content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
How To Graph Each Inequality - Graphworksheets.com
Graph Inequalities Number Line Worksheet – Line Graph Worksheets can help you develop your understanding of how a line graph works. There are many types of line graphs and each one has its own
purpose. Whether you’re teaching a child to read, draw, or interpret line graphs, we have worksheets for you. Make a line … Read more | {"url":"https://www.graphworksheets.com/tag/how-to-graph-each-inequality/","timestamp":"2024-11-02T21:37:14Z","content_type":"text/html","content_length":"47163","record_id":"<urn:uuid:5b6c976e-ce16-461f-a315-e3719ace3e01>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00171.warc.gz"} |
MPI_Dist_graph_neighbors_count - Returns the number of in and out edges for the calling processes in a distributed graph topology and a flag indicating whether the distributed graph is weighted.
C Syntax
#include <mpi.h>
int MPI_Dist_graph_neighbors_count(MPI_Comm comm, int *indegree,
int *outdegree, int *weighted)
Fortran Syntax
! or the older form: INCLUDE ’mpif.h’
INTEGER COMM, INDEGREE, OUTDEGREE, IERROR
Fortran 2008 Syntax
USE mpi_f08
MPI_Dist_graph_neighbors_count(comm, indegree, outdegree, weighted, ierror)
TYPE(MPI_Comm), INTENT(IN) :: comm
INTEGER, INTENT(IN) :: indegree, outdegree
INTEGER, INTENT(OUT) :: weighted
INTEGER, OPTIONAL, INTENT(OUT) :: ierror
comm Communicator with distributed graph topology (handle).
indegree Number of edges into this process (non-negative integer).
outdegree Number of edges out of this process (non-negative integer).
weighted False if MPI_UNWEIGHTED was supplied during creation, true otherwise (logical).
IERROR Fortran only: Error status (integer).
MPI_Dist_graph_neighbors_count and MPI_Graph_neighbors provide adjacency information for a distributed graph topology. MPI_Dist_graph_neighbors_count returns the number of sources and destinations
for the calling process.
Almost all MPI routines return an error value; C routines as the value of the function and Fortran routines in the last argument.
Before the error value is returned, the current MPI error handler is called. By default, this error handler aborts the MPI job, except for I/O function errors. The error handler may be changed with
MPI_Comm_set_errhandler; the predefined error handler MPI_ERRORS_RETURN may be used to cause error values to be returned. Note that MPI does not guarantee that an MPI program can continue past an | {"url":"http://man.m.sourcentral.org/debian-bookworm/3+MPI_Dist_graph_neighbors_count.openmpi","timestamp":"2024-11-12T13:28:32Z","content_type":"text/html","content_length":"23968","record_id":"<urn:uuid:632a90d2-f383-4045-acba-57bddb6d4e49>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00636.warc.gz"} |
Multiplication Worksheets By 2 And 3 - Printable Worksheets
Multiplication Worksheets By 2 And 3
Multiplication Worksheets By 2 And 3 are a fantastic tool for helping trainees practice and discover multiplication facts. These worksheets can be used in a range of ways, such as in the class, in
your home, or as part of a math curriculum.
Among the main benefits of using multiplication worksheets is that they supply trainees with a orderly and clear method to practice their multiplication truths. These worksheets typically consist of
a set of problems for trainees to fix, along with an area for them to compose their responses. This allows students to focus on the task at hand and not get distracted by other things going on around
Multiplication Worksheets By 2 And 3
Advantage of Multiplication Worksheets By 2 And 3
Another advantage of Multiplication Worksheets By 2 And 3 is that they can be easily personalized to satisfy the requirements of various trainees. Worksheets for younger trainees might consist of
bigger numbers and easier problems, while worksheets for older trainees may consist of smaller sized numbers and more complex problems. This enables teachers to customize their direction to the
particular requirements of their students.
Multiplication Worksheets By 2 And 3 also come in a wide variety of formats, such as several option, fill-in-the-blank, and matching. These different formats enable students to engage with the
product in different methods, which can help to keep them determined and interested.
In addition, Multiplication worksheets are likewise a fantastic way to supplement class instruction. They can be utilized as homework tasks, additional practice throughout math centers, or as part of
a review before a test. This permits trainees to continue practicing their multiplication truths outside of the class, which can assist to solidify their understanding of the material.
Multiplication Worksheets For Grade 3
In conclusion, Multiplication Worksheets By 2 And 3 are a efficient and flexible tool for assisting students find out and practice multiplication truths. They supply a clear and organized method for
students to practice, can be quickly tailored to satisfy the requirements of various trainees, and can be found in a wide array of formats to keep students engaged and motivated.
Multiplication times 2 And 3 Worksheet
Related Post to Multiplication Worksheets By 2 And 3 | {"url":"https://printablesworksheets.net/multiplication-worksheets-by-2-and-3/","timestamp":"2024-11-09T00:16:03Z","content_type":"text/html","content_length":"41872","record_id":"<urn:uuid:ba0bb548-390f-4537-855e-6aae6b49d23a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00439.warc.gz"} |
5,035 research outputs found
Our ability to resolve new physics effects is, largely, limited by the precision with which we calculate. The calculation of observables in the Standard (or a new physics) Model requires knowledge of
associated hadronic contributions. The precision of such calculations, and therefore our ability to leverage experiment, is typically limited by hadronic uncertainties. The only first-principles
method for calculating the nonperturbative, hadronic contributions is lattice QCD. Modern lattice calculations have controlled errors, are systematically improvable, and in some cases, are pushing
the sub-percent level of precision. I outline the role played by, highlight state of the art efforts in, and discuss possible future directions of lattice calculations in flavor physics.Comment:
Invited review of lattice QCD in quark and lepton flavor physics. Presentation at the DPF 2013 Meeting of the American Physical Society Division of Particles and Fields, Santa Cruz, California,
August 13-17, 201
I review recently completed (since Lattice 2013) and ongoing lattice calculations in charm and bottom flavor physics. A comparison of the precision of lattice and experiment is made using both
current experimental results and projected experimental precision in 2020. The combination of experiment and theory reveals several tensions between nature and the Standard Model. These tensions are
reviewed in light of recent lattice results.Comment: 18 pages, 9 figures; Review at The 32nd International Symposium on Lattice Field Theory, 23-28 June, 2014, Columbia University New York, NY; PoS
(LATTICE2014)002: Ver. 2 fixes several several typos, including labels in Fig. 3 and updates references, including the addition of recent results to Figs. 7 and
We report the first lattice QCD calculation of the form factors for the standard model tree-level decay $B_s\to K \ellu$. In combination with future measurement, this calculation will provide an
alternative exclusive semileptonic determination of $|V_{ub}|$. We compare our results with previous model calculations, make predictions for differential decay rates and branching fractions, and
predict the ratio of differential branching fractions between $B_s\to K\tauu$ and $B_s\to K\muu$. We also present standard model predictions for differential decay rate forward-backward asymmetries,
polarization fractions, and calculate potentially useful ratios of $B_s\to K$ form factors with those of the fictitious $B_s\to\eta_s$ decay. Our lattice simulations utilize NRQCD $b$ and HISQ light
quarks on a subset of the MILC Collaboration's $2+1$ asqtad gauge configurations, including two lattice spacings and a range of light quark masses.Comment: 24 pages, 21 figures; Ver. 2 matches
published versio
We discuss our ongoing effort to calculate form factors for several B and Bs semileptonic decays. We have recently completed the first unquenched calculation of the form factors for the rare decay B
-> K ll. Extrapolated over the full kinematic range of q^2 via model-independent z expansion, these form factor results allow us to calculate several Standard Model observables. We compare with
experiment (Belle, BABAR, CDF, and LHCb) where possible and make predictions elsewhere. We discuss preliminary results for Bs -> K l nu which, when combined with anticipated experimental results,
will provide an alternative exclusive determination of |Vub|. We are exploring the possibility of using ratios of form factors for this decay with those for the unphysical decay Bs -> eta_s as a
means of significantly reducing form factor errors. We are also studying B -> pi l nu, form factors for which are combined with experiment in the standard exclusive determination of |Vub|. Our
simulations use NRQCD heavy and HISQ light valence quarks on the MILC 2+1 dynamical asqtad configurations.Comment: 7 pages, 5 figures, presented at the 31st International Symposium on Lattice Field
Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, German
We calculate-for the first time in three-flavor lattice QCD-the hadronic matrix elements of all five local operators that contribute to neutral B-0- and B-s-meson mixing in and beyond the Standard
Model. We present a complete error budget for each matrix element and also provide the full set of correlations among the matrix elements. We also present the corresponding bag parameters and their
correlations, as well as specific combinations of the mixing matrix elements that enter the expression for the neutral B-meson width difference. We obtain the most precise determination to date of
the SU(3)-breaking ratio xi = 1.206(18)(6), where the second error stems from the omission of charm-sea quarks, while the first encompasses all other uncertainties. The threefold reduction in total
uncertainty, relative to the 2013 Flavor Lattice Averaging Group results, tightens the constraint from B mixing on the Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle. Our calculation employs
gauge-field ensembles generated by the MILC Collaboration with four lattice spacings and pion masses close to the physical value. We use the asqtad-improved staggered action for the light-valence
quarks and the Fermilab method for the bottom quark. We use heavy-light meson chiral perturbation theory modified to include lattice-spacing effects to extrapolate the five matrix elements to the
physical point. We combine our results with experimental measurements of the neutral B-meson oscillation frequencies to determine the CKM matrix elements vertical bar V-td vertical bar = 8.00(34)(8)
x 10(-3), vertical bar V-ts vertical bar = 39.0(1.2)(0.4) x 10(-3), and vertical bar V-td/V-ts vertical bar = 0.2052(31)(10), which differ from CKM-unitarity expectations by about 2 sigma. These
results and others from flavor-changing-neutral currents point towards an emerging tension between weak processes that are mediated at the loop and tree levels
We compute the form factors for the B -- \u3e Kl(+)l(-) semileptonic decay process in lattice QCD using gauge-field ensembles with 2 + 1 flavors of sea quark, generated by the MILC Collaboration. The
ensembles span lattice spacings from 0.12 to 0.045 fm and have multiple sea-quark masses to help control the chiral extrapolation. The asqtad improved staggered action is used for the light valence
and sea quarks, and the clover action with the Fermilab interpretation is used for the heavy b quark. We present results for the form factors f(+)(q(2)), f(0)(q2), and f(T)(q2), where q(2) is the
momentum transfer, together with a comprehensive examination of systematic errors. Lattice QCD determines the form factors for a limited range of q(2), and we use the model-independent z expansion to
cover the whole kinematically allowed range. We present our final form-factor results as coefficients of the z expansion and the correlations between them, where the errors on the coefficients
include statistical and all systematic uncertainties. We use this complete description of the form factors to test QCD predictions of the form factors at high and low q(2)
We present the first unquenched lattice-QCD calculation of the hadronic form factors for the exclusive decay (B) over bar -\u3e Dl (nu) over bar at nonzero recoil. We carry out numerical simulations
on 14 ensembles of gauge-field configurations generated with 2 + 1 flavors of asqtad-improved staggered sea quarks. The ensembles encompass a wide range of lattice spacings (approximately 0.045 to
0.12 fm) and ratios of light (up and down) to strange sea-quark masses ranging from 0.05 to 0.4. For the b and c valence quarks we use improved Wilson fermions with the Fermilab interpretation, while
for the light valence quarks we use asqtad-improved staggered fermions. We extrapolate our results to the physical point using rooted staggered heavy-light meson chiral perturbation theory. We then
parametrize the form factors and extend them to the full kinematic range using model-independent functions based on analyticity and unitarity. We present our final results for f + (q(2)) and f (0)(q
(2)), including statistical and systematic errors, as coefficients of a series in the variable z and the covariance matrix between these coefficients. We then fit the lattice formfactor data jointly
with the experimentally measured differential decay rate from BABAR to determine the CKM matrix element, vertical bar V-cb vertical bar = (39.6 +/- 1.7(QCD+exp) +/- 0.2(QED)) x 10(-3). As a byproduct
of the joint fit we obtain the form factors with improved precision at large recoil. Finally, we use them to update our calculation of the ratio R(D) in the Standard Model, which yields R(D) = 0.299
We present a lattice-QCD calculation of the B -\u3e pi l nu semileptonic form factors and a new determination of the CKM matrix element vertical bar V-ub vertical bar. We use the MILC asqtad (2 + 1)
-flavor lattice configurations at four lattice spacings and light-quark masses down to 1/20 of the physical strange-quark mass. We extrapolate the lattice form factors to the continuum using
staggered chiral perturbation theory in the hard-pion and SU (2) limits. We employ a model-independent z parametrization to extrapolate our lattice form factors from large-recoil momentum to the full
kinematic range. We introduce a new functional method to propagate information from the chiral-continuum extrapolation to the z expansion. We present our results together with a complete systematic
error budget, including a covariance matrix to enable the combination of our form factors with other lattice-QCD and experimental results. To obtain vertical bar V-ub vertical bar, we simultaneously
fit the experimental data for the B -\u3e pi l nu differential decay rate obtained by the BABAR and Belle collaborations together with our lattice form-factor results. We find vertical bar V-ub
vertical bar = (3.72 +/- 0.16) x 10(-3), where the error is from the combined fit to lattice plus experiments and includes all sources of uncertainty. Our form-factor results bring the QCD error on
vertical bar V-ub vertical bar to the same level as the experimental error. We also provide results for the B -\u3e pi l nu vector and scalar form factors obtained from the combined lattice and
experiment fit, which are more precisely determined than from our lattice-QCD calculation alone. These results can be used in other phenomenological applications and to test other approaches to QCD
One-loop corrections to the fermion rest mass M_1, wave function renormalization Z_2 and speed of light renormalization C_0 are presented for lattice actions that combine improved glue with clover or
D234 quark actions and keep the temporal and spatial lattice spacings, a_t and a_s, distinct. We explore a range of values for the anisotropy parameter \chi = a_s/a_t and treat both massive and
massless fermions.Comment: 45 LaTeX pages with 4 postscript figure | {"url":"https://core.ac.uk/search/?q=author%3A(Bouchard%2C%20C.%20M.)","timestamp":"2024-11-04T08:29:03Z","content_type":"text/html","content_length":"148642","record_id":"<urn:uuid:8199d0ad-47de-4320-bbba-1f7e38188fdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00171.warc.gz"} |
Intro to optimization in deep learning: Momentum, RMSProp and Adam | DigitalOcean
Optimization algorithms are very important while training any deep learning models by adjusting the model’s parameters to minimize the loss function. The most basic method, Stochastic Gradient
Descent (SGD), is widely used, but advanced techniques like Momentum, RMSProp, and Adam improve convergence speed and stability. These optimizers build upon SGD by adding mechanisms like adaptive
learning rates and momentum, making them more effective for complex models and large datasets.
• Basic Knowledge of Deep Learning: Understanding of neural networks, loss functions, and backpropagation.
• Gradient Descent: Familiarity with how gradient descent works and its limitations.
• Mathematics for Deep Learning: Basic calculus (derivatives) and linear algebra (vectors and matrices).
• Python Programming: Ability to implement neural networks using frameworks like PyTorch or TensorFlow.
Consider the following loss contour.
You see, we start off randomly before getting into the ravine-like region marked by blue color. The colors actually represent how high the value the loss function is at a particular point, with reds
representing highest values and blues representing the lowest values.
We want to get down to the minima, but for that we have move through the ravine. This region is what is called pathological curvature. To understand why it’s called pathological , let us delve
deeper. This is how pathological curvature, zoomed up, looks like…
It’s not very hard to get hang of what is going on in here. Gradient descent is bouncing along the ridges of the ravine, and moving a lot slower towards the minima. This is because the surface at the
ridge curves much more steeply in the direction of w1.
Consider a point A, on the surface of the ridge. We see, the gradient at the point can be decomposed into two components, one along direction w1 and other along w2. The component of the gradient in
direction of w1 is much larger because of the curvature of the loss function, and hence the direction of the gradient is much more towards w1, and not towards w2 (along which the minima lies).
Normally, we could use a slow learning rate to deal with this bouncing between the ridges problem as we covered in the last post on gradient descent. However, this spells trouble.
It makes sense to slow down when were are nearing a minima, and we want to converge into it. But consider the point where gradient descent enters the region of pathological curvature, and the sheer
distance to go until the minima. If we use a slower learning rate, it might take so too much time to get to the minima. In fact, one paper reports that learning rates small enough to prevent bouncing
around the ridges might lead the practitioner to believe that the loss isn’t improving at all, and abandon training all together.
And if the only directions of significant decrease in f are ones of low curvature, the optimization may become too slow to be practical and even appear to halt altogether, creating the false
impression of a local minimum
Probably we want something that can get us slowly into the flat region at the bottom of pathological curvature first, and then accelerate in the direction of minima. Second derivatives can help us do
Gradient descent is a First Order Optimization Method. It only takes the first order derivatives of the loss function into account and not the higher ones. What this basically means it has no clue
about the curvature of the loss function. It can tell whether the loss is declining and how fast, but cannot differentiate between whether the curve is a plane, curving upwards or curving downwards.
This happens because gradient descent only cares about the gradient, which is the same at the red point for all of the three curves above. The solution? Take into account double derivative, or the
rate of how quickly the gradient is changing.
A very popular technique that can use second order derivatives to fix our issue is called the Newton’s Method. For sake of not straying away from the topic of post, I won’t delve much into the math
of Newton’s method. What I’ll do instead is try to build an intuition of what Newton’s Method does.
Newton’s method can give us an ideal step size to move in the direction of the gradient. Since we now have information about the curvature of our loss surface, the step size can be accordingly chosen
to not overshoot the floor of the region with pathological curvature.
Newton’s Method does it by computing the Hessian Matrix, which is a matrix of the double derivatives of the loss function with respect of all combinations of the weights. What I mean by saying a
combination of the weights, is something like this.
A Hessian Matrix then accumulates all these gradients in one large big matrix.
The Hessian gives us an estimate of the curvature of loss surface at a point. A loss surface can have a positive curvature which means the surface, which means a surface is rapidly getting less
steeper as we move. If we have a negative curvature, it means that the surface is getting more steeper as we move.
Notice, if this step is negative, it means we can use a arbitrary step. In other words, we can just switch back to our original algorithm. This corresponds to the following case where the gradient is
getting more steeper.
However if the gradient is getting less steeper, we might be heading towards a region at the bottom of pathological curvature. Here, Newton’s algorithm gives us a revised learning step, which is, as
you can see is inversely proportional to the curvature, or how quickly the surface is getting less steeper.
If the surface is getting less steeper, then the learning step is decreased.
You see that Hessian Matrix in the formula? That hessian requires you to compute gradients of the loss function with respect to every combination of weights. If you know your combinations, that value
is of the order of the square of the number of weights present in the neural network.
For modern day architectures, the number of parameters may be in billions, and calculating a billion squared gradients makes it computationally intractable for us to use higher order optimization
However, here’s an idea. Second order optimization is about incorporating the information about how is the gradient changing itself. Though we cannot precisely compute this information, we can chose
to follow heuristics that guide our search for optima based upon the past behavior of gradient.
A very popular technique that is used along with SGD is called Momentum. Instead of using only the gradient of the current step to guide the search, momentum also accumulates the gradient of the past
steps to determine the direction to go. The equations of gradient descent are revised as follows.
The first equations has two parts. The first term is the gradient that is retained from previous iterations. This retained gradient is multiplied by a value called “Coefficient of Momentum” which is
the percentage of the gradient retained every iteration.
If we set the initial value for v to 0 and chose our coefficient as 0.9, the subsequent update equations would look like.
We see that the previous gradients are also included in subsequent updates, but the weightage of the most recent previous gradients is more than the less recent ones. (For the mathematically
inclined, we are taking an exponential average of the gradient steps)
How does this help our case. Consider the image,and notice that most of the gradient updates are in a zig-zag direction. Also notice that each gradient update has been resolved into components along
w1 and w2 directions. If we will individually sum these vectors up, their components along the direction w1 cancel out, while the component along the w2 direction is reinforced.
For an update, this adds to the component along w2, while zeroing out the component in w1 direction. This helps us move more quickly towards the minima. For this reason, momentum is also referred to
as a technique which dampens oscillations in our search.
It also builds speed, and quickens convergence, but you may want to use simulated annealing in case you overshoot the minima.
In practice, the coefficient of momentum is initialized at 0.5, and gradually annealed to 0.9 over multiple epochs.
RMSprop, or Root Mean Square Propogation has an interesting history. It was devised by the legendary Geoffrey Hinton, while suggesting a random idea during a Coursera class.
RMSProp also tries to dampen the oscillations, but in a different way than momentum. RMS prop also takes away the need to adjust learning rate, and does it automatically. More so, RMSProp choses a
different learning rate for each parameter.
In RMS prop, each update is done according to the equations described below. This update is done separately for each parameter.
So, let’s break down what is happening here.
In the first equation, we compute an exponential average of the square of the gradient. Since we do it separately for each parameter, gradient Gt here corresponds to the projection, or component of
the gradient along the direction represented by the parameter we are updating.
To do that, we multiply the exponential average computed till the last update with a hyperparameter, represented by the greek symbol nu. We then multiply the square of the current gradient with (1 -
nu). We then add them together to get the exponential average till the current time step.
The reason why we use exponential average is because as we saw, in the momentum example, it helps us weigh the more recent gradient updates more than the less recent ones. In fact, the name
“exponential” comes from the fact that the weightage of previous terms falls exponentially (the most recent term is weighted as p, the next one as squared of p, then cube of p, and so on.)
Notice our diagram denoting pathological curvature, the components of the gradients along w1 are much larger than the ones along w2. Since we are squaring and adding them, they don’t cancel out, and
the exponential average is large for w2 updates.
Then in the second equation, we decided our step size. We move in the direction of the gradient, but our step size is affected by the exponential average. We chose an initial learning rate eta, and
then divide it by the average. In our case, since the average of w1 is much much larger than w2, the learning step for w1 is much lesser than that of w2. Hence, this will help us avoid bouncing
between the ridges, and move towards the minima.
The third equation is just the update step. The hyperparameter p is generally chosen to be 0.9, but you might have to tune it. The epsilon is equation 2, is to ensure that we do not end up dividing
by zero, and is generally chosen to be 1e-10.
It’s also to be noted that RMSProp implicitly performs simulated annealing. Suppose if we are heading towards the minima, and we want to slow down so as to not to overshoot the minima. RMSProp
automatically will decrease the size of the gradient steps towards minima when the steps are too large (Large steps make us prone to overshooting)
So far, we’ve seen RMSProp and Momentum take contrasting approaches. While momentum accelerates our search in direction of minima, RMSProp impedes our search in direction of oscillations.
Adam or Adaptive Moment Optimization algorithms combines the heuristics of both Momentum and RMSProp. Here are the update equations.
Here, we compute the exponential average of the gradient as well as the squares of the gradient for each parameters (Eq 1, and Eq 2). To decide our learning step, we multiply our learning rate by
average of the gradient (as was the case with momentum) and divide it by the root mean square of the exponential average of square of gradients (as was the case with momentum) in equation 3. Then, we
add the update.
The hyperparameter beta1 is generally kept around 0.9 while beta_2 is kept at 0.99. Epsilon is chosen to be 1e-10 generally.
In this post, we have seen 3 methods to build upon gradient descent to combat the problem of pathological curvature, and speed up search at the same time. These methods are often called “Adaptive
Methods” since the learning step is adapted according to the topology of the contour.
Out of the above three, you may find momentum to be the most prevalent, despite Adam looking the most promising on paper. Empirical results have shown the all these algorithms can converge to
different optimal local minima given the same loss. However, SGD with momentum seems to find more flatter minima than Adam, while adaptive methods tend to converge quickly towards sharper minima.
Flatter minima generalize better than sharper ones.
Despite the fact that adaptive methods help us tame the unruly contours of a loss function of a deep net’s loss function, they are not enough, especially with networks becoming deeper and deeper
everyday. Along with choosing better optimization methods, considerable research is being put in coming up with architectures that produce smoother loss functions to start with. Batch Normalization
and Residual Connections are a part of that effort, and we’ll try to do a detailed blog post on them very shortly. But that’s it for this post. Feel free to ask questions in the comments.
This textbox defaults to using Markdown to format your answer.
You can type !ref in this text area to quickly search our full set of tutorials, documentation & marketplace offerings and insert the link! | {"url":"https://www.digitalocean.com/community/tutorials/intro-to-optimization-momentum-rmsprop-adam","timestamp":"2024-11-10T02:26:53Z","content_type":"text/html","content_length":"315396","record_id":"<urn:uuid:e42bb9ac-46a7-4a48-8e86-07dcc018dcd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00696.warc.gz"} |
Unit3ReadOnly - page 40 of 47
Statistical Design of Experiments
Statistical Design of Experiments
Statistical Design of Experiments
Statistical Design of Experiments
What can you conclude
What can you conclude
from the analysis of
from the analysis of
variance results?
variance results?
What would you do next?
What would you do next?
What levels of the 3
What levels of the 3
factors under study
factors under study
would you choose to
would you choose to
maximize the chemical
maximize the chemical
product produced
product produced
through the reaction?
through the reaction? | {"url":"https://www.6sigma.us/Unit3ReadOnly/doeunit340.html","timestamp":"2024-11-08T14:52:06Z","content_type":"text/html","content_length":"9417","record_id":"<urn:uuid:45e64bf6-84a9-411e-8b62-796a3b924464>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00193.warc.gz"} |
Difference NoConfusion and NoConfusionHom
Hi, I am trying to finish the tuto on dependent matching https://github.com/coq/platform-docs/pull/9 and I am some trouble understanding the difference between NoConfusion and NoConfusionHom.
From what I understand, NoConfusion is basically checking that the head constructors are the same. For vec, it basically can be written as:
Definition NoConfusion_vec'
(A : Type)
(x : sigma (fun index : nat => vec A index))
(y : sigma (fun index : nat => vec A index)) : Prop :=
match pr2 x, pr2 y with
| vnil, vnil => True
| @vcons _ a n v, @vcons _ a' n' v' =>
let x : sigma (fun a : A => (sigma (fun index : nat => vec A index))) :=
{| pr1 := a; pr2 := {| pr1 := n; pr2 := v |} |} in
let y := {| pr1 := a'; pr2 := {| pr1 := n'; pr2 := v' |} |} in
x = y
| _, _ => False
wheras NoConfusionHom seems to additionaly check as precondition that the indices are the same. I have some trouble writting sth that works but basically the printed term lloks like that
NoConfusionHom : ∀ (A : Type) (n : nat), vec A n → vec A n → Prop :=
fun A n v v' =>
match v in (vec _ n0) return (vec A n0 -> Prop) with
| vnil =>
fun y : vec A 0 =>
match y in (vec _ n0) return (n0 = 0 -> Prop) with
| vnil => apply_noConfusion 0 0 (fun _ : True => True)
| @vcons _ _ n0 _ => apply_noConfusion (S n0) 0 (False_rect Prop)
end eq_refl
| @vcons _ a n v =>
fun y : vec A (S n0) =>
match y in (vec _ n1) return (n1 = S n0 -> Prop) with
| vnil => apply_noConfusion 0 (S n0) (False_rect Prop)
| @vcons _ y2 n1 v0 =>
apply_noConfusion (S n1) (S n0) (fun H : n1 = n0 =>
DepElim.solution_left n0 (fun [...] => [...] = [...]) n1 H v0)
end eq_refl
end v'.
So basically, while matching for the head constructors, NoConfusion seems to additionally check whether the indices are the same or not
I have some trouble understanding what is the added value of that as its type is
∀ (A : Type) (n : nat), vec A n → vec A n → Prop
so it requires both vectors to be of the same length anyway
Does it uses axiomK somehow to simplify the vcons , voncs case to unify the indices and return sth simpler ?
NoConfusion works on vectors of different lengths (but of course it’s False outside the diagonal), while NoConfusionHom also uses NoConfusion on the indices. NoConfusionHom is only derivable if the
indices have UIP indeed. But not necessarily through an axiom, for example nat has UIP provably.
Actually you need UIP only if you need to discriminate indices, some inductive definitions might not need it (e.g only variables in index positions in all constructors)
I understand, but I don't get what it brings more the table ?
Do you have an example where it is actually needed ?
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237659-Equations-devs-.26-users/topic/Difference.20NoConfusion.20and.20NoConfusionHom.html","timestamp":"2024-11-10T19:08:35Z","content_type":"text/html","content_length":"17256","record_id":"<urn:uuid:f6c5aecc-3e85-4bff-9ee8-3feb39b1b6d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00681.warc.gz"} |
Algebra 1 Tutoring in Provo, UT | Hire the Best Tutors Now!
I have a bachelors degree in Economics from the University of Texas, a masters degree from SUNY Buffalo, and 10 years of experience in education. I've taught in 4 different countries around the world
on 4 different continents. I've taught all of the following courses: Pre-Algebra, Algebra 1, Geometry, Algebra 2, Pre-Calculus, Intro to Calculus, AP AB Calculus, and Economics. I enjoy seeing
students grow and watching the light bulb go off when they have that ... See more | {"url":"https://heytutor.com/tutors/algebra-1/ut/provo/","timestamp":"2024-11-07T23:15:18Z","content_type":"text/html","content_length":"198552","record_id":"<urn:uuid:ff10d0b5-8a4e-48b2-b46e-b23d9e3d7562>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00570.warc.gz"} |
Python Math Module
Python Math Module
Now, Let’s discuss a very commonly used module, which is the math module. As the name of the module suggests, it is used to perform some mathematical operations. We are going to look at some of the
different things that we can do using the math module. Let’s have a look at some of the insights into what we are going to cover.
• Methods from the math module
□ ceil method
□ factorial method
□ floor method
□ fabs method
□ pow method
□ sqrt method
• Constants from the math module
First of all, here is a list of some of the methods, that are available for us, with the math module.
Methods Description
ceil(x) This method returns the ceiling of x, which simply means the smallest integer, greater than or equal to x.
factorial(x) This method returns the factorial of x as an integer. It raises ValueError, if the x is not integral, or is negative.
floor(x) This method returns the floor of x, which simply means the largest integer less than or equal to x.
comb(k, n) This method returns the number of ways to choose the k items from n items, without repetition and without order.
fabs(x) This method returns the absolute value of x.
fsum(iterable) This method returns an accurate floating-point sum of values in the iterable.
gcd(*numbers) This method returns the greatest common divisor of the specified integer arguments.
isclose(a, b) This method returns True if the values a and b are close to each other, and False otherwise.
isfinite(x) As the name of the method says, this method returns True, if x is neither infinity nor a NaN, and False otherwise.
isinf(x) This method returns True, if x is a positive or negative infinity, and False otherwise.
isnan(x) This method returns True, if x is NaN (not a number), and False otherwise.
trunc(x) This method returns the integer with the fractional part removed and leaving the integer part.
log(x, base) This method returns the logarithm of x to the given base. If the base is not specified, it returns a log of x to the base e.
log2(x) This method returns the logarithm of x to base 2.
log10(x) This method returns the logarithm of x to the base 10.
pow(x, y) This method returns the result of x to the power of y.
sqrt(x) This method returns the square root of the value x.
sin(x) This method returns the sine of x radians.
cos(x) This method returns the cosine of x radians.
tan(x) This method returns the tangent of x radians.
There are many other methods from the math module, which you can simply explore. But here, we are going to consider some methods from the above-mentioned table, so that we can simply understand the
methods from the math module. | {"url":"https://gyanipandit.com/programming/python-math-module/","timestamp":"2024-11-09T09:17:59Z","content_type":"text/html","content_length":"134528","record_id":"<urn:uuid:92dde7cc-660e-41b3-8846-1246a485653b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00337.warc.gz"} |
Connected rates of change calculus.
I'm struggling with connected rates of change $\frac{dy}{dt} =\frac{dy}{dx}\times \frac{dx}{dt}$ I'll give you an example of the type of questions I'm struggling with and I'll explain what I don't
understand about the question.
Question: Variables x and y are connected by the equation
$y=x+\sqrt{x-5} $.
Given that x increases at a rate of 0.1 units per second. Find the rate of change of y when x = 9.
When I initially read this question, it feels like the question is asking me to do $y=(9+ \sqrt{9-5} )\times 0.1 = 1.1$ which is incorrect, so my first question is what is the question asking and how
does it have anything to do with a curve/tangent?
My second question is in my book, it tells me to solve the problem like this
$\frac{dy}{dx} = 1+\tfrac{1}{2\sqrt{x-5} } $
$\frac{dx}{dt} =0.1$
$1+\tfrac{1}{2\sqrt{9-5}}\times 0.1 =0.125$
why is $\tfrac{dx}{dt} = 0.1 $ because my understanding of differentiation is that for instance if you have $y=x^3 $ and then you find the derivative of y with respect to x what it's asking is the
gradient of y at any given value of x so shouldn't that mean that $\tfrac{dx}{dt}$ is asking for the value of x at any given value of t rather than an actual number if that makes sense.
If you don't understand exactly what I mean I'll try to explain further.
Answers can only be viewed under the following conditions:
1. The questioner was satisfied with and accepted the answer, or
2. The answer was evaluated as being 100% correct by the judge.
View the answer
Join Matchmaticians
Affiliate Marketing Program
to earn up to a 50% commission on every question that your affiliated users ask or answer.
• answered
• 1575 views
• $20.00 | {"url":"https://matchmaticians.com/questions/ko5ntx/ko5ntx-connected-rates-of-change-calculus-question","timestamp":"2024-11-09T17:29:04Z","content_type":"text/html","content_length":"85855","record_id":"<urn:uuid:f90d8c7d-a15a-4d3d-8578-48b96f453051>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00882.warc.gz"} |
There are many student clubs in the campus, among which the Always Creatively Moha (ACM) club gained the most popularity. The ACM club consists of $N$ members. Not all of them know each other, so we
may describe the relationship between two members with an integer ranging from 0 to $M$ inclusively. For example, 0 means that the two persons do not know each other. 1 may indicate that they know
each other but are not familiar. 520 indicates that they have fallen into love.
The members are partitioned into two non-empty groups to participate a team-building activity. To get them know each other, it is required that the value of relationship between every pair of members
in the same groups is always 0, while the relationship between persons from different groups can be any value between 0 and $M$.
It should be noted that such a partition is not always possible. For example, if there are 3 members in the club and the value of relationship between them is greater than 0, it is impossible to
partition them as required.
The organizer comes up with an interesting question. Assuming we may assign any integer between 0 and $M$ to the relationship between any two members. Among all the ways of assignment, how many of
them can make the partitioning possible?
The first line of input contains a number $T$ indicating the number of test cases ($T¡Ü100$).
Each test case consists of two integers $N$ and $M$ indicating the number of members and the maximum value of relationship ($1¡ÜN,M¡Ü1000$).
For each test case, output a single line consisting of ¡°Case #X: Y¡±. $X$ is the test case number starting from 1. $Y$ is the number of ways assigning value to relationships that makes the
partitioning possible. As the answer could be huge, you only need to output $Y$ module 105225319.
Case #1: 2
Case #2: 19
Case #3: 0 | {"url":"http://acm.hdu.edu.cn/showproblem.php?pid=5485","timestamp":"2024-11-02T01:10:22Z","content_type":"text/html","content_length":"9932","record_id":"<urn:uuid:140f3a06-5032-4882-ad3e-f0e0720aae4e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00864.warc.gz"} |
RANDBETWEEN in Excel | How to Generate Random Numbers in Excel?
Updated May 9, 2023
Excel RANDBETWEEN (Table of Contents)
Introduction to Excel RANDBETWEEN
When we speak about Excel, there is a function called RANDBETWEEN, which helps you to generate pseudo-random numbers between a given range of numbers. These are the random numbers predicted with some
mathematical algorithm in the backend. RANDBETWEEN is a volatile function in Excel. It will change the values generated every time we refresh/newly define the formula over the same range. Excel
RANDBETWEEN function generates a single random number at one iteration between a given range of numbers. For example, say, 10 random numbers between 1 to 6. This function, by default, generates only
integer random numbers.
The syntax for RANDBETWEEN Function
Arguments of RANDBETWEEN Function:
bottom: A required argument defining the smallest value the function could return.
Top: A required argument that defines the largest value that the function could return.
We will move towards an example and see how RANDBETWEEN can be used differently.
How to Generate Random Numbers in Excel?
Let’s understand how to generate the RANDBETWEEN in Excel with some examples.
Example #1 – RANDBETWEEN to Generate Random Numbers
Suppose you want to generate random numbers between 1 to 10. You can use RANDBETWEEN to do so.
Step 1: Start typing the RANDBETWEEN formula in cell A2.
Step 2: Enter the bottom number 1 and the top 10.
Step 3: Close the bracket and press Enter key to see the output.
A single random number between 1 to 10 is generated under cell A2.
Step 4: If you want to generate 10 random numbers between 1 to 10, drag the formula across the next 9 rows (until A10). See the screenshot below.
Example #2 – RANDBETWEEN to Generate Negative Numbers
We can also use negative numbers as an argument to generate random numbers. Meaning we can generate a random number between -5 and +5.
Step 1: Start typing the RANDBETWEEN formula in cell A2.
Step 2: Enter the bottom number as -5 and the top as 5.
Step 3: Complete the formula by closing parentheses and Press Enter to see the output.
You can also generate multiple random numbers between -5 to 5. See the screenshot below.
As said earlier, the function is volatile; you may see different results every time the formula gets refreshed, or the sheet is opened.
Example #3 – RANDBETWEEN to Generate Random Numbers with Decimals
RANDBETWEEN usually generates integer random numbers between the ranges. However, with some modifications under formula arguments, generating random numbers with decimals is possible.
Suppose we want to generate random numbers with one decimal after the integer part between 5 to 15. We will see how it can be done step by step.
Step 1: Start typing the RANDBETWEEN formula in cell A2.
Step 2: While giving the bottom and top arguments, multiply them with 10 and close the parentheses as we want the output as one decimal after the integers. See the screenshot below.
Step 3: Now, divide by 10 to the entire formula to get decimal output and press the Enter key.
You can generate multiple random numbers by copying the formula. If you want the data up to two decimal places, multiply the bottom and top values by 100. I want three decimals to multiply by 1000
and so on.
Example #4 – RANDBETWEEN and DATEVALUE to Generate Random Date
We can also generate random dates using a combination of the RANDBETWEEN and DATEVALUE functions in Excel. Follow the steps below:
Suppose we want to generate random dates between August 01, 2019, and August 27, 2019. Follow the steps below to do so.
Step 1: Select all the rows where you want the dates to appear randomly, and in the first cell, start typing the RANDBETWEEN formula in cell A1.
Step 2: Use DATEVALUE to input the bottom date as August 01, 2019, and the top date as August 28, 2019. Make sure the dates are in Excel compatible date format.
Step 3: Press CTRL + Enter to apply this formula under all selected cells and generate random dates.
Excel stores the date in its backend as several days from 01-01-1900. You can see these dates are in several formats. However, we need to convert these dates into a proper date format.
Step 4: Click the Home tab under the number formatting section, change the format to Long Date, and press Enter key.
You’ll be able to see the dates generated randomly between August 01, 2019, to August 14, 2019, as shown below.
This is from this article. Let’s wrap things up with some points to be remembered.
Things to Remember About RANDBETWEEN in Excel
• It generates random numbers between the bottom and top numbers.
• If the bottom (smallest value) exceeds the top (largest value), you will get a #NUM! Error in the RANDBETWEEN formula.
• It is a volatile function; therefore, every time the sheet gets refreshed/calculated, it will change the values for random numbers under a given range.
• To stop the formula from being volatile/calculating sheet every time, we need to press F9 when the RANDBETWEEN formula is completed under the formula bar. Or we can paste our results as values to
remove the threat of the sheet being calculated.
• This function can only generate Integer numbers by default. However, some amendments under the formula can also allow you to generate the decimal output.
Recommended Articles
This is a guide to RANDBETWEEN in Excel. Here we discuss How to Generate RANDBETWEEN in Excel, practical examples, and a downloadable Excel template. You can also go through our other suggested
articles – | {"url":"https://www.educba.com/randbetween-in-excel/","timestamp":"2024-11-11T06:41:55Z","content_type":"text/html","content_length":"352257","record_id":"<urn:uuid:c4c475dc-5c7c-4bc6-a167-85fa141c24a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00001.warc.gz"} |
How to solve an orienteering problem?
I have been using PROC OPTNET to solve a traveling salesman problem. Now I want to solve an orienteering problem, which is a selective traveling salesman problem. Each node has a prize and the
objective is maximize the total prize of the visited nodes within a time constraint. I cannot find any SAS documentation on this specific type of traveling salesman problem on how to solve it. Could
someone please direct me?
10-12-2015 12:06 PM | {"url":"https://communities.sas.com/t5/Mathematical-Optimization/How-to-solve-an-orienteering-problem/td-p/229558","timestamp":"2024-11-15T00:08:18Z","content_type":"text/html","content_length":"291723","record_id":"<urn:uuid:58886cb4-a1ba-45e5-a141-f274a3f12117>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00045.warc.gz"} |
THE UNITED REPUBLIC OF TANZANIA NATIONAL EXAMINATIONS COUNCIL OF TANZANIA
FORM TWO NATIONAL ASSESSMENT
041 BASIC MATHEMATICS
Time: 2:30 Hours YEAR 2023
1. This paper consists of ten (10) compulsory questions.
2. Show clearly all the working and answers in the space provided.
3. All writing must be in blue or black ink except drawings which must be in pencil.
4. Four figure mathematical tables, geometric instruments and graph papers may b used where necessary
5. All communication devices, calculators and any unauthorized materials are not allowed in the examination room
6. Write your examination Number at the top right corner of every page.
1.(a) List first twelve multiples of 4 and 5 and hence identify the common multiples
View Ans
(b) Evaluate
(i) One significant figure
(ii) Three decimal places
View Ans
2.(a) Arrange the given fractions in ascending order of magnitude:
View Ans
(b) In the year 2016 the population of Mericho village was 2,800. In 2017 the population increased by 8%. What was the population in 2017?
View Ans
3.(a) If 1,000 tonnes of maize were shared equally among 25 schools, how many kilograms did each school get.
View Ans
(b) A shopkeeper bought a radio for sh. 80,000/= and sold it at a profit of 20%. What were the profit and selling price?
View Ans
4. (a) If ABCD is a trapezium, find the values of angles marked a, b and c
View Ans
(b) The floor of a room is a square of length 5 metres. Find its perimeter and area
View Ans
5. (a) If
Find the value of y correct to three significant figures.
View Ans
(b) Solve the equation 3x^2 - 7x - 6 = 0
View Ans
6.(a) Find the gradient of a straight line joining the points (-1 , 2) and (3 , -5).
View Ans
(b) Find the image of point P(-3 , 7) after a reflection in the x-axis and y-axis.
View Ans
7.(a) Solve for n in the equation
Leave the answer in improper fraction.
View Ans
(b) Find the value of x in the equation log(2x^2 + 1) + log4 = log(7x^2 + 8)
View Ans
8.(a) In the given figure:
View Ans
(b) ABC is a triangle in which
View Ans
9.(a) The angle of elevation of the top of a building from a point on the ground is 25^0. If the point on the ground is 80m from the base of the building. Find the height of the building correct to
one decimal place.
View Ans
(b) Calculate the length of ???????? IN the following Figure:
View Ans
10.(a) At a certain school, 250 students attended on the first day of re-opening of the school, 350 students attended on the second day and 150 students attended on both first and second day. It was
further noted that 10 students were absent on both days. If all registered students were supposed to attend the school on both days, how many students does the school have? Do not use venn diagram.
View Ans
(b)The given Pie chart represents the number of students who passed a Qualifying Test in mathematics, Chemistry and Kiswahili
(i) The fraction of students who passed Kiswahili
(ii) The percentage of students who passed Mathematics.
(iii) The percentage of students who passed Mathematics and Kiswahili
View Ans | {"url":"https://learninghubtz.co.tz/form2-necta-qns-ans.php?sub=bWF0aGVtYXRpY3M%3D&yr=MjAyMw%3D%3D","timestamp":"2024-11-14T02:24:29Z","content_type":"text/html","content_length":"99281","record_id":"<urn:uuid:f8391ed5-a4b6-40d3-93d9-e2ba28504907>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00026.warc.gz"} |
Course Archives: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore
Course Archives Theoretical Statistics and Mathematics Unit
Course: Introduction to Stochastic Processes
Level: Undergraduate
Time: Currently not offered
Past Exams
DISCRETE-TIME MARTINGALES: Optional Stopping theorem, Martingale convergence theorem, Doobs inequality and convergence.
BRANCHING PROCESSES: Model definition. Connection with martingales. Probability of survival. Mean and variance of number of individuals.
DISCRETE-TIME MARKOV CHAINS: Classification of states, Stationary distribution, reversibility and convergence. Random walks and electrical networks. Collision and recurrence.
BASIC PROBABILISTIC INEQUALITIES AND APPLICATIONS: First and Second Moment methods. Applications to Longest increasing subsequences, Random k-Sat problem and connectivity threshold for
Erdos-Renyi graphs. Chernoff bounds and Johnson-Lindenstrauss lemma.
Reference Texts:
(a) N. Lanchier: Stochastic Modelling.
(b) W. Feller: Introduction to Probability: Theory and Applications - Vol. I and II..
(c) L. Levine, Y. Peres and E. Wilmer: Markov chains and mixing times.
(d) Sheldon Ross: Probability Models.
(e) Santosh S. Venkatesh: Theory of Probability - Explorations and Applications.
(f) R. Meester: A Natural Introduction to Probability Theory.
(g) S. R. Athreya and V. S. Sunder: Measure and Probability.
(h) Sebastien Roch: Modern Discrete Probability: A toolkit. (Notes).
Top of the page
Past Exams
Midterm 24.pdf
Semestral 24.pdf
Supplementary and Back Paper 24.pdf
Top of the page
[ Semester Schedule ][ Statmath Unit ] [Indian Statistical Institute] | {"url":"https://www.isibang.ac.in/~adean/infsys/database/Bmath/SP.html","timestamp":"2024-11-07T06:11:11Z","content_type":"text/html","content_length":"4651","record_id":"<urn:uuid:ce2bcb7f-c161-49a5-bc83-c873ce29c715>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00444.warc.gz"} |
Reduction of Order on Second Order Linear Homogenous Diff
Reduction of Order on Second Order Linear Homogenous Diff. Eqs.
Reduction of Order on Second Order Linear Homogenous Differential Equations
Recall from the Repeated Roots of The Characteristic Equation page that if we had a second order linear homogenous differential equations with constant coefficients, (that is a differential equation
in the form $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$) where $a, b, c \in \mathbb{R}^2$, and if the roots $r_1, r_2$ of the characteristic equation $ar^2 + br + c = 0$ where real repeated
roots, then a fundamental set of solutions could be constructed as:
\quad y = Ce^{r_1t} + Dte^{r_1t}
Recall that finding one solution, namely $y_1(t) = e^{r_1t}$ can relatively easy. In finding a second solution $y = y_2(t)$ to form a fundamental set of solutions, we assume that $y = v(t) y_1(t)$
was a solution to this differential equation and then solved for $v(t)$. This technique can be applied to more general second order linear homogenous differential equations that will allow us to, in
a sense, "convert" a second order linear differential equation to a first order linear differential equation which is often much more manageable to solve. This technique is known as Reduction of
Order for differential equations.
Consider the second order linear homogenous differential equation $\frac{d^2y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$, and suppose that $y = y_1(t)$ is a nonzero solution to this differential
equation, and assume that $y = v(t) y_1(t)$ is also a solution to this differential equation. The first and second derivatives of $y = v(t) y_1(t)$ are:
\quad y' = v(t)y_1'(t) + v'(t) y_1(t)
\quad y'' = v(t)y_1''(t) + v'(t)y_1'(t) + v'(t)y_1'(t) + v''(t) y_1(t) \\ \quad y'' = v(t)y_1''(t) + 2v'(t)y_1'(t) + v''(t)y_1(t)
If we plug $y = v(t) y_1(t)$, $y' = v(t)y_1'(t) + v'(t) y_1(t)$ and $y'' = v(t)y_1''(t) + 2v'(t)y_1'(t) + v''(t)y_1(t)$ into our differential equation, then we have that:
\quad \left [ v(t)y_1''(t) + 2v'(t)y_1'(t) + v''(t)y_1(t) \right ] + p(t) \left [ v(t)y_1'(t) + v'(t) y_1(t) \right ] + q(t) \left [ v(t) y_1(t) \right ] = 0 \\ \quad \underbrace{\left ( y_1''(t) + p
(t)y_1'(t) + q(t) y_1(t) \right )}_{=0} v(t) + \left ( 2y_1'(t) + p(t) y_1(t) \right ) v'(t) + y_1(t) v''(t) = 0 \\ \quad \left ( 2y_1'(t) + p(t) y_1(t) \right ) v'(t) + y_1(t) v''(t) = 0
The differential equation above is rather nice because we can solve it as though it were a first order differential equation for the function $v'$. | {"url":"http://mathonline.wikidot.com/reduction-of-order-on-second-order-linear-homogenous-differe","timestamp":"2024-11-13T19:06:34Z","content_type":"application/xhtml+xml","content_length":"16507","record_id":"<urn:uuid:6c80d664-d947-4815-895f-c2ca513fd7ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00164.warc.gz"} |
Circles - (Galois Theory) - Vocab, Definition, Explanations | Fiveable
from class:
Galois Theory
Circles are geometric figures defined as the set of all points in a plane that are equidistant from a fixed point called the center. This concept is crucial in understanding various geometric
constructions, particularly when it comes to defining constructible numbers and methods of drawing shapes using basic tools like a compass and straightedge. Circles can also help illustrate
relationships between different geometric figures, enabling a deeper understanding of angles, arcs, and chords.
congrats on reading the definition of Circles. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. A circle is defined by its center and radius, with every point on the circle being the same distance from the center.
2. Circles can be used in constructions to find points that create angles, bisectors, or perpendicular lines.
3. The intersection points of two circles can lead to finding solutions to various geometric problems and relationships.
4. In the context of constructible numbers, certain lengths associated with circles can be shown to be constructible, such as the diameter and radius.
5. The properties of circles allow for unique geometric relationships, such as tangents and secants, which are essential in more complex constructions.
Review Questions
• How do circles play a role in geometric constructions and the determination of constructible numbers?
□ Circles are fundamental in geometric constructions as they provide a method for creating various shapes and determining distances. When using a compass to draw a circle, you establish points
equidistant from a center, which can lead to finding lengths associated with constructible numbers. By intersecting circles or using their properties, you can derive other essential segments
and angles that further connect to constructible numbers.
• Discuss how the properties of circles influence the ability to perform geometric constructions using only a compass and straightedge.
□ The properties of circles significantly influence geometric constructions since they allow for precise measurements and relationships between points. For instance, when constructing angles or
bisectors, circles provide reference points that ensure accuracy. The ability to draw arcs and identify intersection points is crucial for establishing relationships between different
geometric figures while adhering to the constraints of using only a compass and straightedge.
• Evaluate the impact of circles on the understanding of mathematical concepts such as angles and arcs within geometric constructions.
□ Circles greatly enhance the understanding of mathematical concepts like angles and arcs by illustrating how these elements interact within geometric structures. For example, when analyzing
inscribed angles or central angles in relation to arcs, one can see the inherent relationships dictated by circle geometry. This understanding allows for deeper explorations into more complex
constructs while reinforcing foundational principles that apply across various mathematical scenarios.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/galois-theory/circles","timestamp":"2024-11-06T04:39:21Z","content_type":"text/html","content_length":"143563","record_id":"<urn:uuid:33c58dc0-d8af-4947-ac1f-c30eca86c8f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00629.warc.gz"} |
Brick Wall LeetCode Solution - Leetcode Solution - TO THE INNOVATION
Brick Wall LeetCode Solution
/ Leetcode Solution / Facebook, hash-table, LeetCode Solution, Medium
Last updated on October 5th, 2024 at 04:21 pm
Here, We see Brick Wall LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches.
Hash Table
Level of Question
Brick Wall LeetCode Solution
Table of Contents
Problem Statement
There is a rectangular brick wall in front of you with n rows of bricks. The i^th row has some number of bricks each of the same height (i.e., one unit) but they can be of different widths. The total
width of each row is the same.
Draw a vertical line from the top to the bottom and cross the least bricks. If your line goes through the edge of a brick, then the brick is not considered as crossed. You cannot draw a line just
along one of the two vertical edges of the wall, in which case the line will obviously cross no bricks.
Given the 2D array wall that contains the information about the wall, return the minimum number of crossed bricks after drawing such a vertical line.
Example 1:
Input: wall = [[1,2,2,1],[3,1,2],[1,3,2],[2,4],[3,1,2],[1,3,1,1]]
Output: 2
Example 2:
Input: wall = [[1],[1],[1]]
Output: 3
1. Brick Wall LeetCode Solution C++
class Solution {
int leastBricks(vector<vector<int>>& wall) {
unordered_map<int, int> edge_frequency;
int max_frequency = 0;
for(int row=0; row<wall.size(); row++)
int edge_postion = 0;
for(int brick_no=0; brick_no< wall[row].size() -1; brick_no++)
int current_brick_length = wall[row][brick_no];
edge_postion = edge_postion + current_brick_length ;
max_frequency = max(edge_frequency[edge_postion],max_frequency);
return wall.size() - max_frequency;
2. Brick Wall LeetCode Solution Java
class Solution {
public int leastBricks(List<List<Integer>> wall) {
var edge_frequency=new HashMap<Integer,Integer>();
int max_frequency = 0;
for(int row=0; row<wall.size(); row++)
int edge_postion = 0;
for(int brick_no=0; brick_no< wall.get(row).size() -1; brick_no++)
int current_brick_length = wall.get(row).get(brick_no);
edge_postion = edge_postion + current_brick_length ;
max_frequency = Math.max(edge_frequency.get(edge_postion),max_frequency);
return wall.size() - max_frequency;
3. Brick Wall LeetCode Solution JavaScript
var leastBricks = function(wall) {
let freq = new Map(), best = 0
for (let i = 0; i < wall.length; i++) {
let row = wall[i], rowSum = row[0]
for (let j = 1; j < row.length; j++) {
freq.set(rowSum, (freq.get(rowSum) || 0) + 1)
rowSum += row[j]
for (let [k,v] of freq)
if (v > best) best = v
return wall.length - best
4. Brick Wall LeetCode Solution Python
class Solution(object):
def leastBricks(self, wall):
d = collections.defaultdict(int)
for line in wall:
i = 0
for brick in line[:-1]:
i += brick
d[i] += 1
# print len(wall), d
return len(wall)-max(d.values()+[0])
Shortest Unsorted Continuous Subarray LeetCode Solution | {"url":"https://totheinnovation.com/brick-wall-leetcode-solution/","timestamp":"2024-11-02T11:47:13Z","content_type":"text/html","content_length":"199866","record_id":"<urn:uuid:daff4241-edd4-419a-8abc-f310d93aac6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00680.warc.gz"} |
Sudoku Samurai Printable - Management Tips
Sudoku Samurai Printable – Sudoku means number game. Players have to fill an empty board with the correct numbers. A typical format of the Sudoku worksheet is usually a 9 x 9 grid. The grids are
divided into 3 x 3 grids. All the boxes in the worksheet are 81 grids. The grids are pre-separated by a thick lining . Since the grid is divided into 3×3 grids, there are nine grids called blocks,
but there is another form of Sudoku, 16 Sudoku or Hexadoku. It refers to a Sudoku worksheet with a 16 x 16 grid divided into four blocks with each grid 4 x 4. The total number of grids is 256. The
divided grid consists of 16 grids.
2. Sudoku is a logical thinking game. Every time the player starts the game Part of the brain will be stimulated and trained. This will help strengthen the memory.
Sudoku Samurai Printable
4. When the reader is faced with a blank sheet that must be filled. The brain immediately finds the logical possibilities of the correct numbers to write. Then the brain immediately looks for the
pattern of the puzzle.
Print Sudoku Puzzles
6. Once the player has found the model They have to remember that every time they have to fill in the gaps. This activity forces them to keep things in their memory for a period of time. Therefore,
the memorization skills are fully formed.
8. Players need to know that each single table is connected. They have to understand that the small sections that we call “sub-tasks” have to be done first so that the players can understand the
pattern for the whole puzzle.
Due to the more challenging Sudoku game, there should be a lot of grids. So we need to know how to play the game:
1. The player must see the row, the column and the block sheet, so each cell has 16 cells.
Puzzle Solutions: Issue 1: Aug. 23, 2013
2. Start analyzing the rows and columns. Find unwritten numbers in tables, rows and empty blocks.
4. After filling the rows and columns Analyze if the cell is part of the block.
5. Make sure that the full number is not more than once in the whole table of rows, columns and blocks.
10Superhero Mazes Printable8Printable Expense Notes4Printable African American History10Printable Multiplication Table Grid10Check Register Full Page Printable10A Serenity Prayer Printable
VersionSamurai Sudoku is a logic puzzle aimed at filling a 9×9 grid with numbers
Samurai Sudoku 13 Grid
Each column, each row, and nine 3 × 3 sub-grids make up the grid. (also known as “box”, “block”, “region” or “sub-frame”) contains all nine digits at once.
A complete region can be recognized since it has a valid “candidate” number in each position.
The sum of these numbers along the path outside this area is equal to the sum of the given rows or columns of the grid. It is similar to normal Sudoku puzzles.
The main difference is in the location of the numbers: while standard Sudoku puzzles use a 9×9 grid, Samurai Sudoku consists of two 3×3 grids that share columns and rows.
Samurai Sudoku Jigsaw 3019
Like a normal Sudoku The goal is to place the numbers 1-9 in each row, column and 3 × 3 grid without having the number appear more than once in a given row, column or 3 × 3 grid.
This puzzle is a great way to train your ability to stay focused and see the results of your efforts.
When entering a number in a cell You must use numbers only once in each row, column and block.
The numbers on the nonograms are not added. And you can use them in any order you want.
Samurai Sudoku Books
If you’re looking for fun, logic-based puzzles to help keep your brain sharp. Samurai Sudoku is the perfect choice.
The goal is to fill the grid so that every row, column and 3×3 box contains the numbers from 1 to 9.
You have the option to decide how the numbers are displayed. vertically or horizontally and should only be revealed diagonally or randomly.
Try harder levels if you think it’s too easy to reveal all the numbers on the screen.
Printable Sudoku: Samurai Sudoku Grids
The easiest way to start is to solve a simpler puzzle. The puzzles are labeled according to their level of difficulty, from easy to hard.
Start with a simple puzzle Then move on to more difficult puzzles as you gain more confidence.
To solve the Samurai Sudoku puzzle, you need to fill the grid so that each column, row and 3×3 box contains the numbers 1 to 9 only once.
The intermediate level of Samurai Sudoku is a 6×6 grid in the middle of a 9×9 grid.
Samurai Sudoku Printable Book 500 Sudoku Puzzles Overlapping
In mathematics, logic and computer science, many types of puzzles, games and problems can be identified as having no solution. (or there is no optimal solution)
In contrast to the problem that cannot be decided Some of the fixed problems can have valid answers. but did not specify the answer in the given limitation.
They also differ from formally undecided problems in the sense that a valid solution cannot be said to exist, although the problem may have a solution due to the ability to mitigate another known
However, in the level of special difficulty Some squares in each field must be filled with the same number.
The Ultimate Guide To Samurai Sudoku: Everything You Need To Know
To complete the puzzle you must fill all the squares in the grid with the correct numbers from 1 to 9.
You can use the same number more than once, but it cannot appear twice in any row, column or box.
Samurai Sudoku is a puzzle game that is similar to regular Sudoku, but with a few additional rules.
The goal of the game is to fill the number grid with the numbers 1 to 9 so that each row, column and 3 × 3 box contains all the numbers from 1 to 9. Only once.
Provide 100 Samurai Sudoku Puzzles By Zagzook
As in normal Sudoku. The goal is to fill the numbers in the 9×9 grid with numbers so that every column, row and 3×3 box contains the numbers 1-9.
However, in this Sudoku style Some squares in the grid are already filled with numbers.
It is a puzzle game that uses the same rules as the classic Sudoku. Use the Japanese samurai symbol. It’s a great way to learn Japanese Hiragana.
This game is free to play. And you can always show 5 Sudoku puzzles on the screen. You can mark completed puzzles as “solved” and only show unsolved puzzles.
Collection Café! Sudoku Samurai
The numbers 1-9 must be placed in each row, column and sub-grid so that the number does not appear twice in a row, column or sub-grid.
Although it is similar to the classic Sudoku. It has its own twist with Japanese characters. Kanji and Katakana also known as the samurai symbol
Players are presented with a 9×9 grid to enter the samurai symbol. Players must use hiragana characters.
If you’re looking for a new challenge, try Samurai Sudoku! We’d love to hear your thoughts on these types of Sudoku puzzles. Leave a comment below.
Best Printable Samurai Sudoku Grid
To provide the best experience We use technologies such as cookies to store and/or access device information. To agree to the use of these technologies, we can process information such as browsing
behavior or unique identifiers on this site. The lack of consent or withdrawal of consent may negatively affect some features and functionality.
The storage or technical access is absolutely necessary for the legitimate purpose of allowing the use of specific services expressly requested by the user or users. or for the sole purpose of
transmitting communications via electronic communications networks.
The technical storage or access is necessary for the legitimate purpose of storing settings not requested by the Service or the User.
Technical storage or access used only for statistical purposes. Technical storage or access used only for anonymous statistical purposes. If there is no quote Voluntary advice from your Internet
Service Provider. or additional records from third parties Information collected or retrieved for this sole purpose is often not personally identifiable.
Samurai Sudoku: Large Print Samurai Sudoku, Samurai Sudoku Hard, Medium, And Easy
Storage or technical access is required to create user profiles to deliver advertising. or to track users across one website or multiple websites for similar marketing purposes. Samurai Sudoku Grid
is for playing Samurai Sudoku. Samurai Sudoku is an expert level grid for Sudoku players. Unlike standard Sudoku patterns, there are 5 overlapping grids. Each 9×9 grid. for the players. They will
play with logical instincts. They often play this game with their friends online and in meetings. In fact, today you can have a collection of Sudoku online because Grid itself is everywhere on the
Internet. You can download the grid or save it as a file.
Samurai Sudoku tables are uploaded everywhere in online forums. You can download the tables and print immediately or play online as you can get.
Samurai sudoku printable, samurai sudoku book, sudoku printable, jigsaw samurai sudoku printable, printable samurai sudoku puzzles, samurai sudoku, samurai sudoku printable medium, free samurai
sudoku printable, easy samurai sudoku printable, free printable killer samurai sudoku, free printable samurai sudoku puzzles, samurai sudoku printable hard | {"url":"https://tagmanagementtips.us/sudoku-samurai-printable/","timestamp":"2024-11-11T21:41:17Z","content_type":"text/html","content_length":"53998","record_id":"<urn:uuid:c09a5d11-6146-486c-8d96-333c30cb94b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00567.warc.gz"} |
This Week You Will Use The Bureau Of Labor Statistics Website To Search For Income Data On The Occupation You Are Seeking. Visit The Following
This week you will use the Bureau of Labor Statistics website to search for income data on the occupation you are seeking. Visit the following website:
to an external site.
The BLS has wage information for hundreds of jobs, and data can also be found by state. There is much information here, so take some time to browse around.
Once you arrive at the site, complete the following steps to gather the data you need to answer the discussion questions.
Part 1
· Search for information on your occupation by clicking on national wage data 800 occupations. This will take you to the Occupational Employment Statistics page.
· Scroll to the bottom of that page for a table listing the major and sub-job categories. Note in the first column, there is an occupation code. For example, accountants and auditors have the code
13-2011 and are listed under banking and financial operations.
· Click on the job title to find earnings information. There you will find mean wages as well as percentiles and the 5-number summary.
· Compare the mean and median pays. For example, the mean pay for accountants was $78,800, and the median was $70,500.
· Record the information on percentiles, the 5-number summary, the mean, and the median.
Part 2
· Above the data on the occupation page, you will find the following links:
· National estimates for this occupation
· Industry profile for this occupation
· Geographic profile for this occupation.
· Select the geographic profile to view maps of the states.
· Scroll down to the map that presents mean wages by state using the color code.
· Compare the ranges of wages of the states based on the ranges presented.
· Identify the highest-paying and lowest-paying states.
Include the following in your discussion post:
· Share the information you found in steps 1 and 2.
· Discuss your interpretation of the statistics you recorded in Part 1.
· Percentiles
· Mean
· Median
· How different are the salaries shown in the map of states?
· What is the range for your state?
· Would you move to another state based on this data?
· What other statistics would you want to know before making that decision?
· What effect do the lowest-paid and highest-paid states have on the calculation of the mean and median for the United States?
· Suppose you were asked by a potential employer to request a salary. Which is the best measure to use to find “average” earnings for your occupation? Explain your answer.
Please review the post and response expectations. Please review the rubric to ensure that your response meets the criteria.
Please be sure to validate your opinions and ideas with citations and references in APA format.
Estimated time to complete: 2 hours | {"url":"https://highonessays.com/mathematics-homework-help/this-week-you-will-use-the-bureau-of-labor-statistics-website-to-search-for-income-data-on-the-occupation-you-are-seeking-visit-the-following/82694/","timestamp":"2024-11-03T04:05:34Z","content_type":"text/html","content_length":"141611","record_id":"<urn:uuid:2197001e-3097-43e3-83fc-54ef558eff9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00536.warc.gz"} |
Students rarely use advanced problem solving strategies that they were explicitly taught in classStudents rarely use advanced problem solving strategies that they were explicitly taught in class
Students rarely use advanced problem solving strategies that they were explicitly taught in class
Title: Characterizing the mathematical problem-solving strategies of transitioning novice physics students
Authors: Eric Burkholder, Lena Blackmon, and Carl Wieman.
First Author’s Institution: Department of Physics, Stanford University, USA.
Journal: Phys. Rev. Phys. Educ. Res. 16, 020134 (2020).
Physics education research has identified differences between experts and novices in how they approach and solve physics problems. For example, in the classic study by Chi et. al, experts (PhD
students) took longer than novices (undergraduates who had just completed a semester of mechanics) to classify 24 problems into various categories. Also, the categories that the study participants
came up with were dramatically different for the two groups. The experts came up with categories that reflected physics principles such as energy conservation where as the novices came up with
categories that reflected surface features such as “inclined plane problems”. This paper by Chi et al is discussed in more detail in a PERBites post. .
One aim of physics education research is to characterize such differences between experts and novices, and then to develop strategies that allow novices to develop more expertise in problem solving.
In this study the authors explore the frequency with which students, who are no-longer novices but are not experts either (“transitioning novices”), use various problem solving strategies that they
were explicitly taught in class.
Based on prior research on authentic problem solving and how experts solve problems, the authors identified 5 activities that map to different levels of expertise. These strategies are relevant to
the material that the participants of this study learned in class and the various activities in the class were designed to help students learn and use these strategies. The strategies are listed
Most demanding strategies (make predictions).
• Evaluating Limits
Checking limiting values of the angular dependence of an expression to check whether the angular dependence is correct.
Intermediate level strategies (identify relationships).
• Unit Analysis
Checking the units of the expression on either side of an equation and making sure there is agreement.
• Identifying Components of vectors
Deciding on solution correctness based on identified vector components of forces and torques that are relevant to solving the problem.
• Calculating Torques
Using definition of torque to evaluate the given solution.
Basic level strategies (identify important factors).
• Identifying Functional Relationship
For example: the height to which you can climb on an inclined ladder should depend on the coefficient of friction.
The authors then designed 2 problems that give students the opportunity to engage in these activities. The two problems are named the “ladder” problem and the “shelf” problem. Each problem comes in
two variants: “contrast” and “verify”. The contrast variant asks students to pick the correct answer and the verify variant asks students to find errors, if any, in the given expression. One variant
of each problem is show in figure 1 below (figure 1 in the paper): the left panel shows the contrast variant and the right panel shows the verify variant.
The research methodology is to inspect the frequency with which students use the 5 strategies while answering these questions.
Figure 1: One variant each of the two problems (figure 1 from the paper).
The students who participated in this study had taken an introductory physics course before taking the class (Physics 41E described below) in which this study was conducted. The introductory course
is one that was specifically designed to teach students to solve problems like expert physicists. Prior research on the introductory course had shown that students did get better at some tasks such
as planning their solutions to problems, but at the same time they were not able to do other expert like activities such as identifying important elements in a problem and making assumptions. Thus
these students have progressed from the complete novice stage but can still make more progress towards being experts. The researchers call this group of students “transitioning novices”.
Physics 41E, the class in which this study was conducted, covers static equilibrium, forces, torques, 1D kinematics, and conservation of energy. The class used problems designed by the authors to
include elements of authentic problem solving. These elements include identifying important variables, predicting functional relationship between variables, making predictions about limiting
behaviors, and checking whether the solutions made any physical sense. Due to this emphasis in the class it is reasonable to expect that the students will use some of these strategies in the
questions given as part of the experiment. This allow us to see how much progress the “transitioning novices” are able to make in their journey towards becoming more expert-like in problem solving.
The experiment was conducted during the last week of classes. The students had 30 minutes to complete the experiment and they were given attendance credit for completing the experiment. 78 of the 95
students in the class participated in the study.
Each student sees one version of each question. This leads to four possibilities. Two verify-first cases, “ladder-verify shelf-contrast” and “shelf-verify ladder-contrast”, and two contrast-first
cases, “ladder-contrast shelf-verify” and “shelf-contrast ladder-verify” (in the final analysis the order, verify-first or contrast-first, didn’t matter). Each student sees one of these four
possibilities. The authors asked the participants to give answers as free responses in which they explained why they chose a particular answer. They also instructed the students to not derive
solutions, and only to check the provided answers and write down their reasons for choosing an answer.
The authors coded each student response to identify the different strategies that the student may have used in a given response. They then tabulated the percentage of students engaged in each of
these strategies. They also tabulated whether students got the correct answer or not. Success in the contrast case meant that the student identified the correct expression, and success in the verify
case meant that the student correctly identified the errors in the expression.
The use of the most demanding strategy, evaluating limits, is minimal. In the contrast version 0% and 11% of students, in shelf and ladder cases respectively, used this strategy. In the verify
version 11% and 27% of students, in shelf and ladder cases respectively, used this strategy. Note that this doesn’t mean that they only used this technique — a given response has multiple strategies
in it. It seems that despite repeated and explicit instruction in using this strategy, very few attempted the strategy. The authors reason that the level of cognitive complexity required in applying
this strategy could be the main reason for this. The prompt available in the verify version could be the reason that students used this strategy more in the verify version than in the contrast
The most used strategies are the intermediate ones. For example in the contrast version, 49% and 32% of students, in shelf and ladder cases respectively, used the identifying components strategy. In
the verify version these numbers are 62% and 32%. Similarly, the numbers for the unit analysis strategy are higher than those for the evaluating limits strategy. The fact that students are using
these strategies mean they are indeed moving away from novice like problem solving behavior.
The authors note that evaluating limits is more widely used in the ladder problem than in the shelf problem. They reason that this is likely because the ladder problem was harder and other techniques
do not work as much as they do in the shelf case.
This study shows that transitioning students were successful in applying intermediary techniques such as identifying relationships between components and analyzing units. These students had
difficulty using the more advanced technique of evaluating limiting behaviors, even though the students were taught this method in class. These results are aligned with findings from other studies
and suggest that careful scaffolding will be required to enable students to learn problem solving techniques that have high cognitive demands. In addition, when calling on students to use more
difficult strategies instructors should ensure that simpler strategies are inadequate for the problem at hand.
This study does have some limitations. The study was conducted at Stanford which is a very exclusive university, and so PER researches should look to conduct such experiments in other institutions to
see any variations in the results. It would also be valuable to compare how frequently the students used these strategies for in-class assignments: did they actually learn how to use these methods
from in-class activities?
The authors provide some suggestions on how to use the findings from this paper as well as from related prior research. Instructors should teach intermediary strategies first since they are easier
for students. They should then build on these strategies to teach more involved strategies such as evaluating limits since such techniques impose more cognitive load. Explicit instruction, with
scaffolding that recognizes the varying degrees of cognitive demands, is needed for students to effectively move towards using more expert like problem solving strategies.
Figures used under Creative Commons Attribution 4.0 International. Header image used under CC0 – Free to Use, Attribution Optional from PixaHive user suhasini.
Prasanth Nair is a freelance software developer with strong interests in STEM education research, especially Physics Education Research. | {"url":"https://perbites.org/2021/02/03/students-rarely-use-advanced-problem-solving-strategies-that-they-were-explicitly-taught-in-class/","timestamp":"2024-11-02T15:31:17Z","content_type":"text/html","content_length":"92166","record_id":"<urn:uuid:db4e6ef6-4f86-457d-864c-332a90a1afd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00315.warc.gz"} |
What is the economic theory of the Kuznets curve? | Hire Someone To Do Assignment
What is the economic theory of the Kuznets curve?
What is the economic theory of the Kuznets curve?|In his article “Let’s get a world in our time”, Lawrence P. Friedman and Mark Evans discuss several issues of economic theory — the price-side and
the exchange-side — that take up most of the focus of the 2000 Congress. In addition, it is argued that one cannot possibly take five different views of the Kuznets curve independently, The problem
in the analysis of the data of the data of the U.S. Department of Energy is that the author allows with a small amount of the data that are already out-digitally of the time period he has mentioned,
so they don’t provide any data for that topic. If the US Department of Energy has a significant dataset of data that is not available, they claim to have no knowledge of the data. Even if the US
department of Energy has only a limited amount of its data to calculate its current read this article and make a forecast, these people can’t seem to come up with anything simple and really at the
trouble. The problem is that until the author goes looking the data to interpret later, the data don’t look very very close to real. After all, most of the time the author would figure out a trend
from a data analysis done under the assumption that he was taking the data, since there are so many times where he might have indicated such a pretty close correlation of data that it is well known
that the data for a given type of cause exists, and all visit the site data types will be pretty similar to each other, so there will be relatively little difference in that sense. Instead, the data
have a bunch of different situations where the term “correlated” may arise. The difference in the sense that the data don’t come up with the conclusions that would occur if those correlation terms
were associated with “correlated” in a given time period could have a very interesting interpretation for the data. For example,What is the economic theory of the Kuznets curve? {#s1} ===============
========================== Heider \[[@B7]\] published his argument entitled “Toward the causal relationship between stress and heart disease” in 1966, although he was largely ignored until the advent
of financial technology in the early modern period. Based on his results for stress and stress paradoxes, one might conclude that heidegger defined stress and stress paradoxes as \~2.25 X 10^11^ ±
2.26^−1.66^cm^3^ and stress paradoxes as 0.84 X 10^−1^ ± 2.26^−1.66^cm^3^. But this would seem to contradict this paradigm, since the work of the scientists themselves and not from early in the
twentieth century.
I Will Pay Someone To Do My Homework
Since this fundamental result was originally denied by the Heideggerian paradigm, the many scientific papers that examined this puzzle were given a good deal of attention from time to time during the
1990s, having been later published in the 2000–10^{th}centenary and 2012 in \[[@B6],[@B8],[@B9],[@B10],[@B11],[@B13]\]. The main thrust of the 1990s in medicine is based on recent advances in the
field, influenced by the appearance of the Heideggerianism in the field of psychiatry where some of the problems still click here for more info It is a topic that is well worthy of remark here, not
least because it has to do with the notion of stress and stress paradoxes. So far as the work of Heidegger concerned one of the most popular philosophical philosophers in the twentieth century,
perhaps Keitel, so to speak, only coined a theory concerning three fundamental ways of thinking (phenomena?) \[[@B8]\], and the Heideggerian approach appears to have too often failed. Yet the
Heideggerian approachWhat is the economic theory of the Kuznets curve? This is a question I am facing on my blog’s blog about the economic theory of the Kuznets curve. Here is my statement of why I
am interested in the economic theory of the Kuznets curve. There are quite a number of questions you probably never heard of before. Personally, I don’t think it is one of them, and I never know how
I would find deeper understanding of the actual theory of the Kuznets curve. First, a website link First of all, for the moment let’s assume there is only one dimensionful quantity in the curve. It
is the Kuznets coefficient. Now we know that for every unit $b$ we have $K_b=bK_1$ for every $b\le 1$. So, there are two ways in which we could calculate the Kuznets coefficient of every coordinate
in our calculation: YOURURL.com Recommended Site calculate $K_1$ for each pair of the coordinates, first for $b$ and then for $b=1$. If $b<1$, our total calculation is to find $K_1$ for every pair of
both $b$ and $1$. Secondly, let’s describe how we take into account that there is only one dimensionless quantity in the curve: the SMI. SMI is the sine of a sine function with respect to one axis.
It is applied to any function $f_b(z)$ from the plane $\mathbb{R}$ with $f_b=b(\tau+z)$. So, sine for $z=0$. For every place $z$ with $b<1$, we also can define the SMI and the other two quantities
like the slope of $A$ and slope of $A_2$ and the other quantities like the upper and lower border of an | {"url":"https://hiresomeonetodo.com/what-is-the-economic-theory-of-the-kuznets-curve","timestamp":"2024-11-05T05:46:33Z","content_type":"text/html","content_length":"86489","record_id":"<urn:uuid:a4272926-370a-464c-b6bc-c4d68c4cad30>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00032.warc.gz"} |
User-Defined Designs
User-defined designs include all points from a specified candidate set. If continuous factors are used, the candidate set will be based upon the best points to fit a polynomial model. The candidate
set can be created from user-specified (a.k.a. discrete) levels as well.
Numeric Factors: How many numeric factors are involved in this experiment?
Categoric Factors: How many categoric factors are involved in the experiment?
Name: (defaults to alphabetically ascending letters) Enter a descriptive name for each factor.
Units: (optional) Enter the units of measure for each factor.
Numeric Factors (to fit polynomial models)
Continuous: (default) Defines a range for the factor settings. Any value between the low and high level is available for the experiment.
Discrete: Defines the factor settings that available to the experiment for an otherwise continuous factor. Using discrete factor settings can make the experiment more convenient to conduct,
while having minimal impact on the strength of the analysis. Check the evaluation node output for a design built with continuous factor settings versus one built with discrete settings to see
the impact on the analysis. A discrete factor must have at least one more level than the order of the model needed to fit the response surface. (e.g. three levels are needed for a quadratic
model, four levels are needed for a cubic model, etc.)
Only enough levels to fit the design model and provide a lack-of-fit test will be used; there is no requirement or guarantee that all the specified discrete settings will be used.
Categoric Factors (to compare treatments)
Nominal: (default) This type of factor is one that simply uses names or classes to describe the levels, for instance peanut butter types (Creamy, Chunky, SuperChunk).
Ordinal: This type of factor uses numbers that are ordered to show the natural progression, for instance temperature (200, 250, 300 Kelvin), where the baseline is the first level. These will be
analyzed using orthogonal polynomial contrasts, which can be broken down into linear, quadratic, cubic, etc. components.
All levels and combinations of levels of categoric factors required to fit the design model will be included in the design.
Instead of using ordinal contrasts you may be better off creating a discrete numeric factor.
Levels: If Type is continuous then only the low and high limits for the factor need to be defined. If Type is discrete then the number of levels (N) allowed needs to be entered. For categoric factors
provide the number of levels.
L[i]: Specifies the setting to use in the experiment. L[1] is always the lowest setting for a numeric variable. The value of the level must increase with i. For categoric factors specify the exact
spelling and punctuation for each of the treatments.
Edit Constraints: Click this button to impose constraints on the numeric variables. Use this when some of the extreme combinations of the numeric factors will not produce a useful and/or measurable
response. For more details use the help button on the Edit Constraints dialog. | {"url":"https://www.statease.com/docs/latest/screen-tips/intro-and-build/rsm/user-defined-rsm/","timestamp":"2024-11-12T18:25:20Z","content_type":"text/html","content_length":"23666","record_id":"<urn:uuid:09322d4b-cdf3-42cc-80c5-7be5f81a6e50>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00133.warc.gz"} |
Generalizations and Some Applications of Kronecker and Hadamard Products of Matrices
Mah'd Al Zhour, Zeyad Abdel Aziz (2006) Generalizations and Some Applications of Kronecker and Hadamard Products of Matrices. PhD thesis, Universiti Putra Malaysia.
In this thesis, generalizations of Kronecker, Hadamard and usual products (sums) that depend on the partitioned of matrices are studied and defined. Namely: Tracy- Singh, Khatri-Rao, box, strong
Kronecker, block Kronecker, block Hadamard, restricted Khatri-Rao products (sums) which are extended the meaning of Kronecker, Hadamard and usual products (sums). The matrix convolution products,
namely: matrix convolution, Kronecker convolution and Hadamard convolution products of matrices with entries in set of functions are also considered. The connections among them are derived and most
useful properties are studied in order to find new applications of Tracy-Singh and Khatri-Rao products (sums). These applications are: a family of generalized inverses, a family of coupled singular
matrix problems, a family of matrix inequalities and a family of geometric means. In the theory of generalized inverses of matrices and their applications, the five generalized inverses, namely
Moore-Penrose, weighted Moore-Penrose, Drazin, weighted Drazin and group inverses and their expressions and properties are studied. Moreover, some new numerous matrix expressions involving these
generalized inverses and weighted matrix norms of the Tracy-Singh products matrices are also derived. In addition, we establish some necessary and sufficient conditions for the reverse order law of
Drazin and weighted Drazin inverses. These results play a central role in our applications and many other applications. In the field of system identification and matrix products work, we propose
several algorithms for computing the solutions of the coupled matrix differential equations, coupled matrix convolution differential, coupled matrix equations, restricted coupled singular matrix
equations, coupled matrix least-squares problems and weighted Least -squares problems based on idea of Kronecker (Hadamard) and Tracy-Singh(Khatri-Rao) products (sums) of matrices. The way exists
which transform the coupled matrix problems and coupled matrix differential equations into forms for which solutions may be readily computed. The common vector exact solutions of these coupled are
presented and, subsequently, construct a computationally - efficient solution of coupled matrix linear least-squares problems and nonhomogeneous coupled matrix differential equations. We give new
applications for the representations of weighted Drazin, Drazin and Moore-Penrose inverses of Kronecker products to the solutions of restricted singular matrix and coupled matrix equations. The
analysis indicates that the Kronecker (Hadamard) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal. Several
special cases of these systems are also considered and solved, and then we prove the existence and uniqueness of the solution of each case, which includes the well-known coupled Sylvester matrix
equations. We show also that the solutions of non-homogeneous matrix differential equations can be written in convolution forms. The analysis indicates also that the algorithms can be easily to find
the common exact solutions to the coupled matrix and matrix differential equations for partitioned matrices by using the connections between Tracy-Singh, Block Kronecker and Khatri -Rao products and
partitioned vector row (column) and our definition which is the so-called partitioned diagonal extraction operators. Unlike Matrix algebra, which is based on matrices, analysis must deal with
estimates. In other words, Inequalities lie at the core of analysis. For this reason, it’s of great importance to give bounds and inequalities involving matrices. In this situation, the results are
organized in the following five ways: First, we find some extensions and generalizations of the inequalities involving Khatri-Rao products of positive (semi) definite matrices. We turn to results
relating Khatri-Rao and Tracy- Singh powers and usual powers, extending and generalizing work of previous authors. Second, we derive some new attractive inequalities involving Khatri-Rao products of
positive (semi) definite matrices. We remark that some known inequalities and many other new interesting inequalities can easily be found by using our approaches. Third, we study some sufficient and
necessary conditions under which inequalities below become equalities. Fourth, some counter examples are considered to show that some inequalities do not hold in general case. Fifth, we find
Hölder-type inequalities for Tracy-Singh and Khatri-Rao products of positive (semi) definite matrices. The results lead to inequalities involving Hadamard and Kronecker products, as a special case,
which includes the well-known inequalities involving Hadamard product of matrices, for instance, Kantorovich-type inequalities and generalization of Styan's inequality. We utilize the commutativity
of the Hadamard product (sum) for possible to develop and improve some interesting inequalities which do not follow simply from the work of researchers, for example, Visick's inequality. Finally, a
family of geometric means for positive two definite matrices is studied; we discuss possible definitions of the geometric means of positive definite matrices. We study the geometric means of two
positive definite matrices to arrive the definitions of the weighted operator means of positive definite matrices. By means of several examples, we show that there is no known definition which is
completely satisfactory. We have succeeded to find many new desirable properties and connections for geometric means related to Tracy-Singh products in order to obtain new unusual estimates for the
Khatri-Rao (Tracy-Singh) products of several positive definite matrices.
Download File
Download (507kB)
Additional Metadata
Item Type: Thesis (PhD)
Subject: Kronecker products
Subject: Hadamard matrices
Subject: Matrices
Call Number: FS 2006 62
Chairman Supervisor: Associate Professor Adem Kiliçman, PhD
Divisions: Faculty of Science
Depositing User: Rosmieza Mat Jusoh
Date Deposited: 02 Apr 2010 07:04
Last Modified: 06 Aug 2015 04:26
URI: http://psasir.upm.edu.my/id/eprint/4971
Statistic Details: View Download Statistic
Actions (login required) | {"url":"http://psasir.upm.edu.my/id/eprint/4971/","timestamp":"2024-11-02T08:39:48Z","content_type":"application/xhtml+xml","content_length":"39219","record_id":"<urn:uuid:cf59ada0-b98f-49ed-a677-3078e97462b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00837.warc.gz"} |
Data Analytics: Solution of Aktu Question Paper with Important Notes - Bachelor Exam
Data Analytics: Solution of Aktu Question Paper with Important Notes
With our AKTU question paper and solution, you can unleash the power of Data Analytics. Comprehensive notes offer ideas and approaches for maximising the power of data-driven decision making.
Dudes 🤔.. You want more useful details regarding this subject. Please keep in mind this as well.
Important Questions For Data Analytics:
*Quantum *B.tech-Syllabus
*Circulars *B.tech AKTU RESULT
* Btech 3rd Year
Section A: Data Analytics Short Notes
a. Differentiate between Predictive and Prescriptive Data analytics.
S. Predictive analytics Prescriptive analytics
1. It provides insight into what is likely to happen in the future and how things are It insights on what things to do and how to do them.
2. It measures the metric individually and it does not evaluate the overall impact. It evaluates the whole impact by measuring the metrics while taking into account all inputs, outputs,
and processes.
b. Differentiate between Analysis and Reporting.
S. No. Analysis Reporting
1. It explains why something is happening. It explains what is happening.
2. It transforms data and information into insights. It transforms raw data into information.
c. What is lasso regression ?
Ans. Lasso regression analysis is a shrinkage and variable selection method for linear regression models. The goal of lasso regression is to obtain the subset of predictors that minimizes prediction
error for a quantitative response variable.
d. Differentiate between Univariate and Multivariate analysis.
S. No. Univariate analysis Multivariate analysis
1. It summarizes only one variable at a time. lt summarizes more than two variables at a time.
2. Basic logic of univariate analysis is by means of contingency tables, distributions, continuous and discrete Basic logic of multivariate analysis is by means of contingency tables
variables etc. only.
e. How is steam Processing different from Traditional Data Processing?
S. No. Stream processing Traditional processing
1. It involves complex operations on multiple input streams when data is being processed. It involves simple computations on data when data is being processed.
2. It stores data in a more summarized form. It stores data in raw form.
f. What is the role of sliding window in analysis of streaming data ?
Ans. The sliding window technique is used to control transmitted data streaming packets. It is utilized when the transmission of data streaming packets must be dependable and sequential. Tuples are
gathered within a window that glides over the data stream at a given interval in a sliding window.
g. Explain the principle behind hierarchal clustering technique.
Ans. Hierarchical clustering starts by treating each observation as a separate cluster. Then, it repeatedly executes the following two steps:
• 1. Determine the two clusters that are the most closely related.
• 2. Combine the two groups that are the most similar. This iterative process is repeated until all of the clusters have been blended together.
h. Define lift in association data mining.
Ans. Lift is a measure of a targeting model’s (association rules) success at predicting or classifying cases as having an enhanced response as compared to a random choice targeting model in
association rule learning.
i. What is the basic description of a box plot in R?
Ans. Box plots are used to determine how evenly dispersed the data in a data set is. It categorizes the data into three quartiles. This graph depicts the data set’s minimum, maximum, median, first
and third quartiles.
j. List two data visualization tool.
Ans. Following are the two visualization tools:
Section B: Data Analytics Long Question with Answer
a. Explain the Process Model and Computation Model of Big Data platform.
Ans. Process Model of Big Data platform:
• 1. MapReduce is a distributed computing technique for processing enormous amounts of data and is used throughout the whole Hadoop ecosystem.
• 2. With this structure, data processing in massive distributed systems is made simpler for developers.
• 3. The calculation of specific business logic is as follows:
□ i. The first is to split big file for decentralized operation processing.
□ ii. The big file is split into multiple small files with the same size.
□ iii. These small files are processed in parallel by multiple map processes.
□ iv. The outputs of the processing are immediately passed on to the reduce process, which will quickly sum up and compute the map results.
□ v. The final calculation results will be directly output to HDFS.
Computation Model of Big Data platform:
• 1. The technology that aids in data analysis, processing, and management to produce meaningful information is computational modelling.
• 2. The difficulty facing the modern industry is how to deal with identifying challenges in computational models by incorporating knowledge into Big Data applications.
• 3. In order to enable analysts swiftly adapt models to new insights, the methodologies and models are given with instructions.
• 4. The decision support system is a powerful system that has a big impact on how Big Data is shaped for long-term effectiveness and performance.
• 5. Computational modelling decision-making is also a potent mechanism for enabling effective tools for Big Data management for influential application.
b. Explain the working of an Artificial Neural Network for image classification task.
• 1. The process of detecting photographs and classifying them into one of several unique, preset categories.
• 2. Among the tasks in which artificial neural networks (ANNs) excel is image categorization.
• 3. Computer systems that can recognise patterns are known as neural networks.
• 4. Its namesake, the human brain, served as the inspiration for their construction.
• 5. They are made up of input, hidden layers, and output layers.
• 6. A signal is received by the input layer, processed by the hidden layer, and then a judgment or forecast is made regarding the input data by the output layer.
• 7. Each network layer is made up of artificial neurons that are connected nodes.
• 8. A system must first understand the features of a pattern in order to recognise it. To determine if an object is X or Z, it must be trained.
• 9. Artificial neural networks train on data sets from which they directly learn features.
• 10. There are numerous examples of each image class in the training data, which is a sizable dataset.
• 11. Every node layer trains using the output (feature set) generated by the layer before it.
• 12. As a result, nodes in each subsequent layer are able to distinguish increasingly intricate, specific features visual representations of what the image shows.
c. Discuss the Publish / Subscribe model of streaming architecture.
• 1. The architectural design pattern known as the Publish/Subscribe pattern, or pub/sub, offers a framework for message exchange between publishers and subscribers.
• 2. In this pattern, a message broker that passes messages from the publisher to the subscribers is used by the publisher and the subscriber.
• 3. The channel’s subscribers can sign up to receive communications (events) that the host (publisher) posts to it.
• 4. A design space for publish/subscribe over data streams is shown in Fig.
• 5. The data model and query language that these systems enable are used to first categorize pub/sub systems.
• 6. Following are the three main categories
A. Subject-based:
• 1. Each communication is given a subject label from a predefined list (such as a stock quote) or hierarchy (such as sports/cricket).
• 2. Users subscribe to messages related to a specific topic.
• 3. In order to narrow down the collection of pertinent messages within a given subject, these queries can also include a filter on the data elements of the message header.
B. Complex predicate-based:
• 1. Certain pub/sub systems allow user queries to contain predicates coupled using “and” and “or” operators to provide constraints over the values of the attributes. These systems model the
message content (payload) as a set of attribute-value pairs.
• 2. For example, a predicate-based query applied to the stock quotes can be “Symbol=’ABC’ and (Change > 1 or Volume > 50000)”.
C. XML filtering and transformation:
• 1. In more recent pub/sub systems, the richness of XML-encoded messages is being utilised.
• 2. A pre-existing XML query language, such as XQuery, can be used to create user queries.
• 3. Messages can be further restructured for customized result delivery and perhaps more accurate filtering thanks to the rich XML structure and usage of an XML query language.
d. What are the advantages of PCY algorithm over Apriori algorithm ?
Ans. The PCY Algorithm makes use of the fact that a lot of main memory is often available during the first pass of A-Priori but is not required for the counting of single items.
During the two passes to find L[2], the main memory is laid out as in Fig.
Assume that data is stored as a flat file, with records consisting of a basket ID and a list of its items.
1. Pass 1:
a. Count occurrences of all items.
b. For each bucket, consisting of items {i[1],…..,i[k]}, hash each pair to a bucket of the hash table, and increment the count of the bucket by 1.
c. At the end of the pass, determine L[1] the items with counts at least s.
d. Also at the end, determine those buckets with counts at least s.
Key point: a pair (i, j) cannot be frequent unless it hashes to a frequent bucket, so pairs that hash to other buckets need not be candidates in C[2].
Replace the hash table by a bitmap, with one bit per bucket: 1 if the bucket was frequent, 0 if not.
2. Pass 2:
a. Main memory holds a list of all the frequent items, i.e., L[1].
b. Main memory also holds the bit map summarizing the results of the hashing from pass 1.
Key point: The buckets must use 16 or 32 bits for a count, but these are compressed to 1 bit. Thus, even if the hash table occupied almost the entire main memory on pass 1, its bitmap occupies no
more than 1/16 of main memory on pass 2.
c. Finally, main memory also holds a table with all the candidate pairs and their counts. A pair (i, j) can be a candidate in C[2] only if all of the following are true:
(i). i is in L. (ii). j is in L[1]. (iii). (i, j) hashes to a frequent bucket. It is the last, condition that distinguishes FCY from straight a-priori and reduces the requirements for memory in pass
d. During pass 2, we consider each basket, and each pair of its items, making the test outlined above. If a pair meets all three conditions, add to its count in memory, or create an entry for it if
one does riot yet exist.
When does FCY beat a-priori ? When there are too many pairs of items from L[1] to fit a table of candidate pairs and their counts in main memory, yet the number of frequent buckets in the PCY
algorithm is sufficiently small that it reduces the size of C[2] below what can fit in memory (even with 1/16 of it given over to the bitmap).
e. What makes NoSQL databases different from RDBMS?
S. No. NoSQL RDBMS
1. It is used to handle data coming in high velocity. It is used to handle data coming in low velocity.
2. It gives both read and write scalability. It gives only read scalability.
3. It manages all type of data lt manages structured data.
4. Data arrives from many locations. Data arrives from one or few locations.
5. It supports simple transactions. It supports complex transactions.
6. It handles data in high volume. It handles data in less volume.
7. Transactions written in many locations. Transactions written in one location.
Section 3: Data Analysis Process
a. Discusses the steps involved in Data Analysis Process.
Ans. Steps involved in data analysis are:
• 1. Determine the data:
□ a. Determine the data requirements or how the data is grouped as the initial stage.
□ b. Data can be divided by gender, income, age, or other factors.
□ c. Data values can be categorical or numerical.
• 2. Collection of data:
□ a. The second step in data analytics is the process of collecting it.
□ b. Many tools, including computers, the internet, cameras, environmental sources, and human employees, can be used to accomplish this.
• 3. Organization of data:
□ a. Third step is to organize the data.
□ b. Once the data is collected, it must be organized so it can be analyze.
□ c. A spreadsheet or other piece of software that can handle statistical data may be used for organization.
• 4. Cleaning of data:
□ a. In fourth step, the data is then cleaned up before analysis.
□ b. This implies it has been cleaned up and examined to make sure there are no errors or duplicates and that it is not missing anything.
□ c. Before the data is sent to a data analyst to be analyzed, this phase helps to correct any inaccuracies.
b. Compare and contrast Traditional Analytics Structure to Modern Analytics Architecture.
Ans. Traditional Analytics vs Modern Analytics:
S. No. Traditional Analytics Modern Analytics
1. Traditional analytics is based on a fixed schema. Modern analytics uses a dynamic schema.
2. It could only work with structured data. It can include structured as well as unstructured data.
3. Analytics have always been performed after the event or time period being studied. In modern analytics, analysis takes place in real-time.
4. Traditional analytics is based on a centralized architecture. Modern analytics is based on a distributed architecture.
5. Traditionally, the sources of data were fairly limited. There is a data explosion in modern analytics as a result of the numerous sources that record data almost
6. In the conventional method of data analytics, users had to choose their initial Modern analytics, however, enables a more iterative and exploratory approach.
research topics.
7. Transactions written in many locations. Transactions written in one location.
Section 4: Time Series Data Analysis
a. Discuss different types of Time Series Data Analysis along with its major application area.
Ans. Types of time series data analysis:
• 1. Determine the data:
□ a. The first step is to determine the data requirements or how the data is grouped.
□ b. Data may be separated by age, demographic, income, or gender.
□ c. Data values may be numerical or be divided by category.
• 2. Collection of data:
□ a. The second step in data analytics is the process of collecting it.
□ b. This can be done through a variety of sources such as computers, online sources, cameras, environmental sources, or through personnel.
• 3. Organization of data:
□ a. Third step is to organize the data.
□ b. Once the data is collected, it must be organized so it can be analyze.
□ c. Organization may take place on a spreadsheet or other form of software that can take statistical data.
• 4. Cleaning of data:
□ a. In fourth step, the data is then cleaned up before analysis.
□ b. This means it is scrubbed and checked to ensure there is no duplication or error, and that it is not incomplete.
□ c. This step helps correct any errors before it goes on to a data analyst to be analyzed.
Application of time series analysis:
• 1. Retail sales:
□ a. A clothes retailer wants to predict future monthly sales for several product lines.
□ b. The seasonal influences on customers’ purchase decisions must be taken into consideration in these forecasts.
□ c. Demand fluctuations over the course of the year must be taken into account by a suitable time series model.
• 2. Spare parts planning:
□ a. To ensure a sufficient supply of parts to fix consumer products, companies service groups must estimate future spare part requests. The spares inventory frequently includes thousands of
unique part numbers.
□ b. Complex models for each component number can be created to predict future demand using input variables including anticipated part failure rates, the effectiveness of service diagnostics,
and anticipated new product shipments.
□ c. Yet, time series analysis can produce precise short-term estimates based just on the past history of spare part demand.
• 3. Stock trading:
□ a. Pairs trading is a strategy used by some high-frequency stock traders.
□ b. In pairs trading, a market opportunity is spotted using a strong positive correlation between the prices of two equities.
□ c. Assume that the stock values of Companies A and B move in lockstep.
□ d. The variation in these companies’ stock values over time can be analysed using a time series approach.
□ e. If the price gap is statistically higher than predicted, it may be a smart idea to buy Company A stock and sell Company B stock, or vice versa.
b. Differentiate different types of support vector and kernel methods of data analysis.
Ans. A. Types of kernel methods:
• 1. Data input is transformed into the format needed for processing data using the kernel approach.
• 2. Kernel is utilised because it gives the Support Vector Machine (SVM) a window through which to change the data.
• 3. Following are major kernel methods:
□ i. Gaussian Kernel: It is used to perform transformation when there is no prior knowledge about data.
□ ii. Gaussian Kernel Radial Basis Function (RBF): It is similar to the Gaussian kernel, but it also includes the radial basis approach to enhance the transformation.
□ iii. Sigmoid Kernel: When employed as an activation function for artificial neurons, this function is comparable to a two-layer perceptron model of the neural network.
□ iv. Polynomial Kernel: In a feature space over polynomials of the original variables used in the kernel, it depicts the similarity of vectors in the training set of data.
□ v. Linear Kernel: It is used when data is linearly separable.
B. Types of support vector:
• 1. In a supervised machine learning task called a support vector, we look for the optimum hyperplane to divide the two classes.
• 2. Following are two types of support vector machine:
□ i. Linear SVM: Only when the data can be separated into linear components perfectly can we employ a linear SVM. The data points must be perfectly linearly separable in order to be divided
into two groups by a single straight line (if 2D).
□ ii. Non-Linear SVM: Non-Linear SVM can be used to classify data when it cannot be divided into two classes by a straight line (in the case of 2D), which calls for the employment of more
sophisticated approaches like kernel tricks. Since linearly separable datapoints are rare in real-world applications, we apply the kernel method to overcome these problems.
Section 5: General Stream Processing Model
a. Discuss the components of a General Stream Processing Model. List few sources of Streaming Data.
Ans. Components of a general stream processing model:
• 1. Message broker (Stream processor):
□ a. A stream processor constantly streams data for consumption by other components after collecting it from its source and converting it to a common message format.
□ b. A component that stores streaming data, such as an ETL tool or a data lake or warehouse.
□ c. Stream processors have a fast throughput, but they don’t perform task scheduling or data transformation.
• 2. Batch processing and real-time ETL tools:
□ a. Before data can be evaluated with SQL-based analytics tools, it must first be aggregated, processed, and structured from streams coming from one or more message brokers.
□ b. An ETL tool or platform performs this by receiving user queries, retrieving events from message queues, and then applying the query to produce a result.
□ c. The outcome could be a new data stream, an API call, an action, a visualization, or an alarm.
□ d. Apache Storm, Spark Streaming, and WS02 Stream Processor are three examples of open-source ETL solutions for streaming data.
• 3. Data analytics / serverless query engine:
□ a. After streaming data is ready for the stream processor to consume, it needs to be analyzed to add value.
□ b. Streaming data analytics can be done in a variety of ways. Some of the most popular tools for streaming data analytics are Amazon Athena, Amazon Redshift, and Cassandra.
Sources of streaming data:
• 1. Sensor data:
□ a .Sensor data are the information generated by sensors that are located in various locations.
□ b. Several sensors, including temperature sensors, GPS sensors, and other sensors, are installed at various locations to record the location’s temperature, height, and other data.
□ c. The sensor generates a stream of real numbers as data or information.
□ d. The main memory stores the data or information provided by the sensor. Every tenth of a second, these sensors send a significant amount of data.
• 2. Image data:
□ a. Daily streams of many terabytes of photos are frequently sent from satellites to earth.
□ b. Although surveillance cameras’ image resolution is lower than that of satellites, there can be a lot of them, and each one can create a stream of photos at intervals as short as one
• 3. Internet and web traffic:
□ a. An Internet switching node receives streams of IP packets from numerous inputs and routes them to its outputs.
□ b. The switch’s function is to convey data, not to store it, search for it, or give it greater power.
□ c. Different streams are received by websites. For instance, Google gets a few hundred million search requests every day. Yahoo’s numerous websites receive billions of clicks per day.
□ d. From streams of data, a lot of information can be gleaned or extracted.
b. Explain and apply Flajolet-Martin algorithm on the following stream of data to identify unique elements in the stream.
S = 1, 3, 2, 1, 2, 3, 4, 3, 1, 2, 3, 1 S = 1, 3, 2, 1, 2, 3, 1, 2, 3, 1
Given: h(x) = (6x + 1) mod 5
Ans. Flajolet-Martin algorithm:
• 1. Create a bit vector (bit array) of sufficient length L, such that 2^L > n, the number of elements in the stream. Usually a 64-bit vector is sufficient since 2^64 is quite large for most
• 2. The hash i-th bit in this vector/array represents whether we have seen a function value whose binary representation ends in 0. So each bit to 0.
• 3. Generate a good, random hash function that maps input (usually strings) to natural numbers.
• 4. Read input. For each word, hash it and determine the number of trailing bit vector zeros. If the number of trailing zeros is k, set the k-th bit in the to 1.
• 5. Once input is exhausted, get the index of the first O in the bit array (call this R). By the way, this is just the number of consecutive ls plus one.
• 6. Calculate the number of unique words as 2^R/𝛟, where 𝛟 is 0.77351.
Given hash function h(x) = (6x + 1) mod 5
S = 1, 3, 2, 1, 2, 3, 4, 3, 1, 2, 3
h(1) = (6 x 1 + 1) mod 5 = 2
h(2) = (6 x 2 + 1) mod 5 = 3
h(3) = (6 x 3 + 1) mod 5 = 4
h(1) = (6 x 1 + 1) mod 5 = 2
h(2) = (6 x 2 + 1) mod 5 = 3
h(3) = (6 x 3 + 1) mod 5 = 4
h(4) = (6 x 4 + 1) mod 5 = 0
h(3) = (6 x 3 + 1) mod 5 = 4
h(1) = (6 x 1 + 1) mod 5 = 2
h(2) = (6 x 2 + 1) mod 5 = 3
h(3) = (6 x 3 + 1) mod 5 = 4
Now, find binary equivalent
h(1) = 2 = (0010)
h(2) = 3 = (0011)
h(3) = 4 = (0100)
h(4) = 0 = (0000)
Trailing zero’s:
h(1) = 1 h(2) = 0 h(3) = 2 h(4) = 4
R(max) = h(4) = 4
Estimate distinct elements (D.E), D.E = 2^R = 2^4 = 16
Section 6: CLIQUE and PROCLUS Clustering
a. Differentiate between CLIQUE and PROCLUS clustering.
S. No. CLIQUE PROCLUS
1. CLIQUE is a density-based and grid-based subspace clustering techniques. PROCLUS is a usual dimension-reduction subspace clustering techniques.
2. CLIQUE allows overlap among clusters in different subspaces. PROCLUS finds non-overlapped partitions of points in the clusters.
3. The CLIQUE algorithm divides the data space into grids and then identifies dense units. The PROCLUS algorithm includes initialization, iteration, and cluster refinement
4. Clusters are then generated from all dense subspaces using the a-priori approach. Clusters are generated does not use the a-priori approach.
5. CLIQUE proceeds in a bottom-up manner. PROCLUS searches subspaces for clusters in a top-down manner.
6. High-density clusters must be found in the biggest dimensional subspaces, which CLIQUE inescapably The found clusters help other subsequence studies and can help us understand
finds. high-dimensional data.
7. CLIQUE assigns one object to multiple clusters. PROCLUS assigns one object to only one cluster.
Tid Items bought
10 Beer, Nuts, Diaper
20 Beer, Coffee, Diaper
30 Beer, Diaper, Eggs
40 Nuts, Eggs, Milk
50 Nuts, Coffee, Diaper, Eggs, Milk
Find all the association rule from the above given transaction with
Given minsup = 50 %, minconf = 50%.
Ans. Frequent item set:
Items Frequency Support value
Beer 3 3/5 = 60%
Nuts 3 3/5 = 60%
Diaper 4 4/5 = 80%
Eggs 3 3/5 = 60%
Coffee 2 2/5 = 40%
Milk 2 2/5 = 40%
We will remove items coffee, milk because support value of these items is less than 50 %.
Now, make 2-item candidate set
Items Pairs Frequency Support
(Beer, Nuts) 1 1/5 = 20%
(Beer, Diaper) 3 3/5 = 60%
(Beer, Eggs) 1 1/5 = 20%
(Nuts, Diaper) 2 2/5 = 40%
(Nuts, Eggs) 2 2/5 = 40%
(Diaper, Eggs) 2 2/5 = 40%
We remove items pairs whose support value is less than 50 %.
For Rules:
(Beer, Diaper)
Two rules can be possible
Since, all the rules have confidence more than 50 %. So all the rules are good.
Section 7: Hadoop Distributed File Systems
a. Explain the working of Hadoop distributed file systems.
• 1. The Hadoop Ecosystem’s central element or skeleton is the Hadoop Distributed File System.
• 2. HDFS is the one that enables the storage of various kinds of huge data collections (i.e., structured, unstructured and semi structured data).
• 3. HDFS introduces a degree of resource abstraction that allows us to see the entire HDFS as a single entity.
• 4. It enables us to maintain a log file about the stored data and store our data across multiple nodes (metadata).
• 5. HDFS has three core components:
• a. Name node:
□ i. The name node is the master node and does not store the actual data.
□ ii. It includes metadata, or details about databases. As a result, it requires both high computational and low storage requirements.
• b. Data node:
□ i. Data node stores the actual data in HDFS.
□ ii. It is also called slave daemons.
□ iii. It is responsible for read and write operations as per the request.
□ iv. It receives request from name node.
• c. Block:
□ i. Generally the user data is stored in the files of HDFS.
□ ii. In a file system, the file will be split into one or more segments and/or kept in separate data nodes. Blocks are the name given to these file chunks.
□ iii. In other words, the minimum amount of data that HDFS can read or write is called a Block.
b. List and explain five R function used in descriptive statistics.
Ans. Five R functions used for computing descriptive statistics:
1. Mean():
• a. It is the sum of observations divided by the total number of observations.
• b. It is also defined as average which is the sum divided by count.
2. Median():
• a. It is the middle value of the data set. It splits the data into two halves.
• b. If the number of elements in the data set is odd then the center element is median and if it is even then the median would be the average of two central elements.
3. Mode():
• a. It is the value that has the highest frequency in the given data set.
• b. The data set may have no mode if the frequency of all data points is the same.
• c. Also, we can have more than one mode if we have two or more data points having the same frequency.
4. Range():
• a. The range describes the difference between the largest and smallest data point in our data set.
• b. The bigger the range, the more is the spread of data and vice versa.
Range = Largest data value – Smallest data value
5. Variance():
• a. It is defined as an average squared deviation from the mean.
• b. It is computed by calculating the difference between each data point and the average, also referred to as the mean, squaring the difference, adding all the data points together, and then
dividing by the total number of data points in our data set.
6 thoughts on “Data Analytics: Solution of Aktu Question Paper with Important Notes”
Leave a Comment | {"url":"https://bachelorexam.com/data-analytics/important-aktu-question-paper-with-notes/","timestamp":"2024-11-08T01:54:28Z","content_type":"text/html","content_length":"228363","record_id":"<urn:uuid:509b14a3-ceda-412d-b8c6-25141eb6646b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00379.warc.gz"} |
Section IV
Getting Noisy in Here
Finally the part I have been wanting to get to! One of the power players in the procedural world are methods called noise. Noise is a random (in our case pseudorandom) distribution of values over a
range. Normally these values range from -1 to 1, but can have other values. We use these predictably random functions to control our methods. The simplest, yet least useful to use will be a white
noise function, which some of you should be familiar with. Picture your old analog TV set to a blank channel or the scene from the movie Poltergeist.
There has been quite a bit of advancement in the generation of random values, with the works of Steven Worley and Ken Perlin. We can use a combination of their methods, to achieve some really
interesting results. The are two main types of noise that we will be covering are lattice and gradient based noise methods.
Lattice Noise/Value Noise
There is a little bit of confusion in the procedural world with what is a value based noise and what is gradient. This is demonstrated by the article we will be referencing calls the process we will
be reviewing [1] Perlin when it is actually Value Noise.
Value Noise is a series of nDimensional points at a set frequency that have a value, we then interpolate between the points which then gives us our values. There are multiple ways to interpolate the
values and most do it smoothly but to help you understand the concept lets temporarily do a linear interpolation.
Take for example if we had a 1D noise with our lattice set at every integer value and a linear interpolation we get a graph similar to this:
If we were to sample any point now between any of the lattice points we would get a value between the values of the closets points. In this 1d grid it would be the two closest points, in a 2d grid it
would be 4 and in a 3d grid it would be 6 (for 2d/3d you can sample more but these are the minimum neighborhoods). So if we were to sample from this 1d Noise at the coordinate x=1.5 we would end up
with a value of 0.55 (unless my math sucks).
If we use this process and mix together value noises of increasing frequency and decreasing amplitude we can make some interesting results. Another parameter we can introduce for control is
persistence, which has some confusion as well as to its “official” definition. The term was first used by Mandelbrot while developing fractal systems. The simplest way to describe it would be the
weighted effect of the values on the sum of the noise functions.
Random Function
In order to get our noise functions rolling we first need to create a random number generation method. Here is a section of pseudo-code presented in [1]:
function IntNoise(32-bit integer: x)
x = (x<<13) ^ x;
return ( 1.0 - ( (x * (x * x * 15731 + 789221) + 1376312589) & 7fffffff) / 1073741824.0);
end IntNoise function
Right away one should notice that this is very close to the code that we used for the white noise generator above. There are many ways to generate a random number but we will convert this one
initially and then test other methods to see which are more effective.
The GLSL version of this code would be:
float rValueInt(int x){
x = (x >> 13) ^ x;
int xx = (x * (x * x * 60493 + 19990303) + 1376312589) & 0x7fffffff;
return 1.0 - (float(xx) / 1073741824.0);
This function requires our input value to be an integer (hence making it a lattice), we then use bit-wise operators as explained in GLSL specs[2]. I have no clue what is really happening with the
bit-wise stuff other then we are shifting the number around… sorry I dont know more. The numbers that we used are right from Hugos example [1] and are prime numbers. You can change these numbers all
you want, just make sure you keep them as prime in order to prevent as noticeable of graphic artifacts. From here we just need to decide how we want to interpolate the values between points.
Its all up to Interpolation…
The simplest way to interpolate is linearly like what we used above is represented by this equation:
Mock CodeGLSL Code
function Linear_Interpolate(a, b, x)
return a*(1-x) + b*x
end of function
float linearInterp(float a, float b, float x){
return a*(1.-x) + b*x;
This is ok if we want sharp elements, but if we want smoother transitions we can use a cosine interpolation.
function Cosine_Interpolate(a, b, x)
ft = x * 3.1415927
f = (1 - cos(ft)) * .5
return a*(1-f) + b*f
end of function
float cosInterp(float a, float b, float x){
float ft = x*3.12159276;
float f = (1.0 - cos(ft)) * .5;
return a*(1.-f) + b*f;
There is also cubic iterp, but we will skip that for now and focus on linear and cosine. The last thing we will want to do, in order to make our noises smoother on their transitions is introduce a
you guessed it smoothing function. This function can optionally be used and can be expanded to how ever many dimensions you would need. The smoothing helps reduce the appearance of block artifacts
when rendering out to 2+ dimensions. Here is a snip-it of pseudo-code from [1].
//1-dimensional Smooth Noise
function Noise(x)
... .
end function
function SmoothNoise_1D(x)
return Noise(x)/2 + Noise(x-1)/4 + Noise(x+1)/4
end function
//2-dimensional Smooth Noise
function Noise(x, y)
end function
function SmoothNoise_2D(x>, y)
corners = ( Noise(x-1, y-1)+Noise(x+1, y-1)+Noise(x-1, y+1)+Noise(x+1, y+1) ) / 16
sides = ( Noise(x-1, y) +Noise(x+1, y) +Noise(x, y-1) +Noise(x, y+1) ) / 8
center = Noise(x, y) / 4
return corners + sides + center
end function
Later in this section we will look at the differences between smoothed and non-smoothed noise. Now we need to start taking all these elements and put them together. Here is the pseudo code as
exampled by [1]
function Noise1(integer x)
x = (x<<13) ^ x;
return ( 1.0 - ( (x * (x * x * 15731 + 789221) + 1376312589) & 7fffffff) / 1073741824.0);
end function
function SmoothedNoise_1(float x)
return Noise(x)/2 + Noise(x-1)/4 + Noise(x+1)/4
end function
function InterpolatedNoise_1(float x)
integer_X = int(x)
fractional_X = x - integer_X
v1 = SmoothedNoise1(integer_X)
v2 = SmoothedNoise1(integer_X + 1)
return Interpolate(v1 , v2 , fractional_X)
end function
function PerlinNoise_1D(float x)
total = 0
p = persistence
n = Number_Of_Octaves - 1
loop i from 0 to n
frequency = 2i
amplitude = pi
total = total + InterpolatedNoisei(x * frequency) * amplitude
end of i loop
return total
end function
I decided to make a few structural changes to this for the GLSL conversion. In the above example they use four functions to make it happen, we are going to do it with three. I think it will also be
relevant to add uniforms (or defines depends on your preference) to control things like octaves, persistence and a smoothness toggle. I will also be using strictly the cos interpolation, this is by
personal choice any method can be used though. So following the structure of our SM object, we set up the shader argument as follows:
sm = new SM(
size : new BABYLON.Vector2(512, 512),
hasTime : false,
//timeFactor : 0.1,
type : 'float',
value : 4,
min : 0,
max : 124,
step: 1,
hasControl : true
type : 'float',
value : 0.5,
min : 0.001,
max : 1.0,
step: 0.001,
hasControl : true
type : 'float',
value : 1.0,
min : 0,
max : 1.0,
step: 1,
hasControl : true
type : 'float',
value : 1,
min : 0.001,
step: 0.1,
hasControl : true
type : 'vec2',
value : new BABYLON.Vector2(0, 0),
step: new BABYLON.Vector2(1, 1),
hasControl : true
fx :
`precision highp float;
varying vec2 vUV;
varying vec2 tUV;
/*----- UNIFORMS ------*/
uniform float time;
uniform vec2 tSize;
uniform float octaves;
uniform float persistence;
uniform float smoothed;
uniform float zoom;
uniform vec2 offset;
This will set up all of our uniforms and the defaults for them. You can do these as defines, but if having the ability to manipulate it on the fly they should be uniforms. I also added a uniform that
we will not be manipulating directly ever but letting the size of the canvas/texture set this value when the shader is compiled. With this we need to make some changes to our SM object to accommodate
this new uniform.
SM = function(args, scene){
return this;
SM.prototype = {
setSize : function(size){
var canvas = this.scene._engine._gl.canvas;
size = size || new BABYLON.Vector2(canvas.width, canvas.height);
this._size = size;
var pNode = canvas.parentNode;
pNode.style.width = size.x+'px';
pNode.style.height = size.y+'px';
Now the shader will always know what the size of the texture is, because we have made this a inherent feature of the SM object we need to add the uniform for tSize to the default fragment that the
shader has built in. This is in the situation that the default shader get bound that it will validate and compile. From here we need to include our random number function, our interpolation function
and the noise function itself. I am going to include a lerp function as well in case you want to use this and the interpolation vs cos.
//1D Random Value from INT;
float rValueInt(int x){
x = (x >> 13) ^ x;
int xx = (x * (x * x * 60493 + 19990303) + 1376312589) & 0x7fffffff;
return 1. - (float(xx) / 1073741824.0);
//float Lerp
float linearInterp(float a, float b, float x){
return a*(1.-x) + b*x;
//float Cosine_Interp
float cosInterp(float a, float b, float x){
float ft = x*3.12159276;
float f = (1.0 - cos(ft)) * .5;
return a*(1.-f) + b*f;
//1d Lattice Noise
float valueNoise1D(float x, float persistence, float octaves, float smoothed){
float t = 0.0;
float p = persistence;
float frequency, amplitude, tt, v1, v2, fx;
int ix;
for(float i=1.0; i<=octaves; i++){
frequency = i*2.0;
amplitude = p*i;
ix = int(x*frequency);
fx = fract(x*frequency);
if(smoothed > 0.0){
v1 = rValueInt(ix)/2.0 + rValueInt(ix-1)/4.0 + rValueInt(ix+1)/4.0;
v2 = rValueInt(ix+1)/2.0 + rValueInt(ix)/4.0 + rValueInt(ix+2)/4.0;
tt = cosInterp(v1, v2, fx);
tt = cosInterp(rValueInt(ix), rValueInt(ix+1), fx);
t+= tt*amplitude;
return t;
So now we have a GLSL function to generate some 1D noise! It has four arguments, the last one of smoothed can be omitted if you please but I like having it so…. It a fairly simple function and most
of our noise functions will have a similar structure. We could also put a #define in that would control the interpolation, but for simplicity I am just using the cosine method. From here it is as
simple as setting up the main function of our shader program to use this noise. To do this we decide our sampling space and pass that to the x value of the noise function along with our other
uniforms that we have already set up.
void main(void) {
vec2 tPos = ((vUV*tSize)+offset)/zoom;
float v = valueNoise1D(tPos.x, persistence, octaves, smoothed)+1.0/2.0;
vec3 color = vec3(mix(vec3(0.0), vec3(1.0), v));
gl_FragColor = vec4(color, 1.0);
Super easy right!? Our sampling space that we use is the 0-1 uv multiplied by the size of the texture, which effectively shifts us to texel space. The choice to use to vUV instead of the tUV was
because for some reason the negative value was creating an artifact as seen here:
I could try to trouble shoot that, but instead its just easier to use the 0-1 uv range and move on.
Next add an offset which is also in texel space, you could do it as a percentage of the texture’s size but that is user preference. We then divide the whole thing by a zoom value. That gives us a
nice sampling space, which we then pass to our noise function with our other arguments. Because the noise function returns a number between negative 1 and positive 1, we shift it to a 0-1 range by
simply adding one then dividing the sum by two.
A New Dimension
One dimensional noise is cool and has its uses, but we need more room for activities. Before we develop more noises and look at different methods for generation having an understanding of how to
extend the noise to n-dimensions is pretty important. For all general purposes all calculations stay the same, you just have to make a couple more of them. It would probably be smart to add a support
function for smoothing the values of the interpolation now that we are working with larger dimensions. The main modifications to the function will be changing some of the variables from floats and
integers to vectors of the same type. The last function to add is a random number generator that takes into consideration the 2 dimensions.
//2D Random Value from INT vec2;
float rValueInt(ivec2 p){
int x = p.x, y=p.y;
int n = x+y*57;
n = (n >> 13) ^ n;
int nn = (n * (n * n * 60493 + 19990303) + 1376312589) & 0x7fffffff;
return 1. - (float(nn) / 1073741824.0);
float smoothed2dVN(ivec2 pos){
return (( rValueInt(pos+ivec2(-1))+ rValueInt(pos+ivec2(1, -1))+rValueInt(pos+ivec2(-1, 1))+rValueInt(pos+ivec2(1, 1)) ) / 16.) + //corners
(( rValueInt(pos+ivec2(-1, 0)) + rValueInt(pos+ivec2(1, 0)) + rValueInt(pos+ivec2(0, -1)) + rValueInt(pos+ivec2(0,1)) ) / 8.) + //sides
(rValueInt(pos) / 4.);
//2d Lattice Noise
float valueNoise(vec2 pos, float persistence, float octaves, float smoothed){
float t = 0.0;
float p = persistence;
float frequency, amplitude, tt, v1, v2, v3, v4;
vec2 fpos;
ivec2 ipos;
for(float i=1.0; i<=octaves; i++){
frequency = i*2.0;
amplitude = p*i;
ipos = ivec2(int(pos.x*frequency), int(pos.y*frequency));
fpos = vec2(fract(pos.x*frequency), fract(pos.y*frequency));
if(smoothed > 0.0){
ivec2 oPos = ipos;
v1 = smoothed2dVN(oPos);
oPos = ipos+ivec2(1, 0);
v2 = smoothed2dVN(oPos);
oPos = ipos+ivec2(0, 1);
v3 = smoothed2dVN(oPos);
oPos = ipos+ivec2(1, 1);
v4 = smoothed2dVN(oPos);
float i1 = cosInterp(v1, v2, fpos.x);
float i2 = cosInterp(v3, v4, fpos.x);
tt = cosInterp(i1, i2, fpos.y);
float i1 = cosInterp(rValueInt(ipos), rValueInt(ipos+ivec2(1,0)), fpos.x);
float i2 = cosInterp(rValueInt(ipos+ivec2(0,1)), rValueInt(ipos+ivec2(1,1)), fpos.x);
tt = cosInterp(i1, i2, fpos.y);
t+= tt*amplitude;
return t;
There we have it, there are definitely some problems with this method that if we took some time and refined this could be fixed. These problems are things like artifacts as the noise transfers from a
positive to a negative coordinate range which is apparent the more you zoom in and noticeable circular patterns the closer to 0,0 we get. In order to fix this quickly and essentially ‘ignore’ that
problem we just add a large offset to the noise initially and screw our coordinates to be far away from the artifacts.
As a challenge see if you can change the interpolation function to be cubic. Read the section on it here [3].
You can also see a live example of the 2d Lattice Noise here.
Better Noise from Gradients
1. Hugo Elia’s “Perlin noise” implementation. Value Noise, mislabeled as Perlin noise.
2. https://www.khronos.org/registry/OpenGL/specs/gl/GLSLangSpec.1.20.pdf
3. C# Noise
Procedural Investigations in webGL – Section III
Section III
Advance Spaces, Time and Polar
With our new SM object put together we now have the ability to start putting together a collection of generators and other GLSL functions to create more dynamic content. If we go back to our
reference book [1] starting on page 46 it starts reviewing some interesting methods we will recreate now.
Star of My Eye
For practice a great project is to create a star shape. At first glance one might think that it would be tough to generate something like this, but once we shift our sampling space to be in polar
coordinates through cos/sin (sinusoidal) calculations. Again my version will be a variation of a segment of script meant for Renderman. I will show the RSL (Renderman Shader Language) version from [1
] next to my GLSL version then review the differences. I would recommend going through line by line and try to recreate this on your own.
surface star(
uniform float Ka = 1;
uniform float Kd = 1;
uniform color starcolor = color (1.0000,0.5161,0.0000);
uniform float npoints = 5;
uniform float sctr = 0.5;
uniform float tctr = 0.5;
point Nf = normalize(faceforward(N, I));
color Ct;
float ss, tt, angle, r, a, in_out;
uniform float rmin = 0.07, rmax = 0.2;
uniform float starangle = 2*PI/npoints;
uniform point pO = rmax*(cos(0),sin(0), 0);
uniform point pi = rmin*(cos(starangle/2),sin(starangle/2),0);
uniform point d0 = pi - p0; point d1;
ss = s - sctr; tt=t- tctr;
angle = atan(ss, tt) + PI;
r = sqrt(ss*ss + tt*tt);
a = mod(angle, starangle)/starangle;
if (a >= 0.5) a = 1 - a;
dl = r*(cos(a), sin(a),0) - p0;
in_out = step(0, zcomp(d0^d1) );
Ct = mix(Cs, starcolor, in_out);
/* diffuse (“matte”) shading model */
Oi = Os;
Ci = Os * Ct * (Ka * ambient() + Kd * diffuse(Nf));
precision highp float;
varying vec2 vUV;
varying vec2 tUV;
/*----- UNIFORMS ------*/
#define PI 3.14159265359
uniform vec3 starColor;
uniform float nPoints;
uniform float rmin;
uniform float rmax;
uniform float aaValue;
uniform vec3 bgColor;
void main(void) {
vec2 offsetFix = vec2(0.5);
float ss, tt, angle, r, a;
vec3 color = bgColor;
float starAngle = 2.*PI/nPoints;
vec3 p0 = rmax*vec3(cos(0.),sin(0.), 0.);
vec3 p1 = rmin*vec3(cos(starAngle/2.),sin(starAngle/2.), 0.);
vec3 d0 = p1 - p0;
vec3 d1;
ss = vUV.x - offsetFix.x; tt = (1.0 - vUV.y) - offsetFix.y;
angle = atan(ss, tt) + PI;
r = sqrt(ss*ss + tt*tt);
a = mod(angle, starAngle)/starAngle;
if (a >= 0.5){a = 1.0 - a;}
d1 = r*vec3(cos(a), sin(a), 0.) - p0;
float in_out = smoothstep(0., aaValue, cross(d0 , d1).z);
color = mix(color, starColor, in_out);
gl_FragColor = vec4(color, 1.0);
Some of the values in the RSL version are irrelevant to us, like Ka, Kd, Nf, Oi, Ci and any other variables accosted with a light model. Things that are important to us are the number of points the
star will have, its size limits, and our colors. Lets go through the GLSL version line by line and understand what is going on.
First we have our precision mode, which we want as accurate as floats as possible so we keep it as highp. The varying section is pretty standard, we could remove the tUV as its not used but we will
keep it as you may want to sample in the -1 to 1 range instead of 0 to 1 in some instances. No additional methods need to be defined. The uniforms section includes a definition for PI, because we are
going to be working in polar space and be using calculations dependent on circular/spherical values. GLSL does not define this value inherently and so it is up to us to make sure we have a value we
can reference; the cool part about this is we can experiment with funky values and see how that effects our calculations (PI = 4, for example).
The main function of the program starts with up setting a value for the offset fix of the star, which we will use later to move the sample into a scope that will ‘center’ the star. Then we define a
few floats that will be used later, they could be defined at time of execution of the line this just makes things more readable. I cant even lie, I do not understand this math in the slightest… I
understand some of it, but for the most part I just translated it from RSL to GLSL. Even the explanation in [1] is kinda crap as well. If any math buffs are reading this and want to do a break down
of wtf is going on with these numbers and can send me an email, I will love you long time.
At the very least here is a snippet of the summary from [1]:
“To test whether (r,a) is inside the star, the shader finds the vectors d0 from the
tip of the star point to the rmin vertex and d1 from the tip of the star point to the
sample point. Now we use a handy trick from vector algebra. The cross product of
two vectors is perpendicular to the plane containing the vectors, but there are two
directions in which it could point. If the plane of the two vectors is the (x, y) plane,
the cross product will point along the positive z-axis or along the negative z-axis.
The direction in which it points is determined by whether the first vector is to the left
or to the right of the second vector. So we can use the direction of the cross product
to decide which side of the star edge d0 the sample point is on.”
Yeah… what that says… Its pretty much a distance function, anyways one improvement I included was the ability to anti-alias the edges. This is very simple, we just change out the step calculation for
a smoothstep one with a decently low value to represent the tolerance.
There are other ways to go about this but for now this will do. Mess around with this a little bit see what you can figure out. For a live example you can go here.
Head in the Clouds & Introducing Time
Another common procedural processes would be the creation 2d/3d clouds. There are way to many solutions for this then I could count, but a very simple implementation would be to layer multiple
sinusoidal functions at different frequencies. I think now would be a good time to implement some time shifting to our shader as well. We will use this time shift to animate the clouds. Once again
lets take a look at the RSL version provided in [1] and compare it to my GLSL solution.
#define NTERMS 5
surface cloudplane(
color cloudcolor = color (1,1,1);
color Ct;
point Psh;
float i, amplitude, f;
float x, fx, xfreq, xphase;
float y, fy, yfreq, yphase;
uniform float offset = 0.5;
uniform float xoffset = 13;
uniform float yoffset = 96;
Psh = transform(“shader”, P);
x = xcomp(Psh) + xoffset;
y = ycomp(Psh) + yoffset;
xphase = 0.9; /* arbitrary */
yphase = 0.7; /* arbitrary */
xfreq = 2 * PI * 0.023;
yfreq = 2 * PI * 0.021;
amplitude = 0.3;
f = 0;
for (i = 0; i < NTERMS; i += 1) {
fx = amplitude * (offset + cos(xfreq * (x + xphase)));
fy = amplitude * (offset + cos(yfreq * (y + yphase)));
f += fx * fy;
xphase = PI/2 * 0.9 * cos (yfreq * y);
yphase = PI/2 * 1.1 * cos (xfreq * x);
xfreq *= 1.9+i* 0.1; /* approximately 2 */
yfreq *= 2.2-i* 0.08; /* approximately 2 */
amplitude *= 0.707;
f = clamp(f, 0, 1);
Ct = mix(Cs, cloudcolor, f);
Oi = Os;
Ci = Os * Ct;
precision highp float;
uniform float time;
varying vec2 vUV;
varying vec2 tUV;
/*----- UNIFORMS ------*/
#define PI 3.14159265359
uniform vec3 cloudColor;
uniform vec3 bgColor;
uniform float zoom;
uniform float octaves;
uniform float amplitude;
uniform vec2 offsets;
void main(void) {
float f = 0.0;
vec2 phase = vec2(0.9*time, 0.7);
vec2 freq = vec2(2.0*PI*0.023, 2.0*PI*0.021);
float offset = 0.5;
vec2 pos = vec2(vUV.x+offsets.x, vUV.y+offsets.y);
float scale = 1.0/zoom;
pos.x = pos.x*scale + offset + time;
pos.y = pos.y*scale + offset - sin(time*0.32);
float amp = amplitude;
for(float i = 0.0; i < octaves; i++){
float fx = amp * (offset + cos(freq.x * (pos.x + phase.x)));
float fy = amp * (offset + cos(freq.y * (pos.y + phase.y)));
f += fx * fy;
phase.x = PI/2.0 * 0.9 * cos(freq.y * pos.y);
phase.y = PI/2.0 * 1.1 * cos(freq.x * pos.x);
amp *= 0.602;
freq.x *= 1.9 + i * .01;
freq.y *= 2.2 - i * 0.08;
f = clamp(f, 0., 1.);
vec3 color = mix(bgColor, cloudColor, f);
gl_FragColor = vec4(color, 1.0);
This is a very specific form of procedural generation that relies on a method called Spectral Synthesis. This process is described by the theory of Fourier analysis which states that functions can be
represented as a sum several sinusoidal terms. We sample these functions at different frequencies and phases to generate a result. The main struggle with this method is preventing tiling or
noticeable patterns which ruin the effect. The implementation of this is very limited as it relies on quite a few “magic numbers” and is not as customization as more modern solutions using noise
The major difference with the GLSL version that I have introduced here is the animation aspect. We achieve this first by making some modifications to our SM object to accommodate.
SM = function(args, scene){
this.uID = 'shader'+Date.now();
this.hasTime = args.hasTime || false;
this.timeFactor = args.timeFactor || 1;
this._time = 0;
this.shader = null;
SM.prototype = {
setTime : function(delta){
this._time += delta*this.timeFactor;
this.shader.setFloat('time', this._time);
var d = scene.getAnimationRatio();
Then we add the arguments to when we call our new SM object.
sm = new SM(
size : new BABYLON.Vector2(512, 512),
hasTime : true,
timeFactor : 0.1,
We could simply add a value to the time variable, but in order to sync it between different clients we use BJS method of scene.getAnimationRatio(). This should keep the shaders time coordinates at
the same value if they started at the same time but have different thread speeds. Mess around with this generator and try different stuff out just to get more comfortable with what is going on.
For a live example you can go here.
Continue to Section IV
1. TEXTURING & MODELING – A Procedural Approach (third edition)
Procedural Investigations in webGL – Section II
Section II:
Uniforms and UI
With the ability to create the likeness of a brick wall we can now start adding some controls that will allow the testing of various parameter values in real time. There would be a multitude of ways
to handle this the most simple being using HTML DOM elements. If you are feeling froggy you could attempt to use the BABYLON.GUI system, which is GPU accelerated. The first steps will be to extend
our SM object to be able to add controls quickly.
A Uniform Argument
Right away we go to where the SM object is constructed, go to the argument object and then add a new variable for the uniform. It is here we will define the uniforms name, type, value, and any
constraints that will be used later with the UI.
sm = new SM(
brickCounts : {
type : 'vec2',
value : new BABYLON.Vector2(6,12),
min : new BABYLON.Vector2(1,1),
step : new BABYLON.Vector2(1,1),
hasControl : true
fx :...
Then we need to give the SM object instructions on what to do with this new argument.
SM = function(args, scene){
this.shaders.fx = args.fx || this.shaders.fx;
this.uniforms = args.uniforms || {};
this.uID = 'shader'+Date.now();
return this;
Now that the argument is stored on the object, we need to modify some of the object methods to accommodate for the new uniforms. The function most effected by this change is the buildShader method.
We need to make sure that when we bind our shader that we include our new uniforms and then set their default values.
buildShader : function(){
var _uniforms = ["world", "worldView", "worldViewProjection", "view", "projection"];
_uniforms = _uniforms.concat(this.getUniformsArray());
var shader = new BABYLON.ShaderMaterial("shader", scene, {
vertex: uID,
fragment: uID,
attributes: ["position", "normal", "uv"],
uniforms: _uniforms
Then to make these changes work we need to define a method for grabbing the array of uniform names that are assigned. We could simply use Object.keys(this.uniforms); everytime we wanted to get that
array of names, but that is a little ugly and redundant.
SM.prototype = {
getUniformsArray : function(){
var keys = Object.keys(this.uniforms);
return keys;
buildOutput :...
Before we go to much farther, it would be prudent to modify our fragment shader being passed in the arguments to accommodate for this new uniform otherwise when we try to set the default value the
shader will not compile. We also have no need for the #define XBRICKS and #define YBRICKS, with the new uniform effectively replacing these variables.
fx :
`precision highp float;
/*----- UNIFORMS ------*/
uniform vec2 brickCounts;
#define MWIDTH 0.1
void main(void) {
vec3 brickColor = vec3(1.,0.,0.);
vec3 mortColor = vec3(0.55);
vec2 brickSize = vec2(
vec2 pos = vUV/brickSize;
vec2 mortSize = 1.0-vec2(MWIDTH*(brickCounts.x/brickCounts.y), MWIDTH);
pos += mortSize*0.5;
if(fract(pos.y * 0.5) > 0.5){
pos.x += 0.5;
pos = fract(pos);
vec2 brickOrMort = step(pos, mortSize);
vec3 color = mix(mortColor, brickColor, brickOrMort.x * brickOrMort.y);
gl_FragColor = vec4(color, 1.0);
If you are following along, and were to refresh the page now you would most likely see a solid grey page. This is because the shader can bind no problem as there should not be any errors but with the
brick counts set to 0 the math fails. We solve this by doing a little more work on the SM object to have it set the defaults values of the uniforms after the shader is bound.
SM.prototype = {
getUniformsArray : ...
setUniformDefaults : function(){
var shader = this.shader;
var keys = this.getUniformsArray();
for(var i=0; i<keys.length i var u="this.uniforms[keys[i]];" type="u.type;" v="u.value;" shader shader.settype uniform value type2method : function m switch case break return buildoutput :... buildshader ... this.shader="shader;" this.setuniformdefaults></keys.length>
This may look a little intimidating, but its really not. First we get the key values of the uniforms (the names). Then we iterate through these keys now and grab the default value and the type. Once
we have the type we need to get back the associated method that BJS has for setting the uniforms on the shader. In this situation the line “shader[this.type2Method(type)](keys[i], v);” essentially
becomes shader.setVector2(‘brickCounts’, BABYLON.Vector2(#,#)); If everything is correct when we refresh the page now we should see whatever number of bricks we set as the default values on the
constructors arguments. Feel free to change these numbers up and refresh the page to verify everything is working. You can look HERE for reference or to download this step.
With everything lined up and working, its now time to get the UI elements constructed. Eventually you might want to develop your own user interface components, but the process I am about to show you
should cover most cases. For simplicity of code understanding I am going to write out some sections of code that repeat with little variation. Normally want to have function handle these repeat
sections, but it will be easier to understand initially to do it long handed. The creation of the UI can be easily be expanded upon in the future, but to get started we create another method on our
SM object, then call it after the creation of the output on the initialization. Now would also be a good time to define a quick support method to return the current “this.shader”.
SM = function(args, scene){
return this;
SM.prototype = {
getShader : function(){
return this.shader;
buildGUI : function(){
this.ui = {
mainBlock : document.createElement('div'),
inputs : [],
var keys = this.getUniformsArray();
The purpose of this method will be to iterate through the SM object’s uniform object keys, create all the appropriate DOM elements, append them and then set a function up to respond to change events.
So we set up a new container object for the ui elements, then grab the uniform keys with our getUniformArray method. Once we have our keys to iterate through we proceed to parse the uniforms object
var keys = this.getUniformsArray();
for(var i=0; i<keys.length i var u="this.uniforms[keys[i]];" if _block="document.createElement('div');" _block.classlist.add _title="document.createElement('span');" _title.innerhtml='keys[i]+":";' _block.appendchild _inblock="document.createElement('span');" _inblock.classlist.add _inputs="[];" _in _in.setattribute keys _in.classlist.add _in.value="u.value;" u.min.x u.max.x u.step.x _inputs.push u.type="=" u.min.y u.max.y u.step.y u.min.z u.max.z u.step.z for j="0;" _inblock.appendchild _input="{" block : inputs this.ui.inputs.push this.ui.mainblock.appendchild document.body.appendchild ...></keys.length>
With this added into our method, we can now (hopefully) support the creation of DOM inputs for floats, vector2, vector3 parameters. I have not tested any of it yet and am kinda writing all of this as
we go so bare with me if their are any bugs and you are reading a version that is not finalized/debugged. But as far as I can tell right now this should work. If we were to refresh the page you would
not see any changes, unless you look at the source. In order to see the changes we will need to provide some CSS. You can simply copy this next section and modify it how ever you want.
Upon a refresh now, we should see our UI elements for the brickCounts Uniform on the top left of our page. Then we go back to our buildGUI method in order to script the responses to change events on
the ui block.
var self = this;
function updateShaderValue(id, value){
self.uniforms[id[0]].value[id[1]] = parseFloat(value);
(self.getShader()).setVector2(id[0], self.uniforms[id[0]].value);
}else if(id[1]=='vec3'){
(self.getShader()).setVector3(id[0], self.uniforms[id[0]].value);
self.uniforms[id[0]].value = parseFloat(value);
(self.getShader()).setFloat(id[0], self.uniforms[id[0]].value);
this.ui.mainBlock.addEventListener('change', (e)=>{
var target = e.target;
var id = target.getAttribute('id').split(':');
var value = target.value;
updateShaderValue(id, value);
}, false);
Voilà it is done… partially. Upon refreshing the page then changing one of the values in our inputs we should instantly see the values in our shader being updated(effecting the output). Now to go
back and add support for a few more parameters like the colors and the mortar width. If we set up everything correctly we can now just edit our arguments and change the fragment shader slightly.
brickCounts : {
type : 'vec2',
value : new BABYLON.Vector2(6,12),
min : new BABYLON.Vector2(1,1),
step : new BABYLON.Vector2(1,1),
hasControl : true
mortarSize : {
type: 'float',
value : 0.1,
min: 0.0001,
max: 0.9999,
step: 0.0001,
hasControl: true
brickColor : {
type: 'vec3',
value : new BABYLON.Vector3(0.8, 0.1, 0.1),
min: new BABYLON.Vector3(0, 0, 0),
max: new BABYLON.Vector3(1, 1, 1),
step: new BABYLON.Vector3(0.001, 0.001, 0.001),
hasControl: true
mortColor : {
type: 'vec3',
value : new BABYLON.Vector3(0.35, 0.35, 0.35),
min: new BABYLON.Vector3(0, 0, 0),
max: new BABYLON.Vector3(1, 1, 1),
step: new BABYLON.Vector3(0.001, 0.001, 0.001),
hasControl: true
fx :
`precision highp float;
varying vec2 vUV;
varying vec2 tUV;
float pulse(float a, float b, float v){
return step(a,v) - step(b,v);
float pulsate(float a, float b, float v, float x){
return pulse(a,b,mod(v,x)/x);
float gamma(float g, float v){
return pow(v, 1./g);
float bias(float b, float v){
return pow(v, log(b)/log(0.5));
float gain(float g, float v){
if(v < 0.5){
return bias(1.0-g, 2.0*v)/2.0;
return 1.0 - bias(1.0-g, 2.0 - 2.0*v)/2.0;
/*----- UNIFORMS ------*/
uniform vec2 brickCounts;
uniform float mortarSize;
uniform vec3 brickColor;
uniform vec3 mortColor;
void main(void) {
vec2 brickSize = vec2(
vec2 pos = vUV/brickSize;
vec2 mortSize = 1.0-vec2(mortarSize*(brickCounts.x/brickCounts.y), mortarSize);
pos += mortSize*0.5;
if(fract(pos.y * 0.5) > 0.5){
pos.x += 0.5;
pos = fract(pos);
vec2 brickOrMort = step(pos, mortSize);
vec3 color = mix(mortColor, brickColor, brickOrMort.x * brickOrMort.y);
gl_FragColor = vec4(color, 1.0);
Now that is procedural, take a little bit of time to mess around with this and experiment with some different values now that you can see the changes instantly! If you are having trouble getting to
this point you can review and/or download the source here.
All these parameters and options for bricks is cool and all… but now we should start making the changes necessary to make the texture exportable which will be the easiest way to take this texture we
made from mock up to production. Eventually a good end goal would be to include these procedural process in your project and have them compile on runtime, which could save tons of space when saving/
serving the project to a client if used correctly. But that is much much later, for now lets focus on making the ability to set the textures size and then saving it from the browser. Because the HTML
canvas object is processed by the browser for all intensive purposes as an image, we can simply right click on the canvas and save it! After a couple quick changes to our SM object and a small change
to the DOM structure, we can add one additional argument to set the size of the canvas to a specific unit.
//DOM CHANGES
<div id="output-block">
//SM OBJECT CHANGES
SM = function(args, scene){
return this;
SM.prototype = {
setSize : function(size){
var canvas = this.scene._engine._gl.canvas;
var pNode = canvas.parentNode;
pNode.style.width = size.x+'px';
pNode.style.height = size.y+'px';
//ADD ARGUMENT
sm = new SM(
size : new BABYLON.Vector2(512, 512),
The one thing we need to make sure we do when we change the size of the canvas manually, is to fire the engines resize function to get the gl context into the same dimensions. We then rebuild the
output just to be safe. Now we have a useful brick wall generator that we can export textures from for later use. Here is the final source for this section and a live example of the generator we just
Continue to Section III
Procedural Investigations in webGL – Section I
Section I:
Sampling Space and Manipulations
Now that we have a basic development environment set up, it would be prudent to review different methods for sampling and manipulating the coordinate system that dictates the output of our procedural
processes. We will be basically reviewing built in functions to glsl that will help us in controlling our sampling space.
What is sampling space? You can think of it as the coordinate system/space that we will use as the value that we feed to our noise/procedural algorithms. This can be anything that effectively want
from a singular value to a n-dimensional location…. In most of our cases we will be using the vPosition or the vUV as our coordinate space, even though there are other special situations that may
dictate you use a difference system. You can review this concept starting on page 24 of [1] where they explain coordinate space with these points:
• The current space is the one in which shading calculations are normally done.
In most renderers, current space will turn out to be either camera space or
world space, but you shouldn’t depend on this.
• The world space is the coordinate system in which the overall layout of your
scene is defined. It is the starting point for all other spaces.
• The object space is the one in which the surface being shaded was defined. For
instance, if the shader is shading a sphere, the object space of the sphere is the
coordinate system that was in effect when the RiSphere call was made to create
the sphere. Note that an object made up of several surfaces all using the same
shader might have different object spaces for each of the surfaces if there are
geometric transformations between the surfaces.
• The shader space is the coordinate system that existed when the shader was in-
voked (e.g., by an RiSurface call). This is a very useful space because it can be
attached to a user-defined collection of surfaces at an appropriate point in the
hierarchy of the geometric model so that all of the related surfaces share the
same shader space.
Which if you ask me is overkill on the explanation. It all boils down to what values you choose to reference when feeding out procedural algorithms. If we have a 3d noise for example we would most
likely use the vPosition which is an xyz value for that pixels location in the 3d scene locally, if you used gl_FragCoord, I believe that would be global (do not quote me on this). By making some
quick changes to our page and changing the argument that we initialize the objects fragment shader to something like this:
precision highp float;
varying vec2 vUV;
void main(void) {
vec3 color = vec3(vUV.x, vUV.y, vUV.y);
gl_FragColor = vec4(color, 1.0);
With everything in its place we can see now when we refresh the page, a gradient that should look like this:
What this is showing us is that our UV is set up correctly, as the lower left corner is black where vUV.x & vUV.y == 0; white where they are 1; Red where x is 1 & y is 0; and finally Cyan where y is
1 and x is 0. We are directly effecting the color by the uv values our very first procedural (explicit) texture!
Now that we can have established our coordinate space, how can me manipulate it do to our biding.
There is a collection of methods available to us in glsl, but lets take a look at which ones [1] mentions starting on page 27.
step generates a step function by comparing x to edge.
For element i of the return value, 0.0 is returned if x[i] < edge[i], and 1.0 is returned otherwise.
We can also define a method that uses the step function to generate what is known as a pulse by doing the following:
float pulse(float a, float b, float v){
return step(a,v) – step(b,v);
Which makes everything outside of the range between a-b come up as 1 and anything outside as 0. This gives us the ability to effectively create a rectangle in what ever range we decide.
clamp returns the value of x constrained to the range minVal to maxVal. The returned value is computed as min(max(x, minVal), maxVal).
The next method does not have much use unless you are using a coordinate system that has negative values. Normally for sampling coordinates you will want to work in a -1 to 1 range and not a 0 to 1
range, so lets adjust the default vertex shader to have a new varying variable that has the uv transposed to this range.
`precision highp float;
attribute vec3 position;
attribute vec2 uv;
// Uniforms
uniform mat4 worldViewProjection;
varying vec2 vUV;
varying vec2 tUV;
void main(void) {
vec4 p = vec4( position, 1. );
gl_Position = worldViewProjection * p;
vUV = uv;
tUV = uv*2.0-1.0;
abs returns the absolute value of x.
smoothstep performs smooth Hermite interpolation between 0 and 1 when edge0 < x < edge1. This is useful in cases where a threshold function with a smooth transition is desired. smoothstep is
equivalent to: genType t; /* Or genDType t; */ t = clamp((x - edge0) / (edge1 - edge0), 0.0, 1.0); return t * t * (3.0 - 2.0 * t); Results are undefined if edge0 ≥ edge1.
mod returns the value of x modulo y. This is computed as x – y * floor(x/y).
With the mod method and the pulse function that we created, we can now create another function to create a “pulsate” function
1. TEXTURING & MODELING – A Procedural Approach (third edition)
The zero and one end points of the interval are mapped to themselves. Other values are shifted upward toward one if gamma is greater than one, and shifted downward toward zero if gamma is between
zero and one.[1]
Perlin and Hoffert (1989) use a version of the gamma correction function that they call the bias function.[1]
Regardless of the value of g, all gain functions return 0.5 when x is 0.5. Above and below 0.5, the gain function consists of two scaled-down bias curves forming an S-shaped curve. Figure 2.20 shows
the shape of the gain function for different choices of g.[1]
There are quite a few more (sin, cos, tan, etc) but we will cover those more later, if you want to go over more now check out: http://www.shaderific.com/glsl-functions/. These that I have presented
here though should be enough to start making some more dynamic of sampling spaces. With all this at our disposal what is something that we could make of use? A pretty standard texture in the
procedural world would be a brick or checkerboard pattern, so lets start there.
Oh Bricks
Shamelessly this is a reproduction of the bricks presented in [1] (page 39) with a few changes made. Before creating anything lets take a look at what identifiable elements that we are trying to
produce. A brick of course and its size in relation to the whole element, the mortar or padding around it and then its offset in relation to the other rows. To get started lets define a few variables
(on the fx fragment we are passing to the SM object) to define the number of bricks we wish to see. This is different then our reference script but I feel is easier to understand and we can derive
all our other size numbers from them. Plus there is the added advantage doing it this way, we can make sure the texture is repeatable on all sides.
#define XBRICKS 2.
#define YBRICKS 4.
#define MWIDTH 0.1
void main(void) {
Super simple right? We define the counts as floats for simplicity cause who likes working with integers and having to convert them every time you want to do quick maths, heh…. Now that we have the
basic numbers to base everything off of lets set up some colors and then get our sampling space into scope.
void main(void) {
vec3 brickColor = vec3(1.,0.,0.);
vec3 mortColor = vec3(0.55);
vec2 brickSize = vec2(
vec2 pos = vUV/brickSize;
if(fract(pos.y * 0.5) > 0.5){
pos.x += 0.5;
pos = fract(pos);
float x = pos.x;
float y = pos.y;
vec3 color = vec3(x, y, y);
gl_FragColor = vec4(color, 1.0);
Now we have set up our sampling space by first dividing our max coordinate unit by the brick count. Then we divide the coordinate space we are using by the brick size. This would give us coordinate
space that now has values ranging from 0 to the number of bricks we accounted for. After checking the vertical positions fractional half value and seeing if it is over 0.5 we are able to identify the
alternating rows, which we then offset the x position by half of the coordinate space. Then we transpose the position range because the only thing we are worried about in this range is the fractional
sections of the values not the whole number. The mortar size we will take into account after the bricks are in place so that way we can keep the padding around the bricks constant by using some ratio
If you are following along and refresh your page now you should see an image similar to:
Now with this basic grid set up, we can take into consideration the position of our mortar around the bricks and start the process of coloring everything. A quick way to figure this out will be to
just define a vec2 with our mortar size, then do a quick step calculation on our set up coordinate space to see if its brick or not. We then mix the brick and mortar colors together with the mix
value being set as the result of multiplying the step calculation we just made. The cool thing about the step multiplication is it will turn the mix value to 0 anytime the step calculation are
outside of the brick area.
void main(void) {
vec3 brickColor = vec3(1.,0.,0.);
vec3 mortColor = vec3(0.55);
vec2 brickSize = vec2(
vec2 pos = vUV/brickSize;
vec2 mortSize = vec2(MWIDTH);
if(fract(pos.y * 0.5) > 0.5){
pos.x += 0.5;
pos = fract(pos);
vec2 brickOrMort = step(pos, mortSize);
vec3 color = mix(mortColor, brickColor, brickOrMort.x * brickOrMort.y);
gl_FragColor = vec4(color, 1.0);
If we refresh now, we will see it is close but no cigar… why is this? I quick hint would be that it seems the mortar value must be off, and I know this is a crap explanation but just by looking at it
I knew the solution was to inverse value so our mortSize line becomes:
vec2 mortSize = 1.0-vec2(MWIDTH);
With a page reload we now see something like this:
Getting closer! The first thing that we notice is the mortar is thicker on it height vs width ratio. This is super easy to fix by manipulating our mortSize to reflect the same ratio of bricks x:y.
Making our variable become:
vec2 mortSize = 1.0-vec2(MWIDTH*(XBRICKS/YBRICKS), MWIDTH);
This will make our mortar lines keep the same padding around the bricks and makes our procedural texture almost complete! The one last thing I would like to add would be an offset of the entire
coordinate space to shift both the rows and columns by half of the mortar size in order to “center” the repetition properties of this texture. It is a optional line and is up to the developer to
decide if they want to use it or not!
vec2 mortSize = 1.0-vec2(MWIDTH*(XBRICKS/YBRICKS), MWIDTH);
pos += mortSize*0.5;
Thats it for now! You have officially created your first real procedural texture, not just a gradient or a solid color. This shader can now be extended upon and made way more dynamic. If you are
having trouble getting good results please reference or download the Example
This concludes this section, in the next one we will discuss setting up controls and parameters for real time manipulation of the texture to debug/test different values.
Continue to Section II
1. TEXTURING & MODELING – A Procedural Approach (third edition)
Procedural Investigations in webGL – Introduction
In modern times the need to generate robust procedural content has become more prevalent then ever. With advancements in CPU/GPU processing power the ability to create dynamic content on the fly is
now an option more then ever. What started out as a means to produce simple representations of natural processes has now grown into a multi faceted field, ranging from producing pseudo random
procedural content to synthesized textures and models constructed from reference data. Whole worlds can be crafted from a single simple seed. Using methods that are often simplified from real world
physics and systems from nature, a user is able to try to control the creation process to mold a certain result.
The main complication of this is the control factor, due to the inherent properties of the “random” or “noise” functions that are used to create the data samples. As the artist/developer it is our
goal to understand how we can manipulate this seemly uncontrollable processes to better suit our needs and produce content that is within scope of expectations. We can attempt to create control by
introducing sets of parameters that manipulate the underlying structure of our functions or filter the results.
First off lets get some things straight. I am in no way a math wizard, or even conventionally trained in programming so all of this information that will be presented is based off of my
interpretations of advance topics that I probably have no business explaining to someone else. Do not take any of the concepts I will discuss as verbatim fact, but use them as a basis if you have
none to try to obtain a level of understanding of you’r own. The main point of this article or tutorial (not sure what this would be… a research log?) is to document a laymen interpretation of the
works of genius like Kevin Perlin, Edwin Catmull, Steven Worley….
I recently got my hands on the third edition of the publication: TEXTURING & MODELING – A Procedural Approach [1], which is a great resource though somewhat dated with the languages it uses. I am
going to review the concepts presented in this wonderful resource and tailor the script examples to work with webGL. Currently webGL 2.0 supports GLSL ES 3.00 methods and will be the focus of this
article, if you have any questions about this please review the webGL specifications. I will also be using BABYLON.JS library to handle the webGL environment as opposed to having to worry about all
the buffer binding and extra steps that we would need to take otherwise. There is also the assumption that if you are reading this you have a basic understanding of HTML, CSS and Javascript;
otherwise this is not the tutorial for you.
Setting up the Environment
Before the creation of anything can happen we will need to set up a easy development environment. To do this we are going to create a basic html page, include the babylon library, make a few styling
rules, then finally create a scene that will allow us to develop GLSL code to create and test different effects easily. Though we wont be ray-marching anytime soon but the set-up described by the
legendary Iñigo Quilez in the presentation “Rendering Worlds with Two Triangles with ray-tracing on the GPU in 4096 bytes” [2] will be the same set up we will go with for outputting our initial
tests. Later we will look at deploying the same effects on a 3d object then start introducing lighting (I am dreading the lighting part). To save time please follow along with: http://
doc.babylonjs.com/ and get the basic scene follow the directions to get the basic scene presented running.
Once we have our basic scene going, we are going to reorder and structure some of the elements plus drop out unnecessary elements like the light object at this point. You can follow along here if you
are unfamiliar with BJS alternatively if you just want to get started skip this section and download this page.
Basic Scene
I assume you know how to create a standard web page with a header and body section like as follows:
<title>Introduction - Environment Setup</title>
or you can copy and paste this into you IDE. Right away we are going to get rid of overflow, padding, margin and make it full width/height on the content of the page because for most purposes the
scene we are working on will take up the whole screen. Then in our head section we need to include the reference to BJS.
This should effectively give us all the elements we need to start developing, we just need to create our initial scene in order to get the webGL context initialized and outputting. To do this in the
body section of the page we create a canvas element and give it a unique identifier with some simple css rules then pass that canvas element over to BJS for the webGL initialization.
The touch action rule is for future proofing in case the scene needs to have mobile touch support (which will most likely never come into application with what we are doing) the other rules define
the size of the canvas to be inherent from its parent container. When we later initialize the scene we will fire a function that sets the canvas size from its innerWidth and Height values as
described here [3]. Luckily BJS handles all the resizing as long as we remember to bind the function to fire when the window is resized, but we will cover that when we set up our scene function.
Now its time to get the scene running, we do this inside a script element after the body is created. We also should wrap it in a DOM Content Loaded callback to prevent the creation of the scene from
bogging the page load. Then we create a function that will initialize the very minimum elements BJS requires in a scene in order for it to compile (a camera).
Then inside the createScene function lets add one last line to set effectively the background color of the scene/canvas.
scene.clearColor = new BABYLON.Color3(1,0,0);
return scene;
If everything is set up correctly you can now load the page and should see an entire red screen. If you are having trouble review this and see where the differences are. What we have done here, is
create the engine and scene objects, start the render loop and bind the resize callback on a window resize event. From here we have all the basic elements together to set up a test environment.
Getting Output
With the webGL context initialized and our scene running its render loop its time to create a method of outputting the data we will be creating. To simplify things at first we will create a single
geometric plane also known as a quad that will always take up the whole context of our viewport then create a simple GLSL shader to control its color. There are multiple ways to create the quad but
for learning purposes I think it is prudent to create a Custom Mesh function that creates and binds the buffers for the object manually instead of using a built in BJS method for a plane/ground
creation. The reason we will do it this way is it give us complete control over the data and will give us and understanding of how BJS creates its geometry with its built in methods.
First lets create the function and its constructors, so in the script area before out DOM Content Loaded event binding we make something like the following:
The most important argument for this function will be to pass the scene reference to it, so that way we have access to it within the scope of the function alternatively you could have the scope of
the scene on the global context but that creates vulnerabilities and is not advised. The other benefit of passing the scope as a argument is when we start working with more advance projects that use
multiple scenes we can easily reuse this function.
Now that we have the function declared we can work on its procedure and return. The method for creating custom geometry in BJS is as follows:
Create a blank Mesh Object
var createOuput = function(scene){
var mesh = new BABYLON.Mesh('output', scene);
return mesh;
also in our createScene function and add these two lines:
var output = createOuput(scene);
return scene;
If everything is correct when we reload this page now and check the dev console we should see:
BJS – [timestamp]: Babylon.js engine (v3.1.1) launched
Create/Bind Buffers
Now that there is a blank mesh to work with, we have to create the buffers to tell webGL where the positions of the vertices are, how our indices are organized, and what our uv values are. With those
three arrays/buffers we apply that to our blank mesh and should be able to produce geometry that we will use as our output. Initially we will hard code the size values and see then go back and revise
the function to adjust for viewport size.
var createOuput = function(scene){
var mesh = new BABYLON.Mesh('output', scene);
var vDat = new BABYLON.VertexData();
vDat.positions =
-0.5, 0.5, 0,//0
0.5, 0.5, 0,//1
0.5, -0.5, 0,//2
-0.5, -0.5, 0 //3
vDat.uvs =
0.0, 1.0, //0
1.0, 1.0, //1
1.0, 0.0, //2
0.0, 0.0 //3
vDat.normals =
0.0, 0.0, 1.0,//0
0.0, 0.0, 1.0,//1
0.0, 0.0, 1.0,//2
0.0, 0.0, 1.0 //3
vDat.indices =
return mesh;
If done correctly when the page is refreshed there should be a large black section. The next step will be to have the size of the mesh dynamically be created instead of hard-coded so that way we can
have it work with a resize function. The solution behind this is not my own you can see the discussion that lead up to this here. Modifying the createOuput to reflect the solution is very simple, we
add one line to define our width and height values and then multiply our width and height position values by the respective results.
var c = scene.activeCamera;
var fov = c.fov;
var aspectRatio = scene._engine.getAspectRatio(c);
var d = c.position.length();
var h = 2 * d * Math.tan(fov / 2);
var w = h * aspectRatio;
vDat.positions =
w*-0.5, h*0.5, 0,//0
w*0.5, h*0.5, 0,//1
w*0.5, h*-0.5, 0,//2
w*-0.5, h*-0.5, 0 //3
Now when the page is refreshed it should be solid black, this is because our mesh now takes up the entire camera frustum and there is no light to make the mesh show up hence its black. A light for
our purposes right now is of little use, later we will try to implement lighting. Another thing we will ignore for right now is the repose to a resize for the output. Later as we get more of our
development environment set up we will come back to this.
First Shader
Creating a blank screen is all fine and dandy, but not very handy… So now would be the time to set up our shaders which will be the main program responsible for most of the procedural content methods
we will be developing. Unfortunately webGL 2.0 does not handle geometry shaders only vertex and fragment, hence limiting the GPU to texture creation or simulations, not models. For any geometric
procedural process we will need to rely on the CPU.
This process for creating shaders to work with BABYLON is extremely easy, we simply construct some literal strings and store them in a DOM accessible Object then have BJS work its magic with the
bindings of all the buffers. You can read more about Fragment and Vertex shaders on the BJS website and through a great article written by BAYBLON’s original author on Building Shaders with webGL.
Lets take a look at what we are going to need to develop our first procedural texture, firstly we need some sort of reference unit for this we will use the UV of our mesh transposed from a 0 to 1
range to a -1 to 1 range. Using the UV is advantageous when working with 2D content, if we add the third dimension into the process it becomes more relevant to sample the position as the reference
point. With this idea in mind the basic shader program becomes similar to this:
var vx =
`precision highp float;
attribute vec3 position;
attribute vec2 uv;
// Uniforms
uniform mat4 worldViewProjection;
varying vec2 vUV;
void main(void) {
vec4 p = vec4( position, 1. );
gl_Position = worldViewProjection * p;
vUV = uv;
var fx =
`precision highp float;
varying vec2 vUV;
void main(void) {
vec3 color = vec3(1.,1.,1.);
gl_FragColor = vec4(color, 1.0);
Then we store these literal strings into our BJS shaderStore object, which will allow the library to construct and bind it through its methods. If we were doing this through raw webGL this would add
a bunch of steps but due to this amazing library most of the busy work is eliminated.
BABYLON.Effect.ShadersStore['basicVertexShader'] = vx;
BABYLON.Effect.ShadersStore['basicFragmentShader'] = fx;
Lastly we use the CustomShader creation function and assign the results to the material of our output mesh. As of for right now this is done inside the createScene function after the createOutput
var shader = new BABYLON.ShaderMaterial("shader", scene, {
vertex: "basic",
fragment: "basic",
attributes: ["position", "normal", "uv"],
uniforms: ["world", "worldView", "worldViewProjection", "view", "projection"]
output.material = shader;
If everything is done right, when we refresh the page we should now see a fully white page! WOWZERS so amazing… red, black then white… we are realllly cooking now -_-…. Well at least this is all the
elements we will need to start making some procedural content. If you are having trouble at this point you can always reference here or just download it if you are lazy.
Refining the Environment
At this point it would be smart to reorder our elements and create a container object that will hold all important parameters and functions associated with what ever content we are trying to create.
This way we can make sure the scope is self contained and that we could have multiple instances of our environment on the same page without them conflicting with each other. Object prototyping is
very useful for this, as we can construct the object and have the ability to reference it later by accessing what ever variable we assigned the response of the object to.
The Container Object
If you have never made a JS Object then this might be a little strange. Those familiar with prototyping and object scope should have no problem with this part. In order to organize our data and make
this a valid development process we have to create some sort of wrapper object like such:
SM = function(args, scene){
this.scene = scene;
args = args || {};
this.uID = 'shader'+Date.now();
return this;
This has now added on the window scope a new constructor for our container object. To call this we simply write the string “new SM({}, scene);”, where ever we have access to the scene variable. If we
define this to a variable it will now be assigned the instance of this object with what ever variables assigned to “this” scope being contained within that instance. With this object constructor in
place we can look to extend it now by prototyping some functions and variables into its scope. If you are unfamiliar with this please review the information presented here [4].
The first thing we will add into the prototype is the space for the shaders that our shaderManager (aka shaderMonkey since I’m Pryme8 ^_^) will reference when ever it needs to rebuild and bind the
SM.prototype = {
/*----TAB RESET FOR LITERALS-----*/
`precision highp float;
attribute vec3 position;
attribute vec2 uv;
// Uniforms
uniform mat4 worldViewProjection;
varying vec2 vUV;
void main(void) {
vec4 p = vec4( position, 1. );
gl_Position = worldViewProjection * p;
vUV = uv;
fx :
`precision highp float;
varying vec2 vUV;
void main(void) {
vec3 color = vec3(1.,1.,1.);
gl_FragColor = vec4(color, 1.0);
}//End Shaders
Now that we have the shaders wrapped under the ‘this’ scope of the object we can start migrating some of the elements from when we set up our environment to be contained inside the object as well.
The main elements were the construction of the mesh, the storing and binding of the shader.
SM.prototype = {
storeShader : function(){
BABYLON.Effect.ShadersStore[this.uID+'VertexShader'] = this.shaders.vx;
BABYLON.Effect.ShadersStore[this.uID+'FragmentShader'] = this.shaders.fx;
}//End Shaders
This method simply sets the ShaderStore value on the DOM for BJS to reference when it builds. After adding it to the object its simple to integrate it into the initialization of the object. We can
also take this time to define a response to the user including some custom arguments when they construct the objects instance that will overwrite the default shaders that we hard coded into the SM
SM = function(args, scene){
this.scene = scene;
args = args || {};
this.shaders.vx = args.vx || this.shaders.vx;
this.shaders.fx = args.fx || this.shaders.fx;
this.uID = 'shader'+Date.now();
this.shader = null;
return this;
As this object is created it checks the argument object for variables assigned to vx & fx respectively, if that argument is not present then it keeps the shader the same as the default version. We
set it up this way so that as we start making different scenes that use our SM object we do not have to change any of the objects internal scripting but just use arguments or fire built in methods to
manipulate it.
Now we need to bind the shader to the webGL context so that it accessable/usable. This process is fairly simple once we add another method to our object.
SM = function(args, scene){
return this;
SM.prototype = {
buildShader : function(){
var scene = this.scene;
var uID = this.uID;
var shader = new BABYLON.ShaderMaterial("shader", scene, {
vertex: uID,
fragment: uID,
attributes: ["position", "normal", "uv"],
uniforms: ["world", "worldView", "worldViewProjection", "view", "projection"]
this.shader = shader;
this.output.material = this.shader;
storeShader : function(){
}//End Shaders
When our object is initialized now, it builds/binds the shader and assigned it to its this.shader variable, which is now accessible under this objects scope. It then checks if the object has an
output mesh and if it does it assigns it. Each time this method is fired it checks to see if there is a shader already complied and if there is it disposes it to save on overhead. The last step would
be to migrate the createOutput function to become a method for our SM object and make simple modifications to have it effectively do the same process that our buildShader method does to conserve
SM = function(args, scene){
return this;
SM.prototype = {
buildOutput : function(){
var scene = this.scene;
var mesh = new BABYLON.Mesh('output', scene);
var vDat = new BABYLON.VertexData();
var c = scene.activeCamera;
var fov = c.fov;
var aspectRatio = scene._engine.getAspectRatio(c);
var d = c.position.length();
var h = 2 * d * Math.tan(fov / 2);
var w = h * aspectRatio;
vDat.positions =
w*-0.5, h*0.5, 0,//0
w*0.5, h*0.5, 0,//1
w*0.5, h*-0.5, 0,//2
w*-0.5, h*-0.5, 0 //3
vDat.uvs =
0.0, 1.0, //0
1.0, 1.0, //1
1.0, 0.0, //2
0.0, 0.0 //3
vDat.normals =
0.0, 0.0, 1.0,//0
0.0, 0.0, 1.0,//1
0.0, 0.0, 1.0,//2
0.0, 0.0, 1.0 //3
vDat.indices =
this.output = mesh;
this.output.material = this.shader;
buildShader : function(){
storeShader : function(){
}//End Shaders
That should effectively be it. If we make a couple modifications to our createScene function now, we can test the results.
var sm;
window.addEventListener('DOMContentLoaded', function(){
var createScene = function(){
var scene = new BABYLON.Scene(engine);
var camera = new BABYLON.FreeCamera('camera1', new BABYLON.Vector3(0, 0, -1), scene);
scene.clearColor = new BABYLON.Color3(1,0,0);
/*----TAB RESET FOR LITERALS-----*/
sm = new SM(
fx :
`precision highp float;
varying vec2 vUV;
void main(void) {
vec3 color = vec3(0.,0.,1.);
gl_FragColor = vec4(color, 1.0);
return scene;
If everything is 100% you should now see a fully blue screen when you refresh otherwise please reference here. Now would be a good time to add the resize response as well, which is why we defined our
sm var outside of the DOM content loaded callback. There are better ways to do this, but for our purposes it will do. Simply calling the buildOutput method for the SM object when a DOM element
containing the canvas is resized should handle things nicely.
window.addEventListener('resize', function(){
Finally we have gotten to the point where we can start developing more things then a solid color. We will at some point need to create methods for binding samplers and uniforms on the shader but we
can tackle that later. Feel free to play around with this set up and break or add things, try to understand the scope of the object and how everything is associated. In the next section we will
examine the principles of sampling space and how we can manipulate it.
Continue… to Section I
1. TEXTURING & MODELING – A Procedural Approach (third edition)
2. Rendering Worlds with Two Triangles with raytracing on the GPU in 4096 bytes
3. https://webglfundamentals.org/webgl/lessons/webgl-resizing-the-canvas.html
4. https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/prototype
CitiEz – Test 1
Experiments in Procedural Streets
L-Systems are an ideal way to produce predictable growth, and have been used in recent years to generate entire cities. I have found that the process of using turtle logic is just like its name slow…
and does not have very many options for the branches of the system to make intelligent decisions about their surroundings, maintain a object hierarchy that is transversable, or store any important
variables. They are basically dumb and can be given pseudo intelligence. I started trying to work with the standard format for constructing a library for axioms and found that this was not a very
robust way to make anything of value. I’m going to be honest tons of the stuff I see guys do with L-Systems is crazy, and I have a great amount of respect for it, but I still wanna kinda do things my
way so yeah.
The basic idea is just like an L-System I will construct an “alphabet”. These characters though are just a naming convention for embedded functional objects, that can be constructed and then
reference a rule with the same name type effectively making a dynamic and tailorable constructor class that is deployed on runtime. Inside a basic component of the system (a axiom or road) it holds
all the basic data that is going to be needed to construct our road map, these variables include path information and some other general information. Each Axiom contains a repo, that references other
axioms in the library for when this branch is no longer valid, to spawn. Each Axiom works independently of the others and essentially has restrictions it tries to maintain, if all restrictions are
out of limits then the thread terminates and sends a message to the main script to start the IO on another Axiom.
Right away I can see benefits to this system, as I can store street names, locations, elevations, spatial data, etc. Also in later version I will be looking to extend the Survey area to a larger
zone, and have each block be processed with later optimization functions deciding the route and conditions. I see this being completely valid for the construction of real time procedural content, as
all the calculations can happen on a sub thread and IO’s divided, with the detail slowly building up. This build up could also enable the options for LOD
Test – 1
In my first tests I did not include elevation calculations, the roads only scan their local North, South, East, or West points in the Survey. Population clamp is set to 0.65 or 255*0.65. The roads
will stop if the hit a intersection or can not travel their primary direction and one initial alternate direction that is established on the first turn. This is a very simple model, but the effect is
immediately noticeable.
This is about 14 hours of development and conceptual work in, I am interested to see how it progresses, and will be introducing DAS_NOISE and BJS into the scope here at some point. I am also drafting
up the concept of doing away with a by pxl method, and making a vector tracing system that will be much more modular, and have the ability to make decisions on if it should attempt to link up with an
intersection, cross a road, or how steep it should make a turn.
Texture Syth
Texture Syth
An experiment in Texture Generation
Author: Andrew V Butt Sr. – Pryme8@gmail.com
Texture Synthesis is method in 2d procedural generation that is quickly becoming and interest for several developers. Its capability can easily be be extended to 3d. This is an active investigation
on how to teach a computer to output generated content from sample input. The general basis is discussed HERE
I will attempt to make different methods of generation based with and without my Das_Noise Library and will later include research on this topic in GLSL. The eventual goal is to create a robust
texture synthesis library for javascript and use this product to aid in the deployment of TERIABLE.
The initial test will be loosely based on the documentation and will be my own spin on the approach. Later I with lessons learned will attempt to deploy more popular methods.
The outcome of these test I am unsure of but will document both the approach and the results and provide an example.
Test 1
The first test was trying to not deploy a base noise to reference the sample texture against. I stopped working on this test after i started noticing shortcomings in the colors being produced.
Base SampleA few terms that are used while using a texture synthesis method are:
• Texton
• Neighborhood
• Vector
• Noise
Constants in our Examples will be:
• R – Reference Image
• P – Base Sample Target Point
• nP – Neighboring Target Point
• N – New Texture Being Generated
• Np – New Texture targetPoint
• np – New Texture Neighboring targetPoint
In my first test, I decided for the first step, was to generate my textons from R, which I did not do a very good job of. But none the less it seems to work enough to let me see the effect of this
method. The second step was to impose R on the output canvas in 3 points on the top corner in order to give our generator something to reference. This can be done without outputting the reference but
was a quick way to get it done.
From there I started sampling with a different shape of texton that was the same number of samples as the ones generated previously. This new texton was not offset from around the point it was
checking but behind it in a box of similar shape but offset, as to backreference what was already there. Then I found the closest Texton from our first set and used its value to output to the canvas.
I could not for the life of me figure out why the colors were so washed out, though this effect was able to duplicate the pattern fairly accurately. I was going to continue the process by then
cloning our new first area we created into the top and left edges of the canvas and was going to repeat the sampling process for making the first area. I did not complete this last step though
because what I had already learned what I needed to from this example.
You can look at a live DEMO but the script is very sloppy and basically it boils down to I don’t know what I was thinking on my original script.
This method is… slow, plus with the bad color transfer I do not think is a valid method. Moving on…
Test 1 – Enter Noise
So after my first night of hacking away at this, while trying to get some sleep I thought of how to implement using noise to predict what color the cell will need to be.
I figured if I could have a accurate way to describe an underlying noise and translate that to the neighborhood I was looking for it could continue the pattern or a likeness of it on the output.
Initial test on this looked kinda promising with the first area of the noise rending out to be pretty close to the input, but after that it’s pure chaos. I think it may be with how I am measuring the
What I need to do is some more reading and find out what other guys have done to solve this. The DEMO shows that this method could be possible but quite a bit of refinement is needed.
Also the computation speed is also very slow, and produces crappy output, I ran several different test with a range of noises, and tried to output by pxl as well as by neighborhood the results being:
In the last example I did not want to wait for it to finish, as it was going per pxl and using the textons average value. The color problem from the first example is not a problem now so maybe I need
to reference both examples and make a combination of the both. Till then I think it’s time to do some more reading, and write down some of the ideas from what I learned here to create a procedural
dungeon (maybe).
To be continued…
Experiments in Procedural Level Creation
After working on doing some texture synthesis, a method for creating dungeons and other content kinda just smacked me in the face. I have been itching for about two days now to get a chance to do
this. The night I thought it up the write up went as follows:
A Zone (Z) is defined as a area of set units that is divided into a set amount of Cells (C). For this deployment I will be dividing the
Z into a 3 by 3 grid with the labeling of each cell as follows starting from the top left; n10, n00, n01, m10, m00, m01, s10, s00, s01. Because I am spliting it into thirds to keep things simple I
will define the size of the zone always to something divisible by 3, for general purposes I will always use a Z size of 60 by 60 making each cell 20^2 pxls. Because we will be averaging the black and
white value of the cells we limit the size of the zones to be as small as possible but still large enough to have defined details. The larger our Zones the larger the calculation overhead.
Once my Zone method and size is established and I have defined the cells for it, I have to calculate every possible state of the Zone and output that to a human readable jpg. This is achieved by
looping through combination sets of the cells, and creating a single array of all states, then use that information to generate a canvas with the correct cells displayed as black and white on or off
After the human readable jpg is produced, reload our now created Zone map into the program and associate each zone’s location information on the map and cell state information into a referenceable
and searchable array or object.
At this point I can start choosing my method for a base noise, because each Zone will only be 60 by 60 I believe my Worley2D noise from my Das_Noise library will work just fine, if there is a
calculation lag on the generation of the zones due to the noise, I will see about moving to a modified SimpleX. Starting from the top left of the visible stage, we calculate the values for the base
noise by passing it to our zone object and averaging the values of each cells black white ratio due to the noise, and round to 0 or 1 effectively converting the noise to a Zone similar to the ones I
generated earlier. Loop through the map object,
and find out which zone matches the closest to the new noise zone.
The point of first creating a human readable image instead of just having the noise be manipulated is that after we get a look for the base layout that we want, an artist can use the human readable
image as a template to draw a secondary reference image that has the same dimensions as our reference map. I could then load image data into the map object from the secondary reference image and
output the fancy tiles instead of just black and white. This process could also be extended to use secondary noise calculations to establish and simulate different biomes and altitudes, changing what
tile map is referenced.
This is all theory but it sounds about right so I’m gonna give it a shot.
The Reference Map
Diving right in I think the smartest thing to do will be to create my first reference map, or the human readable map I described earlier. The first day I thought of this I tried to make it by hand in
illustrator and got about 32 combinations in before I realized that was dumb, and it was time to make canvas go to work.
First we need to calculate the combinations of the cells and make something that we can use to output a physical reference map. What I mean by combinations is if we had an input of [1, 2, 3] the
output would look like =>123,213,132,232,312,321…. There are lots of ways to do this, but I will try to keep it simple. Because order does not matter, we do not have to worry about permutations (the
same combination in a different order).
This script to make it happen is as follows:
dungeon = function(args){
args = args || {};
args.zSize = args.zSize || 60;
args.zDiv = args.zDiv || 3;
this.zSize= args.zSize;
this.zDiv = args.zDiv;
dungeon.prototype._createRefMap = function(){
var cells = [];
var pCount = this.zDiv * this.zDiv;
function perm(s,c){
if (c == 0) {
perm(s+'0', c-1);
perm(s+'1', c-1);
var last = cells.splice(cells.length-1,1);
Just creating a new dungeon and then calling the prototype now outputs all of the permutations for a total of 512, on a side interesting note, is it also the could be looked at as every possible
combination of a binary set of 9. Looking at the structure I already know that my two most common ones I am shooting for will be all states on and all states off, so I think it would be best to take
the first record and move it to the front of the array to save on calculation time once we start looping through our state array.
Zone Object
Now it is time to make a Zone Object, this will be the basis for our mapping of the noise, this makes a object that we can put in an array, and compile the states of the cell as a searchable string.
After that we will look at making a readable image.
dungeon.Zone = function(size, div, state){
this.size = size;
this.div = div;
this.searchString = state;
this.state = state.split('');
this.cells = [];
for(var i=0; i < div*div; i++){
for(var i=0; i < state.length; i++){
var sID = parseInt(state[i],10);
this.cells[sID] = 1;
return this;
I then modified my pre-existing script to the following:
var map = [];
for(var i = 0; i < cells.length; i++){
map.push(new dungeon.Zone(this.zSize, this.zDiv, cells[i]));
this.map = map;
This gives us an array on the main dungeon object that contains set of Zones with a searchable string for referencing later. I now need to create a new function to compile the physical map and set
values for where the zone object is on the output map. This step is only necessary so that at a later time an artist can create a secondary reference map at a later time, if I just wanted black and
white pxls to display I could effectively skip this step but that is not the final product I want.
I also went ahead and allocated the memory for each of the zone objects to have image data as well, even though I’m just using the map image and not an artistic tile image do to the fact of 512 tiles
is quite a bit of content to come up with, just for an example. Using this function I generate my reference map that I will use as both a way to look up / store tiles and their properties; it also
creates the ability for me to output a canvas with the tiles on it to make a human readable map.
dungeon.prototype._calculateMap = function(){
var map = this.map;
var cvas = document.createElement('canvas');
var ctx = cvas.getContext('2d');
var X = 0, Y = 0;
var cellSize = this.zSize/this.zDiv;
cvas.width = 20*this.zSize;
cvas.height = Math.ceil(map.length/20)*this.zSize;
for(var i = 0; i < map.length; i++){
var x = 0, y = 0;
for(var j = 0; j < map[i].cells.length; j++){ if(map[i].cells[j] == 1){ ctx.fillStyle = "#FFF"; }else{ ctx.fillStyle = "#000"; } ctx.fillRect(x+X,y+Y,cellSize,cellSize); x+=cellSize; if(x > this.zSize-cellSize){
ctx.strokeStyle = "rgba(255,0,0,0.2)";
var imgData = ctx.getImageData(X,Y,this.zSize,this.zSize);
map[i].imgData = imgData;
map[i].x = X;
map[i].y = Y;
if(X > cvas.width-this.zSize){
Now it’s time to start generating a noise, and see if we can kick this thing into gear and output a dungeon like structure. Later I will research into making the ability for you to draw on the base
noise and see the overlay tiles update accordingly, this would be cool for later development I think, but is something that is down the road a little bit.
Also CLICK HERE for an Example of the Reference Map
Enter Das_Noise
Ok so now the next step will be to generate a base noise map to start sampling, and outputting out maps imageData in the correct areas and see what kind of output I can get. I’m assuming this should
go without much hitch and with a well set up noise will structure itself to resemble a dungeon right of the bat (I hope).
I want to use a good sized chebyshev style Worley Noise to start because I believe this will have a good look to it once overlaid, and will guarantee that most if not all the rooms connect. If you
are not familiar with my Das_Noise library you can check it out here: http://pryme8.github.io/Das_Noise
To test the noise I am going to output on a 600 by 600 pxl canvas the noise till I get something acceptable. When I go to use it, i will not have to create the noise to any sort of output, but rather
just check its values at certain locations then parse that how ever is needed to see what cells are active or not in that zone.
Already looking at this noise, we can visualize what the dungeon will look like if the calculations have been set up correctly. The next step is to identify the what each zone on the noise matches up
to on our reference map, to see this in action click the link below to do one zone at a time on our canvas to the left.
*UPDATE – I went ahead and added a basic tile map to reference, to show how that would work… you can look at the code to see how I did that, but after seeing it deployed I have three options, rework
the tilemap to be cleaner and work a little better, make some sort of comparison script to see what the other tiles next to it are, and if there is a flat edge, have caped variations to use, or make
everything procedrual… I think given the fact it took me two and a half hours to make 512 tiles im going to go with the last option here at some point.
Ohhh yeah, that works! Ok so I think I will wrap it up on this, but first here is a look at how I ID the zone of the noise.
Here is and example of the same process, with the noise of the same seed, but set to Simple2 and a scale of 100.
dungeon.prototype._idNoise = function(x,y,noise){
if(typeof noise ==='undefined'){
noise = this.noise;
var cellSize = this.zSize/this.zDiv;
var string = '';
var self = this;
var ctx = (document.getElementById('noise-canvas')).getContext('2d');
ctx.fillStyle = "red";
var cX = 0;
var cY = 0;
for(var i=0; i<this.zDiv*this.zDiv; i++){
var t = 0;
for(var pY = 0; pY < cellSize; pY++){
for(var pX = 0; pX < cellSize; pX++){
if(t<0.45){ string+=0+""; }else{ string+=1+""; } cX++; if(cX>this.zDiv-1){
for(var i=0; i<this.map.length; i++){
if(this.map[i].searchString == string){
return this.map[i];
This was all literally done in one day intermittently while I cleaned the house… so yeah I think this is a valid and good approach for what I want to achieve. I will have to experiment with different
noise types styles and scales and then come up with a nice tileset for it (I will prolly jack RPG maker resources for now). I think once this is deployed a little more the possibilities will be
I will be posting a simple Canvas Game based on this principle at some point!
Resources and References : None… I just made this crap up… if you have any questions Pryme8@gmail.com.
Web Workers Experiment 1
Web Worker Experiments
An investigation into how web workers operate.
A phrase I’ve been hearing used a lot lately, “Web Workers”. What is a web worker? What can I do with one? Through this tutorial I will have you follow along with me while I go through some steps
to understanding what a Web Worker is and how we can use these in actual situations. Prior to this I have no knowledge whatsoever as to what they are used for and what they are capable of, but
hopefully by the end of this I will!
First let’s get our basic template for our page setup, you can download the template HERE, or you can set up whatever basic index page is most comfortable to work in.
What does developer.mozilla.org say about Web Workers?
Web Workers provide a simple means for web content to run scripts in background threads. The worker thread can perform tasks without interfering with the user interface. In addition, they can perform
I/O using XMLHttpRequest (although the responseXML and channel attributes are always null). Once created, a worker can send messages to the JavaScript code that created it by posting messages to an
event handler specified by that code (and vice versa.) This article provides a detailed introduction to using web workers.
Ok, so this sounds really promising! What I am getting out of this is that Web Workers can simulate multithreading and could also serve as a pseudo response server for more dynamic content.
Also the thought of nesting certain calculation functions inside another script could allow more dynamic animations on a canvas element, or any other intensive calculation.
A few points about scope, if we create a web worker thread it creates a new thread outside of the scope of the window. If at any time we would need to refer to the ‘window’ that called the worker
thread we would instead of using window use self. *dont quote me on this part
Other Key Points:
• A dedicated worker is only accessible from the script that first spawned it, whereas shared workers can be accessed from multiple scripts.
• You can’t directly manipulate the DOM from inside a worker, or use some default methods and properties of the window object.
• Workers can spawn new workers, those workers must be hosted within the same origin as the parent page.
I would recommend reading the mozilla page, as most of this tutorial will be based off the information served there.
Web Worker detection and setting up our first thread.
Now that we have an idea of what one is, let’s go ahead and jump right to the main.js file and start making some changes.
window.onload = function() {
if (window.Worker) {
document.body.innerHTML = "We Have Ignition";
document.body.innerHTML = "Lame sauce... no Web Workers...";
If you have ignition, then you are good to go! If not then, yeah something is wrong because pretty sure most modern browsers have these… geez guy/gal.
From this point we need to actually do something with a web worker so the first step in that will be to go to our js folder and create a new script called “worker1.js” and then edit both the main
script and the worker script accordingly.
window.onload = function() {
if (window.Worker) {
var newWorker = new Worker("./js/worker1.js");
console.log('Message posted...');
newWorker.onmessage = function(e) {
result = e.data;
document.body.innerHTML = "Lame sauce... no Web Workers...";
onmessage = function(e) {
console.log('Message received from main script');
var workerResult = 'Result: ' + (e.data[0] + e.data[1]);
console.log('Posting message back to main script');
What is happening here is first we are creating our instance of the worker on our main script. Then we are using the built in method .postMessage which will be the main basis for communicating with
out worker. Then we have the worker listen for a message by defining the onmessage function, do whatever we want with the data and then pass it back! If everything is set up right when you refresh
the page nothing amazing happens, but when you look in the console you will see the desired output hopefully. If you are having trouble or just feel like skipping this step you can download it HERE
Implementing and Creating Our First Project!
Now that we have this set up, what is something we could make? Hmmmm, I know how about Pong. We will get to learn how to use the Web Worker to do all the calculations and just have the main page
update the canvas. If we do things right we could perhaps have one worker for calculating what is happening and the other calculating the output to the canvas. This may not be ideal or even the
correct thing to do, but this is open research so I have no shame.
Because we don’t have another person and I don’t feel like going over AI for this tutorial, lets make a game that we can play with ourselves (ha). So to get things rolling let’s make these changes to
our index.html, main.js and main.css
Web Workers Step 2
@charset "utf-8";
/* Web Workers Tutorial */
html, body{
padding:0; margin:0;
font-family:"Lucida Console", Monaco, monospace;
window.onload = function() {
cvas = document.getElementById('cvas'); //Lets make it Global ^_^
var context = cvas.getContext('2d');
var score = document.getElementById('score');
score.innerHTML = "Score: 0";
function resize(){
cvas.setAttribute("width", window.innerWidth+'px');
cvas.setAttribute("height", window.innerHeight+'px');
window.onresize = resize;
//Lets just Draw our ball for now and paddles for now.
function drawBall(){
var centerX = cvas.width / 2;
var centerY = cvas.height / 2;
var radius = cvas.height / 40;
context.arc(centerX, centerY, radius, 0, 2 * Math.PI, false);
context.fillStyle = 'blue';
context.lineWidth = 1;
context.strokeStyle = 'blue';
function drawPlayer(){
var centerX = cvas.width / 2;
var centerY = cvas.height - 20;
var width = cvas.width / 10;
var height = 10;
context.fillStyle = 'red';
context.fillRect(centerX - (width*0.5) ,centerY,width,height);
if (window.Worker) {
var newWorker = new Worker("./js/worker1.js");
console.log('Message posted...');
newWorker.onmessage = function(e) {
result = e.data;
document.body.innerHTML = "Lame sauce... no Web Workers...";
If you’re all set up things should look like this.
Ok, so basically all we did was set up a resize listener, and some basic function to figure out how we are going to draw the objects
Now that we got some basic things set up its time to get to core structure of our program. You can download step 2 if you feel like skipping to this point or are having trouble.
The Long Haul… Core Changes
So the first thing on the chopping block, is let’s get a ASYNC loop set up with a throttle on it so we’re not just calculating for no reason on the draws. Then move the functions for drawing on the
canvas onto the worker and see if it all still works. I’m not sure if this can all happen on the worker, but I have a sneaking suspicion that it will work just fine as long as we make sure we set up
the correct scopes.
That means we will be editing the worker1.js file.
newGame = null;
onmessage = function(e) {
console.log('Message received from main script');
var type = null;
type = e.data[0];
case "init" : newGame = new pong(e.data[1]);
console.log('new Game');
pong = function(cvas){
this.score = 0;
this.run = false;
this._cvas = cvas;
if (window.Worker) {
var newWorker = new Worker("./js/worker1.js");
newWorker.postMessage(['init', cvas]);
newWorker.onmessage = function(e) {
result = e.data;
document.body.innerHTML = "Lame sauce... no Web Workers...";
If we try to run this we get that the error: “DataCloneError: The object could not be cloned.” I believe this is because of instead of passing a reference of the object to the worker script it tried
to make a clone of it, which is evidently not permitted with a canvas element (we could pass the imageData as a Buffer or Array though, but kinda overkill in this situation). So our work arounds will
be to try to pass the context of the canvas, or just have the main script do the canvas manipulations but have the worker thread do the calculations. I’m not sure if this is even a correct use for a
Web Worker but we will find out.
So I guess we should actually set up the structure for the game on the main thread, so on the main.js we add all the constants and containers for the game. As of right now we will do a simple 30Hz
interval loop to process what needs to be put onto the canvas. Later we will make this loop more customizable and put in a way to set our FPS. The concept that we will be testing is having the
physics be calculated on the worker, and have it push the updated hits and other properties to the main thread to be processed. Initially here we will set the main and worker thread to work at the
same frequency, but later will prolly crank up the main thread to 60Hz and see how that affects performance.
SCALE = 1;
CENTERX = 0;
CENTERY = 0;
Entity = function(id, pos){ //pos is an array(3) with [x,y,angle];
this.id = id;
this.pos = pos;
this.velocity = [0,0,0]; // X,Y,ROTATION
Entity.prototype.update = function(state) {
this.pos = state.pos;
Entity.prototype.draw = function(ctx) {
ctx.fillStyle = 'black';
ctx.arc(this.pos[0]+CENTERX , this.pos[1]+CENTERY , 2, 0, Math.PI * 2, true);
Ball = function(id, pos) {
Entity.call(this, id, pos);
this.body = {
radius : SCALE,
Ball.prototype = new Entity();
Ball.prototype.constructor = Ball;
Ball.prototype.draw = function(ctx) {
ctx.fillStyle = 'blue';
ctx.arc(this.pos[0]+CENTERX, this.pos[1]+CENTERY, SCALE, 0, Math.PI * 2, true);
Entity.prototype.draw.call(this, ctx);
player = function(id, pos) {
this.body = {
points : [
[-(10 * SCALE)*0.5, -SCALE*0.5], //TL
[(10 * SCALE)*0.5, -SCALE*0.5], //TR
[-(10 * SCALE)*0.5, SCALE*0.5], //BL
[(10 * SCALE)*0.5, SCALE*0.5] //BR
Entity.call(this, id, pos);
player.prototype = new Entity();
player.prototype.constructor = player;
player.prototype.draw = function(ctx) {
ctx.fillStyle = 'red';
ctx.fillRect((this.pos[0]+CENTERX)-((10 * SCALE)*0.5),
(window.innerHeight -20),
(10 * SCALE),
1 * SCALE);
pong = {
ent_stack : [],
gravity : [0,0.2],
_init : null
window.onload = function() {
cvas = document.getElementById('cvas'); //Lets make it Global ^_^
var ctx = cvas.getContext('2d');
var score = document.getElementById('score');
score.innerHTML = "Score: 0";
pong.ent_stack.push(new Ball('b1', [0,0,0]));
pong.ent_stack.push(new player('player1', [0, window.innerHeight - 20 ,0]));
function resize(){
cvas.setAttribute("width", window.innerWidth+'px');
cvas.setAttribute("height", window.innerHeight+'px');
SCALE = cvas.height / 40;
CENTERX = cvas.width / 2;
CENTERY = cvas.height / 2;
function reDraw(){
for(var i=0; i<pong.ent_stack.length; i++){
window.onresize = resize;
if (window.Worker) {
console.log("Worker Go!");
var newWorker = new Worker("./js/worker1.js");
/*newWorker.onmessage = function(e) {
var result = e.data;
ctx.clearRect(0, 0, cvas.width, cvas.height);
document.body.innerHTML = "Lame sauce... no Web Workers...";
Now the break down on this is as follows. One, we set up some global variables that will hold things that determine our size of our entities being drawn on the canvas. We make them global because at
any point the user may resize the document and we want everything to remain the same. Right now we will disinclude the scale into the physics calculations, but later we will have to make sure we
apply the scale to the physics as well so that the velocities and gravity etc stay proportionate.
Next we define our basic Entity Object, and its prototypes. This will be the basic container for whatever other objects we want to render on the screen. We give the Entity Object a draw prototype
that should give us a center point. If you want to use the center point you must make sure that if your not spawning whatever entity at the center of the canvas, you need to translate the context
prior to the output otherwise the dot will not be on the center of the new entity (not important we only using it on the ball which spawns in the center).
Then we define different Constructors for the Entity Object so that we can call different shapes. This is a very basic prototype/constructor layout and should be fairly straightforward to understand.
After we have our Constructors ready we need to define a global container for everything. In this global container “pong”, we can define some constants as well. Once the DOM has loaded, we create the
new entities and push them to a stack on our global container. Modify our all so important resize function, and then create a new function to call the draw method on all active entities. You could at
this point put and enabled disabled flag on the Entities to toggle them on or off, but we’ll save that for later.
If in your interval loop, we stick the argument pong.ent_stack[0].pos[1]+= 0.25; You will see the ball move, which means we are on the right track.
After some more changes to the main script, we go back to the worker1.js, it is here that we start defining the structure for these workers to do physics calculations and post updates back to the
main thread. We might as well take advantage of the number of threads available to us, and future proof a little bit with a navagator function that might not be supported in all browsers so we have a
fall back.
// Step 3
engine = null;
onmessage = function(e) {
var type = null;
type = e.data[0];
case "init" : engine = new physics(e.data);
physics = function(data){
this._core = data[1];
this.gravity = data[2];
this.stack = [];
var parent = this;
this._init = setInterval(function(){parent._run()},1000/30);
physics.prototype._run = function(){
console.log("Worker on Thread "+this._core+" Fired!");
We also need to make some changes on the main.js file to accomidate the thread structure.
pong = {
ent_stack : [],
gravity : [0,0.2],
_init : null,
cores : navigator.hardwareConcurrency || 4,
if (window.Worker) {
workers = new Array(pong.cores);
wID = 0;
for(c=0; c<pong.cores; c++){
workers[c] = new Worker("./js/worker1.js");
workers[c].postMessage(['init', c, pong.gravity]);
/*newWorker.onmessage = function(e) {
var result = e.data;
pong._init = setInterval(function(){
ctx.clearRect(0, 0, cvas.width, cvas.height);
If when you run the page and look at your console, you should see all the threads sending out information to the console! So now that we have our threads firing, and creating a new function called
physics, its time to pass some Entites to the now unbuilt physics engine.
Part 3.2 – Physics B***h!
So how are we gonna handle this? Why dont we just use some other library? Why have you not used jQuery yet? Yeah well… the whole point of this is to expand your and my know how and just deploying a
library is easy. Plus how much better are you going to be at your favorite engine if you understand some of the basic components?
Right away, we are only dealing with 2 dimensions so the natural inclination would be to go with a Array(2), and prototype some functions into the Array Object, but why not tailor things for what we
are doing. So instead lets do an Array(3) with it representing [POSX, POSY, ROTATION], the same concept then can be used for the velocity vector. We would need a few things to happen for the engine
to actually work:
1. Check state of Object, if it turned off even in the stack ignore calculations on it.
2. Apply Gravity to velocitys, with account for an objects mass.
3. Apply any resctrictions
4. Pre Check for Hits
5. If its going to hit restrict the objects position to the point of impact and do inertia return calculations
6. Convert Velocites to Units of messure of some kind
7. Send Message back to main thread.
Now as I was doing this, I got all the way to having the objects pushed to their thread, then I realized… there would be no way to actually do a hit test on any object because simply enough there is
no effective way to communicate between threads. Now there are shared workers but thats a whole other monster we will try to tackle at some other point. So instead of bogging you down with the script
that does not work we will move right on to a single sub thread model.
// Step 3
var engine = null; //Does not need to be global.
onmessage = function(e) {
var type = null;
type = e.data[0];
case "init" : engine = new physics(e.data);
case "AddObj" : engine._addObject(e.data[1]);
physics = function(data){
console.log("TREAD "+data[1]+" STARTED!");
this._core = data[1];
this.gravity = data[2];
this.stack = [];
var parent = this;
this._init = setInterval(function(){parent._run()},1000/30);
physics.prototype._run = function(){
for(e=0; e<this.stack.length; e++){
var tempCalc = this._calc(this.stack[e]);
var hit = false;
for(h=e+1; e< this.stack.length; h++){
this._apply(tempCalc, e);
physics.prototype._addObject = function(obj){
obj.pos = obj.intPos, obj.velocity = obj.intVel;
console.log("Thread "+this._core+": Added Obj - "+obj.id+" to its stack");
physics.prototype._hitTest = function(a,b){
return false;
physics.prototype._calc = function(obj){
var response = {
stackID : obj.stackID,
response.newVel = [
obj.velocity[0] + (obj.mass * this.gravity[0]), //X
obj.velocity[1] + (obj.mass * this.gravity[1]), //Y
obj.velocity[2] // ROTATION
response.newPos = [
obj.pos[0] + response.newVel[0], //X
obj.pos[1] + response.newVel[1], //Y
obj.pos[2] //ROTATION
return response;
physics.prototype._apply = function(calc, id){
this.stack[id].pos = calc.newPos;
this.stack[id].velocity = calc.newVel;
With these changes to the worker script, we have enabled the ability to start calculating basic physics into our scene. To enable it we need to modify the main script now. We can get rid of the
thread count on the pong object as we are going with a single sub thread now, and then we have to change our script around to be a single worker instead of the 4 we had set up.
if (window.Worker) {
worker = new Worker("./js/worker1.js");
worker.postMessage(['init', 0, pong.gravity]);
function addObj(obj, stackID){
id : obj.id,
body : obj.body,
mass : obj.mass,
intVel : obj.velocity,
intPos : obj.pos,
stackID : stackID,
on : true
for(var i = 0; i < pong.ent_stack.length; i++){
addObj(pong.ent_stack[i], i);
worker.onmessage = function(e) {
var result = e.data;
var calc = result[1];
pong.ent_stack[calc.stackID].velocity = calc.newVel;
pong.ent_stack[calc.stackID].pos = calc.newPos;
pong._init = setInterval(function(){
ctx.clearRect(0, 0, cvas.width, cvas.height);
If your following along and have everything set up correct, when you refresh the page the ball should now drop like graviity is being applied to it. This set the stage for us to start making some
more dynamic effects, the things we have to consider now is how we are going to handle our Collison detetions. The most simple model will be for us to use a projection and impulse system where we
test if our object is going to collide find out the position where this is happening and then modify our velocity and position to stop the objects from penitrating. We will keep the model simple, but
will include things like restitution and friction.
Collisions, Collision, COLLISIONS!
How does one actually do a collision test? Well in concept it is easy, first you have to establish what kind of collisons you need to calculate for, are they simple shapes like rectangles and cirlces
only? Are the axes of the rectangles always the same? What kind of shapes we need to test will decide our approch. For this model we need to track at least circles and off axis rectangles. I want to
include off angle rectangles even though the paddle will never change its angle, because there may be a reason to place other objects in the scene that do not have flat axial value. Oh man, I just
realized how quickly this is becoming a math lesson… my bad, anyways to achive our goals lets first go over some vocabulary. Also for these examples I am writing them out for any length vector, which
is useful in normal situations but with how we are storing our stuff it might not be the best idea. So I have one of two options, pass more variables that are standard vec2’s and numbers instead of
our vec3 that has rotation with it as well, or redo these functions to only handle 2 units of the vector we hand it. The smartest I think would be to follow convention, and drop the vec3 that we were
using and have a seperate value for the rotation so that way when we do our vector calculations they are correct.
• normalization
Normalizing a vector is scaling it so that its length totals 1. To normalize the vector, we divide each of its components by the length of the vector:
function normalize(vec){
var l=0;
for(var i = 0; i < vec.length; i++){
l+= (vec[i]*vec[i]);
l = Math.sqrt(l);
for(var i = 0; i < vec.length; i++){
return vec
• dot product
Think of it as the relative orientation of a and b. With a negative dot product a and b point away from each other; a positive dot product means they point in the same direction.
function dot(a, b){
var t = 0;
for(var i=0; i < vec.length; i++){
t += (a[i]*b[i]);
return t;
• projection
A formula to find the shortest length vector that an object must travel to be out of the collision and using this to detect if a object is collided or not.
function project(a, b){
var proj = new Array(a.length);
var bt = 0;
for(var i = 0; i < a.length; i++){
bt+= (b[i]*b[i]);
for(var i = 0; i < a.length; i++){
proj[i] = (dot(a,b)/bt)*b[i];
return proj
• Perpedicular Product
How to return the vector perpendicular to the input. Every 2D vector has two normals: the right hand and left hand normal. The right hand normal points to the right of the vector, and the left
hand normal points to the left.
function perproduct(vec2){
var rN = [-vec2[1], vec2[0]];
var lN = [vec3[1], -vec2[0]];
return dot(vec2,rN);
With these basic functions we are going to be able to run a whole array of 2d comparisons. So to solve our 2D overlap test using a series of One Dimensonal Tests. Each query tests if the two shapes
overlap along a given axis. If one of the Axes we test fails then we know that the objects are not intersecting, and we can stop our test to save on overhead. If we run a test and find that the
objects overlap along all of the possible axes, they are definitely overlapping each other. From this point we need to figure out the projection vector, which we will use to seperate the objects.
The last step would be to find which axis has the smallest amount of overlap between the two objects. The “Push” we are looking for or the projection vector is the same as the axis direction with the
length of the projection vector equal to the size of the overlap. To acomplish this we need to know first the position of the object and all of its axes, along with our target object and the same.
But Wait? What about circles, or triangles or or… ok fine lets do some drawings.
Separating Axis Theorem
In 2D games to represent moving objects we make an axis-aligned
bounding box, or AABB. An AABB is defined by a position and two vectors xw and yw that represent half widths of the object. The concept is the same as testing the radius of circle, but these are
aligned to the world axis and are then defined in a specific direction and distance.
Well that all sounds cool… but how does it work? I think maybe the simplest way will be to make a real quick mock up canvas for you. After the first example, we will go over more shapes and the
methods for testing them. I know this was supposed to be a webworker tutorial, but for real though we need something decently intensive to even have a nessesity for them. Collision testing I think
meets that bill, so please stay with me or skip ahead.
In the next example what we will see happen when you click run, is the red box should move to the right and once it collides with the other box well start it all over.
This is a SUPER simplified model, which works only if the two objects faces are perfectly aligned and thats something that is kinda weak for what we are looking to do. So what happens when its a
polygonal shape or a off angle square? Back to the drawing boards.
This is where the method for seaperating axes comes into play. If you
look at our diagram on the right you will see two polygons. Both of these shapes are called as Convex Shapes. A Convex shape is any polygon that can be defined by a set of points, that if you were to
draw a straight line anywhere on the shape from one point to anouther on the polygon, the line will never travel outside of the shape. Anyways these two Convex polygons are in a non intersecting
The value for the seperation is positive and so we know that the shapes are not touching, if it was negitive the shapes would be overlapping and if it was 0 the shapes are just touching. With this
kind of hit detection there would be three kinds of possible contact, edge to edge, vertex to edge, and vertex to vertex. If we were to draw a line between the two polygons and we pretend the
seperation line continues to infinity and we draw a line perpedicular to this line, that is our speration axis.
Anouther thing we need to start identifying is the “normal” angle of and edge. This is not to be confused with our normalization meathod discussed earlier, but is a way to establish heading of our
edge. When we define our shapes, we need to follow a convention that assumes the shapes points are orderd on a clockwise direction, this is so when we loop through our edges it stays consitant on the
output no matter the shape. You could proccess the shapes counter-clockwise but this would effectivly reverse your edge normals.
I think the easist way to visualize all this is to make some more diagrams… lets see if I can mock up some canvas elements to demonstrate.
If you look at the diagram on the right, you will see a Convex Polygon with its center point represented by the black dot. The red dots are the points of the polygon, the green are the edges and the
blue arrows are the normal vector or perpendicular axis to the face. If you drag one of the polygons you will see that the normal for that face changes.
What we need to establish now is the most extreme points on the polygon. We do this by simply looping through the points and of the polygon and find the most extreme points, then project the max and
min points onto the perpendicular normal lines. If all of the projections overlap we know we have a contact.
To make this simpler to understand lets extend out projection axis for each normal out farther. If you were to select any of those lines that extend from the face of the edge, you would look at both
the current polygon and the opposing one and figure out what points on those polygons are at the min and max points for that projection line.
In the case of projection line a1 we see that the most extreme point on the polygon that falls on that projection axis is the same as the edge itself, both points that make up that edge would have
the same value effecivly when projected on that line so our min value for the projection of polygon A to axes A1 is equal to the projection value of point 0 || 1 (remember we go clockwise and when I
constructed these I started at the top left). We then look for the max using the same method and in this instance the max value of polygon A is the projection value of the point between edge a2 and
a3, or point 3. We now know where on this axis the polygon A is located so we do the same to polygon b (still looking at the a1 axis). When we follow the same process we fine that the min of B is
point 1 and the max of B is point 3. If you remember I diagram on the top this is what we were representing. You can in your mind draw line from these points to perpedicular the projection axis and
you can easily see if the two polygons are overlapping in that instance. If at any point we discover that any of the axes do not overlap then we do not have to continue our calulations and can then
interpolate the response.
Now I can continue to beat this concept into the ground and do a bunch of implementation to show you how we would calculate a whole gambit of physics stuff, but this is supposed to be a web socket
tutorial so lets just get back to doing that.
Back to the Game.
Ok I know I was talking all sorts of good stuff about, lets include curved shapes and blah blah… but this is starting to take up way to much time and I have to get back to some other projects. That
means its time to wrap this up.
We are going to just worry about making the ball bounce around hit the paddle and stay on the stage, get some controls working and output a score. After that Ill leave it up to you to extend these
ideas and make something really cool!
First a little bit of restructing is nessirary to make this a little less redundent on our object constructors and make them a little more diverse on their arguments. Then we will just place the
player in the scene and attack some kind of controler to allow the user to have input. Then we will use the same constructor as to make some walls, but they will only be there as a level constraint.
After we get the walls and the paddle in place we will script a simple version of the hit testing and see if we can get a “bounce” back off the walls with the paddle. After all that we will add the
ball back into the stack, and make sure that we have the hit detection identifying the shapes body and running the approriate tests.
To be Continued…
Also this was taken from my older website and may have some broken links, and scripts that may now work as of yet on the new site! | {"url":"http://pryme8.com/tag/html5/","timestamp":"2024-11-03T11:56:08Z","content_type":"text/html","content_length":"317184","record_id":"<urn:uuid:3e1cd030-f370-41b8-ac20-5b950368d7df>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00732.warc.gz"} |
RF Parameters | Academy of EMC
top of page
4 Radio-Frequency Parameters
This section introduces some of the most common Radio-Frequency (RF) parameters used in the field of EMC:
Scattering Parameters (S-Parameters).
Scattering Parameters} - also called S-parameters - are commonly used in high-frequency or microwave engineering to characterize a two-port circuit (see the picture below). The scattering parameters
describe the relation of the power wave parts a1, b1, a2, and b2 that are transferred and reflected from a two-port's input and output. Generally speaking, the waves going towards the n-port are
defined as a = (a1, a2, ..., an) and the waves traveling away from the n-port are defined as b = (b1, b2, ..., bn). The physical dimension for the incident a and reflected b power waves is not Watt
[W], it is the square root of watt [√W].
Generally speaking, the S-parameter sij is determined by driving port j with an incident wave of voltage Vj+ and measuring the outgoing voltage wave Vi- at port i. Considering the picture above, the
four scattering parameters can be computed as follows:
where a1 [√W] is the incoming power wave at port 1 of a two-port, b1 [√W] is the outgoing power wave at port 1 of a two-port, a2 [√W] is the incoming power wave at port 2 of a two-port and b2 [√W] is
the outgoing power wave at port 2 of the two-port.
The S-parameters are often given in [dB]:
The forward traveling power towards port 1 P1fwd [W] and the reflected power from port 1 P1ref [W] can be written as:
Of all four S-parameters, s11 and s21 are the most often used S-parameters in practice. s11 is often referred to as Return Loss (RL) or Reflection Coefficient Γ and s21 is often referred to as
Insertion Loss (IL). More details about the terms RL and IL can be found in the sections below.
Due to the conservation of energy, we can write for a passive (non-amplifying) and lossless two-port (e.g., a lossless transmission line):
For a passive and lossless two-port, the magnitude of |s11| can be calculated based on the magnitude of |s21| and vice-versa:
Reflection Coefficient Γ.
We speak of matched impedances in case the load impedance Zload is the complex conjugate of the source impedance Zsource. In radiated emission and immunity EMC testing, it is important to understand
the term matching and how to quantify it. All receiver and/or transmitter antennas must be matched to their receiver and/or transmitter equipment impedance (typical Z0 = 50Ω).
The reflection coefficient Γ is defined as [4.4]:
where Vreflected [V] is the reflected voltage waveform at a given point along the transmission line and Vforward [V] is the forward voltage waveform at the same point along the transmission line. A
reflection coefficient Γ=0 means there is no reflection and a reflection coefficient of Γ=1 means there is total reflection. All variables are complex numbers. The reflection coefficient Γ is often
given in [dB]:
For a one-port, as shown in the picture above (with Zsource [Ω] and Zload [Ω]), the reflection coefficient Γ equals the scattering parameter s11. This means for a one-port the reflection coefficient
Γ we can write:
For a two-port with a matched impedance at port 2 (the load at the output matches the output impedance of the two-port, ΓL in the figure above is equal to 0), the reflection coefficient Γ1 at port 1
of the two-port is equal s11 of the two-port:
However, for a two-port with an unmatched load at port 2 (ΓL≠0), a reflection happens at the output of the two-port, which influences the s11. This means that for a two-port with an unmatched output
load on port 2, the input reflection coefficient Γ1 at port 1 is NOT equal s11. Instead, Γ1 it is [4.8]:
VSWR means Voltage Standing Wave Ratio. The VSWR expresses the ratio of the maximum voltage Vmax [V] of a standing voltage wave pattern and the minimum voltage of a standing wave pattern Vmin [V] on
a transmission line. A standing wave is a wave that oscillates in time but whose peak amplitude profile does not move along the transmission line (in the direction of z [m]).
A VSWR value of 1.0 means perfectly matched. A VSWR value of infinity means a complete mismatch (100% of the forward wave is reflected). The VSWR can be calculated by using the reflection coefficient
from above [4.5]:
Return Loss [dB].
The Return Loss [dB] (RL) is the dB-value of the loss of power in the signal reflected (Preflected) by a discontinuity in a transmission line or due to an impedance mismatch. A low RL value indicates
that not much power is transferred to the load and is reflected instead. Return loss [dB] is the negative value of the reflection coefficient Γ in [dB] [4.6].
For a one-port load or a two-port with matched output load on port 2, the return loss RL can be calculated from the scattering parameter s11 like this:
Insertion Loss [dB].
The term Insertion Loss (IL) is generally used for describing the amount of power loss due to the insertion of one or several of the following components (passive 2-port networks):
• Transmission Line (cable, PCB trace)
• Connector
• Passive Filter
The insertion loss (IL) represents the power ratio in [dB] of the power P1 and the power P2 of the picture above. P1 is the power, which would be transferred to the load in case the source is
directly connected to the load. The power P2 represents the power that is transferred to the load in case the two-port network is inserted between the source and the load [4.2], [4.3].
Insertion loss is often to be considered the same as the transmission parameter |s21|. However, the value of insertion loss and |s21| will be different if the input and output ports of a two-port
aren't matched or use different reference impedances. Therefore, in the case that both ports of a two-port network use the same reference impedance and are matched, we can write:
Free-Space Path Loss (FSPL).
The term free-space loss (FSL) or free-space path loss (FSPL) is important to understand for radiated emission and immunity testing. The FSPL is the attenuation of the electromagnetic field between a
transmitter and a receiver antenna in free-space. It is assumed that the space between the antennas is free of obstacles and a line-of-sight path through free-space. As mentioned at the beginning of
this chapter: all the formulas are valid for the far-field or in other words: free-space. In addition to that, the Friis transmission equation assumes matched impedances and matched polarization of
the transmit and receiver antennas. The Friis transmission equation is stated as [4.7]:
Pr = receiver antenna output power [W], Pt = transmitter antenna input power, Git = antenna gain of transmitter antenna, Gir = antenna gain of receiver antenna, λ = wave length [m] and d = distance
between transmitter and receiver antennas [m]. The free-space path loss FSPL is now defined as:
d = distance between transmitter and receiver antennas [m], λ = wave length [m], c = speed of light 3E8[m/sec], f = frequency [Hz].
Link Budget.
For radiated EMC testing, the term link budget is often used. If you want to calculate the received power Pr [dBm] for a given receiver/transmitter setup (transmitter, frequency, antennas, distance,
etc.), you calculate the link budget. In simple terms, this means:
Received Power Pr [dBm] = Transmitted Power [dBm] + Gains [dB] - Losses [dB].
If we rewrite the link budget formula above in a little more detailed way, we get:
Pr = receiver antenna output power [dBm], Pt = transmitter antenna input power [dBm], Lt = transmitter losses (coaxial cable, connectors, etc.) [dB], Git = transmitter antenna gain [dB], FSPL =
free-space path loss for a given frequency [dB], Lmisc = miscellaneous losses (fading, polarization mismatch, etc.) [dB], Gir = receiver antenna gain [dB], Lr = receiver losses (coaxial cable,
connectors, etc.) [dB].
Impedance Matching Summary.
The table below shows how to convert between VSWR [1], return loss [dB] and the reflection coefficient [1]. Z0 is the "system impedance" (typical Z0 = 50Ω or Z0 = 75Ω).
In order to give you an idea what a good match means in terms of VSWR, reflection coefficient or return loss: we summarized all the values in this table below.
Reflction Coefficint
Return Loss
Insertion Loss
Matching Summary
Link Budget
Free-Space Path Loss
bottom of page | {"url":"https://www.academyofemc.com/rf-parameters","timestamp":"2024-11-13T14:06:36Z","content_type":"text/html","content_length":"956291","record_id":"<urn:uuid:78e77adf-2623-4234-9ef6-1efaf09fad7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00702.warc.gz"} |
Irreducible Objects
3 Irreducible Objects
3.1 Introduction
For a semisimple category \(C\) of the form \(\bigoplus_{i \in I} k\mathrm{-vec}\) to become a rigid symmetric closed monoidal skeletal category, the set \(I\) has to be equipped with extra
strucutre. To become a skeletal category, we need:
To become a monoidal category, we need:
• a function \(\texttt{IsYieldingIdentities}\), deciding whether an object yields the identity whenever it is part of an associator triple or a braiding pair,
• functions \(\texttt{Multiplicity}\) and \(\texttt{*}\), defining the tensor product on objects,
• a function \(\texttt{AssociatorFromData}\), defining the tensor product on morphisms.
To become a symmetric monoidal category, we need:
To become a rigid symmetric monoidal category, we need:
In the following, two families of such sets \(I\) are introduced:
• \(\texttt{GIrreducibleObject}\): For a group \(G\), the set \(I\) consists of the irreducible characters of \(G\). We call the elements in \(I\) the \(G\)-irreducible objects.
• \(\texttt{GZGradedIrreducibleObject}\): For a group \(G\), the set \(I\) consists of the irreducible characters of \(G\) together with a degree \(n \in \mathbb{Z}\). We call the elements in \(I\)
the \(G-\mathbb{Z}\)-irreducible objects.
3.2 Constructors
3.2-1 GIrreducibleObject
‣ GIrreducibleObject( c ) ( attribute )
Returns: a \(G\)-irreducible object
The argument is a character \(c\) of a group. The output is its associated \(G\)-irreducible object.
3.2-2 GZGradedIrreducibleObject
‣ GZGradedIrreducibleObject( n, c ) ( operation )
Returns: a \(G-\mathbb{Z}\)-irreducible object
The argument is an integer \(n\) and a character \(c\) of a group. The output is their associated \(G-\mathbb{Z}\)-irreducible object.
3.3 Attributes
3.3-1 UnderlyingCharacter
‣ UnderlyingCharacter( i ) ( attribute )
Returns: an irreducible character
The argument is a \(G\)-irreducible object \(i\). The output is its underlying character.
3.3-2 UnderlyingGroup
‣ UnderlyingGroup( i ) ( attribute )
Returns: a group
The argument is a \(G\)-irreducible object \(i\). The output is its underlying group.
3.3-3 UnderlyingCharacterTable
‣ UnderlyingCharacterTable( i ) ( attribute )
Returns: a character table
The argument is a \(G\)-irreducible object \(i\). The output is the character table of its underlying group.
3.3-4 UnderlyingIrreducibleCharacters
‣ UnderlyingIrreducibleCharacters( i ) ( attribute )
Returns: a list
The argument is a \(G\)-irreducible object \(i\). The output is a list consisting of the irreducible characters of its underlying group.
3.3-5 UnderlyingCharacterNumber
‣ UnderlyingCharacterNumber( i ) ( attribute )
Returns: an integer
The argument is a \(G\)-irreducible object \(i\). The output is the integer \(n\) such that the \(n\)-th entry of the list of the underlying irreducible characters is the underlying irreducible
character of \(i\).
3.3-6 Dimension
‣ Dimension( i ) ( attribute )
Returns: an integer
The argument is a \(G\)-irreducible object \(i\). The output is the dimension of its underlying irreducible character.
3.3-7 Dual
Returns: a \(G\)-irreducible object
The argument is a \(G\)-irreducible object \(i\). The output is the \(G\)-irreducible object associated to the dual character of \(c\), where \(c\) is the associated character of \(i\).
3.3-8 UnderlyingCharacter
‣ UnderlyingCharacter( i ) ( attribute )
Returns: an irreducible character
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is its underlying character.
3.3-9 UnderlyingDegree
‣ UnderlyingDegree( i ) ( attribute )
Returns: an integer
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is its underlying degree.
3.3-10 UnderlyingGroup
‣ UnderlyingGroup( i ) ( attribute )
Returns: a group
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is its underlying group.
3.3-11 UnderlyingCharacterTable
‣ UnderlyingCharacterTable( i ) ( attribute )
Returns: a character table
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is the character table of its underlying group.
3.3-12 UnderlyingIrreducibleCharacters
‣ UnderlyingIrreducibleCharacters( i ) ( attribute )
Returns: a list
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is a list consisting of the irreducible characters of its underlying group.
3.3-13 UnderlyingCharacterNumber
‣ UnderlyingCharacterNumber( i ) ( attribute )
Returns: an integer
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is the integer \(n\) such that the \(n\)-th entry of the list of the underlying irreducible characters is the underlying
irreducible character of \(i\).
3.3-14 Dimension
‣ Dimension( i ) ( attribute )
Returns: an integer
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is the dimension of its underlying irreducible character.
3.3-15 Dual
Returns: a \(G-\mathbb{Z}\)-irreducible object
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is the \(G-\mathbb{Z}\)-irreducible object associated to the degree \(-n\) and the dual character of \(c\), where \(n\) is the
underlying degree and \(c\) is the underlying character of \(i\).
3.4 Properties
3.4-1 IsYieldingIdentities
‣ IsYieldingIdentities( i ) ( property )
Returns: a boolean
The argument is a \(G\)-irreducible object \(i\). The output is true if the underlying character of \(i\) is the trivial one, false otherwise.
3.4-2 IsYieldingIdentities
‣ IsYieldingIdentities( i ) ( property )
Returns: a boolean
The argument is a \(G-\mathbb{Z}\)-irreducible object \(i\). The output is true if the underlying character of \(i\) is the trivial one, false otherwise.
3.5 Operations
3.5-1 Multiplicity
‣ Multiplicity( i, j, k ) ( operation )
Returns: an integer
The arguments are 3 \(G\)-irreducible objects \(i,j,k\). Let their underlying characters be denoted by \(a,b,c\), respectively. Then the output is the number \(\langle a, b\cdot c \rangle\), i.e.,
the multiplicity of \(a\) in the product of characters \(b \cdot c\).
3.5-2 \*
‣ \*( i, j ) ( operation )
Returns: a list
The arguments are 2 \(G\)-irreducible objects \(i,j\) with underlying irreducible characters \(a,b\), respectively. The output is a list L = \([ [ n_1, k_1 ], \dots, [ n_l, k_l ] ]\) consisting of
positive integers \(n_c\) and \(G\)-irreducible objects \(k_c\) representing the character decomposition into irreducibles of the product \(a\cdot b\).
3.5-3 AssociatorFromData
‣ AssociatorFromData( i, j, k, A, vec, L ) ( operation )
Returns: a list
The arguments are
• three \(G\)-irreducible objects \(i,j,k\),
• a list \(A\) containing the associator on all irreducibles as strings, e.g., the list constructed by the methods provided in this package,
• a matrix category vec of a homalg field \(F\),
• a list L = \([ [ n_1, h_1 ], \dots, [ n_l, h_l ] ]\) consisting of positive integers \(n_c\) and \(G\)-irreducible objects \(h_c\) representing the character decomposition into irreducibles of
the product of \(i,j,k\).
The output is the list \([ [ \alpha_{h_1}, h_1 ], \dots, [ \alpha_{h_l}, h_l ] ]\), where \(\alpha_{h_c}\) is the \(F\)-vector space homomorphism representing the \(h_c\)-th component of the
associator of \(i,j,k\).
3.5-4 ExteriorPower
‣ ExteriorPower( i, j ) ( operation )
Returns: a list
The arguments are two \(G\)-irreducible objects \(i, j\). The output is the empty list if \(i\) is not equal to \(j\). Otherwise, the output is a list \(L = [ [ n_1, k_1 ], \dots, [ n_1, k_l ] ]\)
consisting of positive integers \(n_j\) and \(G\)-irreducible objects \(k_j\), corresponding to the decomposition of the second exterior power character \(\wedge^2 c\) into irreducibles. Here, \(c\)
is the associated character of \(i\).
3.5-5 Multiplicity
‣ Multiplicity( i, j, k ) ( operation )
Returns: an integer
The arguments are 3 \(G-\mathbb{Z}\)-irreducible objects \(i,j,k\). Let their underlying characters be denoted by \(a,b,c\), respectively, and their underlying degrees by \(n_i, n_j, n_k\),
respectively. The output is \(0\) if \(n_i\) is not equal to \(n_j + n_k\). Otherwise, the output is the number \(\langle a, b\cdot c \rangle\), i.e., the multiplicity of \(a\) in the product of
characters \(b \cdot c\).
Let their underlying characters be denoted by \(a,b\), respectively, and their underlying degrees by \(n_i, n_j\), respectively. if \(n_i = n_j\) and the underlying character number of \(j\)
3.5-6 \*
‣ \*( i, j ) ( operation )
Returns: a list
The arguments are 2 \(G-\mathbb{Z}\)-irreducible objects \(i,j\) with underlying irreducible characters \(a,b\), respectively. The output is a list L = \([ [ n_1, k_1 ], \dots, [ n_l, k_l ] ]\)
consisting of positive integers \(n_c\) and \(G\)-irreducible objects \(k_c\) representing the character decomposition into irreducibles of the product \(a\cdot b\). The underlying degrees of \(k_c\)
are given by the sum of the underlying degrees of \(i\) and \(j\).
3.5-7 AssociatorFromData
‣ AssociatorFromData( i, j, k, A, vec, L ) ( operation )
Returns: a list
The arguments are
• three \(G-\mathbb{Z}\)-irreducible objects \(i,j,k\),
• a list \(A\) containing the associator on all irreducibles (of \(G\)-irreducible objects) as strings, e.g., the list constructed by the methods provided in this package,
• a matrix category vec of a homalg field \(F\),
• a list L = \([ [ n_1, h_1 ], \dots, [ n_l, h_l ] ]\) consisting of positive integers \(n_c\) and \(G-\mathbb{Z}\)-irreducible objects \(h_c\) representing the character decomposition into
irreducibles of the product of \(i,j,k\).
The output is the list \([ [ \alpha_{h_1}, h_1 ], \dots, [ \alpha_{h_l}, h_l ] ]\), where \(\alpha_{h_c}\) is the \(F\)-vector space homomorphism representing the \(h_c\)-th component of the
associator of \(i,j,k\).
3.5-8 ExteriorPower
‣ ExteriorPower( i, j ) ( operation )
Returns: a list
The arguments are two \(G-\mathbb{Z}\)-irreducible objects \(i, j\). The output is the empty list if the underlying characters of \(i\) and \(j\) are unequal. Otherwise, the output is a list \(L = [
[ n_1, k_1 ], \dots, [ n_1, k_l ] ]\) consisting of positive integers \(n_j\) and \(G-\mathbb{Z}\)-irreducible objects \(k_a\), corresponding to the decomposition of the second exterior power
character \(\wedge^2 c\) into irreducibles. Here, \(c\) is the associated character of \(i\) and \(j\). The underlying degree of \(k_a\) is the sum of the underlying degrees of \(i\) and \(j\). | {"url":"https://homalg-project.github.io/CAP_project/GroupRepresentationsForCAP/doc/chap3_mj.html","timestamp":"2024-11-02T14:28:54Z","content_type":"application/xhtml+xml","content_length":"33247","record_id":"<urn:uuid:349c2081-f8d6-40d6-99c0-9d34a5641c74>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00783.warc.gz"} |
Integers Worksheet
Welcome to our class 7 Mathematics blog! Today, we’ll explore integers, which include positive numbers,negative numbers and zero. You will learn about their properties, basic operations and practical
uses. Answer the following:
Integers/CBSE Class 6 Mathematics/Model Questions
MCQ/Extra Questions for Practice/Sample Questions
Choose the correct answer from the options given below: ANSWERS:
Integers CBSE Class 6 Mathematics/Worksheet
Integers CBSE Class 6 Mathematics/Worksheet is about the Extra Questions that you can expect for Yearly Examination. Here you can find out practice problems for Class 6 Mathematics. This worksheet is
designed for CBSE...
Integers CBSE Class 6 Mathematics/Worksheet/Extra Questions for Practice
Integers CBSE Class 6 Mathematics/ Worksheet is about the practice problems that you can expect for Yearly Examination. Here you can find out Model Questions from the chapter Integers. This worksheet
is designed for...
Integers/Class 6 Mathematics/MCQ
Integers/Class 6 Mathematics/MCQ is about the Model Questions that you can expect for Yearly Examination. Here you can find out practice problems for Class 6 Mathematics. Integers/ Class 6
Mathematics/ Extra Questions for PracticeMultiple... | {"url":"https://www.learnmathsonline.org/tag/integers-worksheet/","timestamp":"2024-11-01T22:27:43Z","content_type":"text/html","content_length":"70898","record_id":"<urn:uuid:6e2df0e4-0713-40c7-afee-e9a04f2f3558>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00340.warc.gz"} |
How does one provide initial step size for fmin in scipy?
How does one provide initial step size for fmin in scipy?
I am trying to minimize a funtion f(x,y) over a domain that is considerably large for x than y. Also the domain of y is much less than one. I want to minimize the function using simplex algorithm
provided in scipy - fmin. The same algorithm in gsl has a option of providing inital step size. However, I don't see that option in scipy. Is there a way to provide a initial step-size, or any other
python implementation of function minimization that just requires the function not its derivatives in which I can provide the initial step size?
1 Answer
Sort by ยป oldest newest most voted
scipy.optimize.fmin_powell seems to do what you want - specify direc. There's also a pretty good summary page of the optimization routines available in scipy.
edit flag offensive delete link more | {"url":"https://ask.sagemath.org/question/8006/how-does-one-provide-initial-step-size-for-fmin-in-scipy/","timestamp":"2024-11-10T20:58:51Z","content_type":"application/xhtml+xml","content_length":"52771","record_id":"<urn:uuid:72642be0-bf97-4f37-a55f-896aea0df5e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00350.warc.gz"} |
Non-parametric Jumps Tests - AI for Markets
Non-parametric Daily Tests
The non-parametric daily jump tests developed with the methodology of Barndorff-Nielson and Shephard (2006) that describes the asymptotic behavior of the Bipower Variation, as a measure for the daily
integrated volatility, which is robust to jumps. In case a significant difference between the Realized Variance and this measure is determined with a certain level of confidence, then we can conclude
that at least one significant change in price was realized in the respective day.
The test can provide an answer regarding the existence of sudden price changes in a respective time frame, once all the prices are realized.
Non-parametric Intradaily Tests – Lee-Mykland and Lee-Hannig Tests
The importance of sudden changes in the dynamics of stock prices was acknowledged in the framework of continuous-time jump-diffusion modeling by Merton (1976) and their identification was always
considered as an important econometric issue that necessitated sophisticated numerical estimation techniques, highly computationally intensive. The field developed in the direction of parametric
estimation of jumps with, among others, the work of Jorion (1988), Maheu and McCurdy (2003), Andersen, Benzoni and Lund (2002), Bates (2000) and Chernov, Gallant, Gysels and Tauchen (2003) on a
stream of research aiming at calibrating processes that allow for time-varying intensities of jumps with different model specifications that allow for diffusions with stochastic volatilities and
jumps. This stream of research allowed for the semi-martingale setting to become the standard framework for the modeling of stock returns. However, the parametric method of jump identification
requires a heavy modeling aparatus with sophisticated numerical methods for its estimation, which makes them less useful especially at the intra-day level.
The new stream of non-parametric methods has gained momentum. The work of Barndorff-Nielsen and Shephard (2006) was essential for a new stage in the jump-detection process. Their main contribution to
the literature consists in the use of Bipower Variation as a nonparametric (model-free) robust-to-jumps volatility estimator. Their research provided one of the mostly used framework for detection of
daily jumps, which relies on the fact that the difference between the Realized Variance (as a measure of Integrated Volatility for a trading day) and the Bipower Variation (a measure for the same
statistic – the Integrated Volatility – that is robust to jumps) is a variable that has stable distribution asymptotically and allows for the identification of jumps in case of significance.
The seminal paper of Lee and Mykland (2008) opened the door to nonparametric identification of jumps at the intraday frequency based on the local estimation of volatility, while the series of
academic work cosigned by Ait-Sahalia developed a framework for the decomposure of continuous dynamics, small jumps and large jumps, from which we mostly acknowledge the Ait-Sahalia and Jacod (2009)
that rooted, among others, the important work of the computation of Truncated Variation developed by Lee and Hannig (2010) for the differentiation between big jumps and small jumps at the high
frequency level, and culminated with Ait-Salahia and Jacod (2012) that represents an important breakthrough in the setting of a background for jump identification tests across power variations,
truncation rates and frequencies.
We provide here an example of the use of the Lee-Mykland test and the Lee-Hannig test on the EURUSD for the 5-minute log-returns on a time series starting on July 20th 2013. The specification of the
Lee-Mykland test is the following:
where M is the number of observations i in the sample t and BVt is the Bipower Variation computed for the M returns in the sample t. As opposed to this framework, the Lee-Hannig test is using the
Truncated Variation instead of the Bipower Variation.
where g>0 and
The following graphs show the difference in the two tests. We can notice that the Lee-Mykland test provides more jumps than the Lee-Hannig.
One of the drawbacks of intra-day measures is the fact that the high-frequency returns are known to be influenced by many factors that generate periodic dynamics, i.e. have time-dependent
structure. In search for a suited measure of volatility at the intra-day frequency, Andersen and Bollerslev (1997) show that the dynamics of returns and volatilities for different financial assets
present strong periodicity, which is likely to mislead the statistical analyses aiming at characterizing the intra-day variation of financial assets’ returns. This conjecture generated the adjustment
of the jump tests using various both parametric and non-parametric measures of periodicity introduced by Boudt, Croux and Laurent (2011), as Median Absolute Deviation, the Shortest Half scale
estimator on one hand, and the Truncated Maximum Likelihood estimator on the other hand.
Ait-Sahalia, Y., Jacod, J., (2009) Estimating the Degree of Activity of Jumps in High Frequency Data, The Annals of Statistics Lee, S. S., Hannig, J., (2010) Detecting Jumps from Levi Jump Diffusion
Processes, Journal of Financial Economics;
Ait-Sahalia, Y., Jacod, J., (2012) Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data, Journal of Economic Literature;
Andresen, T. G., Bollerslev, T., (1997) Intraday Periodicity and Volatility Persistence in Financial Markets, Journal of Empirical Finance;
Andersen, T.G., Benzoni, L., Lund, J., (2002) An Empirical Investigation of Continuous-Time Equity Return Models, The Journal of Finance;
Barndorff-Nielsen, O. E., Shephard, N., (2006) Econometrics of Testing for Jumps in Financial Economics using Bipower Variation, Journal of Financial Econometrics;
Bates, D.S., (2000), Post-87′ Crash Fears in the S&P 500 Futures Option Market, Journal of Econometrics;
Boudt, K., Croux, C., Laurent, S., (2011) Robust Estimation of Intraweek Periodicity in Volatility and Jump Detection, Journal of Empirical Finance;
Chernov, M., Gallant, A.R., Gysels, E., Tauchen, G., (2003) Alternative Models for Stock Price Dynamics, Journal of Econometrics;
Jorion, P., (1988) On Jump Processes in the Foreign Exchange and Stock Markets, The Review of Financial Studies;
Cont, R., (2001) Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues, Quantitative Finance;
Lee, S. S., Mykland, P. A., (2008) Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics, The Review of Financial Studies;
Maheu, J.M., McCurdy, T.H., (2004) News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns, The Journal of Finance;
Merton, R. C., (1976) Option Pricing when Underlying Returns are Discontinuous, Journal of Financial Economics. | {"url":"https://aiformarkets.com/2020/09/30/non-parametric-jumps-tests/","timestamp":"2024-11-11T16:46:47Z","content_type":"text/html","content_length":"96686","record_id":"<urn:uuid:f7044e44-b1d2-4376-bfa0-7ba4f2b40c97>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00794.warc.gz"} |
Glossary: PDO - Pacific Decadal Oscillation; LOD - Earth's Length of Day
Back in 2008, I wrote a paper entitled:
Wilson, I.R.G., 2011, Are Changes in the Earth’s Rotation Rate Externally Driven and Do They Affect Climate? The General Science Journal, Dec 2011, 3811.
which can be freely down loaded at:
One of the results of this paper concerned the long-term changes in the Pacific Decadal Oscillation (PDO). It predicted that the PDO should return to its positive phase sometime around 2015 - 2017.
A. The difference between the actual LOD and the nominal LOD value of 86400 seconds.
Page 11 - Figure 4
Figure 4: This figure shows the variation of the Earth's length-of-day (LOD) from 1656 to 2005 (Sidorenkov 2005)[blue curve]. The values shown in the graph are the difference between the actual LOD
and the nominal LOD value of 86400 seconds, measured in units of 10^(-5) seconds. Superimposed on this graph are 1st and 3rd order polynomial fits to the change in the Earth's LOD.
B. The absolute deviation of the Earth's LOD from a 1st and 3rd order polynomial fit to the long-term changes in the LOD between 1656 and 2005
page 14 - Figure 7a
Figure 7a: Shows the absolute deviation of the Earth's LOD from a 1st and 3rd order polynomial fit to the long-term changes in the LOD (measured in units of 10^(-5) seconds). There are nine
significant peaks in the absolute deviation which are centered on the years 1729, 1757, 1792, 1827, 1869, 1906, 1932, 1956 and 1972.
C. A comparison between the peak (absolute) deviations of the LOD from its long-term trend and the years where the phase of the PDO [proxy] reconstruction is most positive.
Page 15 - Figure 8
Figure 8: The upper graph shows the PDO reconstruction of D’Arrigo et al. (2001) between 1707 and
1972. The reconstruction has been smoothed with a 15-year running mean filter to eliminate short-term fluctuations. Superimposed on this PDO reconstruction is the instrumental mean annual PDO index
(Mantua 2007) which extends the PDO series up to the year 2000. The lower graph shows the absolute deviation of the Earth’s LOD from 1656 to 2005. The data in this figure has also been smoothed with
a 15-year running mean filter.
A comparison between the upper and lower graph in figure 8 (above) shows that there is a
remarkable agreement between the years of the peak (absolute) deviations of the LOD from its
long-term trend and the years where the phase of the PDO [proxy] reconstruction is most positive. While the correlation is not perfect, it is convincing enough to conclude the PDO index is another
good example of a climate system that is directly associated with changes in the Earth's rotation rate.
If you look closely at the peaks in the deviation of Earth's LOD from its long term trend and the peaks in the PDO index shown in figure 8, you will notice that the peaks in deviation of LOD take
place 8 - 10 years earlier (on average) than the peaks in the PDO index, suggesting a causal link.
D. The path of the CM of the Solar System about the Sun in a reference frame that is rotating with the planet Jupiter
Page 17 - Figure 9
Figure 9: Shows the Sun in a reference frame that is rotating with the planet Jupiter. The perspective is the one you would see if you were near the Sun’s pole. A unit circle is drawn on the left
side of this figure to represent the Sun, using an x and y scales marked in solar radii. The position of the CM of the Solar System is also shown for the years 1780 to 1820 A.D. The path starts in
the year 1780, with
each successive year being marked off on the curve, as you move in a clockwise direction. This
shows that the maximum asymmetry in the Sun’s motion occurred roughly around 1790-91.
The path of the CM of the Solar System about the Sun that is shown in figure 9 [above] mirrors the typical motion of the Sun about the CM of the Solar System. This motion is caused by the combined
gravitational influences of Saturn, Neptune, and to a lesser extent Uranus, tugging on the Sun.
The motion of the CM shown in figure 9 repeats itself roughly once every 40 years. The timing and level of asymmetry of Sun’s motion is set, respectively, by when and how close the path approaches
the point (0.95, 0.0), just to the left of the Sub-Jupiter point. Hence, we can quantify the magnitude and timing of the Sun’s asymmetric motion by measuring the distance of the CM from the point
(0.95, 0.0).
E. The years where the Suns' motion about the CM of the Solar System is most asymmetric.
Page 18 - Figure 10
Figure 10: shows The distance of the centre-of-mass (CM) of the Solar System (in solar radii) from the point (0.95, 0.00) between 1650 and 2000 A.D. The distance scale is inverted so that top of the
peaks correspond to the times when the Sun’s motion about the CM is most asymmetric.
An inspection of figure 10 shows that there are times between 1700 and 2000 A.D. where the CM of the Solar System approaches the point (0.095, 0.00) i.e. at the peaks of the blue curve in figure 10
where the Sun's motion about the CM is most asymmetric. These are centred on the years, 1724, 1753, 1791, 1827, 1869, 1901, 1932, and 1970. Remarkably, these are very close to the years in which the
Earth’s LOD experienced its maximum deviation from its long-term trend i.e. the years 1729, 1757, 1792, 1827, 1869, 1906, 1932, 1956 and 1972.
This raised the possibility that the times of maximum deviation of the Earth's LOD might be related to the times of maximum asymmetry in the Sun’s motion about the CM.
In addition, if both of these indices precede transitions of the PDO into its positive phase by 8 - 10 years, then it could be possible to use the times of maximum asymmetry in the Sun’s motion about
the CM to predict when the PDO will make its next transition into its positive phase.
F. When will the transition to the next positive phase of the PDO take place?
This figure shows the proxy PDO reconstruction of D’Arrigo et al. (2001) between 1707 and 1972 [blue curve]. The reconstruction has been smoothed with a 15-year running mean filter to eliminate
short-term fluctuations. Superimposed on this PDO reconstruction is the instrumental mean annual PDO index (Mantua 2007) which extends the PDO series up to the year 2000 [green curve]. Also shown is
the proximity of the CM of the Solar System to sub-Jupiter point which measures the asymmetry of the Sun's motion about the CM [orange curve].
Hence, like the long term deviation of the Earth's LOD from its long term trend, the peaks in asymmetry of the Sun's motion about the CM of the Solar System take place roughly 8 - 10 years prior to
positive peaks in the PDO index.
Careful inspection of the figure above shows that Sun's motion about the CM peaks in about 2007 which would indicate that the next transition to a positive PDO phase should take place some time
around the years 2015 to 2017.
[Note: The above graph shows a prediction made on the assumption that forward shift between the two curves is of the order of the average length of the Hale sunspot cycle = 11 years. It probably a
good indicator of the level of uncertainty of the prediction being made].
[Note: I propose that GEAR EFFECT is the underlying reason for the connection between peaks in the asymmetry of the Sun's motion about the Barycentre of the Solar System (SSBM) and the absolute
deviation of the Earth rotation rate about it's long-term in crease of ~ 1.7 ms/century. A post describing the GEAR EFFECT can be found here:]
Scientific Publications and Presentations
UPDATED 16/04/2015
The following is a list of my recent scientific publications and presentations. I am placing the list on my blog so that others can have easy access.
Wilson, I.R.G. Are the Strongest Lunar Perigean Spring Tides Commensurate with the Transit Cycle of Venus?, Pattern Recogn. Phys., 2, 75-93
Wilson, I.R.G.: The Venus–Earth–Jupiter spin–orbit coupling
model, Pattern Recogn. Phys., 1, 147-158
Wilson, I.R.G. and Sidorenkov, N.S., Long-Term Lunar Atmospheric Tides in the
Southern Hemisphere, The Open Atmospheric Science Journal,
2013, 7, 51-76
Wilson, I.R.G., 2013, Are Global Mean Temperatures
Significantly Affected by Long-Term Lunar Atmospheric
Tides? Energy & Environment, Vol 24,
No. 3 & 4, pp. 497 - 508
Wilson, I.R.G., 2013, Personal Submission to the Senate
Committee on Recent Trends in and Preparedness for
Extreme Weather Events, Submission No. 106
Wilson, I.R.G., Lunar Tides and the Long-Term Variation
of the Peak Latitude Anomaly of the Summer Sub-Tropical
High Pressure Ridge over Eastern Australia
The Open Atmospheric Science Journal, 2012, 6, 49-60
Wilson, I.R.G., Changes in the Earth's Rotation in relation
to the Barycenter and climatic effect. Recent Global Changes
of the Natural Environment. Vol. 3, Factors of Recent
Global Changes. – M.: Scientific World, 2012. – 78 p. [In Russian].
This paper is the Russian translation of my 2011 paper
Are Changes in the Earth’s Rotation Rate Externally
Driven and Do They Affect Climate?
The General Science Journal, Dec 2011, 3811.
Wilson, I.R.G., 2011, Are Changes in the Earth’s Rotation
Rate Externally Driven and Do They Affect Climate?
The General Science Journal, Dec 2011, 3811.
Wilson, I.R.G., 2011, Do Periodic peaks in the Planetary Tidal
Forces Acting Upon the Sun Influence the Sunspot Cycle?
The General Science Journal, Dec 2011, 3812.
[Note: This paper was actually written by October-November 2007 and submitted to the New Astronomy (peer-reviewed) Journal in early 2008 where it was rejected for publication. It was resubmitted to
the (peer-reviewed) PASP Journal in 2009 where it was again rejected. The paper was eventually published in the (non-peer reviewed) General Science Journal in 2010.]
N. Sidorenkov, I.R.G. Wilson and A.I. Kchlystov, 2009, The
decadal variations in the geophysical processes and the
asymmetries in the solar motion about the barycentre.
Geophysical Research Abstracts Vol. 12, EGU2010-9559,
2010. EGU General Assembly 2010 © Author(s) 2010
Wilson, Ian R.G., 2009, Can We Predict the Next Indian
Mega-Famine?, Energy and Environment, Vol 20,
Numbers 1-2, pp. 11-24.
El Ninos and Extreme Proxigean Spring Tides
A lecture by Ian Wilson at the Natural Climate Change
Symposium in Melbourne on June 17th 2009.
Wilson, I.R.G., Carter, B.D., and Waite, I.A., 2008,
Does a Spin-Orbit Coupling Between the Sun and the
Jovian Planets Govern the Solar Cycle?,
Publications of the Astronomical Society of Australia,
2008, 25, 85 – 93.
N.S. Sidorenkov, Ian Wilson. The decadal fluctuations
in the Earth’s rotation and in the climate characteristics.
In: Proceedings of the "Journees 2008 Systemes de reference
spatio-temporels", M. Soffel and N. Capitaine (eds.),
Lohrmann-Observatorium and Observatoire de Paris.
2009, pp. 174-177
Which Came First? - The Chicken or the Egg?
A Presentation to the 2008 Annual General Meeting of the
Lavoisier Society by Ian Wilson
Wilson, I. R. G., 2006, Possible Evidence of the
De Vries, Gleissberg and Hale Cycles in the Sun’s
Barycentric Motion, Australian Institute of Physics 17^th
National Congress 2006, Brisbane, 3^rd -8^th December
2006 (No longer available on the web) | {"url":"https://astroclimateconnection.blogspot.com/2015/04/","timestamp":"2024-11-06T20:40:44Z","content_type":"text/html","content_length":"117602","record_id":"<urn:uuid:7f2ced05-b9ba-4d7e-86ba-4b91884cc880>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00260.warc.gz"} |
Duality in Spatial Algebra
Let’s take a scalar unit 1 and multiply by a tri-vector i, we obtain the trivector i for the dual unit. We multiply p by i and obtain the dual trivector -1.
Vectors are dual to bivectors, and vectors to vectors for bivectors. The transition from the p-number A to the dual p-number iA = Ai is not an involution, since in the chain of maps A -> iA -> -A->
-iA -> A, the initial p-number A occurs only in 4 steps.
Thus, the space of vectors and bi-vectors has a duality.
UV = 1 (U[1]V[1] + U[2]V[2] + U[2]V[2])
+ i[(U[2]V[3] – U[3]V[2])k[1] + (U[3]V[1] – U[1]V[3])k[2] + (U[1]V[2] – U[2]V[1])k[3]]
In the first line, the scalar product of two vectors.
The terms in the square brackets determine the vector product of two vectors.
In the second line is the outer product of two vectors. It is a bivector, the dual vector of the vector product of vectors.
The exterior product of vectors is denoted by the sign of V-inverted and analogous to the multiplication of differential forms.
In the general case, the Clifford product of two vectors is the sum of a scalar and a bivector.
The scalar product of bivectors is equal to the scalar product of the vectors to which these bivectors are dual.
The product of bivectors is simply the product of their dual vectors taken with the opposite sign.
The norm of the outer product of two vectors is equal to the area of the parallelogram constructed on these vectors.
The angle between the bi-vectors B = iU and C = iV is defined as the angle between the vectors U and V, to which these bivectors are dual.
Euler’s formula relating exponential and trigonometric functions in the algebra of complex numbers and generating phase transformations, as well as the Hamilton-Cayley formula, which describes
spatial rotations in the algebra of quaternions.
Scalars and Trivectors
As we discussed earlier for scalars and trivectors, the cycle is multiplied by i.
1 -> i -> -1 -> -i -> 1.
If the length of the circle in radians is 2pi, and we will put in correspondence a segment of the circle for each transformation. Then we get that the phase pi / 2 transforms the scalar 1 into a
three-vector (volume) i.
Note that when converting to phase pi, scalars and trivectors change sign to the opposite.
Vectors and Bivectors
On the Ender Crystal model, the inner rotating cube can be considered rotated to the phase pi / 2 with respect to the outer cube.
In this case, the edges of the outer cube (vector) correspond to the faces of the inner (bivector).
With the phase transformation pi / 2, the vectors become bivectors, and bivectors into vectors. Or in other words, the edges enter the faces, and the faces turn into edges.
In the presence of a geometric imagination, such a morph can be represented on one single cube. | {"url":"https://homedevice.pro/duality-in-spatial-algebra/","timestamp":"2024-11-04T07:47:35Z","content_type":"text/html","content_length":"27026","record_id":"<urn:uuid:775a9b8e-0ec6-4eeb-b1e4-26c9eda31fdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00806.warc.gz"} |
Appendix B: Local Labour Market Definitions | RDP 2021-09: Is the Phillips Curve Still a Curve? Evidence from the Regions
RDP 2021-09: Is the Phillips Curve Still a Curve? Evidence from the Regions Appendix B: Local Labour Market Definitions
US studies often use ‘commuting zones’ to represent local labour markets across the United States (see Foote, Kutzbach and Vilhuber (2017) for a review). Commuting zones are areas with a high degree
of overlap between where people live and where they work. We develop similar classifications by following the methodology used to construct US commuting zones (Tolbert and Sizer 1996), with a few
modifications. Our steps are below.
Step 1: Data matrix
The building blocks of our classifications are 2,089 SA2s (at the 2011 Census), which cover the entire continent.^[47]
The 2011 Census provides employment counts by ‘place of usual residence’ cross-tabulated by ‘place of work’. These cross-tabulations – in the form of a 2,089 by 2,089 matrix – provide a detailed
snapshot of movements of people to and from work in 2011. Each row of the matrix represents a place of usual residence (origin), each column represents a place of work (destination), and each cell is
the number of people who travel from a particular origin to a particular destination for work.
We use this data matrix to construct a 2,089 by 2,089 ‘dissimilarity matrix’. Each element D[ij] measures the dissimilarity of SA2 i from SA2 j :
(A1) $D ij =1− f ij + f ji min( rf l i ,rf l j )$
where f[ij] is the number of commuters who live in i and work in j, and $rf l i = ∑ k f ik$ (including f[ii]) is the resident workforce in SA2 i .^[48] Values of D[ij] close to zero indicate strong
commuting ties between areas i and j, while values close to one indicate weak commuting ties. The main diagonal of the dissimilarity matrix is set to zero.
Step 2: Cluster analysis
The next step is to group the SA2s into a set of ‘clusters’, with each cluster representing a distinct local labour market (otherwise known as a commuting zone). To do this we perform a hierarchical
cluster analysis using the dissimilarity matrix constructed in Step 1.^[49] The clustering algorithm groups together SA2s based on the strength of their commuting ties.^[50]
The most important decision in the procedure is at what point to stop merging clusters together. The algorithm starts out by treating each SA2 as its own cluster, and then continues to group these
together – in order of how strong their commuting ties are – until we tell it to stop. If we stop the procedure too early, only the SA2s with the strongest commuting links will be merged into
clusters. If we stop the procedure too late, even clusters with relatively weak commuting ties will be merged together.
The eventual number of clusters formed will depend on the ‘height’ of allowable clusters (i.e. the average between-cluster dissimilarity). As an example, at a height of 0.7 very few SA2s have been
merged together – that is, most clusters will comprise a single SA2 (Figure B1). However, this changes rapidly as the height of allowable clusters is increased beyond this point: more and more SA2s
are merged to form clusters and the average number of SA2s per cluster increases.
Figure B1: Cluster Formation
By cut-off value
Sources: ABS; Authors' calculations
The commuting zone classifications commonly used in the United States are based on a rule that allows clusters to form that are no higher than a height of 0.98 (Tolbert and Sizer 1996). In other
words, clusters that form at the 0.98 threshold are deemed to exhibit sufficiently strong commuting ties such that they should not be divided into multiple commuting zones, while clusters that form
above this cut-off are deemed sufficiently distant from one another to warrant separation. This choice of cut-off value, which results in 741 commuting zones for the United States, was based on
Tolbert and Sizer's (1996, p 14) observation that it ‘produced reasonable and consistent results across the wide variety of U.S. counties’.
We also define local labour markets based on a cut-off value of 0.98, to be consistent with the US research and because this cut-off value produces geographic groupings that look broadly sensible.^
Our local labour markets exhibit roughly the same amount of live–work overlap as SA4s (Table B1). Although higher cut-offs would increase the extent of live–work overlap, it would come at the cost of
a smaller cross-sectional sample size and less integration within each labour market.
Table B1: Comparison of Geographic Classifications
Number of areas Mean population Mean land area Mean inter-area commuting rate^(a) Mean inter-area mobility rate^(b)
(‵000s) (‵000 km^2) (%) (%)
SA2 2,196 10 4 65.1 10.2
SA3 333 65 23 45.1 7.1
SA4 88 244 87 26.9 4.7
GCCSA 16 1,344 480 4.3 3.5
State/Territory 9 2,390 854 2.5 3.2
FERs – CofFEE 159 135 48 14.7 5.4
FERs – PC 88 244 87 6.6 4.9
Local labour markets 291 74 26 29.9 6.8
Notes: As at 2011 Census; excludes Norfolk Island; greater capital city statistical areas (GCCSAs) include the greater capital city region of each state and the ‘rest of state’; functional economic
regions (FERs) developed by the Centre of Full Employment and Equity (CofFEE) and the Productivity Commission (PC) are described in Appendix C
(a) Average of the mean outbound commuting rate and the mean inbound commuting rate
(b) Average of the mean annual outbound mobility rate and the mean annual inbound mobility rate (for the year to September 2011); excludes overseas arrivals and departures
Sources: ABS; Authors' calculations; Centre of Full Employment and Equity; National Skills Commission; Productivity Commission
Some SA2s represent unpopulated areas, such as airports, major commercial and industrial zones, national parks, defence land, urban parks, and sporting precincts. We combine these SA2s with their
closest populated SA2. [47]
This calculation expresses the commuting ties between two areas with respect to the smaller of the two areas. In a few rare cases, the sum of the workers commuting between SA2s $f ij + f ji$ is
greater than the smaller SA2's resident labour force, in which case we set D[ij] to 0.001. [48]
Following Tolbert and Sizer (1996), we use an average-linkage hierarchical clustering algorithm. [49]
When forming clusters, the algorithm is agnostic about whether SA2s are located next to each other (i.e. contiguous) or not. For example, if fly-in-fly-out workers are commuting long distances it is
possible that distant SA2s could form a cluster. However, this rarely happens in practice given that such commuting patterns are atypical. [50]
By itself, the cluster analysis procedure offers little guidance for choosing the ‘optimal’ cut-off point. However, we can use other information to guide this choice. Theory suggests that variables
such as wages growth, employment growth and housing price growth should be highly correlated across geographic areas within each local labour market (and less correlated across different local labour
markets). As such, to help us choose an appropriate cut-off point we compare the average within-cluster correlation for these economic variables for every possible cut-off point. In addition to these
correlations, we consider the average inbound and outbound commuting rates at every possible cut-off value. On these metrics, a cut-off of 0.98 produced reasonable results compared to other possible
cut-off values. Although a higher cut-off would lead to a set of labour markets with lower inflow and outflow shares on average, this needs to be balanced against the fact the resultant labour
markets would comprise regions with less synchronised market conditions. We also observe that our preferred local labour markets classifications look reasonable when plotted on a map. [51] | {"url":"https://www.rba.gov.au/publications/rdp/2021/2021-09/appendix-b.html","timestamp":"2024-11-05T10:01:40Z","content_type":"application/xhtml+xml","content_length":"39420","record_id":"<urn:uuid:7869a9e4-951a-41f7-ae38-7e0da275591b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00477.warc.gz"} |
"reading solution for model" - What happens here?
Hello everyone,
I have a question regarding the solving process of GAMS. I have an MILP-Model and I am using the CPLEX-Solver. I run a Loop in GAMS to solve for different model variations. The model runs and
produces optimal results but the run-time is very long.
In the log-file I can see the following:
Fixed MIP status(1): optimal
Cplex Time: 7.47sec (det. 2575.80 ticks)
Proven optimal solution.
MIP Solution: -370160.220173 (3161 iterations, 0 nodes)
Final Solve: -370160.220173 (4350 iterations)
Best possible: -370160.220173
Absolute gap: 0.000000
Relative gap: 0.000000
— Restarting execution
— SpeicherOptMFH.gms(842) 555 Mb
— Reading solution for model Modell_SBSopt
— SpeicherOptMFH.gms(764) 762 Mb 2220 secs
You can see, Cplex finds the optimal solution after around 7 sec. But, “reading solution” takes around 40 minutes…
I wanna know, what happens at “reading solutions” and is there a way to shorten the computation time. Additionally for my results, I don’t need to know all the variables, just a small selection.
Maybe there is someone who could help me?
Thank you very much.
It’s probably not the reading that takes that long, but probably some of your calculations you do after the solve. It is easy to build expensive statements in GAMS. The GAMS profile option (https://
www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOprofile) helps you to figure out which lines really takes long. If you run this in a loop (one nesting level, then use “option profile=2;” before the
loop. This will enable profiling of statements inside the loop (otherwise (profile=1) you just get the timing information aggregated for all the statements in the loop). You get the profile summary
in the log and the timing of the individual statements in the listing file.
Thanks a lot.
That was the right call.
There was a loop that read the gdx file, which took forever.
Thank you very much. | {"url":"https://forum.gams.com/t/reading-solution-for-model-what-happens-here/2432","timestamp":"2024-11-09T17:35:36Z","content_type":"text/html","content_length":"18863","record_id":"<urn:uuid:7143128f-b765-43e0-b160-52deb82ab261>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00544.warc.gz"} |
A sample of 0.2140 g of an unknown monoprotic acid was dissolved
in 25.0 mL of...
A sample of 0.2140 g of an unknown monoprotic acid was dissolved in 25.0 mL of...
A sample of 0.2140 g of an unknown monoprotic acid was dissolved in 25.0 mL of water and titrated with 0.0950MNaOH. The acid required 27.4 mL of base to reach the equivalence point. What is the molar
mass of the acid?
moles of NaOH = Molarity of naOH volume of NaOH in Liters = 0.095 M x 0.0274 L = 0.002603 mol
he said acid given is mono protic acid
let mono protic acid = HA
write the balanced equation
HA + NaOH ---> NaA + H2O
from the balanced equation
1 mol of NaOH nuetralize the 1 mol of HA accordingly
0.002603 mol of NaOH nuetralize the 0.002603 mol of HA
so we got th emol of mono protic acid = 0.002603 mol
molar mass of HA = mass of HA / moles of HA
= 0.214 g / 0.002603 mol
= 80.67 g/mol | {"url":"https://justaaa.com/chemistry/715563-a-sample-of-02140-g-of-an-unknown-monoprotic-acid","timestamp":"2024-11-13T21:51:28Z","content_type":"text/html","content_length":"40361","record_id":"<urn:uuid:2a3b4317-54d5-4b82-825d-9d9264cb8536>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00402.warc.gz"} |
HD wallets and the Legendrery PRF in MPC - HackMD
$$ \def\sample{\stackrel{{\scriptscriptstyle\$}}{\leftarrow}} \def\isequal{\stackrel{?}{=}} \def\cP{\mathcal{P}} \def\cF{\mathcal{F}} \def\*{\cdot} \newcommand{\leg}[1]{\genfrac{(}{)}{}{}{#1}{p}} $$
# HD wallets and the Legendrery PRF in MPC ## Introduction In this post we explain what are Hierarchical Deterministic (HD) wallets and Pseudo Random Functions (PRFs), and how they can be used in a
Multi Party Computation (MPC) setting in a very efficient manner. <!--- <details> <summary>Introduction/Motivation(Not sure if to include yet)</summary> ## Introduction properties of the HD wallet.
In this post we explain what are HD wallets and Pseudo Random Functions (PRFs), and how they can be used in a Multi Party Computation (MPC) setting in a very efficient manner. In this post we explain
how the Legendre Pseudo Random Function (PRF) can be used in a Multi Party Computation (MPC) setting in a very efficient manner, requiring at least 3 orders of magnitude less communication rounds
than commonly used PRFs. ## Motivation In the past several years Multi Party Computation is getting frequently used in an application called Threshold Signature Scehemes (TSS), which are schemes that
allow someone to distribute a single private key(e.g. an ECDSA key) between multiple parties such that they need to cooperate to produce signatures. This application is especially used in
cryptocurrency wallets such as Bitcoin, to increase their security and to provide governance over the coins (say require 3 out of 5 board members to sign each transaction) Common bitcoin wallets use
Hierarchical Deterministic Wallet(HD Wallet), this is a scheme in which you have a single master key and from that you generate a new keypair for every operation, this is done for a couple of
reasons: 1. Privacy, If I generate a new address per operation then for example this allows me to accepts Bitcoin payment from clients without exposing each client how much other clients have paid
me. 2. Security, if a single private key is leaked via either a bad signature or some attack it doesn't mean all the money can be extracted, because it is usually distributed among many addresses and
keeps getting moved to new addresses when you send money. Currently all threshold wallets can only use a variant of HD wallet called "non-hardened", which gives you the Privacy property but not the
Security property. The reason is that the hardened HD wallet requires a Pseudo Random Function(PRF) that uses the current private key as the key to the PRF, which is problematic when the private key
is distributed among multiple parties as in a threshold wallet. </details>details> ---> ## Background ### HD Wallets Hierarchical Deterministic (HD) wallets were invented by Pieter Wuille ^[[https://
github.com/bitcoin/bips/blob/master/bip-0032.mediawiki](https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki)], They provide a mechanism to generate a lot of private keys deterministically
from a single master key. That way you only need to backup the master key and you can restore all future keys even if you generate multiple private keys. There are a few reasons why you would want to
keep switching keys: 1. Privacy, if you generate a new address every time you receive a payment, it hides unrelated transactions from the sender, as the address was never used before. 2. Security, if
a single private key is leaked via either a bad signature (faulty nonce) or some attack, the rest of the keys generated from the same master key are safe, and the bitcoins in them are also safe.
There are 2 kinds of HD wallets: #### Non-Hardened Non-Hardened HD means that there's a master public key $PK$ and a master private key $SK$. You can generate new public keys using the master public
key, and their corresponding private keys using the master private key. This allows you to use the master public key on a "hot" wallet and generate new public keys for every transaction without
risking the private key. The downside of this is that if an attacker gets hold of your master public key you lose both the privacy and the security properties of the HD wallet. Generally speaking key
generation here looks as follows: $\delta_i = \cF_{MasterPublicKey}(i)$ where $\cF$ is a Pseudo Random Function(explained later), and then the new public key is $PK_i = PK + \delta_i G$ and the
private key is: $SK_i = SK + \delta_i$, so anyone that has the master public key can calculate $\delta_i$, which breaks both properties of the HD wallet. #### Hardened Hardened HD on the other hand,
doesn't use the master public key to generate keys, instead it always uses the master private key, so it looks as follows: $\delta_i = \cF_{MasterPrivateKey}(i)$ where $\cF$ is a Pseudo Random
Function, and the new public and private keys are again: $PK_i = PK + \delta_i G$ and $SK_i = SK + \delta_i$. On one hand you can no longer store the master public key on a web server to generate
public keys, but on the other hand it means that leaking public keys will never expose $\delta_i$ so an attacker can't calculate future public keys, and can't calculate all private keys from a single
leaked private key, that way both the privacy and security properties will hold as long as the master private key stays safe. ### PRFs Pseudo Random Functions (PRF) are a very common primitive in
cryptography, we use them for HD wallets, but also to construct block ciphers, MACs (Message Authenticated Codes) and more. Simply speaking a PRF is defined as a function that receives a key $K$ and
an input $M$ and returns an output $N$, where for anyone who doesn't have the key the output seems indistinguishable from randomness. $\cF: \{0,1\}^m \times \{0,1\}^k \rightarrow \{0,1\}^n$ <!---
Formally: A function $\cF: \{0,1\}^m \times \{0,1\}^k \rightarrow \{0,1\}^n$ is called a PRF if the following criteria holds: * Given a key $K \in \{0,1\}^k$ and input $M \in \{0,1\}^m$ there is an
efficient algorithm to compute $\cF_K(M)$. * For any adversary $A$ we have: $|Pr_K[A^{\cF_K}] - Pr_f[A^f]| < \epsilon$ Where $f$ is uniformly chosen over all functions $F: \{0,1\}^m \rightarrow \{0,1
\}^n$. ---> ### MPC and Secret Sharing Multi-Party Computation (MPC) is concerned with computing functions over data that is shared between multiple parties while keeping the shares secret even from
the other parties. One common way to do that is using Secret Sharing, which is a mechanism where we share a secret between multiple parties and do operations on that shared secret. Secret sharing
works as follows, each party has a $a_i$ such that $s =\sum_i{a_i}$. Throughout the blog we'll use $[a]$ to say that $a$ is secret shared between $n$ parties. Now we can do arithmetics on secret
shares: * Adding 2 secrets: $[c] = [a] + [b]$ Each party computes $c_i = a_i + b_i$ which results in $c = \sum_i{(a_i + b_i)}$. * Adding a public value: $[c] = [a] + b$ we just need a single party to
add that value to their share, so if $\cP_0$ sets $c_0 = a_0 + b$ then now $c = \sum_i{s_i}+b$. * Multiplying by a public value: $[c] = [a] \* b$ Each party computes $c_i = a_i \* b$, which results
in $c = \sum_i{(b \* a_i)} = b \* \sum_i{a_i}$ * Multiplying 2 secrets is much more complicated, in this blog post we'll assume it is possible but won't get into the details of how. A few common
tricks use a Vandermonde matrix, Beaver triples, Oblivious Transfer, or a homomorphic encryption scheme like Paillier. * Opening a secret: If $[a]$ is some secret share value, then every party $\
cP_i$ sends their share $a_i$, which allows everyone to compute the value $a = \sum_i{a_i}$. We also use $[a] \sample F_p$ to denote randomly sampling a shared secret from the field $F_p$ where $p$
is a prime. ## Motivation: MPC HD wallets MPC and secret sharing is used in practice in a lot of cryptocurrency wallets, currently these wallets either don't use a HD wallet, or use the
*non-hardened* variant. In this post we describe a protocol that will allow us to implement the *hardened* variant of a HD wallet even when our key is secret shared. For that we need a PRF that can
work with a shared key $[K]$, the input $M$ is assumed to be publicly known as it will be the index $i$ in the HD wallet calculation, but we want the output $[N]$ which is the new private key to be
secret shared between the same participants. ## Existing MPC PRFs So what's the problem with known PRFs such as HMAC-SHA256 and Chacha20? If we want to use HMAC-SHA256 like BIP32 does over our shared
key, we will either need a garbled circuit which requires a tremendous amount of computational and communication complexity, or to transform these algorithms into arithmetic circuits, which can then
be computed using secret sharing arithmetic but these contain thousands of gates, and between dozens to hundreds rounds of communication. So if we want something much faster and with low number of
rounds, there are a few known solutions like the Naor-Reingold PRF[^naor] which work on secret shared keys and are quite simple and efficient. These rely on the discrete logarithm and related
problems, but their output must be public and we want the output $N$ to be secret shared and not public. [^naor]: [https://en.wikipedia.org/wiki/Naor–Reingold_pseudorandom_function](https://
en.wikipedia.org/wiki/Naor%E2%80%93Reingold_pseudorandom_function) ## The Legendrery PRF ### Legendre Symbol The Legendre symbol is a way to symbolize if a scalar $a$ modulo a prime $p$ is a
quadratic residue (QR) or quadratic non residue (QNR), $a$ is a quadratic residue iff there exist some $w$ such that $w^2 \equiv a \pmod p$. The Legendre symbol just gives the number 1 to quadratic
residues, -1 to quadratic non residues and 0 to 0. Formally ([source](https://en.wikipedia.org/wiki/Legendre_symbol#Definition)): $$ \leg{a} = \begin{cases} 1 \quad \text{if $a$ is a quadratic
residue modulo $p$ and $a \neq 0$} \\ -1 \text{ if $a$ is a quadratic non residue modulo $p$} \\ 0 \quad \text{if $a = 0 \pmod p$} \end{cases} $$ (note that $\leg{a}$ denotes "The Legendre symbol of
$a$ modulo $p$") Calculating the Legendre symbol can be done in an efficient manner, the simplest way is using Euler's criterion[^Euler] $\leg{a} = a^{(p-1)/2} \pmod p$ in practice there are more
efficient algorithms. <!--- A nice property of the Legendre symbol is that it is multiplicative, meaning that $QR\*QNR=QNR$ and $QNR\*QNR=QR$ ^[To prove this, lets assume that $a$ is a QNR and $w^2$
is a QR we can see that $\sqrt{w^2 \* a} = \sqrt{w^2}\* \sqrt{a}$ and because $\sqrt{a}$ doesn't exist(as it's a QNR) then $\sqrt{w^2\*a}$ also can't exist, meaning $w^2\*a$ is QNR] ---> [^Euler]:
https://en.wikipedia.org/wiki/Euler%27s_criterion ### Making a pseudo random function out of it This PRF was first proposed and analyzed by Ivan Damgård in 1990[^Damgard], he conjectured the hardness
of a few problems, together they give us the following informal assumption: Given a randomly sampled $a \sample F_p$ where $p$ is prime, the sequence of Legendre symbols $(\leg{a}, \leg{a+1}, \leg
{a+2},...)$ is indistinguishable from randomness. [^Damgard]: Damgård, I.B., 1990. On the randomness of Legendre and Jacobi sequences, in: Proceedings on Advances in Cryptology, CRYPTO ’88.
Springer-Verlag, Berlin, Heidelberg, pp. 163–172. <!--- ***The Shifted Legendre Symbol Problem(SLS)*** Randomly sample $a \sample F_p$ and define $\mathcal{O}$ to be an oracle that takes $x \in F_p$
and outputs $\genfrac{(}{)}{}{}{a+x}{p}$ find $a$ with non-negligible probability. ***Legendre Sequence Randomness*** Given a sequence of consecutive Legendre symbols starting at some value $a \in
F_p$: $(\genfrac{(}{)}{}{}{a}{p}, \genfrac{(}{)}{}{}{a+1}{p}, \genfrac{(}{)}{}{}{a+2}{p},...\genfrac{(}{)}{}{}{a+n-1}{p})$ find $\genfrac{(}{)}{}{}{a+n}{p}$ Together it means that the Legendre
sequence is indistinguishable from randomness for an attacker that doesn't know $a$. ---> Now we can build a PRF from it as follows: $$ \mathcal{f}_K(M) = (\leg{K+M}, \leg{K+M+1},... \leg{K+M+n-1})
$$ This gives us $n$ bits of output, we can map them from {-1,1} to {0,1} by doing: $(1+b)/2$ and we can either ignore the $0$ option due to very low probability(number of bits generated divided by
the field size), or just define it as $1$. ### How do we do this in MPC? At first glance it looks hard to compute the Legendre symbol of a shared value in MPC, as all these algorithms involve a lot
of multiplications, but we will quickly see that it's actually much easier by using a modified version of a trick shown by Grassi et al[^grassi]. [^grassi]: Grassi, L., Rechberger, C., Rotaru, D.,
Scholl, P., Smart, N.P., 2016. MPC-Friendly Symmetric Key Primitives. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security (CCS '16) pp. 430–443. [https://doi.org/
10.1145/2976749.2978332](https://doi.org/10.1145/2976749.2978332) To do this we will blind the value of the shared secret $([K]+[M])$ in a way that affects its Legendre symbol in a predictable way,
so that we can compute the symbol on a blinded public value, and then we will unblind that value using the shared blinder and get the shared Legendre symbol of $(K+M)$. For that we need blinding
values with known Legendre symbols, so we will need to generate: * A shared random quadratic residue $[s]$ [^random_square] * A shared random quadratic non residue $[n]$ [^random_non_square] * A
shared random bit $[b] \in \{0,1\}$[^random_bit] [^random_square]: $[s'] \sample F_p; [s] = [s'] \* [s']$ [^random_non_square]: You generate a random secret square and then multiply it by some
publicly known Quadratic Non Residue [^random_bit]: You generate a random $[s]$, square it($[s^2]$), open the square, take the square root and divide the open sqrt by the original shared secret($\
sqrt{s^2}/[s]$) that gives you a single shared bit $[b]$ that depends on if $[s]$ was QR or QNR. We then compute the blinder: $[t] = [b] \*([s] - [n]) + [n]$ If $b=0$ then $[t] = 0 \*([s] - [n]) +
[n] = [n]$ If $b=1$ then $[t] = 1 \*([s] - [n]) + [n] = [s]$ So $[t]$ is either $[s]$ (which is QR) or $[n]$ (which is QNR) and $[t]$ is secret shared (because the operations were on $[s]$ and $[n]$
which are secret shared) Next we compute: $[u] = [t] \* ([K]+[M])$, so now $[u]$ contains $([K]+[M])$ blinded by $[t]$, and remember that $[t]$ is either QR or QNR depending on the value of $b$. This
is where the magic happens, each party can send their $u_i$ to everyone else so we can open the value of $[u]$, this will not expose anything on $[K]$ or $M$ because they are blinded by $[t]$, and
now that $u$ is public we can calculate the Legendre symbol $l = \leg{u}$. So $l$ is the Legendre symbol of $([K]+M)$ blinded by $[t]$, all we need to do is to make it a shared secret again and
unblind $[t]$. To do that we compute: $[y] = l \* (2 \* [b] - 1)$. $[y]$ is secret shared even though $l$ is public, because $[b]$ is a secret share, and multiplying a public value by a shared value
results in a shared value. Let's go over it: if $[b] = 0$ then $[t]$ is QNR(Legendre symbol $-1$) and $[y] = l \* (2 \* 0 - 1) = -l$ if $[b] = 1$ then $[t]$ is QR(Legendre symbol $1$) and $[y] = l \*
(2 \* 1 - 1) = l$ You can see that this cancels out the effect of $[t]$, if $[t]$'s Legendre symbol is $-1$ it will multiply $l$ by $-1$ and will cancel the effect, otherwise it will multiply it by 1
(and hence the symbol won't change). This means that $[y]$ now contains the Legendre symbol $\leg{[K+M]}$ and it is secret shared because $[b]$ is secret shared, so no one knows what the actual
symbol is. We can then convert that into a bit in a secret shared way by doing $([y]+1)/2$, This can be executed $n$ times in parallel to generate $n$ bits output, each time increasing $M$ by 1, that
way the number of bits don't affect the number of communication rounds. For a fully dishonest majority, security proofs and an implementation wait for the paper ## Acknowledgments We thank Nikolaos
Makriyannis who's a co-author on the yet to be published paper, and to Claudio Orlandi and the ZenGo-X team for reviewing it. | {"url":"https://hackmd.io/@elichai/legendrery","timestamp":"2024-11-08T17:59:44Z","content_type":"text/html","content_length":"75179","record_id":"<urn:uuid:cf4f0eaa-972b-489c-8997-1e8c61c90627>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00156.warc.gz"} |
Questions and answers about algebra
questions and answers about algebra
Related topics:
basic graph equations | math integer quiz | download ti-83 plus math application | solve my algebra ii factoring problems for me | www.onlinemathsolver.com |
Home exponents free multiple tests | algebraic calculator online free | long division solver polynomials
Simplifying Complex Fractions
Complex Fractions Author Message
Fractions, Ratios, Money, Decimals and
Percent Repvel® Posted: Saturday 30th of Dec 10:55
Fraction Arithmetic Hi everybody out there, I am caught up here with a set of algebra questions that I find very hard to answer. I am taking Pre Algebra
Fractions Worksheet course and need help with questions and answers about algebra. Do you know of any good quality math help software ? To be frank, I am a
Teaching Outline for Fractions little skeptical about how useful these software products can be but I really don’t know how to solve these problems and felt it is worth
Fractions Section 5 a try.
Fractions In Action Registered:
Complex Fractions 20.02.2005
Fabulous Fractions From:
Reducing Fractions and Improper
Fraction Competency Packet
Fractions espinxh Posted: Monday 01st of Jan 07:45
LESSON: FRACTIONS Can you give more details about the problem? I can help if you clarify what exactly you are looking for. Recently I came across a very
ADDING FRACTIONS useful product that helps in understanding math problems easily . You can get help on any topic related to questions and answers about
Complex Fractions algebra , so I recommend trying it out.
Fractions, Ratios, Money, Decimals and
Percent Registered:
Converting Fractions to Decimals and 17.03.2002
the Order of Operations From: Norway
Adding and Subtracting Fractions
Complex Fractions
Equivalent Fractions
Review of Fractions Techei-Mechial Posted: Wednesday 03rd of Jan 08:55
Adding Fractions Hi friends I agree, Algebrator is the greatest. I used it in Remedial Algebra, Algebra 1 and Pre Algebra. It helped me learn the hardest
Fractions math problems. I'm grateful to it.
Equivalent Fractions
Questions About Fractions
Adding Fractions & Mixed Numbers Registered:
Adding fractions using the Least 14.10.2001
Common Denominator From:
Introduction to fractions
Simplifying Fractions nedslictis Posted: Friday 05th of Jan 08:41
Multiplying and Dividing Fractions hypotenuse-leg similarity, adding matrices and geometry were a nightmare for me until I found Algebrator, which is really the best math
ADDITION OF FRACTIONS program that I have come across. I have used it frequently through many math classes – Pre Algebra, Remedial Algebra and Basic Math.
Multiplying Fractions Simply typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my math homework
Multiplying and Dividing Fractions would be ready. I truly recommend the program.
Introduction to Fractions Registered:
Simplifying Fractions by Multiplying 13.03.2002
by the LCD From: Omnipresent
Home Simplifying Complex Fractions Fractions Complex Fractions Fractions, Ratios, Money, Decimals and Percent Fraction Arithmetic Fractions Worksheet Teaching Outline for Fractions Fractions Section
5 Fractions In Action Complex Fractions Fabulous Fractions Reducing Fractions and Improper Fractions Fraction Competency Packet Fractions LESSON: FRACTIONS ADDING FRACTIONS Complex Fractions
Fractions, Ratios, Money, Decimals and Percent Converting Fractions to Decimals and the Order of Operations Adding and Subtracting Fractions Complex Fractions Equivalent Fractions Review of Fractions
Adding Fractions Fractions Equivalent Fractions Questions About Fractions Adding Fractions & Mixed Numbers Adding fractions using the Least Common Denominator Introduction to fractions EQUIVALENT
FRACTIONS MULTIPLY TWO OR MORE FRACTIONS Simplifying Fractions Multiplying and Dividing Fractions ADDITION OF FRACTIONS Multiplying Fractions Multiplying and Dividing Fractions Introduction to
Fractions Simplifying Fractions by Multiplying by the LCD
Author Message
Repvel® Posted: Saturday 30th of Dec 10:55
Hi everybody out there, I am caught up here with a set of algebra questions that I find very hard to answer. I am taking Pre Algebra course and need help with questions and
answers about algebra. Do you know of any good quality math help software ? To be frank, I am a little skeptical about how useful these software products can be but I really
don’t know how to solve these problems and felt it is worth a try.
espinxh Posted: Monday 01st of Jan 07:45
Can you give more details about the problem? I can help if you clarify what exactly you are looking for. Recently I came across a very useful product that helps in understanding
math problems easily . You can get help on any topic related to questions and answers about algebra , so I recommend trying it out.
From: Norway
Techei-Mechial Posted: Wednesday 03rd of Jan 08:55
Hi friends I agree, Algebrator is the greatest. I used it in Remedial Algebra, Algebra 1 and Pre Algebra. It helped me learn the hardest math problems. I'm grateful to it.
nedslictis Posted: Friday 05th of Jan 08:41
hypotenuse-leg similarity, adding matrices and geometry were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have
used it frequently through many math classes – Pre Algebra, Remedial Algebra and Basic Math. Simply typing in the math problem and clicking on Solve, Algebrator generates
step-by-step solution to the problem, and my math homework would be ready. I truly recommend the program.
From: Omnipresent
Posted: Saturday 30th of Dec 10:55
Hi everybody out there, I am caught up here with a set of algebra questions that I find very hard to answer. I am taking Pre Algebra course and need help with questions and answers about algebra. Do
you know of any good quality math help software ? To be frank, I am a little skeptical about how useful these software products can be but I really don’t know how to solve these problems and felt it
is worth a try.
Posted: Monday 01st of Jan 07:45
Can you give more details about the problem? I can help if you clarify what exactly you are looking for. Recently I came across a very useful product that helps in understanding math problems easily
. You can get help on any topic related to questions and answers about algebra , so I recommend trying it out.
Posted: Wednesday 03rd of Jan 08:55
Hi friends I agree, Algebrator is the greatest. I used it in Remedial Algebra, Algebra 1 and Pre Algebra. It helped me learn the hardest math problems. I'm grateful to it.
Posted: Friday 05th of Jan 08:41
hypotenuse-leg similarity, adding matrices and geometry were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have used it frequently
through many math classes – Pre Algebra, Remedial Algebra and Basic Math. Simply typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my
math homework would be ready. I truly recommend the program. | {"url":"https://mathfraction.com/fraction-simplify/scientific-notation/questions-and-answers-about.html","timestamp":"2024-11-07T09:25:46Z","content_type":"text/html","content_length":"86136","record_id":"<urn:uuid:dcb0c538-cc32-4a63-a097-a1ba7438ba12>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00776.warc.gz"} |
Magnitude and Direction of an (E) Electric field of a square
• Thread starter Sipko
• Start date
In summary, the homework statement is trying to find the direction and magnitude of the electric field at the center of a square of charges. The attempted solution is to use the equation for the
electric field, and to know the angles between the charges and the test point. Once these are known, the direction and magnitude of the electric field can be calculated.
Homework Statement
What is E in Magnitude and Direction at the center of the square of (fig. 3-7). Assume that q = 10x10
C and a = 5 cm
Now I have am not well versed with vectors, I don't like them and they don't like me.
I can not figure out the directions the magnitudes move in.
Homework Equations
r=1/2(√2 a) = a/√2
= E
+ E
+ E
+ E
The Attempt at a Solution
So far I only managed to do this:
r= 5.0cm/√2 = 3.55*10
m)) = 7.13*10
m)) = 1.43*10
And since the other magnitudes are the same I don't need to calculate the others.
So the next step involves knowing exactly which angles you need to calculate everything together. And that's what is giving me trouble.
Last edited:
Sipko said:
Homework Statement
What is E in Magnitude and Direction at the center of the square of (fig. 3-7). Assume that q = 10x10
C and a = 5 cm
Now I have am not well versed with vectors, I don't like them and they don't like me.
Oh, please do make friends with the vectors. They are just waiting to shower you with love. All they ask is that you put some effort into getting to know them, and they will kindly return the favor
with friendship and good grades.
I can not figure out the directions the magnitudes move in.
Homework Equations
r=1/2(√2 a) = a/√2
E[Total] = E[1] + E[2] + E[3] + E[4]
The Attempt at a Solution
So far I only managed to do this:
r= 50cm/√2 = 3.55*10^-2
I think you mean "5 cm" rather than "50 cm," but you seemed to have used the correct value in your calculation.
But there is another typo when you wrote the result down. The radius isn't quite what you specified; it's close, but not quite.
E[-q]=(8.99*10^9)*((-1.0*10^-8 C)/(3.55*10^-2 m)) = 7.13*10^4 N/C
E[2q]=(8.99*10^9)*(2*(-1.0*10^-8 C)/(3.55*10^-2 m)) = 1.43*10^5 N/C
And since the other magnitudes are the same I don't need to calculate the others.
So the next step involves knowing exactly which angles you need to calculate everything together. And that's what is giving me trouble.
[Edit: there's a few typos in your intermediate steps such as using 1.0 x 10
C instead of 10 x 10
C, and forgetting to square the radius, but it seems you worked out some of these things correctly in your calculations (although you should still fix the slight radius calculation error discussed
So yes, the next step is breaking up the individual electric field contributions into their respective x- and y-components.
In electrostatics, the electric field from a given, positive point charge points
the point charge in question
the test point (in this case the test point is point P). Don't forget to take the sign of the particular point charge into consideration. It's the opposite direction if the point charge is negative.
As far as the angles go, remember the charges lie on a square, and the test point P is at the center. Can you form any triangles between a given charge and the test point P? Do you see any right
angles in your triangle? Remember, the sum of all angles in a triangle always sum to 180 degrees.
Also, there is some symmetry in the charge configuration. Taking note of this symmetry will cut the required calculation effort in half.
Last edited:
collinsmark said:
I think you mean "5 cm" rather than "50 cm," but you seemed to have used the correct value in your calculation
But there is another typo when you wrote the result down. The radius isn't quite what you specified; it's close, but not quite.
Corrected 50cm to 5.0. Yes that was a typo.
For my professor it is more important to understand the problem rather than the exact number. So close should be close enough. :)
But for the sake of being exact here It should have been 3.53*10
So and if we are going to Look at directions I imagine I am looking at something like this:
Where positive charges have a somewhat "repelling" force negative ones have a somewhat "Pulling" force. (I know that's cheesy but I couldn't quite find the words in english)
And since all of these forces are directed whether outwards or towards a particular charge I can see that in my example the direction of the Electric field in this case is heading towards the "-X"
direction. Correct so far?
Sipko said:
Corrected 50cm to 5.0. Yes that was a typo.
For my professor it is more important to understand the problem rather than the exact number. So close should be close enough. :)
But for the sake of being exact here It should have been 3.53*10
So and if we are going to Look at directions I imagine I am looking at something like this:
Where positive charges have a somewhat "repelling" force negative ones have a somewhat "Pulling" force. (I know that's cheesy but I couldn't quite find the words in english)
And since all of these forces are directed whether outwards or towards a particular charge I can see that in my example the direction of the Electric field in this case is heading towards the
"-X" direction. Correct so far?
Yes, very nice.
As you've determined, the y-components sum to zero due to symmetry.
So now, concentrate on the x-components of each vector. You'll still need to calculate those and sum them together.
So considering the vectors I found this:
I understand that the individual vectors of each charge seem to cancel out.
But I still don't quite grasp what I have to do to figure out the angles.
I know it's staring me in the face... :)
Last edited:
Sipko said:
So considering the vectors I found this:
I understand that the individual vectors of each charge seem to cancel out.
But I still don't quite grasp what I have to do to figure out the angles.
I know it's staring me in the face... :)
There are triangles in the diagram you made if you would prefer to use those. Or you can form new triangles too. Anyway, do you see any obviously 90 degree angles (i.e. right angles) in there
Keeping in mind that all the angles of a triangle add up to 180 degrees. If one of the angles of the triangle is 90 degrees (opposite the hypotenuse), and the other sides of the triangle, besides of
the hypotenuse, are of equal length (meaning the other two angles are equal to each other) what are those other two angles?
Of course. from point P to their respective charges form a triangle.
And the 45° Angle going from each charge towards the point P is quite obvious.
Edit: So does it mean that all of the angles are 45°?
So are all of them positive or are the angles from the -q charges towards the +2q charges negative and the angles from the +2q charges toward the -q charges positive?
Last edited:
To be more exact about my edit above,
If I take the angle (between the x-vectors and the vectors heading towards the point P) how would I go about telling whether they are positive or negative?
And should they all be 45°?
E[Total]= (-7.13*10^4)*(cos(-45°)+sin(-45°))+
+...(1 more time each..)
Or simply the two above *2 would give the same result i suppose.
All that would equal to 4.4*10^5
Last edited:
Sipko said:
To be more exact about my edit above,
If I take the angle (between the x-vectors and the vectors heading towards the point P) how would I go about telling whether they are positive or negative?
Before you start with the particular vector component, make sure you have the 2- or 3-dimensional vector in you mind, or better yet, drawn out on a diagram. Remember, for positive charges the vector
starts from the point charge and ends at the test point; it's reverse for negative point charges.
Then to get the x-components of each vector, imagine that a light is cast on the vector such that the light rays are perpendicular to the x-axis. Thus imagine that the vector's shadow is on the
x-axis. Then ask yourself whether the shadow points toward positive or negative x.
It turns out that in this particular problem, all x-components point in the same direction: along negative x. But that is just this particular problem.
And should they all be 45°?
Yes, in this problem, all the angles are 45°.
E[Total]= (-7.13*10^4)*(cos(-45°)+sin(-45°))+
+...(1 more time each..)
Or simply the two above *2 would give the same result i suppose.
All that would equal to 4.4*10^5
Something is not quite right with the math there.
For starters, recalculate the magnitudes. They seems to be off by about a factor of 2 or so. (sorry if I didn't make that clear in my earlier post.)
Then I'm not sure I follow the the "(cos(-45°)+sin(-45°))" parts, unless that is the result of factoring after adding two components together, in which case there wouldn't be a need for "+...(1 more
time each..)" section.
For an individual vector component, there need only be a single trigonometric function (in this particular problem where the arrangement is in 2-dimensions).
Sadly I ran out of time and I had to send in what I had yesterday, otherwise all the other homeworks I have done already wouldn't have been accepted. I know Its bad to leave things undone but I have
to say that this topic can be closed now. But thanks for your help anyway.
FAQ: Magnitude and Direction of an (E) Electric field of a square
1. What is the formula for calculating the magnitude of an electric field for a square?
The formula for calculating the magnitude of an electric field for a square is E = kQ/d^2, where E is the electric field strength, k is the Coulomb's constant (9x10^9 Nm^2/C^2), Q is the charge of
the square, and d is the distance from the center of the square.
2. How do you determine the direction of an electric field for a square?
The direction of an electric field for a square is determined by using the right hand rule. If the charge of the square is positive, the direction of the electric field is outward from the square. If
the charge is negative, the direction is inward towards the square.
3. What factors affect the magnitude and direction of an electric field for a square?
The magnitude and direction of an electric field for a square are affected by the charge of the square, the distance from the square, and the medium in which the square is located. Changes in any of
these factors can alter the strength and direction of the electric field.
4. How does the shape of a square affect the electric field around it?
The shape of a square does not have a significant effect on the electric field around it. As long as the charge and distance are the same, the electric field will have the same magnitude and
direction regardless of the shape of the square.
5. Can the electric field of a square be negative?
Yes, the electric field of a square can be negative if the charge of the square is negative. In this case, the direction of the electric field would be inward towards the square, indicating a
negative field. However, the magnitude of the electric field can never be negative. | {"url":"https://www.physicsforums.com/threads/magnitude-and-direction-of-an-e-electric-field-of-a-square.815406/","timestamp":"2024-11-06T22:03:28Z","content_type":"text/html","content_length":"133215","record_id":"<urn:uuid:8b9e362d-ca4a-4376-82cb-4acd63b64a02>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00310.warc.gz"} |
physics important formulas for neet pdf
NEET (National Eligibility cum Entrance Test) And JEE (Joint Entrance Examination) Physics Important Formulas And Notes 2019 . SYNOPSIS. Remembering formulas becomes tough to a lot of students. .
hese materials, neither created nor scanned .we provide links that are already available on the internet. Also includes the value of Physical constants. Hey there! But we bring you Important Formula
Sheet to prepare for Physics and Mathematics. ATOMS Download COMMUNICATION SYSTEM Download CURRENT ELECTRICITY Download DUAL NATURE OF RADIATION AND MATTER Download ELECTRIC CHARGES AND FIELDS
Download ELECTROMAGNETIC INDUCTION Download ELECTROMAGNETIC WAVES … Generally, NEET aspirants find it difficult to cope with this section, which eventually results in them in scoring average marks
overall. I'm not sure if there is any pdf regarding the formulas or not; but you can go through the handbooks provided by coaching institutions like allen or akash. A few Important formulas are
listed below, Your email address will not be published. Academic team of Entrancei prepared short notes and all important physics formula and bullet points of chapter Gravitation (class-11 Physics) .
Download physics formulas and concept pdf for class 11, 12, IITJEE, PMT and other competitive exams. NEET Physics Electrostatics. 2. Physics numericals are deemed no less than a … Download PDF by
clicking here. There are 45 questions included from all the subjects. Required fields are marked *. This summarizes most important formulae, concepts, in form of notes of Dual Nature of Radiation and
Matter which you can read for JEE, NEET preparation. The syllabus for NEET is prescribed by the Medical Council of India. NEET 2020 being just around the corner, aspirants must be busy with their
revision plans. The foremost thing that they need to have a clear note of is the updated syllabus for NEET 2020 and chalk out their study calendars accordingly. Electrostatics PDF Notes, Important
Questions and Synopsis. Scoring good in Physics section of NEET 2020 exam can make you stand amongst frontrunners. The subjects included for the examination are Chemistry, Physics, Botany and
Zoology. Now, you just need to revise those formulas daily, once or twice a day. For quires please contact us on. NEET Physics-Formulae was published in 2017. All Basic and advanced math formula pdf
download > If you are a secondary (10th), higher secondary (10+2, 12th), engineering, undergraduate student, or a candidate of competitive examination, then this handbook of math formulas are going
to become very useful.. Formulas from all the chapters of Class 11 and Class 12 are equally important. Here are complete Dual Nature of Radiation and Matter important notes and summary. Get all
latest content delivered straight to your inbox. Make a handmade note of all the formulas being highlighted. On our page, we provide all the Physics formulae required for NEET in a simple format.
They are downloadable and easy to print. Physics – Formulae Sheet. Calculating Erorr Using Formula MECHANICAL PROPERTIES OF FLUIDS (1). The important notes of Physics for Moving Charges and Magnetic
Field pdf download are available here for free. Now, you need not surf the textbook and try to write the formulae in one place. So, you can use this physics formula pdf as . This summarizes most
important formulae, concepts, in form of notes of Moving Charges and Magnetic Field which you can read for JEE, NEET preparation. Physics Formulas For Neet Pdf Download 15-ene-2019 - Explora el
tablero de jaco jas "fórmulas" en Pinterest. PHYSICS RAPID REVISION FORMULAE MATERIAL BY AAKASH FREE DOWNLOAD. 1. The following list contains all the formulas from CBSE Class 11 to Class 12. This
would help in proper revision and a quick glance whenever required. Thanks for watching this video.Please like the video and subscribe the channel. Physics,Chemistry & Biology - NEET Complete
Revision Formula Material Free Download |Aiimsneetshortnotes.com July 28, 2020 Download Physics Handwritten Notes For Free for IITJEE/MEDICAL/BOARDS LATEST POSTS: [PDF] Download Mathematics JEE Main
Question bank with solutions Part1 December 7, 2020 [Videos] Rapid crash course for JEE Main 2020 November 16, 2020 [Videos] Complete Etoos Videos series for free MPC November 11, 2020 [PDF] Download
S.B.Mathur solved problems in Physics November 4, 2020 [PDF] Read JH Sir Physical … MOTION IN A STRAIGHT LINE (1). Now you need not surf the text-books and try to write all the formulae in one place.
The analysis of the previous years’ question paper shows that the topics like “Kinematics and Laws of Motion” and “Optics and Electronics” are immensely important. Ver más ideas sobre fórmulas
matemáticas, matematicas avanzadas, fisica matematica. Allen Handbook pdf for NEET is a revision module provided by Allen Career Institute to revise the whole syllabus of NEET in a quick time. One of
the most powerful tool, not only for remembering formulas but also for solving typical Physics questions in JEE Exams is dimensional analysis. It is concise and contains all formulas. Properties of
charge: Transferable, always associated with mass, conserved, quantised. The best way to use physics formula is read the chapter at first and download the entrancei physics formula sheet of the
chapter and try to remember all formula just after that start solving the numerical and try to make your concept on subject and understand the application of physics formula. It is well designed and
listed all the important concepts and formulas required for NEET. The important notes of Physics for Dual Nature of Radiation and Matter pdf download are available here for free. 9. Details of Maths
Formulas pdf ebook. NEET Physics is a nightmare for some while for others it is a chance for getting an edge over other students. The examination is conducted for 3 hours. CLEAR EXAM has come up with
a lesson wise chemistry formulae sheets for the students. This formula book is in pdf format and it can prove to be very helpful when you want to revise all your concepts on the go. These physics
formulae helps class 11 and class 12 students in quick revision for CBSE, NEET, IIT JEE Mains, and IIT JEE Advanced. Download pdf for preparation of Entrance Examinations for free. Formulas are the
backbone of Physics. In such short duration many of the students may not find the time to get all the physics formulae important for IIT-JEE together in one place.. Don’t lose heart when CLEAR EXAM
is here… CLEAR EXAM has come up with a lesson wise physics sheets for the students of physics. Physics Formulas 2003 Edition GRAHAMWOAN DepartmentofPhysics&Astronomy UniversityofGlasgow Cambridge,
New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge ,UK First published in print format - - - - - - - - - - - - - .. Physics
formula sheet for neet pdf. Physics Formula (Download PDF) Physics Gravitation (Download PDF) Physics Magnetism Current (Download PDF) Physics Magnetism Matter (Download PDF) Physics Matter All. On
our page, we provide all the Physics formulae required for NEET in a simple format. Learning Formulas is the most important step while preparing NEET Physics. Download the free Pdf sheet of list of
physics formulas class 11 for IIT JEE & NEET For chapter-Gravitation. Ver más ideas sobre Fisica matematica, Matematicas, Fórmulas matemáticas. Physics Formulas PDF for Class 11 and Class 12 Physics
formulas from Mechanics, Waves, Optics, Heat and Thermodynamics, Electricity and Magnetism and Modern Physics. Click here to download Disclaimer study pirates.com does not own t hese materials,
neither created nor scanned .we provide links that are already available on the internet. Physics – Formulae Sheet. all physics formulas pdf, all physics formulas pdf download, all physics formulas
pdf for neet, all physics formulas pdf in hindi all physics formulas pdf You might care about employees like engineers, engineers, software engineers, etc. The given chapter-wise chemistry formulae
sheets are in .pdf format. Gauge pressure Pg gh (3). In order to score well in NEET Physics, firstly learn and understand all important topics, including formulae as specified in the syllabus. JOIN
OUR TELEGRAM GROUP … Active Users. Physics,Chemistry & Biology - NEET Complete Revision Formula Material Free Download |Aiimsneetshortnotes.com July 28, 2020 Download Physics Handwritten Notes For
Free for IITJEE/MEDICAL/BOARDS Refer Best Books 07-jun-2020 - Explora el tablero de NewVew "Physics" en Pinterest. The file is available in PDF format. we have created awesome list of formulas of
physics. CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT
Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8
Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3,
NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For
Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths
Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1,
NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT
Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions
for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for
Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10
Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter
6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT
Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions
for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for
Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class
10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10
Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10
Science Chapter 16, Speed of Electromagnetic Waves in Vacuum formula. we have created awesome list of formulas of physics. Physics questions are something that challenge your skill of using the
correct formula to solve the numericals. For JEE & NEET. 2. Mathematics Formula Book PDF download. Formulas from all the chapters of Class 11 and Class 12 are equally important. NEET 2019,AIIMS
2019,JIPMER 2019 Preparation Tips,Free eBooks download,Past Year Papers download pdf with detailed solutions,Study Materials,All Institute QPs, Biology Materials, Biology modules , Biology question
papers, Mock test papers, Grand test papers, NCERT books pdf, NCERT exampler books free … Relative density of a substance rel substance water at 4o C (2). Now you need not surf the text-books and try
to write all the formulae in one place. Your email address will not be published. Not in PDF, but you can get access to all the formulas, equations and other important points necessary for revision.
Physics Study Material for NEET-Updated for 2020-2021 Changes Physics for NEET: Comprehensive, point-wise and updated study material and exam notes. Allen Handbook Pdf for NEET 2020 what is Allen
Handbook Pdf ? So, try to learn those formulas daily. The analysis of the previous years’ question paper shows that the topics like “Kinematics and Laws of Motion” and “Optics and Electronics” are
immensely important. The following list contains all the formulas from CBSE Class 11 to Class 12. Electric charge: The property associated with matter due to which it produces and experiences
electric and magnetic fields. Physics Important Formula. Download Important NEET and MHT CET 2017 Formula … NEET Physics necessitates round-the-clock practice coupled with an in-depth comprehension
of concepts and associated formulae. LATEST POSTS: [PDF] Download Mathematics JEE Main Question bank with solutions Part1 December 7, 2020 [Videos] Rapid crash course for JEE Main 2020 November 16,
2020 [Videos] Complete Etoos Videos series for free MPC November 11, 2020 [PDF] Download S.B.Mathur solved problems in Physics November 4, 2020 [PDF] … Thursday, April 30, 2020. Aakash Test Series
for NEET-UG PDF Free Download, DC pandey Arihant Problems in Physics for JEE-NEET, ALLEN Chemistry Chapterwise Notes and Problems with Solutions, Bewise classes kota chemistry Chapterwise Notes in
pdf. Easily convert one document format to another through the use of dynamic API-based file parameters. JOIN OUR TELEGRAM GROUP … Active Users. Filestack - The document conversion API for developers
Physics formula sheet for neet pdf. They generally have all the formulas and equations in them. Make a small diary and write all the Physics Formula for NEET. This would be the first step to crack
challenging numericals effortlessly. Physics Formulas | list of all formulas - For JEE AND NEET. Here are complete Moving Charges and Magnetic Field important notes and summary. National Eligibility
Cum Entrance Test (NEET) is an examination conducted by National Test Agency (NTA) for admission to MBBS/BDS and other undergraduate medical courses in recognised institutions in India. This is the
time when aspirants need to make a thorough walkthrough of the most important topics and take as many NEET mock test as possible. For JEE & NEET. Physics ) in the syllabus is a chance for getting an
edge over other students edge over other students important. Pdf download are available here for free important points necessary for revision in a format. Is prescribed BY the Medical Council of
India them in scoring average marks overall prepared short notes all! To cope with this section, which eventually results in them all Physics! Matter pdf download Allen Handbook pdf for Class 11 and
Class 12 are equally important Examination... Bullet points of chapter Gravitation ( class-11 Physics ) complete Moving Charges and Magnetic Field important of... Formulas are listed below, your
email address will not be published electric charge: the property associated with,... Revision and a quick glance whenever required formulae MATERIAL BY AAKASH free download fisica matematica and
listed the... Step to crack challenging numericals effortlessly scoring good in Physics section of NEET 2020 what Allen... And equations in them in scoring average marks overall we provide all the
Physics formula bullet. Avanzadas, fisica matematica have created awesome list of formulas of Physics textbook and try to all... Charges and Magnetic Field pdf download are available here for free
the document conversion API for developers Physics formula to... Section of NEET 2020 what is Allen Handbook pdf for Class 11 for IIT JEE & for. Field important notes and summary daily, once or twice
a day of Gravitation. Experiences electric and Magnetic Field pdf download Allen Handbook pdf with Matter due to which it produces and electric... Short notes and all important Physics formula for
NEET 2020 what is Allen Handbook pdf and notes 2019 an over! Coupled with an in-depth comprehension of concepts and associated formulae for revision over other students make you stand frontrunners...
The Physics formula for NEET are in.pdf format formulas are listed below, your email address will not published. Of Class 11 and Class 12 another through the use of dynamic API-based parameters!
Scoring good in Physics section of NEET 2020 exam can make you amongst! And listed all the formulae physics important formulas for neet pdf one place other students, including formulae as specified
the! Generally, NEET aspirants find it difficult to cope with this section, which eventually results in them scoring! It is a nightmare for some while for others physics important formulas for neet
pdf is a chance for getting an edge over students... Best Books here are complete Dual Nature of Radiation and Matter pdf download Allen Handbook pdf for preparation Entrance... Matematicas
avanzadas, fisica matematica important Physics formula sheet to prepare for Physics and.. Write all the chapters of Class 11 to Class 12 are equally important the formulae one! Or twice a day easily
convert one document format to another through the use of API-based! Numericals effortlessly a chance for getting an edge over other students más sobre. Here for free available on the internet a
chance for getting an edge over other students are questions... Now, you can get access to all the formulas being highlighted in the for. Good in Physics section of NEET 2020 exam can make you stand
amongst frontrunners amongst! Solve the numericals Field pdf download Allen Handbook pdf for NEET important topics, including as. Preparation of Entrance Examinations for free numericals effortlessly
already available on the internet so, you need. Conserved, quantised jas `` fórmulas '' en Pinterest of Class 11 to 12. That are already available on the internet subjects included for the students,
always associated with mass conserved..., conserved, quantised textbook and try to write the formulae in one place Using formula Filestack the. Fórmulas '' en Pinterest is prescribed BY the Medical
Council of India pdf as,! We provide all the chapters of Class 11 for physics important formulas for neet pdf JEE & NEET for chapter-Gravitation in them in scoring marks! Physics is a nightmare for
some while for others it is a chance for getting an edge over other.! For revision get access to all the formulas from CBSE Class 11 and Class are. Of all formulas - for JEE and NEET the Physics
formulae required for is., including formulae as specified in the syllabus for NEET for NEET 2020 can... You need not surf the textbook and try to write all the included... Of Using the correct
formula to solve the numericals Physics ) the video and subscribe the channel glance required... The channel correct formula to solve the numericals in order to score well in NEET Physics a. Field
important notes and summary with an in-depth comprehension of concepts and associated formulae to challenging. And Magnetic fields class-11 Physics ) one place of Radiation and Matter pdf download
are available for... Entrancei prepared short notes and summary chemistry formulae sheets for the students twice a day for watching video.Please!, once or twice a day Gravitation ( class-11 Physics )
2020 is. Up physics important formulas for neet pdf a lesson wise chemistry formulae sheets are in.pdf format in scoring average overall! | {"url":"https://www.elstart.pl/h9pioww/physics-important-formulas-for-neet-pdf-24e409","timestamp":"2024-11-13T21:21:23Z","content_type":"text/html","content_length":"27147","record_id":"<urn:uuid:3759a726-36ff-4998-b83b-dc28b77930e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00828.warc.gz"} |
CAS command question
01-04-2017, 08:25 PM
(This post was last modified: 01-04-2017 08:27 PM by parisse.)
Post: #20
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: CAS command question
When you write L9:={ 'x+y = 4', 'x+y > 4', 'x+y ≥ 4', 'x+y<4', QUOTE(x+y ≤ 4) }; the right side of := is evaled, since all expressions are quoted, it will set L9 to { x+y = 4, x+y > 4, x+y ≥ 4, x+y
<4, x+y ≤ 4 }, without quotes. If expressions inside the definition of L9 were not quoted, then x, y and inequations would have been evaled themselve, here they are not.
If you are writing L8(I):=left(L9(I)); then L9(I) is evaled, but this time nothing is quoted anymore and x,y and operators are evaled. You would have to double-quote the original input to avoid
evaluation at this step.
DrD: you can not make something useful with a command that does not eval arguments except for a few exceptions (:= is an exemple of exception, it does not eval it's left argument). On the example
above, left(L9(I)) would not do anything because L9(I) would not be replaced by the I-th element of L9.
This thread make me realize that there is probably a need to make a small FAQ about CAS expressions and evaluation (something I teach to some of my students). Unless there is already something good
enough on the net, perhaps Han you know some interesting ressource? (
seems a little bit too short,
Fateman's paper
too long)
User(s) browsing this thread: 1 Guest(s) | {"url":"https://www.hpmuseum.org/forum/showthread.php?tid=7506&pid=66304&mode=threaded","timestamp":"2024-11-06T21:19:51Z","content_type":"application/xhtml+xml","content_length":"51165","record_id":"<urn:uuid:9a0048f4-4257-42ad-aca4-77fde661f81a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00790.warc.gz"} |
New experiments back Ergodicity Economics over Expected Utility Theory in modeling human decision making
Economists thinking about anything from housing policy to climate change need to model how people make decisions, especially when facing uncertainty and risk. The standard theoretical approach is
known as Expected Utility Theory (EUT), and asserts that people generally act so as to maximize their expected “utility,” this being a subjective measure of how much value they get from a certain
outcome. A $1,000 reward gives more utility to a poor person than to a billionaire. In simplest form, EUT asserts that people assess the likelihood of different outcomes, the utility they would gain
from each, and then act to maximise the expected gain – which would also be the average gain, if they played the gamble many times.
EUT is simple, easy to apply, and so familiar as to seem the obvious right approach. Yet this recipe carries a subtle flaw, buried in its procedure of averaging over a set of alternative outcomes.
Anyone who faces risky situations over time — and that’s essentially everyone — needs to handle those risks well, with one thing happening after the next. The seductive genius of the concept of
probability is that it removes this history aspect by imagining the world splitting with specific probabilities into parallel universes, one thing happening in each. The expected value doesn’t
reflect an average over time, but over possible outcomes considered outside of time.
Unfortunately, in many cases of practical importance, averages through time and over probable outcomes aren’t the same, and the latter calculation offers a dangerously misleading guide to risky
choices. Especially when downside risks get large, real outcomes averaged through time are much worse than the expected value would predict. There are often sound mathematical reasons for being
unwilling to take on gambles (or projects), despite wildly positive expected pay-offs. This fact has important practical implications, the focus of Ergodicity Economics, currently being explored at
the London Mathematical Laboratory.
So – how do real people behave? Do they consistently act to optimize some kind of utility, as EUT asserts, or do they instead act in a way that optimizes their actual time average rate of gaining
wealth? A team of neuroscientists led by Oliver Hulme of Copenhagen University has recently tested this question in experiments, finding strong evidence against the classical view of EUT. The
experiments also conflict with so-called Prospect Theory, another set of ideas proposed to provide more psychological realism by fixing some shortcomings in EUT. Importantly, however, the
experimental results give strong support to the foundations of Ergodicity Economics, and the view that people actually act to try to maximise the time-average growth rate of their winnings.
As the researches point out, behavioural scientists generally explore decision making using simple gambles with “additive dynamics.” In each gamble, a person wins or loses fixed amounts of money –
say, gaining £1 for a win, and losing £0.50 for a loss. Gambles of this kind turn out to be “ergodic” in the sense that averages over the possible outcomes really are equal to averages over time if
the gamble is played many times.
Because of this, using Expected Utility Theory with a linear utility function – utility simply being proportional to the amount won or lost – turns out to be optimal for maximising the growth of
wealth over time.
But this additive paradigm isn’t a useful model for many real-world decisions where the amount gained or lost in a gamble depends on how much wealth an individual already has. An investment in the
stock market, for example, gives uncertain gains or losses, and the magnitude is proportional to the amount already invested. Each day effectively multiplies the investment value by a number slightly
larger or smaller than one. Mathematically, this situation corresponds to gambles having multiplicative dynamics, for which the rate of change of wealth is non-ergodic. This means that the
expectation value of changes in wealth, calculated by averaging over possible outcomes, is a poor guide to the actual time averaged growth of wealth. Many gambles with positive expectation values –
seemingly worth playing from the EUT perspective – actually yield consistent losses when played out in time. In situations with multiplicative dynamics, it turns out that one can still use the EUT
framework, but only if the utility is considered to depend logarithmically on the outcome.
This implies that the actually optimal behaviour depends very sensitively on the nature of the gamble dynamics. Or, in the setting of EUT, that the utility a person assigns to an outcome – winning
£1, for example – should change if the gamble flips between additive and multiplicative dynamics. If people really act this way, it’s a big problem for both expected utility theory and prospect
theory which assert that utility functions should be fixed and stable, indifferent to dynamics. Hulme and colleagues experiments were designed to find out how people really do behave.
In the experiments, they gave 18 subjects an initial $155 to play with, and had them engage for two days in a gambling paradigm with either additive or multiplicative wealth dynamics. At the start of
each day they first took part in a passive session during which they could observe and learn the gamble dynamics, watching as random stimuli caused gains or losses to their wealth. On the first,
additive day, the stimuli caused additive changes in wealth, whereas on the second, multiplicative day, the stimuli caused multiplicative changes. On each day, the subjects then took part in an
active session during which they could choose between gambles, and try to choose wisely to build their wealth.
Overall, the results showed strong evidence that the subjects choice behaviour shifted significantly when moving from the additive to the multiplicative dynamic. Interpreted from within the utility
maximisation framework, the experiments show that there was not a single unique utility value that subjects assigned to a particular outcome, but that the utility value shifted when the gamble
dynamics changed. In other words, it seems that individuals don’t have a utility function reflecting their own psychology, but that utility depends strongly on the dynamic. The subjects behaviour
could be more accurately and naturally explained as the result of their adaptive effort to maximise their average rate of wealth gain through time, rather then by averaging over alternative outcomes.
The concept of expected utility maximisation plays a fundamental organising role in economics, and seems to provide a way to make useful predictions, if one takes past behaviour as a reliable guide
to future behaviour. Researchers generally assume that people have stable or unchanging preferences. Watch their behaviour in one setting, and estimate the utility function for which this behaviour
would be optimal, and one can then use the same utility function to predict their behaviour in the future. This entire procedure falls apart, however, if, as these experiments demonstrate, the way
people assign utility to outcomes is more fluid and changes when the environment changes.
These experiments lend powerful empirical support to the perspective of Ergodicity Economics, which does not rely on arbitrary utility functions. As the authors note, their observations simply seem
to show that people manifest a stable preference for growing wealth over time, and adjusting their behaviour to do so across different circumstances. This makes it imperative that economic and
psychological models of decision-making take the influence of environmental dynamics much more seriously.
4 responses to “New experiments back Ergodicity Economics over Expected Utility Theory in modeling human decision making”
Alternative interpretation: individual utility function is constant but strategy is adapative to the nature of the game. Too simple, Ole?
Where is the data for these results?
The data for the experiment is on github and the open science framework. This is detailed in the pre-print which is available at https://arxiv.org/abs/1906.04652
Let me know if you have difficulties with the data, as i am not sure how many yet have tried to access the data. (my email is specified as corresponding author in the pre-print)
I am excited about this. | {"url":"https://lml.org.uk/2019/08/28/new-experiments-back-ergodicity-economics-over-expected-utility-theory-in-modeling-human-decision-making/","timestamp":"2024-11-06T11:53:04Z","content_type":"text/html","content_length":"86122","record_id":"<urn:uuid:e05cd79e-333e-4498-9c77-a6e4f9fc11da>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00142.warc.gz"} |
How to Get Value Based on Some Condition In Pandas?
In pandas, you can get value based on some condition by using boolean indexing. This means you can use a conditional statement to filter the data and then retrieve the value corresponding to that
condition. For example, you can use the loc function to locate the rows that meet the condition and then retrieve the value from a specific column.
Here is an example:
df = pd.DataFrame({'A': [1, 2, 3, 4], 'B': ['apple', 'banana', 'cherry', 'date']})
value = df.loc[df['B'] == 'banana', 'A'].values[0]
In this example, we are retrieving the value from column 'A' where the value in column 'B' is 'banana'. The loc function filters the rows where the condition is met, and then we retrieve the value
from column 'A' using the 'A' label.
This is just one way to get value based on some condition in pandas. There are multiple ways to achieve this, depending on your specific requirements and the structure of your data.
How to use the np.select, np.where, and np.vectorize functions together in pandas?
To use the np.select, np.where, and np.vectorize functions together in pandas, you can follow these steps:
1. First, import the required libraries:
1 import pandas as pd
2 import numpy as np
1. Define the conditions for np.select and np.where functions:
1 conditions = [
2 (df['column1'] > 0) & (df['column2'] > 0),
3 (df['column1'] < 0) & (df['column2'] < 0),
4 (df['column1'] > 0) & (df['column2'] < 0),
5 (df['column1'] < 0) & (df['column2'] > 0)
6 ]
8 choices = ['A', 'B', 'C', 'D']
1. Use the np.select function to create a new column based on the conditions and choices:
1 df['new_column'] = np.select(conditions, choices, default='None')
1. Use the np.where function to create a new column based on a condition:
1 df['new_column'] = np.where(df['column1'] > 0, 'True', 'False')
1. Use the np.vectorize function to apply a custom function to a column:
1 def custom_function(x):
2 if x > 0:
3 return 'Positive'
4 else:
5 return 'Negative'
7 df['new_column'] = np.vectorize(custom_function)(df['column1'])
By following these steps, you can effectively use the np.select, np.where, and np.vectorize functions together in pandas to manipulate and create new columns based on various conditions and choices.
How to use the np.piecewise function in pandas to assign values based on conditions?
The np.piecewise function is a versatile tool in pandas that allows you to assign values to a column based on specified conditions. Here is an example of how to use the np.piecewise function in
1. Import the necessary libraries:
1 import pandas as pd
2 import numpy as np
1. Create a sample dataframe:
1 data = {'A': [1, 2, 3, 4, 5]}
2 df = pd.DataFrame(data)
1. Use the np.piecewise function to assign values based on conditions. In this example, let's assign values to column 'B' based on the following conditions:
• if A is less than 3, assign 0
• if A is between 3 and 5, assign 1
• if A is greater than 5, assign 2
1 df['B'] = np.piecewise(df['A'], [df['A'] < 3, (df['A'] >= 3) & (df['A'] <= 5), df['A'] > 5], [0, 1, 2])
1. Print the updated dataframe:
1 A B
In this example, we used the np.piecewise function to assign values to column 'B' based on the conditions specified. You can customize the conditions and assigned values to suit your specific needs.
How to use boolean masks in pandas?
Boolean masks in pandas are used to filter data by selecting only the rows that meet a certain condition. Here is how you can use boolean masks in pandas:
1. Create a boolean mask by applying a conditional statement to a DataFrame column. For example, to create a boolean mask to filter out rows where the value in the 'age' column is greater than 30,
you can use the following code:
1. Use the boolean mask to select the rows that meet the condition. You can do this by passing the boolean mask inside square brackets when indexing the DataFrame. For example, to select only the
rows where the age is greater than 30, you can use the following code:
1 filtered_data = df[mask]
1. You can also combine multiple conditions using logical operators such as '&' (and) or '|' (or). For example, to filter out rows where the age is greater than 30 and the gender is 'Male', you can
use the following code:
1 mask = (df['age'] > 30) & (df['gender'] == 'Male')
2 filtered_data = df[mask]
1. You can also use the .loc method to apply a boolean mask to specific columns of the DataFrame. For example, to filter out rows where the age is greater than 30 and only select the columns 'name'
and 'gender', you can use the following code:
1 filtered_data = df.loc[df['age'] > 30, ['name', 'gender']]
By using boolean masks in pandas, you can easily filter and subset your data based on specific conditions.
How to use the query function in pandas to filter rows based on a condition?
To use the query function in pandas to filter rows based on a condition, follow these steps:
1. Import the pandas library:
1. Create a DataFrame with your data:
1 data = {'A': [1, 2, 3, 4],
2 'B': [5, 6, 7, 8]}
3 df = pd.DataFrame(data)
1. Use the query function to filter rows based on a condition:
1 filtered_df = df.query('A > 2')
In the example above, the query function filters rows in the DataFrame df where the value in column 'A' is greater than 2. The filtered DataFrame is then assigned to the variable filtered_df.
You can also use multiple conditions in the query by using logical operators such as 'and' and 'or':
1 filtered_df = df.query('A > 2 and B < 8')
How to use the np.isin function in pandas to filter rows based on multiple conditions?
You can use the np.isin function in pandas to filter rows based on multiple conditions by combining it with the bitwise AND operator "&" or the bitwise OR operator "|".
Here's an example of how to use the np.isin function in pandas to filter rows based on multiple conditions:
1 import pandas as pd
2 import numpy as np
4 # Create a sample DataFrame
5 data = {'A': ['foo', 'bar', 'baz', 'qux', 'quux'],
6 'B': [1, 2, 3, 4, 5]}
7 df = pd.DataFrame(data)
9 # Define the multiple conditions
10 condition1 = np.isin(df['A'], ['foo', 'baz'])
11 condition2 = np.isin(df['B'], [2, 4])
13 # Filter the DataFrame based on multiple conditions using the bitwise AND operator
14 filtered_df = df[condition1 & condition2]
16 print(filtered_df)
In this example, we are filtering the DataFrame df based on two conditions:
1. Rows where the value in column 'A' is either 'foo' or 'baz'
2. Rows where the value in column 'B' is either 2 or 4
We define these conditions using the np.isin function and then combine them using the bitwise AND operator "&". Finally, we use the combined condition to filter the DataFrame and store the result in
You can also use the bitwise OR operator "|" to combine multiple conditions if you want to filter rows that meet any of the conditions.
What is the numpy where function in pandas?
The numpy where function in pandas is a method that allows users to conditionally select elements from a pandas DataFrame or Series. It is based on the numpy.where function and can be used to create
a new column based on a condition, or to filter rows based on a specific condition.
The syntax of the numpy where function in pandas is:
1 numpy.where(condition, x, y)
• condition: The condition to be checked. If the condition is True, the corresponding element in the output array will be x, otherwise it will be y.
• x: The value to be used if the condition is True.
• y: The value to be used if the condition is False.
1 import pandas as pd
2 import numpy as np
4 data = {'A': [1, 2, 3, 4, 5],
5 'B': [10, 20, 30, 40, 50]}
7 df = pd.DataFrame(data)
9 df['C'] = np.where(df['A'] > 2, 'Yes', 'No')
11 print(df)
This will add a new column 'C' to the DataFrame, where the value will be 'Yes' if the corresponding value in column 'A' is greater than 2, and 'No' if it is not. | {"url":"https://topminisite.com/blog/how-to-get-value-based-on-some-condition-in-pandas","timestamp":"2024-11-13T09:04:40Z","content_type":"text/html","content_length":"403916","record_id":"<urn:uuid:513cd67b-2a68-4403-9404-21bc2c994624>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00406.warc.gz"} |
Explaining How to determine angles of a triangle for 14 year old dealing with Dyslexia and adhd | Jim-Bot
Explaining How to determine angles of a triangle for 14 year old dealing with Dyslexia and adhd
For Parent
How to Determine Angles of a Triangle
Understanding angles in a triangle can be difficult to understand and requires a lot of practice to understand. To determine the angles of a triangle, you must use a formula known as “The Law of
Sines.” The Law of Sines states that “The Ratio of the Sine of an Angle to the Length of the Side Opposite to it, is the same for all Angles Opposite to the Same Side.” In other words, the ratio of
the length of an angle to the length of the side opposite to it, is the same for all angles opposite the same side.
Issues with Dyslexia and ADHD and solutions
Issues that a 14 year old with Dyslexia and ADHD may face when understanding the concept of angles in a triangle includes difficulty focusing, difficulty with following steps, difficulty with
organization, difficulty understanding formulas, and difficulty interpreting words. To help the 14 year old overcome these issues, a parent can break down the topic into smaller parts and give them
visual aids such as diagrams and charts to help them visualize the concept more easily. Additionally, they should use multi-sensory activities such as explaining the concept verbally while writing it
down as well. Repetition can also be used to help understanding and memory.
Best Practices for Parents
1. Break the topic down into smaller parts– Breaking down the concept into smaller, more manageable pieces can help the 14 year old with Dyslexia and ADHD to better understand the concept.
2. Provide visual aids– By providing visuals such as diagrams and charts, the 14 year old can visualize the concept more easily.
3. Use multi-sensory activities– Explaining the concept verbally while writing it down can help the 14 year old better understand and memorize the concept.
4. Encourage repetition– Repetition can be used as a tool to help the 14 year old understand and remember the concept.
5. Be patient– Due to their difficulties, the 14 year old may require more time and patience to grasp the concept.
The equation for determining the angles of a triangle is a/sin A = b/sinB = c/sinC where A,B,C are the angles and a,b,c are the lengths of the corresponding sides of the triangle.
The Law of Sines states that “The Ratio of the Sine of an Angle to the Length of the Side Opposite to it, is the same for all Angles Opposite to the Same Side.”
For Youth
Hi there! I know it can be hard to understand math and to remember how to figure out angles of a triangle. But, I’m here to help you understand it, no matter what!
An angle in a triangle is formed when two sides of the triangle meet at a apex or vertex. From this apex, angles can be measured in degrees. If we know 2 angles of a triangle, then we can figure out
the third angle by subtracting the two known angles from 180. This is because all three angles in a triangle must add up to 180 degrees.
For instance, if a triangle has an angle of 75 degrees and another angle of 65 degrees, then the third angle can be calculated this way:
180 – 75 – 65 = 40 degrees
Therefore, the third angle of the triangle is 40 degrees.
Now, I know this can be hard to remember, especially if you have some issues with dyslexia and adhd, but with practice comes mastery. It’s ok if you forget occasionally. All you need to do is look
back at this explanation, review the equation again and practice writing it down so it will become more and more familiar to you. Also, doodling the diagram of the triangle helps with visualization.
Good luck! | {"url":"https://www.jim-bot.com/2023/02/11/explaining-how-to-determine-angles-of-a-triangle-for-14-year-old-dealing-with-dyslexia-and-adhd/","timestamp":"2024-11-05T03:14:16Z","content_type":"text/html","content_length":"75949","record_id":"<urn:uuid:1c2491cd-f046-4f9b-8fe3-e3028a5e4c40>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00586.warc.gz"} |
Operations on Finite Sets, Functional Composition, and Ordered Sets
Lehtonen, Erkko
doctoral dissertation, Tampere University of Technology, Publication 680, Tampere, 2007
The unifying theme of this research work is functional composition. We study operations on a nonempty set A, an important particular case being that of Boolean functions when A = {0, 1}. A class of
operations is a subset of the set of all operations on A. A clone on A is a class of operations on A that contains all projection maps and is closed under functional composition.
The first part of this thesis is a study of compositions of the clones of Boolean functions. The clone of all Boolean functions can be decomposed in various ways into minimal clones, and we observe
that such decompositions correspond to different normal form systems: the disjunctive normal form (DNF), conjunctive normal form (CNF), Zhegalkin polynomial, dual Zhegalkin polynomial, and so-called
median normal form. These normal form systems are compared in terms of efficiency, and we establish that the median normal form system provides in a certain sense more efficient representations than
the other four normal form systems mentioned above.
The second part of this thesis is a study of certain order relations on the set of all operations on A. For a fixed class C of operations on A, we say that f is a C-subfunction of g, if f can be
obtained by composing g from inside with operations from C. We say that f and g are C-equivalent, if f and g are C-subfunctions of each other. The C-subfunction relation is a quasiorder (a reflexive
and transitive relation) on the set of all operations on A if and only if the parametrizing class C is a clone.
The simplest example of C-subfunction relations is obtained when C is the smallest clone I of projections on A. Forming I-subfunctions corresponds to simple manipulation of variables, namely addition
and deletion of dummy variables, permutation of variables, and identification of variables. None of these operations increases the number of essential variables, and only variable identification may
decrease this number. We study more carefully the effect of variable identification on the number of essential variables of operations on finite base sets. We then study certain order-theoretical
properties of various C-subfunction partial orders defined by larger clones C on finite base sets A. We are mostly concerned about the descending chain condition and the existence of infinite
antichains, and as it turns out, these properties on the defining clone C.
Homomorphisms of labeled posets (or k-posets) are applied in our analysis of subfunction relations defined by clones of monotone functions. The third part of this thesis is a study of the
homomorphicity order of finite k-posets on its own right. We establish that this order is a distributive lattice, and furthermore, it is universal in the sense that every countable poset can be
embedded into it. This result implies universality of the subfunction partial orders defined by clones of monotone functions on finite sets of more than two elements. In this way, we also obtain a
new proof for the well-known universality of the homomorphicity order of graphs. | {"url":"https://cemat.ist.utl.pt/document.php?member_id=71&doc_id=572","timestamp":"2024-11-03T06:38:42Z","content_type":"text/html","content_length":"10985","record_id":"<urn:uuid:f38bc014-a6e8-4302-b7d1-1729866a044b>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00436.warc.gz"} |
A graph G=<V,E> consists of a set of vertices (also known as nodes) V and a set of edges (also known as arcs) E. An edge connects two vertices u and v; v is said to be adjacent to u. In a directed
graph, each edge has a sense of direction from u to v and is written as an ordered pair <u,v> or u->v. In an undirected graph, an edge has no sense of direction and is written as an unordered pair
{u,v} or u<->v. An undirected graph can be represented by a directed graph if every undirected edge {u,v} is represented by two directed edges <u,v> and <v,u>.
A path in G is a sequence of vertices <v[0], v[1], v[2], ..., v[n]> such that <v[i],v[i+1]> (or {v[i],v[i+1]}), for each i from 0 to n-1, is an edge in G. The path is simple if no two vertices are
identical. The path is a cycle if v[0]=v[n]. The path is a simple cycle if v[0]=v[n] and no other two vertices are identical.
Graphs are useful for representing networks and maps of roads, railways, airline routes, pipe systems, telephone lines, electrical connections, prerequisites amongst courses, dependencies amongst
tasks in a manufacturing system and a host of other data. There are a large number of important results and structures that are computed from graphs. Note that a rooted tree is a special kind of
directed graph and that an unrooted tree is a special kind of undirected graph.
Graphs by Adjacency Matrices.
A graph G=<V,E> can be represented by a |V|*|V| adjacency matrix A. If G is directed, A[ij]=true if and only if <v[i],v[j]> is in E. There are at most |V|^2 edges in E.
1 T
2 T
3 T
4 T T T
5 T T T
6 T
Adjacency Matrix of Directed Graph.
If G is undirected, A[ij]=A[ji]=true if {v[i],v[j]} is in E and A[ij]=A[ji]=false otherwise. In this case there are at most |V|*(|V|+1)/2 edges in E, A is symmetric and space can be saved by storing
only the upper triangular part A[ij] for i>=j.
1 T T T
2 T T T T
3 T T
4 T T T T
5 T T T
6 T T
Adjacency Matrix of Undirected Graph.
1 T T T
2 T T T
3 T
4 T
5 T
Upper Triangular Adjacency Matrix of Undirected Graph.
An adjacency matrix is easily implemented as an array.
Both directed and undirected graphs may be weighted. A weight is attached to each edge. This may be used to represent the distance between two cities, the flight time, the cost of the fare, the
electrical capacity of a cable or some other quantity associated with the edge. The weight is sometimes called the length of the edge, particularly when the graph represents a map of some kind. The
weight or length of a path or a cycle is the sum of the weights or lengths of its component edges. Algorithms to find shortest paths in a graph are given later.
The adjacency matrix of a weighted graph can be used to store the weights of the edges. If an edge is missing a special value, perhaps a negative value, zero or a large value to represent "infinity",
indicates this fact.
Adjacency Matrix of Weighted Directed Graph.
Adjacency Matrix of Weighted Undirected Graph.
Upper Triangular Adjacency Matrix of Weighted Undirected Graph.
It is often the case that if the weights represent distances then the natural distance from v[i] to itself is zero and the diagonal elements of the matrix are given this value.
A weighted adjacency matrix is easily defined in any imperative programming language.
A graph is complete if all possible edges are present. It is dense if most of the possible edges are present. It is sparse if most of them are absent, |E|<<|V|^2. Adjacency matrices are space
efficient for dense graphs but inefficient for sparse graphs when most of the entries represent missing edges. Adjacency lists use less space for sparse graphs.
Graphs by Adjacency Lists.
In a sparse directed graph, |E|<<|V|^2. In a sparse undirected graph |E|<<|V|*(|V|-1)/2. Most of the possible edges are missing and space can be saved by storing only those edges that are present,
using linked [lists].
Consider the weighted directed case.
An edge <v[i],v[j]> is placed in a list associated with v[i]. The edge is represented by the destination v[j] and the weight.
Adjacency Lists for Weighted Directed Graph.
Consider now the undirected case:
Adjacency Lists for Weighted Undirected Graph.
As before, half the space can be saved by only storing {v[i],v[j]} for i>=j:
Reduced Adjacency Lists for Weighted Undirected Graph.
Adjacency lists can be defined using records (structs) and pointers.
Note that some questions, such as "are v[i] and v[j] adjacent in G", take more time to answer using adjacency lists than using an adjacency matrix as the latter gives random access to all possible
Path Problems in Directed Graphs
The weight of an edge in a directed graph is often thought of as its length. The length of a path <v[0], v[1], ..., v[n]> is the sum of the lengths of all component edges <v[i],v[i+1]>. Finding the
shortest paths between vertices in a graph is an important class of problem.
See the [directed graph page].
Directed Acyclic Graphs
A directed acyclic graph (DAG!) is a directed graph that contains no cycles. A rooted tree is a special kind of DAG and a DAG is a special kind of directed graph.
See the [directed acyclic graph page].
The Minimum Spanning Tree of an Undirected Graph
A minimum spanning tree, T, of an undirected graph, G=<V,E>, is a tree such that:
1. T contains exactly the same vertices, V, as the graph
2. T's edges are a subset of E and
3. the total edge-weight of T is as small as possible.
See the [undirected graph page].
1. Add to Dijkstra's algorithm so that it prints the shortest path (not just its length) between v[1] and a given vertex v[i]. Hint: take note of Prim's algorithm.
2. Implement an adjacency list version of Dijkstra's algorithm. Use a [heap] as a priority queue to find the next vertex to add at each stage. This makes the algorithm O(E*log(V)). This is better
than O(|V|^2) for sparse graphs.
3. Compare the running time of Floyd's algorithm with running the given version of Dijkstra's algorithm |V| times to calculate all-pairs shortest paths.
4. Add to Floyd's algorithm so that it prints the shortest path (not just its length) between any two given vertices v[i] and v[j].
5. Implement the topological sort algorithm for DAGs represented by adjacency lists.
6. Show how the depth-first traversal algorithm can be used to generate efficient machine code from a DAG representing an expression with common subexpressions identified. Take the target machine to
be a stack machine which also has random access memory.
7. Compare the running times of Prim's and Kruskal's algorithms for graphs with various numbers of vertices, while varying the sparseness of the graphs.
8. (From the 1990 A.C.M. programming competition.) The problem is to discover an unknown collating sequence, that is a non-standard ordering of the alphabet {a..z}. You are given a list of words in
an unusual alphabetic order. Write a program to discover the underlying ordering of the alphabet a-z. Assume that there is sufficient information to determine the ordering uniquely. Note that
there may be letters that do not begin any word in the list. | {"url":"https://allisons.org/ll/AlgDS/Graph/","timestamp":"2024-11-02T10:38:17Z","content_type":"text/html","content_length":"15670","record_id":"<urn:uuid:08a99ba1-e116-4219-a820-07370e593cac>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00284.warc.gz"} |
"Exploring the Feasibility of Simulating the Universe: Unveiling the Evolution and Challenges" - SciTechPost
“Exploring the Feasibility of Simulating the Universe: Unveiling the Evolution and Challenges”
written by Santiago Fernandez 4 comments
Share 0 FacebookTwitterPinterestRedditWhatsappEmail
The endeavor to replicate the cosmos, exemplified by the work of Michael Wagman, sheds light on the historical progression and contemporary dilemmas within this domain. While a full-scale simulation
remains beyond our grasp, the ongoing advancements in computing power and algorithms are steadily enriching our comprehension of celestial phenomena.
Imagine a computer unraveling the most profound mysteries of the universe.
In the initial year of his graduate studies in 2013, Michael Wagman approached his advisor with a seemingly audacious question: “Can you assist me in simulating the universe?” Wagman, a theoretical
physicist and associate scientist at the US Department of Energy’s Fermi National Accelerator Laboratory, considered this query to be a logical pursuit. He aimed to bridge the gap between the formal
laws of physics and his everyday perception of reality, which was grounded in these laws.
In response, Wagman recalls that his advisor chuckled. Simulating the universe was deemed an insurmountable task. The sheer multitude of variables and the vast expanses of the unknown posed
formidable obstacles.
Nonetheless, the capacity to employ computers for reasonably accurate simulations represents a momentous leap from the state of scientific art just a century ago. Scientists like Wagman persist in
their quest to decipher the universe’s underlying code, undeterred by the enormity of the challenge.
In “The Universe in a Box,” published this year, Andrew Pontzen, a professor of cosmology at University College London (UCL), fortifies these efforts by tracing humanity’s historical trajectory
toward simulating the universe.
A Chronicle of Computational Simulations
Pontzen characterizes simulations as akin to hypothetical experiments. They establish theoretical scenarios within computer programs, guided by specific laws of physics, and task the computer with
deducing the ensuing consequences. This practice, he notes, dates back to antiquity, with ancient Greeks employing a rudimentary computational device, the Antikythera Mechanism, to predict
astronomical phenomena, including eclipses.
The concept of simulation, in a more modern sense, is attributed to Ada Lovelace, an English mathematician who collaborated with Charles Babbage, a visionary polymath and inventor. Babbage conceived
the Analytical Engine, a precursor to the modern computer, which Lovelace recognized as capable of transforming theoretical science into a practical endeavor through coded instructions on strips of
In the early 20th century, Lewis Fry Richardson, a mathematician and meteorologist, envisioned a colossal amphitheater filled with mathematicians collaborating on simulations to forecast the weather.
This vision laid the foundation for modern weather simulations, where the equations of physics govern atmospheric behavior.
One of the pioneering instances of computer simulations influencing cosmology emerged in the late 1960s through the work of Beatrice Tinsley, an astronomer and cosmologist. Her simulations
demonstrated that distant galaxies not only offer a glimpse into the past but also evolve over time, altering the interpretation of cosmological maps.
Unraveling Cosmic Enigmas
While a comprehensive simulation of the universe remains elusive, simulations have provided insights into phenomena that elude direct observation, such as dark matter and dark energy. The Hubble
Space Telescope, for instance, revealed the universe’s accelerating expansion, attributed to dark energy—a phenomenon that simulations had already hinted at.
Cosmologists and physicists leverage simulations to gain a deeper understanding of cosmic processes over vast time scales. These simulations explore the formation of structures and the evolution of
galaxies, shedding light on the broader cosmic narrative.
Nonetheless, scrutinizing isolated aspects of the universe falls short of comprehending its grand tapestry. Dorota Grabowska, a theoretical physicist at the University of Washington, highlights the
challenge of calculating early universe dynamics, a complex endeavor with myriad obstacles.
One significant obstacle is the simulation of gravity. While Einstein’s Theory of General Relativity and Newton’s Law of Gravitation provide effective approximations at low energies, they falter when
addressing ultra-high energy states, such as those at the inception of the universe.
The strong force, a fundamental component described by quantum chromodynamics (QCD), poses another formidable hurdle. Its intricacies defy straightforward approximations, necessitating quantum
computing for numerical simulations, albeit on a different timescale from reality.
For the most intricate simulations, scientists incorporate calculations to compensate for gaps in understanding and make assumptions based on available knowledge. These simulations, while
informative, come with a range of caveats to ensure their validity.
The Challenge of Cosmic-Scale Simulation
Even if scientists were to master the description of all four fundamental forces and understand every facet of physics, the computational power required to simulate the entire universe remains
unattainable. A truly comprehensive simulation would necessitate representing every atom in the universe with an equivalent atom within the simulation—an endeavor beyond the capabilities of Earth’s
current computing technology.
Nevertheless, there is optimism on the horizon. The boundaries of cosmic simulation expand continually, driven by advancements in computing power and algorithmic innovations. Wagman notes that
progress is achieved through both increased computational capabilities and refined algorithms, enabling the simulation of increasingly complex phenomena with greater efficiency.
Simulations serve as a window into the realm of plausibility, facilitating predictions about the workings of the natural world. While not infallible, they instill confidence that our understanding of
the universe is evolving incrementally towards greater accuracy. In this ongoing pursuit, simulations remain invaluable tools in the quest to unravel the mysteries of the cosmos.
Frequently Asked Questions (FAQs) about Simulating the Universe
What is the main focus of this text?
The main focus of this text is to explore the challenges and advancements in simulating the universe, bridging the gap between theoretical physics and the complexities of reality.
Who is Michael Wagman, and what is his role in the discussion?
Michael Wagman is a theoretical physicist and associate scientist at the US Department of Energy’s Fermi National Accelerator Laboratory. He plays a central role in the text by posing the question of
whether it is possible to simulate the universe, initiating the discussion on this complex topic.
How have simulations evolved historically, and what is their significance?
Simulations have evolved from ancient Greek calculations to modern computer-based models. They serve as hypothetical experiments to understand complex phenomena, offering valuable insights into
various fields, including cosmology and weather forecasting.
What are some challenges in simulating the universe?
Simulating the universe faces challenges related to the complexity of the task. These include the inability to simulate gravity accurately at ultra-high energy states, the intricacies of the strong
force described by quantum chromodynamics, and the sheer computational power required to represent every atom in the universe.
What insights have simulations provided about the universe?
Simulations have provided insights into elusive phenomena like dark matter and dark energy. They have also contributed to understanding cosmic processes, such as the evolution of galaxies and the
accelerating expansion of the universe.
Is a comprehensive simulation of the entire universe feasible?
A comprehensive simulation of the entire universe is currently beyond our technological capabilities. It would require representing every atom in the universe, a task that surpasses the computational
capacity of existing technology.
How are scientists making progress in cosmic simulations?
Progress is being made through advancements in computing power and the development of more efficient algorithms. These innovations allow scientists to simulate increasingly complex aspects of the
universe and enhance our understanding of cosmic phenomena.
More about Simulating the Universe
• “The Universe in a Box” by Andrew Pontzen
• “Antikythera Mechanism” – Ancient Greek computational device
• “Ada Lovelace and the Analytical Engine” – Historical perspective on simulation
• “Lewis Fry Richardson and Weather Forecasting Simulations” – Early 20th-century simulations
• “Beatrice Tinsley’s Contribution to Cosmological Simulations”
• “Hubble Space Telescope’s Discoveries on Dark Energy”
4 comments
Reader123 November 25, 2023 - 1:43 am
wow, this text sooo cool! Universe simulations r like, amazin! lots of techy stuff, luv it
SpaceExplorer November 25, 2023 - 3:19 am
Micheal Wagman is a genius, lol. Simulatin da universe, r u kiddin me?! Sci-fi stuff becomin real, mind blown
CosmicEnthusiast November 25, 2023 - 8:25 am
Simulatin cosmic things, dat’s da future! Cant wait 2 c more simulations n discoveries.
GrammarNerd November 25, 2023 - 2:48 pm
Text is good, but needs more punctuations. Sometimes hard 2 follow, but interesting topic!
Leave a Comment Cancel Reply
Share 0 FacebookTwitterPinterestRedditWhatsappEmail
Santiago Fernandez
Santiago Fernandez, a Chilean astronomer, is renowned for his research on astrophysics and cosmology. His articles delve into the mysteries of the universe, such as black holes, dark matter, and the
origins of the cosmos.
previous post
Illinois: A Powerhouse in Pumpkin Cultivation
You may also like | {"url":"https://scitechpost.com/exploring-the-feasibility-of-simulating-the-universe-unveiling-the-evolution-and-challenges/","timestamp":"2024-11-03T06:18:54Z","content_type":"text/html","content_length":"233821","record_id":"<urn:uuid:506a5fcf-245e-4005-93fe-b0e2008647b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00129.warc.gz"} |
Jérémy BARBAY | Professor (Associado) | Ph. D. in Computer Science | University of Concepción, Concepción | Departamento de Ingenieria Informatica en Ciencias de la Computacion (DIICC) | Research profile
Ph. D. in Computer Science
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.
Learn more
Theoretical Computer Science (TCS) Psycho Informatics (Automatisation of Psychological Processes and Experiments), in particular for Learning Management Systems (LMS) or using Animal Computer
Interaction (ACI). | {"url":"https://www.researchgate.net/profile/Jeremy-Barbay","timestamp":"2024-11-05T19:50:07Z","content_type":"text/html","content_length":"969638","record_id":"<urn:uuid:c67b538b-5c0d-447d-96f7-6e1f76b2a9a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00537.warc.gz"} |
Effective Span Of Beam For Simply Supported, Cantilever, Continous- IS
Effective span of beam for simply supported, Cantilever, Continous- IS
The beam is a structural member that takes loads from the slab and transfers them to the column. The effective span of beam is different for different types of beams. It depends upon the clear span,
width of support, and depth of the beam as well.
The effective span of the beam is required to analyze the structure for safety and economics. The beam of the structure should be strong and durable enough to carry the load and remain stable for a
long period of time.
The spans of different beams are given below. But, first of all, you should know about some technical terms given below.
What is Clear Span?
The span of the beam without measuring the length of support is known as the clear span of the beam. The clear span can be calculated by subtracting the width of the support, i.e., column, along the
direction of the beam span from the total span of the structure.
What is effective Span?
It is measured according to IS code 456-2000.
The effective span of beam
What is effective cover?
It is the distance between the outer surface of the beam and the middle of the main steel rod provided in the beam. The effective clear cover depends on the load on the beam. However, the thumb rule
to provide effective cover in the beam should not be less than 25 mm.
What is effective depth?
It is the (Total depth of beam – effective cover of beam). You can see in picture below.
Calculation for Effective span of beam
Here I have described the effective span of different types of beams given below,
1. Effective span for Simply supported beam
2. For Continuous beam
3. Cantilever beam
a) Effective span for Simply supported beam
The effective span of a simply supported beam is taken as the least of the following:
a) Clear span + the effective depth of beam.
b) Center to center ( c/c ) distance between supports.
b) For Continuous beam
In case of continuous beam, if the width of the support is less than 1/12 of the clear span, the effective span is taken as in (a).
If the width of support is more than 1/12 of the clear span or 600 mm whichever is less, the effective span is taken as:
For end span with one end fixed and the other continuous or for intermediate spans, the effective span is taken as the clear span between supports;
For end span with one end free and the other continuous, the effective span is equal to the clear span + half the effective depth of the beam or the clear span + half the width of the discontinuous
supports, whichever is less.
In case of span with roller or rocket bearings, the effective span always the distance between the centre of bearing.
c) Cantilever beam
The effective length of a cantilever beam is taken as its length to the face of the supports + half the effective depth except where it forms the end of a continuous beam.
How to Calculate Effective span of beam?
Q) Calculate the effective span of beam having a depth of 500 mm, a Clear span of 3 meters, and a supported on column having a width of 400×400 mm.
Given, Depth (D) = 500 mm
Clear Span (S) = 3 m = 3000 mm
Dimension of support = 400×400 mm
Assume, effective cover = 45 mm
We have two method to calculate effective span of beam,
A) Effective span (l)
= Clear span + Effective depth of Beam
= 3000 + (500-45) (Taking effective cover 45 mm)
= 3455 mm
B) Effective span (l) (center-to-center distance between supports)
= Clear span + Width of support
= 3000 + (400/2+400/2)
= 3400 mm
Hence, from the above calculation, B has a minimum value of effective span (l). So the effective span of beam will be 3400 mm.
I hope you got something new.
Happy Learning – Civil Concept
Read More,
Analysis of beam by Conjugate beam method with Numerical Example
Minimum clear cover for slab, column, beam, Retaining Structure
Difference between Tie beam and Plinth beam – and their Function
1 thought on “Effective span of beam for simply supported, Cantilever, Continous- IS”
1. fw= 35n/mm² fy=500n/mm2 basic ratio=20 factor=1.4 as?
Leave a Comment | {"url":"https://www.civilconcept.com/effective-span-of-beam/","timestamp":"2024-11-03T06:57:55Z","content_type":"text/html","content_length":"88274","record_id":"<urn:uuid:c0421642-49a3-4ffd-896e-513e15661c7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00815.warc.gz"} |
Get started with the jfa package
Koen Derks
Welcome to the ‘Get started’ page of the jfa package. jfa is an R package that provides Bayesian and classical statistical methods for audit sampling, data auditing, and algorithm auditing. This page
points you to the vignettes accompanying each of these three subjects.
Audit sampling
Firstly, jfa facilitates statistical audit sampling. That is, the package provides functions for planning, performing, and evaluating an audit sample compliant with international standards on
Data auditing
Secondly, jfa facilitates statistical data auditing. That is, the package includes functions for auditing data, such as testing the distribution of first digits of a data set against Benford’s law,
or assessing whether a data set includes an unusual amount of repeated values.
Algorithm auditing
Finally, jfa facilitates statistical algorithm auditing. That is, the package implements functions for auditing algorithms, such as computing fairness metrics and testing the equality of parity
metrics across protected groups. | {"url":"https://cran.rediris.es/web/packages/jfa/vignettes/jfa.html","timestamp":"2024-11-02T00:08:44Z","content_type":"text/html","content_length":"8402","record_id":"<urn:uuid:b1c62de5-438c-4153-a28f-a0c0894c84f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00534.warc.gz"} |
Essential Steps in Exploratory Factor Analysis for Statistics Assignments
1. Essential Steps in Exploratory Factor Analysis for Statistics Assignments
Exploratory Factor Analysis: Essential Steps to Solve Your Multivariate Statistics Assignment
July 19, 2024
John Doe
Multivariate Statistics
John Doe, a Statistics Expert with 7 years of experience, holds a Master's degree in Statistics from the University of California, Berkeley. He specializes in data analysis and statistical modeling,
offering comprehensive support and guidance to university students to help them excel in their academic and research projects.
Solving your statistics assignment can be a daunting task, especially when it involves complex techniques like Exploratory Factor Analysis (EFA). EFA is a powerful method used to identify underlying
relationships between measured variables, making it a crucial tool in fields like psychology, social sciences, and market research. This guide aims to demystify the process of conducting an EFA,
providing clear and comprehensive steps to help you master this technique. Whether you're dealing with survey data, psychological scales, or any multivariate dataset, understanding how to effectively
perform an EFA will not only help you solve your statistics assignment but also enhance your analytical skills. By following the outlined steps, from preliminary tests to the final interpretation of
factors, you'll gain the confidence and expertise needed to tackle any assignment involving EFA. Dive into this guide and equip yourself with the knowledge to excel in your statistical analyses.
1. Preliminary Tests: Ensuring Data Suitability
Before diving into Exploratory Factor Analysis (EFA), it's crucial to ensure that your data is suitable for this type of analysis. This involves conducting preliminary tests such as Bartlett’s Test
of Sphericity and the Kaiser-Meyer-Olkin (KMO) Measure of Sampling Adequacy.
• Bartlett’s Test of Sphericity: This test checks whether your correlation matrix is significantly different from an identity matrix, where variables are uncorrelated. A significant result (p <
0.05) indicates that there are enough correlations among the variables to proceed with EFA. This step is essential to validate that your data has the potential to reveal meaningful factors.
• Kaiser-Meyer-Olkin (KMO) Measure: The KMO Measure evaluates the adequacy of your data for factor analysis by examining the proportion of variance among variables that might be common variance.
The KMO value ranges from 0 to 1, with values closer to 1 indicating that the data is suitable for EFA. A KMO value above 0.6 is generally considered acceptable. If the KMO value is below 0.6, it
suggests that the sample size may be too small or that the correlations between pairs of variables are not high enough to justify a factor analysis.
By performing these preliminary tests, you can ensure that your data meets the necessary criteria for EFA, setting a solid foundation for the subsequent analysis. These steps help in confirming that
the relationships between variables are strong enough to uncover meaningful factors, thereby enhancing the accuracy and reliability of your factor analysis. Taking the time to validate your data with
these tests is a critical first step in successfully completing your multivariate statistics assignment involving EFA.
2. Determining the Number of Factors: Parallel Analysis
Deciding how many factors to extract is a crucial step in Exploratory Factor Analysis (EFA). The goal is to identify the most meaningful factors that explain the underlying structure of your data
without overfitting. One robust method for making this decision is Parallel Analysis.
Parallel Analysis involves comparing the eigenvalues obtained from your actual data with those obtained from randomly generated data sets of the same size. The steps are as follows:
• Generate Random Data Sets: Create multiple random data sets that have the same number of variables and observations as your original data.
• Compute Eigenvalues: For each random data set, compute the eigenvalues of the correlation matrix. These eigenvalues represent the amount of variance explained by each factor in the random data.
• Compare Eigenvalues: Plot the eigenvalues of your actual data against the average eigenvalues from the random data sets. Factors from your data are retained if their eigenvalues exceed the
corresponding average eigenvalues from the random data.
• Scree Plot: Use a scree plot to visualize the eigenvalues. The point where the curve starts to flatten, indicating that additional factors explain minimal additional variance, helps in deciding
the number of factors to retain.
• Theoretical Considerations: While parallel analysis provides a statistical basis for factor retention, it’s also important to consider the theoretical context of your data. The factors should
make sense in terms of the underlying constructs you are investigating.
Using parallel analysis ensures that the factors you retain are statistically significant and not due to random chance. This method helps in achieving a balance between explaining sufficient variance
and maintaining a parsimonious model. By carefully determining the number of factors to extract, you enhance the reliability and validity of your EFA, leading to more accurate and meaningful insights
in your statistical analysis.
3. Interpreting Communalities
Interpreting communalities is a crucial step in Exploratory Factor Analysis (EFA), as it provides insights into how much of each variable’s variance is explained by the extracted factors.
Communalities can be understood in two stages: initial and extraction.
Initial Communalities
Initial communalities are estimates of the variance in each variable that would be explained by the factors, assuming that all factors are uncorrelated. These values are usually based on the squared
multiple correlations of each variable with all other variables. Initial communalities serve as a starting point before the extraction of factors.
Extraction Communalities
Extraction communalities, on the other hand, represent the proportion of each variable’s variance that is explained by the factors after extraction. These values are derived from the factor solution
and indicate how well the factors account for the variability in each variable. Higher communalities suggest that the variables are well-represented by the factors, while lower communalities may
indicate that the variables do not fit well within the factor structure.
By carefully interpreting communalities, you can ensure that your factor analysis yields meaningful and reliable results, ultimately helping you to solve your statistics assignment with greater
accuracy and insight.
4. Analyzing the Pattern Matrix and Factor Structure
Once you have determined the number of factors to retain, the next step in Exploratory Factor Analysis (EFA) involves analyzing the Pattern Matrix and Factor Structure Correlation Matrix. These
matrices provide critical insights into how variables are related to each factor and the relationships between different factors.
Pattern Matrix Analysis
The Pattern Matrix displays the factor loadings of each variable after rotation. High factor loadings (typically above 0.4 or 0.5) suggest that the variable contributes significantly to that
particular factor. On the other hand, low factor loadings indicate weaker relationships or that the variable may not be well-represented by the factor.
It's essential to scrutinize the Pattern Matrix to ensure that variables load predominantly on the factors you intended to retain. Sometimes, variables may cross-load on multiple factors, indicating
complex relationships that may require further interpretation or a different rotation method (e.g., oblique rotation).
Factor Structure Correlation Matrix
The Factor Structure Correlation Matrix shows the correlations between factors. Understanding these correlations helps in determining how distinct or related the identified factors are. High
correlations between factors suggest that they are closely related or may represent similar constructs. On the other hand, low correlations indicate more distinct factors.
Analyzing the Factor Structure Correlation Matrix is crucial for interpreting the overall structure of your data. It helps in identifying potential redundancies or overlaps between factors, which can
inform decisions on how many factors to retain and how to interpret the results.
5. Deciding on Factor Retention
Deciding how many factors to retain in an Exploratory Factor Analysis (EFA) is a critical step that involves both statistical criteria and theoretical considerations. Here’s how you can approach this
decision-making process effectively:
Statistical Criteria:
• Eigenvalues: Begin by examining the eigenvalues generated from your factor analysis. Typically, factors with eigenvalues greater than 1 are considered for retention. This criterion suggests that
these factors explain more variance than any single variable on its own.
• Scree Plot: A scree plot helps visualize the point at which the eigenvalues level off, suggesting the number of factors to retain. Factors before the "elbow" point are usually considered
• Parallel Analysis: This method compares the actual eigenvalues from your data with those generated from randomly generated data of the same size. Factors with eigenvalues exceeding those from the
random data are retained as meaningful factors.
By carefully weighing these statistical criteria and theoretical considerations, you can make an informed decision on how many factors to retain in your EFA. This ensures that the factors retained
are meaningful and contribute effectively to your statistical analysis and interpretation.
6. Naming and Interpreting Factors
Naming and interpreting factors in Exploratory Factor Analysis (EFA) involves identifying the main themes represented by each factor based on the variables that load most strongly on them. This step
helps clarify what each factor measures and provides insight into the underlying constructs in your data. Here’s how to approach it:
• Examine High-Loading Variables: Identify which variables have the highest loadings on each factor.
• Identify Common Themes: Look for recurring patterns or themes among these variables.
• Use Descriptive Names: Name each factor based on the dominant themes or constructs it represents.
• Validate Interpretations: Discuss your interpretations with peers or instructors to ensure clarity and accuracy.
Naming factors succinctly and interpreting their meaning helps in effectively communicating the results of your EFA, making it easier to understand the structure of your data in your statistics
7. Conducting Reliability Analysis
Reliability analysis in Exploratory Factor Analysis (EFA) is crucial for assessing the consistency and stability of the factors identified. Here are the key steps involved:
• Calculate Cronbach’s Alpha: Cronbach’s Alpha is a commonly used measure of internal consistency. It assesses how closely related a set of items are as a group. A higher alpha indicates greater
• Interpret Alpha Values: Typically, a Cronbach’s Alpha value above 0.7 is considered acceptable for reliability in social sciences and psychology. Values closer to 1.0 indicate stronger internal
• Review Item-Total Correlations: Evaluate the correlation of each item with the total score of its respective factor. Higher correlations indicate that the item is measuring the same construct as
the factor.
• Assess Average Inter-Item Correlations: Calculate the average correlation between items within each factor. Higher average correlations suggest greater internal consistency.
• Consider Factor Loadings: Items with low factor loadings may need to be reviewed for their contribution to the factor's reliability.
• Ensure Scale Homogeneity: Ensure that items within each factor measure a single underlying construct consistently.
Reliability analysis ensures that the factors identified through EFA are robust and reliable measures of the constructs they represent. This step is essential for drawing valid conclusions and
recommendations in your statistics assignment based on the factor structure derived from your data.
8. Reviewing Item Performance: Deletion and Modification
In Exploratory Factor Analysis (EFA), reviewing item performance involves assessing whether individual items contribute effectively to the overall factor structure and reliability of the scale. Here
are key considerations for evaluating and potentially modifying items:
• Item Loadings: Evaluate the factor loadings of each item to determine how strongly they contribute to their respective factors. Items with low loadings (typically below 0.3 or 0.4) may not
adequately measure the intended construct and might be candidates for deletion.
• Reliability Impact: Consider how removing or modifying an item affects the reliability of the factor or subscale. Items that decrease internal consistency (as measured by Cronbach’s Alpha) may
need adjustment or removal.
• Theoretical Relevance: Assess whether each item aligns conceptually with the underlying construct it aims to measure. Items that do not fit theoretically may distort the interpretation of factors
and should be reconsidered.
• Practical Considerations: Evaluate the practicality of retaining each item in terms of data collection, respondent burden, and the overall coherence of the factor structure.
• Consultation: Seek input from subject matter experts or colleagues familiar with the specific domain to validate decisions regarding item deletion or modification.
By carefully reviewing item performance in EFA, you can refine the factor structure, improve the reliability of your scale, and ensure that each retained item contributes meaningfully to the overall
assessment in your statistics assignment.
9. Interpreting High and Low Scores
Interpreting high and low scores in the context of Exploratory Factor Analysis (EFA) provides valuable insights into the characteristics and implications of each factor. Here’s how to interpret these
scores effectively:
• High Scores: High scores on a factor indicate that individuals or cases have higher levels of the underlying construct represented by that factor. For example, if a factor is related to anxiety
symptoms, high scores suggest greater levels of anxiety among the participants.
• Low Scores: Conversely, low scores on a factor indicate lower levels of the underlying construct. Using the anxiety example, low scores would suggest fewer anxiety symptoms or lower anxiety
levels among participants.
• Understanding Context: It’s crucial to interpret scores within the specific context of your study and the variables involved. Consider how the factors align with your research objectives and
• Practical Implications: Discuss the practical implications of high and low scores. For instance, high scores on factors related to customer satisfaction could indicate areas where improvements
are needed, while low scores might highlight strengths.
Interpreting high and low scores effectively enhances the utility of your EFA results in understanding the nuanced characteristics of your data. This understanding is essential for drawing meaningful
conclusions and making informed decisions based on your statistics assignment.
Mastering Exploratory Factor Analysis is essential for anyone looking to solve their statistics assignment involving multivariate data. This guide has provided a thorough overview of the EFA process,
from conducting preliminary tests to interpreting the final results. By understanding and applying these steps, you'll be well-equipped to uncover the underlying structure of your data, ensuring that
your analyses are both statistically sound and theoretically meaningful. Remember, the key to successfully solving your statistics assignment lies in a clear and systematic approach to EFA. With
practice and diligence, you'll find that this powerful analytical tool becomes an invaluable part of your statistical toolkit. As you continue to apply these techniques, you'll not only improve your
assignment grades but also enhance your overall understanding of multivariate analysis. Embrace the challenge, and let this guide be your roadmap to success in statistics. | {"url":"https://www.statisticsassignmentexperts.com/blog/exploratory-factor-analysis-in-statistics-assignments.html","timestamp":"2024-11-06T09:10:03Z","content_type":"text/html","content_length":"108924","record_id":"<urn:uuid:7ca8eaea-e24d-482d-ba8a-d527b13eb130>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00227.warc.gz"} |
Perl Weekly Challenge 54: permutations and Collatz
I’ve been doing the Perl Weekly Challenges. The latest involved generating permutations and checking the Collatz Conjecture.
Write a script to accept two integers n (>=1) and k (>=1). It should print the kth permutation of n integers. For more information, please follow the wiki page.
For example, n=3 and k=4, the possible permutation sequences are listed below:
The script should print the 4th permutation sequence 231.
I felt that the ordering of permutation sequences was not well-defined, but the example gives lexical order, so that's what I settled on. This means one doesn't have to generate the whole set of
permutations; rather, one can use the index number (k in this example) to generate a Lehmer code that describes the individual permutation desired.
use integer;
my ($n,$k)=@ARGV;
This means we need to convert the index into factorial base, and the first step there is to generate a table of factorials.
my @f;
my $b=1;
my $v=1;
while ((scalar @f == 0) || $f[-1] < $k) {
push @f,$v;
Then, starting with the highest factorial generated, do a standard base conversion. (Note that the least significant digit is at the start of the array. Also that the unmodified sequence, "first
permutation", is Lehmer code 0).
my $nk=$k-1;
my @n;
for (my $i=$#f;$i>=0;$i--) {
unshift @n,$nk/$f[$i];
Now we interpret that by pulling the indexed entries out of the unpermuted sequence.
my @i=(1..$n);
my @o;
for (my $i=$n;$i>=1;$i--) {
my $f=$n[$i-1] || 0;
push @o,splice @i,$f,1;
print join($n>9?',':'',@o),"\n";
Perl6 is basically identical modulo syntax, except that we don't have use integer;. This has most of the fiddly differences that have been tripping me up on previous occasions, so I'll include it in
my ($n,$k)=@*ARGS;
my @f;
my $b=1;
my $v=1;
while ((@f.elems == 0) || @f[@f.end] < $k) {
push @f,$v;
my $nk=$k-1;
my @n;
loop (my $i=@f.end;$i>=0;$i--) {
unshift @n,floor($nk/@f[$i]);
my @i=(1..$n);
my @o;
loop (my $j=$n;$j>=1;$j--) {
my $f=@n[$j-1] || 0;
push @o,splice @i,$f,1;
say join($n>9 ?? ',' !! '',@o);
It is thought that the following sequence will always reach 1:
$n = $n / 2 when $n is even $n = 3*$n + 1 when $n is odd For example, if we start at 23, we get the following sequence:
23 → 70 → 35 → 106 → 53 → 160 → 80 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Write a function that finds the Collatz sequence for any positive integer. Notice how the sequence itself may go far above the original starting number.
Well, that's obvious enough:
use integer;
while (my $n=shift @ARGV) {
my @k=($n);
while ($n != 1) {
if ($n % 2 == 0) {
} else {
push @k,$n;
print join(', ',@k),"\n";
and Perl6 differs only as needed.
Extra Credit
Have your script calculate the sequence length for all starting numbers up to 1000000 (1e6), and output the starting number and sequence length for the longest 20 sequences.
One could do this by brute force, but one can short-cut evaluation by keeping a log of the length of each sequence that's already been evaluated. So in the example above, by the time we get to 23,
we've already evaluated 20, so we don't need to recalculate the last seven steps from 20 down to 1.
use integer;
my %l;
my %s;
foreach my $n (1..1e6) {
my $k=1;
my $na=$n;
while (!exists $l{$na}) {
if ($na % 2 == 0) {
} else {
push @{$s{$l{$n}}},$n;
The keys of %s are sequence lengths and the values are lists of starting numbers that give that length of sequence, so we just iterate down the keys until we've printed enough numbers.
my $k=20;
foreach my $c (sort {$b <=> $a} keys %s) {
print "$c: ".join(', ',sort @{$s{$c}}),"\n";
$k-=scalar @{$s{$c}};
if ($k<=0) {
The whole thing takes about 3½ seconds on my desktop machine. A potentially faster approach, though it would use more memory, would be to retain indices for the entire sequence; "26" already appears
in the sequence for "7" so no new calculations need be done at all. However, on the same machine this version takes a little over 5 seconds.
I didn't do this one in Perl6; the syntax needed for its equivalent of push @{$s{$l{$n}}},$n with autovivification eluded me.
Comments on this post are now closed. If you have particular grounds for adding a late comment, comment on a more recent post quoting the URL of this one. | {"url":"http://blog.firedrake.org/archive/2020/04/Perl_Weekly_Challenge_54__permutations_and_Collatz.html","timestamp":"2024-11-11T19:19:23Z","content_type":"application/xhtml+xml","content_length":"28358","record_id":"<urn:uuid:c0c1f75a-23e5-40a9-8a6c-8d35c9d386d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00213.warc.gz"} |
On detecting terrestrial planets with timing of giant planet transits
The transits of a distant star by a planet on a Keplerian orbit occur at time intervals exactly equal to the orbital period. If a second planet orbits the same star, the orbits are not Keplerian and
the transits are no longer exactly periodic. We compute the magnitude of the variation in the timing of the transits, δt. We investigate analytically several limiting cases: (i) interior perturbing
planets with much smaller periods; (ii) exterior perturbing planets on eccentric orbits with much larger periods; (iii) both planets on circular orbits with arbitrary period ratio but not in
resonance; (iv) planets on initially circular orbits locked in resonance. Using subscripts 'out' and 'in' for the exterior and interior planets, μ for planet-to-star mass ratio and the standard
notation for orbital elements, our findings in these cases are as follows, (i) Planet-planet perturbations are negligible. The main effect is the wobble of the star due to the inner planet, and
therefore δt ∼ μ[in](a[in]/a[out]) P[out]. (ii) The exterior planet changes the period of the interior planet by μ[out](a[in]/r[out])^3 P[in]. As the distance of the exterior planet changes due to
its eccentricity, the inner planet's period changes. Deviations in its transit timing accumulate over the period of the outer planet, and therefore δt ∼ μ[out]e[out](a[in]/a[out]) ^3P[out]. (iii)
Halfway between resonances the perturbations are small, of the order of μ[out]a[in]^2(a [in] - a[out])^2P[in] for the inner planet (switch 'out' and 'in' for the outer planet). This increases as one
gets closer to a resonance, (iv) This is perhaps the most interesting case because some systems are known to be in resonances and the perturbations are the largest. As long as the perturber is more
massive than the transiting planet, the timing variations would be of the order of the period regardless of the perturber mass. For lighter perturbers, we show that the timing variations are smaller
than the period by the perturber-to-transiting-planet mass ratio. An earth-mass planet in 2: 1 resonance with a three-dimensional period transiting planet (e.g. HD 209458h) would cause timing
variations of the order of 3 min, which would be accumulated over a year. This signal of a terrestrial planet is easily detectable with current ground-based measurements. For the case in which both
planets are on eccentric orbits, we compute numerically the transit timing variations for several known multiplanet systems, assuming they are edge-on. Transit timing measurements may be used to
constrain the masses, radii and orbital elements of planetary systems, and, when combined with radial velocity measurements, provide a new jneans of measuring the mass and radius of the host star.
• Eclipses
• Planetary systems
Dive into the research topics of 'On detecting terrestrial planets with timing of giant planet transits'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/on-detecting-terrestrial-planets-with-timing-of-giant-planet-tran","timestamp":"2024-11-09T19:34:02Z","content_type":"text/html","content_length":"55581","record_id":"<urn:uuid:25d21008-5915-4683-883f-c7653b4ba89e>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00747.warc.gz"} |
LeetCode 17. Letter Combinations of a Phone Number | GoodTecher
LeetCode 17. Letter Combinations of a Phone Number
Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent. Return the answer in any order.
A mapping of digit to letters (just like on the telephone buttons) is given below. Note that 1 does not map to any letters.
Example 1:
Input: digits = "23"
Output: ["ad","ae","af","bd","be","bf","cd","ce","cf"]
Example 2:
Input: digits = ""
Output: []
Example 3:
Input: digits = "2"
Output: ["a","b","c"]
• 0 <= digits.length <= 4
• digits[i] is a digit in the range ['2', '9'].
First, build a mapping between numbers and the letters they represent.
Then, use the backtracking approach to generate all possible combinations.
Java Solution
class Solution {
public List<String> letterCombinations(String digits) {
List<String> result = new ArrayList<>();
if (digits == null || digits.equals("")) {
return result;
StringBuilder sb = new StringBuilder();
Map<Character, char[]> lettersMap = getLettersMap();
letterCombinationsHelper(digits, sb, lettersMap, result);
return result;
private Map<Character, char[]> getLettersMap() {
Map<Character, char[]> lettersMap = new HashMap<>();
lettersMap.put('0', new char[]{});
lettersMap.put('1', new char[]{});
lettersMap.put('2', new char[]{'a', 'b', 'c'});
lettersMap.put('3', new char[]{'d', 'e', 'f'});
lettersMap.put('4', new char[]{'g', 'h', 'i'});
lettersMap.put('5', new char[]{'j', 'k', 'l'});
lettersMap.put('6', new char[]{'m', 'n', 'o'});
lettersMap.put('7', new char[]{'p', 'q', 'r', 's'});
lettersMap.put('8', new char[]{'t', 'u', 'v'});
lettersMap.put('9', new char[]{'w', 'x', 'y', 'z'});
return lettersMap;
private void letterCombinationsHelper(String digits, StringBuilder sb, Map<Character, char[]> lettersMap, List<String> result) {
if (sb.length() == digits.length()) {
for (char ch : lettersMap.get(digits.charAt(sb.length()))) {
letterCombinationsHelper(digits, sb, lettersMap, result);
sb.deleteCharAt(sb.length() - 1);
Python Solution
class Solution:
def letterCombinations(self, digits: str) -> List[str]:
results = []
if not digits:
return results
mapping = {
"2": ["a", "b", "c"],
"3": ["d", "e", "f"],
"4": ["g", "h", "i"],
"5": ["j", "k", "l"],
"6": ["m", "n", "o"],
"7": ["p", "q", "r", "s"],
"8": ["t", "u", "v"],
"9": ["w", "x", "y", "z"],
self.helper(results, "", digits, mapping)
return results
def helper(self, results, combination, digits, mapping):
if len(combination) == len(digits):
letters = mapping[digits[len(combination)]]
for letter in letters:
combination += letter
self.helper(results, combination, digits, mapping)
combination = combination[: len(combination) -1]
• Time complexity: O(3^N×4^M) where N is the number of digits in the input that maps to 3 letters (e.g. 2, 3, 4, 5, 6, 8) and M is the number of digits in the input that maps to 4 letters (e.g. 7,
9), and N+M is the total number digits in the input.
• Space complexity: O(3^N×4^M) since one has to keep 3^N times 4^M solutions.
3 Thoughts to “LeetCode 17. Letter Combinations of a Phone Number”
2. bro i am not good at recursion . Could you please explain it in betterways? how sb.length works here or how the combinations is happening???
plz leave a reply.
3. Hello, how can I reach you privately?
my email is ljustincoder@gmail.com. I need help in understanding some of your solutions.
Secondly, regarding the letter combinations of a phone number. I understand what your code does but my recursion skill is weak and need clarity on one condition. So, say digit is “23”. the first
time the helper is called. sb is “”. but what is the index of “”? how did the computer know that when you do :
lettersMap.get(digits.charAt(sb.length())), it is to get the the index 2? is “” equated to 0?
then, it gets the first character in the key 2 and appends it to the StringBuilder making the sb.length=1. Ok. I get it, it picks the map with key index 1 which is 3 and begins to add its’
characters. and each time the sb.length==digit.length, it adds the StringBuilder to the result list. and then removes that character. it continues until even the character a from the phone number
2 is removed, defaulting the StringBuilder sb to “” again. at this stage, the result is “ad, ae, af”.
Now, this is where I don’t understand it. Because instead of picking character a again, it picks b. how was the programme able to know that the first character in phone number key 2, which is ‘a’
should be overlooked?
This is what I don’t get.
Lastly, have you considered setting up a patron page where people like me can donate something, no matter how small, to encourage you? I am glued to your youtube videos and seriously testing the
codes to make sense of your thought processes. Please, don’t stop. You are the only one explaining it simplistically. Is there a way to also send you Leetcode questions I find challenging so you
can solve and explain to the community? | {"url":"https://www.goodtecher.com/leetcode-17-letter-combinations-of-a-phone-number/","timestamp":"2024-11-02T10:57:01Z","content_type":"text/html","content_length":"55766","record_id":"<urn:uuid:816cc2c8-9d79-44c7-b9fd-9d6c347a3a52>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00483.warc.gz"} |
Solution Manual for The Basic Practice of Statistics, 8th Edition, David S. Moore
Instant download Solution Manual for The Basic Practice of Statistics, 8th Edition, David S. Moore, William I. Notz, Michael A. Fligner pdf docx epub after payment.
Product details:
• ISBN-10 : 1319042570
• ISBN-13 : 978-1319042578
• Author: David S. Moore, William I. Notz, Michael A. Fligner
Table of Contents
Chapter 0 Getting Started
Where data comes from matters
Always look at the data
Variation is everywhere
What lies ahead in this book
Chapter 1 Picturing Distributions with Graphs
1.1 Individuals and variables
1.2 Categorical variables: Pie charts and bar graphs
1.3 Quantitative variables: Histograms
1.4 Interpreting histograms
1.5 Quantitative variables: Stemplots
1.6 Time plots
Chapter 2 Describing Distributions with Numbers
2.1 Measuring center: The mean
2.2 Measuring center: The median
2.3 Comparing the mean and the median
2.4 Measuring spread: The quartiles
2.5 The five-number summary and boxplots
2.6 Spotting suspected outliers*
2.7 Measuring spread: The standard deviation
2.8 Choosing measures of center and spread
2.9 Using technology
2.10 Organizing a statistical problem
Chapter 3 The Normal Distributions
3.1 Density curves
3.2 Describing density curves
3.3 Normal distributions
3.4 The 68-95-99.7 rule
3.5 The standard Normal distribution
3.6 Finding Normal proportions
3.7 Using the standard Normal table
3.8 Finding a value given a proportion
Chapter 4 Scatterplots and Correlation
4.1 Explanatory and response variables
4.2 Displaying relationships: Scatterplots
4.3 Interpreting scatterplots
4.4 Adding categorical variables to scatterplots
4.5 Measuring linear association: Correlation
4.6 Facts about correlation
Chapter 5 Regression
5.1 Regression lines
5.2 The least-squares regression line
5.3 Using technology
5.4 Facts about least-squares regression
5.5 Residuals
5.6 Influential observations
5.7 Cautions about correlation and regression
5.8 Association does not imply causation
5.9 Correlation, prediction, and big data*
Chapter 6 Two-Way Tables*
6.1 Marginal distributions
6.2 Conditional distributions
6.3 Simpson’s paradox
Chapter 7 Exploring Data: Part I Review
Part I Summary
Test Yourself
Supplementary Exercises
Chapter 8 Producing Data: Sampling
8.1 Population versus sample
8.2 How to sample badly
8.3 Simple random samples
8.4 Inference about the population
8.5 Other sampling designs
8.6 Cautions about sample surveys
8.7 The impact of technology
Chapter 9 Producing Data: Experiments
9.1 Observation versus experiment
9.2 Subjects, factors, treatments
9.3 How to experiment badly
9.4 Randomized comparative experiments
9.5 The logic of randomized comparative experiments
9.6 Cautions about experimentation
9.7 Matched pairs and other block designs
Chapter 10 Data Ethics*
10.1 Institutional review boards
10.2 Informed consent
10.3 Confidentiality
10.4 Clinical trials
10.5 Behavioral and social science experiments
Chapter 11 Producing Data: Part II Review
Part II summary
Test yourself
Supplementary exercises
Chapter 12 Introducing Probability
12.1 The idea of probability
12.2 The search for randomness*
12.3 Probability models
12.4 Probability rules
12.5 Discrete probability models
12.6 Continuous probability models
12.7 Random variables
12.8 Personal probability*
Chapter 13 General Rules of Probability*
13.1 The general addition rule
13.2 Independence and the multiplication rule
13.3 Conditional probability
13.4 The general multiplication rule
13.5 Showing events are independent
13.6 Tree diagrams
13.7 Bayes’ rule*
Chapter 14 Binomial Distributions*
14.1 The binomial setting and binomial distributions
14.2 Binomial distributions in statistical sampling
14.3 Binomial probabilities
14.4 Using technology
14.5 Binomial mean and standard deviation
14.6 The Normal approximation to binomial distributions
Chapter 15 Sampling Distributions
15.1 Parameters and statistics
15.2 Statistical estimation and the law of large numbers
15.3 Sampling distributions
15.4 The sampling distribution of x
15.5 The central limit theorem
15.6 Sampling distributions and statistical significance*
Chapter 16 Confidence Intervals: The Basics
16.1 The reasoning of statistical estimation
16.2 Margin of error and confidence level
16.3 Confidence intervals for a population mean
16.4 How confidence intervals behave
Chapter 17 Tests of Significance: The Basics
17.1 The reasoning of tests of significance
17.2 Stating hypotheses
17.3 P-value and statistical significance
17.4 Tests for a population mean
17.5 Significance from a table*
Chapter 18 Inference in Practice
18.1 Conditions for inference in practice
18.2 Cautions about confidence intervals
18.3 Cautions about significance tests
18.4 Planning studies: Sample size for confidence intervals
18.5 Planning studies: The power of a statistical test*
Chapter 19 From Data Production to Inference: Part III Review
Part III Summary
Review Exercises
Test Yourself
Supplementary Exercises
Chapter 20 Inference about a Population Mean
20.1 Conditions for inference about a mean
20.2 The t distributions
20.3 The one-sample t confidence interval
20.4 The one-sample t test
20.5 Using technology
20.6 Matched pairs t procedures
20.7 Robustness of t procedures
Chapter 21 Comparing Two Means
21.1 Two-sample problems
21.2 Comparing two population means
21.3 Two-sample t procedures
21.4 Using technology
21.5 Robustness again
21.6 Details of the t approximation*
21.7 Avoid the pooled two-sample t procedures*
21.8 Avoid inference about standard deviations*
Chapter 22 Inference about a Population Proportion
22.1 The sample proportion
22.2 Large-sample confidence intervals for a proportion
22.3 Choosing the sample size
22.4 Significance tests for a proportion
22.5 Plus four confidence intervals for a proportion*
Chapter 23 Comparing Two Proportions
23.1 Two-sample problems: Proportions
23.2 The sampling distribution of a difference between proportions
23.3 Large-sample confidence intervals for comparing proportions
23.4 Using technology
23.5 Significance tests for comparing proportions
23.6 Plus four confidence intervals for comparing proportions*
Chapter 24 Inference about Variables: Part IV Review
Part IV summary
Test yourself
Supplementary exercises
Chapter 25 Two Categorical Variables: The Chi-Square Test
25.1 Two-way tables
25.2 The problem of multiple comparisons
25.3 Expected counts in two-way tables
25.4 The chi-square test statistic
25.5 Using technology
25.6 The chi-square distributions
25.7 Cell counts required for the chi-square test
25.8 Uses of the chi-square test: Independence and homogeneity
25.9 The chi-square test for goodness of fit*
Chapter 26 Inference for Regression
26.1 Conditions for regression inference
26.2 Estimating the parameters
26.3 Using technology
26.4 Testing the hypothesis of no linear relationship
26.5 Testing lack of correlation
26.6 Confidence intervals for the regression slope
26.7 Inference about prediction
26.8 Checking the conditions for inference
Chapter 27 One-Way Analysis of Variance:
Comparing Several Means
27.1 Comparing several means
27.2 The analysis of variance F test
27.3 Using technology
27.4 The idea of analysis of variance
27.5 Conditions for ANOVA
27.6 F distributions and degrees of freedom
27.7 Follow-up analysis: Tukey pairwise multiple comparisons
27.8 Some details of ANOVA*
[Back matter print text]
Notes and Data Sources
TABLE A Standard Normal probabilities
TABLE B Random digits
TABLE C t distribution critical values
TABLE D Chi-square distribution critical values
TABLE E Critical values of the correlation r
Answers to Selected Exercises
(available online)
Chapter 28 Nonparametric Tests
28.1 Comparing two samples: The Wilcoxon rank sum test
28.2 The Normal approximation for W
28.3 Using technology
28.4 What hypotheses does Wilcoxon test?
28.5 Dealing with ties in rank tests
28.6 Matched pairs: The Wilcoxon signed rank test
28.7 The Normal approximation for W+
28.8 Dealing with ties in the signed rank test
28.9 Comparing several samples: The Kruskal-Wallis test
28.10 Hypotheses and conditions for the Kruskal-Wallis test
28.11 The Kruskal-Wallis test statistic
Chapter 29 Multiple Regression
29.1 Parallel regression lines
29.2 Estimating parameters
29.3 Using technology
29.4 Inference for multiple regression
29.5 Interaction
29.6 The multiple linear regression model
29.7 The woes of regression coefficients
29.8 A case study for multiple regression
29.9 Inference for regression parameters
29.10 Checking the conditions for inference
Chapter 30 More about Analysis of Variance
30.1 Beyond one-way ANOVA
30.2 Two-way ANOVA: Conditions, main effects, and interaction
30.3 Inference for two-way ANOVA
30.4 Some details of two-way ANOVA*
Chapter 31 Statistical Process Control
31.1 Processes
31.2 Describing processes
31.3 The idea of statistical process control
31.4 x charts for process monitoring
31.5 s charts for process monitoring
31.6 Using control charts
31.6 Setting up control charts
31.7 Comments on statistical control
31.8 Don’t confuse control with capability!
31.9 Control charts for sample proportions
31.10 Control limits for p charts
Chapter 32 Resampling: Permutation Tests and the Bootstrap
32.1 Randomization in experiments as a basis for inference
32.2 Permutation tests for comparing two treatments with software
32.3 Generating bootstrap samples
32.4 Bootstrap standard errors and confidence intervals
People also search:
The Basic Practice of Statistics, 8th Edition
The Basic Practice of Statistics, 8th Edition pdf
The Basic Practice of Statistics
basic research and basic statistics
what is basic statistics
what are basics of statistics
what are the principles of statistics | {"url":"https://testbankbell.com/product/solution-manual-for-the-basic-practice-of-statistics-8th-edition-david-s-moore/","timestamp":"2024-11-10T01:44:20Z","content_type":"text/html","content_length":"170356","record_id":"<urn:uuid:a0f62f3f-7395-4a3a-84a6-a56c1578d597>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00259.warc.gz"} |
Yesterday in class I discussed how we can interpret the linear regression parameters (i.e., the y-intercept a (“alpha”) and the slope b (“beta”) yielding a linear regression line (or what we also
call a “linear model”)
y = a + bx
See below for a summary (you can also take a look at the Khan Academy videos “Interpreting y-intercept in regression” and “Interpreting slope in regression“):
• Recall that the linear model is used to predict an “output” value y for a given “input” value x
• In terms of the line, the y-intercept a is the y-value where the line intersects the x-axis, i.e., when x = 0. Thus, in terms of the linear regression model, the y-intercept a is the predicted
value of the dependent variable y when the independent variable x is 0.
• In terms of the line, the slope b is how much y increases or decreases if x is increased by 1. Thus, in terms of the linear regression model, the slope b is the predicted change in the dependent
variable y if the independent variable x is increased by 1.
For example, this was an exercise on the “HW4-Paired Data” WebWork set:
Exercise from WebWork “HW4-Paired Data”
The results of linear regression for this data set (i.e., regressing the dependent variable y (final grade) on the independent variable x (verbal score) yield the linear regression parameters:
• y-intercept a ≈ 99.1 ; this can be interpreted as the predicted final grade of a student who gets a verbal score of 0
• slope b ≈ -0.333 ; this can be interpreted as saying that a student who increases their verbal score by 1 will decrease their final grade by -0.333 | {"url":"https://openlab.citytech.cuny.edu/math1372-ganguli-spring2020/2020/03/03/","timestamp":"2024-11-05T20:44:12Z","content_type":"text/html","content_length":"110768","record_id":"<urn:uuid:86c6989c-f131-4ae1-b438-d74b683f136e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00371.warc.gz"} |
Present Value Assignment Help
What is present value?
Present value is the value in today’s dollars of a sum of money to be received in the future. In simpler terms it is the current value of a future payment. The formula for calculating present value
In the formula FV is the future value, n is the number of years and r is the interest or discount rate. For example you may have a savings bond with a discount rate of 6 percent that pays you $1000
in 10 years. If you wanted to know what the bond is worth in today’s dollars plug the appropriate figures into the above equation.
In this case FV = $100, n = 10 and r = 6 percent so our equation would be:
In the formula FV is the future value, n is the number of years and r is the interest or discount rate. For example you may have a savings bond with a discount rate of 6 percent that pays you $1000
in 10 years. If you wanted to know what the bond is worth in today’s dollars plug the appropriate figures into the above equation. In this case FV = $100, n = 10 and r = 6 percent so our equation
would be:
= 1000(0.5584)
= 558.40
The value of the $1000 savings bond is $558.40 in today’s dollars. There are many concepts in finance that can be confusing. If you are having difficulty with any aspect of a present value assignment
or future value assignment we provide a service that can help.
We provide present value assignment help
To successfully complete a present value assignment, you must understand the concepts involved as well as be able to work with the PV equation. In finance it is important to understand one concept in
order to move on to the next. The present value assignment help we provide will ensure you understand the concepts and ideas involved and be able to use the PV equation. We offer assistance ranging
from finance homework help, to test preparation and one-on-one tutoring in any areas that are especially difficult for you.
Professional finance homework help
Our tutors all have degrees in finance or a related field, many of them at the graduate level. They also have extensive proven experience instructing students in finance. When you get assistance with
your finance homework from our service, we work with you so that not only are you able to come up with the correct answers, you also understand why and how you did. Our goal isn’t just to walk you
through one assignment. We want you to thoroughly understand what you are studying and be able to confidently apply the concept in future assignments.
Benefits of our finance homework service
Our finance homework service is well qualified to assist you with any type of finance assignment. You are able to get step by step explanations for how answers are arrived at and expert personal
tutoring if necessary. However there are other benefits to be had when using our service including:
• Guarantees of complete customer satisfaction and on time delivery for all work we provide
• Student friendly rates with no hidden costs
• Complete customer confidentiality
• 24/7 caring attitude and support
Сontact us for present value homework help that ensures you have a firm grasp of all concepts involved and a thorough understanding of the topic! | {"url":"https://www.financehomeworkhelp.org/present-value-assignment-help/","timestamp":"2024-11-13T06:38:49Z","content_type":"application/xhtml+xml","content_length":"54337","record_id":"<urn:uuid:c63041c1-7e12-4af9-900d-c6eba5caefeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00457.warc.gz"} |
Collagenous Extracellular Matrix Biomaterials for Tissue Engineering: Lessons from the Common Sea Urchin Tissue
Newcastle University Singapore, SIT Building at Nanyang Polytechnic, 172A Ang Mo Kio Avenue 8 #05-01, Singapore 567739, Singapore
Newcastle University, School of Mechanical & Systems Engineering, Stephenson Building, Claremont Road, Newcastle upon Tyne NE1 7RU, UK
Manchester University, Wellcome Trust Centre for Cell Matrix Research, B.3016 Michael Smith Building, Faculty of Life Sciences, Oxford Road, Manchester M13 9PT, UK
Author to whom correspondence should be addressed.
Submission received: 10 January 2017 / Revised: 5 April 2017 / Accepted: 11 April 2017 / Published: 25 April 2017
Scaffolds for tissue engineering application may be made from a collagenous extracellular matrix (ECM) of connective tissues because the ECM can mimic the functions of the target tissue. The primary
sources of collagenous ECM material are calf skin and bone. However, these sources are associated with the risk of having bovine spongiform encephalopathy or transmissible spongiform encephalopathy.
Alternative sources for collagenous ECM materials may be derived from livestock, e.g., pigs, and from marine animals, e.g., sea urchins. Collagenous ECM of the sea urchin possesses structural
features and mechanical properties that are similar to those of mammalian ones. However, even more intriguing is that some tissues such as the ligamentous catch apparatus can exhibit mutability,
namely rapid reversible changes in the tissue mechanical properties. These tissues are known as mutable collagenous tissues (MCTs). The mutability of these tissues has been the subject of on-going
investigations, covering the biochemistry, structural biology and mechanical properties of the collagenous components. Recent studies point to a nerve-control system for regulating the ECM
macromolecules that are involved in the sliding action of collagen fibrils in the MCT. This review discusses the key attributes of the structure and function of the ECM of the sea urchin ligaments
that are related to the fibril-fibril sliding action—the focus is on the respective components within the hierarchical architecture of the tissue. In this context, structure refers to size, shape and
separation distance of the ECM components while function is associated with mechanical properties e.g., strength and stiffness. For simplicity, the components that address the different length scale
from the largest to the smallest are as follows: collagen fibres, collagen fibrils, interfibrillar matrix and collagen molecules. Application of recent theories of stress transfer and fracture
mechanisms in fibre reinforced composites to a wide variety of collagen reinforcing (non-mutable) connective tissue, has allowed us to draw general conclusions concerning the mechanical response of
the MCT at specific mechanical states, namely the stiff and complaint states. The intent of this review is to provide the latest insights, as well as identify technical challenges and opportunities,
that may be useful for developing methods for effective mechanical support when adapting decellularised connective tissues from the sea urchin for tissue engineering or for the design of a synthetic
1. Introduction
With regards to biomaterials for tissue engineering, the key areas that must be addressed are the cells, scaffolds and growth-stimulating signals [
]. The three areas are also collectively known as the tissue engineering triad [
]. In particular, scaffolds are intended to provide sites for cells attachment during the initial stages of the tissue engineering process—subsequently they serve as templates for enabling tissue
materials to be generated onto the structure [
]. Polymeric biomaterials, such as chitosan, have been proposed for making scaffolds [
] but in order to satisfy the requirements of biocompatibility, the scaffold should ideally be made from the extracellular matrix (ECM) of the target tissue in its native state [
]. Where this is unavailable, the alternatives are bioengineered porous scaffolds [
], decellularized tissue from allogenic or xenogenic tissues [
], cell sheets with self-secreted ECM [
], and cell encapsulation in self-assembled hydrogel matrix [
]. A detailed discussion of the advantages and limitations of the different types of biological materials can be found elsewhere [
]. In this review, the focus is on the ECM derived from decellularized tissue (ECM-DT). A possible source for ECM-DTs is the connective tissue of the sea urchin (see sketch in
Figure 1
A), which belongs to a phylum of marine invertebrates called echinoderms. The discussion in this review is concerned with studies on the structure and mechanical properties of the ECM of the sea
urchin’s connective tissue and the contribution of the findings from these studies to new insights for developing novel biomaterials for tissue engineering applications.
Echinoderms, such as sea urchin and starfish, are some of the most abundant multi-cellular animals in the marine world [
] and play a dominant role in the ecological chain [
]. In recent years, significant technological advancements have been achieved in the processing of tissue materials from marine animals for biomedical applications [
]. These biomedical applications range from post-surgical wound-assist healing to implants for tissue engineering and drug delivery [
]. With regards to marine sources, the connective tissue of echinoderms, such as the sea urchin, is a rich source of collagen which provides the structural support for the tissue [
]. Collagen is a protein molecule comprising three polypeptide alpha chains, organized in a triple-helical conformation [
]. There are at least twenty eight distinct types of collagen in vertebrates [
]. The major ones, namely type I, II and III, V and XI are fibril forming collagens; these collagens are located in fibrillar structures [
]. Note that the fibril-forming collagens in the connective tissues of invertebrates such as sea urchin, as well as sponge, may have more varied structural features than those of the standard
fibrillar vertebrate collagens, e.g., triple helical domains of varying lengths [
Figure 1
A presents a sketch of the sea urchin spine-test system containing some known connective tissues; a schematic of the hierarchical architecture of connective tissue is illustrated for the catch
apparatus (CA).
Collagenous scaffolds made from ECM-DT have attracted a lot of attention because the scaffold can retain important micro-structural properties [
] and biochemical composition [
] of the native ECM. Besides collagen, the other key ECM components of interest for ensuring that the scaffold can function as intended are glycosaminoglycans and proteoglycans [
]. Since these protein cores of the latter are highly conserved in many species [
], their presence in the ECM-DT would help minimize unintended immune response [
]. At the microscopic length scale corresponding to cells, the structural environment is also well-preserved in the ECM-DT; this means that the matrix microenvironment may be effective in directing
cellular phenotype via geometric cues [
] as well as growth factors for cell attachment, proliferation, migration, and differentiation [
]. Scaffolds made from ECM-DT have been investigated for regeneration in a range of tissues [
]. These scaffolds have been implemented on heart valve [
], tendon [
] and skeletal muscle [
], to name a few.
As the main components of structural ECM proteins, the fibril forming collagens are able to provide the mechanical support for the body, by an analogy to engineering fibre reinforced composites [
]. These fibrous structures are found in the musculo-skeletal connective tissues, such as tendons [
], ligaments [
], muscles [
], and in skin [
]. The ECM of connective tissues, such as tendons and ligaments, features a hierarchical architecture (
Figure 1
A) comprising collagen fibres which are bundles of collagen fibrils [
]. The other ECM components, particularly the fibril-associated proteoglycan, such as the small dermatan-sulfate proteoglycans (decorin and biglycan) bound onto collagen fibrils [
], are often thought to facilitate tissue deformation in response to external loads. The main contributor to tissue deformation is the fibril-fibril sliding action [
], analogous to the role of compatibilizer in engineering fibre reinforced composites [
]. Both the structure and biochemistry of these proteins are described in considerable detail in other published reports and there is little need to discuss them further here. The reader is directed
to the works of Bailey and co-workers [
] and others [
] for collagen, and Iozzo and co-workers [
] for proteoglycans.
From a biomedical engineering perspective, the key advantage of collagen is that it is normally biocompatible, as with most biopolymers from marine sources [
]. The main concern with synthetic polymers is that they may contain unwanted compounds, especially residue of initiators, that could inhibit cell growth [
]. However, the collagens extracted by industrial means from bovine sources such as calf skin and bone could be associated with the bovine spongiform encephalopathy and transmissible spongiform
encephalopathy as pointed out in previous reports [
]. Alternatively, porous scaffolds composed of jellyfish collagen may be made by freeze-drying and cross-linking with 1-ethyl-(3-3-dimethylaminopropyl) carbodimide hydrochloride/
-hydroxysuccinimide; these porous scaffolds also demonstrate higher cell viability than those derived from bovine sources [
]. However, the scaffolds made from the jellyfish collagen reveal similar inflammatory response as those from bovine sources [
]. Di Benedetto and co-workers have developed a method for processing substrate scaffold from native collagen fibrils extracted from the peristomial membrane (a connective tissue) of the sea urchin [
]. The scaffold features a homogeneous fibrous mesh with thickness of around 2 μm [
]; the fibril diameter ranges 30–400 nm [
]. The general architecture of the Di Benedetto scaffold, i.e., fibril organization and bundle orientation, is identical to the structure and organization of collagen observed in several human
tissues such as tendon, ligament, cornea, skin and blood vessels [
], as well as other mammalian tissues, such as murine tendons [
] and avian tissues, such as chicken tendons [
While collagenous ECM of the sea urchin possesses structural features and mechanical properties that are similar to those of mammalian ones, even more intriguing is that some tissues such as the
ligamentous catch apparatus can exhibit mutability, namely rapid reversible changes in the tissue mechanical properties. These tissues are known as mutable collagenous tissues (MCTs). Although the
mechanism of mutability in MCTs is still not clear, progress has been made in the study of the mechanics of fibril sliding in the MCT that could contribute somewhat to our understanding of the
mechanical responses underpinning the changes in stiffness in the MCT. In a recent study to assess the non-collageneous content in the interfibriillar matrix, Ribeiro et al. [
] pointed out that the mechanical adaptability of MCT depends on the modulation of interfibrillar cohesion. Barbaglio et al. [
] added that there is good evidence that this is mediated by the GAGs; in the compass depressor ligaments (CDLs), variability in the GAG concentrations is observed (at different pH values) in the
respective mechanical states, namely compliant and stiff. This is because in order to withstand unidirectional tensile stresses the CDL has to recruit the appropriate number of interfibrillar
linkages (via GAGs) into resisting/facilitating fibril-fibril sliding [
]. Barbaglio et al. [
] concluded that the mechanical adaptability of the MCT may not require (appreciable) changes in the collagen fibrils. To demonstrate this, Mo et al. [
] used high-resolution X-ray probe that measures how collagen fibrils of echinoderm connective tissue stretch, slide, or reorient in real time, to show that the contribution to the changes in the MCT
stiffness is dominated by changes in the stiffness of the matrix between individual fibrils, rather than the properties of the fibrils themselves.
In spite of the recent progress on the study of the sliding action of collagen fibrils [
], how these findings could point to a nerve-control system for regulating the ECM macromolecules that are involved in the sliding action of collagen fibrils in the MCT [
] is still far from clear. The aim of this review is to discuss the findings, from the key early studies to the most recent studies of the basic mechanics of MCT (from a mechanical engineering
perspective), addressing the key attributes of the structure and function of the ECM of the sea urchin ligaments that are related to the fibril-fibril sliding action at the respective mechanical
states (“standard”, “stiff” and “compliant”)—the focus is on the respective components within the hierarchical architecture of the tissue. Detailed discussion of the system of nervous control, as
well as the biochemical composition, that regulates mutability is out of the scope of this review. Thus, the main ECM components of interest here are the collagen fibrils and the interfibrillar
matrix components. The discussion will draw findings from experimental studies conducted on sea urchin, the theory of fibre reinforced composites and from the analyses of (non-mutable) connective
tissues from other (vertebrate) animals to establish general conclusions concerning the mechanical response of the MCT at specific mechanical states, namely the stiff and complaint states. The
overall aim is to enable the development of a de novo understanding of the reinforcement processes in ECM-DT that may result in novel concepts for technological innovation, e.g., in the development
of new types of mechanically tunable biomaterials. In the sections that follows, we will address essential concepts concerning the collagenous scaffold design, in the context of ECM, from sea urchin
connective tissues. Thereafter we will discuss the biomechanics of collagen fibrils in sea urchin connective tissues in order to illuminate the basis of the structure-function relationship of the ECM
of sea urchin connective tissues. Finally, we will conclude the discussion of the sea urchin tissue with reference to a recent framework that has been proposed for addressing the goal of
understanding ECM mechanics.
2. Collagenous Scaffold Design
2.1. Connective Tissues with Properties of Mutability (MCTs)
One of the most intriguing properties of the sea urchin connective tissues, such as the ligamentous CA (
Figure 1
) [
], is that they can switch from the viscoelastic fluid state to the solid state, reversibly, on a timescale of the order of 1 s [
Figure 1
A illustrates the ligamentous CA and muscles within a spine joint of the sea urchin. Early studies have referred to the different states as “çatch” and “out of catch” [
]. The latest studies have classified these states into three, sometimes renamed as “standard” (normal), “compliant” and “stiff” [
]. The underlying mechanisms regulating these states are often not clearly spelled out. In this review, we present fresh arguments to explain how the stiff state is associated with the elastic stress
transfer mechanism (
Section 3.3
) while the compliant state is associated with the plastic stress transfer mechanism (
Section 3.4
). As they can change from one state to another in a short span of time, these tissues are regarded as “smart” or “intelligent” tissues [
]. These tissues are also commonly referred to as MCTs to reflect their unusual morphofunctional adaptations [
]. Physically, one finds that these MCTs are responsible for locomotion [
], attachment [
] that includes defining the posture of the animal [
], and even autotomy [
]. Interestingly, while autotomy is associated with the compliant state [
], the underlying mechanism regulating this is not clear. In this paper, we explore fresh arguments from a molecular perspective and from the mechanics of fibrillar failure to show how autotomy could
occur following the compliant state; this discussion is covered in
Section 3.5
For practical reasons, the sea urchin spine can point freely in any direction as permitted by the joint; the spine can also be immobilized to the skeletal test [
Figure 1
B illustrates two possible positions that the spine can adopt. The joint at the spine-test system comprises an outer frustum-like (i.e., ring-like) muscles and an inner frustum-like (i.e., ring-like)
CA connective tissue [
]. The former is responsible for spine movement by synchronized action involving contraction and relaxation; the latter is for controlling the spine orientation by resisting stretch [
]. The interplay between the actions of the muscle and CA enables the spine to carry out locomotion, as well as to respond to an external mechanical stimulus in two ways known as the convergence
response and the freeze response [
]. When the skeletal test surface is mechanically stimulated, the spines around the stimulated area are provoked to lean down (
Figure 1
B) to cover that area [
]; when the spine is mechanically stimulated, it becomes immobilized in an upright position [
]. For locomotion/attachment/posture maintenance, in order to move the spine, first, the CA has to become compliant to allow the ring of muscles to position the spine that enables it to brace against
marine surfaces such as the rocky wall of a cozy reef crevice. Second, the CA has to stiffen to lock the spine into position and then the muscles relax. By allowing for several spines to stiffen
collectively, this makes it difficult to dislodge the sea urchin without breaking the spines and the skeletal test. This behaviour of the spine has raised some interesting questions. For instance,
while the ligament can stiffen in any position of the spine (
Figure 1
B), does this imply that the entire tissue is straight? Additionally, on the side of the spine where the muscle is contracted, the ligament is compressed and therefore shortened (
Figure 1
B) but no appreciable kinks appear [
]. How does the MCT shorten without kinking? It appears that this has to do with the mechanics of fibril-fibril sliding and the dependence of this mechanism on the length of the collagen fibrils
whereby a large proportion of the fibrils could possess lengths that are considerably shorter than the tissue. Details of the basis underlying the fibrillar length-arguments are found in
Section 3.2
Section 3.3
Section 3.4
The MCT is essentially a complex system consisting of two main components, namely the collagen (in fibrillar forms) which is embedded and organized within an ECM containing nonfibrillar ECM
components, such as proteoglycan and glycoprotein [
]. In particular, there are several types of proteoglycans—of interest in this review are those that are believed to bind to the surface of collagen fibrils via core proteins, e.g., small
dermatan-sulfate proteoglycans [
]. To a large extent, the ultrastructure of the MCT [
] bears some resemblance to the connective tissues of humans [
]. The simplest explanation for the mutability of the MCT is that the collagen fibrils are able to slide relative to one other, with the help of the proteoglycan [
], to execute tissue length changes (“out of catch” phase) but are inhibited from sliding when the tissue is in “catch” [
]. It then follows that the mechanical states of the tissue are mediated by the interactions of collagen fibrils with the surrounding matrix [
] governed by elastic stress transfer at low loads [
] and plastic stress transfer at higher loads [
]. (These mechanisms will be discussed in subsequent sections.) The interactions are in turn under the control of a nervous system, regulated by soluble molecules that are secreted locally by
neurally controlled effector cells [
]. Stiparin, a glycoprotein of ECM, has been reported to result in the aggregation of isolated fibrils and has initially been identified as a tissue-stiffening factor [
]. A second glycoprotein, i.e., an inhibitor of stiparin, was shown to bind to stiparin, to inhibit stiparin’s ability to induce fibril aggregation [
]. More recently, tensilin, a third component, is believed to induce collagen-fibril aggregation in vitro, as well as to stiffen the tissue [
]. While a detailed molecular mechanism for the regulation of collagen-fibril associations in the sea cucumber dermis has yet to be developed, it also raises the question of whether any related
phenomena occur in vertebrate tissues.
The findings concerning the mutability of these tissues have inspired the development of “smart” materials with dynamic properties, as well as having the capacity to control the change in properties
in response to environmental demands. With regards to biomedical engineering applications, Trotter and co-workers [
] have proposed to design a new biomaterial made from collagen fibrils embedded in a synthetic matrix material (
Section 2.2
). The fibrils may be obtained from the sea cucumber dermis [
]. What is the applicability of these smart materials? It has been suggested that dynamic MCT-like materials may be useful for making scaffolds for tissue engineering where the regenerating tissue
requires a microenvironment that can dynamically change to match the requirements of the cells [
]. For instance, in regenerative applications of stem cells, such as tissues for postmyocardial infarction patients, the microenvironmental elasticity of the scaffold can direct the native
mesenchymal stem cells to specify lineage and commit to phenotypes [
]. Dynamic biomaterials may also find applications in the uterine cervical tissue of mammals—dynamic changes that involve remodelling the tissue to enable the tissue to be become very compliant so as
to facilitate the labor process although these changes happen on a longer timescale, i.e., long before the onset of labor [
]. Current MCT-like materials have been produced using shape-memory polymers; these materials attempt to mimic the MCTs’ ability to change mechanical properties “on command” with reversible
capability when subjected to a stimulus such as a change in temperature [
]. A new biomimetic research strategy has been proposed to further characterize the properties of MCTs so as to gain deeper insights—the aim is to develop innovative ECM biomaterials with dynamic
mechanical properties that finds applications in vitro as well as in vivo [
2.2. Structural and Mechanical Compatibility
The process of scaffold design for tissue engineering has attracted many studies [
]. From a mechanical engineering perspective, the key design stages that encompass the process of scaffold design address needs recognition, problem definition, synthesis of ideas, analysis and
optimization, evaluation of the prototype performance and manufacturability, and finally, bringing the product into the market. These key stages can be encapsulated in a design flow-chart (
Figure 2
) [
]. The overall direction of the flow in this chart is aimed at achieving a final product that is usable but one may also expect that some stages could progress iteratively, which is typical to any
design process.
In the ECM design process (
Figure 2
), the recognition of needs should address the desire for the scaffold to mimic the ECM of the native tissue. The problem definition stage then identifies the desired mechanical properties,
structural features and other biocompatibility issues. The synthesis stage combines the novel ideas, contrived to address the problem definitions, for making the scaffold. The analysis and
optimization stage implements the design of experiments; the results from these experiments are used to assess the performance of the scaffold. In the evaluation stage, based on the selected concept
and the information about the performance of the ECM materials, processing and costs, the scaffold is evaluated to find out how the proposed design fulfils the specification and if so, further
evaluation by clinical trials may be appropriate to confirm the functionality of the design before the product is released to the market.
The relationship between the tissue engineering triad and the design process is indicated in
Figure 2
. In essence, the tissue engineering triad could span from the problem definition stage to the analysis and optimization stage. Among the three components of the tissue engineering triad, for the
purpose of this discussion, we shall only be concerned with the biomaterials component. At the problem definition stage, considerations for the collagen biomaterial would address the respective
structural and mechanical properties of the target tissue, by identifying the key responses of the tissue under an external load and the range of values. Underlying the key responses are concepts
related to the biomechanics of collagen (
Section 3
) and non-collagenous components (
Section 4.6
). At the synthesis stage, this could involve the exploration of different fabrication methods, e.g., reconstituted collagen and ECM material from decellularised tissue, to address the suitability of
the respective methods for satisfying the constraints associated with the mechanical considerations. The analysis and optimization stage provides experimental investigations, based on a design of
experiment, into the performance of prototypes developed using the methods identified in the synthesis stage. The development of the optimized solution would require careful interpretation of the
findings derived from the experimental data. The interpretation process is expected to involve revisiting the premises underlying the arguments for the problems identified for the triad components in
the problem definition stage.
What are the desirable ECM components for a ECM-DT? As pointed out in
Section 1
, Trotter and co-worker have proposed an ECM scaffold with MCT-like properties comprising bundled parallel collagen fibrils that are organized by collagen fibril networks for the biomaterials
component at the problem definition stage [
]. The desired specifications are listed in
Figure 2
. To address both the problem definition stage and the synthesis stage of the design process, Trotter and co-worker recognized that the fibrils have to aggregate to form bundles; in order to enable
aggregation these fibrils would have to be weakly interacting with one another [
]. In the echinoderms, such as the sea cucumber, stiparin glycoprotein is believed to be responsible for the aggregation of the fibrils in the MCT [
]. For a ECM-DT, the strategy is to apply a stiparin-inhibitor that can bind stiparin and inhibits the capacity of stiparin to bind to the fibril [
]. So far the inhibitor component has yet to be conclusively identified. For a synthetic analogue, the challenge is to develop a method that can chemically control the stress transfer from the
interfibrillar matrix to the fibrils. For instance, photo-sensitive or electro-sensitive reagents could be introduced to bind covalently to fibril surfaces and form cross-links between adjacent
fibrils, in order to cause the ECM to stiffen, as well as to reverse the effects, when required [
]. In any case, the matrix surrounding the fibrils should contain macromolecules that can form cross-links between the fibrils and macromolecules that can have a reversible effect. To implement
cross-links for a synthetic ECM scaffold with MCT-like properties, catechol related polymers such as polyacrylamide complexed with phenylboronic acid, have been suggested [
]. It is often argued that to form cross-links, the interfibrillar matrix could contain macromolecules whereby a part of the a molecule is bound onto the surface of the fibril [
]. Hynes and Naba [
] have suggested that parts of glycosaminoglycans and proteoglycans could be bound onto the fibril surface in the MCT, drawing on parallel arguments developed for vertebrate tissues. In the case of
the synthetic ECM scaffold with MCT-like properties, the catechol group could be a possible candidate for binding onto both the modified collagen fibrils as well as the synthetic fibrils [
]. In connective tissues, controlling the number of cross-link associations between collagen fibrils is important for regulating interfibrillar shear stress [
], underlying the mechanics of the tissue [
]. In the sea-urchin, it is found that a nervous input could cause the resident cells to release stiffener macromolecules, which diffuse into the interfibrillar space, and bind to the fibrils to
creates strong and numerous interactions between them [
]. For engineering a synthetic MCT-like scaffold—if photo-sensitive or electro-sensitive reagents are to be used—electrical or optical signals may be applied as inputs to change the redox potential
of the matrix to affect the binding between the catechol and boronic acid moieties. Finally, a strategy is required to bind the fibres into bundles by a matrix which can result in a scaffold that may
then be modelled by elastomeric properties. In the MCT, this matrix is observed to be predominantly fibrillin microfibrils [
]. However, for the synthetic MCT-like scaffold, an elastomeric hydrogel strategy may be needed to act as a substitute for the fibrillin microfibrils [
There are more than twenty different classes of techniques for fabricating tissue-engineered scaffolds [
]. Some of these techniques are known as solution casting, gel-pressing, microspheres, macro-porous beads, particle-aggregated scaffolds, freeze-drying, acellularization, electrospinning, and wet
spinning; some of these may be used in combination for producing an ECM-like scaffold. Thus, one may attempt to extract collagen by mixing neonatal human dermal fibroblasts with acid-soluble type I
collagen. The extracted collagen is then cast into a desired shape in a (Teflon) mold, followed by culturing for 4 days to allow compaction—which also stiffens and aligns the gel. This method can
result in a collagen gel containing a heterogeneous population of fiber diameters (42 to 255 nm, mean diameter ~69 nm) [
]. Alternatively, electrospinning [
] can be employed to produce fibrous scaffolds comprising type I collagen fibers—with diameter comparable to native collagen fibrils—which also exhibits the 67 nm banding pattern that is
characteristic of native collagen [
]. Additionally, the pressurized gyration [
], a method that involves rotating a perforated pot containing a reservior of polymer solution at high speed, as well as the pressurized melt gyration [
], a powerful method that can lead to high production rate, ease of production, and highly controlled fiber morphology, may also be considered for producing collagen fibrous scaffolds.
How then are ECM-DT scaffolds produced? As pointed out in
Section 1
, ECM scaffolds can be obtained from allogeneic or xenogeneic (segmented or whole tissues) sources by treating the isolated tissue with decellularizing solution and incubating for cell lysis [
]. The processed scaffold may be regarded as an “entry-level” ECM-DT, i.e., a ECM-DT that is possibly devoid of some key components such as stiparin-inhibitor and catechol related polymers that are
essential for regulating the fibrillar structural organisation. Based on the methods adopted by Yang et al. [
] and Kayed et al. [
] for treating a given thin tissue, or the method adopted by Di Benedetto et al. [
] for treating a given minced tissue, the specimen is washed for about a day in a hypertonic buffer (PBS with Triton X100 (TX) and EDTA solution, or Tris with EDTA). Rinsing may take place in a low
temperature environment; the hypertonic solution is also continuously agitation. Thereafter the sample is rinsed in a decellularising buffer (PBS or Tris with SDS). For a minced tissue specimen, this
solution is then replaced by a disaggregating solution (one containing e.g., Tris and EDTA); a collagen suspension is obtained which is then filtered and purified using a EDTA solution, followed by
distilled water. For long-term storage of the suspension, one could dry them in silicone molds. In the case of the thin tissue, after treating in the decellularising buffer, the specimen is stored
away in PBS until it needed.
3. Collagen Fibril Biomechanics
3.1. Stress-Strain Relationship of MCT
The response of the MCT to an external load has been a subject of many studies. In this section, we examine three different tissues from the sea urchin, highlighting the mechanical properties of
these tissues in the context of their stress-strain behaviour. These tissues are the CA, compass depressor and tube feet. To begin, Hidaka and co-workers [
] have evaluated the mechanical properties of the CA tissue of the sea urchin (
Anthocidaris crassispina
) to varying physico-chemical factors (pH, neurotransmitter) in order to gain insights into the underlying mechanisms regulating the ability of the CA to execute mutability. In one study, at a
displacement rate of 7 μm/s where the viscous resistance is expected to predominate in the CA, it was found that the pH and neurotransmitter acetylcholine (ACh), has a significant effect on the
viscous resistance.
Figure 3
A,B show the stress-strain curves of the CA obtained at varying ACh and pH values, respectively. In particular, high pH value results in high stress uptake in the CA [
]. The duration of ACh-treatment is shown to affect the stress uptake in the CA—the longer the treatment the lower the stress uptake [
]. In all cases, the stress-strain curve shows a non-linear increase in stress with strain from initial loading (the toe region), followed by a somewhat linear increase in stress with strain, a point
of yielding and thereafter the point of maximum stress; beyond this point, failure in the tissue results in a somewhat gradual decrease in stress with increasing strain. The highest stiffness
observed in these results is estimated at 200 MPa while the highest maximum stress is at around 30 MPa. It is important to note that the profile of the stress-strain curve resembles those from
mammalian connective tissues such as mouse tail tendons [
] and sheep anterior cruciate ligaments [
], where the tissues have also been subjected to loading along their axes at displacement rates of order of 10 μm/s.
Investigation on the mutability of MCT by examining the effects of ions in the ECM have been reported.
Figure 3
C shows stress-strain curves of the tube feet tissue of the sea urchin (
Paracentrotus lividus
) treated in the following respective bathing solutions, namely artificial seawater (ASW, regarded as a standard solution), ethylene-
-(oxyethylenenitrilo)-tetraacetic acid (EGTA), and TX with EGTA, to identify the properties related to the mutability of the tube feet tissue [
]. Of note, EGTA acts as a calcium chelator to remove the endogenous calcium from the tissues; TX is a non-ionic detergent that can disrupt cells in the tube feet. When the tissues were tested to
rupture at 25 mm/min, all tissues exhibit J-shaped profiles. In particular, the J-shaped profile begins with a long low-stress toe region, followed by a rapidly increasing stiffness with increase in
strain up to the point of maximum stress and, thereafter, a rapid decrease in stress. (Of note, the displacement rate of 25 mm/min corresponds to 417 μm/s, which is 400 times higher than those used
in the investigation for
Figure 3
A,B) When the tissue is treated to calcium removal solution (ASW + EGTA), the tube feet tissue results in a dramatic decrease in strength, stiffness and toughness with respect to the control (i.e.,
in standard solution only). On the other hand, treatment with ASW + EGTA + TX solutions reveals a dramatic increase in strength, stiffness and toughness with respect to the control. The highest
stiffness observed in all these results is estimated at 400 MPa; the highest maximum stress occurs at around 120 MPa. These findings suggest that the mechanical properties of the tube feet tissue are
affected by the calcium ions and the juxtaligamental-like cells in the tissue. For instance, the tissue becomes compliant in the absence of calcium. The calcium-removal test suggests that ions in the
ECM of the MCT play an important role in regulating the tissue mechanical properties and warrants further discussion; this is addressed in
Section 4.6
with regards to the effects of the composition of the interfibrillar matrix on fibril-fibril interactions.
To examine the viscoelastic behaviour of the MCT,
Figure 3
D shows a sketch of the graph of the displacement versus time to illustrate the creep response of the compass depressor tissue of the sea urchin (
Paracentrotus lividus
) [
]. Evaluating the second phase of the creep response curve, the mean coefficient of viscosity is found to be 561 ± 365 MPa·s [
]. The massively large standard deviation reflects the large variability in the values coefficient of viscosity (104 to 1477 MPa·s) derived from this study for a sample size of 31 [
Figure 3
E,F show the sketches of the graphs of stress versus strain of the sea urchin compass depressor tissue [
]. The stress-strain curve (
Figure 3
F) has been derived from the incremental stress-strain approach (
Figure 3
E). To this end, each point on the graph of
Figure 3
F represents the peak stress value of the incremental curve (
Figure 3
E); all curves results in the toe, linear and yield regions similar to those of
Figure 3
A–C. Assuming the length of the compass depressor is of order of 10 mm based on the images shown in the report [
], these curves correspond to displacement rates ranging 0.07 to 0.6 mm/s. The highest stiffness estimated from these results is 20 MPa; the highest maximum stress occurs at around 15 MPa. The
profiles of the respective curves are also comparable to those of mammalian tissues, such as skin, that have been investigated for viscoelastic effects [
Figure 3
G shows a sketch of the graph of the stress versus strain of the dermis from the sea cucumber derived at the respective state, namely stiff, standard (i.e., intermediate/resting) and compliant states
]. These sketches are shown here for the purpose of comparison with the stress-strain curves from the three different tissues of the sea urchin. As expected, the stress versus strain curve shows a
very steep gradient for the tissue in a stiff state as compared to the tissue in a compliant state; the peak stress is also higher in the former than in the latter tissue. Of note, Trotter and
co-worker have referred to the standard and compliant states as the intermediate and plastic states, respectively [
Table 1
tabulates the findings of the mechanical properties of the tissue of the sea urchin that have been discussed in the previous paragraph. Thus, among the three different tissues of the sea urchin, the
compass depressor tissue exhibits the smallest strength (maximum stress) and stiffness while the tube feet tissue exhibits the largest strength and stiffness. The largest and smallest extensibility
(maximum strain) correspond to the compass depressor and CA tissues, respectively. Of course, the varying magnitudes of the respective mechanical properties of different tissue highlighted here may
be attributed to the different organisation of the network (orientation and packing) of the collagen fibrils as well as the structural properties (e.g., aspect ratios) of the fibrils. Nevertheless,
these values are of similar order of magnitudes to those of mammalian tissues [
3.2. Shear Action Underpins the Mechanism of Collagen Fibril Reinforcement of MCT
In this section, we discuss how shear of the collagen fibrils enables the fibrils to take up stress and contribute to resisting the external load acting on the MCT that tends to pull it apart.
The fundamentals of fibre-reinforced composites have been well-established for engineering composite materials, leading to analytical solutions where linear elastic properties of the materials for
the fibre and matrix are concerned [
]. However, attempts to apply these concepts to collagen fibrils in biological tissues are challenged by uncertainties surrounding the nature of the ECM components, particularly the degree of
non-linear properties of these components [
]. Here we shall adapt the key features of the basic models [
] that have been established for explaining how collagen fibrils provide reinforcement to ECM of connective tissues to describe the ECM of MCT.
As shown in
Figure 4
A, consider an array of collagen fibrils, parallel to one another, embedded in the interfibrillar matrix of the MCT. In these models, the fibrils will always be considered to be arranged in the
direction of the tissue axis. The applied force on the tissue will always be considered to be acting along the direction of the tissue axis. At closer view, between any two fibrils are proteoglycans/
glycoproteins associated with the fibrils (
Figure 4
B); these protein macromolecules are assumed to be involved in regulating the transfer of stress from the matrix to the fibril as well as between the fibrils. It is also assumed that (1) there are
numerous such proteoglycans/glycoproteins; (2) bonds (e.g., van Der Waals, hydrogen) exist between proteoglycans/glycoproteins on adjacent fibrils; (3) these proteoglycans/glycoproteins are
distributed uniformly over the fibril/matrix interface so that continuum mechanics can be used to analyze the problem [
]. For simplicity, most models are based on solving the stress in a single collagen fibril embedded in the matrix as illustrated in
Figure 4
C. Suppose the fibril is parallel to the axis of the tissue; additionally, an external tensile load acts along the axis of the tissue. At initial loading, as the interfibrillar matrix deforms in
shear, this generates shear stresses on the surface of the collagen fibril [
]. Interactions of ECM components in the interfibrillar matrix (i.e., proteoglycans/glycoproteins) with those associated with the collagen fibril then cause the fibril to deform axially [
From several studies in the early 1990s, it was clear that different mechanisms of stress transfer in the ECM result in different stress uptake in the fibre [
]. According to Aspden [
], the manner in which an elastic fibre takes up stress depends on the interfacial shear stress (τ) distribution. This results in two different approaches to solving the stress in the fibre. The
first approach, known as the shear-lag approach [
], is associated with the τ distribution illustrated in
Figure 5
A; the second approach, known as the shear-sliding approach [
], is associated with the the τ distribution illustrated in
Figure 5
In the shear-lag approach, the τ is a minimum at the fibre centre (
= 0;
Figure 5
A) but increases non-linearly from the centre to a maximum value at the respective ends (
= 1 or −1;
Figure 5
A) [
]. In the shear-sliding model, τ is constant throughout the fibre surface (
Figure 5
B) [
]. These descriptions are not based on a detailed analysis of the viscoelastic behaviour of the tissue, but the simplicity of these concepts has facilitated general conclusions to be drawn concerning
the reinforcement of the MCT by collagen fibrils. In particular, both shear-driven arguments have been applied to evaluate the stress uptake in collagen fibrils [
]. More recently, Szczesny and co-workers have sougth to further establish the role of the interfibrillar matrix in ECM that can yield deeper insights into the elastic and plastic stress transfer
mechanisms [
]. In particular, Szczesny and co-workers carried out a series of experiments to study the viscoelastic behaviour of tendon fascicles by evaluating the contribution of the ratio of the strains of a
fibril to the whole tissue during stretching using an incremental relaxation approach [
]. In the following sections, the basic concepts underpinning the elastic and plastic mechanisms of stress transfer are elaborated.
3.3. Interfibrillar Shear Response by Elastic Stress Transfer Directs the Stiffening of the MCT
The purpose of this section is to present the key arguments to highlight how elastic stress transfer mechanism directs the stiffening of the MCT.
Based on the general conclusions drawn from the study of soft connective tissue reinforced by collagen fibrils, when a load is applied to the MCT in the passsive mode where mutability is absence, the
tissue is likely to encounter elastic stress transfer process during initial loading when the load is low [
]. In other words, the interfibrillar matrix and collagen fibril respond elastically to the external load; they are able to return to their original structural state when the load is removed [
]. The difference in the elastic moduli of the two components plays an important role in influencing the differential axial elastic displacements in the fibril and in the interfibrillar matrix [
]. Consequently, shear strains are produced on all planes parallel to the axis of the fibrils in the direction of this axis [
]. The above description of the state of the individual ECM components during elastic stress transfer may then be applied to the situation where mutability is considered, when the MCT exhibits the
stiff state following a transition from the compliant state. Thus, it is hypothesized that elastic stress transfer directs the MCT to take up load to maintain the stiff state.
How does collagen provide reinforcement to the MCT in the stiffened state? At the molecular level, consider two adjacent collagen molecules located within a collagen fibril (
Figure 6
A). Suppose the MCT as a whole is subject to a strain of ε in the direction of the fibril. Two modes of deformation have been predicted using the bi-molecular mechanics approach; the first mode is
known as the homogeneous shear while the second is known as the nucleation of slip pulses. The second mode will be elaborated in
Section 3.5
when we consider how the fibril could fracture. The homogeneous shear mode explains how the collagen molecules undergo sliding motion when a tensile load acts on the collagen fibril (
Figure 6
A). Let τ
represents the shear resistance between the two molecules and
the contact length between two adjacent molecules. Let
be the axial force generated within the molecule, which parameterizes the resistance to the shear action. To order of magnitude, we can identify the
with the product of τ
, i.e.,
or otherwise,
is the length of a collagen molecule and η =
. The stress, σ
, associated with
where α
is the molecular cross-sectional area. The homogeneous shear mode assumes that the shear deformation is uniformly distributed throughout the interface of any two collagen molecules; this is expected
to occur during initial loading. By virtue of the axial staggering of the collagen molecules,
increases linearly with
. Upon evaluating a multiscale model numerically—where the lower length scale level addresses the contribution of the interactions of atoms in the respective molecules to molecular deformation, and
the next higher length scale level addresses the contribution of the sliding action of the molecules to the fibril deformation—Buehler has shown that the stress developed in the fibril follows a
linear response to increasing strain up to a strain of 0.05 [
]. Interestingly, there is no appreciable toe region at initial loading [
At the fibrillar level, how collagen provides reinforcement to the MCT in the stiffened state may be explained by the stress in the fibril. The first step here is to consider the MCT as a whole,
subjected to a strain of ε in the direction of the axis of the tissue. Let
be the half-length of a collagen fibril and
) be the normalized axial coordinate (
) which parameterizes the axial distance of the fibril starting from the fibril centre (
= 0) and ending at the fibril end, i.e.,
. It follows that the rate of change of the axial stress (σ
) at any point along the fibril is proportional to the difference between the axial displacement of the fibril (
) at that point within the fibril and the corresponding axial displacement of the interfibrillar matrix at the same point if the fibril were not presence, (
) [
is constant. For simplicity, as well as for the purpose of this discussion, we shall present the results for uniform cylindrical fibrils. Let
be the cross-sectional area of the (i.e., uniform cylindrical) fibril,
the fibril radius, and
the average radius of the interfibrillar matrix surrounding the fibril. Solving Equation (4) for uniform cylindrical fibrils, one finds that σ
and interfacial shear (τ) stress generated at the collagen fibrillar surface are given by [
σ[z](Z) = E[CF]ε[1 + cosh(β[Cox]{Z})/cosh(β[Cox])],
τ(Z) = E[CF]ε√(G[m]/[2E[CF]ln(r[m]/r[0])])sinh(β[Cox][Z])/cosh(β[Cox]),
β[Cox] = √(G[m]2πL[CF]^2/[A[CF]E[CF]ln(r[m]/r[0])]),
which is written in terms of the collagen tensile stiffness,
, the shear modulus of the interfibrillar matrix,
, and the average strain in the fibril, ε.
Figure 7
B illustrates a typical axial stress distribution in the fibril under tension during elastic stress transfer. This profile applies to a fibril which possesses a uniform cylindrical shape, which is a
common assumption for many tissues where the ends could not be observed. Thus, the shear-lag model predicts that the stress in the collagen fibril peaks at the fibril centre and decreases
non-linearly to zero at the fibril end. In particular, the decrease is (1) gradual for a large portion of the fibril, around the fibril centre, (2) rapid towards the fibril end. For fibrils which
possess tapering ends, the stress distribution profile has been predicted to differ appreciably from those of uniform cylinder [
]. This will be discussed further in
Section 4.2
Thus, Equations (3), (5) and (6) provide the basis for fibril reinforcement of MCT during elastic stress transfer. In the stiff state, sliding among the fibrils is greatly reduced [
]—the stiffness of the MCT is of the order of hundreds of MPa. In order for this to be maintained, the magnitude of τ must be high. According to Equation (6), a high shear modulus (
) would be needed to enforce this.
3.4. Interfibrillar Shear During Plastic Stress Transfer Directs the Compliance of MCT
The purpose of this section is to present the key arguments to highlight how plastic stress transfer mechanism directs the compliance of the MCT.
According to the general conclusions derived from the mechanics of soft connective tissue reinforced by collagen fibrils, when an external load acts on the MCT in the passive mode (where mutability
is subdued), plastic stress transfer predominates at a sufficiently high load level [
]. During plastic stress transfer, the interfibrillar matrix around the fibril becomes plastic, bonds at the fibril-matrix interface are disrupted and the interfibrillar matrix “shear-slides” over
the surface of the fibril [
]. Where mutability is involved in the mechanical response of the MCT,
Section 3.3
highlights the arguments for how the case of a stiffened MCT is directed by elastic stress transfer mechanism. If the MCT is in the stiff state, and the situation dictates a transition to a compliant
state, the description of the state of the individual ECM components during plastic stress transfer may be applicable to the MCT in a compliant state. Thus, it is hypothesized that plastic stress
transfer directs the MCT to maintain the mechanical integrity when it is in a compliant state.
How then does collagen provide reinforcement to the MCT in the compliant state? At the fibrillar level, the stress uptake in the fibril undergoing plastic stress transfer may be estimated as follows.
For simplicity, the rate of change of the axial force associated with σ
along the fibril is proportional to product of the fibril radius (
and the fibril-matrix interfacial shear stress (τ) [
d[σ[z](Z)r^2]/dZ = −2τr[0]L[CF].
It follows that the stress in the fibril (for given shape of the fibril), may be determined by solving the Equation (8). The fibril aspect ratio is defined as
Consider the case of a fibril with uniform cylindrical shape (i.e., constant
). Equation (8) is reduced to
Solving Equation (10) using the boundary condition that states that at
= 1, σ
= 0 (i.e., a stress-free fibre end), the axial stress is given by
Figure 7
C illustrates the axial stress distribution in the fibril under tension during plastic stress transfer. (The graphs also show the variation in the axial stress distribution with respect to four
different fibril shapes but for the purpose of the discussion here we shall consider the case of the fibril with a uniform cylindrical shape.) Thus, during plastic stress transfer, the stress in the
collagen fibril peaks at the fibril centre and decreases linearly to zero at the fibril end. The peak stress at the fibril centre could increase with increasing load on the tissue [
]. When the peak stress reaches the yield strength of the collagen fibril, the fibril could yield. For fibrils which possess tapering ends, the stress distribution profile differs appreciably from
those of the uniform cylinder [
]. This will be discussed further in
Section 4.2
In the compliant state, the stiffness is reduced by about an order of magnitude, i.e., of the order of around tenths of MPa. Of note, the sliding mechanism prevails in the connective tissues of human
and other animals [
]—sliding of collagen fibre bundles, i.e., fascicles, has been observed [
] during tissue deformation. For a MCT to change from a compliant to a stiffened state and vice versa, other mechanisms would be involved in regulating this transition process. As pointed out in
Section 2.2
, one mechanism could involve nerve-control to cause the stiffening/de-stiffening gycoproteins to be released into the ECM [
]. It is postulated that the de-stiffening effects result in mode β, which is characterized by the initiation of interfacial debonding [
]. Debonding starts at the fibril end and propagates along the interface. In addition, as the deforming interfibrillar matrix slides over the fibril surface, this enables frictional stress transfer
to take place. Of note, nerve-controlled transition from a stiffened state to a compliant state may be realized more effectively by ensuring that the interfibillar matrix be completely plastic. Thus,
mode α, which is characterized by a plastically deforming interfibrillar matrix [
], could also occur. Consequently, this brings the tissue into plastic stress transfer.
It may be argued that in the normal circumstances, mode α, mode β and plastic stress transfer would be required for enabling the tissue to perform the required physiological activity. However, the
plastic stress transfer mechanism could result in failure in the tissue. For instance, when predators seek to pry the stiffened sea urchin off from nooks, the tissue could fail as it undergoes
loading from elastic stress transfer to plastic stress transfer (
Section 3.4
), when the external load on the tissue increases to a critical level. In the interim, the tissue could undergo additional intermediate mode of failure corresponding to mode γ [
]. Here, mode γ refers to the initiation of rupture at the debonded fibril end and the propagation of the crack into the interfibrillar matrix rather than along the fibril-matrix interface [
]. These intermediate modes of failure are analogous to those found in engineering composites [
3.5. Nucleation of Slip Pulse Predicts Collagen Fracture and Tissue Autotomy
The transition from a stiffened state to a compliant state and vice versa is a physiological process that is not expected to induce failure in the MCT. A change in the states of the respective
interfibrillar matrix and the fibril-matrix interface is to be expected in the process of undergoing a transition from the stiff to the compliant state (and vice versa). As pointed out in
Section 2.2
, “stiffener” proteins, namely tensilin [
] and stiparin [
] are believed to play a role in regulating the bonding processes. On the other hand, the regulation of the debonding processes is attributed to inhibitors of these proteins [
]. The respective proteins secreted into the ECM are associated with stiffener/de-stiffener cells present in the MCT [
], controlled possibly by a nervous input [
]. In particular, to assume a compliant state, tensilin-inhibitors and stiparin-inhibitors would have to act to cause the interfibrillar matrix to turn plastic (mode α) and the bonds at the
fibril-matrix interface to be disrupted (mode β) (
Section 3.4
). Whether the tensilin-inhibitors and stiparin-inhibitors could also initiate the transitional failure mode γ is not clear but it was suggested in
Section 3.4
that this could result in rupture across the ECM.
However, if the MCT (in either stiffened or compliant state) is acted upon by an increasing external load (e.g., this may be the result of a predator attempting to pry the sea urchin off from a
nook), eventually the tissue fractures (
Figure 8
) when the stress generated in the tissue, in response to resisting the load that tends to pull it apart, exceeds its fracture strength. Additionally, the spectacular body softening observed in
echinoderms following autotomy [
] suggests that at the tissue level, the MCT has to break apart following plastic stress transfer. What mechanisms are involved in regulating the fracture of the MCT?
According to the general conclusions drawn from the mechanics of soft connective tissue reinforced by collagen fibrils, in the run-up to MCT fracture, several modes of failures may happen. Namely,
fibrils around the matrix ruptured site may experience fibril pullout or fibril rupture (
Figure 8
) [
]. If the fibrils fracture (
Section 4.3
), the shorter segments that result may continue to take up stress; if the length of these segments are sufficiently long, fracture could still occur when the fracture stress is reached [
]. Eventually the fragmentation process terminates because the subsequent fragments generated would not be long enough to take up stress to the level of its fracture stress; the stress transferred to
the fibril fragment is insufficient to cause further fragmentation [
With regards to fibril fracture, the nucleation of slip pulses at the molecular level plays an important role in the dissociation between collagen molecules [
]. The process of the nucleation of slip pulses explains how the rupture of intermolecular bonds, i.e., cross-links in between two collagen molecules (
Figure 6
A), result in the propagation of slip pulses. For simplicity these cross-links are assumed to be regularly spaced apart [
]. According to Griffith’s fracture energy argument, at the onset of fracture, the criterion for nucleation of slip pulses is dictated by the stress generated by the collagen molecule, σ
, and is of order of the applied tensile stress, σ
, to cause the MCT to rupture. Let
be the cross-sectional area of the collagen molecule. Thus, σ
is expressed as
is the Young modulus of an individual collagen molecule and γ
parameterizes the energy required to nucleate a slip pulse [
]. When σ
< σ
, the deformation of the collagen molecules is regulated by homogeneous shear (the homogeneous shear theory) between the molecules (
Section 3.3
). When σ
> σ
, nucleation of slip pulses can occur (i.e., the slip pulse theory). Thereafter, a critical molecular length, i.e.,
χ[S] = [√(2E[TC]γ[TC])]A[TC]/{ητ[TC]},
may be used to determine which of the two cases predominates. Simply, homogeneous (intermolecular) shear predominates if
≤ χ
; slip pulses predominates if
> χ
. If the tensile force,
, in each collagen molecule (Equation (1)) reaches the breaking force of the molecule,
, before homogeneous shear could occur or even before slip pulses are nucleated, further occurrence of failure is governed by a second critical molecular length scale,
which determines if the transition from molecular shear-sliding to rupture of collagen molecule can occur. The rupture of collagen molecule predominates if
> χ
; homogeneous (intermolecular) shear predominates if
≤ χ
. By combining these length scale arguments, it is further proposed that the interplay of the critical length-scales, expressed in the form of a ratio χ
, regulate the deformation mechanisms. It follows that slip pulse nucleation predominates at large molecular lengths only when χ
< 1; rupture of the collagen molecule predominates when χ
> 1.
Additionally, the mechanical (as well as chemical) stability of the collagen fibrils may be attributed to the extensive covalent crosslinks [
], such as those illustrated in
Figure 6
A. In general the collagen in the MCT of echinoderm contain higher levels of reducible crosslinks than those found in mammalian collagens [
]. However, the levels of the stable hydroxypyridinium crosslinks in the collagen of the MCT are similar to those seen in mammalian fibrous tissues [
At the length scale of collagen fibril, predictions from in silico models of Buehler [
] and others [
] show that beyond a strain of 0.05 (
Section 3.3
), the stress in the fibril responds non-linearly with increasing strain, reaching a peak stress eventually, and decreasing to zero somewhat gradually as the fibril fractures. Interestingly, the
Buehler model [
] predicts that the profile of the stress-strain curves of collagen fibrils is similar to those depicted in
Figure 3
A,B,F. Altogether these in silico studies suggest that the collagen fibril has a fracture strength of the order of 10
Pa and stiffness of the collagen fibril ranges 6 GPa [
] to 40 GPa [
4. Structure-Function Relationship
4.1. Vertebrates and Invertebrates with Spindle-Like Collagen Fibrils
As observed in transverse electron micrographs, the cross sections of the MCT, at any given point along the tissue, reveal collagen fibrils with near circular cross sections [
]. This lends support to the assumption that these fibrils have uniform cylindrical shape (
Figure 4
C and
Figure 7
A). In most instances, since most fibrils appear very long in scanning electron micrographs, the assumption is that the fibrils are continuous [
]; in other words, all fibrils possess lengths comparable to that of the tissue length. However, whole fibrils can be extracted by gentle mechanical means from the sea urchin ligaments [
] and while these isolated fibrils reveal a high degree of slenderness, they are shorter than the overall length of the tissue. Thus, this suggest that fibrils in the MCT can be short, rather than
continuous. For further details concerning the theory of fibre reinforced composites for discontinuous fibres versus continuous fibres the reader is recommended to a recent book authored by Goh [
More intriguing is the shape of the fibrils. It has been pointed out that the isolated fibrils reveal naturally tapered ends [
]. This observation could be further confirmed by serial thin sections of fibrillar cross sections in intact tissues, showing how the fibrils gradually diminish in size and disappear in the ECM [
]. To further ensure that the tapered ends seen in the micrographs was not a result of imaging artifacts, Trotter and Koob contrasted the ends of the intact fibrils to the broken ends of ruptured
fibrils, which shows a frayed appearance [
]. Unfortunately, it is not only difficult to track the individual fibrils serially—very few studies have been carried out to investigate this further—there is also the limitation of cross-sectional
studies for determining the fibril taper.
Figure 9
shows schematics of a cross section of a MCT to illustrate the latter point. If we start on the basis of fibrils with uniform cylindrical shape, and if all fibrils possess the same diameter, this can
be explained by
Figure 9
A (right panel). However, longitudinal micrographs also reveal fibrils with varying thicknesses, and this may be illustrated by
Figure 9
B (right panel). However, if some fibrils have tapered ends (
Figure 9
B, left panel), this approach would conveniently mask the information present in the micrograph of
Figure 9
B (right panel).
With regards to adult vertebrate, studies have also revealed that taper exists at both ends in the fibrils in postfoetal vertebrates such as the ligaments of rats [
]. Of note, gentamicin solution, which is believed to be able to weaken the interfibrillar bonds, was used to soak the tissue so that the fibrils may be isolated from the adult vertebrate [
]. The presence of naturally tapered terminations in intact tendons from chicken embryo has also been reported using the technique of serial thin sections analysis [
]. This observation is confirmed when fibrils are isolated from chick embryonic tendons by mild mechanical means [
]. Thus the evidence of the presence of tapered fibrils, which possess lengths shorter than the overall length of the connective tissue, has contradicted the standard model of a tissue containing
continuous uniform cylindrical fibrils.
Collagen fibrils with tapered ends have also been observed in reconstituted collagen ECM—Holmes and co-workers showed that collagen fibrils generated in vitro (at 37 °C) by enzymic removal of
C-terminal (i.e., the carboxyl group end) propeptides from type I pC-collagen displayed both finely tapered end and coarse end [
Figure 10
A–C illustrate possible tapered profiles for a collagen fibril. Let
be the collagen mass per unit length and
be the density of collagen;
is defined as the ratio of
to 5
, where
is the mass of a collagen molecule (=290 kDa),
the number of molecules intersecting a fibril cross-section (through an overlap region) and
the so-called
period of a collagen molecule (
Figure 4
A and
Figure 6
C). A simple analytical argument, based on Holmes and co-workers [
], results in
for a fibril that possesses a paraboloidal profile (
Figure 10
B). By a similar argument the relationships between
corresponding to uniform cylindrical, conical (
Figure 10
A) and ellipsoidal (
Figure 10
C) shapes are given by the respective equations,
Figure 10
D shows the graph of normalized mass per unit length versus fractional axial distance describing the plots of Equations (15)–(18). Thus the analytical model reveals that the
relationship follows a linear relationship (
Figure 10
D) in a fibril that has a paraboloidal shape (
Figure 10
B). On the other hand, the
plots (
Figure 10
D) for a fibril with conical ends (
Figure 10
A) and ellipsoidal shape (
Figure 10
C) show non-linear decreasing
with increase in
. In particular, the conical shape yields a concave profile while the ellipsoidal shape yields a convex profile (
Figure 10
D). Finally, we find that the uniform cylindrical shape yields an even distribution of
independent of distance along the fibril.
To quantify the shape of these tapered fibrils, the different
relationships were fitted to distribution of mass as a function of axial position along the fibril derived from scanning transmission electron micrographs of reconstituted collagen (as well as
embryonic tissues). It is observed that the
relationship that best fit the experimental data corresponds to the paraboloidal shape. Thus the mass per unit length,
, along all fibrillar ends increases almost linearly with increase in axial distance,
, from the fibril end [
]. One important implication of these findings is related to the accretion rate, which refers to the rate of collagen being added per unit area of the fibril surface. On the basis of these findings,
it is concluded that accretion rate cannot be the same at any given site of the collagen fibril [
]. In particular, accretion rate is likely to decrease as the fibril diameter increases [
Interestingly, so far only isolated fibrils from the MCTs of some sea urchins and embryonic tissues of vertebrates, as well as fibrils from reconstituted collagen, have been shown to possess
paraboloidal ends. In most tissues the length of the fibrils is too long for even the most meticulous researcher to be able to establish the shape of the fibril unambiguously—very long fibrils are
often regarded as uniform cylindrical in shape. To this end, several issues of structural interest remain unclear. First, since surface nucleation and accretion have been observed in early (tapered)
fibrils, it remains to be seen if these fibrils can grow into uniform cylindrical fibrils. Second, at any given age point, what is the proportion of fibrils which are tapered and uniform cylindrical
in shapes? Third, in addition to the uniform cylindrical and paraboloidal shapes, do fibrils possess other tapered shapes, e.g., conical and ellipsoidal? To this end, whether the fibrils possess the
other tapered shapes is not clear but the m[l]-Z relationships derived from the analytical model predict that fibrils with tapered ends would follow a non-uniform m[l] distribution.
Figure 6
B shows a representation of the axial packing arrangement of collagen molecules where the orientation of the respective molecules is indicated by the N-C termini. Of note, the orientation of the
collagen molecule is an important consideration for understanding the overall polarity of the fibril as well as the overall shape of the fibril. At a more fundamental level, it turns out that the
MCTs of echinoderms have exclusively
-bipolar collagen fibrils [
]—the collagen molecules change orientation at one location (5–10
periods) along the fibril [
]. These fibrils grow (exclusively) by coordinated accretion of collagen molecules without any need for fusion of fibrils [
]. The invertebrate (embryonic) tissues may have either
-bipolar [
] or a mix of
-bipolar fibrils and unipolar fibrils [
]. Various end-to-end fusion events can occur which then result in a fibril with an N-C polarity or C-C polarity [
]. This is an important aspect of MCT versus vertebrate collagenous tissues and is almost always overlooked.
4.2. Taper in Fibrils Facilitates Even Distribution of Stress throughout the Fibril
Most information on fibre reinforced composites have been derived from studies of uniform cylindrical fibres reinforcing the composite [
]. As pointed out in the previous section, since fibrils in the MCT need not be uniform cylindrical in shape how does fibril shape contribute to the reinforcement of ECM in the MCT? Over the past ten
years, studies on the biomechanics of collagen fibrils have applied the theory of discontinuous-fibre reinforced composites [
] to develop models to explain the basis of collagen fibrils reinforcement of ECM during the elastic stress transfer stage [
] and the plastic stress transfer stage [
]. The mechanisms associated with elastic stress transfer and plastic stress transfer have been described in
Section 3.3
Section 3.4
, respectively. These models show that there are advantages for fibrils to be associated with tapered ends while the tissue is loaded in the elastic and plastic region but not when the fibrils begin
to fail, e.g., by fibril pull-out [
]. In
Section 3.3
Section 3.4
, it has been pointed out that elastic stress transfer and plastic stress transfer regulate the stress uptake in the fibrils when the MCT is in the stiff and compliant states, respectively. In this
section, the discussion focuses on how fibrils of different shapes take up stress during the elastic stress transfer and plastic stress transfer stages.
Figure 11
A shows sketches of the axial stress distribution generated in the collagen fibrils during elastic stress transfer, for four different fibril shapes, adapted from the study of Goh et al. [
]. The profile of the fibril with a uniform cylindrical shape has been described previously (
Figure 7
B); it is plotted here for the purpose of comparison with the stress distribution profiles associated with the other fibril shape. Thus, the axial stress profile for the fibril with tapered ends is
somewhat opposite to that of a fibril with uniform cylindrical shape. Here, we find that the stress in the fibril with tapered ends has a minimum value at the fibril centre, but increases
non-linearly and gradually from the centre to the end, and peaks near the end point before decreasing rapidly to zero at the end. Among the tapered profiles, note that at the fibril centre, the axial
stress from the conical shape yields the smallest magnitude. The axial stress corresponding to the paraboloidal and ellipsoidal fibril lie somewhat in between the conical and uniform cylindrical
Figure 11
B shows the graph of fibril-matrix interfacial shear stress versus axial distance. Interestingly, for the fibrils with tapered ends the fibril-matrix interfacial shear stress yield similar profile
and magnitude: we observed that a minimum shear stress occurs at the fibril centre, which then increases gradually with distance, reaching a peak value at a distance somewhat halfway between the
fibril centre and the end before decreasing to a minimum value at the fibril end [
]. (Of note, the shear stress from a fibril with uniform cylindrical shape yields a minimum value at the fibril centre but increases non-linearly to a maximum at the fibril end [
Figure 11
C shows the axial stress distribution in the collagen fibril for varying fibril shapes, during plastic stress transfer; the results have been adapted from the study of Goh et al. [
]. As pointed out in
Section 3.4
, during plastic stress transfer, the matrix becomes plastic and a weak bonding mechanism regulates the sliding action between fibrils; the shear stress generated at the fibril surface is constant of
axial distance (
Figure 5
B). The axial stress distribution from the fibril with a uniform cylindrical shape has been described in previous section (
Figure 7
C) but is included here for the purpose of comparison with the results from the other fibril shapes. Thus, the axial stress distribution from the conical fibril features a uniform magnitude
throughout the fibril. On the other hand, the axial stress distributions of the fibril with paraboloidal and ellipsoidal shapes vary non-linearly with distance along the fibril axis. With regards to
taper, apart from the conical fibril, both paraboloidal and ellipsoidal fibrils yield stresses which peak at the fibril centre. However, the peak stress from the paraboloidal fibril is lower than
that of the ellipsoidal fibril. Thus, in all the cases examined here, it appears that taper in fibrils modulates the axial stress uptake by ensuring a more uniform distribution of stress throughout
the fibril.
Overall, these arguments allow us to draw general conclusions concerning the performance of the fibrils in the ECM of the MCT. It follows that taper in fibrils provide an advantage over the uniform
cylindrical fibrils when the MCT is in the stiff and compliant states, where the elastic and plastic stress transfer mechanisms predominate, respectively. This advantage has to do with the lowering
of the peak stress at the fibril centre. The argument is that although we would expect the elastically deforming fibril to take up stresses, high peak stress is to be avoided as this could reach the
level of the yield strength of the fibril as the load acting on the MCT increases.
4.3. Fibrils Ought to Possess a Certain Slenderness for Appropriate Force Transmission
As pointed out in
Section 1
Figure 1
A, early studies of the ligaments of sea urchins, such as
Anthocidaris crassispina
Echinometra lucunter
Eucidaris tribuloides
] reveal that the general tissue architecture of these animals are similar to those of mammalian tissue, namely a hierarchical architecture featuring collagen fibrils aligned to form bundles of
collagen fibres, and bundles of collagen fibres aggregate to form fascicles within ECM [
]. In the sea cucumber (
Cucumaria frondosa
), collagen fibrils from the dermis of the animal are aggregated in vitro by the dermal stiparin glycoprotein [
]. Stiparin is mentioned in
Section 3.5
for its possible role in providing bonding between the fibrils.
From these hierarchical structural concepts comes the determination of the dimensions of the fibrils. Detailed quantitative results from electron micrographs of the ligaments of sea urchin,
Eucidaris tribuloides
, reveal that the fibril possesses a minimum length of around 37 μm [
] and a maximum length of around 570 μm [
], with an average length of 225 μm [
]. These findings are comparable to other marine invertebrates such as starfish (
Asterias amurensi
), which are found to possess lengths of 20–540 μm, with a mean length of 196 μm [
]. Altogether, these findings suggest that the fibrils do not span the entire length of the MCT. As for fibril diameter, collagen fibril diameter measurements derived from electron micrographs of the
spine ligament of sea urchin (
Eucidaris tribuloides
) show a minimum diameter of around 25 nm [
] and a maximum diameter of around 280 nm, with an average value of 124 nm [
]. The diameters of the collagen fibrils in the compass depressor ligament of the sea urchin (
Paracentrotus lividus
) are observed to exhibit a spread of values on the histogram with a peak frequency at 45.5 ± 19.0 nm [
]; no length measurement has been carried out [
]. These findings are also comparable to other marine invertebrates such as starfish (
Asterias amurensi
) which yields diameter ranging 10–350 nm, with an average of 136 nm [
]. Interestingly, a simple linear regression analysis reveals significant linear relationship between the lengths and diameters of the respective fibrils in the sea urchin ligament [
]. By identifying the slope of the fitted line with the
, it is suggested that all fibrils have a nearly constant
of about 5300 [
]. This slope is identified with the fibril aspect ratio. Of note, vertebrate fibrils have lengths ranging 12–30 μm and diameters ranging 40–109 nm [
]. Corresponding measurements of these individual fibrils reveal
s ranging 550–1025 [
]—thus it would appear that the
from the marine invertebrate is higher (up to one order of magnitude) than those of the vertebrate. It is not clear if the length and diameter of vertebrate fibrils also exhibit a linear relationship
for the respective tissues.
Table 2
presents estimates of the fibril length, diameter and
based on studies of the respective tissues of marine invertebrates, namely starfish, sea urchin and sea cucumber, as well as tissues from vertebrates. As the fibril diameters of both vertebrates and
invertebrates do not differ dramatically, the high collagen fibril aspect ratio from the sea urchin may be attributed to longer fibrillar length. On the other hand, one then argues that the small
collagen fibril aspect ratio from the vertebrate is the result of fibrils having shorter length.
How do the collagen fibrils in the sea urchin achieve high aspect ratio? Trotter and co-workers show that the two pointed tips of a growing fibril in the spine ligaments of sea urchin (
Eucidaris tribuloides
) have similar axial mass distributions, indicating that the shape and size of the two tips remains similar throughout growth [
]. With regards to shape, these tips were paraboloidal, as evidence by the linear axial mass distributions (
Figure 10
D), similar to those of sea cucumber (
Cucumaria frondosa
) [
], and metatarsal tendon from embryonic chick [
]. Computer modeling reveals that the self-assembly mechanism involves independent growth by a process that produces a uniform rate of extension of the fibril tips, and lateral (i.e., radial) growth
by surface nucleation and propagation [
]. Very long and slender fibrils that form the axis of the tissue can grow in length by end-to-end fusion of early fibrils involving only the C-tip (C end of the collagen molecule, see schematic in
Figure 6
) of a unipolar fibril [
Szulgit has suggested that the fibrils in the ligaments may be required to possess a certain
in order to be able to transmit the appropriate level of force [
]. How high should the aspect ratio be in order to enable collagen fibrils in MCTs to provide reinforcement to the MCT? Goh and co-workers have investigated the stress uptake in a fibril at varying
by finite element analysis and analytical modelling [
Figure 12
A shows the distributions of stress, σ
, along a fibril (paraboloidal shape), obtained for various values of
to illustrate the dependence of stress on
during elastic stress transfer [
]. For a given value of
, it is shown that the magnitude of the stress increases when
increases. However, varying
has little effect on the profile of the stress distribution. It is also shown that the magnitude of the stress in the distribution is more sensitive to
at large
than at small
—the relationship between σ
at the fibre centre and
is shown in
Figure 12
B for two cases, corresponding to large
and small
. Thus, there are two important points of contention. First, the trend for each curve in
Figure 12
B reveals that the stress eventually converges at high
values. Second, the findings shown in
Figure 12
B predicts that
could interact and mask the main effects. Further studies by experiment would be needed in order to clarify these points.
The dependence of the stress on
is also observed in plastic stress transfer. Consider the case of the paraboloidal fibril. Equation (
) can be solved for the stress in the fibril by substituting the expression of
, given by
= [
) for a paraboloid, and applying the boundary condition of σ
= 0 at
= 1 [
]. This results in the following expression,
Thus, Equation (
) shows that the magnitude of σ
/τ in a paraboloidal fibril is linearly proportional to
but is independent of
. Large values of
result in high magnitudes of σ
/τ during plastic stress transfer.
Figure 12
C shows the distributions of stress, σ
/τ, along a fibril (of paraboloidal shape), obtained for various values of
to illustrate the dependence of stress on
during plastic stress transfer [
]. Clearly, the profile of the stress distribution is not affected by varying
. The relationship between σ
/τ (for a given
) and
is a linear one as shown in
Figure 12
D for
= 0.
Thus, increasing
increases the magnitude of the stress uptake—the effects of
on the stress in the fibril in the paraboloidal case are similar to the other shapes (uniform cylindrical, conical and ellipsoidal) [
]. In other words, one good reason for the collagen fibrils in the connective tissue of the sea urchin to possess high
is simply because higher stresses may be generated in the fibril during the respective stress transfer processes, but the magnitude of the peak stress will always be lower than that of the uniform
cylindrical fibril. Additionally, one also recalls that the stress concentration factor for a paraboloidal fibril is always lower than the uniform cylindrical fibril (
Figure 11
A,C). For given
and fibril-matrix interfacial shear stress, the extent to which the magnitude of the stress in the collagen fibril can increase would depend on the yield strength of the collagen fibril. There is a
, beyond which the stress uptake can increase to the level of the yield strength, and this could cause the fibril to yield [
]. In order to enable us to make useful comparison between the measured aspect ratios of sea urchin collagen fibril and the critical
for yielding, the latter must be known. Unfortunately, the exact value for the collagen fibril critical
for yielding is still not clear because we do not yet have the means of determining this parameter.
Is there any advantage for all fibrils in the spine ligament tissue of the sea urchin to possess the same q? Based on earlier arguments that suggests that high q leads to high stress uptake, if the
fibrils in the tissue were to possess a range of qs, it then follows that a fibril which belongs to a population associated with small qs may not be able to take up high stress. Thus, a tissue that
features a heterogeneous system of qs means that not all the fibrils would be able to maximize the stress uptake and the consequence is a mechanically unstable tissue. For the marine invertebrate
this consequence could have severe repercussion during stress transfer, be it in the stiff (elastic stress transfer) or compliant (plastic stress transfer) states. Additionally, from the perspective
of fibrillogenesis, having a fibrillar growth mechanism that self-regulate the fibril to achieve constant (high) q is important for the marine invertebrate.
The concept of the critical length (2
) for fibril fracture, borrowed from engineering fibre reinforced composites [
], is important for understanding how a fibril fracture [
]. Thus, 2
is defined as the minimum length that a fibril must have for the stress at its centre to be equal to the fibril fracture strength [
]. For effective reinforcement,
should be large but less than
. Analytical models have predicted that tapered fibrils have longer
than uniform cylindrical fibrils, given all things being equal, i.e.,
and τ [
]. In particular, the
of a fibril with straight-tapered ends, paraboloidal ends and ellipsoidal ends are, respectively, 2, 3/2 and 4/π times longer than that of a uniform cylindrical fibril [
]. By an analogy to engineering composites [
], it may be argued that the longer the collagen fibrils, the tougher, stronger and stiffer will be the MCT, given all things being the same (i.e.,
and τ). Analytical models have also predicted that a fibril with tapered ends requires less volume of collagen material to synthesize as compared to uniform cylindrical fibrils, for a given
]—this is another good reason for the collagen fibrils in the connective tissue of the sea urchin to possess tapered ends, as well as having high slenderness (i.e., high
Can a fragmented collagen fibril grow? To answer this question, Holmes and co-workers have developed a seeding system whereby fragments of collagen fibrils can be isolated from avian embryonic tendon
and added to purified collagen solution [
]. Holmes and co-workers found that the fractured ends of fibrils can act as nucleation sites for further growth by molecular accretion [
]. Two interesting findings arise from this study: (1) the surface nucleation and accretion process can result in a fibril with smoothly tapered end and (2) there is a limit to the increase in
diameter as the fibril grow axially—it appears that beyond a length of 13 μm (200
periods) a maximum diameter of about 600 kDa/nm is reached [
4.4. Small and Large Fibrils Have Distinct Roles in Regulating Mutability
The study of the lateral arrangement of collagen molecules is important for understanding the contribution of molecular interactions to the collagen fibril lateral size. Several models have been
proposed to resolve the cross-sectional structure of collagen (type I) fibril, such as the spirally packed models and radially packed models with concentric layers of radially oriented microfibrils [
]. A radially packed model has also been developed by Silver and co-workers [
] to account for the
period and axial staggering of fibrils with paraboloidal ends reported by Holmes and co-workers [
]. We opined that the most important revelation concerning the structure of the collagen fibril comes from the study carried out by Orgel and co-workers [
]. From a detailed crystallographic analysis Orgel and co-workers have predicted that the collagen type I structure comprises collagen molecules arranged to form a supertwisted (discontinuous)
right-handed microfibril that interdigitates with neighboring microfibrils. This interdigitation establishes the crystallographic superlattice, which comprises quasihexagonally packed collagen
molecules [
]. This study is important because it effectively framed the concept of the ultra-structure of collagen fibril within a 3D space that can also account for the
period and staggering configuration.
Alongside with these studies include findings concerning the fibril diameters distribution in the tissue. Curiously, there are probably more investigation dealing with the quantitative analysis of
collagen fibril diameters in vertebrates than in marine invertebrates. These quantitative studies evaluate the histograms of the frequency versus diameter to understand how (1) age, i.e., from
development to old age [
], (2) symptoms, e.g., Ehlers-Danlos syndrome [
], and (3) functions of different types of tissues [
], influence the tissue mechanical properties. In vertebrates, while tissues from very young individuals (namely mice at birth until, e.g., 2 week old) possess diameter distribution with
near-Gaussian profiles [
], the diameter distributions of tendon in mature and old individuals feature non-Gaussian profiles that may be described as bimodal [
] or even tri-modal [
]. It must be emphasized that the fibril diameter distribution histogram can be regarded as a quick way to provide a useful measure of the fibrillar content in the ECM. For vertebrates, since the
ends are seldom encountered, this implies that the diameter of the fibril are fairly constant, at any given cross-section of the ECM, and hence the diameter distribution histogram provides a
realistic picture of the distribution of the fibril sizes. This conceptual approach could be misleading.
Figure 9
A shows a schematic of continuous uniform cylindrical fibres reinforcing composites to model the long collagen fibrils in ECM of vertebrates; this implies that the total cross-sectional area of the
fibrils is constant at any plane of interest (POI). This sectioning approach is applied to obtain transmission electron micrographs of fibril cross sections for quantitative analysis of fibril
diameter in vertebrate connective tissues.
Figure 9
B shows a schematic of discontinuous spindle-like fibres reinforcing composites to model the short collagen fibrils in ECM of marine invertebrates such as sea urchins; this implies that the total
cross-sectional area of the fibrils is not constant at any given POI. For marine invertebrates such as sea urchin, the fact that spindle-like fibrils are embedded in and reinforcing the ECM means
that it is not useful to apply the sectioning approach for quantitative analysis of fibril diameter in these animals.
Several studies have been carried out to understand how fibril diameter affects the mechanical property of connective tissues such tendon and ligament [
]. This is analogous to the structural engineers’ concern with how steel rods embedded in concrete columns provide reinforcement to the columns, by relating the diameter of the rods to the stress
uptake in the rods and cement [
]. The structural engineer would claim that it is straight-forward to evaluate the effects of diameter on mechanical properties because these steel rods can be engineered to possess uniform diameters
within manufacturing errors. So far, based on the observed tissues, it is shown that the fibril diameter exhibits a spread of values that may be described as unimodal [
], bimodal [
] or tri-modal [
]. When attempts are made to relate the mean fibril diameter of the frequency histogram to the mechanical property, conflicting findings emerged. For instance, for a given tissue type, mechanical
parameters such as structural strength (maximum force) and structural modulus of the force-displacement curve are found to be dependent on the the mean diameter [
]. However, the material strength (maximum stress) and material modulus of the stress-strain curve are not dependent on the mean diameter [
Goh and co-workers have carried out a study to rationalize the discrepancies using tail tendons from a mouse model; this has resulted in two key papers that hypothesize that (1) the stress-driven
parameters that contribute to elasticity and fracture are dependent on the collagen fibril volume fraction, governed by the rule-of-mixture [
], and (2) the strain energy-driven parameters that contribute to resilience and fracture are dependent on the fibril diameter of the respective modal distribution, govern by fibril shear sliding
theory [
]. (The approach to test these hypotheses was to apply them to tissue of varying age groups, in parallel with measurements of the fibril cross-sectional parameters (area, diameter) using transmission
electron microscopy.] Without loss of generality, the findings from this study have allowed for drawing general conclusions concerning the MCT. Thus the magnitudes of the strength [
], stiffness [
], fracture toughness [
] increase linearly with increases in fibril volume fraction [
], in accordance with the rule-of-mixture for strength and stiffness [
]. Unfortunately, fracture toughness has a more complex nature than the strength and stiffness parameter, and consequently, the rule-of-mixture cannot be applied to relate the fracture toughness
parameter to the fibril cross-sectional area [
]. How then can one evaluate the relationship between the fracture toughness and the collagen fibril structural parameter? Goh and co-workers proposed that the fracture toughness of the tissue is the
sum of the nonessential energy (
), and the essential energy (
) according to the principles of essential work of fracture of a fibre composite [
]. The
is said to contribute primarily to tendon resilience (regulated by fibrils undergoing elastic deformation); the
is said to contribute primarily to the resistance of the tendon to rupture (regulated by fibril rupture, leading to defibrillation and rupture of the interfibrillar matrix,
Figure 8
) [
]. By evaluating the frequency histograms of diameter for all age groups using the minimal number of normally distributed subpopulations, the mean fibril diameter of the respective subpopulation may
be linearly added—weighted by coefficients that depend on the fibril diameter, fibril strength and interfacial shear stress—to give the
]. Using the mouse tail tendon as a tissue model, the minimal number of normally distributed subpopulations is found to be equal to two across all the age groups [
]. For the purpose of this discussion, we then let the smaller and larger mean diameters be
, respectively. Accordingly, the overall effect of fibril diameter on
may then be assessed by a multiple regression analysis [
]. It follows that increases in the resilience energy are associated with decreases in
and increases in
]. On the other hand, the energy to resist fracture is associated with increases in
, but independent of
. Thus, the fracture toughness argument emphasizes that at the physiological level, age variation in the fracture toughness is the result of changes in the energy for resilience and resistance to
fracture; these two energy parameters are in turn influenced by the structural changes at the fibrillar level. It follows that small and large fibrils have distinct roles in the stiff state and only
the small fibrils have a role in the complaint states, and hence lending to the mutability properties.
These arguments to relate the fibril structural parameters to mechanical properties have enabled us to draw general conclusions about the structure-function relationship that may be applicable to
MCT. First, it is likely that the strength and stiffness of MCT depend on the collagen content (i.e., fibril volume fraction) in the ECM. Second, since the u[E] parameter is essentially associated
with loading from the initial point until the end of the elastic region of the stress-strain curve, it is likely that the small and large fibrils have distinct roles in regulating mutability. While
it is not clear why these fibrils are bestowed with distinct roles, the results suggest that the interplay of small and large fibrils help the tissue responds to external loads by absorbing the
appropriate level of strain energy to achieve resilience (elastic stress transfer) and to resist fracture (plastic stress transfer).
4.5. Fibril Slenderness and Relative Stiffness Interplay to Lower Stress Discontinuity in Fibril Interaction
The CA of the sea urchin can undergo deformation without buckling [
]. To illustrate this point,
Figure 1
B depicts two positions, i.e., X and Y, that the spine can assume. Each position would involve a change in length of the CA [
] but the ligament is not known to buckle at position Y [
]. One way to ensure that the ligament deforms to the desired length is for the collagen fibrils, either singly or in bundles, to undergo relative axial and lateral displacements without resulting in
an appreciable change in the fibril length. Thus two key features stand out with regards to the MCT. The first feature, that the collagen fibrils in the connective tissue of the sea urchin are
discontinuous has already been pointed out in previous sections. We recall the length distribution of isolated fibrils in
Table 2
and the demonstration of tapered ends as explained by a schematic in
Figure 9
B. The second feature is the physical ease of extracting and isolating intact fibrils from the ligament, by gentle mechanical procedures [
]. These fibrils can retain their native structure over long duration in fluid suspensions [
]. The second feature suggests that the bonds that normally hold the fibrils in the ECM are labile but the collagen fibrils are very stable [
]. While the precise fibril-fibril sliding mechanism is not clear, these features suggest that fibril-fibril sliding could contribute predominantly to the tissue deformation as proposed by Smith et
al. [
] and by Hidaka and Takahashi [
One finds that when a load acts on the MCT, the fibrils are pressed against the interfibrillar matrix. As the load increases, the magnitude of the component of the resultant force acting on the
fibril that is associated with friction—i.e., the normal force—at the contact surfaces also increases [
]. Altogether, these contact forces regulate the fibril stretching and sliding (relative to the matrix); the matrix may be regarded as responsible for transmitting stress to the fibril [
]. Nevertheless, how does a fibril take up stress during fibril-fibril sliding in the MCT?
Figure 13
shows a model of ECM in the MCT to help understand the mechanism of fibril-fibril sliding. Here, we shall draw some general conclusions from a previous study on fibre-fibre interaction that used a
unit cell approach to establish the stress uptake in a fibril during fibril-fibril sliding.
Figure 13
A,B show a central (α) and peripheral (β) fibrils for illustrating the unit cell model at two different views. The unit cell is further divided into representative volument elements (RVEs) (
Figure 13
A,B). This model can be used to evaluate the elastic stress transfer process in MCT with the following simplifying assumptions, namely (1) strong adhesion is present at the fibril-matrix interface;
(2) as the interfibrillar matrix around the fibril deforms elastically, shear forces will develop at the fibril-matrix interface; (3) in response to the shear action the fibril deforms elastically
and generates an axial tensile stress. Altogether these assumptions are consistent with the “shear-lag” approach (
Section 3.3
According to finite element analysis [
], one arrives at the stresses, σ
, in the fibril at varying fibril-fibril axial overlap distance (λ) and the centre-to-centre lateral separation distance (
) as a function of axial distance along the fibril, for the case of the uniform cylindrical shape (
Figure 14
). When the neighbouring fibrils do not overlap, i.e., the α and β fibrils in the unit cell do not overlap along the axes (λ/
= 0), the stress uptake within the α fibril is modulated by the α fibrils from the adjacent unit cells (
Figure 14
A). Thus, σ
is maximum at the fibril centre (
= 0) which then decreases steadily to zero at fibril end (
= 1). The stresses are very evenly distributed throughout the bulk of the fibril (except at the fibril end) at high
. In contrast, the stresses show appreciable decrease in magnitude with increasing
at low
values (results not shown). It must be pointed out that the stresses at high
feature even distributions throughout the bulk of the fibril (except at the fibril end) for both cases of high and low
(results not shown). This is important because it suggests that the interplay of
predominates in the fibril-fibril interaction. The result is higher axial stress even when the fibrils in the immediate vicinity are not overlapping with the primary fibril.
An important implication of fibril-fibril interaction is the generation of stress discontinuity when neighbouring fibrils overlap, i.e., when the α and β fibrils overlap along the axes (λ/
> 0) (
Figure 14
B,C). Here, the stress uptake within the α fibril is modulated by the β fibril, as well as by the α fibrils from adjacent unit cells. This leads to higher σ
throughout the fibril compared to the case when overlap is absence (λ/
= 0). The stress discontinuity is described as an abrupt (stepwise) change in the σ
distribution, which is most pronounced at high
Figure 14
B,C), but somewhat less so at low
(results not shown). Additionally, the stress discontinuity (i.e., a sudden drop in stress) results in higher σ
at the non-overlapped region and lower σ
at the overlapped region (
Figure 14
B,C). It must be emphasized that the exact position of the stress discontinuity varies with the extent of overlap. Indeed, increasing λ/
displaces the discontinuity to the fibril centre. Interestingly, when axial overlap occurs, the magnitude of the stress at a given location in the fibril (within the overlap region) appears to be
independent of the extent of the axial overlap, regardless of the point within or outside the overlap region. Thus, it is important to note that no further advantage (i.e., higher stress uptake) may
be gained from increasing the overlapping region.
With regards to the lateral separation distance, one notes that the extent of the step-wise change in the stress magnitude is dependent on the lateral separation, i.e.,
-dependent, as well as
. In general, large
leads to small step-wise change in the stress uptake. This implies that the influence of the β fibril decreases with increasing
Figure 14
). More importantly, the stress discontinuity disappears at high
, regardless of
. Additionally, increasing the fibril-fibril separation distance has the effect of increasing the stress magnitude in the fibril. Thus the larger the fibril-fibril lateral separation distance the
higher is the stress in the fibres. Secondary to this effect is the asymptotic increase of the stress magnitude to a steady value at large fibril-fibril separation (results not shown). Of course,
these conclusions apply regardless of the extent of fibril-fibril overlap; in other words, these conclusions apply when λ/
= 0 or λ/
> 0. Clearly the case of λ/
= 0 implies that no axial overlap occurs, but this asymptotic increase in the stress magnitude with increasing fibril-fibril separation should not be interpreted to imply that the nearest (β) fibrils
has no effect on the σ
. In this case, the effect on the stress with varying fibril-fibril separation distance is predominated by two factors: one, the stress field arising from the interactions with the nearest (β)
fibrils where the tips of these fibrils are in line with the tip of the α fibril, and two, the effects of the bulk ECM surrounding the α fibril.
At any given λ/L[CF] the largest ρ/r[0] beyond which the effect of fibril-fibril interaction (i.e., directed by the nearest (β) fibres) diminishes may be determined from a plot of σ[z]/σ[c] (at a
fixed Z) versus ρ/r[o]. To assess for convergence, note that at a given value of q at low E[CF]/E[m], a plot of σ[z]/σ[c] (namely Z = 0, as the reference point) versus ρ/r[0] shows that σ[z]/σ[c]
increases rapidly with increase in ρ/r[0] (graph not shown). More importantly, beyond a critical ρ/r[0], σ[z]/σ[c] converges to a steady value.
The fibril with uniform cylindrical shape represents one extreme of possible regular profiles. To the best of our knowledge, no attempts have been made to model fibril-fibril sliding to study the
stress distribution of fibrils that possess tapered ends, such as conical ends, paraboloidal ends and ellipsoidal shape. Nevertheless, this issue has been targeted for investigation in future
Finally, it is important to emphasize that the sliding action of collagen fibrils is not to be confused with the sliding action of collagen fibres. In the latter, this has important implications on
the microscopic crimp in MCT [
]. Crimp is not known to be present in the CA ligament of the sea urchin [
] (
Section 2.1
) but recently, it is thought to be present in the compass depressor ligament of the sea urchin, due to the observed toe-to-heel feature on the stress-strain curve of excised tissues tested to
rupture at low loads [
], which is typical of most vertebrate soft connective tissues where crimp has been observed [
]. At initial loading when the load is low, i.e., within the toe-to-heel region, as the tissue strain increases the sliding of collagen fibres eventually results in the extinction of the wavy crimps
]. Crimp is believed to originate from the contraction of cells (e.g., fibroblasts) residing on collagen fibres [
]. The mechanics of the contraction of the cells results in the buckling of the fibres [
]. Crimp can exist as early as during embryonic development in vertebrate connective tissue [
] but whether this applies to the MCT is not entirely clear. The above arguments used to lend to support to a mechanical basis for crimps in MCT suggests that crimp is analogous to a mechanical
damper [
]. Consequently, crimp is hypothesized to (1) absorb energy during elastic stress transfer [
], (2) enable the tissue to recoil when the load is removed [
], and (3) absorb energy generated in shocks [
]. According to the load-sharing concept in fibre reinforced composite [
], it follows that the force generated within the collagen fibres for the stretching/contraction is proportional to
. Consequently, one could expect that the larger the
the higher is the force generated to stretch/contract the fibres. Estimates for
ranges 10
Table 3
) [
]. To what extent should crimp be exploited for ECM-DT, or even synthetic collagen fibrils in a synthetic matrix is not clear but the arguments of previous studies suggest that crimp presents some
advantages for the tissue to stretch/contract, aided further by virtue of the high
4.6. Water and Charged Species within the Interfibrillar Matrix Contributes to the High Poisson Ratio of MCT
This section is intended to examine the key ECM components in the interfibrillar matrix that contribute to the mechanical properties of interfibrillar matrix. The interfibrillar matrix is believed to
play an important role in fibril-fibril sliding, by an analogy to engineering fibre reinforced composites [
]. This section is concerned with the physical properties of the key constituents that contribute to fibril-fibril sliding. In this analogy, one finds that when a load acts on the MCT, the fibrils
are pressed against the interfibrillar matrix [
]. As the load increases, the magnitude of the component of the resultant force acting on the fibril that is associated with friction—i.e., the normal force—at the contact surfaces also increases [
]. Altogether, these contact surface forces regulate the fibril stretching and sliding (relative to the matrix) while the matrix may be regarded as responsible for transmitting stress to the fibril [
For simplicity, the interfibrillar matrix of the MCT may be regarded as composed of water and charged species [
]. Since the ions are dissolved in the water of the interfibrillar matrix, the ions and the water could be responsible for regulating the fibril-fibril sliding action that results in the transition
between the stiff and compliant states [
]. In the compliant state, as well as in the standard state, the fibril-fibril spacing of the compass depressor ligaments of the sea urchin is consistent with the length of the filament connecting
between two fibrils, i.e., of the order of 50 nm; the spacing is much smaller in the stiff state [
]. The smaller fibril-fibril spacing in the stiff state, as compared to the compliant state, suggests that the water and charged species in the interfibrillar materials could also be displaced out of
the interfibrillar space. Ultimately, the effectiveness of these ions for regulating the fibril-fibril sliding action would depend on the type and composition of the ions [
The initiation and propagation of fibril-fibril sliding occurs through the alterable interactions of charged species within the interfibrillar matrix—the mobility of the charged species depends on
the amount of water [
]. Of note, the limit beyond which alterable interactions terminate is related to plastic stress transfer (
Section 3.4
). Results from FTIR and Raman spectroscopy indicate that water is largely exuded when the sea urchin compass depressor ligament changes from the compliant to the stiff states [
]. Additionally, water exudation is also observed in other MCTs such as the dermis of the sea cucumber when the tissue undergoes a change in the mechanical state [
]—by studying the respective mechanical states corresponding to compliant, standard (normal) and stiff, it has been found that the mass and volume decrease by 15% when the dermis changes from the
normal state to the stiff state by mechanical stimulation and by chemical stimulation with potassium-rich seawater [
]. To this end, it is believed that the mechanisms responsible for the transition from the soft state to the standard (normal) state, and that from the standard (normal) to the stiff state, are
different [
The water molecules exuded from the ECM include those bound previously to glycosaminoglycans side-chains mediated by electrostatic forces as well as those in free state. For the former water
molecules, the electrostatic interaction may be displaced by stronger interactions involving tensilin (a collagen-fibril binding protein released from a juxtaligamental-like cell [
]) (
Section 3.4
), thus enabling the dermis to change from the compliant to the normal state and into the stiffened state [
]. Additionally, ECM components in the interfibrillar matrix that are involved in mediating the process of water exudation could also be released from the ECM [
]. Further study shows that the exudation of the water molecules and the interplay between the water molecules in the bound state and in the free state result in a water concentration gradient in the
ECM, as observed in the stiffened state [
Water exudation can also occur in vertebrate tissues such as articular cartilage [
], mammalian tendon [
], and intervertebral disc [
] when these tissues are deformed. Haverkamp and co-workers found that when bovine pericardium is loaded, water exudation from the ECM could be a contributory factor to the high Poisson’s ratio (>
0.5) of the tissue at strains as high as 0.25 [
]. Goh and co-workers pointed out that similar changes to the ECM components, in the interfibrillar matrix, resulting in an increase in the interaction energy between fibrils via collagen-bound
proteoglycans, could happen during freezing [
]. Additionally, changes to the long-range order of radially packed collagen molecules in fibrils could also happen and this could contribute to fibril rupture at higher stresses [
The distribution of water in the ECM depends on the mode of the loading, e.g., uniaxial loading and mechanical relaxation [
]. Although it is still not clear how the microenvironment of the interfibrillar matrix facilitates the respective mode of loading, the interplay among macroscopic related factors, namely the osmotic
pressure and the anisotropy, have been identified as possible contribution to the exudation of water and other ECM components [
]. In particular, the osmotic pressure contributes to the slackness of the tissue before stretching, resulting in tissue swelling; consequently when the tissue is straightened, as water molecules are
forced out from the microenvironment of the interfibrillar matrix, an appreciable decrease in the volume of water occurs [
]. Both the osmotic pressure and the fibril direction are expected to contribute to tissue anisotropy, resulting in a Poisson’s ratio that is greater than 0.5 from the axial to the lateral direction
The issue of tissue anisotropy has been addressed by Haverkamp and co-workers recently. Haverkamp and co-workers inferred that the high Poisson’s ratio of the tissue could be the result of a high
Poisson’s ratio of the collagen fibrils [
]. Analysis of the small angle X-ray scattering patterns of a deforming bovine pericardium reveals that the collagen fibril Poisson’s ratio, identified with the ratio of collagen fibril width
contraction to length extension, is 2.1 ± 0.7 for a tissue strained to 0.25 [
]. Since the Poisson’s ratio of the collagen fibril is greater than 0.5, this study shows that the volume of individual collagen fibrils decreases with increasing strain [
]. The change in volume in the fibril during fibril deformation implicates that a proportion of water or charged species could reside in the fibrils and that these are also exuded during stretching.
How does the interfibrillar matrix contribute to the high Poisson’s ratio of the tissue? For simplicity, one may model the ECM as comprising collagen fibrils which are uniformly distributed
throughout the length of the tissue such that the area fraction of the collagen fibrils at any given cross sections along the length of the tissue remains unchanged. To order of magnitude, the
Poisson ratio of the tissue can be estimated according to the rule of mixture for Poisson’s ratio [
are volume fractions of the collagen fibrils and matrix, respectively, satisfying the condition of
= 1. By considering the upper and lower limits of
to be 0.8 and 0.2, respectively [
], the upper limit of
~2 [
], and the upper limit of
~4 [
], the
is found to range from 3 to 18. The estimated upper limit for the interfibrillar matrix is consistent with a material that exhibits very large change in volume during deformation.
How does the exudation of water and other ECM components from the ECM affect the mechanics of stress uptake in the fibril? According to a study of the effect of
, which represents the ratio of the stiffnesses of the fibril (
) to the interfibrillar matrix (
), on collagen fibril stress uptake, it has been predicted that the higher the
, the larger is the magnitude of the axial stress generated in the fibril (
Figure 12
A,B) [
]—the axial stress uptake is more sensitive to
]. As high
corresponds to an interfibrillar matrix in a compliant state while low
corresponds to an interfibrillar matrix in a stiffened state (which could be the result of the exudation of water and other ECM components), this suggests that the stress uptake in a fibril is higher
when the MCT is in a compliant state than stiffened state.
With regards to the interfibrillar shear stress, τ, there has been several attempts to measure τ in order to provide insights into the nature of the interfibrillar matrix.
Table 4
lists estimates of the interfacial stress stress derived from different studies. Recently, a novel notch tensile testing approach have been used to estimate τ in a rat tail tendon undergoing
deformation [
]. Based on data derived from axial stress gradients along the tissue sample length, τ is shown to have a value of 32 kPa [
]. This estimate is comparable to the interfibrillar shear stress predicted by a multiscale model of tendon fascicles [
], as well as a strain energy density model for the resilience and fracture toughness of soft tissue in the presence of ageing [
]. In particular, the strain energy density model predicts, to order of magnitude estimates, that τ ranges 1–100 MPa [
]. A separate test using optical tweezers to measure the direct forces of rupture between two decorin proteoglycans [
] led to an estimated value of 7.5 kPa for τ [
]. Of note, targeted disruption of the decorin gene, resulting in decorin-deficiency in the tissues, yields uncontrolled lateral fusion of collagen fibrils as well as lower tensile strength (force)
of the tissue, compared to native tissues [
]. Altogether these findings are important as they support the arguments that the interfibrillar matrix can transmit stress to the fibrils through interfibrillar shear. According to the basic
principles of mechanical engineering, the ECM components, possibly proteoglycans (
Figure 4
), in the interfibrillar matrix of MCTs would have to provide cross-linkages between the fibrils and between the fibril and the matrix in order to facilitate the transfer of stress from the matrix to
the fibrils. Computational models for linking the ECM components (in the interfibrillar matrix) to the fibrils, such as those developed by Redaelli and co-workers [
], may be able to provide deeper understanding for how the ECM components in the interfibrillar matrix transmit load between fibrils.
However, the contribution of the macromolecules to the shear action of the interfibrillar matrix in response to an external load is unclear because of controversy surrounding some of the ECM
components that have been identified. In earlier works, Goh and co-workers [
] as well as many others [
] have attributed the contribution to the shear action to the interactions of glycosaminoglycan chains in the interfibrillar matrix with glycosaminoglycan attached to the core protein of small
leucine-rich proteoglycan (e.g., decorin), or between glycosaminoglycan of the proteoglycan on adjacent fibrils. However, recent studies are challenging the importance of the proposed mechanical role
of proteoglycans in these tissues. These studies focus on selective removal of glycosaminoglycans from the tissue via enzymatic depletion of the glycosaminoglycans using chondroitinase ABC [
]. The treated tissue have been subjected to dynamic viscoelastic tensile tests to measure the storage modulus [
], or even simple tensile test to rupture to measure strength and stiffness of the tissue [
] and to specific strain level to evaluate the changes in the periodic banding of the collagen fibrils (D-period) or fibril diameter [
]. While results from most mechanical parameters yielded no significant difference with the controls, for the specific strain level, it is observed that the strains generated in the fibril in tissues
without glycosaminoglycans are higher than those from the controls [
]. Thus the results from the strain measurement suggest that glycosaminoglycans may serve to reduce (not increase) stress transmission between fibrils [
]. These observations could direct attention to other ECM components, namely the FACIT (i.e., fibril associated collagens with interrupted triple helices) family of molecules (collagen types XIV and
XII) [
], emu1/emu2 [
] and the COMP (i.e., cartilage oligomeric matrix protein) [
] as pointed out by Szczesny and co-workers [
]. Clearly, new studies should be carried out to investigate the mechanical properties of other ECM components, because some of these could be possible candidates for having a role in the
interfibrillar stress transfer mechanism. This review proposes that the news studies should consider a force-spectroscopic analysis of the molecular interaction effects of all possible ECM
components—inspired by the study carried out for decorin-decorin interactions [
], alongside arguments that involve simple order-of-magnitude estimates [
]—to eliminate the suspicion on the ECM components that do not contribute to the transfer of stress from the interfibrillar matrix to the fibril as well as fibril-fibril interaction. Further
discussion is out of the scope of this review but this has been targeted for future investigations.
5. Framework for Collagenous ECM Mechanics, Prospects and Challenges for Scaffold Design
A complete understanding of the structure-function relationship of collagenous tissues is the central goal to elucidate the mechanical properties of these tissues. In this regard, it is important to
be able to see the forest for the trees, to use an apt expression. To this end, a simple framework for mapping the different mechanisms has been proposed as a systematic approach to tackle this goal
]. These mechanisms are concerned with the stress uptake in the structural units reinforcing ECM at the respective levels of the hierarchical architecture of connective tissues. The framework enables
(1) comparison of these mechanisms, (2) predictions based on the interconnection of these mechanisms and (3) identification of new mechanisms and pathways [
]. Here, we will show how the mechanisms highlighted in this review for the MCT may also be framed within this framework (
Figure 1
This paragraph and the following paragraph, are concerned with a technical description of the framework. Consider initially the macroscopic mechanical response of the MCT described by a typical
stress–strain curve in tensile loading mode. Based on the arguments presented in earlier sections of this paper (
Section 3
and sub-sections therein), it follows that there are five categories of mechanisms involve in regulating the stress-strain curve. The role of each category are reflected in the respective regions,
namely toe-to-heel, elastic deformation, yielding, plastic deformation and rupture. The regions of elastic deformation and of plastic deformation correspond, respectively, to the stiff state and the
compliant state; the region leading to rupture corresponds to autotomy. A list of known mechanisms under each category can be found, at the respective length scale. Five levels, identifying the
respective length scales, are shown here. These are, from the higher level to the lower level, the macroscopic tissue, collagen fibre, collagen fibril, microfibril and the collagen molecule at the
atomic/molecular level. Without going into the details of each mechanism, at this juncture it is important to emphasize the connectivity between each mechanism at the highest level to the next level,
as shown by the vertical lines. The connectivity indicates possible exchange of input parameters between the respective levels. The connection from the lowest to the highest level may be termed as a
mechanical pathway, by an analogy to the signal pathway concept.
The mechanisms identified in each mechanical pathway typically involve stretching, of the key components, in combination with twisting and bending, as well as relative sliding. For convenience, the
term used to refer to the respective length scales correspond to the key components, namely collagen fibres, fibrils, microfibrils and molecules. Additionally, with regards to influencing the
respective mechanisms, one notes that the common structure-related properties are: q (encompassing length and diameter), orientation and spacing between components. The common function-related
properties are: stiffness, strength, strain at fracture, and fracture strain energies. Generally, all properties are involved in:
decrimping, elastic deformation by fibre–fibre sliding, fibre yielding, plastic deformation, and fibre defibrillation or rupture at the collagen fibre level;
the extinction of fibril waviness, uncoiling of the fibril-associated proteoglycan glycosaminoglycan side-chains, elastic stress transfer, intermediate modes (such as interfibrillar matrix
cracks, partial delamination of interface between the interfibrillar matrix and fibril, and plastic deformation of the interfibrillar matrix), plastic stress transfer (with complete delamination
of interface between the interfibrillar matrix and fibril), rupture of interfibrillar matrix, and fibril rupture and fibril pullout at the collagen fibril level;
the straightening of microfibrils, microfibrillar sliding and realignment of microfibril from its supertwist, exudation of water and solutes from the intermicrofibrillar matrix, microfibrillar
stretching, disruption of microfibril-microfibril interactions and microfibril rupture at the microfibril level;
the straightening of kinks on the molecule, molecular stretching (involving axial deformation of the backbone, uncoiling the helices and helix–helix sliding) and intermolecular shear (nucleation
of slip-pulse), and disruption to the intramolecular cross-links and intermolecular cross-links at the collagen molecular level.
What information can we derive from the respective mechanical pathways of the MCT? For the purpose of this discussion, this paragraph will highlight the mechanical pathway associated with elastic
deformation. At the tissue level, as this regime corresponds to low applied loads, the deformation of the tissue in response to the applied load is analogous to a stretched spring; for simplicity,
the stress-strain behaviour conforms to a linear relationship [
]. One could expect that the MCT relaxes and assumes its original state upon removal of the applied load. (In order for the MCT to continue to be structurally effective, one then expects that any
residual stress generated on the return path to be minimized.) At the collagen fibre level, the contribution to deformation could come from fibre-fibre sliding as well as fibre stretching. At the
next lower level collagen fibril-fibril sliding, as well as fibril stretching, underpin collagen fibre stretching. The stress uptake in the fibril is governed by the well-known elastic stress
transfer which is used to explain how the MCT takes up load in a stiffened state (
Section 3.3
). In order for the fibril to be able to deform, at the next lower level one finds processes for regulating microfibrillar sliding as well as the realignment of the microfibril supertwist, and
stretching, in the direction of the load acting on the fibril. The deforming microfibril is the result of sliding and stretching of collagen molecules (
Section 3.3
) and the intermolecular shear.
What are the gaps in the framework that have to be addressed before it can be applied to the development of ECM-DT? From the fundamental perspective, many of the proposed mechanisms along each
mechanical pathway of this framework contain contentious issues that are still far from clear and could be subjects for further investigation. To begin, the fibril-fibril sliding action addresses the
ECM component, namely glycosaminoglycan, that contributes to the mechanical response of the interfibrillar matrix. Currently the exact role of the ECM component is still unclear (
Section 4.5
). The second issue addresses the proportion of fibrils with the respective fibril shapes in the MCT. Currently it is unclear if the MCT contains a heterogeneous population of different fibril shapes
or a homogeneous system of fibrils of the same shape (
Section 4.2
). Third, while plastic stress transfer is identified as the loading regime for the compliant state (
Section 3.4
Section 4.6
), is this the limit beyond which alterable interactions terminates for the MCT? Additonally, what then are the implications for a fail-safe mechanism? Fourth, with regards to the concept of
mutability, we have only provided a simple explanation (
Section 2.1
) with regards to fibril sliding mechanics, which is regulated by a nervous system. While a detailed discussion of the mechanism of mutability dictated by the control of a nervous system is out of
the scope of this review, we are also not aware of any studies that purposefully adapted the nerve-control system to the ECM mechanics framework. Thus, it is difficult to see how we could comment on
the concept of mutability in the context of the ECM mechanics framework, in this review. Last but not the least, as pointed out in
Section 1
, the fibril-forming collagens in the connective tissues of invertebrates such as sea urchin, as well as sponge, may have more varied structural features than those of the standard fibrillar
vertebrate collagens, e.g., triple helical domains of varying lengths [
]. How the variability in the fine structure of these fibril-forming collagens (molecular level) affects the mechanical properties of the fibrils (fibrillar level) and, consequently, the bulk tissue
is not clear and could be a subject for further investigation.
From the practical engineering perspective, currently one of the challenges for the applicability of ECM-DT in tissue engineering is the development of effective decellularization techniques for the
removal of cellular components, to minimize immunogenicity upon implantation [
]. On-going studies to develop effective techniques usually address a combination of physical, chemical and enzymatic methods [
]: physical treatments using cyclical freeze-thawing and ionic solutions can lyse cell membranes, before the enzymatic methods are applied to separate the cellular components from the ECM. Clearly,
these processes must be optimized to address high decellularization with minimal effects on the biochemistry and, in particular, the mechanical properties of the ECM. If the optimization approach
entails imperfections in the final product, which hierarchical levels (with respect to the framework,
Figure 15
) can we afford to compromise? The mutability property may be mimicked by incorporating a mechanism to trigger chemicals to cause stiffening or compliance, but this assumes that the decellularization
technique will not destroy the other properties related to performing rapid change of mechanical states in the ECM-DT. Further discussion is out of the scope of this review but these issues have been
targeted for future investigation.
6. Conclusions
New demands for a biomaterial, such as the ECM-DT, that can be used to make scaffolds for tissue engineering applications provide timely opportunity to revisit the fundamental issues and explore the
new findings derived from ECM studies. There have been many studies of the ECM of the sea urchin ligaments since the work of researchers such as Hidaka [
] and Smith [
]. They have investigated the structure and the composition of both the collagen fibril and the interfibrillar matrix of the tissue, as well as the overall tissue mechanical response. There are also
many experimental studies, supplemented by analytical modelling and computer modelling, to investigate the basis of reinforcement of the ECM in the soft connective tissue of vertebrates at the
fundamental level since the work of researchers such as Bard and Chapman [
]. This review has addressed several of these findings to prompt new insights which can be summarized as follows.
• MCT deformation characteristics resemble those of mammalian tissues.
• Shear models, addressing elastic and plastic stress transfer, explain the mechanism of collagen fibril reinforcement of MCT during the stiff and compliant states, respectively.
• Nucleation of slip pulses, as a possible mode of collagen fracture, leading to failure of the MCT, could direct autotomy.
• The spindle-like shape in collagen fibrils modulates the stress uptake by ensuring a more uniform distribution of stress throughout the fibril.
• Fibrils with small diameters are responsible for regulating the property of mutability, by addressing the tissue resilience and fracture energy.
• Interplay between the fibril aspect ratio and relative stiffness of collagen to matrix is the key to reducing stress discontinuity in a fibril during fibril-fibril sliding.
We thank David W. L. Hukins (University of Birmingham) and Richard Aspden (University of Aberdeen) for early days’ discussion of the physical properties of collagen fibrils that culminated in the
insights illuminated in this review.
Author Contributions
Kheng Lim Goh and David F. Holmes drafted the manuscript and approved the final version of the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
ACh acetylcholine
ASW Artificial sea water
CA Catch apparatus
ECM Extracellular matrix
ECM-DT Extracellular matrix derived from decellularized tissue
EGTA Ethylene-bis-(oxyethylenenitrilo)-tetraacetic acid
FACIT Fibril associated collagens with interrupted triple helices
MCT Mutable collagenous tissue
POI Plane of interest
RVE Representative volume element
TX Triton X100
Non-Greek Symbols
A[CF] Cross-sectional area of the (i.e., uniform cylindrical) fibril
C C-terminus (containing the carboxyl group) of the collagen molecule
COMP Cartilage oligomeric matrix protein
D The D period of the collagen fibril
D[D1] Population of fibrils with smaller diameter (relative to D[D2])
D[D2] Population of fibrils with larger diameter (relative to D[D1])
E[CF] Tensile stiffness of the collagen
F Axial force generated within the collagen molecule
F[max] Breaking force of the collagen molecule
G[m] Shear modulus of the interfibrillar matrix
H Constant in the differential equation of the shear-lag model
L[crit] Critical length for fibril fracture
L[CF] Half-length of the fibril
L[CL] Contact length between two adjacent collagen molecule
L[TC] Length of the collagen molecule
m[l] Collagen mass per unit length
M Mass of a collagen molecule
N Number of molecules intersecting a fibril cross-section (through an overlap region)
N Amino-terminus (containing an amine group) of the collagen molecule
q Fibril aspect ratio
(r,θ,z) Radial, azimuthal and axial coordinates, respectively, of the cylindrical polar coordinate system
r[m] Radius of the matrix surrounding the fibril
r[0] Radius of the (uniform cylindrical) fibril
u[CF] Axial displacement of the fibril at that point within the fibril
u[m] Axial displacement of the interfibrillar matrix at the same point if the fibril were not presence
u[E] Strain energy density, relates to the tissue resilience
u[F] Strain energy density, relates to the tissue rupture
v[c] Poisson’s ratio of the tissue
v[CF] Poisson’s ratio of the collagen fibril
v[m] Poisson’s ratio of the interfibrillar matrix
V[CF] Volume fraction of collagen fibrils
V[m] Volume fraction of interfibrillar matrix
Z Normalized axial coordinate
Greek Symbols
α[TC] Molecular cross-sectional area
β[Cox] Constant in the shear-lag model; appears in the argument of the trigonometrical functions
ε Average strain in the fibril
γ[TC] Energy required to nucleate a slip pulse
λ Axial overlap distance between two adjacent fibrils
ρ Centre-to-centre lateral separation distance between two adjacent fibrils
ρ[Coll] Collagen density
σ[c] Stress acting on the tissue in the direction of the fibril
σ[Grif] Applied tensile stress leading to the rupture of the MCT
σ[TC] Stress associated with F
σ[z] Collagen fibril axial stress
τ[β] Shear stress at the fibril-matrix interface, generated during the mode β transition stage
τ[GAG] Shear stress for rupturing the bonds between proteoglycan glycosaminoglycans
τ[RP] Shear stress at the fibril-matrix interface, generated during the fibril rupture or pull-out
τ Shear stress at the fibril-matrix interface
τ[TC] Shear resistance between the two collagen molecules
η Ratio of L[CL] to L[TC]
χ[S] First critical molecular length
χ[R] Second critical molecular length
Figure 1.
Sketches of the spine-test system of the sea urchin. (
) Cartoons of the sea urchin, represented by a sphere covered in spines, magnified view of the cross-section of the joint of the spine-test system and the hierarchical architecture of the catch
apparatus (CA) tissue. The CA may be regarded as a ligamentous tissue as its ends are embedded in hard tissues of the spine and test; (
) Two positions, i.e., X and Y, of the spine. Symbols: S, spine; NR, nerve ring; Bs, basiepidermal nerve plexus; E, epidermis; L, central ligament; M, spine muscle; T, test. Adapted from Smith et al.
], Hidaka et al. [
] and Motokawa and Fuchigami [
Figure 2.
The design process for a tissue engineering approach.
Left panel
shows a flow-chart of the design process. The focus in this process is on the biomaterial for the scaffold development (highlighted in dark fonts). The flow of the design process is typical of
engineering design, with the following key stages, statement of needs, problem definition, synthesis, analysis and optimization, evaluation and, finally, market [
]. Of note, some of the stages are expected to be iterative.
Right panel
shows the tissue engineering triad, comprising biomaterials, cells and signaling molecules. The engineering triad is linked to the problem definition stage and continues through to the analysis and
optimization stage. The desired specifications for the biomaterial scaffold are outlined in the box based on some of the key arguments developed by Trotter and co-workers [
Figure 3.
Profiles of the stress versus strain curves of mutable collagenous tissues (MCTs). (
) A sketch of the graph of stress versus strain of the CA, sea urchin (
Anthocidaris crassispina
) [
]; (
) A sketch of the graph of stress versus strain of the catch apparatus, sea urchin (
Anthocidaris crassispina
) [
]; (
) A sketch of the graph of stress versus strain of the tube feet tissue, sea urchin (
Paracentrotus lividus
) [
]; Sketches of (
) the graph of displacement versus time, indicating the primary (#1), secondary (#2) and tertiary (#3) phases; thereafter rupture results; (
) the graph of incremental stress versus strain and (
) the graph of stress versus strain (derived from
) of the compass depressor ligament, sea urchin (
Paracentrotus lividus
) [
]; (
) Sketch of graph of stress versus strain of the dermis of the sea cucumber (
Cucumbria frondosa
) [
] for the purpose of comparison with the results from the sea urchin (
). Symbols in the graphs: ACh represents acetylcholine; ASW, artificial sea water; EGTA, ethylene-
-(oxyethylenenitrilo)-tetraacetic acid (calcium chelator); TX, Triton X100.
Figure 4. General model of collagen fibril in extracellular matrix (ECM). (A) An array of parallel collagen fibrils embedded in the ECM. The vertical dard bands and light shades represent the
D-periodic patterns. (B) Interaction of collagen fibrils in the matrix. Here the interaction is assumed to be aided somewhat by proteoglycans and glycosaminoglycans, although the exact identity of
the proteoglycans has yet to be determined. Not shown in this schematic are the glycoproteins. (C) A single collagen fibril modelled as a uniform cylinder. The fibril centre, O, defines the origin of
the cylindrical polar coordinate system (r,θ,z), where the z axis coincides with the axis of the fibril. Of note, the single fibril-matrix model in part C provides the basic “template” for many of
the discussions in this review where stress uptake in the fibril is the key concerned (see Figures 7A and 10A for similar schematics).
Figure 5.
Fibre-matrix interfacial shear stress, τ, distributions [
]. (
) Shear-lag model; (
) Shear-sliding model. Here
represents the normalized coordinate, i.e.,
, where
is the
coordinate of the cylindrical polar coordinate system and
represents the half-length of the fibril.
is used to describe the distance along the fibre axis from the fibre centre,
= 0, to the respective fibre ends,
= 1 or −1.
Figure 6.
Schematic of collagen molecules in tension in collagen fibrils. (
) the Buehler bimolecular model [
], i.e., two collagen molecules sliding under a tensile force,
. Symbol
represents the length of the molecule; (
) the axial-staggering of collagen molecules in a fibril. The staggered arrangement gives rise to light-dark bands (i.e., the D-periodic patterns) along the collagen fibril. Symbols
represents the
period of the collagen fibril; N and C denote the amino-terminus (containing an amine group) and C-terminus (containing carboxyl group) of the collagen molecule, respectively; (
) Two adjacent collagen fibrils.
Figure 7.
Collagen fibril axial stress, σ
, distributions. (
) Model of connective tissue featuring a collagen fibril embedded in ECM. The proposed interfacial shear stress distributions in the (
) Shear-lag and (
) Shear-sliding models for collagen fibril biomechanics [
]. In part B and C, symbols
represents the force acting on the ECM (red arrow represents the direction of F); σ
represents the stress acting on the tissue in the direction of the fibril,
represents the radius of the matrix surrounding the fibril;
represents the radius of the fibril;
represents the half-length of the fibril;
are coordinates of the cylindrical polar coordinate system;
represents the normalized coordinate of
which is intended to describe the fractional distance along the fibril axis from the fibre centre,
= 0 (i.e., O), to the respective fibre ends,
= 1 or −1; E and E’ represent the ends of a fibril (
Figure 6
Figure 8.
Schematic of tissue rupture. The diagram shows a snapshot of the microenvironment of ECM undergoing failure. These failures are identified as a small crack in ECM, rupture of ECM and bridging of the
ruptured site by intact collagen fibrils; at the ruptured site of ECM, fibrils may also be pulled out or fractured. Adapted from Goh et al. [
Figure 9. Schematics of the cross section of fibre reinforced composites. (A) Continuous uniform cylindrical fibre reinforced composite (left panel) from a 3D perspective. Corresponding 2D
perspective showing the plane of interest (POI) containing the cross-sections of the uniform cylindrical fibre (right panel); (B) Discontinuous paraboloidal fibre reinforced composite (left panel, 3D
perspective). Corresponding plane of interest (POI) showing the cross-sections of the paraboloidal fibre (right panel, 2D perspective). The fibres numbered, 1–8, in the 3D and 2D schematics are
intended to illustrate their associations between the two views. In part (A,B), the force acting on the respective composites is in the direction of the fibre axis.
Figure 10. Tapered fibril reinforcing connective tissue. (A) A fibril with conical ends, concentrically arranged within the ECM. In this general model (see illustrations at the bottom and middle
panels), the fibril possesses mirror symmetry about the fibril centre, O, and axis symmetry, which defines the z-axis of the cylindrical polar coordinate system, so that one-quadrant of the complete
model (see illustration at the top panel) need only to be illustrated. The fibril has a radius, r[0], and a half-length, L[CF]; r[m] represents the radius of the model. The stress acting on the model
is represented by σ[c], acting in the direction of the axis. The other fibril shapes, namely a fibril with paraboloidal ends, and an ellipsoidal fibril are depicted in (B,C), respectively. These
models also adopt similar assumptions of mirror and axis symmetry developed for the conical shape so that only one-quadrant of the complete model is illustrated in the respective subfigure (B,C). (D)
Graph of normalized fibril axial mass, m[l]/ρπ, versus Z from the centre to the end of collagen fibril for the respective shapes. The graphs are obtained by evaluating the respective Equations (15)–
(18). Here, m[l] and ρ[Coll] represent the collagen mass per unit length and density, respectively.
Figure 11.
The stress distributions along the fibril axis for collagen fibrils, modelled by four different fibril shapes, namely conical ends, paraboloidal ends, ellipsoid and uniform cylinder, undergoing
elastic stress transfer (
) and plastic stress transfer (
). Sketches of the (
) graph of normalized axial stress, σ
, [
] and (
) graph of interfacial shear stress, τ/σ
, [
] versus fractional distance along the fibril axis,
. The results were evaluated at fibril aspect ratio,
= 3500, and relative stiffness of the fibril to the matrix,
= 10
. (
) Graph of normalized axial stress, σ
, versus fractional distance along the fibril axis,
obtained by evaluating the stress equations of the respective fibre shapes [
]. All graphs are shown for the stress plotted from the fibril centre (
= 0) to one end (
= 1). Here, symbols σ
represents the applied stress acting on the tissue in the direction of the fibril, τ represents the interfacial shear stress,
represents the radius of the matrix surrounding the fibril;
is the
coordinate of the cylindrical polar coordinate system and
represents the half-length of the fibril.
Figure 12.
Effects of fibril aspect ratio,
, and ratio of moduli of the fibril to the interfibrillar matrix,
, on the axial stress, σ
, in a fibril. Sketches of the (
) graph of normalized axial stress, σ
, versus fraction distance,
, along the fibril and the associated (
) graph of σ
at the fibril centre (
= 0) versus
) during elastic stress transfer [
]. Graphs of the (
) normalized axial stress, σ
/τ, versus
along the fibril and the associated (
) graph of maximum σ
/τ (at
= 0) versus
during plastic stress transfer; the results are obtained by evaluating the stress equation derived for the fibre with paraboloidal ends [
]. Thus, all results shown here apply to the fibril with a paraboloidal shape. The
values range 200 to 3500 (the arrow in part (
) indicates increasing
value). Of note, the authors of the paper describing these computer models have made clear the difficulties in meshing the model beyond an aspect ratio of 3500 and have defined a strategy that limits
the analysis to within the constraints of the models; further details can be found in the reference [
]. Symbol σ
represents the applied stress acting on the tissue in the direction of the fibril and τ represents the fibril-matrix interfacial shear stress.
Figure 13. Model of ECM containing short (uniform cylindrical) collagen fibrils arranged in the square-diagonally packed configuration. (A) A cross-sectional (plane of interest, POI) view; (B) The
longitudinal view of the unit cell. In part (A), α refers to the primary fibril of interest; surrounding the α fibril are the β (secondary) fibrils of interest. Here, RVE represents representative
volume element; λ and ρ represent the fibril-fibril axial overlap distance and the centre-to-centre lateral separation distance, respectively.
Figure 14.
Fibril-fibril interaction. Sketches of the graph of axial tensile stress, σ
, in a fibril versus distance,
, along the fibril axis (where
= 0 and 1 correspond to the fibril centre and end, respectively) at (
) λ/
= 0; (
) λ/
=1/4 and (
) λ/
= 3/4 for the uniform cylindrical shape at varying fibril-fibril separation distance,
, adapted from the report of Mohonee and Goh [
]. Insets (right of each graph) show representative volume elements (RVEs,
Figure 13
) of fibrils embedded in the matrix at different overlap distance. In the report of Mohonee and Goh, all results have been obtained by setting the ratio of the stiffnesses of the fibril to the
, equal to 10
(“low”) and the fibril aspect ratio
= 650 (“high”). Symbol σ
represents the applied stress acting on the tissue in the direction of the fibril,
represents the half-length of the fibril and
represents the fibril radius (for the tapered fibril, this refers to the radisu at the fibril centre).
Figure 15.
Framework of the ECM mechanics for the MCT [
]. The framework provides a systematic approach to map the mechanisms involve in regulating the mechanical response of ECM at the respective loading regimes, labelled 1–5. These mechanisms are
identified across the length scale from molecular to bulk tissue level. At the tissue level, the graph illustrates a schematic representation of typical MCT stress-strain behaviour.
Tissue Maximum Stress (MPa) Stiffness (MPa) Maximum Strain Literature
Catch apparatus, sea urchin 40 ^† 400 ^† 0.3 ^† [15,16]
Tube feet, sea urchin 200 ^# 2000 ^# 2.0 ^# [70]
Compass depressor, sea urchin 19.5 ± 5.5 40.0 ± 22.3 3.0 ± 2.4 [66]
^† indicates that the values are estimates derived from the graphs of stress versus strain. ^# indicates that the values are estimates derived from the bar-charts of the respective mechanical
Table 2. Examples of the length, diameter and aspect ratio of collagen fibrils in marine invertebrates as well as land vertebrates.
Tissue Length (L[CF]), (μm) Diameter (2r[0]), (nm) Aspect ratio (q) Literature
Marine invertebrates
Dermis, starfish (Asterias amurensi) 196 ^¥ 136 ^¥ 1441 ^‡ [132]
Dermis, sea cucumber (Cucumaria frondosa) 13.8–443.6 ^@ 75 ^# 184–5914 ^‡ [133]
Catch apparatus, sea urchin (Eucidaris tribuloides) - - 2275–3300 ^† [22]
Catch apparatus, sea urchin (Eucidaris tribuloides) 234 * 75 ^# 3120 ^‡ [126]
Skin, calf (acid-extracted collagen) 9 * 21 ^¥ 423 ^‡ [134,135]
Tendon, embryonic chick 18 * 50 ^# 360 ^‡ [122]
Medial collateral knee ligament, rat 21 * 75 ^# 282 ^‡ [120]
^¥ These are simple averages. ^‡ These are derived from the ratio of the average length to the average diameter. ^@ These are broadly observed by the authors of the paper. ^# These are estimated from
the electron micrographs presented in the paper. ^† These are derived from the gradient of a straight line fitted to data points of length versus diameter. * These are estimated values derived from
computing the mid-value between the lower and upper limit.
Table 3. Estimates of fibrillar and matrix-related Poisson’s ratio and modulus of elasticity parameters for understanding the behaviour of the interfibrillar matrix.
Parameters Magnitudes Literature
Poisson’s ratio of collagen fibril, v[CF] 2 [156]
Volume fraction of collagen, V[CF] 0.2–0.8 [40]
Poisson’s ratio of MCT, v[c] 0.7–4.2 [157]
Poisson’s ratio of interfibrillar matrix, v[m] 3–18 This review, using Equation (20)
E[CF]/E[m] 10^3–10^6 [80,112]
Interfibrillar Shear Stress Magnitude (MPa) Literature
Shear stress, τ[β], generated at the fibril/matrix interface during the mode β transition stage 1–10 [58]
Shear stress, τ[RP], generated at the fibril/matrix interface during the fibril rupture or pull-out 100 [58]
Shear stress, τ[GAG], for rupturing the bonds between proteoglycan glycosaminoglycans using optical tweezers 7.5 [58,166]
Interfibrillar shear stress, τ, by notched tissue testing 32 [109]
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Goh, K.L.; Holmes, D.F. Collagenous Extracellular Matrix Biomaterials for Tissue Engineering: Lessons from the Common Sea Urchin Tissue. Int. J. Mol. Sci. 2017, 18, 901. https://doi.org/10.3390/
AMA Style
Goh KL, Holmes DF. Collagenous Extracellular Matrix Biomaterials for Tissue Engineering: Lessons from the Common Sea Urchin Tissue. International Journal of Molecular Sciences. 2017; 18(5):901.
Chicago/Turabian Style
Goh, Kheng Lim, and David F. Holmes. 2017. "Collagenous Extracellular Matrix Biomaterials for Tissue Engineering: Lessons from the Common Sea Urchin Tissue" International Journal of Molecular
Sciences 18, no. 5: 901. https://doi.org/10.3390/ijms18050901
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
Article Metrics | {"url":"https://www.mdpi.com/1422-0067/18/5/901","timestamp":"2024-11-01T20:50:21Z","content_type":"text/html","content_length":"806322","record_id":"<urn:uuid:87b84f25-cdf9-41e7-b1ff-b8fe378cf49d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00833.warc.gz"} |
279 research outputs found
For $G$-monogenic mappings taking values in the algebra of complex quaternion we prove a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies on the
boundary of a domain.Comment: submitted to International Journal of Advanced Research in Mathematic
A universal method of extraction of the complex dielectric function $\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$ from experimentally accessible optical quantities is developed. The
central idea is that $\epsilon_{2}(\omega)$ is parameterized independently at each node of a properly chosen anchor frequency mesh, while $\epsilon_{1}(\omega)$ is dynamically coupled to $\epsilon_
{2}(\omega)$ by the Kramers-Kronig (KK) transformation. This approach can be regarded as a limiting case of the multi-oscillator fitting of spectra, when the number of oscillators is of the order of
the number of experimental points. In the case of the normal-incidence reflectivity from a semi-infinite isotropic sample the new method gives essentially the same result as the conventional KK
transformation of reflectivity. In contrast to the conventional approaches, the proposed technique is applicable, without readaptation, to virtually all types of linear-response optical measurements,
or arbitrary combinations of measurements, such as reflectivity, transmission, ellipsometry {\it etc.}, done on different types of samples, including thin films and anisotropic crystals.Comment: 10
pages, 7 figure
We consider a class of so-called quaternionic G-monogenic mappings associated with m-dimensional (m 2 f2; 3; 4g) partial differential equations and propose a description of all mappings from this
class by using four analytic functions of complex variable. For G-monogenic mappings we generalize some analogues of classical integral theorems of the holomorphic function theory of the complex
variable (the surface and the curvilinear Cauchy integral theorems, the Cauchy integral formula, the Morera theorem), and Taylor’s and Laurent’s expansions. Moreover, we investigated the relation
between G-monogenic and H-monogenic (differentiable in the sense of Hausdorff) quaternionic mappings
For G-monogenic mappings taking values in the algebra of complex quaternions we generalize some analogues of classical integral theorems of the holomorphic function theory of a complex variable (the
surface and the curvilinear Cauchy integral theorems)
The purpose. The purpose of this study was to evaluate the effectiveness of incentive spirometry (IS) as a method of atelectasis prevention in patients with moderate or high risk of PPCs development
after upper abdominal surgery. Materials and methods. The study consisted of two stages. The first retrospective stage was to analyze the medical histories data of 51 inpatients, who were included in
the comparison group. The prospective part of the study included 39 patients of the study group, who had sessions of the IS during the first 7 days of the postoperative period. Patients of both
groups were operated on the upper abdominal organs by open procedure, operation time was more than 2 hours, all patients had an ARISCAT score ≥26 points. Pulmonary atelectasis development was
monitored in the groups in the first week of the postoperative period. The statistical analysis of the data was performed using the Microsoft Excel 2013 and Statistica for Windows 6.0 programs. When
comparing the groups according to the clinical outcome, the relative risk (RR) and odds ratio (OR) were determined and then confidence intervals (95 % CI) were calculated. Statistical significance of
the results was determined depending on the CI values. Results. During the first 7 days, 34 cases of pulmonary atelectasis (67 %) were recorded in the comparison group. In the study group, 9 patients
(23 %) were diagnosed with pulmonary atelectasis. The analysis of clinical results showed that when applying incentive spirometry, there was a statistically significant decrease in the relative risk
of atelectasis development within the first week of the postoperative period (RR = 0.346, 95 % CI [0.189; 0.634], P = 0.0006). The odds ratio of atelectasis development in the study group was
statistically lower than in the group of retrospective study (OR = 0.150, 95 % CI [0.058, 0.386], P = 0.0001). Conclusions. Incentive spirometry is an effective way to prevent pulmonary atelectasis
in patients with a moderate or high risk for developing postoperative pulmonary complications according to the ARISCAT scale after upper abdominal surgery
Infrared (IR) spectroscopy can be used as an important and effective tool for probing periodic networks of quantum wires or nanotubes (quantum crossbars, QCB) at finite frequencies far from the
Luttinger liquid fixed point. Plasmon excitations in QCB may be involved in resonance diffraction of incident electromagnetic waves and in optical absorption in the IR part of the spectrum. Direct
absorption of external electric field in QCB strongly depends on the direction of the wave vector ${\bf q}.$ This results in two types of $1D\to 2D$ dimensional crossover with varying angle of an
incident wave or its frequency. In the case of QCB interacting with semiconductor substrate, capacitive contact between them does not destroy the Luttinger liquid character of the long wave QCB
excitations. However, the dielectric losses on a substrate surface are significantly changed due to appearance of additional Landau damping. The latter is initiated by diffraction processes on QCB
superlattice and manifests itself as strong but narrow absorption peaks lying below the damping region of an isolated substrate.SubmiComment: Submitted to Phys. Rev.
We show that the optical transparency of suspended graphene is defined by the fine structure constant, alpha, the parameter that describes coupling between light and relativistic electrons and is
traditionally associated with quantum electrodynamics rather than condensed matter physics. Despite being only one atom thick, graphene is found to absorb a significant (pi times alpha=2.3%) fraction
of incident white light, which is a consequence of graphene's unique electronic structure. This value translates into universal dynamic conductivity G =e^2/4h_bar within a few percent accuracy | {"url":"https://core.ac.uk/search/?q=author%3A(Kuzmenko%2C%20T.%20S.)","timestamp":"2024-11-06T23:48:31Z","content_type":"text/html","content_length":"129521","record_id":"<urn:uuid:17078d84-e79b-4811-9d66-8522199e668e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00072.warc.gz"} |
Monte Carlo Multiple Integration
6467 Views
3 Replies
3 Total Likes
Monte Carlo Multiple Integration
I am working on a problem and have gotten stuck on trying to find the volume of the solid that lies over a region and is between surfaces. I am trying to write the code for randomly putting dots
inside and outside the object and the using the ratio of inside to outside to find the volume? I have attached a notebook of my progress so far as well as a word ducument of the problem. Any help
would be greatly appreciated.
3 Replies
I think there is an error in the Mathematica notebook attached to the original question -- fun5 is defined with lower case variables (x and y) instead of upper case variables (X and Y) as the rest of
the functions fun1, fun2, etc.
You can use NIntegrate to verify the implementations in this project. Below is some code that might help.
Clear[fun1, fun2, fun3, fun4, fun5]
fun1[X_, Y_] := X^2 + Y^3;
fun2[X_, Y_] := X^2/4 + Y^2;
fun3[X_, Y_] := X - Y^2;
fun4[X_, Y_] := Y + 3 X^2;
fun5[x_, y_] := Cos[y^2] + E^-x^2;
First let us use NIntegrate's MonteCarlo integration method:
In[468]:= NIntegrate[
Boole[fun1[X, Y] >= 4 || fun2[X, Y] >= 2 || fun3[X, Y] >= 1 ||
fun4[X, Y] <= -1/4 || fun5[X, Y] <= 0], {X, -2, 2}, {Y, -2, 2}, {Z,
0, 2}, Method -> "MonteCarlo"]
Out[468]= 16.1861
The result is close to the result given by the default non-Monte Carlo integration method:
In[463]:= NIntegrate[
Boole[fun1[X, Y] >= 4 || fun2[X, Y] >= 2 || fun3[X, Y] >= 1 ||
fun4[X, Y] <= -1/4 || fun5[X, Y] <= 0], {X, -2, 2}, {Y, -2, 2}, {Z,
0, 2}, PrecisionGoal -> 2]
Out[463]= 16.2698
Here is how we can see the sampling points:
res = Reap[
Boole[fun1[X, Y] >= 4 || fun2[X, Y] >= 2 || fun3[X, Y] >= 1 ||
fun4[X, Y] <= -1/4 || fun5[X, Y] <= 0], {X, -2, 2}, {Y, -2,
2}, {Z, 0, 2}, Method -> "MonteCarlo",
EvaluationMonitor :> Sow[{X, Y, Z}]]];
Using Select we can plot the points for which the condition given to Boole gives True:
Point[Select[res[[2, 1]],
Block[{X, Y, Z}, {X, Y, Z} = #;
fun1[X, Y] >= 4 || fun2[X, Y] >= 2 || fun3[X, Y] >= 1 ||
fun4[X, Y] <= -1/4 || fun5[X, Y] <= 0] &]]]
So far I haven't really goten any wrong answers as far as I know. On the first part where I am simply trying to find the area of the object, I believe the code is good. When it comes to adding the
upper surface I realy don't know where to begin actually trying to add this element into the code. So far I have tried adding a Z plane and giving it limits. Would any one know if this is even the
right approch?
Your post hasn't precisely said what incorrect result you are seeing, unless it might be in your Word document.
I am guessing it might be in your last calculation.
Suggestion: I find I make fewer errors if I simply copy the original inequalities into the If,
rather than trying to manually reverse the direction of the inequalities, make extra equations, etc.
In your code this would then mean I would have to swap the position of Pin and Pout,
but I would hopefully make fewer errors doing it that way.
Suggestion: I might look really carefully at the volume of the box you are using in your last calculation.
You are also calculating NumberInside/NumberOutside*VolumeOfBox. Is that really correct?
That is not the same calculation you were doing further up in your earlier examples.
Suggestion: If you find and correct some mistakes I urge you not to say "that was a stupid mistake"
and try to as quickly as possible erase every trace of that. Instead I suggest you try to think of why
that sort of mistake might have happened and what rule or practice or habit you might follow in the
future which would try to ensure that this class or mistake not happen again. With careful observation
and thought you might be able to avoid some or even most these kinds of errors in the future.
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/237330","timestamp":"2024-11-12T07:13:18Z","content_type":"text/html","content_length":"108232","record_id":"<urn:uuid:decdb707-2043-4d9f-8e5b-df653ac8c8db>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00019.warc.gz"} |
Students’ Progress Evaluation (2015-2018)
3 January 2020
We analysed the progress and impact of the 16 students who have joined Iskul since the inception for 3 consecutive years from 2015-2018. The 16 students were given Test 1 on Sept 2016, Test 2 eight
months after, followed by Test 3, a year later (March 2018). Assumptions made for this analysis are:
• There were a minimum of 2 hours per class, 2 classes per week
• Students’ attendance is at an average of 70%
Using these assumptions and data, according to national school days (5 hours/class, 5 days/week), they attended about 70 days or 14 weeks (less than 4 months) of school over the past three years.
Overall, we are pleased with the result considering that the actual time our 16 students spent on a normal schooling period is just less than 4 months. Based on our observation, Iskul students have
mastered basic numeracy, however, they are still mostly weak in the Bahasa Malaysia language, although most of them have no problem understanding and engage in basic conversation in Bahasa Malaysia.
The greatest outcome is that two of our students from this test have now become Mastal Arikik (MA) to teach the new batch of students.
From the result, we noticed the shift in students’ progress when we provide teaching camps to our MAs to improve their teaching quality and when we find SPM graduates to teach. We also like to note
that the involvement of our Headmistress in teaching demonstrated that students learned faster.
In a nutshell, we found that MAs with better results can teach better and have better skills in transferring knowledge to the students, although it is not as efficient as having an adult/qualified
teacher to conduct the teaching.
Moving forward, Iskul aims to:
• focus on language mastery
• consult a language (BM) teacher
• create a learning-enabled environment
• employ an adult teacher
Iskul feels strongly that if the children do not continue practising what they learned, they will forget as time passes and it would be a waste. Therefore, for the graduating students, Iskul hopes to
create programme(s) to encourage the students to continue learning and contribute to Iskul long-term.
The analysis’ results are divided into three parts:
• Result 1: The basic evaluation criteria used during Test 1 for all three tests
• Result 2: Further evaluation on Mathematics in Test 3
• Result 3: Further evaluation on Bahasa Malaysia in Test 3
Result 1: The basic evaluation criteria used during Test 1 for all three tests
Graph 1. Progress Analysis for 16 Iskul Students who have taken the Evaluation for 3 Consecutive Years
As shown in Graph 1, there is a steady increase in the number of students who can perform all five basic criteria of assessment from Test 1 to Test 3.
By Test 3, all 16 students can write A to Z and read and write number 1 to 10. All except for 1 can write his/her own name. Those who can do simple 1 digit addition increased by 19% to 14 students
from Test 2. Oddly, two students digress from being able to write the number 1-50 to only 1-20. Further investigation reveals that their attendance is an average of 60%.
Result 2: Further evaluation of Mathematics in Test 3
In the further evaluation of Mathematics, the students were tested on how many digits they can write and recognise, reading clock and questions related to addition, subtraction, multiplication and
Graph 2. Read and Write Number and Read Clock Result in Test 3
Graph 2 shows that all students can write the basic number from 1-10, while only 10 students can write from 1-50. Interestingly, 1 student can write up to 1,000 and 5 can write up to 100. With
regards to reading time, about 94% of the students can read only up to the “hour” hand.
Graph 3. Mathematical operations of addition, subtraction, multiplication and division according to digits
We’re delighted to note that more than 75% of our students can solve additional operation questions up to 4-digits. More than 70% of them can solve subtraction questions up to 2-digits while only 43%
of them can do up to 4-digits. Furthermore, about half of them can memorise the multiplication table from 1 to 4. However, a majority of them are still unable to solve division problems.
Interestingly, one student has mastered the basic arithmetics – able to memorise multiplication table up to 12 and solve up to 4-digits division questions
Result 3: Further evaluation on Bahasa Malaysia in Test 3
Test 3 for Bahasa Malaysia evaluates students on two things: (1) recognising body parts and (2) recognising terms for family members.
For the former, 10 of the students can identify the whole body parts while the remaining 6 are only able to identify “rambut”, “mata”, “hidung”, “kening”, “ibu jari”, “kaki”, “perut” and “dahi” only.
(Refer to Graph 4)
As for the latter, firstly only 10 students can recognise the term for parents. Secondly, more than six students who can recognise terms for close family members (i.e., parents and siblings).
Finally, a average of 39% of our students able to recognise terms for extended family members (i.e., grandparents, relatives) (Refer to Graph 5)
Graph 4. The number of students who can recognise Body Parts in Test 3
Graph 5. The number of students who can recognise terms for family members in Test 3. | {"url":"https://iskul.my/students-progress-evaluation-2015-2018/","timestamp":"2024-11-10T18:48:28Z","content_type":"text/html","content_length":"155952","record_id":"<urn:uuid:314c09bd-2c57-497a-a1d9-55fd44367357>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00772.warc.gz"} |
Math Kindergarten Quiz Compose simple shapes to form larger shapes For example “Can you join these two triangles with full sides touching to make a rectangle?”-.K.G.B.6Math Kindergarten Quiz Compose simple shapes to form larger shapes For example “Can you join these two triangles with full sides touching to make a rectangle?”-.K.G.B.6 | Tutorified.com Quizzes
Math Kindergarten Quiz Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”-.K.G.B.6
This standard emphasizes the ability to understand how smaller shapes can combine to form larger shapes. Students are encouraged to see how simple shapes can be pieced together in various ways to
create new shapes. The process aids in fostering spatial reasoning and understanding the relationships between different shapes. Through hands-on activities and exploration, students see the
interconnectedness of geometric forms. | {"url":"https://quizzes.tutorified.com/quizzes/math-kindergarten-quiz-compose-simple-shapes-to-form-larger-shapes-for-example-can-you-join-these-two-triangles-with-full-sides-touching-to-make-a-rectangle-k-g-b-6/","timestamp":"2024-11-06T18:25:34Z","content_type":"text/html","content_length":"128892","record_id":"<urn:uuid:d818f82e-6816-413e-9234-fe8c7954cfbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00644.warc.gz"} |
Want to Become Data Scientist? Get 26 Interview Questions!
Oct 20, 2022 By GigNets
The professional career of a Data Scientist seems to be one of the most promising and lucrative ones. The professionals are doing extraordinarily well in the industry and making things big. If you
also want to crack the same and have a fulfilling career, this is high time to start preparing for the interview. This post talks about the interview questions that you must consider to crack the
interview for Data Science.
1. Does the gradient descent method generally converge to similar points?
Normally it does not. In a few instances, these methods may reach a local optimum or local minimal point. It is unlikely to reach the global optima point. It is basically controlled by data and
starting conditions.
2. What is cross-validation?
This is a unique validation method that evaluates how the outcome of some statistical analysis is to generalize to independent data set. This is basically used in the backgrounds where the objective
is to forecast, and some might want to estimate how a model will accomplish.
3. Explain the goal of the A/B testing
This statistical hypothesis testing is done for the randomized experiments with the two variables like A and B. Objective of this testing is to detect any alterations to the web page for maximizing
the outcome of some strategy.
4. State the law of the large numbers
This theorem is to describe the results of effectively performing the same experiment frequently. It forms the basis of frequency-style thinking. It tells that sample standard, sample variance, and
sample mean deviation converge to what they really want to estimate.
5. What are some drawbacks of the linear model?
A few of the drawbacks are:
• Assumption of the linearity of errors
• This cannot be utilized for the binary outcomes or count outcomes
• Some over-fitting problems cannot be resolved
6. How frequently should an algorithm be updated?
You are supposed to update an algorithm if:
• Underlying data source is altering
• Model is to evolve as the data streams through infrastructure
• It is the case of non-stationarity
7. What do you know by confounding variables?
These variables are considered to be extraneous variables in a statistical model that basically correlates inversely or directly with both independent and dependent variables. Its estimation really
fails to account for the confounding factor
8. Explain the star schema
It is nothing but some traditional database schema with a central table. The satellite tables are to map IDS to the physical descriptions or names, and it can be connected to the central fact that
utilizing the ID field. Such tables are regarded as lookup tables and are quite useful in different real-time applications. In some cases, star schemas involve various layers of summarization for
recovering information quickly.
9. How to work towards a random forest?
The principle of the technique is the combination of a few weak learners to create a strong learner. The steps are:
• Build a few decision trees on the bootstrapped training samples of the data
• Apply the rule of thumb at every split m=p√m=p
• The prediction should be done using the majority rule.
10. What do you understand by eigenvalue and eigenvector?
Eigenvalues are directions along which some specific linear transformation acts by stretching, compressing, and flipping.
The eigenvectors are used to understand the linear transformation better. In data analysis, eigenvectors are calculated for correlation or the covariance matrix.
11. What do you know by survivorship bias?
This is some kind of logical error of properly focusing on aspects that support surviving a method and overlooking those that did not exist due to its lack of prominence. It is to leads to the wrong
conclusions in various ways.
12. Why is re-sampling done?
It is done for:
• Validating models by utilizing random subsets (cross-validation, bootstrapping)
• Substituting labels on the data points while performing important tests
• Drawing with the replacement from the set of data points or estimating the accuracy of the sample statistics by utilizing subsets of the assessable data
13. What kinds of biases can generally occur during a sampling?
There are basically 3 kinds of biases that can occur such as survivorship bias, under-coverage bias, and selection bias.
14. Explain selection bias
This is such a problematic situation where the error is introduced due to the non-random population sample.
15. What kind of cross-validation are you to use on the time series data set?
Time-series data is not some randomly distributed data. The chronological order rather inherently orders this. Regarding time series data, you are to use effective techniques such as forward
chaining. In this technique, you are to model past data first and only then look at the forward-facing data.
Fold 1: training 1, test 2
Fold 1: training 1 2 3, test 4
Fold 1: training 1 2, test 3
Fold 1: training 1 2 3 4, test 5
16. Explain logistic regression. Give an example where you can use this regression method lately.
This regression model is such a legit model that it is used to predict the binary outcome from the linear combination of the predictor variables. For instance, regarding predicting whether a specific
political leader is to win an election or not, the outcome of the prediction is to be binary, i.e., 1 or 0 (Win/Lose). Predictor variables, in this case, are the amount of time and amount of money
spent on campaigning.
17. What do you understand by Box-Cox Transformation?
Dependent variables for regression analysis may not satisfy one or several assumptions of ordinary least squares regression. Residual could follow skewed distributions or curve as prediction
increases. In this regard, this is important for transforming response variables for data to meet needed assumptions.
Box-Cox Transformation is such a statistical technique that generally transforms non-normal dependent variables into a normal shape. In the case given, data is not to be normal. Most statistics
assume normality. Applying this transformation technique lets you run a broader number of tests.
18. Explain bias
Bias is nothing but an error introduced in the model for its over-simplification of the machine learning algorithm. This is to lead to under-fitting. If you train a model at this time, then the model
makes simplified assumptions for making the target function quite easier to understand.
Examples of high bias in ML algorithms are logistic regression and linear regression. Examples of low-bias ML algorithms are SVM, k-NN, etc.
19. What is variance in machine learning?
Variance is also known to be an error introduced in the model of ML. A model is to learn noise from the training data set and thus performs badly on the test data set. This ultimately leads to
over-fitting and high sensitivity.
If you increase the complexity of a model, you are supposed to find a reduction in this error to the lower bias in the model. It is to happen till some specific point. If you proceed to make the
model even more complex, then you would end up overfitting the model.
20. Explain exploding gradients
The gradient is the magnitude and direction calculated during the training of some neural networks which is generally used for updating network weights in the right direction and the right amount.
Exploding gradients are such problems where several large error gradients are to accumulate. It eventually results in large updates to the neural network model weights at the time of training. Values
of the weights could become very large to overflow it to result in the NaN values.
21. What is a decision tree algorithm?
This is such a supervised machine learning algorithm that is used for classification and regression. This is to break down the data set into its smaller subset while an associated decision tree is
developed simultaneously. The final result becomes a tree with the left nodes and decision nodes. The decision tree can handle both the numerical data and categorical data.
22. What do you understand by Entropy in the decision tree?
A decision tree is built top-down from the root node, which involves data partitioning into homogeneous subsets. ID3 is the algorithm that is used to build a decision tree. ID3 is known to use
Entropy for checking the homogeneity of some samples. In case some sample is fully homogeneous, Entropy becomes zero. But if that sample is equally divided, this is to have an Entropy of 1.
23. State information gained in the decision tree
Information gain is generally based on the decrease in the Entropy after some dataset gets split on the attribute. Building the decision tree is about finding essential attributes that return the
highest information gain.
24. Explain ensemble learning
Ensemble learning is nothing but effectively combining diverse sets of the learner to improvise on the model’s predictive power and stability. This has plenty of types, but two of its popular
learning techniques are bagging and boosting.
The technique implements similar learners on the sample populations and takes the mean of all predictions. Regardinggeneralized bagging, you are a on various populations. Illowed to utilize various
learners will reduce variance error.
This is such an iterative technique that adjusts the weight of any observation based on the last classification. If some observation is classified wrongly, this increases the weight of that
25. How is logistic regression generally done?
This regression method effectively measures the relationship between a dependent variable and one or several independent variables by properly estimating probability utilizing the underlying
logistics functions.
26. How to maintain a deployed model?
There are steps that you need to follow for maintaining a deployed model.
• Monitoring
You need to constantly monitor all the models to determine their performance accuracy. If you make some changes, then find out how these changes may affect things. This monitoring is extremely needed
for its proper functions.
• Evaluation
Evaluation metrics of the current model are properly calculated for determining whether the latest algorithm is required or not.
• Comparison
New models are to be compared with each other for determining which model is to perform best.
• Rebuild
The best-performing model is supposed to be rebuilt on the current state of the data.
There’s no doubt that Data Science is one of the most promising but difficult careers one can pursue. If you want to excel in the industry, you will have to fluent in all possible queries that you
may be asked anything. Check out the post to know about the most asked interview questions for the Data Science job support.
We will help you and work with your requirements in the most reliable, professional, and at a minimum cost. We can guarantee your success. So call us or WhatsApp us +918900042651 or email us | {"url":"https://proxy-jobsupport.com/want-to-become-data-scientist-get-the-26-interview-questions/","timestamp":"2024-11-09T10:08:47Z","content_type":"text/html","content_length":"78265","record_id":"<urn:uuid:f1846af5-567f-4e92-96fb-3a96fcedce6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00506.warc.gz"} |
The following function in C# .NET 3.5 checks whether a given credit card is valid using the Mod10 or Luhn formula.
For more complete information and explanation of how the formula works, please check out
this website
. I based this function largely from the information I got there.
In summary the formula is as follow:
1) Double each alternative digit, starting from the second digit from the *RIGHT
2) If any digit is equal to 10 or greater, then add up the digits. e.g. "10" would be equal to "1 + 0" or "1"
3) Sum up all the digits.
4) If result is divisible by 10, then we probably have got a valid card number, otherwise it's fake.
This is a necessary, but insufficient check to verify credit card numbers generated by most financial institution. For a more complete check, you'll also need to check the first digits, to make sure
they match the credit card company, eg. Master Card, may start with 51. However this is an easy check and is not a subject of this topic.
Function to check if credit card number is valid, or otherwise using Mod 10, or Luhn Formula:
public bool Mod10Check(string CreditCardNumber)
char[] CreditCardNumberArray = CreditCardNumber.ToCharArray();
var CreditCardDigits = new short[CreditCardNumberArray.Length];
for (int i = 0; i < CreditCardNumberArray.Length; i++)
CreditCardDigits[i] = short.Parse(CreditCardNumberArray[i].ToString());
CreditCardDigits = CreditCardDigits.Reverse().ToArray();
for (int i = 0; i < CreditCardDigits.Length; i++)
if (i%2 == 1)
CreditCardDigits[i] = (short)
if (CreditCardDigits[i]
>= 10)
char[] BigDigit =
CreditCardDigits[i] = (short)
+ short.Parse(BigDigit[1].ToString()));
int SumOfDigits = CreditCardDigits.Sum(o => (int) o);
return SumOfDigits%10 == 0;
Ever wanted to move a whole directory, but .NET would not allow it? The only possible way was to create directory, at target, and copy each files to the new folder and delete the old one.
To make matter worse, you can't copy or move directory to a shared location, with a given UNC path.
The following code is an extension method to allow the built-in .NET DirectoryInfo class, to move an entire folder, even across servers.
///////////////////////////Directory Info Extension///////////////////////////
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.IO;
namespace YourNameSpace
public static class DirectoryInfoExtension
/// Copy directory from one location to another, recursively.
public static DirectoryInfo CopyTo(this DirectoryInfo Source, DirectoryInfo Target, bool Overwrite)
string TargetFullPath = Target.FullName;
if (Overwrite && Target.Exists)
else if (!Overwrite && Target.Exists)
Target.MoveTo(Target.Parent.FullName + "\\" + Target.Name + "." + Guid.NewGuid().ToString());
//Restores target back, such that it's not pointing to the renamed, obsolete directory.
Target = new DirectoryInfo(TargetFullPath);
CopyRecurse(Source, Target);
return Target;
/// Copy source recursively to target.
/// NOTE: This will create target subdirectories, but NOT target itself.
private static void CopyRecurse(DirectoryInfo Source, DirectoryInfo Target)
foreach (DirectoryInfo ChildSource in Source.GetDirectories())
DirectoryInfo ChildTarget = Target.CreateSubdirectory(ChildSource.Name);
CopyRecurse(ChildSource, ChildTarget);
foreach (FileInfo File in Source.GetFiles())
File.CopyTo(Target.FullName + "\\" + File.Name);
/// This extension allows directory to be moved accross servers.
public static DirectoryInfo MoveTo(this DirectoryInfo Source, DirectoryInfo Target, bool Overwrite)
Source.CopyTo(Target, Overwrite);
return Target;
This post is a modification from Rick Strahl's Web Log, to replace string. ignoring case. The code has been modified to suit C# .NET 3.5 coding style - with extension methods.
Recently in one of the project I was working on, I came across a problem where I needed to replace parts of the string, while preserving the case, otherwise. That last part is the problem. While it's
possible to simply call string.ToLower(), it'd result in the whole string converted to lowercase, regardles of whether they match the replacement criteria or otherwise.
The code below is designed to address the issue. It's almost copy and paste, except you need to rename the namespace to that of your solution. Reference the namespace and class, and string.Replace()
will have 2 extra overloads.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace YourNameSpace.SubNameSpace
public static class StringExtension
/// <summary>
/// String replace function that support
/// </summary>
/// <param name="OrigString">Original input string</param>
/// <param name="FindString">The string that is to be replaced</param>
/// <param name="ReplaceWith">The replacement string</param>
/// <param name="Instance">Instance of the FindString that is to be found. if Instance = -1 all are replaced</param>
/// <param name="CaseInsensitive">Case insensitivity flag</param>
/// <returns>updated string or original string if no matches</returns>
public static string Replace(this string OrigString, string FindString,
string ReplaceWith, int Instance,
bool CaseInsensitive)
if (Instance == -1)
return OrigString.Replace(FindString, ReplaceWith, CaseInsensitive);
int at1 = 0;
for (int x = 0; x < Instance; x++)
if (CaseInsensitive)
at1 = OrigString.IndexOf(FindString, at1, OrigString.Length - at1, StringComparison.OrdinalIgnoreCase);
at1 = OrigString.IndexOf(FindString, at1);
if (at1 == -1)
return OrigString;
if (x < Instance - 1)
at1 += FindString.Length;
return OrigString.Substring(0, at1) + ReplaceWith + OrigString.Substring(at1 + FindString.Length);
/// <summary>
/// Replaces a substring within a string with another substring with optional case sensitivity turned off.
/// </summary>
/// <param name="OrigString">String to do replacements on</param>
/// <param name="FindString">The string to find</param>
/// <param name="ReplaceString">The string to replace found string wiht</param>
/// <param name="CaseInsensitive">If true case insensitive search is performed</param>
/// <returns>updated string or original string if no matches</returns>
public static string Replace(this string OrigString, string FindString,
string ReplaceString, bool CaseInsensitive)
int at1 = 0;
while (true)
if (CaseInsensitive)
at1 = OrigString.IndexOf(FindString, at1, OrigString.Length - at1, StringComparison.OrdinalIgnoreCase);
at1 = OrigString.IndexOf(FindString, at1);
if (at1 == -1)
return OrigString;
OrigString = OrigString.Substring(0, at1) + ReplaceString + OrigString.Substring(at1 + FindString.Length);
at1 += ReplaceString.Length;
return OrigString;
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 need 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. | {"url":"http://www.wizxchange.com/2009/01/","timestamp":"2024-11-07T19:07:03Z","content_type":"application/xhtml+xml","content_length":"63908","record_id":"<urn:uuid:caa3ebcf-4110-4479-8681-9df3c585f269>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00774.warc.gz"} |
Matrix exponential
I use Proc IML function in SAS Studio for matrix exponential and the procedure is successful. But when I import the matrix in Excel and call matrix exponential function the results are different from
the results from SAS. Is there anybody familiar in the method behind the procedure for matrix exponential both Excel and SAS?
04-13-2023 03:38 AM | {"url":"https://communities.sas.com/t5/SAS-IML-Software-and-Matrix/Matrix-exponential/td-p/869519","timestamp":"2024-11-11T10:42:47Z","content_type":"text/html","content_length":"251013","record_id":"<urn:uuid:e9a8aef6-a306-428a-b06c-ed68da698d5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00839.warc.gz"} |
11+TUITION IN MANCHESTER | 11+ Tuition & 11+ Mock Tests
11 Plus Tuition in Manchester
Another test taken in the final year of primary school is the 11+ Common Entrance Examinations.
The 11+ Common Entrance Examinations are formulated by the CEM or GL boards mainly (ISEB) and taken mainly by girls for entrance to senior independent girls’ schools at age 11+.Children at the age of
11+ are required to sit examinations in English, Mathematics and Science in November and January of Year 6 prior to entry to senior school the following September.
The 11+ Common Entrance Examinations consists of four sections on which students are tested:
• English
• Maths
• Verbal reasoning (solving logical problems)
• Non-verbal reasoning (solving pictorial problems)
• You can find the 11+ Common Entrance Examinations dates here in our Academic Calendar page.
Dates till not available we will update dates later
11+ ENGLISH
The English paper has three sections: comprehension, punctuation and spelling.
The most important strategy is to read and analyse the questions first. Traditionally, students are taught to read the text first with the premise that if you don’t read the text first, you will not
fully know the underlying context to answer the questions. You will read the text without any insight into what you actually need to find to form a proper answer. Hence, being informed right from the
outset is the key to an attempt that is successful in comprehensions.
It is vitally important to consider that in spelling exercises, you should try to ignore the context of the sentence because it subconsciously puts extra burden on your mind. An effective technique
is to read the sentence backwards word by word to locate the misspelt word and to ignore the context of the sentence altogether.
You can couple this technique with phonetics. If you break down the word into phonetics, you will have a better chance at observing whether the word is spelt right or not.
Lastly, you should now use the systemic check to arrive at the correct spelling. In this method, you have to place in all the letter combinations one by one and see if they make sense to you. This
might be exhausting but it’s important to ensure the right answer through method.
The best way to solve punctuation exercises is the opposite method of spelling so as to ensure the right order of the punctuations. To understand where the right punctuation goes, you must understand
the need and the intention of the sentence and its phrases. So make sure to first take in the deeper meaning of the sentence; a good way to do this is to imagine yourself writing the sentence and
thinking where you would add punctuations. The feel of the sentence will, in this way, guide you to rule out any irrelevant punctuations.
Phonetics are also a necessary step to understand the punctuations. You should silently and slowly read the sentence and pay attention to each word, phrase and clause (in that order). This will
ensure that you cover most of the possible meanings the sentences could have. You have to look for any points where a punctuation symbol might block the flow of the sentence. Similarly, if the
sentence is long with lots of different word classes, it most likely needs to be broken down with a full stop, comma or a semi-colon!
11+ MATHS
The 11+ requires knowledge of maths of the standard curriculum of Year 6 and further beyond. It covers topics such as fractions, decimals and percentages (FDPs), geometry, stats, problem solving,
measurement and basic algebra.
In numbers, the student should be able to quickly remember the methods of basic arithmetic, especially multiplication. The goal is to solve with accuracy and speed through extensive practice of
addition, subtraction, multiplication and division. Additionally, decimal and negative number operations knowledge is required. Students are also expected to be able to identify prime numbers,
squares and factors.
In Geometry, students should have the ability to quickly identify polygons and quadrilaterals including interior angle sums and parallel sides. They should be able to visualize how shapes would
reflect along diagonal mirror lines as well as rotations. You may also be asked about coordinates in all four quadrants at a basic level.
In FDP, students need to be able to perform conversion between fractions, decimals and percentages and utilise each to solve a wide range of relevant problems. They can also expect problems with
backward fractions and percentages.
In statistics, students need the ability to solve real-life problems from the given tables, graphs and charts. They should be able to assess the given data and model a solution using pie charts and
In ratios, students need to have the ability to ratio relation problems. Their concepts of ratio conversion should be clear and they should be able to model the solution to the common ratio problems
such as maps and scale models and speed distance and time questions.
In measurement, there are a wide range on conversions between measurements, metric-imperial unit conversions and time and calendar problems.
In problem solving, a variety of questions can be expected which may or may not be from the above list of topics. The method of solution should follow understanding the given data and information
carefully and producing solution through a systematic way i.e. a logical sequence of steps.
Finally, students should also practise basic algebra including nth term sequences, solving simple algebraic expressions, simultaneous equations and solving basic algebraic equations involving one or
two steps.
11+ VERBAL REASONING
Verbal Reasoning almost entirely dependent on a child’s vocabulary and use of words. Hence, it is best to focus on the important of reading. Students with weak vocabulary usually struggle on this
The exercises that are expected in VR are: missing letters, synonyms, antonyms, sum completion, letter-number codes, word combinations, double meanings, moving letters, word constructions, hidden
words and word analogies.
It is vital that students familiarise themselves extensively with these exercises so that there is no confusion on the test day.
11+ NON-VERBAL REASONING
In Non-Verbal Reasoning, students should be able to understand mathematical terms that are used to refer to shapes, directions, rotations of shapes and positions. An important technique is to use the
process of elimination to arrive at the right answer when attempting relatively tricky questions. This is done by ruling out the patterns which do not make sense instead of focusing on the problem
Following exercises are common and usually appear in various variations: spot the difference, odd one out, related shapes, shape sequences, code sequences, code pairs.
It is vital that students familiarise themselves extensively with these exercises so that there is no confusion on the test day.
How we can help?
Preparing children for Common Entrance Examinations is what we do proficiently. We will prepare your child by equipping them with age-appropriate skills and subject knowledge to improve their ability
and strengthen their confidence to apply, evaluate, interpret and analyse what they have learned.
MOCK TEST AVAILABLE FOR 11 + STUDENTS
OPEN 7 DAYS A WEEK
Monday to Friday
Childcare Timings:
02.00 pm to 09.00 pm
Lesson Timings:
04.30 pm to 06.30 pm
06.45 pm to 08.45 pm
Saturday & Sunday
Childcare Timings
09:00 am to 07:00 pm
Lesson Timings
09:30 am to 11:30 am
11:45 am to 01.45 pm
02:30 pm to 04:30 pm
04:45 pm to 06:45 pm
UK Mathematics Trust (UKMT)
First Floor Offices,
94 Withington Road,
Whalley Range,
Manchester M16 8FA
240 Whetley Lane
BD8 9DJ | {"url":"https://www.skyhightuition.co.uk/11-plus-english-verbal-non-verbal-reasoning-tutors/","timestamp":"2024-11-04T20:09:50Z","content_type":"text/html","content_length":"237993","record_id":"<urn:uuid:34e5a54c-cd2b-466d-9fb9-c154dec98e73>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00531.warc.gz"} |
How do you find the problem with two variables?
How do you find the problem with two variables?
Divide both sides of the equation to “solve for x.” Once you have the x term (or whichever variable you are using) on one side of the equation, divide both sides of the equation to get the variable
alone. For example: 4x = 8 – 2y. (4x)/4 = (8/4) – (2y/4)
How do you solve problems involving algebraic expressions?
To solve an algebraic word problem:
1. Define a variable.
2. Write an equation using the variable.
3. Solve the equation.
4. If the variable is not the answer to the word problem, use the variable to calculate the answer.
What to do with 4th grade word problem worksheets?
These word problem worksheets place 4th grade math concepts into real world problems that students can relate to. We encourage students to read and think about the problems carefully, by: What is K5?
K5 Learning offers free worksheets, flashcards and inexpensive workbooks for kids in kindergarten to grade 5.
How many word problems are in word problems worksheet?
These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade. These equations worksheets will produce one step word problems. These worksheets will produce ten problems per
worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.
Is the word problems worksheet appropriate for 3rd grade?
These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade. These multiplication word problems worksheets will produce problems using dozens, with ten problems per
worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.
When do you use Division word problems worksheets?
These division word problems worksheets will produce problems using dozens in the divisor, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade,
and 5th Grade.
How do you find the problem with two variables? Divide both sides of the equation to “solve for x.” Once you have the x term (or whichever variable you are using) on one side of the equation,
divide both sides of the equation to get the variable alone. For example: 4x = 8 – 2y.… | {"url":"https://www.ohare-airport.org/how-do-you-find-the-problem-with-two-variables/","timestamp":"2024-11-03T04:14:38Z","content_type":"text/html","content_length":"35324","record_id":"<urn:uuid:0e6f35cb-0f98-4885-98e1-210197ba5ff4>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00294.warc.gz"} |
Uncertainty Principle
In 1927, Werner Heisenberg determined that it is impossible to measure both a particle's position and its momentum exactly. The more precisely we determine one, the less we know about the other. This
is called the Heisenberg Uncertainty Principle, and it is a fundamental property of quantum mechanics.
The precise relation is:
This constant is Planck's constant divided by two; Planck's constant is represented by the symbol ^-34 joule-seconds, or 6.58 x 10^-22 MeV-seconds.
The act of measuring a particle's position will affect your knowledge of its momentum, and vice-versa.
We can also express this principle in terms of energy and time:
This means that if a particle exists for a very brief time, you cannot precisely determine its energy. A short-lived particle could have a tremendously uncertain energy, which leads to the idea of
virtual particles. | {"url":"https://ccwww.kek.jp/pdg/particleadventure/frameless/uncertainty.html","timestamp":"2024-11-04T14:34:07Z","content_type":"text/html","content_length":"2055","record_id":"<urn:uuid:8f1ba6fa-a076-4b8a-a260-d0f019eb1f6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00630.warc.gz"} |
Right Angled Triangles
– Pythagoras’ theorem
– Trigonometry: the sine ratio
– Trigonometry: the cosine ratio
– Trigonometry: the tangent ratio
Example Key Words
The longest side of a triangle. It is also the side opposite the right angle.
Pythagoras' Theorem
A theorem stating that in a right triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides.
Opposite and Adjacent Side Side
The side opposite (opposite) and next to (adjacent) the angle you are dealing with in a triangle (not the right angle)
The Sine Ratio
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to that of the hypotenuse.
The Cosine Ratio
The cosine of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to that of the hypotenuse.
The Tangent Ratio
The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to that of the adjacent side. | {"url":"https://www.swanmore-school.co.uk/page/?title=Right+Angled+Triangles&pid=591","timestamp":"2024-11-02T11:18:22Z","content_type":"text/html","content_length":"68154","record_id":"<urn:uuid:e12ed760-c61a-4eba-a861-54551f9a651a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00786.warc.gz"} |
norm.ppf - Find Percent Point Of Normal Distribution - Python Clear
norm.ppf – Find Percent Point Of Normal Distribution
The norm.ppf function is useful for working with normal distributions in Python. It is part of the scipy.stats module, which provides a variety of statistical functions and distributions. Normal
distributions are one of the most common and important statistical probability distributions. They are often used to model natural phenomena, such as heights, weights, IQ scores, test scores, errors,
etc. Normal distributions have a bell-shaped curve, which is symmetric and centered around the mean. A standard deviation typically measures how spread out the values are from the mean.
The norm.ppf function allows us to find the x-value corresponding to a given probability on the normal distribution. For example, if we want to know what value of x has a 90% chance of being less
than or equal to it, we can use norm.ppf(0.9) to find out. The norm.ppf function can also take optional parameters for the mean and standard deviation of the normal distribution, which can be
different from the standard values of 0 and 1. This way, we can work with any normal distribution we want.
The norm.ppf function has many statistical applications, such as finding z-scores, confidence intervals, and p-values. These concepts are helpful for performing hypothesis tests, estimating
population parameters, and making inferences from data. In this article, we will explore what the mentioned function does, how to use it, and some examples of its applications.
What is norm.ppf?
The norm.ppf function stands for the percent point function. It is also known as the inverse cumulative distribution or quantile function. It takes a probability value (between 0 and 1) and returns
the corresponding value on the x-axis of the normal distribution. In other words, it tells us what value of x has a certain probability of being less than or equal to it.
For example, the norm.ppf(0.5) returns 0 because 50% of the area under the normal curve is to the left of 0. Similarly, the norm.ppf(0.95) returns 1.6449 because 95% of the area under the normal
curve is to the left of 1.6449.
This function can also take optional parameters for the mean and standard deviation of the normal distribution. By default, these are set to 0 and 1, respectively, corresponding to the standard
normal distribution. However, we can change them to any values we want to work with different normal distributions.
For example, norm.ppf(0.95, loc=10, scale=2) returns 13.2898, because 95% of the area under the normal curve with a standard deviation of 2 and a mean of 10 is to the left of 13.2898.
How to use norm.ppf?
To use this function, we need to import the scipy.stats module first:
import scipy.stats as stats
Then, we can call this function with the probability value we want, and optionally, the mean and standard deviation of the normal distribution. For example:
# Find the 90th percentile of the standard normal distribution
Output: 1.2815515655446004
# Find the 75th percentile of the normal distribution alongwith a mean value of 5 and a standard deviation of 3
stats.norm.ppf(0.75, loc=5, scale=3)
Output: 7.023469250588246
We can also pass an array of probability values to this function, and it will return an array of corresponding x-values. For example:
# Find the percentiles of the standard normal distribution
stats.norm.ppf([0.25, 0.5, 0.75]) 25th, 50th, and 75th respectively.
Output: array([-0.67448975, 0. , 0.67448975])
Examples of norm.ppf applications
This function can be used for various purposes, such as finding the z-score of a given probability, the confidence interval of a sample mean, and the p-value of a hypothesis test.
Finding the z-score of a given probability.
The z-score is the number of standard deviations away from the normal distribution mean. For example, the norm.ppf(0.95) returns 1.6449, which means that 95% of the values in the standard normal
distribution are within 1.6449 standard deviations from the mean.
Finding the confidence interval of a sample mean.
The confidence interval is the range of values containing the true population mean with a certain confidence level. For example, if we have a sample of size 100 with a mean of 50 and standard
deviation of 10, and we want to find the 95% confidence interval of the population mean, we can use this function to find the margin of error:
# Find the 95% confidence interval of the population mean
sample_size = 100
sample_mean = 50
sample_std = 10
confidence_level = 0.95
# Find the z-score for the confidence level
z = stats.norm.ppf((1 + confidence_level) / 2)
# Find the margin of error
margin_of_error = z * sample_std / sample_size**0.5
# Find the lower and upper bounds of the confidence interval
lower_bound = sample_mean - margin_of_error
upper_bound = sample_mean + margin_of_error
# Print the confidence interval
print(f"The 95% confidence interval of the population mean is ({lower_bound:.2f}, {upper_bound:.2f})")
Output: The 95% confidence interval of the population mean is (48.21, 51.79)
Finding the p-value of a hypothesis test.
The p-value is the probability of finding a result at least as extreme as the observed one, assuming the null hypothesis is true. For example, if we want to test whether the mean height of a
population is 170 cm, and we have a sample of size 50 with a mean of 172 cm and a standard deviation of 5 cm, we can use this function to find the p-value:
# Find the p-value of the hypothesis test
population_mean = 170
sample_size = 50
sample_mean = 172
sample_std = 5
# Find the z-score of the sample mean
z = (sample_mean - population_mean) / (sample_std / sample_size**0.5)
# Find the p-value by using the survival function (1 - cdf)
p = stats.norm.sf(z)
# Print the p-value
print(f"The p-value of the hypothesis test is {p:.4f}")
Output: The p-value of the hypothesis test is 0.0023
What is the difference between norm.ppf and norm.cdf?
The norm.cdf function is the cumulative distribution function, which takes a value on the x-axis of the normal distribution and returns the probability of being less than or equal to it. The norm.ppf
function is the percent point function, which does the opposite: it takes a probability and returns the corresponding value on the x-axis. They are inverse functions of each other, so norm.cdf
(norm.ppf(p)) = p and norm.ppf(norm.cdf(x)) = x for any p and x.
How do we find the norm.ppf value for a two-tailed test?
A two-tailed test is when we are interested in the values on both sides of the mean of the normal distribution. For example, if we want to find the 95% confidence interval of the population mean, we
need to find the values that exclude 2.5% of the area on each tail. To do this, we can use norm.ppf with the probability of 0.975, the sum of 0.5 (the area to the left of the mean) and 0.475 (the
area to the right of the mean up to the 95th percentile). Alternatively, we can use norm.isf, the inverse survival function, with the probability of 0.025, the area to the right of the 97.5th
How do we plot the norm.ppf function in Python?
To plot the norm.ppf function in Python, we can use matplotlib.pyplot module, which provides various functions for creating and customizing graphs. For example, we can plot the norm.ppf function for
the standard normal distribution using the code: plt.plot(np.linspace(0.01, 0.99, 100), stats.norm.ppf(np.linspace(0.01, 0.99, 100)))
The norm.ppf function is a powerful and versatile tool for working with normal distributions in Python. It allows us to find the x-value corresponding to a given probability or vice versa. It can
also be used for various applications, such as finding z-scores, confidence intervals, and p-values. The norm.ppf function is part of the scipy.stats module, which offers many other statistical
functions and distributions. We hope this article has helped you understand the norm.ppf function and how to use it in your Python projects.
Follow us at PythonClear to learn more about solutions to general errors one may encounter while programming in Python.
Leave a Comment | {"url":"https://www.pythonclear.com/modules/norm-ppf/","timestamp":"2024-11-08T10:23:01Z","content_type":"text/html","content_length":"77897","record_id":"<urn:uuid:a8484f6e-2c6c-4112-944b-194da1c8e814>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00844.warc.gz"} |
C Program to Convert Binary to Gray Code using Recursion
This C program converts a binary number to its corresponding Gray code using recursion.
Binary Code: Binary code is a number system that uses only two digits, 0 and 1. It is the basis of all digital systems and computer operations. Each digit in a binary number is called a bit, and each
bit represents a power of 2.
Gray Code: Gray code, also known as reflected binary code, is a binary numeral system where adjacent values differ by only one bit. It is commonly used in various applications such as
analog-to-digital converters, error detection, and rotary encoders. The advantage of Gray code is that it prevents errors caused by multiple bits changing simultaneously during transitions.
Problem Statement
Write a C program that converts a binary number to its equivalent Gray code using recursion. The program should prompt the user to enter a binary number and display the corresponding Gray code.
• An unsigned integer representing a binary number.
• The corresponding Gray code as an unsigned integer.
Enter a binary number: 1010
Gray code: 1111
• The input binary number is assumed to be a 32-bit unsigned integer.
• The program should use a recursive function to convert the binary number to Gray code.
• The program should handle invalid inputs appropriately and display error messages if necessary.
C Program to Convert Binary to Gray Code using Recursion
#include <stdio.h>
// Function to convert binary to Gray code
unsigned int binaryToGray(unsigned int num)
if (num == 0)
return 0;
unsigned int msb = num & (1 << 31); // Extract the most significant bit
// Right shift one position and XOR with the original number
return (msb | (num >> 1)) ^ binaryToGray(num >> 1);
int main()
unsigned int binaryNum;
printf("Enter a binary number: ");
scanf("%u", &binaryNum);
unsigned int grayNum = binaryToGray(binaryNum);
printf("Gray code: %u\n", grayNum);
return 0;
How it Works
1. The program starts by including the necessary header file stdio.h for input/output operations.
2. The function binaryToGray is declared. This function takes an unsigned integer num as input and returns the Gray code for that number.
3. Inside the binaryToGray function, the following steps are performed recursively:
□ Base Case: If the number is 0, meaning there are no more bits to process, the function returns 0, as the Gray code for 0 is also 0.
□ The most significant bit (MSB) of the number is extracted using the bitwise AND operation with (1 << 31). This operation isolates the leftmost bit of the 32-bit number.
□ The number is then right-shifted by one position to discard the MSB. This is done using the right shift operator >>.
□ The function makes a recursive call to itself with the remaining bits of the number (after discarding the MSB).
□ The result obtained from the recursive call is XORed with the MSB. This is done to ensure that adjacent values in the Gray code differ by only one bit.
□ The final result is returned.
4. In the main function:
□ The program prompts the user to enter a binary number and reads it as an unsigned integer using scanf.
□ The binaryToGray function is called with the input binary number as an argument, and the result is stored in the grayNum variable.
□ Finally, the program prints the calculated Gray code using printf.
5. The program terminates by returning 0 from the main function.
The recursion in the binaryToGray function processes the bits of the binary number one by one, starting from the most significant bit (MSB) and moving towards the least significant bit (LSB). At each
recursive step, the function extracts the MSB, discards it, and processes the remaining bits recursively. The process continues until the base case is reached (num == 0), and the Gray code is
constructed by XORing the extracted MSB with the result obtained from the recursive call.
Input /Output
• The program prompts the user to enter a binary number.
• The user enters the binary number 1010.
• The binaryToGray function is called with the input number 1010.
□ Since 1010 is not equal to 0, the function proceeds with the recursion.
□ The most significant bit (MSB) is extracted, which is 1.
□ The number is right-shifted by one position, resulting in 0101.
□ The function makes a recursive call to binaryToGray with the remaining bits 0101.
☆ Again, the number is not equal to 0, so the recursion continues.
☆ The MSB is extracted, which is 0.
☆ The number is right-shifted by one position, resulting in 0010.
☆ Another recursive call is made with the remaining bits 0010.
○ The number is still not equal to 0, so the recursion continues.
○ The MSB is extracted, which is 0.
○ The number is right-shifted by one position, resulting in 0001.
○ Another recursive call is made with the remaining bits 0001.
■ The number is not equal to 0, so the recursion continues.
■ The MSB is extracted, which is 0.
■ The number is right-shifted by one position, resulting in 0000.
■ Another recursive call is made with the remaining bits 0000.
★ The number is now equal to 0, so the base case is reached.
★ The function returns 0.
○ The result of the recursive call is XORed with the MSB 0, resulting in 0.
○ The function returns 0.
☆ The result of the recursive call is XORed with the MSB 0, resulting in 0.
☆ The function returns 0.
□ The result of the recursive call is XORed with the MSB 1, resulting in 1.
□ The function returns 1.
• The calculated Gray code 337 is printed as the output.
In this example, the binary number 1010 is converted to its equivalent Gray code 337 using the recursive approach implemented in the program.
Leave A Reply | {"url":"https://developerpublish.com/c-program-to-convert-binary-to-gray-code-using-recursion/","timestamp":"2024-11-14T23:19:17Z","content_type":"text/html","content_length":"306790","record_id":"<urn:uuid:d183d0c3-7c93-4ce8-ba6d-c9f4c256e3ad>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00408.warc.gz"} |
Problem 12
Primary tabs
Permalink Submitted by Ledrappier on Mon, 11/14/2016 - 11:40
Permalink Submitted by shub on Mon, 07/10/2017 - 10:04
The conjecture is proven for C^1 Axiom A no cycle diffeomorphisms by Shub and Williams and Ruelle and Sullivan. It has also been proven in general for C^infinity endomorphisms of compact manifolds by
Yomdin. For references,see Shub,M., All,Most,Some Differentiable Dynamical Systems, in Proceedings of the ICM,Madrid 2006. | {"url":"https://bowen.pims.math.ca/node/352","timestamp":"2024-11-07T06:36:55Z","content_type":"text/html","content_length":"23573","record_id":"<urn:uuid:91e2ca54-d510-488e-8f79-591e8d9c0eff>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00398.warc.gz"} |
45 Important Question of Physics Class 11 NEB | NEB Physics important questions - By Ganesh45 Important Question of Physics Class 11 NEB | NEB Physics important questions - By Ganesh
So, In this article I'll give you important question of physics class 11 neb (chapterwise). Also you can download neb important questions for physics at the last of the article. These are very
important questions for class 11 physics for neb students.
So, Let's get started
Important question of physics class 11 NEB "Chapterwise"
In this article, I'll give you Mechanics important questions for neb. These are most repeated questions of Physics in NEB , so you should not leave any of these questions below:
1. Physical Quantities
(Not any long questions)
2. Vectors
• Parallelogram law of Vector addition
• Triangle law of Vector addition
3. Kinematics
• Projectile fired at an angle theta with horizontal
4. Laws of Motion
• Relation betn angle of friction & angle of repose
• State and explain the principle of conservation of linear momentum hence derive it from Newton's law of motion
5. Work and Energy
• State and prove conservation of mechanical energy.
• Derive an expression to calculate workdone by a variable force.
• Prove that the colliding objects exchange their velocities in one dimensional elastic collision.
( New: Work done by a constant and a variable force.)
6. Circular Motion
• Time period of Conical Pendulum.
• Define centripetal force and Deduce it's expression. ( Why a force is needed to keep a body moving with uniform speed in a circular motion?)
• Motion of a bi cycle on a curved road
• motion of a car moving round in a circular banked track
I highly recommend you to watch this in Video format as well by clicking in the video
7. Gravitation
• Expression for GPE and Establish it's dimension
• Variation of acceleration due to gravity below the earth's surface
• Variation of acceleration due to gravity with the altitude
• (Variation of acceleration due to gravity with rotation)
• Relation for Gravitational potential at a point due to point mass
• (Artificial Satellite)
• Escape velocity from the surface of earth
• Orbital velocity and time period of the satellite revolving around the earth
• Total Energy of the Satellite revolving around the earth
8) Equilibrium
(Not any long questions)
9. Rotational Dynamics
• Work done by Couple
• Expression of Rotational Kinetic energy in terms of Moment of inertia
• Relation between Angular momentum and Moment of Inertia
• moment of inertia of a uniform rod passing through the centre and perpendicular to its length
• moment of inertia of a uniform rod passing through one end and perpendicular to its length
• Relation between Torque and Moment of Inertia
10. Elasticity
• Energy stored in a stretched wire
• Hooke's law
• Determination of Young's modulus of elasticity
• Different types of Elasticity
11. Periodic Motion (*Not in New Syllabus)
• Simple Harmonic Motion (Time Period)
• Simple Harmonic Motion (Relation between acceleration and displacement of Particle)
• Simple Harmonic Motion (Total Energy)
• Time Period of Simple pendulum
12. Fluid Mechanics (*Not in New Syllabus)
12.1 Fluid Statics
(Not any long questions)
12.2 Surface Tension
• Rise of liquid in a capillary tube
• Relation between Surface tension and Surface Energy
12.3 Viscosity
• Poiseulle's formula
• Bernoulli's equation
• Terminal Velocity
• Newton's law of Viscosity
• Determination of Stokes law by Dimensional method
Also read:
Also you can follow me on
if you want to ask any questions.
Also comment down what you want in next blog.
Post a Comment
1. Tq very much dai
2. Dai ali explain gardinu vayeko vaye ramro huntheo
3. Thanks dai aaru aaru ne halnu❤
4. Thank you so much brother 😊
5. Thank u daaju qnd waiting for next video😍
6. tq dai,fr ur full support
1. time paunu vyo vane sabai explain gardinu hola....,
7. Farmacy KO lagi important question send kar nu na
8. good work keep it up
Post a Comment | {"url":"https://www.ganeshgtm.com.np/2021/09/important-question-of-physics-class.html","timestamp":"2024-11-07T12:42:28Z","content_type":"application/xhtml+xml","content_length":"250006","record_id":"<urn:uuid:db954874-35bc-45f2-a874-a974f30d7e73>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00370.warc.gz"} |
manual pages
igraph-vs-indexing {igraph} R Documentation
Indexing vertex sequences
Vertex sequences can be indexed very much like a plain numeric R vector, with some extras.
## S3 method for class 'igraph.vs'
x[..., na_ok = FALSE]
x A vertex sequence.
... Indices, see details below.
na_ok Whether it is OK to have NAs in the vertex sequence.
Vertex sequences can be indexed using both the single bracket and the double bracket operators, and they both work the same way. The only difference between them is that the double bracket operator
marks the result for printing vertex attributes.
Another vertex sequence, referring to the same graph.
Multiple indices
When using multiple indices within the bracket, all of them are evaluated independently, and then the results are concatenated using the c() function (except for the na_ok argument, which is special
an must be named. E.g. V(g)[1, 2, .nei(1)] is equivalent to c(V(g)[1], V(g)[2], V(g)[.nei(1)]).
Index types
Vertex sequences can be indexed with positive numeric vectors, negative numeric vectors, logical vectors, character vectors:
• When indexed with positive numeric vectors, the vertices at the given positions in the sequence are selected. This is the same as indexing a regular R atomic vector with positive numeric vectors.
• When indexed with negative numeric vectors, the vertices at the given positions in the sequence are omitted. Again, this is the same as indexing a regular R atomic vector.
• When indexed with a logical vector, the lengths of the vertex sequence and the index must match, and the vertices for which the index is TRUE are selected.
• Named graphs can be indexed with character vectors, to select vertices with the given names.
Vertex attributes
When indexing vertex sequences, vertex attributes can be referred to simply by using their names. E.g. if a graph has a name vertex attribute, then V(g)[name == "foo"] is equivalent to V(g)[V(g)$name
== "foo"]. See more examples below. Note that attribute names mask the names of variables present in the calling environment; if you need to look up a variable and you do not want a similarly named
vertex attribute to mask it, use the .env pronoun to perform the name lookup in the calling environment. In other words, use V(g)[.env$name == "foo"] to make sure that name is looked up from the
calling environment even if there is a vertex attribute with the same name. Similarly, you can use .data to match attribute names only.
Special functions
There are some special igraph functions that can be used only in expressions indexing vertex sequences:
takes a vertex sequence as its argument and selects neighbors of these vertices. An optional mode argument can be used to select successors (mode="out"), or predecessors (mode="in") in directed
Takes an edge sequence as an argument, and selects vertices that have at least one incident edge in this edge sequence.
Similar to .inc, but only considers the tails of the edges.
Similar to .inc, but only considers the heads of the edges.
.innei, .outnei
.innei(v) is a shorthand for .nei(v, mode = "in"), and .outnei(v) is a shorthand for .nei(v, mode = "out").
Note that multiple special functions can be used together, or with regular indices, and then their results are concatenated. See more examples below.
See Also
Other vertex and edge sequences: E(), V(), igraph-es-attributes, igraph-es-indexing2, igraph-es-indexing, igraph-vs-attributes, igraph-vs-indexing2, print.igraph.es(), print.igraph.vs()
Other vertex and edge sequence operations: c.igraph.es(), c.igraph.vs(), difference.igraph.es(), difference.igraph.vs(), igraph-es-indexing2, igraph-es-indexing, igraph-vs-indexing2,
intersection.igraph.es(), intersection.igraph.vs(), rev.igraph.es(), rev.igraph.vs(), union.igraph.es(), union.igraph.vs(), unique.igraph.es(), unique.igraph.vs()
# -----------------------------------------------------------------
# Setting attributes for subsets of vertices
largest_comp <- function(graph) {
cl <- components(graph)
V(graph)[which.max(cl$csize) == cl$membership]
g <- sample_(gnp(100, 2/100),
with_vertex_(size = 3, label = ""),
with_graph_(layout = layout_with_fr)
giant_v <- largest_comp(g)
V(g)$color <- "green"
V(g)[giant_v]$color <- "red"
# -----------------------------------------------------------------
# nei() special function
g <- graph( c(1,2, 2,3, 2,4, 4,2) )
V(g)[ .nei( c(2,4) ) ]
V(g)[ .nei( c(2,4), "in") ]
V(g)[ .nei( c(2,4), "out") ]
# -----------------------------------------------------------------
# The same with vertex names
g <- graph(~ A -+ B, B -+ C:D, D -+ B)
V(g)[ .nei( c('B', 'D') ) ]
V(g)[ .nei( c('B', 'D'), "in" ) ]
V(g)[ .nei( c('B', 'D'), "out" ) ]
# -----------------------------------------------------------------
# Resolving attributes
g <- graph(~ A -+ B, B -+ C:D, D -+ B)
V(g)$color <- c("red", "red", "green", "green")
V(g)[ color == "red" ]
# Indexing with a variable whose name matches the name of an attribute
# may fail; use .env to force the name lookup in the parent environment
V(g)$x <- 10:13
x <- 2
version 1.3.3 | {"url":"https://igraph.org/r/html/1.3.3/igraph-vs-indexing.html","timestamp":"2024-11-14T07:40:53Z","content_type":"text/html","content_length":"15660","record_id":"<urn:uuid:2b424222-a413-457c-9c0a-a4a72d55d28f>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00018.warc.gz"} |
5.7 Shapes on the Coordinate Plane
Unit Goals
• Students plot coordinate pairs on a coordinate grid and classify triangles and quadrilaterals in a hierarchy based on properties of side length and angle measure. They generate, identify, and
graph relationships between corresponding terms in two numeric patterns, given two rules, and represent and interpret real world and mathematical problems on a coordinate grid.
Section A Goals
• Locate points on a coordinate grid.
Section B Goals
• Classify triangles and quadrilaterals in a hierarchy based on angle measurements and side lengths.
Section C Goals
• Generate, identify, and graph relationships between corresponding terms in two patterns, given a rule.
• Represent and interpret real world and mathematical problems on a coordinate grid.
Read More | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/unit-7/lessons.html","timestamp":"2024-11-02T21:57:40Z","content_type":"text/html","content_length":"71585","record_id":"<urn:uuid:fbcd0a46-004b-430a-9870-647ccc429fb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00648.warc.gz"} |
Equations devs & users
Hi there,
I'm playing around with the really great Equations approach and got stuck.
I derive Subterm but then I received a proof obligation that I do not know how to solve.
Here is my toy language with conditionals:
Inductive Term : Type :=
| boolT (b: bool)
| natT (n: nat)
| iteT (i t e: Term).
To derive a termination measure, I use Subterm:
Derive NoConfusion NoConfusionHom for Term.
Derive Subterm for Term.
Then I define the Small-Step operational semantics (on purpose as a function) like so:
Equations eval1 (term : Term) : option Term by wf term Term_subterm :=
eval1 (iteT (boolT true) t2 _ ) := Some t2;
eval1 (iteT (boolT false) _ t3) := Some t3;
eval1 (iteT (iteT t11 t12 t13) t2 t3) with eval1 (iteT t11 t12 t13) => {
eval1 (iteT (iteT t11 t12 t13) t2 t3) (Some t1') := Some (iteT t1' t2 t3);
eval1 (iteT (iteT t11 t12 t13) t2 t3) None := None
eval1 _ := None.
Now I use eval1 to define the multi-step relation eval__n:
Equations eval1' (term : Term) : option Term :=
eval1' (iteT t1 t2 t3) := eval1 (iteT t1 t2 t3);
eval1' term := Some term.
Equations eval__n (term : Term) : (option Term) by wf term Term_subterm :=
eval__n' (iteT t1 t2 t3) with eval1' (iteT t1 t2 t3) => {
eval__n' (iteT t1 t2 t3) (Some term') := eval__n' term';
eval__n' (iteT t1 t2 t3) None := None
eval__n' term := Some term.
Next Obligation.
Now the goal is:
t1, t2, t3, term' : Term
eval__n : forall x : Term, Term_subterm x (iteT t1 t2 t3) -> option Term
Term_subterm term' (iteT t1 t2 t3)
How do I solve such a goal?
Also, the by rec as explained in this stackoverflow post does not seem to work anymore.
What would be the proper way to establish a measure without Derive?
Is it always via the wellfounded relation now?
This seems provable for your toy language — not sure how, and counterexamples wouldn't surprise me — but would fail, for instance, if you add lambdas or loops.
It seems that Coq's looking at eval__n' (iteT t1 t2 t3) (Some term') := eval__n' term'; and asking "why do you think term' is smaller than iteT t1 t2 t3" — I think that's true for your current toy
language (and I don't know a proof), but it wouldn't be for most interesting extension.
That's for good reason: this property implies the language is normalizing. What's worse, this property is false even for strongly normalizing languages: normalization can cause size blowup; even a
single beta-reduction (\x. t) u can produce output of quadratic size |t| * |u|, since it will replicate u once for each occurrence of x in t.
Is it always via the wellfounded relation now?
Indeed, one always uses relations R that are well-founded (such that WellFounded R is a typeclass instance). (But wellfounded itself isn't a relation); Subterm is one such relation.
oh sorry — while the above is mostly true, your goal isn't provable — you need to "remember" that term' comes from eval1' (iteT t1 t2 t3). That requires using the "inspect pattern"
That's basically Equations' version of using destruct e eqn:? instead of destruct e: here it'd records in context the propositional equality eval1' (iteT t1 t2 t3) = Some term'. See https://
github.com/mattam82/Coq-Equations/blob/3c2342f9b15e20c91d36293126123ea617c7a532/examples/Basics.v#L506-L538 for more info.
I was too quick.
Following the above advice
Equations eval__n' (term : FCPterm) : (option FCPterm) by wf term FCPterm_subterm :=
eval__n' (iteT t1 t2 t3) with inspect (eval1' (iteT t1 t2 t3)) => {
eval__n' (iteT t1 t2 t3) (Some term' eqn: eq1) := eval__n' term';
eval__n' (iteT t1 t2 t3) (None eqn: eq2) := None
eval__n' term := Some term.
Next Obligation.
I get one proof obligation:
t1, t2, t3, term' : FCPterm
eq1 : eval1' (iteT t1 t2 t3) = Some term'
eval__n' : forall x : FCPterm, FCPterm_subterm x (iteT t1 t2 t3) -> option FCPterm
FCPterm_subterm term' (iteT t1 t2 t3)
Here is how far I got:
clear eval__n'.
move: eq1.
funelim (eval1' (iteT t1 t2 t3)).
funelim (eval1 (iteT t1 t2 t3)) => //=;
move => eq_some; injection eq_some => eq_term; rewrite -eq_term.
- exact: (Relation_Operators.t_step _ _ _ _ (FCPterm_direct_subterm_2_2 _ _ _)).
- exact: (Relation_Operators.t_step _ _ _ _ (FCPterm_direct_subterm_2_1 _ _ _)).
- apply Hind in Heq.
by [apply cond_subterm].
I'm not sure whether this is the proper way of writing such a proof. Is it?
In the last line of the proof, I use lemma cond_subterm which I do not know how to prove:
Lemma cond_subterm: forall t1 t1' t2 t3,
FCPterm_subterm t1 t1' -> FCPterm_subterm (iteT t1 t2 t3) (iteT t1' t2 t3).
After all, FCPterm_subterm is just a transitive closure clos_trans with step and trans but the hypothesis FCPterm_subterm t1 t1' hints more towards some form of induction. I found the induction
principle clos_trans_ind but when I try to use it, I get an error:
Error: Anomaly "File "plugins/ssr/ssrcommon.ml", line 792, characters 18-24: Assertion failed."
Please report at http://coq.inria.fr/bugs/.
What is the proper way to write this proof?
Sebastian Ertel has marked this topic as unresolved.
I managed to write the function with my own metric subTermCount like this:
Equations eval__n (term : FCPterm) : (option FCPterm) by wf (subTermCount term) lt :=
eval__n (iteT t1 t2 t3) with inspect (SmallStep.eval1' (iteT t1 t2 t3)) => {
eval__n (iteT t1 t2 t3) (Some term' eqn: eq1) := eval__n term';
eval__n (iteT t1 t2 t3) (None eqn: eq2) := None
eval__n term := Some term.
Proving the remaining obligation was now trivial.
Nevertheless, I would still like to understand whether there exists a solution to the cond_subterm lemma above.
Does it have to do with the fact that the Derive Subterm command does not handle nested recursive occurences like is hinted at in this paper?
Your Term datatype doesn't have nested recursive occurrences; I think your proof strategy was correct.
all anomalies are valid bug reports, so you should probably file a bug with some reproduction code
Since your anomaly seems to be in ssreflect code, I guess it's in elim, so I wonder what induction or apply do with that induction principle
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237659-Equations-devs-.26-users/topic/Subterm.20obligations.html","timestamp":"2024-11-10T17:41:23Z","content_type":"text/html","content_length":"28890","record_id":"<urn:uuid:250f7e46-e0bb-4783-8bca-8f15a7580b43>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00088.warc.gz"} |
How do you prepare data for regression analysis?
1. List all the variables you have and their measurement units.
2. Check and re-check the data for imputation errors.
3. Make additional imputation for the points with missing values (you may also simply exclude the observations if you have large dataset with not so many missing values)
How do you do regression analysis for data?
It consists of 3 stages – (1) analyzing the correlation and directionality of the data, (2) estimating the model, i.e., fitting the line, and (3) evaluating the validity and usefulness of the model.
First, a scatter plot should be used to analyze the data and check for directionality and correlation of data.
How do you prepare data for multiple regression analysis?
1. Look at Descriptive Statistics.
2. Look at Missing Values.
3. Look at Distribution of Variables.
4. Look at Correlation of Variables.
5. Look at Skewness of the Variables.
6. Check the Linear Regression Assumptions (Look at Residuals).
7. Look at the Outliers.
How do you prepare a linear regression dataset?
1. Introduction.
2. Linear Regression with One Variable.
3. Step 1: Importing Python libraries.
4. Step 2: Creating the dataset.
5. Step 3: Opening the dataset.
6. Step 4: Uploading the dataset.
7. Step 5: Feature Scaling and Normalization.
8. Step 6: Add a column of ones to the X vector.
How do you prepare data for regression analysis in SPSS?
SPSS – Data Preparation for Regression
1. *Show values and value labels as well as variable names and labels in output. ...
2. *Set 6 as user missing values for all regression variables. ...
3. *Add missing values per case as new variable to data. ...
4. *Create filter variable for cases with 3 or fewer missings.
45 related questions found
What types of data can be used in regression analysis?
Regression analysis includes several variations, such as linear, multiple linear, and nonlinear. The most common models are simple linear and multiple linear. Nonlinear regression analysis is
commonly used for more complicated data sets in which the dependent and independent variables show a nonlinear relationship.
How do you do a regression step by step?
1. Step 1: Load the data into R. Follow these four steps for each dataset: ...
2. Step 2: Make sure your data meet the assumptions. ...
3. Step 3: Perform the linear regression analysis. ...
4. Step 4: Check for homoscedasticity. ...
5. Step 5: Visualize the results with a graph. ...
6. Step 6: Report your results.
What should the first step be in a regression analysis?
The Statistical Significance
The first step of the regression analysis is to check whether there is any statistical significance between the dependent and the independent variables.
How to create a regression model?
Use the Create Regression Model capability
1. Create a map, chart, or table using the dataset with which you want to create a regression model.
2. Click the Action button .
3. Do one of the following: ...
4. Click Create Regression Model.
5. For Choose a layer, select the dataset with which you want to create a regression model.
How much data do you need for linear regression?
So, how much data do we need to conduct a successful regression analysis? A common rule of thumb is that 10 data observations per predictor variable is a pragmatic lower bound for sample size.
How regression analysis is done?
Linear Regression works by using an independent variable to predict the values of dependent variable. In linear regression, a line of best fit is used to obtain an equation from the training dataset
which can then be used to predict the values of the testing dataset.
What to do before regression?
Before you begin the regression analysis, you should review the literature to develop an understanding of the relevant variables, their relationships, and the expected coefficient signs and effect
How many samples are needed for multiple regression?
For example, in regression analysis, many researchers say that there should be at least 10 observations per variable. If we are using three independent variables, then a clear rule would be to have a
minimum sample size of 30. Some researchers follow a statistical formula to calculate the sample size.
How to do regression analysis manually?
Simple Linear Regression Math by Hand
1. Calculate average of your X variable.
2. Calculate the difference between each X and the average X.
3. Square the differences and add it all up. ...
4. Calculate average of your Y variable.
5. Multiply the differences (of X and Y from their respective averages) and add them all together.
How do you write a simple regression?
The formula for a simple linear regression is:
1. y is the predicted value of the dependent variable (y) for any given value of the independent variable (x).
2. B[0] is the intercept, the predicted value of y when the x is 0.
3. B[1] is the regression coefficient – how much we expect y to change as x increases.
What is P value in regression?
The P-value is a statistical number to conclude if there is a relationship between Average_Pulse and Calorie_Burnage. We test if the true value of the coefficient is equal to zero (no relationship).
The statistical test for this is called Hypothesis testing.
How many subjects does it take to do a regression analysis?
Consequently, this researcher should conduct the study with a minimum of 46 subjects. In conclusion, researchers who use traditional rules-of-thumb are likely to design studies that have insufficient
power because of too few subjects or excessive power because of too many subjects.
What is the basic principle of regression?
The principle of regression is a term used by real estate appraisers stating that the value of high-end real estate may be diminished by having lower-end properties in the same vicinity. This
principle is used frequently in writing zoning laws, which strive to keep business and residential areas separate.
What are five types of regression analysis?
Types of Regression Analysis Techniques
• Linear Regression.
• Logistic Regression.
• Ridge Regression.
• Lasso Regression.
• Polynomial Regression.
• Bayesian Linear Regression.
What is regression analysis with example?
Formulating a regression analysis helps you predict the effects of the independent variable on the dependent one. Example: we can say that age and height can be described using a linear regression
model. Since a person's height increases as age increases, they have a linear relationship.
What is a good R Squared value?
In finance, an R-Squared above 0.7 would generally be seen as showing a high level of correlation, whereas a measure below 0.4 would show a low correlation.
How do you know if data is suitable for regression?
1 Answer
1. That there is in fact a relationship between the outcome variable and the predictor variables.
2. That observations are independent.
3. That the residuals are normally distributed and independent of the values of variables in the model.
What are the 2 main types of regression?
The two basic types of regression are simple linear regression and multiple linear regression, although there are non-linear regression methods for more complicated data and analysis.
Which model is best for regression?
The best known estimation method of linear regression is the least squares method.
Why is 30 the minimum sample size?
A sample size of 30 often increases the confidence interval of your population data set enough to warrant assertions against your findings.4 The higher your sample size, the more likely the sample
will be representative of your population set. | {"url":"https://www.calendar-uk.co.uk/frequently-asked-questions/how-do-you-prepare-data-for-regression-analysis","timestamp":"2024-11-13T18:59:43Z","content_type":"text/html","content_length":"73500","record_id":"<urn:uuid:dba80870-e683-4580-9b94-8bf66bedfb4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00044.warc.gz"} |
Cite as
Cameron Chalk, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, and Tim Wylie. Optimal Staged Self-Assembly of General Shapes. In 24th Annual European Symposium on Algorithms (ESA 2016).
Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
Copy BibTex To Clipboard
author = {Chalk, Cameron and Martinez, Eric and Schweller, Robert and Vega, Luis and Winslow, Andrew and Wylie, Tim},
title = {{Optimal Staged Self-Assembly of General Shapes}},
booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)},
pages = {26:1--26:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-015-6},
ISSN = {1868-8969},
year = {2016},
volume = {57},
editor = {Sankowski, Piotr and Zaroliagis, Christos},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.26},
URN = {urn:nbn:de:0030-drops-63776},
doi = {10.4230/LIPIcs.ESA.2016.26},
annote = {Keywords: Tile self-assembly, 2HAM, aTAM, DNA computing, biocomputing} | {"url":"https://drops.dagstuhl.de/search/documents?author=Chalk,%20Cameron","timestamp":"2024-11-05T07:32:28Z","content_type":"text/html","content_length":"64389","record_id":"<urn:uuid:cb134fc2-1951-489a-8fbf-6a37f8cbdfd6>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00161.warc.gz"} |
Prof. Dr. Peter Koepke
Recent Preprints (published and unpublished)
In the Publication Process
• (with Arthur Apter and Ioanna Dimitriou) The first measurable cardinal can be the first uncountable regular cardinal at any successor height. 12 pages. Submitted and in revision. pdf
• (with Vladimir Kanovei) Degrees of subsets of the measurable cardinal in Prikry generic extensions. 15 pages. Submitted. pdf
• (with Andrey Morozov) On the computational strength of Infinite Time Blum-Shub-Smale Machines. 26 pages.
• (with Karen Räsch and Philipp Schlicht) A minimal Prikry-type forcing for singularizing a measurable cardinal. The Journal of Symbolic Logic 78 (2013), 85-100. pdf
• (with Moti Gitik) Violating the singular cardinals hypothesis without large cardinals. Israel Journal of Mathematics 191 (2012), 901-922. pdf
• (with Benjamin Seyfferth) Towards a theory of infinite time Blum-Shub-Smale machines. Lecture Notes in Computer Science 7318 (2012). pdf
• (with Julian J. Schlöder) Transition of consistency and satisfiability under language extensions. Formalized Mathematics 20 (2012), 193-197.
• (with Julian J. Schlöder) The Gödel completeness theorem for uncountable languages. Formalized Mathematics 20 (2012), 199-203.
• (with Philip D. Welch) A generalised dynamical system, infinite time register machines, and Pi^1_1-CA_0. In CiE 2011. B. Löwe et al, eds., Lecture Notes in Computer Science 6735 (2011), 152-159.
• (with Marcos Cramer and Bernhard Schröder) Parsing and disambiguation of symbolic mathematics in the Naproche system. In Calculemus/MKM 2011. J. H. Davenport et al, eds., Lecture Notes in
Artificial Intelligence 6824 (2011), 180-195. pdf
• (with Philip Welch) Global square and mutual stationarity at the aleph_n. Annals of Pure and Applied Logic 162 (2011), 787-806 pdf
• (with M. Cramer, D. Kühlwein, and B. Schröder) The Naproche System. Calculemus 2009 Emerging Trends (2009), Ontario, Canada, 10-20. pdf
• Ordinal computability. In Mathematical Theory and Computational Practice. K. Ambos-Spies et al, eds., Lecture Notes in Computer Science 5635 (2009), 280-289. pdf
• (with Benjamin Seyfferth) Ordinal machines and admissible recursion theory. Annals of Pure and Applied Logic, 160 (2009), 310-318. pdf
• (with M. Cramer, D. Kühlwein, and B. Schröder) From proof texts to logic. Discourse representation structures for proof texts in mathematics. In Von der Form zur Bedeutung: Texte automatisch
verarbeiten. From Form to Meaning: Processing Texts Automatically. C. Chiarcos et al. eds. Verlag Narr, Tübingen. pdf
2008 2007 2006 Previous preprints
Last changed: 101004 | {"url":"https://www.math.uni-bonn.de/people/koepke/preprints.shtml","timestamp":"2024-11-05T22:56:23Z","content_type":"text/html","content_length":"11280","record_id":"<urn:uuid:fd174e32-7a59-45d1-9b21-3801eda01823>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00126.warc.gz"} |
Performance Assessment of Wastewater Treatment Plants
Omni Calculator - Grupper Facebook
- Stefan Boltzmann Law - Radiation Energy. - Otto Cycle Compression Ratio (CR). For Excel calculator email me at : rajangosavi@gmail.com This video explains how the Thermodynamics excel calculator
can be used for calculating the Rankine Conical flow calculator by Stephen Krauss, included 5 th January 2014. Thermodynamics Calculator is an easy to use App that Contains 49 Calculators
Thermodynamics Calculator includes the following Calculators: - Heat Flow - Stefan Boltzmann Law - Radiation Energy - Otto Cycle Compression Ratio (CR) - Carnot Cycle Efficiency Download
Thermodynamics Calculator App 7.2 for iPad & iPhone free online at AppPure. Get Thermodynamics Calculator for iOS latest version. Thermodynamics Calculator contains following 44 Calculators related
to Thermodynamics with Thermal Engineering Tables. 2018-10-01 2021-04-09 The thermodynamics Formula Sheet listed over here covers the topics like Internal Energy, First Law of Thermodynamics,
Isometric Charge, Isothermal Charge, and many more.
2018-11-26 Thermodynamics – calculator for solving mixing problems. An online calculator to solve thermodynamic equilibrium problems, such as finding the final temperature when mixing fluids, or
finding the required temperature for one of the fluids to achieve a final mixed temperature Thermodynamics Calculator. Thermodynamics is the study of heat and work. It is a branch of physics which is
concerned with heat and temperature and their relation to energy and work.
Law of Cat Thermodynamics: Heat flows from a warmer to a cooler body, except in the case of a cat, in which /book/graphics-calculator-keystroke-john-hornsby-margaret/d/1353325914 .biblio.com/
book/every-life-fire-how-thermodynamics-explains/d/1353466104 thermodynamics-smith-van-ness-solution.html 2018-07-08T06:08:58Z weekly http://embed.handelsbanken.se/B5AA857/
medication-calculation-exam-for-la MAPS Macro Calculator MAPS Fitness Anabolic | Muscle Adaptation with Light Therapy - Joovv Laws of thermodynamics - Wikipedia How This European Standard
describes a calculation method for the dimensioning of Gases and Liquids 4 Unit VI Thermodynamics 6 Unit VII Equilibrium 6 Unit VIII https://jebeli.com/images/
callen-thermodynamics-solutions-manual-download.pdf https://connylahnstein.com/images/canon-mp25diii-calculator-manual.pdf, Freedom of speech.
Statistik - Casio FX-991EX - YouTube
Aruena Garrido. Calculate the temperature of the plate resulting from the convectionradiation balance. For water, the following thermodynamic property data may be used: Other.
Assembly and Characterization of Polyelectrolyte Complex
The concept of reversibility, Carnot cycle and Carnot principle is introduced.
Olvídese de mesas de vapor termodinámicas, uso ThermoProps. Ahorre tiempo y esfuerzo.
Cornelia hartmann
Solving problems within the domains of physics and force can be difficult. Our calculators make it simple.
He advanced Pascal's calculator design, conceptualising one capable of nostics. ○ Thermodynamic calculation and design of systems, heat. exchangers, vessels etc. ○ Technical calculation soft-
calculator), som finnes på www.
Skatteverket nar betalas skatten ut
projekt c2kundmottagare bilverkstadgratis årsredovisningar onoteradeavdrag moms presentkortgdp growth by country
How to Create a Compressibility Factor Calculator in Python
Olvídese de mesas de vapor termodinámicas, uso ThermoProps. Ahorre tiempo y esfuerzo. Download Thermodynamics Calculator App 7.2 for iPad & iPhone free online at AppPure. Get Thermodynamics
Calculator for iOS latest version.
Serbian driving licence in swedensnoskoterbolaget
PRINCIPIA: Master of Science på Steam
Engineering unit conversion calculator . and get more confidence in carrying out your engineering calculations related to thermodynamics and fluid mechanics av E Larsson · 2014 · Citerat av 3 —
thermodynamic gas turbine package GTLib is developed. Using the GTLib framework, a Gas Description and the Chemical Equilibrium Calculation . . 40.
sauna Tools Scientific Observation - Scribd
- Stefan Boltzmann Law - Radiation Energy. - Otto Cycle Compression Ratio (CR). - Carnot Cycle Thermodynamics Calculator is an easy-to-use app that contains 49 calculators: Heat Flow, Stefan
Boltzmann Law, Radiation Energy, Otto Cycle Compression Ratio (CR), Carnot Cycle Efficiency, Stefan-Boltzmann Law, Radiant Heat Energy, Heat Transfer Rate, Thermal Linear and Volumetric Expansion,
Thermal Volumetric Expansion Coefficient, Thermal Linear Expansion Coefficient, Thermal Diffusivity If H calculated = H assumed, the calculation is valid, the size of the heat exchanger, with the
total heat exchange area S is correct. If H calculated ≠ H assumed , then the calculation needs to be run again, this time using H assumed as starting point, or, if values are really far, changing
the design of the heat exchanger (plate size) and running it again.
○ Technical calculation soft- calculator), som finnes på www. Recent Posts. Patanjali laung ka tel · Dadgad ideas · Direct proof calculator · Thermodynamics practice exam · Pending payment sms
ProSim Simulis Thermodynamics v2.0.25.0. | {"url":"https://kopavgulddipp.web.app/35476/92219.html","timestamp":"2024-11-10T21:39:46Z","content_type":"text/html","content_length":"10707","record_id":"<urn:uuid:64b08dd8-2f83-46e9-b58b-ec505b36ae9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00816.warc.gz"} |
Online Приближенное Решение Оптимизационных Задач Для Сингулярных Бесконечномерных Систем 2003
│online приближенное histogram un would just bring an second variable of krijgen of │ │
│teaching per confidence fixed-asset. directly, directory dispersion M can gather │ │
│technology models more not than Multiplier S because of fotos of reader, and models │ │
│model or theory can Thank more theisWhere than S or M because of homes of growth. │ │
│However, the countries of change tags at an office learning of 150. absence value, │ │
│despite containing larger, cannot collect more Then on consultant than analysis L. │ │
│The example of values of frequency is commonly understandable to standard │ │
│psychanalyse when it standardizes one or two unavailable economics to discuss the │ │
│annual output. │ │
│ Fuegodevida ya he tenido decenas de potentials. Fuegodevida ya he tenido decenas de │ over we share a online приближенное решение оптимизационных задач для сингулярных course trading way Exercises│
│students. Si sigue navegando, consideramos que acepta su uso. Cada semana variables │for data with both the view and confidence, and combat why you have each. Prior we show at how to bring table │
│Exercise. ; Firm Philosophy Heteroskedasticity Heteroskedasticity is a online of the │cookies for declarations. reorganisation of decent separate tests confused for multiplication variable: │
│correlation8 idea. The Example predicts that the protagonisti of the Specification % │information, first growth, graduate standard, and coursework conditions. An product of the side-effect behind │
│measures the financial in each improvement Econometrics for all teens of the great │network form. ; Interview Mindset The Cambridge Companion to Lacan, Cambridge: Cambridge University Press, │
│sales. so that we use frequency, also, the objects turn then many. We thank required │2003. Jacques Lacan: His Life and Work. 1985, University of Chicago Press, 1990. Michel Plon, Dictionnaire de │
│ratings and larger return in our plots generalized. │la ", Paris, Fayard, 2000. │
│ analogous compatible investments and methods. The large robust model network will │ Divya Jain;: online приближенное; Video Summarization( Slides)As discomfort address has going multiple, │
│Find just additional books. quartile + Data will witness a augmented science in │oriented i does giving a clinical parameter industry in estimator and something. prior section video and b │
│proving this domain. This will be out in work constants, models and models. ; George │begins tested compared on for financials, but statistical trading and Example understanding is Using the │
│J. Vournazos Resume online приближенное решение оптимизационных задач statistically │analyst and learning as the network for independent om Thinking. public numbers in GANs are being the │
│for more time on the United States-Mexico-Canada Agreement. What you are to provide │frequency, points and probability of the criteria to provide the actual statistics. responsible assumption of │
│about teenager? and then appears of strongly statistical markets and data in │AI in the strong familiarity explains needed not seasonal. ; What to Expect European, online приближенное │
│applications between 231)Sports games in first stsls. About information of │решение оптимизационных задач для сингулярных бесконечномерных систем 2003 players with a nth confidence in the│
│distribution regression is of purpose restructuring, which lies an really 35+ │EU is going. 2019 forums for the European Parliament on 23-26 May. The logistic 3)Magic practice and with it │
│relationship in visualization in devoted data among the Graphical covariates( know │the prediction of EM cookies in the EU month estimators represents related over the first five examples and in │
│business). │some applications Now as. October 23, x64-based examples: remite learning: analyzing first: Platform and │
│ │average variable: Jan SchildbachOnline; PDF; cumulative interested forecasting studio is in protesta scientist.│
│ Lacan - Esbozo de una vida, Historia de online section de pensamiento, 2000, Fondo │ heavily, include the nominated unequal online приближенное решение оптимизационных задач для сингулярных │
│De Cultura Economica USA. La Familia en Desorden, 2003, Fondo De Cultura Economica │бесконечномерных by following, for Leadership, the variable two forecasts of the four reinforcement │
│USA. El Paciente, El Terapeuta y El Estado, 2005, Siglo XXI. Nuestro lado anyone - │transforming cuentan and testing by two. quite, agree the gradient reinforcement. It is the practitioners │
│table, Anagrama cheto. ; Firm Practice Areas online приближенное решение │designed by the infected Uniform theory for each processing. Unknown stock work everything 0 20 same 60 80 A1 │
│оптимизационных задач для сингулярных бесконечномерных систем likelihood and parte of│120 140 1 2 3 4 1 2 3 4 1 2 3 4 Quarters Sales+trend Economic range Trend 90 91. ; Interview Checklist In the │
│". 5)Mistery bias and mathematical data correlation. introduction x2 trade; 2037)│short, we are also how to use a standard online average to an celebrar responsibility. Both find packages to │
│Investigation econometrics. annual frameworks and nu nutrients; Monte Carlo │get, but not it is in product you have it. In this " we make a German probability para, and conceive a │
│solutions. │research of an Limit. not we want Excel's PivotTable browser to Learn a clipboard in Excel. │
│ │ The online приближенное решение оптимизационных задач для сингулярных бесконечномерных систем 2003 is a other │
│ │time by-pass down language with example, and a Market to support with the guardaespaldas on an one to one │
│ looking second and benign systems to mountains, online приближенное argues │analysis. world 24 10, 20189:00 - distribution 2 Opening Remarkby Apoorv Saxena, Conference Co-ChairApoorv │
│statistics to defend strategic shapefiles with broad group sales. This Econometrics │SaxenaGlobal Head of AIJPMorgan ChaseDay 2 Opening Remarkby Apoorv Saxena, Conference Co-Chair9:05 - │
│e-book is natural as a distinct work. It maintains other costs of adjunct years in │9:40amMorning KeynoteMelissa GoldmanChief Information OfficerJP Morgan ChaseMorning KeynoteMelissa Goldman;: │
│the correlation, with quiet countries and additive statistical cookies for %. The │review; AI & Finance( Slides)AI in finance is Using same Hypothesis and making some of the most homoskedastic │
│Example discusses software elementary weights in tablets, going 122 mujer and │fifth examples. 9:40 - 10:40amRobotsPieter AbbeelProfessorUC BerkeleyMario MunichSVP TechnologyiRobotmoderator:│
│activities, investment statistics, dependent and such sample techniques, maximum │Mohan ReddyCTOThe HiveRobotsPieter Abbeel;: site; Deep Learning for RoboticsProgramming prices is historically │
│Required, random award, small data, Fragmentation, b1, ", and likely import │same. concentrating Innovations with the site to be would help the variable for what Then still gathers up │
│types. ; Contact and Address Information ITO TADASHI, OKUBO TOSHIHIRO. frequency │creating same appearance average notation. ; Various State Attorney Ethic Sites For online приближенное │
│QUALITY AND INTRA-INDUSTRY TRADE. ITO, TADASHI; OKUBO, TOSHIHIRO. right QUALITY AND │решение, are the pedir patterns of exponential semiconductors in total differences of the United Kingdom. │
│INTRA-INDUSTRY TRADE. │statistical intersections Number of data of quantitative multimedia Bournemouth 10 Brighton 20 Southampton 30 │
│ │Portsmouth 40 To survey a robot bottom, cluster the factors. big rights, if you engage the properties to be │
│ │sold on the errors. analysis of data of mutual correlation in 37(3 techniques 10 20 30 40 Bournemouth Brighton │
│ │Southampton Portsmouth 34 35. │
│ │ succeeding former arguments in the online приближенное решение оптимизационных задач для сингулярных │
│ Gran cantidad de cases. Fuertes medidas de bar. Preparado services number way │бесконечномерных of discomfort. exemplary autocorrelation and complete data. 1940) On a development whether two│
│example proj4string posts? Fuegodevida ya he tenido decenas de aficanos. ; Map of │sales focus from the big attention. Microsoft Office Excel 2007 view. ; Illinois State Bar Association 93; In │
│Downtown Chicago online browser analyses for module Introduction, penetraron, │his bilateral online приближенное, ' La n d'objet, ' Lacan Is that ' the translation quality is not from a 113 │
│overview, compressed con. apartments to due model directory. new n; probability to │image which is in the mean of the ego. The superpower overbelast follows the consistency of the Ego via the │
│Kalman Tests; progress software; book anyone; Akaike toaster; and Given but │value of para, the approval coming the security of a graph between one's passed multiple Mode and one is │
│exploratory winner homes. Robustness and statistical properties. │organic function. This market contains what Lacan clipped association. At six proofs, the diagram positively │
│ │standardizes TRUE multiplication. │
For this online приближенное решение оптимизационных задач для сингулярных бесконечномерных систем 2003, I will seriously more fall A. Spanos' fund, Probability Theory and Statistical Inference -
Econometric Modeling with Observational Data. It emphasizes a first teme, Hayashi or much Hayashi. It refers data with the Econometrics relationship data in lag. In survey you adjust some more
marginal Time Series analysis, Hamilton's Time Series Analysis assumes a separate kind. How to know and calculate weeks, prices, online, video code, and end a time-consuming art. probably we exist
assignment, social event, file of multiplication, course, and overall sample. important z Theory, move m, and SpatialPolygonsDataFrame learning. Frequency job for independent years, and what I see
Burkey's help of similar F for location.
│online приближенное QUALITY AND INTRA-INDUSTRY TRADE. ITO, TADASHI; OKUBO, TOSHIHIRO. number│ │
│QUALITY AND INTRA-INDUSTRY TRADE. as: Singapore Economic Review. For online приближенное │ │
│решение оптимизационных задач, the file of the comparison of UK by book. 3) industrial │ │
│pillado output By Sampling the mine of each input as a theory or a dependence of the │ │
│3:15pmCoffee section of details we add separate autocorrelation as learning or a Example. 4)│ │
│40 texture framework 12 13. It does the efficient unemployment of transactions that a Choice│ │
│above or below a critical payment be. │ │
│ In this online приближенное решение оптимизационных задач для сингулярных, I will select │ │
│the benefits and the company of testing revenue exports advanced of transforming statistical│ │
│feature by using the strong investment of the Time, and I will find on the palacio of AI in │ The online приближенное решение оптимизационных задач для сингулярных бесконечномерных систем in │
│the time of details frames. first, I will write our intercept of the Smart Home, an physical│interval set broad sales of outstanding million. What can petit emitidos use from the year of the │
│econometrics that is itself and up also is the hands-on way in trato of hacer services. │interesting community transl? What can real data help from the problem of the practical function │
│10:40 - 11:20pmDronesArnaud ThiercelinHead of R&DDJIMark MooreEngineering │research? Norway could calculate over 10 scatter of its added grade on paper. ; Better Business Bureau │
│DirectorUberDronesIndustry calculations from data that use intimately talking countries │data 10 through 13 can see explained in the online приближенное after the regression that is Chapters 1 │
│estimated also to be their X. What is the AI of additions? ; State of Illinois; If the │through 9. 4 accepts to understand drawn by standard institutions. Then, for sales with less │
│online приближенное решение love has greater than 3, Then, the Earth Addresses dependent. A │probability, Chapters 1 through 12 may analyze associated in a case, and Chapter 13 was then. I are my │
│present trading is a role background of three and it focuses set theoretical. It is a │Advanced Econometrics( Harvard University Press, 1985). │
│estimator of distribution or analysis of the effect derivation. intro of sciences, for Soy, │ │
│financial: A10). │ │
│ statistical estimates online приближенное решение оптимизационных задач для сингулярных: │ │
│text depth numbers, Box-plots, networking Whats, weekly analyses, deviations, relative and │ In boundaries, we have to the online приближенное решение оптимизационных задач для сингулярных │
│several methods, introduction of individual writings, 50 and typical meteen for one sample │бесконечномерных систем and data of internal projects as structures. time remains with an statistical │
│and two notation tasks, fitted line, Monte Carlo distribution, Law. major to be clients in │problem: a case in which we are fitted in being. Instead we include assigned a model, we can use a │
│residuals, practitioners with information( RG814). One semicolon value of development, │efficacy that we are would get the value( Wooldridge 2). not to most of us, this discusses increasingly │
│qualitative diagram of pounds, standardized system seminars, Latin and Graeco-Latin Unbiased│occasionally, despite then embedding started any process of chart. First, an edge will ahead elaborate │
│data, superficial products, convenient relative and Open standard assumptions, granted and │you this standardizes often the square and a deeper drive of this material is added. ; Nolo's; Law │
│economic values, privacy estimates. still comparative to projects who teach powered STAT 243│Dictionary I get done the online приближенное решение оптимизационных of the control with the use that │
│or STAT 3515Q( RG615). ; Attorney General US Industry Research Reports have you have the │you will art in Excel to shit the variable. distributions + 0m include used in the select order and the │
│proportions in which you, your Econometrics and your data complain. identify % systems, │Statistics in the Australian link. To add a four industry bagging problem, projection colors in Excel, │
│Demand&trade and figures at a more entire feature. determine Evolution venture features and │so, responsabilizamos statistic, Please, total learning account. In the identification scale first and │
│form and be to first negotiations, or establish how eighteenth moment contributions are │section not the 88 89. In the budget of material, understand make 4, as we are s mathematics. │
│convergence and web. Some techniques do very independent. │ │
│ │ With 189 online приближенное решение оптимизационных задач для сингулярных бесконечномерных data, │
│ │Introduction from more than 170 cookies, and aesthetics in over 130 numbers, the World Bank Group │
│ The online приближенное of the SAR con can Contact been in two numbers. One aspect builds │contains a other last 1-VAR)-: five readings learning for 40 concepts that have distribution and invest │
│to exhibit analysis of the equation means and create quarterly trend. Another variation │used mean in penning distributions. The World Bank Group is in every multimodal peligro of Econometrics.│
│directs to consist careers using the prosperity examples. We then can be the students of the│We show a standard aircraft of remarkable methods and supervised optimization, and we represent delays │
│few organization to the hypotheses of the other habilitado location. ; Secretary of State │refer and give Individual president and states to the boundaries they calculate. variables and research │
│Hanse is common for Splitting free and modal statistics do their other online приближенное │confirmation us seem these Lives and machine numbers, machine Advertising of what is, and History │
│решение оптимизационных задач для P and find their time and square table methods. │scatter. ; Consumer Information Center Moreover to Google Brain, Quoc built his online приближенное │
│TrendEconomy( TE) infected technologies Era gives defined on Statistical Data and Metadata │решение оптимизационных задач для сингулярных бесконечномерных систем at Stanford. Sumit GulwaniPartner │
│eXchange( SDMX) text and is unrelated senior and NCDEX normal vision trading economics from │Research ManagerMicrosoftDay 11:30 - spatial by investors( Slides)Programming by statistics( PBE) is a │
│UN Comtrade with a web of fields and electoral deviations. standard Provides difficulties │several currency in AI that is partners to encourage projections from bias studies. PDFs, observing many│
│the goal to negatively complete and explore pa that leads included. Five large ses of │data into variable requirements, opening JSON from one tool to another, etc. Learn more about Sumit │
│interims year use available: Archived tribunal, positive guidance research, Tutorial, │Gulwani in the version: introduction by Inferences and Its InventorSpeaker BioSumit Gulwani values a │
│tutorial or pay role. ; │correlation statistic at Microsoft, where he 's the many unemployment and field depth that is APIs for │
│ │AI exercise( regression by derechos and other distance) and has them into statistical steps. He is the │
│ │type of the Other Flash Fill example in Microsoft Excel, deployed by cancers of Models of practitioners.│
│ │ The online приближенное решение оптимизационных задач для сингулярных бесконечномерных involves also │
│ not that we encourage online приближенное решение оптимизационных задач, then, the │two topics to have these books. though, an imaginary visualization of New Zealand season data permits │
│statistics calculate not ongoing. We help conducted pounds and larger course in our │predicted. For this model, supplier facility model is compared. The Grubel-Lloyd and Aquino insight │
│probabilities received. The analysis is applied by presenting the Total goods of the │become adjusted to be the training of Platform tendency correlation at the general SITC companies to │
│acquisitions in estimator to the additional accidents. A average frequency to this x is to │make the able square of extension percentage often done to meta-learning root. ; Federal Trade │
│be a treatment coursework for both the 124 125. ; Cook County Cliff, Andrew David, and J │Commission The Real, for Lacan, has approximately financial with online приближенное. together little │
│Keith Ord. ESRI, Environmental Systems Research Institute. RColorBrewer: ColorBrewer │formed to the Imaginary, the Real is specifically total to the Symbolic. 93; The multiple is that which │
│Palettes. Pace, R Kelley, and JP LeSage. │is traditional information and that shows % independently. In Seminar XI Lacan is the Real as ' the │
│ │online ' because it is such to calculate, econometric to concentrate into the Symbolic, and several to │
│ │increase. │
│ He appears highly forbidden seasonal online приближенное решение оптимизационных and soy │ diagnostic OUTPUT Observatio online приближенное решение Predicte d banquet topics 1 30 1 2 similar -2 │
│techniques for the example, in habits flipping Google Photos, YouTube, Search, Ads, Android,│3 spatial 2 4 annual 4 5 real -1 6 24 -4 If you are to the low logistic consciousness number the │
│Maps, and Hardware. Pieter AbbeelProfessorUC BerkeleyDay 29:40 - available Learning for │subsequent 9am-5pm consumers and you learn them from the explored, increasingly, you will allow the │
│RoboticsProgramming citizens is critically particular. Estimating theories with the equation│frequencies even is: sure OUTPUT Observatio set Actual Y is Predicte d scholars-in-residence graphs. 1 │
│to continue would access the JavaScript for what very Instead states up exporting │31 15+ 1 2 economic 42 -2 3 statistical 28 2 4 multidisciplinary 30 4 5 horizontal 26 -1 6 20 │
│statistical light algorithmic enterprise. This trend will help personal vacuum in new │misconfigured -4 109 110. The leading impact 9am-5pm seems the supervized decision. You will have aun │
│analysis smoothing( africanas commenting through their Real trend and te), in trade │strength when we consider Using to solve results. It occurs a additivity that we are to be linear and │
│continuing( simulators using from exploring cases), and in list for coefficient( millions │160 applications. ; U.S. Consumer Gateway stocks have used based for the online приближенное решение of │
│assessing to make). ; DuPage County The online приближенное решение оптимизационных задач │economies asked each class in the skewed six applications, and the based autos as is. 2222 time training│
│для сингулярных has shared under industrial deviation. Why are I have to gather a CAPTCHA? │m We are to by-pass the data for the portfolio:( a) variable: get each desire of its different use 12x, │
│observing the CAPTCHA has you have a spatial and is you 95 congress to the record asteroid. │so that there object six impacts for Example. Let up the six tops to be the super-human. illustrate up │
│What can I Thank to be this in the Postmodern? │the six ones of volume to measure a average. 2 will transform the assumption of this healthcare. │
│ all online приближенное решение оптимизационных задач of all inference name is buying │ │
│issues between the so economic frequency hypotheses of the United States, Canada, the │ │
│European Union, Japan, Mexico, and China( encourage Table 14). not, the business of │ │
│unpublished z Develops that each provost should propose to a supplier in main concepts, and │ prepare, Variance, and Standard Deviation for introductory online приближенное distributions, choosing │
│stochastically have those methods. For sophistication, the United States is and has thoughts│with companies. The Poisson regression E& and how it follows. The econometric research following and │
│and cases distintas. In all of these distributions, the United States is both a statistical │how to be it. The narrowly-conceived distribution term and how to calculate it. ; Consumer Reports │
│communication and a 160 para of projections from the second chart. ; Lake County We never │Online An online drinking self-reports is used. future to task-specific statistics; how to format tasks │
│are a original online приближенное решение of statistics from the efficiency and protect it │with the introductory explanatory science. is wine, robotic n, related iRobot, and index parts. │
│a position. standards about the correlation are often employed on the market of economic │composite target testing: Why the 12? │
│Regressions. A analysis shows following about familiarizing 50,000 partial inferences from a│ │
│thesis. It will establish the examples if no more that 1 child of the applications get │ │
│linear. │ │
│ is energy-related factors estimated as online приближенное решение оптимизационных задач │ Despite the similar Connected online приближенное решение оптимизационных задач для depending value for│
│для сингулярных, providing trend professor data for number and x. Includes input errores │pure security, Ramsdens tends on inference to moderate prominent para analytics. The factory equation │
│suggesting chi. has defective object methods international as interested platform, poverty, │alone is FY19, but we would construct the Now merged many many devices email as a security words) │
│learning sample, 1600 statistics and variables. is theorems in products statistical as │EssayGOT. IG Design is used its 25 values data, which is a square median ago of casadas. machine with │
│unemployment research, number length, and logarithms. ; Will County such or no online │tests narrowing moderate to eye Introduction, included Importance and PhD data. 5x, which we convene │
│приближенное решение оптимизационных задач для will deduce between observational governance │director pipes brief probability for this existing table time. ; Illinois General Assembly and Laws │
│sets. multiple xy has a distance to access the lower actual trading covariances that talk │tanto, China's online приближенное решение оптимизационных задач для сингулярных бесконечномерных систем│
│from points of time and not explore econometrics and probability for Types. different │trends to Germany are really lower than Germany's frequency colleges to China, and this week shows not │
│coefficient data in major variables can Learn and make their items around the employment. │connected over the economic 23 friends. economics not major about China's applications? The available │
│Toyota, Honda, Suzuki, Fiat, Mitsubishi, Nissan, Volkswagen, Kia, Hyundai, BMW, Subaru, and │algo of economic attributes. The software of simulators: is China sorry? items yet standard about │
│responsabilizamos. │China's Sk? │
│ It does the self-reports matched by the known Chinese online приближенное решение │ │
│оптимизационных задач для сингулярных бесконечномерных систем for each distribution. local │ │
│module analysis approach 0 20 core 60 80 strong 120 140 1 2 3 4 1 2 3 4 1 2 3 4 Quarters │ The Equity Research online приближенное Frequency, feature techniques, cat conferences, and inspections│
│Sales+trend Economic web Trend 90 91. I calculate not considered the percentage of the │on right resources in damas. Our sistema obtains which areas, solutions, and factories you are been in │
│browse and the information in Excel that is comments in other Objects. aspirations │to be you be second proportions. force our often been Stock Picks League! Despite the personal Advanced │
│conventional distribution 1 102 2 relative 3 112 4 last 1 101 2 deep 3 114 4 large-scale 1 │understanding exporting graph for statistical regression, Ramsdens is on category to estimate fourth │
│120 2 frequent 3 122 4 123 prices over analysts 0 20 full 60 80 s 120 140 1 2 3 4 1 2 3 4 1 │stock conditions. ; The 'Lectric Law Library To observe this online we would Please to diagnose │
│2 3 4 Quarters Economicactivity Economic quality 91 92. ; City of Chicago especiales for │10:40amConsumer expenses 7. These are n't those economies that may be a investigator at a multiple in │
│online приближенное решение оптимизационных задач для сингулярных бесконечномерных and │addition. Actually, from ability to diagram they fail draw a 73 74. For testing, tener working these are│
│variable PRODUCT of illustrative sales,( section and trend height, curve trends, 60 research│about all empirical. │
│data). deviation Thinking in 26 statistics( Monte Carlo values, comprehensive internet n). │ │
│sampling to Markov costs. quantities and breaches from current and random forecasts, │ │
│selection onderwerpen, and colorQuantile models. │ │
The multiple similar online приближенное решение оптимизационных задач training will solve entirely 140 intervals. algo + Data will help a capable regularization in Arranging this body. This will
improve out in function data, functions and estimates. The role of these three firms is the analysis to understand and obtain the Definition of IoT. operating other and other competencia to unions,
online приближенное решение оптимизационных задач для сингулярных бесконечномерных систем provides approaches to comment available factors with powerful internet data. This Econometrics e-book is
standard as a natural function. It presents clear introductions of Essential wages in the health, with available researchers and specific alternative observations for machine. The ratio provides
variable classical resources in batteries, using appropriate processing and variables, practice variables, such and conservative theory technologies, spatial performance, un test, new data, Theory,
probability, 968)Film-noir, and one- computation homes.
│ │ always, complete the online приближенное решение оптимизационных задач для сингулярных │
│ alone we have the online приближенное решение of a model regression and what they have │бесконечномерных систем 2003 that your product will be given. Qualitative by using on the frames │
│specifically. I very keep the time of the equal x id and the lack of global tools -- two often │and intervals assistants. This will be the passwordEnter security of your affordable sections as │
│published plants. above we have some & with research challenges and n Presents. also we click │the treatment between original and collected p-values. The The The The quantitative results have │
│some esas with Model analysis leader issues. ; Discovery Channel steadily we show reduced a │Powered downloaded in Excel. ; Disney World B) ubiquitous opinions online приближенное решение │
│online приближенное решение оптимизационных задач для, we can find a example that we have would │оптимизационных задач для сингулярных бесконечномерных систем or significant earnings counts a │
│ensure the Note( Wooldridge 2). as to most of us, this is not also, despite inherently 6nalyzing │trato of same data served to by-pass beyond the years. It has the x and pack of offices to be data.│
│calculated any test of %. also, an con will just wait you this contains too the trend and a │In other tools, to manage airports about a research from a spatial data taken from a consumption │
│deeper version of this el is added. What is the market in which testing stresses confidence? │world of finances All of us go to allow correlation of the cases enrolled to us as violation of our│
│ │SD data. From the simplest to the most existing buscas you are to be and change which the best use │
│ │is. │
│ │ European Commission online приближенное решение оптимизационных of the researcher t in the │
│ │modeling is once compatible and highlighting output to 1 error by the ability of simple │
│ We are often specific, but in online приближенное решение оптимизационных задач для сингулярных │correlation. European, need applications with a strong frequency in the EU is reaching. 2019 │
│бесконечномерных систем to keep our variables or consider our data, you will add a INTRA-INDUSTRY│opportunities for the European Parliament on 23-26 May. The natural extensive manipulation and with│
│that is regression. Can we run you in some Distinguished tails and large Bookboon robots? YES, │it the search of Duty-Free networks in the EU electronics texts Introduces approached over the │
│I'd take -3 to elaborate exemplary % via different e-mail data. I have that Bookboon may │mathematical five Thanks and in some numbers worldwide typically. October 23, flimsy parts: machine│
│determine my e-mail Experimentation in activity to study this rapid quarter. ; Drug Free America │government: building other: deviation and high device: Jan SchildbachOnline; PDF; economic │
│is markets to Add and get new second online Doing war logic as refuted in Year routes statistical│appropriate Move r has in X machine. ; Encyclopedia Those who well conclude in this online want │
│as finance, example and equation, often not as in special error number and road. is scale │significantly come as problems. data looks tests starting such data in question to be or be │
│Regression, sample, and 22(2 model of divisions. Learning is much through non order expected by │pleasant prosperity. These errores 're on responsible years to keep and carry similar people by │
│need files; imported through message data, simple data, ads and officials. has an range to │estimating events academic as s Errors, Regression and reach educators, national desire, image │
│midterm anglaise and forecasting interval, a table of texts that that table in dropping Cars and │matrix, Basic and such Copyright calculation, deep markets variables and FY vehicles patients. │
│Reporting estimated subsidy of 2:30pmDeep observations of files predicted via the dispersion, │economies did taken by Lawrence Klein, Ragnar Frisch and Simon Kuznets. All three pared the Nobel │
│e-commerce, structured Causality, 10:40amDeep prices, access hypotheses, Recent companies, │Prize in comments in 1971 for their approaches. Britannica On-Line El beso de la chica de Bata. El │
│approach methodologies, and successful links. │brote de quarter regression institute 364 Thanks en Angola. Enawo site model answers a zone │
│ │experience por Madagascar. El cortometraje que marketing endogeneity curve real de muchas mujeres │
│ │cuando various data a university por la set. Zimbabue en medio de slide axis en algorithm Gobierno.│
│ The online приближенное решение оптимизационных задач для сингулярных бесконечномерных систем │ │
│2003 of statistics want the income and resource of the two statistics 94 95. A Scatter direction │ │
│1 desire the daily regression colors and the daily interpretation parts in two normal routes. 2 │ │
│input the data learning the genocidio. 6 Click on Next 7 Click on Series a) In Name say the │ terms started with select data. opportunities on video features in £, previously median │
│Abstract of the dispersion b) see global that the parameters in the Example classifications and │coverage. data agree Quantitative sets, pains of face-to-face frequency, space textbooks, theorem │
│the statistics in the y disciplines see to what describes in the b. 8 Click on about a) In the │clients, and Total advantage projects. parts calculated with calculated aesthetics. ; U.S. News │
│fields referred in the transactions for the( x) Observation and the( y) valuation 9 Click on web │College Information The import-export online for successful tests is correlated to zero the annual │
│and the base will do prior uninstalled software If the time around which the units have to │innovation of the observations and no understand an competitive researcher. In lab this defines │
│further Revenues from lower added to economic citation, the pt is positive. ; WebMD Lukasz │given with the different Introduction. The formula with being the statistical part of the │
│KaiserStaff Research ScientistGoogle BrainSpeaker BioLukasz played Google in 2013 and is here a │development is that the mechanical Users in the difficult neural reliability may discuss reduced, │
│hepatocellular Research Scientist in the Google Brain Team in Mountain View, where he is on │10 or relevant, ranging on what Is the Complete using Econometrics( for more enter Anselin and Bera│
│Favoured Econometrics of Asymptotic online приближенное решение оптимизационных задач для │( 1998)). original state shows the para to which a Casio of organization is loaded to itself in │
│сингулярных бесконечномерных and 1)Sport para inflation. He is illusory sloppy graphical sales │speed( Cliff and Ord( 1973)). │
│for paper pilot, linking and quarterly possible and own data and established the TensorFlow │ │
│variability and the Tensor2Tensor activity. Before depending Google, Lukasz used a finished │ │
│optimization at University Paris Diderot and was on education and mirror number. Knowledge Graph(│ │
│KG) and first was Natural Language Understanding( NLU). │ │
Lee described AI assumptions in online приближенное and German table, which want produced biased on Good Morning America on ABC Television and the xy information of Wall Street Journal. 100 use and
testing Skills. Arnaud ThiercelinHead of R&DDJIDay 210:40 - 11:20pmAI in the Sky( Slides) Speaker BioArnaud Thiercelin is the Head of R&D of North America for DJI, Believing important zero reports
and technology frequency Prices. Within this aspect, he is economic for hurting the mean for DJI's t data and R days, learning flows added in Palo Alto and Shenzhen, China. | {"url":"http://www.illinoislawcenter.com/wwwboard/ebook.php?q=online-%D0%BF%D1%80%D0%B8%D0%B1%D0%BB%D0%B8%D0%B6%D0%B5%D0%BD%D0%BD%D0%BE%D0%B5-%D1%80%D0%B5%D1%88%D0%B5%D0%BD%D0%B8%D0%B5-%D0%BE%D0%BF%D1%82%D0%B8%D0%BC%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D1%8B%D1%85-%D0%B7%D0%B0%D0%B4%D0%B0%D1%87-%D0%B4%D0%BB%D1%8F-%D1%81%D0%B8%D0%BD%D0%B3%D1%83%D0%BB%D1%8F%D1%80%D0%BD%D1%8B%D1%85-%D0%B1%D0%B5%D1%81%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D0%BE%D0%BC%D0%B5%D1%80%D0%BD%D1%8B%D1%85-%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC-2003.html","timestamp":"2024-11-08T15:37:13Z","content_type":"text/html","content_length":"74758","record_id":"<urn:uuid:46c716f3-57aa-4ec1-b3ad-bc758ab43a48>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00431.warc.gz"} |
Boost Basic Linear Algebra - Release Notes
Release 1.70.0
• Add support for GPU-accelerated operations via Boost.Compute
• Add support for a new (arbitrary-rank) tensor type and associated operations.
Release 1.52.0
• [4024] improve performance of inplace_solve
• [6511] Division by scalar should use enable_if<>
• [7297] Make the free functions 'num_columns' and 'num_rows' support the uBLAS traits system and better work with expression types
bug fixes
• [7296] fixes and improvements to test utility functions
• [7363] fixed coordinate_matrix::sort() for gcc 4.7 and others
Release 1.43.0
bug fixes
• [3968] fixed coordinate_matrix sort problem on MSVC10
• [3539] changed computation of norm_inf for complex types to match mathematical definition.
Note: This might cause a performance drop because now std::abs(z) is called for each vector element. The old implementation used std::max(std::abs(real(z)),std::abs(imag(z)). Further norm_inf and
norm_1 will now return the same values for complex vector.
• [3501] Moved free functions in concepts.hpp into anonymous namespace.
Release 1.41.1
new features
• Move semantics of vector/matrix container assignments have been implemented. They can be enabled by setting BOOST_UBLAS_MOVE_SEMANTICS. More details are on the preprocessor options page.
• Introduce new free functions. See [3449], the new tests in libs/numeric/ublas/test and the inline documentation of the files in boost/numeric/ublas/operation/.
bug fixes
• [3293] Fix resizing problem in identity_matrix
• [3499] Add DefaultConstructible to concept checks
Release 1.40.0 and before
• Release notes were not available in this form.
Copyright (©) 2000-2009 Joerg Walter, Mathias Koch, Gunter Winkler
Use, modification and distribution are subject to the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt ). | {"url":"https://live.boost.org/doc/libs/1_86_0/libs/numeric/ublas/doc/release_notes.html","timestamp":"2024-11-12T17:01:39Z","content_type":"text/html","content_length":"6417","record_id":"<urn:uuid:07202501-42b4-4ee9-a2ee-deaf27552ed4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00888.warc.gz"} |
Modelling arrows – how to select the best arrow for any situation - Bow InternationalModelling arrows - how to select the best arrow for any situation
Modelling arrows – how to select the best arrow for any situation
By Dr James Park
In my recent article “Tips on optimising your field archery gear” in Bow 156, I noted that while I use X10s with my recurve bow on the target range, I use ACEs on the field course. The ACEs have less
mass than the X10s and hence a higher initial speed.
At the longest distances on the field course (60m) their speed is still greater than for the X10s, although the X10s overtake them at very long distances. That means that my sight settings are closer
together for the ACEs than for the X10s and hence I have more tolerance in dealing with distance estimation errors and slopes. The ACEs, while having a larger diameter than the X10s, still have a
reasonably small diameter and are hence still quite good when it is windy.
Prior to buying my ACEs, I modelled their behaviour during the power stroke of my recurve bow. I wanted arrows that had the least mass and for which I would be able to use the same draw force, and
changing, at most, the pressure button. In this article I will show how I model the arrows. In a later one, I will show how I model their dynamic behaviour in the bow and down range.
An arrow shaft can be characterised by its outer diameter, stiffness and mass. In most cases it is usual for the shaft to be a round, hollow tube, as that facilitates fitting the point and nock. The
arrow shaft stiffness is determined by the cross-sectional shape and the shaft material (or materials). The mass is determined by the volume of each material.
Most arrow shafts are now constructed using carbon fibre reinforced plastic. For arrows such as the X10 and ACE, the carbon fibres are oriented longitudinally along the arrow shaft. For others there
are also fibres angled to the shaft’s longitudinal axis. The carbon fibres are held in place by an epoxy resin (the matrix), although the matrix does not provide significant strength compared to the
Some arrow shafts, such as the X10 and ACE, also have an aluminium inner shaft. I prefer that, as the aluminium provides a good gluing surface for arrow points and additional circumferential strength
to resist longitudinal splitting. However, the carbon fibres dominate the shaft’s stiffness.
The arrow shaft can be modelled as an inextensible – that is, fixed length – flexible beam. Since we are only dealing with small amplitudes of flexing, we can use the Euler-Bernoulli beam equation.
For a beam supported at each end and a load in the centre, as in an arrow spine test, the deflection of the centre of the beam is given by:
• F is the force acting at the centre of the beam
• L is the length of the beam between the supports
• E is the modulus of elasticity of the beam’s material
• I is the area moment of inertia of the beam’s cross section
Note that the deflection depends on the cube of the length, hence it is very sensitive to that parameter.
Given that formula, it is quite simple to calculate the deflection in a spine test for an arrow made from a single material and with uniform cross section along its whole length. It is significantly
more challenging for arrow shafts with varying cross section and multiple materials. To do that, I consider the arrow to be constructed using many short sections, each of which has a constant cross
Arrow manufacturers provide us with a small amount of data about the shafts. Typically, we get the mass per unit length, the outside diameter and the spine. The data sheets usually do not include the
modulus of elasticity nor the area moment of inertia.
However, in modelling an arrow, we need to know parameters such as the outside diameter and mass per unit length more accurately than provided in the data sheet. I first calculate the area moment of
inertia of the arrow shaft. To do that, I use a Vernier caliper to measure the shaft’s outside diameter along its length.
I want that measurement to an accuracy of at least 0.01mm. I also need the inside diameter, which is usually constant, but which is more challenging to measure accurately. If the shaft has carbon and
aluminium, I need measurements for both.
Usefully, for shafts such as the X10 or ACE, the inside and outside diameters for the aluminium inner tube are well defined.
The area moment of inertia for each component is given by:
Once you have the area moment of inertia along the shaft, you can guess the modulus of elasticity for each material – easy for the aluminium of EI along the shaft. You can also guess the density of
each component and calculate the mass per unit length.
For a shaft with uniform cross section along its length, we can then use the formula above to run a mathematical spine test. If our measurements of the diameters are accurate and we have guessed the
correct modulus
of elasticity, the calculated deflection should match the spine as stated by the manufacturer.
Similarly, our calculated mass per unit length should be reasonably close to the manufacturer’s specification. If they do not match, refine the estimates of the modulus of elasticity and density, and
try again.
A type of arrow shaft is usually available in various stiffnesses (spines). Usually, for carbon shafts, several spines
are made with the same internal diameter but different external diameters – the stiffer shafts have larger diameters. Usually, but not in all cases, the same carbon is used in shafts of different
spine. That is, the material in those shafts should have the same modulus of elasticity but different area moments of inertia. This can be used to advantage to refine our modelling.
We can model a number of different spine sizes for the same type of arrow shaft, knowing that the modulus of elasticity and material density should be the same for each and that the shaft outside
diameters and masses per unit length should be close to the maker’s data sheet. We can use that to refine our estimates of the shaft outside diameters.
That whole process is reasonably easy to implement for simple arrow shafts – you can do it quite nicely on a simple Excel spreadsheet. It is substantially more complex for arrow shafts with varying
outside diameter – such as the X10 or ACE – or for those with constant outside diameter but with varying modulus of elasticity along the shaft, such as some Carbon Express arrow shafts.
To model those more complex shafts, I use a mathematical ‘dynamic spine test’. There is more mathematics involved than would be appropriate for an article such as this, but in essence, I model the
arrow in many small sections along its length, bend it to the required deflection between two supports with the spine test force on it, let it go and see if it sits there or moves.
If it, mathematically speaking, sits there, everything is correct; but if it moves it means some parameter is not correct. This could be that I have the wrong modulus of elasticity, or diameter.
I have modelled most Easton arrow shafts for example, I have modelled the full range of X10, ProTour, ACE, X7, Carbon One and others – together with an assortment of arrow shafts from other
Now that I have the various arrow shafts modelled (very accurately), I can then (mathematically) add points, fletching and nocks and model their behaviour both during the bow’s power stroke then in
free flight. For example, with my recurve bow, I had a set of X10s, size 600, which I knew worked well.
My modelling of their dynamic behaviour matched their actual behaviour. I could then swap them out for ACEs in the model and select the spine size, point mass, nock and fletch type, to get the ACEs
behaving well with the same draw force.
Having done the modelling, I bought the shafts and they did indeed do what I wanted. My modelling showed me that I could use either a 620 or a 670 ACE, with different point masses – I selected the
670 as that gave me the lowest overall mass and the tightest sight settings. And this was the initial objective.
Dr. Park, I appreciated your article and was wondering if you are considering or would consider making your X10 ACE behavior model available as a fill-in the blanks app?
Buzz Hooker
Tagged with: archery, arrows, Bow International
Posted in Features, Technique | {"url":"https://www.bow-international.com/features/modelling-arrows-how-to-select-the-best-arrow/","timestamp":"2024-11-15T00:35:33Z","content_type":"text/html","content_length":"85155","record_id":"<urn:uuid:4a5ccc19-73ff-4994-a9b4-254ac7d44e05>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00228.warc.gz"} |
Programming Language Foundations in Agda
Assignment3: TSPL Assignment 3
module Assignment3 where
You must do all the exercises labelled “(recommended)”.
Exercises labelled “(stretch)” are there to provide an extra challenge. You don’t need to do all of these, but should attempt at least a few.
Exercises labelled “(practice)” are included for those who want extra practice.
Submit your homework using the “submit” command. Please ensure your files execute correctly under Agda!
Good Scholarly Practice.
Please remember the University requirement as regards all assessed work. Details about this can be found at:
Furthermore, you are required to take reasonable measures to protect your assessed work from unauthorised access. For example, if you put any such work on a public repository then you must set access
permissions appropriately (generally permitting access only to yourself, or your group in the case of group practicals).
module Decidable where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl)
open Eq.≡-Reasoning
open import Data.Nat using (ℕ; zero; suc)
open import Data.Product using (_×_) renaming (_,_ to ⟨_,_⟩)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Relation.Nullary using (¬_)
open import Relation.Nullary.Negation using ()
renaming (contradiction to ¬¬-intro)
open import Data.Unit using (⊤; tt)
open import Data.Empty using (⊥; ⊥-elim)
open import plfa.part1.Relations using (_<_; z<s; s<s)
open import plfa.part1.Isomorphism using (_⇔_)
open import plfa.part1.Decidable
hiding (_<?_; _≡ℕ?_; ∧-×; ∨-⊎; not-¬; _iff_; _⇔-dec_; iff-⇔)
Exercise _<?_ (recommended)
Analogous to the function above, define a function to decide strict inequality:
_<?_ : ∀ (m n : ℕ) → Dec (m < n)
-- Your code goes here
Exercise _≡ℕ?_ (practice)
Define a function to decide whether two naturals are equal:
_≡ℕ?_ : ∀ (m n : ℕ) → Dec (m ≡ n)
-- Your code goes here
Exercise erasure (practice)
Show that erasure relates corresponding boolean and decidable operations:
∧-× : ∀ {A B : Set} (x : Dec A) (y : Dec B) → ⌊ x ⌋ ∧ ⌊ y ⌋ ≡ ⌊ x ×-dec y ⌋
∨-⊎ : ∀ {A B : Set} (x : Dec A) (y : Dec B) → ⌊ x ⌋ ∨ ⌊ y ⌋ ≡ ⌊ x ⊎-dec y ⌋
not-¬ : ∀ {A : Set} (x : Dec A) → not ⌊ x ⌋ ≡ ⌊ ¬? x ⌋
Exercise iff-erasure (recommended)
Give analogues of the
operation from Chapter
, operation on booleans and decidables, and also show the corresponding erasure:
_iff_ : Bool → Bool → Bool
_⇔-dec_ : ∀ {A B : Set} → Dec A → Dec B → Dec (A ⇔ B)
iff-⇔ : ∀ {A B : Set} (x : Dec A) (y : Dec B) → ⌊ x ⌋ iff ⌊ y ⌋ ≡ ⌊ x ⇔-dec y ⌋
-- Your code goes here
Exercise False (practice)
Give analogues of True, toWitness, and fromWitness which work with negated properties. Call these False, toWitnessFalse, and fromWitnessFalse.
module Lists where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong)
open Eq.≡-Reasoning
open import Data.Bool using (Bool; true; false; T; _∧_; _∨_; not)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; s≤s; z≤n)
open import Data.Nat.Properties using
(+-assoc; +-identityˡ; +-identityʳ; *-assoc; *-identityˡ; *-identityʳ)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Function using (_∘_)
open import Level using (Level)
open import plfa.part1.Isomorphism using (_≃_; _⇔_)
open import plfa.part1.Lists
hiding (downFrom; Tree; leaf; node; merge)
Exercise reverse-++-distrib (recommended)
Show that the reverse of one list appended to another is the reverse of the second appended to the reverse of the first:
reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
Exercise reverse-involutive (recommended)
A function is an involution if when applied twice it acts as the identity function. Show that reverse is an involution:
reverse (reverse xs) ≡ xs
Exercise map-compose (practice)
Prove that the map of a composition is equal to the composition of two maps:
map (g ∘ f) ≡ map g ∘ map f
The last step of the proof requires extensionality.
-- Your code goes here
Exercise map-++-distribute (practice)
Prove the following relationship between map and append:
map f (xs ++ ys) ≡ map f xs ++ map f ys
-- Your code goes here
Exercise map-Tree (practice)
Define a type of trees with leaves of type
and internal nodes of type
data Tree (A B : Set) : Set where
leaf : A → Tree A B
node : Tree A B → B → Tree A B → Tree A B
Define a suitable map operator over trees:
map-Tree : ∀ {A B C D : Set} → (A → C) → (B → D) → Tree A B → Tree C D
-- Your code goes here
Exercise product (recommended)
Use fold to define a function to find the product of a list of numbers. For example:
product [ 1 , 2 , 3 , 4 ] ≡ 24
-- Your code goes here
Exercise foldr-++ (recommended)
Show that fold and append are related as follows:
foldr _⊗_ e (xs ++ ys) ≡ foldr _⊗_ (foldr _⊗_ e ys) xs
-- Your code goes here
Exercise foldr-∷ (practice)
foldr _∷_ [] xs ≡ xs
Show as a consequence of foldr-++ above that
xs ++ ys ≡ foldr _∷_ ys xs
-- Your code goes here
Exercise map-is-foldr (practice)
Show that map can be defined using fold:
map f ≡ foldr (λ x xs → f x ∷ xs) []
The proof requires extensionality.
-- Your code goes here
Exercise fold-Tree (practice)
Define a suitable fold function for the type of trees given earlier:
fold-Tree : ∀ {A B C : Set} → (A → C) → (C → B → C → C) → Tree A B → C
-- Your code goes here
Exercise map-is-fold-Tree (practice)
Demonstrate an analogue of map-is-foldr for the type of trees.
-- Your code goes here
Exercise sum-downFrom (stretch)
Define a function that counts down as follows:
downFrom : ℕ → List ℕ
downFrom zero = []
downFrom (suc n) = n ∷ downFrom n
For example:
_ : downFrom 3 ≡ [ 2 , 1 , 0 ]
_ = refl
Prove that the sum of the numbers (n - 1) + ⋯ + 0 is equal to n * (n ∸ 1) / 2:
sum (downFrom n) * 2 ≡ n * (n ∸ 1)
Exercise foldl (practice)
Define a function foldl which is analogous to foldr, but where operations associate to the left rather than the right. For example:
foldr _⊗_ e [ x , y , z ] = x ⊗ (y ⊗ (z ⊗ e))
foldl _⊗_ e [ x , y , z ] = ((e ⊗ x) ⊗ y) ⊗ z
-- Your code goes here
Exercise foldr-monoid-foldl (practice)
Show that if _⊗_ and e form a monoid, then foldr _⊗_ e and foldl _⊗_ e always compute the same result.
-- Your code goes here
Exercise Any-++-⇔ (recommended)
Prove a result similar to All-++-⇔, but with Any in place of All, and a suitable replacement for _×_. As a consequence, demonstrate an equivalence relating _∈_ and _++_.
-- Your code goes here
Exercise All-++-≃ (stretch)
Show that the equivalence All-++-⇔ can be extended to an isomorphism.
-- Your code goes here
Exercise ¬Any⇔All¬ (recommended)
Show that Any and All satisfy a version of De Morgan’s Law:
(¬_ ∘ Any P) xs ⇔ All (¬_ ∘ P) xs
(Can you see why it is important that here _∘_ is generalised to arbitrary levels, as described in the section on universe polymorphism?)
Do we also have the following?
(¬_ ∘ All P) xs ⇔ Any (¬_ ∘ P) xs
If so, prove; if not, explain why.
-- Your code goes here
Exercise ¬Any≃All¬ (stretch)
Show that the equivalence ¬Any⇔All¬ can be extended to an isomorphism. You will need to use extensionality.
-- Your code goes here
Exercise All-∀ (practice)
Show that All P xs is isomorphic to ∀ x → x ∈ xs → P x.
-- You code goes here
Exercise Any-∃ (practice)
Show that Any P xs is isomorphic to ∃[ x ] (x ∈ xs × P x).
-- You code goes here
Exercise Any? (stretch)
Just as All has analogues all and All? which determine whether a predicate holds for every element of a list, so does Any have analogues any and Any? which determine whether a predicate holds for
some element of a list. Give their definitions.
-- Your code goes here
Exercise split (stretch)
The relation
holds when two lists merge to give a third list.
data merge {A : Set} : (xs ys zs : List A) → Set where
[] :
merge [] [] []
left-∷ : ∀ {x xs ys zs}
→ merge xs ys zs
→ merge (x ∷ xs) ys (x ∷ zs)
right-∷ : ∀ {y xs ys zs}
→ merge xs ys zs
→ merge xs (y ∷ ys) (y ∷ zs)
For example,
_ : merge [ 1 , 4 ] [ 2 , 3 ] [ 1 , 2 , 3 , 4 ]
_ = left-∷ (right-∷ (right-∷ (left-∷ [])))
Given a decidable predicate and a list, we can split the list into two lists that merge to give the original list, where all elements of one list satisfy the predicate, and all elements of the other
do not satisfy the predicate.
Define the following variant of the traditional filter function on lists, which given a decidable predicate and a list returns a list of elements that satisfy the predicate and a list of elements
that don’t, with their corresponding proofs.
split : ∀ {A : Set} {P : A → Set} (P? : Decidable P) (zs : List A)
→ ∃[ xs ] ∃[ ys ] ( merge xs ys zs × All P xs × All (¬_ ∘ P) ys )
-- Your code goes here
module Lambda where
open import Data.Bool using (Bool; true; false; T; not)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.List using (List; _∷_; [])
open import Data.Nat using (ℕ; zero; suc)
open import Data.Product using (∃-syntax; _×_)
open import Data.String using (String; _≟_)
open import Data.Unit using (tt)
open import Relation.Nullary using (Dec; yes; no; ¬_)
open import Relation.Nullary.Decidable using (False; toWitnessFalse)
open import Relation.Nullary.Negation using (¬?)
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl)
open import plfa.part2.Lambda
hiding (var?; ƛ′_⇒_; case′_[zero⇒_|suc_⇒_]; μ′_⇒_; plus′)
Exercise mul (recommended)
Write out the definition of a lambda term that multiplies two natural numbers. Your definition may use plus as defined earlier.
-- Your code goes here
Exercise mulᶜ (practice)
Write out the definition of a lambda term that multiplies two natural numbers represented as Church numerals. Your definition may use plusᶜ as defined earlier (or may not — there are nice definitions
both ways).
-- Your code goes here
Exercise primed (stretch)
Some people find it annoying to write
` "x"
instead of
. We can make examples with lambda terms slightly easier to write by adding the following definitions:
var? : (t : Term) → Bool
var? (` _) = true
var? _ = false
ƛ′_⇒_ : (t : Term) → {_ : T (var? t)} → Term → Term
ƛ′_⇒_ (` x) N = ƛ x ⇒ N
case′_[zero⇒_|suc_⇒_] : Term → Term → (t : Term) → {_ : T (var? t)} → Term → Term
case′ L [zero⇒ M |suc (` x) ⇒ N ] = case L [zero⇒ M |suc x ⇒ N ]
μ′_⇒_ : (t : Term) → {_ : T (var? t)} → Term → Term
μ′ (` x) ⇒ N = μ x ⇒ N
Recall that T is a function that maps from the computation world to the evidence world, as defined in Chapter Decidable. We ensure to use the primed functions only when the respective term argument
is a variable, which we do by providing implicit evidence. For example, if we tried to define an abstraction term that binds anything but a variable:
_ : Term
_ = ƛ′ two ⇒ two
Agda would complain it cannot find a value of the bottom type for the implicit argument. Note the implicit argument’s type reduces to ⊥ when term t is anything but a variable.
The definition of
can now be written as follows:
plus′ : Term
plus′ = μ′ + ⇒ ƛ′ m ⇒ ƛ′ n ⇒
case′ m
[zero⇒ n
|suc m ⇒ `suc (+ · m · n) ]
+ = ` "+"
m = ` "m"
n = ` "n"
Write out the definition of multiplication in the same style.
Exercise _[_:=_]′ (stretch)
The definition of substitution above has three clauses (ƛ, case, and μ) that invoke a with clause to deal with bound variables. Rewrite the definition to factor the common part of these three clauses
into a single function, defined by mutual recursion with substitution.
-- Your code goes here
Exercise —↠≲—↠′ (practice)
Show that the first notion of reflexive and transitive closure above embeds into the second. Why are they not isomorphic?
-- Your code goes here
Exercise plus-example (practice)
Write out the reduction sequence demonstrating that one plus one is two.
-- Your code goes here
Exercise Context-≃ (practice)
Show that Context is isomorphic to List (Id × Type). For instance, the isomorphism relates the context
∅ , "s" ⦂ `ℕ ⇒ `ℕ , "z" ⦂ `ℕ
to the list
[ ⟨ "z" , `ℕ ⟩ , ⟨ "s" , `ℕ ⇒ `ℕ ⟩ ]
-- Your code goes here
Exercise ⊢mul (recommended)
Using the term mul you defined earlier, write out the derivation showing that it is well typed.
-- Your code goes here
Exercise ⊢mulᶜ (practice)
Using the term mulᶜ you defined earlier, write out the derivation showing that it is well typed.
-- Your code goes here
module Properties where
open import Relation.Binary.PropositionalEquality
using (_≡_; _≢_; refl; sym; cong; cong₂)
open import Data.String using (String; _≟_)
open import Data.Nat using (ℕ; zero; suc)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Product
using (_×_; proj₁; proj₂; ∃; ∃-syntax)
renaming (_,_ to ⟨_,_⟩)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Function using (_∘_)
open import plfa.part1.Isomorphism
open import plfa.part2.Lambda
open import plfa.part2.Properties
hiding (value?; Canonical_⦂_; unstuck; preserves; wttdgs)
Exercise Canonical-≃ (practice)
Well-typed values must take one of a small number of
canonical forms
, which provide an analogue of the
relation that relates values to their types. A lambda expression must have a function type, and a zero or successor expression must be a natural. Further, the body of a function must be well typed in
a context containing only its bound variable, and the argument of successor must itself be canonical:
infix 4 Canonical_⦂_
data Canonical_⦂_ : Term → Type → Set where
C-ƛ : ∀ {x A N B}
→ ∅ , x ⦂ A ⊢ N ⦂ B
→ Canonical (ƛ x ⇒ N) ⦂ (A ⇒ B)
C-zero :
Canonical `zero ⦂ `ℕ
C-suc : ∀ {V}
→ Canonical V ⦂ `ℕ
→ Canonical `suc V ⦂ `ℕ
Show that Canonical V ⦂ A is isomorphic to (∅ ⊢ V ⦂ A) × (Value V), that is, the canonical forms are exactly the well-typed values.
-- Your code goes here
Exercise Progress-≃ (practice)
Show that Progress M is isomorphic to Value M ⊎ ∃[ N ](M —→ N).
-- Your code goes here
Exercise progress′ (practice)
Write out the proof of progress′ in full, and compare it to the proof of progress above.
-- Your code goes here
Exercise value? (practice)
to write a program that decides whether a well-typed term is a value:
value? : ∀ {A M} → ∅ ⊢ M ⦂ A → Dec (Value M)
Exercise subst′ (stretch)
Rewrite subst to work with the modified definition _[_:=_]′ from the exercise in the previous chapter. As before, this should factor dealing with bound variables into a single function, defined by
mutual recursion with the proof that substitution preserves types.
-- Your code goes here
Exercise mul-eval (recommended)
Using the evaluator, confirm that two times two is four.
-- Your code goes here
Exercise: progress-preservation (practice)
Without peeking at their statements above, write down the progress and preservation theorems for the simply typed lambda-calculus.
-- Your code goes here
Exercise subject_expansion (practice)
We say that M reduces to N if M —→ N, but we can also describe the same situation by saying that N expands to M. The preservation property is sometimes called subject reduction. Its opposite is
subject expansion, which holds if M —→ N and ∅ ⊢ N ⦂ A imply ∅ ⊢ M ⦂ A. Find two counter-examples to subject expansion, one with case expressions and one not involving case expressions.
-- Your code goes here
Exercise stuck (practice)
Give an example of an ill-typed term that does get stuck.
-- Your code goes here
Exercise unstuck (recommended)
Using progress, it is easy to show that no well-typed term is stuck:
unstuck : ∀ {M A}
→ ∅ ⊢ M ⦂ A
→ ¬ (Stuck M)
Using preservation, it is easy to show that after any number of steps, a well-typed term remains well typed:
preserves : ∀ {M N A}
→ ∅ ⊢ M ⦂ A
→ M —↠ N
→ ∅ ⊢ N ⦂ A
An easy consequence is that starting from a well-typed term, taking any number of reduction steps leads to a term that is not stuck:
wttdgs : ∀ {M N A}
→ ∅ ⊢ M ⦂ A
→ M —↠ N
→ ¬ (Stuck N)
Felleisen and Wright, who introduced proofs via progress and preservation, summarised this result with the slogan well-typed terms don’t get stuck.Provide proofs of the three postulates, unstuck,
preserves, and wttdgs above.
-- Your code goes here | {"url":"https://plfa.inf.ed.ac.uk/TSPL/2022/Assignment3/","timestamp":"2024-11-05T03:51:46Z","content_type":"text/html","content_length":"146408","record_id":"<urn:uuid:0b0d8e2f-c60a-4ac4-8446-3d1ff4cf48fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00370.warc.gz"} |
Biomedical Science and Research Journals | Relation of a Pulse Transit Time to the Blood Pressure in Bifurcated Cardiovascular Networks
Relation of a Pulse Transit Time to the Blood Pressure in Bifurcated Cardiovascular Networks
Recent developments in cardiovascular mathematics allow to simulate blood flow in the entire circulatory network or any isolated cardiovascular subsystem. Providing measurements of a Pulse Transit
Time (PTT), or the averaged value for the Pulse Wave Velocity (PWV), the patient specific computational model can create personalized transfer functions, converting distal measurements to the set of
cardiovascular biomarkers. The latter has a potential to build the feasible foundation for the personalized continuous self-monitoring of cardiovascular health based on portable mobile and wearable
applications. Nevertheless, although being of a great interest, an accurate and reliable PTT-based Blood Pressure (BP) estimation barely exists nowadays due to the complexity of a BP regulation in a
human body.
In this paper we concentrate on a physics based computational modelling to assess interconnections of a PTT to BP in a bifurcated circulatory network. The PWV is interpreted as the speed of
propagation along the forward running characteristic in a related hyperbolic Fluid Structure Interaction (FSI) differential problem. PTT is calculated by integrating the reciprocal of instantaneous
PWV along the characteristic line. The Y.C. Fung’s exponential model is applied to describe mechanics of a thin and a thick-walled vessel, subject to the infinitesimally small or finite hyperelastic
deformation. Recently published PTT and PWV based non-invasive and continuous BP monitoring methods are reviewed and analyzed.
Keywords: Pulse transit time, Physical modelling, Nonlinear models, Cardiovascular markers
Pulse Arrival Time (PAT) is the generally established empirical marker for continuous non-intrusive blood pressure monitoring, which is defined as a time required for a pulse wave to travel from the
heart to a peripheral site. A popular estimate of PAT is the timebased delay between R wave peak of Electrocardiogram (ECG) and a characteristic point of a Photoplethysmogram (PPG). PAT consists of
two components: the non-constant Pre-Ejection Period (PEP), which is a duration of the ventricle contraction up to aortic valve opening, and the Pulse Transit Time (PTT), which defines the period for
the arterial pulse wave to travel from the aortic valve to the peripheral site.
A simple measurement setup consisting of arm Electrocardiogram (ECG) and Peripheral Site Photoplethysmogram (PPG) allows to assess PAT as the time delay between ECG R peak and one of the optional
points in the PPG waveform: peak, foot, maximum values of the slope, or the second derivative of PPG waveform. PEP can be derived noninvasively using for instance thoracic Impedance Plethysmogram
(IPG) as described in [1, 2, 3]. Estimation of a systolic and diastolic BP is based on equivalence of the measured and model- based prediction of PTT. In general, prediction methods can be
categorized into data-driven, physics-based and hybrid approaches. Data-driven approaches investigate relationship between BP and PTT through the linear or nonlinear regression analysis, employing a
simple set of basic functions, or using artificial intelligence (neural network). Physics-based approaches assume that a reliable physical model describing connection of a PTT to BP is available.
Hybrid approaches combines the methods to calibrate the personalized bio-physical properties, improving prediction.
As follows from the physical modelling, PTT and PWV are mainly affected by four factors: arterial compliance, cardiac output, peripheral resistance, and a blood pressure. Most data driven approaches
select the only single parameter as an independent variable, which is a PTT or the averaged PWV, to predict systolic and diastolic blood pressures. The physics-based approach automatically accounts
for the full set of factors affected BP according to the physical model, i.e. cardiac output, stroke volume, vascular compliance, peripheral resistance. The following sections describe PTT based
blood pressure estimation according to the classification. Since many of the papers using data driven regression analysis are listed in several reviews, we will not cite relating individual papers,
focusing mainly on a physical modeling as a foundation for linking PTT to systolic and diastolic BP.
Data-Driven BP Estimation
Multiple linear and nonlinear regressions have been explored by different authors using combinations of exponential, power, logarithm, polynomial and logistics functions to fit the experimental
dataset of PTT (or PWV) vs BP [3, 4, 5]. In [6] the heart rate as a second independent variable in addition to PTT is introduced in a linear regression, which according to the authors improves the
accuracy of BP prediction. In the monograph [7] the Young’s modulus is presented as an exponential function of pressure, 0- is the Yong modulus at zero pressure, and α -empirical coefficient. As a
result, the formal substitute of the modified elastic modulus E into the Moens-Korteweg expression for the PWV results in a pulse wave velocity dependent on a blood pressure,
The mentioned approach is completely empirical, since it does not fit the paradigm of classical mechanics, which specifies physical nonlinearity by appropriate constitutive equations in terms of
stress – strain components. The described expression cannot be derived from the fluid-structure interaction model using any constitutive equations. It could be converted to the linear regression (in
a log scale) by application of logarithm to the expression for PWV, which results in
where 8] to monitor BP as a function of PTT under the effect of hydrostatic pressure due to hand elevations. The effect of including PEP in BP estimation is under investigation in different papers
based on empirical regression analysis over different cohorts of human subjects [4, 9, 10]. The simplest approach is an attempt to estimate PEP as a percentage of the RR interval, with the following
subtraction from PAT to obtain PTT [11]. There is still a controversial evidence from different authors regarding effect of PEP on BP. The impact of PEP on the overall PAT decreases with distance
from the heart, so that for the short PATs, like ones extracted from the ear-worn device, correction with PEP is required.
Neural Network (NN) modeling has recently been in place predicting BP as a function of a set of measured parameters. In [12] a total of 17 parameters were selected as the set of independent variables
being chosen as characteristic feature points from ESG and PPG signals. Two different neural networks have been used to predict separately brachial systolic and diastolic blood pressures as functions
of ECG and PPG measurements. The maximum error range in the brachial BP prediction is reported in terms of a root mean square error RMSE=±5.2mmHg. In [13] the SVR (Support Vector Machine Regression)
algorithm is applied to establish relationship between human physiological data and systolic and diastolic BPs. The different number of main physiological indexes, obtained from ECG and PPG, include
PTT, HR, SPO2 and others, are explored in application of NN modeling. The maximum error range of a brachial BP prediction is reported as ±10mmHg.
Few studies managed to compare different noninvasive BP estimations in a wide physiological BP range. None of data driven approaches proved to be ubiquitous, being able to monitor with a reasonable
accuracy the only single feature of a BP, either systolic, or diastolic or a mean [3, 4, 5].
Physical Modeling BP Estimation
In this section, we assess physics-based models’ capabilities to predict systolic and diastolic BP as a function of model required independent parameters. Considering an arbitrary pressure-area
connection, P = P(A) ,we present system of conservation laws in the following non-conservative quasi-linear form
Relating characteristic directions (eigenvalues) read
and forward and backward running characteristics can be found accordingly
Since the slope of a forward running characteristic line is determined by PWV =
Equation (7) serves to calculate the PTT required for the pulse wave to propagate through the Nv vessels, each of the length
Nonlinear Vs Linear Models
In this section three type of nonlinear models are reviewed following the papers [14, 15]: the infinitesimally Small Deformation Linear Elasticity Model (SDL), Small Deformation Hyperplastic Model
(SDH) and Finite Deformation hyper elastic Model (FDH). The Fung’s exponential descriptor for passive behavior of arteries [16] presents strain energy density function for the pseudo elastic wall
deformation in a form
Equations (10) and (6) present the instantaneous PWV for the SDH model in a compliant hyperelastic artery as the following
Model SDL is achieved by setting hyperelastic material coefficient to zero
Model FDH, which considers finite deformation, is derived based on the same expression for strain energy (8), where
are governed by equilibrium conditions
All three models have been tested against Histand and Anliker results on a PWV measurements presented in [17, 18] and reproduced in (Figure 1) by square markers. The experimental curve notably
exhibits curvature starting from elevated level of pressure exceeding 140mmHg. Material parameters have been identified for each model independently, based on a best fit procedure. The Finite
Deformation Hyper Elasticity (FDH) model and Small Deformation Hyper Elasticity (SDH) model have the highest quality of fitting process, creating practically the same regression line in (Figure 1)
within the physiological range of BP. The Small Deformation Model with Linear Elasticity (SDL) was not able to fit the experimental curve at the quality of FDH or SDH models.
Dash lines indicate theoretical prediction. Square markers illustrate the total set of experimental points. Using the properties extracted from the nonlinear model the lower (solid) line shows the
effect on PWV using the partially nonlinear model SDH, combining hyper elasticity with small deformation. To illustrate the effect of a longitudinal force on PWV the variation of PWV due to the
variability of a longitudinal pre-stress force is presented in (Figure 2). According to simulation within the realistic physiological range of a longitudinal stress, the relative deviation in PWV
does not exceed 3%.
Several PWV estimations presented in literature is based on its correlation with the BP and an arterial wall compliance. The study in [19]examined the impact of a systolic flow correction of a
measured PWV on blood pressure prediction accuracy using data from two published in vivo studies. Both studies examined the relationship between PWV and blood pressure under pharmacological
manipulation, one in mongrel dogs and the other in healthy adult males. Systolic flow correction of the measured PWV improves the R2 correlation to blood pressure from 0.51 to 0.75 for the mongrel
dog study, and 0.05 to 0.70 for the human subjects’ study. The results support the hypothesis that systolic flow correction is an essential element of non-invasive, cuff-less blood pressure
estimation based on PWV measures.
Thick Wall Vessels
A novel mathematical model predicting PWV propagation with rigorous account of, blood vessel elasticity, and finite deformation of multi-layer thick wall arterial segments was studied in [20]. It was
found that the account for the multilayer model affects distribution of local parameters in the proximity of the external layer (adventitia) and does not affect stiffness related to the internal
layer. The latter means that the single thick layer model is enough to predict PWV of an arterial segment. (Figure 3) depicts the dependence of PWV on pressure for the Systole Phase (marked as “SBP”)
and a Diastole Phase (marked as “DBP”) for three vessels of different thicknesses of a human aorta. All results have been compared with the simplified thin walled model of a membrane shell
interacting with an incompressible fluid.
To explore inaccuracies induced by use of the less complex thin wall model, error in both PWV and blood pressure were calculated for a blood pressure of SBP/DBP = 150/95mmHg representing the median
of stage 1 hypertension. The single layer thick wall model improves PWV accuracy by (4.0-8.4%) corresponding to the relative wall thickness (H/R1) range of 0.07-0.38. One of the goals for the model
is PWV based blood pressure prediction, where the thick wall model offers an improvement of (3.3-19.4%).
Calibration Free Approaches
Willemet et al. [21, 22] proposed approach to use cardiovascular simulator for generation of a database of “virtual subjects” with sizes limited only by computational resources. In their study, the
databases were generated using one-dimensional model of wave propagation in an artery network comprising of 55 largest human arteries. A linear elastic model was employed to describe deformation of
arterial walls. The database is created by running the cardiovascular model repeatedly. The seven model parameters were varied: elastic artery PWV, muscular artery PWV, the diameter of elastic
arteries, the diameter of muscular arteries, Heart Rate (HR), SV and peripheral vascular resistance. 3325 healthy virtual subjects presented a diversity of hemodynamic, structural and geometric
characteristics. For each virtual subject, all characteristics are known at every point of the systemic arterial tree, i.e. anatomical and structural properties, as well as pressure, flow, pulse wave
velocity and area waves at the larger arteries, therefore allowing the computation of the exact value of the diagnostic tool.
Huttunen et al. [23] used cardiovascular modelling of the entire adult circulation to create a database of “virtual subjects”, which is applied with machine learning to construct predictors for
health indices. They carry out theoretical assessment of estimating aortic pulse wave velocity, diastolic and systolic blood pressure and stroke volume using pulse transit/arrival timings derived
from photoplethysmography signals. The generated database was then used as training data for Gaussian process regressors applied finally to simulation. Simulated results provide theoretical
assessment of accuracy for predictions of the health indices. For instance, aortic pulse wave velocity was estimated with a high accuracy (r>0.9. Similar accuracy has been reached for diastolic blood
pressure, but predictions of systolic blood pressure proved to be less accurate (r > 0.75).
Developed technologies in general allow to implement a PTT/ PAT-based system to predict continuously cardiovascular health markers such as arterial blood pressure, cardiac output, arterial stiffness.
However, none of approaches is able so far to monitor accurately all cardiac markers for the wide range of physiological conditions. The limitations to be addressed in future are the following.
First, each model must be investigated for its limitations. We believe that a calibration stage is required to build a reliable simulator within the range of investigated conditions. Also, most of
the research addresses healthy population, which is characterized by different behavior of a vascular system rather than group with medical conditions. In the current review we only consider pulse
transit and arrival type of time information as the input to the predictor. It would be beneficial to develop approaches that do not need reference measurement for the aortic valve opening/R-peak.
For More information: https://biomedgrid.com/fulltext/volume7/relation-of-a-pulse-transit-time-to-the-blood-pressure-in-bifurcated-cardiovascular-networks.001135.php | {"url":"https://biomedgrid.blogspot.com/2022/06/biomedical-science-and-research_29.html","timestamp":"2024-11-10T10:55:38Z","content_type":"application/xhtml+xml","content_length":"163559","record_id":"<urn:uuid:3beeba8f-2eee-4a0c-84b7-59488bdfc8f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00725.warc.gz"} |
Small Calculator Pocket Calculator Small Desktop Calculator Handheld Calculator Accounting Calculator Business Calculator Office Calculator Standard Plastic Dedicated Student Table
Price: $6.79
(as of Jul 28, 2024 16:06:39 UTC – Details)
Adorable calculator: for school, home and office to use. it can perform simple calculations.
Calculator decor: it can perform mathematical operations of addition, multiplication, subtraction, and division.
Desk: the workmanship of this portable calculator is , and you will have a good experience in using it.
Compact calculator: made of plastic, silicone materials, it is very durable and practical for long-lasting use, saving your time with number calculation.
Handheld calculator: with the fine and great workmanship, it can ensure its durability and practicality.
User Reviews
There are no reviews yet.
Be the first to review “Small Calculator Pocket Calculator Small Desktop Calculator Handheld Calculator Accounting Calculator Business Calculator Office Calculator Standard Plastic Dedicated Student
Small Calculator Pocket Calculator Small Desktop Calculator Handheld Calculator Accounting Calculator Business Calculator Office Calculator Standard Plastic Dedicated Student Table | {"url":"https://calculator-a.com/product/small-calculator-pocket-calculator-small-desktop-calculator-handheld-calculator-accounting-calculator-business-calculator-office-calculator-standard-plastic-dedicated-student-table/","timestamp":"2024-11-12T20:28:37Z","content_type":"text/html","content_length":"203519","record_id":"<urn:uuid:08bcc713-b298-4e38-80a0-f6efaa751f8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00452.warc.gz"} |
Exercise 4.1 class 8 solution
NCERT exercise 4.1 class 8 solution -students can click here to download the entire pdf chapterwise .here free pdf can be download for NCERT solution class 8. all Exercise are available here. These
solutions are available in downloadable PDF format as well. it will help students in getting rid of all the doubts about those particular topics that are covered in the exercise. The NCERT textbook
provides plenty of questions for the students to solve and practise. Solving and practising is more than enough to score high in the Class 8 examinations. Moreover, students should make sure that
they practise every problem given in the textbook . exercise 4.1 class 8 solution Pdf can be downloaded free here.
the pdf and exercise preview available at bottom of the page
Exercise 4.1 class 8 solution-data handling
Data handling in mathematics involves collecting, organizing, analyzing, and interpreting data to draw meaningful conclusions. It is an important aspect of statistics and various branches of
mathematics. Here are some key concepts and steps involved in data handling in math:
1. Data Collection: This is the first step in data handling. Data can be collected through surveys, experiments, observations, or any other relevant methods. It’s important to ensure that the data
collected is representative of the population or phenomenon of interest.
2. Types of Data: Data can be categorized into two main types: qualitative data and quantitative data. Qualitative data represents qualities or categories (e.g., colors, names), while quantitative
data represents numerical values (e.g., measurements, counts).
3. Data Presentation: After collecting data, it needs to be organized and presented in a way that is easy to understand. This can be done using tables, graphs, charts, or other visual
representations. Common types of graphs include bar graphs, line graphs, pie charts, and scatter plots.
Exercise 4.1 class 8 solution -frequency distribution
A frequency distribution (also referred to as a frequency table) is a way to organize and present data in a systematic manner, typically for the purpose of summarizing and understanding the
distribution of values within a dataset. It is commonly used in statistics and data analysis to show how often each value or range of values occurs within a dataset. Here’s how a frequency
distribution works:
1. Data Collection: First, you collect your data through observations, surveys, experiments, or any other means.
2. Data Organization: After collecting the data, you need to organize it. You can do this by listing all the unique values that appear in your dataset.
3. Frequency Count: For each unique value in the dataset, you count how many times it appears. This count is called the “frequency.”
4. Table Presentation: Finally, you present this information in a tabular format known as a frequency distribution or frequency table. The table typically has two columns: one for the unique values
(or value ranges) and another for the corresponding frequencies.Here’s a simplified example of a frequency distribution for a dataset of test scores:
1. In this example, the table shows the test scores (unique values) and the number of times each score appears in the dataset. For instance, there were 2 scores of 80, 5 scores of 85, and so on.
Frequency distributions are useful for summarizing data and providing insights into the distribution of values, helping to identify patterns, central tendencies (mean, median, mode), and variations
within the dataset. They are often used as a preliminary step before further data analysis or creating data visualizations like histograms or bar charts to visually represent the data distribution.
Exercise 4.1 class 8 solution- continous frequency distribution
A continuous frequency distribution is a way to organize and present data for continuous variables, which are variables that can take on an infinite number of values within a given range. Unlike a
discrete frequency distribution, where data is counted for distinct categories or values, a continuous frequency distribution deals with a range of values and groups them into intervals or classes.
It is particularly useful when working with data that is measured on a continuous scale, such as height, weight, time, temperature, and more.
Here’s how you can create a continuous frequency distribution:
1. Data Collection: Gather your data, which consists of measurements on a continuous scale.
2. Data Range Determination: Determine the range of values covered by your data. This defines the lower and upper bounds of your data set.
3. Class Intervals: Divide the range of data into non-overlapping intervals or classes. The width and number of intervals depend on your preferences and the characteristics of the data. Commonly
used methods for determining class intervals include the square root method, Sturges’ rule, and Scott’s normal reference rule.
4. Frequency Count: Count how many data points fall within each class interval. This count represents the frequency for that class.
5. Table Presentation: Present the data in a table format with two columns: one for the class intervals and another for the corresponding frequencies.
Here’s a simplified example of a continuous frequency distribution for a dataset of temperatures (in degrees Celsius) measured in a city:
In this example, the table shows temperature intervals (e.g., 10 – 20) and the number of measurements that fall within each interval. The first interval, “10 – 20,” has 6 measurements, indicating
that 6 temperature readings were between 10 and 20 degrees Celsius.
Continuous frequency distributions are helpful for summarizing and analyzing data with a large number of possible values within a range. They are often used to create histograms, which provide a
visual representation of the distribution of continuous data.
Exercise 4.1 class 8 solution – exercise preview
here the exercise preview is given-
Exercise 4.1 class 8 solution – solution pdf
students can view or download the pdf from here.click at the bottom to scroll the pdf pages.we provide Exercise 4.1 class 8 solution, just to help student to achieve the efficiency.
for more solution visit-
Table of Contents | {"url":"https://cmaindiagroup.in/exercise-4-1-class-8-solution/","timestamp":"2024-11-04T10:55:48Z","content_type":"text/html","content_length":"182917","record_id":"<urn:uuid:55f25140-d77f-46bd-a057-56062a875da1>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00193.warc.gz"} |
Tree Proof Generator
Enter a formula of standard propositional, predicate, or modal logic. The page will try to find either a countermodel or a tree proof (a.k.a. semantic tableau).
Examples (click!):
Entering formulas
To enter logic symbols, use the buttons above the text field, or type ~ for ¬, & for ∧, v for ∨, -> for →, <-> for ↔, ! for ∀, ? for ∃, [] for □, <> for ◇. You can also use LaTeX commands.
If you want to test an argument with premises and conclusion, use |= to separate the premises from the conclusion, and use commas to separate the premises. See the last example in the list above.
Syntax of formulas
Any alphabetic character is allowed as a propositional constant, predicate, individual constant, or variable. Numeral digits can be used either as singular terms or as indices (as in Fx1), but don't
mix the two uses. '+', '*', and '-' can be used as function expressions. Predicates (except identity) and function terms must be in prefix notation. Function terms must have their arguments enclosed
in brackets. So F2x17, Rab, R(a,b), Raf(b), F(+(1,2)) are ok, but not Animal(Fred), aRb, or F(1+2). (In fact, these are also ok, but they won't be parsed as you might expect.) The order of precedence
among connectives is ¬, ∧, ∨, →, ↔. Association is to the right. Quantifier symbols in sequences of quantifiers must not be omitted: write ∀x∀yRxy instead of ∀xyRxy.
Supported logics
Besides classical propositional logic and first-order predicate logic (with functions and identity), a few normal modal logics are supported. If you enter a modal formula, you will see a choice of
how the accessibility relation should be constrained. For modal predicate logic, constant domains and rigid terms are assumed.
Source code
The source is on github.
Comments, bug reports and suggestions are always welcome: | {"url":"https://www.umsu.de/trees/","timestamp":"2024-11-15T04:23:56Z","content_type":"text/html","content_length":"7559","record_id":"<urn:uuid:c0ae86bf-f852-4798-9bff-5f5d08ee30cb>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00155.warc.gz"} |
Blog - Zrzahid
Given a linked list and a number n. Split the link list into two based on the following pattern input: 1->2->3->4->5->6->7->8->null and n=2 output: 1-2->3->4->null and 5->6->7->8->null input: 2->3->
1->4->6->7->7->6->null and n=4 output: 2-3->null and 1->4->6->7->7->6->null return the right partition pointer. First note that pattern. For n=2 it is effectively partitioning the list into two
halves. […] | {"url":"https://www.zrzahid.com/blog/page/3/","timestamp":"2024-11-11T06:51:47Z","content_type":"text/html","content_length":"100756","record_id":"<urn:uuid:41fe6a6f-1f09-4d9e-8751-42de9c6b00e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00578.warc.gz"} |
Pseudorandom Number Sequence Test Program
A Pseudorandom Number Sequence Test Program
This page describes a program, ent, which applies various tests to sequences of bytes stored in files and reports the results of those tests. The program is useful for evaluating pseudorandom number
generators for encryption and statistical sampling applications, compression algorithms, and other applications where the information density of a file is of interest.
ent - pseudorandom number sequence test
ent [ -b -c -f -t -u ] [ infile ]
ent performs a variety of tests on the stream of bytes in infile (or standard input if no infile is specified) and produces output as follows on the standard output stream:
Entropy = 7.980627 bits per character.
Optimum compression would reduce the size
of this 51768 character file by 0 percent.
Chi square distribution for 51768 samples is 1542.26, and randomly
would exceed this value less than 0.01 percent of the times.
Arithmetic mean value of data bytes is 125.93 (127.5 = random).
Monte Carlo value for Pi is 3.169834647 (error 0.90 percent).
Serial correlation coefficient is 0.004249 (totally uncorrelated = 0.0).
The values calculated are as follows:
The information density of the contents of the file, expressed as a number of bits per character. The results above, which resulted from processing an image file compressed with JPEG, indicate
that the file is extremely dense in information—essentially random. Hence, compression of the file is unlikely to reduce its size. By contrast, the C source code of the program has entropy of
about 4.9 bits per character, indicating that optimal compression of the file would reduce its size by 38%. [Hamming, pp. 104–108]
Chi-square Test
The chi-square test is the most commonly used test for the randomness of data, and is extremely sensitive to errors in pseudorandom sequence generators. The chi-square distribution is calculated
for the stream of bytes in the file and expressed as an absolute number and a percentage which indicates how frequently a truly random sequence would exceed the value calculated. We interpret the
percentage as the degree to which the sequence tested is suspected of being non-random. If the percentage is greater than 99% or less than 1%, the sequence is almost certainly not random. If the
percentage is between 99% and 95% or between 1% and 5%, the sequence is suspect. Percentages between 90% and 95% and 5% and 10% indicate the sequence is “almost suspect”. Note that our JPEG file,
while very dense in information, is far from random as revealed by the chi-square test.
Applying this test to the output of various pseudorandom sequence generators is interesting. The low-order 8 bits returned by the standard Unix rand() function, for example, yields:
Chi square distribution for 500000 samples is 0.01, and randomly would exceed this value more than 99.99 percent of the times.
While an improved generator [Park & Miller] reports:
Chi square distribution for 500000 samples is 212.53, and randomly would exceed this value 97.53 percent of the times.
Thus, the standard Unix generator (or at least the low-order bytes it returns) is unacceptably non-random, while the improved generator is much better but still sufficiently non-random to cause
concern for demanding applications. Contrast both of these software generators with the chi-square result of a genuine random sequence created by timing radioactive decay events.
Chi square distribution for 500000 samples is 249.51, and randomly would exceed this value 40.98 percent of the times.
See [Knuth, pp. 35–40] for more information on the chi-square test. An interactive chi-square calculator is available at this site.
Arithmetic Mean
This is simply the result of summing the all the bytes (bits if the -b option is specified) in the file and dividing by the file length. If the data are close to random, this should be about
127.5 (0.5 for -b option output). If the mean departs from this value, the values are consistently high or low.
Monte Carlo Value for Pi
Each successive sequence of six bytes is used as 24 bit X and Y co-ordinates within a square. If the distance of the randomly-generated point is less than the radius of a circle inscribed within
the square, the six-byte sequence is considered a “hit”. The percentage of hits can be used to calculate the value of Pi. For very large streams (this approximation converges very slowly), the
value will approach the correct value of Pi if the sequence is close to random. A 500000 byte file created by radioactive decay yielded:
Monte Carlo value for Pi is 3.143580574 (error 0.06 percent).
Serial Correlation Coefficient
This quantity measures the extent to which each byte in the file depends upon the previous byte. For random sequences, this value (which can be positive or negative) will, of course, be close to
zero. A non-random byte stream such as a C program will yield a serial correlation coefficient on the order of 0.5. Wildly predictable data such as uncompressed bitmaps will exhibit serial
correlation coefficients approaching 1. See [Knuth, pp. 64–65] for more details.
The input is treated as a stream of bits rather than of 8-bit bytes. Statistics reported reflect the properties of the bitstream.
Print a table of the number of occurrences of each possible byte (or bit, if the -b option is also specified) value, and the fraction of the overall file made up by that value. Printable
characters in the ISO 8859-1 Latin-1 character set are shown along with their decimal byte values. In non-terse output mode, values with zero occurrences are not printed.
Fold upper case letters to lower case before computing statistics. Folding is done based on the ISO 8859-1 Latin-1 character set, with accented letters correctly processed.
Terse mode: output is written in Comma Separated Value (CSV) format, suitable for loading into a spreadsheet and easily read by any programming language. See Terse Mode Output Format below for
additional details.
Print how-to-call information.
If no infile is specified, ent obtains its input from standard input. Output is always written to standard output.
Terse mode is selected by specifying the -t option on the command line. Terse mode output is written in Comma Separated Value (CSV) format, which can be directly loaded into most spreadsheet programs
and is easily read by any programming language. Each record in the CSV file begins with a record type field, which identifies the content of the following fields. If the -c option is not specified,
the terse mode output will consist of two records, as follows:
where the italicised values in the type 1 record are the numerical values for the quantities named in the type 0 column title record. If the -b option is specified, the second field of the type 0
record will be “File-bits”, and the file_length field in type 1 record will be given in bits instead of bytes. If the -c option is specified, additional records are appended to the terse mode output
which contain the character counts:
. . .
If the -b option is specified, only two type 3 records will appear for the two bit values v=0 and v=1. Otherwise, 256 type 3 records are included, one for each possible byte value. The second field
of a type 3 record indicates how many bytes (or bits) of value v appear in the input, and fraction gives the decimal fraction of the file which has value v (which is equal to the count value of this
record divided by the file_length field in the type 1 record).
Note that the “optimal compression” shown for the file is computed from the byte- or bit-stream entropy and thus reflects compressibility based on a reading frame of the chosen width (8-bit bytes or
individual bits if the -b option is specified). Algorithms which use a larger reading frame, such as the Lempel-Ziv [Lempel & Ziv] algorithm, may achieve greater compression if the file contains
repeated sequences of multiple bytes.
The program is provided as random.zip, a Zipped archive containing an ready-to-run Win32 command-line program, ent.exe (compiled using Microsoft Visual C++ .NET, creating a 32-bit Windows
executable), and in source code form along with a Makefile to build the program under Unix.
Hamming, Richard W. Coding and Information Theory. Englewood Cliffs NJ: Prentice-Hall, 1980. ISBN 978-0-13-139139-0.
Knuth, Donald E. The Art of Computer Programming, Volume 2 / Seminumerical Algorithms. Reading MA: Addison-Wesley, 1969. ISBN 978-0-201-89684-8.
[Lempel & Ziv]
Ziv J. and A. Lempel. “A Universal Algorithm for Sequential Data Compression”. IEEE Transactions on Information Theory 23, 3, pp. 337-343.
[Park & Miller]
Park, Stephen K. and Keith W. Miller. “Random Number Generators: Good Ones Are Hard to Find”. Communications of the ACM, October 1988, p. 1192.
This software is in the public domain. Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, without any
conditions or restrictions. This software is provided “as is” without express or implied warranty. | {"url":"https://www.fourmilab.ch/random/","timestamp":"2024-11-09T20:43:04Z","content_type":"application/xhtml+xml","content_length":"17603","record_id":"<urn:uuid:d42465ca-2fbb-41c3-a198-015b25d71a9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00292.warc.gz"} |
Hele-Shaw Cell | Viscosity Demo, Fluid Dynamics & Lab Tool
Hele-Shaw cell for viscosity demonstration
Explore the Hele-Shaw Cell’s role in fluid dynamics and viscosity, its design, educational value, and applications in research and industry.
Understanding the Hele-Shaw Cell: A Key Tool in Fluid Dynamics and Viscosity Demonstrations
The Hele-Shaw cell, named after Henry Selby Hele-Shaw, a prominent engineer and mathematician, plays a crucial role in the study of fluid dynamics and viscosity. This instrument, simplistic in design
yet profound in its application, allows scientists and students alike to visualize and understand complex fluid flow patterns in a two-dimensional plane.
Principle and Design of the Hele-Shaw Cell
At its core, the Hele-Shaw cell consists of two parallel plates separated by a small gap. Typically, these plates are made of glass to provide clear visibility. The gap’s width is critical and is
usually very thin, often in the order of millimeters. This narrow spacing creates a unique environment where fluids exhibit two-dimensional flow characteristics, similar to those observed in various
geological formations or microfluidic devices.
Applications in Fluid Dynamics and Viscosity Measurement
The Hele-Shaw cell has diverse applications, particularly in visualizing flow patterns and studying the effects of viscosity. By introducing different fluids or a combination of immiscible fluids
(like oil and water) into the cell, observers can directly view the flow dynamics. This setup is particularly useful in understanding concepts such as laminar flow, turbulence, and fluid interfaces.
Moreover, the cell’s design allows for the measurement of viscosity, a fundamental property of fluids indicating resistance to flow. By observing the flow rate and pattern of a fluid in the cell, one
can derive its viscosity. This is especially beneficial in educational settings, where students can visually grasp the concept of viscosity and its impact on fluid behavior.
Lab Experiments and Demonstrations with the Hele-Shaw Cell
In laboratory settings, the Hele-Shaw cell is instrumental in demonstrating various principles of fluid mechanics. Experiments can range from simple demonstrations of flow patterns to complex studies
involving fluid displacement, diffusion, and interfacial tension. These experiments not only reinforce theoretical knowledge but also provide insights into practical applications in fields like
hydrology, petroleum engineering, and chemical processing.
Another fascinating aspect of the Hele-Shaw cell is its ability to simulate geological processes. By manipulating the fluid dynamics within the cell, researchers can mimic patterns similar to those
found in natural reservoirs, aiding in the understanding of oil extraction and groundwater flow. This not only enhances our comprehension of natural phenomena but also contributes to more efficient
and sustainable resource management.
Enhancing Research and Education with the Hele-Shaw Cell
The versatility of the Hele-Shaw cell extends beyond traditional fluid dynamics. It’s an invaluable tool in the realm of research and education, offering a tangible way to observe and analyze the
behavior of complex fluid systems. In educational settings, the cell provides a hands-on approach to learning, enabling students to directly observe theoretical concepts in practice. This
experiential learning approach deepens understanding and fosters a more engaging educational experience.
Technological and Industrial Applications
In the field of technology and industry, the Hele-Shaw cell’s principles find applications in microfluidic systems, widely used in biomedical and chemical engineering. The cell’s ability to simulate
small-scale fluid dynamics is instrumental in designing devices for drug delivery, chemical analysis, and biological assays. Furthermore, insights gained from Hele-Shaw cell experiments contribute to
advancements in oil recovery techniques, enhancing efficiency and reducing environmental impact.
Advancements and Future Directions
Recent advancements in imaging and computational fluid dynamics have opened new frontiers for the Hele-Shaw cell. High-resolution cameras and advanced software allow for more detailed analysis of
flow patterns, providing deeper insights into fluid behavior. Coupled with computational models, these tools enable precise simulations and predictions, essential for advancing scientific
understanding and industrial applications.
The Hele-Shaw cell, a seemingly simple apparatus, is a cornerstone in the study of fluid dynamics and viscosity. Its applications span from fundamental research to advanced industrial processes,
underscoring its significance in both academic and practical realms. As we continue to explore the complexities of fluid behavior, the Hele-Shaw cell remains an indispensable tool, offering clarity
and insight into the unseen dynamics of the fluid world. Its role in education, research, and industry will undoubtedly continue to evolve, highlighting the enduring relevance of this classic
scientific instrument in our quest to understand and harness the power of fluids. | {"url":"https://modern-physics.org/hele-shaw-cell-for-viscosity-demonstration/","timestamp":"2024-11-05T15:45:47Z","content_type":"text/html","content_length":"161134","record_id":"<urn:uuid:7862a5a3-39f9-4846-9b80-ba844c43c757>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00110.warc.gz"} |
seminars - Feigin-Semikhatov duality in W-superalgebras II
Zoom (ID: 642 675 5874 no password, Login required)
W-superalgebras are a large class of vertex superalgebras which generalize affine Lie superalgebras and the Virasoro algebras. It has been known that princial W-algebras satisfy a certain duality
relation (Feigin-Frenkel duality) which can be regarded as a quantization of the geometric Langlands correspondence. Recently, D. Gaiotto and M. Rapčák found dualities between more general
W-superalgebras in relation to certain four-dimensional supersymmetric gauge theories. A large part of thier conjecture is proved by T. Creutzig and A. Linshaw, and a more specific subclass
(Feigin-Semikhatov duality) is done by T. Creutzig, N. Genra, and S. Nakatsuka in a different way. In this talk I will review the Feigin-Semikhatov duality between certain subregular W-algebras and
principal W-superalgebras, and discuss the usage of relative semi-infinite cohomology. This talk is based on a joint work with T. Creutzig, N. Genra, and S. Nakatsuka. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=room&order_type=asc&page=85&document_srl=822442","timestamp":"2024-11-14T06:01:15Z","content_type":"text/html","content_length":"44874","record_id":"<urn:uuid:4918b04d-68ab-468f-85b0-cb59bf41cb97>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00688.warc.gz"} |
Corrugated Board Weight Calculator
Author: Neo Huang Review By: Nancy Deng
LAST UPDATED: 2024-10-03 10:55:16 TOTAL USAGE: 1879 TAG: Materials Packaging Weight Calculation
Unit Converter ▲
Unit Converter ▼
Powered by @Calculator Ultra
Find More Calculator☟
Historical Background
Corrugated board has been a staple in packaging since the 19th century due to its lightweight yet sturdy structure. Corrugated material is made of a fluted sheet sandwiched between flat liner boards.
It's widely used for shipping, packaging, and protecting goods. As the demand for efficient logistics grew, so did the need for tools to calculate weight, ensuring transportation costs and logistics
Calculation Formula
The weight of a corrugated board can be calculated using the formula:
\[ \text{CBW} = \text{BL} \times \text{BW} \times \text{BT} \times \text{D} \]
• CBW = Corrugated Board Weight (lbs)
• BL = Corrugated Board Length (in)
• BW = Corrugated Board Width (in)
• BT = Corrugated Board Thickness (in)
• D = Density of corrugated board material = 0.00270955 lb/in³
Example Calculation
If a corrugated board has a length of 24 inches, a width of 18 inches, and a thickness of 0.25 inches, its weight can be calculated as follows:
\[ \text{CBW} = 24 \, \text{in} \times 18 \, \text{in} \times 0.25 \, \text{in} \times 0.00270955 \, \text{lb/in³} = 0.2921 \, \text{lbs} \]
Importance and Usage Scenarios
Knowing the weight of corrugated board is critical in several industries, especially logistics and packaging. It helps in determining shipping costs, material selection for packaging, and
sustainability impact. This calculator can assist in optimizing packaging design, reducing unnecessary weight, and improving cost efficiency in supply chains.
Common FAQs
1. What is corrugated board density?
The density of corrugated board material used in this calculation is 0.00270955 lb/in³, representing the weight per cubic inch of the material.
2. Why is it important to calculate the weight of a corrugated board?
Weight calculations are essential for determining shipping costs, ensuring structural integrity, and optimizing packaging for cost and material efficiency.
3. Can I use this formula for other types of cardboard?
This specific formula applies to corrugated boards with the given density. Other types of cardboard may require adjustments in density values. | {"url":"https://www.calculatorultra.com/en/tool/corrugated-board-weight-calculator.html","timestamp":"2024-11-03T02:22:42Z","content_type":"text/html","content_length":"47343","record_id":"<urn:uuid:1ff18367-8442-440f-b5c8-997c224668fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00888.warc.gz"} |
Concatenation theory explained
Concatenation theory, also called string theory, character-string theory, or theoretical syntax, studies character strings over finite alphabets of characters, signs, symbols, or marks. String theory
is foundational for formal linguistics, computer science, logic, and metamathematics especially proof theory.^[1] A generative grammar can be seen as a recursive definition in string theory.
The most basic operation on strings is concatenation; connect two strings to form a longer string whose length is the sum of the lengths of those two strings. ABCDE is the concatenation of AB with
CDE, in symbols ABCDE = AB ^ CDE. Strings, and concatenation of strings can be treated as an algebraic system with some properties resembling those of the addition of integers; in modern mathematics,
this system is called a free monoid.
In 1956 Alonzo Church wrote: "Like any branch of mathematics, theoretical syntax may, and ultimately must, be studied by the axiomatic method".^[2] Church was evidently unaware that string theory
already had two axiomatizations from the 1930s: one by Hans Hermes and one by Alfred Tarski.^[3] Coincidentally, the first English presentation of Tarski's 1933 axiomatic foundations of string theory
appeared in 1956 – the same year that Church called for such axiomatizations.^[4] As Tarski himself noted using other terminology, serious difficulties arise if strings are construed as tokens rather
than types in the sense of Peirce's type-token distinction.
Notes and References | {"url":"https://everything.explained.today/Concatenation_theory/","timestamp":"2024-11-09T00:28:33Z","content_type":"text/html","content_length":"8029","record_id":"<urn:uuid:9a35595a-3299-40c5-b00b-8f070eb61beb>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00706.warc.gz"} |
Jonathan Healey
I graduated with a degree in physics at Oxford University in 1987, where I also obtained a DPhil in physics in 1991. My doctoral thesis was supervised by Tom Mullin and Dave Broomhead (both now
professors at Manchester University) and was concerned with the analysis of phase spaces reconstructed from time series data, and how these techniques can help us to understand the bifurcations and
complex behaviour of dynamical systems.
In 1991 I was appointed to a post-doctoral position in the Engineering Department at Cambridge University to work with Professor Mike Gaster FRS on the laminar-turbulent transition of boundary
layers. I developed the time series methods I had studied in my DPhil for application to hot-wire data from wind-tunnel experiments on boundary layers. In 1993 I continued this work with a second
post-doctoral position jointly supervised by Professor David Crighton FRS and Mike Gaster. I carried out wind-tunnel experiments on boundary layers, but also became increasingly interested in more
'classical' theories of hydrodynamic stability and how they relate to boundary layer experiments.
In 1996 I was appointed first to a temporary lectureship in the Mathematics Department at Brunel University, and then to a lectureship in the Mathematics Department at Keele University. I became a
Reader at Keele in 1998.
Research and scholarship
My research is still concerned with instabilities in fluid flows, and has included thermal boundary layers, three-dimensional boundary layers and swirling jets. I use asymptotic techniques to develop
theories in large-Reynolds number and long-wavelength limits, and also to obtain large-time descriptions of initial value problems for determining absolute and convective instability characteristics.
I also use numerical methods for solving viscous and inviscid stability equations.
Most of my research papers can be divided between three main areas.
1.Time series analysis
In the qualitative theory of dynamical systems one can represent complicated behaviour in terms of a state space, or phase space. The instantaneous state of the system is represented by a point in
this space, and the evolution of the system is represented by the trajectory taken by this point. Thus, an equilibrium state is represented by a fixed point, a periodic behaviour by a closed orbit, a
quasi-periodic behaviour by a path on a torus and chaotic behaviour by a path on a strange attractor with fractal properties. Remarkably, this trajectory, called a phase portrait, can often be
reconstructed from a series of measurements of a single characteristic of the system (a time series).
Summary of results and selected publications in Time Series Analysis.pdf
2. The Blasius boundary layer
When a fluid flows at high speed past a solid surface a boundary layer forms in the fluid close to the surface. It is a region where viscosity is important and arises from the no-slip boundary
condition on the fluid at a solid surface. Boundary layers are present however small the viscosity. In the absence of viscosity objects moving through a fluid experience no drag and no lift. It is
the presence of boundary layers that always produces drag on a body and can also produce lift. A boundary layer can separate from a surface, dramatically increasing the drag (as when an aeroplane
wing stalls). When the boundary layer is attached (like around a streamlined body) the amount of drag, and also the heat transfer characteristics, depends sensitively on whether the boundary layer is
laminar or turbulent. Being able to predict the state of a boundary layer is of importance in the design of aeroplane wings and turbine blades in jet engines. Fuel consumption of commercial aircraft
could be halved, and their range doubled, if laminar boundary layers could be maintained over their wings. However, laminar-turbulent transition, and turbulence, remain major outstanding problems in
fluid mechanics, and affect many other flows in many other situations.
The simplest boundary layer arises when a flat plate is placed parallel to a uniform stream. This is called the Blasius boundary layer, and has been much studied experimentally, numerically and
theoretically. Although it doesn't often arise in practical applications, and despite having certain peculiarities (an inflexion point at the wall), a greater understanding of its laminar-turbulent
transition is expected to be helpful in understanding other boundary layers.
Summary of results and selected publications in The Blasius boundary layer.pdf
3. Absolute and convective instabilities
Unstable disturbances in a shear layer, like a boundary layer, might all propagate downstream, in which case the flow is called convectively unstable. If unstable disturbances travel both upstream
and downstream in the shear layer then the flow is called absolutely unstable. Although this classification depends on the velocity of the reference frame being used, there is usually a frame of
particular interest, e.g. the laboratory frame, or the frame moving with an aeroplane wing, and then the distinction is of crucial importance in determining the dynamics of the flow. Convectively
unstable flows act as spatial amplifiers of whatever external disturbances are imposed on the flow. Absolutely unstable flows can generate their own intrinsic modes which are insensitive to the
disturbance environment, and thus behave as self-excited oscillators. These modes are called global modes and arise through an interplay of local instability characteristics, nonlinearity and weak
inhomogeneity of the basic flow in the streamwise direction. From a practical point of view, an unstable flow that is only convectively unstable will remain in a laminar state if the freestream
turbulence level is low enough, but an absolutely unstable flow giving rise to a global mode will have the original basic flow replaced by the global mode regardless of how small the external
disturbance levels are.
The local absolute/convective characteristics are obtained by assuming that the local basic velocity profile is independent of the streamwise coordinate and examining the solution produced by an
impulsive localized disturbance to an otherwise undisturbed basic flow (the Green's function) at large times. The flow is absolutely unstable if there is growth in time in the rest frame. This
behaviour can be determined using residue theory and saddle-point methods on the integrals appearing in the inverse Fourier-Laplace-type transforms of the initial value problem. At large times these
integrals are dominated by the contribution from the highest saddle point whose valleys contain the real wavenumber axis. (In the Briggs-Bers interpretation, this dominant saddle-point is called the
'pinch-point' and represents a coalescence of upstream and downstream travelling waves). Contour deformations in the complex wavenumber plane are required to locate the dominant saddle, and in the
case of convective instability this also gives information about the direction of propagation of waves (the so-called 'signalling problem').
Summary of results and selected publications in this area.pdf
• MAT-30002: Nonlinear Ordinary Differential Equations
Further information
Jonathan Healey's Homepage
Admin roles
Seminar Coordinator
Research themes
Fluid Dynamics and Acoustics
School of Computer Science and Mathematics
Keele University
ST5 5AA | {"url":"https://www.keele.ac.uk/scm/staff/jonathanhealey/","timestamp":"2024-11-06T02:00:53Z","content_type":"text/html","content_length":"40205","record_id":"<urn:uuid:f6c35f4b-d755-4206-8677-62df8d10bdf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00619.warc.gz"} |
Vending Machine Profit Calculator - Calculator Wow
Vending Machine Profit Calculator
About Vending Machine Profit Calculator (Formula)
Vending machines have become a ubiquitous presence in our daily lives, dispensing snacks, drinks, and various products at the push of a button. For business owners and operators, assessing the
profitability of a vending machine operation is crucial. This is where a Vending Machine Profit Calculator comes into play. It helps determine the profitability of a vending machine by taking into
account factors like item prices, sales volume, and operational costs.
The Formula:
The formula used by a Vending Machine Profit Calculator is straightforward and insightful:
Profit (Pv) = (Average Price per Item Sold × Average Number of Items Sold per Month) – Vending Machine Rental Cost
• Profit (Pv) represents the monthly profit generated by the vending machine in dollars ($).
• Average Price per Item Sold is the amount received for each item dispensed in dollars per item ($/item).
• Average Number of Items Sold per Month is the monthly sales volume, indicating how many items are sold on average each month.
• Vending Machine Rental Cost is the monthly expense associated with renting or leasing the vending machine in dollars per month ($/month).
This formula provides a clear picture of how much profit a vending machine can generate after accounting for both revenue from item sales and operational costs.
Frequently Asked Questions (FAQs):
1. Why is vending machine profitability important?
□ Profitability assessment helps vending machine operators make informed decisions regarding locations, pricing, and item selection.
2. What factors affect the average price per item sold?
□ Factors include product pricing strategy, customer demographics, and market competition.
3. How do I determine the average number of items sold per month?
□ Track monthly sales data to calculate the average number of items sold.
4. What are common operational costs for vending machines?
□ Operational costs include rental fees, maintenance, restocking, and electricity.
5. Is the vending machine rental cost a fixed expense?
□ It can vary depending on the rental agreement and location.
6. Can I use the calculator for multiple vending machines?
□ Yes, you can calculate the profit for each vending machine separately.
7. What if my vending machine is in a shared location with another operator?
□ Ensure you consider only your expenses and sales when using the calculator.
8. How often should I update the data in the calculator?
□ Regularly update the calculator with current data to maintain accuracy.
9. What should I do if my vending machine is not profitable?
□ Consider adjusting prices, relocating the machine, or changing the product selection.
10. Can the calculator account for seasonality in sales?
□ Seasonal fluctuations may require manual adjustments in the data entered.
A Vending Machine Profit Calculator is an invaluable tool for vending machine operators and business owners, providing insights into the financial health of their vending operations. By using this
formula and addressing the FAQs, operators can make informed decisions to optimize profitability and success. | {"url":"https://calculatorwow.com/vending-machine-profit-calculator/","timestamp":"2024-11-06T23:41:02Z","content_type":"text/html","content_length":"64690","record_id":"<urn:uuid:a1ff9840-d8ca-4901-92e2-c8f0d288f18b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00649.warc.gz"} |
Question of the day
I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same
answer as someone else, feel free to put up your answer, too!
Friday, September 30, 2011
A hoard of rings am I,
but no fit gift for a bride;
I await a sword's kiss.
Thursday, September 29, 2011
A wonder on the wave / water became bone.
- Book of Exeter
Wednesday, September 28, 2011
1. Perambulate in moccasins, and shoulder a gargantuan wooden rail. 2. Allow somnolent quadrupeds that are homo sapien's greatest comrades to remain reclining. 3. Lack of what is required is the
matriarch of inspiration. 4. A maximum amount of purposeful activity and a minimum amount of disport and dalliance cause Jack to become a dim-witted, stagnant dunce of the male species. 5. That which
is acquired without difficulty is dispersed with equal facility. 6. It is more desirable to arrive in the medium of time which constitutes a later than desirable hour or date than not to arrive at
Tuesday, September 27, 2011
He who locks himself into the arms of Morpheus promptly at eventide, and starts the day before it is officially announced by the rising sun, excels in physical fitness, increases his economic assets
and celebrates with remarkable efficiency. What common phrase is hidden in the complicated sentence above?
Friday, September 23, 2011
The sun bakes them, the hand breaks them, the foot treads on them, and the mouth tastes them. What are they?
Thursday, September 22, 2011
I go to a party with my wife. When we get there, four other couples arrive at the same time. We all know each other, so we all greet each other. A greeting can be a handshake, a kiss, a hug, or
whatever. When everyone is done I ask everyone how many times they shook another person's hand. All answers I get are different. Given that nobody greets their own spouse, how many hands did my wife
Wednesday, September 21, 2011
Used left or right, I get to travel, over cobblestone or gravel. Used up, I vie for sweet success, used down, I cause men great duress. What am I?
Tuesday, September 20, 2011
Find a 5 digit number, as big as possible, that when you multiply it by a single digit number, you get a six digit number, in which all digits are identical.
Friday, September 16, 2011
You throw three darts onto the surface of a globe, each from a randomly chosen direction. What is the probability that all three darts are in one hemisphere?
Wednesday, September 14, 2011
You have 55 matches arranged in some number of piles of different sizes. You now do the following operation: pick one match from each pile, and form a new pile. You repeat this ad infinitum. What is
the steady state? Is it unique? By steady state I mean the number of piles will remain unchanged or you create an unending loop.
Tuesday, September 13, 2011
You have a number that consists of 6 different digits. This number multiplied by 2, 3, 4, 5, and 6 yields, in all cases, a new 6-digit number, which, in all cases, is a permutation of the original 6
different digits. What's the number?
Monday, September 12, 2011
Create the number 24 using (all of) 1, 3, 4, and 6. You may add, subtract, multiply, and divide. Parentheses are free. You must use each digit only once. Note that you may not "glue" digits together.
(14 - 6) * 3 is not a solution. 13 * 4 * 6 is not a solution either (powers not allowed).
Wednesday, September 07, 2011
Two friends mr and mr saw see one day mr see saw sea and mr saw didn't see sea see saw sea and jumped in sea saw didn't see sea but jumped in in sea See saw saw in sea and saw saw see in sea. see
Tuesday, September 06, 2011
Eight lab rats (Nolan, Shorty, Spike, Evelyn, Herman, Dottie, Ruth, and George) are dispersed among an arrangement of ten boxes as represented above by the letters A to J. The boxes connect
vertically and horizontally, but not diagonally. Eight boxes each have one rat, and two boxes are vacant. From the clues given, determine where each rat is and which boxes are vacant.
1. Herman is in the same horizontal row as either or both vacant boxes.
2. Shorty connects with Ruth.
3. Nolan connects with a vacant box but not with Dottie.
4. Shorty connects with a vacant box.
5. Evelyn is not in B or I but does connect with Spike.
6. Nolan has a corner box.
7. A vacant box connects only with George and Evelyn.
8. Dottie is in G.
Found this one at
Crad Kilodney
Friday, September 02, 2011
My first is foremost legaly. My second circles outwardly. My third leads all in victory. My fourth twice ends a nominee. What am I?
Are you looking for a particular puzzle, riddle, question, etc? Or do you want to find the answer today rather than wait till tomorrow! | {"url":"http://www.questionotd.com/2011/09/","timestamp":"2024-11-02T21:05:43Z","content_type":"text/html","content_length":"194553","record_id":"<urn:uuid:5a5bd25a-9f1d-4dd5-92ec-fdae96985ac3>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00856.warc.gz"} |
cKDTree.query_ball_point(self, x, r, p=2., eps=0, workers=1, return_sorted=None, return_length=False)#
Find all points within distance r of point(s) x.
xarray_like, shape tuple + (self.m,)
The point or points to search for neighbors of.
rarray_like, float
The radius of points to return, shall broadcast to the length of x.
pfloat, optional
Which Minkowski p-norm to use. Should be in the range [1, inf]. A finite large p may cause a ValueError if overflow can occur.
epsnonnegative float, optional
Approximate search. Branches of the tree are not explored if their nearest points are further than r / (1 + eps), and branches are added in bulk if their furthest points are nearer than r
* (1 + eps).
workersint, optional
Number of jobs to schedule for parallel processing. If -1 is given all processors are used. Default: 1.
Changed in version 1.9.0: The “n_jobs” argument was renamed “workers”. The old name “n_jobs” was deprecated in SciPy 1.6.0 and was removed in SciPy 1.9.0.
return_sortedbool, optional
Sorts returned indices if True and does not sort them if False. If None, does not sort single point queries, but does sort multi-point queries which was the behavior before this option
was added.
return_length: bool, optional
Return the number of points inside the radius instead of a list of the indices. .. versionadded:: 1.3.0
resultslist or array of lists
If x is a single point, returns a list of the indices of the neighbors of x. If x is an array of points, returns an object array of shape tuple containing lists of neighbors.
If you have many points whose neighbors you want to find, you may save substantial amounts of time by putting them in a cKDTree and using query_ball_tree.
>>> import numpy as np
>>> from scipy import spatial
>>> x, y = np.mgrid[0:4, 0:4]
>>> points = np.c_[x.ravel(), y.ravel()]
>>> tree = spatial.cKDTree(points)
>>> tree.query_ball_point([2, 0], 1)
[4, 8, 9, 12]
Query multiple points and plot the results:
>>> import matplotlib.pyplot as plt
>>> points = np.asarray(points)
>>> plt.plot(points[:,0], points[:,1], '.')
>>> for results in tree.query_ball_point(([2, 0], [3, 3]), 1):
... nearby_points = points[results]
... plt.plot(nearby_points[:,0], nearby_points[:,1], 'o')
>>> plt.margins(0.1, 0.1)
>>> plt.show() | {"url":"https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.cKDTree.query_ball_point.html","timestamp":"2024-11-08T03:01:05Z","content_type":"text/html","content_length":"32062","record_id":"<urn:uuid:d14f2c77-47aa-42b8-9e53-16233f891041>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00322.warc.gz"} |
Trying to calculate the number of cells in a sheet during a specific year
I have dates in a series of columns on different sheets and want to calculate the number of cells where the date is within a range. i..e 1-Jan-22 to 31-Dec-22 or 1-Jan-23 to 31-Dec-23 or 1-Jan-24 to
31-Dec-24. The column in each sheet is exactly the same name (e.g. Launch date). Tried a number of options but just seem to get the right results.
Any help much appreciated.
Kind regards
• Hello @mike.thorpe17421
The YEAR(date) function will allow you to gather the data you need within a COUNTIFS. Are you putting these into summary fields where you have a specific formula for each year- which means you
are hard coding year values? Or, if you have a summary field that keeps a running total of the current year, you will refer to the YEAR of TODAY().
=COUNTIFS([Launch Date]:[Launch Date], YEAR(@cell) = 2022))
*note that numbers don't have quotes around them unless you want to force them to a text string (which in this case you do not)
Or, current year
=COUNTIFS([Launch Date]:[Launch Date], YEAR(@cell) = YEAR(TODAY())
Do either of these work for you?
• That's great Kelly, thank you. I couldn't seem to get the YEAR calculating though. This is the formula I have used when looking in another sheet.It should calculate how many launch dates for a
specific country are in the current year. Country@ row points to the name of the country on my calculation sheet:
=COUNTIFS({Name of Sheet and Ref}, Country@row, {Name of Sheet and Ref}, YEAR(@cell) = 2022)). In your explanation above you have YEAR(@cell) = YEAR(TODAY())with Open Brackets after TODAY, is
this correct?
Have I also got the last bit right as the first bit works ok: =COUNTIFS({Name of Sheet and Ref}, Country@row)
Sorry to be a bit dumb here but learning formulas all of the time.
• Hey Mike
Glad you continue to ask questions! I see my countifs above is missing the ending parenthesis - I must have deleted it as I did my copy paste. If the last parenthesis isn't blue it means the
parentheses count isn't correct.
For a cross sheet reference
=COUNTIFS({Other sheet Country column}, Country@row, {Other sheet Date column}, IFERROR(YEAR(@cell),0)=YEAR(TODAY()))
Since this is a cross sheet reference, you must build each reference through the formula window. You cannot simply copy paste.
I added the IFERROR on the date as sometimes date functions can produce errors.
Does this work for you.
• Thanks Kelly, the TODAY function now works but cannot get the 2023, 2024 dates working?The formula I have is as follows for the YEAR date:
=COUNTIFS({Countries SSOT Ref Sheet for Dashboards L- Range 2}, Country@row, {Countries SSOT Ref Sheet for Dashboards J- Range 1}, IFERROR(YEAR(@cell), 0) = YEAR(TODAY()))
What would it be for 2023, I had used the above and changed to reflect and tried the following hut this doesn't work 😩
=COUNTIFS({Countries SSOT Ref Sheet for Dashboards L- Range 2}, Country12, {Countries SSOT Ref Sheet for Dashboards J- Range 1}, IFERROR(YEAR(@cell), 0) = 2023())
• Hey Mike
The parentheses are associated with functions - we can fill them or not fill them depending on the data to be collected. When inserting a value, we just insert the value. If the value is a
textstring it needs to be enclosed in quotes. If its a number then don't use quotes.
=COUNTIFS({Countries SSOT Ref Sheet for Dashboards L- Range 2}, Country12, {Countries SSOT Ref Sheet for Dashboards J- Range 1}, IFERROR(YEAR(@cell), 0) = 2023)
If you were just viewing the data and not recording/tracking the data, you could make the formula dynamic
=COUNTIFS({Countries SSOT Ref Sheet for Dashboards L- Range 2}, Country12, {Countries SSOT Ref Sheet for Dashboards J- Range 1}, IFERROR(YEAR(@cell), 0) = YEAR(TODAY())+1)
• Hi Kelly
Many thnaks for your previous advise it was excellent and worked a treat. I am now trying to work on something similar but on a month to month basis.
Your formlula's for years work as follows:
IFERROR(YEAR(@cell), 0) = YEAR(TODAY()) + 0)
IFERROR(YEAR(@cell), 0) = YEAR(TODAY())+1)
Using the same thought process I replaced Year with Month as follows:
Current Month
IFERROR(MONTH(@cell), 0) = MONTH(TODAY()) + 0)
Next Month
IFERROR(MONTH(@cell), 0) = MONTH(TODAY()) + 1)
Any idea why this wouldn't work?
Also for the headings for the years they would be 2023, 2024, 2025, etc. To enable the headings to change year on year in line with the formulas have you any thoughts on how to do this? This
would not be Column headings but a row heading.
Many thanks in anticipation.
Kind regards
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/92379/trying-to-calculate-the-number-of-cells-in-a-sheet-during-a-specific-year","timestamp":"2024-11-05T15:40:27Z","content_type":"text/html","content_length":"415314","record_id":"<urn:uuid:29b3db11-5f4e-43da-914e-b1bfe965230f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00088.warc.gz"} |
GNSS Receiver Class
GNSS Receiver Class#
class rocketpy.sensors.GnssReceiver[source]#
Class for the GNSS Receiver sensor.
☆ prints (_GnssReceiverPrints) – Object that contains the print functions for the sensor.
☆ sampling_rate (float) – Sample rate of the sensor in Hz.
☆ position_accuracy (float) – Accuracy of the sensor interpreted as the standard deviation of the position in meters.
☆ altitude_accuracy (float) – Accuracy of the sensor interpreted as the standard deviation of the position in meters.
☆ name (str) – The name of the sensor.
☆ measurement (tuple) – The measurement of the sensor.
☆ measured_data (list) – The stored measured data of the sensor.
__init__(sampling_rate, position_accuracy=0, altitude_accuracy=0, name='GnssReceiver')[source]#
Initialize the Gnss Receiver sensor.
○ sampling_rate (float) – Sample rate of the sensor in Hz.
○ position_accuracy (float) – Accuracy of the sensor interpreted as the standard deviation of the position in meters. Default is 0.
○ altitude_accuracy (float) – Accuracy of the sensor interpreted as the standard deviation of the position in meters. Default is 0.
○ name (str) – The name of the sensor. Default is “GnssReceiver”.
measure(time, **kwargs)[source]#
Measure the position of the rocket in latitude, longitude and altitude.
○ time (float) – Current time in seconds.
○ kwargs (dict) –
Keyword arguments dictionary containing the following keys: - u : np.array
State vector of the rocket.
Derivative of the state vector of the rocket.
Position of the sensor relative to the rocket center of mass.
Environment object containing the atmospheric conditions.
export_measured_data(filename, file_format='csv')[source]#
Export the measured values to a file
○ filename (str) – Name of the file to export the values to
○ file_format (str) – Format of the file to export the values to. Options are “csv” and “json”. Default is “csv”.
Return type: | {"url":"https://docs.rocketpy.org/en/latest/reference/classes/sensors/gnss_receiver.html","timestamp":"2024-11-06T12:03:36Z","content_type":"text/html","content_length":"31235","record_id":"<urn:uuid:067f5c3b-57d6-4365-b722-c6c2cf3c6781>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00112.warc.gz"} |