods of anomalous
signals and many periods of continuous, good quality registration occurring in all of the DMAs. Van Thienen & Montiel (2014) have presented a non-censored list of registered leaks, and the results
obtained by the application of the CFPD block analyses (non-automatized, so no application of CuBOid) to these data. In most cases, the leaks could be recovered. Presently, we applied the CuBOid
algorithm to the same data in order to retrieve the anomalies from the same leakage list. Different sets of parameter values controlling the CuBOid algorithm were tested and results were compared.
For all of the tested combinations the algorithm was able to identify almost all of the registered leaks (success of identification and its practical meaning are discussed below). The differences
consisted of the estimated amplitudes and the number of identified blocks describing each anomaly. The best results, in terms of amplitudes and number of blocks, were obtained for the following set
of parameters: number of steps = 3, number of clusters = 5, L[x] = 0.7, f[1] = 0.01, f[2] = 0.7. The results obtained for this set are shown in Table 7.
As can be seen in Table 7, for most cases, the CuBOid algorithm has succeeded in autonomously detecting the anomalies. The algorithm failed to detect four of the 22 registered leaks, namely the leaks
at the DMAs of Belleville Réservoir, Cité Universitaire, Plaine Vaugirard (1) and Sorbonne. The registered leak at Belleville Réservoir is a single day event, harder to detect by the algorithm. In
the case of Cité Universitaire, data gaps prevented the CuBOid algorithm from finding good solutions. Incomplete event registrations of anomalies at Plaine Vaugirard (1) and Sorbonne hinder the
interpretation of results, although in the latter case, the anomaly which is detected seems unrelated.
For the identified anomalies the results were assessed in two ways: accuracy of identified start and end-dates and estimated intensity. Regarding the start and end-dates three situations were
identified: good agreement of a single block, good agreement of combination of blocks and identification of leakage repair. The first situation refers to anomalies that are identified by a single
block and for which the start and/or end-dates are the same, or within 1 day difference, of the corresponding reported dates. Since analysis were carried out on a monthly basis, in some cases the
end-date matches the last day of a month. This happens, for instance, for Belleville. The start-date is 1 day from the registered date, but the end-date corresponds to the last day of the period of
analysis (30-4-2011). To overcome this issue the analysis could be repeated considering a 2 month period. The second situation refers to anomalies that were found not as one single block, but as a
succession of blocks. This is probably related to the noisy character of the dataset (in the sense that many things are going on). Even though it would have been more elegant for the algorithm to
find these as a single block, for operational purposes it does not really make a difference. A special case of this type is presented by Chapelle. The corresponding signal shows a huge anomaly,
apparently unrelated to the leak, of more than 4,000 m^3/h which drops by approximately 300 m^3/h at the reported date of the fixing of the leak. The third situation refers to blocks that identify
not the leakage, but the leakage repair. In these cases, the start-date of the block is closer to the end-date of the registered anomaly. For instance, for Plaine Vaugirard (2), the algorithm
identifies a block starting at 26-04-2011, 1 day earlier than the end-date of the registered anomaly. In this case the estimated intensity of the anomaly is also negative, due to the reduction in
measured flow. This leads us to the estimated intensities: results were classified using different shades of green (or grey), representing the relative deviation from the registered value. In many
cases, the amplitude matches the amplitude estimated by Eau de Paris quite well. Note, however, that a mismatch in the start date or amplitude may also be due to an inaccuracy in the original
Figure 8
illustrates the obtained graphical results for Courcelles (month of February 2012), Maine (month of May 2012) and Vaugirard (month of June 2011). It is visible that the inherent noise of the data
make human interpretation of these block diagrams more difficult, while the algorithm performs well on clearly identifying the anomalies, emphasizing the capability and usefulness of the algorithm.
As mentioned above, some natural anomalies have a smooth rather than an abrupt initiation (a leak with growing flow rate over time). In an extension of this work, these could also be included in the
analyses with a separate type of block function, with two non-zero segments, the first linearly rising from 0 to 1 and the second a constant 1.
Operational application of the CFPD method and the CuBOid algorithm will clearly not focus on the rapid detection of large bursts. More suitable methods exist, e.g. monitoring for combined flow
increases and pressure drops above threshold levels. As CFPD depends on duration of anomalies for their detection, it is more appropriate for detecting smaller, less urgent leakages which
nevertheless may represent a significant amount of water lost over longer periods of time. As such, an accurate determination of the amplitude of anomalies is more important than an accurate
determination of the start and end dates. Also, representation by the method of a single anomaly as a succession of multiple blocks rather than a single block, as sometimes seen in our results, does
not present a problem. The method can be implemented as part of a monitoring system for relatively small leakages, identifying anomalies, e.g. one per week or month and sending suspect anomalies (for
which a grading or classification may need to be developed) to human operators for further analysis.
In this paper, we presented the CuBOid algorithm for the automated detection of anomalies in CFPD block diagrams.
The automated recognition of features in CFPD block diagrams has several advantages. The tests which have been performed demonstrate clearly that the method works well to objectively identify
anomalies in synthetic data, with automated estimation of start and end dates as well as amplitudes. Successful application of the method to real flow data from Paris, showing autonomous detection of
82% of known anomalies, shows that the CuBOid algorithm can also perform well in operational conditions. However, a broader application to different datasets and distribution systems is required to
generalize this conclusion. This algorithm can remove the need for human interpretation of matrices of a and b-factors in the CFPD block analysis method. This means that analysis time is reduced and
greater objectivity and reproducibility of the analyses are achieved. Moreover, it opens the possibility of application to automatized alarms. Therefore, the logical next step would be application in
a real distribution network as part of the operational framework.
Even though the CuBOid algorithm has been shown to provide a useful addition to the CFPD algorithm, it will fail to recognize anomalies with amplitudes significantly below system noise levels (e.g.
stochastic variability). This is a limitation of the CFPD method rather than the CuBOid algorithm, which is investigated in more detail in Van Thienen (2013), and is a limitation of other leak
detection methods as well. Also, the main power of the CFPD method is in recognizing events which last multiple days. The CuBOid algorithm does not change this, as this issue is intrinsic in the CFPD
method. For the rapid detection of anomalies within minutes or hours, more suitable methods exist.
There is, however, room for improvement in the CuBOid algorithm in the sense that events with a less block-like shape, such as slowly increasing leakage rates, can be included in the future by
defining specific shape functions for these.
Fine tuning the algorithms' parameters is important to obtain better results. At this point, the need for setting the adequate values for these several parameters might be a drawback of the presented
method. This paper provides some insights on the influence of these parameters on the outcoming results. For practical applications it would be easier to provide some rules of thumb for the choice of
these parameters. Deriving these rules requires more extensive tests, considering series of water flow data from several distribution systems with different characteristics. That is why future
developments should also include: (1) a more extensive investigation on the influence of the algorithms' parameter values on subsequent results, including combinations not considered in the present
paper (Table 1); (2) tests on real flow data coming from water distribution systems with different characteristics and containing different types of anomalies.
Considerate comments resulting from thorough reviews by three anonymous reviewers have helped to significantly improve the quality and clarity of the paper. These are gratefully acknowledged. The
authors would also like to acknowledge W-SMART and the utilities participating in and sponsoring its INCOM project (Eau de Paris and Eaux du Nord), Frank Montiel of Eau de Paris for providing the
analyzed data, Cédric Auliac of CEA for fruitful discussions and thoughtful comments on an early version of the paper, and also the European Water Innovation fund of the Dutch water companies for
additional funding. Part of the work was performed in the SmartWater4Europe project. This project has received funding from the European Union's Seventh Programme for research, technological
development and demonstration under grant agreement number 619024. | {"url":"https://iwaponline.com/jh/article/18/3/514/3516/Automated-feature-recognition-in-CFPD-analyses-of","timestamp":"2024-11-10T20:54:57Z","content_type":"text/html","content_length":"332474","record_id":"<urn:uuid:4af1a831-e37f-45ad-af02-cd51e27c49d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00181.warc.gz"} |
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• What stays the same during a collision?
Unit 8: Energy (unit content) More info about Unit 8 suitable for High School Courses: Physical Science / Physics First / Physics
Building on knowledge from previous units, students connect forces and motion to energy. Students design their own experiments to study different types of energy storage, energy transfers and
transformations. Questions answered by this eUnit:
• What are some different types of energy storage?
• How can one represent energy transfers and transformations using pie charts and bar graphs?
• How is energy related to work?
• How can we calculate the energy stored in a system?
• How do you mathematically calculate kinetic energy, elastic potential energy and gravitational potential energy?
• How can you use the conservation of energy theorem to find position or velocity of an object?
Read more information for teachers | {"url":"https://exploringphysics.com/about-the-app/","timestamp":"2024-11-06T20:04:11Z","content_type":"text/html","content_length":"80761","record_id":"<urn:uuid:9f29f12b-eb64-4c1b-bc65-45c0d9f166b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00543.warc.gz"} |
(GEN-13-14) Subject: Federal Pell Grant Duration of Eligibility and Lifetime Eligibility Used
Publication Date: May 16, 2013
Subject: Federal Pell Grant Duration of Eligibility and Lifetime Eligibility Used
Summary: This letter provides additional information on Federal Pell Grant Duration of Eligibility provisions of the HEA and the use of Lifetime Eligibility Used in its implementation.
Dear Colleague:
In Dear Colleague Letter GEN-12-01, posted on January 18, 2012, we provided information on the provisions of the Consolidated Appropriations Act, 2012 (Public Law 112-74) that impacted the federal
student aid programs authorized under Title IV of the Higher Education Act of 1965, as amended (HEA). One of those provisions limited, effective with the 2012-2013 award year, the duration of a
student’s eligibility to receive a Federal Pell Grant to 12 semesters (or its equivalent) [see HEA section 401(c)(5)]. In addition to Dear Colleague Letter GEN-12-01, we have posted several
operational electronic announcements and technical specification documents on the implementation of the Pell Grant duration of eligibility provision (see below for a listing of those postings).
The information below provides additional guidance regarding the Department’s earlier announcement to use a calculated Pell Grant “Lifetime Eligibility Used” (LEU) in the implementation of the Pell
Grant duration of eligibility provision. It also provides important information on how schools determine the amount of a student’s Pell Grant award when the student has limited duration of
eligibility (i.e.; an LEU of greater than 500 percent). Finally, this letter provides information on how a student or institution may dispute the accuracy of Pell Grant LEU information in the
Department’s Common Origination and Disbursement (COD) System.
How does the Department determine the "equivalent" of 12 semesters?
The 12 semester duration limit in the new HEA provision is the equivalent of six years of Pell Grant funding. The Pell Grant rules establish for any year in which a student receives Pell Grant
funding a “Scheduled Award” ^1 . A student whose actual disbursement of Pell Grant funds for an award year was equal to his or her Scheduled Award would have used 100% of the Scheduled Award for the
award year. Thus, under the new limitation, the maximum duration of Pell Grant funding for a student is 600%. A student who enrolled less-than full-time or not for the full academic year, or both,
would receive less than his or her Scheduled Award and would have used less than 100 percent of that award year’s Scheduled Award.
To determine the percent for any award year, divide the total of the student’s actual disbursements for the award year by the student’s Scheduled Award for that award year.
The COD System uses Pell Grant disbursement information reported by institutions since the beginning of the program^2 to calculate a student’s Pell Grant LEU by adding together each of the annual
percentages of the student’s Pell Grant Scheduled Award that was actually disbursed to the student.
The following is an example of how an LEU is calculated for a student whose receipt of Pell Grant funds varied over two award years.
The student had a 2011-2012 expected family contribution (EFC) of 0 and therefore had a 2011-2012 Pell Grant Scheduled Award of $5,550. If that student was only enrolled for the Fall 2011 semester
and therefore only received $2,775 for her full-time enrollment in one semester, she would have used 50% of her 2011-2012 Scheduled Award. If, for the 2012-2013 award year, the same student had an
EFC of 1000, with a Scheduled Award of $4,600 and enrolled as a three-quarter time student for both semesters, she would have used 75% of her Scheduled Award for the 2012-2013 award year.
If the student had not received a Pell Grant for any other award years, her total Pell Grant LEU, after the end of the 2012-2013 award year, would be 125% (50% from the 2011-2012 award year and 75%
from the 2012-2013 award year), leaving 475% before the student would reach the maximum 600% Pell duration of eligibility limit.
How Does an Institution Determine a Student’s Eligibility for a Pell Grant Award?
• LEU of 500% or Less: A student with an LEU of 500% or less is eligible to receive up to 100 percent of the full Scheduled Award for the award year, since the student has at least 100% LEU
• LEU of 600% or More: A student whose Pell LEU is 600% or more may not receive additional Pell Grant funds.
• LEU of Greater Than 500% But Less Than 600%: A student with an LEU of greater than 500% but less than 600% is not eligible to receive a full Scheduled Award but may receive a partial Pell Grant
award of the difference between 600% and the student’s LEU.
How Do Institutions Calculate Reduced Pell Grant Awards?
To determine the Pell Grant award for a student with an LEU of greater than 500% but less than 600%, an institution should follow the same procedures it would for a transfer student who received a
Pell Grant disbursement at another institution for the same award year (See Volume 3 of the Federal Student Aid Handbook, Chapter 3: Calculating Pell and Iraq & Afghanistan Service Grant Awards).
Rounding Rules - The COD System calculates a student’s LEU to the third decimal point. Institutions may not round that three-decimal LEU percentage because doing so could result in the student either
not receiving his or her full Pell Grant eligibility, or exceeding the statutory 600% limitation.
In the calculation of a student’s Pell Grant Annual Award, institutions should truncate at the cents place (e.g., $1,233.567 truncated to $1,233.56). If an institution only awards Pell Grants in
whole dollars, the award must be truncated down to the next whole dollar (e.g., $1,233.56 truncated to $1,233).
Note: If the institution only awards Pell Grants in whole dollars, a first disbursement may be increased to the next higher dollar as long as a subsequent disbursement is reduced by the next lower
These rounding requirements are demonstrated in the examples below.
Example 1
Background -
• LEU prior to the 2013-2014 award year: 534.255%.
• Program Type: Semester based credit hour program
• EFC: 0
• 2013-2014 Scheduled Award: $5,645
• Enrollment Status: Full-time for both semesters
• Annual Award: $5,645.
Award Calculation -
• Subtract the student’s LEU from 600.000%: 600.000% - 534.255% = 65.745%
• Multiply the result by the Scheduled Award: 65.745% x $5,645 = $3,711.30525 = truncated to $3,711.30
Payment Period Distribution -
• First semester: $ 2,822.50, which is the lesser of one-half of the Annual Award of $5,645 ($2,822.50), or the student’s remaining eligibility of $3,711.30.
• Second semester: $888.80, which is the lesser of one-half of the Annual Award of $5,645 ($2,822.50) or the student’s remaining eligibility after prior payment periods of ($3,711.30 less $2,822.50
= $888.80)
Example 2
Background -
• LEU prior to the 2013-2014 award year: 566.425%.
• Program Type: Semester based credit hour program
• EFC: 525
• 2013-2014 Scheduled Award: $5,095
• Enrollment Status: Full-time for both semesters
• Annual Award: $5,095.
Award Calculation -
• Subtract the student’s LEU from 600.000%: 600.000% - 566.425% = 33.575%
• Multiply the result by the Scheduled Award: 33.575% x $5,095 = $1,710.64625 = truncated to $1,710.64
Payment Period Distribution -
• First semester: $ 1,710.64, which is the lesser of one-half of the Annual Award of $5,095 ($2,547.50) or student’s remaining eligibility of $1,710.64.
• Second semester: $0, which is the lesser of one-half of the Annual Award of $5,645 ($2,822.50) or the student’s remaining eligibility after prior payment periods of ($1,710.64 less $1,710.64 =
$0). Student is no longer eligible for additional Pell Grants
Example 3
Background -
• LEU prior to the 2013-2014 award year: 555.500%.
• Program Type: Semester based credit hour program
• EFC: 1550
• 2013-2014 Scheduled Award: $4,095
• Enrollment Status: Three-quarter time the first semester, and full-time for the second semester
• Annual Award: $3,583.00 (the total of one half of the three-quarter time Annual Award for the first semester, and one half of the full-time Annual Award for the second semester or $1,535.50 +
$2,047.50 = $3,583.00).
Award Calculation -
• Subtract the student’s LEU from 600.000%: 600.000% - 555.500% = 44.500%
• Multiply the result by the Scheduled Award: 44.500% x $4,095 = $1,822.275 = truncated to $1,822.27
Payment Period Distribution -
• First semester: $ 1,535.50, which is the lesser of one-half of the three-quarter time Annual Award of $3,071 ($1,535.50) or the student’s remaining eligibility of $1,822.27.
• Second semester: $286.77, which is the lesser of one-half of the full-time Annual Award of $4,095 ($2,047.50) or the student’s remaining eligibility after prior payment periods ($1,822.27 less
$1,535.50 = $286.77).
Example 4
Background -
• LEU prior to the 2013-2014 award year: 566.425%.
• Program Type: Semester based credit hour program
• EFC: 525
• 2013-2014 Scheduled Award: $5,095
• Enrollment Status: Half-time for both semesters
• Annual Award: $2,548.
Award Calculation -
• Subtract the student’s LEU from 600.000%: 600.000% - 566.425% = 33.575%
• Multiply the result by the Scheduled Award: 33.575% x $5,095 = $1,710.64625 = truncated to $1,710.64
Payment Period Distribution -
• First semester: $ 1,274.00, which is the lesser of one-half of the Annual Award of $2,548 ($1,274.00), not to exceed the student’s remaining eligibility of $1,710.64.
• Second semester: $436.64, which is the lesser of one-half of the Annual Award of $2,548 ($1,274.00) or the student’s remaining eligibility after prior payment periods ($1,710.64 less $1,274.00 =
Example 5
The student’s LEU prior to the 2013-2014 award year is 550.000%. The student has a 0 EFC and thus a $5,645 Scheduled Award for 2013-2014. The student will be enrolled in a 900 clock hour program with
only 22 weeks of instructional time. Because this is a clock hour program, the Annual Award is also $5,645.
Background -
• LEU prior to the 2013-2014 award year: 550.000%.
• Program Type: 900 clock hour program in 22 weeks of instructional time
• EFC: 0
• 2013-2014 Scheduled Award: $5,645
• Enrollment Status: full-time in clock hour program
• Annual Award^3 : $4,776.54
Award Calculation -
• Subtract the student’s LEU from 600.000%: 600.000% - 550.000% = 50.000%
• Multiply the result by the Scheduled Award (not by the Annual Award): 50.000% x $5,645 = $2,822.50
Payment Period Distribution -
• First payment period: $2,388.26, which is the lesser of one-half of the Annual Award of $4,776.53 ($2,388.26) or the student’s remaining eligibility of $2,822.50.
• Second payment period (after the student has successfully completed the first payment period of 450 clock hours and 11 weeks): $434.24, which is the lesser of the student’s remaining eligibility
after prior payment periods ($2,822.50 less $2,388.26 = $434.24) or one-half of the Annual Award of $4,776.53 ($2,388.26).
Minimum Pell Grant Awards –
There is no de minimus award amount for purposes of determining a student’s award because of the 600% LEU limitation. For example, for the 2013-2014 award year, a student with an EFC of 0 and an LEU
of 593.000% would be eligible for the remaining 7.000% which is $395.15. Even a student with a very small remaining LEU is eligible to receive the calculated amount of Pell Grant. For example, a
student with an EFC of 2550 and an LEU of 599.500% would be eligible for the remaining 0.500% which is $15.475 truncated to $15.47 or, if necessary, rounded down to $15.00.
As noted, for further information on determining how to award and disburse a Pell Grant please reference Volume 3 of the Federal Student Aid Handbook, Chapter 3: Calculating Pell and Iraq &
Afghanistan Service Grant Awards.
Liability for Exceeding the 600% LEU Maximum
Prior to each disbursement, institutions are required to review the student’s records to ensure that the student is eligible for the financial aid disbursement. If the institution has information
that indicates that the student is not eligible for all or part of that disbursement because of the LEU limit, it must cancel or reduce, as appropriate, the student’s Pell Grant award. If an
institution disbursed Pell Grant funds beyond the student’s eligibility because it failed to follow regulatory and operational procedures, the institution is liable for the overpayment and must make
the necessary COD System adjustments.
Institutional Reporting Liability
On February 28, 2013, we published a Federal Register notice that reduced from 30 days to 15 days the timeframe for when an institution must submit to the COD System, Pell Grant disbursement
information, including adjustments to previously reported disbursements. The reduced 15 day timeframe applies to any disbursement or adjustment made on or after April 1, 2013. This reduction in the
reporting timeframe was made to limit those circumstances where a student might be required to repay a Pell Grant overpayment as a result of having exceeded the new Pell Grant LEU limitation. This
timelier reporting is especially important in the event that a student transfers from one institution to another.
An institution that does not report Pell Grant disbursement within the required 15 day timeframe may be liable for any overpayment that results from another institution disbursing Pell Grant funds
with incomplete information because of the late reporting.
Student Liability
In instances where all involved institutions were in compliance with all disbursement and reporting requirements, the student would be liable for the overpayment pursuant to the Federal Pell Grant
regulations at 34 CFR 690.79. Thus, to mitigate the possibility of a Pell Grant overpayment, we urge institutions to report Pell Grant disbursement (and adjustment) information to the COD System as
early as possible.
Disputing the Accuracy of Pell Grant LEU Information
While the COD System has the most up to date information about students’ Pell Grant disbursements, there may be circumstances where a student disputes the accuracy of the information in the COD
System. Under Public Law 112-74, the Secretary does not have the statutory authority to “waive” a student’s Pell Grant eligibility limitation. It is the responsibility of the institution where the
student is attempting to receive a Pell Grant to assist in resolving a student’s assertion that the information in the COD System is in error. Generally, confirmation or rejection of the student’s
assertion will be based on documentation obtained from one or more of the institutions reported in the COD System as having disbursed Pell Grant funds to the student.
Note that if the amount of the Pell Grant LEU percentage being disputed would not, if corrected, make a student eligible for additional Pell Grant funding, the institution should not escalate the
matter. For example, if a student’s reported LEU is 850% and he is disputing one award year’s percentage, the institution should explain to the student that even if the disputed amount is resolved in
the student’s favor, he would still remain ineligible for additional Pell Grant funding because he would be at 750% LEU, well over the 600% statutory limit.
An example of a disputed LEU would be if the student claimed that he never attended one of the institutions that reported Pell Grant disbursement information. Acceptable documentation would be a
written statement from the previous institution confirming that the student never attended, or at least never received Pell Grant funds from that institution for the award year in question.
If, based on available documentation, the institution believes that the COD System information may be incorrect, the institution, not the student, must contact Federal Student Aid’s COD School
Relations Center at 800/474-7268. The Department, after its review of the student’s assertion and any submitted information, will provide the institution with a response and instructions as to how to
Additional details on the documentation required to support a student’s claim that their Pell LEU is inaccurate will be provided a future electronic announcement to be posted to the IFAP Web site.
Contact Information
If you have questions regarding the information included in this announcement, contact the COD School Relations Center at 800/474-7268 or by e-mail at CODSupport@ed.gov.
We appreciate your help in the implementation of the Pell LEU limit and the assistance you provide to students and their families who benefit from the federal student aid programs.
Jeff Baker, Director
Policy Liaison and Implementation
Federal Student Aid
U.S. Department of Education
Federal Pell Grant LEU Operational and Technical Documents
2012-01-18 - (Dear Colleague Letter) Subject: Changes Made To The Title IV Student Aid Programs By The Recently Enacted Consolidated Appropriations Act, 2012.
2012-02-17 - (Grants) Subject: Preliminary Information - Implementation of the 12 Semester Lifetime Limit for Federal Pell Grants
2012-04-06 - (COD System) Subject: Additional COD System Implementation for 2012-2013 Award Year
2012-05-10 - (Grants) Subject: Reminder - Retrieving the Pell Grant LEU Report Through the School's Reporting Newsbox on the COD Web site
2012-06-14 - (Grants) Subject: Pell Grant LEU Information - Additional 2012-2013 COD System Update
2012-06-29 - (Grants) Subject: Pell Grant LEU Information - July 2012 System Updates for Lifetime Limit for Federal Pell Grants
2012-08-13 - (General) Subject: Pell Grant Lifetime Eligibility Used: Importance of Timely Reporting
A student’s Scheduled Award is based on the maximum Pell Grant award amount for the award year, the student’s Expected Family Contribution (EFC), and the student’s Cost of Attendance (COA) and is the
amount the student would receive if enrolled full-time for an entire academic year.
The Pell Grant Program was formerly known as the Basic Educational Opportunity Grant Program and began with the 1973-1974 award year.
See Volume 3, Chapter 3 of the Federal Student Aid Handbook for more information on how to calculate a Pell Grant award for a clock-hour program. Because the student’s program is only 22 weeks in
duration and not the full statutory minimum of 26 weeks of instructional time, the payment period amounts must be calculated under Pell Grant Formula 4 by multiplying the Scheduled Award by the
lesser of the fraction provided in 34 CFR 690.63(e)(2)(i) and (ii). The student’s annual award is 22 weeks divided by 26 weeks times the Scheduled Award of $5,645 = $4,776.5384 truncated to | {"url":"https://fsapartners.ed.gov/knowledge-center/library/dear-colleague-letters/2013-05-16/gen-13-14-subject-federal-pell-grant-duration-eligibility-and-lifetime-eligibility-used","timestamp":"2024-11-09T03:26:51Z","content_type":"text/html","content_length":"42537","record_id":"<urn:uuid:7058a295-c0f6-4b5a-a535-fdacbf04b08a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00020.warc.gz"} |
Linear and nonlinear stability of acoustics with nonuniform entropy in chambers with mean flow
The linear and quadratic coupling of small fluctuating quantities of entropy in the acoustic field of a chamber are studied in order to help predict stability in ramjet combustion chambers. In the
linear case, the entropy and acoustics are decoupled in the interior of the chamber. All linear coupling occurs at the boundary conditions. For cases where the entropy fluctuations are of the same
order of magnitude as the pressure oscillations and the coupling is of order one, the linear stability of the acoustic field is strongly dependent upon the entropy fluctuations. In the nonlinear
case, the acoustic-entropy interactions are much smaller than the acoustic-acoustic interactions in the limit of the assumptions.
American Institute of Aeronautics and Astronautics Conference
Pub Date:
June 1987
□ Acoustic Instability;
□ Combustion Stability;
□ Entropy;
□ Flow Chambers;
□ Frequency Stability;
□ Coupled Modes;
□ Flow Distribution;
□ Pressure Oscillations;
□ Ramjet Engines;
□ Sound Waves;
□ Acoustics | {"url":"https://ui.adsabs.harvard.edu/abs/1987aiaa.confR....H/abstract","timestamp":"2024-11-07T03:21:48Z","content_type":"text/html","content_length":"35168","record_id":"<urn:uuid:04df330b-158e-4bff-a4c6-e2a48e9a4dd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00364.warc.gz"} |
ManPag.es -
sgerfs.f −
subroutine SGERFS (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
Function/Subroutine Documentation
subroutine SGERFS (characterTRANS, integerN, integerNRHS, real, dimension( lda, * )A, integerLDA, real, dimension( ldaf, * )AF, integerLDAF, integer, dimension( * )IPIV, real, dimension( ldb, * )B,
integerLDB, real, dimension( ldx, * )X, integerLDX, real, dimension( * )FERR, real, dimension( * )BERR, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
SGERFS improves the computed solution to a system of linear
equations and provides error bounds and backward error estimates for
the solution.
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= ’N’: A * X = B (No transpose)
= ’T’: A**T * X = B (Transpose)
= ’C’: A**H * X = B (Conjugate transpose = Transpose)
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A is REAL array, dimension (LDA,N)
The original N-by-N matrix A.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF.
LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B is REAL array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
On exit, the improved solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR is REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters:
ITMAX is the maximum number of steps of iterative refinement.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 185 of file sgerfs.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://manpag.es/SUSE131/3+sgerfs.f","timestamp":"2024-11-11T17:07:02Z","content_type":"text/html","content_length":"22496","record_id":"<urn:uuid:7058a419-5618-4a96-9301-2ee8e7b5b384>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00161.warc.gz"} |
How do you convert fractions to decimals?. In this lesson you will learn how to convert fractions to decimals to the tenths place by using visual aids. - ppt download
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CS250: Python for Data Science
This unit will introduce you to the field of data science. Before delving into the programming aspects of the course, it is important to have a clear view of what data science is. There are many
techniques and computational methodologies for dealing with data science problems. The goal of this unit is to help put the rest of the course in context and help you understand how to conceptually
organize various facets of the field.
When attempting to solve a data science problem, the overarching goal is to derive inferences and draw conclusions based on existing data sets. Such inferences are made through statistical,
computational, and visualization techniques. Furthermore, even before computations can be made, data sets must often be curated and refined. This unit will help you to order and categorize your
thinking to understand the flow of data science processes.
Completing this unit should take you approximately 7 hours. | {"url":"https://learn.saylor.org/course/view.php?id=504§ion=1","timestamp":"2024-11-05T03:28:57Z","content_type":"text/html","content_length":"451496","record_id":"<urn:uuid:1045e0b1-f5eb-4ab0-a119-58626e0df99f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00653.warc.gz"} |
On Success runs of a fixed length defined on a q-sequence of binary
On Success runs of a fixed length defined on a q-sequence of binary trials
We study the exact distributions of runs of a fixed length in variation which considers binary trials for which the probability of ones is geometrically varying. The random variable E_n,k denote the
number of success runs of a fixed length k, 1≤ k ≤ n. Theorem 3.1 gives an closed expression for the probability mass function (PMF) of the Type4 q-binomial distribution of order k. Theorem 3.2 and
Corollary 3.1 gives an recursive expression for the probability mass function (PMF) of the Type4 q-binomial distribution of order k. The probability generating function and moments of random variable
E_n,k are obtained as a recursive expression. We address the parameter estimation in the distribution of E_n,k by numerical techniques. In the present work, we consider a sequence of independent
binary zero and one trials with not necessarily identical distribution with the probability of ones varying according to a geometric rule. Exact and recursive formulae for the distribution obtained
by means of enumerative combinatorics. | {"url":"https://api.deepai.org/publication/on-success-runs-of-a-fixed-length-defined-on-a-q-sequence-of-binary-trials","timestamp":"2024-11-13T19:11:32Z","content_type":"text/html","content_length":"154278","record_id":"<urn:uuid:5463dcf8-5db9-4aed-b1c9-c43e3201b0e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00858.warc.gz"} |
Thermodynamics of Computation
This category uses the form Reference Material.
Semantic Drilldown
Reference group property = Reference group
Author property = Author name
Type property = Type
Source property = Source name, requires = Type
Volume property = Volume, requires = Source
Year property = Year
?Author name;?Source name;?Volume;?Pages;?Year;mainlabel=-;format=template;template=Reference Material Display | {"url":"https://centre.santafe.edu/thermocomp/w/index.php?title=Category:Reference_Materials&veaction=edit","timestamp":"2024-11-10T13:03:22Z","content_type":"text/html","content_length":"78448","record_id":"<urn:uuid:6b70983b-fdaf-4e07-b00f-24d4205107da>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00131.warc.gz"} |
Go to the source code of this file.
subroutine zlat2c (UPLO, N, A, LDA, SA, LDSA, INFO)
ZLAT2C converts a double complex triangular matrix to a complex triangular matrix.
Function/Subroutine Documentation
subroutine zlat2c ( character UPLO,
integer N,
complex*16, dimension( lda, * ) A,
integer LDA,
complex, dimension( ldsa, * ) SA,
integer LDSA,
integer INFO
ZLAT2C converts a double complex triangular matrix to a complex triangular matrix.
Download ZLAT2C + dependencies
[TGZ] [ZIP] [TXT]
ZLAT2C converts a COMPLEX*16 triangular matrix, SA, to a COMPLEX
triangular matrix, A.
RMAX is the overflow for the SINGLE PRECISION arithmetic
ZLAT2C checks that all the entries of A are between -RMAX and
RMAX. If not the convertion is aborted and a flag is raised.
This is an auxiliary routine so there is no argument checking.
UPLO is CHARACTER*1
[in] UPLO = 'U': A is upper triangular;
= 'L': A is lower triangular.
N is INTEGER
[in] N The number of rows and columns of the matrix A. N >= 0.
A is COMPLEX*16 array, dimension (LDA,N)
[in] A On entry, the N-by-N triangular coefficient matrix A.
LDA is INTEGER
[in] LDA The leading dimension of the array A. LDA >= max(1,N).
SA is COMPLEX array, dimension (LDSA,N)
Only the UPLO part of SA is referenced. On exit, if INFO=0,
[out] SA the N-by-N coefficient matrix SA; if INFO>0, the content of
the UPLO part of SA is unspecified.
LDSA is INTEGER
[in] LDSA The leading dimension of the array SA. LDSA >= max(1,M).
INFO is INTEGER
= 0: successful exit.
[out] INFO = 1: an entry of the matrix A is greater than the SINGLE
PRECISION overflow threshold, in this case, the content
of the UPLO part of SA in exit is unspecified.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 112 of file zlat2c.f. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d6/dff/zlat2c_8f.html","timestamp":"2024-11-03T18:31:00Z","content_type":"application/xhtml+xml","content_length":"12180","record_id":"<urn:uuid:50503e85-f659-4d3a-9888-5a715c280bc7>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00032.warc.gz"} |
American Mathematical Society
Models and Methods for Sparse (Hyper)Network Science in Business, Industry, and Government
Sinan G. Aksoy
Aric Hagberg
Cliff A. Joslyn
Bill Kay
Emilie Purvine
Stephen J. Young
The authors of this piece are organizers of the AMS 2022 Mathematics Research Communities summer conference Models and Methods for Sparse (Hyper)Network Science, one of four topical research
conferences offered this year that are focused on collaborative research and professional development for early-career mathematicians. Additional information can be found at https://www.ams.org/
programs/research-communities/2022MRC-HyperNet. Applications are open until February 15, 2022.
The authors are hosting an AMS sponsored Mathematics Research Community (MRC) focusing on two themes that have garnered intense attention in network models of complex relational data: (1) how to
faithfully model multi-way relations in hypergraphs, rather than only pairwise interactions in graphs; and (2) challenges posed by modeling networks with extreme sparsity. Here we introduce and
explore these two themes and their challenges. We hope to generate interest from researchers in pure and applied mathematics and computer science.
The Rise of Network Science
Graph theory has been driven by applied questions, from its apocryphal roots in the Seven Bridges of Königsberg problem to modern day network analysis. What had been cast in its infancy as a
collection of recreational puzzles has evolved into an expansive and diverse discipline. Graph analyses are now common across nearly all areas of science. Accordingly, modern graph theory has evolved
to engage methods from probability, topology, linear algebra, mathematical logic, computer science, and more.
Relational phenomena involving specific patterns of linkage between entities can afford natural representations as graphs. Prime examples are network systems, often massive, which arise in fields
such as molecular biology, social systems, cyber systems, materials science, infrastructure modeling (e.g., Figure 1), and high performance computing.
Figure 1.
Synthetic Texas power transmission network generated from publicly availableFootnote^1 https://egriddata.org/dataset/activsg2000-2000-bus-synthetic-grid-geolocated-texas^✖ test data.
As elucidated by Chung in a 2010 Notices article Chu10, despite coming from disparate domains, such networks exhibit “amazing coherence” in their shared empirical properties. Such hallmarks include
sparsity (the number of edges is linear in the number of vertices), the small world phenomenon (any two vertices are connected by a short path, local neighborhoods are typically dense), and
heavy-tailed degree distributions (the number of degree vertices is roughly ). Researchers have addressed fundamental questions surrounding these networks—how they evolve, which structures are
critical to their function, which graph invariants capture meaningful properties, and so on—in an area commonly referred to as “network science” NBW06.
Since Chung’s Notices article 12 years ago, the scope of network science has continued to grow beyond emphasizing small-world ubiquity, and into studying richer classes of mathematical structures
that better reflect the nuances of real-world networks. In part due to the increasing widespread availability of complex relational data sets, researchers have coalesced around a new class of applied
questions where the properties of the relational data are, in and of themselves, driving the questions.
Relations Beyond Graphs
Over the past several decades, there has been an increasing realization within network science that multi-way interactions can play a critical role in networked systems. For instance, as highlighted
by COVID-19 spread, the interactions at group gatherings can have a cumulative effect that can be obscured when reduced to pairwise interactions. In order to faithfully capture these multi-way
interactions, it is valuable to move beyond the standard graph structure consisting of vertices and edges , to the richer framework of hypergraphs, where the edge set is a subset of , the power set
of .
Where graphs can represent only pairwise relations natively, hypergraphs naturally code for multi-way interactions. Nonetheless, it is routine to resort to analyzing systems and data exhibiting
multi-way interactions via “auxiliary graphs” produced from multi-way data, such as the line graph (which encodes intersections between pairs of hyperedges) or the 2-section graph (which replaces
hyperedges with graph cliques). However, as illustrated by Figure 2, two non-isomorphic hypergraphs may have the same line graph. Similarly, two non-isomorphic hypergraphs may have the same
2-sections as well. Simply put, these most natural encodings of hypergraphs by auxiliary graphs fail to retain some pertinent information. Despite hopes that incorporating weights into the auxiliary
graphs would allow faithful representation of hypergraphs via graphs, recent work by Kirkland Kir18 shows that this is not the case.
And while hypergraphs are bijective to bipartite graphsFootnote^2 Technically “bicolored,” there are caveats here in the case of disconnected hypergraphs.^✖ in which one of the parts is labeled as
vertices and the other as edges, naive deployment of graph methods against them will not necessarily reveal the “set”-valued properties of the original hypergraph. The resulting algorithms are at
best cumbersome to phrase and study in this framework, and at worst simply recapitulate the corresponding hypergraph-native methods. Thus, it is apparent that hypergraphs require their own analytical
tool set to avoid the information loss inherent to graph reduction or bipartite approaches.
Figure 2.
Non-isomorphic hypergraphs with the same line graph.
While shifting from modeling pairwise to multi-way interactions may seem like a subtle change, the implications are far-reaching and profound. For example, in a hypergraph the natural generalization
of a walk of length is a sequence , …, of hyperedges such that for all we have . However, unlike in graphs these pairwise intersections have a non-trivial notion of “width,” i.e., . This allows the
set of hypergraph walks to be partitioned by functions of their width, such as the minimum or mean width of intersections. In contrast with graphs, these width-based partitions induce non-trivial
filtrations on a set of hypergraph walks. Since walks are foundational to defining many network science concepts, these filtrations in turn induce filtrations on component structure, connectivity,
diameter, centrality, etc., which can be used to provide further insight into the network structure AJM20.
Building off this increased expressivity, a number of analytical tools have been developed to study hypergraphs, ranging from walk and centrality based methods AJM20Ben19, motif and subgraph pattern
analysis LKS20, and dynamical processes on hypergraphs dATM21LR20. Additionally, hypergraphs interact strongly with important structures from computational topology such as abstract simplicial
complexes (hypergraphs that include all possible subedges), and there is a burgeoning movement to join network science to analytical approaches bridging to these higher-order mathematical fields
Challenges of (Hyper)Analytics
Rather than attempt a methods survey, here we discuss thematic challenges associated with hypergraph spectral methods that reflect common issues facing hypernetwork science. Hypergraph Laplacians and
associated spectral methods are commonly used to obtain embeddings, rank entities, and cluster data across domain areas, ranging from partitioning circuit netlists in VLSI, grouping term-document
data in natural language processing, and performing image segmentation. How to optimally define hypergraph matrix and tensor representations to better enable such analyses has emerged as a central
Despite a plethora of proposals over the past several decades, there is little consensus as to which notion of hypergraph Laplacian is most appropriate. Furthermore, such proposals are starkly
different, depending on whether or not one assumes uniformly sized hyperedges. For example, in the uniform case, Chung Chu93 took a homological approach, Lu and Peng LP11 introduced a so-called
higher-order generalized Laplacian rooted in hypergraph random walks, while Cooper and Dutle CD12 utilize multilinear-algebraic techniques to study multidimensional arrays they call hypermatrices.
Unfortunately, there appears to be no obvious way to extend these notions to non-uniform hypergraphs. Accordingly, these methods likely have limited applicability to hypergraphs arising from real
data, which are almost always naturally non-uniform.
While proposed non-uniform hypergraph Laplacians are applicable to real, messy hypergraph data, whether they effectively capture higher order structure present in hypergraphs but absent in graphs is
disputed. As shown by Agarwal ABB06, a number of non-uniform hypergraph Laplacians are related, via trivial shifts and scalings, to graph Laplacians associated with the auxiliary graphs mentioned
above like the two-section (clique expansion). To mitigate the information loss inherent in such reductions Kir18, one approach is to study hypergraph matrices associated with non-reversible random
walks CR19HAPP20, while other recent work advocates non-uniform hypergraph adjacency tensors BCM17. However, these and other “hypergraph-native” approaches often come with caveats, underscoring the
difficulty of devising practical yet faithful hypergraph methods: in this case, the former approach requires external weights to be effective, while the high-dimensionality of the tensor suggested in
the latter poses computational challenges.
Modeling Sparsity
In addition to developing hypergraph analytic tools, network science is also grappling with how to develop models that capture the unusual combination of extreme properties exhibited by many complex
networks. From the very first attempts to develop a robust theory of random graphs, it was recognized that the models being developed were, at best, imperfect representations of the real world.
Indeed, Erdős and Rényi pointed this out in ER61:
“The evolution of random graphs may be considered as a (rather simplified) model of the evolution of certain real communication-nets, e.g. the railway-, road- or electric network system of a
country or some other unit, or of the growth of structures of anorganic or organic matter, or even of the development of social relations. Of course, if one aims at describing such a real
situation, our model of a random graph should be replaced by a more complicated but more realistic model.”
Since then numerous random graph models have been developed to capture various underlying structural or mechanistic properties; including approaches to capture the degree sequence (either exactly or
probabilistically), network self-similarity, structural restrictions, network evolution based on preferences or biological mechanisms, and others.
Despite the proliferation of random graph models, there are significant structural features of data for business, industrial, and governmental (BIG) applications that still are not captured. For
instance, while many of the networks important in BIG applications exhibit both connectivity and extreme sparsity, random graph models typically require an average degree of (for models with more
edge independence) or at least 3 (for models with less edge independence) in order to ensure connectivity. However, for systems such as the power grid (see Figure 1) or networks built from
communication traces, connectivity is present a priori despite an average degree between 1 and 2.
In addition, many BIG applications are driven by experimental data that are essentially correlational in nature. Examples include correlated gene expression across multiple experimental conditions or
macroscale structural properties of novel materials across a variety of microscale properties. These data sources are naturally represented in terms of a weighted hypergraph, and yet, many of the
current analytical methods applied to these data sources rely on graph (as opposed to hypergraph) models. While there are many reasons for this discrepancy, one of the major contributing factors is a
relative lack of random hypergraph models which can be meaningfully parameterized to be reflective of observed correlational data. While random bipartite graph models exist, they suffer from the
problems described above. Between the need for connected random models exhibiting extreme sparsity, the increasing relevance of hypergraph data sources, and other peculiarities of BIG data sources,
there is a significant opportunity to develop novel random hypergraph models driven by a new class of applications.
An Invitation
The authors of this article are organizing an AMS MRC in the summer of 2022 on these topics. We will be exploring the way that graphs and hypergraphs can be employed in real-world scenarios such as
those in biology, computer science, social science, and power engineering. The goal of this collaborative workshop is to bring together researchers from multiple different domains including
mathematics, computer science, and application domains to develop and extend graph-theoretical concepts that are rooted in problems of national significance, including:
In critical infrastructure systems, such as the power grid or natural gas distribution system, it is often necessary to understand the combinatorial structure of the system to understand
macroscale system behavior.
Computer network data represents point to point information exchanges such as emails, network traffic, or process logs. This type of data is frequently modeled as a rapidly changing dynamic graph
with the goal of discovering behavioral patterns and anomalies in the system.
In the case of *-omics data from biology, much of the data is pairwise, or multi-way, rate of expression under various environmental conditions. This naturally leads to a variety of graphical
structures, from directed hypergraphs to undirected graphs, depending on the choice of data representation.
A key factor in the understanding of the behavior of microbial communities is the directed graph of reinforcing interactions, i.e., the presence of microbe A increases with the increase of
microbe B.
In blogging and social networks such as Twitter, users interact with external content by posting links, thereby forming a user-content hypergraph whose structure affects information spread.
We invite early-career applicants from all domains to join us. The most crucial characteristic of the applicants is the desire to build a community that is willing to teach and learn about other
disciplines and to form true interdisciplinary teams. The organizers have identified several deep theoretical problems and will provide guidance and resources as participants tackle them. In addition
to the technical collaborations, there will be opportunities to learn about research in the national laboratory system and in industry, expand networks, and participate in other professional
development activities. We invite you to apply and join us in exploring this topic of theoretical interest and practical significance.
This manuscript has been authored in part by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the US Department of Energy. The United States Government retains and the publisher, by
accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of
this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance
with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
This work was supported in part by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National
Nuclear Security Administration of US Department of Energy (Contract No. 89233218CNA000001).
Pacific Northwest National Laboratory is a multi-program national laboratory operated for the US Department of Energy (DOE) by Battelle Memorial Institute under Contract No. DE-AC05-76RL01830.
Figures 1 and 2 were created by Sinan Aksoy. | {"url":"https://www.ams.org/journals/notices/202202/noti2424/noti2424.html?adat=February%202022&trk=2424&pdfissue=202202&pdffile=rnoti-p287.pdf&cat=none&type=.html","timestamp":"2024-11-13T22:14:28Z","content_type":"text/html","content_length":"135597","record_id":"<urn:uuid:5db98b65-250c-4238-901e-eb8b279c0007>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00236.warc.gz"} |
Degrees of Freedom Calculator
Use this degrees of freedom calculator to find out the crucial variable of one and two sample t tests and chi-square test and also ANOVA.
What Are Degrees of Freedom?
The possible values in a dataset that can be altered to get the proper estimation of the data are called degrees of freedom.
How To Find Degrees of Freedom?
No doubt the best way to calculate the statistical variable is by using free degree of freedom calculator. But you must also comprehend the manual calculations that are possible only if you take into
consideration the following expressions:
Degrees of Freedom Formula:
Let’s have a look at the following statistical tests and their related formulas for degrees of freedom calculation:
1-Sample t-Test:
For this test, you can calculate dof by following the equation below:
df = N - 1
N = Total values present in a dataset
df = Degrees of Freedom
2-Sample t-Test:
Here we have a suitable partition for equal and unequal variances:
Equal Variances:
In case of equal dispersion of the data set, the degree of freedom is calculated by this formula:
df = N₁ + N₂ - 2
N₁ = First sample entities
N₂ = Second sample entities
Unequal Variances:
In case of unequal data expansion, the degree of freedom formula is given as:
df = (σ₁/N₁ + σ₂/N₂)2 / [σ₁2 / (N₁2 * (N₁-1)) + σ₂2 / (N₂2 * (N₂-1))],
σ = Variance (for calculations, tap variance calculator)
For this statistical procedure, we have the following degrees of freedom equations:
Between Groups:
df = k - 1,
Within Groups:
df = N - k,
Overall DOF:
df = N - 1
Chi-Square Test:
The degrees of freedom statistics for Chi Squared test can be analysed by subjecting to the formula as given below:
df = (rows - 1) * (columns - 1)
For quick and better approximations, start using this best degrees of freedom calculator.
How To Calculate Degrees of Freedom?
Let’s move ahead and resolve a couple of examples to clarify the concept in more depth!
Example # 01:
How to find degrees of freedom for t Test with data values as 23?
Here we have:
N = 23
Calculating degrees of freedom:
df = N-1
df = 23 -1
df = 22
Example # 02:
How to determine degrees of freedom for a Chi Square table representing the marital status by education below:
Status Middle or Lower School (%) High School (%) Bachelor’s (%) Master’s (%) PhD (%) Total (%)
Single 46 40 25 17 18 30
Married 31 40 54 67 64 50
Divorced 15 10 11 6 9 10
Widowed 8 10 11 11 9 10
Total 100 100 100 100 100 100
Here we have:
Number of column = 5
Number of rows = 4
Performing degree of freedom calculation:
df = (rows - 1) * (columns - 1)
df = (4 - 1) * (5 - 1)
df = 3 * 4 df = 12
How Our Calculator Works?
Let's learn together how you can swiftly find degree of freedom in a couple of clicks with this free dof calculator. Stay with it!
• From first drop-down list, select for which test you wish to find this particular variable
• After you make a selection, do enter all required elements in their designated fields
• At last, tap the calculate button
• Degree of freedom for selected test type
• T-Statistics
• Standard Deviations
From the source of Wikipedia: Degrees of freedom, Applications, Mechanics From the source of Study.com: Degrees of Freedom, critical values | {"url":"https://calculator-online.net/degrees-of-freedom-calculator/","timestamp":"2024-11-04T05:28:41Z","content_type":"text/html","content_length":"68935","record_id":"<urn:uuid:99adc1cc-88dc-4176-8a3a-19d74b3e7f64>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00748.warc.gz"} |
Isometric Immersions and Embeddings of Locally Euclidean Metrics
£ 40.00
I. Kh. Sabitov, Moscow State University, Russia
The aim of this volume is to review the results on isometric immersions of locally Euclidean metrics in Euclidean spaces along with the description of the extrinsic geometry of these immersions.
2009 Pbk ISBN; 976-1-904868-62-0 270pp
The review begins with the consideration of a problem specific only for constant curvature metrics, namely the problem of “natural” realization of locally Euclidean metrics by Euclidean-space domains
of corresponding dimension with the standard Euclidean metric and then studies their isometric immersions in Euclidean spaces of greater dimension with emphasis on the problems of smoothness
There are no reviews yet.
Only logged in customers who have purchased this product may leave a review. | {"url":"https://cambridgescientificpublishers.com/product/isometric-immersions-and-embeddings-of-locally-euclidean-metrics","timestamp":"2024-11-12T20:22:28Z","content_type":"text/html","content_length":"144247","record_id":"<urn:uuid:7751e7e5-932e-4fdf-8982-b9cd94a96e39>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00655.warc.gz"} |
Where is the ring structure for real numbers declared?
I would like to define a different power function to be used on real number formulas, starting with the ring and ring_simplify tactics. I wish to just re send the Add Ring command for R, but I don't
find where this Add Ring is in the sources. Can somebody point me to the right place?
I think it's implicit in Add Field Rfield, in RealField.v
That seems logical, the Rfield structure contains a copy of RTheory
see also https://github.com/coq/coq/pull/10734
Add Field does force a Add Ring behavior. Each addition overwrites the effect of the previous one.
The Add Field command that takes effect for final users that called Require Import Reals. is actually found in theories/Reals/RIneq.v line 2689.
I am frustrated that exponent iin power expressions is processed through full computation. As result, if my exponent is 10, it is expanded to a large ugly expression:
((R1 + R1) * (R1 + (R1 + R1) * (R1 + R1)))
I tried to replace IZR by an inert function, which would have to be removed at post-processing time, but I don't manage to make the post-processing to work. And I was unable to find examples of such
post-processing in the sources of Coq.
My code is visible here
I must have made a mistake in my previous search. postprocessing is used in theories/setoid_ring/ArithRing.v
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/Where.20is.20the.20ring.20structure.20for.20real.20numbers.20declared.3F.html","timestamp":"2024-11-05T15:37:59Z","content_type":"text/html","content_length":"7959","record_id":"<urn:uuid:f29fec91-1b9b-4618-a2c9-c16a65890c31>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00207.warc.gz"} |
How to use RANDBETWEEN function in ExcelHow to use RANDBETWEEN function in Excel
How to use RANDBETWEEN function in Excel
In this article, we will learn How to use the RANDBETWEEN function in Excel.
Random number generator means to get some random numbers between given numbers. Let's say if a user needs to get random numbers between 1-6 (roll of dice), the user doesn't need a dice for it. Excel
provides you with function. Let's understand how function works and an example illustrates its use. You can also get random numbers in decimal.
RANDBETWEEN function in Excel
The RANDBETWEEN function in excel randomizes some numbers based on lower limit and upper limit.
RANDBETWEEN function Syntax
top : number cannot be above this number.
bottom : number cannot be lower than this number.
Note: RANDBETWEEN function is volatile and changes its value very soon. To record the numbers copy and paste only values.
Example :
All of these might be confusing to understand. Let's understand how to use the function using an example. Here we have some examples to see how to get random numbers between 10 and 50.
Use the formula
Just copy and paste this formula to the required number of times you need the result.Here only 6 numbers are generated. The RANDBETWEEN function only returns the whole number not any decimal number.
But we can use it to convert the random decimal number.
Random decimal numbers with RANDBETWEEN functionÂ
To get Random decimal numbers between two values let's say 1 to 10. So what can be the possible outcome 0.1, 1.234, 5.94, 3.5, 9.3222. It can be upto 1 decimal digit or 2 decimal digits. Now you
don't need to worry. Let's say we have x and y two numbers. And we need to get 2 decimal digit numbers. So multiple both numbers with 100.
x -> 100x (new bottom number)
y -> 100y (new top number)
New formula
Let's get the random numbers between 10 and 50 upto 1 decimal digit.
Use the formula
As you can see now we decimal random numbers between 10 and 50. This simple method will solve the decimal number. Now similarly we can get the 2 decimal digit random numbers.
Use the formula
As you can clearly see random numbers between 2 numbers.
RANDOM TEXT values using RANDBETWEEN function
Now use the INDEX formula, this is more helpful than the CHOOSE formula as it doesn't require to input individual values. You can provide the fixed array using the naming the list using named range.
Use the formula :
list : named range used for A3:A14.
Explanation :
1. ROWS(table) returns the number of rows in the table which will be the last index of value in the list which is 12.
2. RANDBETWEEN function returns a random number from 1 to 12.
3. INDEX function will return the value corresponding to the returned number in the first column as list has only one column.
This is much simpler and easier. Now we will use the above formula for the value in the table.
RANDBETWEEN refreshes every time when something gets changed in the workbook. So when you are satisfied with the random data. Copy and paste values using the Paste special shortcut to get fixed
values. You can learn how to get the random numbers or random date values in Excel
Here are all the observational notes using the formula in Excel
Notes :
1. The function refreshes every time when something changes in the workbook.
2. The RANDBETWEEN function returns an error, if the first number argument ( bottom ) is larger than the second number argument ( top ).
3. The RANDBETWEEN function returns error, if the argument to the function is non - numeric.
4. The function is mostly used to randomize a list in Excel or generate a random whole number.
Hope this article about How to use the RANDBETWEEN function in Excel is explanatory. Find more articles on generating values and related Excel formulas here. If you liked our blogs, share it with
your friends on Facebook. And also you can follow us on Twitter and Facebook. We would love to hear from you, do let us know how we can improve, complement or innovate our work and make it better for
you. Write to us at info@exceltip.com.
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Sigma Notation Worksheets
Sigma notation is a method used to write out a long mathematical expression or sequence in a concise way. There are situations where students are required to add an infinite quantity, and as you
know, it is impossible to add a quantity infinite times, not something humanly possible. But, math always comes to rescue; even, in this case, there is an easy way to do that. This is when we can
bring the concept of 'series' in the light. It is a major part of calculus and something that we can help us in our day-to-day activities. Series is the process or operation of adding infinitely many
quantities to a given first quantity. While we discuss series, there is one important concept that you need to be well-aware of, and that is the sigma notation of series. So, what is the sigma
notation of a series? Sigma is a Greek letter, denoted by Σ { , } which is used to represent a 'sum.'
In these worksheets, your students will learn to use sigma notation to express the value of sequences. They will simplify expressions using sigma notation. They will find the difference between two
sequences expressed in sigma notation. There are 6 worksheets in this set. These are moderately complex problems and a sound understanding of trigonometry is required in order for students to be
successful with these worksheets. This set of worksheets contains step-by-step solutions to sample problems, both simple and more complex problems, a review, and a quiz. It also includes ample
worksheets for students to practice independently. Most worksheets contain between eight and ten problems. When finished with this set of worksheets, students will be able to use sigma notation to
express the value of sequences. These worksheets explain how to use sigma notation to express the value of sequences. Sample problems are solved and practice problems are provided.
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Denier to Ne Calculator - Textile Calculator
Denier to Ne Calculator
Denier to Ne Calculator
The formula of Denier to Ne Calculator
The formula to convert denier to ne, where denier is a measure of the linear mass density of fibers and ne is a measure of yarn count, is as follows:
Ne= 5315/denier
This formula allows you to convert denier to Ne, where Ne represents the yarn count and denier is the measure of linear mass density of fibers.
If you’re in the textile industry, you’ve likely come across terms like “denier” and “Ne” (pronounced as “Nee”). These are essential metrics used in determining the characteristics of textile fibers
and yarns. But what exactly do they mean, and how are they related? In this comprehensive guide, we’ll delve into the world of denier and Ne, and how you can use a Denier to Ne Calculator to
streamline your textile calculations.
What is Denier?
Denier is a unit of measurement that refers to the linear mass density of fibers. It quantifies the mass in grams per 9000 meters of the fiber. Essentially, denier tells you how thick or thin a fiber
is. The higher the denier value, the thicker the fiber.
Calculate Denier to Ne
How Denier is Calculated
To calculate denier, you divide the mass of the fiber (in grams) by its length (in meters) and then multiply by 9000. The formula is:
Denier=Mass (in grams)/Length (in meters)×9000
What is Ne?
Ne, on the other hand, stands for “Number English” and is a measure of yarn count. It indicates the number of 840-yard lengths (or “hanks”) of yarn that weigh one pound. In simpler terms, Ne tells
you how fine or coarse a yarn is. A higher Ne value corresponds to a finer yarn.
How Ne is Calculated
To calculate Ne, you use the formula:
Using a Denier to Ne Calculator
Now, you might be wondering how to perform these calculations quickly and accurately. That’s where a Denier to Ne Calculator comes in handy. These online tools allow you to input the denier value of
a fiber and instantly get the corresponding Ne value, saving you time and effort.
Features of Denier to Ne Calculators
• User-Friendly Interface: Most calculators have a simple interface that makes it easy to input denier values and obtain Ne results.
• Instant Conversion: With just a click of a button, you can convert denier to Ne within seconds.
• Accurate Results: Denier to Ne calculators are built on precise mathematical formulas, ensuring accurate conversions every time.
Benefits of Understanding Denier and Ne
Understanding denier and Ne is crucial for various aspects of the textile industry, including:
• Product Development: Manufacturers use denier and Ne values to develop textiles with specific characteristics, such as thickness and softness.
• Quality Control: Textile companies rely on denier and Ne measurements to maintain consistent quality standards across their products.
• Cost Efficiency: By optimizing yarn count and fiber thickness, businesses can minimize material waste and production costs.
In conclusion, denier and Ne are important metrics in the textile industry that help quantify the characteristics of fibers and yarns. By using a Denier to Ne Calculator, you can effortlessly convert
between these two units, enabling smoother operations and informed decision-making in textile production and development.
FAQs (Frequently Asked Questions)
1. Can I use a Denier to Ne Calculator for any type of fiber?
Yes, Denier to Ne Calculators are versatile tools that can be used for various types of fibers, including synthetic and natural fibers.
2. Are there any limitations to using Denier to Ne Calculators?
While Denier to Ne Calculators provide accurate results, it’s essential to ensure that the input denier values are correct for precise conversions.
3. Can I use Denier to Ne Calculators on mobile devices?
Yes, most Denier to Ne Calculators are optimized for mobile use and can be accessed through web browsers on smartphones and tablets.
4. Are there any alternative methods for converting denier to Ne?
While manual calculations are possible using the formulas provided, Denier to Ne Calculators offer a quicker and more convenient solution.
5. Where can I find a reliable Denier to Ne Calculator?
There are several online platforms and websites that offer Denier to Ne Calculators. It’s essential to choose one from a reputable source to ensure accurate results. | {"url":"https://textilecalculator.com/denier-to-ne-calculator/","timestamp":"2024-11-05T16:37:58Z","content_type":"text/html","content_length":"176600","record_id":"<urn:uuid:a6aef803-4e5b-4c5e-b80d-a325382171fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00074.warc.gz"} |
Finding the Integration of a Function Involving an Exponential Function Using Integration by Parts
Question Video: Finding the Integration of a Function Involving an Exponential Function Using Integration by Parts Mathematics • Third Year of Secondary School
Determine β «(9π ₯ + 7)/π ^(5π ₯) dπ ₯.
Video Transcript
Determine the integral of nine π ₯ plus seven over π to the power of five π ₯ dπ ₯.
So now the first thing we want to do before we integrate this is weβ re actually gonna rewrite it. Iβ ve actually rewritten it as the integral of nine π ₯ plus seven multiplied by π to the
power of negative five π ₯ dπ ₯. So what Iβ ve done is Iβ ve actually made π to the power of negative five π ₯ instead of one over π to the power five π ₯ because thatβ s actually the
same thing.
Okay, great! So now what Iβ m gonna do is actually use integration by parts to actually help us to integrate this. And what integration by parts tells us is that if we actually have something in the
form of the integral of π ’ dπ £ dπ ₯ dπ ₯, and therefore itβ s gonna be equal to π ’π £ minus the integral of the π £ dπ ’ dπ ₯ dπ ₯. Okay, so now weβ ve actually got this letβ s use it
to actually determine what the integral of our expression is going to be.
So first of all, we need to decide what are π ’ and what are dπ £ dπ ₯ are going to be. So our π ’ is going to be nine π ₯ plus seven, and our dπ £ dπ ₯ is going to be equal to π to the
power of negative five π ₯. So okay, great! So now what we want to do is actually to differentiate π ’ to find dπ ’ dπ ₯, and then we want to integrate dπ £ dπ ₯ to actually find π £. So
therefore if we differentiate nine π ₯ plus seven, we just get nine.
And thatβ s because nine π ₯ differentiates to nine because actually itβ s the coefficient, which is nine, multiplied by the exponent, which is one, which gives us nine. And then we reduce
exponent of our π ₯ so it goes to zero. So weβ re just left with nine. And then seven if we differentiate that, it goes to zero. So dπ ’ dπ ₯ equals nine and we can say that π £ is equal to
negative π to the power of negative five π ₯ over five. And we got that by integrating π to the power of negative five π ₯. Thatβ s because we know that actually if you integrate π to the
power of π π ₯ then what you actually gonna get is one over π π to the power of π π ₯. So we applied that and we got π £ is equal to negative π to the negative five π ₯ over five.
Okay, great! So now we got π ’, π £, dπ ’ dπ ₯, and dπ £ dπ ₯. So now we can actually apply what weβ ve seen in the integration by parts. So we can say that actually the integral that weβ re
looking for is equal to negative nine π ₯ plus seven multiplied by π to the power of negative five π ₯ over five, because this is our π ’ multiplied by our π £, and then minus the integral of
negative nine π to the power of negative five π ₯ over five dπ ₯. And this is because this is our π £ dπ ’ dπ ₯.
Okay, great! So now letβ s integrate the second part. So therefore, weβ re gonna have negative nine π ₯ plus seven multiplied by π to power of negative five π ₯ over five, and then plus nine
over 25 π to the power of negative five π ₯. Weβ ve got that because we actually integrated our negative nine π to the power of negative five π ₯ over five using the same method weβ d used
So just a quick recap what weβ ve done. So we integrate our negative nine π to the power of negative five π ₯ over five dπ ₯. What we get is that this is equal to well we have one over our π
and our π was our coefficient of dπ ₯ in the exponent, so one over negative five, multiplied by negative nine π to the power of negative five π ₯ over five. And when we did that, we actually
have the denominators multiplied to get us negative five multiplied by five, which gives us negative 25. Because it was a negative nine π to the power of negative five π ₯ over five becomes
positive, so we get positive nine π to the power of negative five π ₯ over 25.
Okay, great! So weβ ve now done all the integrating. What we need to do is just simplify. Itβ s worth noting also this point that what we will need is actually a constant of integration. So now
what we want to do is actually complete the next final steps, which is actually to simplify what weβ ve got. And what Iβ ve done here is Iβ ve included our constant of integrations, so Iβ ve got
plus π Ά. So the first thing we want to do is actually well we want to add our fractions. And to enable us to do that, what we need to do is actually have the same denominator.
So Iβ ve multiplied the first fraction numerator and denominator by five. So we have negative five multiplied by nine π ₯ plus seven π to the power of negative five π ₯ plus nine π to the
power of negative five π ₯ over 25. So therefore, if we actually expand the parentheses, what we get is negative 45π to the power of negative five π ₯ π ₯ plus 35π to the power of negative
five π ₯ plus nine π to the negative five π ₯ over 25.
So therefore, if we actually collect terms on numerator, we get negative 45π to the power of negative five π ₯ π ₯ plus 44π to the negative five π ₯ over 25. And then what we can do is
actually take out π to negative five π ₯ as a factor. So we get negative 45π ₯ plus 44 multiplied by π to negative five π ₯ over 25, so which leads us to our final answer where weβ ve
actually taken negative nine over five out as a factor. So we can say that the integral of nine π ₯ plus seven over π to the power of five π ₯ dπ ₯ is equal to negative nine over five
multiplied by π ₯ plus 44 over 45 multiplied by π to the power of negative five π ₯, then plus π Ά because we donβ t forget our constant of integration. And that is our final answer. | {"url":"https://www.nagwa.com/en/videos/964159762123/","timestamp":"2024-11-11T11:35:02Z","content_type":"text/html","content_length":"257235","record_id":"<urn:uuid:718b1786-abdb-43c5-b662-1bee562f1a4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00196.warc.gz"} |
[Solved] If the harmonic mean of 60 and x is 48, then what is the val
If the harmonic mean of 60 and x is 48, then what is the value of x?
This question was previously asked in
NDA 02/2021: Maths Previous Year paper (Held On 14 Nov 2021)
View all NDA Papers >
Answer (Detailed Solution Below)
Option 3 : 40
Electric charges and coulomb's law (Basic)
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10 Questions 10 Marks 10 Mins
Formula used:
For n terms, a[1], a[2], a[3], ……., a[n].
Harmonic mean = \(\rm \frac{n}{\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + ....... \frac{1}{a_{n}} }\)
If b is the Harmonic mean of a and b then
b = 2ab/(a + b)
Given, The harmonic mean of 60 and x is 48
According to the formula used
\(\rm \frac{2\times 60x}{x + 60} = 48\)
⇒ 120x = 48(x + 60)
⇒120x = 48x + 2880
⇒ 120x - 48x = 2880
⇒ x = 40
∴ The value of x is 40.
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Data Science Minor
Data Science
Data Science Minor
Data science is an interdisciplinary field that combines computer science, statistics, and business intelligence. A Data Science minor gives students data handling, modeling, and interpretation
Data science partners with fields such as Accounting, Banking and Finance, Biology, Computer Science, Criminology, Economics, Health Care, Government, Management, Manufacturing, Mathematical
Sciences, Media, Advertising, and Marketing, Physics and Astronomy, Psychology, Retail, Sports, and Tech Startups.
Required Courses (16 hours)
MATH 1530 – Applied Statistics
3 credit hours
Prerequisites: Two years of high school algebra and a Math Enhanced ACT 19 or greater or equivalent. Descriptive statistics, probability, and statistical inference. The inference unit covers means,
proportions, and variances for one and two samples, and topics from one-way ANOVA, regression and correlation analysis, chi-square analysis, and nonparametrics. TBR Common Course: MATH 1530
TBC: Quantitative Literacy
BIA 2610 – Statistical Methods
3 credit hours
The application of collecting, summarizing, and analyzing data to make business decisions. Topics include measures of central tendency, variation, probability theory, point and interval estimation,
correlation and regression. Computer applications emphasized.
MATH 2050 – Probability and Statistics
3 credit hours
Prerequisite: MATH 1810 or MATH 1910. Data analysis, probability, and statistical inference. The inference material covers means, proportions, and variances for one and two samples, one-way ANOVA,
regression and correlation, and chi-square analysis. TBR Common Course: MATH 2050
CSCI 1170 – Computer Science I
4 credit hours
Prerequisite: MATH 1730 or MATH 1810 with a grade of C or better or Math ACT of 26 or better or Calculus placement test score of 73 or better. The first of a two-semester sequence using a high-level
language; language constructs and simple data structures such as arrays and strings. Emphasis on problem solving using the language and principles of structured software development. Three lecture
hours and two laboratory hour.
DATA 1500 – Introduction to Data Science
3 credit hours
(Same as BIA 1500.) Introduces basic principles and tools as well as its general mindset in data science. Concepts on how to solve a problem with data include business and data understanding, data
collection and integration, exploratory data analysis, predictive modeling, descriptive modeling, data product creation, evaluation, and effective communication.
DATA 3500 – Data Cleansing and Feature Engineering
3 credit hours
Prerequisite: CSCI 1170. Techniques and applications used to collect and integrate data, inspect the data for errors, visualize and summarize the data, clean the data, and prepare the data for
modeling for various data types.
DATA 3550 – Applied Predictive Modeling
3 credit hours
(Same as STAT 3550.) Prerequisite: CSCI 1170. An overview of the modeling process used in data science. Covers the ethics involved in data science, data preprocessing, regression models,
classification models, and presenting the model.
Contact Us
Data Science Institute
MTSU Box 0499
1301 East Main Street
Murfreesboro, TN 31732 | {"url":"https://datascience.mtsu.edu/minor/","timestamp":"2024-11-03T05:55:54Z","content_type":"text/html","content_length":"58212","record_id":"<urn:uuid:c7f31d4c-c158-40ca-9611-8ba24b118244>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00524.warc.gz"} |
Investment property holding costs hits an all-time low
The cost to hold an investment property hits an all-time low
Over the last few weeks, lenders have aggressively cut fixed rates, particularly for investors that borrow on an interest only basis. Three and five year fixed rates now range between 3.18% and 3.40%
p.a. This means the cost to hold an investment property is as low as it’s ever been.
This doesn’t mean we all should run out and buy an investment property.
Historic investment property holding costs
The graph below charts the annual after-tax holding cost of a median value house (average of Melbourne & Sydney) expressed in today’s dollars. As you can see, a property’s after-tax holding costs
have typically ranged between $10,000 and $30,000 per annum over the past 40 years.
The red line is the estimated annual after-tax holding costs based on current fixed rates.
A $800k apartment will cost $500 per month to hold
Let’s look at the cost to hold an $800,000 investment property (apartment) using actual data as an example.
Therefore, this property, for example will cost you circa $505 per month (after-tax) to hold.
Low rates will likely inflate property values
It is a commonly accepted economic principal that lower interest rates typically lead to an increase in asset values (i.e. the value of equities and property rise). The reason being is that the lower
cost of debt means higher profits to owners which means assets are worth more.
The graph below charts three variables:
• The rolling average capital growth rate over 20 years for median houses in Melbourne and Sydney; and
• The cost to hold an investment property (as charted above). This is calculated as the annual after-tax holding cost of a median house based on prevailing interest rates at that time, expressed in
today’s dollars; and
• The average rolling 20 year growth rate between 2000 and end of 2019.
This chart demonstrates that periods of higher capital growth have tended to follow periods of time where holding costs were below average.
It may cost you less cash flow to generate similar capital growth rates
As you can see from the chart above, the rolling 20 year capital growth rates have ranged between 4% p.a. and 9% p.a. It’s a big range because of the particular periods of time. For example, the low
growth in 2009 measures how property values changed just prior to the early 1990’s recession and during the midst of the GFC – two unfortunate points in history. Similarly, the peak in 2003 measures
growth from the early 1980’s when property boomed.
Perhaps the best long-term indicator is the average rate of 7% p.a. The average inflation rate since year 2000 is circa 2.5% p.a., so the real growth rate (i.e. excluding inflation) has been 4.5%
p.a. In today’s terms, that equates to a growth rate of circa 6% p.a., assuming inflation will continue to hover at around 1.5% p.a.
Investing in an asset that generates a growth rate of 6% p.a. that only costs $500 per month to hold could produce tremendous financial outcomes.
• Cost flow cost in today’s dollars over 20 years = $100,000
• Value uplift in today’s dollars after 20 years = over $1.3 million
But interest rates will surely rise one day
Of course, the above calculation is academic and as such could be misleading. Lots of things could change, which would change the above calculation. Capital growth rates could be lower, interest
rates could rise, the property might require capital improvements and so on.
I freely acknowledge all these factors. The point I’m trying to make is that if capital growth rates remain the same, which I think is likely given population growth and lower interest rates, then
the lower holding costs will lead to better overall investment returns. How much “better” will those returns be? Only time will tell. No one really knows.
Low rates can encourage mistakes
Asset mispricing is more likely to occur in lower interest rate markets as a result of the inefficient allocation of capital. Put simply, people, businesses and institutions take higher and maybe
less diligent risks because the cost of money is so low. This can include overpaying for assets. Relating this to the property market, it is possible that most property prices will rise i.e. good and
bad properties alike.
And, if lower rates lead to higher values, the reverse can be true also. When rates eventually rise, values can fall, particularly for lower-quality assets that don’t have strong fundamentals.
The best way to mitigate these risks (i.e. being caught in an overinflated market) is to level up on quality. This means investing in the highest quality property that your budget will allow.
Focusing on quality is vital in all markets, but arguably even more important in a lower interest rate market.
What to do next
As I said above, you should not take this blog as an indicator that I believe everyone should rush out and invest in property. No. I think you should develop your own investment strategy that suits
your personal circumstances and risk profile. Then implement that strategy without being too distracted by market conditions.
The point of this blog is to point out that maybe the stars have aligned for property investors due to: (1) all-time low property holding costs (2) low interest rates will which likely lead to higher
asset values and (3) improved sentiment in the property market will further stimulate price growth. And if your personal strategy includes investing in property, now could be the time to do it. As
always, if you have any questions or need any assistance, we are here to help. | {"url":"https://prosolution.com.au/investment-property-holding-costs/","timestamp":"2024-11-15T00:37:09Z","content_type":"text/html","content_length":"108353","record_id":"<urn:uuid:5d307088-7a70-479d-abf2-bba02e54bf7a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00823.warc.gz"} |
Using Spearman’s Rank Coefficient Technique To Analyze Survey Data - SurveyPoint
Named after Charles Spearman, Spearman’s rank coefficient is a non-parametric technique for finding the relationship between two variables.
This technique is primarily used for data analysis and is denoted by the Greek letter’ p’ (Rho).
In short, Spearman’s correlation is used to measure the direction and strength between two ranked variables. But, before we get into this, we must understand the Person’s correlation.
What Is Pearson’s Correlation?
Statistically, Pearson’s correlation is a measure to determine the strength of two paired data having a linear relationship. Moreover, to calculate and test these ranking variables, it is essential
for the data to meet the following assumptions:
• They must be linearly connected.
• They are distributed bivariant.
• They have an interval level.
RELATED: Inferential Statistics: An Introductory Guide
When To Use Spearman’s Rank Coefficient?
Before you use Spearman’s rank coefficient technique, you must ensure that your data meets the requirements. Explicitly speaking, Spearman’s correlation can be used if the data on your graph exhibits
a continuous flow. As you can see from the chart, they have a monotonic relationship and follow a line.
When it comes to monotonic relationships, the variables are closely linked to one another. This emphasizes that if one variable increases, the other tends to increase or decrease, not specifically in
a straight line.
In short, the Spearman correlation can be used to study curvilinear relationships. However, the variables must tend to change in a particular direction. Below are some monotonic relation patterns
where this technique can be used:
• Positive Monotonic: With one variable increasing, another tends to increase. However, the line may not be in a linear fashion.
• Negative Monotonic: With one variable decreasing, another must decrease, but not necessarily linearly.
The Spearman rank coefficient technique can be a great choice if you are studying ordinal data. To be exact, ordinal data contains at least three categories in a natural order. For instance, ordinal
data are first, second, and third in a rally race.
Rho can also analyze the correlation between items on a Likert scale.
Calculating Spearman’s Rho
As a first step, make sure your data is ordinal. It doesn’t need to be changed if it is already ordinal. In contrast, if the data you are dealing with is continuous, you must convert it into ranks
before calculating the Rho.
Spearman’s rank coefficients rank from -1 to +1. The signs indicated whether the variables share a positive or negative monotonic relationship. A positive sign would mean that with one variable
increasing, the other would increase too. On the other hand, a negative sign implies that with one variable increasing, the other would decrease.
Moreover, if there is no change between the variables, the Spearman rank will indicate a relationship, hence showing Zero.
RELATED: Standard Deviation and Standard Error: Concept and Difference
Formula & Calculations
Here is the formula for calculating Spearman’s rank:
n = number of data points in the variables
di = rank difference of the “ith” element
In this technique, p (Rho) can rank from +1 to -1.
• If p is equal to +1, there is a perfect rank association.
• If p is equal to -1, there is a negative association between the rank.
• If p is equal to 0, there is no relation between the variables.
Moreover, the closer p is to 0, the weaker the association between the ranks being analyzed.
You must also note that comparing the values at every level is mandatory.
Calculating Spearman’s Rank Coefficient: An Example
The table below shows nine students’ scores in two subjects, history, and geography. The table is ordianlly created. Here rank shows ordinal ranks and implies the difference in the ranks of History
and Geography for every student. d square indicates square terms for d.
History Rank Geography Rank d d square
After you have set your data in ordinal forms, like done in the table above, your next step is (∑d square), which is adding all d square values. In this case, ∑d square equals 12.
Simply add all the values using the formula given above.
In this case,
p = 1-(6*12)/(9(81-1))
Here, p is near +1, indicating a perfect association between the ranks.
Merits of Calculating Spearman’s Rank Coefficients
Here are some reasons why this technique can help you analyze the data better:
• Compared to Pearson’s correlation technique, this method is simple to understand and implement. It, however, gives the same results as Pearson’s method, given that none of the ranks are repeated.
• This technique can be helpful if you are analyzing qualitative variables like intelligence, honesty, beauty, or efficiency. In short, this method does not require you to deal with any numerics.
• In contrast, Pearson’s correlation assumes that the parent population is normal based on the sample observations. However, if this assumption is violated, you would need a non-parametric measure
to analyze data. This measure makes no such assumptions. Spearman’s technique can help you with that.
• Spearman’s is the only technique to analyze such data if you are dealing with data that can not be measured quantitatively but arranged serially.
• This technique can also be used to analyze numerical data. The data can be converted into an ordinal form in either ascending or descending order to calculate the degree of relation between the
Demerits of Calculating Spearman’s Rank Coefficient
While this method is great for calculating the degree of relationship between multiple variables, it has certain drawbacks. Here are some of them:
• This method can only be used if you are willing to assume the relationship between the variables is linear.
• In case the value of your two variables exceeds 30, you would need to give this technique a lot of time and effort. In short, this is an excellent choice for small data, but this method might not
be a great choice if you are dealing with loads of it.
• This method is unreliable if you are measuring the relationship between two variables whose distribution is a grouped frequency distribution.
Let Us Do The Maths for You
Basic Statistics + Comparative Analysis = Tangible Solutions
While it’s simple to spot a disparity between two figures, it requires more work to ascertain whether or not that disparity holds statistical significance. It can be difficult if your research topic
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Applied Maths Honours Seminar
SMS scnews item created by Martin Wechselberger at Mon 10 Sep 2007 1645
Type: Seminar
Distribution: World
Expiry: 14 Sep 2007
Calendar1: 14 Sep 2007 1400-1600
CalLoc1: Carslaw 251
CalTitle1: Applied Maths Honours Seminar -- Applied Maths Honours Seminar
Auth: wm@p6283.pc.maths.usyd.edu.au
Applied Maths Honours Seminar
The order of presentations of this year’s Applied Maths Honours Students is as follows:
2.00 to 2.30 Prudence Philp
2.30 to 3.00 Shannon He
3.00 to 3.30 Angus Liu
3.30 to 4.00 Dhruv Saxena
Time: 2.00 to 2.30
Speaker: Prudence Philp
Title: Modelling the effects of vaccination in populations
Abstract: Vaccination causes changes in the dynamics of disease in a population. I will
use a simple disease model to derive some effects of introducing an infant vaccine into
a population. Some of these effects are not ideal, and more realistic modelling can
show that for some diseases, vaccinating below threshold levels can have perverse
consequences for the population.
Many childhood diseases are characterised by seasonal fluctuations. Modelling
seasonality is important for developing vaccination schemes since periodic behaviour can
be exploited by vaccinating in pulses. I will discuss some of the implications of
including seasonality in models, specifically focussing on recent attempts at using the
same disease model to explain several seasonal childhood diseases with widely different
patterns of behaviour.
Time: 2.30 to 3.00
Speaker: Shannon He
Title: Cellular Automata Modelling of HIV Infection
Abstract: The dynamics of the long-lasting, latent phase of the three-stage HIV-1
infection is not well understood even to this day. Many theories have been proposed to
explain the phenomenon, one of which is the notion of deceptive imprinting, or original
antigenic sin. The essence of this idea is that immune cells produced in response to an
initial viral infection may in fact suppress the creation of new immune cells in
response to newly evolved viral strains. Hence a chronic infection that involves viral
strains capable of undergoing point mutations cannot be easily overcome as the immune
system fails to create new immune cells promptly. We incorporate this idea of immune
competition in a Cellular Automata (CA) model and investigate its impact on the dynamics
of both the viral and immune system. Our findings are presented with reference to
previous work.
Time: 3.00 to 3.30
Speaker: Angus Liu
Title: Chaos in the Solar System - a study of the motion of Pluto
Abstract: When Newton formulated the laws of gravitation and motion in the 17th Century,
it was thought that all physical phenomena could be entirely predicted, given that at
some instant we knew the position and motion of all the particles in the universe.
Nowhere was this more evident than in the clockwork-like motions of the planets and
other bodies in our Solar System. In 1988, two theorists from MIT, Sussman and Wisdom
showed with accurate numerical schemes that the motion of Pluto was in fact chaotic with
a predictable timescale of only around 200 million years. This came as a great shock,
with the further consequence that the chaotic motion of Pluto could cause chaotic motion
in the other planets due to its gravitational perturbations, and so the whole solar
system could be ultimately chaotic.
Time: 3.30 to 4.00
Speaker: Dhruv Saxena
Title: Level Set Method for surface minimisation
Abstract: Periodic minimal surfaces are ubiquitous in nature, many of them occuring
within cell membranes and intercellular structures. In order to study them numerically,
it is necessary to have an effective method for constructing and analysing minimal
surfaces. In this study we use a level set approach to model the Schwartz P surface in
terms of Fourier series. We derive the differential equations for the Fourier
coefficients, from which the minimal surface is constructed. The talk will introduce
ideas from Level Set Theory and discuss the results of this new approach. | {"url":"https://www.maths.usyd.edu.au/s/scnitm/wm-AppliedMathsHonoursSemina?Clean=1","timestamp":"2024-11-09T10:40:48Z","content_type":"text/html","content_length":"4949","record_id":"<urn:uuid:d818892d-2947-4689-92f6-41d986da05b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00111.warc.gz"} |
Factorization Structures and Their Applications in Discrete Geometry: A Study of Cones and Polytopes
How can the computational aspects of factorization structures be leveraged to develop efficient algorithms for polytope analysis and manipulation?
Factorization structures offer a potent framework for analyzing and manipulating polytopes, particularly those compatible with them. This computational power stems from several key aspects:
Generalized Gale's Evenness Condition: This condition, inherent to factorization structures, provides an efficient way to determine the facial structure of compatible polytopes. Instead of checking
all possible subsets of vertices, we can leverage this condition to directly identify facets and, consequently, lower-dimensional faces. This leads to faster algorithms for tasks like facet
enumeration and vertex enumeration. Explicit Face Descriptions: Factorization structures allow for explicit descriptions of faces in terms of intersections of hyperplanes defined by the structure's
defining tensors (as seen with the φtΣj,ℓ hyperplanes). This enables direct computation of face properties like normals, volumes, and adjacency relations, crucial for many polytope algorithms.
Vandermonde Identities: The generalized Vandermonde identities arising from factorization structures provide a powerful tool for analyzing the interaction of compatible polytopes with lattices. This
is particularly relevant for problems involving lattice point enumeration (Ehrhart theory) and integer programming, where these identities can lead to more efficient computational methods. Projective
Transformations: The inherent compatibility of factorization structures with projective transformations allows for efficient manipulation of polytopes. We can simplify computations by projecting a
polytope to a lower-dimensional space while preserving its combinatorial structure, thanks to the factorization structure. By integrating these computational aspects, we can develop efficient
algorithms for: Face Enumeration: Quickly determine all faces of a compatible polytope. Vertex Enumeration: Efficiently find all vertices of a polytope defined by inequalities. Lattice Point
Enumeration: Count lattice points inside dilations of a polytope. Polytope Intersection: Compute the intersection of two compatible polytopes. Polytope Projection: Project a polytope onto a
lower-dimensional space while preserving its combinatorial type. These algorithms can be further optimized by exploiting the specific properties of different factorization structures, like the
Segre-Veronese or the product structures.
Could the concept of factorization structures be extended to non-Euclidean geometries, and what implications might this have for understanding geometric structures in those settings?
Extending factorization structures to non-Euclidean geometries is a tantalizing prospect with potentially profound implications. While the current framework is deeply rooted in the projective
geometry of ℝm or ℂm, several avenues for generalization exist: Ambient Space: Instead of projective spaces, we could consider other homogeneous spaces as ambient spaces, such as spheres, hyperbolic
spaces, or more general Riemannian symmetric spaces. This would require adapting the notion of linear inclusion and tensor products to the appropriate geometric setting. Curves: The concept of
factorization curves, central to the current definition, could be generalized to other geometric objects like geodesics, circles, or more general submanifolds. The defining condition would then
involve intersections with appropriate families of these objects. Algebraic Structures: The use of tensor products hints at a possible connection with representation theory. Exploring factorization
structures from a representation-theoretic perspective might offer insights into generalizing them to settings with richer algebraic structures, like Lie groups or quantum groups. Such extensions
could have significant implications: New Geometric Structures: They could lead to the discovery and classification of novel geometric structures in non-Euclidean settings, analogous to how
factorization structures provide a framework for understanding toric varieties and their associated geometric objects. Canonical Metrics: Just as factorization structures are linked to canonical
metrics in Kähler geometry, their generalizations might provide insights into the existence and properties of canonical metrics in non-Euclidean geometries. Discrete Analogues: The connection between
factorization structures and polytopes suggests the possibility of defining discrete analogues of these structures in non-Euclidean spaces, potentially leading to new combinatorial objects and
theorems. However, extending factorization structures to non-Euclidean geometries presents significant challenges. Defining appropriate analogues of key concepts like linear inclusion, tensor
products, and factorization curves while preserving the essential properties of the original framework requires careful consideration.
In what ways does the study of factorization structures and their associated geometric objects mirror or diverge from the study of symmetry groups and their representations in mathematics?
While seemingly distinct, the study of factorization structures and symmetry groups share intriguing parallels and illuminating divergences: Similarities: Classification: Both areas are concerned
with classifying objects up to isomorphism. Factorization structures are classified based on their dimension and defining tensors, while symmetry groups are classified by their structure and
representations. Geometric Realizations: Both areas seek to understand abstract algebraic structures through their geometric realizations. Factorization structures manifest as geometric objects like
polytopes and cones, while symmetry groups act on geometric spaces, revealing symmetries and invariants. Decomposition: Both areas utilize decomposition techniques to understand complex objects.
Factorization structures can be decomposed into products of simpler structures, while representations of symmetry groups can be decomposed into irreducible representations. Differences: Focus:
Factorization structures primarily focus on specific linear inclusions and their interplay with projective geometry, leading to a concrete geometric framework. Symmetry groups, however, have a
broader scope, encompassing the study of transformations and their algebraic properties in various mathematical contexts. Construction: Factorization structures are constructed by specifying linear
inclusions satisfying certain intersection properties. Symmetry groups, on the other hand, arise from the inherent symmetries of mathematical objects or spaces. Applications: Factorization structures
have found significant applications in toric geometry, Kähler geometry, and discrete geometry, particularly in constructing and analyzing specific geometric structures. Symmetry groups have a wider
range of applications across mathematics and physics, including crystallography, quantum mechanics, and coding theory. Interplay: Despite their differences, the two areas are not entirely separate.
The use of tensor products in defining factorization structures hints at a potential connection with representation theory, suggesting that factorization structures could be viewed as specific
representations of certain algebraic structures. Exploring this connection might lead to a deeper understanding of both factorization structures and symmetry groups. In conclusion, the study of
factorization structures mirrors the study of symmetry groups in their shared emphasis on classification, geometric realization, and decomposition. However, they diverge in their specific focus,
construction methods, and range of applications. Investigating the potential interplay between these areas could unlock new insights and connections. | {"url":"https://linnk.ai/insight/scientificcomputing/factorization-structures-and-their-applications-in-discrete-geometry-a-study-of-cones-and-polytopes-w4vXBNqQ/","timestamp":"2024-11-03T19:26:14Z","content_type":"text/html","content_length":"366496","record_id":"<urn:uuid:912d33d2-cfa9-4dc3-b1bb-a4985d8986fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00846.warc.gz"} |
Using the Momentum In Fortran?
In Fortran, the momentum of an object can be calculated using the formula p = m * v, where p is the momentum, m is the mass of the object, and v is the velocity of the object. To use the momentum in
Fortran, you first need to define the variables for mass and velocity, and then use the formula to calculate the momentum. This can be done within a Fortran program by writing the necessary code to
perform the calculation. The momentum can then be used in further calculations or displayed as output to the user. By utilizing the momentum in Fortran, you can easily incorporate physics concepts
into your programming projects and simulations.
How to visualize the impact of momentum on gradient descent in Fortran?
To visualize the impact of momentum on gradient descent in Fortran, you can create a simple 2D example where you try to minimize a function using gradient descent with and without momentum.
Here is a step-by-step guide to visualizing the impact of momentum on gradient descent in Fortran:
1. Define a simple 2D function that you want to minimize, for example, a quadratic function such as f(x, y) = x^2 + y^2.
2. Implement the gradient descent algorithm without momentum in Fortran. This involves calculating the gradient of the function at each iteration and updating the parameters x and y accordingly.
3. Implement the gradient descent algorithm with momentum in Fortran. This involves adding a momentum term to the update rule that takes into account the previous step size and direction.
4. Create a plot of the function and overlay the trajectory of the parameter updates from both gradient descent algorithms.
5. Run the algorithms and observe how the trajectories evolve over time. You should see that gradient descent with momentum converges faster and with fewer oscillations compared to gradient descent
without momentum.
Here is a simple Fortran code snippet to get you started:
1 program gradient_descent
3 implicit none
4 real :: x, y, lr, momentum, beta
5 integer :: i
6 integer :: max_iter
7 real :: grad_x, grad_y
8 real :: prev_grad_x, prev_grad_y
10 ! Initialize parameters and hyperparameters
11 x = 5.0
12 y = 5.0
13 lr = 0.01
14 momentum = 0.9
15 beta = 0.9
16 max_iter = 100
18 ! Perform gradient descent without momentum
19 do i = 1, max_iter
20 grad_x = 2.0*x
21 grad_y = 2.0*y
23 x = x - lr * grad_x
24 y = y - lr * grad_y
25 end do
27 ! Perform gradient descent with momentum
28 x = 5.0
29 y = 5.0
30 prev_grad_x = 0.0
31 prev_grad_y = 0.0
32 do i = 1, max_iter
33 grad_x = 2.0*x
34 grad_y = 2.0*y
36 x = x - lr * grad_x + momentum * prev_grad_x
37 y = y - lr * grad_y + momentum * prev_grad_y
39 prev_grad_x = lr * grad_x
40 prev_grad_y = lr * grad_y
41 end do
43 end program gradient_descent
You can use a plotting library such as gnuplot or matplotlib in Python to visualize the trajectories of the parameter updates from both gradient descent algorithms. Just save the updates to a file
and plot them using the library of your choice.
What is the role of momentum in deep learning algorithms in Fortran?
In deep learning algorithms in Fortran, momentum is used as a technique to help accelerate the convergence of the optimization process. Momentum is a hyperparameter that controls the contribution of
the previous weight update in the current update step.
The role of momentum in deep learning algorithms in Fortran is to smooth out variations in the update process, therefore helping to avoid local minima and saddle points and speeding up convergence to
the global minimum. It does this by adding a fraction of the previous weight update to the current update, which can help the weights move in a more consistent and stable direction.
Overall, momentum in deep learning algorithms in Fortran helps to improve the training process by reducing oscillations and speeding up convergence, leading to faster and more stable optimization of
the neural network.
How to compare different momentum strategies in Fortran?
To compare different momentum strategies in Fortran, you can follow these steps:
1. Implement each momentum strategy as a separate subroutine or function in Fortran. Each strategy should take as input a list of stock prices and return the final result, such as the average return
or cumulative return.
2. Create a main program in Fortran that will call each momentum strategy subroutine or function with the same input data (stock prices).
3. Run the main program and record the results of each momentum strategy. Compare the results based on criteria such as average return, cumulative return, volatility, and maximum drawdown.
4. Use statistical analysis techniques to further compare the performance of each momentum strategy. You can calculate standard deviation, Sharpe ratio, and other metrics to evaluate risk-adjusted
5. Perform sensitivity analysis by varying the parameters of each momentum strategy to see how they impact performance.
6. Visualize the results using plots or graphs to compare the performance of each momentum strategy visually.
By following these steps, you can effectively compare different momentum strategies in Fortran and determine which strategy performs best based on your criteria and objectives.
What is the difference between momentum and other optimization techniques in Fortran?
Momentum is a technique used in optimization algorithms to help accelerate learning and convergence towards the optimal solution. It is specifically designed to help overcome local minima and other
obstacles in the optimization process.
Other optimization techniques in Fortran, such as gradient descent or stochastic gradient descent, do not incorporate momentum. These techniques rely on updating the parameters of the model based on
the gradient of the loss function at each iteration. While effective in many cases, they can sometimes struggle to converge if the loss function has a complex or non-convex shape.
In contrast, momentum algorithms introduce an additional term that takes into account the history of previous parameter updates. This helps smooth out the optimization process and prevent the model
from getting stuck in local minima. By incorporating momentum, the algorithm is able to gain inertia and speed up convergence towards the global minimum.
How to incorporate momentum in various machine learning models in Fortran?
In Fortran, you can incorporate momentum in various machine learning models, such as neural networks, by adding a momentum term to the parameter update equation. Here is an example of how to
incorporate momentum in a simple neural network implemented in Fortran:
1. Define the momentum term:
1 real, parameter :: momentum = 0.9
2 real, dimension(:,:), allocatable :: delta_weights
4 ! Initialize delta_weights to zero
5 allocate(delta_weights(shape(weights)))
6 delta_weights = 0.0
1. Update the weights with momentum:
1 ! Calculate the gradient of the loss function
2 ! dLoss_dweights is the gradient of the loss function with respect to the weights
4 ! Update the delta_weights with momentum
5 delta_weights = momentum * delta_weights - learning_rate * dLoss_dweights
7 ! Update the weights using the momentum term
8 weights = weights + delta_weights
1. Update the model parameters based on the momentum-enhanced gradient descent algorithm.
By incorporating momentum in this way, the model will benefit from the additional damping effect on oscillations in the parameter updates, resulting in faster convergence and potentially better
performance on the training data.
How to adjust momentum based on the characteristics of the optimization problem in Fortran?
To adjust momentum based on the characteristics of the optimization problem in Fortran, you can follow these steps:
1. Evaluate the sensitivity of the optimization problem to changes in momentum. This involves analyzing how the convergence and performance of the optimization algorithm are affected by different
values of the momentum parameter.
2. Experiment with different values of momentum in the optimization algorithm. Try out a range of values and observe how each value impacts the convergence speed, stability, and overall performance
of the optimization algorithm.
3. Adjust the momentum parameter based on the specific characteristics of the optimization problem. For example, if you have a highly non-convex or ill-conditioned problem, you may need to use a
higher momentum to help the optimization algorithm escape local minima.
4. Consider using adaptive momentum techniques. Some optimization algorithms, such as stochastic gradient descent with momentum, offer techniques for adaptively adjusting the momentum parameter
based on the dynamics of the optimization problem.
5. Fine-tune the momentum parameter through iterative experimentation and validation. Continuously monitor the performance of the optimization algorithm with different momentum values and adjust the
parameter accordingly until you find an optimal setting for your specific problem.
By following these steps and adjusting the momentum parameter based on the characteristics of the optimization problem, you can effectively optimize the performance of your Fortran program. | {"url":"https://topminisite.com/blog/using-the-momentum-in-fortran","timestamp":"2024-11-07T07:40:49Z","content_type":"text/html","content_length":"387508","record_id":"<urn:uuid:65def471-3d20-457b-b95f-8b893ae9c88b>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00328.warc.gz"} |
50 + Time and Work Questions and Answers | Quantitative Aptitude50 + Time and Work Questions and Answers | Quantitative Aptitude - Leverage Edu
Time and Work questions and answers are fundamental concepts in quantitative aptitude, playing an important role in various competitive exams. This topic evaluates an individual’s ability to
understand and manage time efficiently, a skill vital for success in today’s fast-paced world.
This blog provides a list of 50+ time and work questions that are part of the quantitative aptitude section of competitive exams.
Most competitive exams, ranging from banking to government services and management entrance tests, include questions on time and work to assess candidates’ problem-solving abilities.
What are Time and Work Questions?
Time and Work questions involve calculating the amount of work done by individuals or groups in a specified period. These questions typically include scenarios where people work together or alone to
complete a task, and the goal is to determine the time required for completion. Understanding the relationships between time, work, and efficiency is key to solving these problems.
Must Read: Classification Reasoning Questions | Verbal Reasoning
Basic Level For Practice
1. A takes 10 days to complete a project alone. If he is assisted by B every second day, how many days will it take to finish the project?
a) 15 days
b) 18 days
c) 20 days
d) 25 days
Answer: (b) 18 days
2. If C can complete a task in 15 days, how many days will it take for C and D working together to finish the same task if D is twice as efficient as C?
a) 5 days
b) 7.5 days
c) 10 days
d) 12.5 days
Answer: (c) 10 days
3. E takes 6 days to complete a job. If F is 20% less efficient than E, how many days will F take to complete the same job alone?
a) 7.2 days
b) 8 days
c) 9 days
d) 10 days
Answer: (a) 7.2 days
4. G and H together can complete a task in 8 days. If G takes 12 days to complete the task alone, how long will it take H to complete the task individually?
a) 12 days
b) 16 days
c) 20 days
d) 24 days
Answer: (d) 24 days
5. I can finish a job in 5 days. If J is 25% less efficient than I, how many days will it take for J to complete the same job?
a) 6 days
b) 7.5 days
c) 8 days
d) 10 days
Answer: (b) 7.5 days
6. K can complete a task in 15 days. If L is 60% as efficient as K, how long will it take for L to finish the same task?
a) 24 days
b) 30 days
c) 36 days
d) 40 days
Answer: (b) 30 days
7. M takes 8 days to complete a work. N is 50% more efficient than M. How many days will N take to complete the same work?
a) 4 days
b) 5 days
c) 6 days
d) 7 days
Answer: (b) 5 days
8. O and P can complete a project in 12 days. If P alone takes 18 days to complete the project, how long will it take for O to complete it individually?
a) 24 days
b) 30 days
c) 36 days
d) 48 days
Answer: (b) 30 days
9. Q can complete a job in 10 days. If R is 20% less efficient than Q, how many days will it take for R to complete the same job?
a) 12 days
b) 15 days
c) 18 days
d) 20 days
Answer: (b) 15 days
10. S takes 12 days to complete a task. If T is 25% more efficient than S, how many days will it take for T to finish the same task?
• a) 8 days
• b) 9 days
• c) 10 days
• d) 11 days
Answer: (c) 10 days
11. U can finish a job in 15 days. If V is 30% less efficient than U, how many days will it take for V to complete the same job?
a) 18 days
b) 20 days
c) 22 days
d) 24 days
Answer: (a) 18 days
12. W can complete a project in 8 days. If X is 40% more efficient than W, how many days will it take for X to finish the same project?
a) 4 days
b) 5 days
c) 6 days
d) 7 days
Answer: (a) 4 days
13. Y takes 10 days to complete a task. If Z is 50% less efficient than Y, how many days will it take for Z to complete the same task?
a) 15 days
b) 18 days
c) 20 days
d) 25 days
Answer: (b) 18 days
14. A and B together can complete a work in 16 days. If A takes 24 days to complete the work alone, how many days will it take for B to complete the work individually?
a) 32 days
b) 36 days
c) 40 days
d) 48 days
Answer: (d) 48 days
15. C takes 20 days to complete a project. If D is 30% less efficient than C, how many days will it take for D to complete the same project?
a) 26 days
b) 28 days
c) 32 days
d) 40 days
Answer: (b) 28 days
16. E and F together can finish a task in 10 days. If E alone takes 15 days to complete the task, how many days will it take for F to complete the work individually?
a) 15 days
b) 20 days
c) 25 days
d) 30 days
Answer: (c) 25 days
17. G takes 14 days to complete a project. If H is 25% more efficient than G, how many days will it take for H to complete the same project?
a) 9 days
b) 10 days
c) 11 days
d) 12 days
Answer: (a) 9 days
18. I and J together can complete a task in 18 days. If I takes 24 days to complete the task alone, how many days will it take for J to complete the work individually?
a) 36 days
b) 42 days
c) 48 days
d) 54 days
Answer: (b) 42 days
19. K takes 16 days to finish a job. If L is 20% less efficient than K, how many days will it take for L to complete the same job?
a) 18 days
b) 20 days
c) 22 days
d) 25 days
Answer: (b) 20 days
20. M and N together can complete a task in 15 days. If M alone takes 25 days to complete the task, how many days will it take for N to complete the work individually?
a) 35 days
b) 40 days
c) 45 days
d) 50 days
Answer: (c) 45 days
21. O takes 30 days to complete a project. If P is 15% more efficient than O, how many days will it take for P to complete the same project?
a) 20 days
b) 25 days
c) 28 days
d) 32 days
Answer: (b) 25 days
Tips To Solve Time and Work Questions and Answers
Here we have mentioned certain tips and tricks to solve questions related to time and work:
• Understand the concept of efficiency and how it influences work rates.
• Establish the relationship between time, work, and efficiency.
• Break down complex scenarios into smaller, more manageable steps.
• Use LCM (Least Common Multiple) to find common work rates.
• Pay attention to units of time and work to ensure consistency in calculations.
Also Read: Questions of Syllogism Reasoning | Verbal Reasoning
Practice Advanced Level Questions on Time and Work
If A can complete a work in 10 days and B can complete the same work in 15 days, how many days will they take together to complete the work?
A) 5 days
B) 6 days
C) 7 days
D) 8 days
Answer: B) 6 days
If 6 men or 8 women can do a piece of work in 17 days, how many days will 10 men and 6 women take to complete the work?
A) 9 days
B) 10 days
C) 11 days
D) 12 days
Answer: A) 9 days
A and B can complete a work together in 12 days. If A alone can complete the work in 20 days, then how many days will B alone take to complete the work?
A) 24 days
B) 30 days
C) 36 days
D) 40 days
Answer: D) 40 days
12 men complete a work in 15 days. 8 men start the work and after 5 days, 4 more men join them. In how many days will the remaining work be completed?
A) 5 days
B) 6 days
C) 7 days
D) 8 days
Answer: B) 6 days
A can do a piece of work in 20 days which B can do in 30 days. If they work on it together for 12 days, then what fraction of the work is left?
A) 1/4
B) 1/6
C) 1/8
D) 1/10
Answer: C) 1/8
A can complete a work in 24 days and B can complete the same work in 36 days. In how many days will they complete the work together?
A) 10 days
B) 12 days
C) 14 days
D) 16 days
Answer: B) 12 days
If 6 men can do a piece of work in 18 days, how many days will 9 men take to complete the same work?
A) 8 days
B) 10 days
C) 12 days
D) 14 days
Answer: B) 10 days
A can do a piece of work in 10 days which B can do in 15 days. If they work on it together for 6 days, then what fraction of the work is left?
A) 1/5
B) 1/6
C) 1/10
D) 1/15
Answer: A) 1/5
10 men can complete a work in 15 days. After working for 5 days, 5 more men joined them. In how many days will the remaining work be completed?
A) 6 days
B) 7 days
C) 8 days
D) 9 days
Answer: A) 6 days
A can complete a work in 15 days and B can complete the same work in 10 days. If they work together for 6 days, then what fraction of the work is left?
A) 1/5
B) 1/6
C) 1/10
D) 1/15
Answer: B) 1/6
If 20 men can complete a work in 30 days, in how many days will 15 men complete the same work?
A) 40 days
B) 45 days
C) 50 days
D) 60 days
Answer: D) 60 days
A and B together can complete a work in 8 days. B and C together can complete the same work in 12 days. If A, B, and C together can complete the work in 6 days, then in how many days will C alone
complete the work?
A) 12 days
B) 16 days
C) 24 days
D) 32 days
Answer: C) 24 days
If 15 men can do a piece of work in 12 days, then how many men are required to do the same work in 8 days?
A) 20 men
B) 24 men
C) 30 men
D) 36 men
Answer: B) 24 men
A and B together can complete a work in 15 days. If they start the work together and A leaves after 6 days, in how many days will B complete the remaining work?
A) 6 days
B) 7.5 days
C) 8 days
D) 9 days
Answer: C) 8 days
A and B can do a piece of work in 20 days which B and C can do in 15 days. If C alone can complete the work in 30 days, then in how many days will A, B, and C together complete the work?
A) 4 days
B) 6 days
C) 8 days
D) 10 days
Answer: B) 6 days
If 12 men can do a piece of work in 15 days, how many days will 24 men take to complete the same work?
A) 5 days
B) 7.5 days
C) 10 days
D) 12.5 days
Answer: A) 5 days
A can complete a work in 18 days and B can complete the same work in 24 days. If they work together for 6 days, then what fraction of the work is left?
A) 1/6
B) 1/8
C) 1/9
D) 1/12
Answer: D) 1/12
If 9 men can do a piece of work in 15 days, how many days will 12 men take to complete the same work?
A) 8 days
B) 10 days
C) 12 days
D) 15 days
Answer: B) 10 days
A can complete a work in 24 days and B can complete the same work in 36 days. If they start the work together and A leaves after 10 days, then in how many days will B complete the remaining work?
A) 8 days
B) 9 days
C) 10 days
D) 12 days
Answer: C) 10 days
A and B can complete a work in 20 days which B and C can do in 15 days. If A alone can complete the work in 30 days, then in how many days will B alone complete the work?
A) 24 days
B) 30 days
C) 36 days
D) 40 days
Answer: A) 24 days
If 15 men can complete a work in 10 days, how many days will 10 men take to complete the same work?
A) 15 days
B) 20 days
C) 25 days
D) 30 days
Answer: B) 20 days
A can complete a work in 25 days and B can complete the same work in 35 days. If they start the work together and work alternately for a day each, then in how many days will the work be completed?
A) 35 days
B) 37 days
C) 39 days
D) 41 days
Answer: C) 39 days
If 12 men can complete a work in 20 days, how many men are required to complete the same work in 15 days?
A) 16 men
B) 18 men
C) 20 men
D) 24 men
Answer: A) 16 men
A can complete a work in 30 days and B can complete the same work in 20 days. If they start the work together and work alternately for a day each, then in how many days will the work be completed?
A) 30 days
B) 32 days
C) 34 days
D) 36 days
Answer: B) 32 days
If 15 men can do a piece of work in 18 days, how many days will 10 men take to complete the same work?
A) 24 days
B) 27 days
C) 30 days
D) 36 days
Answer: C) 30 days
A can complete a work in 12 days and B can complete the same work in 15 days. If they start the work together, then in how many days will they complete half of the work?
A) 3 days
B) 3.5 days
C) 4 days
D) 4.5 days
Answer: D) 4.5 days
If 8 men can do a piece of work in 12 days, then how many days will 12 men take to complete the same work?
A) 6 days
B) 8 days
C) 9 days
D) 10 days
Answer: A) 6 days
A and B can complete a work in 10 days which B and C can do in 15 days. If A alone can complete the work in 20 days, then in how many days will C alone complete the work?
A) 30 days
B) 36 days
C) 40 days
D) 45 days
Answer: B) 36 days
If 10 men can do a piece of work in 15 days, then how many days will 20 men take to complete the same work?
A) 7.5 days
B) 9 days
C) 10 days
D) 12 days
Answer: B) 9 days
A and B can complete a work in 20 days which B and C can do in 15 days. If A alone can complete the work in 30 days, then in how many days will C alone complete the work?
A) 40 days
B) 45 days
C) 50 days
D) 60 days
Answer: B) 45 days
How can I improve my efficiency in solving Time and Work questions?
Practice is key. Regularly solve a variety of time and work problems to strengthen your understanding of the concepts. Additionally, focus on time management during practice sessions to enhance your
overall speed and accuracy.
Are there shortcuts for solving Time and Work questions?
Yes, identifying patterns and using shortcuts can significantly speed up your problem-solving. Look for common factors, use LCM wisely, and practice mental calculations to save time during exams.
Can Time and Work concepts be applied in real-life scenarios?
Answer: Absolutely. The principles of time and work are applicable in everyday situations, whether it’s managing projects, dividing tasks among team members, or estimating the time required for
completing a task.
This was all about the “Time and Work Questions and Answers”. For more such informative blogs, check out our Maths Section, or you can learn more about us by visiting our Study Material Section | {"url":"https://leverageedu.com/discover/indian-exams/exam-prep-time-and-work-questions-and-answers/","timestamp":"2024-11-12T22:04:47Z","content_type":"text/html","content_length":"299942","record_id":"<urn:uuid:d9a33e8a-fcb7-4c11-ae37-b3b0b3e78157>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00766.warc.gz"} |
Quadrature of the lunes
A lune is the area left when part of a circle is cut off by another circle. Can you work out the area?
This resource is from Underground Mathematics.
A lune is the area left when part of a circle is cut off by another circle, as in the following problems. It is called a lune because it looks a bit like the moon.
1. In the following figure, two semicircles have been drawn, one on the side $AB$ of the triangle, and the other on the side $AC$ of the triangle (with centre $O$). What is the area of the blue
(shaded) lune which is bounded by the two semicircles?
As a bonus, can you construct a square on the diagram with the same area as the blue lune, using only a straight edge (ruler) and compasses? This is called the quadrature (making into a square)
of the lune.
2. In the following figure, three semicircles have been drawn, one on each of the sides of the right-angled $6$-$8$-$10$ triangle. What is the total area of the two coloured (shaded) lunes in the
This is an Underground Mathematics resource.Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide
free web-based resources that support the teaching and learning of post-16 mathematics.Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and
teacher notes to help with their use in the classroom.
Student Solutions
Michael, from Exeter Mathematics School, explained how to do question 1.
To find the area of the lune, we can find the area of the semicircle with diameter $AB$, and subtract from this the difference between the areas of sector $OAB$ an
d triangle $OAB$.
Using Pythagoras' theorem on $AOB$, we have $AB = \sqrt{2^2+2^2} = 2 \sqrt{2}$, so the radius of the outer circle of the lune is $\sqrt{2}$.
• Semicircle area: $\frac{\pi r^2}{2} = \frac{\pi \left( \sqrt{2} \right)^2}{2} = \pi $
• Sector area: $\frac{\pi r^2}{4} = \frac{\pi \left( 2 \right) ^ 2}{4} = \pi $
• Triangle area: $\frac{1}{2}bh = \frac{1}{2} \times 2 \times 2 = 2$
Therefore the lune area is $\pi - \left( \pi - 2 \right) = 2$.
He also demonstrates how to construct the quadrature of the lune:
The quadrature of the lune must have area $2$, so must have side lengths $\sqrt{2}$.
Let $M_1$ and $M_2$ be the midpoints of $AB$ and $AC$ respectively. Then the line $OM_1$ bisects $AB$ at right angles, and likewise for $OM_2$ and $BC$, since they are chords of the circle.
This is then a square, as the angles are all right angles (we know that $M_1BM_2$ is a right angle because it is subtended in a semicircle) and $M_1B = M_2B = \sqrt{2}$.
This square has the same area as the lune, and therefore is the quadrature of the lune.
Kristian, from Maidstone Grammar School, was able to use the same method to answer question 2. Here is his solution:
Let $D$ be the midpoint of $AC$, this is distance $5$ from $A$, $B$ and $C$.
The angle $B\hat{C}A$ can be calculated using trigonometry, to be $B\hat{C}A = \mathrm{cos}^{-1}\left(\frac{8}{10}\right) = 36.86...^\circ$.
Then, using the circle theorem that the angle subtended at the centre is twice that subtended at the circumference, $B\hat{D}A = 2B\hat{C}A = 73.73...^\circ$.
Blue Lune
The sector $ADB$ has area $\frac{A\hat{D}B}{360}\pi r^2 = \frac{73.73...}{360} \times \pi \times 5^2 = 16.08...$.
The triangle $ADB$ has area $\frac{1}{2} \times 5 \times 5 \times \mathrm{sin}\left( 73.73... \right) = 12$, using the formula for the area of a triangle.
The segment between $A$ and $B$ has area $16.08... - 12 = 4.08...$.
Then, the semicircle with diameter $AB$ has area $\frac{1}{2} \times \pi \times 3^2 = 14.13...$.
This means: $$\text{Area of Lune} = \text{Area of Semicircle} - \text{Area of Segment} = 14.13... - 4.08... = 10.049... $$
Green Lune
The sector $CDB$ has area $\frac{C\hat{D}B}{360}\pi r^2 = \frac{106.26...}{360} \times \pi \times 5^2 = 23.18...$.
The triangle $CDB$ has area $\frac{1}{2} \times 5 \times 5 \times \mathrm{sin}\left( 106.26... \right) = 12$, using the formula for the area of a triangle.
The segment between $B$ and $C$ has area $23.18... - 12 = 11.18...$.
Then, the semicircle with diameter $BC$ has area $\frac{1}{2} \times \pi \times 4^2 = 25.13...$.
This means: $$\text{Area of Lune} = \text{Area of Semicircle} - \text{Area of Segment} = 25.13... - 11.18... = 13.95... $$
Therefore, the total lune area is: $$\text{Total Lune Area} = \text{Blue Lune Area} + \text{Green Lune Area} = 10.049... + 13.95... = 24$$
Joe, from Leventhorpe School, found the result in a very nice way, which explains why the area that Kristian found was an integer. Joe was able to show that the area of the two lunes adds up to give
the area of the original triangle. Here is his working:
If a right angled triangle has sides $a$, $b$ and $c$, where $c$ is the hypotenuse, Pythagoras' Theorem states that $a^2+b^2=c^2$. This can be applied to the areas of squares constructed on the
sides, but this also applies to any shape, as long as those constructed on the three sides are similar. Suppose we do this with semicircles. In the diagram, the areas of semicircles A and B add to
give that of semicircle C.
The triangle formed in this diagram is certainly right-angled, as it is subtended in a semi-circle.
Rotating the triangle by $180^\circ$ in the circle, the segments labelled as $x$ are the same, as are those labelled as $y$.
Subtracting this from the equality of areas established above, this gives that $A + B = C$ in the diagram to the right. Since rotating does not change the area, this says that the original triangle
had the same area as the two lunes.
Joe was then able to use this to establish that the total area of the lunes in question 2 was $24$, the area of the triangle. He also adapted this to question 1:
The two lunes in the diagram have the same area, as the right angled triangle is isosceles. The triangle has area $\frac{1}{2} \times 2 \times 4 = 4$, so each lune has area $2$.
Thank you and well done to everyone who contributed their solutions to this problem | {"url":"https://nrich.maths.org/problems/quadrature-lunes","timestamp":"2024-11-11T01:46:24Z","content_type":"text/html","content_length":"49573","record_id":"<urn:uuid:6fe99435-cea9-4929-97ad-56886767b34a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00461.warc.gz"} |
How to Calculate The Capacity Of TSX Vibrating Screen?
First, the one-way method of linear vibration will be explained, and then the rotation method of circular vibration will be explained.
When screening materials with a weight of 1.6t/m, the allowable depth of the bed shall not exceed 5 times of the screen, and when screening materials with a weight of 0.8t/m, the allowable depth of
the bed shall not exceed four times of the screen hole.
In general, attention should be paid to:
The following formula is often used to calculate the capacity of vibrating screen
The total formula is: VSC=D * W * V * C
VSC is the capacity of vibrating screen
D is the depth of the riverbed
It is easy to calculate the bed depth to determine the optimal screen width. If the depth of the bed is too deep, the material will not contact the screen surface. The depth of the fine screen bed is
very important, especially when cleaning, to ensure that the sprayed water penetrates the entire depth of the material. Cohesive, clayey material, UG 2, gypsum requires high-frequency linear
horizontal to slightly inclined screens.
Finally, all the parameters needed to calculate the material flow rate on the screen are obtained. Now, you can easily calculate the traffic through the formula defined below.
These units are typically tons and hours.
All these calculation steps are provided as examples, and all values used here are randomly selected. You can follow the same calculation steps in the calculation, or you can change the type of
calculation according to the exact value you have. All operations discussed here are reversible, which means that if the flow rate is known or a specific flow rate is required, the material speed,
vibration motor running speed or system stroke can be easily calculated in the same way. | {"url":"https://www.tsxscreen.com/blog/how-to-calculate-the-capacity-of-tsx-vibrating-screen.html","timestamp":"2024-11-13T08:18:38Z","content_type":"text/html","content_length":"35343","record_id":"<urn:uuid:be069ae7-86b9-4c9b-9bdb-c3b9cb2f70fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00820.warc.gz"} |
Solstat: A statistical approximation library | Primitive Blog
Solstat: A statistical approximation library
Smart Contracts,Mathematics,Technical
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Solstat: A statistical approximation library
Smart Contracts,Mathematics,Technical
<h2 id="numerical-approximations">Numerical Approximations</h2> <p>There are many useful mathematical functions that engineers use in designing applications. This body of knowledge can be more widely
described as <em>approximation theory</em> for which there are many <a href="https://xn--2-umb.com/22/approximation/">great resources</a>. An example of functions that need approximation and are also
particularly useful to us at Primitive are those relating to the <em>Gaussian</em> (or <em>normal</em>) distribution. Gaussians are fundamental to statistics, probability theory, and engineering
(e.g., <a href="https://en.wikipedia.org/wiki/Central_limit_theorem">the Central Limit Theorem</a>).</p> <p>At Primitive, our RMM-01 trading curve relies on the (standard) Gaussian probability
density function (PDF) $\phi(x)=\frac{1}{\sqrt{2\pi}} e^\frac{-x^2}{2}$, the cumulative probability distribution (CDF) $\Phi$, and its inverse $\Phi^{-1}$. These specific functions appear due to how
<a href="https://en.wikipedia.org/wiki/Brownian_motion">Brownian motion</a> appears in the pricing of <a href="https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model">Black-Scholes European
options</a>. The Black-Scholes model assumes <a href="https://en.wikipedia.org/wiki/Geometric_Brownian_motion">geometric Brownian motion</a> to get a concrete valuation of an option over its
maturity.</p> <p><code>solstat</code> is a Solidity library that approximates these Gaussian functions. It was built to achieve a high degree of accuracy when computing Gaussian approximations within
compute constrained environments on the blockchain. Solstat is open source and available for anyone to use. An interesting use case being showcased by the team at <a href="https://asphodel.io/">
asphodel</a> is for designing drop rates, spawn rates, and statistical distributions that have structured randomness in onchain games.</p> <p>In the rest of this article we'll dive deep into function
approximations, their applications, and methodology.</p> <h2 id="approximating-functions-on-computers">Approximating Functions on Computers</h2> <p>The first step in evaluating complicated functions
with a computer involves determining whether or not the function can be evaluated "directly", i.e. with instructions native to the processing unit. All modern processing units provide basic binary
operations of addition (possibly subtraction) and multiplication. In the case of simple functions like $f(x)=mx+b$ where $m$, $x$, and $b$ are integers, computing an output can be done <em>
efficiently</em> and <em>accurately</em>.</p> <p>Complex functions like the Gaussian PDF $\phi(x)$ come with their own unique set of challenges. These functions cannot be evaluated directly because
computers only have native opcodes or logical circuits/gates that handle simple binary operations such as addition and subtraction. Furthermore, integer types are native to computers since their
mapping from bits is canonical, but decimal types are not ubiquitous. There can be no exponential or logarithmic opcodes for classical bit CPUs as they would require infinitely many gates. There is
no way to represent arbitrary real numbers without information loss in computer memory.</p> <p>This begs the question: How can we compute $\phi(x)$ with this restrictive set of tools? Fortunately,
this problem is extremely old, dating back to the human desire to compute complicated expressions by hand. After all, the first "computers" were people! Of course, our methodologies have improved
drastically over time.</p> <p>What is the optimal way of evaluating arbitrary functions in this specific environment? Generally, engineers try to balance the "best possible scores" given the
computation economy and desired accuracy. If constrained to a fixed amount of numerical precision (e.g., max error of $10^{-18}$), what is the least amount of:</p> <ul> <li><strong>(Storage)</strong>
How much storage is needed (e.g., to store coefficients)?</li> <li><strong>(Computation)</strong> How many total clock cycles are available for the processor to perform?</li> </ul> <p>What is the
best reasonable approximation for a fixed amount of storage/computational use (e.g., CPU cycles or bits)?</p> <ul> <li><strong>(Absolute accuracy)</strong> Over a given input domain, what is the
worst-case in the approximation compared to the actual function?</li> <li><strong>(Relative accuracy)</strong> Does the approximation perform well over a given input domain relative to the magnitude
of the range of our function?</li> </ul> <p>The above questions are essential to consider when working with the Ethereum blockchain. Every computational step that is involved in mutating the
machine's state will have an associated gas cost. Furthermore, DeFi protocols expect to be accurate down to the <code>wei</code>, which means practical absolute accuracy down to $10^{-18}$ ETH is of
utmost importance. Precision to $10^{-18}$ is near the accuracy of an "atom's atom", so reaching these goals is a unique challenge.</p> <h2 id="our-computers-toolbox">Our Computer's Toolbox</h2> <p>
Classical processing units deal with binary information at their core and have basic circuits implemented as logical opcodes. For instance, an <code>add_bits</code> opcode is just a realization of
the following digital circuit:</p> <p><img src="/assets/blog/solstat/full_adder.png" alt=""></p> <p>These gates are adders because they define an addition operation over binary numbers. These full
adders can be strung together with a carry-out pin for higher adders. For example, a <a href="https://en.wikipedia.org/wiki/Adder_(electronics)#Ripple-carry_adder">ripple carry adder</a> can be
implemented this way, and extended to an arbitrary size, such as the 256bit requirements in Ethereum.</p> <p>Note that adders introduce an error called <em>overflow</em>. Using <code>add_4bits</code>
to add <code>0001</code> to <code>1111,</code> the storage space necessary to hold a large number is exhausted. This case must be handled within the program. Fortunately for Ethereum 256bit numbers,
this overflow is far less of an issue due to the magnitude of the numbers expressed ($2^{256}\approx 10^{77}$). For perspective, to overflow 256bit addition one would need to add numbers on the order
of the estimated number of atoms in the universe ($\approx 10^{79}$). Furthermore, the community best practices for handling overflows in the EVM are well understood.</p> <p>At any rate, repeated
addition can be used to build multiplication and repeated multiplication to get integer powers. In math/programming terms:</p> <p>$$ 3\cdot x =\operatorname{multiply}(x,3)=\underbrace{\operatorname
{add}(x,\operatorname{add}(x,x))}_{2\textrm{ additions}} $$</p> <p>and for powers:</p> <p>$$ x^3=\operatorname{pow}(x,3)=\underbrace{\operatorname{multiply}(x,\operatorname{multiply}(x,x))}_{2\textrm
{ multiplications}}. $$</p> <p>Subtraction and division can also be defined for gate/opcode-level integers. Subtraction has similar overflow issues to addition. Division behaves in a way that returns
the quotient and remainder. This can be extended to integer/decimal representations of rational numbers (e.g., fractions) using floating-point or fixed-point libraries like <a href="https://
github.com/abdk-consulting/abdk-libraries-solidity">ABDK</a>, and the library in <a href="https://github.com/transmissions11/solmate/blob/ed67feda67b24fdeff8ad1032360f0ee6047ba0a/src/utils/
FixedPointMathLib.sol">Solmate</a>. Depending on the implementation, division can be more computationally intensive than multiplication.</p> <h3 id="more-functionality">More Functionality</h3> <p>
With extensions of division and multiplication, negative powers can be constructed such that:</p> <p>$$ x^{-1}=\frac{1}{x}=\operatorname{divide}(1,x). $$</p> <p>None of these abstractions allow
computers to express infinite precision with arbitrarily large accuracy. There can never be an exact binary representation of irrational numbers like $\pi$, $e$, or $\sqrt{2}$. Numbers like $\sqrt{2}
$ <em>can</em> be represented precisely in <a href="https://en.wikipedia.org/wiki/Computer_algebra_system">computer algebra systems (CAS)</a>, but this is unattainable in the EVM at the moment.</p>
<p>Without computer algebra systems, quick and accurate algorithms for computing approximations of functions like $\sqrt{x}$ must be developed. Interestingly $\sqrt{x}$ arises in the infamous
approximation from <a href="https://www.youtube.com/watch?v=p8u_k2LIZyo">Quake lll</a>, which is an excellent example of an approximation optimization yielding a significant performance improvement.
</p> <h3 id="rational-approximations">Rational Approximations</h3> <p>The EVM provides access to addition, multiplication, subtraction, and division operations. With no other special-case assumptions
as in the Quake square root algorithm, the best programs on the EVM can do is work directly with sums of <em>rational functions</em> of the form:</p> <p>$$ P_{m,n}(x)=\frac{\alpha_0 +\alpha_1 x + \
alpha_2 x^2 + \cdots + \alpha_m x^m}{\beta_0 + \beta_1 x + \beta_2 x^2 + \cdots + \beta_n x^n}. $$</p> <p>The problem is that most functions are not rational functions! EVM programs need a way to
determine the coefficients $\alpha$ and $\beta$ for a rational approximation. A good analogy can be made to polynomial approximations and power series.</p> <h2 id="using-our-small-toolbox">Using our
Small Toolbox</h2> <p>When dealing with approximations, an excellent place to start is to ask the following questions: Why is an approximation needed? What existing solutions already exist, and what
is the methodology they employ? How many digits of accuracy are needed? The answers to these questions provide solid baseline to formulate approximation specifications.</p> <p>Transcendental or
special functions are analytical functions not expressed by a rational function with finite powers. Some examples of transcendental functions are the exponential function $\exp(x)$, the inverse
logarithm $\ln(x)$, and exponentiation. However, if the target function being approximated has some nice properties (e.g., it is differentiable), it can be locally approximated with a polynomial.
This is seen in the context of <a href="https://en.wikipedia.org/wiki/Taylor%27s_theorem">Taylor's theorem</a> and more broadly as the <a href="https://en.wikipedia.org/wiki/
Stone%E2%80%93Weierstrass_theorem">Stone-Weierstrass theorem</a>.</p> <h3 id="power-series">Power Series</h3> <p>Polynomials (like $P_N(x)$ below) are a useful theoretical tool that also allow for
function approximations.</p> <p>$$ P_N(x)=\sum_{n=0}^N a_nx^n=a_0+a_1x+a_2x^2+a_3x^3+\cdots + a_N x^N $$</p> <p>Only addition, subtraction, and multiplication are needed. There is no need for
division implementations on the processor. More generally, an infinite polynomial called a <em>power series</em> can be written by specifying an infinite set of coefficients ${a_0,a_1,a_2,\dots}$ and
combining them as:</p> <p>$$ \sum_{n=0}^\infty a_n x^n. $$</p> <p>A specific way to get a power series approximation for a function $f$ around some point $x_0$ is by using Taylor's theorem to define
the series by:</p> <p>$$ \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n = f(x_0) + f'(x_0)x + \frac{f''(x_0)}{2!}x^2 + \frac{f'''(x_0)}{3!}x^3 +... $$</p> <p>Intuitively, the Taylor series
approximations are built by constructing the best "tangent polynomial," for example, the 1st order Taylor approximation of $f$ is the tangent line approximation to $f$ at $x_0$</p> <p>$$ f(x)\approx
f(x_0)+f'(x_0)(x-x_0)=\underbrace{f'(x_0)}<em>{\textrm{slope}}x+\underbrace{f(x_0)-f'(x_0)x_0}</em>{y-\textrm{intercept}}. $$</p> <p>For $\exp(x)$, there is the resulting series</p> <p>$$ \exp(x)=\
sum_{n=0}^\infty \frac{x^n}{n!} $$</p> <p>when approximating around $x_0=0$.</p> <p>Since polynomials can locally approximate transcendental functions, the question remains where to center the
approximations.</p> <p>The infinite series is precisely equal to the function $\exp(x)$, and by truncating the series at some finite value, say $N$, there is a resulting <em>polynomial approximation
</em>:</p> <p>$$ \exp(x)\approx\sum_{n=0}^N \frac{x^n}{n!} = 1+x+\frac{x^2}{2}+\cdots + \frac{x^N}{N!}. $$</p> <p>The function $\phi(x)$ can be written by scaling the input and output of $\exp$ by</
p> <p>$$ \sqrt{2\pi}\phi(\sqrt{2}x)=\exp(-x^2)=\sum_{n=0}^N \frac{(-x)^n}{n!}. $$</p> <p>This demonstrates what various orders of polynomial approximation look like compared to the function itself.</
p> <p><img src="/assets/blog/solstat/polynomial_approx.png" alt=""></p> <p>This solves the infinity problem and now these polynomials can be obtained procedurally at least for functions that are $N$
times differentiable. In theory the Taylor polynomial can be as accurate as needed. For example, increase $N$. However, there are some restrictions to keep in mind. For instance, since factorials $!$
grow exceptionally fast, there may not be enough precision to support numbers like $\frac{1}{N!}$. $20!>10^{18}$, so for tokens with 18 digits, the highest order polynomial approximation for $\exp(x)
$ on Ethereum can only have degree 19. Furthermore, polynomials have some unique properties:</p> <ol> <li>(No poles) Polynomials never have vertical asymptotes.</li> <li>(Unboundedness) Non-constant
polynomials always approach either infinity or negative infinity as $x\to \pm\infty$.</li> </ol> <p>An excellent example of this failure is the function $\phi(x)$ which asymptotically approaches 0 as
$x\to \pm \infty$. Polynomials don't do well approximating this! In the case of another even simpler function $f(x)=\frac{1}{x}$, this function can be approximated by polynomials away from $x=0$, but
doing so is a bit tedious. Why do this when division can be used to compute $f(x)$? It's more expensive and decentralized application developers must be frugal when using the EVM.</p> <h3 id=
"laurent-series">Laurent Series</h3> <p>Polynomial approximations are a good start, but they have some problems. Succinctly, there are ways to more accurately approximate functions with <em>poles</
em> or those that are <em>bounded</em>.</p> <p>This form of approximation is rooted in complex analysis. In most cases, any real-valued function $f(x)=y$ can instead allow the inputs $z$ and outputs
to be complex $f(z)=w$. This small change enables the <a href="https://en.wikipedia.org/wiki/Laurent_series"><em>Laurent series</em></a> expression for functions $f(z)$. A Laurent series includes
negative powers, and, in general, looks like:</p> <p>$$ f(z) = \sum_{n=-\infty}^{\infty}a_nz^n = \cdots+ a_{-2}\frac{1}{z^2} + a_{-1} \frac{1}{z} + a_{0} + a_1z + a_2z^2+ \cdots $$</p> <p>For a
function like $f(x)=\frac{1}{x}$ the Laurent series is specified by the list of coefficients $a_{-1}=1$ and $a_n=0$ for $n\neq 1$. Whatever precision intended is the precision of the division
algorithm if we implement $f(z)$ as a Laurent series!</p> <p>The idea of the Laurent series is immensely powerful, but it can be economized further by writing down an approximate form of the function
slightly differently.</p> <h3 id="rational-approximations-1">Rational Approximations</h3> <p>"If you sat down long enough and thought about ways to rearrange addition, subtraction, multiplication,
and division in the context of approximations, you would probably write down an expression close to this":</p> <p>$$ P_{m,n}(x)=\frac{\alpha_0 + \alpha_1 x + \alpha_2 x^2 + \cdots + \alpha_m x^m}{\
beta_0+\beta_1 x + \beta_2 x^2 + \cdots +\beta_n x^n}. $$</p> <p>Specific ways to arrange the fundamental operations can benefit particular applications. For example, there are ways to determine
coefficients $\alpha$ and $\beta$ that do not run into the issue of being smaller than the level of precision offered by the machine. Fewer total operations are needed, resulting in less total
storage use for the coefficients simultaneously.</p> <p>Aside from computational efficiency, another benefit of using rational functions is the ability to express degenerate function behavior such as
singularities (poles/infinities), boundedness, or asymptotic behavior. Qualitatively, the functions $\exp(-x^2)=\sqrt{2\pi}\phi(\sqrt{2}x)$ and $\frac{1}{1+x^2}$ look very similar on all of $\R$ and
the approximation fares far better than $1-x^2$ outside of a narrow range. See the labeled curves in the following figure.</p> <p><img src="/assets/blog/solstat/rational_vs_polynomial.png" alt=""></
p> <h3 id="continued-fraction-approximations">Continued fraction approximations</h3> <p>The degree of accuracy for a given approximation should be selected based on need and with respect to
environmental constraints. The approximations in <code>solstat</code> are an economized continued fraction expansion:</p> <p>$$ a_0+\frac{x}{a_1+\frac{x}{a_2+\frac{x}{a_3+\frac{x}{~~~\ddots}}}} $$</
p> <p>This is typically a faster way to compute the value of a function. It's also a way of definining the Golden Ratio (the <strong>most</strong> irrational number):</p> <p>$$ \varphi = 1+\frac{1}
{1+\frac{1}{1+\frac{1}{1+\frac{1}{~~~\ddots}}}} $$</p> <p>Finding continued fraction approximations can be done analytically or from Taylor coefficients. There are some specific use cases for
functions that have nice recurrence relations (e.g., factorials) since they work well algebraically with continued fractions. The implementation for <code>solstat</code> is based on these types of
approximations due to some special relationships defined later.</p> <h3 id="finding-and-transforming-between-approximations">Finding and Transforming Between Approximations</h3> <p>Thus far this
article has not discussed getting these approximations aside from the case of the Taylor series. In each case, an approximation consists of a list of coefficients (e.g., Taylor coefficients $
{a_0,a_1,\dots}$) and a map to some expression with finitely many primitive function calls (e.g., a polynomial $a_0+a_1x+\cdots$).</p> <p>The cleanest analytical example is the Taylor series since
the coefficients for well-behaved functions can be found by computing derivatives by hand. When this isn't possible, results can be computed numerically using finite difference methods, e.g., the
first-order central difference, in order to extract these coefficients:</p> <p>$$ f'(x)\approx \frac{f(x+h/2)-f(x-h/2)}{h} $$</p> <p>However, this can be impractical when the coefficients approach
the machine precision level. Also, Laurent series coefficients are determined the same way.</p> <p>Similarly, the coefficients of a rational function (or <a href="https://en.wikipedia.org/wiki/
Pad%C3%A9_approximant">Pade</a>) approximation can be determined using an iterative algorithm (i.e., akin to <a href="https://en.wikipedia.org/wiki/Newton%27s_method">Newton's method</a>). Choose the
order $m$ and $n$ of the numerator and denominator polynomials to find the coefficients. From there, many software packages have built-in implementations to find these coefficients efficiently, or a
solver can be implemented to do something like <a href="https://mathworld.wolfram.com/WynnsEpsilonMethod.html">Wynn's epsilon algorithm</a> or the <a href="https://en.wikipedia.org/wiki/
Minimax_approximation_algorithm">minimax approximation algorithm</a>.</p> <p>All of the aforementioned approximations can be transformed into one another depending on the use case. Most of these
transformations (e.g., turning a polynomial approximation into a continued fraction approximation) amount to solving a linear problem or determining coefficients through numerical differentiation.
Try different solutions and see which is best for a given application. This can take some trial and error. Theoretically, these algorithms seek to determine the approximation with a minimized maximal
error (i.e., minimax problems).</p> <h3 id="breaking-up-the-approximations">Breaking up the approximations</h3> <p>Functions $f\colon X \to Y$ also come along with domains of definition $X$.
Intuitively, the error for functions with bounded derivatives has an absolute error proportional to the domain size. When trying to approximate $f$ over all of $X$, the smaller the set $X$, the
better. It only takes $n+1$ points to define a polynomial of degree $n$. This means a domain $X$ with $n+1$ points can be perfectly computed with a polynomial.</p> <p>For domains with infinitely many
points, reducing the measure of the region approximated over is still beneficial especially when trying to minimize absolute error. For more complicated functions (especially those with large
derivative(s)), breaking up the domain $X$ into $r$ different subdomains can be helpful.</p> <p>For example, in the case of $X=[0,1]$, a 5th degree polynomial approximation for $f\colon [0,1]\to \R$
has max absolute error $10^{-4}$. After splitting the domain into $r=2$ even-sized pieces, the result is $f_1\colon [0,1/2]\to \R$ and $f_2\colon [1/2,1] \to \R$, which is used in the original
algorithms to determine two distinct sets of coefficients for their approximations. In the domain of interest, $f_1$ and $f_2$ only have $10^{-6}$ in error. Yet, if extended $f_1$ outside of $[0,1/2]
$, the error will have increased to $10^{-2}$. Each piece of the function is optimized purely for its reduced domain.</p> <p>Breaking domains into smaller pieces allows for piecewise approximations
that can be better than any non-piecewise implementation. At some point, piecewise approximations require so many conditional checks that it can be a headache, but this can also be incredibly
efficient. Classic examples of piecewise approximations are piecewise linear approximations and <a href="https://en.wikipedia.org/wiki/Spline_(mathematics)">(cubic) splines</a>.</p> <h2 id=
"ethereum-environment">Ethereum Environment</h2> <p>In the Ethereum blockchain, every transaction that updates the world state costs gas based on how many computational steps are required to compute
the state transition. This constraint puts pressure on smart contract developers to write efficient code. Onchain storage itself also has an associated cost!</p> <p>Furthermore, most tokens on the
blockchain occupy 256 bits of total storage for the account balance. Account balances can be thought of as <code>uint256</code> values. Fixed point math is required for accurate pricing to occur on
smart contract based exchanges. These libraries take the <code>uint256</code> and recast it as a <code>wad256</code> value which assumes there are 18 decimal places in the integer expansion of the
<code>uint256</code>. As a result, the most accurate (or even "perfect") approximations onchain are always precise to 18 decimal places.</p> <p>Consequently, it is of great importance to be
considerate of the EVM when making approximations onchain. All of the techniques above can be used to make approximations accurate near $10^{-18}$ in precision and economical simultaneously. To get
full $10^{-18}$ precision, the computation for rational approximations would need coefficients with higher than 256bit precision and the associated operations.</p> <h2 id="solstat-implementation">
Solstat Implementation</h2> <p>A continued fraction aproximation of the Gaussian distribution is performed in <a href="https://github.com/primitivefinance/solstat/blob/main/src/Gaussian.sol">
Gaussian.sol</a>. <a href="https://github.com/transmissions11/solmate/blob/ed67feda67b24fdeff8ad1032360f0ee6047ba0a/src/utils/FixedPointMathLib.sol">Solmate</a> is used for fixed point operations
alongside a custom library for units called <a href="https://github.com/primitivefinance/solstat/blob/main/src/Units.sol">Units.sol</a>. The majority of the logic is located in <a href="https://
github.com/primitivefinance/solstat/blob/main/src/Gaussian.sol">Gaussian.sol</a>.</p> <p>First, a collection of constants used for the approximation is defined alongside custom errors. These
constants were found using a special technique to obtain a continued fraction approximation that is for a related function called the <a href="https://en.wikipedia.org/wiki/Gamma_function">gamma
function</a> (or more specifically, the <a href="https://en.wikipedia.org/wiki/Incomplete_gamma_function">incomplete gamma function</a>). By changing specific inputs/parameters to the incomplete
gamma function, the <a href="https://en.wikipedia.org/wiki/Error_function">error function</a> can be obtained. The error function is a shift and scaling away from being the Gaussian CDF $\Phi(x)$.</
p> <h3 id="gaussian">Gaussian</h3> <p>The gaussian contract implements a number of functions important to the gaussian distributions. Importantly all of these implementations are for a mean $\mu = 0$
and variance $\sigma^2 = 1$.</p> <p>These implementations are based on the <a href="https://e-maxx.ru/bookz/files/numerical_recipes.pdf">Numerical Recipes</a> textbook and its C implementation. <a
href="https://e-maxx.ru/bookz/files/numerical_recipes.pdf">Numerical Recipes</a> cites the original text by Abramowitz and Stegun, <a href="https://personal.math.ubc.ca/~cbm/aands/
abramowitz_and_stegun.pdf">Handbook of Mathematical Functions</a>, which can be read to understand these functions and the implications of their numerical approximations more thoroughly. This
implementation is also differentially tested with the <a href="https://github.com/errcw/gaussian">javascript Gaussian library</a>, which implements the same algorithm.</p> <h3 id=
"cumulative-distribution-function">Cumulative Distribution Function</h3> <p>The implementation of the CDF aproximation algorithm takes in a random variable $x$ as a single parameter. The function
depends on helper functions known as the error function <code>erf</code> which has a special symmetry allowing for the approximation of the function on half the domain $\R$.</p> <p>$$ \operatorname
{erfc}(-x) = 2 - \operatorname{erfc}(x) $$</p> <p>It is important to use symmetry when possible!</p> <p>Furthermore, it has the other properties:</p> <p>$$ \operatorname{erfc}(-\infty) = 2 $$</p> <p>
$$ \operatorname{erfc}(0) = 1 $$</p> <p>$$ \operatorname{erfc}(\infty) = 0 $$</p> <p>The reference implementation for the error function can be found on p221 of Numerical Recipes in section C 2e. <a
href="https://mathworld.wolfram.com/Erfc.html">This page</a> is a helpful resource.</p> <h3 id="probability-density-function">Probability Density Function</h3> <p>The library also supports an
approximation of the Probability Density Function(PDF) which is mathematically interpeted as $Z(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x - \mu)^2}{2\sigma^2}}$. This implementation has a maximum
error bound of of $1.2\cdot 10^{-7}$ and can be refrenced <a href="https://mathworld.wolfram.com/ProbabilityDensityFunction.html">here</a>. The Gaussian PDF is even and symmetric about the $y$-axis.
</p> <h3 id="percent-point-function--quantile-function">Percent Point Function / Quantile Function</h3> <p>Aproximation algorithms for the Percent Point Function (PPF), sometimes known as the inverse
CDF or the quantile function, are also implemented. The function is mathmatically defined as $D(x) = \mu - \sigma\sqrt{2}\operatorname{ierfc}(2x)$, has a maximum error of $1.2\cdot 10^{-7}$, and
depends on the inverse error function <code>ierf</code> which is defined by</p> <p>$$ \operatorname{ierfc}(\operatorname{erfc}(x)) = \operatorname{erfc}(\operatorname{ierfc}(x))=x $$</p> <p>and has a
domain in the interval $0 < x < 2$ along with some unique properties:</p> <p>$$ \operatorname{ierfc}(0) = \infty $$</p> <p>$$ \operatorname{ierfc}(1) = 0 $$</p> <p>$$ \operatorname{ierfc}
(2) = - \infty $$</p> <h3 id="invariant">Invariant</h3> <p><code>Invariant.sol</code> is a contract used to compute the invariant of the RMM-01 trading function such that $y$ is computed in:</p> <p>
$$ y = K\Phi(\Phi^{⁻¹}(1-x) - \sigma\sqrt{\tau}) + k $$</p> <p>This can be interpretted graphically with the following image:</p> <p><img src="/assets/blog/solstat/rmm.png" alt=""></p> <p>Notice the
need to compute the normal CDF of a quantity. For a more detailed perspective on the trading function, take a look at the <a href="/papers/Whitepaper.pdf">RMM-01 whitepaper</a>.</p> <h2 id=
"solstat-versions">Solstat Versions</h2> <p>Solstat is one of Primitive's first contributions to improving the libraries availible in the Ethereum ecosystem. Future improvements and continued
maintenance are planned as new techniques emerge.</p> <h2 id="differential-testing">Differential Testing</h2> <p>Differential testing by Foundry was critical for the development of Solstat. A popular
technique, differential testing seeds inputs to different implementations of the same application and detects differences in the execution. Differential testing is an excellent complement to
traditional software testing as it is well suited to detect semantic errors. This library used differential testing against the javascript <a href="https://github.com/errcw/gaussian">Gaussian library
</a> to detect anomalies and varying bugs. Because of differential testing we can be confident in the performance and implementation of the library.</p> | {"url":"https://www.primitive.xyz/posts/solstat","timestamp":"2024-11-01T20:35:14Z","content_type":"text/html","content_length":"82671","record_id":"<urn:uuid:db24e990-4ea0-4b30-930b-42cab2656d36>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00268.warc.gz"} |
Section: New Results
On the maximal number of real embeddings of minimally rigid graphs in ${ℝ}^{2}$, ${ℝ}^{3}$ and ${S}^{2}$
Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space ${ℝ}^{d}$ or on a sphere and other manifolds which in addition satisfy certain edge length
constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely
related to the classification of rigid graphs according to their maximal number of real embeddings. In [17], we are interested in finding edge lengths that can maximize the number of real embeddings
of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of
real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain
embeddings in ${ℝ}^{3}$. One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration. Our results
include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in ${ℝ}^{2}$ and ${ℝ}^{3}$, while in the case of ${S}^{2}$ we
achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum
number of realizations. | {"url":"https://radar.inria.fr/report/2019/ouragan/uid103.html","timestamp":"2024-11-08T02:08:29Z","content_type":"text/html","content_length":"40426","record_id":"<urn:uuid:dc53d1a8-b5e2-4d88-accf-4ce69c8871c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00185.warc.gz"} |
Math, Grade 6, Ratios, Finding Percents
Jan’s and Martin’s Ideas
Work Time
Jan's and Martin's Ideas
• Jan said that the same percent can represent different quantities. Is she correct? Explain.
• Martin said that a single quantity can be represented by different percents. Is he correct? Explain.
Ask yourself:
• When considering Jan’s statement, think about 50% of two different “wholes."
• When considering Martin’s statement, think about this situation: you have $100 and your friend has $20, and each of you contributes $10 to your school library fundraiser. | {"url":"https://openspace.infohio.org/courseware/lesson/2087/student/?section=6","timestamp":"2024-11-12T05:11:31Z","content_type":"text/html","content_length":"34874","record_id":"<urn:uuid:3186c019-d349-449f-97fd-0ad1edfc96d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00110.warc.gz"} |
Understanding Kolmogorov-Smirnov (KS) Tests for Data Drift on Profiled Data
• Data Quality
• ML Monitoring
TLDR: We experimented with statistical tests, Kolmogorov-Smirnov (KS) specifically, applied to full datasets as well as dataset profiles and compared results. The results allow us to discuss the
limitations of data profiling for KS drift detection and the pros and cons of the KS algorithm for different scenarios. We also provide the code for you to reproduce the experiments yourself.
Data drift is a well-known issue in ML applications. If unaddressed, it can degrade your model significantly and make your model downright unusable. The first step to address those issues is to be
able to detect and monitor for data drift.
There are multiple approaches to monitoring data drift in production. It is very common to use statistical tests to get a drift detection value and monitor it over time. Traditional drift detection
algorithms usually need the full original data to calculate these values, but for large-scale systems, having complete access to historical data might be infeasible due to storage or privacy
concerns. A possible alternative is to sample your data beforehand, which also comes with disadvantages: you might lose important information, such as rare events and outliers, through aggregation,
damaging the result’s reliability.
A third approach is to profile your data before applying your drift detection algorithm. Profiles capture key statistical properties of data, such as distribution metrics, frequent items, missing
values, and much more. We can then use those statistical properties to apply adapted versions of drift detection techniques. Of course, since there is no such thing as a free lunch, this strategy has
its downsides. A profile is an estimate of the original data and, as such, using it for drift detection will generate approximations of the actual drift detection value that you would get if you had
used the complete data.
But how exactly does the profiling process work with the drift detection algorithms, and how much do we lose by doing it?
In this blog post, we’ll limit ourselves to numerical univariate distributions, and choose one specific algorithm to run the experiment: the Kolmogorov-Smirnov (KS) test. We’ll also get to have
some nice insights into the suitability of the KS test itself for different scenarios.
Here’s what we’ll cover in this blog post:
• What is the KS test?
• What is data profiling?
• Experiment Design
• The Experiments
- Experiment #1 — Data volume
- Experiment #2 — No. of buckets
- Experiment #3 — Profile size
You can check the code used in this blog post or even run the experiment yourself by accessing the experiment’s Google Colab notebook.
The Kolmogorov-Smirnov Test
The KS test is a test of the equality between two one-dimensional probability distributions. It can be used to compare a sample with a reference probability distribution or compare two samples. Right
now, we are interested in the latter. When comparing two samples, we are trying to answer the following question:
“What is the probability that these two sets of samples were drawn from the same probability distribution?”
The KS test is nonparametric, which means we don’t need to rely on assumptions that the data are drawn from a given family of distributions. This is good, since we often won’t know the underlying
distribution beforehand in the real world.
The statistic
The KS statistic can be expressed as:
D = supₓ|F₁(x) — F₂(x)|
where F1 and F2 are the two cumulative distribution functions of the first and second samples, respectively.
Another way to put it is that the KS statistic is the maximum absolute difference between the two cumulative distributions.
The image below shows an example of the statistic, depicted as a black arrow.
The two-sample KS statistic. Source: Wikipedia[1]
The Null Hypothesis
The null hypothesis used in this experiment is that both samples are drawn from the same distribution. For instance, a small p-value would indicate that the data is unlikely if all assumptions
defining our statistical model are true (including our test hypothesis). In other words, we can interpret the p-value as a measure of compatibility between the data and the underlying assumptions
that define our statistical model, with 0 representing complete incompatibility and 1 representing complete compatibility*[2].
To calculate this value, the KS statistic is taken into account along with the sample size of both distributions. Typical thresholds for rejecting the null hypothesis are 1% and 5%, implying that any
p-value less than or equal to these values would lead to the rejection of the null hypothesis.
(*) Errata by the author — The original sentence in the The Null Hypothesis section was: “For example, a p-value of 0.05 would mean a 5% probability of both samples being from the same distribution.”
This is a misconception and does not represent the correct definition of the p-value, as stated in the paper Statistical tests, P values, confidence intervals, and power: a guide to
Data Profiling
Profiling a dataset means collecting statistical measurements of the data. This enables us to generate statistical fingerprints, or summaries, of our data in a scalable, lightweight, and flexible
manner. Rare events and outlier-dependent metrics can be accurately captured.
To profile our data, we’ll use the open-source data logging library whylogs. Profiling with whylogs is done in a streaming fashion, requiring a single pass over the data, and allows for
parallelization. Profiles are also mergeable, allowing you to inspect your data across multiple computing instances, time periods, or geographic locations. This is made possible with a technique
called sketching, pioneered by Apache DataSketches.
Precisely for this example, we’ll leverage the profile’s distribution metrics. To calculate the KS statistic, we need to generate an approximation for the sample’s cumulative distribution function.
This is made possible with a technique called data sketching, pioneered by Apache DataSketches.
Experiment Design
First, we need the data. For this experiment, we will take two samples of equal size from the following distributions:
• Normal: Broad class of data. Unskewed and peaked around the center
• Pareto: Skewed data with long tail/outliers
• Uniform: Evenly sampled across its domain
In this blog post, we’ll show the results for normal distribution only, but you can find the same experiments for Pareto and Uniform distributions directly in the example notebook here. The
overall conclusions drawn from the normal distribution case can also be applied to the remaining distributions.
Drift Injection
Next, we’ll inject drift into one sample (which we’ll call the target distribution) to compare it to the reference, unaltered, distribution.
We will inject drift artificially by simply shifting the data’s mean according to a parameter. We chose to use the ratio of the distribution’s interquartile range. Here’s what it looks like for the
normal distribution case:
Image by author
The idea is to have four different scenarios: no drift, small drift, medium drift, and large drift. The magnitude classification and the ideal process of detecting/alerting for drifts can be very
subjective, depending on the desired sensitivity for your particular application. In this case, we are assuming that the small-drift scenario is small enough for it to be safely ignored. We are also
expecting that the medium and large drift scenarios should result in a drift alert since both would be cases for further inspection.
Applying the KS test
As the ground truth, we will use scipy’s implementation of the two-sample KS test with the complete data from both samples. We will then compare those results with the profiled version of the test.
To do so, we’ll use whylogs’ approximate implementation of the same test, which uses only the statistical profile of each sample.
The distribution metrics contained in the profiles are obtained from a process called sketching, which gives them many useful properties but adds some amount of error to the result. For this reason,
the KS test result can be different each time a profile is generated. We’ll profile the data 10 times for every scenario, and compare the ground truth to statistics such as the mean, maximum, and
minimum of those runs.
Experiment Variables
Our main goal is to answer:
“How does whylogs’ KS implementation compare to scipy’s implementation?”
However, this answer depends on several different variables. We will run three separate experiments to better understand the effect of each variable: data volume, number of buckets, and profile size.
The first one relates to the number of data points in each sample, whereas the last two relate to whylogs internal, tunable parameters.
Experiment #1 — Data Volume
The number of data points in a sample affects not only the KS test in general but also the profiling process itself. It is reasonable, then, to investigate how it affects the results.
We compared the p-values for both implementations with varying sample sizes (for both target and reference distributions): 500, 1k, 5k, 10k, and 50k.
Image by author
You’ll notice that we don’t have error bars for the ground truth. For a given sample size and drift magnitude, scipy’s result is deterministic, since we’re always using the complete data, whereas,
for whylogs, the error bars represent the maximum and minimum values found in the 10 runs.
Note that, for medium and large drift cases, both y-axis are really close to 0, so even for a sample size of 500, both implementations result in a p-value of effectively 0, indicating that our data
is highly incompatible with the null hypothesis. For the no drift and small drift scenarios, we can see that both implementations yield very similar results when comparing the mean p-value of the
sketch-based implementation, but with some difference for specific runs, especially for large sized samples. However, for almost all cases, the ground truth lies somewhere in between the range of the
profiled case. It is also worth noting that, at a 95% confidence interval, both implementations would yield the same conclusion for all points in all scenarios.
KS test is really sensitive, and its sensitivity increases according to the sample size: in the small-drift scenario, for sample sizes greater than or equal to 5k, we reject the null hypothesis. Even
though this is not technically wrong, we initially considered this case to be so small that it could be safely ignored.
At this point, we should ask ourselves whether this test is actually telling us what we care about. A p-value smaller than 0.05 would lead to rejecting the null hypothesis, but it doesn’t tell us
anything about the effect size. In other words, it tells us that there is a difference, but not how much of a difference there is. There might be statistical significance, but not an actual practical
significance to it.
Experiment #2 — No. of Buckets
To get a discrete cumulative distribution, we first need to define the number of buckets. The sketch-based KS test will then use those buckets to calculate the statistics. We will run experiments
with equally spaced bins of sizes: 5, 10, 50, and 100. For each of the 10 runs, we will calculate the absolute error between the exact and the sketch-based whylogs’ implementation and plot the mean,
along with the error bars representing the minimum and maximum errors found. We will show those errors according to the sample size and drift magnitude, just like in the previous experiment.
whylogs’ current version has 100 as the default number of buckets, and that is also the value used in the previously shown results.
Image by author
Since some of the values in the graphs are much higher than the remaining ones, we’re breaking the y axis in some cases, to better visualize all the bars in the plot. Even so, some of the bars are
still too small to be seen. The errors for the medium and large drift scenarios are very close to 0, meaning that both implementations get similar results.
Overall, the error’s mean seems to decrease when increasing the number of buckets. However, the variance of the errors increases for higher sample sizes, which is due to the increasing estimation
errors in the profiling process.
The experiments so far show some degree of randomness for the no-drift scenario, for both implementations. Since the KS test relies solely on the maximum absolute difference between distributions,
any slight changes resulting from the sampling process will affect the no-drift scenario.
Experiment #3 — Profile Size
As previously stated, in a profile we have an approximate distribution in the format of a data sketch. A data sketch is configured with a parameter K, which dictates the profile’s size and its
estimation error [3]. The higher this parameter, the lower the estimation error will be. All of the previous experiments were run with a K=1024, but now we want to see how the errors get affected
with varying numbers of K.
This time, we will fix the sample size to 100k and the number of buckets to 100 and vary the K parameter to the following values: 256, 512, 1024, 2048, and 4096.
Image by author
We have omitted charts for drift sizes 0.4 and 0.75 due to the consistently small amounts of errors making visualization unnecessary.
The X-axis is shown according to the profile’s size when serialized: K values of 256, 512, 1024, 2048, and 4096 yield approximate profile sizes of 6KB, 11KB, 22KB, 43KB, and 83KB, respectively.
As seen before, any drifted scenario just shows how sensitive the KS test is. The bars can’t be seen for the medium and large drift scenarios because their values are effectively 0, but in the
no-drift scenario we can see that the error is inversely proportional to the profile size, and by extension to the K parameter. By increasing K, the errors due to profiling decrease, approximating
both implementation’s results.
We can also verify that, for this scenario, the errors are quite small. But if we are interested in minimizing those errors, we can sacrifice profile space for better results.
Let’s summarize some key takeaways from these experiments:
• Performing the KS test on data profiles is possible, and the results are very close to the standard implementation. However, the results are non-deterministic.
• KS test is very sensitive, and it tends to get even more sensitive with higher sample sizes. When testing solely under the null hypothesis, it might tell us something about the difference between
distributions, but is insensitive to how much of a difference there is.
• We can tune internal parameters for better results for the whylogs’ implementation. In particular, we can increase the profile size to get results closer to the ground truth.
We hope this helps in building intuition on how the KS test works along with data profiling. We also got to have a better understanding of KS test limitations. Motivated by that, we at whylogs are
already implementing additional similarity measures! For instance, Hellinger distance is already implemented in whylogs, so stay tuned for more experiments and benchmarks!
Thank you for reading, and feel free to reach out if you have any questions/suggestions! If you’re interested in exploring whylogs in your projects, consider joining our Slack community to get
support and also share feedback!
“Understanding Kolmogorov-Smirnov (KS) Tests for Data Drift on Profiled Data” was originally published by Towards Data Science.
[1] — Kolmogorov–Smirnov test. (2022, October 29). In Wikipedia. https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
[2] — Greenland, S., Senn, S.J., Rothman, K.J. et al. Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations. Eur J Epidemiol 31, 337–350 (2016).
[3] — Karnin, Z., Lang, K., & Liberty, E. (2016). Optimal Quantile Approximation in Streams. arXiv. https://doi.org/10.48550/arXiv.1603.05346
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[Solved] How long does it take an automobile trave | SolutionInn
How long does it take an automobile traveling in the left lane at 60.0 km/h to pull
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initially 100 m apart?
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The Sum of A Square and B Square - Dien Mayson Ganh
Imagine you have two square tiles, one with an area represented by A and the other with an area represented by B. If you were to sum the areas of these two squares, the total space covered would be
referred to as the sum of A square and B square. This concept plays an essential role in various mathematical fields, including algebra, geometry, and even physics.
Understanding Square Numbers
Before delving into the sum of A square and B square, let’s establish a fundamental understanding of square numbers. A square number results from multiplying an integer by itself. For example, 4 is a
square number as 2 x 2 equals 4. Similarly, 9 is a square number as 3 x 3 equals 9. The process of squaring a number involves raising it to the power of 2.
The Sum of Two Squares
When dealing with two square numbers, say A and B, the sum of their areas can be expressed algebraically as A^2 + B^2. This expression represents the total combined area covered by the two squares.
Geometric Interpretation
Visually, if you were to represent square A and square B on a graph, with sides of lengths √A and √B respectively, the total area covered by both squares would indeed be A^2 + B^2, which is the sum
of the individual areas.
Expanding the Expression
The sum of A square and B square can be further expanded using algebraic formulas to yield more insights into the relationships between variables. The expanded form of (A + B)^2, known as FOIL
(First, Outer, Inner, Last), results in:
(A + B)^2 = A^2 + 2AB + B^2
Example Applications
The concept of the sum of A square and B square finds practical applications in various areas. For instance, in physics, the Pythagorean theorem states that in a right-angled triangle, the square of
the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be expressed mathematically as:
a^2 + b^2 = c^2
Where a and b are the lengths of the two shorter sides of the triangle, and c is the length of the hypotenuse.
Properties and Patterns
Exploring the sum of A square and B square reveals interesting properties and patterns. Some notable aspects include:
• Even Squares: The sum of two even square numbers is always even.
• Odd Squares: The sum of two odd square numbers is always even.
• One Even, One Odd: The sum of an even square and an odd square is always odd.
• Special Cases: Some numbers can be expressed as the sum of two square numbers in multiple ways, such as 25 = 3^2 + 4^2 = 5^2.
Applications in Number Theory
The sum of A square and B square also plays a significant role in number theory, especially in the field of sums of squares. Mathematicians have long been fascinated by representing numbers as the
sum of two or more square numbers. This area of study has led to intriguing discoveries and conjectures, such as Fermat’s Theorem on sums of two squares.
Frequently Asked Questions (FAQs)
Q1: What is the sum of squares formula?
A: The sum of squares formula is represented as A^2 + B^2, where A and B are the areas of two square tiles.
Q2: How is the sum of A square and B square geometrically interpreted?
A: Geometrically, the sum of A square and B square represents the total area covered by two square tiles with areas A and B, respectively.
Q3: What are some properties of the sum of A square and B square?
A: Some properties include even squares yielding an even sum, odd squares yielding an even sum, and one even and one odd square yielding an odd sum.
Q4: In what field of mathematics is the concept of the sum of squares frequently used?
A: The sum of squares is commonly applied in algebra, geometry, physics, and number theory.
Q5: Can a number be expressed as the sum of two square numbers in more than one way?
A: Yes, certain numbers can be represented as the sum of two square numbers in multiple ways, such as the number 25.
In conclusion, the sum of A square and B square is a fundamental concept that finds applications in various mathematical disciplines. Understanding this concept not only provides insights into
algebra and geometry but also opens doors to exploring advanced mathematical theories and patterns. Whether you’re solving geometric problems or diving into number theory, the sum of A square and B
square serves as a cornerstone for deeper mathematical exploration. | {"url":"https://dienmaysonganh.com/the-sum-of-a-square-and-b-square/","timestamp":"2024-11-04T18:21:16Z","content_type":"text/html","content_length":"62408","record_id":"<urn:uuid:edfcd485-4a51-4619-a04d-a609c38fa452>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00260.warc.gz"} |
Download CBSE Class 10 Sanskrit 2024 PYQs PDF
Download CBSE Class 10 Sanskrit 2024 Past Year Papers PDF
If we are talking about the ancient language that is widely known, the majority of the students would answer Sanskrit. CBSE has included the Sanskrit language in its curriculum for students to get
familiar with this language. The previous year papers for CBSE Class 10 Sanskrit are essential study material for students preparing for the board exams.
Downloading the class 10 sanskrit question paper with answers for the last year can help students understand the exam pattern and get familiar with the types of questions they can find in the
sanskrit exam. Students can download the sanskrit past year paper for 2024 from the table given below and include it in their exam preparation.
Sanskrit Class 10 Previous Year Question Paper: Pattern
CBSE has increased the percentage of competency-based questions for the upcoming board exams. However, the pattern is expected to remain the same. Based on the class 10 sanskrit previous year
question paper PDF, the paper includes four sections.
Section A includes one unseen reading comprehension weighing 10 marks. Students have to answer the given questions related to the passage.
Section B evaluates the student’s creative writing skills and weighs 15 marks. The questions are based on letter writing, article writing, description writing, and much more. Section C checks the
student’s grammatical skills, whereas Section D is all about literary skills.
How to Download Sanskrit Class 10 Previous Year Question Paper
Class 10 Sanskrit Question Papers with Answers are released every year on the official CBSE website. Students can click the required link to download the previous year's papers. The papers can also
be downloaded from the links given in the table above, and you can browse to this website to download the previous year papers for CBSE Class 10 and other subjects.
Why to Download Sanskrit Class 10 Previous Year Question Paper
There are already many study materials for every subject available on the market, and adding another one seems like a tedious task. Including the class 10 sanskrit previous year question paper PDF in
the exam preparation routine can help students' scores significantly.
• The papers will include chapter-wise questions as per their allotted mark weights, and students can predict the important topics after analysing these papers.
• There is a high probability that many questions may reoccur in the upcoming exams, so practicing them will prepare students beforehand.
• Students can modify their exam preparation strategy based on their performance after solving these sanskrit class 10 question papers from the question papers from the previous year.
How to Start Solving the Sanskrit Class 10 Previous Year Question Paper
Time management is an important aspect of performing well in the board exams. Regardless of any study material, understand how to start preparing from that resource.
• The foremost thing to do before starting to use any resource is to understand the syllabus. Go through the CBSE Class 10 Sankrit Syllabus and decide the time for attempting these questions.
• Don’t refer to the solution before you have attempted the question. Try to attempt it by yourself and cross-check with the given marking scheme. This will help in understanding the weak areas,
and you can start working on improving them.
• Complete the syllabus using the Class 10 Sanskrit NCERT before trying to attempt previous-year questions. It will significantly increase efficiency.
Include past 10-year questions for the CBSE Class 10 Sanskrit exam to score above 90. Understand how to answer in the correct way to avoid losing minimal marks. Now, you can download the files to
your computer for offline access at any time. These links are made available for free.
Extra 10% Discount
on Educart books via email. | {"url":"https://www.educart.co/previous-year-question-paper/cbse-class-10-sanskrit-previous-year-question-paper-2024","timestamp":"2024-11-04T20:50:09Z","content_type":"text/html","content_length":"201507","record_id":"<urn:uuid:2ffdfabd-d611-4dab-80da-6b97c0a8f463>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00439.warc.gz"} |
Solve the equation and choose the BEST answer: (frac{mathrm{9} }{mathrm{b}} = frac{mathrm{b} }{mathrm{4}})
A 6
Cross-multiply: \(9 \times 4 = b \times b\)
Then, find the square root of both sides:
The actual square root of \(b^2\) is \(\pm b\) and the actual square root of 36 is \(\pm b\), so:
\(b= \pm 6\)
Of the answer choices for this question, 6 is the best choice since it is the only one that is true about the equation.
Related Information
Solve the equation and choose the BEST answer: \(\frac{\mathrm{9} }{\mathrm{b}} = \frac{\mathrm{b} }{\mathrm{4}}\)
I think this is a good app to use to review for the ATI TEAS.
I feel confident that adding this app to my studying regimen will enable me to score well on the TEAS.
This app is great practice. | {"url":"https://teas-prep.com/question/solve-the-equation-and-choose-the-best-answer-7194w46h39png-6249204242448384/","timestamp":"2024-11-08T09:15:34Z","content_type":"text/html","content_length":"82120","record_id":"<urn:uuid:244b2248-3e0f-4f85-9c2f-c48e6cf31f41>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00192.warc.gz"} |
\l ojasiewicz gradient inequality
Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so–called {\em tangency variety} of $f$ at $\bar
{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of … Read more | {"url":"https://optimization-online.org/tag/l-ojasiewicz-gradient-inequality/","timestamp":"2024-11-10T01:45:36Z","content_type":"text/html","content_length":"83131","record_id":"<urn:uuid:3da70d28-1e03-4262-b2a8-7226d6f75595>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00202.warc.gz"} |
PROC SURVEYMEANS computes degrees of freedom df to obtain the % confidence limits for means, proportions, totals, ratios, and other statistics. The degrees of freedom computation depends on the
variance estimation method that you request. Missing values can affect the degrees of freedom computation. See the section Missing Values for details.
Taylor Series Variance Estimation
For the Taylor series method, PROC SURVEYMEANS calculates the degrees of freedom for the t test as the number of clusters minus the number of strata. If there are no clusters, then the degrees of
freedom equal the number of observations minus the number of strata. If the design is not stratified, then the degrees of freedom equal the number of PSUs minus one.
If all observations in a stratum are excluded from the analysis due to missing values, then that stratum is called an empty stratum. Empty strata are not counted in the total number of strata for the
table. Similarly, empty clusters and missing observations are not included in the total counts of cluster and observations that are used to compute the degrees of freedom for the analysis.
If you specify the MISSING option, missing values are treated as valid nonmissing levels for a categorical variable and are included in computing degrees of freedom. If you specify the NOMCAR option
for Taylor series variance estimation, observations with missing values for an analysis variable are included in computing degrees of freedom.
Replicate-Based Variance Estimation
When there is a REPWEIGHTS statement, the degrees of freedom equal the number of REPWEIGHTS variables, unless you specify an alternative in the DF= option in a REPWEIGHTS statement.
For BRR or jackknife variance estimation without a REPWEIGHT statement, by default PROC SURVEYMEANS computes the degrees of freedom by using all valid observations in the input data set. A valid
observation is an observation that has a positive value of the WEIGHT variable and nonmissing values of the STRATA and CLUSTER variables unless you specify the MISSING option. See the section Data
and Sample Design Summary for details about valid observations.
For BRR variance estimation (including Fay’s method) without a REPWEIGHTS statement, PROC SURVEYMEANS calculates the degrees of freedom as the number of strata. PROC SURVEYMEANS bases the number of
strata on all valid observations in the data set, unless you specify the DFADJ method-option for VARMETHOD=BRR. When you specify the DFADJ option, the procedure computes the degrees of freedom as the
number of nonmissing strata for an analysis variable. This excludes any empty strata that occur when observations with missing values of that analysis variable are removed.
For jackknife variance estimation without a REPWEIGHTS statement, PROC SURVEYMEANS calculates the degrees of freedom as the number of clusters (or number of observations if there are no clusters)
minus the number of strata (or one if there are no strata). For jackknife variance estimation, PROC SURVEYMEANS bases the number of strata and clusters on all valid observations in the data set,
unless you specify the DFADJ method-option for VARMETHOD=JACKKNIFE. When you specify the DFADJ option, the procedure computes the degrees of freedom from the number of nonmissing strata and clusters
for an analysis variable. This excludes any empty strata or clusters that occur when observations with missing values of an analysis variable are removed.
The procedure displays the degrees of freedom for the t test if you specify the keyword DF in the PROC SURVEYMEANS statement. | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_surveymeans_details14.htm","timestamp":"2024-11-14T09:07:15Z","content_type":"application/xhtml+xml","content_length":"18300","record_id":"<urn:uuid:020cb9f5-0cc5-46c7-889d-d0fd79491025>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00854.warc.gz"} |
Renormalization-group improvement in ha
SciPost Submission Page
Renormalization-group improvement in hadronic $τ$ decays in 2018
by D. Boito, P. Masjuan, F. Oliani
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Diogo Boito
Submission information
Preprint Link: scipost_201811_00036v1 (pdf)
Date submitted: 2018-11-22 01:00
Submitted by: Boito, Diogo
Submitted to: SciPost Physics Proceedings
Proceedings issue: The 15th International Workshop on Tau Lepton Physics (TAU2018)
Ontological classification
Academic field: Physics
Specialties: • High-Energy Physics - Phenomenology
Approach: Theoretical
One of the main sources of theoretical uncertainty in the extraction of the strong coupling from hadronic tau decays stems from the renormalization group improvement of the series. Perturbative
series in QCD are divergent but are (most likely) asymptotic expansions. One needs knowledge about higher orders to be able to choose the optimal renormalization-scale setting procedure. Here, we
discuss the use of Pad\'e approximants as a model-independent and robust method to extract information about the higher-order terms. We show that in hadronic \boldmath $\tau$ decays the fixed-order
expansion, known as fixed-order perturbation theory (FOPT), is the most reliable mainstream method to set the scale. This fully corroborates previous conclusions based on the available knowledge
about the leading renormalon singularities of the perturbative series.
Current status:
Has been resubmitted
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2018-12-9 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_201811_00036v1, delivered 2018-12-09, doi: 10.21468/SciPost.Report.725
This is an interesting exercise and the contribution is well written. It reports on results presented in Ref. [21], to which the authors refer for further details.
I have one major concern, and I would like the authors to address it in their contribution.
Eq. (15) is strictly valid only in the large-$\beta_0$ limit.
This means that its derivation is not forced to reproduce the two-loop universality of the QCD beta function, i.e., the fact that the first two coefficients of the QCD perturbative beta function are
renormalisation-scheme independent. Let me notice that the two-loop universality has crucial physical consequences, because it determines exactly the existence of the nontrivial IR zero of QCD at
large $N$ and $N_f$.
Thus my question is: how may your analysis and in particular Eq.(15) and Figure 2 change when two-loop universality of the QCD beta function is correctly accounted for in the derivation?
I would like the authors to explicitly discuss this potential issue, how and where their analysis could be affected and why eventually its numerical effects could be negligible in a certain range of
$N$ and $N_f$. Most importantly, could this change the comparison between FOPT and CIPT?
Requested changes
The authors should address the issue of neglected two-loop universality, see report.
answer to question
Answer to referee's comments:
First of all, we would like to thank the referee for the careful
reading and the comments on our contribution to the proceedings.
The referee points out correctly that Eq. (15) is valid only in the
large-beta_0 limit. Therefore, effects due to the two-loop coefficient
of the beta function are absent. The modifications of this equation
when one departs from the large-beta_0 limit are two. First, the
renormalon singularities of the Adler function become branch cuts and
are no longer simple or double poles. This is discussed in detail in
our Refs. [15] and [19]. Second, the prefactor of Eq. (15) which plays
a role in the simplification of the analytic structure will change.
Actually, we are, at present, actively working on modifications to
this prefactor due to the two-loop running of alpha_s. It can be
shown, using the asymptotic formula for the running of alpha_s, that
corrections to this result are proportional to beta_2/beta_1^2.
Therefore, since the running of alpha_s is dominated by the leading
term in the beta-function, it is correct to say that in QCD Eq. (15)
is correct as a first approximation and that the main consequences of
this prefactor remain valid in QCD, i.e. the Borel transform of
delta^{0} is less singular than the Borel transform of the Adler
We should point out, however, that the results in large-beta_0 are
used in our work only as a guide, a sort of laboratory for our
strategy. The results we obtain in QCD, discussed in Sec. 5, do not
rely on the large-beta_0 limit, and are obtained from the coefficients
computed at five-loops in full QCD, with no approximation. In this
sense, our final results are not affected by the limitations and
(over) simplifications that exist in the large-beta_0 limit.
Therefore, the comparison between CIPT and FOPT uses Eq. (15) only as
a guide to the general structure of the Borel transform of
delta^{(0)}, and is not affected by the details of Eq. (15).
We believe that this was not completely clear in the proceedings
because we had to shorten the discussion compared to the original
work. We have now added in Sec. 5 on p. 11 (in red) explanations on
Eq. (15) and how it is used in our final results in QCD. We believe
this new version is improved and we hope the referee will agree. | {"url":"https://scipost.org/submissions/scipost_201811_00036v1/","timestamp":"2024-11-09T07:50:10Z","content_type":"text/html","content_length":"37448","record_id":"<urn:uuid:57044209-ad7e-449f-8829-f18d934e2042>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00485.warc.gz"} |
Jump to navigation Jump to search
Function Plots
The simplest way to plot a function is to give Plot the description of the function and the domain, and let Plot decide where to evaluate the function. You do this by giving the function instead of
the data as the last variable in the right argument to plot or pd. The plot may be a 2-D plot or a 3-D plot. An example is
plot 0 10 ; 'sin'
The character string instead of data tells Plot to evaluate the sin function over the interval [0,10]. One advantage of a function plot, besides its simplicity, is that it has the X-axis information
so can label that axis correctly. Compare the above example to the equivalent numeric "plot sin 5+i:5j99". In this case, "plot" does not have the X values available so it labels that axis with
integers from 0 to 99 as seen here, whereas the function plot labels the X-axis with values 0-10, as given to the function.
The independent variable(s) are given either as intervals or as lists of points. An interval is indicated by 2 or 3 numbers, specifying start value,end value,number of steps. A list of points is
indicated by a boxed argument containing the points, or by an unboxed list of more than 3 points. Multiple intervals or point-lists are allowed.
If the number of steps is omitted or 0, Plot will pick an appropriate number of points to use. It does so by repeatedly subdividing the interval until the curve is smooth or it decides that the curve
is discontinous, in which case it plots continuous sections separately.
The subdivision is controlled by the plot options Cfuncres and singtoler. Cfuncres (C is x or y) gives the subdivision resolution: an interval smaller than 1/Cfuncres of the screen will not be split.
Cfuncres defaults to twice the pixel resolution of the plot. singtoler is used when the display has singularities, and controls how much of the heading-off-to-infinity tail of the curve will be shown
at the singularity. You can experiment to find a good value for singtoler for your application; the default is 10 and higher numbers cause more of the tail to be displayed.
The function(s) to be displayed can be given as a list of gerunds, one for each verb to be drawn, or as a string where the verb-specifiers are separated by the ` character (use doubled ` as an escape
if your verb contains a ` character). Each verb-specifier can be in either tacit or explicit form: if it contains the words y or y. it is assumed to describe an explicit verb, otherwise a tacit one.
The verbs are invoked with lists as arguments and should be able to return a list of results. If you use pd , note that the verbs are not executed until pd 'show' is processed, so the values of any
public variables that are referred to by an explicit verb will use the values in effect when the pd 'show' is executed. Public variables referred to in a tacit verb are frozen (using f.) when the pd
for the function is issued.
Examples of function plots:
plot _10 10 ; '%' NB. reciprocal: has a discontinuity
plot _10 10 ; 'sin`cos' NB. two curves
plot 0.001 0.1 ; 'sin % y' NB. sin(1/x), a busy function
3D plot example:
f=: 4 : '(cos r) % 1 + r =. x +&:*: y'
plot _4 4 100 ; _4 4 100 ; 'f'
Re-working Function Definition from Monadic to Dyadic
Sometimes it may be necessary to change the form of a function's definition to accommodate a function plot.
For instance, say we have the standard sombrero function defined to take an argument specifying the points along one side of the base of the sombrero (a single vector argument):
sombrero0=: [: (1&o. % ]) [: %: [: +/~ *:
So, a straightforward plot might look like this:
'surface' plot sombrero0 i:20j99
However, since this function takes only a single argument, it generates the grid of points by orthogonally adding the squares of the vector argument with +/~ so it isn't in a dyadic form. Note that a
dyadic form is more general since we specify the grid by two sets of points to be combined orthogonally instead of using the single set twice.
A dyadic form might look like this:
dyasombrero=: (4 : '(1&o. % ]) %:+/*:x,y')"0/
where we put "0/ in the definition to work on the scalar elements of one vector versus each scalar element of the other vector.
So, we can make a non-square sombrero like this:
plot _25 25 100; _15 15 100; 'dyasombrero'
Here's another way to make a sombrero and save it as a .PNG file. Note that we also replace the default palette with our own by re-assigning RGCLR_jzplot:
load '~User/code/bmpPal.ijs'
RGCLR_jzplot_=: ADJPAL
sombrero=: 4 : '(cos % >:)x +&:*: y'
'mesh 0' plot _4 4 100; _4 4 100; 'sombrero'
pd 'save png C:\amisc\pix\sombreroCos.png' | {"url":"https://code.jsoftware.com/wiki/Plot/Function","timestamp":"2024-11-11T00:53:14Z","content_type":"text/html","content_length":"23428","record_id":"<urn:uuid:6572d0d6-e65a-4380-81df-f61d2f206644>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00663.warc.gz"} |
Chapter 9: Arrays
In this chapter, we’ll be examining arrays. An array is a collection of objects or variables of the same type. Each object or variable in an array is an element of that array. Each element of the
array is assigned an index, or a numerical value starting from zero. Elements of arrays are accessed with the array indexing operator, two square brackets [].
Note: The plural form of index is indices or indexes. Both are grammatically correct. You may see these words used interchangeably throughout various math, science, and programming fields.
Creating Arrays
Declaring Arrays
Arrays, like any variable, must be declared before you can use them. In Java, you declare an array by placing square brackets next to the variable type.
Declaring Array
int[] myArray;Code language: CSS (css)
In plain English, this line is saying “create a new reference variable for an array of ints and call it myArray.”
Instantiating Arrays
When we initialize variables, we are allocating memory for them. We must do the same for arrays. Arrays, however, are considered to be collections. Collections are objects, not variables. Recall that
when we allocate memory for an object, we are instantiating it. When we allocate memory for a variable, we’re initializing it. You may see the terms instantiate and initialize used interchangeably in
your studies.
When you instantiate an array, you must tell the JVM how much memory to allocate for that array. Does this array need 5 spaces in memory? Ten? Twenty? To instantiate the array after declaring it, we
need to say it’s a new array with a certain size. Once the size or length of the array is set, it cannot change. That would look like this:Instantiate Array
myArray = new int[5];Code language: JavaScript (javascript)
The above line is saying “create instance myArray as a new array of integers with 5 elements. Set aside 5 locations in memory for ints.”
Visualization of a 5 element array of ints
It is important that you do not create arrays with more indices than you need to use. If you do this, you’re wasting space in the computer’s memory.
We can now set and access values from any index in the array. Indexing starts from zero, so if the array has 5 elements, the indices go from 0 to 4, not 1 to 5.
In the sample code below, we set index 0 of the array to 43 and print it to the screen.
Assign, Instantiate, Use Array
int[] myArray;
myArray = new int[5];
myArray[0] = 43;
System.out.println(myArray[0]);Code language: JavaScript (javascript)
We can set any element in the array to any value permitted by the int type. This will become especially useful later, when we combine arrays with loops. We can loop through the indices and do all
sorts of fun things.
Setting Values For More Elements.
int[] myArray;
myArray = new int[5];
myArray[0] = 43;
myArray[1] = 24;
myArray[2] = 69;
System.out.println(myArray[2]);Code language: JavaScript (javascript)
We can also declare and initialize our arrays in one line.Declare and Instantiate Array
int[] myArray = new int[5];Code language: JavaScript (javascript)
IndexOutOfBounds Exception
If you run out of space in your array, or attempt to access an index beyond the length you originally set, you will get an index out of bounds exception. You will likely become well acquainted with
this error. When you see it, you need to review your logic and array size. Usually you’ll only be off by one number in an index, or a loop will go one step too far.
int[] myArray;
myArray = new int[5];
myArray[5] = 43;Code language: JavaScript (javascript)
Exception in thread "main" java.lang.ArrayIndexOutOfBoundsException: Index 5 out of bounds for length 5
Array Creation Shortcut
If you already know the values you need in your array, you can declare, instantiate, and assign values to it all in one line.
Here’s an example that adds five values to an int array.
int[] myArray = { 5, 10, 15, 20, 25};
You declare the array as you normally would, then add a comma separated list of values in curly braces. There is no need to say how many elements are in the array, it will automatically figure that
part out.
Array Length
The number of elements an array can hold is the length of the array. You can find the length of an array by accessing the .length variable.
int myArray[] = {5, 10, 15, 20, 25};
System.out.printf("The length of the array is: %d", myArray.length);Code language: JavaScript (javascript)
The length of the array is: 5
Finding the length of an array is helpful if you’re creating an algorithm that might deal with arrays of unknown or different lengths.
Calculating The Average – Example
The Problem
Let’s take a look at a simple example. Say you want to create a program that calculates the average of five different values entered in by the user. The program would look like this when run:
Enter 5 numbers to find the average.
Calcululating Average...
The average of the terms you entered is: 5.2
If you wanted to make a program like this using only variables, it wouldn’t be too hard. You could use an input scanner and set five different variables equal to the numbers the user enters, then add
them up and find the average. This would involve a bit of repetitive code. One of our goals when designing things is to make methods as generic or reusable as possible. If we code a program like
this, it will only be able to ever find the average of 5 values.
There is a way to find the mean of multiple terms using loops without arrays, but we will ignore that option for demonstration.
Example Program (No Arrays)
Scanner input = new Scanner(System.in);
System.out.println("Enter 5 numbers to find the average.");
double a = input.nextDouble();
double b = input.nextDouble();
double c = input.nextDouble();
double d = input.nextDouble();
double e = input.nextDouble();
System.out.println("Calculating average...");
double average = (a+b+c+d+e)/5.0;
System.out.println("The average of the terms you entered is: " + average);Code language: JavaScript (javascript)
If we modified it to support 10 values, we would have to create 5 additional variables and manually code each term into calculating the average. That’s not too hard. But what if we wanted to modify
it to support 20, 30, or even 100 different entries? It would quickly become a tedious nightmare. This is where arrays come in to save the day.
Arrays allow us to use a single variable identifier to store multiple elements. Soon, we’ll learn how to make a program that calculates the average of any number of terms using an array.
The Solution (With Arrays)
Now, we’re going to use an array and loops to calculate the average from a given set of terms.
• The program will ask how many numbers we are finding the average from. This will be the length of our array.
• We will loop through the indices of the array, assigning each index a value the user enters.
• Loop through all the indices to find the sum of all the values.
• Calculate the average.
Start by importing the Scanner class and creating a new scanner. Then, ask the user how many numbers we’re finding the average of.
import java.util.Scanner;
public class Main {
public static void main(String[] args) {
System.out.println("How many numbers are you entering?");
Scanner input = new Scanner(System.in);
int terms = input.nextInt();
}Code language: JavaScript (javascript)
Next, we need to create an array to hold the values. I have decided to use doubles as my variable type. Declare and instantiate the array with the number the user entered. Then, ask the user to enter
the numbers.
double[] values = new double[terms];
System.out.println("Enter " + terms + " numbers to find the average.");Code language: JavaScript (javascript)
Create a loop that iterates through each index of the array, starting from zero. In this case, we’re going from zero to the number of elements in the array (our terms variable).
In the body of the loop, record the number the user enters for that index of the array.
I have decided to use a for loop for this.
for(int i = 0; i < terms; i++){
values[i] = input.nextDouble();
}Code language: HTML, XML (xml)
Once this loop is finished, we have assigned values to each index of the array. We can close the scanner.
input.close();Code language: CSS (css)
Now, we need to find the sum of all the values that have been entered. Create a variable called sum that will store the total sum of all the values entered. Iterate through the loop again, and add
each value to the sum.
//find the average
System.out.println("Calculating average...");
double sum = 0;
for(int i = 0; i < terms; i++){
sum = sum + values[i];
}Code language: JavaScript (javascript)
We now have the sum and the number of terms. We can find the average.
double average = sum / terms;
System.out.println("The average of the terms you entered is: " + average);Code language: JavaScript (javascript)
Now, when you run the program, you should be able to figure out the average of any number of items.
me@kevinsguides:~$ avg_calculator.java
How many numbers are you entering?
Enter 3 numbers to find the average.
Calculating average...
The average of the terms you entered is: 5.333333333333333
Extended Calculations
Now, let’s use the same program above, but calculate some additional information from the values. Let’s find the maximum value entered.
To find the maximum value, we will set a variable to zero. Assume the user only enters positive values. We can loop through each index of the array and check the current value against the recorded
maximum. If the value at this index is greater than what’s currently recorded, it becomes the new maximum.
//calculate the max
double max = 0;
//iterate through all values
for (int i = 0; i < terms; i++){
//if this value is greater than the recorded max, it's the new max
if (values[i]>max){
System.out.println("The largest number you entered (maximum) is:" + max);Code language: JavaScript (javascript)
How many numbers are you entering?
Enter 3 numbers to find the average.
Calculating average...
The average of the terms you entered is: 32.333333333333336
The largest number you entered (maximum) is:53.0
The program now finds the average and the maximum value entered.
Try Yourself: Can you create another loop that finds the minimum value?
Basic Algorithm – Sorting
Now that we have a basic foundation for using loops and arrays together, consider the following question. How could we make an algorithm that sorts the elements of an array of integers in order from
smallest number to greatest?
First, let’s consider the logic. If you were given a list of five numbers: 5, 9, 7, 2, and 1, you could easily sort them at a glance, likely with little thought required. The order from smallest to
greatest is 1, 2, 5, 7, then 9. How can we get a computer to figure this out? We need to break the problem down into instructions a computer can understand. There are multiple ways we could achieve
this with a loop.
Sorting An Array From Smallest to Greatest
One way we could rearrange this list of numbers is to iterate through each number. If this number is larger than the next number in the sequence, we will swap the number with the next number. Each
time the loop executes, the number is moved to the next index if it’s greater than that number.
• Given Values: 5, 9, 7, 2, 1
• Check if the first value is greater than the second.
• 5 is not greater than 9, leave it as is.
• Check if the second value is greater than the third.
• 9 is greater than 7. Swap 9 and 7.
• Values: 5, 7, 9, 2, 1
• 9 has moved one slot to the right.
• Check if the third value is greater than the fourth.
• 9 is greater than 2, so swap 9 and 2.
• Values: 5, 7, 2, 9, 1
• Check if the fourth value is greater than the fifth.
• 9 is greater than 1, so swap 9 and 1.
• Values: 5, 7, 2, 1, 9
Now, we have 9 in the proper position. It has moved to the end of the array. We repeat this process three more times. Each time, the next highest number is moved as far to the right as possible.
Given: 5, 9, 7, 2, 1
Outer Loop 1 Finishes: 5, 7, 2, 1, 9
Outer Loop 2 Finishes: 5, 2, 1, 7, 9
Outer Loop 3 Finishes: 2, 1, 5, 7, 9
Outer Loop 4 Finishes: 1, 2, 5, 7, 9
Here is a visual representation of the steps involved, if that helps with understanding.
Examine the code for the completed program below. It takes an array of the 5 values above and sorts them from smallest to greatest, then prints out the values.
public class Main {
public static void main(String[] args) {
int sortMe[] = new int[5];
sortMe[0] = 5;
sortMe[1] = 9;
sortMe[2] = 7;
sortMe[3] = 2;
sortMe[4] = 1;
//outer loop four times
for(int a = 0; a < 4; a++){
//inner loop moves largest to the right
for(int i = 0; i < 4; i++){
//if this value is greater than the next value
if (sortMe[i] > sortMe[i+1]){
//create a temporary value to hold the current value, so we don't lose it
int tmp = sortMe[i];
//swap the values
sortMe[i] = sortMe[i+1];
sortMe[i+1] = tmp;
//print out the sorted array
for(int i = 0; i < 5; i++){
}Code language: JavaScript (javascript)
The inner loop checks each value in the array and moves the largest number found to the end. The outer loop executes 3 more times to move the largest number to the right each time. Both loops run 4
times and not 5, because there are 4 swaps that can occur the way we have this set up. In fact, if we ran the inner loop 5 times, on the fifth time, we’d be checking the 4th index value against the
5th index value. There is no 5th index value so we’d get an IndexOutOfBounds Exception.
Notice that when I swapped the values I created a temporary variable to store one of the values. This was necessary to swap the values.
Consider what would’ve happened if I just swapped the values using the indexes.
//wrong way to swap
sortMe[i+1]=sortMe[i];Code language: JavaScript (javascript)
When the program runs, it does something like this. Let’s say we’re at the beginning where sortMe[1] = 9 and sortMe[2] = 7.
sortMe[2] = 7;
//wrong way to swap
sortMe[1]=sortMe[2]; //sortMe[1] now equals 7
sortMe[2]=sortMe[1]; //sortMe[1] was turned into 7 in the last step, so sortMe[2] now equals 7.
//the value 9 has disappearedCode language: JavaScript (javascript)
When I first set sortMe[i] to sortMe[i+1], I am overwriting the value of sortMe[i]. If I do this, then when I set sortMe[i+1] to sortMe[i] after setting sortMe[i] to sortMe[i+1], it would be setting
both indexes to the same value. The temporary variable stores one of the values, so I can use it again immediately. We want to SWAP the values, not set them equal to the same number.
When we use the temporary value, this is what happens:
//right way to swap
int tmp = sortMe[1]; //tmp is set to 9
sortMe[1] = sortMe[2]; //sortMe[1] is set to 7
sortMe[2] = tmp; //sortMe[2] is set to 9Code language: JavaScript (javascript)
Code Efficiency Note
The way we created the inner loop in the above section to sort values works, but is it as efficient as possible?
Consider the fact that the inner loop is set to run a set number of times, every time. The first time it runs through, it moves the largest number to the right. The outer loop then moves onto the
second execution of the inner loop, and the inner loop is run again, moving the second largest number to the second space from the end.
When the inner loop runs its second time, it still checks if the second last value is higher than the last value. We already know this will be false, since we moved the highest number to the end the
first time the inner loop ran. Therefore, there is no need to check if this last swap needs to be performed.
If we’re dealing with an array of 5 elements, then the first time the outer loop runs, we need to check all 4 swaps with the inner loop. The second time, we only need to check the first 3 swaps. The
third time, the first two swaps, and so on.
We can prevent these extra unnecessary checks from occurring by simply adding -a to the condition for the inner loop. So the first time it runs, it runs all 4 times. The second time it runs, it runs
4-1 times, the third time, 4-2 times, and so on.
More Efficient Inner Loop Condition
for(int i = 0; i < 4-a; i++){
Add -a to the inner loop condition, and test the code again. It should still sort just like before, but now the code is more efficient. It’s not wasting as much CPU resources.
The key takeaway here is that just because it works, doesn’t mean it’s the most efficient way of working. Carefully consider the logic behind the algorithms you write. On a small scale, it won’t make
a huge difference. But if we were sorting a thousand numbers, or a million numbers, these inefficiencies could result in significant amounts of wasted CPU time.
Passing Arrays To Methods
Arrays, like variables, can be passed to methods as arguments. This is important for making reusable code. The previous sorting example is hard coded to only sort an array with five elements. Let’s
modify the algorithm that sorts numbers from smallest to greatest so it can work with any array. We’ll do this by making a sortLeastGreatest method, which sorts any array of ints from smallest to
The method definition will look like this:
public static void sortLeastGreatest(int[] sortMe){
}Code language: JavaScript (javascript)
Overflow Exceptions
If you get an overflow exception when looping, check the end condition! In many cases, you may only need to decrease the conditional variable by one index.
The method is public, it can be accessed anywhere in the program. The static keyword is required so we can call it from the main method. The method is of type void. This method does not return
anything, it only modifies an existing array. The parameter is an int array called sortMe, which is the array that will be sorted.
Like all objects in Java, arrays are passed by value. In the case of arrays, the value being passed is a reference variable corresponding to the original array. So when we modify the reference
variable in the method, it actually updates the values of the original array. No new array is being created in the method.
Now we can largely use the same code as before, we just need to modify the loops to work with an array of any length, since we don’t know the length of the arrays we’re passing to this method. The
condition of the loops will make them stop at the array length minus one.
If you do this correctly, it will look like this:
/** * You can put this after the the main method within the same class
* @param sortMe the array to sort
public static void sortLeastGreatest(int[] sortMe){
//outer loop
for(int a = 0; a < sortMe.length; a++){
//inner loop moves largest to the right
for(int i = 0; i < sortMe.length-1-a; i++){
//if this value is greater than the next value
if (sortMe[i] > sortMe[i+1]){
//create a temporary value to hold the current value, so we don't lose it
int tmp = sortMe[i];
//swap the values
sortMe[i] = sortMe[i+1];
sortMe[i+1] = tmp;
}Code language: PHP (php)
Once it’s done sorting, it must return the sorted array sortMe.
Now, let’s test out the method we created. I make a test array, fill it with values, and use the method to sort it. I print this sorted array to the console.
//replace your main method with this
public static void main(String[] args) {
int[] arr = {5, 10, 9, 7, 2, 1, 4};
//print sorted array
for(int i = 0; i < arr.length; i++){
System.out.print(arr[i] + ", ");
}Code language: JavaScript (javascript)
1, 2, 4, 5, 7, 9, 10,
It clearly works for this array. But never test something just once, create some other test arrays with random values and make sure they’re all properly sorted.
int[] testArrayB = {5, 6, 23, 43, 24, 54, 63, 632, 132, 4234, 54, 0, 92, 1, 4, 5, 4, 2, 55, 234};
//print sorted array
for(int i = 0; i < testArrayB.length; i++){
System.out.print(testArrayB[i] + ", ");
}Code language: PHP (php)
0, 1, 2, 4, 4, 5, 5, 6, 23, 24, 43, 54, 54, 55, 63, 92, 132, 234, 632, 4234,
Now I know it works with longer arrays, and it can properly sort a variety of numbers. The logic behind the algorithm works and no unexpected behavior occurred, even with duplicate values.
Returning New Arrays/Copying
Suppose instead of sorting an existing array from least to greatest, we want to take the values from an original array, and return a new sorted array. That way, the original array doesn’t change. To
do this, we need to create a copy of the original array, modify sort copy, and return the reference to the sorted copy.
The example code below copies the values of the original array into a new array, then sorts and returns it.
public static int[] sortLeastGreatest(int[] sortMe){
//create a new array to store the copy
int copy[] = new int[sortMe.length];
//copy sortMe into a new array
for(int i = 0; i < sortMe.length; i++){
copy[i] = sortMe[i];
//sort array
for(int a = 0; a < copy.length; a++){
for(int i = 0; i < copy.length-1-a; i++){
if (copy[i] > copy[i+1]){
int tmp = copy[i];
copy[i] = copy[i+1];
copy[i+1] = tmp;
return copy;
}Code language: PHP (php)
Now, when we want to sort arrays, calling the sortLeastGreatest method will return a new array of ints.
public static void main(String[] args) {
int[] unsorted = {5, 9, 3, 5, 1};
int[] sorted = sortLeastGreatest(unsorted);
//print out the arrays
System.out.println("Original array: ");
for(int i = 0; i < unsorted.length; i++){
System.out.print(unsorted[i] + ", ");
System.out.println("\nSorted array: ");
for(int i = 0; i < sorted.length; i++){
System.out.print(sorted[i] + ", ");
}Code language: PHP (php)
Original array:
5, 9, 3, 5, 1,
Sorted array:
1, 3, 5, 5, 9,
Now if we need the original array for some reason later, we still have it to work with.
For-each Loops
For-each loops are a useful shortcut to quickly examine the contents of an array. They allow you to loop through every value in an array using a variable that matches the data type of the array. The
variable declared in the for-each loop becomes each value of the array in the loop body. There’s no need to use indexes with for-each loops. They just go through the array from index zero to the end
one step at a time. Each time the loop body is executed the variable declared takes the place of whatever value is at that index.
The structure of a for each loop is as follows. Notice that we use a colon instead of a semicolon.
for (type variable : array){
//do something with variable
}Code language: PHP (php)
Functionally, the above code does almost the same thing as:
for (int i = 0; i < array.length; i++){
//do something with array[i]
}Code language: PHP (php)
For-each loops use the for keyword just like regular for loops.
Here is an example of a for-each loop in action. This for-each loop simply traverses an array of ints and prints each one out.
int[] array = {5, 10, 15, 20, 25};
for(int val : array){
System.out.print(val + ", ");
}Code language: PHP (php)
This is like saying “for each int in my array, print out the value.” The variable I named value has taken the place of each value in the array.
5, 10, 15, 20, 25,
Since for-each loops only access the values of an array without telling us the index we’re at on each iteration, they don’t provide a direct way to set values. You will need to use a normal for loop
with indices if you want to modify the values of an array. The variable in the for-each loop containing each value in the array is not tied to the array itself. If you modify this variable, the array
itself will remain the same.
Additionally, because for-each loops only start from index zero, you can only use them if you need to go through a loop from start to finish. If you want to skip indices, or go from the end of the
array to the beginning, you’ll need to use a different type of loop.
Example: Find The Lowest Number
Let’s find the lowest number in an array of doubles using a for-each loop. To do this, we’ll declare and initialize a variable to keep track of the lowest number (called lowest). We’ll set this
number equal to the first element of the array to start. Then, in the for-each loop, we’ll check each value. If the value found is lower than the last lowest found number, it becomes the new lowest
Once the for-each loop is done executing, the variable will contain the lowest number.
double[] randomNumbers = {54.9, 53, 213.5, 40.1, 8.89, 65, 31, 65, 76};
double lowest = randomNumbers[0];
for(double val : randomNumbers){
//if the number at this value is lower than what was set before, it's the new lowest number.
if (val < lowest){
lowest = val;
System.out.println("The lowest number is: " + lowest);Code language: JavaScript (javascript)
The lowest number is: 8.89
Here’s a summary of what happens on each iteration of the for-each loop if you need help understanding what happened:
1. lowest = 54.9, val = 54.9, 54.9 is not less than 54.9, continue
2. lowest = 54.9, val = 53, 53 is less than 54.9, 53 becomes the new lowest number, lowest is set to 53
3. lowest = 53, val=213.5, 213.5 is not less than 53, continue
4. lowest = 53, val=40.1, 40.1 is less than 53, 40.1 becomes the new lowest number, lowest is set to 40.1
5. lowest = 40.1, val=8.89, 8.89 is less than 40.1, 8.9 becomes the new lowest number, lowest is set to 8.9
6. lowest = 8.89, val=65, 65 is not less than 8.89, continue
7. lowest = 8.89, val=31, 31 is not less than 8.89, continue
8. lowest = 8.89, val=65, 65 is not less than 8.89, continue
9. lowest = 8.89, val=76, 76 is not less than 8.89, we have reached the end of the loop and found the lowest number
Array Expansion
The size of an array is final and cannot be changed. That said, it is possible to “expand” an array by copying it into a new, larger array, and referencing the new, larger array.
Suppose you have an array with five elements. The original array might look like this:
//an int array with 5 elements
int[] arr = {10, 20, 30, 40, 50};Code language: JavaScript (javascript)
Now lets expand the array to fit ten elements. The example below shows how to do this.
//an int array with 5 elements
int[] myArray = {10, 20, 30, 40, 50};
//create a reference to the original array
int[] originalArray = myArray;
//have myArray reference a new array with 10 elements
myArray = new int[10];
//copy the first 5 values into the new array
for (int i = 0; i < originalArray.length; i++){
myArray[i] = originalArray[i];
//get rid of (dereference) the original array
originalArray = null;Code language: JavaScript (javascript)
Array Expansion Visualization
After enlarging the array, we can access elements 5-9 of the new, larger array without throwing an IndexOutOfBounds exception.
//now we can access elements 5-9 of the new array
myArray[5] = 60;
myArray[6] = 70;
myArray[7] = 80;
myArray[8] = 90;
myArray[9] = 100;
for(int i: myArray){
System.out.print(i + ", ");
}Code language: PHP (php)
10, 20, 30, 40, 50, 60, 70, 80, 90, 100,
It worked! We essentially turned an array with 5 elements into an array with 10 elements. Remember, technically we replaced the original array with a new, larger array and got rid of the original.
The original array never changed in size. The variable is referencing a completely new array with the elements of the original.
Computationally, this is an expensive task. Imagine if we had an array with a thousand elements, and needed to expand it to support 1001 elements. If we just expand the array by 1 element each time
we need to add a new value, it’s going to have to go through the entire original array and copy it into a new array every single time we add one new item. That becomes very inefficient.
If there’s a possibility you’ll need to add more items to an array after creating it, you should expand the length of the array by a multiple of the original. That way, your program will not need to
perform as many expansion operations. Instead of just expanding an array’s length by one element, expand it by double or triple the original length, depending on how often you think you’ll need to
expand it. This will use more memory, but a lot less CPU time. It’s OK to reserve a little extra memory if you think you’ll need it later.
In the next chapter, we will learn how to use another collection object called ArrayList which expands array-like collections for us.
Assignment: Mean, Median, Mode Calculator
Create a program that asks the user how many values they want to enter, then have them enter that amount of values. Place all these values into an array of doubles. Then, print out the numbers sorted
from least to greatest. Calculate the mean, median, and mode of all these numbers and print the results.
The program must have four methods, in addition to the main method.
• Methods:
□ sortLeastGreatest(double[] arr) – the method to sort the array
□ double findMean(double[] arr) – a method that returns the mean of an array of doubles
□ double findMode(double[] arr) – a method that returns the mode of an array of doubles – return -1 if no mode was found
□ double findMedian(double[] arr) – a method that returns the median of an array of doubles
You may assume the array has already been sorted from least to greatest before using the findMean, findMedian, and findMode methods (no need to call sortLeastGreatest from these methods).
Math Refresher
The mean is the average of a set of numbers. It is found by taking the sum of all the numbers and dividing that number by the number of terms.
mean=sum of numbersamount of numbers
If you take a set of numbers that are in order, and find the number in the middle, that number is the median.
For example, if given this set of data: 1, 5, 9, 10, 12
The median would be 9, since it’s the number in the middle.
If there’s an even number of items, the median is the mean of the middle two numbers.
For example, if given this set of data: 1, 5, 9, 10, 12, 16
The median would be 9.5, because 9+102=9.5
The mode is the number that occurs most often in the data set. Sets of data can contain multiple modes (if there’s five twos and five fours, for example).
For the sake of this assignment, assume that the data set only contains one possible mode.
• Data: 1, 4, 5, 9, 9, 10
• Mode: 9 (occurs twice)
Hints & Methodology
You can use the template to get started, if needed. It contains only the method definitions you will need. Everything else, you should fill in yourself.
• To get started, create a new project with a class file and a main method. Import and instantiate the scanner, then ask the user how many values they’re entering.
• Create an array of doubles double[] values;. Set the length of the array to the number the user specified. values = new double[totalValues];
• Use a loop and fill the array with the values the user enters.
• Sort the array from least to greatest using the sortLeastGreatest method defined in this chapter. Modify it to use doubles instead of ints.
• Use a loop to print out the sorted array as a comma separated list of values System.out.print(values[i] + ", ");.
• Create methods to find the mean, median and mode. Find the results and print them to the screen.
It may help to visualize how each method needs to work by writing a sample set of data on a piece of paper, then figure out each step you need to take to find each value.
The Mean Method
We already learned how to find the mean with an array of ints in this chapter. Use the same logic for this method, but with doubles. Then return the mean as a double.
The Median Method
After sorting the array, you can find the median. We will assume that the sort method has been executed before the getMedian method is executed, so there is no need to re-sort the array in the
getMedian method.
You will have to first determine if you’re working with an even set of numbers, or an odd set of numbers.
To find if you have an even or odd set, you can use the modulus operator. Remember, this operator returns the remainder after dividing two numbers. If length of array2 has a remainder of 1, then the
length must be odd. If the remainder is zero, the length must be even.
if(arr.length % 2 == 1) can be used to determine if there are an odd number of items in the array.
If dealing with an odd number of items, the median is the number in the middle. Divide the total length of the array by 2 to find the index of the middle item.
If dealing with an even number of items, the median is the average of the two middle numbers. Divide the length of the array by two to find the index of the greater middle item. Subtract this number
by 1 to find the index of the lesser middle item. Add the values at these indices together and divide by 2.0 to find the median.
Remember, the length of the array is the total number of elements. The indices of an array start at zero. If an array has 5 elements, its length is 5, but its indices run from 0 to 4.
If the array has 5 elements, the 3rd element is the one in the middle. When we divide length/2 the answer is 2.5, but since it’s an int, 5/2 will equal 2. So we can use length/2 to find the middle
index. Even though it 5/2 returns 2, 2 is the proper index of the third, middle element.
If the array has 6 elements, then the 3rd and 4th terms are the two middle numbers we need to find the average of. When we divide 6/2, we get 3. Since the indices start at zero, index 3 will give us
the fourth item in the sequence (the higher middle number). We must subtract the index by 1 to find the index of the lower middle number.
Return the median once you find it.
The Mode Method
We’re going to assume our data set only has one mode, or no modes. We will ignore the situation of multiple modes. If we entered the terms 5, 5, 6, 6, 7, and 8, we would have two modes – 5 and 6.
We’re going to ignore this possibility.
Create a variable to keep track of the mode. It should be a double. Create a variable to keep track of the number of times this number shows up in the array. It should be an int.
Now, you need to figure out how many times each number appears in the array. You will have to loop through each value in the array.
Create an inner loop that again goes through each value in the array. It should check if the current value found matches the value we’re checking for. If they match, then increment the number of
occurrences found by 1.
Check if the amount of times that this number appears is greater than the amount of times we have recorded for the last mode found. If we found more of this number than we did for another number,
then this must be the new mode. Update the mode with the current found value and set the total amount of occurrences to the occurrences found for this number.
Return the mode you found. If the total number of occurrences found is 1, then only 1 of each number must exist. So return -1 in this case.
Sample Output
When you run the program, it should work something like this:
How many values are you entering?
Please begin entering your values.
Sorted values:
2.5, 3.0, 3.0, 4.0, 5.0,
The mean is: 3.5
The median is: 3.0
The mode is: 3.0
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Apps For
Awesome Apps For Anyone Who Struggles With Math
Math homework doesn’t have to be a struggle, thanks to modern technology. Mathematics is the toughest subject for most students. Unfortunately, this subject is incorporated in almost every course.
This makes running away from math almost impossible. However, there are awesome apps that any student that struggles with math can use to make life easier. Here are some of these apps.
This is a free, clever photo calculator application that solves almost any math equation faster. This app does the magic of solving a math problem in two steps. You launch the application and aim the
Smartphone at the math equation. Align the problem that you wish to solve with the frame brackets of the app and watch the animated dots as they sparkle. After a few seconds, the app will generate
the answer on the screen automatically. It’s that easy.
Math Formulas Free
Many students seek help with homework when unable to recall formulas when solving math problems. Basically, trying to recall all mathematics formulas is not easy. Fortunately, you can easily get help
with this challenge. Math Formulas Free is an app that makes looking for complicated and easy formulas easy. It is ideal for university and high school students as well as engineers. This app covers
apps for geometry, differentiation, algebra, integration, statistics, and matrix.
Mathematics Dictionary
If you are one of the students that struggle to differentiate millimeter from milliliter, you should use this app. Mathematics Dictionary app extends beyond pure math definitions and terms to include
meanings and phrases that are relevant and useful to learners in courses that link to math. Thus, this app will be useful to you even if you are studying diverse things like oceanography or stock
Math Expert
There are times when you turn to a math homework helper when you can remember most of the equation or formula but you can’t put fingers on the missing function or term. In that case, you can use Math
Expert. This app stores many physics and math formulas. However, it includes a feature that allows you to fill in areas that you can recall. Once you do that, the app checks the possible calculations
Math Tricks
Shortcuts are very important when it comes to solving math problems. The Math Tricks app presents 23 tricks. These include simple multiplication and addition tips as well as complex stuffs like
multiplication for numbers with a one at the end and finding squares for numbers.
GMAT Math Flashcards
Before you assume that you can’t find an answer for a math problem, use GMAT Math Flashcards. This is an app that will easily help you with math revisions. Try this app when practicing or revising
math homework. It includes up to 425 flash cards by Graduate Management Admission Test tutors. The wide range of questions as well as solutions that the app presents will enable you to practice
solving different math problem at different difficulty levels.
Basically, you need math skills to excel in fields like accounting, engineering, and meteorology. If you struggle with math, try these apps to boost your skills and grades. If unable to improve after
using these awesome apps, consider getting math homework help online. Dissertation writers for hire an help you as well. | {"url":"https://www.boothbayplayhouse.com/awesome-math-apps","timestamp":"2024-11-12T22:23:38Z","content_type":"text/html","content_length":"13327","record_id":"<urn:uuid:a923a068-5cb2-462f-89d5-d5c37fe6daaf>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00658.warc.gz"} |
Non Verbal Reasoning
Non Verbal Reasoning - Classification - Discussion
Discussion Forum : Classification - Section 2 (Q.No. 48)
Directions to Solve
In each problem, out of the five figures marked (1), (2), (3), (4) and (5), four are similar in a certain manner. However, one figure is not like the other four. Choose the figure which is different
from the rest.
Choose the figure which is different from the rest.
(1) (2) (3) (4) (5)
All other figures can be rotated into each other.
6 comments
Ramanujam said: 7 years ago
According to me, The Answer is B.
Sahithy said: 7 years ago
Answer is A.
Figure (2) and (5) can be rotated into each other and figure (3) and (4) can be rotated into each other.
The odd one is figure (1).
So the answer is A.
Sahithy said: 7 years ago
(2) and (5) can be rotated into each other. (3) and (4) can be rotated into each other.
So the answer is (1).
Leni said: 8 years ago
Anser is 5. Because all are radial while 5 is perpendicular.
HEMRAJ said: 8 years ago
I cant't understand this, please explain anyone.
Ashish kumar seth said: 8 years ago
How answer is A? Please explain in details.
Quick links
Quantitative Aptitude
Verbal (English)
Placement Papers | {"url":"https://www.indiabix.com/non-verbal-reasoning/classification/discussion-234-1","timestamp":"2024-11-01T20:56:07Z","content_type":"text/html","content_length":"43748","record_id":"<urn:uuid:22439a32-17b8-4474-b764-596bd3db4a94>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00298.warc.gz"} |
One Man Back: Step Up or Not? - The Gammon Press
Cash game, center cube.
White to play 3-2.
Suppose you’re an intermediate-level backgammon player (a little vague, to be sure, but you get the idea) and you’d like to improve. What’s the best way to study the game in a systematic manner?
Different players have different approaches. Here’s what I like to do as a training regimen:
1) Play a daily practice match against Extreme Gammon (XG). There are other bots, but XG is the best. I like five or ten-game cash sessions, since even a short session will usually produce a good
amount of interesting study material.
2) Let the bot analyze the session when finished. The bot will highlight a number of errors.
3) Perform rollouts on the errors to make sure they are, in fact, errors. I like 1296 trials, 3-ply. It’s not perfect, but it’s relatively quick and will produce more accurate results than the bot’s
raw evaluation. You can always do longer and more rigorous rollouts later if the position seems to warrant extra attention.
4) When you’ve found a real error, try to categorize the position. Think about why you liked your original play, and try to see why the bot’s play might be better. Keep a notebook of different
categories of positions. Print out your new position and stick it in the notebook. Your goal is to find categories where you’re making a lot of errors, and try to refine your understanding of that
group of positions.
5) Keep track of your error rate for the session in a spreadsheet. Individual sessions will have huge variance, so average your results over 20-session samples. Over time, you’d like that error rate
to drop slowly.
6) Repeat.
A key skill in this whole process is the ability to categorize positions effectively. Weak players tend to see positions in terms of broad, mushy categories: ‘middle games’, ‘back games’, ‘endings’,
and so forth. Better players see the game in terms of large numbers of narrow but well-defined categories: ‘1-3 backgame with adequate timing’, or ‘5-point anchor game with a third checker back’.
Being able to see the game in terms of such narrow categories is advantageous because narrow categories may have strategies and heuristics which apply well within the category but don’t apply once we
change the position slightly. The doubling strategies which govern a pure 5-point anchor position, for instance, don’t apply well once the defender has a third checker back.
I like to categorize positions is two broad ways:
1) By position type. “Position type” just refers to the broad outline of the structure of the position. Typical examples are “2-4 backgame” or “5-prime versus 4-prime”.
2) By tactics. A tactical category is just a description of the choices available. “Slot versus split”, “run out or build point”, and “escape prime or extend prime” are examples of tactical
Sometimes we may have to combine both a positional type and a tactical type to make a useful category. For instance, “slotting versus splitting” tends to obey one set of rules in the very early game
and a different set somewhat later. So we might have an “Early game: slotting versus splitting” category and a “Middle game: slotting versus splitting” category, with different strategies and
Now let’s look at this position. Before we start discussing the pros and cons of the possible plays, our first job is to categorize the position. It’s a very common type that I call “One Man Back”.
One side (Black in this case) has escaped both his back checkers, while the other side (White) still has one checker back in a deep position.
In “One Man Back” positions, Black generally has a small lead in the race, something like 5-10 pips. Black won’t have a big lead in the race because generally no hitting has occurred yet, but he
should have some racing edge because he’s rolled big enough to get his back checkers out, while White hasn’t.
What sort of interesting problems do we get in “One Man Back” positions? While many plays are routine, the interesting ones generally fall into one of three groups:
1) White’s checker plays. White’s problems arise when he can move his back checker but isn’t sure if he should. If he can run into the outfield and try to disengage, he should almost always do so. If
all he can do is advance in Black’s board, then we get positions like this one. Should he move up where Black can point on him, thereby getting a chance to escape, or not?
2) Black’s checker plays. In general Black is bringing down builders and trying to make inner points. His good checker play problems tend to be one of three types:
a) Leaving indirect shots: If Black can leave an indirect shot but get an extra builder, should he do so?
b) Make a point or pick and pass: Suppose White plays 24/21 in this position and Black then rolls 2-1 or 3-1. Should he make the 5-point or pick and pass?
c) Hit loose: If White advances and Black can play safe or hit loose, should he hit?
3) Cube decisions. When should Black be doubling? In general, he needs less of a race lead for doubling than required in a straight race, since he also has the vigorish of winning with a prime.
Now let’s look at this position, and see just what White should do with a 3-2.
White has three choices. If he wants to advance his back checker with 24/21, then his deuce will be 6/4. If he leaves his back checker alone, he can play either 6/3 5/3, making the 3-point, or 6/4 6/
3, leaving two blots.
Let’s look at the last two plays first. They’re close, but making the point is slightly better. Its advantage comes in the variations where White runs into the outfield next turn and Black hits. In
those variations, White will sometimes have return shots, and if he hits a shot, the inner board blots are a real liability. For instance, here’s a typical such sequence:
White 3-2: Plays 6/4 6/3
Black doubles
White takes
Black 5-2: Plays 13/6
White 6-4: Runs with 24/14
Black 2-1: Hits with 13/11*/10.
While these sequences are unlikely, they should be very strong for White. They’re much weaker, however, with two blots floating around in the home board.
Eliminating 6/4 6/3 leaves us with a simple choice: stepping up in the board or making the 3-point. The race is relatively close (White trails by six, 120-114 after the roll) and that fact points to
the solution to the problem. White should move up and try to escape, even though he’ll be escaping into a race where he’s an underdog. Consider these arguments:
(1) If White can get to a race, Black has only one way to win, namely the race. If White stays back, Black has two ways to win: either in the race, or by building a prime. Unfortunately for White,
the small numbers that work poorly for Black in the race allow Black to fill in his 4-point and 5-point, building a prime and making the race essentially irrelevant.
(2) Getting pointed on after moving up hardly hurts at all, because the pointing numbers are crushing in any event. Suppose you knew that Black’s next roll was going to be 3-2. If you play 24/21 6/4
and Black points on you, you’re about 20% to win from the bar. If you stay back and make the 3-point instead, you’re still only about 22% to win. Better to be sure, but not by enough to matter.
(3) Moving up gains enormously if Black’s next roll is 2-1 or 3-1. If you stay back, Black makes his 5-point with these numbers and you’re about 20% to win. If you come up, Black’s best play is to
pick and pass with each number (6/4*/3 or 7/4*/3), but then you’re about 30% to win. That’s a big gain on 11% of Black’s possible throws.
(4) Moving up or staying back doesn’t affect the cube action next turn. In either case Black has a clear double and White has a big take. His take is easier if he moves up, but it’s very easy in any
This is a key position because this type of decision (move up under pressure or stay back) arises frequently and is often misplayed. Moving up to reach an inferior race seems counter-intuitive to
many players, but if the race isn’t too bad it’s mostly the right play. And in these “one man back” positions, the race is almost never too bad because of the way the positions come about. | {"url":"https://thegammonpress.com/feb-01-2024-one-man-back-step-up-or-not/","timestamp":"2024-11-09T06:39:12Z","content_type":"text/html","content_length":"245220","record_id":"<urn:uuid:80eb0d67-b448-45a2-adde-9ae709d9683a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00867.warc.gz"} |
Creating a gaussian random number in SQL Server 2005
I've been looking into this for a while, but so far I couldn't find anyone having posted a working sample of Gaussian random number generator ( and I need it for my probject)
As a bonus I also show how to modify the box-muller algorithm so that the resulting random numbers will conform to a given mean and standard deviation. It's very simple although it wasn't as
intuitive as I wished, personally.
-------------------------CODE BELOW---------------------------
-- =============================================
-- Author: Alwyn Aswin
-- Create date: 01/02/2009
-- Description: Generate a normally distributed random number.
-- NOTE: Please leave the author's attribution, if you copy this code.
-- =============================================
CREATE PROCEDURE BoxMullerRandom
@Mean float = 0
,@StdDev float = 1
,@BMRand float out
--@choice is the variable used to store the random number to return
declare @choice float, @store float, @choiceid uniqueidentifier
--checks to see if a box muller random number was already cached from previous call.
select top 1 @choiceid = randomid, @choice = random from boxmullercache
if(@choice is not null) -- if we do, delete that entry, since it's useable only once.
print 'loading from cache'
delete from boxmullercache
where randomid = @choiceid
else --otherwise, generate a pair of box muller random number.
print 'generate new ones'
declare @MethodChoiceRand float
set @MethodChoiceRand = rand()
--We re-roll if we get a 0, and use 0.5 as the cutoff point.
while @MethodChoiceRand = 0
set @MethodChoiceRand= rand()
-- Reroll if @MethodChoiceRand = 0, this will ensure that the interval, may be divided into 2 groups with equal number of members.
-- AND it has the advantage of removing the problematic ln(0) error from the Box-Muller equation.
declare @rand1 float, @rand2 float
select @rand1 = rand(), @rand2 = rand()
while @rand1 = 0 or @rand2 = 0
select @rand1 = rand(), @rand2 = rand()
declare @normalRand1 float, @normalRand2 float
SELECT @normalRand1 = sqrt(-2 * log(@rand1)) * cos(2*pi()*@rand2)
,@normalRand2 = sqrt(-2 * log(@rand1)) * sin(2*pi()*@rand2)
print 'box muller no 1:' + convert(varchar,@normalRand1) + ', box muller no 2:' + convert(varchar,@normalRand2)
--RandomlySelects which one to store, which one to save.
if @MethodChoiceRand <= 0.5
print 'choice 1'
select @choice = @normalRand1, @store = @normalRand2
else if @MethodChoiceRand > 0.5
print 'choice 2'
select @choice = @normalRand2, @store = @normalRand1
--stores the other pair into the cache to be retrieved during subsequent call to this method.
insert into boxmullercache (randomid, random)
values (newid(),@store)
--fix up the random number, so that it should have the correct mean and standard deviation.
set @BMRand = @choice * @stddev + @mean
I leave out the creation of the cachetable to the reader. You
to create a table to hold the other of the 2 values created via the Box-Muller algorithm! It should be fairly straight forward, it just needs an ID and a float column, which can be deduced from the
code above.
Please feel free to comment or suggest ways on improving the code.
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1.1.4 AP Calculus Exam Problem int sec x tan x dx
1.1.4 AP Calculus Exam Problem int sec x tan x dx
• MHB
• Thread starter karush
• Start date
In summary, the conversation discusses an integral question involving secant and tangent functions. The answer to the question is secant plus a constant, and the conversation also provides a helpful
trick for solving similar integrals using the derivatives of the multiple-choice answers. The problem is considered to be medium-easy difficulty.
$\displaystyle\int \sec x \tan x \: dx =$
(A) $\sec x + C$
(B) $\tan x + C$
(C) $\dfrac{\sec^2 x}{2}+ C$
(D) $\dfrac {tan^2 x}{2}+C $
(E) $\dfrac{\sec^2 x \tan^2 x }{2}+ C$
$(\sec x)'=\sec x \tan x$ so the answer is $\sec x +C$ (A)
Last edited:
karush said:
$\displaystyle\int \sec x \tan x \: dx =$
(A) $\sec x + C$
(B) $\tan x + C$
(C) $\dfrac{\sec^2 x}{2}+ C$
(D) $\dfrac {tan^2 x}{2}+C $
(E) ${\sec^2 x \tan^2 x }{2}+ C$
$(\sec x)'=\sec x \tan x$ so the answer is $\sec x +C$ (A)
Yup! (Muscle)
A neat trick to do is to differentiate each of the multiple-choice answers in turn until you get the expression to be integrated.
Instead of memorizing a "secant-tangent" integral rule, I would use the fact that $tan(x)= \frac{sin(x)}{cos(x)}$ and $sec(x)= \frac{1}{cos(x)}$ so that $\int sec(x)tan(x)dx= \int \frac{sin(x)}{cos^2
(x)}dx$. Let [tex]u= cos(x)[/tex] so that [tex]du= -sin(x)dx[/tex] and the integral becomes $-\int\frac{du}{u^2}= -\int u^{-2}du= -(-u^{-1})+ C= \frac{1}{cos(x)}+ C= sec(x)+ C$.
HallsofIvy said:
Instead of memorizing a "secant-tangent" integral rule, I would use the fact that $tan(x)= \frac{sin(x)}{cos(x)}$ and $sec(x)= \frac{1}{cos(x)}$ so that $\int sec(x)tan(x)dx= \int \frac{sin(x)}
{cos^2(x)}dx$. Let [tex]u= cos(x)[/tex] so that [tex]du= -sin(x)dx[/tex] and the integral becomes $-\int\frac{du}{u^2}= -\int u^{-2}du= -(-u^{-1})+ C= \frac{1}{cos(x)}+ C= sec(x)+ C$.
btw how would you rate this problem :
easy, medium, hard
I would consider it about "medium-easy".
FAQ: 1.1.4 AP Calculus Exam Problem int sec x tan x dx
1. What is the purpose of the "1.1.4 AP Calculus Exam Problem int sec x tan x dx"?
The purpose of this problem is to test a student's understanding of integration techniques, specifically the use of trigonometric identities and substitution, in solving definite integrals.
2. How does this problem relate to the AP Calculus exam?
This problem is a sample question that may appear on the AP Calculus exam, specifically on the multiple-choice section. It is designed to assess a student's knowledge and skills in calculus, as
outlined in the AP Calculus curriculum.
3. What is the significance of the "int sec x tan x dx" in this problem?
The "int sec x tan x dx" represents the definite integral that needs to be solved in this problem. It is a common type of integral that requires the use of trigonometric identities and substitution
to solve.
4. What strategies can be used to solve this problem?
Some strategies that can be used to solve this problem include using the trigonometric identity sec x = 1/cos x, substituting u = cos x, and using the power rule for integration.
5. How can I prepare for this type of problem on the AP Calculus exam?
To prepare for this type of problem, it is important to have a strong understanding of the fundamentals of calculus, including integration techniques and trigonometric identities. Practice solving
similar problems and familiarize yourself with the format and expectations of the AP Calculus exam. | {"url":"https://www.physicsforums.com/threads/1-1-4-ap-calculus-exam-problem-int-sec-x-tan-x-dx.1041669/","timestamp":"2024-11-14T01:05:27Z","content_type":"text/html","content_length":"104860","record_id":"<urn:uuid:c4fe9efa-d696-4af8-8587-26c7ab388175>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00766.warc.gz"} |
Warm-up: Estimation Exploration: Large Denominators (10 minutes)
The purpose of this estimation exploration is for students to reason about the size of a complex fraction sum with large denominators. Students can see that 1 is a good estimate because one fraction
is small and the other is close to 1. In the synthesis they refine this estimate to explain why the value of the sum is a little larger than 1.
• Groups of 2
• Display the expression.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
Student Facing
What is the value of the sum?
Record an estimate that is:
│ too low │ about right │ too high │
│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│
Activity Synthesis
• “How do you know that the sum is greater than 1?” (\(\frac{17}{19}\) is \(\frac{2}{19}\) short of a whole. Since 17ths are bigger than 19ths, adding \(\frac{3}{17}\) makes it greater than 1.)
Activity 1: Priya’s Salad Dressing (20 minutes)
The purpose of this activity is for students to add and subtract fractions and estimate sums and differences of fractions using the context of a recipe. Students may have different responses and
reasoning for the estimation questions. In both cases, they can calculate and compare fractions but they may have different thoughts about how these differences would affect the recipe or what
exactly it means for the recipe to make “about \(1\frac{1}{2}\) cups.” In the synthesis, students discuss the reasonableness of the estimates and how to make precise calculations (MP6). When students
relate their calculations to Priya's salad dressing they reason abstractly and quantitatively (MP2).
Reading: MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the questions. “We are going to read this question 3 times.” After the 1st Read: “Tell your
partner what this situation is about.” After the 2nd Read: “List the quantities. What can be counted or measured?” Reveal the question(s). After the 3rd Read: “What strategies can we use to solve
this problem?”
Advances: Reading, Representing
• Groups of 2
• “What kind of ingredients do you like to put in your salad?” (lettuce, cabbage, beans, seeds, beets, tomatoes, cheese)
• “What kinds of dressings do you put on your salad?” (homemade, Italian, blue cheese, tamari)
• 1–2 minutes: quiet think time
• 6–8 minutes: small-group work time
• Monitor for students who:
□ estimate to determine that Priya’s recipe will make about \(1\frac{1}{2}\) cups of dressing
□ add \(\frac{3}{4} + \frac{1}{3}+\frac{1}{2}\) to determine the precise amount of dressing Priya’s recipe will make
Student Facing
Priya’s Salad Dressing Recipe
• \(\frac{3}{4}\) cup olive oil
• \(\frac{1}{3}\) cup lemon juice
• \(\frac{1}{2}\) cup mustard
• Pinch of salt and pepper
1. Priya has \(\frac{2}{3}\) cup of olive oil. She is going to borrow some more from her neighbor. How much olive oil does she need to borrow to have enough to make the dressing?
2. 1 tablespoon is equal to \(\frac{1}{16}\) of a cup. Priya decides that 1 tablespoon of olive oil is close enough to what she needs to borrow from her neighbor. Do you agree with Priya? Explain or
show your reasoning.
3. Priya says her recipe will make about \(1\frac{1}{2}\) cups of dressing. Do you agree? Explain or show your reasoning.
Activity Synthesis
• “If Priya borrows a tablespoon of olive oil from her neighbor and uses it to make dressing, will she be putting in more or less olive oil than the recipe calls for?” (\(\frac{1}{16}\) is smaller
than \(\frac{1}{12}\) so she will be putting in less olive oil.)
• “Do you think 1 tablespoon is close enough?”
• Poll the class.
• “How might Priya’s decision to use 1 tablespoon of olive oil change the salad dressing?” (It won’t make a difference because the difference is so small. It might taste more lemony or more
mustardy because there is not as much oil. It might affect the consistency of the dressing a little.)
• Ask previously selected students to share their estimates for the amount of salad dressing in the given order.
• “Why might Priya estimate that the recipe makes \(1\frac{1}{2}\) cups of salad dressing?” (\(\frac{3}{4}\) is \(\frac{1}{4}\) away from 1 and \(\frac{1}{3}\) is close to \(\frac{1}{4}\).)
• “Does the recipe make more or less than \(1\frac{1}{2}\) cups? How do you know?” (More because \(\frac{1}{3}\) is more than \(\frac{1}{4}\).)
• “How many cups does Priya’s recipe make? How do you know?” (\(1\frac{7}{12}\), I added \(\frac{1}{3}\), \(\frac{3}{4}\), and \(\frac{1}{2}\).)
Activity 2: More Problems to Solve (15 minutes)
The purpose of this activity is for students to solve multi-step problems involving the addition and subtraction of fractions with unlike denominators. Students work with both fractions and mixed
numbers and can use strategies they have learned such as adding on to make a whole number. When students connect the quantities in the story problem to an equation, they reason abstractly and
quantitatively (MP2).
Representation: Access for Perception. Read both problems aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Conceptual Processing, Language
• Groups of 2
• “You and your partner will each choose a different problem to solve and then you will discuss your solutions.”
• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion
Student Facing
1. Choose a problem to solve.
Problem A:
Jada is baking protein bars for a hike. She adds \(\frac{1}{2}\) cup of walnuts and then decides to add another \(\frac{1}{3}\) cup. How many cups of walnuts has she added altogether?
If the recipe requires \(1\frac{1}{3}\) cups of walnuts, how many more cups of walnuts does Jada need to add? Explain or show your reasoning.
Problem B:
Kiran and Jada hiked \(1 \frac{1}{2}\) miles and took a rest. Then they hiked another \(\frac{4}{10}\) mile before stopping for lunch. How many miles have they hiked so far?
If the trail they are hiking is a total of \(2\frac{1}{2}\) miles, how much farther do they have to hike? Explain or show your reasoning.
2. Discuss the problems and solutions with your partner. What is the same about your strategies and solutions? What is different?
3. Revise your work if necessary.
Activity Synthesis
• “How were the problems the same? How were they different?” (For both problems I had to add fractions first and then subtract that total from another number. There were mixed numbers in both
• “How did you use equivalent fractions to solve these problems?” (All the fractions we worked with had different denominators so we had to find equivalent fractions with the same denominators in
order to add or subtract.)
Lesson Synthesis
“Today we solved problems that required adding and subtracting fractions.”
Display Priya‘s salad dressing recipe.
“What strategy did you use to find out how much salad dressing Priya’s recipe makes?” (The denominators for the fractions are 2, 3 and 4 so I used 12 because I know that it is a multiple of 2, 3, and
4. I put the half and fourths together first since I could use 4 as a common denominator and then I used 12 to add the fourths and third.
Display: \(\frac{1}{12} - \frac{1}{16}\)
“What strategy did you use to find this difference for the olive oil?” (I knew that 48 is \(4 \times 12\) and \(3 \times 16\) so I used that as a common denominator. I used \(12 \times 16\) as a
common denominator.)
“How do you decide which strategy to use?” (It depends on the numbers. If I know a small common multiple of the denominators, I use that. If I don’t, I can always use the product of the
Cool-down: Evaluate Expressions (5 minutes) | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/unit-6/lesson-12/lesson.html","timestamp":"2024-11-03T07:42:31Z","content_type":"text/html","content_length":"84690","record_id":"<urn:uuid:aee0e66a-a6b6-4614-88b5-77195c751cb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00453.warc.gz"} |
[Solved] As per IS 456:2000, which grade of concrete has a tensile st
As per IS 456:2000, which grade of concrete has a tensile stress of 3.2 N/mm^2 under limit state method?
Answer (Detailed Solution Below)
Option 3 : M25
Environmental Engineering for All AE/JE Civil Exams Mock Test
5 K Users
20 Questions 20 Marks 20 Mins
Characteristic strength of concrete (fck):
The compressive strength of concrete is given in terms of the characteristic compressive strength of 150 mm size cubes tested after 28 days
The characteristics strength is defined as the strength of concrete below which not more than 5% of the test results are expected to fall
Flexural strength of concrete:
The theoretical maximum flexural tensile stress occurring in the extreme fibres of RC beam, which causes cracking is referred to as the modulus of rupture (fCr)
The clause 6.2.2 of IS 456 gives the modulus of rupture of flexural tensile strength as
\({{\rm{f}}_{{\rm{cr}}}} = 0.7\sqrt {{{\rm{f}}_{{\rm{ck}}}}} \)
fcr = 3.2 N/mm2
\(3.2 = 0.7\sqrt {{\rm{f_{ck}}}}\)
f[ck] = (3.2/0.7)^2 = 20.897 ≈ 21 N/mm^2
Hence, the most appropriate answer is option 3.
Latest HPCL Engineer Updates
Last updated on Jul 4, 2024
-> HPCL Engineer 2024 Exam Date has been released. The exam will be conducted on 18th August 2024.
-> A total of 158 vacancies have been announced for the HPCL Engineer post in various disciplines.
-> Interested candidates had applied online from 5th to 30th June 2024.
-> Candidates with a full-time engineering discipline in the relevant stream are eligible to apply.
-> The selection will be based on a Computer Based Test Group Task and/or Interview. Prepare for the exam using HPCL Engineer Previous Year Papers. | {"url":"https://testbook.com/question-answer/as-per-is-4562000-which-grade-of-concrete-has-a--64ff0b5a3f0f75f892856057","timestamp":"2024-11-12T03:15:32Z","content_type":"text/html","content_length":"226163","record_id":"<urn:uuid:0692bc81-cfe3-4b38-93ef-d1df84533e8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00272.warc.gz"} |
These tax changes to pensions - Tim Worstall
Contrary to the widespread belief that only City fat cats on massive bonuses need worry, savers being targeted by the taxman range from members of public sector pension schemes…
Is it the public sector schemes, at least the top end of them, which will be hit hardest?
I\’ll probably garble this but, so, we\’ve got one major change which will hit them.
So there\’s this maximum pension limit, the maximum value of a pension you can have while still getting tax relief. £1.5, 1.8 million, something like that.
When you\’re making contributions to a fund this is easy enough to value. But what about those public sector pensions which aren\’t so funded, which are final salary ones?
It\’s been assumed so far that you take the annual pension and multiply it by 10 to give what the value of the fund is. This is to now go up to 16, maybe even 20.
Seems fair enough really, given that life expectancy at pension age is between 15 and 20 years.
So, fat cat civil servant, on £100k a year pension (not that there are all that many of those but….), under the old rules was estimated to have a fund of £1 million. Under the new, £2 million and he
gets to pay tax on that pension….when it\’s earned, not just when it\’s paid out to him.
Have I got that right?
And if I have, is that really what this is all about? Taxing back some part of those overly generous pensions granted to parts of the public services without actually breaching the contracts under
which such pensions have been granted?
9 thoughts on “These tax changes to pensions”
“Taxing back some part of those overly generous pensions granted to parts of the public services without actually breaching the contracts under which such pensions have been granted?”
Never thought of that. As Bob Peck said to the velociraptor, “clever girl”.
He would only be taxed on the bit of it that’s over the maximum (which is going to be cut to £1.5 million, so he will be taxed now, while he’s working, on £500,000 (the £2m value minus the £1.5m
limit). But yes, that looks about right.
But they’ll be hit in another way too. There isn’t just a limit on the capital value of the pension (the £1.5 million), there’s also a limit on the annual increase in value, which is going to be
So if our public sector employee is on a scheme that promises him a pension of 1/60th of his final salary for each year that he is so employed, say he’s been there for 20 years, was earning
£60,000 and gets a promotion that takes his pay up to £75,000. Lets assume they go for 20 as the multiple (keeps the maths easier):
Year 1, his pension if he retired straight away would be (20/60) x £60,000 = £20,000. The capital value of that accrued pension right, using the 20 multiplier, is therefore £20,000 x 20 =
Year 2, he’s worked an extra year and his salary has gone up, both of which increase the potential pension, so his accrued pension rights would be worth (21/60) x £75,000 x 20 = £525,000
So the value of his accrued pension has increased by £125,000 in one year. That’s more than the £50,000 annual contribution limit, so he’ll be taxed on £125,000 – £50,000 = £75,000. Since he’s
paying 40% tax, that’s £30,000 tax.
That’s what they’re bleating about.
But actually he probably won’t have to pay that much tax, for two reasons:
1) He’s allowed inflation on last year’s value (just as a funded scheme isn’t taxed on its investment returns). If inflation is 3%, then 3% x £400,000 = £12,000 of that increase in value would be
tax free. So he’s only taxed on £63,000 rather than £75,000.
2) If you don’t use your £50,000 annual limit, you will be allowed to carry the remainder forward for future years (from memory for 3 years). Assuming he didn’t get an above-inflation pay rise in
the previous 3 years, his annual increase would have just been because he’d clocked up an extra year’s employment. So the previous year the value of his pension would have been (19/60) x £60,000
x 20 = £380,000. From that to £400,000 is only an increase of £20,000, so he would have had £30,000 of his £50,000 annual limit left to carry forward. Three years of that would be more than
enough to cover him.
So actually they are only going to taxed if they either have huge pensions (which, as you say, isn’t many of them) or they regularly get pay rises that are far above inflation.
Sorry for long and boring accountant’s comment, but people here might actually understand it.
Richard: My understanding is that they use the 20x factor for the lifetime allowance calculations (which came into being in 2006, and are nothing new – all they’re doing now is taking it back to
the 2006 level of £1.5m) but that they use a 10x factor for the annual allowance calculation. At least they always did use a 10x factor, and I’ve seen nothing so far to say that that’s changing.
Just keep an eye on early retirement/voluntary severance/redundancy before 06.04.2011. Presumably it’ll become more attractive.
RA – nope; I’ve just looked it up, and they are going to increase the multiplier for the annual increase as well as the lifetime allowance. It wouldn’t make sense otherwise, because in both cases
you’re estimating the capital value of the future pension.
Although it seems HMRC’s current assumption is that the factor (for both purposes) will be 16 rather than 20.
See here:
dearime – civil servants resigning? That would be an added benefit for the public finances!
(Provided it’s early retirement on standard terms, rather than enhanced or redundancy).
They’ll be using 16 for the LTA as well? That means that in real terms, for people with a DB benefit the allowance is going up, not down.
Current allowance is £1.8m and the factor is 20, meaning that a pension of £90k is at the limit and anything over that is taxed.
New allowance is £1.5m and the factor is 16, then a pension of £93,750 is at the limit and anything over that will be taxed…
“There isn’t just a limit on the capital value of the pension”
For DC schemes, the lifetime limit is the amount contributed, not the actual valuation of the fund after investments have gone up/down, yes? | {"url":"https://www.timworstall.com/2010/10/these-tax-changes-to-pensions/","timestamp":"2024-11-11T06:31:24Z","content_type":"text/html","content_length":"196927","record_id":"<urn:uuid:9d298d60-ea6b-4348-b8ea-57c2d8388e9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00057.warc.gz"} |
Distributive Property Worksheet Doc
Distributive Property Worksheet Doc. Edit your distributive property worksheet pdf online type text, add images, blackout confidential details, add comments, highlights and more. Web distributive
property worksheets the sums given on the below free and printable worksheets test you on your knowledge of distribute property as you perform basic.
Distributive Property Equations Worksheet from www.onlineworksheet.my.id
Web distributive property worksheets the sums given on the below free and printable worksheets test you on your knowledge of distribute property as you perform basic. Some of the worksheets displayed
are the distributive property, using the distributive property. Students explicitly rewrite and answer multiplication equations using the distributive property. | {"url":"http://studydblamb123.s3-website-us-east-1.amazonaws.com/distributive-property-worksheet-doc.html","timestamp":"2024-11-03T04:01:25Z","content_type":"text/html","content_length":"26684","record_id":"<urn:uuid:b7bf1188-ed7d-4ec1-84f7-7ad2c56da837>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00275.warc.gz"} |
Load Path Analysis
Load Path Analysis: Methods, Importance, and Applications
Introduction to Load Path Analysis
Load path analysis determines how forces move through a structure, from the point of application to the foundation. This process is crucial in structural engineering to ensure that buildings and
other structures are stable and safe. If forces are not properly distributed, uneven stress can lead to failure or even collapse.
This page explores the significance of understanding load paths, the methods engineers use, and how load path analysis impacts structural design.
What is Load Path Analysis?
Load path analysis focuses on tracing how different forces move within a structure. These forces include dead loads, live loads, and wind loads, among others. The analysis ensures that forces are
transferred through structural elements such as beams, columns, and walls to the foundation.
Understanding load paths helps engineers design safe structures. A continuous load path ensures all parts work together to resist forces without overstressing any single component.
Importance of Load Path Analysis
Load path analysis is essential for maintaining structural integrity. Engineers must ensure that forces are properly transferred through all structural elements to prevent weak points or overloads.
This process ensures that no part of the structure bears excessive stress.
Ensuring Structural Safety
The primary goal is to ensure the building or structure can support all forces acting on it. By analyzing load paths, engineers can prevent local failures, which can compromise the entire structure.
Avoiding Overloading
Load path analysis identifies areas where forces might accumulate, allowing engineers to reinforce those points. This prevents any single component from carrying more force than it was designed to
handle, reducing the risk of overloading.
Optimizing Materials
Proper analysis of load paths ensures efficient use of materials. Engineers can distribute materials strategically, using stronger elements where needed while avoiding over-reinforcement elsewhere.
This leads to more cost-effective and sustainable designs.
Methods of Load Path Analysis
Several methods are available for analyzing load paths, from manual calculations to advanced simulation tools. Engineers choose the method based on the complexity of the project and the precision
Manual Calculations
For simple structures, manual calculations can be sufficient. Engineers apply principles of statics and mechanics to calculate how forces transfer through beams, columns, and walls. This method works
well for smaller buildings or when only basic analysis is required.
Finite Element Analysis (FEA)
Finite element analysis (FEA) provides more detailed insights into how forces move through a structure. Engineers use FEA to simulate stress distribution across small elements, allowing for more
accurate analysis. FEA is especially useful for large or complex projects, such as bridges and high-rise buildings.
Structural Modeling Software
Engineers often use structural modeling software to visualize how forces travel through a structure. These tools allow for a 3D analysis of load distribution, making it easier to identify potential
problem areas and adjust the design accordingly.
Applications of Load Path Analysis
Load path analysis applies to a wide range of engineering projects, ensuring the safety and efficiency of structures under various forces. Its applications span residential buildings, bridges,
seismic design, and retrofitting.
Building Design
In building design, load path analysis ensures that every component—beams, columns, walls—properly carries and transfers loads to the foundation. This process is especially important for high-rise
buildings, where the load paths become more complex due to wind and seismic forces.
Bridge Engineering
Bridges carry dynamic loads from traffic and environmental forces. Engineers must analyze the load paths within the bridge to ensure that the deck, piers, and abutments work together to distribute
the forces safely to the foundation.
Seismic Design
In regions prone to earthquakes, load path analysis helps ensure that seismic forces are transferred safely throughout the building. By understanding these paths, engineers can reinforce critical
areas to reduce the risk of failure during seismic events.
Retrofitting Structures
Load path analysis is also crucial when retrofitting older buildings. Engineers analyze existing load paths and determine whether additional reinforcement is necessary to improve the structure’s
capacity to carry modern loads.
Challenges in Load Path Analysis
Load path analysis poses challenges, particularly in complex structures. Engineers must consider factors like unpredictable forces and the interaction between different materials to ensure a safe,
effective design.
Handling Unpredictable Loads
Unpredictable forces, such as those from wind or seismic activity, complicate load path analysis. Engineers need to account for varying loads and ensure that the structure can withstand both expected
and unexpected forces without failure.
Complex Load Paths
For structures like skyscrapers or bridges, load paths can become complicated. Engineers must ensure that forces move efficiently through each component without overloading any part of the structure.
This requires detailed analysis using advanced tools.
Material Behavior
Materials like steel, concrete, and composites behave differently under stress. Engineers must consider how each material responds to bending, shear, and torsion while ensuring the load transfers
correctly through each element.
Innovations in Load Path Analysis
Advances in technology are improving how engineers conduct load path analysis. These innovations allow for more accurate simulations, better material usage, and improved overall safety.
3D Modeling Tools
Engineers now use 3D modeling tools to simulate load paths with greater accuracy. These tools help them visualize the movement of forces and adjust designs early in the process to prevent future
Artificial Intelligence (AI)
AI is increasingly used to optimize load path analysis. AI algorithms can analyze vast datasets to recommend the most efficient load paths, improving safety while reducing material waste.
High-Performance Materials
Engineers are also exploring the use of high-performance materials like fiber-reinforced polymers and ultra-high-strength concrete. These materials allow for more flexibility in design, improving
load distribution and reducing the risk of structural failure.
Conclusion: The Role of Load Path Analysis
Load path analysis plays a crucial role in structural engineering by ensuring that forces are safely and efficiently distributed throughout a structure. With the help of modern tools and methods,
engineers can design safer, more cost-effective buildings and infrastructure. The continued development of analysis tools and new materials will further enhance the ability to create structures that
stand up to the challenges of modern engineering. | {"url":"https://turn2engineering.com/electrical-engineering/power-systems-engineering/load-path-analysis","timestamp":"2024-11-07T00:20:41Z","content_type":"text/html","content_length":"211343","record_id":"<urn:uuid:e42623c3-b316-424e-b157-7cc25d8471f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00853.warc.gz"} |
"E" Day – February 7, 2025
E-Day is an annual American holiday dedicated to the Euler number. Celebrated on February 7th. This mathematical constant has no exact origin. It is a numerical sequence and is used to calculate
logarithmic expressions.
The first time the number “e” was discussed was about three centuries ago. The first descriptions of this mathematical phenomenon appeared in the works of John Napier, a mathematician from Scotland.
He hinted and talked about a certain numerical sequence that he managed to deduce. However, the mathematician never named the value, although he claimed that with its help he managed to solve a
series of logarithms.
In 1683, this magic number was deduced by Jacob Bernoulli. The common answer to the logarithmic expression was the number “e” (the author used the designation with the letter b in his works). In
1731, the numerical sequence was studied by Leonhard Euler, who gave it a name, taking the first letter from his last name.
There are still many rumors and conjectures about the Euler number, and mathematicians from all over the world continue to attempt to discover new applications and properties of this number.
Interesting facts
• In 2018, a funny coincidence was discovered. If we represent the record of the number (2.718281828) as 2/71 (by the first digits), then we can talk about the date October 2, 1971. It is from this
day that the Day of e-mail is. And for this service, the designation e-mail is often used.
• The Euler number has a “relative” – the number Pi, which is equal to 3.14 and is also an infinite value.
How to celebrate
Solve several logarithmic examples on E-Day using this numerical value. Read more interesting information online about the creation of the Euler number.
Tell about the holiday on social networks. Ask other users how often they have used the Euler number in their lives (perhaps to calculate something).
When is “E” Day in 2025?
“E” Day is observed on February 7 each year.
Weekday Month Day Year
Friday February 7 2025
Saturday February 7 2026
Sunday February 7 2027
Monday February 7 2028
Wednesday February 7 2029 | {"url":"https://weirdholiday.com/e-day/","timestamp":"2024-11-03T02:41:33Z","content_type":"text/html","content_length":"106632","record_id":"<urn:uuid:ff0880ba-5c9d-4fcc-ac16-8cd82f903539>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00004.warc.gz"} |
Simplicial Manifold
This came to me in a dream: I saw space fractured into a zillion pieces. And not just random shards laying on the ground like a smashed crystal wine glass, no, they were ordered, all the diverse
sizes and shapes imaginable intricately arranged.
simplex is easy to visualize. For example, a 2-simplex can be thought of as a closed triangle with three vertices and three line segments--or edges--acting as its boundaries connecting the
vertices. But, topologically, it can take any cell-shape, a cell being a generalization of a simplex. Each 2-dimensional cell is homeomorphic--maps invariants to invariants--to the Euclidean
space R^2. Topologically, an n-simplex is equivalent to an n-ball.
Convention states that points on the boundaries of a surface do not have neighborhoods. The space must be open, in other words, for all points to have neighborhoods. Why? Visualize a 'point' on
the boundary of a figure or an end 'point' of an interval. And also visualize a 'nbhd' as completely encapsulating, enclosing, enfolding that point as a cocoon would. A point on the extreme outer
edge, beyond which there is nothing, no space, not even infinitesimal, cannot be contained within the interior of a nbhd.
However, according to an article in Wikipedia: "Every boundary point has a neighborhood homeomorphic to the "half" n-ball." It hinges on treating the interior and the boundary as separate
manifolds: "If M is a manifold with boundary of dimension n, then Int M is a manifold (without boundary) of dimension n and ∂M is a manifold (without boundary) of dimension n-1." So what you end
up with is a union of manifolds instead of one continuous surface. In much the same way, a closed simplex can be considered as the union of its interior and the boundary.
all of one piece, a continuum. That is to say, it has no sub-partitions joined together as does a polyhedron with its polygonal faces. Each point/neighborhood of a manifold corresponds to a local
flat Euclidean plane; so, in a sense, considering that a manifold is composed of points, it can be said to be Euclidean-ized.
In brief, a (real) n-dimensional manifold is a topological space M for which every point x ∈ M has a neighborhood homeomorphic to Euclidean space R^n. From Wikipedia, we have: "A simplicial
manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold."
So we have: closed simplex equals cell equals manifold. By the use of barycentric coordinates we can further sub-divide the cell into simple quotient spaces to create a polyhedron of space. With
each factoring we create a composition series, further refining the cell's detail. The cell, it may be pointed out, is susceptible to continuous deformations. In the process, its interior
morphology rearranges its play of symmetries in a dynamic and self-organizing manner. And as the interior of a cell is sensivitve to its boundary shape, it is said to be shape-aware, analogous in
regulatory capacity or methodology to the morphogenetic field of a biological cell. An important point that shouldn't be underestimated concerns the orientation -- spin -- imposed on a simplex,
infusing it with a causal asymmetric tendency effecting ripples and contours. This feature eliminates random inconsistencies.
→ Click image to the right for larger version
A change of basis takes place; we zoom in. Barycentric coordinates expand the vertices onto a smaller, more refined scale, adding degrees of freedom, revealing underlying dimensions. As the
space grows in complexity, each vertex, boundary and interior point transforms into a linear combination -- weighted by the barycentric coordinates acting as coefficients -- of the newly defined
vertices. Essentially, the original coordinate system is relabeled. Also, any number of lines can be drawn through the interior of our cell, intersecting an edge at a point. For instance, for the
triangle or 2-D simplex, a line can be drawn from each vertex to a point on its opposite side (edge), this point is then written in terms of the basis of three weighted vectors -- our
barycentric-ized vertices:
point = a[1]v[1] + a[2]v[2] + a[3]v[3].
So let each cell of any particular dimension be increasingly refined with lines drawn to points along its edges, further dissecting and contorting its volume, further detailing the enclosed
(convex) compacted space. The areas and volumes of faces and simplices of our complex, it should be pointed out, are quantized according to elementary Planck units. Lines of partition [field
lines, if you will] intersect at the center of mass -- the source point of the cell. Individual particles of matter and radiation are thus created.
For example, an electron can correspond to a factored network of quotient spaces in a fluctuating state, stuck in a loop due to the particular configuration, compressed into a bounded condition.
Increasing and decreasing refinement within a certain range corresponds to its everchanging position and momentum. As the space of that cell increases in detail due to fluctuating factoring, a
photon is absorbed. As it decreases, a photon is expelled. This represents a probability distribution as a result of, or corresponding to, the quantum partitioning. The electron can be localized
by the barycentroid (or multiple barycentroids) of the cell, insensitive to noise and deformations.
Looking at our cell-complex from another perspective, continually varying spatial vortices collectively forming an ever-shifting strange attractor, an emergent phenomenon, manifests in the
macro-world as spacetime, the fabric of it, to use a hackneyed mataphor. Piecewise, a strange attractor renders the overall appearance of the cell as it goes through its fluctuations,
fractalizing space. A collection of these cells (simplex, node), interfacing at their boundaries (face, surface), on any required dimension, induces neighboring individual -- quantum level --
cells to arrange their interior structures in sympatico, either by means of extensions of partitioning lines from the source point crossing respective boundaries, or through resonation of
sub-vortical frequencies.
barycenter means center of mass or gravity]. Space, taffy-like, creates the surface appearance of curvature due to its contraction and stretching, to its folding and squeezing. The underlying
cause may be attributed to the increasing refinement -- or factoring -- of the sections of simple quotient spaces -- factor groups of denser symmetry -- spontaneously generating, and nested in a
hierarchy of sub-domains. Three-dimensionally, we can think of a tetrahedron as homeomorphic to a solid ball, an image that seems to be more agreeable for its natural symbol of centering.
bottom or center, bubbling up new micro-states from nano-moment to nano-moment, evolving fresh pseudo-stable surfaces through some kind of force convection.
Simple refers to non-divisibility. Each factor group -- each sub-domain of compacted space -- is a whole unto itself, locked in a preferred arrangement or quantized set of arrangements. The mesh
size of any specific simplex net varies relatively, but must be no greater than a suitable Planck designation or limit based on the compact nature of the associated Lie group in Hilbert Space. An
appropriate distance-metric can be imposed from without but understood only as an artifact for purposes of calculation. Being topological, the underlying field has no inherent metric; in a
topological space distance has no meaning. Objects exist in and of themselves without reference to any background coordinate system. Furthermore, thinking of space as a fluid medium allows us to
forgo the associated notions of rigidity and flatness of space inherent in our concept of manifold. It bends and twists and can reform into any conceivable topologically valid shape. Time and
motion are thus seen as effects of this constant partitioning. Particles as constricted loops, trapped in a web of symmetrically bound energy. Forces as changes in the loop structure due to
fluctuation of the spatial components -- the factors. | {"url":"http://adriandorn.com/math/manifold.htm","timestamp":"2024-11-03T22:59:57Z","content_type":"text/html","content_length":"11005","record_id":"<urn:uuid:d4d08ed8-120a-4be8-88cb-8c8ee188eea2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00717.warc.gz"} |
Here is a calculation of Wolfram Alpha. How can i caculate it with Sage?
Here is a calculation of Wolfram Alpha. How can i caculate it with Sage?
asked 2013-12-19 01:36:56 +0100
This post is a wiki. Anyone with karma >750 is welcome to improve it.
Here is a calculation of Wolfram Alpha. How can i caculate it with Sage?
1 Answer
Sort by ยป oldest newest most voted
answered 2013-12-29 23:08:04 +0100
This post is a wiki. Anyone with karma >750 is welcome to improve it.
many example: http://sagenb.mc.edu/pub/
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Calculus Tutorials
Algebra Review
The material in this tutorial is reprinted with permission of the authors and publisher of How to Ace Calculus by Colin Adams, Abigail Franklin, and Joel Hass.
Here we will list some of the most common mistakes that we see on exams. If you can avoid these, then at least your mistakes will be uncommon. Most of the mistakes that occur repeatedly involve
algebra, rather than calculus. They can be avoided by being careful and checking your work. Others involve common misunderstandings about various aspects of calculus.
1. $ (x+y)^2 = x^2 + y^2 $. MISTAKE!
Powers don’t behave this way. The correct way to expand this expression gives $$ (x+y)^2 = x^2 + 2xy + y^2. $$
2. $\displaystyle{ \frac{1}{x+y} = \frac{1}{x} + \frac{1}{y} }$. MISTAKE!
The rule for adding fractions gives $$ \frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy}. $$
3. $\displaystyle{ \frac{1}{x+y} = \frac{1}{x} + y }$. MISTAKE!
This very common error comes from carelessness about what’s in the denominator. It can be avoided by careful handwriting or frequent use of parentheses.
4. $ \sqrt{x+y} = \sqrt{x} + \sqrt{y} $. MISTAKE!
There is no simplified way to write $\sqrt{x + y}$. You just have to live with it as is.
5. $ x < y $ so $ kx < ky $ where $k$ is a constant. MISTAKE!
This is true when $k$ is a POSITIVE constant. If $k$ is negative you need to reverse the inequality. If $k$ is zero all bets are off. For example, if $ x < y $ then $ -x > -y $.
6. Forgetting to simplify fractions in limits: MISTAKE!
It is not correct to say $\displaystyle{ \lim_{x \to 1} \frac{x^2 – 1}{x-1} = \frac{0}{0} }$ and therefore the limit is undefined. Even worse would be to cancel the zeroes and say the limit
equals one.
Any time you get $\frac{0}{0}$ for a limit, it is a BIG WARNING SIGN that says YOU HAVE MORE WORK TO DO! In this case, $$ \lim_{x \to 1} \frac{x^2 – 1}{x-1} = \lim_{x \to 1} \frac{(x+1)(x-1)}
{x-1} = \lim_{x \to 1} x + 1 = 2. $$
7. $ \sin{2x}/x = \sin{2} $. MISTAKE!
You can only cancel terms in the numerator and denominator of a fraction if they are not inside anything else and are just multiplying the rest of the numerator and denominator. The function $\
sin 2x$ is NOT $\sin{2}$ multiplied by $x$. If the fraction had been written as $$ \frac{\sin{(2x)}}{x} $$ it would be harder to make such an error.
8. $ ax = bx $ therefore $ a = b $. MISTAKE!
This is a more subtle mistake. The cancellation is correct IF $x$ is not $0$. For example $2x = 3x$ forces $x = 0$. You cannot cancel the $x = 0$ and conclude that $2 = 3$. Not in this universe,
9. $ \frac{d}{dx} \left[ 2^x \right] = x 2^{x-1} $. MISTAKE!
The correct answer is $ 2^x \ln{2} $. The power rule only applies if the base is a variable and the exponent is a constant, as in $x^3$.
10. $ \frac{d}{dx} \left[ \sin{(x^2 + 1)} \right] = \cos{(2x)} $. MISTAKE!
This is a typical example of the kind of mistakes made when applying the chain rule. The correct answer is $$ \frac{d}{dx} \left[ \sin{(x^2 + 1)} \right] = \cos{(x^2 + 1)} \times 2x. $$
11. $ \frac{d}{dx} \left[ \sin{(x^2 + 1)} \right] = \cos{(x^2 + 1)} + \sin{(2x)} $. MISTAKE!
Another common way in which the chain rule is misapplied. This time the product rule has been used in a setting where the chain rule was the way to go.
12. $ \frac{d}{dx} \left[\cos{x}\right] = \sin{x}$. MISTAKE!
The answer should be $-\sin{x}$. Extremely common error costing students over 10 million points a year on exams around the world.
13. $\displaystyle{ \frac{d}{dx} \left[ \frac{f}{g} \right] = \frac{f g’ – g f’}{g^2} }$. MISTAKE!
This is backwards! It should be $$ \frac{d}{dx} \left[ \frac{f}{g} \right] = \frac{g f’ – f g’}{g^2}. $$
14. $ \frac{d}{dx} \left[ \ln 3 \right] = \frac{1}{3} $. MISTAKE!
The quantity $\ln{3}$ is a constant, so $\frac{d}{dx} \left[ \ln 3 \right] = 0$. The same is true for ALL constants. So $\frac{d}{dx} \left[e\right] = 0$ and $\frac{d}{dx} \left[ \sin \left (\
frac{\pi}{2} \right) \right] = 0$ as well.
15. $\displaystyle{ \int x \, dx = \frac{x^2}{2} }$. MISTAKE!
The correct answer is $\displaystyle{ \int x \, dx = \frac{x^2}{2} + C }$. Picky profs penalize points pedantically.
16. $\displaystyle{ \int \frac{1}{x} \, dx = \frac{x^0}{0} + C }$. MISTAKE!
The power rule for integration does not apply to $x^{-1}$. Instead, $$ \int \frac{1}{x} \, dx = \ln \left\vert x \right\vert + C. $$
17. $ \int \tan{x} \, dx = \sec^2 x + C $. MISTAKE!
It’s the other way around. $\frac{d}{dx} \left[ \tan{x} \right] = \sec^2 x$. The correct answer is $$ \int \tan x \, dx = \ln \left\vert \sec{x} \right\vert + C $$ as can be found by
$u$-substitution with $u = \cos{x}$.
18. Forgetting to simplify: MISTAKE!
For example $$ \int x \sqrt{x} \, dx $$ is easy if you notice that $x \sqrt{x} = x^{3/2}$ and then apply the power rule for integration. But if you try to do it using integration by parts or
substitution, you will find your self in outer space without a space suit.
19. Not substituting back to the original variable: MISTAKE!
$$ \int 2 x e^{x^2} \, dx $$ does not equal $e^u + C$. It equals $e^{x^2} + C$.
20. Misreading the problem: MISTAKE!
If asked to find an area, don’t find a volume. If asked to find a derivative, don’t find an integral. If asked to use calculus to solve a problem, don’t do it in your head using algebra. Although
it seemes silly to include this item in our list, billions of points have been taken off exams for mistakes of this type. After you finish a problem on the exam, go back and read the question
again. Check to make sure you answered the question that was asked.
21. Thinking you’re prepared when you’re not. MISTAKE!
This mistake is perhaps the most important, so we’ll put it in even though it pushes us over the 20 mistakes limit. The worst mistake many students make is to think they know the material better
than they really do. It’s easy to fool yourself into thinking you can solve a problem when you’re looking at the answer book or at a worked out solution. Test your knowledge by trying problems
under exam conditions. If you can do them under that restriction, the exam should be a breeze. | {"url":"https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-tutorials/precalculus/algebra-review/","timestamp":"2024-11-03T09:40:45Z","content_type":"text/html","content_length":"43287","record_id":"<urn:uuid:99b5c271-2245-4fad-abb3-a7e464df8391>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00075.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2022.12
URN: urn:nbn:de:0030-drops-161728
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16172/
Bannach, Max ; Fleischmann, Pamela ; Skambath, Malte
MaxSAT with Absolute Value Functions: A Parameterized Perspective
The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a
propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost
solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative
weights. In this work we study the problem with negative weights and observe that the problem becomes computationally harder - which we formalize from a parameterized perspective in the sense that
various variations of the problem become W[1]-hard if negative weights are present.
Allowing negative weights also introduces new variants of the problem: Instead of maximizing the sum of weights of satisfied clauses, we can maximize the absolute value of that sum. This turns out to
be surprisingly expressive even restricted to monotone formulas in disjunctive normal form with at most two literals per clause. In contrast to the versions without the absolute value, however, we
prove that these variants are fixed-parameter tractable. As technical contribution we present a kernelization for an auxiliary problem on hypergraphs in which we seek, given an edge-weighted
hypergraph, an induced subgraph that maximizes the absolute value of the sum of edge-weights.
BibTeX - Entry
author = {Bannach, Max and Fleischmann, Pamela and Skambath, Malte},
title = {{MaxSAT with Absolute Value Functions: A Parameterized Perspective}},
booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-236-5},
ISSN = {1868-8969},
year = {2022},
volume = {227},
editor = {Czumaj, Artur and Xin, Qin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16172},
URN = {urn:nbn:de:0030-drops-161728},
doi = {10.4230/LIPIcs.SWAT.2022.12},
annote = {Keywords: parameterized complexity, kernelization, weighted maximum satisfiability, absolute value maximization}
Keywords: parameterized complexity, kernelization, weighted maximum satisfiability, absolute value maximization
Collection: 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)
Issue Date: 2022
Date of publication: 22.06.2022
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=16172","timestamp":"2024-11-10T14:48:41Z","content_type":"text/html","content_length":"7233","record_id":"<urn:uuid:d7214755-e108-4a9a-a285-f4d6918e14c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00491.warc.gz"} |
9 Free Multiplication and Division Fact Families Worksheets
Get 150+ Free Math Worksheets!
These multiplication and division fact families worksheets will help to visualize and understand multiplication and division fact families and number systems. 3rd grade students will learn basic
multiplication and division using fact families and can improve their basic math skills with our free printable multiplication and division fact families worksheets.
9 Exciting Multiplication and Division Fact Families Worksheets
Please download the following fact family worksheets and practice multiplication and division on the pages.
Introduction to Multiplication and Division Fact Families
Multiplication and division are fun things to do. We apply the basics of multiplication when we need to do some repeated additions. And to sort or arrange something in smaller groups, we use
Before jumping on to the main activities, we will first learn what a fact family is. Look at the word fact family. We know that a family is a group of members who have some relations among them.
The same condition applies to the members of a fact family. In a fact family, we will see some relationships form among the three numbers. These relationships can be formed based on either addition
and subtraction or multiplication and division.
The three numbers that are assigned in a multiplication and division fact family will have two relationships for multiplication and two relationships for division.
For example, in a conference, 16 people sit in eight chairs, which are sorted into 2 groups. From the above information, we can form four relationships, which are,
8 x 2 = 16
2 x 8 = 16
16 ÷ 2 = 8
16 ÷ 8 = 2
We can also explain the multiplication and division fact families using a part-part-whole theory. In this explanation, 16 will be the whole number, and 8 and 2 will be its two parts. When we multiply
the two parts by each other, we will get the whole number.
On the other hand, if we divide the whole number by any of the parts, then we will get the other part as the result.
Completing Facts Worksheets
In the following worksheet, you will find three numbers given in each triangle. The number in the upper axis is the product or the whole, and its two parts are given at the base axes.
Your job is to find the multiplication and division relationship between the product, or the whole, and its two parts, or factors. Read the discussion at the beginning of this article if you become
confused about how to find all the relationships.
Fill Missing Numbers of Fact Family Worksheets
In this worksheet, you will see several multiplication and division fact family bubbles. All these bubbles are made with the basic facts for the numbers 2-12.
But you can notice that some of the bubbles are incomplete. Find the missing factors or the products of that bubble and write them down to fill up the empty places.
Number Bond Fact Family Worksheets
A number bond is another great activity to practice multiplication and division fact family worksheets.
• Here, the product of multiplication is placed in the middle of the number bond. And around the product, you can see some empty spaces.
• Count the number of empty spaces, which is the first factor of the product.
• Divide the product by that factor, and the result is the second factor or the second part of the product.
• Write the result to fill all the empty spaces around the product.
• In case you have to determine the product, then count the number of spaces around the empty product, and this time multiply the number of spaces with the number given inside them.
• That way, you will find the product instantly.
Street Made of Fact Family Worksheet
We have made a whole street with some fact families. All the buildings in the street represent a certain fact family. Applying the techniques discussed in the above sections, complete the empty
spaces of all the buildings to make them usable for living.
Fact Family of Arrays Worksheets
In this discussion, we will practice multiplication and division fact families of arrays. Multiplication and division arrays are the sequenced or planned arrangements of some rows and columns that
will lead to multiplication and division equations. What do you have to do on these worksheets?
• First, inspect the arrays carefully and count all the items in the array that will be the product.
• Count the number of rows and columns, and these will be the two factors of the product.
• Then, form the multiplication and division equations with those numbers.
Fact Family Finding Products or Factors Worksheet
Another simple multiplication and division fact family activity. Some rectangles are given on each page of the following worksheet. Each rectangle consists of three rooms, where the bigger room is
for the product or the whole and the smaller two rooms are for its parts or factors.
Find the missing products or factors as per the problems and write down all the missing numbers to fill the rooms.
Fact Family Sorting Worksheet
Some sets of numbers are given in the following worksheet. Observe those numbers carefully and try to figure out the multiplication and division relations between them.
That means you have to find if all the numbers of a set are from a single fact family or not. Go through all the sets to find that. Then sort them into the table provided in the worksheet as per
Fact Family Triangle and Circle Worksheets
This activity of today’s discussion will be filling in missing products and factors in some multiplication and division fact family triangles and circles. See the given numbers in each of the shapes
and figure out what you have to find.
The product or the factors. Then, using multiplication or division as required, find all the missing numbers.
Simple Word Problems with Multiplication and Division Fact Families
In our last activity, we will solve some simple word problems regarding multiplication and division fact families. Here, after reading each word problem carefully, you have to find the fact families
for the given factors and solve the problem.
Download Free PDF Worksheets
Download the following combined PDF and enjoy your practice session.
So today, we’ve discussed multiplication and division fact families worksheets using the concepts of multiplication and division, facts of numbers 1-12, and some interactive activities like filling
gaps, number bonds, finding products, fact family street, word problems, etc. Download our free worksheets, and after practicing these worksheets, students will surely improve their mathematical
skills and have a better understanding of multiplication and division fact families.
Hello, I am Md. Araf Bin Jayed. I have completed my B.Sc in Industrial and Production Engineering from Ahsanullah University of Science and Technology. Currently I am working as a Content Developer
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and hard work, I want to contribute to the overall growth of this organization. | {"url":"https://youvegotthismath.com/multiplication-and-division-fact-families-worksheets/","timestamp":"2024-11-05T04:21:57Z","content_type":"text/html","content_length":"362474","record_id":"<urn:uuid:cc4dcc64-5073-4d3c-b8fc-70130897cc76>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00329.warc.gz"} |
Parkfield Curriculum - Year 6 Mathematics
The long term plan below shows the order in which units are taught and approximately how many weeks are spent on each unit.
These are broken down further into the small stepsfor each unit of work. All small steps involve an element of reasoning and problem solving and link to the National Curriculum.
Step 1 Numbers to 1,000,000
Step 2 Numbers to 10,000,000
Step 3 Read and write numbers to 10,000,000
Step 4 Powers of 10
Step 5 Number line to 10,000,000
Step 6 Compare and order any integers
Step 7 Round any integer
Step 8 Negative numbers
National Curriculum Links:
Pupils should be taught to:
• read, write, order and compare numbers up to 10,000,000 and determine the value of each digit
• round any whole number to a required degree of accuracy
• use negative numbers in context, and calculate intervals across 0
• solve number and practical problems that involve all of the above
Addition, Subtraction, Multiplication and Division
Step 1 Add and subtract integers
Step 2 Common factors
Step 3 Common multiples
Step 4 Rules of divisibility
Step 5 Primes to 100
Step 6 Square and cube numbers
Step 7 Multiply up to a 4-digit number by a 2-digit number
Step 8 Solve problems with multiplication
Step 9 Short division
Step 10 Division using factors
Step 11 Introduction to long division
Step 12 Long division with remainders
Step 13 Solve problems with division
Step 14 Solve multi-step problems
Step 15 Order of operations
Step 16 Mental calculations and estimation
Step 17 Reason from known facts
National Curriculum Link:
Pupils should be taught to:
• multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication
• divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as
appropriate for the context
• divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context
• perform mental calculations, including with mixed operations and large numbers
• identify common factors, common multiples and prime numbers
• use their knowledge of the order of operations to carry out calculations involving the 4 operations
• solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why
• solve problems involving addition, subtraction, multiplication and division
• use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy
Step 1 Equivalent fractions and simplifying
Step 2 Equivalent fractions on a number line
Step 3 Compare and order (denominator)
Step 4 Compare and order (numerator)
Step 5 Add and subtract simple fractions
Step 6 Add and subtract any two fractions
Step 7 Add mixed numbers
Step 8 Subtract mixed numbers
Step 9 Multi-step problems
National Curriculum Links:
Pupils should be taught to:
• use common factors to simplify fractions; use common multiples to express fractions in the same denomination
• compare and order fractions, including fractions >1
• add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions
• multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 1/4 × 1/2 = 1/8 ]
• divide proper fractions by whole numbers [for example, 1/3 ÷ 2 = 1/6 ]
• associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8 ]
• identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places
• multiply one-digit numbers with up to 2 decimal places by whole numbers
• use written division methods in cases where the answer has up to 2 decimal places
• solve problems which require answers to be rounded to specified degrees of accuracy
• recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
Step 1 Multiply fractions by integers
Step 2 Multiply fractions by fractions
Step 3 Divide a fraction by an integer
Step 4 Divide any fraction by an integer
Step 5 Mixed questions with fractions
Step 6 Fraction of an amount
Step 7 Fraction of an amount – find the whole
National Curriculum Links:
Pupils should be taught to:
• multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, 1/4 × 1/2 = 1/8 ]
• divide proper fractions by whole numbers [for example, 1/3 ÷ 2 = 1/6 ]
• associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8 ]
• recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
Step 1 Metric measures
Step 2 Convert metric measures
Step 3 Calculate with metric measures
Step 4 Miles and kilometres
Step 5 Imperial measures
National Curriculum Links:
Pupils should be taught to:
• solve problems involving the calculation and conversion of units of measure, using decimal notation up to 3 decimal places where appropriate
• use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal
notation to up to 3 decimal places
• convert between miles and kilometres
• recognise that shapes with the same areas can have different perimeters and vice versa
• recognise when it is possible to use formulae for area and volume of shapes
• calculate the area of parallelograms and triangles
• calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km³]
Step 1 Add or multiply?
Step 2 Use ratio language
Step 3 Introduction to the ratio symbol
Step 4 Ratio and fractions
Step 5 Scale drawing
Step 6 Use scale factors
Step 7 Similar shapes
Step 8 Ratio problems
Step 9 Proportion problems
Step 10 Recipes
National Curriculum Links:
Pupils should be taught to:
• solve problems involving the relative sizes of 2 quantities where missing values can be found by using integer multiplication and division facts
• solve problems involving the calculation of percentages [for example, of measures and such as 15% of 360] and the use of percentages for comparison
• solve problems involving similar shapes where the scale factor is known or can be found
• solve problems involving unequal sharing and grouping using knowledge of fractions and multiples
Step 1 1-step function machines
Step 2 2-step function machines
Step 3 Form expressions
Step 4 Substitution
Step 5 Formulae
Step 6 Form equations
Step 7 Solve 1-step equations
Step 8 Solve 2-step equations
Step 9 Find pairs of values
Step 10 Solve problems with two unknowns
National Curriculum Links:
Pupils should be taught to:
• use simple formulae
• generate and describe linear number sequences
• express missing number problems algebraically
• find pairs of numbers that satisfy an equation with 2 unknowns
• enumerate possibilities of combinations of 2 variables
Step 1 Place value within 1
Step 2 Place value – integers and decimals
Step 3 Round decimals
Step 4 Add and subtract decimals
Step 5 Multiply by 10, 100 and 1,000
Step 6 Divide by 10, 100 and 1,000
Step 7 Multiply decimals by integers
Step 8 Divide decimals by integers
Step 9 Multiply and divide decimals in context
National Curriculum Links:
Pupils should be taught to:
• associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, 3/8 ]
• identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places
• multiply one-digit numbers with up to 2 decimal places by whole numbers
• use written division methods in cases where the answer has up to 2 decimal places
• solve problems which require answers to be rounded to specified degrees of accuracy
• recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
Fractions, decimals and percentages
Step 1 Decimal and fraction equivalents
Step 2 Fractions as division
Step 3 Understand percentages
Step 4 Fractions to percentages
Step 5 Equivalent fractions, decimals and percentages
Step 6 Order fractions, decimals and percentages
Step 7 Percentage of an amount – one step
Step 8 Percentage of an amount – multi-step
Step 9 Percentages – missing values
National Curriculum Links:
Pupils should be taught to:
• Use common factors to simplify fractions; use common multiples to express fractions in the same denomination
• Associate a fraction with division and calculate decimal fraction equivalents for a simple fraction
• Recall and use equivalences between simple fractions, decimals and percentages, including in different contexts
• Compare and order fractions, including fractions >1
• Solve problems involving the calculation of percentages and the use of percentages for comparison
Area, perimeter and volume
Step 1 Shapes – same area
Step 2 Area and perimeter
Step 3 Area of a triangle – counting squares
Step 4 Area of a right-angled triangle
Step 5 Area of any triangle
Step 6 Area of a parallelogram
Step 7 Volume – counting cubes
Step 8 Volume of a cuboid
National Curriculum Links:
Pupils should be taught to:
• solve problems involving the calculation and conversion of units of measure, using decimal notation up to 3 decimal places where appropriate
• use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal
notation to up to 3 decimal places
• convert between miles and kilometres
• recognise that shapes with the same areas can have different perimeters and vice versa
• recognise when it is possible to use formulae for area and volume of shapes
• calculate the area of parallelograms and triangles
• calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm³) and cubic metres (m³), and extending to other units [for example, mm³ and km³]
Step 1 Line graphs
Step 2 Dual bar charts
Step 3 Read and interpret pie charts
Step 4 Pie charts with percentages
Step 5 Draw pie charts
Step 6 The mean
National Curriculum Links:
Pupils should be taught to:
• interpret and construct pie charts and line graphs and use these to solve problems
• calculate and interpret the mean as an average
Step 1 Measure and classify angles
Step 2 Calculate angles
Step 3 Vertically opposite angles
Step 4 Angles in a triangle
Step 5 Angles in a triangle – special cases
Step 6 Angles in a triangle – missing angles
Step 7 Angles in a quadrilateral
Step 8 Angles in polygons
Step 9 Circles
Step 10 Draw shapes accurately
Step 11 Nets of 3-D shapes
National Curriculum Links:
Pupils should be taught to:
• draw 2-D shapes using given dimensions and angles
• recognise, describe and build simple 3-D shapes, including making nets
• compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
• illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius
• recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles
Step 1 The first quadrant
Step 2 Read and plot points in four quadrants
Step 3 Solve problems with coordinates
Step 4 Translations
Step 5 Reflections
National Curriculum Links:
Pupils should be taught to:
• describe positions on the full coordinate grid (all 4 quadrants)
• draw and translate simple shapes on the coordinate plane, and reflect them in the axes | {"url":"https://curriculum.parkfieldprimary.com/mathematics/year-6-mathematics","timestamp":"2024-11-07T00:44:33Z","content_type":"text/html","content_length":"265566","record_id":"<urn:uuid:b34a079d-6545-4651-b6e9-96af479c81d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00415.warc.gz"} |
What is BoB Software? BoB Biomechanics
BoB/Teaching is a package specifically designed for teaching biomechanics and bringing the subject to life.
BoB/Teaching contains a human musculoskeletal model and is pre-loaded with examples illustrating many biomechanical principles. BoB/Teaching has all of the analysis and graphics capabilities of BoB/
Research, but cannot import new data.
The BoB/Teaching package also contains editable a worksheets for students to follow and complete. These exercises will guide the students through many biomechanical examples illustrating principles
and applications.
BoB/Teaching enables every student to have a musculoskeletal model on their own PC or Mac.
BoB/Teaching is a self-contained package for students of biomechanics enabling them to investigate, observe and learn biomechanical analysis. The package contains examples of biomechanical activities
ranging from simple, idealised movements to real-life motion captured. BoB/Teaching has a short learning curve and includes many tutorial videos.
BoB/Teaching encourages the student to experiment, enquire and gain a deep understanding of the principals employed in biomechanical analysis with a very short learning curve and minimal
BoB/Teaching contains all of the sophisticated graphics and analysis functions of BoB. BoB/Teaching also includes a 3-dimensional viewer for an interactive interface and generates output as graphs,
numeric data, videos and images to engage the students’ interest.
BoB/Teaching can analyse and display:
Anatomical trajectories
Body instances
Point position/vel/acc
Joint angles
Joint range of motion
Distance/angles between points
Muscle length
Segmental orientation
Velocity vectors
Torques at joints
Externally applied force
Ground reaction forces
Joint contact forces
Muscle labelling
Muscle forces
Muscle energy/power
Total muscle energy/power
Muscle torque decomposition
The BoB/Teaching package contains examples of:
Static hand weight
Knee flexing
Arm curl
Single foot balance
Moving hand weight
Climbing stairs
Jogging with EMG
Ballet fouette
Cycling with EMG
Inverse dynamics
Gait with EMG
Baseball pitching
Manual handling
Tennis serve | {"url":"https://www.bob-biomechanics.com/bob-teaching/","timestamp":"2024-11-11T18:37:18Z","content_type":"text/html","content_length":"92894","record_id":"<urn:uuid:00bdea32-bbf7-497c-8c15-0f4fdfdf65ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00244.warc.gz"} |
Integrable Systems, Topology, and Physicssearch
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Integrable Systems, Topology, and Physics
Edited by: Martin Guest : Tokyo Metropolitan University, Tokyo, Japan
eBook ISBN: 978-0-8218-7899-6
Product Code: CONM/309.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
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Integrable Systems, Topology, and Physics
Edited by: Martin Guest : Tokyo Metropolitan University, Tokyo, Japan
eBook ISBN: 978-0-8218-7899-6
Product Code: CONM/309.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
• Contemporary Mathematics
Volume: 309; 2002; 324 pp
MSC: Primary 35; 37; 53; 58; 70
Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry,
symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the
University of Tokyo in July 2000. It brought together scientists in all of the areas influenced by integrable systems. This book is the second of three collections of expository and research
This volume focuses on topology and physics. The role of zero curvature equations outside of the traditional context of differential geometry has been recognized relatively recently, but it has
been an extraordinarily productive one, and most of the articles in this volume make some reference to it. Symplectic geometry, Floer homology, twistor theory, quantum cohomology, and the
structure of special equations of mathematical physics, such as the Toda field equations—all of these areas have gained from the integrable systems point of view and contributed to it.
Many of the articles in this volume are written by prominent researchers and will serve as introductions to the topics. It is intended for graduate students and researchers interested in
integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.
The first volume from this conference, also available from the AMS, is Differential Geometry and Integrable Systems, Volume 308 in the Contemporary Mathematics series. The forthcoming third
volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.
Graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 309; 2002; 324 pp
MSC: Primary 35; 37; 53; 58; 70
Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic
geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of
Tokyo in July 2000. It brought together scientists in all of the areas influenced by integrable systems. This book is the second of three collections of expository and research articles.
This volume focuses on topology and physics. The role of zero curvature equations outside of the traditional context of differential geometry has been recognized relatively recently, but it has been
an extraordinarily productive one, and most of the articles in this volume make some reference to it. Symplectic geometry, Floer homology, twistor theory, quantum cohomology, and the structure of
special equations of mathematical physics, such as the Toda field equations—all of these areas have gained from the integrable systems point of view and contributed to it.
Many of the articles in this volume are written by prominent researchers and will serve as introductions to the topics. It is intended for graduate students and researchers interested in integrable
systems and their relations to differential geometry, topology, algebraic geometry, and physics.
The first volume from this conference, also available from the AMS, is Differential Geometry and Integrable Systems, Volume 308 in the Contemporary Mathematics series. The forthcoming third volume
will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.
Graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CONM/309","timestamp":"2024-11-04T07:39:56Z","content_type":"text/html","content_length":"93866","record_id":"<urn:uuid:00aed71b-5544-4a88-b64f-cd78813bd5bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00804.warc.gz"} |
Programmers’ Patch
I have always disliked iconv. Both iconv and icu are designed to convert text in various character encodings into other encodings. iconv is normally installed on your computer (in OSX or Linux etc.),
but icu is much easier to use. The problem is that iconv doesn't provide any easy way to compute the length of the destination buffer, whereas in icu this is trivial. For example, if I want to
compute how long a buffer to contain text will be when I convert it from utf-16 all I do is pass 0 as the destination length and a NULL buffer and it tells me how long to make it:
Having called that function, I simply allocate a buffer of exactly that length, pass it into the conversion function again, and Bob's your uncle. The way to do this in iconv is to guess how big to
make it, then reallocate the destination buffer as often as needed during the chunk by chunk conversion. Then you can read the number of characters converted. What a messy way to do it! I
particularly like the fact that icu does NOT require you to specify a locale, as iconv does for some obscure reason. That limits the conversions you can do to those locales installed on your machine,
and you have to guess which of them is appropriate for the current conversion. That's just nuts.
A suffix tree is a data structure that facilitates the finding of any substring of length M in a text of length N. A substring of length 6 (M) can then be found in a text of a million characters (N)
in time proportional to M. This is much faster than the best string-searching algorithms, which take time proportional to the length of the text. Building a suffix tree does take time and space
proportional to N, but this only needs to be done once. This makes suffix trees a very useful data structure for bioinformatics and various other textual applications.
Section 1 gives an overview of how the algorithm works. The remaining sections describe the various components of the algorithm: the phases, extensions, finding the suffix of the previous phase,
suffix links, skipping extensions and completing the tree. The discussion is backed up by working C code that includes a test suite, a tree-printing module, and gnuplot files for precisely
documenting cpu and memory usage.
1. Overview
Suffix trees used to be built from right to left. For the string "banana" you would create an empty tree, then insert into it "a", then "na", "ana", "nana", "anana" and finally "banana". Ukkonen's
idea was to build the tree left to right by adding all the suffixes of progressively longer prefixes of the string. So first he would insert all the suffixes of "b", then all the suffixes of "ba",
"ban", "bana", banan" and finally "banana". This looks a lot less efficient than the right to left method, because it multiplies the number of tree-insertions by a factor of N. However, by using a
number of tricks the time complexity reduces from N^3 to N.
The algorithm divides the building into N phases, each containing a number of extensions. The phases correspond to building the implicit (unfinished) suffix tree for each prefix of the overall
string. The extensions correspond to inserting each of the suffixes of each prefix into the tree.
Implicit tree of "banana" after phase 2
Listing 1
The first implicit tree called I[0] is constructed manually. The empty root node is created and a leaf constructed containing the string "b". The algorithm then proceeds through N additional phases
in each of which the tree is expanded to the right by one character. The terminal NULL character (written "$") is added as a unique suffix at the end so we can distinguish the suffix "anana$" from
"ana$" (otherwise "ana" would be a prefix of "anana"). The set_e function will be described in Section 7 below.
2. Phases
Listing 2
A phase is just a succession of extensions. The global e variable represents the index of the last character represented in the tree. So if e equals 5 this means that all leaves end in the "a" at the
end of "banana", and 6 means that the tree is complete with its terminal $. And if e equals 0 this means that the 0th character ("b") is in the tree. So for each phase we increment e and update the
length of every leaf automatically.
The phase algorithm calls the extension algorithm with successively shorter suffixes of the current prefix (j..i). The loop starts with the value of old_j, which is the last value that j had in the
previous phase, starting with 0. This optimisation is explained in section 6 below.
If the extension function returns 0 then the rest of the current phase can be skipped. How this works will be explained in the next section.
3. Extensions
To understand the extension algorithm you have to be clear about how the tree's data structure works:
1. A character range j..i means the character str[j] up to and including the character str[i].
2. Since there is only one incoming edge per node in a tree we can represent edges as belonging to their nodes. The edges in nodes are defined by their start position in the string and their length.
So a start position of 1 in "banana" is the character "a" and if its length was 3 this would represent the string "ana". This way of storing strings has 3 advantages:
a. Each node has the same size
b. The text can be of any type: even wide UTF-16 characters.
c. Since the text is not copied into the nodes the overall storage is proportional to N.
3. A particular character in the tree is uniquely identified by a pointer to its node and an index into the string. I call this a "pos" or position.
Listing 3
The first step is to find the position of the suffix j..i-1 which was inserted during the previous phase. Each such substring, called β in Gusfield, is now extended by one character, so that the
substring j..i is added to the tree. There are three possibilities:
1. If the substring β is at the end of some leaf we just extend the leaf by one. (This step is automatic when we update via the e global).
2. β ends at a node or in the middle of an edge and the next, or ith character, is not yet in the tree. If it ends at the start of a node we create a new leaf and attach it to that node as a child.
If it ends in the middle of an edge we split the edge by adding a new internal node and attaching the leaf to it.
3. β ends at a node or in the middle of an edge and the next, or ith character, is already in the tree. Since we were proceeding left to right, it follows that all the remaining suffixes in this
phase are also suffixes of that earlier copy of this substring and must already be extended. So there is nothing more to do now and we can abort the rest of the phase.
The update_old_beta function is explained in Section 6, and update_current_link is explained in Section 4.
4. Suffix links
Navigating the tree between branches instead of down from the root converts the basic algorithm from time proportional to N^3 to N^2. To make this possible suffix links must be created. These record
the path followed by the extension algorithm as it moves through the tree during a phase. They are then used as short-cuts when constructing the extensions of the following phase. A suffix link is
set to point to the current internal node from the last such node created or found by rules 2 or 3 (see Listing 3). When rule 3 ends the phase prematurely there must already be a path leading back to
the root from that point in the tree. The following suffix links are defined for the suffix tree of "banana$":
link: ana -> na
link: na -> a
link: a -> [R]
| | |
| | -$
| |
| -$
| |
| -$
In Listing 4 the update_current_link function sets the link of the last seen internal node "current" to the value of the next internal node.
Listing 4
5. Finding β
For each iteration of the extension algorithm the correct position for the new suffix j..i is at the end of the existing suffix j..i-1, or β. This string can be found naively by walking down from the
root starting at character offset j. However, this requires navigating through a maximum of N nodes for each extension. A shortcut that takes constant time is needed, and can be concocted by
following the suffix links.
Figure 1: Walking across the tree
The last position of i-1 in each extension can be used to find its position in the next extension by following the nearest suffix link. The node immediately above the last position will not in fact
contain a suffix link, because this hasn't been set yet. We must therefore go one node further up the tree (see Figure 1) to the first node with a suffix link, or to the root. In doing so we trace a
path called γ. After arriving at the end of the link we then walk down the branch, where we will find an exact copy of γ, to the new position for the next extension. The journey is complicated by the
use of indices of characters, not the characters themselves. Also, we may encounter multiple nodes on our journey down the next branch. Since the length of the journey is determined by the local
distances between nodes and not the size of the tree, informally it is clear that the time required will be constant with respect to N.
Listing 5
Listing 5 shows an implementation of the algorithm. There are four possibilities:
1. If this is the first extension in the phase we just use the last value of β, extended by one character, from the previous phase.
2. A range where j > i indicates the empty string. (Recall that in find_beta the value of i is that of the previous phase). In this case we are trying to extend the root by a single character.
3. If the suffix is the entire string (starting at 0) this means the longest leaf, which is always pointed to by f.
4. In all other cases we walk across the tree by first locating the position of the previous j..i substring. Then we walk up at least one node or to the root, follow the link and walk down following
the same textual path. (If we do reach the root we must discard γ, because it will be incorrect. In this case we just walk down naively from the root.) Walking up does not require us to make any
choices at each node since there is always only one parent, but on the way down we require a path to follow so that the correct children are selected at each node. So we save the "path" (A simple
data type containing an offset and a length) during the up-walk, and destroy it once we have walked down the other side.
6. Skipping extensions
We have already established in the previous section that the time taken for each extension is constant. However, the number of extensions per phase is still proportional to N. Linear time complexity
is attained by reducing this to a constant also.
A true suffix tree has exactly N leaves for a text of length N. Since the only rule in the extension algorithm that creates leaves is rule 2, and since "once a leaf always a leaf" it follows that on
average rule 2 must be applied exactly once per phase. Similarly, rule 3 can at most be applied once per phase. We have already observed that the use of the e global makes all applications of rule 1
redundant. So, informally, each phase will take constant time if we can just skip all the leaf extensions and start with the first necessary application of rule 2.
An examination of the program's execution reveals that the rules are applied in order for each phase: first a number of rule 1s, then rule 2 and finally rule 3 (if at all). The applications of rules
2 and 3 for the string "banana" are:
applying rule 2 at j=1 for phase 1
applying rule 2 at j=2 for phase 2
applying rule 3 at j=3 for phase 3
applying rule 3 at j=3 for phase 4
applying rule 3 at j=3 for phase 5
applying rule 2 at j=3 for phase 6
applying rule 2 at j=4 for phase 6
applying rule 2 at j=5 for phase 6
applying rule 2 at j=6 for phase 6
So we only have to remember the position of the last inserted suffix after each application of rule 2 or 3 and this can then be used instead of β at the start of the next phase. Also the value of j
can be the last value it had in the previous phase. This trick allows us to skip most of the extensions and reduce their number per phase to a constant value.
Listing 6
We remember the last position of j..i in the previous phase by extending the position of β by the ith character, as shown in Listing 6.
6. Finalising the tree
Leaf-nodes are extended automatically by setting their length to "infinity", which for practical purposes, can be INT_MAX in C (2147483647). Whenever we request the end of such a node the answer will
then be the current value of e. However, this is inconvenient for a finished tree, in which the lengths of all nodes should be correctly set. We can do this simply by recursing down the tree, looking
for leaves and setting their lengths to e-node_start(v)+1. The time cost is proportional to N, but in addition to that already incurred, so overall the algorithm remains O(N).
Listing 7
7. Demonstration of linearity
Ukkonen's original algorithm did not specify how to represent the multi-branching nodes of the tree. The choice is linked lists or hashtables. It turns out that hashtables are not much better than
lists, even for plain text, and use more memory. Running the test program for either pure list nodes or a mixture of hashtables (for branches > 6) or lists for smaller nodes confirms that the time
taken does indeed increase linearly with file size:
Here is the memory usage comparing plain lists and a hashtable when the node size exceeds 6:
If you set MAX_LIST_CHILDREN to INT_MAX in tree.c and recompile you will get the list representation.
E. Ukkonen, 1995. Online construction of suffix trees. Algorithmica 14.3, 249–260. http://www.cs.helsinki.fi/u/ukkonen/SuffixT1withFigs.pdf
D. Gusfield, 1997. Linear-time construction of suffix trees in Algorithms on strings, trees and sequences, Cambridge University Press. http://www.stanford.edu/~mjkay/gusfield.pdf
D. Schmidt, 2013. A C implementation of Ukkonen's suffix tree-building algorithm, with test suite and tree print.
Using hash tables to improve scalability of Ukkonen's algorithm | {"url":"https://programmerspatch.blogspot.com/2013/02/","timestamp":"2024-11-02T22:12:32Z","content_type":"text/html","content_length":"75260","record_id":"<urn:uuid:1def41f4-6aba-4e5f-b2de-7fae9604b2cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00467.warc.gz"} |
shortest_path_length(G, source=None, target=None, weight=None)[source]¶
Compute shortest path lengths in the graph.
• G (NetworkX graph) –
• source (node, optional) – Starting node for path. If not specified, compute shortest path lengths using all nodes as source nodes.
Parameters: • target (node, optional) – Ending node for path. If not specified, compute shortest path lengths using all nodes as target nodes.
• weight (None or string, optional (default = None)) – If None, every edge has weight/distance/cost 1. If a string, use this edge attribute as the edge weight. Any edge attribute
not present defaults to 1.
length – If the source and target are both specified, return the length of the shortest path from the source to the target.
If only the source is specified, return a dictionary keyed by targets whose values are the lengths of the shortest path from the source to one of the targets.
If only the target is specified, return a dictionary keyed by sources whose values are the lengths of the shortest path from one of the sources to the target.
If neither the source nor target are specified return a dictionary of dictionaries with path[source][target]=L, where L is the length of the shortest path from source to target.
Return int or dictionary
Raises: NetworkXNoPath – If no path exists between source and target.
>>> G=nx.path_graph(5)
>>> print(nx.shortest_path_length(G,source=0,target=4))
>>> p=nx.shortest_path_length(G,source=0) # target not specified
>>> p[4]
>>> p=nx.shortest_path_length(G,target=4) # source not specified
>>> p[0]
>>> p=nx.shortest_path_length(G) # source,target not specified
>>> p[0][4]
The length of the path is always 1 less than the number of nodes involved in the path since the length measures the number of edges followed.
For digraphs this returns the shortest directed path length. To find path lengths in the reverse direction use G.reverse(copy=False) first to flip the edge orientation.
See also
all_pairs_shortest_path_length(), all_pairs_dijkstra_path_length(), single_source_shortest_path_length(), single_source_dijkstra_path_length() | {"url":"https://networkx.org/documentation/networkx-1.10/reference/generated/networkx.algorithms.shortest_paths.generic.shortest_path_length.html","timestamp":"2024-11-06T20:43:34Z","content_type":"text/html","content_length":"20466","record_id":"<urn:uuid:e28dd3f5-4529-441e-a470-c2e9ded240c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00776.warc.gz"} |
cross-lags in single times series of multiple variables
I have collected data on 3 variables X, Y, Z on the same single subject over time.The model I would like to fit in Lavaan includes both cross-and autolags.
The problem that I have is that I do not know how to construct such a model to fit on a single subject's timeseries of X,Y,Z.
I do know how to fit the model in Lavaan on data from multiple subjects with timeseries of X,Y,Z, I also know how to fit the SEM in RStan such that it can deal with both single subject and multiple
subject timeseries. Can anyone provide a hint how to adapt my lavaan model structure so it can deal with single subject's timeseries?
In Rstan my model looks like:
Y[2:N] ~ normal(beta[1] + beta[2] * X[2:N], sigma1);
X[2:N] ~ normal(beta[3] + beta[4] * X[1:(N - 1)] .* Y[1:(N - 1)] + beta[5] * X[1:(N - 1)] .* Z[1:(N - 1)], sigma2);
Z[2:N] ~ normal(beta[6], sigma3);
where N is time-series length (and .* is just how RStan multiple two vectors)
So in this model Y~Xlag1, X~Xlag1*Ylag1+Xlag1*Zlag1 and Z~1.
I have tried to fit this model in lavaan using the growth function, which works fine as long as I analyze timeseries of multiple subjects simultaneously, but it does not work for single subjects :
My data I rearranged into wide format: e.g. variable Y is divided into N variables (y1, y2, etc., where y1 is the y-value at timestep 1).
subject y1 y2 ...
1 2.3 0.3 ...
2 1.4 4.9 ...
My model looks like
... series continues till N
x2~beta4*y1:x1 +beta5*z1:x1
x3~beta4*y2:x2 +beta5*z2:x2
... series continues till N
... series continues till N
... series continues till N
... series continues till N
If I fit his lavaan model to a dataset containing timeseries of mutliple subjects (output = growth(model, widedata)), the outcomes are as expected (I simulated data with known effect sizes).
However, when I fit this model to a single timeseries (output = growth(model, widedata[1,]), the growth function gives the following error:
Error in lav_data_full(data = data, group = group, cluster = cluster, :
lavaan ERROR: some variables have only 1 observation or no finite variance
Apparently, and maybe not unsurprisingly, the growth function cannot handle datasets with only one observation per variable, even if regression parameters (beta2, etc.) are shared among equations.
I assume I should be able to code the model in Lavaan a different and smarter way, but am not sure how to do it? Is there a way to do it similar as in the above Stan model?
Any help is much appreciated,
cheers Martijn | {"url":"https://groups.google.com/g/lavaan/c/K9yfEkJyexE","timestamp":"2024-11-04T18:53:37Z","content_type":"text/html","content_length":"773364","record_id":"<urn:uuid:3c04f188-9c13-4aef-8f23-75d38462e61f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00761.warc.gz"} |
Relative velocity and projectile motion problem solving
• Thread starter SAM31
• Start date
In summary, the conversation revolves around a complete solution provided by the speaker for a specific question. They are seeking feedback and clarification on their solution, which is deemed
professional. The solution involves using kinematic equations to calculate the time of flight and the final results seem to be accurate, despite minor arithmetic errors. The speaker expresses a
preference for using algebra rather than dealing with long decimal numbers.
Homework Statement
Question 2:A golf ball is hit from 4.3 m above a golfing fairway with an initial velocity of 30.0 m/s at an angle of 35° above the horizontal.
A. Determine the time of flight for the ball.
B. Determine the range for the golf ball.
C. Determine the velocity for the golf ball the instant before the ball impacts the ground.
Relevant Equations
Pythagorean theorem equations
It seems that you have a complete solution. Is there a question you wish to ask or are you posting a solution for the benefit of others?
kuruman said:
It seems that you have a complete solution. Is there a question you wish to ask?
I was wondering if someone could look over the steps to ensure I'm on the right track and provide me with any feedback!
Is this your solution or someone else's? If yours, it looks professional. I have not put in the numbers, but it looks right.
kuruman said:
Is this your solution or someone else's?
This is my own attempt at a solution for the question asked above
See edited post #4. If you wish to check your work, plug in the time of flight in the kinematic equations and see if the ball lands where you thought it would be.
Science Advisor
Homework Helper
Gold Member
Arithmetic errors creep in here:
but they do not affect the final results to 2sf, and otherwise it looks good.
SAM31 said:
This is my own attempt at a solution for the question asked above
All those long decimal numbers don't look good to me. This is why algebra was invented.
FAQ: Relative velocity and projectile motion problem solving
1. What is relative velocity and how is it calculated?
Relative velocity is the velocity of an object in relation to another object. It is calculated by subtracting the velocity of the first object from the velocity of the second object.
2. How do you solve projectile motion problems?
To solve projectile motion problems, you will need to use equations of motion, such as the kinematic equations. You will also need to break down the motion into horizontal and vertical components,
and consider factors such as gravity and air resistance.
3. What is the difference between absolute and relative velocity?
Absolute velocity is the velocity of an object with respect to a fixed point, while relative velocity is the velocity of an object with respect to another moving object.
4. How can you determine the angle of projection in a projectile motion problem?
The angle of projection in a projectile motion problem can be determined by using the trigonometric functions of sine, cosine, and tangent. You will need to know the initial velocity and the
horizontal and vertical components of the motion.
5. How does air resistance affect projectile motion?
Air resistance, also known as drag, can affect projectile motion by slowing down the object and changing its trajectory. This is because air resistance acts in the opposite direction of the motion,
causing a decrease in velocity and a change in the direction of the object's path. | {"url":"https://www.physicsforums.com/threads/relative-velocity-and-projectile-motion-problem-solving.1045715/","timestamp":"2024-11-03T13:54:34Z","content_type":"text/html","content_length":"112460","record_id":"<urn:uuid:92ebf978-e826-4a95-9940-18ca40024a9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00240.warc.gz"} |
5.4: Two-Way Tables (2 of 5)
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Learning Objectives
• Calculate marginal, joint, and conditional percentages and interpret them as probability estimates.
In the previous section, we used the information in a two-table to examine the relationship between two categorical variables. Our goal was to answer the big question: Are the variables related?
In this section, we continue to work with two-way tables, but we ask a different set of questions.
Community College Enrollment
The following table summarizes the full-time enrollment at a community college located in a West Coast city. There are a total of 12,000 full-time students enrolled at the college. The two
categorical variables here are gender and program. The programs include academic and vocational programs at the college. Assume that a student can enroll in only one program.
Arts-Sci Bus-Econ Info Tech Health Science Graphics Design Culinary Arts Row Totals
Female 4,660 435 494 421 105 83 6,198
Male 4,334 490 564 223 97 94 5,802
Column Totals 8,994 925 1,058 644 202 177 12,000
Let’s consider a few preliminary questions to get familiar with this new data set.
• 1. What proportion of the total number of students are male students?
$\frac{\mathrm{number\; of\; male\; students}}{\mathrm{total\; number\; of\; students}}=\frac{\mathrm{5,802}}{\mathrm{12,000}}=\mathrm{0.4835}(\mathrm{or\; 48.35\%})$
• 2. What proportion of the total number of students are Bus-Econ students?
$\frac{\mathrm{number\; of\; Bus-Econ\; students}}{\mathrm{total\; number\; of\; students}}=\frac{925}{\mathrm{12,000}}=\mathrm{0.077}(\mathrm{or\; 7.7\%})$
Note that to calculate this proportion, we used two numbers in the margin that relate to just one of the categorical variables (program). This calculation is therefore called a marginal proportion.
Note: This proportion does not help us determine if gender is related to program because it involves only one of the variables.
Now consider the following question:
• If we choose one student at random from among all 12,000 students at the college, how likely is it that this student will be in the Bus-Econ program?
From our previous calculation, we know that only about 8% (7.7%) of the students at the college are in the Bus-Econ program. That’s a fairly low number, so it is not very likely that our random
student will be a Bus-Econ student.
One way to state our conclusion is to say:
• There is about an 8% chance of picking a Bus-Econ major.
This means that if we selected 100 students at random, we would expect on average that 8 of them would be in the Bus-Econ program.
Here is another way to state this conclusion:
• There is about an 0.08 probability of picking a Bus-Econ major.
Because this probability is exactly the same as the marginal proportion we calculated earlier, we call it a marginal probability.
Note: P for Probability
It is customary to use the capital letter P to stand for probability. So instead of writing “The probability that a student is in Bus-Econ program equals 0.08,” we can write P(student is in Bus-Econ)
= 0.08.
The following table is used for the next Try It and Did I Get This? activities.
Arts-Sci Bus-Econ Info Tech Health Science Graphics Design Culinary Arts Row Totals
Female 4,660 435 494 421 105 83 6,198
Male 4,334 490 564 223 97 94 5,802
Column Totals 8,994 925 1,058 644 202 177 12,000
Conditional Probability
Here is the same community college enrollment data.
Arts-Sci Bus-Econ Info Tech Health Science Graphics Design Culinary Arts Row Totals
Female 4,660 435 494 421 105 83 6,198
Male 4,334 490 564 223 97 94 5,802
Column Totals 8,994 925 1,058 644 202 177 12,000
Here is our first question:
• If we select a female student at random, what is the probability that she is in the Health Sciences program?
Answer Of the 6,198 female students at the college, 421 are enrolled in Health Sciences. (Find these numbers in the table.) The probability we are looking for is:
$\frac{\text{421}}{\text{6,198}}\approx \text{0.07}$
Therefore, the probability that a female student is in the Health Sciences program is approximately 0.07.
Focus on Language
We need to pause here and be very careful about the language we use in describing this situation.
Note that we start with a female student and then ask what is the probability that this female student is in the Health Sciences department.
In this case, our starting point is that the student is a female. This information sets the conditions for calculating the probability. Once the condition (student is female) is set, we focus on the
female student population. In terms of the two-way table, it means that the only numbers we will be using are in the Female row: 421 and 6,198.
What Is a Conditional Probability?
The probability we calculated earlier is an example of a conditional probability. In general, a conditional probability is one that is based on a given condition. Here the given condition is that the
student is female.
Here is the notation we use for a conditional probability:
• Original question: If we select a female student at random, what is the probability that she is in the Health Sciences program?
• Notation: P(student is in Health Sciences given that student is female).
• We also write this as P(Health Sciences given female).
An even shorter way of writing this is to use a vertical bar | in place of given:P(Health Sciences | female).
The following table is used for the next Try It and Did I Get This? activities.
Arts-Sci Bus-Econ Info Tech Health Science Graphics Design Culinary Arts Row Totals
Female 4,660 435 494 421 105 83 6,198
Male 4,334 490 564 223 97 94 5,802
Column Totals 8,994 925 1,058 644 202 177 12,000
Contributors and Attributions
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Introduction To Trigonometry Class 10 Exercise 82 – Online degrees
Introduction To Trigonometry Class 10 Exercise 82
Ncert Solutions For Class 10 Maths Chapter 8 Exercise 8 2
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Ncert Solutions For Class 10 Maths Chapter 8 Exercise 8 2
Ncert Solutions For Class 10 Maths Chapter 8 Introduction To
Ncert Solutions For Class 10 Maths Chapter 8 Exercise 8 2
Ncert Solutions For Class 10 Maths Chapter 8 Introduction To
Ncert Solutions For Class 10 Maths Chapter 8 Exercise 8 2
Ncert Solutions For Class 10 Maths Chapter 8 Introduction To
Ncert Solutions For Class 10 Maths Chapter 8 Exercise 8 2
Ncert Solutions For Class 10 Maths Chapter 8 Introduction To
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Ncert Solutions For Class 10 Maths Chapter 8 Exercise 8 2
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Facts About Machine Learning You Should Know
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In 2018, the increase in machine learning made a big splash. It has the potential to be a powerful way to make people smarter, so it is no longer something from the future. Businesses that sell to
both businesses and consumers have found it helpful to drive better business results, such as creating influential content, increasing paid conversions, and lowering marketing costs.
Machine learning means allowing computers to do things without help from people. The technology concentrates on creating computer programs that can access and learn from data for future use.
Let’s look at how to get good at Machine Learning Technology.
Before placing a Machine Learning Model into production, it must be possible to train, test, and validate it. Getting data ready for analytics speeds up machine learning and data science projects,
giving business customers a more immersive business experience. This automates the pipeline from data to insights, which comprises the six steps listed below.
Data gathering
Collecting data in Machine Learning is very important because the amount and quality of the data will determine how nice the analytical model will be. The different files must be put together into
one single file. The information is put into a table and given the name Training Data.
Filter Data
The step involves putting the data correctly and getting it ready for use. Randomizing the order of data is done to ensure that the order doesn’t change the predicted results.
Analyze the data
The data cleaning up is then looked at to see if it can be used for machine learning. Later, the data is split into training sets and evaluating sets. This step is about eliminating duplicates,
fixing errors and missing values, normalizing the data, changing the data type, etc.
Train the Models
A particular algorithm is made to do a certain job. This step is very important because it involves making the very important choice of which algorithm to use for a model. The model has been trained
so that it gives accurate results. The goal of training the model is to answer a question or make a prediction as often as possible. The iteration process describes every move of the training.
Evaluate Model
Several metrics are used to measure how well the model works. The model is tested on data that has never been used before. This helps tune the model better.
Parameter Tuning
After evaluating the algorithm, the parameters are changed to make it better. It has several training steps, a learning rate, initial values, a distribution, and a learning rate.
Make Forecasts
The last step is to make a prediction, which answers a few questions. You can finally determine if the ML model predicts what will happen. It gives a rough idea of how well the model will work in the
real world.
Commonly Used Algorithms for Machine Learning
As the world moves toward digital transformation, technology has made it possible for big tech companies to compete for the best data scientists. The main target is to let computers learn
independently, without human help, and change their actions accordingly. Every year, more and more money is put into technology. Several algorithms in the technology can be used to solve almost any
kind of data problem.
Let’s look closely at the different Machine Learning algorithms.
Linear Regression
The Supervised Learning algorithm of Machine Learning is what Linear Regression is based on. Based on a continuous variable, the algorithm estimates true values like the house cost, the number of
calls, etc. (S). Most of the time, it is used to find the relationships between variables by trying to fit the best line. This line is called a regression line, shown by the equation Y = a * X + b.
The following variables are used to train the model:
• X: input training data
• Y: labels to data (supervised learning)
During training, a model fits the best line to anticipate the value of y given a value of x. Find the values of a and b to find the best regression line.
• a: coefficient of X
• b: intercept
Logistic Regression
Logistic Regression is not a regression formula but a monitored classification algorithm. It helps figure out discrete values like 0/1, yes/no, and true/false based on a set of independent variables
(s). For a given set of values for the input variable x, the output vector y can only predict discrete values.
It is also called logit regression because it fits information to a logit function to determine how likely something will happen. Its output is a number between 0 and 1, showing how likely something
is. A sigmoid function is used to model the data.
Logistic Regression is known as:
• Binomial: The value of the target variable can only be “0” or “1.”
• Multinomial: Variable has 3 or even more variables. It means that the numbers don’t matter.
• Ordinal: The categories in the target variables are in order. For example, “very poor,” “poor,” “good,” “very good,” and “excellent” are all ways to describe a performance score.
The Decision Tree
A decision tree is a managed learning algorithm often used to sort things into groups. It can be used for dependent variables that are both categorical and continuous. The algorithm is shown as a
tree, where each leaf node portrays a class label, and each internal node represents an attribute. It can show how the Boolean function works.
To be made assumptions while utilizing a decision tree.
• At first, the entire training set is thought of as a root.
• The feature values are considered definite, and the continuous values are broken up into discrete ones before the model is built.
• Records are passed around in a loop based on the attributes’ values.
• Statistics determine the order of attributes as origins or internal nodes.
kNN (k- Nearest Neighbors)
Many people use the algorithm to solve classification problems, but it can also be used to solve regression problems. It is an easy algorithm that stores all available cases and sorts new cases based
on how their k neighbors vote. Distance functions like Euclidean, Manhattan, Minkowski, and Hamming distance are used to determine the K-nearest neighbor. If K = 1, the case is put in the class of
the case that is closest to it.
Things to consider when picking kNN:
• It isn’t easy to figure out what to do.
• Variables must be made equal.
• Works better in the pre-processing phase before moving on to the kNN outlier, noise removal.
K-means is a form of an unsupervised algorithm that is used to solve the clustering problem. The next step is to group the given data set into several clusters. The data points in the cluster are
alike and different from those in the peer groups. In K-means, each group has its center. The total square value for that cluster is the total of the squares of the variance between the centreline
and the data points. The cluster solution is found by adding up the square values for each cluster.
Creating clusters:
• For each centroid, the algorithm chooses k points.
• Each point of data forms a group with the closest centroid.
• The center of each cluster is found based on an existing member of the cluster. There are new center points.
• After a new centroid has been formed, steps 2 and 3 are done again. Please find out how far away each data point is from the new centers and link it to a new k-cluster. Repeat the steps until the
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72 Types of Data Visualization for Storytelling and Visual Analytics
Data Storytelling 101: 72 Types of Data Visualization to Design Stories
Reading Time: 14 mins
Design and visualization are an important part of data stories. They help users to identify insights quickly. There are multiple charts, and graphs available to make informative and meaningful data
stories. However, in this guide, we share 72 different types of data visualization to create different types of data stories.
What are Data Visualization?
Data visualization is the pictorial or graphical representation of data. It communicates the hidden information in data with images and charts. They are used to make important insights visually
obvious. Data as visual narratives bridges the gap between data consumption and decision making.
Visualizations are also a vital part of a data story. They make the non-obvious insights visible on the screen and help in decision-making. In our earlier data storytelling blogs, we talked about
easy steps on how to create data stories and tips to structure data stories. In this episode, we’ll show different types of data visualization that you can use to design your data story.
Why are Data Visualization Important?
Every dataset is different, and certain visualizations suit certain types of data. For instance, a line chart is perfect to show variations in data with respect to time. A pie or bar chart is the
most suited to show categorical data.
Data visualizations allow users to identify patterns and trends in a single chart or a series of charts rather than exploring thousands of rows and columns in an Excel sheet. Even though data
scientists and analysts explore meaningful insights from spreadsheets, it gets difficult to communicate them to stakeholders. That’s where data visualizations pitch in.
Types of Data Visualization for Different Data Stories
There are multiple ways to visualize data – how do you know which one to pick? Below are some categories to decide which data relationship is most important in your data story. We are classifying the
types of data visualization for each category. You can compare and see which one works best for you. We believe this list can be a useful starting point for you to make informative and meaningful
data visualizations.
Types of Data Visualization to Show Deviations
Visualizing deviations in data is common. Variations, be it positive or negative, are compared with a reference point, which is usually zero. However, reference points can also be a target or a
long-term average. You can use the following types of data visualization charts to capture trends (positive/neutral/negative).
Diverging Bar Visualization
This is a simple standard bar chart that visualizes both negative and positive magnitude values.
Diverging Stacked Bar Visualization
In stacked bar chart visualizations, some parts of the data are represented in either adjacent (horizontal bars) or stacked (vertical bars) manner. This data visualization is perfect for presenting
survey results that involve sentiments (e.g. disagree, neutral, agreed).
Spine Visualization
It is a type of horizontal bar chart that is used to show comparison. This data visualization splits a single value into two contrasting components (e.g. Male/Female).
Surplus/Deficit Filled Line Visualization
Use this data visualization if you want to illustrate numbers against a reference point or show a balance between two series.
Surplus/deficit filled Area Visualization
This visualization is the same as the surplus/deficit filled chart but has an added feature of illustrating areas as shades.
Types of Data Visualization to Show Correlations
This type of data visualization shows the relationship between two or more variables. Scatter plots, bar charts, and XY heatmaps are a few types of data visualization in this category.
A scatterplot is also called a scatter chart, scattergram, or scatter diagram. Scatterplots show the relationship between two variables, aligned on different axes.
Column + Line timeline
Use this visualization to show the periodic relationship of any value with another one with respect to time.
Connected scatterplot
Use this data visualization to show the changing relationships between two variables over time.
This data visualization is similar to a scatterplot. However, you can consider a third variable and can embed additional details by sizing the circles.
XY Heatmap
A good way of showing the patterns between two categories of data, not so suited for showing fine differences in amounts.
Types of Data Visualization to Show Rankings
Ordered lists or rankings are useful to quickly identify top or bottom performers. For example, your website data can give you insights about the most viewed pages by ranking them from highest to
lowest or vice-versa.
Ordered Bar
These are standard bar charts that show the ranks of values easily when ordered. These are horizontally aligned.
Ordered Column
The vertically aligned column bars display the data organized as categories and heights or lengths proportional to the values.
Ordered Proportional Symbol
This one is similar to scatterplots. You can use this data visualization chart to show significant variations between two or more variables.
Dot Strip Plot
Dot strip plots represent rankings as dots plotted on the axes. This is a space-efficient method of ranking across different categories of variables.
Slope charts show the rate of change in the ranks of a particular category over time.
This is a chart where the bars are in the form of lollipops. They are visually appealing bar/column charts and are a catchy way of showing rankings.
Bump charts are a cross-over type of visualization for the changing ranks of categories with respect to dates or time. For large datasets, consider grouping lines using color.
Types of Data Visualization to Show Distributions
Histogram visualizations use bars of different heights to show the distribution of variables in data. Each group bins the number into ranges and the size of the bar represents the volume of the data.
Keep the gaps between columns small to highlight the ‘shape’ of the data.
Dot Plot
A simple way of showing the range (min/max) of data across multiple categories by plotting the variables as circles.
Dot Strip Plot
Dot strip plots represent rankings as dots plotted on the axes. This is a space-efficient method of ranking across different categories of variables.
Barcode Plot
Barcode plots show all the data in a table and are useful to highlight individual values.
Box plot
A box plot is also known as a whisker plot and is majorly used in explanatory data analysis. It displays the numerical data through their quartiles and summarises multiple distributions by showing
the median (center) and the range of the data.
Violin Plot
The violin plot is similar to a box plot but in the shape of a violin. This chart is more effective with complex distributions where data can’t be summarised with a simple average.
Population Pyramid
In the population pyramid, histogram-like bars are placed horizontally to show the age and sex breakdown of population distribution.
Cumulative Curve
Similar to a line chart, a cumulative curve chart is a good way of showing how unequal a distribution is. Here, the y-axis is always a cumulative frequency and x-axis is always a measure.
Frequency Polygons
Frequency polygons are used to display multiple distributions of data. The insights are discoverable if the data sets are limited (3 or 4).
Types of Data Visualization to Show Change Over Time
Using the following types of data visualization you can show the trends in a data set. The trends can go back centuries. However, to display non-obvious insights and context to read, choose the
correct time period.
Line Charts
Line chart is one of the traditional visualizations to show a changing time-series. If data is irregular, consider markers to represent data points.
Column Charts
These works well for showing change over time, but usually best with only one series of data at a time.
Column + Line Timeline
It is a mixture of the above two charts. It shows a relationship between amount and rate listed on the two axes.
Slope charts can display change over time precisely if the data is simplified into 2 or 3 points.
Area Chart
Use with care. Area charts are good at showing changes to total, but seeing a change in components can be very difficult.
You can use candlestick visualizations to show the change in the intensity of an activity. For example, for any daily activity, you can show the opening/closing or high/low points of the activity.
Fan Chart
You can use fan charts to show the uncertainty in future projections. So, this chart, along with giving you the projections, also highlights the aberrations in the projection.
Connected Scatterplot
A connected scatterplot is a good way of showing deviations in data for two variables whenever there is a relatively clear pattern of progression.
Calendar Heatmap
A calendar heatmap is visually similar to a treemap. It is a great way of showing temporal patterns (daily, weekly, monthly), at the expense of showing precision in quantity.
Priestley Timeline
Priestley timeline charts are good to visualize the date and duration in a data story. Choose this only when these two parameters have a huge role to play in your story.
Circle Timeline
Circle timelines are another variation of dot plots. They are good for showing discrete values of varying sizes across multiple categories. For example, earthquakes by continents.
Vertical Timeline
Vertical timelines present time on the Y-axis and are similar to line charts. These charts give a good experience on the mobile view.
You can use seismogram visualization when there are significant variations in data. They are a good alternative to circle timeline charts.
Streamgraph visualizations are similar to area charts. You can use them when visualizing changing proportions over time is more important than individual values.
Types of Data Visualization to Show Huge Magnitudes
You can use the following set of charts to show size comparisons in data. These charts are good to show counted numbers rather than a value such as changing rate or percentage.
Column Charts
Column charts are a standard way to compare the size of things. You must always start at 0 on the axis and each column represents the size of a category.
Bar Charts
Bar charts are the same as column charts but in this case, they are aligned horizontally to the x-axis. Good when the data are not time series and labels have long category names.
Paired Column
Paired columns are also called as coupled columns. Here, more than one bar is used to represent one category but in different timelines. The chart can become tougher to read with more than 2 series.
Horizontal Paired Bar Charts
Paired bars are the same as paired columns but are aligned horizontally parallel to the X-axis. The chart can become tougher to read with more than 2 series.
Marimekko visualization can show the size and proportion of different data packed in a square or rectangular form.
Proportional Symbol
Proportional symbol charts are similar to scatter plots and are used when there are big variations between values. You can use this chart if visualizing small differences between data is not part of
your data story.
Isotype (pictogram)
The acronym Isotype stands for the International System of Typographic Picture Education. Isotype visualizations represent data in the form of icons. An excellent solution to represent whole numbers.
Radar visualizations are a space-efficient way of showing the value of multiple variables. Setting the visualization in a way that it makes sense to the reader is very important here.
Parallel Coordinates
It is an alternative to radar charts and sets up a relationship between multiple categories by linking them to each other.
Bullet charts are a good alternative for bar charts and can organize data in a space-efficient way, vertically and horizontally.
Grouped Symbol
Grouped symbol charts are an alternative to bar/column charts when being able to count data or highlight individual elements is useful. Best to be used for limited data points.
Types of Data Visualization to Show Part to Whole Data
The following set of charts can visualize the breakdown of a single entity into multiple components.
Pie Charts
Pie-chart is one of the classic types of data visualization to show part-to-whole data. However, there can be small segments of categories if the data is huge. Thus, it becomes difficult to
accurately compare the size of the segments.
A donut chart is similar to a pie chart. However, the blank center can be a good space to include more information about the data.
A treemap visualization is suitable when it comes to highlighting the hierarchical structure and quantity of the categories in terms of size.
A Voronoi visualization is a plane partitioned into segments to visualize different sizes of categories in a data set.
Arc diagrams are majorly used to visualize political data such as election results.
Grid plots are good for showing percentages. They work best when used on whole numbers and work well in multiple layout forms.
Venn Diagrams
Venn diagrams show the relationship between a finite collection of different categories in a data set.
Stacked Column
A stacked column chart is similar to a bar chart but components stacked over each other. They can be difficult to read with more than a few components
Waterfall visualization can be useful for showing part-to-whole relationships where some of the components are negative.
Types of Data Visualization to Show Spatial Data
Map data visualizations are good to plot election data, census data, and any other type of data related to population. You can use the following types of data visualization when you have the data for
precise locations or you want to find insights from geographical patterns. You can create all these charts using the structure of maps, such as country maps.
Basic Choropleth (rate/ratio)
Choropleth is a type of thematic map where the area or regions are shaded in proportion to a given data variable. The focus of this chart is to visualize rates rather than totals data based on
Proportional Symbol (count/magnitude)
Proportional symbol charts are the opposite of basic choropleth charts. Here you can focus on displaying totals rather than rates. The chart will display significant differences but small differences
will be hard to notice.
Flow Map
A Flow Map geographically shows the unambiguous movement of information or objects from one location to another.
Contour Map
This type of map visualization creates a geometrical contour share for showing areas of equal value on a map. You can use deviation color schemes for showing +/- values.
Equalised Cartogram
Equalized cartogram maps are good for representing voting regions with equal share.
Scaled Cartogram (value)
You can use scaled cartogram maps to size areas on the map according to particular values.
Dot Density
Dot density maps are a simple and effective way to display density differences in geographic distributions across a landscape.
Heat Maps
Grid-based data values mapped with an intensity color scale. They are like a choropleth map – but not snapped to an admin/political unit.
Types of Data Visualization to Show Data Flows
You can use the following set of data visualizations to indicate flows in data in terms of volumes or intensity of movement between two or more states or conditions. These might be logical sequences
or geographical locations.
Sankey Diagram
A Sankey diagram is a visualization used to depict a flow from one set of values to another by connecting lines called nodes. The width of the lines represents the flow rate.
Designed to show the sequencing of data through a flow process, typically budgets. Can include +/- components
Chord Diagram
Chord diagrams visualize the relationships in data. It is a complex but powerful diagram that can illustrate 2-way flows in a matrix.
Network Diagram
Network diagrams are used to display the interconnectedness of relationships of varying types in a data set.
Pratap Vardhan, our principal data scientist created this list. This data visualization repository is inspired by Financial Times’ Visual Vocabulary & Andy Kriebel’s ft. and is hosted on Github.
There are hundreds of modern-age data visualizations. Developers and information designers can go beyond just bar graphs and line charts to display insights as data stories.
We also conduct workshops for businesses that want to leverage data storytelling for making successful decisions.
Check out the agenda of our data storytelling workshop. We use real-time examples and hands-on experience to make you exercise data storytelling techniques.
We hope that these multiple types of data visualization will help you create wonderful data stories. Finally, do share this resource with your colleagues and fellow analysts, designers, and
developers. Share on social media and tag us (@gramener).
Leave a Comment | {"url":"https://blog.gramener.com/types-of-data-visualization-for-data-stories/","timestamp":"2024-11-09T04:26:38Z","content_type":"text/html","content_length":"538954","record_id":"<urn:uuid:78296c5e-5e6e-45ad-a980-8d0c73c942d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00198.warc.gz"} |
Rencontre, March 13, 2020
Amina Doumane (CNRS, ENS de Lyon), Completeness for Identity-free Kleene Lattices
• March 12, 2020, 15:30 - 16:30
We provide a finite set of axioms for identity-free Kleene lattices, which we prove sound and complete for the equational theory of their relational models. Our proof builds on the completeteness
theorem for Kleene algebra, and on a novel automata construction that makes it possible to extract axiomatic proofs using a Kleene-like algorithm.
We provide a finite set of axioms for identity-free Kleene lattices, which we prove sound and complete for the equational theory of their relational models. Our proof builds on the completeteness
theorem for Kleene algebra, and on a novel automata construction that makes it possible to extract axiomatic proofs using a Kleene-like algorithm. | {"url":"https://chocola.ens-lyon.fr/events/meeting-2020-03-12/talks/doumane/","timestamp":"2024-11-10T06:08:06Z","content_type":"application/xhtml+xml","content_length":"4091","record_id":"<urn:uuid:5330f7b7-e8c8-4f43-86f1-044cd20a30f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00642.warc.gz"} |
Turing Machine
Singletape and Multitape
Singletape Turing Machine
basic calculating operations with JS and html
• select operation: addition (+), subtraction (-), multiplication (*), division (/), factorial (!):
- all calulcations are based on the unary system
Multitape Turing Machine
basic calculating operations with JS and html
• select operation: addition (+), subtraction (-), multiplication (*), factorial (!):
- all calulcations are based on the unary system
This div shows the description of the content being loaded by javascript:Singletape, Turingmachine simulation: addition, subtraction, multiplication and factorial unary built in javascript and html,
Multitape Turing Machine,
• added direction of tapes, 22. March 15
• added direction of tape to transition table, 25. March 15
, turing machine, factorial, multiplication, css, js, javascript, project, zhaw, unary, This turing machine calculates addition, subtraction, multiplication and factorial. Built with javascript (js)
and html. Showing the state diagram and the state table of the calculation. This turingmachine is using multiple tapes., basic calculating operations with JS and html, • select operation: addition
(+), subtraction (-), multiplication (*), factorial (!):
- all calulcations are based on the unary system, operation, integer, • Tape, Reset, Step, Run, Pause, Proceed, Halted, current Step #, • Transition table, State #, Read, Write, Head Direction,
nextState, a project by , ZHAW exercise (Modul Informatik-I, Kurs Informatik-2: Aufgabenserie-5), solution by Stefan Sidler and
Roman Lickel
, 28.04.2012, return to Tape, bring diagram to front, • Diagram, proceed long calculation?, calculating...
• added direction of tapes, 22. March 15
• added direction of tape to transition table, 25. March 15
Extract of wikipedias article (
Wikipedia: Turing machine
A Turing machine is a device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer
algorithm, and is particularly useful in explaining the functions of a CPU inside a computer.
The "Turing" machine was described in 1936 by Alan Turing^[1] who called it an "a-machine" (automatic machine). The Turing machine is not intended as practical computing technology, but rather as a
hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.
Turing gave a succinct definition of the experiment in his 1948 essay, "Intelligent Machinery". Referring to his 1936 publication, Turing wrote that the Turing machine, here called a Logical
Computing Machine, consisted of:
...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed. At any moment there is one symbol in the machine; it
is called the scanned symbol. The machine can alter the scanned symbol and its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behaviour of
the machine. However, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an
innings.^[2] (Turing 1948, p. 61)
A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine (UTM, or simply a universal machine). A more mathematically oriented definition with a similar
"universal" nature was introduced by Alonzo Church, whose work on lambda calculus intertwined with Turing's in a formal theory of computation known as the Church–Turing thesis. The thesis states that
Turing machines indeed capture the informal notion of effective method in logic and mathematics, and provide a precise definition of an algorithm or 'mechanical procedure'.
Studying their abstract properties yields many insights into computer science and complexity theory. | {"url":"https://turingmaschine.klickagent.ch/","timestamp":"2024-11-11T09:23:39Z","content_type":"text/html","content_length":"11932","record_id":"<urn:uuid:8ad11645-4062-4f74-9523-17ea3adf7918>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00222.warc.gz"} |
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ACF and PCF
ACF stands for Auto Correlation, whereas PACF stands for Partial Auto Correlation. Before we go into the details, let's define Correlation, which exists in both the ACF and the PACF.
Correlation refers to the connection between two variables or qualities. Suppose we have two characteristics to deal with: weight and BMI. If we plot them in a scatterplot, we can observe that the
BMI increases with each addition of weight. Then we may conclude that weight and BMI are connected or have a high correlation.
We assess this association using the Pearson association Factor, which ranges from -1 to 1. A value close to 1 indicates a strong positive connection, whereas a value near -1 indicates a negative
positive correlation.
But in Time Series Analysis, we frequently have to work with a single feature. We observe previous data to identify patterns and then utilize those patterns to estimate what will happen in the
future. And now comes ACF and PACF. These two words reflect the correlation between the values of a single characteristic, whereas the correlation is between two features.
Let us now take a closer look at ACF. Assume we are dealing with a stock price dataset. The correlation between the current stock price and the past stock price is referred to as ACF. ACF indicates
how strongly they are connected with one another.
But what if the correlation between two data points at different periods is altered by additional data points? Here comes PACF as a rescuer. Let me explain with an example.
Assume t, t-1, and t-2 are the stock prices from today, yesterday, and the day before yesterday, respectively. Now, t can be associated with t-2, as can t-1. The PACF of t-1 is the true correlation
between t and t-1 after removing the impact of t-2.
Application of ACF and PACF
Selecting the ideal model in Machine Learning is a time-consuming process. Though we must use the trial and error technique to determine the optimal model, it would be preferable if we could
beforehand predict which model would perform best with our unique dataset.
And here come ACF and PACF as saviors. They are mostly used to select between the auto-regressive (AR) and moving average (MA) models. ACF and PACF not only assist us in selecting the model but also
indicate which lagged value will perform best.
Use Cases of ACF and PACF
We need to understand when to use ACF and PACF while working with them.
• ACF: If we use the Moving Average Model, we will compute our lagged value using ACF. The ACF plot will include a Horizontal Threshold Line that represents the degree of significance. The vertical
lines that cross this horizontal line form a meaningful relationship and should be used.
• PACF: PACF will be utilized to determine the lag value of the Auto Regressive Model. The selection process is the same as the ACF.
Now we will plot ACF and PACF on Ice Cream Production
Importing Libraries
Reading the Dataset
EDA (Exploratory Data Analysis)
Plot the Data
Now we will plot the data that ice creams were produced over the period of time.
ACF Plot
Here are things that we need to observe from the above plot:
• To observe the long-term effect, we will have to set the "lags" parameter to a higher value.
• The "Blue shade area" is called the "Error Band". Anything inside the error band isn't statistically significant.
PACF Plot
Here are things that we need to observe from the above plot:
• There's a strong lag at 1. As it's just the time series with itself that's why it will always be 1 2.
• Based on PACF, we can build an Auto-Regressive Model with lagged values 1, 2, 3, 8, and 13.
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Multiplication instructions
Opcode P/U Category Description
DSL user ALU: multiply double shift left
MH user ALU: multiply multiply high
MHL user ALU: multiply multiply high and low
MHL0 user ALU: multiply multiply high and low, tribble 0
MHL1 user ALU: multiply multiply high and low, tribble 1
MHL2 user ALU: multiply multiply high and low, tribble 2
MHL3 user ALU: multiply multiply high and low, tribble 3
MHL4 user ALU: multiply multiply high and low, tribble 4
MHL5 user ALU: multiply multiply high and low, tribble 5
MHNS user ALU: multiply multiply high no shift
ML user ALU: multiply multiply low
DSLDouble shift left
Register Signedness
All ignored
1 opcode only
Flag Set if and only if
N bit 35 of the result is set
Z all result bits are zero
T flag does not change
R flag does not change
DSL (double shift left) is a critical instruction for long multiplication, providing in one CPU instruction what would otherwise take four instructions. Sample code is available under 36-bit
DSL adds the T flag with wrapping to b, and then shifts the sum left six bits. The six vacated bits are filled using the six leftmost bits of a. The result is written to c.
N and Z are set as if the destination is a signed register. The N and Z flags have no purpose in the long multiplication application for which DSL was designed, but I chose to update them in case
someone invents a use for this information at a later date. T and R do not change.
This documentation does not match what the dissertation says about DSL, in that the left and right operands have since been interchanged. This switch was made so that DSL can directly obtain the
correct register copy after an MHL5 instruction in long multiplication.
MHMultiply high
Register Signedness
All ignored
1 opcode only
Flag Set if and only if
N never; flag is cleared
Z c = 0
T c mod 64 ≠ 0
R T is set or R is already set
This is a key instruction for unsigned “short” multiplication where one of the factors fits into six bits, and the product fits into 36 bits. The smaller of the factors must be copied into all of the
tribbles via CX or an assembler constant. MH multiplies the tribbles of a and b pairwise, but the six 12-bit results cannot fit the 6-bit spaces afforded by the tribbles of c. Instead, MH retains
only the six most significant bits of each 12-bit result. ML is the complementary instruction that retains the six least significant bits of each.
To meaningfully add the output of MH and ML, their place values must be aligned consistently, meaning that MH needs a 6-position left shift, and that the result can spill to as many as 42 bits (which
will not fit in a 36-bit register) as a result of that shift. The solution is that instead of shift, MH rotates its result six bits left. If the six bits rotated into the rightmost places are not all
zeros, the T and R flags are set because the eventual product will not fit in 36 bits. Otherwise T is cleared, R is left unchanged, and the output of MH can be directly added to ML to obtain the
36-bit product. Z will be set if the output of MH is all zeros. N is always cleared.
Here is an unsigned short multiplication example with full range checking, and an always-accurate Z flag at the end whether or not overflow occurs. Four instructions are needed. When multiplying by a
small constant, the CX can be optimized out by hand.
unsigned big small t result ; will multiply big * small
t = cx small ; copy small into all tribbles
result = big mh t ; high bits of product
t = big ml t ; low bits of product
result = result + t ; result is now big * small
Like other macros, CX is not yet available. Although it may be tempting to use SWIZ in place of CX like this:
t = small swiz 0
SWIZ will not check to verify small is between 0 and 63. CX will have this verification and set T and R if small is out of range.
Note about replacing MH with MHL
The MHL family of instructions cannot improve over the performance of MH and ML for short multiplication, because MH is able to include a 6-bit shift that MHL and its derivatives cannot. (The issue
is that only the beta RAMs can shift six bits, and only the gamma RAMs can split registers.) The MHL family is for long multiplication.
MHLMultiply high and low
Register Signedness
All ignored
1 opcode only
MHL is the flagship of the MHL family of instructions for long multiplication and is the most flexible, although ordinarily the MHL0 through MHL5 instructions are used instead. MHL is a simultaneous
execution of MHNS and ML, where the left and right copies of register c are allowed to desynchronize. Specifically, the MHNS result is stored in the left copy of register c, and the ML result is
stored in the right copy.
For a drawing that shows the two copies of the register file in relation to the architecture, see page 200 of the dissertation. A short discussion of register splitting can be found on page 187;
however, that discussion assumes the presence of a fast hardware multiplier that stores an entire multiplication result in a split register. MHL stores a partial result.
MHL multiplies the tribbles of a and b pairwise. In order to fit the six 12-bit results into c, the six most significant bits of each product are written to the left copy of c, and the six least
significant bits of each product are written to the right copy of c. These writes are done simultaneously. To preclude any semantic confusion as to whether flags follow the left or right copy of a
result, none of the MHL instructions change any CPU flags at all.
The differing values in the left and right copies of c can be selected in subsequent instructions by using the left and right operand positions of subsequent ALU instructions. Certain ALU
instructions such as shifts are not symmetric in their left and right operands, so very unusual code may require an intervening instruction to transfer a value from one copy of the register file to
the other copy. There are also two instructions that you probably don’t need to worry about after MHL, namely BOUND and WCM, where the syntactic left operand is actually the electrically right
operand and vice versa. Also, most assignment instructions place the electrically left operand on the right side of the = sign.
MHL0Multiply high and low, tribble 0
Register Signedness
All ignored
1 opcode only
MHL0 replicates tribble 0 (bits 0–5) of a across all six subwords, and then multiplies them pairwise with the tribbles of b. In order to fit the six 12-bit results into c, the six most significant
bits of each product are written to the left copy of c, and the six least significant bits of each product are written to the right copy of c. No flags are changed. See also MHL.
Except that only one instruction is required for MHL0, it is equivalent to:
t = a swiz 000000000000`o
c = t mhl b
The MHL0–MHL5 instructions can be seen in action under 36-bit multiplication.
MHL1Multiply high and low, tribble 1
Register Signedness
All ignored
1 opcode only
MHL1 replicates tribble 1 (bits 6–11) of a across all six subwords, and then multiplies them pairwise with the tribbles of b. In order to fit the six 12-bit results into c, the six most significant
bits of each product are written to the left copy of c, and the six least significant bits of each product are written to the right copy of c. No flags are changed. See also MHL.
Except that only one instruction is required for MHL1, it is equivalent to:
t = a swiz 010101010101`o
c = t mhl b
The MHL0–MHL5 instructions can be seen in action under 36-bit multiplication.
MHL2Multiply high and low, tribble 2
Register Signedness
All ignored
1 opcode only
MHL2 replicates tribble 2 (bits 12–17) of a across all six subwords, and then multiplies them pairwise with the tribbles of b. In order to fit the six 12-bit results into c, the six most significant
bits of each product are written to the left copy of c, and the six least significant bits of each product are written to the right copy of c. No flags are changed. See also MHL.
Except that only one instruction is required for MHL2, it is equivalent to:
t = a swiz 020202020202`o
c = t mhl b
The MHL0–MHL5 instructions can be seen in action under 36-bit multiplication.
MHL3Multiply high and low, tribble 3
Register Signedness
All ignored
1 opcode only
MHL3 replicates tribble 3 (bits 18–23) of a across all six subwords, and then multiplies them pairwise with the tribbles of b. In order to fit the six 12-bit results into c, the six most significant
bits of each product are written to the left copy of c, and the six least significant bits of each product are written to the right copy of c. No flags are changed. See also MHL.
Except that only one instruction is required for MHL3, it is equivalent to:
t = a swiz 030303030303`o
c = t mhl b
The MHL0–MHL5 instructions can be seen in action under 36-bit multiplication.
MHL4Multiply high and low, tribble 4
Register Signedness
All ignored
1 opcode only
MHL4 replicates tribble 4 (bits 24–29) of a across all six subwords, and then multiplies them pairwise with the tribbles of b. In order to fit the six 12-bit results into c, the six most significant
bits of each product are written to the left copy of c, and the six least significant bits of each product are written to the right copy of c. No flags are changed. See also MHL.
Except that only one instruction is required for MHL4, it is equivalent to:
t = a swiz 040404040404`o
c = t mhl b
The MHL0–MHL5 instructions can be seen in action under 36-bit multiplication.
MHL5Multiply high and low, tribble 5
Register Signedness
All ignored
1 opcode only
MHL5 replicates tribble 5 (bits 30–35) of a across all six subwords, and then multiplies them pairwise with the tribbles of b. In order to fit the six 12-bit results into c, the six most significant
bits of each product are written to the left copy of c, and the six least significant bits of each product are written to the right copy of c. No flags are changed. See also MHL.
Except that only one instruction is required for MHL5, it is equivalent to:
t = a swiz 050505050505`o
c = t mhl b
The MHL0–MHL5 instructions can be seen in action under 36-bit multiplication.
MHNSMultiply high no shift
Register Signedness
All ignored
1 opcode only
Flag Set if and only if
N never; flag is cleared
Z all result bits are zero
T flag does not change
R flag does not change
MHNS is a former key instruction for unsigned long multiplication, where two 36-bit factors are multiplied as 6-bit tribbles and eventually sum to produce a 72-bit result. MHNS multiplies the
tribbles of a and b pairwise, but the six 12-bit results cannot fit the 6-bit spaces afforded by the tribbles of c. Instead, MHNS retains only the six most significant bits of each result. The
tribbles are output in their original positions, instead of being rotated left as with MH. The Z flag is set if the outcome of MHNS is all zeros, and cleared otherwise. N is always cleared, and T and
R do not change.
MHNS has been supplanted by the MHL family of instructions, allowing the number of instructions required for long multiplication to be reduced from 47 to 35. But the MHL opcodes require a little more
hardware and firmware loader support, due to their register splitting. Architectures derived from Dauug|36 which either do not split registers or have reduced ALU memory may benefit from using MHNS
to multiply. For sample code showing how this used to be done, see page 113 of the dissertation.
MLMultiply low
Register Signedness
All ignored
1 opcode only
Flag Set if and only if
N never; flag is cleared
Z all result bits are zero
T flag does not change
R flag does not change
ML is a key instruction for unsigned “short” multiplication where one of the factors fits into six bits, and the product fits into 36 bits. It was also a key instruction for unsigned long
multiplication until being supplanted by the MHL family.
The smaller of the factors must be copied into all of the tribbles via CX or an assembler constant. ML multiplies the tribbles of a and b pairwise, but the six 12-bit results cannot fit the 6-bit
spaces afforded by the tribbles of c. Instead, ML retains only the six least significant bits of each 12-bit result. The Z flag is set if the outcome of ML is all zeros, and cleared otherwise. N is
always cleared, and T and R do not change. See MH and MHNS for more information and sample code. | {"url":"https://dauug.cs.wright.edu/User_instructions/ALU_multiply","timestamp":"2024-11-07T07:17:30Z","content_type":"application/xhtml+xml","content_length":"29600","record_id":"<urn:uuid:9894ec9a-9b10-43a2-8f00-f8540105512b>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00723.warc.gz"} |
Solving Multi- Step Equations. And we don’t know “Y” either!! - ppt download
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(PDF) Bespoke geometric glazing design for building energy performance analysis
... From the four path- ways explored, the NMT pathway using Topolog- icEnergy was able to model and handle complex ge- ometry and produce reliable results, while benefit- ting from the advantages of
NMT. As shown in figure 6, the workflow consisted of (a) modelling the exter- nal envelope and the glazing design on a flat surface, (b) mapping the glazing unto the curved wall, (c) sub- dividing
and planarizing the wall and mapped glaz- ing into a set of wall panels and windows, (d) slicing the building into multiple stories, and finally (e) send- ing the model to OpenStudio/EnergyPlus for
energy analysis ( Wardhana et al. 2018) Modelling the external envelope and the glazing design on a flat surface. We start by creating a series of circular wires that are then lofted into a surface. | {"url":"https://www.researchgate.net/publication/327891457_Bespoke_geometric_glazing_design_for_building_energy_performance_analysis","timestamp":"2024-11-07T16:30:28Z","content_type":"text/html","content_length":"578631","record_id":"<urn:uuid:6065d657-f027-4b0d-8a2f-c10b88766468>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00592.warc.gz"} |
maglance.com -
Today we will talk about Baccarat Grind Formula. Which is a smart way of making money Better than using our own cents. Walking around money depending on the mood. This formula was invented since
1965. It’s been a long time, but people still use it today. The origin or history of this formula has disappeared. We have to try it. I’ll explain.
Explain how to use the formula.
Grind is a simple payment system. It is used in the game Baccarat because Baccarat has a form interest. The chances of it happening are equal. In addition, the returns on both sides if we bet are
very close. (Banker side wins 0.95 times, Player side wins 1 time)
1. Let the bet count as a round (Cycle) . The round will end. Must have a profit equal to 1 unit only, more than 2, 3… units. For example, I bet 5 baht per unit (1 unit) if I win in the eye. first I
will get about 5 baht (1 unit) back. which means that I have finished the round.
Each betting round should have. The value per unit is the same. For example, 1 unit equals 5 baht so keep doing this until you stop playing.
2. If the bet wins, we will increase the betting unit 1 unit at a time, but if it is found that if added then the next turn Found that if winning, it will get more than 1 profit, don’t need to add
more units.
3. If bet and lose, let us bet the same number of units.
Analysis of Baccarat Grind Formula (Oscar’s Grind)
The events that cause the formula to work
1. Some wins, some loses, alternately. But overall, winning must be a little more.
2. Consecutive wins, e.g. 3 times in a row, 2 times in a row
3. Win a lot, rarely lose.
bad performance event
1. Losing some, winning some, alternating with each other, but overall, the loss has to be a little more.
2. Losing 3 times in a row, 2 times sometimes (not very negative)
3. Losing a lot really lucky (If found, it will be very negative)
Recipe overview
After trying this formula Overview after playing a large set The result is a steady profit that goes on slowly. Gradually increase by 1 unit, but if it’s lost, it’s not a lot of loss. It’s considered
less. Anyone who wants to earn a large sum of money at once may not be able to rely on this formula. And don’t let too many negatives. I recommend setting a limit per day. If today you feel that you
have lost more than this amount of xxx baht, stop playing the next day.
Formula Risks
This formula is a low risk formula (Risk 2/5). If you interested membership with us UFABET | {"url":"https://maglance.com/football/baccarat-grind-formula-smart-ways-to-make-money-with-the-oscar-grind-system/","timestamp":"2024-11-04T16:59:57Z","content_type":"text/html","content_length":"38765","record_id":"<urn:uuid:06ba83ff-7b0f-4d6b-ac07-19dbdf9deaad>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00374.warc.gz"} |
In mathematics, the
of a subset
of a totally or partially ordered set
is the least element of
that is
greater than or equal to
all elements of
. Consequently, the supremum is also referred to as the
least upper bound
. If the supremum exists, it is unique, meaning that there will be only one supremum. If
contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to
. For instance, the negative real numbers do not have a greatest element, and their supremum is 0 .
The above text is a snippet from Wikipedia: Supremum
and as such is available under the Creative Commons Attribution/Share-Alike License.
1. (of a subset) the least element of the containing set that is greater or equal to all elements of the subset. The supremum may or may not be a member of the subset.
The above text is a snippet from Wiktionary: supremum
and as such is available under the Creative Commons Attribution/Share-Alike License.
Need help with a clue?
Try your search in the crossword dictionary! | {"url":"https://www.crosswordnexus.com/word/SUPREMUM","timestamp":"2024-11-04T00:59:47Z","content_type":"application/xhtml+xml","content_length":"10527","record_id":"<urn:uuid:e5d4c929-fd99-4954-8d29-b42052eb2a24>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00305.warc.gz"} |
Regression case study part one: New Zealand's greenhouse gas emissions
Python Statistics and data science
31 January, 2021
Check out part two here.
This is the first of two posts describing the development and evaluation of a regression model in Python. In this first section, I'll describe the data itself—data on New Zealand's greenhouse gas
emissions from 1990 to 2016—and in the second section, we'll look at how to construct a simple linear regression model. The second post in this series delves into multiple regression, and more
advanced ways to evaluate model fit and specification.
Part one: the data
This project uses the New Zealand’s greenhouse gas emissions 1990–2016 dataset. The data were sourced from the Greenhouse Gas Inventory of New Zealand's Ministry for the Environment (MfE), and were
released as part of MfE's Environment Aotearoa 2019 report. The Greenhouse Gas Inventory is produced by MfE every year, and forms part of New Zealand's reporting obligations under the United Nations
Framework Convention on Climate Change and the Kyoto Protocol.
The dataset contains measurements of gases added to the atmosphere by human activities in New Zealand—that is, excluding natural sources of greenhouse gases, such as volcanic emissions and biological
processes—from 1990 to 2016. The data were aggregated into sector-specific categories, and comprise 306 distinct numerical measurements of greenhouse gas emissions, spread across 17 years and five
main sectors. The entire dataset is used for this project.
The data were obtained experimentally, by measuring the amount of greenhouse gas emissions that different sectors in New Zealand produce. The data were collected from 1990 to 2016, across New Zealand
and by different reporting bodies.
While MfE compiles the Greenhouse Gas Inventory, the responsibility for reporting greenhouse gas emissions falls upon different government agencies, according to sector:
• The Ministry for the Environment reports on industrial processes and other product use, waste and land use and forestry;
• Energy emissions are reported by the Ministry of Business, Innovation and Employment;
• and Agriculture emissions are reported by the Ministry for Primary Industries.
Finally, MfE provide further guidance on the measurement and reporting of greenhouse gas emissions in New Zealand.
The dataset comprises four variables:
• Gas (type of gas emitted; categorical variable with primary levels of CO[2] (carbon dioxide), CH[4] (methane) and N[2]O (nitrogen dioxide). The sum of the measured gases by sector was also
reported as All gases.
• Source (source of gas emissions by sector). Emissions were grouped into five major sectors according to the process that generates the emissions. The sectors are Agriculture, Energy, Industrial
processes, Waste and Land use;
• Year (1990 to 2016);
• and carbon dioxide equivalent units.
The data were obtained in a wide format with the "Year" variable spread across columns. The numerical measure of greenhouse gas emissions is expressed in carbon dioxide equivalent (CO
-e) units, which measures how much global warming a given type and amount of greenhouse gas causes. This is used to enable us to compare the effect of different types of greenhouse gas in a
consistent way, so that we can compare the relative contribution of different gases to the greenhouse effect, regardless of the actual amounts emitted. The values ranged from less than –30,000 CO
-e to over 80,000 CO
-e units. Negative values of CO
-e were attributed to the
Land use
sector due to the carbon sequestration effects of forestry. The emission sector and the year were treated as potential explanatory variables, while the CO
-e unit values were treated as the response variable.
Part two: simple regression
First steps with Python
Before we construct our model, let's have a quick look at the data, which were downloaded on 2 February 2020.
import pandas as pd
import seaborn as sns
import statsmodels.formula.api as smf
dat = (
var_name='Year', value_name='Units')
'Year': 'int32',
'Gas': 'category',
'Source': 'category'
We've used pandas.melt to pivot the data to a long format, giving us Gas, Source and Year explanatory variables. Our Units variable contains the measured CO[2]-e units for each observation.
One thing you might notice straight away is that there is a negative number in the Units field. This occurs when Source is equal to "Land use, land-use change and forestry". Also known by its
abbreviation of LULUCF, this category represents both emissions and removal of emissions from the atmosphere due to land-use changes. In New Zealand, for this period, the values are negative,
indicating that LULUCF was a net remover of greenhouse gas emissions, rather than producer.
Keeping that in mind, I thought it'd be a good idea to whip up a quick plot showing the change of net greenhouse gas emissions (i.e., including LULUCF and the role of forestry in removing emissions
from the atmosphere) compared to gross greenhouse gas emissions (ignoring removal) over the period of the data we have available:
summary = dat.query("Gas == 'All gases' & Source.isin(['All sources, Net (with LULUCF)', 'All sources, Gross (excluding LULUCF)'])")
summary.Source = summary.Source.cat.remove_unused_categories()
sns.lineplot(data=summary, x='Year', y='Units', hue='Source')
It's clear that LULUCF really does play an important role in mediating New Zealand's greenhouse gas emissions. But back to the point of this post, which is to demonstrate simple regression with
Python—let's construct a model that uses Year as the explanatory variable, and net emissions as the response variable. Basically, we'll see if that trend that we see with our own two eyes—emissions
increasing over time—is statistically significant in a linear regression model.
I should add at this point that, of course, Year is absolutely not a real explanatory variable. Arbitrarily changing the year shouldn't change anything about greenhouse gas emissions. Instead, Year
will be correlated with the true explanatory variables, like increases in number of cars on the road, increases in number of dairy farms, etc.
The model
Actually creating a simple linear regression model is really straightforward. We use the statsmodels.formula.api function ols (for ordinary least-squares), and fit the model on the data for net
greenhouse gas emission units:
net_summary = summary[summary.Source == "All sources, Net (with LULUCF)"]
reg = smf.ols('Units ~ Year', data=net_summary).fit()
Printing the output of the model gives us a bunch of useful (and at first confusing) information:
• Model and Method remind us that we've made an ordinary least-squares model.
• The R-squared value of 0.791 tells us that about 79% of the variance in greenhouse gases can be explained by the Year variable. (Let's be careful here again -- Year doesn't actually have any real
relationship with greenhouse gas emissions, it's completely arbitrary, right?)
• The F-statistic of 94.88 tells us that our model, taking into account the Year variable, fits better than a model with only the intercept (i.e., no explanatory variables). I'm going to defer to
Jim's explanation of the F-statistic in the context of regression models here, if you'd like a more in-depth explanation.
• The Prob (F-statistic) value of 5.44 x 10 tells us that our F-statistic is almost certainly not so high by chance, and therefore that our model (showing Year as a good predictor of greenhouse gas
emissions) is statistically significant.
Underneath these statistics is the coefficient table, with two entries: Intercept and Year.
• The coef (coefficient) of 894 for the Year variable tells us that each year tends to see an increase in greenhouse gas emissions of about 894 units. This is important because the whole point of
running this regression analysis is to see whether the coefficient of our explanatory variable(s) is different from zero—i.e., has an effect on the response variable—and of course, whether this
difference is due to chance or not.
• The t-statistic is simply the coefficient divided by the standard error. I personally find it difficult to interpret the t-statistic in a regression context, particularly coming from a background
of using t-tests to compare two population means. There is a very math-heavy explanation here, and quite a good overview of all the outputs of a regression model, including the t-statistic, here.
However, just looking at its definition—the coefficient divided by the standard error—we can maybe think intuitively here. If the standard error is nearly as large as the coefficient, then the
coefficient probably isn't going to be that useful, as the explanatory variable (Year for us) isn't going to be able to predict the response variable without a large amount of error. But really,
we can skip over all this and just look at the p-value:
• The P>|t| value of 0.000 (or, I suppose, less than 0.001) tells us that the t-statistic is statistically significant (there is a less than 0.1% chance that we would have collected these data if
the null hypothesis were true, that Year has no effect on greenhouse gas emissions).
• Finally, we can see the two-tailed 95% confidence interval values for the coefficient ([0.025 and 0.975, respectively).
And that's basically it for our first model. Our fit suggests that Year is a pretty good predictor of net greenhouse gas emissions, with emissions tending to increase by between around 700 to 1100
CO2-[e] units per year!
Next, we're going to look at multiple regression and evaluating model fit.
You can find all the code used in this post here.
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Warshall Algorithm
Graphs are one of the most important topics from an interview perspective. Questions related to graphs are frequently asked in interviews of all the major product-based companies. The Floyd Warshall
algorithm is for finding the shortest path between all the pairs of vertices in a weighted graph; the algorithm works for both directed and undirected graphs. This algorithm is asked directly in an
interview, or the interviewer may give you a real-life situation and will ask you to solve it using graphs.
Before moving on to the algorithm, it is recommended to study Graphs first. This article will describe Floyd Warshall's algorithm with a detailed explanation and code in Java.
Floyd Warshall Algorithm
Before moving to the jargon of the algorithm, let's understand the problem statement with a real-life example.
Imagine that you(A) and four of your friends (B, C, D, and E) live in different cities/locations. The distance between each of these cities is known. You want to know the optimal path, i.e., the
least distance between each of your friend's places. The problem can be solved by representing each of your friend's places and your place by vertices of directed and weighted graphs, as shown below.
The weights between the edges are the distance between vertices of the graph.
Using the Floyd Warshall algorithm, the above problem can be solved very easily. Floyd Warshall will tell you the optimal distance between each pair of friends and will also tell you that the
quickest path from friend D to A’s house is via B.
Floyd Warshall algorithm is used to find the shortest path between all the vertices of a directed or undirected weighted graph with no negative cycles.. It is also known as Floyd's algorithm, the
Roy-warshall algorithm, the Roy-Floyd algorithm.
This algorithm follows the dynamic programming approach to find the shortest paths. The algorithm doesn't construct the path itself, but it can reconstruct it with a simple modification. | {"url":"https://www.naukri.com/code360/library/floyd-warshall-algorithm","timestamp":"2024-11-12T02:01:48Z","content_type":"text/html","content_length":"436771","record_id":"<urn:uuid:f287091e-7ee7-4edb-a71d-d34bcc87d672>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00795.warc.gz"} |
Lesson 8
Rising and Falling
8.1: Notice and Wonder: A Bouncing Curve (5 minutes)
The purpose of this warm-up is to elicit the idea that some graphs have repeated behavior, which will be useful when students graph periodic motion in a later activity. While students may notice and
wonder many things about the image, the repeating pattern of the outputs of the graph is the important discussion point.
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss
the things they notice and wonder with their partner, followed by a whole-class discussion.
Student Facing
What do you notice? What do you wonder?
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses
have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification,
or point out contradicting information.
If the repetitive behavior of the graph does not come up during the conversation, ask students to discuss this idea. If none of the students notice or wonder anything about repeating, here are some
possible prompts for discussion:
• “How would you describe the graph to someone who cannot see it?”
• “Are there any parts of this graph that happen more than once?”
• “Does anyone notice anything that repeats?”
8.2: What is Happening? (15 minutes)
In this activity, students examine relationships shown in graphs. They consider different types of events that cause repeated outputs (MP2). They examine one situation which is periodic (the distance
of a car from the start line as it goes around a racetrack) and two situations which are not periodic, but do rise and fall at regular intervals. The long-term goal in this unit is to develop
mathematical tools which can help model these complex phenomena.
Monitor for students who connect the repetitive nature of these graphs to the context of the clock hands or Ferris wheel from earlier lessons.
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2 of 3 to complete. Chunking this task into more manageable parts may also
support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Here are some relationships that produce graphs that have a repetitive nature. For each situation, describe the dependent and independent variable. How does the dependent variable change? What might
cause this change?
1. This is the graph of the distance of a race car from the starting line as it goes around a track.
2. This is the graph of the temperature in a city in Australia over 21 days.
3. This is the graph of two populations over time.
Student Facing
Are you ready for more?
A ladybug is resting at the tip of a clock’s minute hand, which is 1 foot long. When it is 12:15, the ladybug is 10 feet above the ground.
1. Calculate how far above the ground the ladybug is at 12:00, 12:30, 12:45, and 1:00.
2. Estimate how far above the ground the ladybug is at 12:10, 12:20, and 12:40.
3. Plot the distances of the ladybug from the ground from 12:00 to 1:00.
4. If the ladybug stays on the minute hand, predict how its distance from the ground will change over the next hour (from 1:00 to 2:00). What about from 2:00 to 3:00?
Anticipated Misconceptions
Some students may be unsure how to read the axes for all but the first relationship, as they look different than what students have seen so far. Encourage these students to focus on a specific
coordinate on a graph to help them make sense of the axis labels as a way to get started.
Activity Synthesis
Begin the discussion by selecting previously identified students to share their observations about connections between the repeating nature of these graphs and the clock hands or Ferris wheel
contexts from earlier lessons. If no groups make this connection, invite students to do so now—for example, by comparing the period of the race car to the period of the minute hand on a clock or by
contrasting how the height of a point on a Ferris wheel always moves between the same high and low while the city high and low temperatures change from day to day.
Display the graphs for all students to see. Here are some questions for discussion:
• “What similarities do you see between the graphs representing these situations?” (There are highs and lows in all the graphs, and these highs and lows come at pretty regular intervals.)
• “Do these graphs all represent functions? How can you tell?” (Yes, in these situations for each time, there is a single value.)
• “What do you think it means for a function to be periodic?” (The output values follow a repeating pattern.)
• “What other words could you use to describe this type of behavior?” (repeating, cyclic)
Conclude the discussion by reminding students that previously, we named functions whose values repeat at regular intervals periodic functions. The graph of the race track is an example of what a
periodic function can look like. The other two situations shown, temperature and population over time, are not quite periodic functions since the output values are not the same over and over again,
but they have a definite periodic pattern to them.
8.3: Card Sort: Graphs of Functions (15 minutes)
The purpose of this activity is for students to compare different function types with a focus on periodic and non-periodic functions. The card sort allows students to compare a variety of graphs,
helping students to construct their understanding of what the graphs of periodic functions can look like in preparation for future lessons that focus on the graphs of cosine and sine.
Monitor for different ways groups choose to categorize the graphs, but especially for categories that distinguish between function types.
Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the graphs, they should work with their partner to come
up with categories. Select 2–3 groups to share the categories they chose, then instruct students to choose new categories and sort their cards again.
Conversing: MLR8 Discussion Supports. In their groups of 2, students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all
to see: “_____ and _____ make a category because . . .”, and “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree. This will help students clarify
their reasoning of different function types that are and are not periodic.
Design Principle(s): Support sense-making; Maximize meta-awareness
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. As students work with their partner, display sentence frames to support conversation such as:
“I noticed _____ so I . . .”, “Why did you . . . ?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Your teacher will give you a set of cards that show graphs.
1. Sort the cards into categories of your choosing. Be prepared to describe your categories.
Pause for a whole-class discussion.
2. Sort the cards into new categories in a different way. Be prepared to describe your new categories.
Activity Synthesis
Select groups to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes
between function types (for example, linear, quadratic, exponential, periodic). Attend to the language that students use to describe their categories and graphs, giving them opportunities to describe
their graphs more precisely. Highlight the use of terms like linear, exponential, quadratic, and periodic.
Lesson Synthesis
Tell students that any function that repeats its values at regular intervals is called a periodic function, not just functions about moving in circles like clock hands. Invite students to suggest
situations that would have some kind of periodic graph like the examples discussed in the lesson. Ask, “What kinds of things cause values that repeat periodically?” (Height of the end of a clock hand
over time, distance of the end of a clock hand from the center of the clock over time, sound waves, energy, or water use over the course of a year.) If time allows, invite students to sketch graphs
to illustrate their thinking about the periodic nature of a situation.
8.4: Cool-down - Measuring the Hours (5 minutes)
Student Facing
Many familiar things go through cycles:
• The sun (and the moon) rise and fall regularly each day.
• The seasons repeat regularly every year.
• Tides come in and go out at regular intervals.
Some of these kinds of events are measurable, and we can create functions to model and study them. For example, imagine there is a spot of paint on a bike tire with a 26 inch diameter. We can measure
the height of the spot above the ground as the bike moves. Here is what a table and graph of the relationship between the distance the bike has traveled and the height of the spot would look like:
│ distance bike │height of paint │
│travels (inches) │ spot (inches) │
│0 │0 │
│10 │3.66 │
│20 │12.58 │
│30 │21.74 │
│40 │25.97 │
│50 │22.90 │
│60 │14.26 │
│70 │4.90 │
│80 │0.11 │
│90 │2.57 │
│100 │10.91 │
We can make some observations about the situation using the graph. For example, the height of the paint spot will never be less than 0 or greater than 26. This makes sense for the situation since the
tire will not go below ground, and it can only reach 26 inches high, the wheel’s diameter. The height of the paint spot will go up and down in a repeating pattern as the tire rotates. This kind of
function is called periodic because it represents something that happens over a certain interval and then repeats. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/3/6/8/index.html","timestamp":"2024-11-04T12:09:04Z","content_type":"text/html","content_length":"112194","record_id":"<urn:uuid:7baec045-4a3f-41d7-aab2-aadbedc0b9e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00163.warc.gz"} |
Brain Trigger #38 - Sample Number puzzle question - Math Shortcut Tricks
Almost every competitive exams have Number puzzle quetions. Here we will discuss few of them. Do these type of Number puzzle questions. These will help you for your competitive exams. Your math
skills is very much needed to solve this kind of problems. You can also apply your learning skills of Shortcut Tricks. | {"url":"https://www.math-shortcut-tricks.com/brain-trigger-38-sample-number-puzzle-question/","timestamp":"2024-11-06T18:40:25Z","content_type":"text/html","content_length":"199892","record_id":"<urn:uuid:677b135e-6093-47f7-b6a5-95c211fce1b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00864.warc.gz"} |
Rate of Return - Explained
What is a Rate of Return?
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What is a Rate of Return?
Rate of Return (RoR) refers to the net profit or loss of an investment over a period expressed as a proportion of the original cost of an investment. Profits are the income realized from the sale of
an investment plus the capital gains. A loss is described as negative returns when the amount invested is greater than zero. Rate of return is usually expressed as a percentage.The Rate of return can
be calculated on any investment or asset like vehicle, real estate, shares, stocks among others as long as the asset was bought at one point and sold in future.
Calculating Rate of Return
The simple rate of return is also called growth rate of return on investment (ROI). Taking into consideration the time, value of money and inflation effects, the rate of returns can also be expressed
as the net amount of discounted cash flow realized from an investment once the adjustment of inflation is done. Rate of return can be considered for over a single period of time. This can be for any
length of time. However, the period can instead be divided into sub-periods. In this case, there is more than one period. The end of one sub-period marks the beginning of another sub-period. Where we
have multiple connecting sub groups, the rate of return of the whole period can be calculated by adding the returns of each sub-period. The formula below is used to calculate the rate of return:
Formula for Rate of Return
Rate of return (RoR) = (Current value Initial Value) 100 Initial value Example: By taking an example of buying a home so as to understand how rate of return can be calculated. Lets say you purchase a
house for $365,000 in cash. Years later, you plan to sell your house and you are able to sell the house at $492,750 after deducting the real estate agents fee and commission plus any tax. The rate of
return will be: Current price - $492,750 Initial Price - $365,000 (492,750-365,000)/365,000 x 100 = 35% Now lets say for instance you sold the house at a less price than the amount you used to
purchase it, for instance at a cost of $292,000, the same formula is used to calculate the rate of return which in this case will be a loss or negative return. RoR = (292,000 365,000)/365,000 X 100=
Rate of Returns for stocks and Bonds
When calculating the rate of returns for stocks and bonds are a little different. Assuming an investor purchases a stock for $75 per share, stays with the stock for five years and receives a total
dividend of $10. Later, the investor sells the stock for $95 per share, this means that the gain earned per share is $95 - $75=$20. He has also earned $10 the dividend income making the gain to be
$20+$10= $30. The rate of return for the stock is therefore $30 per share. This is divided by $75 which is the initial cost. This makes 0.4, multiplied by 100 to make 40%. In case 2, consider an
investor who pays $2,500 for 5% bond. The investment makes $100 interest income yearly. Lets say the investor sells the bond after two years for $3,000 premium value earning $500 plus $200 total
interest. The rate of return in this case will be $500 gin plus $200 interest income, divided by the initial cost of $2,500, resulting into 28%. ($3,000 - $2,500) + $200 X 100 = 28% $2,500
Real versus Nominal Rate of Return
The simple rate of return discussed above of buying a house is known as nominal rate of return. This is because it does not put into account the inflation effect over time. Inflation lowers the
buying power of money. The value we talked about of $492,750 will never be the same 6 years later. This is due to inflation effect. In cases where the effect of inflation is considered, the rate of
return is known as real rate of return.
Rate of return versus Compound annual growth rate (CAGR)
CAGR factors in the growth over several periods unlike simple rate of return. CAGR is the average rate of return of an investment over a stated period of time which is longer than one year. Compound
annual growth rate is also known as Annualized Rate of Return. While rate of return expresses the loss or profit of an investment over a random period of time, annualized RoR or CAGR describes the
return of an investment over every year. To calculate compound annual growth rate, its the value of an investment at the end of the period, divided by the value at the start of that period, raise
result to the power of one, divided by the number of years and minus one from the result. This is also written as CAGR = (Value of an investment at the end) 1/no.of years 1 Beginning value of an
investment Lets use an example to fully understand the difference. We will use the above case of a simple rate of return where an investor purchased a house at $365,000 and sold it $492,750. To
calculate the CAGR, it will be: [(492,750/365,000)1/6 -1] x 100 = 5% per year Someone would just calculate the simple rate of return, which was 35% and divide it by six years. This would be = 5.83%.
There is a difference because CAGR level returns such that they are the same each year then compounds them.
Discounted cash flow
It is a method used to value an investment based its cash flow in future. It takes the proceeds from an investment and discounts each of the cash flow according to the discount rate. Discount rate
represents the lowest rate of return an investor can accept.
Internal Rate of Return (IRR)
IRR is the interest that makes the net present value of all the cash flows from a particular investment equals to zero. It is used to assess how attractive a project or an investment is in terms of
profitability. If the IRR of a new project goes below the desired rate of return, the project is rejected. However, if it exceeds the required rate, the new project is embraced. IRR is an important
tool in a company that plans to take multiple new development projects. The management is able to know the projected return of each project.
Uses of Rate of Return
• Rate of returns can be used to make investment decisions.
• Financial analysts use rate of return to compare the performance of a company over a specified period of time.
• Used to compare the performance between companies.
• Companies use it compare internal rate of returns of different projects and decide which project to pursue and which one will bring more returns in the company. | {"url":"https://thebusinessprofessor.com/business-personal-finance-valuation/rate-of-return-definition","timestamp":"2024-11-09T01:11:37Z","content_type":"text/html","content_length":"101429","record_id":"<urn:uuid:b8c91323-52da-4a16-b618-f8999071cde2>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00288.warc.gz"} |
Regression model (Time Series) (RTS) | Jason Siu
Applied forecasting
This section builds up:
• How to make a regression forecast
• Metrics to decide good regression model
• Correlation, causation and forecasting behind Regression
Multiple regression and forecasting
• yt is the variable we want to predict: the “response” variable
• Each xj,t is numerical and is called a “predictor”. They are usually assumed to be known for all past and future times.
• The coefficients β1, . . . , βk measure the effect of each predictor after taking account of the effect of all other predictors in the model. I.E., the coefficients measure the marginal effects
Some useful predictors for linear models
Here we introduce 4 ways of dealing with predictors. They are Dummy Variable, Fourieer, Intervention variable, and distributed lag.
Dummy variables
Uses of dummy variables
Fourier series
We know how to add dummy variables to the regression model, but the problem is that there will be too many terms (
). For example,
• To find quarterly data, we set , then we get 3 dummy variables.
• To find yearly data, we set , then we get 11 dummy variables.
• To find hourly data, we set , then we get 23 dummy variables.
• To find weekly data, we set , then we get 51 dummy variables.
• To find daily data, we set to get the annual seasonality, then we get 364 dummy variables.
Which is way too many.
Solution: Fourier series - (i.e., harmonic regression)
We need fewer predictors than with dummy variables, especially when
is large.
• Particularly useful for weekly data, for example, where m≈52. For short seasonal periods (e.g., quarterly data), there is little advantage in using Fourier terms over seasonal dummy variables.
What is K
K is the parameter we set to specify how many pairs of sin and cos terms to include.
e.g., if we set model(TSLM(Beer ~ trend() + fourier(K = 2))) , then that means the model has 2 pairs of sin and cos
What if only the first two Fourier terms are used ( and )?
The seasonal pattern will follow a simple sine wave.
A regression model containing Fourier terms is often called a harmonic regression because the successive Fourier terms represent harmonics of the first two Fourier terms.
Example1: Harmonic regression: beer production
recent_production %>%
f1 = TSLM(Beer ~ trend() + fourier(K = 1)),
f2 = TSLM(Beer ~ trend() + fourier(K = 2)),
season = TSLM(Beer ~ trend() + season())
Example2: Harmonic regression: eating-out expenditure
Looking at the cafe data, it shows a seasonality.
R Code for Fourier term
fit <- aus_cafe %>%
K1 = TSLM(log(Turnover) ~ trend() + fourier(K = 1)),
K2 = TSLM(log(Turnover) ~ trend() + fourier(K = 2)),
K3 = TSLM(log(Turnover) ~ trend() + fourier(K = 3)),
K4 = TSLM(log(Turnover) ~ trend() + fourier(K = 4)),
K5 = TSLM(log(Turnover) ~ trend() + fourier(K = 5)),
K6 = TSLM(log(Turnover) ~ trend() + fourier(K = 6))
This is a yearly dataset, so , we can set k up to 6, having up to 6 pairs of cos and sin.
Interpretation of the plot
• When , it's only going to be 11 coefficients. This is because the sin and cos for the last coefficient is redundant. You've got the sine of pi times t is equal to 0.
• When , the seasonality is actually just a sine wave.
• When , it gets a smaller AICc, and my pattern is now starting to a bit more complicated by adding in one more harmonic.
• There is a tiny difference between and . The AICc of 6 is not as good as 5. So the best model is
Rule of thumb:
• As we add extra terms, the shape of the seasonality gets more complicated. It tries to match what's going on in the data so that's the genius of fourier terms!
• WE can model any type of periodic pattern by having enough terms
What is the benefit of Fourier term?
• To find quarterly data, we set , then we get 3 dummy variables.
• To find yearly data, we set , then we get 11 dummy variables.
• To find weekly data, we set , then we get 51 dummy variables.
We would not bother using quarterly data, as m is small. But when it comes to long periods of seasonality like
• monthly data, we can sometimes save one or two degrees of freedom.
• daily data, with an annual pattern, then , which is large. So, instead of having 364 coefficients, we might use 10, 12 or some small number of fourier terms.
• hourly data, we are modeling a time of day pattern, then , or we are modelling a half-hourly day pattern where , So, instead of having 23 or 47coefficients, we might use 3 or 8 fourier terms.
• weekly data, , because of the extra days after the end of the 52nd week. So, you can have a non-integer .
Can k be non-integer
Of course yes! For weekly data, , because of the extra days after the end of the 52nd week.
So, you can have a non-integer . All you're doing is computing things like 2 pi kt on m, you can do that when m is not an integer there's no problem and so you can handle non-integer seasonality.
Intervention variables
Example: ad expense to the sales
We regress sales on our usual variable ad expenditures and we created a dummy variable to capture whether the sales are taking place during the intervention or not.
We increased our ad expenditures during red framed period of time, for example start giving coupons.
Our dummy variable takes a value of 1 whenever the time period or sales are occurring in a period with our intervention.
The value of show us the effect of that particular intervention on the sales value as compared with the the omitted category that is without the intervention
As this example goes, we can have 3 cases:
Case 1: Spikes
We have the output variable against time.
• Our variable was having a slope of and then suddenly we intervene in the market in the mid-period where our ad expenditures went up. Suddenly, we saw that the value of the sales variable went up
as well and then the increase in the ad expenditures last until our sales went back to our normal sales.
• When we say normal, here we assume
• The slope changes during the intervention with which the dummy variable = 1.
Case 2: Steps
Slicing our data into parts, we will denote the time period before the intervention by 0 and the period with the “
” intervention denoted by 1.
Case 3: Change of slope
By increasing our ad expenditures, our happens before the intervention
After the intervention is much steeper .
1. Case 1 and case 2 are linear model, whereas case 3 is non-linear.
We can also make the dummy when variables there are 1) Outliers, 2) Holidays, 3) Trading days, and so on
Distributed lags
Say our and .
We don't necessarily expect when advertising goes up, sales go up straight away. It might take a little while before people actually make the sale, especially for expensive goods. So sometimes you'll
want to put in lagged variable of how much did we ever spend on advertising last month and how's that affecting sales this month.
Nonlinear trend
Residual diagnostics
For forecasting purposes, we require the following assumptions:
We use 2 types of Residual plots to
1. spot outliers.
2. decide whether the linear model was appropriate.
1. Scatterplot of residuals εt against each predictor xj,t
If a pattern is observed, there may be “heteroscedasticity” in the errors. This means that
1. the variance of the residuals may not be constant.
2. the relationship is nonlinear
2. Scatterplot residuals against the fitted values yˆt
If a plot of the residuals vs fitted values shows a pattern, then there is heteroscedasticity in the errors. (Could try a transformation.)
If this problem occurs, a transformation of the forecast variable such as a logarithm or square root may be required
Selecting predictors and forecast evaluation
Comparing regression models
We have AIC, , BIC, to compare the model.
It is not so useful because
1. does not allow for “degrees of freedom’ ’.
1. Adding any variable tends to increase the value of R^2, even if that variable is irrelevant.
ADJUSTED R^2
Solution to this is to use ADJUSTED R^2
Akaike’s Information Criterion (AIC)
where L is the likelihood and k is the number of predictors in the model.
• AIC penalizes terms more heavily than R^2.
• Minimizing the AIC is asymptotically equivalent to minimizing MSE via leave-one-out cross-validation (for any linear regression).
AIC has a caveat because when T (# obs) is too low, the AIC tends to select too many predictors.
Bayesian Information Criterion (SBIC / BIC / SC)
where L is the likelihood and k is the number of predictors in the model.
BIC penalizes terms more heavily than AIC.
Minimizing BIC is asymptotically equivalent to leave-v-out cross-validation when v = T[1 − 1/(log(T) − 1)].
• We don’t use BIC because it's not optimizing any predictive property of the mode
To calculate MSE, we use CV too!
Traditionally you've got training sets and test sets time series cross validation is where you fit lots of training sets and you're predicting one step ahead.
Leave-one-out cross-validation
We use all of the data, except for one on one points; we predict that one point.
Then, we use all of the data except for a different point to predict that point and so on.
So the test set is always one observation in the data, and we got t possible test sets, where t is the length of the data set.
What are the step of LOOCV?
1. Remove observation t from the data set, and fit the model using the remaining data. Then compute the error (e∗t=yt−^yt) for the omitted observation. (This is not the same as the residual because
the tth observation was not used in estimating the value of ^yt.)
2. Repeat step 1 for t=1,…,T.
3. Compute the MSE from e∗1,…,e∗T. We shall call this the CV.
Best Subset Selection
Fit all possible regression models using one or more of the predictors.
The overall best model is chosen from the remaining models.
The best model is chosen through cross-validation or some other method that chooses the model with the lowest measure of test error. ETC3550, we use CV, AIC, AICc
In this final step, a model cannot be chosen based on R-squared, because R-squared always increases when more predictors are added. The model with the lowest K-Fold CV test errors is the best model.
It is computationally expensive and as the # of predictors increases, the combination grows exponentially.
• For example, 44 predictors leads to 18 trillion possible models!
Backward selection (or backward elimination),
Start with a model containing all variables.
Try subtracting one variable at a time.
Keep the model if it has lower CV or AICc. Iterate until no further improvement.
In another words, Starts with all predictors in the model (full model), iteratively removes the least contributive predictors and stops when you have a model where all predictors are statistically
improvement is determined by metrics like RSS,CV or adjusted R square; ETC3550 uses CV or AICc.
1. Computational power is very similar to forwarding Selection.
2. Stepwise regression is not guaranteed to lead to the best possible model.
3. Inference on coefficients of final model will be wrong.
Forecasting with regression
When using regression models for time series data, we have the different types of forecasts that can be produced, depending on what is assumed to be known when the forecasts are computed.
A comparative evaluation of ex-ante forecasts and ex-post forecasts can help to separate out the sources of forecast uncertainty. This will show whether forecast errors have arisen due to poor
forecasts of the predictor or due to a poor forecasting model.
Ex-ante forecasts
Those that are made using only the information that is available in advance.
• For example, ex-ante forecasts for the percentage change in US consumption for quarters following the end of the sample, should only use information that was available up to and including 2019
• These are genuine forecasts, made in advance using whatever information is available at the time. Therefore in order to generate ex-ante forecasts, the model requires forecasts of the predictors.
• To obtain these we can use one of the simple methods introduced in Section 5.2 or more sophisticated pure time series approaches that follow in Chapters 8 and 9. Alternatively, forecasts from
some other source, such as a government agency, may be available and can be used.
Ex-post forecasts
Those that are made using later information on the predictors.
• For example, ex-post forecasts of consumption may use the actual observations of the predictors, once these have been observed.
• These are NOT genuine forecasts, but are useful for studying the behaviour of forecasting models.
• The model from which ex-post forecasts are produced should not be estimated using data from the forecast period.
• That is, ex-post forecasts can assume knowledge of the predictor variables (the xx variables), but should not assume knowledge of the data that are to be forecast (the yy variable).
Scenario based forecasting
• When we don’t know the information that is available in advance, we assumes possible scenarios for the predictor variables known in advance.
□ For example, a US policy maker may be interested in comparing the predicted change in consumption when there is a constant growth of 1% and 0.5% respectively for income and savings with no
change in the employment rate, versus a respective decline of 1% and 0.5%, for each of the four quarters following the end of the sample.
• Prediction intervals for scenario based forecasts do NOT include the uncertainty associated with the future values of the predictor variables.
□ The resulting forecasts are calculated below and shown in Figure 7.18.
R code for Scenario based forecasting
#1. make future_scenarios
future_scenarios <- scenarios(
Increase = new_data(us_change, 4) %>% ## we set up 4 peroids ahead
# assuming income increase 1%, Savings by .5%, Unemployment by 0%, Production by 0%)
mutate(Income = 1, Savings = 0.5, Unemployment = 0, Production = 0),
Decrease = new_data(us_change, 4) %>%
mutate(Income = -1, Savings = -0.5, Unemployment = 0, Production = 0),
names_to = "Scenario"
#2. Make predictions
fc <- forecast(fit_consBest, new_data = future_scenarios)
Building a predictive regression model
Correlation, causation and forecasting
Correlation is not causation
• When x is useful for predicting y, it is not necessarily causing y. e.g., predict number of drownings y using number of ice-creams sold x.
• Correlations are useful for forecasting, even when there is no causality.
• Better models usually involve causal relationships (e.g., temperature x and people z to predict drownings y)
It occurs when :
• Two predictors are highly correlated (i.e., the correlation between them is close to ±1).
• A linear combination of some of the predictors is highly correlated with another predictor.
• A linear combination of one subset of predictors is highly correlated with a linear combination of another subset of predictors
• Author:Jason Siu
• Copyright:All articles in this blog, except for special statements, adopt BY-NC-SA agreement. Please indicate the source! | {"url":"https://www.jason-siu.com/article/fbbf5c53-02c0-4a13-a445-c3cfb7eb7c5c","timestamp":"2024-11-10T03:14:40Z","content_type":"text/html","content_length":"487547","record_id":"<urn:uuid:f6b4daae-1233-4476-87dc-a2add139c696>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00314.warc.gz"} |
RE: Lie group E8
Exotic symmetry seen in electrons“An exotic type of symmetry - suggested by string theory and theories of high-energy particle physics, and also conjectured for electrons in solids under certain
conditions - has been observed experimentally for the first time.
An international team, led by scientists from Oxford University, report in a recent article in Science how they spotted the symmetry, termed E8, in the patterns formed by the magnetic spins in
crystals of the material cobalt niobate, cooled to near absolute zero and subject to a powerful applied magnetic field.
The material contains cobalt atoms arranged in long chains and each atom acts like a tiny bar magnet that can point either 'up' or 'down'.
When a magnetic field is applied at right angles to the aligned spin directions, the spins can 'quantum tunnel' between the 'up' and 'down' orientations. At a precise value of the applied field these
fluctuations 'melt' the ferromagnetic order of the material resulting in a 'quantum critical' state. ”Read more
Title:Authors:Read more
“The "exceptionally simple theory of everything," proposed by a surfing physicist in 2007, does not hold water, says Emory mathematician Skip Garibaldi.
Garibaldi, a rock climber in his spare time, did the math to disprove the theory, which involves a mysterious structure known as E8. The resulting paper, co-authored by physicist Jacques Distler of
the University of Texas, will appear in an upcoming issue of Communications in Mathematical Physics.”Read more
Title:Authors:Read more
Title: Spontaneous B-L Breaking as the Origin of the Hot Early Universe
Authors: Wilfried Buchmüller, Valerie Domcke, Kai Schmitz
The decay of a false vacuum of unbroken B-L symmetry is an intriguing and testable mechanism to generate the initial conditions of the hot early universe. If B-L is broken at the grand unification
scale, the false vacuum phase yields hybrid inflation, ending in tachyonic preheating. The dynamics of the B-L breaking Higgs field and thermal processes produce an abundance of heavy neutrinos whose
decays generate entropy, baryon asymmetry and gravitino dark matter. We study the phase transition for the full supersymmetric Abelian Higgs model. For the subsequent reheating process we give a
detailed time-resolved description of all particle abundances. The competition of cosmic expansion and entropy production leads to an intermediate period of constant 'reheating' temperature, during
which baryon asymmetry and dark matter are produced. Consistency of hybrid inflation, leptogenesis and gravitino dark matter implies relations between neutrino parameters and superparticle masses, in
particular a lower bound on the gravitino mass of 10 GeV.
Read more (1221kb, PDF)
Golden ratio discovered in a quantum world
Researchers from the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB), in cooperation with colleagues from Oxford and Bristol Universities, as well as the Rutherford Appleton Laboratory,
UK, have for the first time observed a nanoscale symmetry hidden in solid state matter. They have measured the signatures of a symmetry showing the same attributes as the golden ratio famous from art
and architecture. The research team is publishing these findings in Science on the 8. January.
On the atomic scale particles do not behave as we know it in the macro-atomic world. New properties emerge which are the result of an effect known as the Heisenberg's Uncertainty Principle. In order
to study these nanoscale quantum effects the researchers have focused on the magnetic material cobalt niobate. It consists of linked magnetic atoms, which form chains just like a very thin bar
magnet, but only one atom wide and are a useful model for describing ferromagnetism on the nanoscale in solid state matter.
When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal
The system reaches a quantum uncertain - or a Schrödinger cat state. This is what we did in our experiments with cobalt niobate. We have tuned the system exactly in order to turn it quantum critical
" - Prof. Alan Tennant, the leader of the Berlin group.
Read more
The Point of E_8 in F-theory GUTs
Jonathan J. Heckman, Alireza Tavanfar, Cumrun Vafa
We show that in F-theory GUTs, a natural explanation of flavour hierarchies in the quark and lepton sector requires a single point of E_8 enhancement in the internal geometry, from which all Yukawa
couplings originate. The monodromy group acting on the seven-brane configuration plays a key role in this analysis. Moreover, the E_8 structure automatically leads to the existence of the additional
fields and interactions needed for minimal gauge mediated supersymmetry breaking, and almost nothing else. Surprisingly, we find that in all but one Dirac neutrino scenario the messenger fields in
the gauge mediated supersymmetry breaking sector transform as vector-like pairs in the 10 + 10* of SU(5). We also classify dark matter candidates available from this enhancement point, and rule out
both annihilating and decaying dark matter scenarios as explanations for the recent experiments PAMELA, ATIC and FERMI. In F-theory GUT models, a 10-100 MeV mass gravitino remains as the prime
candidate for dark matter, thus suggesting an astrophysical origin for recent experimental signals.
Read more
(92kb, PDF)
Credit Garrett Lisi
Credit Garrett Lisi
F-theory GUTs
RE: Lie group E8 | {"url":"https://astronomy.activeboard.com/t10792179/lie-group-e8/","timestamp":"2024-11-07T03:29:45Z","content_type":"application/xhtml+xml","content_length":"96626","record_id":"<urn:uuid:3a6d517b-918a-4c3e-aaa4-a2feb55ce6ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00626.warc.gz"} |