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SPOJ.com - Problem MARBLES
MARBLES - Marbles
Hänschen dreams he is in a shop with an infinite amount of marbles. He is allowed to select n marbles. There are marbles of k different colors. From each color there are also infinitely many marbles.
Hänschen wants to have at least one marble of each color, but still there are a lot of possibilities for his selection. In his effort to make a decision he wakes up. Now he asks you how many
possibilites for his selection he would have had. Assume that marbles of equal color can't be distinguished, and the order of the marbles is irrelevant.
The first line of input contains a number T ≤ 100 that indicates the number of test cases to follow. Each test case consists of one line containing n and k, where n is the number of marbles Hänschen
selects and k is the number of different colors of the marbles. You can assume that 1 ≤ k ≤ n ≤ 1000000.
For each test case print the number of possibilities that Hänschen would have had. You can assume that this number fits into a signed 64 bit integer.
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nodiveg: 2024-10-19 18:11:06
Can anyone explain why my idea is wrong for tc 2 where I chose all 7 marbles among 30 of different kinds and now I have left 23 marbles as I already chose at least one of each kind by choosing 7
marbles among 30 now for each marble I have 7 possibilities so why the answer is not 7^23 ?? can anyone explain
eyad9090: 2023-11-13 09:56:44
if you coudn't understand how to get this formula watch this two videos
this really helps me
tarun_28: 2020-04-20 10:01:49
AC with a O(t*(n-1)*(k-1)) DP solution;)
ajaygupta007: 2020-04-07 14:37:34
stars and bar problem
hetp111: 2019-10-14 22:18:25
use r=min(r,n-r) to avoid overflow.
ajaytec227: 2019-09-26 13:05:31
Just find the value it's ok if it overflows.
The right answer for
is -18446780961959
nitin_uniyal21: 2019-06-21 12:58:37
If you don't know how to proceed. Please see https://www.youtube.com/watch?v=UTCScjoPymA
This will teach you how you can derive the formula (by analyzing the patterns).
toolatetostart: 2019-05-28 11:09:26
find out n-1 C k-1,carefully ,choose iteration as min(n-k,k-1) as nCr can be written as nCn-r
vritta: 2019-05-02 18:24:44
if you are stuck then this - <snip> might help (contains explanation along with code).
Also please don't post unproductive comments like "A.C. in one go" etc. Nobody cares about your A.C.
Last edit: 2023-01-29 20:55:28
harry_shit: 2019-01-14 09:42:28
i am in love with spoj!!
Added by: Adrian Kuegel
Date: 2004-06-19
Time limit: 1s
Source limit: 10000B
Memory limit: 1536MB
Cluster: Cube (Intel G860)
Languages: All except: NODEJS PERL6 VB.NET
Resource: own problem | {"url":"https://www.spoj.com/problems/MARBLES/","timestamp":"2024-11-09T04:55:29Z","content_type":"text/html","content_length":"28085","record_id":"<urn:uuid:3f87282e-7b51-49f8-a35b-1b175e4fe5eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00115.warc.gz"} |
Matrix view of the helix
Next: CAUSALITY AND SPECTAL FACTORIZATION Up: FINITE DIFFERENCES ON A Previous: FINITE DIFFERENCES ON A
Physics on a helix can be viewed thru the eyes of matrices and numerical analysis. This is not easy because the matrices are so huge. Discretize the (x,y)-plane to an 12).
The two-dimensional matrix of coefficients for the Laplacian operator is shown in (12), where, on a cartesian space, h=0, and in the helix geometry, h=-1. (A similar partitioned matrix arises from
packing a cylindrical surface into a h=-1. With the partitioning thus invisible, the matrix simply represents one-dimensional convolution and we have an alternative analytical approach,
one-dimensional Fourier Transform. We often need to solve sets of simultaneous equations with a matrix similar to (12). The method we use is triangular factorization.
If you will allow me some truncation approximations, I now claim that the laplacian represented by the matrix in equation (12) is factored into two parts 12). Recall that triangular matrices allow
quick solutions of simultaneous equations by backsubstitution. That is what we do with our deconvolution program.
Next: CAUSALITY AND SPECTAL FACTORIZATION Up: FINITE DIFFERENCES ON A Previous: FINITE DIFFERENCES ON A Stanford Exploration Project | {"url":"https://sepwww.stanford.edu/sep/prof/toc_html/gee/hlx/paper_html/node9.html","timestamp":"2024-11-05T15:11:00Z","content_type":"text/html","content_length":"7302","record_id":"<urn:uuid:86e02d7e-2e43-40d8-a74a-a4abe2f673c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00199.warc.gz"} |
Gamma Function in Scipy.Special
I was playing with the Gamma Distribution Function in python. As typically, I used a combination of [DEL:bumpy:DEL] numpy arrays and the scipy.special functions for Gamma and incomplete gamma
functions. At one point I realized that the Gamma distribution function (bottom panel on the figure below) was not reaching unity for large x values and for some parameters (k=0.63, theta=0.05). In
the given case, the deviation was about 0.3 in probability space, which is significant.
Note: The Gamma Distribution Function (its cdf or pdf) is not the same thing as the Gamma function!
The solution I came up with is to use the module mpmath. When I do, everything looks fine.
I googled only briefly, but did not find any description of this error in scipy.special. Also I am not sure, if using mpmath is the best/fastest/most elegant solution. However, at least it seems to
4 Responses to 'Gamma Function in Scipy.Special'
I get the same strange plots if I define the cdf of the gamma distribution as:
gamma_cdf = lambda x, alpha, beta: sp.special.gammainc(alpha, beta * x) / sp.special.gamma(alpha)
or, if you like the other formulation better:
gamma2 = lambda x, k, beta: gamma_cdf(x, k, 1 / beta)
which is the obvious thing to do. However in scipy.special, gammainc is defined as:
“1 / gamma(a) * integral(exp(-t) * t**(a-1), t=0..x)”
So, dividing by sp.special.gamma(alpha) is the straw that broke the gamma’s back. This will bring you up to 1:
gamma_cdf = lambda x, alpha, beta: sp.special.gammainc(alpha, beta * x)
By the way: are “bumpy arrays” more uneven than numpy ones? SCNR (I am also sorry for the url)
well, gamma gamma, thanks for pointing out what breaks ones neck.. 😉
Things may get even hairier when you use more complex distributions in scipy.stats.
One can get quite interesting insights into some distributions – after the headaches of transforming parameters from textbooks or wikipedia into loc, scale, etc. of the stats functions to achieve
the same results, have gone away.
I think scipy.stats.gamma is one of those candidates.
By the way, why don’t you use that one?
thanks for pointing towards the functions integrated into scipy.stats — I guess I wanted to see things more directly… 😉 | {"url":"https://www.planetwater.org/2013/08/19/gamma-function-in-scipy-special/","timestamp":"2024-11-02T23:43:23Z","content_type":"application/xhtml+xml","content_length":"33538","record_id":"<urn:uuid:910e22fc-2e69-4da0-bc1b-be9c7c6639d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00696.warc.gz"} |
Reproducible blogging | R-bloggersReproducible blogging
[This article was first published on
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
As a fact-based blog, the posts here contain very often diagrams and data tables. To enable you to reproduce the results and insights, I include the computations as computer code.
Most blogposts I write are markdown text combined (or weaved) with computer code written in the R language. I created a small package mdtools that puts the tools together and smoothes the workflow.
This post gives an short introduction to the mdtools package: how to install it, the first post, caveats, and future directions.
Read more » | {"url":"https://www.r-bloggers.com/2011/07/reproducible-blogging/","timestamp":"2024-11-09T16:33:14Z","content_type":"text/html","content_length":"84521","record_id":"<urn:uuid:12c44520-476f-4e1e-bba3-0f13285cb58c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00803.warc.gz"} |
Riemannian Ricci curvature lower bounds in metric measure spaces with
Accepted Paper
Inserted: 20 jul 2012
Last Updated: 17 feb 2013
Journal: Transactions of the AMS
Year: 2012
In prior work of the first two authors with Savaré, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces $(X,d,m)$ was introduced, and the corresponding
class of spaces denoted by $RCD(K,\infty)$. This notion relates the $CD(K,N)$ theory of Sturm and Lott-Villani, in the case $N=\infty$, to the Bakry-Emery approach. In the aforementioned paper, the
$RCD(K,\infty)$ property is defined in three equivalent ways and several properties of $RCD(K,\infty)$ spaces, including the regularization properties of the heat flow, the connections with the
theory of Dirichlet forms and the stability under tensor products, are provided. But only finite reference measures $m$ have been considered. The goal of this paper is twofold: on one side we extend
these results to general $\sigma$-finite spaces, on the other we remove a technical assumption concerning a strengthening of the $CD(K,\infty)$ condition. This more general class of spaces includes
Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.
Tags: GeMeThNES
Keywords: Optimal Mass Transportation, Ricci curvature, Entropy | {"url":"https://cvgmt.sns.it/paper/1886/","timestamp":"2024-11-03T07:31:19Z","content_type":"text/html","content_length":"9362","record_id":"<urn:uuid:0d28a5d1-f163-437d-84b6-30785d9f7968>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00888.warc.gz"} |
What is the LCM of 2 and 3?
The LCM of 2 and 3 is the smallest number that is a multiple of both 2 and 3. In this case, the LCM of 2 and 3 is 6. The concept of LCM is often used in math problems, especially those that involve
Knowing the LCM of two numbers is essential when adding or subtracting fractions with different denominators. By finding the LCM, you can convert the fractions to have the same denominator, making
them easier to add or subtract.
But how can you find the LCM of two numbers? One method is to list the multiples of each number and look for the smallest multiple that they have in common. However, this can be time-consuming and
A faster and more efficient way to find the LCM of two numbers is to use prime factorization. By breaking each number down into its prime factors, you can easily find their LCM.
In the case of 2 and 3, their prime factorizations are:
2 = 2
3 = 3
To find the LCM, we need to take the highest power of each prime factor. In this case, the highest power of 2 is 1, and the highest power of 3 is also 1. Therefore, the LCM of 2 and 3 is 2 x 3 = 6.
In conclusion, understanding the concept of LCM is important in mathematics, especially when dealing with fractions. By knowing how to find the LCM of two numbers, you can make math problems easier
to solve. So the LCM of 2 and 3 is 6.
More LCM Questions
Leave a Comment | {"url":"https://conversiongalaxy.com/lcm-of-numbers/lcm-of-2-and-3/","timestamp":"2024-11-06T22:00:10Z","content_type":"text/html","content_length":"94548","record_id":"<urn:uuid:285ffb53-aa19-4416-8564-a69a929600eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00682.warc.gz"} |
Unlock the magic of numbers with free math courses from MIT
Explore 19 popular online courses, including linear algebra, probability, calculus, and computability.
Photo: iStock
By Sara Feijo
Math isn’t just about numbers and equations. It’s a universal language that tells stories of logic, patterns, theorems, and more. In honor of Math Storytelling Day on Sept. 25 — established to
encourage people to tell stories through math — discover the magic behind the numbers with 19 popular and free online courses from MIT Open Learning.
Explore how matrix theory and linear algebra can be useful in disciplines beyond mathematics — from physics, economics, and social sciences, to natural sciences and engineering.
Get an elementary introduction to probability and statistics with applications.
Gain a deep understanding of the principles that underpin statistical inference: estimation, hypothesis testing, and prediction.
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Discover the ins and outs of the derivative: what it is, how to compute it, and when to apply it to solve real-world problems.
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Understand differentiation and integration of functions of one variable.
Tune into this series of videos introducing how calculus works and why it’s important.
Take a deeper dive with the Calculus Online Textbook, the most-viewed and most-downloaded individual file in MIT OpenCourseWare’s collection.
Explore the derivative in higher dimensions, and learn how to apply it to solve real-world problems.
Further your understanding of rates of change with differential, integral, and vector calculus for functions of more than one variable.
Grasp the fundamentals of mathematical analysis, from convergence of sequences and series, continuity, and differentiability, to Riemann integral, sequences and series of functions, uniformity, and
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Learn the equations and techniques most useful in science and engineering.
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Embark on an intensive review of undergraduate-level mathematics for prospective and beginning graduate students in science and engineering.
These courses are available through MIT OpenCourseWare and MITx, which are part of MIT Open Learning. OpenCourseWare offers free, online, open educational resources from more than 2,500 courses that
span the MIT undergraduate and graduate curriculum. MITx offers high-quality massive open online courses adapted from the MIT classroom for learners worldwide. | {"url":"https://medium.com/open-learning/unlock-the-magic-of-numbers-with-free-math-courses-from-mit-f778ec7a4b06","timestamp":"2024-11-03T22:35:06Z","content_type":"text/html","content_length":"139636","record_id":"<urn:uuid:ddf6554b-d106-4ccc-a035-e2c173490bed>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00067.warc.gz"} |
Explicit binary tree codes with polylogarithmic size alphabet
This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size. For every constant < 1 we give an explicit binary tree code
with distance and alphabet size poly(log n), where n is the depth of the tree. This is the first improvement over a two-decade-old construction that has an exponentially larger alphabet of size poly
(n). As part of the analysis, we prove a bound on the number of positive integer roots a real polynomial can have in terms of its sparsity with respect to the Newton basis—a result of independent
Original language English
Title of host publication STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
Editors Monika Henzinger, David Kempe, Ilias Diakonikolas
Pages 1074-1087
Number of pages 14
ISBN (Electronic) 9781450355599
State Published - 20 Jun 2018
Externally published Yes
Event 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States
Duration: 25 Jun 2018 → 29 Jun 2018
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
Conference 50th Annual ACM Symposium on Theory of Computing, STOC 2018
Country/Territory United States
City Los Angeles
Period 25/06/18 → 29/06/18
• Explicit constructions
• Sparse polynomials
• Tree codes
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'Explicit binary tree codes with polylogarithmic size alphabet'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/explicit-binary-tree-codes-with-polylogarithmic-size-alphabet","timestamp":"2024-11-14T10:44:36Z","content_type":"text/html","content_length":"43547","record_id":"<urn:uuid:25cf7ff9-5868-47f5-b31a-4f7e45dd2335>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00390.warc.gz"} |
Time sequence prediction with FNN-LSTM
At the moment, we decide up on the plan alluded to within the conclusion of the latest Deep attractors: Where deep learning meets
chaos: make use of that very same approach to generate forecasts for
empirical time sequence knowledge.
“That very same approach,” which for conciseness, I’ll take the freedom of referring to as FNN-LSTM, is because of William Gilpin’s
2020 paper “Deep reconstruction of unusual attractors from time sequence” (Gilpin 2020).
In a nutshell, the issue addressed is as follows: A system, recognized or assumed to be nonlinear and extremely depending on
preliminary situations, is noticed, leading to a scalar sequence of measurements. The measurements aren’t simply – inevitably –
noisy, however as well as, they’re – at greatest – a projection of a multidimensional state area onto a line.
Classically in nonlinear time sequence evaluation, such scalar sequence of observations are augmented by supplementing, at each
time limit, delayed measurements of that very same sequence – a way known as delay coordinate embedding (Sauer, Yorke, and Casdagli 1991). For
instance, as a substitute of only a single vector X1, we might have a matrix of vectors X1, X2, and X3, with X2 containing
the identical values as X1, however ranging from the third remark, and X3, from the fifth. On this case, the delay can be
2, and the embedding dimension, 3. Varied theorems state that if these
parameters are chosen adequately, it’s doable to reconstruct the entire state area. There’s a downside although: The
theorems assume that the dimensionality of the true state area is thought, which in lots of real-world purposes, gained’t be the
That is the place Gilpin’s thought is available in: Prepare an autoencoder, whose intermediate illustration encapsulates the system’s
attractor. Not simply any MSE-optimized autoencoder although. The latent illustration is regularized by false nearest
neighbors (FNN) loss, a way generally used with delay coordinate embedding to find out an enough embedding dimension.
False neighbors are those that are shut in n-dimensional area, however considerably farther aside in n+1-dimensional area.
Within the aforementioned introductory post, we confirmed how this
approach allowed to reconstruct the attractor of the (artificial) Lorenz system. Now, we wish to transfer on to prediction.
We first describe the setup, together with mannequin definitions, coaching procedures, and knowledge preparation. Then, we let you know the way it
From reconstruction to forecasting, and branching out into the true world
Within the earlier publish, we skilled an LSTM autoencoder to generate a compressed code, representing the attractor of the system.
As typical with autoencoders, the goal when coaching is similar because the enter, that means that total loss consisted of two
parts: The FNN loss, computed on the latent illustration solely, and the mean-squared-error loss between enter and
output. Now for prediction, the goal consists of future values, as many as we want to predict. Put otherwise: The
structure stays the identical, however as a substitute of reconstruction we carry out prediction, in the usual RNN approach. The place the standard RNN
setup would simply instantly chain the specified variety of LSTMs, we’ve got an LSTM encoder that outputs a (timestep-less) latent
code, and an LSTM decoder that ranging from that code, repeated as many instances as required, forecasts the required variety of
future values.
This after all implies that to judge forecast efficiency, we have to examine in opposition to an LSTM-only setup. That is precisely
what we’ll do, and comparability will become fascinating not simply quantitatively, however qualitatively as properly.
We carry out these comparisons on the 4 datasets Gilpin selected to display attractor reconstruction on observational
data. Whereas all of those, as is obvious from the pictures
in that pocket book, exhibit good attractors, we’ll see that not all of them are equally suited to forecasting utilizing easy
RNN-based architectures – with or with out FNN regularization. However even people who clearly demand a special method enable
for fascinating observations as to the impression of FNN loss.
Mannequin definitions and coaching setup
In all 4 experiments, we use the identical mannequin definitions and coaching procedures, the one differing parameter being the
variety of timesteps used within the LSTMs (for causes that may grow to be evident once we introduce the person datasets).
Each architectures had been chosen to be simple, and about comparable in variety of parameters – each principally consist
of two LSTMs with 32 models (n_recurrent will probably be set to 32 for all experiments).
FNN-LSTM seems to be practically like within the earlier publish, aside from the truth that we cut up up the encoder LSTM into two, to uncouple
capability (n_recurrent) from maximal latent state dimensionality (n_latent, stored at 10 identical to earlier than).
# DL-related packages
# going to want these later
encoder_model <- perform(n_timesteps,
title = NULL) {
keras_model_custom(title = title, perform(self) {
self$noise <- layer_gaussian_noise(stddev = 0.5)
self$lstm1 <- layer_lstm(
models = n_recurrent,
input_shape = c(n_timesteps, n_features),
return_sequences = TRUE
self$batchnorm1 <- layer_batch_normalization()
self$lstm2 <- layer_lstm(
models = n_latent,
return_sequences = FALSE
self$batchnorm2 <- layer_batch_normalization()
perform (x, masks = NULL) {
x %>%
self$noise() %>%
self$lstm1() %>%
self$batchnorm1() %>%
self$lstm2() %>%
decoder_model <- perform(n_timesteps,
title = NULL) {
keras_model_custom(title = title, perform(self) {
self$repeat_vector <- layer_repeat_vector(n = n_timesteps)
self$noise <- layer_gaussian_noise(stddev = 0.5)
self$lstm <- layer_lstm(
models = n_recurrent,
return_sequences = TRUE,
go_backwards = TRUE
self$batchnorm <- layer_batch_normalization()
self$elu <- layer_activation_elu()
self$time_distributed <- time_distributed(layer = layer_dense(models = n_features))
perform (x, masks = NULL) {
x %>%
self$repeat_vector() %>%
self$noise() %>%
self$lstm() %>%
self$batchnorm() %>%
self$elu() %>%
n_latent <- 10L
n_features <- 1
n_hidden <- 32
encoder <- encoder_model(n_timesteps,
decoder <- decoder_model(n_timesteps,
The regularizer, FNN loss, is unchanged:
loss_false_nn <- perform(x) {
# altering these parameters is equal to
# altering the power of the regularizer, so we maintain these fastened (these values
# correspond to the unique values utilized in Kennel et al 1992).
rtol <- 10
atol <- 2
k_frac <- 0.01
okay <- max(1, floor(k_frac * batch_size))
## Vectorized model of distance matrix calculation
tri_mask <-
form = c(tf$forged(n_latent, tf$int32), tf$forged(n_latent, tf$int32)),
dtype = tf$float32
num_lower = -1L,
num_upper = 0L
# latent x batch_size x latent
batch_masked <-
tf$multiply(tri_mask[, tf$newaxis,], x[tf$newaxis, reticulate::py_ellipsis()])
# latent x batch_size x 1
x_squared <-
tf$reduce_sum(batch_masked * batch_masked,
axis = 2L,
keepdims = TRUE)
# latent x batch_size x batch_size
pdist_vector <- x_squared + tf$transpose(x_squared, perm = c(0L, 2L, 1L)) -
2 * tf$matmul(batch_masked, tf$transpose(batch_masked, perm = c(0L, 2L, 1L)))
#(latent, batch_size, batch_size)
all_dists <- pdist_vector
# latent
all_ra <-
tf$sqrt((1 / (
batch_size * tf$vary(1, 1 + n_latent, dtype = tf$float32)
)) *
batch_masked - tf$reduce_mean(batch_masked, axis = 1L, keepdims = TRUE)
), axis = c(1L, 2L)))
# Keep away from singularity within the case of zeros
#(latent, batch_size, batch_size)
all_dists <-
tf$clip_by_value(all_dists, 1e-14, tf$reduce_max(all_dists))
#inds = tf.argsort(all_dists, axis=-1)
top_k <- tf$math$top_k(-all_dists, tf$forged(okay + 1, tf$int32))
# (#(latent, batch_size, batch_size)
top_indices <- top_k[[1]]
#(latent, batch_size, batch_size)
neighbor_dists_d <-
tf$collect(all_dists, top_indices, batch_dims = -1L)
#(latent - 1, batch_size, batch_size)
neighbor_new_dists <-
tf$collect(all_dists[2:-1, , ],
top_indices[1:-2, , ],
batch_dims = -1L)
# Eq. 4 of Kennel et al.
#(latent - 1, batch_size, batch_size)
scaled_dist <- tf$sqrt((
tf$sq.(neighbor_new_dists) -
# (9, 8, 2)
tf$sq.(neighbor_dists_d[1:-2, , ])) /
# (9, 8, 2)
tf$sq.(neighbor_dists_d[1:-2, , ])
# Kennel situation #1
#(latent - 1, batch_size, batch_size)
is_false_change <- (scaled_dist > rtol)
# Kennel situation 2
#(latent - 1, batch_size, batch_size)
is_large_jump <-
(neighbor_new_dists > atol * all_ra[1:-2, tf$newaxis, tf$newaxis])
is_false_neighbor <-
tf$math$logical_or(is_false_change, is_large_jump)
#(latent - 1, batch_size, 1)
total_false_neighbors <-
tf$forged(is_false_neighbor, tf$int32)[reticulate::py_ellipsis(), 2:(k + 2)]
# Pad zero to match dimensionality of latent area
# (latent - 1)
reg_weights <-
1 - tf$reduce_mean(tf$forged(total_false_neighbors, tf$float32), axis = c(1L, 2L))
# (latent,)
reg_weights <- tf$pad(reg_weights, list(list(1L, 0L)))
# Discover batch common exercise
# L2 Exercise regularization
activations_batch_averaged <-
tf$sqrt(tf$reduce_mean(tf$sq.(x), axis = 0L))
loss <- tf$reduce_sum(tf$multiply(reg_weights, activations_batch_averaged))
Coaching is unchanged as properly, aside from the truth that now, we regularly output latent variable variances along with
the losses. It’s because with FNN-LSTM, we’ve got to decide on an enough weight for the FNN loss element. An “enough
weight” is one the place the variance drops sharply after the primary n variables, with n thought to correspond to attractor
dimensionality. For the Lorenz system mentioned within the earlier publish, that is how these variances regarded:
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
0.0739 0.0582 1.12e-6 3.13e-4 1.43e-5 1.52e-8 1.35e-6 1.86e-4 1.67e-4 4.39e-5
If we take variance as an indicator of significance, the primary two variables are clearly extra necessary than the remaining. This
discovering properly corresponds to “official” estimates of Lorenz attractor dimensionality. For instance, the correlation dimension
is estimated to lie round 2.05 (Grassberger and Procaccia 1983).
Thus, right here we’ve got the coaching routine:
train_step <- perform(batch) {
with (tf$GradientTape(persistent = TRUE) %as% tape, {
code <- encoder(batch[[1]])
prediction <- decoder(code)
l_mse <- mse_loss(batch[[2]], prediction)
l_fnn <- loss_false_nn(code)
loss <- l_mse + fnn_weight * l_fnn
encoder_gradients <-
tape$gradient(loss, encoder$trainable_variables)
decoder_gradients <-
tape$gradient(loss, decoder$trainable_variables)
encoder_gradients, encoder$trainable_variables
decoder_gradients, decoder$trainable_variables
training_loop <- tf_function(autograph(perform(ds_train) {
for (batch in ds_train) {
tf$print("Loss: ", train_loss$outcome())
tf$print("MSE: ", train_mse$outcome())
tf$print("FNN loss: ", train_fnn$outcome())
mse_loss <-
tf$keras$losses$MeanSquaredError(discount = tf$keras$losses$Discount$SUM)
train_loss <- tf$keras$metrics$Imply(title = 'train_loss')
train_fnn <- tf$keras$metrics$Imply(title = 'train_fnn')
train_mse <- tf$keras$metrics$Imply(title = 'train_mse')
# fnn_multiplier needs to be chosen individually per dataset
# that is the worth we used on the geyser dataset
fnn_multiplier <- 0.7
fnn_weight <- fnn_multiplier * nrow(x_train)/batch_size
# studying charge can also want adjustment
optimizer <- optimizer_adam(lr = 1e-3)
for (epoch in 1:200) {
cat("Epoch: ", epoch, " -----------n")
test_batch <- as_iterator(ds_test) %>% iter_next()
encoded <- encoder(test_batch[[1]])
test_var <- tf$math$reduce_variance(encoded, axis = 0L)
print(test_var %>% as.numeric() %>% round(5))
On to what we’ll use as a baseline for comparability.
Vanilla LSTM
Right here is the vanilla LSTM, stacking two layers, every, once more, of measurement 32. Dropout and recurrent dropout had been chosen individually
per dataset, as was the educational charge.
lstm <- perform(n_latent, n_timesteps, n_features, n_recurrent, dropout, recurrent_dropout,
optimizer = optimizer_adam(lr = 1e-3)) {
mannequin <- keras_model_sequential() %>%
models = n_recurrent,
input_shape = c(n_timesteps, n_features),
dropout = dropout,
recurrent_dropout = recurrent_dropout,
return_sequences = TRUE
) %>%
models = n_recurrent,
dropout = dropout,
recurrent_dropout = recurrent_dropout,
return_sequences = TRUE
) %>%
time_distributed(layer_dense(models = 1))
mannequin %>%
loss = "mse",
optimizer = optimizer
mannequin <- lstm(n_latent, n_timesteps, n_features, n_hidden, dropout = 0.2, recurrent_dropout = 0.2)
Knowledge preparation
For all experiments, knowledge had been ready in the identical approach.
In each case, we used the primary 10000 measurements accessible within the respective .pkl information provided by Gilpin in his GitHub
repository. To avoid wasting on file measurement and never depend upon an exterior
knowledge supply, we extracted these first 10000 entries to .csv information downloadable instantly from this weblog’s repo:
geyser <- download.file(
electrical energy <- download.file(
"https://uncooked.githubusercontent.com/rstudio/ai-blog/grasp/docs/posts/2020-07-20-fnn-lstm/knowledge/electrical energy.csv",
"knowledge/electrical energy.csv")
ecg <- download.file(
mouse <- download.file(
Must you wish to entry the entire time sequence (of significantly higher lengths), simply obtain them from Gilpin’s repo
and cargo them utilizing reticulate:
Right here is the info preparation code for the primary dataset, geyser – all different datasets had been handled the identical approach.
# the primary 10000 measurements from the compilation offered by Gilpin
geyser <- read_csv("geyser.csv", col_names = FALSE) %>% choose(X1) %>% pull() %>% unclass()
# standardize
geyser <- scale(geyser)
# varies per dataset; see beneath
n_timesteps <- 60
batch_size <- 32
# remodel into [batch_size, timesteps, features] format required by RNNs
gen_timesteps <- perform(x, n_timesteps) {
perform(i) {
begin <- i
finish <- i + n_timesteps - 1
out <- x[start:end]
) %>%
n <- 10000
practice <- gen_timesteps(geyser[1:(n/2)], 2 * n_timesteps)
check <- gen_timesteps(geyser[(n/2):n], 2 * n_timesteps)
dim(practice) <- c(dim(practice), 1)
dim(check) <- c(dim(check), 1)
# cut up into enter and goal
x_train <- practice[ , 1:n_timesteps, , drop = FALSE]
y_train <- practice[ , (n_timesteps + 1):(2*n_timesteps), , drop = FALSE]
x_test <- check[ , 1:n_timesteps, , drop = FALSE]
y_test <- check[ , (n_timesteps + 1):(2*n_timesteps), , drop = FALSE]
# create tfdatasets
ds_train <- tensor_slices_dataset(list(x_train, y_train)) %>%
dataset_shuffle(nrow(x_train)) %>%
ds_test <- tensor_slices_dataset(list(x_test, y_test)) %>%
Now we’re prepared to take a look at how forecasting goes on our 4 datasets.
Geyser dataset
Folks working with time sequence could have heard of Old Faithful, a geyser in
Wyoming, US that has regularly been erupting each 44 minutes to 2 hours for the reason that 12 months 2004. For the subset of knowledge
Gilpin extracted,
geyser_train_test.pkl corresponds to detrended temperature readings from the primary runoff pool of the Previous Devoted geyser
in Yellowstone Nationwide Park, downloaded from the GeyserTimes database. Temperature measurements
begin on April 13, 2015 and happen in one-minute increments.
Like we stated above, geyser.csv is a subset of those measurements, comprising the primary 10000 knowledge factors. To decide on an
enough timestep for the LSTMs, we examine the sequence at varied resolutions:
It looks as if the habits is periodic with a interval of about 40-50; a timestep of 60 thus appeared like a superb attempt.
Having skilled each FNN-LSTM and the vanilla LSTM for 200 epochs, we first examine the variances of the latent variables on
the check set. The worth of fnn_multiplier similar to this run was 0.7.
test_batch <- as_iterator(ds_test) %>% iter_next()
encoded <- encoder(test_batch[[1]]) %>%
as.array() %>%
encoded %>% summarise_all(var)
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
0.258 0.0262 0.0000627 0.000000600 0.000533 0.000362 0.000238 0.000121 0.000518 0.000365
There’s a drop in significance between the primary two variables and the remaining; nevertheless, in contrast to within the Lorenz system, V1 and
V2 variances additionally differ by an order of magnitude.
Now, it’s fascinating to match prediction errors for each fashions. We’re going to make a remark that may carry
via to all three datasets to return.
Maintaining the suspense for some time, right here is the code used to compute per-timestep prediction errors from each fashions. The
similar code will probably be used for all different datasets.
calc_mse <- perform(df, y_true, y_pred) {
(sum((df[[y_true]] - df[[y_pred]])^2))/nrow(df)
get_mse <- perform(test_batch, prediction) {
comp_df <-
test_batch[[2]][, , 1] %>%
as.array()) %>%
rename_with(perform(title) paste0(title, "_true")) %>%
prediction[, , 1] %>%
as.array()) %>%
rename_with(perform(title) paste0(title, "_pred")))
mse <- purrr::map(1:dim(prediction)[2],
paste0("X", varno, "_true"),
paste0("X", varno, "_pred"))) %>%
prediction_fnn <- decoder(encoder(test_batch[[1]]))
mse_fnn <- get_mse(test_batch, prediction_fnn)
prediction_lstm <- mannequin %>% predict(ds_test)
mse_lstm <- get_mse(test_batch, prediction_lstm)
mses <- data.frame(timestep = 1:n_timesteps, fnn = mse_fnn, lstm = mse_lstm) %>%
collect(key = "sort", worth = "mse", -timestep)
ggplot(mses, aes(timestep, mse, coloration = sort)) +
geom_point() +
scale_color_manual(values = c("#00008B", "#3CB371")) +
theme_classic() +
theme(legend.place = "none")
And right here is the precise comparability. One factor particularly jumps to the attention: FNN-LSTM forecast error is considerably decrease for
preliminary timesteps, at the start, for the very first prediction, which from this graph we anticipate to be fairly good!
Curiously, we see “jumps” in prediction error, for FNN-LSTM, between the very first forecast and the second, after which
between the second and the following ones, reminding of the same jumps in variable significance for the latent code! After the
first ten timesteps, vanilla LSTM has caught up with FNN-LSTM, and we gained’t interpret additional growth of the losses based mostly
on only a single run’s output.
As a substitute, let’s examine precise predictions. We randomly decide sequences from the check set, and ask each FNN-LSTM and vanilla
LSTM for a forecast. The identical process will probably be adopted for the opposite datasets.
given <- data.frame(as.array(tf$concat(list(
test_batch[[1]][, , 1], test_batch[[2]][, , 1]
axis = 1L)) %>% t()) %>%
add_column(sort = "given") %>%
add_column(num = 1:(2 * n_timesteps))
fnn <- data.frame(as.array(prediction_fnn[, , 1]) %>%
t()) %>%
add_column(sort = "fnn") %>%
add_column(num = (n_timesteps + 1):(2 * n_timesteps))
lstm <- data.frame(as.array(prediction_lstm[, , 1]) %>%
t()) %>%
add_column(sort = "lstm") %>%
add_column(num = (n_timesteps + 1):(2 * n_timesteps))
compare_preds_df <- bind_rows(given, lstm, fnn)
plots <-
purrr::map(sample(1:dim(compare_preds_df)[2], 16),
perform(v) {
ggplot(compare_preds_df, aes(num, .knowledge[[paste0("X", v)]], coloration = sort)) +
geom_line() +
theme_classic() +
theme(legend.place = "none", axis.title = element_blank()) +
scale_color_manual(values = c("#00008B", "#DB7093", "#3CB371"))
plot_grid(plotlist = plots, ncol = 4)
Listed here are sixteen random picks of predictions on the check set. The bottom reality is displayed in pink; blue forecasts are from
FNN-LSTM, inexperienced ones from vanilla LSTM.
What we anticipate from the error inspection comes true: FNN-LSTM yields considerably higher predictions for instant
continuations of a given sequence.
Let’s transfer on to the second dataset on our record.
Electrical energy dataset
It is a dataset on energy consumption, aggregated over 321 completely different households and fifteen-minute-intervals.
electricity_train_test.pkl corresponds to common energy consumption by 321 Portuguese households between 2012 and 2014, in
models of kilowatts consumed in fifteen minute increments. This dataset is from the UCI machine learning
Right here, we see a really common sample:
With such common habits, we instantly tried to foretell a better variety of timesteps (120) – and didn’t should retract
behind that aspiration.
For an fnn_multiplier of 0.5, latent variable variances seem like this:
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
0.390 0.000637 0.00000000288 1.48e-10 2.10e-11 0.00000000119 6.61e-11 0.00000115 1.11e-4 1.40e-4
We undoubtedly see a pointy drop already after the primary variable.
How do prediction errors examine on the 2 architectures?
Right here, FNN-LSTM performs higher over an extended vary of timesteps, however once more, the distinction is most seen for instant
predictions. Will an inspection of precise predictions verify this view?
It does! Actually, forecasts from FNN-LSTM are very spectacular on all time scales.
Now that we’ve seen the simple and predictable, let’s method the bizarre and troublesome.
ECG dataset
Says Gilpin,
ecg_train.pkl and ecg_test.pkl correspond to ECG measurements for 2 completely different sufferers, taken from the PhysioNet QT
How do these look?
To the layperson that I’m, these don’t look practically as common as anticipated. First experiments confirmed that each architectures
aren’t able to coping with a excessive variety of timesteps. In each attempt, FNN-LSTM carried out higher for the very first
That is additionally the case for n_timesteps = 12, the ultimate attempt (after 120, 60 and 30). With an fnn_multiplier of 1, the
latent variances obtained amounted to the next:
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
0.110 1.16e-11 3.78e-9 0.0000992 9.63e-9 4.65e-5 1.21e-4 9.91e-9 3.81e-9 2.71e-8
There is a spot between the primary variable and all different ones; however not a lot variance is defined by V1 both.
Other than the very first prediction, vanilla LSTM exhibits decrease forecast errors this time; nevertheless, we’ve got so as to add that this
was not constantly noticed when experimenting with different timestep settings.
Taking a look at precise predictions, each architectures carry out greatest when a persistence forecast is enough – in truth, they
produce one even when it’s not.
On this dataset, we definitely would wish to discover different architectures higher in a position to seize the presence of excessive and low
frequencies within the knowledge, reminiscent of combination fashions. However – had been we pressured to stick with one among these, and will do a
one-step-ahead, rolling forecast, we’d go together with FNN-LSTM.
Talking of blended frequencies – we haven’t seen the extremes but …
Mouse dataset
“Mouse,” that’s spike charges recorded from a mouse thalamus.
mouse.pkl A time sequence of spiking charges for a neuron in a mouse thalamus. Uncooked spike knowledge was obtained from
CRCNS and processed with the authors’ code with a purpose to generate a
spike charge time sequence.
Clearly, this dataset will probably be very exhausting to foretell. How, after “lengthy” silence, are you aware {that a} neuron goes to fireside?
As typical, we examine latent code variances (fnn_multiplier was set to 0.4):
Again, we don’t see the first variable explaining much variance. Still, interestingly, when inspecting forecast errors we get
a picture very similar to the one obtained on our first, geyser, dataset:
So here, the latent code definitely seems to help! With every timestep “more” that we try to predict, prediction performance
goes down continuously – or put the other way round, short-time predictions are expected to be pretty good!
Let’s see:
In fact on this dataset, the difference in behavior between both architectures is striking. When nothing is “supposed to
happen,” vanilla LSTM produces “flat” curves at about the mean of the data, while FNN-LSTM takes the effort to “stay on track”
as long as possible before also converging to the mean. Choosing FNN-LSTM – had we to choose one of these two – would be an
obvious decision with this dataset.
When, in timeseries forecasting, would we consider FNN-LSTM? Judging by the above experiments, conducted on four very different
datasets: Whenever we consider a deep learning approach. Of course, this has been a casual exploration – and it was meant to
be, as – hopefully – was evident from the nonchalant and bloomy (sometimes) writing style.
Throughout the text, we’ve emphasized utility – how could this technique be used to improve predictions? But, looking at
the above results, a number of interesting questions come to mind. We already speculated (though in an indirect way) whether
the number of high-variance variables in the latent code was relatable to how far we could sensibly forecast into the future.
However, even more intriguing is the question of how characteristics of the dataset itself affect FNN efficiency.
Such characteristics could be:
• How nonlinear is the dataset? (Put differently, how incompatible, as indicated by some form of test algorithm, is it with
the hypothesis that the data generation mechanism was a linear one?)
• To what degree does the system appear to be sensitively dependent on initial conditions? In other words, what is the value
of its (estimated, from the observations) highest Lyapunov exponent?
• What’s its (estimated) dimensionality, for instance, when it comes to correlation
Whereas it’s straightforward to acquire these estimates, utilizing, for example, the
nonlinearTseries package deal explicitly modeled after practices
described in Kantz & Schreiber’s basic (Kantz and Schreiber 2004), we don’t wish to extrapolate from our tiny pattern of datasets, and go away
such explorations and analyses to additional posts, and/or the reader’s ventures :-). In any case, we hope you loved
the demonstration of sensible usability of an method that within the previous publish, was primarily launched when it comes to its
conceptual attractivity.
Thanks for studying!
Gilpin, William. 2020.
“Deep Reconstruction of Unusual Attractors from Time Sequence.” https://arxiv.org/abs/2002.05909
Grassberger, Peter, and Itamar Procaccia. 1983.
“Measuring the Strangeness of Unusual Attractors.” Physica D: Nonlinear Phenomena
9 (1): 189–208. https://doi.org/
Kantz, Holger, and Thomas Schreiber. 2004. Nonlinear Time Sequence Evaluation. Cambridge College Press.
Sauer, Tim, James A. Yorke, and Martin Casdagli. 1991.
“Embedology.” Journal of Statistical Physics
65 (3-4): 579–616. | {"url":"http://thefutureofworkinstitute.xyz/2023/05/28/time-series-prediction-with-fnn-lstm/","timestamp":"2024-11-12T13:18:15Z","content_type":"text/html","content_length":"177278","record_id":"<urn:uuid:fd062d94-42b9-47be-b8ff-22c0d5e791b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00160.warc.gz"} |
The Elements of Spherical Trigonometry
James Hann
Popular passages
... principles untouched. The only method of avoiding this error is to confide to men, who are masters of their respective subjects, the task of drawing up Popular Introductions to the several
branches of Science. The Publisher trusts that the following list of names will be a sufficient guarantee to the Public that what he proposes to attempt in the cause of Popular Instruction will be
done well, and that these little treatises will fully answer the purpose for which they are intended, namely, to...
JOHN \YEALK (of which Prospectuses have been extensively issued), have realised the anticipated success from that portion of the public who seek the attainment of those objects of Science which
belong to the business of life, and the highest and most useful subjects in the Elements of Art and Science. Pursuing the same path, to render further aid to public instruction, and to direct the
attention of the Heads and Principals of the several Colleges and Mr.
The surface of a spherical triangle is measured by the excess of the sum of its three angles above two right angles, multiplied by the tri-rectangular triangle.
CONSTRUCTING CRANES for the Erection of Buildings and for Hoisting Goods, by JOSEPH GLYNN, FRS, CE , TREATISE ON THE STEAM ENGINE, by DR. LARDNER, LL.D., Editor of the " Cabinet Cyclopaedia
... cos a = cos b cos с + sin b sin с cos A ; (2) cos b = cos a cos с + sin a sin с cos в ; ^ A. (3) cos с = cos a cos b + sin a sin b cos C.
... inestimable value, and too many attempts cannot be made to render them perfect and complete." To carry out this new Series successfully and methodically, the most eminent men in scholastic
erudition and elementary instruction have been selected, under the able management and editing of Mr. JAMES...
Popular treatises are to Science what boats are to large ships ; they assist people in getting aboard ; but as no one would trust himself to a weak or inefficient boat, so no one ought to begin the
study of Science with an imperfect guide. It sometimes happens that popular treatises are made to appear easy by the omission of those very details which are most essential to be known : they
Course for the easy comprehension of the leading principles of various Sciences. It has been remarked that " those who are in the ship of Science ought to remember that the disciples cannot arrive
without the aid of boats.
BRIDGES, tec., more particularly the Conway and Britannia Bridges, describing the Experiments made to determine their form, strength, and efficiency, together with the construction of the same, the
floating and raising the tubes, *c 2 li.
He did not even understand the rule I made use of for finding the excess of the sum of the three angles of a spherical triangle above...
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Byte Swapping in NumPy
Bytes are the unit of data which is 8 bits in length that is basically used to store the data in the memory of a computer depending on which architecture the CPU uses. In Python, there is a library
called Numpy which is basically used for working with arrays, and there is a method provided by Numpy called byteswap() which can be used for byte swapping in Numpy.
In the article “Byte Swapping in NumPy”, we will discuss the need for byte swapping, how byte swapping can be done, and lastly, and implementation of byte swapping.
Numpy Library in Python
In the article “Byte Swapping in NumPy”, we will get some idea about the Numpy library because we are going to use the Numpy library in the concept of byte swapping. So, Numpy is the library provided
by Python that basically works on the array operations, which results in faster execution time than the generic Python lists.
We are going to see an example in which we will create the Numpy array and reverse it because we will need to create it while doing byte swapping and we are doing reversing because internally byte
swapping does the same:
In the above code, the array() function is used from the library Numpy where the list is passed of 4 elements. Then reversing the array is done by creating two variables, i and j, for the start and
end of the array. Then an iteration is done till i is lesser than j, and swapping of the element at i and j is done. Then i and j are incremented and decremented by 1. Lastly, we printed the reversed
Numpy array. | {"url":"https://www.naukri.com/code360/library/byte-swapping-in-numpy","timestamp":"2024-11-14T14:53:46Z","content_type":"text/html","content_length":"418328","record_id":"<urn:uuid:097ae1f1-fc47-409f-acd2-75ad0dffeaf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00314.warc.gz"} |
Guide to F#
Written by Mike James
Wednesday, 29 December 2010
F# is the only language to be added to Visual Studio for a very long time. What makes it so special?
There is a growing trend to include elements of functional programming in “mainstream” languages such as C#. If functional programming is so good this raises the question of why we don’t just move to
a functional language.
After all, there is an ideal candidate, F# which is a true .NET based language.
So, is F# very different?
Let’s find out.
What is functional programming?
It’s almost too easy to say that functional programming is all about using functions but there are many ways of doing this.
Functional programming attempts to make programming more like mathematics by making programming functions more like mathematical functions.
In maths a function is a set of rules that given an input set of values produces a single result – the value of the function. This initially sounds very much like the sort of function we use in
programming, but there are some important differences.
For example a mathematical function doesn’t change any of its input values and certainly doesn’t change any value that isn’t part of the function.
This is usually expressed by stating that mathematical functions don’t have side effects whereas programming functions often do. A typical programmed function might well change its input parameters
and pass them back as secondary results and often changes variables, for example within the user interface, that are not really part of the function. In fact for many functions the side effects are
what are important.
The key ideas in functional programming are “no side effects” and “immutability” – although many a keen functional programmer would be horrified by this oversimplification.
Immutability relates to the idea that once a function has been evaluated it shouldn’t change its value – mathematical functions don’t.
In many ways it is the requirement for immutability that is usually the most difficult to swallow by an experienced procedural programmer.
Consider the familiar statement:
which means take the current value stored in x, add one to it and store the result back in x. In functional programming you really should consider x to be a function and the value stored in it is the
value of the function x. Hence by the requirement for immutability its value can’t change and so any instruction like x=x+1 is complete nonsense.
It is like writing:
which is even more clearly nonsense.
The reason why this is so difficult for a traditional programmer to accept is that x=x+1 is the point where programming really diverges from mathematics. From the viewpoint most of us are familiar
with, programming is about describing “procedure”, i.e. what happens, and it is perfectly normal to take the value stored in x, add one to it and store it back in x.
Most of what we have achieved has been via procedural programming even if we have been using functions. Functional programming is about non-procedural programming. It’s about static relationships and
things that change as little as possible. If this sounds a little unpromising remember that a function can still be “active” in the sense that it “works out a result” – but with no side effects and
once the result is derived it doesn’t change.
You might very well at this point be doubting that functional programming can work – how can it possibly banish procedure from programming?
I have to admit that in practice functional languages have to let a bit of “procedure” sneak in. Just as procedural languages such as C# have elements of functional programming, so function languages
such as F# have elements of procedural programming.
What is important is to minimise their use and be aware of what you are doing. Many F# functions are used just for their side effects – usually called “imperative” programming. In practice working
with F# is a mix of functional and imperative programming.
You might also find some of the mechanisms that functional programming languages “invent” contrived, and as much a transgression of the basic principles as a dropping back to procedural semantics,
but it’s all an effort to keep the purity of the approach.
Added to this is the fact that F# is overtly a “mixed mode” language, attempting to fuse functional programming with other approaches. In this article the focus is on the functional as this is the
least well-known of the approaches but lookout for others!
To understand the advantages on offer, you have to ask the question what exactly is wrong with a procedural approach that a functional approach fixes?
The simple answer is that if you don’t specify a procedure then the machine is free to decide exactly how to implement your code. In theory, functional programming makes threading, and parallelism in
general, very easy and fairly safe.
A functional program is also supposed to be easier to prove correct, debug, and so on, than a procedural program and all of this is true – but this doesn’t mean that it is impossible to write a bad
functional program.
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The ABOVE element
Permitted Context: %math
Content Model: %math
The <ABOVE> element is used to draw a line, arrow, curly bracket, or accent above the expression enclosed by this element. Stretchy symbols should be stretched to match the width of the enclosed
expression. For example:
<above>X + Y</above> giving X + Y
<above sym=equals>X + Y</above> giving X + Y
You can also place an expression centered above the line or arrow with the SUP element or its shortref form, for example:
<above sym=cub>n(n - 1)(n - 2)&dots;(n - m + 1)</above>
<sup><text>total of m factors</text></sup>
which would be rendered as (within limits of ascii art):
total of m factors
n(n - 1)(n - 2) ... (n - m + 1)
Permitted Attributes
An entity name for a symbol, e.g. cub for a curly bracket (brace). Defaults to line. The other choices are: larr (left arrow), rarr (right arrow), hat and tilde.
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JUST FOR NERDS | Caledonia Wilson
top of page
Me and the monochromator
The nerd runs deep within me. I love the analytical challenge of mathematics, and see it as a beautiful search for truth, not rote memorization of methods to complete calculations. Surprisingly,
studying math has helped me sharpen my craft of acting. Both a mathematical proof and a scene have the same goal: sharing an inherent truth in such a way that others can engage with the work.
The preparations begin with the same questions:
• What do I know already? How much of this information will help me right now?
• How can I break this material down into its simplest chunks?
• What weird stuff am I going to have to try to crack this one?
Being a mathematician helps me be a better actor. I can break things down into their simplest bits and put them back together again in a way that tells others something new about the world. Here's a
little bit more about my technical background in STEM:
In addition to studying theatre, dance, Arabic, and anthropology in college, I loved taking my math and physics courses. During two summers, I was lucky enough to study at Research Experiences for
Undergraduates funded by the National Science Foundation. One of my projects was at Rensselaer Polytechnic Institute, retrofitting a monochromator with modern mass-produced technology and optics lab
equipment. The next summer, I studied combinatorics at the University of Minnesota, with a dimer interpretation of cluster algebras, which you can read about here.
I graduated from Mount Holyoke College summa cum laude in 2019, where I majored in mathematics and minored in Arabic. I earned highest honors for my senior thesis in fluid dynamics, and was inducted
into Phi Beta Kappa upon graduation. After graduation, I headed to Budapest on a Fulbright grant, where I studied math at Budapest Semesters in Mathematics and published research in convex geometry
in the journal Discrete Mathematics. You can read the paper here.
Math reminds me to think problems through, step by step. The logical scaffolding of problem solving techniques continues to inform my work as an actor.
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Gallery Quotes (7 quotes)
A grove of giant redwoods or sequoias should be kept just as we keep a great or beautiful cathedral. The extermination of the passenger pigeon meant that mankind was just so much poorer; exactly as
in the case of the destruction of the cathedral at Rheims. And to lose the chance to see frigate-birds soaring in circles above the storm, or a file of pelicans winging their way homeward across the
crimson afterglow of the sunset, or a myriad terns flashing in the bright light of midday as they hover in a shifting maze above the beach—why, the loss is like the loss of a gallery of the
masterpieces of the artists of old time.
Art gallery? Who needs it? Look up at the swirling silver-lined clouds in the magnificent blue sky or at the silently blazing stars at midnight. How could indoor art be any more masterfully created
than God’s museum of nature?
Dewar’s rule in his laboratory was as absolute as that of a Pharaoh, and he showed deference to no one except the ghost of Faraday whom he met occasionally all night in the gallery behind the lecture
Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] … an historical role … in the development of pure mathematics. A great
revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid
and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor’s set theory and Peano’s space-filling curve. Historically, the revolution was forced by the discovery of
mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded … as “pathological,” .… as a “gallery of monsters,” akin to the cubist paintings and
atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of
pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended
completely the limitations imposed by its natural origins.
Now, as Mandelbrot points out, … Nature has played a joke on the mathematicians. The 19th-century mathematicians may not have been lacking in imagination, but Nature was not. The same pathological
structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.
Gold is found in our own part of the world; not to mention the gold extracted from the earth in India by the ants, and in Scythia by the Griffins. Among us it is procured in three different ways; the
first of which is in the shape of dust, found in running streams. … A second mode of obtaining gold is by sinking shafts or seeking among the debris of mountains …. The third method of obtaining gold
surpasses the labors of the giants even: by the aid of galleries driven to a long distance, mountains are excavated by the light of torches, the duration of which forms the set times for work, the
workmen never seeing the light of day for many months together.
The mind of a young man (his gallery I mean) is often furnished different ways. According to the scenes he is placed in, so are his pictures. They disappear, and he gets a new set in a moment. But as
he grows up, he gets some substantial pieces which he always preserves, although he may alter his smaller paintings in a moment.
To a person uninstructed in natural history, his country or sea-side stroll is a walk through a gallery filled with wonderful works of art, nine-tenths of which have their faces turned to the wall.
Teach him something of natural history, and you place in his hands a catalogue of those which are worth turning around. Surely our innocent pleasures are not so abundant in this life, that we can
afford to despise this or any other source of them. | {"url":"https://todayinsci.com/QuotationsCategories/G_Cat/Gallery-Quotations.htm","timestamp":"2024-11-07T06:48:27Z","content_type":"text/html","content_length":"91967","record_id":"<urn:uuid:c193f609-a550-48ec-8433-a6214b4cca45>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00889.warc.gz"} |
Electrical Technology Solutions - AdvancementCapacitance Formula | Capacitance | Capacitance Units | Capacitance Equation | What is Capacitance Capacitance Formula | Capacitance | Capacitance Units | Capacitance Equation | What is Capacitance
Capacitance Formula | Capacitance | Capacitance Units | Capacitance Equation | What is Capacitance
A capacitor is an electrical component that stores energy in an electric field. It is a fundamental component used in electronic circuits to regulate voltage, filter noise, and control timing. In
this article, we will explore the working principle of capacitors, how to calculate capacitance, and its applications.
What is a Capacitor?
A capacitor is an electrical component that stores energy in an electric field. It consists of two conductive plates separated by a dielectric material. When a voltage is applied across the plates,
an electric field is created, causing a charge to build up on each plate. The capacitor's capacitance is the measure of its ability to store charge.
How Does a Capacitor Work?
When a capacitor is connected to a power source, it charges up to the source voltage. Once fully charged, the capacitor blocks the flow of current. When the capacitor is connected to a load, it
discharges, releasing its stored energy to power the load. Capacitors are commonly used in electronic circuits for filtering noise, timing, and voltage regulation.
How Do You Determine the Value of Capacitance?
The value of capacitance is determined by the physical characteristics of the capacitor, including the surface area of the plates, the distance between the plates, and the permittivity of the
dielectric material. The formula for capacitance varies depending on the type of capacitor used.
Standard Units of Capacitance
The standard unit of capacitance is the farad (F), named after Michael Faraday, a pioneering scientist in electromagnetism. Other units of capacitance include microfarads (μF) and picofarads (pF).
Capacitance of a Parallel Plate Capacitor
The capacitance of a parallel plate capacitor is given by C = εA/d, where ε is the permittivity of the dielectric material, A is the surface area of the plates, and d is the distance between the
Capacitance of a Spherical Capacitor
The capacitance of a spherical capacitor is given by C = 4πεab / (b-a), where ε is the permittivity of the dielectric material, a is the radius of the inner sphere, and b is the radius of the
outer sphere.
Factors Affecting Capacitance
The capacitance of a capacitor is affected by the distance between the plates, the surface area of the plates, and the permittivity of the dielectric material. Temperature, humidity, and the
frequency of the applied voltage also affect capacitance.
Applications of Capacitors
Capacitors are used in a wide range of electronic applications, including power supplies, filters, timing circuits, and signal processing. They are also used in electric vehicles, aerospace
technology, and medical equipment.
Capacitor Fundamentals: Capacitance, Voltage, Charge, Reactance, Quality and Dissipation Factors
Capacitors are electronic components used in many different circuits and devices to store electrical energy. They consist of two conductive plates separated by a dielectric material, which can be a
vacuum or an insulating material. Capacitors come in various shapes and sizes, and their characteristics are determined by factors such as capacitance, voltage, reactance, quality factor, dissipation
factor, and energy storage.
Capacitance of a Capacitor
The capacitance of a capacitor is defined as the ability of the capacitor to store electrical charge. It is measured in Farads (F) and is determined by the physical characteristics of the capacitor,
such as plate area, plate separation distance, and dielectric constant. The capacitance of a capacitor can be calculated using the following formula:
C = Q/V
where C is the capacitance, Q is the charge stored on the capacitor, and V is the voltage across the capacitor.
Charge Stored in a Capacitor
The charge stored in a capacitor is directly proportional to the voltage applied across the capacitor and the capacitance of the capacitor. The formula for the charge stored in a capacitor is:
Q = CV
where Q is the charge stored in the capacitor, C is the capacitance of the capacitor, and V is the voltage applied across the capacitor.
Voltage of the Capacitor
The voltage of a capacitor is the potential difference between the two plates of the capacitor. When a capacitor is connected to a voltage source, it charges up to the voltage of the source. The
voltage of a capacitor can be calculated using the following formula:
V = Q/C
where V is the voltage across the capacitor, Q is the charge stored on the capacitor, and C is the capacitance of the capacitor.
Reactance of the Capacitor
The reactance of a capacitor is the opposition of the capacitor to the flow of alternating current (AC). It is measured in Ohms and is determined by the frequency of the AC and the capacitance of the
capacitor. The formula for the reactance of a capacitor is:
Xc = 1/(2Ï€fC)
where Xc is the reactance of the capacitor, f is the frequency of the AC, and C is the capacitance of the capacitor.
Quality Factor of Capacitor
The quality factor of a capacitor is a measure of the efficiency of the capacitor. It is defined as the ratio of the energy stored in the capacitor to the energy lost in the capacitor per cycle. The
formula for the quality factor of a capacitor is:
Q = 1/(2Ï€fRC)
where Q is the quality factor, f is the frequency of the AC, R is the resistance of the circuit, and C is the capacitance of the capacitor.
Dissipation Factor of Capacitor
The dissipation factor of a capacitor is a measure of the losses in the capacitor. It is defined as the ratio of the energy lost in the capacitor to the energy stored in the capacitor per cycle. The
formula for the dissipation factor of a capacitor is:
D = tanδ
where D is the dissipation factor and δ is the phase angle between the voltage and current in the capacitor.
Energy Stored in a Capacitor
The energy stored in a capacitor is the work done to charge the capacitor. It is equal to the product of the capacitance, voltage, and half the square of the voltage. The formula for the energy
stored in a capacitor is:
E = 1/2CV²
where E is the energy stored in the capacitor, C is the capacitance of the capacitor, and V is the voltage across the capacitor
Average Power of Capacitor
The average power of a capacitor is the power dissipated by the capacitor over a period of time. The formula for the average power of a capacitor is:
P = V²/R
where P is the power dissipated by the capacitor, V is the voltage across the capacitor, and R is the resistance of the circuit.
Capacitor Voltage During Charge / Discharge
When a capacitor is charged or discharged, its voltage changes over time. The voltage of a capacitor during charging and discharging can be calculated using the following formulas:
During Charging:
V(t) = V₀(1 - e^(-t/RC))
where V(t) is the voltage of the capacitor at time t, V₀ is the initial voltage of the capacitor, R is the resistance of the circuit, C is the capacitance of the capacitor, and e is the natural
logarithm base.
During Discharging:
V(t) = V₀e^(-t/RC)
where V(t) is the voltage of the capacitor at time t, V₀ is the initial voltage of the capacitor, R is the resistance of the circuit, C is the capacitance of the capacitor, and e is the natural
logarithm base.
Capacitance Formulas
Capacitance of a Plate Capacitor Formula:
C = ε₀A/d
where C is the capacitance of the plate capacitor, ε₀ is the permittivity of free space, A is the area of the plates, and d is the distance between the plates.
Self Capacitance of a Coil (Medhurst Formula):
C = (Ï€²DL)/(ln(4L/d) - 0.5)
where C is the self capacitance of the coil, D is the diameter of the coil, L is the length of the coil, and d is the wire diameter.
Self Capacitance of a Sphere Formula:
C = 4πε₀r
where C is the self capacitance of the sphere, ε₀ is the permittivity of free space, and r is the radius of the sphere.
Self Capacitance of a Toroid Inductor Formula:
C = (2Ï€²Îµ₀h)/(ln(r₂/r₁))
where C is the self capacitance of the toroid inductor, ε₀ is the permittivity of free space, h is the height of the toroid, r₁ is the inner radius of the toroid, and r₂ is the outer radius of the
Ohm’s Law for Capacitor
Ohm’s law for a capacitor relates the current flowing through the capacitor to the voltage across the capacitor and the capacitance of the capacitor. The formula for Ohm's law for a capacitor is:
I = C(dV/dt)
where I is the current flowing through the capacitor, C is the capacitance of the capacitor, and dV/dt is the rate of change of voltage with respect to time.
Capacitors are essential components in many electronic circuits and devices. They store electrical energy, have various characteristics such as capacitance, voltage, reactance, quality factor, and
dissipation factor. These characteristics are essential to understand to design circuits and devices that use capacitors efficiently. Understanding the formulas and equations that govern the behavior
of capacitors can help in designing circuits and devices that use capacitors effectively.
What is the formula of the capacitor?
The formula for a capacitor is given by:
C = Q/V
Where C represents capacitance, Q represents charge, and V represents voltage.
What is capacitance and its unit?
Capacitance is a measure of a capacitor's ability to store an electrical charge. It is defined as the ratio of the amount of charge stored on each plate to the voltage applied across the plates. The
unit of capacitance is the farad (F), named after Michael Faraday.
One farad is defined as the capacitance of a capacitor that will store a charge of one coulomb (C) when a voltage of one volt (V) is applied across it. However, the farad is a very large unit of
capacitance, so capacitors are usually measured in smaller units like microfarads (µF), nanofarads (nF), or picofarads (pF).
Why capacitance is Q by V?
Capacitance is defined as the ratio of the amount of charge stored on each plate of a capacitor to the voltage applied across the plates. Therefore, the capacitance formula C = Q/V represents this
relationship between charge and voltage.
In other words, if you apply a voltage V to a capacitor, it will store a charge Q on each plate, and the amount of charge stored will be directly proportional to the voltage applied. The
proportionality constant is the capacitance C, which is a measure of the capacitor's ability to store charge.
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Mathsspin deluxe
A challenging game which will improve your mental agility - now with improved graphics
You might think that maths games can never be fun, but just try it - the addition of movement, immediate scoring and a time-limit makes a surprising difference.
And after developing your skill at this game your mental algebra will be devastating!
Instructions and controls
In this game you have to match the value given in the center to one of the values in the orbiting objects. You need to clear all the answers before time runs out.
Just use the mouse pointer to click on your choice. The first level should let you get the hang of this, but don't go away! It gets more tricky and fun.
You get 10 points for a correct answer but lose 20 for an incorrect choice. In addition, you lose a second for every wrong answer in the level so far. So the first mistake costs you one second, the
next two, the third a further three seconds, and so on.
You get points for the time remaining at the end of each level. There may be several identical answers available - you can pick which of these to use.
There are nine levels, which work through basic addition and subtraction, multiplication, fractions (this is hard), precidence of the basic operations, powers and finally, all of these together.
You can submit your score to a high-score table to compare it against the rest of the internet.
If you find the graphics distracting, you can still play the original version which has the same mathematical content. | {"url":"http://lysisgames.com/games/mathsspin/Mathsspin2.html","timestamp":"2024-11-03T18:34:15Z","content_type":"text/html","content_length":"5561","record_id":"<urn:uuid:e97f2a09-70e8-435d-abf8-78f5c79e3bdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00170.warc.gz"} |
Some Properties and Applications of Spaces of Modular Forms With ETA-Multiplier
Date of Award
Spring 2022
Document Type
Open Access Dissertation
First Advisor
Matthew Boylan
This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite
{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0
\pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if $\bfrac{n}{5} = 1$ and $\bfrac{n}{7} = -1$}.$$
It follows that for all natural numbers $n$ and for $k \in \{6,19,24,26,31,34\}$, $$a_7(5\cdot7^{t+1}\cdot n + 7^t\cdot k - 2) \equiv 0 \pmod 5.$$
In the second part of the dissertation, we give results on where Hecke operators map spaces of modular forms which arise as multiples of eta-quotients. Let ${N\in \{1, 2, 3, 4, 5, 6, 8, 9\}}$ and let
$f(z)$ be a level $N$ holomorphic eta quotient with integer weight. Then we precisely describe how $T_n$ with $\gcd(n, 6) = 1$ permute subspaces of the form $$\{f(Dz)F(Dz) : F(z) \in M_w(\Gamma_0(N),
\chi)\}.$$ Subspaces of this type play a significant role in recent works \cite{A1, A2, B, BB, G2, Y1, Y2, ZZ}, primarily for $N = 1$ and with applications, for example, to congruences for partition
© 2022, Cuyler Daniel Warnock
Recommended Citation
Warnock, C. D.(2022). Some Properties and Applications of Spaces of Modular Forms With ETA-Multiplier. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/6853 | {"url":"https://scholarcommons.sc.edu/etd/6853/","timestamp":"2024-11-08T11:37:56Z","content_type":"text/html","content_length":"34747","record_id":"<urn:uuid:88cf3af7-a8d3-4141-a3a1-a71b06087beb>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00186.warc.gz"} |
Factoring Trinomials Worksheet Answers
Factoring Trinomials Worksheet Answers
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Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries
Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:775-803, 2021.
We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed
exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural
characterization of clusters, called $(\beta,\gamma)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not
needed). Using $(\beta,\gamma)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $R^d$) into a graph-theoretic
notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(\beta,\gamma)$-convex clusters using $O(k^2 \log n + k^2 (\frac{6}{\beta\gamma})^{dens
(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is
necessary, and we also show that, if we are allowed to make $O(k^2 + k \log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary
scales, even when the scale of each cluster is unknown.
Cite this Paper
Related Material | {"url":"http://proceedings.mlr.press/v134/bressan21a.html","timestamp":"2024-11-05T09:30:28Z","content_type":"text/html","content_length":"16794","record_id":"<urn:uuid:c069300f-5acd-4090-a68c-13b28374f60c>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00297.warc.gz"} |
Venn Diagram Printables
Venn Diagram Printables
Teachers may create a venn diagram during a lesson as part of their presentation and they may also instruct students to create the diagrams. There are many advantages to using a venn diagram to help
display and organize information for students.
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Hyphal growth segmentation
Hyphal growth is the driving force of fungal bloodstream infections due to their ability to pierce through epithelial tissue. Investigating the growth of a hypha via image analysis is often hindered
by crossing or touching hyphae from different fungal branches and the quality of the image. The 3D-HyTracer can, to some extent, overcome common problems resulting from, e.g., regional lack of
signal, low signal-to-noise ratio, or crowded hyphal structures. It can handle gaps in disrupted hyphal segmentations and detect and correct atypical hyphal traces for the underlying segmentation.
Similarly, hyphal structures originating from different spores can be dissected by the use of 3D-HyTracer.
Experimental Collaborators
Visualized processing steps of the 3D-HyTracer on simulated hyphae
From left to right, we have a segmented hyphal structure that is then traced using 3D-HyTracer in the middle panel. Not how a part of hyphae is missing at the green arrow in the left-most image, and
this is corrected in the tracing step; see green arrow in the middle panel. In the middle and right-most panel, red arrows, we see how errors are corrected when a mistake in tracing occurred by
taking the most probable branching angles and structures.
Technical details
The 3D-Hytracer creates a low-memory and easily accessible spatial graph in NetworkX format for given hyphal segmentations. The very core functionality of the 3D-HyTracer is based on steepest descent
foreground tracing, a method adapted from the neuron tracing tool Rivulet2. It requires a scanned hyphal structure and the positions of fungal spores. In preparation for the tracing, the 3D-Hytracer
assigns every voxel with a cost value based on the distance transform of the binarized initial image and a distance value, equating values on the shortest path between the voxel and the spore in the
cost map. The underlying shortest-path problem is solved by approximating the Eikonal equation using the marching cubes algorithm. The resulting distance values allow the hyphal tracing even on
poorly segmented images. This is done for every spore individually before the automated correction of hyphal traces. The correction of hyphal traces is done using known properties of hyphal growth,
e.g. the calculation of branching rates and angles, turning angles, hyphal orientation, length, and diameter. The correction step of hyphal tracing can readily be adjusted for the entire graph or
parts of the graph if, for example, a different fungus with different branching patterns is analyzed. In fungal pellets, a collection of clustered spores with a dense core of inseparable hyphae,
fully tracing a hypha back to the corresponding spore might not be possible. To enable an analysis of such samples, the 3D-Hytracer is extended by a module that first distinguishes between pellet
core and periphery and then traces/analyses the hyphal periphery only. | {"url":"https://asb.hki-jena.de/project/3d-hytracer/?query-6d674543-page=12&cst&query-a874f4e5-page=3","timestamp":"2024-11-05T10:45:25Z","content_type":"text/html","content_length":"103702","record_id":"<urn:uuid:6f56be5c-ed59-4001-b618-de94e040732b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00438.warc.gz"} |
A trait has two alleles represented by p and q if p equals .35 what is q? - Answers
What does q equal to in an alleleA trait has two alleles represented by p and q If p equals 0.68 what is q?
This is a Hardy Weindburg situation P represents the percentage of the population that has a dominant allele... now there can only be two alleles one is dominant and one is recessive... q is the
recessive allele This means that p+q=1 and so q has to be equal to 0.32 If you do the square of p (p^2) then that gives you the number of people who are homozygous dominant If you do the square of q
(q^2) then that gives you the number of people who are homozygous recessive If you do 2*(p*q) then that will give you the number of people who are heterozygous Hope this helps... | {"url":"https://math.answers.com/math-and-arithmetic/A_trait_has_two_alleles_represented_by_p_and_q_if_p_equals_.35_what_is_q","timestamp":"2024-11-12T13:02:14Z","content_type":"text/html","content_length":"158265","record_id":"<urn:uuid:ecf793b6-9401-47a5-a890-4fe68430ecbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00707.warc.gz"} |
Wave ripples
From Coastal Wiki
Sea waves shape the bottom and generate different morphological patterns, which are characterized by a wide range of length scales. The ripples are the smallest bedforms but, notwithstanding their
relatively small size, they play a prominent role in many transport processes. Indeed, usually, the flow separates at their crests and vortices are generated which increase momentum transfer,
sediment transport and, in general, mixing phenomena.
Even though the ripples generated by sea waves (wave ripples) appear to be similar to the ripples generated by steady currents/slowly varying tidal currents, they have different characteristics since
they are the result of a different mechanism of formation. What follows concerns wave ripples, which hereinafter are simply named 'ripples'. The ripples generated by steady currents are not
considered in the present article.
Ripple geometry
The geometry of the most common ripples is almost two-dimensional and similar to that of a wave with a crest and a trough (see figure 1). However, the wavelength [math]\lambda[/math] of the ripples
is of the order of magnitude of the amplitude of the fluid displacement oscillations close to the sea bottom and it turns out to be of [math]O [/math] (10 cm) , while the height of the ripples is of
a few centimetres.
The estimate of the geometrical characteristics of the ripples is important for many reasons. First, ripple presence enhances the suspended sediment transport. Indeed, when the steepness of the
ripples becomes larger than approximately 0.1 ^[1], the oscillatory bottom boundary layer separates from the crest of the bedforms and a vortex is generated, twice per wave cycle, on the lee side of
the ripple by the roll-up of the free shear layer shed from the ripple crest. This vortex picks-up the sediments from the bottom and at flow reversal, when it is convected in the opposite direction
by the free stream, it ejects the sediments far from the bottom. Later, the vortex decays and the sediments settle down. Moreover, because of the vortices they shed, the ripples significantly
increase the bottom friction and the dissipation of energy. Finally, ripples may act as a source of nutrients. In fact, for high values of the bottom shear stress, ripples are washed out and the
nutrients are released into the water column at a rate which depends on the time development of their geometrical characteristics.
Empirical formulae to predict ripple wavelength and height are used for practical purposes. The physical quantities which affect the ripple geomety can be assumed to be i) the density [math]\rho[/
math] of the water, ii) the kinematic viscosity of the water [math]\nu[/math], iii) the period [math]T[/math] of the velocity oscillations induced close to the sea bed by the surface waves or,
alternatively, the angular frequency [math]\omega=2 \pi /T[/math], iv) the amplitude [math]U_0[/math] of the velocity oscillations ([math]U_0=a\omega/\sinh (kh)[/math], [math]k=2 \pi/L[/math] being
the wavenumber of the surface wave, [math]a[/math] being its amplitude and [math]h[/math] the local water depth), v) the sediment size [math]d[/math], vi) the density [math]\rho_s[/math] of the
sediment and vii) gravity acceleration [math]g[/math]. By applying dimensional arguments, the wavelength and height of the ripples turn out to depend on four dimensionless parameters, i.e. the
relative density [math]s=\rho_s / \rho[/math], the ratio [math] d /\delta[/math] between the grain size and the viscous length [math]\delta=\sqrt{2 \nu / \omega}[/math], a sediment Reynolds number
[math]R_p = \sqrt{ (s-1)g d^3} / \nu[/math] and a flow Reynolds number [math]R_{\delta} = U_0 \delta / \nu[/math]. Of course, these parameters can be replaced by their combinations and in the
literature it is common to encounter other parameters such as the mobility number [math]\psi= U_0^2 / ((s-1)gd)[/math], the Reynolds number of the sediment defined by [math]R_d = U_0 d / \nu[/math]
and the flow Reynolds number [math]Re = U_0^2 / (\nu \omega) = R^2_\delta / 2[/math].
Both a simple dimensional analysis and idealized models, based on linear stability analyses (see the article Wave ripple formation), show that the geometrical characteristics of ripples can not be
predicted from the knowledge of just one dimensionless parameter. However, the empirical formulae which can be found in the literature use one parameter for simplicity. Hence, the first question to
be addressed is: which is the independent parameter that mainly controls ripple geometry ?
The plethora of predictors using different parameters, along with the significant differences among the predictions they provide, suggest that the problem of predicting ripple characteristics is far
to be definitively solved. An exhaustive description of all the predictors and the advantages/disadvantages of each of them is beyond the aim of the present article. The paper of Nelson et al. ^[2]
describes and discusses some of the predictors commonly used. In the following, to give an idea of these predictors and of their performances, we describe only a few of them.
Ripple wavelength
Laboratory and field data indicate that the wavelengths of the ripples generated by regular surface waves are somewhat different from those generated by irregular waves. However, Soulsby and
Whitehouse ^[3] and Nelson et al. ^[2] proposed a single predictor to be used under both regular and irregular waves and they related the ratio between the ripple wavelength [math]\lambda[/math] and
the amplitude [math]U_0/\omega[/math] of the fluid displacement oscillations close to the bottom to the parameter [math] U_0 / ( \omega d)[/math]:
[math]\Large \frac{\lambda}{U_0/\omega} \normalsize =\Large \frac{1}{ a_1 + b_1 \frac{U_0}{\omega d} \left[1-e^{-\left( c_1 \frac{U_0}{\omega d} \right)^{d_1}} \right]} \normalsize , \qquad(1) [/
where the constants suggested by ^[1] are
[math]a_1=1, \ \ \ \ \ b_1= 1.87 \times 10^{-3}, \ \ \ \ \ c_1=2.0 \times 10^{-4}, \ \ \ \ \ d_1=1.5 , \qquad (2) [/math]
while the constants suggested by ^[2] are
[math]a_1=0.72, \ \ \ \ \ b_1= 2.00 \times 10^{-3}, \ \ \ \ \ c_1=1.57 \times 10^{-4}, \ \ \ \ \ d_1=1.15 \qquad (3) [/math]
Figure 2 shows measured ripple wavelengths along with the values provided by equation (1) with the values of the constants suggested by both Nelson et al. ^[2] and Soulsby and Whitehouse ^[3].
The blue points of figure 2 are extracted from the database collected by ^[2] for regular waves or oscillatory flows. Of course when data for irregular waves (red points) are added by considering the
flow generated by the significant wave (red points), the scatter of the data slightly increases. The experimental data obtained using an oscillating tray are discarded because Miller and Komar ^[4]
concluded that the results of oscillating bed experiments are different from water tunnel, wave channel and field results. Moreover, some of the data of ^[2] are not plotted in figure 2 because they
refer to field data or data obtained in large flume facility where the average characteristics of the surface waves change over time. Even though Nelson et al. ^[2] introduced heuristic criteria to
consider only ripples which attained a morphological equilibrium, it might be that some of the data refer to relic ripples the geometry of which does not depend on the actual characteristics of the
surface waves.
On the other hand, Inman ^[5] suggested that, for an assigned sediment size, the ripple wavelength depends on [math]2 U_0 / \omega[/math], i.e. twice the amplitude of the fluid displacement
oscillations close to the bottom. In particular Inman ^[5], by analysing experimental data, showed that the ripple wavelength is directly proportional to [math]2 U_0 / \omega[/math] up to a critical
value whereupon the crest-to-crest distance becomes inversely proportional to [math]2 U_0 / \omega[/math] and finally it attains a constant value. Figure 3 shows that the values of [math]\lambda/d[/
math] plotted versus [math](2 U_0/\omega)/d[/math], for the data already considered in figure 2. The laboratory measurements dominate the left hand side of the plot while the field measurements
dominate the right hand side. Later, Clifton ^[6] classified ripples as orbital, anorbital and suborbital ripples. Orbital ripples are characterized by a wavelength proportional to the amplitude of
the fluid displacement oscillations
[math]\lambda \simeq 0.65 \Large \frac{2 U_0}{\omega} \normalsize . \qquad (4) [/math]
Anorbital ripples appear for large values of [math]2 U_0/(\omega d)[/math] and their wavelength is almost independent of [math]2 U_0 / \omega[/math] and ranges between [math]400 d[/math] and [math]
600 d[/math], even though the measurements show a large scatter which does not allow to obtain a more precise value. The critical point is to predict what type (orbital/anorbital) of ripple appears
for given hydrodynamic and morphodynamic parameters. Moreover, suborbital ripples exist too, which have a wavelength which depends on both [math]2 U_0/ \omega[/math] and the grain size [math]d[/math]
. This problem is not present if formula (1) is used to predict ripple wavelength, however also figure 2 shows that a significant number of observations (values of [math]U_0/(\omega d)[/math] in the
range [math](10^3,10^4)[/math]) deviate from the trend predicted by relationship equation (1)
Nielsen ^[7] proposed to predict the spacing of the ripples as function of the sediment mobility number
[math]\psi=\Large \frac{U_0^2}{\left( s-1 \right) g d} \normalsize \qquad (5) [/math]
and he suggested
[math]\Large \frac{\lambda}{U_0/\omega} \normalsize = \exp\left(\Large \frac{693-0.37 \ln^8\psi}{1000 +0.75 \ln^7 \psi} \normalsize \right) \qquad(6) [/math]
for ripples observed in the field and
[math]\Large \frac{\lambda}{U_0/\omega} \normalsize = 2.2 - 0.345 \psi^{0.34} \qquad (7) [/math]
for the ripples generated by a regular oscillatory flow. The values of the ripple wavelength predicted by means of (6) and (7) are plotted as function of the mobility number in figure 4, along with
the experimental data.
Ripple height
Once ripple wavelength is estimated, the height [math]\eta[/math] can be obtained from the prediction of ripple steepness. Soulsby and Whitehouse ^[3] suggest to predict the ripple steepness [math]\
eta/\lambda[/math] as function of the parameter [math]U_0/(\omega d)[/math]
[math]\Large \frac{\eta}{\lambda} \normalsize = 0.15 \left[ 1 - \exp\left[ -( \Large \frac{5.0 \times 10^{3} } {U_0/\omega d} )^{3.5} \normalsize \right] \right] . \qquad (8) [/math]
On the other hand, Nelson et al. ^[2] suggest to predict the ripple steepness as function of the ripple wavelength according to the formula
[math]\Large \frac{\eta}{\lambda} \normalsize =0.12 \lambda^{-0.056} , \qquad (9) [/math]
where [math]\lambda[/math] should be in metres. The predictor equation (9) has the disadvantage of predicting the ratio [math]\eta/\lambda[/math] as function of a dimensional quantity. Moreover, to
predict the ripple steepness, it is necessary either to know the wavelength or to predict the value of [math]\lambda[/math] by using equations (1) and (3). Figure 5 shows a comparison between the
results provided by equation (8) and the experimental measurements for regular and and irregular waves.
Ripple symmetry index
Even though the profile of the ripples is almost symmetric with respect to their crest, a small degree of asymmetry is invariably generated by the steady streaming, which is present under a
propagating wave because of nonlinear effects ^[10], and by the difference between the forward fluid velocity, which takes place under the crests of the surface wave, and the backward velocity under
the troughs. Hence, a symmetry index of the ripple can be defined as the ratio between the length [math]l_2[/math] of the gentle (up-current) side to the length [math]l_1[/math] of the steep
(down-current) side of the bottom forms. Figure 6 shows the values of [math]\; (l_2 / l_1)-1 \;[/math] plotted versus the strength of the steady streaming for the experimental data of Inman ^[5],
Tanner ^[11] and Blondeaux et al. ^[9]. In figure 6, the value of the steady velocity component [math]U_s[/math] is estimated by means of the theory of Longuet-Higgins ^[10] ([math]U_s = 3\pi a^2\
omega /[2L\sinh^2(2\pi h/L)][/math]). Hence, the abscissa of figure 6 is equal to [math]3\pi a /[2L\sinh(2\pi h/L)][/math]. As expected, the results plotted in figure 6 indicate that ripples tend to
become more asymmetric as the mass transport velocity increases.
Three-dimensional ripples
Often, the ripple profile is two-dimensional but depending on sediment and flow characteristics, other ripple shapes are observed. For example, figure 7a shows the brick-pattern ripples observed by
Sleath ^[1] during a laboratory experiment. Brick pattern ripples have the crests perpendicular to the direction of the fluid oscillations, as two-dimensional ripples. However, the crests are joined
by equally spaced bridges of smaller amplitude which are parallel to the direction of fluid oscillations and shifted by half a wavelength between adjacent sequences. It follows that the overall
bottom topography resembles a wall made by bricks.
Other three-dimensional ripples do exist and photos can be found in the books of Sleath ^[1] and Allen ^[8].
Friction factor for ripples
In large scale hydrodynamic problems, it is not possible to compute the flow field with the spatial resolution required to compute the flow around each ripple. Hence, these small scale bedforms are
usually modelled as a roughness of the bed of appropriate size. Experimental measurements indicate that the size of the roughness is related to the ripple height. Van Rijn ^[12] suggests that the
roughness size ranges between one and three ripples heights. These values are supported by the data shown in figure 3.6.7 of Nielsen's book ^[13]. However, experimental measurements carried out for
low flow intensities indicates that also the shape of the ripples, and in particular their steepness [math]\eta/\lambda[/math], affects the equivalent roughness size. In the literature, it is
suggested that the equivalent roughness size [math]k_s[/math] can be given values close to [math]c_1 \eta^2/\lambda[/math], where [math]c_1[/math] is a constant. Nielsen ^[13] proposed [math]c_1=8[/
math] while Grant and Madsen ^[14] and Li et al. ^[15] proposed [math]c_1=28[/math] and Van Rijn ^[12] suggested [math]c_1 =20[/math], even though in a previous work Van Rijn ^[16] assumed [math]k_s=
1.1 \eta \left(1-e^{-25 \eta/\lambda} \right)[/math]
For large flow intensities, when the coherent vortices shed by the ripples pick-up a lot of sediments from the bed and put them into suspension, Nielsen ^[13] suggests to estimate the equivalent
roughness by means of
[math] k_s= 8 \Large \frac{\eta^2}{\lambda} \normalsize +170 d \sqrt{\theta -0.05} , [/math]
where [math]\theta=\tau / ((\rho_s-\rho) g d)[/math] is the Shields parameter evaluated by using the skin friction [math]\tau[/math].
Grain-sorting over ripples
In the field, the sediment is often a mixture of particles having different sizes and ripples give rise to sorting phenomena. Foti and Blondeaux ^[17] made laboratory experiments with a sediment
mixture characterized by a bimodal grain size distribution and observed that the coarser fraction oscillated around the crests of the bedforms while the fine fraction tended to move towards the
troughs (see figure 8). Moreover, they found that the sorting phenomena affect the dynamics of the ripples. Indeed, the ripples generated by a well sorted sediment turn out to be shorter than those
generated by a poorly sorted sediment.
Related articles
The main authors of this article are Paolo Blondeaux and Giovanna Vittori
Please note that others may also have edited the contents of this article.
Citation: Paolo Blondeaux; Giovanna Vittori; (2024): Wave ripples. Available from http://www.coastalwiki.org/wiki/Wave_ripples [accessed on 2-11-2024]
• For other articles by this author see Category:Articles by Paolo Blondeaux
• For other articles by this author see Category:Articles by Giovanna Vittori
• For an overview of contributions by this author see [[Special:Contributions/{{{AuthorName1}}}]]
• For an overview of contributions by this author see [[Special:Contributions/{{{AuthorName2}}}]] | {"url":"https://www.coastalwiki.org/wiki/Wave_ripples","timestamp":"2024-11-02T21:22:35Z","content_type":"text/html","content_length":"60750","record_id":"<urn:uuid:d91a0696-5949-4cc4-87a1-bdde5185533b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00585.warc.gz"} |
A computer program to evaluate the NVM propagator for rigid asymmetric tops for use in path integral simulations of rigid bodies
A computer program to evaluate the NVM propagator for rigid asymmetric tops for use in path integral simulations of rigid bodies
Published: 1 March 2013| Version 1 | DOI: 10.17632/d6j6nxgch7.1
Carl McBride, Eva G. Noya, Carlos Vega
Abstract Here we provide FORTRAN source code to facilitate the calculation of the “Noya–Vega–McBride” (NVM) rotational propagator for asymmetric tops [E.G. Noya, C. Vega, C. McBride, J. Chem. Phys.
134 (2011) 054117] for a given value of P T and A , B and C , where P is the number of beads, T is the temperature, and A , B and C are the rotational constants for the system in question. The
resulting NVM propagator calculated by the code provided can then be used to obtain the quantum rotational energy d... Title of program: NVM Catalogue Id: AEOA_v1_0 Nature of problem Calculation of
the NVM rotational propagator Versions of this program held in the CPC repository in Mendeley Data AEOA_v1_0; NVM; 10.1016/j.cpc.2012.10.025 This program has been imported from the CPC Program
Library held at Queen's University Belfast (1969-2018)
Physical Chemistry, Molecular Physics, Computational Physics | {"url":"https://data.mendeley.com/datasets/d6j6nxgch7/1","timestamp":"2024-11-14T14:22:39Z","content_type":"text/html","content_length":"103847","record_id":"<urn:uuid:b1773335-0976-4f23-bbb0-6af730e6f3c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00050.warc.gz"} |
Math Story : Division By One And Division By Itself Property - Fun2Do LabsMath Story : Division By One And Division By Itself Property
Math Story : Division By One And Division By Itself Property
How Will Cirha Protect The Puppies?
A heavy storm hit Samper Town last night. It caused severe damage, and everybody is sad.
“This is heartbreaking”, says Cirha looking at the people and their pain. Suddenly she hears a whining sound. As she turns around to find out, she sees some little puppies crying. They look
miserable. Cirha’s heart sinks seeing them.
“At least I have a place to stay and protect myself. These little puppies do not even have anything. I should do something to help them”, she thinks. After a while, she decides to protect and feed
them. But how will she do this?
She decides to take them home. Cirha is so generous indeed. “Haha! First, protect yourself, then think about protecting the puppies. Haha!” says a strange voice. Who do you think it is?
It’s Asquarho. Taking the advantage of the current situation, he has come to trouble people and make them even sadder. Immediately, he runs to attack Cirha. Will she fight him back?
“You thought troubling me was this easy? Here I come”, she says and runs to attack him back. Bang, boom, crash! She fights him bravely.
Asquarho runs to save himself. Cirha quickly checks on the puppies and walks toward her home. She is not only generous but also brave and strong. Do you agree?
Soon Cirha reaches her home. “They must be starving. Let me feed them all”, she says. Quickly she goes to her kitchen and gets some buns.
“I have 9 buns, and there are 9 puppies altogether. How many buns will each get when I divide them equally?” she thinks. Can you help her?
“I can divide 9 from 9 only once. That means each puppy will get one bun to eat. Nice!”, she says. To crosscheck, she keeps one bun near every puppy. “Perfect! Dividing 9 buns among the 9 puppies
each gets one bun”, she says. The puppies are happily eating.
After eating, some of the puppies fall asleep, and some are up. “Playing might help them feel better”, she thinks. She quickly goes and gets some balls. “There are 5 puppies awake now, and I have
5 balls. How many balls will each get when I divide them equally?” she questions.
We can divide 5 by 5, only once, “When I divide 5 balls among the 5 puppies, each gets one ball. Amazing!” she says.
“9 divided by 9 is equal to 1. 5 divided by 5 is also equal to 1. Does that mean, that whenever we divide the number by itself, the answer is 1? Oh yes! It is an amazing trick”, she says. Cirha is
delighted to have discovered something new today.
All the puppies are tired now. They also fall asleep. Cirha notices that some of them are shivering. “Oh no! They must be feeling cold. Let me cover them with the blankets”, she says. She runs
upstairs to find some blankets.
As she enters her room, she is shocked. “Oh no! So much water. Everything is drenched! What will I do now?” she worries. How will she take care of herself and the puppies now?
Cirha left the window open last night! All the water from the storm had come inside. But she decides to not give up. “Cirha! Where are you?” she hears a voice suddenly. Has someone come to visit
It is Triho. He has come to see if Cirha is safe and doing fine after the dreadful storm. Not able to see her down, he decides to check upstairs. He finds Cirha sad and worried. She explains
everything to him. ”Do not worry, I will get the blankets for the puppies. Together, we will also clean up this area. Relax!” he says.
Triho is back with the blankets. “There are 3 blankets. If I give 1 to each of the puppies, will all of them get the blanket?” he questions. To find the answer, Cirha suggests he use division.
“Hurray! Each gets a blanket”, he says.
The puppies feel better and are sleeping peacefully. Triho and Cirha decide to clean the upstairs now.
“Phew! This looks great now”, says Triho. Cirha is elated to see her neat and tidy room. She thanks Triho and decides to treat him with something. “Ice cream time!” she says excitedly and
rushes to get some ice cream.
“I have 2 chocolate ice creams. If I give 1 to each of us, will both of us get equal ice creams?” she questions. Can you guess if each got equal ice creams?
Of course Yes! When 2 is divided by 1, the answer is 2. So both of them get ice cream. But wait! It looks like Triho has realized something.“When I divided 3 by 1, the answer was 3. Now when you
divided 2 by 1, the answer is 2. That means when we divide a number by 1, the result is number itself. Wow!” he says.
Two discoveries on division, safe puppies and happy Cirha and Triho. What a wonderful day it has been! No act of kindness, no matter how small or big, is ever wasted! Do you agree?
We Learnt That…
• When we divide a number by itself, the answer is always 1.
• When we divide a number by 1, the answer is the number itself.
• No act of kindness, no matter how small or big, is ever wasted.
Let’s Discuss
• Why was everybody sad?
• “At least I have a place to stay and protect myself. These little puppies do not even have anything.” why did I say so?
• What happens when we divide a number by itself?
• What happens when we divide a number by 1?
• “No act of kindness, no matter how small or big, is ever wasted.” Do you agree?
Please refer this guide by Fun2Do Labs for teaching division properties to kids : | {"url":"https://www.fun2dolabs.com/math-story-division-properties/how-will-cirha-protect-the-puppies/","timestamp":"2024-11-13T17:37:14Z","content_type":"text/html","content_length":"50804","record_id":"<urn:uuid:753bd767-7a3d-455b-a189-e52dd625167f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00681.warc.gz"} |
How to Perform Statistical Analysis In MATLAB?
Performing statistical analysis in MATLAB involves several key steps and functions. Here is an overview of the process:
1. Import Data: Start by importing your data into MATLAB. You can read data from various file formats such as CSV, Excel, or text files using functions like readtable or xlsread.
2. Data Preprocessing: Clean and preprocess your data as per your requirements. This step may involve removing missing values, outliers, or transforming data for better analysis.
3. Descriptive Statistics: Calculate basic descriptive statistics to gain insights about your data. MATLAB provides functions like mean, median, var, std to compute summary statistics.
4. Statistical Tests: MATLAB offers a wide range of statistical tests to perform hypothesis testing or compare different groups of data. Some commonly used tests include t-tests (ttest2 for two
samples, ttest for one sample), ANOVA (anova1, anova2, anova, etc.), chi-square test (chi2test), and more.
5. Regression Analysis: MATLAB has a comprehensive set of functions to perform linear regression and other regression analyses. You can use the fitlm, regress, or polyfit functions to model
relationships between variables and assess their significance.
6. Multivariate Analysis: MATLAB supports advanced multivariate statistical techniques such as principal component analysis (PCA), factor analysis, cluster analysis, and discriminant analysis. These
methods are useful for dimensionality reduction, pattern recognition, and exploring relationships between variables.
7. Data Visualization: MATLAB offers powerful visualization tools to create plots and graphs for better understanding and presentation of your statistical analysis results. Functions like plot,
histogram, boxplot, scatter, and heatmap help in visualizing data distributions, relationships, and patterns.
8. Custom Analysis: MATLAB is a flexible environment that allows you to implement custom statistical analyses beyond built-in functions. You can write your own scripts or functions using MATLAB's
programming capabilities, making it suitable for a wide range of statistical research and analysis tasks.
By following these steps and utilizing MATLAB's extensive statistical toolbox, you can conduct in-depth statistical analyses and gain valuable insights from your data.
How to conduct analysis of variance (ANOVA) in MATLAB?
To conduct analysis of variance (ANOVA) in MATLAB, you can use the built-in function called anova1. Here is the general syntax and steps to perform the ANOVA analysis:
1 p = anova1(data, groups);
1. Prepare your data: Make sure you have your data organized in a matrix or vector form, where each column represents a different group or treatment.
2. Call the anova1 function: Pass your data and specify the groups as input arguments to the anova1 function. This function performs a one-way ANOVA analysis.
3. Get the ANOVA table: The output of the anova1 function is an ANOVA table, which contains various statistics such as the sum of squares, degrees of freedom, and p-values.
4. Interpret the results: The p-value in the ANOVA table represents the significance level of the differences between the groups. If the p-value is less than a certain threshold (e.g., 0.05), it
indicates that there are significant differences between the groups.
1 % Generate example data
2 group1 = [1, 2, 3, 4, 5];
3 group2 = [2, 4, 6, 8, 10];
4 group3 = [3, 6, 9, 12, 15];
5 data = [group1', group2', group3'];
7 % Perform ANOVA analysis
8 p = anova1(data);
10 % Interpret the results
11 if p < 0.05
12 disp('There are significant differences between the groups.');
13 else
14 disp('There are no significant differences between the groups.');
15 end
In this example, the ANOVA analysis is performed on three groups (group1, group2, and group3) with five data points in each group. The p-value is then checked to determine if there are significant
differences between the groups.
What is the role of regression analysis in statistical modeling?
Regression analysis plays a crucial role in statistical modeling as it helps in understanding the relationship between variables. It allows researchers to examine how a dependent variable (outcome
variable) is affected by one or more independent variables (predictor variables).
The role of regression analysis in statistical modeling includes:
1. Predictive Modeling: Regression analysis helps in predicting future values or outcomes based on historical data. By identifying relationships between variables, it enables researchers to estimate
the value of the dependent variable for different values of the independent variables.
2. Hypothesis Testing: Regression analysis provides a framework for testing hypotheses and making statistical inferences. Researchers can test if a particular independent variable has a
statistically significant impact on the dependent variable, helping in evaluating the significance of variables in the model.
3. Model Fitting and Selection: Regression analysis helps in finding the best-fitting model by evaluating the goodness of fit. Various methods like Ordinary Least Squares, Ridge regression, or Lasso
regression can be employed to select the most appropriate model that explains the relationship between variables.
4. Understanding Variable Relationships: Regression analysis allows researchers to quantify the relationship between variables. It helps in identifying the strength and direction of the
relationships, whether they are positive, negative, or non-linear.
5. Control for Confounding Factors: Regression analysis enables researchers to control for confounding factors by including additional variables in the model. This helps in isolating the effect of a
specific independent variable on the dependent variable, removing the influence of other variables.
6. Assumptions and Robustness Checks: Regression analysis requires certain assumptions about the data, such as linearity, independence, and homoscedasticity. By conducting various diagnostics and
robustness checks, researchers can assess whether these assumptions are met and identify any potential issues in the model.
Overall, regression analysis is an essential tool in statistical modeling as it helps in understanding, predicting, and interpreting relationships between variables, providing valuable insights for
decision-making, policy formulation, and scientific research.
What is the concept of probability density functions (PDF) in MATLAB?
In MATLAB, a Probability Density Function (PDF) represents the probability distribution of a continuous random variable. It provides the relative likelihood of different outcomes occurring within a
specified range. The concept of PDF in MATLAB is implemented using various built-in functions.
The primary function used for PDF in MATLAB is "pdf," which returns the probability density values for a given distribution at specified points. This function takes two input arguments: the
distribution object and the values where the PDF needs to be evaluated.
For example, to obtain the PDF values of a normal distribution with mean 0 and standard deviation 1 at points ranging from -3 to 3, the following code can be used:
1 x = -3:0.1:3; % Points at which PDF needs to be evaluated
2 mu = 0; % Mean of the normal distribution
3 sigma = 1; % Standard deviation of the normal distribution
5 pdf_values = normpdf(x, mu, sigma); % Compute PDF values
7 plot(x, pdf_values); % Plot PDF
8 xlabel('x');
9 ylabel('PDF');
10 title('Normal Distribution PDF');
This code calculates the PDF values for the normal distribution using the "normpdf" function and then plots the PDF using the "plot" function.
Other function-specific PDF functions are also available for different distributions, such as "exppdf" for the exponential distribution, "unifpdf" for the uniform distribution, "chi2pdf" for the
chi-square distribution, etc.
Probability density functions are crucial for various statistical analyses, hypothesis testing, generating random numbers from specific distributions, and probability calculations in MATLAB.
What is the process of conducting principle component analysis (PCA) in MATLAB?
To conduct Principle Component Analysis (PCA) in MATLAB, you can follow these steps:
1. Prepare your data: Make sure you have your dataset ready. The data should be arranged as a table or matrix, where each row represents a data point, and each column represents a feature or
2. Load the data in MATLAB: Import your data into MATLAB using the appropriate function such as csvread or xlsread.
3. Standardize the data: Standardizing the data is important for PCA since it scales the variables to the same range. You can use the zscore function in MATLAB to standardize the data.
4. Compute the covariance matrix: Calculate the covariance matrix of the standardized data using the cov function in MATLAB. The covariance matrix indicates the relationships between the variables.
5. Compute the eigenvectors and eigenvalues: Use the eig function in MATLAB to calculate the eigenvalues and eigenvectors of the covariance matrix. The eigenvectors represent the principal
components, and the eigenvalues indicate the amount of variance explained by each principal component.
6. Sort the eigenvalues and eigenvectors: Sort the eigenvalues and eigenvectors in descending order based on the magnitude of the eigenvalues. This will rank the principal components in terms of
their importance.
7. Select the number of principal components: Choose the number of principal components to retain based on a certain threshold or the amount of variance you want to explain. You can look at the
cumulative explained variance or plot the scree plot to aid in the decision.
8. Compute the transformed data: Transform the standardized data into the space defined by the selected principal components by multiplying it with the corresponding eigenvectors.
The above steps summarize the basic process of conducting PCA in MATLAB. Additionally, you can perform various visualizations and analyses on the transformed data to gain insights into the underlying
structure of your dataset.
How to perform resampling methods like bootstrapping in MATLAB?
To perform bootstrapping in MATLAB, you can follow these steps:
1. Load or generate your dataset. Let's assume you have a vector or matrix data with your data.
2. Choose the number of iterations for your bootstrap. The higher the number, the more accurate the result, but it will also take longer to compute.
3. Set up a loop that will iterate the chosen number of times:
1 numIterations = 1000; % Set the number of iterations
2 n = length(data); % Get the length of your data
4 for i = 1:numIterations
5 % Perform resampling
6 end
1. Inside the loop, generate a random sample of size n (with replacement) from your data. You can use datasample or randsample functions in MATLAB:
1 bootstrapSample = datasample(data, n); % Replace 'datasample' with 'randsample' if using an older version of MATLAB
1. Perform the desired analysis on the bootstrap sample. This can involve any statistical analysis, such as calculating the mean, median, standard deviation, or any other estimator of interest. For
example, to calculate the mean:
1 bootstrapMean(i) = mean(bootstrapSample);
1. After the loop, you will have an array (or matrix if performing multiple analyses) containing the bootstrap estimates. You can then use these estimates to compute confidence intervals, hypothesis
tests, or other statistical measures.
Note: The above steps provide a basic outline for performing bootstrapping in MATLAB. Depending on your specific analysis and requirements, you might need to adapt the code accordingly. | {"url":"https://infervour.com/blog/how-to-perform-statistical-analysis-in-matlab","timestamp":"2024-11-15T01:26:14Z","content_type":"text/html","content_length":"394393","record_id":"<urn:uuid:7e44954f-6f60-4b18-9160-87f6a093de41>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00846.warc.gz"} |
pdg-control Training Problem II update
Basically we’ve done some simple stuff and now would be a good time to think about tools for some heavy-lifting computational challenges standing in the way. Michael & Jim have been a great help on
this, and Jake has also helped with theoretical underpinnings.
Statement of problem
We consider the impact of uncertainty in the harvesting of fisheries which involve alternative stable state dynamics.
Current Findings
• Adding uncertainty in current state means optimal escapement isn’t the optimal solution (contrary to Reed model) (Sethi et. al. 2005)
• We are considering uncertainty/variability entering at three levels: population dynamics (next yr’s stock, Reed case), stock assessment (this yr’s stock), and implementation of quotas, in the
context of alternative stable states.
• When the parameterization of these uncertainties is correctly known, the optimal solution surprisingly doesn’t face many more sudden crashes than the unharvested dynamics. (Casino “paradox” –
income generated by a casino is highly predictable since the rules of uncertainty are well known).
• Errors in the parameterization of the uncertainty or the biological parameters can have dramatic effects in the allee threshold model
Questions for further study
• Is the optimal strategy risk adverse?
• Can we add parameter uncertainty / Bayesian learning about parameters?
• In particular, can we implement the case in which stock is not assessed directly but estimated from harvest and knowledge of harvest effort, under parameters we learn about?
Technical capacity and challenges
• We’ve developed anR package for stochastic dynamic programming that allows a user to quickly experiment with different levels & types of uncertainty on different biological and economic models
from a library of options and create visualizations of the solutions over an ensemble of realizations.
• Curse of dimensionality is a major challenge to go beyond the 1D training problem into the 3D parrotfish model or the learning model. We investigated Heuristic Sampling Nicol & Chadès, 2011 as a
possible solution but this far this seems to scale rather poorly with the size of the control space.
• Sethi G, Costello C, Fisher A, Hanemann M and Karp L (2005). “Fishery Management Under Multiple Uncertainty.” Journal of Environmental Economics And Management, 50. ISSN 00950696, https://
• Beyond Stochastic Dynamic Programming: A Heuristic Sampling Method For Optimizing Conservation Decisions in Very Large State Spaces, Sam Nicol, Iadine Chadès, (2011) Methods in Ecology And
Evolution, 2 10.1111/j.2041-210X.2010.00069.x | {"url":"https://www.carlboettiger.info/2012/01/24/pdg-control-training-problem-ii-update.html","timestamp":"2024-11-10T17:14:53Z","content_type":"text/html","content_length":"16187","record_id":"<urn:uuid:b34bbb52-e20a-4d6e-83a4-788a10ec7047>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00550.warc.gz"} |
Convert Watt Hours to Newton Meters (whr to Nm)
Watt Hours to Newton Meters Converter
Watt Hours
Watt Hours
= 0
Newton Meters
Watt Hours To Newton Meters
Conversion Table
Unit Conversion Value
1 Watt Hours 0.00 Newton Meters
2 Watt Hours 0.00 Newton Meters
5 Watt Hours 0.00 Newton Meters
10 Watt Hours 0.00 Newton Meters
20 Watt Hours 0.01 Newton Meters
50 Watt Hours 0.01 Newton Meters
100 Watt Hours 0.03 Newton Meters
200 Watt Hours 0.06 Newton Meters
500 Watt Hours 0.14 Newton Meters
1000 Watt Hours 0.28 Newton Meters
About Watt Hours
Understanding Watt Hours: A Comprehensive Guide
Introduction to Watt Hours
Watt hours (Wh) are a unit of energy commonly used to quantify the amount of electricity consumed or produced over time. It is an essential concept in the fields of electrical engineering, energy
management, and sustainability. The watt hour measures how much power (in watts) is used over a period of one hour. This unit is crucial for determining the efficiency and capacity of batteries,
solar panels, and various electronic devices.
Defining Key Concepts
To grasp the concept of watt hours, it’s important to understand the following foundational terms:
• Watt (W): A watt is a unit of power that measures the rate at which energy is used or generated. One watt is equivalent to one joule of energy transferred per second. Mathematically, it can be
expressed as:
[ \text{Power (W)} = \frac{\text{Energy (J)}}{\text{Time (s)}} ]
• Joule (J): A joule is a derived unit of energy in the International System of Units (SI). It represents the energy transferred when a force of one newton moves an object one meter.
• Time (t): In terms of watt hours, time is measured in hours, highlighting the duration over which power consumption occurs.
The Calculation of Watt Hours
The formula to calculate watt hours is straightforward:
[ \text{Watt Hours (Wh)} = \text{Power (W)} \times \text{Time (h)} ]
For example, if a device consumes 100 watts of power and operates for 3 hours, the total energy consumed can be calculated as follows:
[ \text{Wh} = 100 , \text{W} \times 3 , \text{h} = 300 , \text{Wh} ]
This calculation helps consumers and businesses understand their energy usage more effectively.
Applications of Watt Hours
1. Battery Operations:
□ Batteries are often rated in watt hours to indicate how much energy they can store and deliver. For instance, a 500 Wh battery can deliver 500 watts of power for one hour, or 250 watts for
two hours. This measurement helps in comparing the performance and longevity of different batteries in devices like smartphones, laptops, and electric vehicles.
2. Solar Energy Systems:
□ In solar energy applications, watt hours are used to gauge the energy output of solar panels. For example, a solar panel rated at 300 watts generating power for five hours will produce 1500
Wh, or 1.5 kWh, indicating the amount of energy harvested from sunlight. This metric is vital for evaluating the efficiency of solar energy systems and understanding household energy needs.
3. Home Energy Consumption:
□ Home appliances are often rated by their wattage, and calculating their total consumption in watt hours helps homeowners manage their energy bills. For instance, if an electric heater
operates at 1500 watts for four hours, it consumes 6000 Wh, or 6 kWh. Monitoring these figures can lead to more informed decisions about energy use and conservation strategies.
4. Electric Vehicles (EVs):
□ In the context of electric vehicles, watt hours are critical in determining the range and efficiency of a vehicle. The battery capacity in electric vehicles is often expressed in kilowatt
hours (kWh), where 1 kWh equals 1000 Wh. Understanding how many watt hours an EV uses per mile can help potential buyers assess its efficiency and suitability for their needs.
Converting Watt Hours
It might be necessary to convert watt hours into other energy units depending on the application. Here are some common conversions:
• Kilowatt Hours (kWh):
Since 1 kWh = 1000 Wh, to convert watt hours to kilowatt hours, simply divide by 1000:
[ \text{kWh} = \frac{\text{Wh}}{1000} ]
• Joules:
Using the conversion factor (1 Wh = 3600 J since there are 3600 seconds in one hour):
[ \text{J} = \text{Wh} \times 3600 ]
Importance of Understanding Watt Hours
1. Energy Efficiency:
□ Understanding watt hours can empower individuals and organizations to make better choices regarding energy consumption, leading to reduced bills and a lower carbon footprint. By tracking
energy use in watt hours, users can identify which appliances are energy hogs and seek out more efficient alternatives.
2. Renewable Energy Integration:
□ As society moves towards renewable energy sources, comprehending how watt hours work aids in optimizing the use of resources such as wind and solar. It enables better planning for energy
storage solutions and anticipating energy needs based on available sunlight or wind conditions.
3. Informed Purchasing Decisions:
□ Consumers benefit from understanding watt hours when purchasing electronics and appliances. Devices with lower energy consumption ratings in watt hours may be more appealing due to their
long-term savings on energy costs.
4. Environmental Sustainability:
□ Reducing energy consumption directly impacts global efforts to fight climate change. By monitoring watt hours and implementing conservation practices, households and businesses contribute to
a more sustainable future.
Watt hours are a fundamental concept in energy management that reflect how we consume and produce energy over time. From battery usage to solar energy systems and everyday appliances, understanding
watt hours enables informed decisions that promote energy efficiency and sustainability. As technology evolves and our reliance on electricity continues to grow, grasping the nuances of watt hours
will be increasingly essential for maximizing our energy use while minimizing negative environmental impacts. Whether you are a consumer, engineer, or policymaker, the significance of watt hours
cannot be overstated, encapsulating both the challenges and opportunities in today’s energy landscape.
About Newton Meters
Newton Meters: Understanding the Unit of Torque
Introduction to Newton Meters
The Newton meter (Nm) is the SI unit of torque, which is a measure of the rotational force applied about an axis. Torque is crucial in various fields such as physics, engineering, and mechanics, as
it describes how a force can cause an object to rotate. The concept of torque is fundamental in understanding how machines work, from simple tools to complex systems like engines and robotics.
Definition of Torque
Torque ((\tau)) is calculated as the product of the force ((F)) applied and the distance ((r)) from the point of rotation (or pivot point) at which the force is applied. Mathematically, this
relationship is expressed as:
[ \tau = r \times F ]
• (\tau) is the torque measured in Newton meters (Nm).
• (r) is the distance from the pivot point to where the force is applied, measured in meters (m).
• (F) is the applied force measured in Newtons (N).
Breakdown of the Unit
1. Newton: The newton is the standard unit of force in the International System of Units (SI). It is defined as the force required to accelerate a mass of one kilogram at the rate of one meter per
second squared ((1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2)).
2. Meter: The meter is the base unit of length in the SI system. It is defined as the distance light travels in a vacuum in (1/299,792,458) seconds.
Therefore, when we say "newton meter," we are referring to the amount of torque produced by applying a force of one newton at a perpendicular distance of one meter from the axis of rotation.
Applications of Newton Meters
1. Mechanical Systems
In mechanical systems, torque plays a critical role in the functioning of machines. For instance, in vehicles, the torque generated by the engine is transmitted to the wheels via the drivetrain,
influencing the vehicle's acceleration and ability to perform work against resistance (such as climbing a hill).
2. Engineering Design
Engineers often need to calculate the torque requirements for screws, bolts, and other fasteners to ensure that connections are secure without damaging materials. Specifications for components will
typically include a torque range, usually specified in Newton meters, which should be followed during assembly.
3. Funicular Forces
In applications such as cranes or lifting equipment, torque calculations are essential to determine the load capacity and the stability of the structure while lifting loads. This ensures safety and
efficiency in operations.
4. Sports and Fitness
In sports, understanding torque can enhance performance. For example, athletes may study torque in relation to their movements to improve techniques in activities like throwing, swinging, or jumping.
Calculating Torque Examples
To understand how to calculate torque in practical scenarios, let’s consider a couple of examples:
Example 1: Simple Lever
Suppose you have a lever that is 2 meters long. If you apply a force of 10 Newtons at the end of the lever, the torque exerted about the pivot point is calculated as follows:
[ \tau = r \times F = 2 \text{ m} \times 10 \text{ N} = 20 \text{ Nm} ]
This means that a torque of 20 Newton meters is being applied at the pivot.
Example 2: Wrench Application
Imagine using a wrench to tighten a bolt. If the length of the wrench is 0.3 meters (30 centimeters) and you apply a force of 50 Newtons perpendicular to the wrench, the torque is:
[ \tau = 0.3 \text{ m} \times 50 \text{ N} = 15 \text{ Nm} ]
In this case, you are applying a torque of 15 Newton meters to the bolt.
Significance of Direction
Torque has both magnitude and direction, making it a vector quantity. The direction is determined by the right-hand rule: if you curl the fingers of your right hand in the direction of the force
applied, your thumb points in the direction of the torque vector. This directional aspect is vital in mechanical systems where multiple torques may interact.
Measurement Tools
Torque can be measured using various tools, including:
1. Torque Wrenches: These tools allow you to apply a specific torque to a fastener. They often have a scale marked in Newton meters to help users achieve the desired torque.
2. Dynamometers: Used in more advanced applications, dynamometers can measure torque output from engines and motors.
3. Torque Sensors: These electronic devices can measure torque in real-time in various industrial applications, providing feedback for automated systems.
The Newton meter is a fundamental unit in mechanical physics, representing the concept of torque and its importance in the real world. From the performance of vehicles to the safety of structures,
understanding and calculating torque in Newton meters is crucial for engineers, designers, and technicians. By comprehensively studying torque and its applications, one gains insights into the forces
that govern motion and stability in numerous systems. With ongoing advancements in technology and engineering practices, the relevance of the Newton meter remains pivotal in the continual evolution
of mechanical design and application.
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Convert whr to NmConvert whr to Newton MetersConvert Watt Hours to Newton MetersConvert Watt Hours to Nm | {"url":"https://www.internettoolwizard.com/convert-units/power/whr/Nm","timestamp":"2024-11-02T21:47:56Z","content_type":"text/html","content_length":"93411","record_id":"<urn:uuid:3e2ea349-3e05-4f57-a6f5-0e9981f6da28>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00846.warc.gz"} |
The iron core of the inductor should be removed
To reduce the resonant frequency in an LCR series circuit with a generator
The generator frequency should be reduced
Another capacitor should be added in parallel to the first
The iron core of the inductor should be removed
Dielectric in the capacitor should be removed
The correct Answer is:B
We know that resonant frequency in an L-C-R circuit is given by
Now to reduce V0 either we can increase L or we can increase C
To increase capacitance, we must connect another
capacitor parallel to the first. | {"url":"https://www.doubtnut.com/qna/649445898","timestamp":"2024-11-02T08:20:02Z","content_type":"text/html","content_length":"243817","record_id":"<urn:uuid:593a976b-63c2-45e7-967b-b7239a7b5be9>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00278.warc.gz"} |
Activation functions in neural networks [Updated 2024] | SuperAnnotate
The astounding success of artificial neural networks can be attributed in part to the fact that they are able to estimate complex, non-linear functions that are often present in real-world data. This
is achieved through activation functions, which introduce non-linearity into the neural networks and facilitate them to find a better fit for the input data. Thus, it can be said that activation
functions are crucial for the effectiveness of neural networks.
This article will try to provide a relatively comprehensive, but not overly technical overview of activation functions in neural networks. By the end, you'll have a firm grasp of the following:
• What activation functions are and why to use them
• How activation functions help neural networks learn
• Why activation functions need to be differentiable
• What the most widely used activation functions are, their pros and cons
• How to choose an activation function when training a neural network
What is activation function?
Activation function in neural networks is a mathematical function that determines the output of a neuron based on its input. As the name suggests, it is some kind of function that should "activate"
the neuron. Whether it will be convolutional neural networks or recurrent neural networks, the activation function decides how to proceed. Just as neurons in the brain receive signals from the body
and make decisions on how to process them, neurons in artificial neural networks work in a similar manner. They act as transfer functions, receiving input values and producing corresponding output
How do activation functions work?
Before discussing modern and widely used activation functions, it's a good idea to get a solid understanding of how they work in a neural network. Regardless of network architecture, an activation
function will take the values generated by a given network layer (in a fully connected network, this would be the sum of weights and biases) and apply a certain transformation to these values to map
them to a specific range.
Here's a useful illustration of the role an activation function plays in a neural network.
After taking a weighted sum of the inputs plus the bias (W₁X₁ + W₂*X₂ + … + W𝚗*X𝚗+ b), we pass this value to some activation function ⨍, which then gives us the output of the given neuron. In this
case, each of the Xᵢ values is the output of a neuron from the previous layer, while Wᵢ is our neuron's weights assigned to each input Xᵢ.
Why use an activation function
While not all activation functions are non-linear, the overwhelming majority are, and for a good reason. Nonlinear activation functions help introduce additional complexity into neural networks and
facilitate them to “learn” to approximate a much larger swathe of functions. If not for nonlinear activation functions, neural networks would only be able to learn linear and affine functions since
the layers would be linearly dependent on each other and would just comprise a glorified affine function.
Another important aspect of activation functions is that they allow us to map an input stream of unknown distribution and scale it to a known one (e.g., the sigmoid function maps any input to a value
between 0 and 1). This helps stabilize the training of neural networks and also helps map the values to our desired output in the output layer (for non-regression tasks).
Why should an activation function be differentiable
The most important feature that the activation function should have is to be differentiable. Artificial neural networks learn using an algorithm called backpropagation. This algorithm essentially
uses the model's incorrect predictions to adjust the network in a way that will make it less incorrect, thereby improving the network's predictive capabilities. This is done through differentiation.
Activation functions and their derivatives. Image source
Therefore, in order for a network to be trainable, all its elements need to be differentiable, including the activation function. However, it doesn't mean that the neural network will do its best
during the training. There are more barriers that need to be passed to reach the goal. The differentiability creates other problems, especially in deep learning, such are "vanishing" and "exploding"
In the "vanishing" gradient case from one hidden layer to another, the values of the gradient can be smaller and smaller that eventually become zero. The "exploding" gradient is the other side of the
problem where from one hidden layer to another the values become bigger and bigger and reach infinity.
Simple activation functions
With this in mind, what does a real-world activation function look like? Perhaps the simplest activation function one could think of is the identity activation function in which case the input and
output values will be the same.
Using this linear activation function doesn't add any complexity to the neural networks and neural networks become similar to the linear regression model.
Of course, this wouldn't be of much use as it literally doesn't do anything, and so we would still face the aforementioned problem of an unpredictable distribution of values, destabilizing the
training of our deep neural networks.
Step function
A somewhat more effective activation function than linear activation, but still a super simple way to tackle this problem is the binary step function:
Binary step function
As one can see, all the step activation function does is take the input, and assign it to either 0 or 1, depending on whether the input is larger or smaller than 0. While this fixes the issue of
having a more predictable distribution of values, it's almost never used, because you lose a lot of information by squishing all nuance out of the neural network.
Non-linear activation functions
Now that we have a solid grasp of what activation functions do, let's discuss some non-linear activation functions that are actually used in practice. There has been a hefty amount of research
regarding non-linear activation functions in recent years, which has introduced new and improved activation functions and, thus, affected the popularity of old ones. However, tried-and-true
activation functions are still used often and have their place.
Sigmoid / Logistic activation function
The sigmoid activation function or logistic activation function is a very popular non-linear activation function that maps input data to the output range (0, 1). Unlike the step function, the sigmoid
function doesn't just output 0 or 1, but instead numbers in that range (not including 0 and 1 themselves). Here's an illustration of the sigmoid activation function and its first derivative:
Sigmoid function (red) and first derivative(blue)
In comparison to the linear function or binary step function, the derivative of the sigmoid is not a constant value. The derivative is a well-defined function and will be possible to pass value, for
any given value. However, while the sigmoid activation function is better than the ones discussed before and does have its place (especially in tasks like binary classification), it has somewhat
major drawbacks. The activation function is close to zero when the input values are too big or too small, that is the problem that we described above as a "vanishing" gradient problem.
All these saturated neurons “kill” the gradients. Another drawback is that since the range is (0, 1), the output of the sigmoid activation function is not 0-centered, which also causes problems
during backpropagation (a detailed discussion of these phenomena is out of this article's scope, though). Finally, exponential functions are a bit expensive computationally, which can slow down the
neural network training process.
Softmax function
The softmax activation function is similar to the sigmoid function. It is common to use on output layer to represent output values as probabilities. The mathematical expression is presented below.
The expression is mathematically defined for all x in the range (-inf to inf), but computationally it has some limitations. Exponents can be very large numbers and cause some problems during
calculation. Due to that, in the first step of calculation the maximum input value will be subtracted from all x[i] and only after that use softmax function as it is presented above will be used to
calculate output.
Sigmoid/softmax activation functions. Image source
The softmax function is mainly used on the output layer as some kind of transfer function to represent output layer values as probabilities, usually for classification tasks.
Hyperbolic tangent function
The tanh activation function is somewhat similar to the sigmoid in the sense that it also maps the input values to an s-shaped curve, but in this case, the output range is (-1, 1) and is centered at
0, which solves one of the problems with the sigmoid function. Tanh stands for the hyperbolic tangent, which is just the hyperbolic sine divided by the hyperbolic cosine, similar to the regular
tangent. Here's an illustration along with the formula:
Tangent hyperbolic function(red) and first derivative (blue)
While the tanh activation function can be more effective than the sigmoid activation function, it still encounters the same problems as the sigmoid function by causing issues during backpropagation.
In the case of very large or very small values, the derivative of the tanh function gets closer and closer to zero and makes the neural network harder to train. As an exponential function, it is
computationally costly. However, the tanh function is a handy activation function to use in the hidden layers to pass better input values to the next hidden layer.
Inverse tangent function
The inverse tan function is another non-linear activation function. Similar to the sigmoid function and tanh function it has an 'S'-shape and with the derivatives similar shapes as well. Inverse
tangent function outputs in the range of (-π/2, π/2).
Inverse tangent function (red) and first derivative(blue)
Again has the same "killing"/ "vanishing" gradient problem, but as there is no exponent in the calculation of the gradient, it is relatively faster than tanh or sigmoid functions.
Rectified linear unit (ReLU) function
ReLU activation function is a more modern and widely used activation function. It stands for Rectified Linear Unit and looks like this:
The beauty of the ReLU activation function lies partly in its simplicity. As one can see, all it does is replace negative values with 0 and keep positive ones as is. This avoids the problem of
“killing” the gradients of large and small values, while also being much faster computationally as it involves simpler mathematical operations. Also, in practice neural networks using ReLU tend to
converge about six times faster than those with sigmoid and tanh.
However, ReLU still faces some problems. First off, it's not 0-centered, which can cause problems during training. Most importantly though, it does not deal with negative inputs in a particularly
meaningful way. During the backpropagation process neural network updates weights with the gradients. Some neurons, that have negative input values, will have zero gradient and will not be updated.
There is the possibility that some neurons will not be updated during the whole process of neural network training at all. Those neurons are called "dead" neurons.
Modern activation functions tend to take ReLU and try to fix these problems. A lot of variations of the ReLU function were developed to avoid this problem in neural network model training.
Parametric ReLU function
The Parametric ReLU activation function builds on top of ReLU by trying to handle negative values in a more meaningful way. More specifically, instead of replacing negative values with 0, it
multiplies them by some user-defined number between 0 and 1, which essentially lets some of the information contained in the negative inputs be used in the neural network model training. The
disadvantage of this activation function is that the parameter is not learnable and the user should define it very carefully in the neural network architecture as results can vary depending on that
Parametric ReLU (red) and first derivative (blue). a = 0.2 in the figure.
Leaky ReLU function
The leaky ReLU activation function is the specific case for the parametric ReLU activation function, where a=0.01. The leaky ReLU functions in neural networks can cause problems such as updating
weights slower as the parameter is very small. It is preferable to use the parametric ReLU function in the neural network model.
Leaky ReLU function. Image source
Exponential linear units (ELU) function
Exponential linear units(ELU) is yet another non-linear activation function as an alternative to the ReLU function. Positive inputs have the same output for both of them, but the negative values are
smoother with the ELU due to the exponent, but it also has its disadvantage of making it computationally costly for this type of activation function. Fig. 9 shows the function and first derivative
Exponential Linear Unit (ELU) function (red) and first derivative(blue).
All of the above-mentioned activation functions (ReLU, Leaky ReLU, Parametric ReLU, and ELU) have one common problem. All of them are the same activation function for positive input data (output as a
linear function) and the gradient is constant and equals "1" for all positive values. If the weights of the hidden layers are big values they can start to multiply together and get bigger and bigger,
which will cause exploding gradient problem.
Gaussian error linear unit (GELU) function
GELU is one of the newest activation functions. It uses a standard Gaussian cumulative distribution function to weigh the input data. The GELU has major differences compared to ReLU, ELU and etc. All
the activation functions' output values depend on the input value sign. The GELU activation function uses the value of the input rather which makes it more efficient. The neuron input multiplies by m
∼ Bernoulli(Φ(x)), where Φ(x) = P (X ≤ x), X ∼ N (0, 1) is the cumulative distribution function of the standard normal distribution:
The GELU, ReLU and ELU comperision. Image source
The computational cost for this activation function is high so some approximations are done to make it easier to calculate.
Swish function
The swish activation function is the multiplication of input data and parametrized sigmoid function output for the input data. The parameter "a" in the swish function in the vast majority of neural
network model cases is 1. The function in that case is called sigmoid linear unit (SiLU).
The swish function shows the advantages of deep learning. The activation function is mostly used when the number of hidden layers is big.
Siwsh activation function(red) and frist derivative(blue).
How to choose an activation function
Here's the million-dollar question in machine learning: how to actually choose the right activation function when training a neural network from scratch? Different activation functions have different
advantages and disadvantages and depending on the type of the artificial neural network the outcome may be different.
The starting point can be to choose one of the ReLU-based activation functions (including ReLU itself) since they have empirically proven to be very effective for almost any task. After it tries to
choose other activation functions for hidden layers may be different activation functions for multiple layers to see how the performance changes.
Neural network architecture, machine learning tasks, and many others have an impact on activation function selection. For example, if the task is binary classification then the sigmoid activation
function is a good choice, but for the multi-class classification softmax function is better as it will output probability representation for each class.
In convolutional neural networks, activation functions can be ReLU-based to increase the convergence speed. However, some architectures require specific activation functions. For example, recurrent
neural network architectures and Long Short Term Memory architectures utilize the sigmoid function and tanh function, and their logic gate-like architecture wouldn't work with ReLU.
Summing up
To recap, activation functions are crucial for modern neural networks because they facilitate the learning of a much wider set of functions and can, thus, make the model learn to perform all sorts of
complex tasks. The impressive advances in computer vision, natural language processing, time series, and many other fields would be nearly impossible without the opportunities created by non-linear
activation functions. While exponent-based activation functions like the sigmoid and tanh functions have been used for decades and can yield good results, more modern ones like ReLU work better in
most applications. As a rule of thumb, when training a neural network from scratch, one can simply use ReLU, leaky ReLU, or GELU and expect decent results. | {"url":"https://www.superannotate.com/blog/activation-functions-in-neural-networks","timestamp":"2024-11-05T10:27:32Z","content_type":"text/html","content_length":"57621","record_id":"<urn:uuid:d1a347b2-8c08-4c3c-b4e7-a54179697c36>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00575.warc.gz"} |
Introduction To Lenses - The Base Curve - The Lost Contacts
In Lesson 1 of this course, you learned about the lens index. In this lesson you will learn about another property of lenses called the base curve.
Remember that to fully understand the concepts covered in this lesson, it is best to have read through the entire The Optics of Vision course.
The True Shape of Lenses
Up until now, I have been simplifying the shape of lenses in order to make obvious the distinction between plus and minus lenses.
Here is the shape of lenses that you are use to seeing:
Here is a more realistic depiction what these lenses look like when they are manufactured for glasses:
Quite different, right? But don’t worry, it’s still very simple. Let’s break it down.
The Front and Back Curve
Every lens has a front and a back surface curvature. The front curvature is always convex and the back curvature is concave.
The Lens Power
The overall lens power is simply the sum of the front and back curvature.
• The front curve is always positive
• The back curve is always negative
In order to do this you must be comfortable with adding positive and negative numbers.
For example, here an example of a +2.00 lens.
This is a +2.00 lens because the front curve is +6.00 and back curve is -4.00. If you sum these together, you get +2.00.
+6.00 + (-4.00) = +2.00
What is the Base Curve?
In practice, we call the front curve the base curve.
Now, of course there are several different ways to combine base (front) curves and back curves in order to arrive a +2.00 lens.
Other options would be:
Base Curve Back Curve
+2.00 0.00
+3.00 -1.00
+4.00 -2.00
+5.00 -3.00
But as it turns out, only specific combinations of base curves and back curves are used for any given desired lens power. The goal is to create a lens that is as comfortable as possible to see
through by using the most appropriate combination of base and back curves.
For Example, the following lenses are both +2.00 lenses, but one has a base curve of +6.00 while the other has a base curve of +12.00. Which one would you rather have in your glasses?
Obviously, the lens with the +6.00 base curve is much more desirable.
How To Determine The Optimal Base Curve
The most optimal base curve for any prescription can be calculated using these equations:
For Plus Prescriptions
Base Curve = Spherical Equivalent + 6.00
For Minus Prescriptions
Base Curve = [ 1/2 * (Spherical Equivalent) ] + 6.00
No worries though, as with everything else we don’t actually work with formulas. We make charts.
This is a simple base curve selection chart. Note: This chart does not apply to all lens indexes and lens designs.
There are many reasons why base curves have to be carefully selected for the strength of each prescription. Here are just a few:
The wrong base curve can…
• cause unnecessary distortions in vision
• cause inability to adapt to a new prescription
• cause lenses to be thicker than they need be
How To Measure The Base Curve?
The base curve of any lens can be measured with a tool called a radius gauge, also known as a lens clock.
A lens clock has three prongs that can measure the curvature of lenses (and other surfaces).
When those 3 prongs are place against a flat surface, the gauge should read zero. The following image shows a lens clock held against a flat counter top.
A lens clock read zero (more or less) when held up to a flat surface.
When a lens clock is held up against a convex surface, the black numbers indicate the curvature of that surface.
A lens clock measuring a +5.00 front/base curvature.
When a lens clock is held up against a concave surface, the red numbers indicate the curvature of that surface.
A lens clock measuring -2.00 back curvature.
In the example above, the base curve is +5.00 and the back curve is -2.00. Hence, this lens has an overall power of +3.00 (+5.00 + -2.00).
The same can be done on lenses already mounted into glasses.
The process I’ve illustrated here is somewhat simplified and assumes there is only Sphere power in the lenses. In reality, many lenses you encounter will have both Sphere and Cylinder power. It is
rare to use a lens clock to determine the cylinder power of glasses, but if you’re interested in learning more about that, click here.
Who Deals With Base Curve?
Optometrists do not write down the base curve on their patient’s prescriptions, so does it fall upon the optician to order the correct base curve? Yes, most of time.
Despite how critical the correct base curve is to the overall finished pair of glasses, it is not typically the biggest concern when dispensing glasses. In most optical stores, the customer’s lens
order is sent to a lens manufacturing laboratory who’s skilled technicians select the most appropriate base curve for every lens order.
Of course, if a specific base curve is required, the optometrist or optician can specify this on the prescription/lens order. Often times when the base curve is specified, it is done so in order to
match the base of curve of the lenses in the customer’s old glasses.
Case Scenario:
A customer gets her eye exam and it is determined that her prescription has not changed at all since she got her last pair of glasses. Even though her current glasses are still in good shape, she
decides to get new glasses anyway. You make her new glasses with the exact same prescription as her old glasses. The customer returns to the store one week later saying that she can’t see as well
with her new glasses as with her old ones.
What could be the problem?
Possibility #1
It could be the base curve. Using a lens clock, measure the base curve of the old and the new pair of glasses. If the base curves are different between the two pairs, it could be very difficult for
the customer to adapt to the new glasses (even though the prescriptions are exactly the same).
Possibility #2
If the base curves are the same between the two pairs, another possibility could be incorrect PD (pupillary distance) and OC (ocular center) height measurements.
If you don’t know what a PD and an OC height are, that’s because I haven’t taught it to it you yet, but that is the topic of the next lesson! | {"url":"https://thelostcontacts.com/introduction-to-lenses-the-base-curve/","timestamp":"2024-11-12T12:35:30Z","content_type":"text/html","content_length":"279928","record_id":"<urn:uuid:1988b012-6515-4324-a906-6a43dfa877e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00854.warc.gz"} |
Volume of a Prism - Formula, Derivation, Definition, Examples - Grade Potential Mountain View, CA
Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The figure’s name is originated from the fact that it is made by taking into account a polygonal base and extending its sides until it intersects the opposite
This article post will discuss what a prism is, its definition, different types, and the formulas for volume and surface area. We will also take you through some instances of how to employ the data
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, called bases, which take the shape of a plane figure. The additional faces are rectangles, and their count
relies on how many sides the similar base has. For instance, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there will be five sides.
The characteristics of a prism are interesting. The base and top both have an edge in common with the other two sides, creating them congruent to each other as well! This implies that all three
dimensions - length and width in front and depth to the back - can be broken down into these four parts:
1. A lateral face (signifying both height AND depth)
2. Two parallel planes which make up each base
3. An fictitious line standing upright across any provided point on any side of this figure's core/midline—usually known collectively as an axis of symmetry
4. Two vertices (the plural of vertex) where any three planes join
Types of Prisms
There are three main kinds of prisms:
• Rectangular prism
• Triangular prism
• Pentagonal prism
The rectangular prism is a regular kind of prism. It has six sides that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. It looks close to a triangular prism, but the pentagonal shape of the base makes it apart.
The Formula for the Volume of a Prism
Volume is a calculation of the sum of area that an item occupies. As an important figure in geometry, the volume of a prism is very important for your studies.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Finally, considering bases can have all kinds of shapes, you will need to know a few formulas to figure out the surface area of the base. Despite that, we will touch upon that later.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we are required to observe a cube. A cube is a 3D item with six faces that are all squares. The formula for the volume of a cube is V=s^3,
V = Volume
s = Side length
Now, we will get a slice out of our cube that is h units thick. This slice will make a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula implies the base area of
the rectangle. The h in the formula refers to height, which is how thick our slice was.
Now that we have a formula for the volume of a rectangular prism, we can use it on any type of prism.
Examples of How to Utilize the Formula
Since we understand the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, now let’s use them.
First, let’s work on the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches.
V=432 square inches
Now, consider another problem, let’s work on the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.
V=450 cubic inches
Provided that you have the surface area and height, you will figure out the volume with no problem.
The Surface Area of a Prism
Now, let’s talk about the surface area. The surface area of an item is the measurement of the total area that the object’s surface consist of. It is an important part of the formula; thus, we must
learn how to find it.
There are a few different ways to work out the surface area of a prism. To figure out the surface area of a rectangular prism, you can utilize this: A=2(lb + bh + lh), where,
l = Length of the rectangular prism
b = Breadth of the rectangular prism
h = Height of the rectangular prism
To compute the surface area of a triangular prism, we will use this formula:
b = The bottom edge of the base triangle,
h = height of said triangle,
l = length of the prism
S1, S2, and S3 = The three sides of the base triangle
bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh
We can also use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Finding the Surface Area of a Rectangular Prism
Initially, we will determine the total surface area of a rectangular prism with the ensuing information.
l=8 in
b=5 in
h=7 in
To figure out this, we will plug these numbers into the corresponding formula as follows:
SA = 2(lb + bh + lh)
SA = 2(8*5 + 5*7 + 8*7)
SA = 2(40 + 35 + 56)
SA = 2 × 131
SA = 262 square inches
Example for Computing the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will figure out the total surface area by following same steps as before.
This prism will have a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Hence,
SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)
SA = (40*7) + (2*60)
SA = 400 square inches
With this data, you should be able to work out any prism’s volume and surface area. Check out for yourself and observe how simple it is!
Use Grade Potential to Improve Your Mathematical Skills Now
If you're having difficulty understanding prisms (or any other math concept, consider signing up for a tutoring class with Grade Potential. One of our experienced teachers can assist you learn the
[[materialtopic]187] so you can ace your next test. | {"url":"https://www.mountainviewinhometutors.com/blog/volume-of-a-prism-formula-derivation-definition-examples","timestamp":"2024-11-02T15:40:19Z","content_type":"text/html","content_length":"78665","record_id":"<urn:uuid:b304b98d-981b-4193-a581-35409e0201ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00054.warc.gz"} |
Multiplication Math Coloring Worksheets
Math, especially multiplication, creates the keystone of numerous scholastic techniques and real-world applications. Yet, for many students, understanding multiplication can present an obstacle. To
address this hurdle, teachers and moms and dads have actually accepted a powerful device: Multiplication Math Coloring Worksheets.
Introduction to Multiplication Math Coloring Worksheets
Multiplication Math Coloring Worksheets
Multiplication Math Coloring Worksheets -
Get Your FREE Set of Multiplication Coloring Worksheets Math fun rocks You re on your way to great math fun for your kids This free printable PDF set includes 4 pages of Color by Product pages 4
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Multiplication Coloring We hope you like these multiplication worksheets If you enjoy them check out Coloring Squared Multiplication and Division It collects our basic and advanced multiplication and
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Multiplication Coloring Pages At GetDrawings Free Download
Multiplication Coloring Pages At GetDrawings Free Download
Practice Multiplication Facts with these Color by Number Worksheets Third grade and fourth grade students who are learning their multiplication facts will have a great time completing these fun
coloring pages They also make for a fun art activity for students in later grades This collection of worksheets is growing and I ll continue adding
Printable Color by Code Worksheets Color by number activities are a fun way for kids of all ages to practice basic multiplication skills A coloring sheet is definitely more fun than repetitive math
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From traditional pen-and-paper exercises to digitized interactive styles, Multiplication Math Coloring Worksheets have actually evolved, satisfying varied learning styles and choices.
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4th Grade Math Worksheets Multiplication Color By Number Times Tables Worksheets
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How to Complete the Multiplication Pages There are five coloring pages for our kiddos to color Each page has a number written at the top of the page All the students have to do is find the
expressions that equal that number For example one of the pages has a 12 at the top So the students will go on a hunt for 12 x 1 2 x 6 and 3 x 4
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Relating multiplication to everyday circumstances adds importance and practicality to workouts.
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15 Best Halloween Multiplication Coloring Printables Printablee
Halloween Math Coloring Multiplication Division Worksheets Teaching Math And More
Check more of Multiplication Math Coloring Worksheets below
Multiplication Worksheet color by number 1 Hess Un Academy
Free Printable Math Multiplication Coloring Worksheets FREE PRINTABLE
Free Printable Color By Number Multiplication Worksheets Free Printable
3rd Grade Multiplication Coloring Worksheets AlphabetWorksheetsFree
Color By Number Multiplication Worksheets Times Tables Worksheets
Free Math Coloring Pages Multiplication Download Free Math Coloring Pages Multiplication Png
Multiplication Coloring Coloring Squared
Multiplication Coloring We hope you like these multiplication worksheets If you enjoy them check out Coloring Squared Multiplication and Division It collects our basic and advanced multiplication and
division pages into an awesome coloring book Super Multiplication and Division 50 puzzles 14 95
Free Multiplication Coloring Worksheets Printables
The multiplication problems on these worksheets are simple making them best for 3rd grade and 4th grade students These hidden picture coloring worksheets feature 1 digit by 1 digit multiplication
problems and sometimes 2 digit by 1 digit multiplication problems These multiplication coloring pages include the times table facts 1 12 giving
Multiplication Coloring We hope you like these multiplication worksheets If you enjoy them check out Coloring Squared Multiplication and Division It collects our basic and advanced multiplication and
division pages into an awesome coloring book Super Multiplication and Division 50 puzzles 14 95
The multiplication problems on these worksheets are simple making them best for 3rd grade and 4th grade students These hidden picture coloring worksheets feature 1 digit by 1 digit multiplication
problems and sometimes 2 digit by 1 digit multiplication problems These multiplication coloring pages include the times table facts 1 12 giving
3rd Grade Multiplication Coloring Worksheets AlphabetWorksheetsFree
Free Printable Math Multiplication Coloring Worksheets FREE PRINTABLE
Color By Number Multiplication Worksheets Times Tables Worksheets
Free Math Coloring Pages Multiplication Download Free Math Coloring Pages Multiplication Png
math coloring Worksheet multiplication Addition And Subtraction Worksheets Multiplication
Math Coloring Pages Multiplication Coloring Home
Math Coloring Pages Multiplication Coloring Home
Multiplication Worksheets Multiplication worksheets Multiplication Facts worksheets
Frequently Asked Questions (Frequently Asked Questions).
Are Multiplication Math Coloring Worksheets appropriate for any age groups?
Yes, worksheets can be tailored to different age and skill degrees, making them adaptable for numerous students.
Exactly how typically should pupils practice utilizing Multiplication Math Coloring Worksheets?
Consistent technique is key. Regular sessions, preferably a few times a week, can yield substantial renovation.
Can worksheets alone improve mathematics abilities?
Worksheets are a valuable tool however ought to be supplemented with diverse knowing techniques for detailed ability growth.
Exist on the internet platforms using free Multiplication Math Coloring Worksheets?
Yes, several educational internet sites use open door to a wide range of Multiplication Math Coloring Worksheets.
Exactly how can parents support their children's multiplication practice in your home?
Encouraging constant technique, providing support, and producing a favorable knowing environment are beneficial actions. | {"url":"https://crown-darts.com/en/multiplication-math-coloring-worksheets.html","timestamp":"2024-11-04T07:57:52Z","content_type":"text/html","content_length":"29529","record_id":"<urn:uuid:c571615f-3017-4d77-be52-31c6df429f0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00568.warc.gz"} |
topmodels 0.3-0
• Entire package now leverages distributions3 for object-oriented computations on distributions fitted/predicted by various kinds of models.
• In particular, procast() first obtains prodist() (probability distribution) and then applies the standard methods for computing densities, probabilities, quantiles, and moments.
• Similarly, proresiduals() obtains the predicted distributions and compares with the newresponse() to obtain (randomized) quantile residuals by default. Alternatively, PIT residuals or Pearson
residuals as well as raw response residuals are available. The function proresiduals() also replaces both pitresiduals() and qresiduals() which were provided by earler versions of topmodels.
• Via the same approach proscore() implements various kinds of scoring rules, in particular log-score (negative log-likelihood), (continuous) ranked probability score (CRPS), mean absolute error
(MAE), mean squared error (MSE), and the Dawid-Sebastiani-Score (DSS). The standard log-likelihood (without sign change) is also available.
• For the CRPS one can either leverage the functions from the scoringRules package (if available) or the new crps.distribution() method for numeric approximation/numeric integration to calculate
the CRPS for univariate distributions. This is also used when no analytic solution is available in the scoringRules package.
• The graphical functions rootogram(), pithist(), qqrplot(), wormplot(), and reliagram() are also switched to the new infrastructure based on distributions3, notably via procast() and proresiduals
• The pointwise and simultaneous confidence intervals in rootogram() now rely on the exact PoissonBinomial() distribution (now available in distributions3) rather than its binomial approximation.
• In addition to pointwise and simultaneous confidence intervals for rootogram(), "tukey" confidence intervals are now available which simply correspond to limits of -1 and 1 for hanging or
suspended rootograms. For other flavors of rootograms these limits are transformed correspondingly.
• New distribution/model interfaces were added first in topmodels but some subsequently moved to other packages: GAMLSS() is now in gamlss.dist, BAMLSS() is now in bamlss, and Empirical() is still
in topmodels for now.
• New wrapper function promodel() that adds the class "promodel" (for probabilistic model) to an existing model object so that predict() dispatches to procast() and residuals() dispatches to
proresiduals(). This facilitates using model functionality based on the standard predict() and residuals() methods like marginaleffects.
topmodels 0.2-0
• New version, presented at DAGStat 2022 and at useR! 2022 (together with distributions3).
• Some conceptual changes in the generation of graphical evaluation tools for both base R and ggplot2 style graphics.
• autoplot() builds now on newly written geom_*() and stat_*() functions.
topmodels 0.1-0
• First version, presented at useR! 2021.
• Diagnostic graphics for Q-Q plots of randomized residuals, PIT (probability integral transform) histograms, reliability diagrams, wormplots, and rootograms. All graphical evaluations can be
rendered both in base R graphics and ggplot2.
• Basic probabilistic forecasting infrastructure for lm, crch, disttree and glm model classes. Not all families and forecasting types are fully supported yet. | {"url":"https://topmodels.r-forge.r-project.org/topmodels/NEWS.html","timestamp":"2024-11-05T04:18:14Z","content_type":"application/xhtml+xml","content_length":"29996","record_id":"<urn:uuid:03e67204-07d7-4b90-8ba9-4228c7bac5a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00050.warc.gz"} |
Sword finger offer (difficult)
JZ19 clockwise print matrix
code analysis
public class Solution {
public ArrayList<Integer> printMatrix(int[][] matrix) {
ArrayList<Integer> ret = new ArrayList<>();
int r1 = 0, r2 = matrix.length - 1, c1 = 0, c2 = matrix[0].length - 1;
while (r1 <= r2 && c1 <= c2) {
// upper
for (int i = c1; i <= c2; i++)
// right
for (int i = r1 + 1; i <= r2; i++)
if (r1 != r2){
// lower
for (int i = c2 - 1; i >= c1; i--)
if (c1 != c2){
// Left
for (int i = r2 - 1; i > r1; i--)
r1++; r2--; c1++; c2--;
return ret;
Special case: when there is only one line, the following two if judgments are required.
Z25 copy of complex linked list
Method 1 (map)
public class Solution {
public RandomListNode Clone(RandomListNode pHead) {
if(pHead == null) return null;
RandomListNode cur = pHead;
Map<RandomListNode, RandomListNode> map = new HashMap<>();
// 3. Copy each node and create a Map mapping of "original node - > new node"
while(cur != null) {
map.put(cur, new RandomListNode(cur.label));
cur = cur.next;
cur = pHead;
// 4. Build the next and random points of the new linked list
while(cur != null) {
map.get(cur).next = map.get(cur.next);
map.get(cur).random = map.get(cur.random);
cur = cur.next;
// 5. Return the header node of the new linked list
return map.get(pHead);
Algorithm flow:
1. If the header node head is empty, nullnull is returned directly;
2. Initialization: hash table map, node cur points to the header node;
3. Copy linked list:
1. Create a new node and add key value pairs to dic (original cur node, new cur node);
2. cur traverses to the next node in the original linked list;
4. The reference to build a new linked list points to:
1. Build the next and random references of the new node;
2. cur traverses to the next node in the original linked list;
5. Return value: the head node map[cur] of the new linked list;
Source: LeetCode
Get a dictionary corresponding to the new and old nodes. You need a node and get it from the value of the dictionary
Method 2: splicing + splitting
public class Solution {
public RandomListNode Clone(RandomListNode pHead) {
if(pHead == null) return null;
RandomListNode cur = pHead;
// 1. Copy each node and build a splicing linked list
while(cur != null) {
RandomListNode tmp = new RandomListNode(cur.label);
tmp.next = cur.next;
cur.next = tmp;
cur = tmp.next;
// 2. Build the random direction of each new node
cur = pHead;
while(cur != null) {
if(cur.random != null)
cur.next.random = cur.random.next;
cur = cur.next.next;
// 3. Split two linked lists
cur = pHead.next;
RandomListNode pre = pHead;//Old linked list pointer
RandomListNode res = pHead.next;//New linked list pointer
while(cur.next != null) {
pre.next = pre.next.next;
cur.next = cur.next.next;
pre = pre.next;
cur = cur.next;
pre.next = null; // Handle the original end node of the linked list separately
return res; // Return to the new chain header node
Wrong, cannot used original node of list
Therefore, the original tail node of the linked list is processed separately
Maximum value of JZ64 sliding window
Violent solution
import java.util.*;
public class Solution {
public ArrayList<Integer> maxInWindows(int [] num, int size) {
ArrayList<Integer> list=new ArrayList<Integer>();
int max=0;
if(num.length == 0 || size > num.length || size==0){
return list;
for(int i=0;i <= num.length - size;i++){
for(int j=i;j<size + i;j++){
if(max < num[j]){
return list;
Monotone queue
import java.util.*;
public class Solution {
public ArrayList<Integer> maxInWindows(int [] nums, int k) {
//Monotone queue
//Here are the points to note:
//The queue is placed from large to small
//If the first value (i.e. the maximum value) is not in the window interval, delete the first value
//If the added value is less than the value at the end of the queue, it will be added to the end of the queue
//If the new value added is greater than the value at the end of the queue, delete the value smaller than the new value added in the queue. If the new value added is added to the queue
//If the new value-added is greater than all the values in the queue, delete all, and then put the new value-added at the top of the queue to ensure that the queue is from large to small
ArrayList<Integer> list = new ArrayList<>();
if(k <= 0|| k > nums.length)return list;
Deque<Integer> deque = new LinkedList<>();
//Window interval not formed
for (int i = 0; i < k; i++) {
//When the queue is not empty, the current value is compared with the value at the end of the queue. If it is greater than, the value at the end of the queue is deleted
//It is deleted circularly until the value in the queue is greater than the current value, or until the queue is empty
while (!deque.isEmpty() && nums[i] > deque.peekLast()) deque.removeLast();
//After executing the above loop, the queue is either empty or the value is greater than the current value, and then the current value is added to the queue
//After the window interval is just formed, the first value of the queue is added to the queue
//After the window is formed, we need to add the first row of the queue to the array, and the following loop directly skips this step, so we need to add it directly
//Window interval formation
for (int i = k; i < nums.length; i++) {
//I-k is already outside the interval. If the first value is equal to nums[i-k], it means that the first value is no longer in the interval and needs to be deleted
if (deque.peekFirst() == nums[i - k]) deque.removeFirst();
//Deletes a value in the queue that is greater than the current value
while (!deque.isEmpty() && nums[i] > deque.peekLast()) deque.removeLast();
//Adds the current value to the queue
//Adds the first value of the queue to the arr array
return list;
The maximum value comes first
JZ56 delete duplicate nodes in linked list
public class Solution {
public ListNode deleteDuplication(ListNode pHead) {
// Recursive exit: when the "input node is empty" or "there is no next node", it returns directly
if (pHead == null || pHead.next == null) return pHead;
if (pHead.val != pHead.next.val) {
// If the values of "current node" and "next node" are different, the current node can be retained
pHead.next = deleteDuplication(pHead.next);
return pHead;
} else {
// If the current node is the same as the next node, you need to skip a continuous segment with the same value
ListNode tmp = pHead;//For comparison
while (tmp != null && tmp.val == pHead.val) tmp = tmp.next;
return deleteDuplication(tmp);
public ListNode deleteDuplication(ListNode pHead) {
ListNode h = new ListNode(0), p = h;
h.next = pHead;
while (pHead != null) {
if (pHead.next != null && pHead.val == pHead.next.val) {
p.next = null;
} else if (pHead.next != null && p.next == null) {
p.next = pHead.next;
} else {
p = p.next;
pHead = pHead.next;
return h.next;
pHead is a pointer
p is the pointer
If the current node repeats with the next node, p.next is not empty. Otherwise, if p.next is empty, the current node is skipped (representing that the current node is the last node of the same kind)
Non recursive binary
class Solution {
public ListNode deleteDuplication(ListNode pHead) {
ListNode dummy = new ListNode(-1);
ListNode tail = dummy;
while (pHead != null) {
// When entering the loop, ensure that the pHead is not the same as the previous node
if (pHead.next == null || pHead.next.val != pHead.val) {
tail.next = pHead;
tail = pHead;
// If the pHead is the same as the next node, skip the same node (reach the last bit of "continuous same segment")
while (pHead.next != null && pHead.val == pHead.next.val) pHead = pHead.next;
pHead = pHead.next;
tail.next = null;
return dummy.next;
Dummy: dummy node. Save the correct header node
tail: connect the correct node
pHead: trial and error
When to use dummy nodes
It can avoid dealing with the boundary problem where the header node is empty and reduce the possibility of code execution exceptions.
This perfectly solves the problem that once the head node is deleted, the element cannot be returned.
Dummy nodes are generally used to save the of the head node, so they are used for non recursive problems that may lead to the absence of the head node.
JZ27 string arrangement
public ArrayList<String> Permutation(String s) {
c = s.toCharArray();
//Recursion from the first level
//Converts a String array ArrayList to an array of type String
return list1;
public void dfs(int x) {
//Recursive exit. There is only one character at the end. There is no need to exchange
if (x == c.length - 1) {
//Converts a character array to a string
//In order to prevent repeated elements in the same layer of recursion
HashSet<Character> set = new HashSet<>();
//It's very clever here. The first layer can be a, B and C, so there are three cases. Here i = x, dfs(0) happens to start with i = 0
// When there are only two cases in the second layer, dfs(1) i = 1 starts
for (int i = x; i < c.length; i++) {
//Pruning occurs. When this element is included, skip directly
if (set.contains(c[i])) {
//Exchange elements, which is very clever here. When in the second layer dfs(1),x = 1, then i = 1 or 2, either exchange 1 and 1, or exchange 1 and 2
swap(i, x);
//Go to the next level of recursion
dfs(x + 1);
swap(i, x);
public void swap(int i, int x) {
char temp = c[i];
c[i] = c[x];
c[x] = temp;
Like the picture,
The first layer is a b c, the first for loop.
(x == c.length - 1) recursive end condition
Each layer will have a set storage, so the first element of each layer cannot be repeated,
Ergodic recursion
The first swap(i, x): it is responsible for replacing the following and the first element of each layer
dfs(x + 1); Will go to the next layer: X is the number of layers
The second swap is used to make the order of the character array return to the state before recursion, so as not to affect the external traversal order. Because when entering the recursive operation
after the first exchange, the order of the character array changes.
Next spread
public ArrayList<String> Permutation2(String str) {
ArrayList<String> list = new ArrayList<String>();
if (str == null || str.length() == 0) {
return list;
char[] chars = str.toCharArray();
int len = chars.length;
while (true) {
int l = len - 1;
int r;
while (l >= 1 && chars[l - 1] >= chars[l]) {
if (l == 0)
r = l;
while (r < len && chars[r] > chars[l - 1]) {
swap(chars, l - 1, r - 1);
reverse(chars, l);
return list;
private void swap(char[] cs, int i, int j) {
char temp = cs[i];
cs[i] = cs[j];
cs[j] = temp;
private void reverse(char[] chars, int k) {
if (chars == null || chars.length <= k)
int len = chars.length;
for (int i = 0; i < (len - k) / 2; i++) {
int m = k + i;
int n = len - 1 - i;
if (m <= n) {
swap(chars, m, n);
A full permutation can be regarded as a string, which can have prefix and suffix.
Generate the next permutation of a given full permutation. The so-called next of one is that there is no other between this one and the next.
This requires that this one and the next have a common prefix as long as possible, that is, the change is limited to the suffix as short as possible.
[example] 839647521 is an arrangement of 1 – 9. In the arrangement of 1-9, the front is 123456789 and the rear is 987654321,
If the scanning from right to left is increased, it will reach 987654321, and there will be no next one. Otherwise, find the position where the descent occurs for the first time.
[example] how to get the next one of 346987521
1. Find the position of the first p (i-1) < p (I) from the tail
3 4 6 <- 9 <- 8 <- 7 <- 5 <- 2 <- 1
Finally, find that 6 is the first smaller number and record the position i-1 of 6
2. Find the last number greater than 6 from position i: chars [R] > chars [L - 1]. Because it is while, r-1 meets the condition
3 4 6 -> 9 -> 8 -> 7 5 2 1
Finally find the position of 7 and record the position as m
3. Exchange the values of positions i-1 and m
4. All data after i position in reverse order
Then 347125689 is the next arrangement of 346987521
while (true) {
int i = chars.length - 1;
int j = chars.length - 1;
while (i > 0 && chars[i-1] >= chars[i ]) {//In order to ensure that it does not cross the boundary and can be judged to the greatest extent, only
if (i==0){
while (j > i && chars[i-1] >= chars[j]) {
swap(chars, i-1, j);
reverse(chars, i );
In the second step, you can also find the first number greater than i from back to front
While (I > 0 & & chars [I-1] > = chars [i]) {/ / in order to ensure that it does not cross the boundary and can be judged at the maximum, only chars [I-1] > = chars [i]
Next spread
public char[] nextPermutation(String str) {
char[] nums = str.toCharArray();
int i = nums.length - 1;
while (i > 0 && nums[i-1] >= nums[i ]) {
if (i > 0) {
int j = nums.length - 1;
while (j >= 0 && nums[i] >= nums[j]) {
swap(nums, i, j);
reverse(nums, i );
return nums;
In order for 321 to become 123, i has a boundary value of 0,.
Path in matrix
code analysis
public boolean hasPath (char[][] board, String word) {
// write code here
char[] words = word.toCharArray();
for(int i = 0; i < board.length; i++) {// Traverse the board, regard all elements as the starting node and start searching
for(int j = 0; j < board[0].length; j++) {
if(dfs(board, words, i, j, 0)) return true;// If it is found once, it will directly return true
return false;// false returned when not found
boolean dfs(char[][] board, char[] word, int i, int j, int k) {
//If the array subscript is out of bounds or board [i] [J]= Word [depth], it does not match and returns false;
if(i >= board.length || i < 0 || j >= board[0].length || j < 0 || board[i][j] != word[k]) return false;
if(k == word.length-1 ) return true;//Description has all been matched
board[i][j] = '\0'; // That is, it represents an empty character, which will never be found in the word
boolean res = dfs(board, word, i + 1, j, k + 1) || dfs(board, word, i - 1, j, k + 1) ||
dfs(board, word, i, j + 1, k + 1) || dfs(board, word, i , j - 1, k + 1);
board[i][j] = word[k];// Backtracking: restores the marker bit
return res; | {"url":"https://www.fatalerrors.org/a/sword-finger-offer-difficult.html","timestamp":"2024-11-09T13:17:18Z","content_type":"text/html","content_length":"29722","record_id":"<urn:uuid:19272623-54e5-4a1d-9c5f-c67ee6310901>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00489.warc.gz"} |
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Dumre Rs.11700 Rs.20475 Rs.23400 Rs.29250 Rs.35100 Rs.17550 Rs.17550
Hetauda, Makwanpur Rs.18720 Rs.32760 Rs.37440 Rs.46800 Rs.56160 Rs.28080 Rs.28080
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Khurkot Pashupatinagar Rs.41500 Rs.72625 Rs.83000 Rs.103750 Rs.124500 Rs.62250 Rs.62250
Trishuli, Nuwakot Rs.10000 Rs.17500 Rs.20000 Rs.25000 Rs.30000 Rs.15000 Rs.15000
Syabrubesi Rs.13781 Rs.24116.75 Rs.27562 Rs.34452.5 Rs.41343 Rs.20671.5 Rs.20671.5
Namobuddha Rs.7000 Rs.12250 Rs.14000 Rs.17500 Rs.21000 Rs.10500 Rs.10500
Bhakunde Beshi Rs.8000 Rs.14000 Rs.16000 Rs.20000 Rs.24000 Rs.12000 Rs.12000
Nepalthok Rs.10000 Rs.17500 Rs.20000 Rs.25000 Rs.30000 Rs.15000 Rs.15000
Palanchok Bhagwati Rs.8000 Rs.14000 Rs.16000 Rs.20000 Rs.24000 Rs.12000 Rs.12000
Melamchi Rs.10000 Rs.17500 Rs.20000 Rs.25000 Rs.30000 Rs.15000 Rs.15000
Daman, Simbhanjyang Rs.8269 Rs.14470.75 Rs.16538 Rs.20672.5 Rs.24807 Rs.12403.5 Rs.12403.5
Chitwan(via Daman) Rs.17199 Rs.30098.25 Rs.34398 Rs.42997.5 Rs.51597 Rs.25798.5 Rs.25798.5
Barabise (Barabishe) Rs.10000 Rs.17500 Rs.20000 Rs.25000 Rs.30000 Rs.15000 Rs.15000
Border Land Rs.7718 Rs.13506.5 Rs.15436 Rs.19295 Rs.23154 Rs.11577 Rs.11577
Kodari Rs.12000 Rs.21000 Rs.24000 Rs.30000 Rs.36000 Rs.18000 Rs.18000
Charikot Rs.12000 Rs.21000 Rs.24000 Rs.30000 Rs.36000 Rs.18000 Rs.18000
Jiri Rs.19803 Rs.34655.25 Rs.39606 Rs.49507.5 Rs.59409 Rs.29704.5 Rs.29704.5
Charaudi Rs.6689 Rs.11705.75 Rs.13378 Rs.16722.5 Rs.20067 Rs.10033.5 Rs.10033.5
Fishling Rs.9500 Rs.16625 Rs.19000 Rs.23750 Rs.28500 Rs.14250 Rs.14250
Dhading Beshi Rs.9500 Rs.16625 Rs.19000 Rs.23750 Rs.28500 Rs.14250 Rs.14250
Malekhu, Dhading Rs.7000 Rs.12250 Rs.14000 Rs.17500 Rs.21000 Rs.10500 Rs.10500
Kurintar, Manakamana Rs.9999 Rs.17498.25 Rs.19998 Rs.24997.5 Rs.29997 Rs.14998.5 Rs.14998.5
Nilkantha, Dhading Rs.8000 Rs.14000 Rs.16000 Rs.20000 Rs.24000 Rs.12000 Rs.12000
Kuringhat Rs.8085 Rs.14148.75 Rs.16170 Rs.20212.5 Rs.24255 Rs.12127.5 Rs.12127.5
Mugling Rs.8820 Rs.15435 Rs.17640 Rs.22050 Rs.26460 Rs.13230 Rs.13230
Aabu Kahaireni, Tanahu Rs.11592 Rs.20286 Rs.23184 Rs.28980 Rs.34776 Rs.17388 Rs.17388
Gorkha Rs.11700 Rs.20475 Rs.23400 Rs.29250 Rs.35100 Rs.17550 Rs.17550
Dumre Rs.11700 Rs.20475 Rs.23400 Rs.29250 Rs.35100 Rs.17550 Rs.17550
Besisahar Rs.15500 Rs.27125 Rs.31000 Rs.38750 Rs.46500 Rs.23250 Rs.23250
Bandipur Bazaar Rs.12500 Rs.21875 Rs.25000 Rs.31250 Rs.37500 Rs.18750 Rs.18750
Naya Pool Rs.20500 Rs.35875 Rs.41000 Rs.51250 Rs.61500 Rs.30750 Rs.30750
Baglung Rs.22500 Rs.39375 Rs.45000 Rs.56250 Rs.67500 Rs.33750 Rs.33750
Beni, Myagdi Rs.24000 Rs.42000 Rs.48000 Rs.60000 Rs.72000 Rs.36000 Rs.36000
Waling, Syangja Rs.25625 Rs.44843.75 Rs.51250 Rs.64062.5 Rs.76875 Rs.38437.5 Rs.38437.5
Syangja, Nepal Rs.24125 Rs.42218.75 Rs.48250 Rs.60312.5 Rs.72375 Rs.36187.5 Rs.36187.5
Putalibazar, Syangja Rs.22617 Rs.39579.75 Rs.45234 Rs.56542.5 Rs.67851 Rs.33925.5 Rs.33925.5
Chitwan Rs.15925 Rs.27868.75 Rs.31850 Rs.39812.5 Rs.47775 Rs.23887.5 Rs.23887.5
Chitwan, Sauraha Rs.14000 Rs.24500 Rs.28000 Rs.35000 Rs.42000 Rs.21000 Rs.21000
Chitwan Jugedi Rs.10658 Rs.18651.5 Rs.21316 Rs.26645 Rs.31974 Rs.15987 Rs.15987
Chitwan Jungle Lodge Rs.16500 Rs.28875 Rs.33000 Rs.41250 Rs.49500 Rs.24750 Rs.24750
Chitwan Machan Rs.16500 Rs.28875 Rs.33000 Rs.41250 Rs.49500 Rs.24750 Rs.24750
Chitwan Ice Land Rs.14333 Rs.25082.75 Rs.28666 Rs.35832.5 Rs.42999 Rs.21499.5 Rs.21499.5
Chitwan Tharu Village Rs.16500 Rs.28875 Rs.33000 Rs.41250 Rs.49500 Rs.24750 Rs.24750
Chitwan Temple Tiger Rs.16500 Rs.28875 Rs.33000 Rs.41250 Rs.49500 Rs.24750 Rs.24750
Chitwan Meghauli Rs.16500 Rs.28875 Rs.33000 Rs.41250 Rs.49500 Rs.24750 Rs.24750
Changunarayan Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Narayanghat Rs.13000 Rs.22750 Rs.26000 Rs.32500 Rs.39000 Rs.19500 Rs.19500
Narayani Safari/Machan/Paradise Rs.16400 Rs.28700 Rs.32800 Rs.41000 Rs.49200 Rs.24600 Rs.24600
Ram Janaki Temple, Janakpur Rs.30190 Rs.52832.5 Rs.60380 Rs.75475 Rs.90570 Rs.45285 Rs.45285
Gaur, Rautahat Rs.26500 Rs.46375 Rs.53000 Rs.66250 Rs.79500 Rs.39750 Rs.39750
Malangwa, Sarlahi Rs.28000 Rs.49000 Rs.56000 Rs.70000 Rs.84000 Rs.42000 Rs.42000
Lahan Rs.35000 Rs.61250 Rs.70000 Rs.87500 Rs.105000 Rs.52500 Rs.52500
Kakarvitta Rs.49000 Rs.85750 Rs.98000 Rs.122500 Rs.147000 Rs.73500 Rs.73500
Dharan Rs.43750 Rs.76562.5 Rs.87500 Rs.109375 Rs.131250 Rs.65625 Rs.65625
Dhankutta Rs.47575 Rs.83256.25 Rs.95150 Rs.118937.5 Rs.142725 Rs.71362.5 Rs.71362.5
Hile Rs.24000 Rs.42000 Rs.48000 Rs.60000 Rs.72000 Rs.36000 Rs.36000
Bhairahawa Sunauli Rs.24000 Rs.42000 Rs.48000 Rs.60000 Rs.72000 Rs.36000 Rs.36000
Lumbini Rs.25778 Rs.45111.5 Rs.51556 Rs.64445 Rs.77334 Rs.38667 Rs.38667
Tansen, Palpa Rs.25000 Rs.43750 Rs.50000 Rs.62500 Rs.75000 Rs.37500 Rs.37500
Krishna Nagar Rs.29000 Rs.50750 Rs.58000 Rs.72500 Rs.87000 Rs.43500 Rs.43500
Nepalgunj, Banke Rs.42000 Rs.73500 Rs.84000 Rs.105000 Rs.126000 Rs.63000 Rs.63000
Dang Rs.35500 Rs.62125 Rs.71000 Rs.88750 Rs.106500 Rs.53250 Rs.53250
Bardiya National Park Rs.46500 Rs.81375 Rs.93000 Rs.116250 Rs.139500 Rs.69750 Rs.69750
Mahendranagar Rs.56500 Rs.98875 Rs.113000 Rs.141250 Rs.169500 Rs.84750 Rs.84750
Gokarneshwor Rs.2500 Rs.4375 Rs.5000 Rs.6250 Rs.7500 Rs.3750 Rs.3750
Gokarna Forest resort Rs.2500 Rs.4375 Rs.5000 Rs.6250 Rs.7500 Rs.3750 Rs.3750
Kapan Rs.2500 Rs.4375 Rs.5000 Rs.6250 Rs.7500 Rs.3750 Rs.3750
Bhaktapur Rs.2500 Rs.4375 Rs.5000 Rs.6250 Rs.7500 Rs.3750 Rs.3750
Kritipur Chovar Rs.2600 Rs.4550 Rs.5200 Rs.6500 Rs.7800 Rs.3900 Rs.3900
Bungmati Khokana Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Godawari Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Vajrabarahi chapagaun Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Dakshinkali Rs.3150 Rs.5512.5 Rs.6300 Rs.7875 Rs.9450 Rs.4725 Rs.4725
Hattiban Resort Rs.4000 Rs.7000 Rs.8000 Rs.10000 Rs.12000 Rs.6000 Rs.6000
Sundarijal Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Shivapuri, Sundarijal Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Sankhu Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Sanga vanjyang (Shiva Temple) Rs.2468 Rs.4319 Rs.4936 Rs.6170 Rs.7404 Rs.3702 Rs.3702
Shiva Temple, Sanga (Kailashnath Mahadev Statue) Rs.4000 Rs.7000 Rs.8000 Rs.10000 Rs.12000 Rs.6000 Rs.6000
Dhulikhel, Kavrepalanchok Rs.3150 Rs.5512.5 Rs.6300 Rs.7875 Rs.9450 Rs.4725 Rs.4725
Dhulikhel Picnic Spot Rs.3150 Rs.5512.5 Rs.6300 Rs.7875 Rs.9450 Rs.4725 Rs.4725
Panauti Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Dhulikhel, Kawa Rs.3150 Rs.5512.5 Rs.6300 Rs.7875 Rs.9450 Rs.4725 Rs.4725
Dhulikhel, Panauti Rs.4800 Rs.8400 Rs.9600 Rs.12000 Rs.14400 Rs.7200 Rs.7200
Bhaktapur and Nagarkot Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Nagarkot Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Lakuri Vanjyang Rs.6000 Rs.10500 Rs.12000 Rs.15000 Rs.18000 Rs.9000 Rs.9000
Bhaktapur, Changunarayan Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Nagarjun Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Nagarjun and Balaju Rs.5200 Rs.9100 Rs.10400 Rs.13000 Rs.15600 Rs.7800 Rs.7800
Jamacho Gumba, Nagarjun Rs.7000 Rs.12250 Rs.14000 Rs.17500 Rs.21000 Rs.10500 Rs.10500
Kakani Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Bhaktapur, Dhulikhel Rs.4000 Rs.7000 Rs.8000 Rs.10000 Rs.12000 Rs.6000 Rs.6000
Phulchoki and Godawari Rs.6000 Rs.10500 Rs.12000 Rs.15000 Rs.18000 Rs.9000 Rs.9000
Phulchoki Rs.7150 Rs.12512.5 Rs.14300 Rs.17875 Rs.21450 Rs.10725 Rs.10725
Bhaktapur, Changunarayan & Nagarkot Rs.5250 Rs.9187.5 Rs.10500 Rs.13125 Rs.15750 Rs.7875 Rs.7875
Pathibhara, Nallu Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Shivapuri, Budanilkantha Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Lele Manakamana Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Chandragiri Rs.5000 Rs.8750 Rs.10000 Rs.12500 Rs.15000 Rs.7500 Rs.7500
Rasuwagadhi Rs.17000 Rs.29750 Rs.34000 Rs.42500 Rs.51000 Rs.25500 Rs.25500
Daman Rs.10238 Rs.17916.5 Rs.20476 Rs.25595 Rs.30714 Rs.15357 Rs.15357
Kulekhani Rs.8000 Rs.14000 Rs.16000 Rs.20000 Rs.24000 Rs.12000 Rs.12000
Halesi, Khotang Rs.18500 Rs.32375 Rs.37000 Rs.46250 Rs.55500 Rs.27750 Rs.27750
Damauli, Tanahun Rs.11393 Rs.19937.75 Rs.22786 Rs.28482.5 Rs.34179 Rs.17089.5 Rs.17089.5
Jomsom, Mustang Rs.42000 Rs.73500 Rs.84000 Rs.105000 Rs.126000 Rs.63000 Rs.63000
Chitwan Gaighat Rs.12000 Rs.21000 Rs.24000 Rs.30000 Rs.36000 Rs.18000 Rs.18000
Chitwan Hetauda Rs.18720 Rs.32760 Rs.37440 Rs.46800 Rs.56160 Rs.28080 Rs.28080
Chitwan Birgunj Rs.23058 Rs.40351.5 Rs.46116 Rs.57645 Rs.69174 Rs.34587 Rs.34587
Siraha Rs.30503 Rs.53380.25 Rs.61006 Rs.76257.5 Rs.91509 Rs.45754.5 Rs.45754.5
Chitwan Malangwa Rs.28000 Rs.49000 Rs.56000 Rs.70000 Rs.84000 Rs.42000 Rs.42000
Phidim, Panchthar Rs.55052 Rs.96341 Rs.110104 Rs.137630 Rs.165156 Rs.82578 Rs.82578
Kulekhani Birgunj Rs.18000 Rs.31500 Rs.36000 Rs.45000 Rs.54000 Rs.27000 Rs.27000
Kulekhani Gaur Rs.18375 Rs.32156.25 Rs.36750 Rs.45937.5 Rs.55125 Rs.27562.5 Rs.27562.5
Kulekhani Malangwa Rs.26000 Rs.45500 Rs.52000 Rs.65000 Rs.78000 Rs.39000 Rs.39000
Ghorahi, Dang Rs.35500 Rs.62125 Rs.71000 Rs.88750 Rs.106500 Rs.53250 Rs.53250
Pyuthan Rs.31605 Rs.55308.75 Rs.63210 Rs.79012.5 Rs.94815 Rs.47407.5 Rs.47407.5
Chitwan Janakpur Rs.30190 Rs.52832.5 Rs.60380 Rs.75475 Rs.90570 Rs.45285 Rs.45285
Gaighat, Udayapur Rs.36650 Rs.64137.5 Rs.73300 Rs.91625 Rs.109950 Rs.54975 Rs.54975
Sindhuli Rs.26901 Rs.47076.75 Rs.53802 Rs.67252.5 Rs.80703 Rs.40351.5 Rs.40351.5
Kulekhani Hetauda Rs.12000 Rs.21000 Rs.24000 Rs.30000 Rs.36000 Rs.18000 Rs.18000
Manthali, Ramechap Rs.13000 Rs.22750 Rs.26000 Rs.32500 Rs.39000 Rs.19500 Rs.19500 | {"url":"https://www.carrentnepal.com/place/kathmandu","timestamp":"2024-11-15T04:12:50Z","content_type":"text/html","content_length":"438339","record_id":"<urn:uuid:7410169c-3b17-48dc-bd43-2fa16962f32d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00143.warc.gz"} |
[M08] Obscurity
Language can be used to mislead and confuse, or to make certain ideas seem more profound than they really are. One main task of critical thinking is to identify these linguistic pitfalls. Let us
start with the first major pitfall - obscurity.
"Obscurity" here refers to unclear meaning. A concept or a linguistic expression can be unclear for various reasons. One reason is that it might be ambiguous, i.e. having more than one meaning. The
other reason is that it might be vague. A term is said to be vague if there are borderline cases where it is indeterminate as to whether it applies or not. Finally, a term might also have an unclear
meaning in that its meaning is incomplete. Let us look at these cases one by one.
§1. Ambiguity
There are actually different kinds of ambiguity:
Lexical ambiguity
This is a single word or term having more than one meaning in the language. For example, the word "deep" can mean profoundity ("What you have said is very deep."), or it can be used to describe
physical depth ("This hole is very deep"). Similarly for words like "young" (inexperienced or young of age), "bank" (river bank or financial institution), etc.
Referential ambiguity
It is not clear which thing or group is being referred to. This often arises when the context does not make it clear what a pronoun or quantifier is referring to.
• "Ally hit Georgia and then she started bleeding." Who is hurt? Ally or Georgia?
• "Everybody is coming to the party." Certainly "everybody" does not refer to every human being in the whole world. But then which group of people are we talking about? Of course in normal
situations the speaker usually has some specific group of people in mind.
• Many people like to make very general statements, such as "All politicians are corrupt". Literally, this statement implies that there is no politician who is not corrupted. But of course we can
think of many counterexamples to such a claim. So the person who makes the statement might say "I don't really mean each and every politician." But then who exactly are the people referred to?
Syntactic ambiguity
This means having more than one meaning because there is more than one way to interpret the grammatical structure. This can happen even when it is clear what the meanings of the individual words are.
• "We shall be discussing violence on TV." - It might mean the discussion will be conducted during a television programme, or it might mean violence on TV is the topic to be discussed.
When dealing with ambiguous language the thing to do is of course to clarify the meaning of the expression, for example by listing out all the different possible interpretations. This process of
removing ambiguity is call "disambiguation".
§2. Vagueness
An term is vague if it has an imprecise boundary. This means that there are cases where it is indeterminate whether the term applies or not. For example, a small but closed room with no windows or
doors and no light inside is certainly dark. If we switch on a 100W lightbulb inside it will become bright. But we turn on the dimmer for the light and dim the light slowly until it goes out, then
the room will gradually change from a bright room to a dark one. But there is no precise point at which the room suddenly ceases to be bright. Similarly, there is no precise point at which the room
suddenly becomes dark. The terms "dark" and "bright" do not have clear boundaries of applications in this situation, and we say that these terms are vague.
The term "a tall person" is also vague in that there are certain cases where it is hard to say whether a person is tall or not, but this indecision is not due to lack of knowledge about that person's
height. You might know exactly how tall that person is, but still you don't know whether he is tall or not. This is because the meaning of the term is not precise enough. Other examples of vague
terms : "heavy", "dark", "mountain", "clever", "cheap".
Notice that we should make a distinction between vagueness and ambiguity. A word can be vague even though it is not ambiguous, and an ambiguous term having more than one meaning would not be said to
be vague if the different meanings it has are very precise.
Vague terms can be useful in everyday life because often we do not have to be too precise. How precise we should be depends of course on the context.
Here is a form of bad argument about vagueness which we often encounter. The argument's conclusion is that there is really no difference between X and Y, and the reason is that there is no sharp
difference between them.
• Example : "There is really no such thing as objective truth or falsity. Whether something is true or false is often hard to say."
This is a bad argument because even though a distinction might have borderline cases, it does not follow that the distinction is not real. For example, it might sometimes be unclear whether a room is
dark or bright. But (a) there is still a real distinction between dark and bright rooms, and (b) there can be clear cases where we have one but not the other.
Vagueness should be avoided when we want to speak precisely, as vagueness decreases the informational content of a claim. For example, compare these sentences :
• He is quite old, actually exactly eighty years old.
• He is quite old, actually about eighty years old.
• He is quite old.
Many students often like to ask questions such as :
• Is there going to be a lot of homework for this course?" "Is the final exam going to be difficult?"
But of course words like "difficult" and "a lot" are vague.
Vague terms can make a claim vague and impossible to confirm or disprove.
• Horoscope predictions for example : "Be prepared for a change of direction this week as something crops up." - SCMP Sunday Post Magazine.
• "This piece of news is going to affect the market somewhat."
But of course one might try to use vagueness to one's advantage in order to be non-committal or imprecise.
• "As a minister I agree that to some extent I am responsible."
• "The government will deal with this problem in an appropriate manner when the right time comes."
§3. Incomplete Meaning
A term has an incomplete meaning if the property or relation it expresses depends on some further parameter to be specified by the context, either explicitly or implicitly. This includes terms such
as "useful", "important", "similar" and "better". Practically all objects are useful and important only in some respects but not others. For example, is love more important than money? Well, it
depends. If you are starving to death, then money is more important. But if you are trying to determine which of the two contributes more to a happy and fulfilling life, then the answer might be
So just saying that something is useful or important is empty unless it is made clear in what way it is so. This is also necessary if we want to evaluate whether what is said is true or not.
• "The education director shall visit Scotland to study their educational system because it is similar to the one in Hong Kong."
• "Will this year's final exam be similar to the one last year?"
• "It is better to be beautiful than to be good. But . . . it is better to be good than to be ugly." Oscar Wilde (1854 - 1900)
• "Art never improves, but . . . the material of art is never quite the same." T. S. Eliot (1888 - 1965)
See if you can identify the ways in which these examples are ambiguous.
1. For sale - an antique table suitable for lady with thick legs.
2. For sale - ten puppies from an Australian terrier and a Boston terrier.
3. He left the bomb fifty yards to the right of the car in front of the house.
4. Mary loves Peter and Paul and Susan loves him too.
5. It is not advisable to take aspirin and alcohol after a meal.
6. I saw her duck.
7. The teacher hit the student with a stick.
8. Tiffany worries about annoying taxi drivers.
9. The old men and women sat at the front of the hall.
10. The CEO about his biggest fears, global warming and bitcoins.
How would you improve the precision or clarity of these claims?
1. It's going to take a really long time to complete this project.
2. We predict that a lot of people will come to the party.
§4. Bullshit
There is a lot of pseudo-aphorisms which seem to be profound, but close to being nonsensical. They have little content because of their obscurity, but are not quite nonsensical because they are
grammatical and contain a lot of buzzwords. Here are some examples:
• Intuition requires exploration. Intuition is the knowledge of life-force, and of us.
• You can go to that ultimate ground of creation and introduce an intention, and just by introducing the intention, you activate the field of infinite correlation.
• Synchronicity requires exploration.
• Truth is the consciousness that wants to know.
• This life is nothing short of a refining rebirth of holistic rejuvenation.
• We are being called to explore the nexus itself as an interface between spacetime and self-actualization.
Some of these quotes were actually randomly generated by a computer (see http://sebpearce.com/bullshit/). Two are actual quotes from a popular new-age guru Deepak Chopra. Can you tell them apart?
Some scientists have found that people who are more receptive to this kind of statements are less analytical and reflective. See Pennycook et al. (2015) On the reception and detection of
pseudo-profound bullshit. Judgment and Decision Making, Vol. 10, No. 6, pp. 549–563. | {"url":"https://philosophy.hku.hk/think/meaning/pitfalls1.php","timestamp":"2024-11-06T01:34:02Z","content_type":"text/html","content_length":"21214","record_id":"<urn:uuid:a58851c7-86ab-4b41-a98f-b64fd0cb973b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00736.warc.gz"} |
1037 - Cow Contest
N (1 <= N <= 100) cows, conveniently numbered 1..N, are participating
in a programming contest. As we all know, some cows code better
than others. Each cow has a certain constant skill rating that is
unique among the competitors.
The contest is conducted in several head-to-head rounds, each between
two cows. If cow A has a greater skill level than cow B (1 <= A <=
N; 1 <= B <= N; A != B), then cow A will always beat cow B.
Farmer John is trying to rank the cows by skill level. Given a list
the results of M (1 <= M <= 4,500) two-cow rounds, determine the
number of cows whose ranks can be precisely determined from the
results. It is guaranteed that the results of the rounds will not
be contradictory.
* Line 1: Two space-separated integers: N and M
* Lines 2..M+1: Each line contains two space-separated integers that
describe the competitors and results (the first integer, A, is
the winner) of a single round of competition: A and B
* Line 1: A single integer representing the number of cows whose ranks
can be determined
sample input
sample output
USACO JAN08 | {"url":"http://hustoj.org/problem/1037","timestamp":"2024-11-13T15:15:21Z","content_type":"text/html","content_length":"8404","record_id":"<urn:uuid:2b9e45ff-dca9-4400-95a6-e1def2fc1182>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00422.warc.gz"} |
The Overlapping Generations Model
26. The Overlapping Generations Model#
In this lecture we study the famous overlapping generations (OLG) model, which is used by policy makers and researchers to examine
• fiscal policy
• monetary policy
• long-run growth
and many other topics.
The first rigorous version of the OLG model was developed by Paul Samuelson [Samuelson, 1958].
Our aim is to gain a good understanding of a simple version of the OLG model.
26.1. Overview#
The dynamics of the OLG model are quite similar to those of the Solow-Swan growth model.
At the same time, the OLG model adds an important new feature: the choice of how much to save is endogenous.
To see why this is important, suppose, for example, that we are interested in predicting the effect of a new tax on long-run growth.
We could add a tax to the Solow-Swan model and look at the change in the steady state.
But this ignores the fact that households will change their savings and consumption behavior when they face the new tax rate.
Such changes can substantially alter the predictions of the model.
Hence, if we care about accurate predictions, we should model the decision problems of the agents.
In particular, households in the model should decide how much to save and how much to consume, given the environment that they face (technology, taxes, prices, etc.)
The OLG model takes up this challenge.
We will present a simple version of the OLG model that clarifies the decision problem of households and studies the implications for long-run growth.
Let’s start with some imports.
import numpy as np
from scipy import optimize
from collections import namedtuple
import matplotlib.pyplot as plt
26.2. Environment#
We assume that time is discrete, so that \(t=0, 1, \ldots\).
An individual born at time \(t\) lives for two periods, \(t\) and \(t + 1\).
We call an agent
• “young” during the first period of their lives and
• “old” during the second period of their lives.
Young agents work, supplying labor and earning labor income.
They also decide how much to save.
Old agents do not work, so all income is financial.
Their financial income is from interest on their savings from wage income, which is then combined with the labor of the new young generation at \(t+1\).
The wage and interest rates are determined in equilibrium by supply and demand.
To make the algebra slightly easier, we are going to assume a constant population size.
We normalize the constant population size in each period to 1.
We also suppose that each agent supplies one “unit” of labor hours, so total labor supply is 1.
26.3. Supply of capital#
First let’s consider the household side.
26.3.1. Consumer’s problem#
Suppose that utility for individuals born at time \(t\) takes the form
\[ U_t = u(c_t) + \beta u(c_{t+1})\]
• \(u: \mathbb R_+ \to \mathbb R\) is called the “flow” utility function
• \(\beta \in (0, 1)\) is the discount factor
• \(c_t\) is time \(t\) consumption of the individual born at time \(t\)
• \(c_{t+1}\) is time \(t+1\) consumption of the same individual
We assume that \(u\) is strictly increasing.
Savings behavior is determined by the optimization problem
\[ \max_{c_t, c_{t+1}} \, \left \{ u(c_t) + \beta u(c_{t+1}) \right \} \]
subject to
\[ c_t + s_t \le w_t \quad \text{and} \quad c_{t+1} \le R_{t+1} s_t \]
• \(s_t\) is savings by an individual born at time \(t\)
• \(w_t\) is the wage rate at time \(t\)
• \(R_{t+1}\) is the gross interest rate on savings invested at time \(t\), paid at time \(t+1\)
Since \(u\) is strictly increasing, both of these constraints will hold as equalities at the maximum.
Using this fact and substituting \(s_t\) from the first constraint into the second we get \(c_{t+1} = R_{t+1}(w_t - c_t)\).
The first-order condition for a maximum can be obtained by plugging \(c_{t+1}\) into the objective function, taking the derivative with respect to \(c_t\), and setting it to zero.
This leads to the Euler equation of the OLG model, which describes the optimal intertemporal consumption dynamics:
\[ u'(c_t) = \beta R_{t+1} u'( R_{t+1} (w_t - c_t))\]
From the first constraint we get \(c_t = w_t - s_t\), so the Euler equation can also be expressed as
\[ u'(w_t - s_t) = \beta R_{t+1} u'( R_{t+1} s_t)\]
Suppose that, for each \(w_t\) and \(R_{t+1}\), there is exactly one \(s_t\) that solves (26.4).
Then savings can be written as a fixed function of \(w_t\) and \(R_{t+1}\).
We write this as
\[ s_t = s(w_t, R_{t+1})\]
The precise form of the function \(s\) will depend on the choice of flow utility function \(u\).
Together, \(w_t\) and \(R_{t+1}\) represent the prices in the economy (price of labor and rental rate of capital).
Thus, (26.5) states the quantity of savings given prices.
26.3.2. Example: log preferences#
In the special case \(u(c) = \log c\), the Euler equation simplifies to \(s_t= \beta (w_t - s_t)\).
Solving for saving, we get
\[ s_t = s(w_t, R_{t+1}) = \frac{\beta}{1+\beta} w_t\]
In this special case, savings does not depend on the interest rate.
26.3.3. Savings and investment#
Since the population size is normalized to 1, \(s_t\) is also total savings in the economy at time \(t\).
In our closed economy, there is no foreign investment, so net savings equals total investment, which can be understood as supply of capital to firms.
In the next section we investigate demand for capital.
Equating supply and demand will allow us to determine equilibrium in the OLG economy.
26.4. Demand for capital#
First we describe the firm’s problem and then we write down an equation describing demand for capital given prices.
26.4.1. Firm’s problem#
For each integer \(t \geq 0\), output \(y_t\) in period \(t\) is given by the Cobb-Douglas production function
\[ y_t = k_t^{\alpha} \ell_t^{1-\alpha}\]
Here \(k_t\) is capital, \(\ell_t\) is labor, and \(\alpha\) is a parameter (sometimes called the “output elasticity of capital”).
The profit maximization problem of the firm is
\[ \max_{k_t, \ell_t} \{ k^{\alpha}_t \ell_t^{1-\alpha} - R_t k_t -w_t \ell_t \}\]
The first-order conditions are obtained by taking the derivative of the objective function with respect to capital and labor respectively and setting them to zero:
\[ (1-\alpha)(k_t / \ell_t)^{\alpha} = w_t \quad \text{and} \quad \alpha (k_t / \ell_t)^{\alpha - 1} = R_t\]
26.4.2. Demand#
Using our assumption \(\ell_t = 1\) allows us to write
\[ w_t = (1-\alpha)k_t^\alpha \]
\[ R_t = \alpha k_t^{\alpha - 1} \]
Rearranging (26.10) gives the aggregate demand for capital at time \(t+1\)
\[ k^d (R_{t+1}) := \left (\frac{\alpha}{R_{t+1}} \right )^{1/(1-\alpha)}\]
In Python code this is
def capital_demand(R, α):
return (α/R)**(1/(1-α))
def capital_supply(R, β, w):
R = np.ones_like(R)
return R * (β / (1 + β)) * w
The next figure plots the supply of capital, as in (26.6), as well as the demand for capital, as in (26.11), as functions of the interest rate \(R_{t+1}\).
(For the special case of log utility, supply does not depend on the interest rate, so we have a constant function.)
26.5. Equilibrium#
In this section we derive equilibrium conditions and investigate an example.
26.5.1. Equilibrium conditions#
In equilibrium, savings at time \(t\) equals investment at time \(t\), which equals capital supply at time \(t+1\).
Equilibrium is computed by equating these quantities, setting
\[ s(w_t, R_{t+1}) = k^d(R_{t+1}) = \left (\frac{\alpha}{R_{t+1}} \right )^{1/(1-\alpha)}\]
In principle, we can now solve for the equilibrium price \(R_{t+1}\) given \(w_t\).
(In practice, we first need to specify the function \(u\) and hence \(s\).)
When we solve this equation, which concerns time \(t+1\) outcomes, time \(t\) quantities are already determined, so we can treat \(w_t\) as a constant.
From equilibrium \(R_{t+1}\) and (26.11), we can obtain the equilibrium quantity \(k_{t+1}\).
26.5.2. Example: log utility#
In the case of log utility, we can use (26.12) and (26.6) to obtain
\[ \frac{\beta}{1+\beta} w_t = \left( \frac{\alpha}{R_{t+1}} \right)^{1/(1-\alpha)}\]
Solving for the equilibrium interest rate gives
\[ R_{t+1} = \alpha \left( \frac{\beta}{1+\beta} w_t \right)^{\alpha-1}\]
In Python we can compute this via
def equilibrium_R_log_utility(α, β, w):
R = α * ( (β * w) / (1 + β))**(α - 1)
return R
In the case of log utility, since capital supply does not depend on the interest rate, the equilibrium quantity is fixed by supply.
That is,
\[ k_{t+1} = s(w_t, R_{t+1}) = \frac{\beta }{1+\beta} w_t\]
Let’s redo our plot above but now inserting the equilibrium quantity and price.
R_vals = np.linspace(0.3, 1)
α, β = 0.5, 0.9
w = 2.0
fig, ax = plt.subplots()
ax.plot(R_vals, capital_demand(R_vals, α),
label="aggregate demand")
ax.plot(R_vals, capital_supply(R_vals, β, w),
label="aggregate supply")
R_e = equilibrium_R_log_utility(α, β, w)
k_e = (β / (1 + β)) * w
ax.plot(R_e, k_e, 'o',label='equilibrium')
26.6. Dynamics#
In this section we discuss dynamics.
For now we will focus on the case of log utility, so that the equilibrium is determined by (26.15).
26.6.1. Evolution of capital#
The discussion above shows how equilibrium \(k_{t+1}\) is obtained given \(w_t\).
From (26.9) we can translate this into \(k_{t+1}\) as a function of \(k_t\)
In particular, since \(w_t = (1-\alpha)k_t^\alpha\), we have
\[ k_{t+1} = \frac{\beta}{1+\beta} (1-\alpha)(k_t)^{\alpha}\]
If we iterate on this equation, we get a sequence for capital stock.
Let’s plot the 45-degree diagram of these dynamics, which we write as
\[ k_{t+1} = g(k_t) \quad \text{where } g(k) := \frac{\beta}{1+\beta} (1-\alpha)(k)^{\alpha} \]
def k_update(k, α, β):
return β * (1 - α) * k**α / (1 + β)
α, β = 0.5, 0.9
kmin, kmax = 0, 0.1
n = 1000
k_grid = np.linspace(kmin, kmax, n)
k_grid_next = k_update(k_grid,α,β)
fig, ax = plt.subplots(figsize=(6, 6))
ymin, ymax = np.min(k_grid_next), np.max(k_grid_next)
ax.plot(k_grid, k_grid_next, lw=2, alpha=0.6, label='$g$')
ax.plot(k_grid, k_grid, 'k-', lw=1, alpha=0.7, label='$45^{\circ}$')
ax.legend(loc='upper left', frameon=False, fontsize=12)
ax.set_xlabel('$k_t$', fontsize=12)
ax.set_ylabel('$k_{t+1}$', fontsize=12)
26.6.2. Steady state (log case)#
The diagram shows that the model has a unique positive steady state, which we denote by \(k^*\).
We can solve for \(k^*\) by setting \(k^* = g(k^*)\), or
\[ k^* = \frac{\beta (1-\alpha) (k^*)^{\alpha}}{(1+\beta)}\]
Solving this equation yields
\[ k^* = \left (\frac{\beta (1-\alpha)}{1+\beta} \right )^{1/(1-\alpha)}\]
We can get the steady state interest rate from (26.10), which yields
\[ R^* = \alpha (k^*)^{\alpha - 1} = \frac{\alpha}{1 - \alpha} \frac{1 + \beta}{\beta} \]
In Python we have
k_star = ((β * (1 - α))/(1 + β))**(1/(1-α))
R_star = (α/(1 - α)) * ((1 + β) / β)
26.6.3. Time series#
The 45-degree diagram above shows that time series of capital with positive initial conditions converge to this steady state.
Let’s plot some time series that visualize this.
ts_length = 25
k_series = np.empty(ts_length)
k_series[0] = 0.02
for t in range(ts_length - 1):
k_series[t+1] = k_update(k_series[t], α, β)
fig, ax = plt.subplots()
ax.plot(k_series, label="capital series")
ax.plot(range(ts_length), np.full(ts_length, k_star), 'k--', label="$k^*$")
ax.set_ylim(0, 0.1)
If you experiment with different positive initial conditions, you will see that the series always converges to \(k^*\).
Below we also plot the gross interest rate over time.
R_series = α * k_series**(α - 1)
fig, ax = plt.subplots()
ax.plot(R_series, label="gross interest rate")
ax.plot(range(ts_length), np.full(ts_length, R_star), 'k--', label="$R^*$")
ax.set_ylim(0, 4)
ax.set_ylabel("gross interest rate")
The interest rate reflects the marginal product of capital, which is high when capital stock is low.
26.7. CRRA preferences#
Previously, in our examples, we looked at the case of log utility.
Log utility is a rather special case of CRRA utility with \(\gamma \to 1\).
In this section, we are going to assume that \(u(c) = \frac{ c^{1- \gamma}-1}{1-\gamma}\), where \(\gamma >0, \gamma\neq 1\).
This function is called the CRRA utility function.
In other respects, the model is the same.
Below we define the utility function in Python and construct a namedtuple to store the parameters.
def crra(c, γ):
return c**(1 - γ) / (1 - γ)
Model = namedtuple('Model', ['α', # Cobb-Douglas parameter
'β', # discount factor
'γ'] # parameter in CRRA utility
def create_olg_model(α=0.4, β=0.9, γ=0.5):
return Model(α=α, β=β, γ=γ)
Let’s also redefine the capital demand function to work with this namedtuple.
def capital_demand(R, model):
return (α/R)**(1/(1-model.α))
26.7.1. Supply#
For households, the Euler equation becomes
\[ (w_t - s_t)^{-\gamma} = \beta R^{1-\gamma}_{t+1} (s_t)^{-\gamma}\]
Solving for savings, we have
\[ s_t = s(w_t, R_{t+1}) = w_t \left [ 1 + \beta^{-1/\gamma} R_{t+1}^{(\gamma-1)/\gamma} \right ]^{-1}\]
Notice how, unlike the log case, savings now depends on the interest rate.
def savings_crra(w, R, model):
α, β, γ = model
return w / (1 + β**(-1/γ) * R**((γ-1)/γ))
26.7.2. Equilibrium#
Equating aggregate demand for capital (see (26.11)) with our new aggregate supply function yields equilibrium capital.
Thus, we set
\[ w_t \left [ 1 + \beta^{-1/\gamma} R_{t+1}^{(\gamma-1)/\gamma} \right ]^{-1} = \left (\frac{R_{t+1}}{\alpha} \right )^{1/(\alpha - 1)}\]
This expression is quite complex and we cannot solve for \(R_{t+1}\) analytically.
Combining (26.10) and (26.21) yields
\[ k_{t+1} = \left [ 1 + \beta^{-1/\gamma} (\alpha k^{\alpha - 1}_{t+1})^{(\gamma-1)/\gamma} \right ]^{-1} (1-\alpha)(k_t)^{\alpha}\]
Again, with this equation and \(k_t\) as given, we cannot solve for \(k_{t+1}\) by pencil and paper.
In the exercise below, you will be asked to solve these equations numerically.
26.8. Exercises#
Solve for the dynamics of equilibrium capital stock in the CRRA case numerically using (26.22).
Visualize the dynamics using a 45-degree diagram.
Solution to Exercise 26.1
To solve for \(k_{t+1}\) given \(k_t\) we use Newton’s method.
\[ f(k_{t+1}, k_t) = k_{t+1} \left[ 1 + \beta^{-1/\gamma} \left ( \alpha k^{\alpha-1}_{t+1} \right )^{(\gamma-1)/\gamma} \right] - (1-\alpha) k^{\alpha}_t =0\]
If \(k_t\) is given then \(f\) is a function of unknown \(k_{t+1}\).
Then we can use scipy.optimize.newton to solve \(f(k_{t+1}, k_t)=0\) for \(k_{t+1}\).
First let’s define \(f\).
def f(k_prime, k, model):
α, β, γ = model.α, model.β, model.γ
z = (1 - α) * k**α
a = α**(1-1/γ)
b = k_prime**((α * γ - α + 1) / γ)
p = k_prime + k_prime * β**(-1/γ) * a * b
return p - z
Now let’s define a function that finds the value of \(k_{t+1}\).
def k_update(k, model):
return optimize.newton(lambda k_prime: f(k_prime, k, model), 0.1)
Finally, here is the 45-degree diagram.
kmin, kmax = 0, 0.5
n = 1000
k_grid = np.linspace(kmin, kmax, n)
k_grid_next = np.empty_like(k_grid)
for i in range(n):
k_grid_next[i] = k_update(k_grid[i], model)
fig, ax = plt.subplots(figsize=(6, 6))
ymin, ymax = np.min(k_grid_next), np.max(k_grid_next)
ax.plot(k_grid, k_grid_next, lw=2, alpha=0.6, label='$g$')
ax.plot(k_grid, k_grid, 'k-', lw=1, alpha=0.7, label='$45^{\circ}$')
ax.legend(loc='upper left', frameon=False, fontsize=12)
ax.set_xlabel('$k_t$', fontsize=12)
ax.set_ylabel('$k_{t+1}$', fontsize=12)
The 45-degree diagram from the last exercise shows that there is a unique positive steady state.
The positive steady state can be obtained by setting \(k_{t+1} = k_t = k^*\) in (26.22), which yields
\[ k^* = \frac{(1-\alpha)(k^*)^{\alpha}} {1 + \beta^{-1/\gamma} (\alpha (k^*)^{\alpha-1})^{(\gamma-1)/\gamma}} \]
Unlike the log preference case, the CRRA utility steady state \(k^*\) cannot be obtained analytically.
Instead, we solve for \(k^*\) using Newton’s method.
Solution to Exercise 26.2
We introduce a function \(h\) such that positive steady state is the root of \(h\).
\[ h(k^*) = k^* \left [ 1 + \beta^{-1/\gamma} (\alpha (k^*)^{\alpha-1})^{(\gamma-1)/\gamma} \right ] - (1-\alpha)(k^*)^{\alpha}\]
Here it is in Python
def h(k_star, model):
α, β, γ = model.α, model.β, model.γ
z = (1 - α) * k_star**α
R1 = α ** (1-1/γ)
R2 = k_star**((α * γ - α + 1) / γ)
p = k_star + k_star * β**(-1/γ) * R1 * R2
return p - z
Let’s apply Newton’s method to find the root:
k_star = optimize.newton(h, 0.2, args=(model,))
print(f"k_star = {k_star}")
k_star = 0.25788950250843484
Generate three time paths for capital, from three distinct initial conditions, under the parameterization listed above.
Use initial conditions for \(k_0\) of \(0.001, 1.2, 2.6\) and time series length 10.
Solution to Exercise 26.3
Let’s define the constants and three distinct intital conditions
ts_length = 10
k0 = np.array([0.001, 1.2, 2.6])
def simulate_ts(model, k0_values, ts_length):
fig, ax = plt.subplots()
ts = np.zeros(ts_length)
# simulate and plot time series
for k_init in k0_values:
ts[0] = k_init
for t in range(1, ts_length):
ts[t] = k_update(ts[t-1], model)
ax.plot(np.arange(ts_length), ts, '-o', ms=4, alpha=0.6,
label=r'$k_0=%g$' %k_init)
ax.plot(np.arange(ts_length), np.full(ts_length, k_star),
alpha=0.6, color='red', label=r'$k^*$')
ax.set_xlabel(r'$t$', fontsize=14)
ax.set_ylabel(r'$k_t$', fontsize=14) | {"url":"https://intro.quantecon.org/olg.html","timestamp":"2024-11-12T02:26:41Z","content_type":"text/html","content_length":"110660","record_id":"<urn:uuid:f279c280-9a94-4a22-b2ef-b56b8b96fb48>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00358.warc.gz"} |
Effect of microstructural anisotropy on the fluid-particle drag force and the stability of the uniformly fluidized state
Lattice-Boltzmann simulations of fluid flow through sheared assemblies of monodisperse spherical particles have been performed. The friction coefficient tensor extracted from these simulations is
found to become progressively more anisotropic with increasing Péclet number, Pe= ̇γ d ^2, where ̇γ is the shear rate, d is the particle diameter, and d is the particle self-diffusivity. A model is
presented for the anisotropic friction coefficient, and the model constants are related to changes in the particle microstructure. Linear stability analysis of the two-fluid model equations including
the anisotropic drag force model developed in the present study reveals that the uniformly fluidized state of low Reynolds number suspensions is most unstable to mixed mode disturbances that take the
form of vertically travelling waves having both vertical and transverse structures. As the Stokes number increases, the transverse-to-vertical wavenumber ratio decreases towards zero; i.e. the
transverse structure becomes progressively less prominent. Fully nonlinear two-fluid model simulations of moderate to high Stokes number suspensions reveal that the anisotropic drag model leads to
coarser gas-particle flow structures than the isotropic drag model.
All Science Journal Classification (ASJC) codes
• Condensed Matter Physics
• Mechanics of Materials
• Mechanical Engineering
• Applied Mathematics
• complex fluids
• instability
• suspensions
Dive into the research topics of 'Effect of microstructural anisotropy on the fluid-particle drag force and the stability of the uniformly fluidized state'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/effect-of-microstructural-anisotropy-on-the-fluid-particle-drag-f","timestamp":"2024-11-07T03:40:00Z","content_type":"text/html","content_length":"54880","record_id":"<urn:uuid:fc00cb11-dfef-4bd0-aef6-9c05b9e97079>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00677.warc.gz"} |
Not tested, the offset values. python coding challenges - Py.CheckiO
import math
def checkio(a, b, c):
a_a = (b * b + c * c - a * a)/(2 * b * c)
a_a = math.acos(a_a)
b_b = float(a * a + c * c - b * b) / (2 * a * c)
b_b = math.acos(b_b)
ang.append(180 - ang[0] - ang[1])
return ang
return [0,0,0]
Attached screen which shows that in the fourth step, the result of why that is swapped. On my computer the result is in the proper order. Somebody please explain what is the problem? (Sorry for bad
english) To solve the problem using the cosine theorem
err_1- there is an error when checking
err_2 - Reverse the order of the variables and the right decision
my_comp - correct information on my computer
Created at: 2016/09/30 07:42; Updated at: 2016/09/30 09:06 | {"url":"https://py.checkio.org/forum/post/10218/not-tested-the-offset-values/","timestamp":"2024-11-13T04:59:27Z","content_type":"text/html","content_length":"24959","record_id":"<urn:uuid:7afd47b9-bb40-4b98-8b9c-ee362ddcadc9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00623.warc.gz"} |
Internal Fluctuations in a Population of Deer Mice with Hantavirus Infection
Special Atricle - Hantavirus
Austin J Infect Dis. 2016; 3(1): 1019.
Internal Fluctuations in a Population of Deer Mice with Hantavirus Infection
Reinoso JA* and de la Rubia JF
Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia (UNED), Spain
*Corresponding author: José A. Reinoso, Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia (UNED), Spain
Received: May 04, 2016; Accepted: May 25, 2016; Published: May 26, 2016
We study the role of internal fluctuations and the thermodynamic limit in the population dynamics of deer mice, and describe the evolution of infected mice with Sin Nombre virus. This virus is the
main cause of Hantavirus Pulmonary Syndrome (HPS) among humans in North America. In this way, we try to support those features observed in phenomenological models as the critical carrying capacity, K
[c] and the delay between population of mice and infected ones. We introduce the underlying processes, in particular the delayed maturation process, and derive from the master equation the mean field
description for the thermodynamic limit. It matches the phenomenological model. Then we compare the model with the numerical Gillespie algorithm for the long-term phenomenon related to El Ni&nTilde;o
southern oscillations. Internal fluctuations are able to drive the infection to extinction, mostly in the scenario of El Ni&nTilde;o, for both the transient and the steady state. We also study
analitically the steady state. On the other hand, the thermodynamic limit plays the opposite role, and supports the infection. In general, we see how those features observed in the phenomenological
description are where recovered both in the scenario related to La Ni&nTilde;a and in the thermodynamic limit.
The population dynamics of deer mouse is central to the study of Hantavirus Pulmonary Syndrome (HPS), and is the subject of intense research since in 1993 deer mouse was identified as the host of
SinNombre virus, which causes HPS [1].
Consequently, HPS cases are related to population of infected mice. We study this relation in terms of the basic epidemiological theory that suggests a link between HPS cases and contagion events. In
particular humans get infected mainly through the contact with mice, or the inhalation of an aerosolized mixture of virus, feces and dried urine particles. Nowadays the mortality rate due to HPS is
40% [2].
At the same time contagion events are correlated with available resources. In the long term phenomena they depend on the climate variations and in particular on El Ni&nTilde;o southern oscillations.
On the other side, the virus remains inside the mouse without causing its death and propagating among mice horizontally, i.e., from mouse to mouse, mainly through direct contact [2]. In this
direction several studies have pointed out how the number of infected mice is sensitive to El Ni&nTilde;o southern oscillations. During adverse periods the population of mice drastically decreases
and the virus may even disappear. While on the contrary, when conditions improve, there is a big increase of population, high enough to cause an outbreak of infection [2,3].
In order to study the infection in deer mice at long-term, several simple models have been proposed [4-6].
The first model corresponds to Abramson-Kenkre (AK model), and describes the dynamics in terms of 2 variables, susceptible and infected mice [4]. The fundamental parameter of the model is the
carrying capacity, K, that accounts for the amount of resources available for mice, and which value depends on the different scenarios related to El Ni&nTilde;o southern oscillations. When the
scenario corresponds to El Ni&nTilde;o the amount of resources is high and consequently K increases, together with the population. In this case when K is bigger than a critical value, K[c], the
infection spreads. While in La Ni&nTilde;a period, there are less resources and the scenario is related to a low value of K, and consequently the decrease in the number of mice. If K goes under K[c],
the infection disappears.
In a new model, developed by the authors, we introduce a slightly different scheme to take into account a division in terms of age [5]. It is based on field studies that claim young mice do not
contract the virus [7,8]. The model has 3 variables: young mice, susceptible adults and infected adults. It shows a characteristic time given by the maturation term, T, which produces a delay in both
the outbreak and disappearance of infection in relation with the population of mice.
These phenomenological models are also extended to other climatic variations as the climate change mainly through the amount of available resources. An estimation of those resources, described by K,
is crucial for the prediction and control of infection in areas where climate change is significant. As for El Ni&nTilde;o, good conditions are correlated with outbreaks while bad conditions are
correlated with the reduction or the eradication of the infection.
While these models are deterministic, real systems are discrete and the number of related mice finite. This approach requires a better description in order to see the relevance of internal
fluctuations and its relation with phenomenological models [9-11]. In particular, if those feature seen before for the phenomenological model in the long-term are supported by a more fundamental
description [5].
In section II we consider the analytical approach given by the master equation. After that, in section III, we compare it with the exact numerical description for both above and below the
thermodynamic limit. The numerical description is introduced by the modified Gillespie algorithm that considers non-markovian processes. In section IV, we first study fluctuations in the steady state
with a perturbative method (subsection IV A) and later numerically with the modified Gillespie algorithm (subsection IV B). We also compare both approaches. Finally, conclusions summarize the
Analytical Results
Due to the stochasticity of the system, one has to rely on statistics and try to determine in a more solid description those features already seen in the phenomenological description. We first start
writting down a general approach corresponding to the master equation. It describes the temporal evolution for the probability of the variables. In particular, we work with 3 variables: young mice,
Y, susceptible adults, S, and infected adults, I. In compact form they look as following:
X = (Y, S, I) and X’ = (Y', S', I’)
$dP(X,t) dt = ∑ X " ( ω X, X " P( X " ,t)− ω X " ,X P(X,t)) (1) MathType@MTEF@5@5@+=
In our case, the master equation is built on several processes that account for the different ingredients introduced in [5]. They are represented through the transition rates,$ω X, X "
Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaeqyYdC3aaSbaaSqaaiaadIfacaGGSaGaamiwamaaCaaameqabaGaai4jaaaaaSqabaaaaa@3B39@$ and $ω X " ,X
Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaeqyYdC3aaSbaaSqaaiaadIfadaahaaadbeqaaiaacEcaaaWccaGGSaGaamiwaaqabaaaaa@3B39@$ , and consist in
births, deaths, competition, contagion and maturation. They are all markovian processes except the maturation that lasts a finite time.
$Y → τ S (2) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=
In order to go further and be able to derive the master equation, it is necessary to study in depth the maturation process. In this way, we divide it into more manageable sub processes.
Among them, the first sub process corresponds to a birth. Second, a period that describes the time mouse overcomes youth, T. And third, how mouse becomes adult, Y → S. The probability of the whole
process is described as follows.
$∑ Y",S", I " P(Y,S,I,t+Δt;Y+1,S−1,I,t;Γ; ϒ Y",S", I " ) (3) MathType@MTEF@5@5@+=
We analyze each process in a more precise and mathematical form [12].
The probability starts with the summation of all possible initial states corresponding to births (Y ', S ', I ' ) . This probability is represented by $ϒ Y",S", I " MathType@MTEF@5@5@+=
yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaeuO0de6aaSbaaSqaaiaadMfacaGGNaGaaiilaiaadofacaGGNaGaaiilaiaadMeadaahaaadbeqaaiaacEcaaaaaleqaaaaa@3E3C@$, and corresponds to
the following expression:
$P( ϒ Y " S " I " )=b( S " + R " +( Y " −1))ΔtP( Y " , S " , I " ,t−τ) MathType@MTEF@5@5@+=
Once the mouse is born it enters in the maduration period, represented by G. It describes how the mouse becomes adult and approached by e^-?t, where t is the maduration period and ? the difficulty to
passing from youthhood to adulthood.
Finally, when the mouse arrives at (Y + 1, S - 1, I, t) it becomes a susceptible adult (Y, S, I, t + Δt). This last stage always happens when the other conditions fulfill.
At this point, we are able to write down the master equation in a more suitable form. In particular, we present it in terms of creation and destruction operators.
$Ef(X,t)=f(X+1,t) E -1 (X,t)=f(X-1,t) (4) MathType@MTEF@5@5@+=
The final master equation for all the processes [5,9], reads as follows.
$dP dt =( E Y −1 −1)b(Y+S+I)P+( E Y −1)cYP +( E S −1)cSP+ E I −1)cIP +( E Y −1) 1 2k Y(Y−1+S+I)P +( E S −1) 1 2k S(Y+S−1+I)P +( E I −1) 1 2k I(Y+S+I−1)P +( E s E I −1 −1)aSIP +( E S −1 E Y −1) ∑ Y "
S " I " P(Y,S,I,t/Γ; ϒ Y " S " I " ) e −γτ b( S " + I " +( Y " −1))P( Y " , S " , I " ,t−τ) (5) MathType@MTEF@5@5@+=
This expression describes the time evolution for the probability of the 3 variables (Y,S,I). However, the set of equations is not closed and cannot be solved directly. On the other hand it is
possible to get insight looking at different moments. In particular we study the first moment. For this case we approximate $X i X j ¯ = X i ¯ X j ¯ MathType@MTEF@5@5@+=
, where i and j indicate the different variables
$d Y ¯ dt =b M ¯ −c Y ¯ − Y ¯ ( M ¯ −1) 2k −b e −λt ( M ¯ (t−τ)−1) ( 6 ) MathType@MTEF@5@5@+=
$d S ¯ dt =b e −λτ ( M ¯ (t−τ)−1−c S ¯ − S ¯ ( M ¯ −1) 2k −a S ¯ I ¯ ( 7 ) MathType@MTEF@5@5@+=
$d I ¯ dt =−c I ¯ − I ¯ ( M ¯ −1) 2k +a S ¯ I ¯ ( 8 ) MathType@MTEF@5@5@+=
This new description corresponds to the mean values of the probability. We can go a step further and consider the thermodynamic limit as a particular case. If N → ∞ and Ω → ∞ keeping constant N/Ω,
and considering the density instead of the number of mice, the remaining expressions are
$d M Y dt =bM−c M Y − M Y M K −b e −λτ M(t−τ) (9) MathType@MTEF@5@5@+=
$d M As dt =b e −λτ (t−τ)−c M As − M As M K −a M As M Ai (10) MathType@MTEF@5@5@+=
$d M Ai dt =−c M Ai − M Ai M K −a M As M Ai (11) MathType@MTEF@5@5@+=
Where, Y As Ai Y As Ai $M Y = Y ¯ Ω , M S = S ¯ Ω , M I = I ¯ Ω , and M= M Y + M S + M I MathType@MTEF@5@5@+=
Ù Ù Ù Parameters do not change, while a=aΩ and K=2k/Ω.
This mean field approach is in consonance with the phenomenological model introduced in [5]. In this case, the phenomenological description and its features are capture in the thermodynamic limit.
However, we still do not know if those features are also valid in regions where internal fluctuations are significant.
Numerical Studies Comparision
The delayed Gillespie algorithm has been studied recently [13- 15]. Following the exact scheme developed by Cai, we have adapted it to our system by introducing a probability in the process that
governs the transition to adulthood.
To study the role of fluctuations we identify two different scenarios depending on climatic conditions as the El Ni&nTilde;o southern oscillations [5]. Scenario A corresponds to favorable conditions
(El Ni&nTilde;o) for the increase of population and the subsequent outbreak of infection. And scenario B where the conditions are harsh (La Ni&nTilde;a), and consequently the number of mice decrease
together with the infection.
We will describe the system in both, in and out the thermodynamic limit. However, computational capabilities constrain our simulations to real domain (finite Ω). In order to get the thermodynamic
limit we introduce fluctuations in the number of infected mice i.e., mice coming from adjacent niches. We introduce these fluctuations as a minimum source of infected mice.
A. Scenario A
In scenario A, the phenomenological model corresponds to K > K[c], is favorable to the increase of population [4,5]. As the virus spreads among adults, there is a delay between the population growth
generated by the increase of youth, and the infection, which occurs when youth mice become adults. It is given by the maturation time, t, and characterizes the system [5]. We see how the evolution of
the system comes into 2 different and consecutive time intervals. First, from (0, τ), the system evolves towards the absence of infection, and second, in (τ, ∞), the system evolves towards the
outbreak of infection.
In (Figures 1a & b), we see the evolution of the mean value for those realizations above (dashed (red) line) and below thermodynamic limit (dash-dotted (magenta) line). In (Figure 1a), for the case
of infected mice, the dash-dotted line does not follow the phenomenological model (solid (blue) line). It is due to the 0 absorbent state, reachable mainly in the interval (0, τ). In this scenario
infection can disappear, and the outbreak of infection may not happen.
Figure 1: (Color online) Comparison of the temporal evolution for the phenomenological model in solid (blue) line and the mean value of realizations, in dash-dotted (magenta) line below the
thermodynamic limit, while the dashed (red) line above it. Both set of realizations are obtained from the modified Gillespie algorithm. It describes scenario A. In (a) & (b), it is depicted M[ and M
respectively. The carrying capacity is K = 15. K > K[c].]
Figure 1: (Color online) Comparison of the temporal evolution for the phenomenological model in solid (blue) line and the mean value of realizations, in dash-dotted (magenta) line below the thermodynamic limit, while the dashed (red) line above it. Both set of realizations are obtained from the modified Gillespie algorithm. It describes scenario A. In (a) & (b), it is depicted M[ and M respectively. The carrying capacity is K = 15. K > K[c].
The dashed (red) line corresponds to the thermodynamic limit, and evolves in consonance with the phenomenological description. In this case the absence of infection is not reachable. For the period
(τ, ∞), the system evolves towards the outbreak of infection.
B. Scenario B
When climatic conditions are harsh, the phenomenological model is described by K < K[c], and the population decreases (see Figures 2a & b) following general trends [4,5]. The infection tends to
disappear after a period of persistence given by the maturation time (see Figures 2a) [5]. This persistence is a serious thread that could lead to HPS cases.
In this scenario, (Figures 2a & b), fluctuations do not play any fundamental role since realizations evolve from finite values of infected mice in (0, τ) towards the lack of infection in the interval
(τ, ∞), and both kind of realizations, below and above the thermodynamic limit fit qualitatively well with the phenomenological model.
It is also worth mentioning how the evolution of the number of mice in regimes, scenario A (Figure 1b) and B (Figure 2b), and the phenomenological model fit well and are described by the logistic
equation [5].
Figure 2: (Color online) Comparison of the temporal evolution as in figure 1 for scenario B. Both situations, above and below the thermodynamic limit, evolve together. The carrying capacity takes the
following value, K = 7.5. K < K[c].
Figure 1: (Color online) Comparison of the temporal evolution as in figure 1 for scenario B. Both situations, above and below the thermodynamic limit, evolve together. The carrying capacity takes the following value, K = 7.5. K < K[c].
Steady State
In this section we study the probability density for the steady state in scenario A. First we introduce the analytical perturbative method and later we study numerically the steady state through the
modified Gillespie algorithm. Finally, we compare both approaches.
A. Analytical approach
It is possible to develop a stochastic model based on the mean field description from which we can derive an approximation to the stationary probability for infected mice [9]. Let us decompose M in
its steady mean value and its internal fluctuations: M + δM.
$M(t)=K(b−c)+δM(t) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=
xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaamytaiaacIcacaWG0bGaaiykaiabg2da9iaadUeacaGGΩaGaamOyaiabgkHiTiaadogacaGGPaGaey4kaSIaeqiTdqMaamytaiaacIcacaWG0bGaaiykaaaa@44AC@$ (12)
Where δM (t) is a random variable whose probability is given by the following expression with a white noise,ξ . [9].
$dδM(t) dt =−(b−c)δM+ 2Kb(b−c)ξ(t) MathType@MTEF@5@5@+=
$<δM>=0 < (δM) 2 >=Kb <δM(t)δM( t " )>=Kb e −(b−c)| t− t " | (14) MathType@MTEF@5@5@+=
We start from the mean field description (see equations. 9,10 and 11), and consider that δM is independent of δM (t-τ). Through some calculations, we arrive at a stochastic description for MAi.
$d M Ai dt =(aK(b−c) e −λτ −b) M Ai −a M Ai 2 − M Ai δZ (15) MathType@MTEF@5@5@+=
$<δZ>=0 < (δZ) 2 >= C 2 <δZ(t)δZ( t " )>= C 2 e −(b−c)| t− t " | ( 16 ) MathType@MTEF@5@5@+=
$C 2 =Kb( ( K −1 −a) 2 + a 2 R 2 c(2b−c) +a( K −1 −a)(c(1− e −λτ )( 1 c + 1 (2b−c) MathType@MTEF@5@5@+=
This is a stochastic equation with colored noise. The approximate Fokker-Planck equation [16,17] which describes the process is the following one.
$∂P( M Ai ,t) ∂t =− ∂ ∂ M Ai G( M Ai )P( M Ai ,t)+ ∂ ∂ M Ai g( M Ai ) ∂ ∂ M Ai g( M Ai )D( M Ai )P( M Ai ,t) MathType@MTEF@5@5@+=
Where the different terms in the stochastic equation correspond to:
$G( M Ai )=(aK(b−c) e −λτ −b) M Ai −a M Ai 2 (19) MathType@MTEF@5@5@+=
$g( M Ai )=− M Ai MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=
xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4zaiaacIcacaWGnbWaaSbaaSqaaiaadgeacaWGPbaabeaakiaacMcacqGH9aqpcqGHsislcaWGnbWaaSbaaSqaaiaadgeacaWGPbaabeaaaaa@3F98@$ (20)
$D( M Ai )= C 2 (b−c)(b−c+a M Ai ) MathType@MTEF@5@5@+=
Now we look for the stationary probability density of M[Ai], considering that the boundary condition at ∞ is natural. N corresponds to the normalization constant.
$P( M Ai )=N(1+ a (b−c) M Ai ) M Ai (−1− b (b−c) 2 c + a(b−c) (b−c) 2 e −λτ K c ) e - (b−c) e −λτ (a M Ai 2 (2a(−b+c)K+ e λτ (4b−2c+a M Ai ))) 2c M Ai MathType@MTEF@5@5@+=
We see how the stationary probability density has a singularity at 0. In some cases it can be normalizable. In particular when K is above the curve $K c = b e λτ a(b−c) . MathType@MTEF@5@5@+=
This curve is identified in (Figure 3) with the solid (black) line. There are also 2 more transitions. The region between the curves K[c] and $K c * MathType@MTEF@5@5@+=
yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaaaaaaa@3885@$ in dashed (blue) line still have a singularity at 0 together with a finite
distribution for low values of infected mice. Above a second curve, $K c ** MathType@MTEF@5@5@+=
yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaiaacQcaaaaaaa@3933@$ in dash-dotted (red) line, a local maximum appears. And finally, for
values above $K c * MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=
Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaaaaaaa@3885@$ the singularity at 0 disappears.
Figure 3: Phase diagram for the theoretical stationary probability density of infected mice. We discuss it in terms of K and a, leaving constant the rest of parameters: b = 2, c = 0.6, ?=0.42 and, t=
2. The solid (black) line corresponds to Kc, the dashed (blue) line to the transition given by $K c * MathType@MTEF@5@5@+=
yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaaaaaaa@3885@$ and the dash-dotted (red) line to $K c ** MathType@MTEF@5@5@+=
yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaiaacQcaaaaaaa@3933@$ .Points labeled from (a) to (d) represent the values for the stationary
probability density depicted in figure 4.
Figure 3: Phase diagram for the theoretical stationary probability density of
infected mice. We discuss it in terms of K and a, leaving constant the rest of
parameters: b = 2, c = 0.6, ?=0.42 and, t=2. The solid (black) line corresponds
to Kc, the dashed (blue) line to the transition given by $K c * MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaaaaaaa@3885@$ and the dash-dotted (red) line to $K c ** MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaiaacQcaaaaaaa@3933@$
.Points labeled from (a) to (d) represent the values for the
stationary probability density depicted in figure 4.
Thus, we find following the analysis developed in [9], that the steady state under internal fluctuations presents a general transition characterized by K[c]. Behind it; internal fluctuations can
drive the system to the 0 fixed point. However, as MAi increases together with K this chance decreases, K > K[c], and eventually the 0 fixed point becomes unreachable (above $K c *
Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabaqaamaabaabaaGcbaGaam4samaaDaaaleaacaWGJbaabaGaaiOkaaaaaaa@3885@$ or a bottleneck to access the 0 fixed point for
high values of K).
B. Numerical approach. Comparison
We study the numerical probability density for the steady state of scenario A with the modified Gillespie algorithm and compare it with the previous analytical results.
Since we only compute a small number of realizations (1000 realizations) in a finite time, we may describe partially the probability distribution. In (Figure 4) we see in (red) histograms the
numerical approach. When K increases, the absorbent 0 fixed point becomes less reachable until it is finally unattainable (Figure 4 from (a) to (d)). At this point, (Figure 4d), the system is above
the thermodynamic limit.
Figure 4: Stationary distribution density for infected mice corresponding to those parameters label in figure 3. We consider internal fluctuations both in the theoretical distribution in solid (blue)
line and in the numerical (red) histogram. The phenomenological steady state is drawn with a vertical (blue) line. Displayed numerical distributions are taken over 1000 realizations at a finite time.
The zoom in (d) shows the probability for the 0 absorbent state.
Figure 4: Stationary distribution density for infected mice corresponding to those parameters label in figure 3. We consider internal fluctuations both in the theoretical distribution in solid (blue) line and in the numerical (red) histogram. The phenomenological steady state is drawn with a vertical (blue) line. Displayed
numerical distributions are taken over 1000 realizations at a finite time. The zoom in (d) shows the probability for the 0 absorbent state.
We now compare these results with theoretical probability distribution density in the steady state (solid (blue) line). In Figure 4, we show the spreading of the steady state due to internal
fluctuations. For K < K[c] there is no infection as we see in (a). However, from panels (b) to (d), K > K[c], we see how the system evolves towards a maximum for finite values of infection, and tends
to leave unreachable the 0 fixed point. Following this evolution we see how in panel (c) the numerical description shows already a local maximum in contrast to the analytical description, where it
has not appeared yet. The infection is more robust in the numerical distribution. Finally, in panel (d), we show how the system enters in the thermodynamic limit. Numerically, where the 0 fixed point
becomes unreachable, and analytically, in those cases where there is a bottleneck that makes very difficult to reach the 0 fixed point.
In all, this suggests that the infection could get extincted in the stationary state out of the thermodynamic limit.
The mean field of these distributions coincides with the mean value of the phenomenological model when we consider the thermodynamic limit (vertical (blue) line).
Considering the finiteness and discreteness of our system, we have studied under the same conceptual framework as in [5], new analytical and numerical approaches in order to support those features
observed in the phenomenological model for El Ni&nTilde;o southern oscillations. Throughout the article we have seen how in both, the temporal evolution and the steady state, the absent of infection
is an absorbent state reachable out of the thermodynamic limit. In particular internal fluctuations are relevant in scenario A, unlike scenario B, where infection evolves to extinction. Mainly above
the thermodynamic limit, we observe K[c] and the delay among the total number of mice and those infected.
J. A. R. acknowledges support from grant BES-2008-003398. | {"url":"https://austinpublishinggroup.com/infectious-diseases/fulltext/ajid-v3-id1019.php","timestamp":"2024-11-05T00:26:47Z","content_type":"text/html","content_length":"162118","record_id":"<urn:uuid:1a26f061-7978-4a55-8ad3-ad27a4deaa1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00188.warc.gz"} |
[Solved] In which of these forecasting techn
In which of these forecasting techniques are past staffing levels used to project future human resource requirements?
Answer (Detailed Solution Below)
Option 2 : Time series analysis
MP Vyapam Group 2 - Previous Year Paper Qs Mixed
2.2 K Users
10 Questions 10 Marks 10 Mins
The correct answer is Time series analysis.
Key Points
Time series analysis:
• In this, past staffing levels used to project future human resource requirements.
• It is a specific way of analyzing a sequence of data points collected over an interval of time.
• It is an advanced area of data analysis that focuses on processing, describing, and forecasting time series, which are time-ordered datasets.
• The data is recorded at consistent intervals over a set period of time rather than just recording the data points intermittently or randomly.
• Since, we have past data, the future can be accordingly predicted.
Additional Information
Regression analysis
• It is a statistical technique that predicts the level of one variable (the “dependent” variable) based on the level of another variable (the “independent” variable).
• It is a quantitative demand forecasting technique.
• It presupposes that a linear relationship exists between one or more independent variables, which are predicted to affect the dependent variable. For instance- future HR demand for personnel.
Nominal group method
• It is a tool used to help groups generate ideas and reach a consensus.
• It is defined as a structured method for group brainstorming that encourages contributions from everyone.
• It facilitates quick agreement on the relative importance of issues, problems, or solutions.
Latest MP Vyapam Group 2 Updates
Last updated on Feb 5, 2024
MP Vyapam Recruitment Revised Result has been released. Earlier, MP Vyapam Group 2 Result was withdrawn. The MP Vyapam Group 2 Notification 2023 had been released for recruitment to 1946 vacancies.
The vacancies were announced for posts like Rural Agriculture Development Officer, Lab Technician, Agriculture Director, and others. The selection process includes a written examination. To prepare
well for the exam practice using the MP Vyapam Group 2 Previous Year Papers. | {"url":"https://testbook.com/question-answer/inwhichoftheseforecasting--63a1d5e1b0790e1f4f33bd4c","timestamp":"2024-11-09T23:08:49Z","content_type":"text/html","content_length":"217701","record_id":"<urn:uuid:892f1582-040d-403d-9713-9eb6c1852944>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00551.warc.gz"} |
Pseudorandom generators with optimal seed length for non-boolean poly-size circuits
A sampling procedure for a distribution P over {0, 1}^ℓ is a function C : {0, 1}^n → {0, 1}^ℓ such that the distribution C(U[n]) (obtained by applying C on the uniform distribution U[n]) is the
"desired distribution" P. Let n > r ≥ ℓ = n^Ω(1). An ∈-nb-PRG (defined by Dubrov and Ishai [2006]) is a function G : {0, 1}^r → {0, 1}^n such that for every C : {0, 1}^n → {0, 1}^ℓ in some class of
"interesting sampling procedures," C′(U[r]) = C(G(U[r])) is ∈-close to C(U[n]) in statistical distance. We construct poly-time computable nb-PRGs with r = O(ℓ) for poly-size circuits relying on the
assumption that there exists β > 0 and a problem L in E = DTIME(2^O(n)) such that for every large enough n, nondeterministic circuits of size 2^βn that have NP-gates cannot solve L on inputs of
length n. This assumption is a scaled nonuniform analog of (the widely believed) EXP ≠ Σ^P[2], and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and
Ishai have r = Ω(ℓ^2) and are based on very strong cryptographic assumptions or, alternatively, on nonstandard assumptions regarding incompressibility of functions on random inputs. When restricting
to poly-size circuits C : {0, 1}^n → {0, 1}^ℓ with Shannon entropy H(C(U[n])) ≤ k, for ℓ > k = n^Ω(1), our nb-PRGs have r = O(k). The nb-PRGs of Dubrov and Ishai use seed length r = Ω(k^2) and
require that the probability distribution of C(U[n]) is efficiently computable. Our nb-PRGs follow from a notion of "conditional PRGs," which may be of independent interest. These are PRGs where G(U
[r]) remains pseudorandom even when conditioned on a "large" event {A(G(U[r])) = 1}, for an arbitrary poly-size circuit A. A related notion was considered by Shaltiel and Umans [2005] in a different
setting, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai. We also give an unconditional construction of poly-time computable nb-PRGs for poly(n)-size, depth d circuits
C : {0, 1}^n → {0, 1}^ℓ with r = O(ℓ · log^d+O(1)n). This improves upon the previous work of Dubrov and Ishai that has r ≥ ℓ^2. This result follows by adapting a recent PRG construction of Trevisan
and Xue [2013] to the case of nb-PRGs. We also show that this PRG can be implemented by a uniform family of constant-depth circuits with slightly increased seed length.
Bibliographical note
Funding Information:
This research was supported by BSF grant 2010120, ISF grant 864/11, and ERC starting grant 279559. A preliminary version of this article appeared in STOC 2014.
Publisher Copyright:
© 2017 ACM.
• Hardness versus randomness
• Pseudorandom generators
• Pseudorandomness
• Randomness complexity of sampling
ASJC Scopus subject areas
• Theoretical Computer Science
• Computational Theory and Mathematics
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Popular Science Monthly/Volume 69/November 1906/The Value of Science: The Notion of Space III - Wikisource, the free online library
Chapter III. The Notion of Space
IN the articles I have heretofore devoted to space I have above all emphasized the problems raised by non-Euclidean geometry, while leaving almost completely aside other questions more difficult of
approach, such as those which pertain to the number of dimensions. All the geometries I considered had thus a common basis, that tri-dimensional continuum which was the same for all and which
differentiated itself only by the figures one drew in it or when one aspired to measure it.
In this continuum, primitively amorphous, we may imagine a network of lines and surfaces, we may then convene to regard the meshes of this net as equal to one another, and it is only after this
convention that this continuum, become measurable, becomes Euclidean or non-Euclidean space. From this amorphous continuum can therefore arise indifferently one or the other of the two spaces, just
as on a blank sheet of paper may be traced indifferently a straight or a circle.
In space we know rectilinear triangles the sum of whose angles is equal to two right angles; but equally we know curvilinear triangles the sum of whose angles is less than two right angles. The
existence of the one sort is not more doubtful than that of the other. To give the name of straights to the sides of the first is to adopt Euclidean geometry; to give the name of straights to the
sides of the latter is to adopt the non-Euclidean geometry. So that to ask what geometry it is proper to adopt is to ask, to what line is it proper to give the name straight?
It is evident that experiment can not settle such a question; one would not ask, for instance, experiment to decide whether I should call AB or CD a straight. On the other hand, neither can I say
that I have not the right to give the name of straights to the sides of non-Euclidean triangles because they are not in conformity with the eternal idea of straight which I have by intuition. I
grant, indeed, that I have the intuitive idea of the side of the Euclidean triangle, but I have equally the intuitive idea of the side of the non-Euclidean triangle. Why should I have the right to
apply the name of straight to the first of these ideas and not to the second? Wherein does this syllable form an integrant part of this intuitive idea? Evidently when we say that the Euclidean
straight is a true straight and that the non-Euclidean straight is not a true straight, we simply mean that the first intuitive idea corresponds to a more noteworthy object than the second. But how
do we decide that this object is more noteworthy? This question I have investigated in 'Science and Hypothesis.'
It is here that we saw experience come in. If the Euclidean straight is more noteworthy than the non-Euclidean straight, it is so chiefly because it differs little from certain noteworthy natural
objects from which the non-Euclidean straight differs greatly. But, it will be said, the definition of the non-Euclidean straight is artificial; if we for a moment adopt it, we shall see that two
circles of different radius both receive the name of non-Euclidean straights, while of two circles of the same radius one can satisfy the definition without the other being able to satisfy it, and
then if we transport one of these so-called straights without deforming it, it will cease to be a straight. But by what right do we consider as equal these two figures which the Euclidean geometers
call two circles with the same radius? It is because by transporting one of them without deforming it we can make it coincide with the other. And why do we say this transportation is effected without
deformation? It is impossible to give a good reason for it. Among all the motions conceivable, there are some of which the Euclidean geometers say that they are not accompanied by deformation; but
there are others of which the non-Euclidean geometers would say that they are not accompanied by deformation. In the first, called Euclidean motions, the Euclidean straights remain Euclidean
straights, and the non-Euclidean straights do not remain non-Euclidean straights; in the motions of the second sort, or non-Euclidean motions, the non-Euclidean straights remain non-Euclidean
straights and the Euclidean straights do not remain Euclidean straights. It has, therefore, not been demonstrated that it was unreasonable to call straights the sides of non-Euclidean triangles; it
has only been shown that that would be unreasonable if one continued to call the Euclidean motions motions without deformation; but it has at the same time been shown that it would be just as
unreasonable to call straights the sides of Euclidean triangles if the non-Euclidean motions were called motions without deformation.
Now when we say that the Euclidean motions are the true motions without deformation, what do we mean? We simply mean that they are more noteworthy than the others. And why are they more noteworthy?
It is because certain noteworthy natural bodies, the solid bodies, undergo motions almost similar.
And then when we ask: Can one imagine non-Euclidean space? that means: Can we imagine a world where there would be noteworthy natural objects affecting almost the form of non-Euclidean straights, and
noteworthy natural bodies frequently undergoing motions almost similar to the non-Euclidean motions? I have shown in 'Science and Hypothesis' that to this question we must answer yes.
It has often been observed that if all the bodies in the universe were dilated simultaneously and.in the same proportion, we should have no means of perceiving it, since all our measuring instruments
would grow at the same time as the objects themselves which they serve to measure. The world, after this dilatation, would continue on its course without anything apprising us of so considerable an
event. In other words, two worlds similar to one another (understanding the word similitude in the sense of Euclid, Book VI.) would be absolutely indistinguishable. But more; worlds will be
indistinguishable not only if they are equal or similar, that is, if we can pass from one to the other by changing the axes of coordinates, or by changing the scale to which lengths are referred; but
they will still be indistinguishable if we can pass from one to the other by any 'point-transformation' whatever. I will explain my meaning. I suppose that to each point of one corresponds one point
of the other and only one, and inversely; and besides that the coordinates of a point are continuous functions, otherwise altogether arbitrary, of the corresponding point. I suppose besides that to
each object of the first world corresponds in the second an object of the same nature placed precisely at the corresponding point. I suppose finally that this correspondence fulfilled at the initial
instant is maintained indefinitely. We should have no means of distinguishing these two worlds one from the other. The relativity of space is not ordinarily understood in so broad a sense; it is
thus, however, that it would be proper to understand it.
If one of these universes is our Euclidean world, what its inhabitants will call straight will be our Euclidean straight; but what the inhabitants of the second world will call straight will be a
curve which will have the same properties in relation to the world they inhabit and in relation to the motions that they will call motions without deformation. Their geometry will, therefore, be
Euclidean geometry, but their straight will not be our Euclidean straight. It will be its transform by the point-transformation which carries over from our world to theirs. The straights of these men
will not be our straights, but they will have among themselves the same relations as our straights to one another. It is in this sense I say their geometry will be ours. If then we wish after all to
proclaim that they deceive themselves, that their straight is not the true straight, if we still are unwilling to admit that such an affirmation has no meaning, at least we must confess that these
people have no means whatever of recognizing their error.
All that is relatively easy to understand, and I have already so often repeated it that I think it needless to expatiate further on the matter. Euclidean space is not a form imposed upon our
sensibility, since we can imagine non-Euclidean space; but the two spaces, Euclidean and non-Euclidean, have a common basis, that amorphous continuum of which I spoke in the beginning. From this
continuum we can get either Euclidean space or Lobachevskian space, just as we can, by tracing upon it a proper graduation, transform an ungraduated thermometer into a Fahrenheit or a Réaumur
And then comes a question: Is not this amorphous continuum that our analysis has allowed to survive a form imposed upon our sensibility? If so, we should have enlarged the prison in which this
sensibility is confined, but it would always be a prison.
This continuum has a certain number of properties, exempt from all idea of measurement. The study of these properties is the object of a science which has been cultivated by many great geometers and
in particular by Riemann and Betti and which has received the name of analysis situs. In this science abstraction is made of every quantitative idea and, for example, if we ascertain that on a line
the point B is between the points A and C, we shall be content with this ascertainment and shall not trouble to know whether the line ABC is straight or curved, nor whether the length AB is equal to
the length BC, or whether it is twice as great.
The theorems of analysis situs have, therefore, this peculiarity that they would remain true if the figures were copied by an inexpert draftsman who should grossly change all the proportions and
replace the straights by lines more or less sinuous. In mathematical terms, they are not altered by any 'point-transformation' whatsoever. It has often been said that metric geometry was
quantitative, while projective geometry was purely qualitative. That is not altogether true. The straight is still distinguished from other lines by properties which remain quantitative in some
respects. The real qualitative geometry is, therefore, analysis situs.
The same questions which came up apropos of the truths of Euclidean geometry, come up anew apropos of the theorems of analysis situs. Are they obtainable by deductive reasoning? Are they disguised
conventions? Are they experimental verities? Are they the characteristics of a form imposed either upon our sensibility or upon our understanding?
I wish simply to observe that the last two solutions exclude each other. We can not admit at the same time that it is impossible to imagine space of four dimensions and that experience proves to us
that space has three dimensions. The experimenter puts to nature a question: Is it this or that? and he can not put it without imagining the two terms of the alternative. If it were impossible to
imagine one of these terms, it would be futile and besides impossible to consult experience. There is no need of observation to know that the hand of a watch is not marking the hour 15 on the dial,
because we know beforehand that there are only 12, and we could not look at the mark 15 to see if the hand is there, because this mark does not exist.
Note likewise that in analysis situs the empiricists are disembarrassed of one of the gravest objections that can be leveled against them, of that which renders absolutely vain in advance all their
efforts to apply their thesis to the verities of Euclidean geometry. These verities are rigorous and all experimentation can only be approximate. In analysis situs approximate experiments may suffice
to give a rigorous theorem and, for instance, if it is seen that space can not have either two or less than two dimensions, nor four or more than four, we are certain that it has exactly three, since
it could not have two and a half or three and a half.
Of all the theorems of analysis situs, the most important is that which is expressed in saying that space has three dimensions. This it is that we are about to consider, and we shall put the question
in these terms: When we say that space has three dimensions, what do we mean?
3. The Physical Continuum of Several Dimensions
I have explained in 'Science and Hypothesis' whence we derive the notion of physical continuity and how that of mathematical continuity has arisen from it. It happens that we are capable of
distinguishing two impressions one from the other, while each is indistinguishable from a third. Thus we can readily distinguish a weight of 12 grams from a weight of 10 grams, while a weight of 11
grams could neither be distinguished from the one nor the other. Such a statement, translated into symbols, may be written:
${\displaystyle A=B,B=C,A<C}$.
This would be the formula of the physical continuum, as crude experience gives it to us, whence arises an intolerable contradiction that has been obviated by the introduction of the mathematical
continuum. This is a scale of which the steps (commensurable or incommensurable numbers) are infinite in number, but are exterior to one another instead of encroaching on one another as do the
elements of the physical continuum, in conformity with the preceding formula.
The physical continuum is, so to speak, a nebula not resolved; the most perfect instruments could not attain to its resolution. Doubtless if we measured the weights with a good balance instead of
judging them by the hand, we could distinguish the weight of 11 grams from those of 10 and 12 grams, and our formula would become:
${\displaystyle A<B,B<C,A<C}$.
But we should always find between ${\displaystyle A}$ and ${\displaystyle B}$ and between ${\displaystyle B}$ and ${\displaystyle C}$ new elements ${\displaystyle D}$ and ${\displaystyle E,}$ such
${\displaystyle A=D,D=B,A<B;B=E,E=C,B<C,}$
and the difficulty would only have receded and the nebula would always remain unresolved; the mind alone can resolve it and the mathematical continuum it is which is the nebula resolved into stars.
Yet up to this point we have not introduced the notion of the number of dimensions. What is meant when we say that a mathematical continuum or that a physical continuum has two or three dimensions?
First we must introduce the notion of cut, studying first physical continua. We have seen what characterizes the physical continuum. Each of the elements of this continuum consists of a manifold of
impressions; and it may happen either that an element can not be discriminated from another element of the same continuum, if this new element corresponds to a manifold of impressions not
sufficiently different, or, on the contrary, that the discrimination is possible; finally it may happen that two elements indistinguishable from a third, may, nevertheless, be distinguished one from
the other.
That postulated, if ${\displaystyle A}$ and ${\displaystyle B}$ are two distinguishable elements of a continuum ${\displaystyle C,}$ a series of elements may be found, ${\displaystyle E_{1}}$, ${\
displaystyle E_{2}}$, ⋅⋅⋅, ${\displaystyle E_{n}}$, all belonging to this same continuum ${\displaystyle C}$ and such that each of them is indistinguishable from the preceding, that ${\displaystyle
E_{1}}$ is indistinguishable from ${\displaystyle A}$ and ${\displaystyle E}$ n indistinguishable from ${\displaystyle B.}$ Therefore we can go from ${\displaystyle A}$ to ${\displaystyle B}$ by a
continuous route and without quitting ${\displaystyle C.}$ If this condition is fulfilled for any two elements ${\displaystyle A}$ and ${\displaystyle B}$ of the continuum ${\displaystyle C,}$ we may
say that this continuum ${\displaystyle C}$ is all in one piece. Now let us distinguish certain of the elements of ${\displaystyle C}$ which may either be all distinguishable from one another, or
themselves form one or several continua. The assemblage of the elements thus chosen arbitrarily among all those of ${\displaystyle C}$ will form what I shall call the cut or the cuts.
Take on ${\displaystyle C}$ any two elements ${\displaystyle A}$ and ${\displaystyle B.}$ Either we can also find a series of elements ${\displaystyle E_{1}}$, ${\displaystyle E_{2}}$, ⋅⋅⋅, ${\
displaystyle E_{n}}$, such: (1) that they all belong to ${\displaystyle C;}$ (2) that each of them is indistinguishable from the following, ${\displaystyle E_{1}}$ indistinguishable from ${\
displaystyle A}$ and ${\displaystyle E^{u}}$ from ${\displaystyle B;}$ (3) and besides that none of the elements E is indistinguishable from any element of the cut. Or else, on the contrary, in each
of the series ${\displaystyle E_{1}}$, ${\displaystyle E_{2}}$, ⋅⋅⋅, ${\displaystyle E_{n}}$ satisfying the first two conditions, there will be an element ${\displaystyle E}$ indistinguishable from
one of the elements of the cut. In the first case we can go from ${\displaystyle A}$ to ${\displaystyle B}$ by a continuous route without quitting ${\displaystyle C}$ and without meeting the cuts; in
the second case that is impossible.
If then for any two elements ${\displaystyle A}$ and ${\displaystyle B}$ of the continuum ${\displaystyle C,}$ it is always the first case which presents itself, we shall say that ${\displaystyle C}$
remains all in one piece despite the cuts.
Thus, if we choose the cuts in a certain way, otherwise arbitrary, it may happen either that the continuum remains all in one piece or that it does not remain all in one piece; in this latter
hypothesis we shall then say that it is divided by the cuts.
It will be noticed that all these definitions are constructed in setting out solely from this very simple fact, that two manifolds of impressions sometimes can be discriminated, sometimes can not be.
That postulated, if, to divide a continuum, it suffices to consider as cuts a certain number of elements all distinguishable from one another, we say that this continuum is of one dimension; if, on
the contrary, to divide a continuum, it is necessary to consider as cuts a system of elements themselves forming one or several continua, we shall say that this continuum is of several dimensions.
If to divide a continuum ${\displaystyle C,}$ cuts forming one or several continua of one dimension suffice, we shall say that ${\displaystyle C}$ is a continuum of two dimensions; if cuts suffice
which form one or several continua of two dimensions at most, we shall say that ${\displaystyle C}$ is a continuum of three dimensions; and so on.
To justify this definition it is proper to see whether it is in this way that geometers introduce the notion of three dimensions at the beginning of their works. Now, what do we see? Usually they
begin by defining surfaces as the boundaries of solids or pieces of space, lines as the boundaries of surfaces, points as the boundaries of lines, and they affirm that the same procedure can not be
pushed further.
This is just the idea given above: to divide space, cuts that are called surfaces are necessary; to divide surfaces, cuts that are called lines are necessary; to divide lines, cuts that are called
points are necessary; we can go no further, the point can not be divided, so the point is not a continuum. Then lines which can be divided by cuts which are not continua will be continua of one
dimension; surfaces which can be divided by continuous cuts of one dimension will be continua of two dimensions; finally space which can be divided by continuous cuts of two dimensions will be a
continuum of three dimensions.
Thus the definition I have just given does not differ essentially from the usual definitions; I have only endeavored to give it a form applicable not to the mathematical continuum, but to the
physical continuum, which alone is susceptible of representation, and yet to retain all its precision. Moreover, we see that this definition applies not alone to space; that in all which falls under
our senses we find the characteristics of the physical continuum, which would allow of the same classification; that it would be easy to find there examples of continua of four, of five, dimensions,
in the sense of the preceding definition; such examples occur of themselves to the mind.
I should explain finally, if I had the time, that this science, of which I spoke above and to which Riemann gave the name of analysis situs, teaches us to make distinctions among continua of the same
number of dimensions and that the classification of these continua rests also on the consideration of cuts.
From this notion has arisen that of the mathematical continuum of several dimensions in the same way that the physical continuum of one dimension engendered the mathematical continuum of one
dimension. The formula
${\displaystyle A>C,A=B,B=C,}$
which summed up the data of crude experience, implied an intolerable contradiction. To get free from it it was necessary to introduce a new notion while still respecting the essential characteristics
of the physical continuum of several dimensions. The mathematical continuum of one dimension admitted of a scale whose divisions, infinite in number, corresponded to the different values,
commensurable or not, of one same magnitude. To have the mathematical continuum of ${\displaystyle n}$ dimensions, it will suffice to take ${\displaystyle n}$ like scales whose divisions correspond
to different values of ${\displaystyle n}$ independent magnitudes called coordinates. We thus shall have an image of the physical continuum of ${\displaystyle n}$ dimensions, and this image will be
as faithful as it can be after the determination not to allow the contradiction of which I spoke above.
It seems now that the question we put to ourselves at the start is answered. When we say that space has three dimensions, it will be said, we mean that the manifold of points of space satisfies the
definition we have just given of the physical continuum of three dimensions. To be content with that would be to suppose that we know what is the manifold of points of space, or even one point of
Now that is not as simple as one might think. Every one believes he knows what a point is, and it is just because we know it too well that we think there is no need of defining it. Surely we can not
be required to know how to define it, because in going back from definition to definition a time must come when we must stop. But at what moment should we stop?
We shall stop first when we reach an object which falls under our senses or that we can represent to ourselves; definition then will become useless; we do not define the sheep to a child; we say to
him: See the sheep.
So, then, we should ask ourselves if it is possible to represent to ourselves a point of space. Those who answer yes do not reflect that they represent to themselves in reality a white spot made with
the chalk on a blackboard or a black spot made with a pen on white paper, and that they can represent to themselves only an object or rather the impressions that this object made on their senses.
When they try to represent to themselves a point, they represent the impressions that very little objects made them feel. It is needless to add that two different objects, though both very little,
may produce extremely different impressions, but I shall not dwell on this difficulty, which would still require some discussion.
But it is not a question of that; it does not suffice to represent one point, it is necessary to represent a certain point and to have the means of distinguishing it from an other point. And in fact,
that we may be able to apply to a continuum the rule I have above expounded and by which one may recognize the number of its dimensions, we must rely upon the fact that two elements of this continuum
sometimes can and sometimes can not be distinguished. It is necessary therefore that we should in certain cases know how to represent to ourselves a specific element and to distinguish it from an
other element.
The question is to know whether the point that I represented to myself an hour ago is the same as this that I now represent to myself, or whether it is a different point. In other words, how do we
know whether the point occupied by the object
at the instant ${\displaystyle \alpha }$ is the same as the point occupied by the object ${\displaystyle B}$ at the instant ${\displaystyle \beta }$, or still better, what this means?
I am seated in my room; an object is placed on my table; during a second I do not move, no one touches the object. I am tempted to say that the point ${\displaystyle A}$ which this object occupied at
the beginning of this second is identical with the point ${\displaystyle B}$ which it occupies at its end. Not at all; from the point ${\displaystyle A}$ to the point ${\displaystyle B}$ is 30
kilometers, because the object has been carried along in the motion of the earth. We can not know whether an object, be it large or small, has not changed its absolute position in space, and not only
can we not affirm it, but this affirmation has no meaning and in any case can not correspond to any representation.
But then we may ask ourselves if the relative position of an object with regard to other objects has changed or not, and first whether the relative position of this object with regard to our body
has changed. If the impressions this object makes upon us have not changed, we shall be inclined to judge that neither has this relative position changed; if they have changed, we shall judge that
this object has changed either in state or in relative position. It remains to decide which of the two. I have explained in 'Science and Hypothesis' how we have been led to distinguish the changes of
position. Moreover, I shall return to that further on. We come to know, therefore, whether the relative position of an object with regard to our body has or has not remained the same.
If now we see that two objects have retained their relative position with regard to our body, we conclude that the relative position of these two objects with regard to one another has not changed;
but we reach this conclusion only by indirect reasoning. The only thing that we know directly is the relative position of the objects with regard to our body. A fortiori it is only by indirect
reasoning that we think we know (and, moreover, this belief is delusive) whether the absolute position of the object has changed.
In a. word, the system of coordinate axes to which we naturally refer all exterior objects is a system of axes invariably bound to our body, and carried around with us.
It is impossible to represent to oneself absolute space; when I try to represent to myself simultaneously objects and myself in motion in absolute space, in reality I represent to myself my own self
motionless and seeing move around me different objects and a man that is exterior to me, but that I convene to call me.
Will the difficulty be solved if we agree to refer everything to these axes bound to our body? Shall we know then what is a point thus defined by its relative position with regard to ourselves? Many
persons will answer yes and will say that they 'localize' exterior objects.
What does this mean? To localize an object simply means to represent to oneself the movements that would be necessary to reach it. I will explain myself. It is not a question of representing the
movements themselves in space, but solely of representing to oneself the muscular sensations which accompany these movements and which do not presuppose the preexistence of the notion of space.
If we suppose two different objects which successively occupy the same relative position with regard to ourselves, the impressions that these two objects make upon us will be very different; if we
localize them at the same point, this is simply because it is necessary to make the same movements to reach them; apart from that, one can not just see what they could have in common.
But, given an object, we can conceive many different series of movements which equally enable us to reach it. If then we represent to ourselves a point by representing to ourselves the series of
muscular sensations which accompany the movements which enable us to reach this point, there will be many ways entirely different of representing to oneself the same point. If one is not satisfied
with this solution, but wishes, for instance, to bring in the visual sensations along with the muscular sensations, there will be one or two more ways of representing to oneself this same point and
the difficulty will only be increased. In any case the following question comes up: Why do we think that all these representations so different from one another still represent the same point?
Another remark: I have just said that it is to our own body that we naturally refer exterior objects; that we carry about everywhere with us a system of axes to which we refer all the points of
space, and that this system of axes seems to be invariably bound to our body. It should be noticed that rigorously we could not speak of axes invariably bound to the body unless the different parts
of this body were themselves invariably bound to one another. As this is not the case, we ought, before referring exterior objects to these fictitious axes, to suppose our body brought back to the
initial attitude. | {"url":"https://en.m.wikisource.org/wiki/Popular_Science_Monthly/Volume_69/November_1906/The_Value_of_Science:_The_Notion_of_Space_III","timestamp":"2024-11-14T00:47:46Z","content_type":"text/html","content_length":"119065","record_id":"<urn:uuid:fa90d63b-12ac-4a87-a662-4ea948e794ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00302.warc.gz"} |
Smerat, Sebastian (2011): Ground state and dynamical properties of the finite Kondo lattice model and transport through carbon based nanodevices: a numerical study. Dissertation, LMU München: Faculty
of Physics
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The first topic of this thesis is the study of many-body effects in an one-dimensional strongly correlated electronic system - the Kondo lattice model. This system is tackled numerically by means of
the density matrix renormalization group, since analytic method, i.e., perturbation the- ory fail due to competing coupling constants. The Kondo lattice model consists of a conduction band of
electrons which couple via a spin exchange coupling to a localized spin lattice. We study the spectral properties of the one-dimensional Kondo lattice model as a function of the exchange coupling,
the band filling, and the quasimomentum in the ferromagnetic and paramagnetic phases. We compute the dispersion relation of the quasiparticles, their lifetimes, and the Z factor. The exact ground
state and the quasiparticle-dispersion relation of the Kondo lattice model with one conduction electron are well known. The quasiparticle could be identified as the spin polaron. Our calculations of
the dispersion relation for partial band fillings give a result similar to the one-electron case, which suggests that the quasiparticle in both cases is the spin polaron. We find that the
quasiparticle lifetime differs by orders of magnitude between the ferromagnetic and paramagnetic phases and depends strongly on the quasimomentum. Further- more, we study the effects of the Coulomb
interaction on the phase diagram, the static magnetic susceptibility and electron spin relaxation. We show that onsite Coulomb interaction supports ferromagnetic order and nearest neighbor Coulomb
interaction drives, depending on the elec- tron filling, either a paramagnetic or ferromagnetic order. Furthermore, we calculate electron quasiparticle life times, which can be related to electron
spin relaxation and decoherence times, and explain their dependence on the strength of interactions and the electron filling in order to find the sweet spot of parameters where the relaxation time is
maximized. We find that effective exchange processes between the electrons dominate the spin relaxation and decoherence rate. In the second topic of this thesis, we numerically calculate the electron
transport through carbon nanotube based quantum dot devices. We use a master equation’s approach in first order of the tunneling rate to the leads and an extended constant interaction model to model
the carbon nanotube system. This work has been done in collaboration with two experimental groups and we compare their respective experimentally obtained data to our numerical calculations. In both
collaborations striking similarity between the numerical data and the experimental data is found. In the first collaboration transport through a carbon nanotube peapod, i.e, a carbon nanotube filled
with fullerenes, has been measured. We identify a small hybridization between a fullerene molecule and the surrounding carbon nanotube to be of crucial importance for the understanding of the
transport data. In the second collaboration, electron transport through a carbon nanotube rope, i.e., a bundle of carbon nanotubes has been measured. Also here, hybridization between the different
nanotubes plays a crucial role. Furthermore, an external magnetic field is applied, which enables the identification of specific spin states of the compound quantum dot system. This might be
important for future applications of such devices in spin-dependent electronics.
Item Type: Theses (Dissertation, LMU Munich)
Keywords: physics, Kondo lattice model, carbon nanotubes, electron transport
Subjects: 500 Natural sciences and mathematics
500 Natural sciences and mathematics > 530 Physics
Faculties: Faculty of Physics
Language: English
Date of oral examination: 25. March 2011
1. Referee: Schollwöck, Ulrich
MD5 Checksum of the PDF-file: 9e67cf55ce4454e3544429f0c6284129
Signature of the printed copy: 0001/UMC 19367
ID Code: 12941
Deposited On: 13. Apr 2011 12:02
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Transactions Online
Fanxin ZENG, Zhenyu ZHANG, "Construction of Multi-Dimensional Periodic Complementary Array Sets" in IEICE TRANSACTIONS on Fundamentals, vol. E93-A, no. 7, pp. 1392-1395, July 2010, doi: 10.1587/
Abstract: Multi-dimensional (MD) periodic complementary array sets (CASs) with impulse-like MD periodic autocorrelation function are naturally generalized to (one dimensional) periodic complementary
sequence sets, and such array sets are widely applied to communication, radar, sonar, coded aperture imaging, and so forth. In this letter, based on multi-dimensional perfect arrays (MD PAs), a
method for constructing MD periodic CASs is presented, which is carried out by sampling MD PAs. It is particularly worth mentioning that the numbers and sizes of sub-arrays in the proposed MD
periodic CASs can be freely changed within the range of possibilities. In particular, for arbitrarily given positive integers M and L, two-dimensional periodic polyphase CASs with the number M^2 and
size L L of sub-arrays can be produced by the proposed method. And analogously, pseudo-random MD periodic CASs can be given when pseudo-random MD arrays are sampled. Finally, the proposed method's
validity is made sure by a given example.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.1392/_p
author={Fanxin ZENG, Zhenyu ZHANG, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Construction of Multi-Dimensional Periodic Complementary Array Sets},
abstract={Multi-dimensional (MD) periodic complementary array sets (CASs) with impulse-like MD periodic autocorrelation function are naturally generalized to (one dimensional) periodic complementary
sequence sets, and such array sets are widely applied to communication, radar, sonar, coded aperture imaging, and so forth. In this letter, based on multi-dimensional perfect arrays (MD PAs), a
method for constructing MD periodic CASs is presented, which is carried out by sampling MD PAs. It is particularly worth mentioning that the numbers and sizes of sub-arrays in the proposed MD
periodic CASs can be freely changed within the range of possibilities. In particular, for arbitrarily given positive integers M and L, two-dimensional periodic polyphase CASs with the number M^2 and
size L L of sub-arrays can be produced by the proposed method. And analogously, pseudo-random MD periodic CASs can be given when pseudo-random MD arrays are sampled. Finally, the proposed method's
validity is made sure by a given example.},
TY - JOUR
TI - Construction of Multi-Dimensional Periodic Complementary Array Sets
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1392
EP - 1395
AU - Fanxin ZENG
AU - Zhenyu ZHANG
PY - 2010
DO - 10.1587/transfun.E93.A.1392
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E93-A
IS - 7
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - July 2010
AB - Multi-dimensional (MD) periodic complementary array sets (CASs) with impulse-like MD periodic autocorrelation function are naturally generalized to (one dimensional) periodic complementary
sequence sets, and such array sets are widely applied to communication, radar, sonar, coded aperture imaging, and so forth. In this letter, based on multi-dimensional perfect arrays (MD PAs), a
method for constructing MD periodic CASs is presented, which is carried out by sampling MD PAs. It is particularly worth mentioning that the numbers and sizes of sub-arrays in the proposed MD
periodic CASs can be freely changed within the range of possibilities. In particular, for arbitrarily given positive integers M and L, two-dimensional periodic polyphase CASs with the number M^2 and
size L L of sub-arrays can be produced by the proposed method. And analogously, pseudo-random MD periodic CASs can be given when pseudo-random MD arrays are sampled. Finally, the proposed method's
validity is made sure by a given example.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.1392/_p","timestamp":"2024-11-05T21:53:58Z","content_type":"text/html","content_length":"61928","record_id":"<urn:uuid:d213f82b-fcde-4687-b8d5-a98423fd1a1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00603.warc.gz"} |
Numeracy 2 E-Portfolio For Spring 2018 Semester - Paper Answers
This is your Numeracy 2 e-portfolio for the semester commencing February 2018 (Spring 2018). Please save a copy on your computer and back it up regularly (e.g. by saving it on your computer / in the
cloud (e.g. Google Drive) / emailing it to yourself. You should print a working copy and bring it to all lectures and tutorials. However, at the end of the course, you will need to submit a completed
electronic copy.
Please read carefully the module handbook, the marking criteria and the grade descriptors.
You are responsible for ensuring you understand the policy and regulations about academic misconduct. You must:
•Complete this work alone except where required or allowed by this assignment briefing paper and ensure it has not been written or composed by or with the assistance of any other person.
•Make sure all sentences or passages quoted from other people’s work in this assignment (with or without trivial changes) are in quotation marks, and are specifically acknowledged by reference to the
author, work and page.
Section 1 is worth 75% of the final mark and consists of 8 questions (70%) and periodic Skills Audit (carrying 5%).
Section 2 consists of 3 tasks. Combined they are worth 25% of the final mark.
Task 1 – Two Real life examples (8%)
Task 2 – Online Activity (10%)
Task 3 – Reflective log (7%)
Week / Content Section 1 Question Learning Outcome Page
Section 1
1.Recap numeracy 1. Introduction. Powers. Use of calculator 1 * 1,2
2.Powers, root, logarithms. Use of calculator 2 * 1,2
3.Simple & compound interest 1 3,4 * 1,2
4.Linear relationships. Scatter plots. 5 * 1,2,3
5.Further linear relationships 5 * 1,2,3
6.The future value of money. Net present value. 6 * 1,2
7.Presentation of data. Histograms. 7 * 1,2,3
8.Probability. 8* 1,2
9.Revision None 1,2,3
Section 2
10.Real-Life Examples N/A 1,3
11.Online Activity N/A 1,2,3
12.Reflective Log N/A 1,2,3
* Also assessed in the online quiz, Section 2, Task 3
Section 1
This section should be filled in as you acquire the skills required for each question.
Answer all questions. Please show your workings and/or explain your results as required.
Marks will be awarded for good presentation. Please evaluate your progress using the skills audits provided.
You may use your calculator as required.
You must show your working.
QUESTION 1 [6 marks]
Powers and Roots:
a) Simplify (2 marks)
b) Simplify (2 marks)
c) Evaluate (2 marks)
QUESTION 2 [8 marks]
a) Express the power 100 1/2 using the root notation and evaluate. (2 marks)
b) Evaluate (2 marks)
c) Simplify 7 (2 marks)
d) Scientific notation allows one to express large or small numbers in a simpler form. Express the UK population of 65,648,000 in a scientific notation (2 marks)
SKILLS AUDIT: WEEKS 1 – 2
I know how to…. I can do well I need practice I’m not sure I can’t do
13.I understand what a power is ? ? ? ?
14.I can perform calculations and simplifications using power ? ? ? ?
15.I understand what a root is ? ? ? ?
16.I can perform calculations and simplifications using roots, using a scientific or financial calculator if required ? ? ? ?
QUESTION 3 [10 marks]
Ann Miller invests £150,000 at an interest rate of 6% p.a.
Calculate the final balance after 5 years.
a) Using simple interest? (1 mark)
b) Using interest compounded annually? (3 marks)
c) Using interest compounded semi-annually? (3 marks)
d) Using interest compounded quarterly? (3 marks)
QUESTION 4 [10 marks]
a) Eliza invests £22,000 at a 2% interest rate annually.
Compounding the interest annually, how long will it take her to receive the balance of £33,000?
b) Using Rule 72, calculate how long will it take Eliza to double her investments?
c) Mr Ramsbottom invests £32,000 in a bank savings account and after 10 years his balance is £45,200.20.
Calculate the compound interest rate he received and round your answer to the second decimal place.
Dividing both sides of the equation by 22000 and simplifying we obtain
Introducing logs on both sides of the equation we obtain
Divide both sides by 32000 to obtain
Introducing logs to both side of the equation we obtain
WEEKS 3 – 4
I know how to…. I can do well I need practice I’m not sure I can’t do
17.I understand the idea of simple interest ? ? ? ?
18.I can perform simple interest calculations ? ? ? ?
19.I understand the idea of compound interest ? ? ? ?
20.I can perform compound interest calculations using a calculator if required ? ? ? ?
21.I understand the Rule of 72 (or 69 or 70) and can apply it. ? ? ? ?
a) Find the value of x if (1 mark)
b) Solve the equation X + 20 = 70 (1 mark)
c) Solve the equation = 10 (1 marks)
d) To plot the linear graph of y = 3x + 10 complete the following table:
x – 8 -5 0 7 12 24
y -14 -5 10 31 46 82
WEEK 5
I know how to…. I can do well I need practice I’m not sure I can’t do
22.I understand the idea of a linear relationship between two variables ? ? ? ?
23.I can manipulate a linear equation to solve for a variable ? ? ? ?
24.I can construct a scatter plot from a set of data (a linear relationship applies) and apply a line of best fit. ? ? ? ?
25.I understand the y-intercept and slope (gradient) of a graph and their meaning to real situations (). ? ? ? ?
26.I can use the scatter plot produced in part (12) to derive a linear relationship between two variables (). ? ? ? ?
27.I can use the relationship from part (14) to extrapolate and interpolate ? ? ? ?
Question 6 [10 marks]
Sarah Hair Saloon is considering an investment project to purchase and run a Hair Saloon business. The initial cost is £55,000. The annual cash inflows (income) are projected to be as follows:
Year 1 Year 2 Year 3 Year 4
£15,000 £25,000 £45,000 £15,000
The discount rate for this investment is 8% p.a., compounded annually.
a) Work out the Net Present Value (NPV) of this investment. (8 marks)
b) Should Sarah proceed with this project? Explain your reasoning. (2 marks)
Sarah should proceed with the project. The project is viable since the NPV is positive.
WEEK 6
I know how to…. I can do well I need practice I’m not sure I can’t do
28.I understand the idea of the future value of money ? ? ? ?
29.I understand the idea the net present value (NPV) of a project ? ? ? ?
30.I can complete a net present value calculation, using a calculator if required ? ? ? ?
Question 7 [10 marks]
A set of test scores, marked out of 100, is as follows:
a) Produce a tally of this data set suitable for the production of a histogram (3 marks)
b) Draw a histogram of this data set (6 marks)
c) Comment on the distribution of these marks. (1 marks)
Part a
We group the data in Excel using the data analysis function to obtain the table below
Part b
Then, drawing the histogram we obtain
Part c
The histogram above shows that most of the scores lie between 60-74 implying that the median lies there.
WEEK 7
I know how to…. I can do well I need practice I’m not sure I can’t do
31.I understand the idea of frequency distribution ? ? ? ?
32.I can read and interpret a histogram ? ? ? ?
33.I can construct a histogram from a set of data ? ? ? ?
Question 8 [8 marks]
Probability is a measure of the likelihood and can be stated as a ratio, percentage or generally as a number between zero and one.
a) What is the probability when the likelihood is impossible? (1 mark)
b) What is the probability when the likelihood is certain? (1 mark)
c) Express the probability of 0.06 as a % (2 marks)
d) Josiah tossed a coin and thrown a die at the same time (simultaneously). Work out the probability of getting a head on the coin and a 5 on the die.
Part a
The probability when the likelihood is impossible equals zero.
Part b
The probability when the likelihood is certain equals 1.
Part c
Probability of 0.06 as a percentage equals
Part d
Table 1 below shows the possible outcomes when tossing a coin and throwing a dice simultaneously. H represents a head while T stands for the Tail. Also the numbers 1-6 represent the number that
appears on top when a dice is thrown.
Table 1: Total possible outcomes
WEEK 8
I know how to…. I can do well I need practice I’m not sure I can’t do
34.I understand simple probabilities ? ? ? ?
35.I can perform probability calculations, using a calculator if required ? ? ? ?
36.I understand and can perform exchange rate calculations ? ? ? ?
Section 2
Task 1 – Two Real life examples (100 words each) [8 marks]
Give two real-life situations or problems in businesses that involve the topics studied in this module (e.g. powers and roots, simple and compound interests, linear relationships, graphs,
probabilities and Net Present values (NPV)).
[TYPE YOUR ANSWERS TO TASK 1 HERE]
(1)Net Present Value (4 marks)
NPV is a capital budgeting technique that takes into account the time value of money when making calculations. Investors use the method as the basis for selecting or rejecting a project. As a result,
NPV can be positive, negative or zero. Positive NPV implies that cash inflows are higher than cash outflows meaning that the project is viable and should be accepted. A zero NPV denotes an equal
amount of cash inflows and cash outflows. A project may be considered to be acceptable when it has zero NPV. On the other hand, a project with negative NPV should be ignored since it brings losses.
(2) Linear relationships (4 marks)
Linear relations use one or more variables where one depends on another. Almost every situation in life with an unknown quantity can be represented using linear relationships. For instance,
calculating mileage rates and predicting profit. Besides, linear equations can be applied in calculating variable costs. For example, if a taxi charges $8 to pick a person from a hotel and another
$0.12 per kilometre travelled. One can get a linear equation to find the total cost of the taxi over a given distance. That is, setting x to represent distance covered and setting y to represent the
total cost. Therefore, the linear relationship will be y=0.12x+8
Task 2 – Online Activities [10 marks]
This relates to the quiz. Please complete and pass all three relevant quiz/activity; screenshot and save the result’s screen ready to be pasted on the portfolio.
Ensure the followings are visible before the screenshot:
Your full names on the top right-hand corner of the screen
Your test result is any score from 40% to 100%
Task 3 – Reflective Log (150 words) [7 marks]
This reflective log should develop as the course proceeds, and may be the last part to be completed. Reflect honestly on your experiences throughout the semester. Start your reflective log from week
one by completing the skills audits and by writing personal weekly notes after each topic. Please ask for your Tutor’s support if needed.
You may wish to consider the following points when providing your reflective comments:
Actually, before the semester I started I was clueless about how the semester could unfold, bearing in mind of what I overheard from our predecessors. They could instil fear in us that the course is
tough. However, I was amazed to realize how simple and informative the course was. Learning about powers, simple and compound interest, probability and linear relationships in the first few classes
of the semester was a bit challenging. Particularly, probability. However, after concerted efforts through topical tutorials, I was able to understand everything from the topic. This made me feel
more confident about the topic. Truly, I was amazed to know that these topics are applicable in real life. Especially, the simple and compound interests which are crucial tools in financial
Throughout the next couple of classes, I was thrilled to see how simpler the classes became. I think this is attributed to the positive attitude and high self-esteem I developed with time. I could
perform calculations more easily and interpret real-life word problems based on the understanding of the topics. However, I need to improve my speed, especially when using a calculator to compute the | {"url":"https://paperanswers.com/numeracy-2-e-portfolio-for-spring-2018-semester/","timestamp":"2024-11-09T07:45:51Z","content_type":"text/html","content_length":"71832","record_id":"<urn:uuid:d1bc7285-482c-4212-bb19-30666b2dfa22>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00597.warc.gz"} |
Truth Values
Calculating the Truth Value of a Compound Proposition
The truth or falsity of a proposition is called its truth value. The truth value of a compound proposition can be calculated from the truth values of its components, using the following rules:
• For a conjunction to be true, both conjuncts must be true.
• For a disjunction to be true, at least one disjunct must be true.
• A conditional is true except when the antecedent is true and the consequent false.
• For a biconditional to be true, the two input values must be the same (either both true or both false).
• A negation has the opposite value of the negated proposition.
We can use these rules to calculate the truth value of any compound proposition, beginning with the truth values of its simple components (the sentence letters) then calculating the truth value of
each connective in the order that the connectives are used to join the component propositions.
Suppose A, B, and C are all true. What is the truth value of the following compound proposition?
((A • B) ⊃ ~C)
To figure it out, first take note of the order in which connectives are used to join the component propositions. If we were constructing the above WFF according to the
rules of syntax
, we would start by joining A and B with “•” to make
(A • B)
, and we’d prefix C with “~” to make ~C. Then we would join
(A • B)
and ~C with “⊃” to make
((A • B) ⊃ ~C)
. We follow this same order when calculating the truth value of the compound proposition:
1. (A • B) is true, because both conjuncts are true.
2. ~C is false, because C is true.
3. ((A • B) ⊃ ~C) is false, because the antecedent (A • B) is true and the consequent ~C is false.
The last connective to be calculated is the main connective. The truth value of the main connective is the truth value of the compound proposition as a whole. (As you may recall, the main connective
represents the logical structure of the compound proposition as a whole.) In the above example, the main connective is “⊃”, so the proposition is a conditional. Since the “⊃” is false, the
proposition as a whole is false.
Using numerals to represent truth values
Calculating the truth value of a compound proposition can be challenging when the proposition is very complex. To make things easier, we can write the truth values beneath each of the letters and
connectives in a compound proposition, using the numeral “1” to represent true and “0” to represent false, as shown in the example below.
In many logic textbooks, truth values are represented using the letter “T” for true and “F” for false. This is merely a matter of convention, but there are advantages to using numerals “1” and “0” to
represent truth values (as frequently done in computer science) rather than letters. Using letters to represent truth values can be confusing when “T” and “F” are also used as sentence letters. To
avoid that problem, lowercase “t” and “f” are sometimes used instead; but then the truth values are more difficult to read, especially in
truth tables
, because “t” and “f” look similar at a glance. Both problems can be avoided by using numerals to represent truth values.
Suppose A, B, and C are all true, but D is false. What is the truth value of
((A • B) ⊃ (~C ∨ D))
Step 1. The truth values of A, B, C, and D are given, so we write them beneath the sentence letters:
(( A • B ) ⊃ ( ~ C ∨ D ))
Step 2. The values of the conjunction and negation can be calculated from the sentence letters, so we write those next:
(( A • B ) ⊃ ( ~ C ∨ D ))
Step 3. The value of the disjunction can now be calculated. In order for a disjunction to be true, at least one of its disjuncts must be true. But neither ~C nor D is true, so the disjunction is
(( A • B ) ⊃ ( ~ C ∨ D ))
Step 4. Finally, the value of the conditional can be calculated. Its antecedent (the conjunction) is true, and its consequent (the disjunction) is false; so the conditional is false:
(( A • B ) ⊃ ( ~ C ∨ D ))
Since “⊃” is the main connective, its truth value is the same as the truth value of the proposition as a whole: the proposition is false.
Calculating Truth Values When Some Components are Unknown
It is often possible to calculate the truth value of a compound proposition even when the truth values of some components are unknown, as illustrated in the following example.
Suppose P is true, but the truth values of Q and R are unknown. What is the truth value of
~(P ∨ (Q ≡ R))
Step 1. The truth value of P is given:
Step 2. The disjunction must be true, because at least one of the disjuncts is true:
Step 3. The negation must be false, since the negated proposition (the disjunction) is true:
Since “~” is the main connective, the proposition is false. | {"url":"https://www.skillfulreasoning.com/propositional_logic/truth_values.html","timestamp":"2024-11-10T06:21:38Z","content_type":"text/html","content_length":"8916","record_id":"<urn:uuid:bd275629-40fc-4ae7-81e7-a38a92bb1185>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00735.warc.gz"} |
What is an example of a line chart?
A line graph, also known as a line chart, is a type of chart used to visualize the value of something over time. For example, a finance department may plot the change in the amount of cash the
company has on hand over time. The line graph consists of a horizontal x-axis and a vertical y-axis.
How do you read data on a line graph?
Interpreting Line Charts The changing slope of the line segments emphasizes changes, trends, and patterns. For a single series of data, assess the changes in the line to identify trends and patterns.
When you have multiple metrics, compare their lines to determine whether they have the same trend and patterns.
How do you make a line graph with data?
Create a line chart
1. Copy the example worksheet data into a blank worksheet, or open the worksheet that contains the data that you want to plot into a line chart.
2. Select the data that you want to plot in the line chart.
3. Click the Insert tab, and then click Insert Line or Area Chart.
4. Click Line with Markers.
How do you create a data chart?
Create a chart
1. Select the data for which you want to create a chart.
2. Click INSERT > Recommended Charts.
3. On the Recommended Charts tab, scroll through the list of charts that Excel recommends for your data, and click any chart to see how your data will look.
4. When you find the chart you like, click it > OK.
What is a line chart in Excel?
A line chart is a built-in Excel chart type, with each data series plotted as a separate line. Line charts are a good way to show change or trends over time. In contrast to column or bar charts, line
charts can handle more categories and more data points without becoming too cluttered.
What is line chart in statistics?
A line chart is a visual comparison of how two variables—shown on the x- and y-axes—are related or vary with each other. It shows related information by drawing a continuous line between all the
points on a grid.
What does a line graph look like?
A line graph shows how a value changes, usually over time. Most line graphs look like a jagged line going across the page. How high the line is above a time marked on the axis tells you how high the
value is. A dieter may use a line graph to track how their weight fluctuates as time goes by.
How does line graph look like?
What is a good question for a line graph?
Understanding a Line Graph
QUESTION ANSWER
1. What is the title of this graph? Value of Sarah’s Car
2. What is the range of values on the horizontal scale? 2001 to 2007
3. What is the range of values on the vertical scale? 0 to 25,000
4. How many points are in the graph? 7
How do I select data for a chart in Excel?
Follow these steps:
1. On the Insert tab, select the chart type you want.
2. On the Chart Design tab, select Select Data.
3. Click in the Chart data range box, and then select the data in your worksheet.
How do you make a line graph with two sets of data?
Below are steps you can use to help add two sets of data to a graph in Excel:
1. Enter data in the Excel spreadsheet you want on the graph.
2. Select the data you want on the graph.
3. Click the “Insert” tab and then look at the “Recommended Charts” in the charts group.
4. Choose “All Charts” and click “Combo” as the chart type. | {"url":"https://www.sweatlodgeradio.com/what-is-an-example-of-a-line-chart/","timestamp":"2024-11-09T16:45:37Z","content_type":"text/html","content_length":"130263","record_id":"<urn:uuid:bec0c096-bf4a-4a7b-93f9-5662b63edebb>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00646.warc.gz"} |
Build Your Own Model
Build Your Own Model¶
One important feature of orbit is to allow developers to build their own models in a relatively flexible manner to serve their own purpose. This tutorial will go over a demo on how to build up a
simple Bayesian linear regression model using Pyro API in the backend with orbit interface.
Orbit Class Design¶
In version 1.1.0, the classes within Orbit are re-designed as such:
1. Forecaster
2. Model
3. Estimator
Forecaster provides general interface for users to perform fit and predict task. It is further inherited to provide different types of forecasting methodology:
1. [Stochastic Variational Inference (SVI)]
2. Full Bayesian
The discrepancy on these three methods mainly lie on the posteriors estimation where MAP will yield point posterior estimate and can be extracted through the method get_point_posterior(). Meanwhile,
SVI and Full Bayesian allow posterior sample extraction through the method get_posteriors(). Alternatively, you can also approximate point estimate by passing through additional arg such as
point_method='median' in the .fit() process.
To make use of a Forecaster, one must provide these two objects:
1. Model
2. Estimator
Theses two objects are prototyped as abstract and next subsections will cover how they work.
Model is an object defined by a class inherited from BaseTemplate a.k.a Model Template in the diagram below. It mainly turns the logic of fit() and predict() concrete by supplying the fitter as a
file (CmdStanPy) or a callable class (Pyro) and the internal predict() method. This object defines the overall inputs, model structure, parameters and likelihoods.
Meanwhile, there are different APIs implement slightly different ways of sampling and optimization (for MAP). orbit is designed to support various APIs such as CmdStanPy and Pyro (hopefully PyMC3,
Numpyro in the future!). The logic separating the call of different APIs with different interface is done by the Estimator class which is further inherited in PyroEstimator and StanEstimator.
Diagram above shows the interaction across classes under the Orbit package design.
Creating a Bayesian Linear Regression Model¶
The plan here is to build a classical regression model with the formula below:
\[y = \alpha + X \beta + \epsilon\]
where \(\alpha\) is the intercept, \(\beta\) is the coefficients matrix and \(\epsilon\) is the random noise.
To start with let’s load the libraries.
import pandas as pd
import numpy as np
import torch
import pyro
import pyro.distributions as dist
from copy import deepcopy
import matplotlib.pyplot as plt
import orbit
from orbit.template.model_template import ModelTemplate
from orbit.forecaster import SVIForecaster
from orbit.estimators.pyro_estimator import PyroEstimatorSVI
from orbit.utils.simulation import make_regression
from orbit.diagnostics.plot import plot_predicted_data
from orbit.utils.plot import get_orbit_style
%matplotlib inline
Since the Forecaster and Estimator are already built inside orbit, the rest of the ingredients to construct a new model will be a Model object that contains the follow:
• a callable class as a fitter
• a predict method
Define a Fitter¶
For Pyro users, you should find the code below familiar. All it does is to put a Bayesian linear regression (BLR) model code in a callable class. Details of BLR will not be covered here. Note that
the parameters here need to be consistent .
class MyFitter:
max_plate_nesting = 1 # max number of plates nested in model
def __init__(self, data):
for key, value in data.items():
key = key.lower()
if isinstance(value, (list, np.ndarray)):
value = torch.tensor(value, dtype=torch.float)
self.__dict__[key] = value
def __call__(self):
extra_out = {}
p = self.regressor.shape[1]
bias = pyro.sample("bias", dist.Normal(0, 1))
weight = pyro.sample("weight", dist.Normal(0, 1).expand([p]).to_event(1))
yhat = bias + weight @ self.regressor.transpose(-1, -2)
obs_sigma = pyro.sample("obs_sigma", dist.HalfCauchy(self.response_sd))
with pyro.plate("response_plate", self.num_of_obs):
pyro.sample("response", dist.Normal(yhat, obs_sigma), obs=self.response)
log_prob = dist.Normal(yhat[..., 1:], obs_sigma).log_prob(self.response[1:])
{"log_prob": log_prob}
return extra_out
Define the Model Class¶
This is the part requires the knowledge of orbit most. First we construct a class by plugging in the fitter callable. Users need to let the orbit estimators know the required input in addition to the
defaults (e.g. response, response_sd etc.). In this case, it takes regressor as the matrix input from the data frame. That is why there are lines of code to provide this information in
1. _data_input_mapper - a list or Enum to let estimator keep tracking required data input
2. set_dynamic_attributes - the logic define the actual inputs i.e. regressor from the data frame. This is a reserved function being called inside Forecaster.
Finally, we code the logic in predict() to define how we utilize posteriors to perform in-sample / out-of-sample prediction. Note that the output needs to be a dictionary where it supports components
class BayesLinearRegression(ModelTemplate):
_fitter = MyFitter
_data_input_mapper = ['regressor']
_supported_estimator_types = [PyroEstimatorSVI]
def __init__(self, regressor_col, **kwargs):
self.regressor_col = regressor_col
self.regressor = None
self._model_param_names = ['bias', 'weight', 'obs_sigma']
def set_dynamic_attributes(self, df, training_meta):
self.regressor = df[self.regressor_col].values
def predict(self, posterior_estimates, df, training_meta, prediction_meta, include_error=False, **kwargs):
model = deepcopy(posterior_estimates)
new_regressor = df[self.regressor_col].values.T
bias = np.expand_dims(model.get('bias'),-1)
obs_sigma = np.expand_dims(model.get('obs_sigma'), -1)
weight = model.get('weight')
pred_len = df.shape[0]
batch_size = weight.shape[0]
prediction = bias + np.matmul(weight, new_regressor) + \
np.random.normal(0, obs_sigma, size=(batch_size, pred_len))
return {'prediction': prediction}
Test the New Model with Forecaster¶
Once the model class is defined. User can initialize an object and build a forecaster for fit and predict purpose. Before doing that, the demo provides a simulated dataset here.
Data Simulation¶
x, y, coefs = make_regression(120, [3.0, -1.0], bias=1.0, scale=1.0)
df = pd.DataFrame(
np.concatenate([y.reshape(-1, 1), x], axis=1), columns=['y', 'x1', 'x2']
df['week'] = pd.date_range(start='2016-01-04', periods=len(y), freq='7D')
│ │ y │ x1 │ x2 │ week │
│0│2.382337 │0.345584 │0.000000 │2016-01-04 │
│1│2.812929 │0.330437 │-0.000000│2016-01-11 │
│2│3.600130 │0.905356 │0.446375 │2016-01-18 │
│3│-0.884275│-0.000000│0.581118 │2016-01-25 │
│4│2.704941 │0.364572 │0.294132 │2016-02-01 │
test_size = 20
train_df = df[:-test_size]
test_df = df[-test_size:]
Create the Forecaster¶
As mentioned previously, model is the inner object to control the math. To use it for fit and predict purpose, we need a Forecaster. Since the model is written in Pyro, the pick here should be
model = BayesLinearRegression(
blr = SVIForecaster(
<orbit.forecaster.svi.SVIForecaster at 0x2a6164950>
Now, an object blr is instantiated as a SVIForecaster object and is ready for fit and predict.
2024-03-19 23:37:55 - orbit - INFO - Using SVI (Pyro) with steps: 501, samples: 100, learning rate: 0.1, learning_rate_total_decay: 1.0 and particles: 100.
2024-03-19 23:37:56 - orbit - INFO - step 0 loss = 27333, scale = 0.077497
INFO:orbit:step 0 loss = 27333, scale = 0.077497
2024-03-19 23:37:58 - orbit - INFO - step 100 loss = 12594, scale = 0.0092399
INFO:orbit:step 100 loss = 12594, scale = 0.0092399
2024-03-19 23:38:00 - orbit - INFO - step 200 loss = 12591, scale = 0.0095592
INFO:orbit:step 200 loss = 12591, scale = 0.0095592
2024-03-19 23:38:03 - orbit - INFO - step 300 loss = 12593, scale = 0.0094199
INFO:orbit:step 300 loss = 12593, scale = 0.0094199
2024-03-19 23:38:06 - orbit - INFO - step 400 loss = 12591, scale = 0.0092691
INFO:orbit:step 400 loss = 12591, scale = 0.0092691
2024-03-19 23:38:10 - orbit - INFO - step 500 loss = 12591, scale = 0.0095463
INFO:orbit:step 500 loss = 12591, scale = 0.0095463
<orbit.forecaster.svi.SVIForecaster at 0x2a6164950>
Compare Coefficients with Truth¶
estimated_weights = blr.get_posterior_samples()['weight']
The code below is to compare the median of coefficients posteriors which is labeled as weight with the truth.
print("True Coef: {:.3f}, {:.3f}".format(coefs[0], coefs[1]) )
estimated_coef = np.median(estimated_weights, axis=0)
print("Estimated Coef: {:.3f}, {:.3f}".format(estimated_coef[0], estimated_coef[1]))
True Coef: 3.000, -1.000
Estimated Coef: 2.956, -0.976
Examine Forecast Accuracy¶
predicted_df = blr.predict(df)
_ = plot_predicted_data(train_df, predicted_df, 'week', 'y', test_actual_df=test_df, prediction_percentiles=[5, 95])
Additional Notes¶
In general, most of the diagnostic tools in orbit such as posteriors checking and plotting is applicable in the model created in this style. Also, users can provide point_method='median' in the fit()
under the SVIForecaster to extract median of posteriors directly. | {"url":"https://orbit-ml.readthedocs.io/en/latest/tutorials/build_your_own_model.html","timestamp":"2024-11-10T01:23:05Z","content_type":"text/html","content_length":"52552","record_id":"<urn:uuid:27a7a60a-4337-42ff-b5a4-5ce6a1d269d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00051.warc.gz"} |
Exercise-2 (Break-even analysis of a multiproduct company) - Accounting For Management
Exercise-2 (Break-even analysis of a multiproduct company)
PQR Company sells two products – product A and product B. The total fixed expenses of the company are $1,197,000. The monthly data of PQR is as follows:
Product A:
Product B:
1. Prepare contribution margin income statement for the company.
2. Calculate break-even point in dollars.
(1) Income statement:
(2) Computation of break-even point:
The PQR company sells two products. Its break-even point can be easily computed by dividing the total fixed expenses by overall contribution margin ratio (CM ratio).
Fixed expenses/Overall CM ratio
= 1,197,000/.63
= $1,900,000
Help us grow by sharing our content ♡ | {"url":"https://www.accountingformanagement.org/exercise-2-cvapr/","timestamp":"2024-11-05T13:09:17Z","content_type":"text/html","content_length":"42772","record_id":"<urn:uuid:7d572ae4-aebd-4f85-bcdf-d8c4484769f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00198.warc.gz"} |
Forecasting with Python and Power BI - AbsentData
Forecasting with Python and Power BI
We are using an airline passenger dataset that you can get from Kaggle, which shows us the data from 1949 to 1960 for airline passengers. Please find the PBIX file on my Github
Load in my dependencies, which are pandas and numpy and Matplotlib. I'm just going to import those in. And then once I have that, I'm going to read in the CSV that has our airline passenger data.
While I'm reading that in, by bringing that file name, I'm going to set the index column as the date column, which is called month in this data set. I'm going to ensure that we can parse the dates.
Watch the Video
If I check the head of that data set, you can see that the head has the month as the index, and then the number of passengers as a row. The data starts at 1949 and then the tail is from 1960. If I
look at the info, we also can see that same data as 144 months. You can see it spans from 1949 to 1960.
Triple Exponential Smoothing Forecast
The model that we're going to be using is triple exponential smoothing the Holt-Winters triple exponential smoothing model for our forecast. We're going to get that model from statsmodel.
from statsmodel.tsa.holtwinters import ExponentialSmoothing
We're going to use that as a function, but we need to set the parameters as that function and fit our data to that model.
Remember, our data looks like it's growing exponentially instead of in a linear fashion. So I'm going to use the multiplicative model instead of the additive model for defining our data. If you look
at these two images here, you can see the additive seasonality is more linear and then multiplicative is a little bit more on the exponential side of things.
Create a training and test set
Load in your machine learning model
So once I've defined that model, Fit it to our training data using the fit function. Plot the test data and plot the forecast using the forecast function to forecast 14 months forward from my
training data,
If we plot both of those, we can see that the model adheres quite well to our data. Now that we know that we have a decent model, we can also adjust how many periods forward we are forecasting using
that forecast function.
So once we have this, we can create a dashboard that does the same thing but let's look how we use that code in Power BI. Access the Python scripting witn Transform Data . Next and go over to run
Python. The same steps from Jupyter Notebook are required . You can see the code below somewhat truncated, Simply copy and paste it in.
import pandas as pd
import numpy as np
from statsmodels.tsa.holtwinters import ExponentialSmoothing
model = ExponentialSmoothing(dataset['#Passengers'],trend='mul',seasonal='mul',seasonal_periods=12).fit()
range = pd.date_range('01-01-1961',periods=36,freq='MS')
predictions = model.forecast(36)
predictions_range = pd.DataFrame({'#Passengers':predictions,'Month':range})
4 Comments
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4 years ago
• Great Solution using Python, could you please pbix file.
3 years ago
Great presentation.
If I get it right you are doing the model training right in the PBI Transform Data step.
I assume for some complex models and large training sets it is better to train the model outside of PBI, persist it into a file and in the PBI Transform just load the model and calculate the
Is it possible to do this in an automated way?
2 years ago
do you have a link to the jupyter notebook? Id like to use this on my PowerBI file.. Thank you! | {"url":"https://absentdata.com/power-bi/forecasting-with-python-and-power-bi/","timestamp":"2024-11-11T15:57:13Z","content_type":"text/html","content_length":"494281","record_id":"<urn:uuid:db4ca944-5ce3-4536-8df9-344085ee3a8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00891.warc.gz"} |
Problem 2 – Hint 3
As we increase the voltage, the temperature of the wire is increasing. Due to tiny temperature variations, the phase transition threshold is not exceeded simultaneously along the whole wire, but at a
single point. At that point, the heat dissipation rate is suddenly increased so that not only will the temperature of that point start raising, but the heat will propagate to the neighboring regions
triggering there a phase transition, too. This process will continue until a thermal quasi-equilibrium is reached: with an equilibrium temperature profile along the wire, the temperature at the phase
separation point equals to the critical temperature, corresponding neither to an expansion nor to a contraction of the high-resistivity region.
Note: no new correct solution submitted during this week, so the intermediate results page is not updated. The number of registered participants is now 1185.
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next hint for the Problem 1 will be published at 13:00 GMT, 15th January 2023, together with the next updates of the
intermediate results. | {"url":"https://physicscup.ee/problem-2-hint-2-5/","timestamp":"2024-11-03T22:52:30Z","content_type":"text/html","content_length":"50965","record_id":"<urn:uuid:98106468-86fd-48c1-99fb-8a0954444f3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00786.warc.gz"} |
DSpace Arşivi :: by Yazar "Kara, Hasan" değerine göre listeleniyor
Yazar "Kara, Hasan" seçeneğine göre listele
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• Bounds for the Error in Approximating a Fractional Integral by Simpson's Rule
(Mdpi, 2023) Budak, Hueseyin; Hezenci, Fatih; Kara, Hasan; Sarikaya, Mehmet Zeki
Simpson's rule is a numerical method used for approximating the definite integral of a function. In this paper, by utilizing mappings whose second derivatives are bounded, we acquire the upper
and lower bounds for the Simpson-type inequalities by means of Riemann-Liouville fractional integral operators. We also study special cases of our main results. Furthermore, we give some examples
with graphs to illustrate the main results. This study on fractional Simpson's inequalities is the first paper in the literature as a method.
• Conformable fractional versions of Hermite-Hadamard-type inequalities for twice-differentiable functions
(Springer, 2023) Hezenci, Fatih; Kara, Hasan; Budak, Huseyin
In this paper, new inequalities for the left and right sides of the Hermite-Hadamard inequality are acquired for twice-differentiable mappings. Conformable fractional integrals are used to derive
these inequalities. Furthermore, we provide our results by using special cases of obtained theorems.
• Fractional Simpson-Type Inequalities for Twice Differentiable Functions
(Univ Maragheh, 2023) Budak, Hueseyin; Kara, Hasan; Hezenci, Fatih
In the literature, several papers are devoted to inequal-ities of Simpson-type in the case of differentiable convex functions and fractional versions. Moreover, some papers are focused on
in-equalities of Simpson-type for twice differentiable convex functions. In this research article, we obtain an identity for twice differentiable convex functions. Then, we prove several
fractional inequalities of Simpson-type for convex functions.
• Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
(De Gruyter Poland Sp Z O O, 2022) Vivas-Cortez, Miguel J. J.; Kara, Hasan; Budak, Hüseyin; Ali, Muhammad Aamir; Chasreechai, Saowaluck
In this article, we introduce the notions of generalized fractional integrals for the interval-valued functions (IVFs) of two variables. We establish Hermite-Hadamard (H-H) type inequalities and
some related inequalities for co-ordinated convex IVFs by using the newly defined integrals. The fundamental benefit of these inequalities is that these can be turned into classical H-H
inequalities and Riemann-Liouville fractional H-H inequalities, and new k k -Riemann-Liouville fractional H-H inequalities can be obtained for co-ordinated convex IVFs without having to prove
each one separately.
• Generalized fractional midpoint type inequalities for co-ordinated convex functions
(University of Nis, 2023) Hezenci, Fatih; Budak, Hüseyin; Kara, Hasan; Sarıkaya, Mehmet Zeki
In this research paper, we investigate generalized fractional integrals to obtain midpoint type inequalities for the co-ordinated convex functions. First of all, we establish an identity for
twice partially differentiable mappings. By utilizing this equality, some midpoint type inequalities via generalized fractional integrals are proved. We also show that the main results reduce
some midpoint inequalities given in earlier works for Riemann integrals and Riemann-Liouville fractional integrals. Finally, some new inequalities for k-Riemann-Liouville fractional integrals are
presented as special cases of our results. © 2023, University of Nis. All rights reserved.
• Generalized Hermite-Hadamard inclusions for a generalized fractional integral
(Rocky Mt Math Consortium, 2023) Budak, Hueseyin; Kara, Hasan; Hezenci, Fatih
We introduce new generalized fractional integrals for interval-valued functions. Then we prove generalized Hermite-Hadamard type inclusions for interval-valued convex functions using these newly
defined generalized fractional integrals. We also show that these results generalize several results obtained in earlier works.
• Hermite-Hadamard, Trapezoid and Midpoint Type Inequalities Involving Generalized Fractional Integrals for Convex Functions
(Univ Maragheh, 2023) Kara, Hasan; Erden, Samet; Budak, Hüseyin
We first construct new Hermite-Hadamard type in-equalities which include generalized fractional integrals for convex functions by using an operator which generates some significant fractional
integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Mid-point type results involving generalized fractional integrals for func-tions
whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in
the earlier works.
• Hermite-Hadamard-Mercer type inclusions for interval-valued functions via Riemann-Liouville fractional integrals
(Scientific Technical Research Council Turkey-Tubitak, 2022) Kara, Hasan; Ali, Muhammad Aamir; Budak, Hüseyin
In this research, we first establish some inclusions of fractional Hermite???Hadamard???Mercer type for interval -valued functions. Moreover, by special cases of our main results, we show that
our main results reduce several inclusions obtained in the earlier works.
• Hermite-Hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals
(Wiley, 2021) Kara, Hasan; Ali, Muhammad Aamir; Budak, Huseyin
In this paper, we define interval-valued left-sided and right-sided generalized fractional double integrals. We establish inequalities of Hermite-Hadamard like for coordinated interval-valued
convex functions by applying our newly defined integrals.
• New Extensions of the Parameterized Inequalities Based on Riemann-Liouville Fractional Integrals
(Mdpi, 2022) Kara, Hasan; Budak, Hüseyin; Hezenci, Fatih
In this article, we derive the above and below bounds for parameterized-type inequalities using the Riemann-Liouville fractional integral operators and limited second derivative mappings. These
established inequalities generalized the midpoint-type, trapezoid-type, Simpson-type, and Bullen-type inequalities according to the specific choices of the parameter. Thus, a generalization of
many inequalities and new results were obtained. Moreover, some examples of obtained inequalities are given for better understanding by the reader. Furthermore, the theoretical results are
supported by graphs in order to illustrate the accuracy of each of the inequalities obtained according to the specific choices of the parameter.
• New midpoint type inequalities for generalized fractional integral
(Univ Tabriz, 2022) Budak, Hüseyin; Kara, Hasan; Kapucu, Rabia
In this paper, we first establish two new identities for differentiable function involving generalized fractional integrals. Then, by utilizing these equalities, we obtain some midpoint type
inequalities involving generalized fractional integrals for mappings whose derivatives in absolute values are convex. We also give several results as special cases of our main results.
• New parameterized inequalities for twice differentiable functions
(Univ Nis, Fac Sci Math, 2023) Budak, Hüseyin; Kara, Hasan; Hezenci, Fatih; Sarıkaya, Mehmet Zeki
The present paper first establishes that an identity involving generalized fractional integrals is proved for twice differentiable functions by using a parameter. By using this equality, we
obtain some parameterized inequalities for the functions whose second derivatives in absolute value are convex. Finally, we show that our main results reduce to trapezoid, midpoint Simpson and
Bullen-type inequalities which are proved in earlier published papers.
• New Simpson type inequalities for twice differentiable functions via generalized fractional integrals
(Amer Inst Mathematical Sciences-Aims, 2022) You, Xue Xiao; Hezenci, Fatih; Budak, Hüseyin; Kara, Hasan
Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also
investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type
inequalities for functions whose second derivatives in absolute value are convex.
• New version of fractional Simpson type inequalities for twice differentiable functions
(Springer, 2021) Hezenci, Fatih; Budak, Huseyin; Kara, Hasan
Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also
investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for
twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a
new version of fractional Simpson type inequalities for twice differentiable functions.
• Novel results of Milne-type inequalities involving tempered fractional integrals
(Springer, 2024) Hezenci, Fatih; Budak, Huseyin; Kara, Hasan; Bas, Umut
In this current research, we focus on the domain of tempered fractional integrals, establishing a novel identity that serves as the cornerstone of our study. This identity paves the way for the
Milne-type inequalities, which are explored through the framework of differentiable convex mappings inclusive of tempered fractional integrals. The significance of these mappings in the realm of
fractional calculus is underscored by their ability to extend classical concepts into more complex, fractional dimensions. In addition, by using the Holder inequality and power-mean inequality,
we acquire some new Milne-type inequalities. Moreover, the practicality and theoretical relevance of our findings are further demonstrated through the application of specific cases derived from
the theorems.
• On extensions of Hermite-Hadamard type inclusions for interval-valued convex functions
(Univ Nis, Fac Sci Math, 2023) Kara, Hasan; Budak, Huseyin; Hezenci, Fatih
In this work, by using weighted Jensen inclusion, we establish some new weighted Hermite- Hadamard type inclusions involving two real parameters for interval-valued convex functions. In addition,
some extensions of Hermite-Hadamard inclusion are obtained by special choices of parameters. Moreover, we give some examples to illustrate the main results of this work.
• On Fejer Type Inclusions for Products of Interval-Valued Convex Functions
(Univ Nis, Fac Sci Math, 2021) Budak, Hüseyin; Kara, Hasan; Erden, Samet
We first get some new Fejer type inclusions for products of interval-valued convex mappings. The most important feature of our work is that it contains Fejer type inclusions for both
interval-valued integrals and interval-valued fractional integrals.
• On Fejer type inequalities for co-ordinated hyperbolic rho-convex functions
(Amer Inst Mathematical Sciences-Aims, 2020) Kara, Hasan; Budak, Huseyin; Kiris, Mehmet Eyup
In this study, we first establish some Hermite-Hadamard-Fejer type inequalities for coordinated hyperbolic rho-convex functions. Then, by utilizing these inequalities, we also give some
fractional Fejer type inequalities for co-ordinated hyperbolic rho-convex functions. The inequalities obtained in this study provide generalizations of some result given in earlier works.
• On Fractional Newton Inequalities via Coordinated Convex Functions
(Mdpi, 2022) Kösem, Pınar; Kara, Hasan; Budak, Hüseyin; Ali, Muhammad Aamir; Nonlaopon, Kamsing
In this paper, firstly, we present an integral identity for functions of two variables via Riemann-Liouville fractional integrals. Then, a Newton-type inequality via partially differentiable
coordinated convex mappings is derived by taking the absolute value of the obtained identity. Moreover, several inequalities are obtained with the aid of the Holder and power mean inequality. In
addition, we investigate some Newton-type inequalities utilizing mappings of two variables with bounded variation. Finally, we gave some mathematical examples and their graphical behavior to
validate the obtained inequalities.
• On generalized Ostrowski, Simpson and Trapezoidal type inequalities for co-ordinated convex functions via generalized fractional integrals
(Springer, 2021) Budak, Huseyin; Hezenci, Fatih; Kara, Hasan
In this study, we prove an identity for twice partially differentiable mappings involving the double generalized fractional integral and some parameters. By using this established identity, we
offer some generalized inequalities for differentiable co-ordinated convex functions with a rectangle in the plane R2. Furthermore, by special choice of parameters in our main results, we obtain
several well-known inequalities such as the Ostrowski inequality, trapezoidal inequality, and the Simpson inequality for Riemann and Riemann-Liouville fractional integrals. | {"url":"https://acikerisim.duzce.edu.tr/browse/author?value=Kara,%20Hasan","timestamp":"2024-11-05T06:37:14Z","content_type":"text/html","content_length":"749983","record_id":"<urn:uuid:da14feaa-8b62-4627-bc5d-45e44453cdde>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00409.warc.gz"} |
Modular Arithmetic
In Quadratic Residues we learnt what it means to take the square root modulo an integer. We also saw that taking a root isn't always possible.
In the previous case when $p = 29$, even the simplest method of calculating the square root was fast enough, but as $p$ gets larger, this method becomes wildly unreasonable.
Lucky for us, we have a way to check whether an integer is a quadratic residue with a single calculation thanks to Legendre. In the following, we will assume we are working modulo a prime $p$.
Before looking at Legendre's symbol, let's take a brief detour to see an interesting property of quadratic (non-)residues.
Quadratic Residue * Quadratic Residue = Quadratic Residue
Quadratic Residue * Quadratic Non-residue = Quadratic Non-residue
Quadratic Non-residue * Quadratic Non-residue = Quadratic Residue Want an easy way to remember this? Replace "Quadratic Residue" with $+1$ and "Quadratic Non-residue" with $-1$, all three results are
the same!
So what's the trick? The
Legendre Symbol
gives an efficient way to determine whether an integer is a quadratic residue modulo an odd prime $p$.
Legendre's Symbol: $(a / p) \equiv a^{(p-1)/2} \mod p$ obeys:
$(a / p) = 1$ if $a$ is a quadratic residue and $a \not\equiv 0 \mod p$
$(a / p) = -1$ if $a$ is a quadratic non-residue $\mod p$
$(a / p) = 0$ if $a \equiv 0 \mod p$
Which means given any integer $a$, calculating $a^{(p-1)/2} \mod p$ is enough to determine if $a$ is a quadratic residue.
Now for the flag. Given the following 1024 bit prime and 10 integers, find the quadratic residue and then calculate its square root; the square root is your flag. Of the two possible roots, submit
the larger one as your answer.
So Legendre's symbol tells us which integer is a quadratic residue, but how do we find the square root?! The prime supplied obeys $p = 3 \mod 4$, which allows us easily compute the square root. The
answer is online, but you can figure it out yourself if you think about Fermat's little theorem.Challenge files:
You have solved this challenge! View solutions
You must be logged in to submit your flag. | {"url":"https://cryptohack.org/courses/modular/root1/","timestamp":"2024-11-10T19:10:48Z","content_type":"text/html","content_length":"23036","record_id":"<urn:uuid:182ee44a-0322-4591-9c85-7142e3c197dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00443.warc.gz"} |
Powerball Calculator
Created by Tibor Pál, PhD candidate
Reviewed by Dominik Czernia, PhD and Jack Bowater
Based on research by
Cipra T. Financial and Insurance Formulas; 2006See 1 more source
Last updated: Sep 02, 2024
Did you win a prize in the Powerball lottery? With the Powerball calculator (which is actually a Powerball payout calculator or a lottery lump sum vs. annuity calculator), you can estimate how much
money you will receive and compare the Powerball lump sum vs. annuity payouts to make the best financial decision on your winnings.
Since you can estimate the taxes deducted from your prize, you can also apply our tool as a Powerball tax calculator. Still, you may check our lottery tax calculator if you want to learn more about
taxes deducted from a lottery prize.
In the present article, you can read about the following:
• How Powerball annuity works;
• How is the Powerball annuity calculated?
• How is the Powerball annuity paid out?
• How much is the prize of Powerball after taxes?
Powerball lump sum vs. annuity
If you are lucky enough to win the lottery, you need to make an important decision on how to collect your prize. In general, there are two ways the Powerball pays out: through a lottery annuity or as
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In general, if you would like to receive all of your money as early as possible, the lump-sum Powerball payout is the best option for you as you receive all of your jackpot at once (after taxes have
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How does the Powerball lump sum work?
As we mentioned already, the winner may claim their Powerball jackpot in its cash value, or, in other words, as a lump sum. When choosing this option, the winner is paid in one lump sum. However, the
cash value payout of the jackpot is much less than the one offered for an annuity Powerball payout. So, you may ask, "How much do I get if I win the Powerball?" It is about 52 percent of the total
jackpot amount (before taxes). For example, if the Powerball jackpot is at $100 million, the cash value would be around $52 million.
Despite the considerable initial loss, it has its advantages, which might be attractive enough for turning to this option. For example:
• Probably the biggest advantage is that you can receive the money immediately. As the well-known proverb says — time is money.
• By obtaining your prize quickly, you can cover immediate financial expenses and spend the money immediately.
• If you're thinking about retirement or have a good investment plan, the lump-sum payment might be a reasonable choice.
How does the Powerball annuity payout work?
The other option for Powerball winners is to receive the prize in annual payments called an annuity. How many years is the Powerball annuity payout for? Well, this alternative pays the winner a
specific amount annually for 30 years until the total jackpot is fully deployed.
The annuity value is paid through government bonds purchased with the jackpot's cash value. These bonds earn revenue over the annual payments, making up for the difference between the cash value and
the advertised annuity jackpot value.
Powerball after taxes
It is clear that the winner cannot simply receive the total advertised amount of the Powerball jackpot immediately. However, its final amount is greatly affected by the taxes you are obliged to pay
after winning a jackpot.
As for the lump-sum payment, before all other fees, it is subject to a federal withholding tax of 24%, which is then complemented by the remaining federal taxes according to the marginal rates (37%
in the top tax brackets), corrected by possible deductions. Besides federal taxation, the prize is also subject to any applicable state tax depending on the winner's residence. The state tax on
lottery can range from zero to around 10 percent.
In the case of annuity payouts, the amount paid annually will be added to the winner's income tax return each year and paid at tax time. Keep in mind, however, that tax rates are not set in stone and
may alter over time.
How to use the Powerball payout calculator?
There are only a couple of variables required to run the Powerball payout calculator:
1. In the field "How much did you win?", set the Powerball jackpot you would like to analyze.
2. Select how you file your taxes for "What is your filing status?".
3. Choose your state of residence from the drop-down list asking "Where do you live?" to determine the applicable state tax rate.
4. You'll find a tabulated overview of the gross payout, federal taxes, state taxes, and net payout that applies to the winnings if you choose a lump sum payout or an annuity payout. In addition,
you can find the yearly annuity payouts in the payout schedule as well.
How many years is the Powerball annuity payout?
The Powerball annuity payout offers winners 30 payments spread over 29 years, where payment amounts increase each year by around 5 percent.
What is the federal tax on Powerball winnings?
The federal tax on the lottery is based on the federal marginal rates, which is 37 percent in the highest bracket. In practice, there is a 24 percent federal withholding tax of the gross prize plus
the remaining tax, based on your filing status.
For example, if your gross prize is $1,000,000, you need to pay $334,072 in total tax ($240,000 federal withholding plus the remaining $94,072 for single filing status in 2021).
Do I need to pay always state taxes if I win on Powerball lottery?
No. The applicable state tax range from zero to around 10 percent. Thus, lottery prizes are not subject to state taxes in states like California, Texas, or Florida.
How much do I get if I win the Powerball?
Of course, your Powerball winnings depends on the prize you win, but also greatly depends on how you opt for receiving the winning (lump-sum or annuity payout) and the applicable federal and state
taxes. Use our Powerball calculator to see how much you can get with different options.
How do I calculate Powerball annuity?
You need to follow the below to estimate the annuity payments of a Powerball jackpot:
1. Use the following growing annuity formula to compute the payout in a given year (n):
Payout in year n = -Gross payout / [(1 − 1.05^30) / 0.05] × 1.05^n−1
2. Deduct federal tax, which is about 37% of the given annuity payout.
3. Deduct state tax, if applicable.
Powerball calculator disclaimer
You should consider the Powerball calculator as a model for financial approximation. All payment figures, balances, and tax figures are estimates based on the data you provided in the specifications
that are, despite our best efforts, not exhaustive.
Without claiming completeness, please note the following:
• The federal taxes are approximated based on the 2024 marginal tax tables published by the , without taking account of possible deductions;
• All state taxes are estimated with fixed-rate calculation applicable in October 2024 (without the effect of filing status, the possible existence of graduated-rate brackets, or modification in
• Potential additional local taxes are not considered; and
• If you are not a U.S. resident, you will typically have a flat 30% federal withholding, and state taxes may differ from what is listed above.
For this reason, we created the calculator for instructional purposes only. Still, if you experience a relevant drawback or encounter any inaccuracy, we are always pleased to receive useful feedback
and advice.
What is your filing status?
│ │Lump sum** │Annuity │Difference │
│Gross Payout │$52,000,000│$100,000,000 │$48,000,000│
│Fed. Tax 24% │$12,480,000│$24,000,000 │$11,520,000│
│Add. Fed Tax* │$6,718,188 │$11,745,633 │$5,027,445 │
│State Taxes │$1,300,000 │$2,500,000 │$1,200,000 │
│Net Payout │$31,501,812│$61,754,368 │$30,252,556│
* Additional Federal Tax (up to 37%, tax amount based on filing status minus the withheld 24% Federal Tax).
** The gross payout for lump sum payment is approximated by 52% of the total lottery prize - you can alter the lump sum payout percentage in the advanced mode.
Yearly annuity payouts
Year Gross Payment Federal Taxes State Taxes Net Payment
1 1,505,144 515,091 37,629 952,424
2 1,580,401 542,936 39,510 997,955
3 1,659,421 572,173 41,486 1,045,762
4 1,742,392 602,873 43,560 1,095,959
5 1,829,511 635,107 45,738 1,148,667
6 1,920,987 668,953 48,025 1,204,009
7 2,017,036 704,491 50,426 1,262,119
8 2,117,888 741,806 52,947 1,323,135
9 2,223,782 780,987 55,595 1,387,201
10 2,334,972 822,127 58,374 1,454,470
11 2,451,720 865,324 61,293 1,525,103
12 2,574,306 910,681 64,358 1,599,267
13 2,703,022 958,306 67,576 1,677,140
14 2,838,173 1,008,312 70,954 1,758,907
15 2,980,081 1,060,818 74,502 1,844,761
16 3,129,085 1,115,949 78,227 1,934,909
17 3,285,540 1,173,837 82,138 2,029,564
18 3,449,817 1,234,620 86,245 2,128,951
19 3,622,307 1,298,441 90,558 2,233,308
20 3,803,423 1,365,454 95,086 2,342,883
21 3,993,594 1,435,817 99,840 2,457,937
22 4,193,274 1,509,699 104,832 2,578,743
23 4,402,937 1,587,275 110,073 2,705,589
24 4,623,084 1,668,729 115,577 2,838,778
25 4,854,238 1,754,256 121,356 2,978,626
26 5,096,950 1,844,059 127,424 3,125,467
27 5,351,798 1,938,353 133,795 3,279,650
28 5,619,388 2,037,361 140,485 3,441,542
29 5,900,357 2,141,320 147,509 3,611,528
30 6,195,375 2,250,476 154,884 3,790,014 | {"url":"https://www.omnicalculator.com/finance/powerball","timestamp":"2024-11-06T23:17:24Z","content_type":"text/html","content_length":"282045","record_id":"<urn:uuid:ff60309c-44c1-4eb0-a438-3d1f209e2247>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00797.warc.gz"} |
Order of Operations in context of resolution of instrument formula
31 Aug 2024
Here is an academic article on the Order of Operations (BODMAS) in the context of resolving instrument formulas:
Title: The Importance of Order of Operations in Instrument Formula Resolution: A Study on the Application of BODMAS
The resolution of instrument formulas is a crucial step in various scientific and engineering applications. However, the complexity of these formulas often leads to errors if not approached
correctly. This article highlights the significance of the Order of Operations (BODMAS) in resolving instrument formulas. We demonstrate how the application of BODMAS can lead to accurate results by
providing examples of instrument formulas in both ASCII and BODMAS formats.
Instrument formulas are mathematical expressions that describe the behavior of physical instruments, such as sensors, transducers, and amplifiers. These formulas often involve complex calculations
involving multiplication, division, addition, and subtraction operations. The Order of Operations (BODMAS) is a set of rules that dictate the order in which these operations should be performed to
ensure accurate results.
The BODMAS Rules:
The BODMAS acronym stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. These rules are applied in the following order:
1. Brackets: Evaluate any expressions within parentheses or brackets first.
2. Orders: Evaluate any exponential operations (e.g., squaring or cubing) next.
3. Division and Multiplication: Perform these operations from left to right.
4. Addition and Subtraction: Finally, perform these operations from left to right.
Example 1: Simple Instrument Formula
Consider the following instrument formula:
Vout = (2 × Vin) + (3 - 1)
In ASCII format, this formula would be written as:
Vout = (2*Vin) + (3-1)
To resolve this formula using BODMAS, we follow the rules:
1. Evaluate expressions within brackets: (2 × Vin) and (3 - 1)
2. Perform multiplication: 2 × Vin
3. Perform subtraction: 3 - 1 = 2
4. Add the results: (2 × Vin) + 2
The final result is:
Vout = (2*Vin) + 2
Example 2: Complex Instrument Formula
Consider the following instrument formula:
Vout = ((Vin × R1) / (R2 + R3)) - (4 × Vref)
In ASCII format, this formula would be written as:
Vout = (((Vin*R1)/(R2+R3))-4*Vref)
To resolve this formula using BODMAS, we follow the rules:
1. Evaluate expressions within brackets: ((Vin × R1) / (R2 + R3)) and (4 × Vref)
2. Perform multiplication: Vin × R1
3. Perform division: (Vin × R1) / (R2 + R3)
4. Perform subtraction: (4 × Vref)
5. Add the results: ((Vin × R1) / (R2 + R3)) - (4 × Vref)
The final result is:
Vout = (((Vin*R1)/(R2+R3))-4*Vref)
In this article, we have demonstrated the importance of the Order of Operations (BODMAS) in resolving instrument formulas. By applying the BODMAS rules, we can ensure accurate results and avoid
errors that may arise from incorrect order of operations. The examples provided illustrate how the application of BODMAS can lead to correct solutions for complex instrument formulas.
1. “Order of Operations” by Math Open Reference (2022)
2. “Instrument Formulas” by Instrumentation, Measurement & Control (2019)
Note: The article is written in a formal academic tone and includes references to support the information presented.
Related articles for ‘resolution of instrument formula’ :
• Reading: Order of Operations in context of resolution of instrument formula
Calculators for ‘resolution of instrument formula’ | {"url":"https://blog.truegeometry.com/tutorials/education/df9fe9b8331b70baab0f31cc6c77bf8b/JSON_TO_ARTCL_Order_of_Operations_in_context_of_resolution_of_instrument_formula.html","timestamp":"2024-11-12T21:32:02Z","content_type":"text/html","content_length":"19400","record_id":"<urn:uuid:9e3b1fb7-f1e1-4a3c-8dc3-ca358c202f1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00040.warc.gz"} |
Is part of the Bibliography
Magnetic nanoparticles (MNP) are widely investigated for biomedical applications in diagnostics (e.g. imaging), therapeutics (e.g. hyperthermia) and general biosensing. For all these applications,
the MNPs’ unique magnetic relaxation mechanism in an alternating magnetic field (AFM) is stimulated to induce desired effects. Whereas magnetic fluid hyperthermia (MFH) and magnetic particle imaging
(MPI) are the most prominent examples for biomedical application, we investigate the relatively new biosensing application of frequency mixing magnetic detection (FMMD) from a fundamental
perspective. Generally, we ask how specific MNP parameters (core size, magnetic anisotropy) influence the signal, specifically we predict the most effective MNP core size for signal generation. In
FMMD, simultaneously two AFM are applied: a low-frequency magnetic driving field, driving MNP close to saturation, and a high-frequency excitation field that probes MNP susceptibility: . Resulting
from the nonlinear magnetization of the MNP, harmonics of both individual incident frequencies as well as intermodulation products of these frequencies are generated. In this work, we present
numerical Monte-Carlo(MC)-based simulations of the MNP relaxation process, solving the Landau-Lifshitz-Gilbert (LLG) equation to predict FMMD signals: As Figure 1 shows for the first four
intermodulation signals , with , we can clearly see that larger core sizes generally increase the signal intensity. Same trend is predicted by a simple Langevin-function based thermal equilibrium
model. Both predictions include a lognormal size distribution. The effect of core size distribution presumably dominates the effect of magnetic anisotropy. The findings are supported by comparison
with experimental data and help to identify which MNP are best suited for magnetic biosensing applications using FMMD.
Dual frequency magnetic excitation of magnetic nanoparticles (MNP) enables enhanced biosensing applications. This was studied from an experimental and theoretical perspective: nonlinear sum-frequency
components of MNP exposed to dual-frequency magnetic excitation were measured as a function of static magnetic offset field. The Langevin model in thermodynamic equilibrium was fitted to the
experimental data to derive parameters of the lognormal core size distribution. These parameters were subsequently used as inputs for micromagnetic Monte-Carlo (MC)-simulations. From the hysteresis
loops obtained from MC-simulations, sum-frequency components were numerically demodulated and compared with both experiment and Langevin model predictions. From the latter, we derived that
approximately 90% of the frequency mixing magnetic response signal is generated by the largest 10% of MNP. We therefore suggest that small particles do not contribute to the frequency mixing signal,
which is supported by MC-simulation results. Both theoretical approaches describe the experimental signal shapes well, but with notable differences between experiment and micromagnetic simulations.
These deviations could result from Brownian relaxations which are, albeit experimentally inhibited, included in MC-simulation, or (yet unconsidered) cluster-effects of MNP, or inaccurately derived
input for MC-simulations, because the largest particles dominate the experimental signal but concurrently do not fulfill the precondition of thermodynamic equilibrium required by Langevin theory.
Biomedical applications of magnetic nanoparticles (MNP) fundamentally rely on the particles’ magnetic relaxation as a response to an alternating magnetic field. The magnetic relaxation complexly
depends on the interplay of MNP magnetic and physical properties with the applied field parameters. It is commonly accepted that particle core size is a major contributor to signal generation in all
the above applications, however, most MNP samples comprise broad distribution spanning nm and more. Therefore, precise knowledge of the exact contribution of individual core sizes to signal
generation is desired for optimal MNP design generally for each application. Specifically, we present a magnetic relaxation simulation-driven analysis of experimental frequency mixing magnetic
detection (FMMD) for biosensing to quantify the contributions of individual core size fractions towards signal generation. Applying our method to two different experimental MNP systems, we found the
most dominant contributions from approx. 20 nm sized particles in the two independent MNP systems. Additional comparison between freely suspended and immobilized MNP also reveals insight in the MNP
microstructure, allowing to use FMMD for MNP characterization, as well as to further fine-tune its applicability in biosensing.
Frequency mixing magnetic detection (FMMD) is a sensitive and selective technique to detect magnetic nanoparticles (MNPs) serving as probes for binding biological targets. Its principle relies on the
nonlinear magnetic relaxation dynamics of a particle ensemble interacting with a dual frequency external magnetic field. In order to increase its sensitivity, lower its limit of detection and overall
improve its applicability in biosensing, matching combinations of external field parameters and internal particle properties are being sought to advance FMMD. In this study, we systematically probe
the aforementioned interaction with coupled Néel–Brownian dynamic relaxation simulations to examine how key MNP properties as well as applied field parameters affect the frequency mixing signal
generation. It is found that the core size of MNPs dominates their nonlinear magnetic response, with the strongest contributions from the largest particles. The drive field amplitude dominates the
shape of the field-dependent response, whereas effective anisotropy and hydrodynamic size of the particles only weakly influence the signal generation in FMMD. For tailoring the MNP properties and
parameters of the setup towards optimal FMMD signal generation, our findings suggest choosing large particles of core sizes dc > 25 nm nm with narrow size distributions (σ < 0.1) to minimize the
required drive field amplitude. This allows potential improvements of FMMD as a stand-alone application, as well as advances in magnetic particle imaging, hyperthermia and magnetic immunoassays. | {"url":"https://opus.bibliothek.fh-aachen.de/opus4/solrsearch/index/search/searchtype/authorsearch/author/Ahmed+Shalaby","timestamp":"2024-11-05T19:26:48Z","content_type":"application/xhtml+xml","content_length":"40454","record_id":"<urn:uuid:94cc3ca7-ee23-4fd5-8e12-8585d420ea27>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00039.warc.gz"} |
Fractions Practice Worksheet - UnicMinds
Fractions Practice Worksheet
Fractions can represent parts of a group or parts of a whole. The terms that we use for fractions are the “numerator” and the “denominator.” The numerator is the number of parts that we have and the
denominator is the total number of parts that make up the whole.
Let’s look at an example:
We have different shapes and we’ll divide each one of them into a number of equal parts, which is the denominator.
The number of colored parts is the numerator.
Main types of Fractions
Question 1:
Simplify 15/16 – 11/12
Question 2:
A bucket contains 24 ¾ litres of water. How many ¾ litre of jugs can be filled from the bucket to get it emptied.
Question 3:
A carton contains 40 boxes of nails and each box weighs 3 ¾ Kgs. How much would a carton of nails weigh?
Question 4:
Which of the following numbers are equal?
A). -9/12 and 8/-12
B). -16/20 and 20/-25
C). -7/21 and 3/-9
D). -8/-14 and 13/21
Question 5:
Arrange them in descending order 1/5, 3/7, 7/10, and 13/28.
Question 6:
If 60/75 is equivalent to 3/x, then what is the value of x?
Question 7:
If 5/7 = x/28, find the value of x
Question 8:
Write 5/8 as an equivalent fraction with a denominator of 24.
Hint: Equivalent fractions are the fractions that have different numerators and denominators but are equal to the same value.
Hope this is useful, thank you.
You may like to read: Decimals Practice Worksheet, Benefits of Math Tuitions, and Best CBSE Math Books for Class 10. | {"url":"https://unicminds.com/fractions-practice-worksheet/","timestamp":"2024-11-10T18:47:11Z","content_type":"text/html","content_length":"431045","record_id":"<urn:uuid:f8f0bbbc-d145-46f8-9566-637f0f12f261>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00025.warc.gz"} |
Bayesian Networks and Certainty Factors
A Bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic independencies. For example, a Bayesian network could represent
the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.
Bayesian Networks are also called : Bayes nets, Bayesian Belief Networks (BBNs) or simply Belief Networks. Causal Probabilistic Networks (CPNs).
A Bayesian network consists of :
a set of nodes and a set of directed edges between nodes.
the edges reflect cause-effect relations within the domain.
The effects are not completely deterministic (e.g. disease -> symptom).
the strength of an effect is modeled as a probability.
Bayesian Networks
We have applied Bayesian probability theory, in earlier three examples (example 1, 2, and 3) , to relate two or more events. But this can be used to relate many events by tying them together in a
Consider the previous example 3 - Clinic trial
The trial says, the probability of the patients having HIV virus is 0.15.
A blood test done on patients :
If patient has virus, the test is +ve with probability 0.95.
If the patient does not have the virus, the test is +ve with probability 0.02.
This means given : P(H) = 0.15 ; P(P|H) = 0.95 ; P(P|¬H) = 0.02
Imagine, the patient is given a second test independently of the first; means the second test is done at a later date by a different person using different equipment. So, the error on the first test
does not affect the probability of an error on the second test.
In other words the two tests are independent. This is depicted using the diagram below :
A simple example of a Bayesian Network.
Event H is the cause of the two events P1 and P2.
The arrows represent the fact that H is driving P1 and P2.
The network contained 3 nodes.
If both P1 and P2 are +ve then find the probability that patient has the virus ? In
other words asked to find P(H|P1 ∩ P2) .
How to find ?
â– Bayes Theorem
As worked before for P(P) which is the probability of a +ve result, here again break this into two separate cases:
patient has virus and both tests are +ve
patient not having virus and both tests are +ve
As before use the second axiom of probability
P(P1 ∩ P2) = P(P1 ∩ P2 |H) P(H) + P(P1 ∩ P2 |¬H) P(¬H)
‡ Because the two tests are independent given H we can write :
P(P1 ∩ P2) = P(P1|H) P(P2|H) P(H) + P(P1|¬H) P(P2|¬H) P(¬H)
0.95 × 0.95 × 0.15 + 0.02× 0.02 × 0.85
Substitute this into Bayes Theorem above and obtain
Note : The results while two independent HIV tests performed Previously we calculated the probability, that the patient had HIV
given one +ve test, as 0.8934.
Later second HIV test was performed. After two +ve tests, we see that the probability has gone up to 0.99749.
So after two +ve tests it is more certain that the patient does have the HIV virus.
The next slide : a case where one tests is +ve and other is -ve.
Case where one tests is +ve and other is -ve.
This means, an error on one of the tests but we don‟t know which one; it may be any one.
P(H| P1 ∩ ¬P2).
The issue is - whether the patient has HIV virus or not ?
‡ We need to calculate
Following same steps for the case of two +ve tests, write Bayes Theorem
‡ Note :
Belief in H, the event that the patient has virus, has increased.
Prior belief was 0.15 but it has now gone up to 0.299.
This appears strange because we have been given two contradictory pieces of data. But looking closely we see that probability of an error in each case is not equal.
The probability of a +ve test when patient is actually -ve is 0.02. The probability of a -ve test when patient is actually +ve is 0.05. Therefore we are more inclined to believe an error on the
second test and this slightly increases our belief that the patient is +ve.
More Complicated Bayesian Networks
The previous network was simple contained three nodes. Let us look at a slightly more complicated one in the context of heart disease.
Given the following facts about heart disease.
Either smoking or bad diet or both can make heart disease more likely.
Heart disease can produce either or both of the following two symptoms:
high blood pressure
an abnormal electrocardiogram
Here smoking and bad diet are regarded as causes of heart disease. The heart disease in turn is a cause of high blood pressure and an abnormal electrocardiogram.
â– An appropriate network for heart disease is represented as
Here H has two causes S and D.
Find probability of H, given each of the four possible
combinations of
A medical survey gives us the following data :
P(S) = 0.3 P(D) = 0.4
P(H| S ∩ D) = 0.8
P(H| ¬S ∩ D) = 0.5
P(H| S ∩ ¬D) = 0.4
P(H| ¬S ∩ ¬D) = 0.1
P(B|H) = 0.7 P(B|¬H) = 0.1
P(E|H) = 0.8 P(E|¬H) = 0.1
Given these information, an answer to the question concerning this network :
what is the probability of heart disease ?
[Note : The interested students may try to the find answer.] | {"url":"https://www.brainkart.com/article/Bayesian-Networks-and-Certainty-Factors_8590/","timestamp":"2024-11-09T05:48:47Z","content_type":"text/html","content_length":"64418","record_id":"<urn:uuid:fee8c7d2-e66a-45dc-9c4b-539c6d36d543>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00153.warc.gz"} |
Gauss reciprocity law
From Encyclopedia of Mathematics
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A relation connecting the values of the Legendre symbols (cf. Legendre symbol) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. Quadratic reciprocity law). In addition to the
principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation
$$\left(\frac pq\right)\left(\frac qp\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$
there are two more additions to this law, viz.:
The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to
give a complete proof of the law [1]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime.
Attempts to establish the reciprocity law for cubic and biquadratic residues led Gauss to introduce the ring of Gaussian integers.
[1] C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)
[2] I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)
[3] H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)
Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of algebraic number theory and class field
theory. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law.
How to Cite This Entry:
Gauss reciprocity law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_reciprocity_law&oldid=35697
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Gauss_reciprocity_law&printable=yes","timestamp":"2024-11-10T08:12:15Z","content_type":"text/html","content_length":"16067","record_id":"<urn:uuid:2a03b5e0-167b-4b63-a157-29edcf79af51>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00267.warc.gz"} |
What is the probability of drawing an ace from a standard deck of cards?
What is the probability of drawing an ace from a standard deck of cards?
The probability of picking up an ace in a 52 deck of cards is 4/52 since there are 4 aces in the deck. The odds of picking up any other card is therefore 52/52 – 4/52 = 48/52.
What is the probability of drawing a 10 from a deck of 52 cards?
A Statistics tutor answered So, the probability of drawing a 10 and then heart with replacement is 1/4 * 1/13 = 1/52.
What is the probability of drawing a black card or a ten in a deck of cards?
Out of 52 cards in a deck, there are 2 black 10 cards. So the probability of drawing a black 10 is =2/52=1/26.
What is the probability of spade or an ace?
You should know that a standard 52-card deck is made up of four suits of 13 cards each. This means that there are 13 spades and 4 aces (the Ace of Spades is both). So the probability of drawing any
spade is 1 in 4, and the probability of drawing any ace is 1 in 13.
What is the probability of drawing a jack or red card from a standard deck of cards?
1 Expert Answer. Here to help with Math! The answer is A. The probability that you draw a red card is 26/52 or 1/2, since half the cards in the deck are red.
What is the probability of drawing a 3 from a deck of cards?
A standard deck of playing cards has four suits — each suit has 3 face cards. That means a standard deck already contains twelve face cards, so the probability of getting three is 100%. Or did you
mean “What is the chance of getting 3 face cards by picking only three cards without replacement?”
When two cards are drawn from a standard deck What is the probability of drawing a face card or an ace?
When you draw first card the probability of it being an Ace is 4/52. Since you have already drawn a card which is an Ace then there will be remaining 3 aces in 51 cards. So probability of second card
being an Ace is 3/51. | {"url":"https://yourgametips.com/miscellaneous/what-is-the-probability-of-drawing-an-ace-from-a-standard-deck-of-cards/","timestamp":"2024-11-08T04:51:12Z","content_type":"text/html","content_length":"122580","record_id":"<urn:uuid:2db18b01-94fe-4cb7-8962-fad09ceb9f11>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00214.warc.gz"} |
I. Introduction
One of the most important problems in natural language processing is text classifications which are applicable in so many areas of modern technology and innovation. With text as one of the most
common categories of data available, more than 80% of data are unstructured. This makes it difficult and challenging to comprehensively analyze them, hence many businesses are unable to exploit its
full potential for business benefit [
]. The inability to exploit full potential of billions of unstructured data brings about the suitability of machine learning algorithm for training model for text classification and sentiment
analysis in areas like web search [
], spam detection [
], topic classification [
], news classification, sentiment classification [
] to the spotlight, as machine learning algorithms can be use in Natural Language Processing for text classification and sentiment [
] related activities such as spam filtering and other use of text classification as a core technique for machine learning process.
The aim of this research is evaluate and compare performances of each of the three of the most important classifier commonly used for state of the art natural language processing text classifier and
sentiment analysis. We experimented this by implementing each of Linear Support Vector Machine, deep learning through Convolutional Neural Network (CNN), and multinomial variant of Bayesian
classifier for text classification, after evaluation and comparison of the accuracy and performances of each classifier, we also dived into causes of variation and differences in their performances.
Convolusional Neural Network (deep learning based)
Artificial neural network works to mimic functionality of human brain, input in artificial neural network represents the dendrites find in human brain while the different axion terminals then
represent output from the neural network. In deep learning, a typical network contains one or more several hidden layers called deep neural network (
Figure 1
), and it works by performing computation on input data fed to the network to give an output.
Convolution Neural Networks(CNNs) are complex multi-layered artificial neural networks with the ability to detect complex features such as extraction of features in images and text in dataset. They
are very efficient in computer vision task and so are commonly used for image segmentation, object detection, image classification, and text classification tasks [
] for natural language processing.
Convolutional neural network has convolution layer and pooling layer (
Figure 2
) as the two major layers for separate purposes, and while the convolution layer obtains features from data, pooling layer reduces sizes of the feature map from the data.
During convolution, features are obtained from data, features that are obtained during convolution operation are fed to CNN. Outputs from this convolution operation are the most important features
and they are called feature map, a convolved feature, or an activation map. The output is computed by applying feature detector which is also known as kernel or filter to the input data. The next
stage of the process is computation of the feature map by multiplication of the kernel and the matrix representation of the input data together, this process ensures that the feature map that is
passed to the CNN is smaller but contains all essential features. This process is done by filter by going step by step process known as strides through every element in the input data.
SUPPORT VECTOR MACHINE (SVM)
Support Vector Machine (SVM) is a supervised machine algorithm used for both classification and regression task [
]. It works by looking for a hyper-plane (
Figure 3
) that creates a boundary between two classes of data so as to properly classify them, it determines the best decision boundary between categories, hence they can be applied to vector which can
encode data, and so text are classified into vector during text classification tasks by SVM algorithm.
Once the algorithm determined the decision boundary for each of the category in the dataset for which we want to analyze, we can proceed to obtain the representations of each of all the texts we want
to classify in our NLP [
] and check for the side of the boundary that those representations fall into.
Theories supporting suitability of SVM for text classification
High dimensional input space: when training a classifier for text classification task, it is common to deal with several features, but the fact that SVM tries to prevent overfitting by using
overfitting protective measure which are independent on the number of features in the data gives SVM the potential to handle feature spaces in dataset, and hence their suitability for tasks relating
to natural language processing.
Linearly separable properties of text categorization: Virtually, all of categories involved in text classification are linearly separable. And so, the fact that SVM works by finding linear separator
between each of the categories in the textual dataset makes SVM suitable for tasks relating to NLP such as text classification, sentiment analysis, cyber troll and so on.
Multinomial Naïve Bayes (MNB)
Multinomial variant of Bayesian classifier is an algorithm commonly used for text classification related tasks and also problems with multiple classes [
]. In order to understand how Bayesian classifier works, it is important to understand the basic concept of Bayes theorem.
Tosin Ige, and Sikiru Adewale [
] successfully use multinomial Naïve Bayes algorithm to developed a machine learning based model along with an automated chatbot that can identify and intercept bully messages in an online chat
conversation to protect potential victim with 92% accuracy. The accuracy of their model is a big discovery in Natural language processing field of artificial intelligence.
In Bayes theorem, the probability of an event occurring based on the prior knowledge of conditions related to an event is calculated based on the formula:
P(A|B) = P(A) * P(B|A)/P(B).
Where we are calculating the probability of class A when predictor B is already provided.
• P(B) = prior probability of B
• P(A) = prior probability of class A
• P(B|A) = occurrence of predictor B given class A probability
Base on the principle of this calculation, we can automate the computation of tags in text and categorize them.
III. Research Methodology
1) Dataset:
For each of the implementations in this research, we use stack Overflow questions and tags dataset which is publicly available at:
and can also be query from Google BigQuery platform. The dataset contains two main columns, the first column contains the question for all non-deleted Stack Overflow questions in the database while
the second column is Tag which contains the tags on each of these questions.
2) Pre-processing and Feature Engineering: Since our dataset is a collection of several posts and tags from stackoverflow dataset, it is imperative to clean the data of several unwanted characters.
This necessitated the need for pre-procesing and feature engineering before we could actually use the data. As part of the pre-processing step, we removed unwanted characters from text, remove
punctuation mark, stopwords, search for and decoding HTML, and so on, and then we finally split the dataset into training and validation dataset as part of our final data pre-processing steps.
After pre-processing followed feature engineering during which we converted each of the text to matrix of token count by CountVectorizer. The count matrix is further converted to a normalized tf-idf
representation also known as tf-idf transformer. Haven completed both pre-processing and feature engineering process, we then proceeded to train our model on SVM, Multinomial, and CNN classifiers.
We used pipeline which serves as a compound classifier in scikit-learn, each unique word was assigned an index while we used tokenizer to count. Parameter for number of words was passed to the
tokenizer to ensure that our vocabulary is limited to only top words. Haven done this; we were able to use our tokenizer along with texts_to_matrix method to create proper training data that could be
passed to the model. After feeding our model with one hot encoded vector, and the transformation of features and labels to a format such that it could be read by keras deep learning python library.
We trained the model by passing training data and labels, batch size and epochs to the fit() method. As for the SVM implementation, we useCountVectorizer() method, TfidfTransformer() method, after
which we called SGDClassifier with the following configuration argument
In the case of training our model with Convolutional Neural Network which requires larger amount of training dataset for optimal performance, we set the validation_split to 0.1 to ensure that 90% of
the dataset is use for training while the remaining 10% is use for validation, we set the batch size to 4 and number of epochs to 50, and then monitor both the result and performance of each of the
IV. Result and Discussion
First and foremost, we observed some notable differences in training time for each of the classifiers. We got different performances and results for each of the experimental implementation of the
three classifiers. Our first attention was drawn to the performance of Multinomial Naïve Bayes (
Table 1
) as it has the worst performance for each of the evaluation criteria among the three (3). In order to investigate this we, we used another dataset with few data to train the model, it was at this
stage that we got a better result. This clearly proof to us that Multinomial variants of Bayesian classifier is good for small dataset, the reason for this is not far fetch, since multinomial naïve
Bayes assume independency between features in the dataset, as the data gets bigger, the assumption of independency tends not to be true for some of the features in the dataset, hence the drop in
performances with large dataset.
As for Support Vector Machine (SVM) classifier whose performance is in-between that of Multinomial and CNN classifier (
Figure 4
). SVM works better than Multinomial for two reasons, the first reason is because it is not based on any independence assumption among the features, the second reason is because of its ability to
look for a hyperplane that creates a boundary between two classes of data so as to properly classify them, as this enables it to determine the best decision boundary between the categories. Although,
SVM works better than Multinomial Naïve Bayes for Natural Language Processing text classification tasks, it is also not suitable for a very large dataset due to complexity in training.
For each of all the evaluation parameters for our experimental implementation, Convolutional Neural Network (CNN) has the best overall performance, with average precision (78%), average recall (76%),
average F1-Score (76%), average accuracy (77%). We believed that deep learning convolutional neural network work best among the three (3) for natural language processing text classification task
because it contains filter/ kernels which can help to identify patterns in text data, also the fact that these filters are translational invariant means that they can detect patterns regardless of
their position in the sentence. Here, the convolutional architecture can identify -gram that could be use for prediction with the need to pre-specify any embedding vector for ngram.
V. Evaluation
Since each of the evaluation techniques have its own drawback, so in order to have better evaluation of our model and to ensure effective comparison of performances of each of the classifier on
Natural Language Processing text classification task, we combined different evaluation techniques to effectively evaluate the performance of each of the models and compare result. We used average
precision score, average recall score, average F1-Score, and average accuracy as our evaluation parameters for comparison;
This is the actual ratio of the number of correct predictions to the total number of predictions. It represents the most fundamental of all the evaluating metrics used to evaluate model performance.
The formula is given by.
Accuracy = (TP+TN)/(TP+TN+FP+FN)
As fundamental as it is, it performs poorly in the presence of an imbalance dataset. Suppose a model classifies that most of the data belongs to the major class label. It yields higher accuracy. But
in general, the model cannot classify on minor class labels and has poor performance.
Precision is the ratio of true positives to the summation of true positives and false positives. It basically analyses the positive predictions.
The drawback of Precision is that it does not consider the True Negatives and False Negatives.
Recall is the ratio of true positives to the summation of true positives and false negatives. It analyses the number of correct positive samples.
The drawback of Recall is that often it leads to a higher false positive rate.
F1 SCORE
This is the harmonic mean of precision and recall. It is well known that there is precision-recall trade-off such that if we increase the precision, recall decreases and vice versa. F1 score
evaluation technique combines both the precision and recall score to give harmonic mean to better evaluate model performance.
F1 score = (2×Precision×Recall)/(Precision+Recall)
VI. Conclusion
In this research, we confirmed that of the three (3) of the most popular classifier for NLP text classification task, Convolutional Neural Network work best and even with all parameters for metric
evaluation when we have enough training dataset, CNN is good for text classification in the presence of enough data because due to the presence of filter/ kernels which help to indentify patterns in
text data regardless of their position in the sentence.
In the absence of enough training data, Support Vector Machine (SVM) works best which we believed to be due to its ability to look for a hyperplane that creates a boundary between different classes
of data so as to properly classify them. Multinomial Naive Bayes works based on independent assumption between features which are sometimes not valid in real data, we believe this is the reason for
its least performance among the three, and we believed that Multinomial Bayes classifiers must not be trusted for state of the Natural Language Processing (NLP) text classification task.
VII. Future Research Direction
Bayesian classifier works best when two conditions are met. The first is the generic condition of independence that all features are independent of each other which rarely holds in real life. The
fact that this condition rarely holds in real life limits the applicability of naïve Bayes in real-world use cases as they become unsuitable where there is any full or partial dependency between any
of the features in the data. The second condition is based on individual assumption of each variant of Bayesian classifier which does not always hold. The workings of each of the existing variants of
Bayesian classifier are based on different assumption which is the single most important factor that influences their performance and accuracy. This explains why each of the existing variants of
Bayesian classifier is suitable for different classification tasks depending on the nature of distribution of the dataset if it agrees with their assumption. Each of the current variants of Bayesian
classifiers works well if only the distribution of the data agrees with their assumption and the generic condition of independence holds.
In order to improve on the performance of Naïve bayes classifier, additional research work is needed to lower each of the two levels aof assumption of the Bayesian variants to the barest minimum.
Lowering these assumptions will significantly improve the performances of naïve bayes.
1. Ige, T., & Kiekintveld, C. (2023). Performance Comparison and Implementation of Bayesian Variants for Network Intrusion Detection. arXiv preprint arXiv:2308.11834.
2. Sentiment Analysis of Internet Posts," 2019 IEEE Eurasia Conference on IOT, Communication and Engineering (ECICE), Yunlin, Taiwan, 2019, pp. 154-155. [CrossRef]
3. J. Zhao, L. Dong, J. Wu, K. Xu. MoodLens: An Emoticon-Based Sentiment Analysis System for Chinese Tweets in Weibo. KDD 2012.).
4. Y. Chen and Z. Zhang, "Research on text sentiment analysis based on CNNs and SVM," 2018 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), Wuhan, China, 2018, pp. 2731-2734.
5. Park D S, Chan W, Zhang Y, et al. Specaugment: A simple data augmentation method for automatic speech recognition[J]. arXiv preprint arXiv:1904.08779, 2019.
6. Faris H, Ala’M A Z, Heidari A A, et al. An intelligent system for spam detection and identification of the most relevant features based on evolutionary random weight networks[J]. Information
Fusion, 2019, 48:67-83.
7. Watanabe K, Zhou Y. Theory-driven analysis of large corpora: Semisupervised topic classification of the UN speeches[J]. Social Science Computer Review, 2020: 0894439320907027.
8. Gao Z, Feng A, Song X, et al. Target-dependent sentiment classification with BERT[J]. IEEE Access, 2019, 7: 154290-154299.
9. Ige, T., & Adewale, S. (2022a). Implementation of data mining on a secure cloud computing over a web API using supervised machine learning algorithm. International Journal of Advanced Computer
Science and Applications, 13(5), 1–4. [CrossRef]
10. Ige, T., & Adewale, S. (2022b). AI powered anti-cyber bullying system using machine learning algorithm of multinomial naïve Bayes and optimized linear support vector machine. International
Journal of Advanced Computer Science and Applications, 13(5), 5–9. [CrossRef]
11. Park D S, Chan W, Zhang Y, et al. Specaugment: A simple data augmentation method for automatic speech recognition[J]. arXiv preprint arXiv:1904.08779, 2019.
12. Faris H, Ala’M A Z, Heidari A A, et al. An intelligent system for spam detection and identification of the most relevant features based on evolutionary random weight networks[J]. Information
Fusion, 2019, 48: 67-83.
13. Watanabe K, Zhou Y. Theory-driven analysis of large corpora: Semisupervised topic classification of the UN speeches[J]. Social Science Computer Review, 2020: 0894439320907027.
14. Gao Z, Feng A, Song X, et al. Target-dependent sentiment classification with BERT[J]. IEEE Access, 2019, 7: 154290-154299.
15. Yong Z, Youwen L, Shixiong X. An improved KNN text classification algorithm based on clustering[J]. Journal of computers, 2009, 4(3): 230-237.
16. Sang-Bum Kim, Kyoung-Soo Han, Hae-Chang Rim and Sung Hyon Myaeng, "Some Effective Techniques for Naive Bayes Text Classification," in IEEE Transactions on Knowledge and Data Engineering, vol. 18,
no. 11, pp. 1457-1466, Nov. 2006. [CrossRef]
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18. Wang Z Q, Sun X, Zhang D X, et al. An optimal SVM-based text classification algorithm[C]//2006 International Conference on Machine Learning and Cybernetics. IEEE, 2006: 1378-1381.
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20. Wang S, Huang M, Deng Z. Densely connected CNN with multi-scale feature attention for text classification[C]//IJCAI. 2018: 4468-4474.
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Average Average Average Average
Classifier Precision Recall F1-Score Accuracy
Multinomial Naïve Bayes (MNB) 0.72 0.69 0.68 0.69
Support Vector Machine (SVM) 0.77 0.77 0.76 0.76
Deep Learning (CNN) 0.78 0.76 0.77 0.77
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How to Sort by Number in Excel (4 Techniques) - Excelgraduate
How to Sort by Number in Excel (4 Techniques)
We often need to work with number lists where the numbers are randomly arranged. But arranging a number list from smaller to larger, or from larger to smaller numbers will help us to find the data
way more easily. This article will show you how to sort by number in Excel in 4 easy ways.
Sort by Number in Excel Using the Sort Command
To sort by number in Excel with Sort command, follow these steps below:
1. Select the numbers first.
2. Go to the Data tab.
3. Click on the A to Z or Z to A icon to sort by number in Excel.
Shortcut: If you are looking for keyboard shortcut keys to sort by number in Excel,
• Press ALT> A > S > A sequentially to sort in A to Z order (ascending).
• Press ALT > A > S > D sequentially to sort in Z to A order (descending).
Sort by Numbers in Excel with the A-Z or Z-A button
Use A-Z and Z-A Commands from the Data tab
You can also use the A-Z or Z-A buttons to sort in Excel by number. For that,
1. Click on any single cell from the number column.
2. Then click on the Data tab.
3. Now from the Sort & Filter group, click on the A-Z button or Z-A button to sort numbers in an ascending or descending order, respectively.
I sorted the numbers in ascending order and the resulting data set is below:
Use A-Z and Z-A Commands from the Home tab
You can also find the A-Z and Z-A commands in the Home tab.
1. Hit the Home tab.
2. Under the Editing group, click on the Sort & Filter drop-down list
3. Then click on the Sort A to Z/Sort Z to A command.
Sort by Number in Excel Applying the Filter Command
Use Filter Command from Data tab
To sort in Excel by number using the Filter command, go through these steps below:
1. Select any cell from the table.
2. Click on the Data tab from the ribbon.
3. Then select the Filter command under the Sort & Filter group.
Small arrows will pop up beside each of the headers of the data set.
4. Click on the small arrow of the column containing numbers.
5. Now from the drop-down, click on either the Sort Smallest to Largest option or the Sort Largest to Smallest option.
Use Filter Command from Home tab
You can also find the Filter command in another direction.
1. Go to the Home tab.
2. Then under the Editing group, click on the Sort & Filter drop-down list.
3. Then select the Filter option.
Use Filter Command from the Context Menu
If a column contains names (not numbers), you can also filter the column alphabetically using the Filter command. To sort the names in Excel, click on the small arrow beside the header that contains
names, in the drop-down Sort A to Z and Sort Z to A commands will show up.
Sort by Number in Excel Using the Custom Sort Command
Use the Custom Sort command to sort by number in Excel. Now, go through the procedure below:
1. Select the entire table.
2. Then click on the Home tab.
3. Then click on the Sort & Filter drop-down list under the Editing group.
4. Hit the Custom Sort command.
Sort dialog box will appear in Excel.
5. From the Column section, click on the Sort by drop-down.
6. Select your preferred column which contains numbers.
7. Then click on the Order drop-down.
8. Now select the Smallest to Largest command or Largest to Smallest command.
9. Finally, click OK.
Sort by Number in Excel with the Expanding Range
Ascending Order
SMALL(data_range,ROWS (expanding_range))
• SMALL Function: Sets the lower numerical value in ascending position in a data set.
• data_range: It is a range of cells that will be studied to find the smallest value.
• ROWS Function: The total number of rows in a given array.
• expanding_range: It is the expanding range that defines the difference from the smallest value to any cell.
Formula Explanation
Here, $B$2:$B$10 secures the range from B2 to B10. The expanding_range shows the difference between the B2 cell to other cells by $B$2:B2.
To sort by number in Excel with the expanding range in ascending order, follow these steps:
1. Double-click on an empty cell where you want to place the ascending ordered list.
2. Then write down the formula according to data_range & expanding_range in that cell.
For example, in my data set the numbers start from the B2 cell and end with the B10 cell. So the formula was: =SMALL($B$2:$B$10,ROWS($B$2:B2))
3. Press ENTER after writing the formula.
4. Then, hold the cursor on the Fill Handle and drag it from cell C2 to C10.
Descending Order
Sorting numbers in a downward order using a descending formula is as simple as the previous ascending formula. Just replace the SMALL function with the LARGE function.
LARGE(data_range,ROWS (expanding_range))
To sort by number in Excel with the expanding range in descending order, go through these steps:
1. Double-click on an empty cell where you want to place the descending order list.
I have selected the D2 cell.
2. Then, write down the formula with the following data_range & expanding_range in that cell.
For instance, according to my data table, the formula was: =LARGE($B$2:$B$10,ROWS($B$2:B2))
3. Press ENTER after writing the formula.
4. Then hold the cursor on the Fill Handle and drag it to the end of the column.
How to Sort by Number in rows in Excel
Sort by number in rows in Excel, here are the steps below:
1. Select the entire table.
2. Go to the Home tab.
3. Under the Editing group, click on the Sort & Filter drop-down.
4. Select Custom Sort.
Sort dialog box will pop up.
5. Click on the Options command.
6. Choose the orientation as Sort Left to Right.
7. Press OK.
8. Select your preferred row number from the Sort By drop-down list. I have selected row 10.
9. Then, change the Order from the drop-down. I have selected the Smallest to Largest option.
10. Hit OK.
Here is the outcome after sorting the rows from the smallest to largest order.
So, I tried to cover as many approaches as possible to arrange numbers in ascending and descending orders in Excel. I hope all of the techniques will seem easy to apply. I am hoping this article will
be useful for you. Feel free to leave a comment if you find anything difficult. Also, let us know which technique you have found the easiest.
Frequently Asked Questions
How do you sort in Excel by number and keep rows together?
To sort in Excel by number while keeping rows together, follow these steps:
1. Highlight the columns containing the numbers you want to sort, including any related data in other columns that you want to move together.
2. Press ALT + D + S sequentially to open the Sort dialog box.
3. In the Sort dialog box, select the column by which you want to sort under Sort by.
4. Choose Values from the drop-down menu to indicate that you are sorting by numerical values.
5. Choose the desired sorting order, either Smallest to Largest or Largest to Smallest.
6. If you have additional columns you want to sort by, click on Add Level in the dialog box and repeat steps 3-4 for each level.
7. Check the Preserve cell formatting box if you want to maintain the formatting of the cells while sorting.
8. Once you’ve configured your sorting options, click OK to apply the sort.
This will arrange the rows based on the numerical values in the specified column while keeping related data together. By following these steps, you can sort your data in Excel by number and ensure
that the rows stay together, maintaining the relationships between different sets of data.
What is the shortcut for sort by number in Excel?
In Excel, the shortcut for sorting by number is:
1. Select the Range: Highlight the column of numbers you want to sort.
2. Apply the Shortcut: Press ALT + H + S + N sequentially.
This shortcut activates the Sort dialog box and automatically selects the Sort by Number option, allowing you to quickly sort the selected range numerically in ascending order.
Can you sort rows in numbers?
Yes, you can sort rows in Numbers. Follow these steps:
1. Open your Numbers spreadsheet.
2. Select the rows you want to sort.
3. Click on the Table menu.
4. Choose Sort Rows.
5. Specify the sorting options such as column and order (ascending or descending).
Numbers will then rearrange the selected rows based on your chosen criteria, allowing you to organize your data efficiently. | {"url":"https://excelgraduate.com/sort-by-number-in-excel/","timestamp":"2024-11-08T11:21:59Z","content_type":"text/html","content_length":"164811","record_id":"<urn:uuid:341fbae4-39db-4058-8199-d74f76facb65>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00013.warc.gz"} |
Space Complexity - (Bioinformatics) - Vocab, Definition, Explanations | Fiveable
Space Complexity
from class:
Space complexity measures the amount of memory space required by an algorithm to execute as a function of the size of the input data. It includes both the space needed for the inputs as well as the
space required for auxiliary structures used during computation. Understanding space complexity is crucial because it helps in evaluating the efficiency of algorithms, especially in environments with
limited memory resources.
congrats on reading the definition of Space Complexity. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Space complexity is expressed as a function of input size, typically denoted using Big O notation, such as O(n) or O(n^2).
2. Dynamic programming often requires more memory for storing intermediate results, leading to higher space complexity compared to simpler algorithms.
3. Heuristic algorithms usually trade off optimality for reduced space requirements, making them more efficient in scenarios with limited memory.
4. The overall space complexity can vary significantly between recursive and iterative implementations of an algorithm, with recursion typically consuming more stack space.
5. In addition to memory used for variables and data structures, space complexity also accounts for constant factors, which can impact performance on large inputs.
Review Questions
• How does understanding space complexity enhance your ability to choose between different algorithms for a given problem?
□ Understanding space complexity allows you to evaluate how much memory an algorithm will require based on your input size. When selecting an algorithm, considering its space requirements
alongside time complexity can help you make informed decisions about performance trade-offs. For example, if memory is limited, you might opt for a heuristic algorithm even if it doesn't
guarantee an optimal solution, prioritizing lower space usage instead.
• Compare the space complexity of dynamic programming approaches versus heuristic algorithms and discuss their implications for real-world applications.
□ Dynamic programming typically has higher space complexity due to its need to store intermediate results in tables or matrices, making it less suitable for problems with large input sizes or
limited memory. In contrast, heuristic algorithms usually have lower space complexity since they may simplify the problem or focus on finding approximate solutions. This difference means that
while dynamic programming can provide optimal results, it may not be feasible in environments where memory usage is a critical constraint, such as embedded systems or mobile applications.
• Evaluate how different implementations (recursive vs. iterative) affect the space complexity of an algorithm and provide examples.
□ The implementation style of an algorithm significantly impacts its space complexity. Recursive algorithms often use additional stack space proportional to their depth of recursion, leading to
higher space requirements compared to iterative counterparts that typically only use a fixed amount of memory for variables. For instance, calculating Fibonacci numbers recursively has
exponential space complexity due to numerous function calls piling up in memory. In contrast, an iterative solution only requires linear space for storing results, demonstrating how choice of
implementation can directly affect efficiency and feasibility in resource-limited environments.
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Facebook contest data set explanation - Physics of Risk
Facebook contest data set explanation
During summer hiatus I have promised to look into a data set I have extracted from one particular Facebook contest. While many Facebook contests are not based on any logical reasoning, this contest
appeals to me as it appears to require at least some thought or expertise.
Last time I have briefly explored the data set. Now I will try to build models for the event-space observations.
The simple random model
For a first model let us simply consider that probability to get a comment with an answer 5 is fixed, \( p \).
As one can see in Fig. 3 the best \( p \) is around \( 0.88 \). Here I measure goodness as a sum of log-probabilities of the events that occurred. As I operate in the event-space, I can ignore the
inter-event time.
The simple herding model
Next lets consider a bit more sophisticated model. Let us assume that the probability to get a comment with an answer 5 depends on the current number of comments with an answer 5:
$$p ( X_5 \rightarrow X_5+1 ) = \frac{\varepsilon + X_5}{2 \varepsilon + X_5 + X_o}.$$
Note that the equation above also includes \( X_o \) which represent the number of other comments. Hence we have to define the probability for an increase in \( X_o \):
$$p ( X_o \rightarrow X_o+1 ) = \frac{\varepsilon + X_o}{2 \varepsilon + X_5 + X_o}.$$
As we can see in Fig. 4 the model seems to work best with \( \varepsilon =1.03 \). So which model works better? The simple random model at its best produces goodness measure of \( -88.95 \), while
the simple herding model at its best produces goodness measure of \( -93.92 \). Though the difference appears small, the simple random model seems to outperform the simple herding model.
The simple local herding model
Lets build another simple herding model, but now let the probabilities to be proportional to the respective fractions of comments:
$$p ( X_5 \rightarrow X_5+1 ) = \frac{\varepsilon + \frac{X_5}{N}}{2 \varepsilon + \frac{X_5 + X_o}{N}},$$
$$p ( X_o \rightarrow X_o+1 ) = \frac{\varepsilon + \frac{X_o}{N}}{2 \varepsilon + \frac{X_5 + X_o}{N}}.$$
Because of this form of the transition probabilities I will refer to this model as the simple local herding model.
As we can see in Fig. 5 the model seems to work best with \( \varepsilon =0.081 \). This model outperforms the simple random model as it produces a better goodness measure of \( -88.3 \). Though once
again the difference appears to be small.
One could add further sophistication to the model, such as introducing asymmetry into the model. Yet such sophistication no longer bring significant improvements. In case of asymmetry, different \( \
varepsilon \) values for "Guess 5" and "other", goodness measure increases to \( -88.21 \).
Interim conclusion
Though I cannot claim statistical significance, I am inclined to conclude that herding behavior is strong in this data set as there are few reasons to prefer 5 over other possible answers besides
initial dominance of the comments with answer 5.
Another important notice is that local herding model works better than global herding model. This basically mean that people read only a few comments before copying the prevalent answer. Thus it is
not very probable that mathematically literate comments (the ones pointing out that there are infinitely many answers) reached the broader audience.
Next time
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Colsher's inversion formula
Next: Conical tilt geometry Up: 3D reconstruction formulas Previous: Orlov's inversion formula
Assume again that g = Pf is known for
We want to derive an inversion procedure similar to the one in section 2.1. With the backprojection
we have again
provided that
which corresponds to (2.2). As in (2.2) we express the relationship
see Colsher (1980). In order to get an inversion formula for P we have to determine v such that
A solution
Filters such as the Colsher filter (3.10) do not have small support. This means that g in (3.8) has to be known in all of g is only available in part of
where 3.11) is constant in the vertical direction, v is a 3.8) reduces to an integral over horizontal lines in 3.11) does not quite satisfy
Frank Wuebbeling
Thu Sep 10 10:51:17 MET DST 1998 | {"url":"https://www.uni-muenster.de/AMM/num/Preprints/1998/natterer_1/paper.html/node18.html","timestamp":"2024-11-11T11:06:16Z","content_type":"text/html","content_length":"6912","record_id":"<urn:uuid:e93b6061-966f-458a-878c-cc05f44356fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00626.warc.gz"} |
Frequency dependence of ionic conductivity of electrolyte solutions
Chandra, Amalendu ; Bagchi, Biman (2000) Frequency dependence of ionic conductivity of electrolyte solutions Journal of Chemical Physics, 112 (4). pp. 1876-1886. ISSN 0021-9606
PDF - Publisher Version
Official URL: http://link.aip.org/link/?JCPSA6/112/1876/1
Related URL: http://dx.doi.org/10.1063/1.480751
A theory for the frequency dependence of ionic conductivity of an electrolyte solution is presented. In this theory contributions to the conductivity from both the ion atmosphere relaxation and the
electrophoretic effects are included in a self-consistent fashion. Mode coupling theory, combined with time-dependent density functional theory of ion atmosphere fluctuations, leads to expressions
for these two contributions at finite frequencies. These expressions need to be solved self-consistently for the frequency dependence of the electrolyte friction and the ion conductivity at varying
ion concentrations. In the limit of low concentration, the present theory reduces exactly to the well-known Debye-Falkenhagen (DF) expression of the frequency-dependent electrolyte friction when the
non-Markovian effects in the ion atmosphere relaxation are ignored and in addition the ions are considered to be pointlike. The present theory also reproduces the expressions of the
frequency-dependent conductivity derived by Chandra, Wei, and Patey when appropriate limiting situations are considered. We have carried out detailed numerical solutions of the self-consistent
equations for concentrated solutions of a 1:1 electrolyte by using the expressions of pair correlation functions given by Attard. Numerical results reveal that the frequency dependence of the
electrolyte friction at finite concentration can be quite different from that given by the DF expression. With the increase of ion concentration, the dispersion of the friction is found to occur at a
higher frequency because of faster relaxation of the ion atmosphere. At low frequency, the real part of the conductivity shows a small increase with frequency which can be attributed to the
well-known Debye-Falkenhagen effect. At high frequency, the conductivity decreases as expected. The extensions of the present theory to treat frequency-dependent diffusivities of charged colloid
suspensions and conductivity of a dilute polyelectrolyte solution are discussed.
Item Type: Article
Source: Copyright of this article belongs to American Institute of Physics.
ID Code: 4240
Deposited On: 13 Oct 2010 09:09
Last Modified: 16 May 2016 14:55
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MATH 412 Introduction to Analysis I
MATH 412 Introduction to Analysis I
Catalog Description
MATH 412 Introduction to Analysis I (4)
Introduction to concepts and methods basic to real analysis. Topics such as the real number
system, sequences, continuity, uniform continuity and differentiation. 4 lectures. Prerequisite:
MATH 306 or consent of instructor.
Required Background or Experience
Math 306, completion of a calculus sequence which includes functions of several variables,
and sufficient mathematical maturity.
Learning Objectives
Students should re-emphasize and obtain a deeper understanding of the definition of function in
the context of this course. Additionally, students should obtain an understanding of the limiting
processes basic to functions of a single variable. This understanding will make much of the
literature of mathematics accessible, and will provide a deeper insight into computational
processes with which students are somewhat familiar.
Text and References
See course supervisor. Typical text choices are ones by Bartle, Rudin or Goldberg. For winter
quarter sections of Math 412, Fundamental Ideas of Analysis, by Reed, is a nice choice as well.
Minimum Student Materials
Paper, pencils, and notebook.
Minimum University Facilities
Classroom with ample chalkboard space for class use.
Content and Method
The real number system
Topology of R1
Continuity and uniform continuity
Methods of Assessment
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Out: 2 February 2003
Problem Set 3: L-System Fractals Due: 11 February 2003, before class
Collaboration Policy - Read Carefully
For this problem set, you may work alone or with a partner of your choice. If you work with a partner, you and your partner should turn in one assignment with both of your names on it and both
people must participate equally in all of the work. You should read the whole problem set yourself and think about the questions before beginning to work on them with your partner.
You may discuss this assignment with other students in the class and ask and provide help in useful ways. You may consult any outside resources you wish including books, papers, web sites and
people except for materials from last year's CS200 course. If you use resources other than the class materials, indicate what you used along with your answer.
□ Get more practice with recursive definitions and procedures
□ Explore the power of rewrite rules
□ Learn to use lists
□ Write recursive functions that manipulate lists
□ Make a better CS200 logo than "The Great Lambda Tree of Infinite Knowledge and Ultimate Power"
│ Reading: Before doing this problem set, you should read SICP 2.1 and 2.2 (you may skip section 2.2.4). │
│ Download: Download ps3.zip to your machine and unzip it into your home directory J:\cs200\ps3. │
│ │
│ This file contains: │
│ │
│ • ps3.ss — A template for your answers. You should do the problem set by editing this file. │
│ • graphics.ss — Scheme code for drawing curves. This is similar to graphics.ss from PS2, except we have made some improvements (as described later) that will help you draw better L-System │
│ Fractals. │
│ • lsystem.ss — provided incomplete code for producing L-System Fractals │
In this problem set, you will explore a method of creating fractals known as the Lindenmayer system (or L-system). Aristid Lindemayer, a theoretical biologist at the University of Utrecht,
developed the L-system in 1968 as a mathematical theory of plant development. In the late 1980s, he collaborated with Przemyslaw Prusinkiewicz, a computer scientist at the University of Regina,
to explore computational properties of the L-system and developed many of the ideas on which this problem set is based.
The idea behind L-system fractals is that we can describe a curve as a list of lines and turns, and create new curves by rewriting old curves. Everything in an L-system curve is either a forward
line (denoted by F), or a right turn (denoted by Ra where a is an angle in degrees clockwise). We can denote left turns by using negative angles.
We create fractals by recursively replacing all forward lines in a curve list with the original curve list. Lindemayer found that many objects in nature could be described using regularly
repeating patterns. For example, the way some tree branches sprout from a trunch can be described using the pattern: F O(R30 F) F O(R-60 F) F. This is interpreted as: the trunk goes up one unit
distance, a branch sprouts at an angle 30 degrees to the trunk and grows for one unit. The O means an offshoot — we draw the curve in the following parentheses, and then return to where we
started before the offshoot. The trunk grows another unit and now another branch, this time at -60 degrees relative to the trunk grows for one units. Finally the trunk grows for one more unit.
The branches continue to sprout in this manner as they get smaller and smaller, and eventually we reach the leaves.
We can describe this process using replacement rules:
Start: (F)
Rule: F ::= (F O(R30 F) F O(R-60 F) F)
Here are the commands this produces after two iterations:
Iteration 0: (F)
Iteration 1: (F O(R30 F) F O(R-60 F) F)
Iteration 2: (F O(R30 F) F O(R-60 F) F O(R30 F O(R30 F) F O(R-60 F) F) F O(R30 F) F O(R-60 F) F O(R-60 F O(R30 F) F O(R-60 F) F) F O(R30 F) F O(R-60 F) F)
Here's what that looks like:
Iteration 0 Iteration 1 Iteration 2
Iteration 5 The Great Lambda Tree of Infinite Knowledge and Ultimate Power
Note that L-system command rewriting is similar to the replacement rules in a BNF grammar. The important difference is that with L-system rewriting, each iteration replaces all instances of F in
the initial string instead of just picking one to replace.
We can divide the problem of producing an L-system fractal into two main parts:
1. Produce a list of L-system commands that represents the fractal by rewriting according to the L-system rule; and
2. Drawing a list of L-system commands.
We will first work on producing the list of L-system commands, and then work on how to draw a list of L-system commands.
Representing L-System Commands
Here is a BNF grammar for L-system commands:
1. CommandSequence ::= ( CommandList )
2. CommandList ::= Command CommandList
3. CommandList ::=
4. Command ::= F
5. Command ::= RAngle
6. Command ::= OCommandSequence
7. Angle ::= Number
│ Question 1: Show that (F O(R-60 F) F) is a string in the language defined by our BNF grammar. To do this, you should start with CommandSequence, and show a sequence of replacements that │
│ follow the grammar rules that produce the target string. You can use the rule numbers above to identify the rules. │
We need to find a way to turn strings in this grammar into objects we can manipulate in a Scheme program. We can do this by looking at the BNF grammar, and converting the non-terminals into
Scheme objects.
;;; CommandSequence ::= ( CommandList )
(define make-lsystem-command list)
;;; We represent the different commands as pairs where the first item in the
;;; pair is a tag that indicates the type of command: 'f for forward, 'r for rotate
;;; and 'o for offshoot. We use quoted letters to make tags, which evaluate to the
;;; letter after the quote. The tag 'f is short for (quote f).
;;; Command ::= F
(define (make-forward-command) (cons 'f #f)) ;; No value, just use false.
;;; Command ::= RAngle
(define (make-rotate-command angle) (cons 'r angle))
;;; Command ::= OCommandSequence
(define (make-offshoot-command commandsequence) (cons 'o commandsequence))
│ Question 2: It will be useful to have procedures that take L-system commands as parameters, and return information about those commands. Define the following procedures in ps3.ss: │
│ │
│ • (is-forward? lcommand) evaluates to #t if the parameter passed is a forward command (indicated by its first element being a 'f tag). │
│ • (is-rotate? lcommand) │
│ • (is-offshoot? lcommand) │
│ • (get-angle lcommand) evaluates to the angle associated with a rotate command. Produces an error if the command is not a rotate command (see below for how to produce an error). │
│ • (get-offshoot-commands lcommand) evaluates to the offshoot command list associated with an offshoot command. Produces an error if the command is not an offshoot command. │
You will find the following procedures useful:
□ (car lst) evaluates to the first element of the list parameter
□ (eq? v1 v2) evaluates to #t if v1 and v2 are exactly the same; otherwise evaluates to false. For example, (eq? 's 's) evaluates to #t and (eq? 's 't) evaluates to #f.
□ (error message) produces an error with message a string given as the first parameter. For example, (error "Yikes! Attempt to get-angle for a command that is not an angle command") would
display the message in red and stop execution. It is useful to use error in your code so you will more easily identify bugs.
If you define these procedures correctly, you should produce these evaluations:
> (is-forward? (make-forward-command))
> (is-forward? (make-rotate-command 90))
> (get-angle (make-rotate-command 90))
> (get-angle (make-forward-command))
Yikes! Attempt to get-angle for a command that is not an angle command
You should be able to make up similar test cases yourself to make sure the other procedures you defined work.
Rewriting Curves
The power of the L-System commands comes from the rewriting mechanism. Recall how we described the tree fractal:
Start: (F)
Rule: F ::= (F O(R30 F) F O(R-60 F) F)
To produce levels of the tree fractal, we need a procedure that takes a list of L-system commands and replaces each F command with the list of L-system commands given by the rule.
So, for every command in the list:
□ If the command is an F command, replace it with the replacement commands
□ If the command is an RAngle command, keep it unchanged
□ If the command is an OCommandSequence command, recursively rewrite every command in the offshoot's command list the same way
One slight complication is that the replacement commands are a list of L-system commands, and we want to end up with a flat list of L-System commands.
For example, consider a simple L-System rewriting:
Start: (F)
Rule: F ::= (F R30 F)
We want to get:
Iteration1: (F R30 F)
Iteration2: (F R30 F R30 F R30 F)
but if we just replace F's with (F R30 F) lists, we would get:
Iteration1: ((F R30 F))
Iteration2: ((F R30 F) R30 (F R30 F))
The easiest way to fix this problem is to flatten the result. The code should look similar to many recursive list procedures you have seen (this code is provided in lsystem.ss):
(define (flatten-commands ll)
(if (null? ll) ll
(if (is-lsystem-command? (car ll))
(cons (car ll) (flatten-commands (cdr ll)))
(flat-append (car ll) (flatten-commands (cdr ll))))))
(define (flat-append lst ll)
(if (null? lst) ll
(cons (car lst) (flat-append (cdr lst) ll))))
│ Question 3: Define a procedure rewrite-lcommands in ps3.ss that takes a list of L-system commands as its first parameter. The second parameter is a list of L-system commands that should │
│ replace every forward command in the first list of commands in the result. │
│ │
│ Here's the easy part: │
│ │
│ (define (rewrite-lcommands lcommands replacement) │
│ (flatten-commands │
│ (map │
│ ; Procedure to apply to each command │
│ lcommands))) │
│ │
│ Complete the definition of rewrite-lcommands. │
To make interesting L-system curves, we will need to apply rewrite-lcommands many times. We will leave that until the last question. First, we will work on turning sequences of L-system commands
into curves we can draw.
Improving Our Curve Drawing
To help you draw better L-system Curves, we have provided a new version of the curve drawing code from PS2. You don't need to modify this code, but should understand the changes described below.
In PS2, we represented points using a procedure:
(define (make-point x y)
(lambda (selector) (if selector x y)))
(define (x-of-point point) (point #t))
(define (y-of-point point) (point #f))
Now that you know about cons pairs, it would be more natural to represent points using a cons pair:
(define (make-point x y) (cons x y))
(define (x-of-point point) (car point))
(define (y-of-point point) (cdr point))
Note that we can change the way we represent points, and all the old PS2 code will still work without changing anything else (except it will run faster now). This is data abstraction, and is very
important for building large programs.
Our pictures will be more interesting if points can have color too. We represent a colored point using a list of three values: x, y and color:
(define (make-colored-point x y c) (list x y c))
(define (is-colored-point? point) (= (length point) 3))
We have defined colored points so the old colorless points still work (and appear black). We changed the definition of make-point to produce a list of two values, instead of a cons pair, since
that way the x-of-point and y-of-point procedures will work for both colored and uncolored points:
(define (make-point x y) (list x y))
(define (x-of-point point) (car point))
(define (y-of-point point) (cadr point)) ;; (cadr x) = (car (cdr x))
;;; Regular points are black. Colored points have a color.
(define (color-of-point point)
(if (is-colored-point? point)
(caddr point)
(make-color 0 0 0)))
To make our curves appear in color, we need to change the way we draw points on the window to pass in the color also:
(define (window-draw-point point)
((draw-pixel window) (convert-to-position point) (color-of-point point)))
Manipulating Curves
The good thing about defining curves as functions is it is easy to modify and combine then in interesting ways.
For example, the procedure rotate-ccw takes a curve and rotates it 90 degrees counter-clockwise by swapping the x and y points:
(define (rotate-ccw curve)
(lambda (t)
(let ((ct (curve t)))
(- (y-of-point ct))
(x-of-point ct)
(color-of-point ct)))))
Note that (rotate-ccw c) evaluates to a curve. The function rotate-ccw is a proceture that takes a procedure (a curve) and returns a procedure that is a curve.
Predict what (draw-curve-points (rotate-ccw mid-line) 1000) and (draw-curve-points (rotate-ccw (rotate-ccw mid-line)) 1000) will do. Confirm your predictions by trying them in your Interactions
Here's another example:
(define (shrink curve scale)
(lambda (t)
(let ((ct (curve t)))
(* scale (x-of-point ct))
(* scale (y-of-point ct))
(color-of-point ct)))))
Predict what (draw-curve-points (shrink mid-line .5) 1000) will do, and then try it in your Interactions window.
The shrink procedure doesn't produce quite what we want because in addition to changing the size of the curve, it moves it around. Make sure you understand why this happens. Try shrinking a few
different curves to make sure.
One way to fix this problem is to center our curves around (0,0) and then translate them to the middle of the screen. We can do this by adding or subtracting constants to the points they produce:
(define (translate curve x y)
(lambda (t)
(let ((ct (curve t)))
(+ x (x-of-point ct))
(+ y (y-of-point ct))
(color-of-point ct)))))
Now we have translate, it makes more sense to define mid-line this way:
(define (horiz-line t) (make-point t 0))
(define mid-line (translate horiz-line 0 0.5))
To check you understand everything so far, use translate, horiz-line and shrink to draw a line half the width of the window that is centered in the middle of the display window.
In addition to altering the points a curve produces, we can alter a curve by changing the t values it will see. For example,
(define (first-half curve)
(lambda (t) (curve (/ t 2))))
is a function that takes a curve and produces a new curve that is just the first half of the passed curve.
Predict what each of these expressions will do:
□ (draw-curve-points (first-half mid-line) 1000)
□ (draw-curve-points (first-half (first-half mid-line)) 1000))
Try evaluating them in your Interactions window to check if you were right. (Remember to use (clear-window) to clear the display window so you can see the new curve without the old one.)
The provided code includes several other functions that transform curves including:
□ (scale-x-y curve x-scale y-scale) — evalutes to curve stretched along the x and y axis by using the scale factors given
□ (scale curve scale) — evaluates to curve stretched along the x and y axis by using the same scale factor
□ (rotate-around-origin curve degrees) — evaluates to curve rotated counterclockwise by the given number of degrees.
You should be able to understand the code in graphics.ss that defines these functions.
It is also useful to have curve transforms where curves may be combined. An example is (connect-rigidly curve1 curve2) which evaluates to a curve that consists of curve1 followed by curve2. The
starting point of the new curve is the starting point of curve1 and the end point of curve2 is the ending point of the new curve. Here's how connect-rigidly is defined:
(define (connect-rigidly curve1 curve2)
(lambda (t)
(if (< t (/ 1 2))
(curve1 (* 2 t))
(curve2 (- (* 2 t) 1)))))
Predict what (draw-curve-points (connect-rigidly vertical-mid-line mid-line) 1000) will do. Is there any difference between that and (draw-curve-points (connect-rigidly mid-line
vertical-mid-line) 1000)? Check your predictions in the Interactions window.
Distributing t Values
The draw-curve-points procedure does not distribute the t-values evenly among connected curves, so the later curves appear dotty. This isn't too big a problem when only a few curves are combined;
we can just increase the number of points passed to draw-curve-points to have enough points to make a smooth curve. In this problem set, however, you will be drawing curves made up of thousands
of connected curves. Just increasing the number of points won't help much, as you'll see in Question 4.
The way connect-rigidly is defined above, we use all the t-values below 0.5 on the first curve, and use the t-values between 0.5 and 1.0 on the second curve. If the second curve is the result of
connecting two other curves, like (connect-rigidly c1 (connect-rigidly c2 c3)) then 50% of the points will be used to draw c1, 25% to draw c2 and 25% to draw c3.
│ Question 4: Define a procedure num-points that determines the approximate number of t-values that will be used for the n^th curve when drawing │
│ │
│ (connect-rigidly c1 (connect-rigidly c2 (connect-rigidly curve3 (... cn)))) │
│ │
│ Think about this yourself first, but look in ps3.ss for a hint if you are stuck. │
Your num-points procedure should produce results similar to:
> (exact->inexact (num-points 1000 10))
> (exact->inexact (num-points 1000 20))
> (exact->inexact (num-points 1000000 20))
This means if we connected just 20 curves using connect-rigidly, and passed the result to draw-curve-points with one million as the number of points, there would still be only one or two points
drawn for the 20^th curve. If we are drawing thousands of curves, for most of them, not even a single point would be drawn!
To fix this, we need to distribute the t-values between our curves more fairly. We have provided a procedure connect-curves-evenly in graphics.ss that connects a list of curves in a way that
distributes the range of t values evenly between the curves.
The definition is a bit complicated, so don't worry if you don't understand it completely. You should, however, be able to figure out the basic idea for how it distributed the t-values evenly
between every curve in a list of curves.
(define (connect-curves-evenly curvelist)
(lambda (t)
(let ((which-curve
(if (>= t 1.0) (- (length curvelist) 1)
(inexact->exact (floor (* t (length curvelist)))))))
((get-nth curvelist which-curve)
(* (length curvelist)
(- t (* (/ 1 (length curvelist)) which-curve)))))))
It will also be useful to connect curves so that the next curve begins where the first curve ends. We can do this by translating the second curve to begin where the first curve ends. To do this
for a list of curves, we translate each curve in the list the same way using map:
(define (cons-to-curvelist curve curvelist)
(let ((endpoint (curve 1.0))) ;; The last point in curve
(cons curve
(map (lambda (thiscurve)
(translate thiscurve (x-of-point endpoint) (y-of-point endpoint)))
Drawing L-System Curves
To draw an L-system curve, we need to convert a sequence of L-system commands into a curve. We defined the connect-curves-evenly procedure to take a list of curves, and produce a single curve
that connects all the curves. So, to draw an L-System curve, we need a procedure that turns an L-System Curve into a list of curve procedures.
Below is code for converting a list of L-System commands with some parts missing (it is explained below, but try to understand it yourself before reading further; if you don't understand cond,
review SICP 1.1.6.):
(define (convert-lcommands-to-curvelist lcommands)
(cond ((null? lcommands)
;;; We make a leaf with just a single point of green:
(lambda (t)
(make-colored-point 0.0 0.0 (make-color 0 255 0)))
((is-forward? (car lcommands))
(convert-lcommands-to-curvelist (cdr lcommands))))
((is-rotate? (car lcommands))
;;; If this command is a rotate, every curve in the rest
;;; of the list should should be rotated by the rotate angle
;; L-system turns are clockwise, so we need to use - angle
((rotate-angle (- (get-angle (car lcommands)))))
(lambda (curve)
;;; Question 5: fill this in
;;; Question 5: fill this in
((is-offshoot? (car lcommands))
;;; Question 6: fill this in
(#t (error "Bad lcommand!"))))
We define convert-lcommands-to-curvelist recursively. The base case is when there are no more commands (the lcommands parameter is null). It evaluates to the leaf curve (for now, we just make a
point of green — you may want to replace this with something more interesting to make a better fractal). Since convert-lcommands-to-curvelist evaluates to a list of curves, we need to make a list
of curves containing only one curve.
Otherwise, we need to do something different depending on what the first command in the command list is. If it is a forward command we draw a vertical line. The rest of the fractal is connected
to the end of the vertical line using cons-to-curvelist. The recursive call to convert-lcommands-to-curve produces the curve list corresponding to the rest of the L-system commands. Note how we
pass (cdr lcommands) in the recursive call to get the rest of the command list.
│ Question 5: Fill in the missing code for handling rotate commands (marked as Question 5 in ps3.ss). You will want to use (rotate-around-origin curve rotate-angle) somewhere in your code to │
│ rotate every curve after the rotate command by the rotate-angle. │
You can test your code by drawing the curve that results from any list of L-system commands that does not use offshoots. For example, evaluating
(make-lsystem-command (make-rotate-command 150)
(make-rotate-command -120)
0.3 0.7)
0 .5)
should produce a "V".
│ Question 6: Fill in the missing code for handling offshoot commands (marked as Question 6 in ps3.ss). │
We have provided the position-curve procedure to make it easier to fit fractals onto the graphics window:
(position-curve curve startx starty) evaluates to a curve that translates curve to start at (startx, starty) and scales it to fit into the graphics window maintaining the aspect ratio (the
x and y dimensions are both scaled the same amount)
The code for position-curve is in curve.ss. You don't need to look at it, but should be able to understand it if you want to.
Now, you should be able to draw any l-system command list using position-curve and the convert-lcommands-to-curvelist function you completed in Questions 5 and 6. Try drawing a few simple
L-system command lists before moving on to the next part.
│ Question 7: Define a procedure make-lsystem-fractal in ps3.ss that takes three parameters: replace-commands, a list of L-system commands that replace forward commands in the rewriting; start, │
│ a list of L-system commands that describes the starting curve; level, the number of iterations to apply the rewrite rule. │
│ │
│ Hint: You should use the rewrite-lcommands you defined in Question 4. You may also find it useful to use the n-times function you defined in PS2. │
You should be able to draw a tree fractal using make-tree-fractal and draw-lsystem-fractal (these and the tree-commands list of L-system commands are defined in lsystem.ss):
(define (make-tree-fractal level)
(make-lsystem-fractal tree-commands
(make-lsystem-command (make-forward-command)) level))
(define (draw-lsystem-fractal lcommands)
(connect-curves-evenly (convert-lcommands-to-curvelist lcommands))
0.5 0.1)
For example, (draw-lsystem-fractal (make-tree-fractal 3)) will create a tree fractal with 3 levels of branching.
│ Question 8: Draw some fractals by playing with the L-system commands. Try changing the rewrite rule, the starting commands, level and leaf curve (in convert-lcommands-to-curvelist) to draw an │
│ interesting fractal. You might want to make the branches colorful also. Try an make a fractal picture that will make a better course logo than the current Great Lambda Tree Of Infinite │
│ Knowledge and Ultimate Power. The best pictures will appear on the course web page and will be rewarded with untold fame, invaluable fortune and maybe even a double gold star on this problem │
│ set. Turn in the code you used, as well as a printout of the display window. If your fractal deserves to be seen in color, email an image of it to cs200-staff@cs.virginia.edu. │
Credits: This problem set was originally created for CS200 Spring 2002 by Dante Guanlao, Jon Erdman and David Evans, and revised for CS200 Spring 2003 by Jacques Fournier and David Evans, and
revised for CS200 Spring 2004 by David Evans. | {"url":"http://www.cs.virginia.edu/cs200/problem-sets/ps3/","timestamp":"2024-11-04T00:43:32Z","content_type":"text/html","content_length":"33266","record_id":"<urn:uuid:594ed4fe-2b99-4c78-b2fc-ef039947a20a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00122.warc.gz"} |
Elementary Statistics
Decent Advice On Where To Find Elementary Statistics Homework Help
Homework help of any kind can be found in a variety of places. It is important when you are looking for help that you know what to look for to make sure you are getting accurate information that will
help instead of confuse your issues. If you are looking for help for statistics for the elementary school age student, you are probably trying to help your child so he will also need help choosing
where to get the help from. Here is some decent advice on where to find the help you are looking for:
1. Once place you can get help for your child would be his teacher. Of course, the teacher would know the information but he may not be available at times that are convenient for you or your child.
Also, if your child is having a problem with concepts, maybe he needs it explained to him from a different person than his teacher.
2. You may also find help from your child’s textbook and help your child understand how things are explained. Sometimes your child may just need a little help with different wording of his book in
order to understand it easier.
3. The internet is one of the best places to find the help your child needs for his assignment help. There are two different types of help you can receive from the internet, passive and active.
□ Passive help can be found almost anywhere on the internet. These are sites that give research information. These places will offer explanation on any topic you put in the search engine. Some
will give examples and others will just explain it differently. Usually there will be step-by-step help so it is easier for your child to understand.
□ Active sites are great because they are run by live people and you can get one-on-one information with immediate feedback. You can find forums that will explain the answers to your child or
you can actually talk live to some tutors and get any help with elementary statistics that you need. An example, you can get assistance from this website.
Any of these choices will help with the statistic homework you are looking for but the internet will probably give you the greatest opportunity for your child to succeed. Find a site that customers
have given excellent reviews and chances are it is a great site to work with. | {"url":"https://www.educaterowan.org/elementary-statistics-assignment","timestamp":"2024-11-05T07:09:00Z","content_type":"application/xhtml+xml","content_length":"17905","record_id":"<urn:uuid:f7d73d71-cbbf-4ab4-bd1b-72606d08c711>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00022.warc.gz"} |
Ball remotal subspaces of Banach spaces
Ball remotal subspaces of Banach spaces
We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely
remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable
remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).
Pradipta Bandyopadhyay, Bor-Luh Lin, and T. S. S. R. K. Rao. "Ball remotal subspaces of Banach spaces." Colloquium Mathematicae 114.1 (2009): 119-133. <http://eudml.org/doc/283714>.
abstract = {We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is
densely remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a
separable remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).},
author = {Pradipta Bandyopadhyay, Bor-Luh Lin, T. S. S. R. K. Rao},
journal = {Colloquium Mathematicae},
keywords = {farthest points; remotal sets; densely remotal sets; Banach spaces; strictly convex spaces; locally uniformly rotund spaces; Radon-Nikodym property; Asplund spaces; Lebesgue-Bochner
spaces; spaces of compact operators},
language = {eng},
number = {1},
pages = {119-133},
title = {Ball remotal subspaces of Banach spaces},
url = {http://eudml.org/doc/283714},
volume = {114},
year = {2009},
TY - JOUR
AU - Pradipta Bandyopadhyay
AU - Bor-Luh Lin
AU - T. S. S. R. K. Rao
TI - Ball remotal subspaces of Banach spaces
JO - Colloquium Mathematicae
PY - 2009
VL - 114
IS - 1
SP - 119
EP - 133
AB - We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely
remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable
remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).
LA - eng
KW - farthest points; remotal sets; densely remotal sets; Banach spaces; strictly convex spaces; locally uniformly rotund spaces; Radon-Nikodym property; Asplund spaces; Lebesgue-Bochner spaces;
spaces of compact operators
UR - http://eudml.org/doc/283714
ER -
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Inner Product - (Quantum Cryptography) - Vocab, Definition, Explanations | Fiveable
Inner Product
from class:
Quantum Cryptography
The inner product is a mathematical operation that takes two vectors in a Hilbert space and produces a complex number, reflecting the notion of geometric angles and lengths. This operation helps
define important concepts such as orthogonality and distance in quantum mechanics. In the context of quantum states, the inner product is crucial for understanding how different states relate to one
another, including their probabilities and overlaps.
congrats on reading the definition of Inner Product. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The inner product is denoted by the angle brackets notation, written as \langle \psi | \phi \rangle for two quantum states |\psi\rangle and |\phi\rangle.
2. The result of an inner product can reveal whether two quantum states are orthogonal; if the inner product is zero, the states are orthogonal.
3. The inner product is not only a measure of similarity but also provides information about probabilities when measuring one state in the basis of another.
4. In finite-dimensional spaces, the inner product corresponds to the standard dot product, while in infinite dimensions, it requires more rigorous definitions.
5. The inner product must satisfy certain properties: it should be linear in its first argument, conjugate symmetric, and positive definite.
Review Questions
• How does the inner product relate to the concepts of probability and measurement in quantum mechanics?
□ The inner product plays a vital role in determining probabilities when measuring quantum states. Specifically, if we have two states |\psi\rangle and |\phi\rangle, the square of the absolute
value of their inner product \(|\langle \psi | \phi \rangle|^2\) gives the probability of obtaining state |\psi\rangle if the system is initially prepared in state |\phi\rangle. This
relationship underscores how different quantum states can overlap and influence measurement outcomes.
• Discuss how orthogonality in quantum states is determined through the inner product and its implications for state measurements.
□ Orthogonality in quantum states is determined using the inner product; if two states |\psi\rangle and |\phi\rangle are orthogonal, then their inner product \langle \psi | \phi \rangle equals
zero. This property indicates that measuring one state will yield no information about the other state. Orthogonal states are crucial for defining distinct outcomes in quantum measurements
and form the basis for constructing quantum bits (qubits) used in quantum computing.
• Analyze how the properties of the inner product influence the structure of Hilbert spaces and their applications in quantum mechanics.
□ The properties of the inner product significantly shape the structure of Hilbert spaces by ensuring they adhere to essential mathematical requirements such as completeness and the ability to
define orthonormal bases. These properties allow for meaningful physical interpretations in quantum mechanics, facilitating concepts like superposition and entanglement. For instance, because
Hilbert spaces can support an infinite number of dimensions through their inner products, they enable complex system representations essential for advanced applications like quantum
cryptography and quantum computing.
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Lowey’s Old 446 Exams
Please note that this course changes from year to year. Thus this year we may cover connections on Quiz A, while last year Quiz A covered tension members. Sometimes, material is completely removed
from the course and something else takes its place. Thus, don’t get upset if you never heard of something you see on one of these old quizzes. It’s possible that we just no longer cover it, or that
we will cover it on a different exam. Yes, I realize that some of the exams say “See the board for the figures.” Sorry, I forgot to bring the board back to my office back in 2000. Yes I realize that
some of the material on the exam is missing, or incorrect. It was corrected on the board during the exam – the same board that I forgot to bring back with me after the exam. Yes, I realize that the
method of calculating Cb has changed since I last taught the material. We will use the new way of calculating it from now on.
Note that in the Miscellaneous Exams, Solutions, etc, I have thrown all kinds of junk from my files, half of which you can’t read, nor can I. I really hesitate to give you these, but look them over
if you like. Many are so old that the LRFD Code methods have changed dramatically. For example, the equation for Cb has changed, as have other things. BE CAREFUL! NOT FOR THE CASUAL STUDENT! In fact,
during the Fall 2006 semester the whole code changed, and most of the solutions shown here would now be considered incorrect. Thus use these simply as a guide to the types of questions you will
probably be asked.
What you see is what you get.
Of highest importance to your continued use of this material: DO NOT bring any of these problems by my office and ask me to solve them for you, or they will be immediately removed from distribution.
You can take them to the tutor or to your friends, but I simply do not have time for each of you to individually bring each of these problems by my office and solve them for you. There simply is not
enough time during the semester to have each of you come by with each problem and ask that you be shown how to work it individually. No way. I am happy to let you see my old exams, such as they are,
to determine the types of problems I gave in the past, and you are welcome to get together and work them out. But the first person who asks me ANYTHING about any one of these exams will cause this
resource to immediately disappear, and their name will be posted here:
On __________________, Mr/Ms ____________________ brought an old exam to me and asked if one of the numbers shown was a 1 or a 5. For this reason the old exams have been removed and are henceforth
considered contraband. Possession of this material is now considered illegal under the Digital Millennium Copyright Act. They will be reinstated in 10 years, when it is hoped that the students are
better at following the rules.
Grading Example – why I know I am being consistent on what to take off after 20 hours of grading.
Spring of 2000
Summer of 1998 Fall of 1998 • Quiz A© Fall of 2000 Miscellaneous Exams, Solutions, etc.1© 2© 3© 4© 5© 6© 7© 8© 9© 10© 11© 12© 13©
• Final Exam©
Fall of 2004 Fall of 2005 Fall of 2006 Fall of 2007 Spring of 2008
Spring of 2009 Fall of 2009 Spring of 2011 Spring of 2012 Fall of 2012
Spring of 2017
Spring of 2013 • Quiz A© Spring of 2018
Spring of 2014 Fall of 2014 Also see this semester’s .mp4 class #16 video at 35 minutes.
• Quiz B©
• Final Exam© both exams were similar with a few changed values.
Spring of 2019
Quiz A .pdf
Quiz A .mpg
Video failed
Quiz B .pdf
Quiz B .mpg
Quiz F student solutions | {"url":"https://lowery.engr.tamu.edu/2021/08/02/loweys-old-446-exams/","timestamp":"2024-11-06T07:40:50Z","content_type":"text/html","content_length":"31535","record_id":"<urn:uuid:d9e19b3f-bb44-41e9-b1e5-3d19e29f401e>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00312.warc.gz"} |
Printable Maths Puzzles Year 4 - Printable Crossword Puzzles
Printable Maths Puzzles Year 4 – printable maths puzzles for year 6, printable maths puzzles year 4, Who does not understand about Printable Maths Puzzles Year 4? This multimedia is traditionally
used to teach expression. In almost any part of this world, this multimedia will need to have been quite familiar for most people. No less than, people could possibly have ever seen it at school.
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As for college students, this is probably not a whole new thing anymore. This press is very familiarized for use in educating and studying pursuits. You can find things you might need to know
associated with the crossword puzzle. Have you been considering understanding more? Now, let us have a look at the data below.
What you ought to Learn about Printable Maths Puzzles Year 4
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various entertaining actions. Effectively, Printable Maths Puzzles Year 4 might be one of the activities. Here is the way you fix the puzzles.
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there are always recommendations of what terms that you need to find in the puzzle. This list may contain over 5 terms to find. All depends in the puzzle creator, even though.
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studying is not the sole one that uses this mass media. It is very feasible to use in other subjects.
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It is rather very easy to create the Printable Maths Puzzles Year 4, appropriate? You don’t must commit lots of your time and effort which makes it by using a help of the device creator. Printable
Maths Puzzles Year 4 | {"url":"https://crosswordpuzzles-printable.com/printable-maths-puzzles-year-4/","timestamp":"2024-11-13T14:47:28Z","content_type":"text/html","content_length":"52962","record_id":"<urn:uuid:0a3edca0-b6ca-482d-b472-a2c866aca340>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00172.warc.gz"} |
St. Petersburg Electrotechnical University 'LETI'
Enrollment in this course is by invitation only
About the Course
The course is dedicated to studying the main theoretical concepts of algebra and their application to solving various types of tasks that occur in different fields of mathematics. The theoretical
part of the course includes basic concepts of algebra, including rational and irrational polynomials, modulus, powers and logarithms, trigonometry, plane and analytic geometry. The practical part of
the course includes examples of tasks corresponding to these themes, as well as the methods and techniques used to solve them. This knowledge and skills allow students to start successfully the
university course in calculus.
The online course is developed for use in distance and blended learning.
• The study of basic concepts of algebra, including rational and irrational polynomials, modulus, powers and logarithms, trigonometric functions, plane and analytic geometry, progressions and
vector operations.
• Formation of the ability to solve the most widespread types of problems related to these concepts.
Common Course Outline
• Module 1. Algebraic equations, inequalities and systems
□ Lesson 1.1 Basic concepts and definitions: the dictionary of algebra
□ Lesson 1.2 Linear and quadratic equations
□ Lesson 1.3 Systems of equations
□ Lesson 1.4 Inequalities and systems of inequalitiess
• Module 2. Modulus (absolute value)
□ Lesson 2.1 Definition of modulus
□ Lesson 2.2 Equations including absolute values, and the method of intervals
• Module 3. Irrational equations and inequalities
□ Lesson 3.1 Basic concepts: irrational numbers and functions. Radicals
□ Lesson 3.2 Irrational equations. Extraneous roots
□ Lesson 3.3 Irrational inequalities
• Module 4. Powers and logarithms
□ Lesson 4.1 Concepts and basic properties of power and exponential function
□ Lesson 4.2 Logarithms: concepts and basic properties
□ Lesson 4.3 Exponential and logarithmic equations
• Module 5. Trigonometry
□ Lesson 5.1 Main concepts: the trigonometric circle, measurement of angles, trigonometric functions
□ Lesson 5.2 Basic trigonometric formulae
□ Lesson 5.3 Trigonometric equations and inequalities
• Module 6. Arithmetic and geometric progressions
□ Lesson 6.1 Definition and basic properties of progressions
□ Lesson 6.2 Basic formulae and tasks concerning progressions
• Module 7. Plane geometry
□ Lesson 7.1 Basic concepts: point and line, line segment, circle
□ Lesson 7.2 Triangles: definition, special cases, properties and formulae
□ Lesson 7.3 Quadrilaterals: definition, special cases, properties and formulae
• Module 8. Analytic geometry
□ Lesson 8.1 Basic concepts: designation of points, lines and circles. Analytic representation of lines and circles
□ Lesson 8.2 Conditions for parallel and intersecting lines. Examples of tasks in analytic geometry
• Module 9. Vector algebra
□ Lesson 9.1 Basic concepts and definitions
□ Lesson 9.2 Vector operations
• Module 10. Tasks containing parameters
□ Lesson 10.1 The concept of parameter, forming a family of equations (inequalities)
Information about attestation
For the attestation, the student should:
• pass entry quiz (10% of the final grade);
• pass control tests for the course modules (50% of the final grade);
• complete lesson training tasks (10% of the final grade);
• pass examination testing (30% of the final mark).
Rating system
The results of tests and completed assignments are evaluated according to the rating system, the total amount of accrued interest for all types of activities is converted into an assessment on a
four-point scale:
• «excellent» – at least 90% of successfully performed;
• «good» – at least 70%, less than 90%;
• «satisfactory» – at least 60%, less than 70%;
• «unsatisfactory» – less than 60%.
Entry requirements and target audience
The course is designed for school students of pre-graduate years and bachelors of the 1st year of study.
If you pass an introduction test for this course you will be accepted on a course.
Course authors
Goryainov Victor Sergeevich
PhD, Assoсiate professor of Photonics Department
St.-Petersburg Electrotechnical University «LETI»
Zhikorentseva Polina Aleksandrovna
Assistant professor of Humanities and Bioethics Department
St.-Petersburg Pediatric Medical University | {"url":"https://open.etu.ru/courses/course-v1:etu+Math101+fall_2022/about","timestamp":"2024-11-09T06:40:28Z","content_type":"text/html","content_length":"23757","record_id":"<urn:uuid:802f974f-0ca8-4bf1-9ce6-e664e3dad7d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00358.warc.gz"} |
Crosstown LRT | ?m | ?s | Metrolinx | Arcadis
Member Bio
Apr 24, 2007
Reaction score
For vehicle. The constraint is the line, not a given vehicle. The situation you describe is simply mitigated by adding additional vehicles.
You can add vehicles when the trains are going faster too. All else being equal, speed does increase capacity.
Member Bio
Nov 10, 2007
Reaction score
You can add vehicles when the trains are going faster too. All else being equal, speed does increase capacity.
True - but only a barrier if you are up against the ultimate capacity of the line. You are already in deep trouble if you need to rely on that to keep the line moving (which I suppose is where the
Yonge subway was before Covid). It's certainly not going to be a Line 5 issue in the near future.
Member Bio
Mar 23, 2008
Reaction score
You can add vehicles when the trains are going faster too. All else being equal, speed does increase capacity.
Depends on what constraints you set. If you decide on the # of vehicles per hour, and can correctly predict the speed, you can calculate the total number of trains needed. Lower speed will results in
a larger required number of trains and higher operating expenses. But if you are willing to pay that price, you still can provide the desired frequency and capacity.
If your total number of trains is fixed and you can't get more, then your statement is correct; higher speed results in a higher frequency and capacity, because each train makes more trips per day.
Member Bio
Apr 24, 2007
Reaction score
Depends on what constraints you set. If you decide on the # of vehicles per hour, and can correctly predict the speed, you can calculate the total number of trains needed. Lower speed will
results in a larger required number of trains and higher operating expenses. But if you are willing to pay that price, you still can provide the desired frequency and capacity.
If your total number of trains is fixed and you can't get more, then your statement is correct; higher speed results in a higher frequency and capacity, because each train makes more trips per
Yeah that's pretty much it. If you have higher speeds then you can carry more people with fewer trains. Sure you could add more trains to a slower line, that will cost extra.
Member Bio
Oct 16, 2014
Reaction score
I'm not sure why we are revisiting the design basis for this line - great theory, but water under the bridge at this point.
If the Crosstown as opened proves inadequate, there are at least two things that can be done.
One is to deal head on with the traffic signalling issue - there is no reason that the line in its as built form can't have this retrofitted, probably mostly through the as built signalling. There
may be more leverage for this after we have a few thousand people riding every hour....if the line proves too slow, and the right things were done to create political awareness and pressure, the
penny may drop with Council.
The other is to replace the fleet (it will happen sooner than we think, the oldest cars in the fleet are already five years old) with higher capacity cars. A three car Flexity train has a lot of
wasted space - long couplers, rounded ends, cabs at the end of every car including middle cars. I'm sure there could be a 15% capacity improvement just by addressing that. (I wonder what the longest
single carbody segment that will fit in the curve geometry is.... three Flexities is an awful lot of axles)
The signalling changes can happen immediately, there is likely no rush for the fleet solution but it may be needed some day.
- PUl
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May 7, 2007
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The only scenario which is worth modelling significant change against is with a constructed Ontario Line, surely? Any modelling done with no DRL, or DRL only to Pape, is going to be superceded. Now,
Ontario Line which removing some demand toward Line 1 will also create some net extra, so this isn’t a statement that it will drive down overall ridership, but my assumption is that it will shorten
average Crosstown ride lengths because some travellers will converge on Science Centre rather than go all the way to Yonge-Eglinton. Hopefully the easternmost part of the line will also see at least
some demand head [S:west:S] east to Kennedy and a frequent service on GO, once all the LSE track is fully back in service and a fit for purpose track arrangement/grade separation is constructed
between Scarborough Junction and Kennedy.
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(it will happen sooner than we think, the oldest cars in the fleet are already five years old
I really can't see this happening, not unless the cars end up being lemons. In those 5 years they've sustained very little wear and tear. They'll just keep them to age 35 instead of 30.
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Oct 16, 2014
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I really can't see this happening, not unless the cars end up being lemons. In those 5 years they've sustained very little wear and tear. They'll just keep them to age 35 instead of 30.
It will be interesting to see if aging is lessened for this fleet versus the downtown trams.... one would think that salt intrusion etc would be less, but the whole fleet has been sitting out in the
rain and snow....
And at the end of the day - rust never sleeps.
- Paul
Are they all parked outside? There's no indoors storage?
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Sep 13, 2011
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Are they all parked outside? There's no indoors storage?
Unless they're in for repairs/servicing, cleaning, or stored in the mainline underground, they are outside.
Unless they're in for repairs/servicing, cleaning, or stored in the mainline underground, they are outside.
That changes things. How utterly bizzare that the rolling stock was allowed to sit outside for 5 years (and counting!) without ever carrying a paying passenger.
What is it with this town and its refusal to store most of its rail rolling stock indoors? Surely the cars would fare a lot better long term not exposed to the elements.
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Apr 24, 2007
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That changes things. How utterly bizzare that the rolling stock was allowed to sit outside for 5 years (and counting!) without ever carrying a paying passenger.
What is it with this town and its refusal to store most of its rail rolling stock indoors? Surely the cars would fare a lot better long term not exposed to the elements.
The service bays will only hold so many cars and where do you plan to store the rest of them that has cover for them??
Very few places in Europe have inside storage for their fleet with most out in the yard. Buffalo store their cars indoors with three others being outside in the US. All of TTC fleet is outdoors.
The service bays will only hold so many cars and where do you plan to store the rest of them that has cover for them??
Very few places in Europe have inside storage for their fleet with most out in the yard. Buffalo store their cars indoors with three others being outside in the US. All of TTC fleet is outdoors.
You store the rest of them in cover that is built.
What places in Europe are these? Certainly not in central Europe, where tram yards without covered storage are the exception, not the norm.
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Dec 24, 2016
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Could a case for subsidized housing be built above the yard such that the yard is covered? I know soundproofing would be a challenge, but it could make use of dead air space.
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Apr 24, 2007
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You store the rest of them in cover that is built.
What places in Europe are these? Certainly not in central Europe, where tram yards without covered storage are the exception, not the norm.
Belgium, Frankfurt, Nice come to mind and did not see every yard for them, other than Nice back in 2012.
Could a case for subsidized housing be built above the yard such that the yard is covered? I know soundproofing would be a challenge, but it could make use of dead air space.
Housing can be built over yards, and you only have to look at TTC Davisville yard that has been on the books for decades of having development over it. Look at what is taking place in NYC where yards
are being covered up. | {"url":"https://urbantoronto.ca/forum/threads/toronto-crosstown-lrt-m-s-metrolinx-arcadis.11782/page-1603","timestamp":"2024-11-06T07:07:39Z","content_type":"text/html","content_length":"151564","record_id":"<urn:uuid:25b3f94c-6c5e-41c6-9665-0da2ae14eeab>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00186.warc.gz"} |
C-ID Descriptor
Descriptor Details
• Trigonometry
• Not Identified
• 851
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• 3.0
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The study of trigonometric functions, their inverses and their graphs, identities and proofs related to trigonometric expressions, trigonometric equations, solving right triangles, solving triangles
using the Law of Cosines and the Law of Sines, polar coordinates, and introduction to vectors.
1. Rectangular coordinates, angles and circular/radian measure;
2. Definitions of the six trigonometric functions according to the right triangle, the unit circle, and the rectangular coordinate system;
3. Applications of the right triangle;
4. Simplification of trigonometric expressions;
5. Proofs of trigonometric identities;
6. Graphs of trigonometric functions: period, amplitude, phase shift, asymptotes;
7. Inverse trigonometric functions and their graphs;
8. Trigonometric equations;
9. Solving Triangles: Law of Sines and Law of Cosines;
10. Polar coordinates and equations; and
11. DeMoivre’s Theorem and applications
12. Introduction to vectors.
Upon successful completion of the course, students will be able to:
1. Identify special triangles and their related angle and side measures;
2. Evaluate the trigonometric function of an angle in degree and radian measure;
3. Manipulate and simplify a trigonometric expression;
4. Solve trigonometric equations, triangles, and applications;
5. Graph the basic trigonometric functions and apply changes in period, phase and amplitude to generate new graphs;
6. Evaluate and graph inverse trigonometric functions;
7. Prove trigonometric identities;
8. Convert between polar and rectangular coordinates and equations;
9. Graph polar equations;
10. Calculate powers and roots of complex numbers using DeMoivre’s Theorem; and
11. Represent a vector (a quantity with magnitude and direction) in the form <a,b> and ai+bj.
Tests, examinations, homework or projects where students demonstrate their mastery of the learning objectives and their ability to devise, organize and present complete solutions to problems.
A college level text designed for science, technology, engineering and math majors, and supporting the learning objectives of this course.
• No
• Course does not transfer to UC’s
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Algebra: Completing the Square and the Quadratic Formula
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Adam Panagos / Engineer / Lecturer
Completing the Square and the Quadratic Formula
A few algebra videos I made for my son to help him review.
9.5 Completing the Square: When a = 1
We solve quadratic equations by completing the square for the special case when a = 1.
9.5 Completing the Square: Perfect Square Trinomials
We complete the square to find the value for c to make each expression a perfect-square trinomial.
9.5 Completing the Square: The Vertex Of A Parabola
Given the equation for a parabola, we complete the square to easily identify the parabola vertex.
9.5 Completing the Square: When a is not 1
We solve quadratic equations by completing the square when the coefficient a is NOT equal to 1.
9.6 The Quadratic Formula: Solving Equations
We use the quadratic formula to solve quadratic equations.
9.6 The Quadratic Formula: Appropriate Solutions
When working word problems that involve using the quadratic formula, one should be careful to choose a final solution that is appropriate.
9.6 The Quadratic Formula: Discriminant and Number of Solutions
The discriminant of a quadratic equation is b^2 - 4ac. Depending on the value of the discriminant, the equation has either 2 real-valued solutions, 1 real-valued solution, or no real-valued
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(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v₀ and ω₀ a “reverse” spin of angular
speed (see Fig. 11–41). A kinetic friction force acts on the ball as it initially skids across the table.
(d) If ω₀ is 10% larger than ,w_C i.e.,ω₀ = 1.10w_C, determine the ball’s cm velocity v_CM when it starts to roll without slipping. [Hint: The ball possesses two types of angular momentum, the first
due to the linear speed v_CM of its cm relative to point O, the second due to the spin at angular velocity ω about its own cm. The ball’s total L about O is the sum of these two angular momenta.] | {"url":"https://www.pearson.com/channels/physics/explore/angular-momentum/conservation-of-angular-momentum?chapterId=0214657b","timestamp":"2024-11-14T17:17:05Z","content_type":"text/html","content_length":"503819","record_id":"<urn:uuid:431ab005-d6d2-4ca7-ad5f-a89e981a786e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00702.warc.gz"} |
Finding Angles In Triangles - Angleworksheets.com
Finding Angle Measures Of Triangles Worksheet – There are many resources that can help you find angles if you’ve been having trouble understanding the concept. These worksheets will help you
understand the different concepts and build your understanding of these angles. Using the vertex, arms, arcs, and complementary angles postulates, students will learn how to … Read more
Finding Missing Angles In Triangles Worksheet Pdf Grade 8
Finding Missing Angles In Triangles Worksheet Pdf Grade 8 – There are many resources that can help you find angles if you’ve been having trouble understanding the concept. These worksheets will help
you understand the different concepts and build your understanding of these angles. Students will be able to identify unknown angles using the vertex, … Read more | {"url":"https://www.angleworksheets.com/tag/finding-angles-in-triangles/","timestamp":"2024-11-08T22:24:40Z","content_type":"text/html","content_length":"52980","record_id":"<urn:uuid:ed940fb9-135a-4060-8783-192e53d42f92>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00615.warc.gz"} |
How much does a gram weigh on a digital scale? [Solved]
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
How much does a gram weigh on a digital scale?
A digital scale is generally more accurate in terms of readings than an analog scale. A gram is a unit of mass equal to one-thousandth of a kilogram.
Answer: Accuracy and unit decide how much a gram is measured on a digital scale.
Go through the explanation to understand better.
The quantity measured on a digital scale depends on two important factors:
1) Accuracy of the digital scale
A 95% accurate digital scale would measure 1 gram equal to somewhere between 0.95 - 0.98 grams.
2) Unit in which it measures that particular quantity
A digital scale that measures mass in ounces will measure 1 gram as 0.0353 ounces.
Thus, accuracy and unit decide how much a gram is measured on a digital scale.
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/questions/how-much-does-a-gram-weigh-on-a-digital-scale/","timestamp":"2024-11-07T22:19:42Z","content_type":"text/html","content_length":"196920","record_id":"<urn:uuid:8956420a-ea95-41ff-a58f-192de97858a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00188.warc.gz"} |
Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are mathematical expressions which comprises of one or several terms, all of which has a variable raised to a power. Dividing polynomials is a crucial working in algebra that includes
working out the remainder and quotient as soon as one polynomial is divided by another. In this blog, we will examine the various techniques of dividing polynomials, including long division and
synthetic division, and provide instances of how to apply them.
We will further discuss the significance of dividing polynomials and its applications in various fields of math.
Significance of Dividing Polynomials
Dividing polynomials is an essential function in algebra that has many utilizations in many domains of math, involving calculus, number theory, and abstract algebra. It is utilized to work out a wide
range of challenges, including finding the roots of polynomial equations, calculating limits of functions, and calculating differential equations.
In calculus, dividing polynomials is applied to figure out the derivative of a function, that is the rate of change of the function at any time. The quotient rule of differentiation consists of
dividing two polynomials, that is utilized to figure out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is applied to study the characteristics of prime numbers and to factorize huge values into their prime factors. It is further applied to study algebraic
structures such as fields and rings, that are fundamental concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in many
domains of mathematics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The method is on the basis of
the fact that if f(x) is a polynomial of degree n, subsequently the division of f(x) by (x - c) gives a quotient polynomial of degree n-1 and a remainder of f(c).
The synthetic division algorithm consists of writing the coefficients of the polynomial in a row, applying the constant as the divisor, and performing a series of calculations to figure out the
quotient and remainder. The result is a streamlined structure of the polynomial which is easier to function with.
Long Division
Long division is an approach of dividing polynomials that is applied to divide a polynomial with any other polynomial. The approach is based on the reality that if f(x) is a polynomial of degree n,
and g(x) is a polynomial of degree m, where m ≤ n, then the division of f(x) by g(x) offers uf a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm consists of dividing the highest degree term of the dividend by the highest degree term of the divisor, and further multiplying the result with the whole divisor. The
answer is subtracted of the dividend to get the remainder. The process is repeated as far as the degree of the remainder is less compared to the degree of the divisor.
Examples of Dividing Polynomials
Here are some examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could utilize synthetic division to simplify the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Thus, we can express f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's say we have to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 by the polynomial g(x) = x^2 - 2x + 1. We can apply long division to simplify the expression:
First, we divide the highest degree term of the dividend with the highest degree term of the divisor to attain:
Subsequently, we multiply the entire divisor by the quotient term, 6x^2, to attain:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to obtain the new dividend:
6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)
which simplifies to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, by the largest degree term of the divisor, x^2, to achieve:
Then, we multiply the entire divisor with the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that streamline to:
10x^2 + 2x + 3
We recur the procedure again, dividing the largest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to obtain:
Then, we multiply the total divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies to:
13x - 10
Thus, the answer of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Ultimately, dividing polynomials is a crucial operation in algebra which has many uses in multiple domains of math. Comprehending the different approaches of dividing polynomials, such as long
division and synthetic division, can guide them in solving complex challenges efficiently. Whether you're a student struggling to get a grasp algebra or a professional operating in a field which
involves polynomial arithmetic, mastering the ideas of dividing polynomials is important.
If you need help understanding dividing polynomials or anything related to algebraic concept, contemplate reaching out to Grade Potential Tutoring. Our experienced teachers are available online or
in-person to offer personalized and effective tutoring services to support you succeed. Call us right now to schedule a tutoring session and take your math skills to the next stage. | {"url":"https://www.sanantonioinhometutors.com/blog/dividing-polynomials-definition-synthetic-division-long-division-and-examples","timestamp":"2024-11-15T03:35:25Z","content_type":"text/html","content_length":"78832","record_id":"<urn:uuid:a7cf7af2-b552-414b-9467-b78886ab4cad>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00068.warc.gz"} |
how to draw a star without lifting your pencil
on Introduction. In shape 2, there are four vertices of odd degree and one vertex of even degree, so it does not have any Euler path or Euler circuit. If we go back to shape 1, it does not matter if
you start from 5 or 6 since the solutions are mirror images of each other. As it turns out, there’s much more to this puzzle than meets the eye. “If a graph has an Euler circuit, then all of its
vertices must be even vertices.”. A non-bridge ALWAYS has priority over a bridge. ( Log Out / This might work better if you're in one of those family restaurants with free crayons for kids. Now we
are looking for a path (or a cycle) in the graph that visits every edge exactly once. Mess around with it for a bit and you’ll get it. If we follow the path 6-5-4-3-6-4-2 and delete all the edges
that we travel, the final graph would look like this: If we continue with vertex 5, which is a bridge, we would get stuck there, so we would have to lift our pencil to draw this shape. Answer: No.
From there, draw a diagonal line upward and to the left, crossing your first line. 13 years ago Then we could hand that information over to someone else who could redraw that image however they
liked. In this graph, vertices A, B, C, D, E and F are all even, so we will find an Euler circuit. I bet you could tell me the answer without me asking the question. We're dropping it off one
surface, down one edge and back onto the other surface. on Introduction. It is possible to use graphs to solve deep problems in geometry and topology. 13 years ago As a subscriber, you are not only a
beneficiary of our work but also its enabler. It’s a support for truth and fairness in journalism. 9 years ago How to Draw 3D Art - Easy Line Paper Trick - Duration: 2:25. carefully drop your pencil
back onto the proper side of the paper oops, to drop your pencil it must have been lifted. on Introduction, or use your eraser to move or slide the pencil without touching the lead on there, You
wouldn't lift your pencil if you were to DROP it, right? 10 years ago We have been keeping you up-to-date with information on the developments in India and the world that have a bearing on our health
and wellbeing, our lives and livelihoods, during these difficult times. At this difficult time, it becomes even more important that we have access to information that has a bearing on our health and
well-being, our lives, and livelihoods. They could move the points around, or draw curved connecting edges, or squiggles. It’s worth noticing that the walking tour of Königsberg is quite different on
the surface than the “house” image in puzzle 1. Being curious as to what the catch is, why I'd take the bet. (Notice that we started and ended with vertex B, as we were supposed to do.). The story,
related by anthropologist Emil Torday, goes like this: the children were drawing complicated networks in the sand. It's my passion 2,102 views. To draw this without lifting the pen and without
tracing the same line more than once. Draw a STAR without lifting pen/pencil. It's good for parties and for taking money from unsuspecting dupes. on Introduction, 12 years ago The fundamental
observation that solves all these puzzles is that there are two special points in a path drawn with a single, unbroken, pencil line: the beginning and the end. Then it has, uh 2! Let’s apply these on
our examples. Removing this edge from the graph would make it disconnected. I’m sure he will be the coolest blogger around!
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IM 2 - Ch 14
Chapter 14 - Solving Quadratic Equations and Inequalities
"This chapter introduces the quadratic formula and emphasizes choosing an appropriate method to solve quadratic equations. Quadratic inequalities are solved using a coordinate plane, and then an
algebraic strategy is introduced. Systems of equations involving one or more quadratic equations are solved."
Carnegie Textbook
Lesson 14.2 - Using a CalculatorBased Ranger to Model Quadratic Motion
"This lesson provides an opportunity for students to use a calculator-based ranger to model the trajectory of a ball." - Carnegie Textbook | {"url":"https://www.msstevensonmath.com/im-2---ch-14.html","timestamp":"2024-11-09T04:30:30Z","content_type":"text/html","content_length":"86466","record_id":"<urn:uuid:daf0c548-b286-4f96-9d3a-e5bba05c7d69>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00792.warc.gz"} |
Improbable Wins
What is going on with the win probabilities in NBA games?
After a long couple of months, the NBA playoffs will be wrapping up soon. The finals are currently underway with the Dallas Mavericks facing the Boston Celtics.
Unfortunately, despite an impressive showing from the Mavericks last Friday night (winning 122 to 84!), the odds of them winning the series are most definitely NOT in their favor. Prior to Friday
night, without a single win, they were down 3 games in the series to the Celtics. Historically, in NBA playoff history, out of the 150+ teams that have gone down 0-3 in a series, not a single one has
come back to win the series.
But zooming out from just the finals, by all accounts, the entire playoffs this year were entertaining, with no shortage of games that seemed to come down to the last few minutes. One series in
particular was the Celtics vs. the Pacers in the Eastern Conference finals, which was recently the focus of the weekly Fiddler on Proof Substack.
At the end of the series, ESPN put out the following graphic below showing that during Games 1, 3, and 4, there was a point when the Pacers had a 90% chance of winning, and yet, somehow, the Celtics
ended up sweeping the Pacers, winning each of those 3 games, plus Game 2.
The win probability charts for those 3 respective games are below (Celtics in green and Pacers in dark blue), which gives an indication as to how exciting they were to watch, coming right down to a
dramatic finish at the end.
For reference, the win probability over the course of game 2 is shown below. Clearly a more “normal” game…
Seeing the Celtics come back from a 90% probability for the Pacers to win, not just once, but 3 times, naturally led some people to question the reported win probabilities.
Simple Analysis
Coming back to the Fiddler on the Proof post on this, is it possible to try and understand this using some kind of analysis? To start, what if we first look at a very simple scenario proposed by the
• Let’s assume there are only 5 possessions in a game (don’t worry, we will revisit this assumption)
• During a given possession, there is a 50% chance that your team will score (relatively representative of NBA teams)
• If your team doesn’t score, then assume the other team scores instead
• A score is worth 1 point to keep it simple
• Analyze only the games where the opponent had a 75% chance of winning at some point in the game and then determine how often your team actually ended up winning
Common sense would tell us that if at any point the opponent had a 75% chance of winning, then your team should have had a 25% chance of winning.
With only 5 possessions, the possible enumerations of scenarios can be solved analytically, but I find it easier to approach these through computer simulations (and that will pay off when the
analysis gets more complicated).
We can easily simulate a game here by randomly drawing five times from a Bernoulli Distribution with a probability parameter of 0.5. Taking the inverse of each value we return then corresponds to the
opponent’s (Team 2) score. Accumulating both sets of values gives us the running score after each possession. In the example below, Team 1 wins, being the first to reach 3 points.
For each possession along the game, we can calculate the probability of Team 2 winning, which returns the plot below for this example.
In this example, after possession 3, there was actually a 75% chance that Team 2 would win, however, Team 1 ultimately came back to win.
If we scale this approach up, generating 100’s of games at a time, and then take only the games where Team 2 had at least a 75% chance of winning, we can then look at how often Team 1 came back to
win it.
As mentioned above, we would expect this to be ~25% intuitively, but in fact it’s lower than that with an average of ~19%. The distribution of probabilities of Team 1 winning after Team 2 had at
least a 75% chance of winning at some point in the game is shown below. While it varies from 5% to 35%, the average is clearly centered below 20% and nowhere near the expected 25%.
More Complicated Analysis
Were the results above just a result of the low possession count? Let’s try and find out by making the analysis more similar to an actual NBA game and in particular the Celtics/Pacers series:
• Increase the number of possessions to 101, which is reflective of an actual NBA game
• Maintain the scoring at 50% for each time while in reality the probability of any team scoring will be different than 50%
• Analyze the Celtics / Pacers game by determining the probability of the Celtics winning in games when the Pacers had at least a 90% chance of winning at some point in the game
Just like the analysis above, we can do this again using simulation and most of the same logic. Here the use of simulation is important because the analytical approach given the combinations of
potential scores on each possessions is huge.
We’ll start first with creating a single game to walk through the steps, this time with 101 possessions. It’s worth noting that unlike a real game where there is no limit on the score, in our simple
example the winner is the first to reach a score of 51.
And the corresponding moving chart of win probability for Team 2 now starts to look more similar to some of the ESPN charts.
Just like in the simpler analysis, if we then generate 100’s of games at a time using these conditions and take only those where Team 2 had a probability of winning of at least 90% at some point in
the game, we can calculate the fraction of games where Team 1 came back to win.
The distribution of probabilities that Team 1 comes back to win is shown below indicating an average of around 8%. Again, just like we saw in the simpler analysis, it is actually lower than what we
would intuitively expect (10%).
Going into this analysis, I was expecting to find that if anything, the in-game probabilities were somehow easier to overcome than we might intuitively expect. This would explain why the Celtics
seemingly did this 3 different times in just a 4 game stretch.
It’s entirely possible (potentially even likely) that this simplified approach was too simple, but what it showed was that it was actually harder to overcome the in-game probabilities. Potentially as
the number of possessions increase, we see evidence that we may converge to the intuition i.e. with more possessions a 90% chance of Team 2 winning would correspond to a 10% chance of Team 1 winning.
In either case, it would appear that what the Celtics did against the Pacers was in fact close to a 1 in a 1000 type event and therefore pretty amazing!
Awesome breakdown, Andrew! Appreciate you explaining how those ESPN projections work. I'm a big Pacers fan, watched most of their games this season, so it was a tough read. Ha!
As I think about where analytics and reality collide... While I think the analytics captured Game I correctly (the one we coughed up), at no point in those other two games did I feel the Pacers
have close to a 9 in 10 chance of winning. As the better team, Boston's ability to ramp up their game (which they did) in the minutes that matter is a real outcome that's hard to nail in a
make-miss model. All that aside... that the Knicks had a 0% chance of winning the Pacers-Celtics series is the only statistic that mattered! :)
Expand full comment
1 more comment... | {"url":"https://news.pontemanalytics.com/p/improbable-wins","timestamp":"2024-11-12T20:28:45Z","content_type":"text/html","content_length":"222211","record_id":"<urn:uuid:73cbd6e4-71ff-4795-8bf5-c27df53ce192>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00324.warc.gz"} |
What is Multiplication? - IM CERTIFIED® BLOG
Multiplication is vexation,
Division is as bad;
The Rule of Three doth puzzle me,
And Practice drives me mad.
(old nursery rhyme.)
Some people might answer that multiplication is repeated addition. For example, $5 \times 7$ is 7 added 5 times: $7 + 7 + 7 + 7 + 7 = 35$. One problem with this is that it is possible to get confused
about the number of additions. There are only 4 plus signs in the equation, so did I really repeat the addition 5 times? This confusion is particularly acute when I am trying to explain why $0 \times
7 = 0$. According to multiplication-is-repeated-addition, this is 7 added to itself 0 times. Well, okay, I’m not going to do any additions, but don’t I still have the 7 that I started with? Wouldn’t
that make $0 \times 7 = 7$? Or would it be 0 because I don’t have any 7s? This confusion is about how to do the multiplication; it could be cleared up by a conceptualization of what multiplication
is. (For those who are inclined to abstract mathematics, Jason Zimba has written a tongue-in-cheek proof that multiplication is not repeated addition.)
In the standards, students learn in grade 3 to “interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each.” This is the equal groups way of thinking about multiplication.
There is a big difference between this and repeated addition: equal groups is a way of thinking, whereas repeated addition is a way of doing. That is, repeated addition tells you how to calculate $5
\times 7$. But it doesn’t really tell you why you are doing the calculation except “that’s what I told you to do.” By contrast, the equal groups way of thinking is not a calculation, it is a
conceptualization. And it is a fairly natural one. Equal groups appear everywhere: eggs in egg cartons, crayons in boxes, or arrays of windows on the side of a building. The array is a natural way to
arrange equal groups, and leads to all sorts of useful facts about multiplication. For example, I can see a $5 \times 7$ array as 5 rows of 7 or as 7 columns of 5 objects.
Thus 5 groups of 7 is 7 groups of 5, so $5 \times 7 = 7 \times 5$. This is not at all obvious with repeated addition. It’s not easy to see why 7 + 7 + 7 + 7 + 7 = 5 + 5 + 5 + 5 + 5 + 5 + 5. Of
course, doing some calculation will show you that is true. But it won’t show you why it is true.
The equal groups way of thinking about multiplication clarifies the role of repeated addition. Starting in kindergarten, students understand addition as putting together; $2 + 3$ is the number of
things you get when you put a set of 2 things together with a set of 3 things. When they understand multiplication in terms of equal groups, they can call on their prior understanding of addition and
put all the groups together. Starting with $5 \times 7$ as the number of things in 5 groups of 7 things each, I can put the 5 groups together and get $7 + 7 + 7 + 7 + 7$. Note also that the equal
groups way of thinking also copes quite nicely with $0 \times 7$. If I have 0 groups of 7 I don’t have anything at all, so $0 \times 7 = 0$.
Perhaps the most profound advantage of the equal groups way of thinking is that it can be extended to fraction multiplication. If I want to understand $\frac53 \times 7$, repeated addition is no
help. What does it mean to add 7 to itself $\frac53$ times? On the other hand, you can make sense of $\frac53$ groups of 7. You have to work at it a bit; you have to decide what “$\frac13$ groups”
means. But students in grade 2 used “one third of” for partitions of circles and rectangles, so it is not unnatural to suggest that “$\frac13$ groups” is one part of a group when you divide the group
into 3 equal parts. A progression of representations leads from equal groups, for multiplication of whole numbers, to arrays, to area representations of multiplication that can be used for
multiplying fractions:
Multiplication does not need to be vexation. A clear conceptual picture of what multiplication is can help students to navigate the various strategies for carrying it out, and to recognize situations
in which multiplication can be used to solve a problem.
Next Step
For a more extended discussion of the progression of representations, see Kristin Umland’s paper “The Role of Models in Mathematics Teaching and Learning.” (Note that the paper uses the word model
for what I have called representations here.) | {"url":"https://illustrativemathematics.blog/2018/12/11/what-is-multiplication/","timestamp":"2024-11-07T04:11:11Z","content_type":"text/html","content_length":"87630","record_id":"<urn:uuid:fdf8a11a-d50e-4a30-b6de-18e8a17cb883>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00857.warc.gz"} |
Categories, Logic and Physics
Some Related Work
Other axiomatic approaches to the foundations of quantum mechanics
Pioneers in this area were Garreth Birkhoff & John von Neumann, George Mackey, Joseph Jauch & Constantin Piron, Dave Foulis & Charles Randall and Gunther Ludwig. Survey's are:
The founding paper was:
• Birkhoff, G. and von Neumann, J. (1936), The Logic of Quantum Mechanics, Annals of Mathematics 37, 823—843.
There are also the tracts:
• Mackey, G.W. (1963), Mathematical Foundations of Quantum Mechanics, W.A. Benjamin Inc.
• Piron, C. (1976), Foundations of Quantum Physics, W.A. Benjamin Inc.
• Ludwig, G. (1985; 1987), An Axiomatic Basis of Quantum Mechanics: 1. Derivation of Hilbert Space; 2. Quantum Mechanics and Macrosystems, Springer-Verlag.
Some current developments are:
Categories in logic and foundations of computing
A Milestone:
Formally, this logical development stands orthogonal to Birkhoff/von Neumann quantum logic. It is rather this logic and not Birkhoff/von Neumann quantum logic which provides the logical foundation
for the Monoidal approach, making the ability to copy and delete premisses explicit. Although not yet as articulated and exploited as Girrard's Linear Logic, one could argue that Linear Logic was
already present in Jim Lambek's earlier work on mathematical linguistics (1956) and categorical logic (1970's). The categorical semantics of Linear Logic based on Mike Barr's *-autonomous categories
is in:
More recent computer science motivated developments are:
Earlier work on categories in foundations of physics
Categories in mathematical physics
General category theory resources
General physics resources | {"url":"http://categorieslogicphysics.wikidot.com/papers:some-related-work","timestamp":"2024-11-09T05:53:11Z","content_type":"application/xhtml+xml","content_length":"28714","record_id":"<urn:uuid:c493348e-dc62-4ff8-b584-dd3c1a6d7dff>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00284.warc.gz"} |
What sxs 4 seater is best for trails/rocks/forest?
I live in the Yukon and need a 4 seater sxs for the family to go play in, there is no sand here so that doesn't need to factor into the make or model. I'll be doing trails along with going off the
beaten path in the forest and the tundra. Lots of rocks up here also so i need to be able to climb those. Thanks for any and all input.
Sep 22, 2008
Reaction score
The only 4 seater I would look at for a trail utv, is the kawi Teryx 4 seater. The best 4 seater on the market ...imo.
All the rest are just to long....
Oct 18, 2009
Reaction score
X2 on the Kawi! The others are all very low and long which makes them get high centred all the time.
Sent from my iPhone using Tapatalk
X3 on the Kwai...I know 3 families running them. Great machine.
Wow, guess u guys are coming to the light after all
Must be all the broken axles
Wow, guess u guys are coming to the light after all
Must be all the broken axles
We are talking 4 seater family machine.
Jan 15, 2008
Reaction score
We are talking 4 seater family machine.
yup cheap on parts too, they seem to be laying all along the trails, from a previous passerby.
We are talking 4 seater family machine.
Best of both worlds , family goes along , makes it back every time, no broken axles to piss with
Win.win as I see it
The short wheel travel or suspention and low ground clearences concern me. What are they both 8.5"? I was kinda looking at the rzr 4 900. Also how reliable are they?
Last edited:
The short wheel travel or suspention and low ground clearences concern me. What are they both 8.5"? I was kinda looking at the rzr 4 900. Also how reliable are they?
My buddy put in a 2 inch lift.
28 inch tires.
No clearance issues.
Now what about a 2 seater? My wife is thinking she will just get her own 2 seater next season since she wont be coming out much this year she is like 7 months pregnant. So that really opens up the
options? Would a rzr 900s be best for 2 seaters?
The kawi 2 seater is nice too, where ur legs would go in the 4 seater is all storage , awsome idea, same frame
New 800 engine is a good one too I hear
Sep 22, 2008
Reaction score
You keep moving the target.... If your talking 2 seater that is a whole new ball game.... And PoPo rules the 2 seat market by a mile.... Kawi does not make a sport utv. Tuff to beat the variety and
performance in the rzr line up....
what about the can am mavericks? It just when i go to the rzr forums it seems everyone posting about **** wrong with their rzrs. Like sand in engine, air filter crazy dirty in the 2015 900 series.
Flattening out of something your belt is on or some ****(i'm not very vehicle inclined) ect... ect... They mavericks interior looks far superior. But how are they for reliability?
Oct 18, 2009
Reaction score
Go price them out and you will be looking at the rzr's! Most of the time the biggest problem with a machine is the lose nut behind the steering wheel. People are stupid and wreck things, then go onto
the internet and complain. Polaris has the best line up of sxs
Sent from my iPhone using Tapatalk
Go price them out and you will be looking at the rzr's! Most of the time the biggest problem with a machine is the lose nut behind the steering wheel. People are stupid and wreck things, then go
onto the internet and complain. Polaris has the best line up of sxs
Sent from my iPhone using Tapatalk
Sounds correct.
If I option the commander xtp and a 900 to have the same stuff the price difference is like 150$.
Jan 15, 2008
Reaction score
go buy a friggin honda then. sheesh. do you have any real idea what you want, 4 seater, 2seater, teryx, ranger/rzr or a brp?
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Just getting a 2 seater, the wife is going to get her own. 2 is better then when!!! Sorry for all the confusion. I still need a 2 seater that can go over the stuff I said, the extra room if I go
hunting would be convenient. But I could just remove the passanger seat in a 900 s. | {"url":"https://www.snowandmud.com/threads/what-sxs-4-seater-is-best-for-trails-rocks-forest.99495/","timestamp":"2024-11-11T03:19:08Z","content_type":"text/html","content_length":"170940","record_id":"<urn:uuid:55c4e0f7-9ca2-42eb-80f0-d963633b1704>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00488.warc.gz"} |
Learning complexity vs. communication complexity
This paper has two main focal points. We first consider an important class of machine learning algorithms - large margin classifiers, such as Support Vector Machines. The notion of margin complexity
quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of
discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas. In the same way that matrix rigidity is related to rank, we introduce the notion of
rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin
complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether P SPACE can be separated from the polynomial hierarchy
in communication complexity. There are numerous known relations between the field of learning theory and that of communication complexity [6, 9, 25, 15], as one might expect since communication is an
inherent aspect of learning. The results of this paper constitute another link in this rich web of relations. This link has already proved significant as it was used in the solution of a few open
problems in communication complexity [19, 17, 28].
Original language English
Title of host publication Proceedings - 23rd Annual IEEE Conference on Computational Complexity, CCC 2008
Pages 53-63
Number of pages 11
State Published - 2008
Event 23rd Annual IEEE Conference on Computational Complexity, CCC 2008 - College Park, MD, United States
Duration: 23 Jun 2008 → 26 Jun 2008
Publication series
Name Proceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print) 1093-0159
Conference 23rd Annual IEEE Conference on Computational Complexity, CCC 2008
Country/Territory United States
City College Park, MD
Period 23/06/08 → 26/06/08
Dive into the research topics of 'Learning complexity vs. communication complexity'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/learning-complexity-vs-communication-complexity-13","timestamp":"2024-11-03T06:43:34Z","content_type":"text/html","content_length":"50744","record_id":"<urn:uuid:9551c77c-c40d-4ce9-91f9-66a1350a9332>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00837.warc.gz"} |
Memory-efficient Accurate Sampling for Counting LOcal Triangles in Graph Streams
We propose local triangle counting algorithms in a graph stream based on edge sampling: MASCOT for a single graph stream, and MultiBMASCOT and MultiWMASCOT for a multigraph stream. To develop MASCOT,
We first devise two edge sampling based naive algorithms MASCOT-C and MASCOT-A which have the advantages of memory-efficiency and low variance. Our proposed algorithm MASCOT takes both advantages by
the strategy of "unconditional counting before sampling"
For a multigraph stream, MultiBMASCOT provides binary triangle counting, i.e. ignoring the effect of duplicated edges. This is the same as counting local triangles for the correponding simple graph.
MultiWMASCOT provides weighted triangle counting where a weight of an edge is determined by the number of its repeated occurences. Concretely, MultiWMASCOT counts each triangle as the product of the
weights of the participating three edges.
Here is a download link for our
is described in the following paper.
• MASCOT: Memory-efficient and Accurate Sampling for Counting Local Triangles in Graph Streams
Yongsub Lim, and U Kang
21st ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), 2015
[PDF] [BIBTEX]
• Memory-efficient and Accurate Sampling for Counting Local Triangles in Graph Streams: From Simple to Multigraphs
Yongsub Lim, Minsoo Jung, and U Kang
ACM Transactions on Knowledge Discovery from Data (TKDD), vol. 12, issue 1, Feburuary 2018.
[PDF] [BIBTEX]
Simple Graphs
Name # Nodes # Edges Description Name # Nodes # Edges Description
Advogato 5,155 39,285 Trust network NotreDame 325,729 1,090,108 Web graph of Notre Dame
Enron 36,692 183,831 Enron email excahnges Petster 623,766 15,699,276 Social websites for cat and dog owners
Wiki-Conflict 116,836 2,027,871 Edit confliction BerkStan 685,230 6,649,470 Web graph of Berkeley and Stanford
Gowalla 196,591 950,327 Online social network DBLP 1,314,050 5,362,414 Co-author network in DBLP
Movies9 253,045 6,611,899 Co-reviewed movies in Amazon LiveJournal 4,846,609 42,851,237 LiveJournal online social network
Stanford 281,903 1,992,636 Web graph of Stanford.edu
Name # Nodes # Edges Description Name # Nodes # Edges Description
Actor 382,219 33,115,812 Actor collaboration in movies ItWiki 1,703,605 86,548,398 Hyperlinks in Italian Wikipedia
Baidu 415,641 3,284,387 "related to" links in encyclopedia Baidu ChinWiki 1,930,275 3,284,387 Hyperlinks in Chinese Wikipedia
DBLP 1,314,050 18,986,618 Co-author network in DBLP
• Yongsub Lim (Dept. of Computer Science and Engineering, Seoul National University)
• Minsoo Jung (Dept. of Computer Science and Engineering, Seoul National University)
• U Kang (Dept. of Computer Science and Engineering, Seoul National University) | {"url":"https://datalab.snu.ac.kr/mascot/","timestamp":"2024-11-13T08:22:21Z","content_type":"text/html","content_length":"14566","record_id":"<urn:uuid:b4c6c2f8-b1f4-43a2-880c-52363b92af25>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00627.warc.gz"} |
Inferential Statistics - Reviews & Coupon - Java Code Geeks
Inferential Statistics
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8.6/10 (Our Score)
Product is rated as #14 in category R
This course covers commonly used statistical inference methods for numerical and categorical data. You will learn how to set up and perform hypothesis tests, interpret p–values, and report the
results of your analysis in a way that is interpretable for clients or the public. Using numerous data examples, you will learn to report estimates of quantities in a way that expresses the
uncertainty of the quantity of interest. You will be guided through installing and using R and RStudio (free statistical software), and will use this software for lab exercises and a final project.
The course introduces practical tools for performing data analysis and explores the fundamental concepts necessary to interpret and report results for both categorical and numerical data Duke
University has about 13,000 undergraduate and graduate students and a world–class faculty helping to expand the frontiers of knowledge. The university has a strong commitment to applying knowledge in
service to society, both near its North Carolina campus and around the world.
Instructor Details
Mine Cetinkaya-Rundel
Votes: 3
Courses : 3
Mine Cetinkaya-Rundel is an Assistant Professor of the Practice at the Department of Statistical Science at Duke University. She received her Ph.D. in Statistics from the University of California,
Los Angeles, and a B.S. in Actuarial Science from New York University's Stern School of Business. Dr. Cetinkaya-Rundel is primarily interested in innovative approaches to statistics pedagogy. Some of
her recent work focuses on developing student-centered learning tools for introductory statistics courses, teaching computation at the introductory statistics level with an emphasis on
reproducibility, and exploring the gender gap in self-efficacy in STEM fields. Her research interests also include spatial modeling of survey, public health, and environmental data. She is a
co-author of OpenIntro Statistics and a contributing member of the OpenIntro project, whose mission is to make educational products that are open-licensed, transparent, and help lower barriers to
education. She is also a co-editor of the Citizen Statistician blog and a contributor to the Taking a Chance in the Classroom column in Chance Magazine.
Specification: Inferential Statistics
Duration 25 hours
Year 2016
Level Beginner
Certificate Yes
Quizzes No
53 reviews for Inferential Statistics
Show all Most Helpful Highest Rating Lowest Rating
Amit C –
The course is very well explained I had to refer other materials for ANOVA technique to understand it better hence that part can be either improved OR more reference material be provided
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priyesh s –
This course is super and explained so well by the professor. I would recommend this course to anyone who has interest in data science
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Amarendra S –
Had a great learning experience with in depth knowledge of statistics, inference and hypothesis. Structure of the course helped me grasp things in an organized way. The use of real time data to
explain concepts had a great impact in making things easier to understand and relate to things around us.
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Richard M –
Generally a great course, but would benefit from a better explanation at times of how to use R effectively.
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Chen N –
Much better than the course offered by John Hopkins University on the same subject. Concepts are clearly explained with detailed examples. Nice course to solidify your statistics skills. And BTW,
really cute professor 🙂
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Akther H –
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Galin D –
Very relevant to modern day needs of a data scientist/statistician. Easy to understand as a relative beginner.
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Afzal A –
Expertly designed course, Useful.
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Lalu P L –
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Jacob T –
The best online course I have taken so far. It teaches you all the statistical methods you need to do for inference. The lessons are well taught and organized in a way where each lesson builds
off the previous. The final project is also a great way to put everything you learned throughout together.
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Rui Z –
Professor Mine is terrific. I’m sure she has a great depth of knowledge and grateful that she’s able to deliver her knowledge out to listeners. She uses meaningful examples all along the course,
no dry pure mathematical cases at all. That helps a ton to digest concepts. And she constantly repeat some core concepts and how to interpret a statistic right. I didn’t realize how important
this was until I was challenged with questions, then I came back and hear again her interpretation, and the whole thing became clearer. She’s one of the best professors I’ve ever listened to, and
I’ve been through grad school, met so many professors. The current mentor Rolf was great at supporting. He answers a lot of questions in the forum. He’s very responsible and supportive. So if
you’re considering on taking this course, take it now as mentor will change! I haven’t finished the course yet, but the enrollment rate seems to be quite decent, so I wouldn’t expect it to take
too long to get final project reviewed and get certificate. I assume this is an important issue for any course takers. The only downside is that there could be more R code teaching, especially on
complicated simulations. That way it may be more friendly to R beginners. I know it’s important to do research ourselves for codes, but beginners could lack of proper terminology or vision by
nature to do the research on Google. Especially when I’m physically in the Main Land of China, where it takes some efforts to even get on Google, so doing code research took a lot of my time and
was a little frustrating towards the end. But again, the overall course and support are great! If this is not a 5 start course, I can hardly give out my highest mark to any other courses. It
helped me to understand inferencial statistics, practice R, and think more like a statistician.
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Ondina F P –
Very good explained course, with lot of useful exercise, so you can be sure to understand the theory. Th practical examples in R are well designed and explained. This is definitely a must for
someone interested in statistics, with beginning concepts that you need to keep in mind for further coursers. The teaches is also excellent, explanation and examples are very good. Recommended!
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Natalie R –
Well–taught, but they need to provide more resources to help people learn R. R is not a user–friendly app and I needed to google how to do a lot of the things they’re asking us to do. Needless to
say, I can google how to work in R on my own without paying Coursera a fee.
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Amruta G –
It’s a great course! Helped me identify and clarify lots of concepts which I had understood just halfway in class.
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Diego R G –
A very good introduction to the fundamentals of inference and NHST. It’s very important that you do all excercises and readings or you will not learn as much. Also, the course won’t provide a lot
of information on how to use R, but if you spend a good amount of time on your project and make sure that it’s good you will learn enough. I had to review a lot of R projects that were not very
good, which suggests that some students aren’t learning what they should. If you want to learn statistics or have limited knowledge on the topic, and also want to learn a bit about how to use R,
take the course. If you already know statistics and you only want to learn R, then this might not be the course for you, as the emphasis is on statistics per se.
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Ruben D S P –
great class, I improved many Statistics skills and learned R at the same time
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schlies –
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Evren O –
At times it feels lazy how it is put together. The examples are confusing (rather than clarifying) and there is close to no teaching of R, but the assignments are meant to be done on it. In fact
in the forums it is endorsed by mentors to learn R somewhere else. Likewise, I saw one comment where the student mentioned how they got confused by a core concept (p–value) and could finally wrap
their head around it by watching a Khan Academy video. And sadly, this was also endorsed by a mentor. Overall, I found the effort put into this course insufficient for people who are new to
Statistics or R. Therefore, the name of the entire specialization becomes misleading as it suggests that we were going to be taught how to use R in statistics. I had high hopes for this course
but sadly I will abandon it and spend my money on an alternative course/specialization.
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Heungbak C –
Very useful and meaningful lectures! I learned many things from this course. Thank you.
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Nandkishore –
seeking statistics inference through example in very convenient and easier way
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Nandkishore S –
seeking statistics inference through example in very convenient and easier way
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Robin M –
Great course, more difficult than the first module. The code was super useful to learn.
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David B J –
Very nice job of explaining the material. I love the diverse set of examples used in the lectures and labs.
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Harkeerat S T –
Very rigorous coursework. Loved the material.
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Daniel H –
An overview of inference, light on the math, light on the theory, and with an unfortunate failure to reinforce what may be the most important part of practice: what should be done when conditions
for a particular method are not met. When you teach students how to evaluate the conditions required for certain methods, but then walk through those methods even when the conditions aren’t met,
you reinforce poor practice. If you want to use an example where the conditions aren’t met, STOP once you find out the conditions aren’t met. STOP and REINFORCE the fact that you cannot use a
method without meeting conditions. It is not a valuable exercise to walk through the plug and chug calculations anyway. STOP, discuss why you can’t proceed, and then move on to another example if
you want to give your students an opportunity to practice taking the method through to its conclusion.
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Mit P –
Great learning experience. Very well crafted course. Thank you Dr. Rundel and the entire team of instructors!
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gerardo r g –
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Eduardo M –
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farzad s –
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Tran T H –
It is very helpful to me.
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dumessi –
It is a great course, while some underlying logics are not clearly explained. And the quiz has some unexplained context, which is confused.
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fatima s –
Thank you Dr. Mine Cetinkaya–Rundel, you are the best teacher I ever had.
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Veliko D –
Amazing course. Perfect balance between theory and practice!
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Parab N S –
An excellent course by Professor Rundel on Inferential Statistics.
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Majeed K –
An excellent course. Thank you Ms. Rundel
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Charlotte C –
This is the second in the series with professor Cetinkaya–Rundel. She explains everything very well and makes the subject fascinating.
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Jingyi Y –
No tutor answering questions in the discussion platform.
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Hao C –
Teaching: I really like the clear and concise teaching style of lecturer and the wide range of simple real–life example used to explain the course content. I’m a social science student. Although
I’ve studied quantitative research methods before, this course gives me some new insights into inferential statistics. I think I will never forget the statistical meaning of p–value after this
course! Course Structure: The course structure is well organized with clear focus in each week. The first and second weeks are easy to follow, but the third and fourth weeks are more challenging.
Textbook: The textbook used in this course is a good supplementary material, although it is not necessary to read the textbook. Course videos have already explained everything that we need to
know at intro level. However, it is worth reading the textbook for the third and fourth weeks. Assessment: The assessment of quiz in each week is relatively easy. The exploratory data analysis
required in peer–reviewed assignment is slightly challenging, because it might be hard for beginners to touch every required point.
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Giulia T –
Nice follow up from the previous course in the specialisation. The teacher is clear and the main concepts are reminded throughout the course and explained in good depth
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Nikhil K –
This course was simply amazing.
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Bibek D –
very informative course which was very easy to learn…..
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Soumya A –
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Aaron M –
A good course for learning statistical inference, though I found that more than a week per module was required to really absorb the content.
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Topon C R –
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Ghada S –
I think it is a little bit difficult for someone who knows nothing about probability or R.
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Amy W –
The course is well designed, and the examples given in each lesson are informative and interesting. For the final project, I wanted to group some categories from one variable together in a new
variable, but I did not have the code I needed to do it. It would have been very helpful to have that information in one of the labs prior to doing the final project.
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Volodymyr F –
Excellent course! I enjoy the whole Statistics with R specialization.
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Nimish B –
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Cynthia J J –
This is a wonderful course with a very good instructor. Her explanations and observations are clear, concise and on point. I am so glad I am taking this course because now the mystery of
statistics is over for me. I finally understand this logic.
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Daniel G P C –
It was good I learned a lot.
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John C –
Excellent course to learn about Inferential Statistics and R!
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Eduardo B D S –
Great course. I think questions could be harder, but overall I learned a lot, especially with the final project.
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Bruno A –
Nice refresher course on inference testing, with a broad coverage of the types of variables / analyses. Not heavy at all on the math side. Students have to find their tips for R coding on the
Internet for the most part. It is a good way to learn! But we could use more standard tips from the course itself, especially on EDA.
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Special thanks to Alistair Stewart, Oana Ciobotaru and Sergey Vasilyev (authors of the “Accountable Light Client Systems for PoS Blockchains” paper) for the helpful discussions and feedback that were
instrumental in making this protocol possible.
We present a protocol for efficiently verifying the Ethereum Beacon chain's Casper FFG consensus proofs using a SNARK-based approach. With this scheme, computationally constrained environments, such
as on-chain or off-chain consensus clients, can securely follow the Casper FFG protocol and benefit from the crypto-economic security provided by the over 17 million ETH ($34 billion at the time of
writing) staked on the Beacon chain. This protocol offers full node-level security that is orders of magnitude more secure than the sync committee, and is fully Byzantine fault-tolerant.
The sync committee was introduced to the beacon chain in the Altair hard-fork and it consists of a randomly selected subset of 512 validators from the full validator set. The motivation for this
committee was consensus proofs that could be verified cheaply. Unfortunately, this protocol introduces new security assumptions that are completely orthogonal to the security of the beacon chain.
More specifically, it has much lower crypto-economic security, as well as a lack of slashing for byzantine behavior, which is critical to the safety of POS consensus.
Therefore, bridges and off-chain consensus clients that rely on the beacon chain consensus proofs must trust that the sync committee will not collude to perform eclipse or data withholding attacks,
even when there are no consequences for such actions. We find this blind-faith security model to be completely unacceptable. Consequently, we have opted for the more ambitious approach of directly
verifying the Casper FFG consensus proofs.
We let $e$ be a bilinear pairing function such that $e : \mathbb{G}_1 \times \mathbb{G}_2 \rightarrow \mathbb{G}_T$. All groups have some prime-order $p$. Let $\textmd{g}_1$ and $\textmd{g}_2$ be the
generators for $\mathbb{G}_1$ and $\mathbb{G}_2$ respectively.
Next we define the hash function $H_1: \mathcal{M} \rightarrow \mathbb{G}_1$. This function simply takes an arbitrary length message and maps it to an element of the $\mathbb{G}_1$ group.
BLS Signatures
BLS signatures$^{[1]}$ enable consensus proofs that are very efficient to verify as it supports both public key and signature aggregation. So a verifier only needs to verify a single aggregate
signature rather than $n$ signatures.
$KeyGen():$ Choose a random $s \leftarrow \mathbb{F}_p$ and output $pk = \textmd{g}_2^s$ and $sk = s$
$Sign(sk, m):$ outputs $\sigma = H_1(m)^{sk}$. This signature is a single group element in $\mathbb{G}_1$.
$AggregateSignature(\sigma_1, \dots, \sigma_n) :$ This reduces a set of signatures $\sigma_1, \dots\sigma_n$ to a single group element $\tilde\sigma = \sum^{n-1}_{i=0}\sigma_i$. Outputs $\tilde\
sigma$ .
$AggregateKeys(pk_i, \dots, pk_n) :$ This reduces a set of public keys $pk_1, \dots, pk_n$ to a single group element $apk = \sum^{n-1}_{i=0}pk_i$. Outputs $apk$.
$Verify(pk, m, \sigma):$ Checks the equality of the pairing:
$e(H_1(m)^{sk}, \textmd{g}_2) = e(H_1(m), pk)$
This works because our pairing is bilinear:
$e(H_1(m), \textmd{g}_2)^{sk} = e(H_1(m), \textmd{g}_2)^{sk}$
This also extends to aggregate signatures:
$\begin{equation} e(\textmd{g}_1, \tilde\sigma ) = e(apk, H_1(m)) \tag{1} \end{equation}$
Homomorphic KZG Commitments
We’ve previously reviewed the KZG commitment scheme here. One great feature of KZG commitments is that they are homomorphic. What this means is that we can update the values in a commitment to some
polynomial without needing the full polynomial. Recall that a KZG commitment $C$ to any set $V$ is of the form:
$\textmd{g}^{\phi(s)} = \sum^{n}_{i=0} (\textmd{g}^{s^i})^{\phi_i}$
Where $s$ is the secret value, $n = |V| - 1$, and $\phi(x)$ is the polynomial that interpolates all the coordinates $(x_i, v_i)$ derived from the Lagrange basis:
$\begin{split} \phi(x) &= \sum_{i \in [0, n)}^n v_i \cdot \mathcal{L}_i(x)\\ &= \sum_{i \in [0, n)}^n v_i \cdot (\prod\limits_{j \space = \space 1, \space \\ { i \spacee \space j}}^{n} (\frac{x-x_j}
{x_i - x_j})) \end{split}$
Notice that we can rewrite our commitment $C$ as:
$\textmd{g}^{\phi(s)} = \sum_{i \in [0, n)} (\textmd{g}^{\mathcal{L}_i(s)})^{v_i}$
So that updating a value in this commitment from $v_i \rightarrow v_i^\prime$ can be seen as
$\begin{equation} \textmd{g}^{\phi ^\prime (s)} = \textmd{g}^{\phi(s)} + (\textmd{g}^{\mathcal{L}_i(s)})^{\delta_i} \tag{2} \end{equation}$
where $\delta_i$ is given as:
$\delta_i = v^\prime_i - v_i\\$
This works because:
$\begin{split} \phi^\prime(x_i) &= \phi(x_i) + \mathcal{L}_i(x_i) \cdot \delta_i \\ &= v_i + \delta_i \\ &= v_i + v^\prime_i - v_i\\ &= v^\prime_i \end{split}$
Unfortunately for the verifier, naively using equation $(2)$ to update it’s commitment requires computing the Lagrange base $\mathcal{L}_i(x)$ which has a runtime complexity of $O(2(deg(\phi)-1))$.
This complexity comes from evaluating the terms for both the numerator and denominator. An optimization that can be made here is to have the prover compute and provide the value $\textmd{g}^{\mathcal
{L}_i(s)}$ instead, which the verifier can use to compute the update. But how can the verifier trust the correctness of this value? i.e $\mathcal{L}_i(x_i) = 1$. This is where KZG proofs come in.
First we define the polynomial $L(x)$ as the sum of all Lagrange bases in $\phi(x)$:
$L(x) = \sum^{n-1}_{i = 0} \mathcal{L}_i(x)$
This allows us create a KZG commitment to this polynomial $\textmd{g}^{L(s)}$ as part of the KZG set up ceremony. Since we know that $L(x_i) - \mathcal{L}_i(x_i) = 0$, thus the prover can can compute
a KZG proof for the $x_i$-th coordinate using the quotient.
$\psi(x) = \frac{L(x) - \mathcal{L}_i(x_i)}{(x - x_i)}$
so that the verifier can verify $\textmd{g}^{\mathcal{L}_i(s)}$ by using the pairing check:
$e(\frac{\textmd{g}^{L(s)}}{\textmd{g}^{\mathcal{L}_i(s)}}, \textmd{g}) = e(\textmd{g}^{\psi(s)}, \frac{\textmd{g}^s}{\textmd{g}^{x_i}})$
But what if the prover wants to update multiple points in the commitment? Then they’ll have to submit the terms $(\textmd{g}^{\mathcal{L}_i(s)}, \textmd{g}^{\delta_i}) \space \forall i \in I$ where
$I$ is the set of all points to be updated. So updating our commitments becomes
$\begin{equation} \textmd{g}^{\phi ^\prime (s)} = \textmd{g}^{\phi(s)} + \sum^{|I|}_{i \in I}(\textmd{g}^{\mathcal{L}_i(s)})^{\delta_i} \tag{3} \end{equation}$
What about verifying these terms? This is possible with KZG multi-proofs$^{[3]}$. Dankrad's article on KZG commitments outlines that to verify batch KZG proofs, the verifier needs to compute some
Lagrange bases themselves. However, the complexity for the bases is now reduced to $O(2(I - 1))$. In our case, the prover has already supplied those Lagrange bases. Let's define $I(x)$ as follows:
$I(x) = \sum^{|I|}_{i \in I} \mathcal{L}_i(x) \tag{4}$
Using the polynomial in $(4)$, We can come to the following conclusions:
\begin{align*} \forall i \in I, \space L(x_i) - I(x_i) &= 0 \\ L(x) - I(x) &= \psi(x) \cdot \prod^{|I|}_{i \in I} (x - x_i) \\ L(x) - I(x) &= \psi(x) \cdot Z_I(x) \end{align*}
Since the prover provides the individual terms $\textmd{g}^{\mathcal{L}_i(s)} \space \forall i \in I$, the verifier can use them to compute:
$\textmd{g}^{I(s)} = \sum^{|I|}_{i \in I}\textmd{g}^{\mathcal{L}_i(s)}$
Finally the prover provides the KZG multi-proof $\textmd{g}^{\psi(s)}$ which is still a single group element, but allows us to verify multiple points using the pairing check:
$e(\frac{\textmd{g}^{L(s)}}{\textmd{g}^{I(s)}}, \textmd{g}) = e(\textmd{g}^{\psi(s)}, \textmd{g}^{Z_I(s)}) \tag{5}$
Casper FFG
The Casper FFG consensus protocol defines the finality rule for the Ethereum beacon chain$^{[4]}$. It does this by introducing what it refers to as "source" and "target" checkpoints. These
checkpoints are 32 slots apart and correspond to the epoch boundaries of the beacon chain. This means that Casper FFG finalizes whole epochs, rather than arbitrary block sequences.
An epoch according to the protocol goes through three stages: unjustified, justified, and finalized. The genesis block is a special case, as it is already finalized by the protocol rules. Moreover,
an epoch (source) can only be considered finalized if there exists a direct descendant epoch (target) that has been justified by a supermajority of the authority set.
Blocks b1 & b2 are finalized, while b3 is only justified.
As stated in my article on the sync committee, there are simply too many authorities in the Ethereum beacon chain (Currently 560k and rising). Passing around a epoch checkpoint to sign would degrade
the network. To solve this issue, the authorities are split up into committees with a maximum count of 2048 per committee. These attestation committees produce signed Casper FFG votes. The Casper FFG
protocol uses BLS signature which allows signatures from individual committee members to be aggregated into a single signature per committee.
Attestation messages are published in the beacon chain blocks and contains: Casper FFG votes, a bitlist of the validators in the committee who signed the attestation and a BLS signature over the
AttestationData. The BLS signatures for the attestation messages use the $\mathbb{G}_1$ group for public keys, while it’s signatures are in the $\mathbb{G}_2$ group. A consensus client observing the
attestation messages can conclude that some epoch is justified (and it’s parent finalized) if they collect enough messages from a supermajority of the validator set confirming the epoch.
struct Attestation {
aggregation_bits: Bitlist<MAX_VALIDATORS_PER_COMMITTEE>
data: AttestationData
signature: BlsSignature
struct AttestationData {
slot: u64
index: u64
// LMD GHOST vote
beacon_block_root: H256
// FFG vote
source: Checkpoint
target: Checkpoint
Validators can join the validator set by locking up 32 ETH. The beacon chain adds their BLS public key as well as other protocol metadata (such as their activation_epoch, exit_epoch and a boolean
flag that tracks if they’ve been slashed ) to the “validator registry” an ssz list object on the [BeaconState](https://github.com/ethereum/consensus-specs/blob/dev/specs/phase0/beacon-chain.md#
beaconstate) . Once added to the validator set, their initial activation_epoch and exit_epoch is set to a constant FAR_FUTURE_EPOCH $(2^{64}-1)$ [source]. This triggers the epoch transition function
to schedule them for activation for the next epoch once the current epoch ends. [source]. It's worth noting that the beacon chain does NOT remove any validators from its registry. Instead, it updates
their activation_epoch, exit_epoch, or slashed values.
If a validator is found to have either proposed two competing beacon blocks for the same height [source] or signed an attestation that violates casper FFG's rules [source]. Then their
validator.slashed will be updated to true immediately, meaning they can no longer propose blocks or sign attestations. Validators may choose to voluntarily exit the active set after some minimum
period, after which their exit_epoch will be changed from FAR_FUTURE_EPOCH to some near future epoch. [source]. Hence, all inactive validators will satisfy the condition exit_epoch < current_epoch <
activation_epoch or slashed = true.
It is critical to note that both the BeaconState and the "validator registry" list are SSZ objects that can be merkleized. As a result, it is possible to obtain SSZ merkle multi-proofs of validator
state changes, such as deposits (new deposits), exits, and slashings, which can be verified against the BeaconState root. This will be important later on.
Aggregate Public Key Proofs
The Accountable Light Client Systems for PoS Blockchains$^{[5]}$ paper by Web3 Foundation researchers presents a SNARK that can verify whether the constituent public keys in an aggregate BLS public
key exist as a subset of a list of potential signers. Using this SNARK the verifier only needs to maintain a KZG commitment to the BLS public keys of all potential signers.
More formally, Given a set of potential signers $\{pk_i\} \forall i \in$ $T$. The verifier holds a commitment $C$ to the list of public keys in $T$. The prover can then send a bitlist $b$ which
represents a subset $S$ of the signers in $T$, an aggregate BLS signature $\sigma$, aggregate public key $apk = \sum_{i = 0}^{|S|} pk_i$ and a succint proof $\pi$ that $apk = \sum_{i = 0}^{|T|} b_i \
cdot pk_i$.
After verifying the aggregated public key SNARK proof $\pi$, the verifier can simply perform the naive aggregate BLS signature verification. This SNARK construction removes the requirement for the
verifier to know the individual public keys in $T$ making it perfect for truly light clients.
The SNARK circuit itself simply performs elliptic curve affine additions of the BLS public keys and constrains these additions using a custom PLONK gate.
The SNARK requires a pair of pairing-friendly elliptic curves: one for the BLS signature (the inner curve) and one for the SNARK itself (the outer curve). The paper recommends using the curves
BLS12-377 and BW6-761. However, for the BLS12-381 keys used in the Casper FFG protocol, we must use the BW6-767 curve for the outer curve to avoid non-native field arithmetic that would significantly
increase the number of constraints and, consequently, proving times. The protocol for aggregated public key proofs is formally defined below:
$APK.Setup(t, s):$ This outputs the proving and verification keys $\langle srs_{pk}, srs_{vk} \rangle$ using the powers of $s$ and a SNARK preprocessing algorithm, which can be used to commit and
prove a maximum number of $t$ signers.
$APK.Commit(srs_{pk}, T):$ Given a set of public keys $T = \{pk_i\}_{i = 1}^{t}$, outputs a commitment to the public keys $C$.
$APK.Prove(srs_{pk}, C, \{pk_i\}^{|S|}_{i = 1}, b):$ Computes the SNARK proof for the aggregation of the given public keys and a bitlist $b = \{bit_i\}^t_{i = 1}$ that indicates their positions in
the original set $T$. Outputs $\pi_{apk}$ .
$APK.Verify(srs_{vk}, C, apk, \pi_{apk}, b):$ Given the commitment, aggregated public key, SNARK proof and bitlist. This verifies that $apk = \sum_{i = 0}^{|S|} b_i \cdot pk_i$. Outputs $1/0$.
With the preliminaries out of the way, let's take a look the zkCasper protocol. Our approach is to begin by establishing a trusted commitment to the list of all validators in the Beacon chain.
However, the validators in the Beacon chain will not sign this commitment as the statuses of the individual validators in the set changes. Instead, they sign the block roots of epoch boundaries.
Fortunately, these block roots contain a merkle commitment to all the validators in the validator registry. Therefore, the verifier can leverage the homomorphic property of KZG commitments in order
to update their commitment after verifying the merkle proofs of the validator set changes. In this way, the verifier can securely track all validator set changes.
Armed with this commitment, the verifier only needs to know the aggregate of public keys that signed a committee's attestation messages. Using the apk SNARK, the verifier can confirm that the
aggregate of these aggregate public keys, along with a bitlist and a SNARK proof, corresponds to the commitment it has to the public keys of the full validator set. This enables the verifier verify
as many attestation messages at once in order to take advantage of the performance benefits of equation $(1)$.
An issue that arises is that the beacon chain uses a combination of Casper FFG and LMD Ghost$^{[6]}$. A consequence of this decision is that signed attestations do not need to reach super-majority
participation before they are published in beacon chain blocks. This means that there may be multiple signed attestation messages from a committee with overlapping participants. The aggregate of
these overlapping signers is unfortunately incompatible with the apk SNARK’s constraints. However this just means we cannot verify a supermajority of the beacon chain’s attestations in one go, we can
instead prove $n$ batches of attestations that have no overlapping committee signatures. Where $n$ is the maximum number of times a single committee produced an attestation.
The protocol is formally defined below:
$Setup(s, t, V):$ Performs the SNARK set up, $\langle srs_{pk}, srs_{vk} \rangle \rarr APK.Setup(t, s)$. Computes the commitment $C = APK.Commit(srs_{pk}, V)$ where $V$ is the set of all validator
public keys. Computes the update key $srs_{uk} = g^{L(s)}$. Outputs $\langle srs_{uk}, srs_{pk}, srs_{vk} \rangle$ .
$ProveAttestations(srs_{pk}, C, A):$ Given a set of $n$ Casper FFG committee attestations with non-overlapping signatures $A = (m_i, \tilde\sigma_i, (\{pk_j\}^{|P|}_{j=1})_i) \forall i \in [1, n)$
where $P$ is the set of participating public keys in each committee attestation. The prover computes the bitlist $b$ for the participating public keys of all the individual validators, then finally
they compute the apk proof for the aggregation of these public keys, and outputs $\langle A, \pi_{apk}, b \rangle$. Where:
$\pi_{apk} = APK.Prove(srs_{pk}, C, \sum^{|P| \cdot n}_{i=1} pk_i , b)$
$VerifyAttestations(srs_{vk}, C, A, S, \pi_{apk}, b):$ Where, $A = (m_i, \tilde\sigma_i, apk_i) \forall i \in [1, n)$, $m_i = \langle \mathcal{E}_s, \mathcal{E}_t \rangle$. First the verifier
computes $b = d \oplus b$, where $d$ is the bitlist of disabled validators, next they verify the apk proof for the participating public keys:
$APK.verify(srs_{vk}, C, \sum^n_{i=0} apk_i , \pi_{apk}, b) \in \{true, false\}$
finally they verify the BLS signatures for each committee’s attestations using equation $(1)$
$e(\textmd{g}_1, \tilde\sigma_i ) = e(apk_i, H_1(m_i))$
If all signature verifications pass, the verifier updates the bitlist for all the signers seen so far by computing $S = S \lor b$. Outputs $1/0$. A source epoch boundary $\mathcal{E}_s$ should be
considered final if $Hamming(S) \ge \frac{2}{3}(|C| - |d|) + 1$.
$ProveValidatorUpdates(C, I, srs_{uk}):$ The prover computes the merkle multi-proof $\pi_{merkle}$ for all validators $v_i \in I$ whose status (joined, exit_epoch, activated_epoch, slashed) have
changed with respect to the latest finalized epoch block root $\mathcal{E}_s$. Next they compute the KZG proof $\pi_{kzg} = g^{\psi(s)}$ for the points that pass through $I(x)$ defined in $(4)$.
Outputs $\langle I, \pi_{kzg}, \pi_{merkle} \rangle$.
$UpdateValidatorSet(srs_{uk}, C, I, \pi_{merkle}, \pi_{kzg}, \mathcal{E}_s) :$ First the verifier verifies the ssz merkle proof for the validator statuses via $ssz.Verify(\mathcal{E}_s, \pi_{merkle},
\{v_i \space \forall i \in I\})$, where $v_i$ represents the validator struct in the beacon state for the block root of the finalized epoch $\mathcal{E}_f$. If the Merkle verifications pass, the
verifier then proceeds to verify $I(x), \pi_{kzg}$ using the equation in $(5)$. If the KZG proof is valid, then the verifier can update their commitment $C$ with the set of all new validators in $I$
using equation $(3)$. Finally they compute $d = (d \lor r) \oplus a$ , where $r, a \sub I$, $r$ is the bitlist of all validators who have been disabled (exited, slashed) and $a$ is the bitlist of all
validators who have been activated. Outputs the updated commitment $C^{\prime}$.
$^{[1]}$ Dan Boneh; Ben Lynn & Hovav Shacham. "Short Signatures from the Weil Pairing".
$^{[2]}$ Dan Boneh, Manu Drijvers, and Gregory Neven. "Compact Multi-Signatures for Smaller Blockchains"
$^{[3]}$ Dankrad Feist, Kate Polynomial Commitments
$^{[4]}$ Vitalik Buterin and Virgil Griffith. "Casper the Friendly Finality Gadget"
$^{[5]}$ Oana Ciobotaru, Fatemeh Shirazi, Alistair Stewart , and Sergey Vasilyev. "Accountable Light Client Systems for PoS Blockchains"
$^{[6]}$ Combining Ghost and Casper | {"url":"https://docs.hyperbridge.network/protocol/consensus/casper-ffg","timestamp":"2024-11-14T07:51:50Z","content_type":"text/html","content_length":"441889","record_id":"<urn:uuid:23536f10-dfbc-427d-b345-973767740643>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00316.warc.gz"} |
Anyons in Quantum Many-Body Systems
In this work, we describe the characterization of fracton phases in terms of their gapped excitations. In particular, we describe fusion and statistical processes in Abelian fracton phases. As
compared to more conventional states, there are two key new features. First is the restricted mobility of excitations, which implies that statistical processes need not always take the form of
familiar braiding processes. The fusion theory we develop encodes the mobility of excitations, which allows us to use it as a starting point to describe statistical processes. Second, the number of
distinct excitation types in fracton phases in infinite, in contrast to conventional phases with intrinsic topological ordered (iTO). Moreover, if one considers excitations supported in a region with
linear size $L$, the number of excitation types supported in the region grows exponentially with $L$. This strongly suggests that, in order to get a manageable theory, we need to impose some
structure beyond what is present in the theory of conventional iTO phases. To build a theory that incorporates these features, we consider lattice translation symmetry. If we ignore translation
symmetry, the fusion of excitations in an Abelian fracton phase is described by an infinite Abelian group, whose elements correspond to distinct excitation types. Translation symmetry acts on this
Abelian group, giving it more structure and making it into a more manageable object to work with. Moreover, this action directly allows us to describe the mobility of excitations at the level of the
fusion theory, which then forms the basis for a description of statistical processes. | {"url":"https://www.pks.mpg.de/de/anyons-in-quantum-many-body-systems/poster-contributions","timestamp":"2024-11-09T19:32:39Z","content_type":"text/html","content_length":"136063","record_id":"<urn:uuid:820ca2d1-5255-4f8e-ae3e-2c8de202dafc>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00859.warc.gz"} |
get next enum - SourceTrail
Problem Statement
Given an enum constant, the objective is to find the next enum value in the sequence. If the given constant is the last one in the list, then the first enum value should be returned as the next
To solve this problem, let’s start by creating an enum class called DaysOfWeek as an example. We will then proceed to add a method called getNext() that returns the next enum value in the sequence.
Here’s a step-by-step walk-through of the code:
1. First, create an enum class called DaysOfWeek and define its enum constants.
public enum DaysOfWeek {
MONDAY, TUESDAY, WEDNESDAY, THURSDAY, FRIDAY, SATURDAY, SUNDAY;
2. Next, add the getNext() method inside the DaysOfWeek enum class. This method will return the next enum value based on the current enum value.
public enum DaysOfWeek {
MONDAY, TUESDAY, WEDNESDAY, THURSDAY, FRIDAY, SATURDAY, SUNDAY;
public DaysOfWeek getNext() {
if (this.ordinal() == DaysOfWeek.values().length - 1) {
return MONDAY;
return DaysOfWeek.values()[this.ordinal() + 1];
3. The getNext() method works as follows:
– We first check if the current enum value is the last one in the list (by comparing the ordinal value with the length of the values array minus 1).
– If the current enum value is the last one, we return the first enum value (i.e. MONDAY).
– Otherwise, we return the next enum value using the ordinal value to index into the values array.
We can now use the getNext() method to fetch the next enum value for any given enum constant.
Utilizing Libraries and Functions
Java provides several libraries and functions that come in handy when working with enums. Here are a couple of examples:
• Java’s EnumSet is a specialized Set implementation designed specifically for enums. It is very efficient and provides methods to perform bulk operations, such as complement, union, intersection,
and difference.
• EnumMap is another specialized collection designed specifically for enum keys. Much like EnumSet, it is highly efficient and offers significant performance improvements over a standard HashMap
with enum keys. EnumMap also maintains the natural order of its keys.
In conclusion, enums in Java offer a type-safe, efficient, and convenient way of working with related constants. The solution provided in this article demonstrates how to retrieve the next enum value
given a specific enum constant. Additionally, we explored some powerful Java libraries and functions that can help streamline our work with enums, such as EnumSet and EnumMap.
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Tutorial: Introduction to Deep Learning
March 31, 2023
Tutorial: Introduction to Deep Learning
This tutorial provides an introduction to deep learning algorithms and their applications in various fields. We will cover the fundamentals of deep learning, including its underlying workings, neural
network architectures, and popular frameworks used for implementation. Additionally, we will discuss some of the most common types of deep learning models and explore real-world applications of these
techniques to solve complex problems.
Deep learning is an essential tool for data science and machine learning, as it allows for the uncovering of hidden patterns in large datasets. Understanding the fundamentals of deep learning
algorithms enables the identification of appropriate problems that can be solved with deep learning, which can then be applied to your own projects or research.
Acquiring knowledge of deep learning can be incredibly beneficial for professionals. Not only can they use these skills to stay competitive and work more efficiently, but they can also leverage deep
learning to identify new opportunities and create innovative applications. With the rapid advancement of technology, it is becoming increasingly important for professionals to stay up-to-date with
emerging trends in order to stay ahead of the competition. Deep learning is an invaluable skill that can help professionals achieve this goal.
This tutorial will introduce you to the fundamentals of deep learning, including its underlying workings and neural network architectures. You will also learn about different types of deep learning
models and their applications in various fields. Additionally, you will gain hands-on experience building deep learning models using TensorFlow.
About this tutorial
This tutorial is aimed at anyone interested in understanding the fundamentals of deep learning algorithms and their applications. It is suitable for beginner to intermediate level readers, and no
prior experience with deep learning or data science is necessary.
What is Deep Learning?
Deep learning is a cutting-edge machine learning technique based on representation learning. This powerful approach enables machines to automatically learn high-level feature representations from
data. Consequently, deep learning models achieve state-of-the-art results on challenging tasks, such as image recognition and natural language processing.
Deep learning algorithms use an artificial neural network, a computing system that learns high-level features from data by increasing the depth (i.e., number of layers) in the network. Neural
networks are partially inspired by biological neural networks, where cells in most brains (including ours) connect and work together. Each of these cells in a neural network is called a neuron.
Shallow and Deep Neural Network
A neural network is comprised of the following components:
1. Input Layer: This is where the training observations are fed through the independent variables.
2. Hidden Layers: These are the intermediate layers between the input and output layers. This is where the neural network learns about the relationships and interactions of the variables fed in the
input layer.
3. Output Layer: This is the layer where the final output is extracted as a result of all the processing which takes place within the hidden layers.
4. Node: A node, also called a neuron, in a neural network is a computational unit that takes in one or more input values and produces an output value.
A shallow neural network is a neural network with a small number of layers, often comprised of just one or two hidden layers. Shallow neural networks are typically used for simple tasks, such as
regression or classification. A simple shallow neural network with one hidden layer is shown below. The two response variables x1 and x2 feed into the two nodes n1 and n2 of the single hidden layer,
which then generate the output.
In contrast to shallow neural networks, a deep (dense) neural network consist of multiple hidden layers. Each layer contains a set of neurons that learn to extract certain features from the data. The
output layer produces the final results of the network. The image below represents the basic architecture of a deep neural network with n-hidden layers.
The additional hidden layers in a deep neural network enable it to learn more complex patterns than a shallow neural network. Consequently, deep neural networks are more accurate but also more
computationally expensive to train than shallow neural networks. Therefore, deep neural networks are preferable for complex, real-time, real-world applications such as multivariate time series
forecasting, natural language processing, real-time forecasting, or predictive lead times.
How does Deep Learning Work?
At its simplest level, deep learning works by taking input data and feeding it into a network of artificial neurons. Each neuron takes the input from the previous layer of neurons and uses that
information to recognize patterns in the data. The neurons then weight the input data and make predictions about the output. The output can be a class or label, such as in computer vision, where you
might want to classify an image as a cat or dog.
Important Components of a Deep Neural Network:
1. Forward Propagation: In this process, input is passed forward from one layer of the network to the next until it passes through all layers and reaches the output.
2. Backpropagation: This is an iterative process that uses a chain rule to determine the contribution of each neuron to errors in the output. The error values are then propagated back through the
network, and the weights of each neuron are adjusted accordingly.
3. Optimization: This technique is used to reduce errors generated during backpropagation in a deep neural network. Various algorithms, such as gradient descent and stochastic gradient descent, can
be used to optimize the network.
4. Activation Functions: Activation functions are used to convert inputs into an output that can be recognized by the neural network. There are several types of activation functions, including
linear, sigmoid, tanh, and ReLu (Rectified Linear Units).
5. Loss Functions: These functions are used to measure how well a neural network has performed after backpropagation and optimization. Common loss functions include mean squared error (MSE) and
By combining all of these components, deep learning can take complex inputs and produce accurate predictions for a variety of tasks.
Deep Learning Algorithms
The three most popular deep learning algorithms are convolutional neural networks (CNNs), recurrent neural networks (RNNs), and long short-term memory networks (LSTMs). CNNs are used for image
recognition, object detection, and classification. RNNs are used for sequence modeling, such as language translation and text generation. LSTMs use a special type of memory cell that enables them to
remember longer sequences and are used for tasks such as recognizing handwriting and predicting stock prices.
Some less common, but still powerful deep learning algorithms include generative adversarial networks (GANs), autoencoders, reinforcement learning, deep belief networks (DBNs), and transfer learning.
• GANs can be used for image generation, text-to-image synthesis, and video colorization.
• Autoencoders are helpful for data compression and dimensionality reduction.
• Reinforcement learning is a type of machine learning in which agents learn to perform tasks by interacting with the environment.
• DBNs are primarily used for unsupervised feature learning.
• Transfer learning allows models trained on one problem to be reused for another.
With the ability to process large amounts of data and create accurate models, these deep learning algorithms are revolutionizing the way we use artificial intelligence.
Implementation in TensorFlow
It is not possible to cover all deep learning algorithms in a single tutorial, as that would require an entire book or set of books. However, we will provide an overview of the process by
implementing one of the popular deep neural networks in this tutorial: Convolutional Neural Networks (CNNs).
CNNs are a type of deep learning architecture that is particularly suitable for image processing tasks. They require large datasets to be trained on, and one of the most popular datasets is the MNIST
dataset. This dataset consists of a set of hand-drawn digits and is used as a benchmark for image recognition tasks.
Implementing a convolutional neural network (CNN) on the MNIST dataset has several advantages. The dataset is popular and easy to understand, making it an ideal starting point for those beginning
their journey into deep learning. Additionally, since the goal is to accurately classify images of handwritten digits, CNNs are a natural choice. In the following sections, we will provide a
step-by-step guide for implementing CNNs on the MNIST dataset using TensorFlow.
First, let's import the necessary libraries:
import tensorflow as tf
from tensorflow.keras.layers import Conv2D, MaxPooling2D, Flatten, Dense
from tensorflow.keras.models import Sequential
Next, we will load the MNIST dataset and normalize its values such that they fall between 0 and 1. Since pixel values range from 0 to 255, we can normalize our data by dividing our datasets by 255.0.
Dividing by 255.0 instead of 255 ensures our results are returned as decimal values and not integers.
(X_train, y_train), (X_test, y_test) = tf.keras.datasets.mnist.load_data()
X_train, X_test = X_train/255.0, X_test/255.0
We then reshape the input data into 4D arrays to feed into the CNN.
X_train = X_train.reshape(60000, 28, 28, 1)
X_test = X_test.reshape(10000, 28, 28, 1)
Now we will define the model architecture of our CNN. To do this, we will use the Sequential class from TensorFlow and add layers to our network.
We will add the layers to our model in the following order:
• The first layer is a convolutional layer, with 32 filters of size 3x3 each and an activation function of ReLU (Rectified Linear Unit). This layer takes as input the image data in the shape of
28x28 pixels with 1 color channel.
• The second layer is a max pooling layer, which reduces the number of parameters by taking the maximum value in each 2x2 pixel window.
• The third layer is a flattening layer, which converts the pooled image data into a single-dimensional vector.
• The fourth and fifth layers consist of dense layers with 128 and 10 neurons each. They use ReLU and softmax activation functions, respectively. The output of the last layer is the predicted label
for each image in the dataset.
model = tf.keras.Sequential()
model.add(tf.keras.layers.Conv2D(32, (3, 3), activation='relu', input_shape=(28, 28, 1)))
model.add(tf.keras.layers.MaxPooling2D((2, 2)))
model.add(tf.keras.layers.Dense(128, activation='relu'))
model.add(tf.keras.layers.Dense(10, activation='softmax'))
Now that the model is defined, we need to compile it by specifying our optimizer and loss function.
model.compile(optimizer='adam', loss='sparse_categorical_crossentropy', metrics=['accuracy'])
Next, let's train our model for two epochs. The number of epochs is generally kept on the higher side for better performance, but since it can be computationally intensive, we'll use two epochs for
this tutorial.
model.fit(X_train, y_train, epochs=2)
Epoch 1/2
1875/1875 [==============================] - 35s 18ms/step - loss: 0.1506 - accuracy: 0.9550
Epoch 2/2
1875/1875 [==============================] - 33s 18ms/step - loss: 0.0518 - accuracy: 0.9846
<keras.callbacks.History at 0x7f6c7d317760>
We can now evaluate the accuracy of our model on the test dataset.
test_loss, test_acc = model.evaluate(X_test, y_test)
print(f'Test accuracy: {test_acc}')
313/313 [==============================] - 2s 7ms/step - loss: 0.0472 - accuracy: 0.9833
Test accuracy: 0.983299970626831
After completing the training, we can use the model to make predictions on new, unseen data. We have successfully implemented a CNN on the MNIST dataset using TensorFlow and achieved a dependable
accuracy on unseen data.
This tutorial covered the basics of deep learning algorithms and their various components and their applications to various tasks. Additionally, it provides a step-by-step guide to implementing a
convolutional neural network (CNN) on the MNIST dataset using TensorFlow.
In conclusion, deep learning algorithms are revolutionizing the way computers learn. Understanding how to implement them is essential for anyone working in Artificial Intelligence or Machine
Learning. By mastering these skills, you can be at the forefront of developing complex and powerful models with a wide range of applications.
If you want to enhance your understanding of deep learning algorithms, Dataquest is the perfect place for you! Our comprehensive courses provide an in-depth exploration of the fundamentals and
applications of deep learning. Sign up for the Introduction to Deep Learning in TensorFlow course to develop a solid foundation in this exciting field. Our interactive platform and engaging content
will help you elevate your understanding of these complex topics to new heights. Sign up for Dataquest's courses today and become a master of deep learning algorithms!
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How to Insert Error Bars in Google Sheets (3 Practical Examples)
Suppose you are working on the projection of next year’s valuation of your company. You need to calculate this year’s data for this projection. But you cannot get accurate data with this calculation
as this is a projection and you will get the realized value at the end of next year. In that case, you can add error bars in your google sheets. So, your projection dataset will show the error
percentage. Here, we will learn how to insert error bars in google sheets.
Moreover, the overview of this article is shown above. You will learn more once you go through the total article.
A Sample of Practice Spreadsheet
You may copy the spreadsheet below and practice by yourself.
What Is Error Bar in Google Sheets
Sometimes we need to work on a dataset where we need to predict the data. So, keeping an uncertainty range or variability range is a smart move. If you make a chart from your dataset then add error
bars in the chart to keep the variability of the data to keep the errors of the prediction. For instance, you are working on next season’s profit projection. So here you don’t know the actual value.
But you can calculate the range based on this year’s profit. Once you convert your data into a bar chart you can insert error bar in the chart to add the uncertainty of the dataset.
Types of Error Bars in Google Sheets
There are 3 types of error bars in google sheets. You can insert error bars as per your requirement.
Percentage Error Bar: The percentage error bar is a process where you can insert the error percentage in the chart. Say, you want to keep the uncertainty of your value at 10%. Now you can insert 10%
error bars and the percentage will be added to the value of that particular range.
Constant Error Bar: The constant error bar is like a percentage error bar but here you need to insert the exact uncertainty value. For example, you want to keep a window open for 5 million on your
next year’s profit projection. So, you will insert 5 million error bars in your chart.
Standard Deviation Error Bar: This error bar is applied when you need to insert the standard deviation in the chart. Suppose you need to keep a window open exactly the same value as the standard
deviation. Then you insert the standard deviation error bar in the chart.
3 Practical Examples to Insert Error Bars in Google Sheets
The dataset below contains Company Name, Profit (in Million), and the months we will calculate the profit January, and February. The dataset represents the companies’ profits from the month of
January and February. We will add error bars of the profit margin using different processes. Suppose the profit margin is 6%. There we insert 6% error bars. Or if you convert this percentage into
numbers then this 6% is 2 million in total profit. So we can also add 2 million in the chart as error bars. So, let’s start.
1. Applying Percentage Error Bars
Now we will apply percentage error bars in the chart. As said earlier, if we want to keep a window open on percentage form then use percent error bars. Follow the steps below to execute this process.
📌 Steps:
• This dataset represents the profit for the last two months. First, we calculate the average value of the last two month’s profit using the AVERAGE function and then make a chart with the average
• Now, drag down the fill handle to copy the formula in the blank cells so that we get every average values to make a chart.
• Now, hold Ctrl key and select range B6:B10 and E6:E10 as X axis and Y axis.
• Here, select Insert >> Chart to get a Bar Chart as below.
• Here is the new chart with the required information.
• Consequently, Chart editor window will appear with the bar chart and select Customize >> Series from the window.
• Once you select Series from Customize group, Error bars option will appear.
• After that, select the Error bars option and a drop-down list will appear.
• Select Percent from the drop-down list and add the required percentage of error in your dataset.
• Here, add a 5% error in the chart. So, the Bar chart shows 5% error bars in the chart.
• Lastly, here is the final output.
Read More: How to Insert Formula in Google Sheets for Entire Column
2. Adding Constant Error Bars
Here, inserting constant error bars will show you a number that is the open window of the errors. Follow the steps below to complete the process.
📌 Steps:
• Here, you can add constant error bars instead of a percent from the Error bars window.
• Initially, create a bar chart using the process already shown before.
• Therefore, select Constant from the drop-down list of the Error bars and write 2 as we will keep 2 million errors in the error bar.
• In the end, the final output is below.
Read More: How to Insert a Legend in Google Sheets (With Easy Steps)
Similar Readings
3. Using Standard Deviation Error Bars
Another way is inserting the standard deviation error bars. Let’s complete the process following the steps below.
📌 Steps:
• In the beginning, create the bar chart with the steps already shown before.
• After that, select cell H4 and calculate the total standard deviation of the average value using the STDEV function of the dataset.
• Now, following the previous methods, select Standard Deviation From the drop-down bar of Error bars.
• Last, here is the final output below.
Read More: How to Insert a Drop-Down List in Google Sheets (2 Easy Ways)
How to Insert Individual Error Bars in Google Sheets
We can also insert customized error bars selecting multiple datasets at a time. Suppose, you need data projection for 3 months in a row but working on 3 different charts can take a lot of work. Now
you can get 3 different error percentages in a single chart using the below process. Now follow the steps below to insert the error bars for multiple datasets using the below dataset.
📌 Steps:
• In the beginning, create a bar chart following the steps earlier.
• Afterward, calculate the standard deviation of these two months as below.
• Then, drag down the fill handle to copy this formula in the blank cells so that we get standard deviation for individual companies.
• Moreover, select the range B5:B10, and C5:D10 in the Data range option to get the standard deviation of the profit in January, and February of these two months.
• Then, select Switch rows/columns and Use column B as headers so that the company name is in the X axis.
• Therefore, select Customize >> Series and select the company name from the drop-down menu. Here, first, select XYM Motors from the drop-down list and add error bars.
• Then, select Standard Deviation from Type group and write down 1.41 as the specific standard deviation of this company is 1.41.
• Lastly, enter the information of the other companies using the similar way already shown in XYM Motors. For instance, if you want to insert the standard deviation of Nike shoes then select Nike
Shoes from Series and insert the standard deviation of this company in the Error bars.
• Repeat the procedures for all other companies.
• In the end, the final output is below.
Things to Remember
• Select the dataset carefully. If your dataset has merged columns then the chart won’t show you the proper axis name. In that case, double-click on the name and manually add the actual name.
• We can add more than one dataset under one individual error bar.
In this article, we explained how to insert error bars in google sheets using different options. Hopefully, these processes will help you apply this method to your dataset. Please let us know in the
comment section if you have further queries or suggestions. You may also visit our OfficeWheel blog to explore more Google Sheets-related articles.
Related Articles
We will be happy to hear your thoughts
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ANONYMOUS wrote:
> Hi, > > If an ml file has the expression; "3 / 2". or for example: > > function divide(a, b) > x <- a/b > return x > > and then this is called as divide(3, 2). This will obviously be a problem when
translated to C, as 3 and 2 will be treated as integers not doubles. > > Simply adding (double)("Expression") is also not suitable as there may be many operations in an expression. > > Can we assume
that this will not occur, as part of the No invalid expressions condition provided by Chris? Or do we need to find a way to manage this.
You can't pass "integers" to a function; the arguments are always doubles, so even if you pass 2, because the argument was defined to be a double, it will become 2.0. Every time you see a constant
number, just put (double) in front of it or alternatively a decimal point and a zero, e.g. 2 becomes 2.0. I don't think you can ignore this issue as Chris explicitly uses constant integer values
expecting the program to interpret them as doubles. However, Amitava said you can make assumptions and list them into a report, so maybe you don't anymore. | {"url":"https://secure.csse.uwa.edu.au/run/help2002?p=np&opt=B411&year=2024","timestamp":"2024-11-03T09:48:39Z","content_type":"application/xhtml+xml","content_length":"21412","record_id":"<urn:uuid:ce9b0d6d-60f1-4dda-8afa-07523bdd45c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00502.warc.gz"} |