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Adding Radical Expressions
You can only add radicals that have the same radicand (the same expression inside the square root).
Here's how to add them:
1) Make sure the radicands are the same. If not, you cannot add them.
2) Add the outside numbers
3) Keep the radicand (the inside) the same.
Let's examine a few examples.
Since the radicands match, all we have to do is add 3+4 and keep the
There is an understood one in front of
Don't let negatives through you off. -2-8=-10 and
This one cannot be added since the radicands do not match. This is the final answer.
At first glance, it appears this one cannot be added. However,
Practice: Add the radicals.
Answers: 1)
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Ciclo de Palestras – Segundo Semestre de 2017
Ciclo de Palestras – Segundo Semestre de 2017
As palestras ocorrerem no Auditório do Laboratório de Sistemas Estocásticos (LSE), sala I-044b, as 15:30 h, a menos de algumas exceções devidamente indicadas.
Lista completa (palestras previstas para datas futuras podem sofrer alterações)
Atypical observations, called outliers, are common in many real datasets. Classical procedures do not contemplate their existence and therefore their application may lead to wrong conclusions. For
instance, the sample mean or the least-squares fit of a regression model, can be very adversely influenced by outliers, even by a single one. Robust methods arise to cope with these atypical
observations, mitigating their impact in the final analysis. The median is, probably, the most popular example of a robust procedure to summarize a univariate dataset. In this talk we discuss on the
use of the median as a systematic tool to construct robust procedures. Robust double protected estimators of the quantiles of the distribution of a random response in a missing at random setting are
discussed. Also prediction intervals in a sufficient dimension reduction context are presented.
In order to classify or divide in clusters a large set of data, statistical procedures sort the data by the size of some characteristic parameters. The classical mean/variance paradigm fits nicely
into a setup with Gaussian distributions, but the results can be sensitive on small deviations from the assumed model. In order to improve robustness, one can use alternative measures of centrality,
like medians in scalar data. In a multidimensional setup we can start with so called statistical depth functions and define a median set as a set of deepest points. The talk is focused on the best
known and the most popular Tukey’s depth (J. W. Tukey). Computation of depth is a hard problem, and although there exist exact algorithms, they do not work in really high dimensions. We will present
a new approximate algorithm that was used in solving a real problem with acoustic signals (joint work with Milica Bogićević, doctoral student of Applied Mathematics at University of Belgrade,
Serbia), and can be used in the Big Data context, with very large data sets and high dimensions. Incidentally, J. W. Tukey is one of a few pioneers in what is now meant by Data Science, and some
interesting details from his life will be mentioned in the talk.
Real-life problems are often modeled in an uncertain setting, to better reflect unknown phenomena, specific to the application. For problems when decisions are taken prior to observing the
realization of underlying random events, probabilistic constraints are an important modelling tool if reliability is a concern, in particular because they give a physical interpretation to risk.
Probabilistic constraints arise in many real-life problems, for example electricity network expansion, mineral blending, chemical engineering. Typically, these constraints are used when an inequality
system involves random parameters considered as critical for the decision making process.
A key concept to numerically dealing with probabilistic constraints is that of p-efficient points. By adopting a dual point of view, we develop a solution framework that includes and extends various
existing formulations. The unifying approach is built on the basis of a recent generation of bundle methods called with on-demand accuracy, characterized by its versatility and flexibility.
The methodology is illustrated on the optimal management of a chain of cascaded reservoirs coupled with turbines producing electrical energy.
Joint work with W. van Ackooij, V. Berge and W. de Oliveira
There has been considerable advances in understanding the properties of the LASSO procedure in sparse high-dimensional models. Most of the work is, however, limited to the independent and identically
distributed setting whereas most time series extensions consider independent and/or Gaussian innovations. Kock and Callot (2016, Journal of Econometrics) derived equation-wise oracle inequalities for
Gaussian vector autoregressive models. We extend their work to a broader set of innovation processes, by assuming that the error process is non-Gaussian and conditionally heteroskedastic. It is of
particular interest for financial risk modeling and covers several multivariate-GARCH specifications, such as the BEKK model, and other factor stochastic volatility specifications. We apply this
method to model and forecast large panels of daily realized volatilities.
Fixation in finite populations: discrete and continuous views
Max Oliveira de Souza (UFF)
We will present two different viewpoints on fixation: In the first part of the talk, we identify a general class of evolutive processes which share many features with the classical Moran and
Wright-Fisher (WF) processes—and include both of them. We also identify when a process in this class will have a strictly increasing fixation, and show that (WF) processes may have a decreasing
fixation, contrary to the Moran processes. We also show that WF is universal from the point of view of fixation: given almost any fixation vector, there is at least one WF process that realises it.
In the second part, we show how to construct continuous approximations of the fixation probability for birth-death processes that are valid beyond the weak-selection limit. Using this approximation
we give continuous restatements of two classical concepts in the discrete setting: (i) the ESS$_N$ and (ii) risk dominant strategies. In particular, we obtain an asymptotic definition in the
quasi-neutral regime of the celebrated 1/3 law. This is joint work with FACC Chalub.
Para alguns autores o século XX foi o fim das grandes certezas na História das Ciências, o fim da era da inocência. Na Economia, infelizmente, chegamos ao século XXI sem que as grandes certezas
fossem minimamente abaladas, ainda que empiricamente não tenham qualquer sustentação. A Crise de 2008/2009 levou a um intenso debate e economistas renomados foram chamados ao Congresso Americano para
explicar por que não foram capazes de prever a Grande Crise. As ciências sociais, em geral, carecem de um método comum e aceito por todos de refutação de teorias o que permite a coexistência de
teorias antagônicas para explicar os fenômenos. No entanto, da mesma forma que os argumentos teóricos, as implicações derivadas para política econômica são igualmente antagônicas. Na palestra, será
apresentado um referencial teórico e metodológico distinto do mainstream econômico e serão discutidas as diferentes implicações para política econômica, procurando trazer a análise para a atual
situação da economia brasileiras e as propostas alternativa existentes.
Existe praticamente um consenso de que a proporção de suicídios cometidos com armas de fogo em relação ao suicídio total é a melhor maneira de medir indiretamente a prevalência dessas armas. No
entanto, essa proxy não é precisa para localidades com baixa densidade populacional, tendo em conta que os suicídios são eventos raros. Para contornar esse problema, exploramos as características
socioeconômicas das vítimas de suicídio, de modo a propor uma nova proxy para prevalência de arma de fogo. Avaliamos o nosso indicador com microdados de suicídio do Ministério da Saúde (MS), entre
2000 e 2010.
We discuss an “operational”‘ approach to testing convex composite hypotheses when the underlying distributions are heavy-tailed. It relies upon Euclidean separation of convex sets and can be seen as
an extension of the approach to testing by convex optimization. In particular, we show how one can construct quasi-optimal testing procedures for families of distributions which are majorated, in a
certain precise sense, by a sub-spherical symmetric one and study the relationship between tests based on Euclidean separation and “potential-based tests.” We apply the promoted methodology in the
problem of sequential detection and illustrate its practical implementation in an application to sequential detection of changes in the input of a dynamic system. (Joint work with Anatoli Juditsky
and Arkadi Nemirovski)
Inferência em população finita é uma área da estatística notadamente reconhecida por sua importância prática e sua aparente dicotomia com outras áreas da inferência estatística. Contudo a inferência
em modelos de superpopulação possui objetivos bastante semelhantes à inferência estatística. Nesta palestra, os fundamentos teóricos da inferência em populações finitas baseada em modelos serão
abordados e suas diferenças com relação à inferência usual serão ressaltadas. O papel do plano amostral na inferência de parâmetros de interesse também será discutido. Finalmente, apresentaremos
algumas aplicações e desenvolvimentos recentes na área.
Brazil has a dual higher education market with the coexistence of public no-tuition institutions and private tuition-funded enterprises. About 3/4 of enrollments are in private higher education
institutions (HEI). Still heavily regulated, since the market liberalization in 1997, private institutions can merge and acquire (M&A) other private HEI. We provide an overview of the recent growth
of this sector and the significant role of mergers and aquisitions. We show that entry rates are small and fastest growing HEIs exploited M&A extensively. We evaluate the effects of mergers on
employment. Using Difference-in-Difference analysis, we estimate smaller faculty size and a proportional to a reduction in enrollment after a merger, on average.
The main goal of ERICA – Study of Cardiovascular Risks in Adolescents, was to estimate the prevalence of cardiovascular risk factors in adolescents from 12 to 17 years who attended public and private
schools in Brazilian cities with more than 100 thousands inhabitants. The study will also enable the investigation of several associations involving sociodemographic characteristics, cardiovascular
risk factors and metabolic changes. Besides the questionnaire filled out by 85,000 adolescents, weight, height, waist circumference, and blood pressure were measured. Also, in subsample of
approximately 42,000 adolescents studying in the morning term, blood was drawn for measuring lipids, glucose, insulin, and glycated hemoglobin.
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sormrz - Linux Manuals (3)
sormrz (3) - Linux Manuals
sormrz.f -
subroutine sormrz (SIDE, TRANS, M, N, K, L, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
Function/Subroutine Documentation
subroutine sormrz (characterSIDE, characterTRANS, integerM, integerN, integerK, integerL, real, dimension( lda, * )A, integerLDA, real, dimension( * )TAU, real, dimension( ldc, * )C, integerLDC,
real, dimension( * )WORK, integerLWORK, integerINFO)
SORMRZ overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q * C C * Q
TRANS = 'T': Q**T * C C * Q**T
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by STZRZF. Q is of order M if SIDE = 'L' and of order N
if SIDE = 'R'.
SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.
TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T.
M is INTEGER
The number of rows of the matrix C. M >= 0.
N is INTEGER
The number of columns of the matrix C. N >= 0.
K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
L is INTEGER
The number of columns of the matrix A containing
the meaningful part of the Householder reflectors.
If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.
A is REAL array, dimension
(LDA,M) if SIDE = 'L',
(LDA,N) if SIDE = 'R'
The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
STZRZF in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
TAU is REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by STZRZF.
C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The dimension of the array WORK.
If SIDE = 'L', LWORK >= max(1,N);
if SIDE = 'R', LWORK >= max(1,M).
For optimum performance LWORK >= N*NB if SIDE = 'L', and
LWORK >= M*NB if SIDE = 'R', where NB is the optimal
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
Further Details:
Definition at line 189 of file sormrz.f.
Generated automatically by Doxygen for LAPACK from the source code.
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12 Matching Annotations
1. Jun 2017
Who is Mistaken?Benjamin EysenbachMITbce@mit.eduCarl VondrickMITvondrick@mit.eduAntonio TorralbaMITtorralba@csail.mit.eduFigure 1: Can you determine who has a false belief about this
scene? In this paper, we study how to recognize when a person in a short sequence is mistaken. Above, the woman is mistaken about the chair being pulled away from her.TimeFigure 1:Can you
determine who believes something incorrectly in this scene?In this paper, we study how to recognizewhen a person in a scene is mistaken. Above, the woman is mistaken about the chair being
pulled away from her in the thirdframe, causing her to fall down. Thered arrowindicates false belief. We introduce a new dataset of abstract scenes to studywhen people have false beliefs.
We propose approaches to learn to recognizewhois mistaken andwhenthey are mistaken.AbstractRecognizing when people have false beliefs is crucial forunderstanding their actions. We
introduce the novel prob-lem of identifying when people in abstract scenes have in-correct beliefs. We present a dataset of scenes, each visuallydepicting an 8-frame story in which a
character has a mis-taken belief. We then create a representation of characters’beliefs for two tasks in human action understanding: pre-dicting who is mistaken, and when they are
mistaken. Ex-periments suggest that our method for identifying mistakencharacters performs better on these tasks than simple base-lines. Diagnostics on our model suggest it learns
importantcues for recognizing mistaken beliefs, such as gaze. We be-lieve models of people’s beliefs will have many
The analysis showsthat, although they are superficially similar, NCE is a general parameter estimation technique that is asymp-totically unbiased, while negative sampling is best
understood as a family of binary classification modelsthat are useful for learning word representations but not asa general-purpose estimator
I think NCE is slightly different from CE. Unfortunately, Chris sort of ignores Noah's work on CE in this explanation. Although, the connection between NCE and NS is nicely explained.
We present an extension to Jaynes’ maximum entropy principle that handles latent variables. Theprinciple oflatent maximum entropywe propose is different from both Jaynes’ maximum entropy
principleand maximum likelihood estimation, but often yields better estimates in the presence of hidden variablesand limited training data. We first show that solving for a latent maximum
entropy model poses a hardnonlinear constrained optimization problem in general. However, we then show that feasible solutions tothis problem can be obtained efficiently for the special
case of log-linear models—which forms the basisfor an efficient approximation to the latent maximum entropy principle. We derive an algorithm thatcombines expectation-maximization with
iterative scaling to produce feasible log-linear solutions. Thisalgorithm can be interpreted as an alternating minimization algorithm in the information divergence, andreveals an intimate
connection between the latent maximum entropy and maximum likelihood principles.To select a final model, we generate a series of feasible candidates, calculate the entropy of each,
andchoose the model that attains the highest entropy. Our experimental results show that estimation basedon the latent maximum entropy principle generally gives better results than
maximum likelihood whenestimating latent variable models on small observed data samples.
Towards intelligent negative sampling
Wang et al. (2002) discuss the latent maximumentropy principle. They advocate running EM manytimes and selecting the local maximum that maxi-mizes entropy. One might do the same for the
localmaxima of any CE objective, though theoretical andexperimental support for this idea remain for futurework.
Interesting proposal, quite similar to the neg. sampling with 'exploration / exploitation'.
Definitely, worth atleast a couple reads!
One can envision amixedobjective function that tries to fit the labeledexamples while discriminating unlabeled examplesfrom their neighborhoods.
Interesting - a mixed objective function -> this seems like a multi-task framework!
--> Re-read and understand
We have presentedcontrastive estimation, a newprobabilistic estimation criterion that forces a modelto explain why the given training data were betterthan bad data implied by the positive
This is again an interesting way to see it: "... forces a model to explain why the given training data were better than bad data implied by the positive examples."
Viewed as a CE method, this approach (though ef-fective when there are few hypotheses) seems mis-guided; the objective says to move mass to each ex-ample at the expense of all other
training examples
A very cool remark and makes sense!!
An alternative is to restrict theneighborhood to the set of observed training exam-ples rather than all possible examples (Riezler, 1999;Johnson et al., 1999; Riezler et al., 2000):
This equation is reminiscent of the equation proposed by Nickel et al., 2017 - the Poincare Embeddings paper. Especially, look for Negative Sampling.
Implementational issues in batch norm.
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In a standard linear model, we assume that $\mathbb{E}(Y\vert X=x)=\beta_0+\beta_1 x$. Alternatives can be considered, when the linear assumption is too strong.
A natural extension might be to assume some polynomial function,
$\mathbb{E}(Y\vert X=x)=\beta_0+\beta_1 x+\beta_2 x^2 +\cdots +\beta_k x^k$
Again, in the standard linear model approach (with a conditional normal distribution using the GLM terminology), parameters $\boldsymbol{\beta}=(\beta_0,\beta_1,\cdots,\beta_k)$ can be obtained using
least squares, where a regression of $Y$ on $\boldsymbol{X}=(1,X,X^2,\cdots,X^k)$ is considered.
Even if this polynomial model is not the real one, it might still be a good approximation for $\mathbb{E}(Y\vert X=x)=h(x)$. Actually, from Stone-Weierstrass theorem, if $h(\cdot)$ is continuous on
some interval, then there is a uniform approximation of $h(\cdot)$ by polynomial functions.
Just to illustrate, consider the following (simulated) dataset
xr = seq(0,n,by=.1)
yr = sin(xr/2)+rnorm(length(xr))/2
db = data.frame(x=xr,y=yr)
with the standard regression line
reg = lm(y ~ x,data=db)
Consider some polynomial regression. If the degree of the polynomial function is large enough, any kind of pattern can be obtained,
But if the degree is too large, then too many ‘oscillations’ are obtained,
and the estimation might be be seen as no longer robust: if we change one point, there might be important (local) changes
Actually, if our interest is to have locally a good approximation of $h(\cdot)$, why not use a local regression?
This can be done easily using a weighted regression, where, in the least square formulation, we consider
$\min\left\{ \sum_{i=1}^n \omega_i [Y_i-(\beta_0+\beta_1 X_i)]^2 \right\}$
(it is possible to consider weights in the GLM framework, but let’s keep that for another post). Two comments here:
• here I consider a linear model, but any polynomial model can be considered. Even a constant one. In that case, the optimization problem is
$\min\left\{ \sum_{i=1}^n \omega_i [Y_i-\beta_0]^2 \right\}$which can be solve explicitly, since
$\widehat{\beta}_0=\frac{\sum \omega_i Y_i}{\sum \omega_i}$
• so far, nothing was mentioned about the weights. The idea is simple, here: if you can a good prediction at point $x_0$, then $\omega_i$ should be proportional to some distance between $X_i$ and
$x_0$: if $X_i$ is too far from $x_0$, then it should not have to much influence on the prediction.
For instance, if we want to have a prediction at some point $x_0$, consider $\omega_i\propto \boldsymbol{1}(\vert X_i-x_0 \vert<1)$. With this model, we remove observations too far away,
Actually, here, it is the same as
A more general idea is to consider some kernel function $K(\cdot)$ that gives the shape of the weight function, and some bandwidth (usually denoted h) that gives the length of the neighborhood, so
$\omega_i = K\left(\frac{x_0-X_i}{b}\right)$
This is actually the so-called Nadaraya-Watson estimator of function $h(\cdot)$.
In the previous case, we did consider a uniform kernel $K(x)=\boldsymbol{1}(x\in[-1/2,+1/2])$, with bandwith $2$,
But using this weight function, with a strong discontinuity may not be the best idea… Why not a Gaussian kernel,
This can be done using
fitloc0 = function(x0){
On our dataset, we can plot
Here, we want a local regression at point 2. The horizonal line below is the regression (the size of the point is proportional to the wieght). The curve, in red, is the evolution of the local
Let us use an animation to visualize the construction of the curve. One can use
but for some reasons, I cannot install the package easily on Linux. And it is not a big deal. We can still use a loop to generate some graphs
for(i in 1:length(vx0)) graphloc(i)
and then, in a terminal, I simply use
convert -delay 25 /home/freak/local-reg-1*.png /home/freak/local-reg.gif
Of course, it is possible to consider a linear model, locally,
fitloc1 = function(x0){
or even a quadratic (local) regression,
fitloc2 = function(x0){
Of course, we can change the bandwidth
To conclude the technical part this post, observe that, in practise, we have to choose the shape of the weight function (the so-called kernel). But there are (simple) technique to select the
“optimal” bandwidth h. The idea of cross validation is to consider
$\min\left\{ \sum_{i=1}^n [Y_i-\widehat{Y}_i(b)]^2 \right\}$
where $\widehat{Y}_i(b)$ is the prediction obtained using a local regression technique, with bandwidth $b$. And to get a more accurate (and optimal) bandwith $\widehat{Y}_i(b)$ is obtained using a
model estimated on a sample where the ith observation was removed. But again, that is not the main point in this post, so let’s keep that for another one…
Perhaps we can try on some real data? Inspired from a great post on http://f.briatte.org/teaching/ida/092_smoothing.html, by François Briatte, consider the Global Episode Opinion Survey, from some TV
show, http://geos.tv/index.php/index?sid=189 , like Dexter.
file = "geos-tww.csv"
html = htmlParse("http://www.geos.tv/index.php/list?sid=189&collection=all")
html = xpathApply(html, "//table[@id='collectionTable']")[[1]]
data = readHTMLTable(html)
data = data[,-3]
Let us reshape the dataset,
data$no = 1:96
data$mu = as.numeric(substr(as.character(data$Mean), 0, 4))
data$se = sd(data$mu,na.rm=TRUE)/sqrt(as.numeric(as.character(data$Count)))
data$season = 1 + (data$no - 1)%/%12
data$season = factor(data$season)
data$no,data$mu+1.96*data$se,col="light blue")
As done by François, we compute some kind of standard error, just to reflect uncertainty. But we won’t really use it.
for(s in 1:8){reg=lm(mu~no,data=db,subset=season==s)
lines((s-1)*12+1:12,predict(reg)[1:12],col="red") }
Henre, we assume that all seasons should be considered as completely independent… which might not be a great assumption.
db = data
NW = ksmooth(db$no,db$mu,kernel = "normal",bandwidth=5)
We can try to look the curve with a larger bandwidth. The problem is that there is a missing value, at the end. If we (arbitrarily) fill it, we can run a kernel regression,
NW = ksmooth(db$no,db$mu,kernel = "normal",bandwidth=12)
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Techno Press
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Volume 5, Number 3, July 2020
In order to evaluate the performance of three heuristic optimization algorithms, namely, simulated annealing (SA), genetic algorithm (GA) and particle swarm optimization (PSO) for optimal stacking
sequence of laminated composite plates with respect to critical buckling load and non-dimensional natural frequencies, a multi-objective optimization procedure is developed using the weighted
summation method. Classical lamination theory and first order shear deformation theory are employed for critical buckling load and natural frequency computations respectively. The analytical critical
buckling load and finite element calculation schemes for natural frequencies are validated through the results obtained from literature. The comparative study takes into consideration solution and
computational time parameters of the three algorithms in the statistical evaluation scheme. The results indicate that particle swarm optimization (PSO) considerably outperforms the remaining two
methods for the special problem considered in the study.
Key Words
Benchmarking; Heuristic optimization algorithms; structural optimization; laminated composites; buckling load; fundamental frequencies
Mustafa Akbulut: TUBITAK Marmara Research Center, Kocaeli 41400, Turkey
Abdulhamit Sarac: TUBITAK National Metrology Institute, Kocaeli 41400, Turkey
Ahmet H. Ertas: Department of Mechanical Engineering, Faculty of Engineering & Natural Sciences, Bursa Technical University, Bursa 16330, Turkey
The main purpose to introduce model based controller in proposed control technique is to provide better and fast learning of the floating dynamics by means of fuzzy logic controller and also
cancelling effect of nonlinear terms of the system. An iterative adaptive dynamic programming algorithm is proposed to deal with the optimal trajectory-tracking control problems for autonomous
underwater vehicle (AUV). The optimal tracking control problem is converted into an optimal regulation problem by system transformation. Then the optimal regulation problem is solved by the policy
iteration adaptive dynamic programming algorithm. Finally, simulation example is given to show the performance of the iterative adaptive dynamic programming algorithm.
Key Words
complex systems; fuzzy models; delay-dependent robust stability criterion; parallel distributed compensation
Tim Chen: AI Lab, Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Safiullahand Khurram:Department of Computer Science, Kunduz University, Kunduz, Afghanistan
Joelli Zoungrana:School of Intelligent Science, Colinas University of Boé, Avenida 14 de Novembro Entrada do Bairro de Hafia Boe C.P. 1340 Bissau Guinea-Bissau
Lallit Pandey and J.C.Y. Chen: Department of Soil Science, Patuakhali Science and Technology University,
Dumki 8602, Patuakhali, Bangladesh
The reasonable layout of vacuum cleaner can effectively improve the collection efficiency of iron filings generated in the process of steel production. Therefore, in this study, the CFD-DEM coupling
model and two-fluid model are used to calculate the iron filings collection efficiency of vacuum cleaner with different inclination/cross-sectional area, pressure drop and inlet angle. The results
are as follows: The CFD-DEM coupling method can truly reflect the motion mode of iron filings in pneumatic conveying. Considering the instability and the decline of the growth rate of iron filings
collection efficiency caused by high pressure drop, the layout of 75° inclination is suggested, and the optimal pressure drop is 100Pa. The optimal simulation results based on two-fluid model show
that when the inlet angle and pressure drop are in the range of 45°~65° and 70Pa~100Pa, larger mass flow rate of iron filings can be obtained. It is hoped that the simulation results can offer some
suggestion to the layout of vacuum cleaner in the rolling mill.
Key Words
CFD-DEM; iron filings collection efficiency; optimization analysis; two-fluid model; mass flow rate
School of Civil Engineering, Wuhan University, Wuhan, 430072, Hubei, China
This article describes the technology for constructing of a multiply-connected three-dimensional area's finite element representation. Representation of finite-element configuration of an area is
described by a discrete set that consist of the number of nodes and elements of the finite-element grid, that are orderly set of nodes' coordinates and numbers of finite elements. Corresponding
theorems are given, to prove the correctness of the solution method. The adequacy of multiply-connected area topology's finite element model is shown. The merging of subareas is based on the
criterion of boundary nodes' coincidence by establishing a simple hierarchy of volumes, surfaces, lines and points. Renumbering nodes is carried out by the frontal method, where nodes located on the
outer edges of the structure are used as the initial front.
Key Words
modeling; finite element; grid; numbering; ordering; node; vertex; face; front; algorithm; connect; area
Askhad M. Polatov, Akhmat M. Ikramov: Department of Mathematics, National University of Uzbekistan,4, University Street,
Tashkent 100174, Uzbekistan
Daniyarbek D. Razmukhamedov: Turin Polytechnic University in Tashkent, 17, Kichik halqa yo'li Street, Tashkent 100000, Uzbekistan
In this paper, we develop models to design the airfoil using Multilayer Feed-forward Artificial Neural Network (MFANN) and Support Vector Regression model (SVR). The aerodynamic coefficients
corresponding to series of airfoil are stored in a database along with the airfoil coordinates. A neural network is created with aerodynamic coefficient as input to produce the airfoil coordinates as
output. The performance of the models have been evaluated. The results show that the SVR model yields the lowest prediction error.
Key Words
support vector regression model; neural networks; airfoil design; inverse design; backpropagation
K. Thinakaran: Computer Science Engeneering., Saveetha School of Engineering, SIMATS, Chennai 600 077 TN, India
R. Rajasekar: Aeronautical Engineering, MVJ Engineering College, Bangalore, India
K. Santhi: Sreenivasa Institute of Technology and Management Studies, Chittoor, India
M. Nalini: Computer Science Engeneering., Saveetha School of Engineering, SIMATS, Chennai 600 077 TN, India
In this paper is presented the solution method for three-dimensional problem of transversely isotropic body's elastoplastic deformation by the finite element method (FEM). The process of problem
solution consists of: determining the effective parameters of a transversely isotropic medium; construction of the finite element mesh of the body configuration, including the determination of the
local minimum value of the tape width of non-zero coefficients of equation systems by using of front method; constructing of the stiffness matrix coefficients and load vector node components of the
equation for an individual finite element's state according to the theory of small elastoplastic deformations for a transversely isotropic medium; the formation of a resolving symmetric-tape system
of equations by summing of all state equations coefficients summing of all finite elements; solution of the system of symmetric-tape equations systems by means of the square root method; calculation
of the body's elastoplastic stress-strain state by performing the iterative process of the initial stress method. For each problem solution stage, effective computational algorithms have been
developed that reduce computational operations number by modifying existing solution methods and taking into account the matrix coefficients structure. As an example it is given, the problem solution
of fibrous composite straining in the form of a rectangle with a system of circular holes.
Key Words
modeling; algorithm; grid; front; FEM; transversal isotropy; fiber; composite; hole; elastoplastic; strain, stress
Askhad M. Polatov, and Akhmat M. Ikramov: Department of Mathematics, National University of Uzbekistan, 4, University Street,Tashkent 100174, Uzbekistan
Abduvali A. Khaldjigitov : Samarkand branch of Tashkent University of Information Technologies, 47A,
Shohruh Mirzo Str., Samarkand, 140100, Uzbekistan
The enhancements of bioenergy production effectiveness require the comprehensively experimental study of several parameters affecting these bioprocesses. The interpretation of the obtained
experimental results and the estimation of optimum yield are extremely complicated such as misinterpreting the results of an experiment. The use of mathematical modeling and statistical experimental
designs can consistently supply the predictions of the potential yield and the identification of defining parameters and also the understanding of key relationships between factors and responses.
This paper summarizes several mathematical models used to achieve an adequate overall and maximal production yield and rate, to screen, to optimize, to identify, to describe and to provide useful
information for the effect of several factors on bioenergy production processes. The usefulness, the validity and, the feasibility of each strategy for studying and optimizing the bioenergy-producing
processes were discussed and confirmed by the good correlation between predicted and measured values.
Key Words
system modeling; experimental design methods; neural network design, identification, optimization, bioenergy
Hamza Akroum, and Abderrezak Aibeche: Laboratoire d'Automatique Appliquée, Université M'Hamed Bougara de Boumerdès, 1 Av. de l'indépendance 35000 Boumerdés, Algeria
Dahbia Akroum-Amrouche : Département de chimie, faculté des sciences,Université M'Hamed Bougara de Boumerdes Av. de l'independance 35000 Boumerdes, Algeria
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Lesson 18
Subtraction in Equivalent Expressions
Lesson Narrative
Previously in this unit, students solved equations of the form \(px+q=r\) and \(p(x+q)=r.\) Sometimes, work has to be done on a more complicated expression to get an equation into one of these forms.
And sometimes, it is desirable to rewrite an expression in an equivalent form to understand how the quantities it represents are related. This work has some pitfalls when the expression has negative
numbers or subtraction. For example, it is common for people to rewrite \(6x-5+2x\) as \(4x+5\) by reading “\(6x\) minus” and so subtracting the \(2x\) from the \(6x.\) Another example is rewriting
an expression like \(5x-2(x+3)\) as \(5x-2x+6.\) Students do not see expressions as complicated as these in this lesson (they are coming in the next few lessons), but this lesson is meant to
inoculate students against errors like these by reminding them that while subtraction is not commutative, addition is, and subtraction can be rewritten as adding the opposite. So in our example, \
(6x-5+2x\) can be rewritten \(6x + \text-5 + 2x\) and then rearranged \(6x+2x+\text-5.\) Likewise, \(5x-2(x+3)\) can be rewritten \(5x + \text-2(x+3)\) before distributing -2.
Learning Goals
Teacher Facing
• Explain (orally, in writing, and using other representations) how the distributive and commutative properties apply to expressions with negative coefficients.
• Justify (orally and in writing) whether expressions are equivalent, including rewriting subtraction as adding the opposite.
Student Facing
Let's find ways to work with subtraction in expressions.
Student Facing
• I can organize my work when I use the distributive property.
• I can re-write subtraction as adding the opposite and then rearrange terms in an expression.
CCSS Standards
Building On
Building Towards
Glossary Entries
• term
A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression \(5x + 18\) has two terms. The first
term is \(5x\) and the second term is 18.
Additional Resources
Google Slides For access, consult one of our IM Certified Partners.
PowerPoint Slides For access, consult one of our IM Certified Partners.
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How to Make an Expanding Spawn Area for Prefabs?
Hello, everyone. I am making a game where I want some prefab cubes to be continuously spawned around a bigger cube; kind of like the rings of a planet. The problem is: the bigger cube is continuously
growing in size, therefore more objects must be spawned around a larger area/circumference. How would I do this? (possibly an expanding ring shaped spawn area?) Thanks!
Define your spawn region in polar coordinates (radius, angle) around the center of your cube. Spherical coordinates if you’re working in 3D. You can easily expand the radius to be either the size of
the cube times some factor or plus some factor. You can spawn at random angles or according to some pattern.
You’ll need to include a conversion from polar/spherical coordinates to Cartesian coordinates, though.
Vector2 PolarToCart (float r, float theta) {
return new Vector2(r * Mathf.Cos(theta),
` r* Mathf.Sin(theta));
Vector3 SphericaltoCart (float r, float theta, float phi) {
return new Vector3(r * Mathf.Cos(theta)*Mathf.Sin(phi),
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Chapter 12: Sample Surveys Terms and Notes Sample: - hardwareChapter 12: Sample Surveys Terms and Notes Sample: - hardware
Chapter 12: Sample Surveys Terms and Notes Sample:
Chapter 12: Sample Surveys Terms and Notes Sample:
Chapter 12: Sample Surveys
Terms and Notes
Sample: a subset of a population that is examined in order to determine information about the entire
Types of Samples: Note that all statistical sampling approaches have the common goal that chance,
rather than human choice, is used to select the sample.
Cluster Sample: a sampling approach in which entire groups (i.e., clusters) are chosen at
random; a census is taken of each cluster. Each cluster should be representative of the entire
population. All clusters should be heterogeneous and similar to each other. The problem with
cluster samples is that the clusters are often not homogeneous and representative.
Convenience Sample: a sample of individuals who are conveniently available. Convenience
samples often fail to be representative.
Multistage Sample: a sampling approach that combines several sampling methods. Example:
stratify the country by geographic region; randomly select cities from each region; interview a
cluster of residents from each city. Care should be taken at each step not to introduce bias.
Simple Random Sample (SRS): a sample of size 𝑛 in which each set of 𝑛 elements has an equal
chance of being selected. This is the standard against which other sampling methods are
Stratified Random Sample: the population is divided into subgroups (i.e., strata), and random
samples are taken from each subgroup. This is better than a simple random sample if the strata
are relatively homogeneous and different from each other. It results in reduced sampling
variability, and can point out differences in responses among groups.
Systematic Sample: individuals are selected systematically from a sampling frame (e.g., every
10th person). Can be representative if there is no relationship between the order of the
sampling frame and the variables of interest.
Randomization: each member of a population is given a fair, random chance of selection in the sample.
This reduces bias in a sample.
Biased Sample: one that over- or under-emphasizes some characteristics of the population. It is caused
by poor design and is not reduced as sample size increases.
Types of Bias
Voluntary Response Bias: occurs when sample participants are self-selected volunteers (i.e.,
those willing to participate).
Undercoverage Bias: occurs when some members of the population are inadequately covered
in a sample.
Nonresponse Bias: occurs when respondents to a survey differ in meaningful ways from nonrespondents.
Response Bias: occurs when the question is asked in such a way that it influences the response.
Sample Size: the number of individuals in a sample.
Required sample size does NOT depend on the size of the population (as long as the population is large
enough and our sample is less than 10% of the population).
Representative Sample: A sample whose statistics accurately reflect the corresponding population
Sampling Frame: a list of individuals from which the sample is drawn.
Sampling Variability: the natural tendency of randomly drawn samples to differ from one another.
Note: sampling variability is not a problem.
Pilot: A small trial run of a survey used to determine if the questions are clear.
Population: the entire group of individuals that we hope to learn about.
Census: examination of information about every member of a population. This is the best approach
when the population is small and accessible.
Why not do a census all the time?
Difficult or expensive to complete.
Populations rarely stand still. A census takes time and the population changes during it.
A census is more complex than a sample.
Parameter: a descriptive measure (using a numerical value) of the population, e.g., 𝜇, 𝜎. Also called a
population parameter.
Statistic: a descriptive measure (using a numerical value) of a sample, e.g., 𝑥̅ , 𝑠. Also called a sample
Key Statistics and Parameters
Sample Statistic
Population Parameter
𝜇 (mu)
Standard Deviation
Regression Coefficient
The Valid Survey
𝜎 (sigma
𝜌 (rho)
𝛽 (beta)
What do I want to know?
Am I asking the right respondents (i.e., do I have the right sampling frame)?
Am I asking the right questions? Ask only questions that help you learn what you want to know.
Be specific. In each question, either give a set of alternative answers (i.e., multiple choice) or
ask for a numerical response, if possible. Ask questions in a neutral way (i.e., avoid bias).
What will I do with the answers: will they address what I want to know?
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The method of fundamental solutions applied to boundary value problems on the surface of a sphere
Alves, Carlos J. S.; Antunes, Pedro R. S.
Computers and Mathematics with Applications, 75 (7) (2018), 2365-2373
In this work we propose using the method of fundamental solutions (MFS) to solve boundary value problems for the Helmholtz-Beltrami equation on a sphere. We prove density and convergence results that
justify the proposed MFS approximation. Several numerical examples are considered to illustrate the good performance of the method.
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PreCalc WK 5 Sem 2
Monday 2-7-22
• Building and using Sinusoidal Models
• Definition parameters of phase shift models from given information
□ Max and Min
☆ these should imply amplitude and midline
□ where data starts (min or max) matters PRACTICE
☆ min (use reflected cosine)
☆ max (use cosine)
□ where data starts matters (middle)
☆ did it start at midline and go UP (use sine) (this is subject to change, so check frequently)
☆ did it start at midline and go DOWN (use reflected sine)
Tuesday 2-8-22
• Work on examples from Paul Forester Word Problem Packet
• Show how to use regression feature of Desmos (time permits) 2-9-22 NO CLASS
Wednesday 2-9-22 2-10-22 Review for quiz on friday
• NO CLASS SIP DAY (2nd Hour roundtable) 2-11-22
Thursday 2-10-22
• Review for Quiz on Building and Using Trigonometric Models
Friday 2-11-22 Summative Assessment
• Building and Using Trigonometric Models
Building and Using Sinusoidal Models Digital Resources
Building Sine and Cosine Models
Length 17:05
Building a Sinusoidal Model from a table of coordinates
Length: 8:24
Ferris Wheel Problem Examples
Length 18:30
Tide Problems
Length 18:30
Application of Sine and Cosine Models
Length 23:00
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Shelf Test Methodology
The Shelf Test is designed to study purchasing preferences through a realistic shopping experience.
Shelf Metrics visualization:
• Shelf Heatmap displays the density of clicks using 2d kernel density estimate. There are 9 layers—the bottom layer represents an area where we expect to find 99% of the clicks, and the top layer
50%. Note: a click in this context is a literal click on the location on the screen, whether to select a product for purchase, or just to view its details. Product location randomization isn’t
considered—the heatmap illustrates coordinates of the clicks on whatever product a particular respondent happened to see on that position.
• Product Heatmap visualizes total product quantities with a color scale ranging from the minimum (green) to the maximum (red) quantity.
• Clicks view shows individual clicks made by respondent. A click in this context is a literal click on the location on the screen, whether to select a product for purchase, or just to view its
details. Product location randomization isn’t considered - the heatmap illustrates coordinates of the clicks on whatever product a particular respondent happened to see on that position.
• Shelf view has no additional overlays so that the original shelf setup can be viewed without obstructions. If product randomization was involved, displayed image is one of the possible
realizations of the algorithm, and individual experience of a respondent may have varied.
Total summary statistics:
• Average cart spend is the geometric mean of respondent-level sum of money spent.
Base: respondents who purchased at least one product.
• Average cart size is the simple mean of the respondent-level quantity of purchased products.
Base: respondents who purchased at least one product.
• Average number of unique products in cart is the simple mean of the respondent-level number of unique purchased product.
Base: respondents who purchased at least one product.
• Average time at shelf is the geometric mean of time spent on the question.
Base: respondents who saw the question.
• Average attention span per product is the simple mean across product-level Attention Span aggregates. NA option isn’t considered.
• Total products purchased is the sum of all products purchased by all respondents throughout the survey duration. NA option isn’t considered.
• Total product spend is the sum of money spent on products purchased by all respondents. NA option isn’t considered.
Product metrics:
• Purchase selection represents proportion of respondents who purchased the product.
Base: respondents who saw the question.
• Purchase quantity is the simple mean of the quantities of that product purchased by respondents who decided to purchase that product.
Base: respondents who purchased at least one unit of that product.
• Purchase amount is the geometric mean of the amount of money spent of that product by respondents who decided to purchase that product.
Base: respondents who purchased at least one unit of that product.
• Attention span is the geometric mean of the amount of time spent viewing that product's detail view.
Base: respondents who purchased at least one unit of that product.
• Time to cart is the geometric mean of the amount of time between starting the task and selecting that product.
Base: respondents who purchased at least one unit of that product.
Geometric Mean
When analyzing the results of a Shelf Test, the geometric mean is often used as a way to account for outliers in the data. The geometric mean is most frequently used in economics and finance but can
also be used when dealing with values that don't have a finite range, such as money. Unlike the standard arithmetic mean, the geometric mean does a better job at trying to account for and not be as
influenced by extreme outliers.
In the case of the Shelf Test - because there could be so much behavioral variation where some respondents might fill up their carts - rather than removing them as outliers altogether, the geometric
means acknowledges this could be real life behavior but does not let it skew the results.
For example, in the shelf test, average cart spend is calculated using geometric mean because it is a more robust metric that avoids the extreme influence of outliers.
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In this talk I advertise chaotic strings as a pure mathematical model which promise to give answers to present open questions in elementary particle physics and cosmology. Unexplained are the mass
spectrum of elementary particles, the occurence of dark energy, and the development of usual four-dimensional space-time as a part of a 11-dimensional manifold in the course of the fundamental
process well-known as Big Bang. This process is looked at under a new, philosophical point of view as being not in space-time, but instead space-time generating. The talk of course cannot dwell on
every point in detail but hopefully will give an overview and the taste of the subject which is exciting and encouraging for the author.
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NUMBERVALUE Excel
What Is NUMBERVALUE Excel Function?
The NUMBERVALUE Excel function is a mathematical tool that allows users to convert text that represents a number into an actual numerical value. This function is particularly useful when dealing
with data imported from external sources, where numbers may be stored as text. By utilizing the NUMBERVALUE function, users can eliminate any formatting issues and perform various calculations
and analyses accurately. The function takes a text argument and evaluates it for any valid numbers; if it identifies a numeric value within the text, it returns the corresponding numerical
In the following example, we will demonstrate how to calculate the date and time value from a given value using the NUMBERVALUE function in Excel. The data is presented in a well-structured table,
where Column A represents the value and Column B represents the Result.
To perform this calculation, you need to input the complete formula =NUMBERVALUE(A2). After entering the formula, press the Enter key. To apply this formula to the remaining values, effortlessly drag
the cursor to cell B3, as shown in the image below.
Key Takeaways
• NUMBERVALUE function in Excel can handle different number formats, such as decimal separators or currency symbols, making it versatile for international datasets.
• The NUMBERVALUE function assists professionals in seamlessly incorporating textual data into numerical operations. It enhances accuracy in financial modeling, statistical analysis, or any other
tasks relying on precise calculations within the Excel environment.
• The occurrence of a #VALUE! The error can be attributed to the following scenarios:
□ When the group separator is positioned after the decimal separator within the text argument.
□ In case of invalid arguments being provided.
□ If the decimal_separator is used more than once.
□ Any empty spaces within the text argument will be disregarded. For example, “30 0 0” will be returned as 3000.
□ When the decimal_separator or group_separator contains multiple characters, only the first character will be considered.
The NUMBERVALUE function utilizes the following arguments:
Text – This is the required argument. This refers to the text that will undergo conversion into a numerical value.
Decimal_separator – This is the optional argument. This character is employed to separate the integer and fractional parts of the resulting number.
Group_separator – This is the optional argument. It serves the purpose of separating number groupings, such as millions from thousands, thousands from hundreds, and so on.
How To Use NUMBERVALUE Function In Excel?
To effectively utilize the NUMBERVALUE function in Excel, follow these steps.
#1 – Access From The Excel Ribbon
Choose the empty cell which will contain the result.
Go to the Formulas tab and click it.
Select the Text option from the menu.
Select NUMBERVALUE from the drop-down menu.
A window called Function Arguments appears.
As the number of arguments, enter the value in the text, decimal_separator, and group_separator.
Select OK.
#2 – Enter The Worksheet Manually
Select an empty cell for the output.
Type =NUMBERVALUE() in the selected cell. Alternatively, type =N and double-click the NUMBERVALUE function from the list of suggestions shown by Excel.
Press the Enter key.
Example #1
In the following example, we will demonstrate how to calculate the numerical value from a given value using the NUMBERVALUE function in Excel. The data is organized in a table, with Column A
containing the value and Column B containing the Result.
To calculate the numerical value using the NUMBERVALUE function in Excel, please follow these steps:
Step 1: Select the column where you want to enter the formula and calculate the result. For this example, we will use cell B2.
Step 2: Enter the Excel NUMBERVALUE formula in cell B2.
Step 3: Enter the value of the text argument as A2, which represents the entered value in cell A2.
Step 4: The complete formula entered should be =NUMBERVALUE(A2).
Step 5: After entering each value in the preceding step, press the Enter key. We can see the result in cell B2.
Step 6: Press enter and drag the cursor to cell B5, as shown in the image below.
Example #2 – Calculating Sales With NUMBERVALUE Function
In the following example, we will demonstrate how to calculate the numerical sales from a given value using the NUMBERVALUE function in Excel. The data is organized in a table with Column A
containing the Months, Column B containing the value and Column C containing the Result.
To calculate the numerical value using the NUMBERVALUE function in Excel, please follow these steps:
Step 1: Select the column where you want to enter the formula and calculate the result. For this example, we will use cell C2.
Step 2: Enter the Excel NUMBERVALUE formula in cell C2.
Step 3: Enter the value of the text argument as B2, which represents the entered value in cell B2.
Step 4: The complete formula entered should be =NUMBERVALUE(B2).
Step 5: After entering each value in the preceding step, press the Enter key. We can see the result in cell C2.
Step 6: Press enter and drag the cursor to cell C13, as shown in the image below.
Example #3 – Calculating Average With NUMBERVALUE Function
In the following example, we will demonstrate how to calculate the average value from a given value using the NUMBERVALUE function in Excel. The data is organized in a table, with Column A containing
the value and Cell A8 containing the Result.
To calculate the numerical value using the NUMBERVALUE function in Excel, please follow these steps:
Step 1: Select the column where you want to enter the formula and calculate the result. For this example, we will use cell A8.
Step 2: Enter the Excel NUMBERVALUE formula in cell A8.
Step 3: Enter the value of the serial number argument as A2:A7, which represents the entered value in cell A2:A7.
Step 4: The complete formula entered should be {=AVERAGE(NUMBERVALUE(LEFT(A2:A7)))}.
Step 5: After entering each value in the preceding step, press CTRL + SHIFT + the Enter key. The result in cell A8 will be displayed as 3.5.
NUMBERVALUE Function vs VALUE Function
The NUMBERVALUE function and the VALUE function are both powerful tools in Excel that allow users to convert text into a numerical value. However, there are some key differences between the two
The NUMBERVALUE function is useful when dealing with text that includes numbers and symbols such as commas or currency symbols. It can handle these symbols and extract the numeric value from the text
effectively. On the other hand, the VALUE function is more straightforward and converts text into a number without considering any specific format or symbol usage. While it may seem like both
functions perform similar tasks, understanding their distinctions is crucial for accurate data manipulation and analysis in Excel spreadsheets.
Therefore, selecting the appropriate function based on the specific requirements of your data set will ensure precise conversions without compromising data integrity or generating errors within your
Important Things To Note
• In the absence of specified decimal_separator and group_separator arguments, the function will automatically employ the separators based on the current locale settings.
• By providing an empty string (“”) as the text argument, we will obtain a result of 0.
• The presence of a #NUM error indicates that the group_separator appears after the decimal separator.
• When the group_separator is positioned before the decimal separator.
Frequently Asked Questions (FAQs)
1. Explain the use of NUMBERVALUE with an example.
Consider the below example.
To perform this calculation, you need to enter the complete formula =NUMBERVALUE(A2). Once you have entered the formula, press the Enter key. To apply this formula to the subsequent values, drag the
cursor to cell B3, as depicted in the image below.
2. Are there any limitations or considerations to keep in mind when using the NUMBERVALUE function in Excel?
• Firstly, this function can only convert numbers written in standard numeric formats, such as “123” or “-45.67”. It fails to recognize numbers with non-numeric characters, including currency symbols
or thousands of separators.
• Additionally, regional differences might affect the conversion process as some countries utilize decimal commas instead of decimal points.
Another critical point is that NUMBERVALUE cannot handle fractions or percentages directly; one must first remove any non-numeric characters or symbols before conversion.
3. Where is the NUMBERVALUE function in Excel?
One can activate the NUMBERVALUE function in Excel using the following steps:
• Choose the empty cell which will contain the result.
• Go to the “Formulas” tab and click it.
• Select the “Text” group.
• Select “NUMBERVALUE” from the drop-down menu.
• A window called “Function Arguments” appears.
• As the number of arguments, enter the value in the “text,” “decimal_separator,” and “group_separator.”
• Select OK.
Download Template
This article must help understand the NUMBERVALUE Excel Function’s formula and examples. You can download the template here to use it instantly.
Recommended Articles
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Paper justifying AR4’s use of a uniform prior for estimating climate sensitivity shown to be faulty
Paper justifying AR4’s use of a uniform prior for estimating climate sensitivity shown to be faulty
Originally posted on Jul 30, 2014 – 10:41 AM at Climate Audit
In July 2004 the IPCC held a Working Group 1 (WG1) Workshop on climate sensitivity, as part of the work plan leading up to AR4. In one session, Myles Allen of Oxford university and a researcher in
his group, David Frame, jointly gave a presentation entitled “Observational constraints and prior assumptions on climate sensitivity”. They developed the work presented into what became an
influential paper, Frame et al 2005,[1] here, with Frame as lead author and Allen as senior author.
Frame and Allen pointed out that climate sensitivity studies could be – whether or not they explicitly were – couched in a Bayesian formulation. That formulation applies Bayes’ theorem to produce a
posterior probability density function (PDF), from which best estimates and uncertainty ranges are derived. The posterior PDF represents, at each value for climate sensitivity (ECS), and of any other
parameters (fixed but uncertain variables) being estimated, the product of the likelihood of the observations at that value and the “prior” for the uncertain parameters that is also required in
Bayes’ theorem.
Obviously, the posterior PDF, and hence the best estimate and upper uncertainty bound for ECS, depend on the form of the prior. Both the likelihood and the prior are defined over the full range of
ECS under consideration. The prior can be viewed as a weighting function that is applied to the likelihood (and can be implemented by a weighted sampling of the likelihood function), but in terms of
Bayes’ theorem it is normally viewed as constituting a PDF for the parameters being estimated prior to gaining knowledge from the data-based likelihood.
Frame et al 2005 stated that, unless warned otherwise, users would expect an answer to the question “what does this study tell me about X, given no knowledge of X before the study was performed”.
That is certainly what one would normally expect from a scientific study – the results should reflect, objectively, the data used and the outcome of the experiment performed. In Bayesian terms, it
implies taking an “Objective Bayesian” approach using a “noninformative” prior that is not intended to reflect any existing knowledge about X, rather than a “Subjective Bayesian” approach – which
involves the opposite and produces purely personal probabilities.
Frame and Allen claimed that the correct prior for ECS – to answer the question they posed – depended on why one was interested in knowing ECS, and that the prior used should be uniform (flat) in the
quantity in which one was interested. Such a proposal does not appear to be supported by probability theory, nor to have been adopted elsewhere in the physical sciences. Although for some purposes
they seem to have preferred a prior that was uniform in TCR, their proposal implies use of a uniform in ECS prior when ECS is the target of the estimate. AR4 pointed this out, and adopted the Frame
et al 2005 proposal of using a uniform in ECS prior when estimating ECS. Use of a uniform prior for ECS resulted in most of the observational ECS estimates given in Figure 9.20 and Table 9.3 of AR4
having very high 95% uncertainty bounds.
Consistent with the foregoing thesis, Frame et al 2005 stated that “if the focus is on equilibrium warming, then we cannot rule out high sensitivity, high heat uptake cases that are consistent with,
but nonlinearly related to, 20th century observations”. Frame and Allen illustrated this in their 2004 presentation with ECS estimates derived from a simple global energy balance climate model, with
forcing from greenhouse gases only. The model had two adjustable parameters, ECS and Kv – here meaning the square root of effective ocean vertical diffusivity. The ‘observable’ variables – the data
used, errors in which are assumed to be independent – were 20th century warming attributable to greenhouse gases (AW), as estimated previously using a pattern-based detection and attribution
analysis, and effective heat capacity (EHC) – the ratio of the changes in ocean heat content and in surface temperature over a multidecadal period.
Frame and Allen’s original graph (Figure 1) showed that use of a uniform prior in ECS gives a very high 95% upper bound for climate sensitivity, whereas a uniform prior in Feedback strength (the
reciprocal of ECS) – which declines with ECS squared – gives a low 95% bound. A uniform prior in the observable variables (AW and EHC) also gives a 95% bound under half that based on a uniform in ECS
prior; using a prior that is uniform in transient climate response (TCR) rather than in AW, and is uniform in EHC, gives an almost identical PDF.
However, the Frame et al 2005 claim that high sensitivity, high heat uptake cases cannot be ruled out is incorrect: such cases would give rise to excessive ocean warming relative to the observational
uncertainty range. It follows that Frame and Allen’s proposal to use a uniform in ECS prior when it is ECS that is being estimated does not in fact answer the question they posed, as to what the
study tells one about ECS given no prior knowledge about it. Of course, I am not the first person to point out that Frame and Allen’s proposal to use a uniform-in-ECS prior when estimating ECS makes
no sense. James Annan and Julia Hargreaves did so years ago.
Frame et al 2005 was a short paper, and it is unlikely that many people fully understood what the authors had done. However, once Myles Allen helpfully provided me with data and draft code relating
to the paper, I discovered that the analysis performed hadn’t actually used likelihood functions for AW and EHC. The authors had mistakenly instead used (posterior) PDFs that they had derived for AW
and EHC, which are differently shaped. Therefore, the paper’s results did not represent use of the stated priors. And although, I am told, the Frame et al 2005 authors had no intention of using an
Objective Bayesian approach, the PDFs they derived for AW and EHC do appear to correspond to such an approach.
Now, it is simple to form a joint PDF for AW and EHC by multiplying their PDFs together. Having done so, the model simulation runs can be used to perform a one-to-one translation from AW–EHC to ECS–
Kv coordinates, and thereby to convert the PDF for AW–EHC into a PDF for ECS–Kv using the standard transformation-of-variables formula. That formula involves multiplication by the ‘Jacobian’
[determinant], which converts areas/volumes from one coordinate system to another. The standard Bayesian procedure of integrating out an unwanted variable, here Kv, then provides a PDF for ECS. The
beauty of this approach is that conversion of a PDF upon a transformation of variables gives a unique, unarguably correct, result.
What this means is that, since Frame and Allen had started their ‘Bayesian’ analysis with PDFs not likelihood functions, there was no room for any argument about choice of priors; priors had already
been chosen (explicitly or implicitly) and used. Given the starting point of independent estimated PDFs for AW and EHC, there was only one correct joint PDF for ECS and Kv, and there was no dispute
about obtaining a marginal PDF for ECS by integrating out Kv. The resulting PDF is what the misnamed black ‘Uniform prior in Observable variables’ curve in Figure 1 really represented.
Even when, unlike in Frame and Allen’s case, the starting point is likelihood functions for the observable variables, there are attractions in applying Bayes’ theorem to the observable (data)
variables (in some cases after transforming them), at which point it is often obvious which prior is noninformative, thereby obtaining an objective joint PDF for the data variables. A transformation
of variables can then be undertaken to obtain an objective joint posterior PDF for the parameters. I used this approach in a more complicated situation in a 2013 climate sensitivity study,[2] but it
is not in common use.
After I discovered the fundamental errors made by the Frame et al 2005 authors, I replicated and extended their work, including estimating likelihood functions for AW and EHC, and wrote a paper
reanalysing their work. As well as pointing out the errors in Frame et al 2005 and, more importantly, its misunderstandings about Bayesian inference, the case provided an excellent case-study for
applying the transformation of variables approach, and for comparing estimates for ECS using:
1. a Bayesian method with a uniform in ECS (and Kv) prior, as Frame and Allen advocated;
2. an Objective Bayesian method with a noninformative prior;
3. a transformation of variables from the joint PDF for (AW, EHC); and
4. a non-Bayesian profile likelihood method.
All except method 3. estimate ECS directly from likelihood functions for AW and EHC. Since those two likelihood functions were not directly available, I estimated each of them from the related PDF. I
did so by fitting to each of those PDFs a parameterised probability distribution for which I knew the corresponding noninformative prior, and then dividing it by that prior. This procedure
effectively applies Bayes’ theorem in reverse, and seems to work well provided the parameterised probability distribution family chosen offers a close match to the PDF being fitted.
The profile likelihood method– an objective non-Bayesian method not involving any selection of a prior – provides approximate confidence intervals. Such intervals are intended to reflect long-run
frequencies on repeated testing, and are conceptually different from Bayesian probability estimates. However, noninformative priors for Objective Bayesian inference are often designed so that the
resulting posterior PDFs provide uncertainty ranges that closely replicate confidence intervals.
The ECS estimates resulting from the various methods are shown in Figure 2, a slightly simplified version of Figure 5 in my paper.
Figure 2. Estimated marginal PDFs for climate sensitivity (in K or °C) derived on various bases. The box plots indicate boundaries, to the nearest grid value, for the percentiles 5–95 (vertical bar
at ends), 10-90 (box-ends), and 50 (vertical bar in box: median), and allow for off-graph probability lying between ECS = 10°C and ECS = 20°C. (The cyan box plot shows confidence intervals, the
vertical bar in the box showing the likelihood profile peak).
Methods 2 and 3 [the red and black lines and box plots in Figure 2] give identical results – they logically must do in this case. The green line, from Frame et al 2005, is an updated version of the
black line in Figure 1, using a newer ocean heat content dataset. The green line’s near identity to the black line confirms that it actually represents a transformation of variables approach using
the Jacobian. Method 4 [the cyan box plot in Figure 2], profile likelihood, gives very similar results. That similarity strongly supports my assertion that methods 2 and 3 provide objectively-correct
ECS estimation, given the data and climate model used and the assumptions made. Method 1, use of a uniform prior in ECS (and in Kv), [blue line in Figure 2] raises the median ECS estimate by almost
50% and overestimates the 95% uncertainty bound for ECS by a factor of nearly three. The dashed grey line shows the result of Frame et al 2005’s method of estimating ECS that claimed to use a uniform
prior in ECS and Kv, but which in fact equated to using the transformation of variables method without including the required Jacobian factor.
For the data used in Frame et al 2005, the objective estimation methods all give a best (median) estimate for ECS of 2.4°C. Correcting for an error in Frame et al 2005’s calculation of the ocean heat
content change reduces the best estimate for ECS to 2.2°C, still somewhat higher than other estimates I have obtained. That is very likely because Frame et al 2005 used an estimate of attributable
warming based on 20th century data, which has been shown to produce excessive sensitivity estimates.[3]
The noninformative prior used for method 2 is shown in Figure 3. The prior is very highly peaked the in low ECS, low Kv corner, and by an ECS of 5°C is, at mid-range Kv, under one-hundredth of its
peak value . What climate scientist using a Subjective Bayesian approach would choose a joint prior for ECS and Kv looking like that, or even include any prior like it if exploring sensitivity to
choice of priors? Most climate scientists would claim I had chosen a ridiculous prior that ruled out a priori the possibility of ECS being high. Yet, as I show in my paper, use of this prior produces
identical results to those from applying the transformation of variables formula to the PDFs for AW and EHC that were derived in Frame et al 2005, and almost the same results as using the
non-Bayesian profile likelihood method.
Figure 3: Noninformative Jeffreys’ prior for inferring ECS and Kv from the (AW, EHC) likelihood. (The fitted EHC distribution is parameterised differently here than in my paper, but the shape of the
prior is almost identical.)
Use of a uniform prior for ECS in Bayesian climate sensitivity studies has remained common after AR4, with the main alternative being an ‘expert prior’ – which tends to perpetuate the existing
consensus range for ECS. The mistake many scientists using Bayesian methods make is thinking that the shape of a prior simply represents existing probabilistic knowledge about the value of the
parameter(s) concerned. However, the shape of a noninformative prior – one that has minimal influence, relative to the data, on parameter estimation – represents different factors. In particular, it
reflects how the informativeness of the data about the parameters varies with parameter values, as the sensitivity of the data values to parameter changes alters and data precision varies. Such a
prior is appropriate for use when either there is no existing knowledge or – as Frame et al 2005 correctly imply is normal in science – parameter estimates are to be based purely on evidence from the
study, disregarding any previous knowledge. Even when there is existing probabilistic knowledge about parameters and that knowledge is to be incorporated, the prior needs to reflect the same factors
as a noninformative prior would in addition to reflecting that knowledge. Simply using an existing estimated posterior PDF for the parameters as the prior distribution will not in general produce
parameter estimates that correctly combine the existing knowledge and new information.[4]
Whilst my paper was under review, the Frame et al 2005 authors arranged a corrigendum to Frame et al 2005 in GRL in relation to the likelihood function error and the miscalculation of the ocean heat
content change. They did not take the opportunity to withdraw what they had originally written about choice of priors, or their claim about not being able to rule out high ECS values based on 20th
century observations. My paper[5] is now available in Early Online Release form, here. The final submitted manuscript is available on my own webpage, here.
Nicholas Lewis
^[1] Frame DJ, BBB Booth, JA Kettleborough, DA Stainforth, JM Gregory, M Collins and MR Allen, 2005. Constraining climate forecasts: The role of prior assumptions. Geophys. Res. Lett., 32, L09702
^[2] Lewis, N., 2013. An objective Bayesian improved approach for applying optimal fingerprint techniques to estimate climate sensitivity. Journal of Climate, 26, 7414-7429.
^[3] Gillett et al, 2012. Improved constraints on 21st-century warming derived using 160 years of temperature observations. Geophys. Res. Lett., 39, L01704
^[4] Lewis, N., 2013. Modification of Bayesian Updating where Continuous Parameters have Differing Relationships with New and Existing Data. arXiv:1308.2791 [stat.ME].
^[5] Lewis N, 2014. Objective Inference for Climate Parameters: Bayesian, Transformation of Variables and Profile Likelihood Approaches. Journal of Climate, doi:10.1175/JCLI-D-13-00584.1
James Annan had a blog post about his and Julia Hargreaves’ efforts to get their criticisms of the use of a uniform prior for ECS estimation published, here. Their paper, “On the generation and
interpretation of probabilistic estimates of climate sensitivity”, Climatic Change, 2011, 104, 3-4, pp 423-436, is available here.
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About sensitivity analysis of constraint shadow price | AIMMS Community
I would like to know how to get the sensitivity analysis of a constraint shadow price.
For example, say we have the following constraint: x+y ≤ c
and its shadow price "SP".
How do I get the relationship between c and SP ? In other words, the sensitivity analysis of SP according to c.
I would appreciate if you can show me how to achieve this in AIMMS. Thank you very much for your assistance!
|
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Arrays—Data Structures & Algorithms for Data ScientistsArrays—Data Structures & Algorithms for Data Scientists
Datascience in Towards Data Science on Medium,
Arrays—Data Structures & Algorithms for Data Scientists
Arrays—Data Structures & Algorithms for Data Scientists
How dynamic and static arrays work under the hood
Photo by Caspar Camille Rubin on Unsplash
As data scientists, we rarely get asked LeetCode-style questions, so the need for us to learn data structures and algorithms is less than for software engineers.
However, being able to write efficient code is a great multiplier for your data science career. Imagine you can be a data scientist who knows how to implement ML models but also understands the best
practices when it comes to writing code and has an appreciation and knowledge of software engineering?
You suddenly become very valuable and almost a unicorn in the market. That’s why I have started taking a data structures and algorithms course, from which I plan to share what I have learned.
This post will be dedicated to arrays, how they work under the hood, and their different types.
Data Structures
A data structure is a convenient way to store information inside a computer. As Wikipedia defines it:
A data structure is a data organization and storage format that is usually chosen for efficient access to data. More precisely, a data structure is a collection of data values, the relationships
among them, and the functions or operations that can be applied to the data.
The simplest data structure is an array, which is essentially a list of values of characters. Below is an example in Python.
arr = [1, 2, 3, 4, 5]
arr = ["e", "g", "o", "r"]
Under the hood, each element is held in random access memory (RAM) with a given memory address.
Integers take up 4 bytes (32 bits) of memory, whereas characters take up only 1 byte (8 bits). So, the object type in the array will dictate the memory size allocated to it.
Static Arrays
As data scientists mainly work with Python, we hardly ever encounter static arrays in our work. However, in languages like Java, C++, and Fortran, static arrays are the default.
When we say “static” array, we refer to an array with a fixed predefined size and type by the user, for example, in C++.
// define type and length of array
int nums[12] = {31,28,31,30,31,30,31,31,30,31,30,31};
To access an element from the array, all we have to do is pass in the index to the array,
// pass in some index i
This is an O(1) operation regarding time complexity, as the computer already knows where to look to get that element. The time to get the element from the index does not increase as the array get’s
To loop over an array, is an O(n) operation as the time increases as the length of the array increases.
for (int i = 0; i < 12; i++) {
cout << nums[i] << "\n";
If we wanted to delete from an array, this is also an O(n) operation in the worst-case scenario.
This is because if we wanted to delete the first element in an array, we would need to shift every other element once to the left. This process would increase as the array increased, so it’s an O(n)
However, if we were deleting the last element, then it would be a O(1) as no shifting is involved.
delete_element(int arr[], int i, int length) {
for (int index = i; index < length - 1; index++) {
arr[index] = arr[index + 1];
Similarly, inserting a value at some given index is also an O(n) operation because it may require shifting all elements once to the right.
However, again if we were inserting the last element, then it would be a O(1) as no shifting is involved.
insert_element(int arr[], int i, int n, int& length) {
for (int index = length - 1; index >= i; index--) {
arr[index + 1] = arr[index];
arr[i] = n;
Dynamic Arrays
Dynamic arrays are the default in Python. We can freely append to arrays as much as we want without any worry if we break any allocated size.
In Python, this is easily achieved through the “append” operation for lists, which are essentially dynamic arrays.
arr = [1, 2, 3, 4]
However, when adding elements to a dynamic array, it does not actually grow the array itself. What happens under the hood is that a new array (double the capacity) is created, and the original
elements are copied in along with the new elements we want to add.
The diagram below illustrates this process.
Diagram by author.
After the new array is created, the old array is then deleted from memory. One can see this process is actually quite inefficient!
Adding to a dynamic array has an O(1) amortised time complexity. Amortised refers to the average complexity of an operation. If we increased the array size every time, then this would be an O(n)
operation, but this is not the case on average.
The reason we double is nuanced and is answered terrifically in this quora thread. However, the main reason is to avoid the time consuming process of always resizing when wanting to add an element to
the array.
Inserting or deleting from a dynamic array is the same time complexity as a static array. We would have to shift elements to the right or left, so the worst case is O(n). Likewise, accessing a
dynamic array is also O(1), the same as a static array.
A quick note: With time complexity, we care about asymptotic analysis. So, an operation that is O(2n) and an operation that is O(n) are seen as the same in terms of time complexity. Any
multiplier or constant is disregarded.
A stack is a specific dynamic array type that doesn’t exist natively in many programming languages, but you create your own stack type if need be.
In a stack, you can only add and remove the top element; you can’t perform operations on any elements in the middle of the stack. This process is called Last In, First Out (LIFO).
Diagram by author.
There are two primary operations:
• Push: Add an element to the top of the stack. In the above scenario, it will go on top of the 4.
• Pop: Remove the most recently added element. In the above scenario, it will remove the 4.
There is also the Peek operation, which shows you the last element added without removing it. In the above scenario, it will just return the value of 4.
As all these operations involve the element at the end of the array, they are all O(1) operations.
Summary & Further Thoughts
Arrays are probably the most fundamental and accessible data structure to work with, however you must use them efficiently. I hope this article gave you some insight into the different types of
arrays and how their complexity of operation works.
Another Thing!
I have a free newsletter, Dishing the Data, where I share weekly tips and advice as a practising data scientist. Plus, when you subscribe, you will get my FREE data science resume and short PDF
version of this AI roadmap!
Dishing The Data | Egor Howell | Substack
Connect With Me!
Arrays—Data Structures & Algorithms for Data Scientists was originally published in Towards Data Science on Medium, where people are continuing the conversation by highlighting and responding to
this story.
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Rounding and Precision
Last updated
Rounding and precision are important aspects with any numerical or financial system, and Zuora is no exception. Understanding how rounding and precision are defined within Zuora will give you insight
into how Zuora handles precision, rounding and calculations.
Rounding a numeric value is the act of replacing one numeric value with one that is approximately equal but has a shorter, simpler, or more explicit representation. For example, replacing USD
$65.8476 with USD $65.85. Rounding in Zuora takes place with units of measure (UOM) and currency values.
• UOM (unit of measure): Examples include quantity on a price plan, quantity entered in a subscription, and usage records uploaded in a month.
• Currency: Examples include price, invoice amount, MRR, and tax total.
Precision of a numeric value describes the number of digits that are used to express that value, including digits to both the left and the right of any decimal point. For example 4.520 has a
precision of 4. Zuora supports up to 13 digits to the left of the decimal place, and up to 9 digits to the right.
Currency rounding rules
Currency rounding is defined by the currency itself.
Zuora includes rounding options that you can apply to any individual currency.
• Rounding Mode: Allows you to define whether a currency should be rounded Down, Up, or Half Up. By default, Zuora uses Half Up rounding on all currencies (for example, where 3.49 is rounded to 3
and 3.50 is rounded to 4). The Down and Up options are useful if you need to use a different rounding mode for a particular currency (for example, in Japan, it is often the case that currency
values are rounded Down to the nearest Yen).
• Rounding Increment: Allows you to define the currency increment used for rounding. This is useful if you are working with currencies that do not round to the nearest cent (for example, Swiss
francs are typically rounded to the nearest 5 cents).
By default, any currency that has a greater number of decimal places than defined by the currency will be rounded based on the near rule (half up), with values of 5 or greater being rounded up (using
the Half Up setting).
For example, because the Japanese Yen (JPY) does not have a cent in circulation, the lowest value is 1 JPY. When a user enters an amount with decimal places, Zuora will round it to the nearest
non-decimal value. For example, if a user enters 15.67, Zuora Billing automatically convert this value to 16.00.
The same is true when activating or editing a currency from within Zuora Billing Settings Customize Currencies: the Decimal Places values are pre-defined based on the decimal places supported by each
Currency rounding on invoices
Zuora rounds the Invoice Line Item Amount (pre-tax) based on the List Price of the product and on the rounding rules that you have set for the currency. Tax is also rounded off in this manner, but is
rounded from the value of the Invoice Line Item Amount (pre-tax).
For example:
• List Price: 454.5454545 (product rate plan charge)
• Invoice Amount (pre-tax): 454.55
□ This is rounded from the List Price.
• Tax: 45.46
□ This is rounded from the Invoice Amount (pre-tax)
• Extended Price: 500.01
If you want to use a List Price that has decimal places, you can remove the 0.01 value offset by processing an adjustment when the invoice has been generated.
Usage rounding rules
A value is rounded based on its predefined rounding rule, and each currency and UOM has a rounding rule associated with it. UOM rounding rules are user-defined, while currency rounding rules are
defined by the currencies themselves.
Zuora supports the following rounding rules:
• Round up (usage only): Any usage amount that has a greater number of decimal places than supported by the UOM will be rounded up to the number of supported decimal places. For example, if the
UOM, “Gigabyte,” supports two decimal places and the usage record is created with quantity of 2.334 Gigabytes, when rounded up, the rounded numeric value is 2.34 Gigabytes.
• Round down (usage only): Any usage amount that has a greater number of decimal places than supported by UOM will be rounded down to the number of supported decimal places. For example, if the UOM
“Users” supports no decimal places and a usage record is created with a quantity of 2.334 Users, when rounded down, it will be 2 Users.
Data type rounding rules
There are five data types in Zuora that are either currency or UOM values. Each data type has its own rules for when rounding occurs.
Data Examples Rounding Based On
Price Also known as rate, unit price from product catalog, unit price Prices are never rounded, even if precision exceeds currency. For example, a unit price of $3.1235/GB
is always stored and calculated as $3.1235.
Amount Extended price (for example, $4.99/user * 4 users), adjustment amount, invoice total, Rounding is based on currency’s defined rounding rules.
invoice balance, total tax amount.
Metrics MRR, TCV Rounding is based on currency’s defined rounding rules.
Tax "Tax line items", individual taxation items Tax line items are not rounded. Summed tax line items result in total tax, defined as an amount.
Quantity Default quantity in product catalog, quantity entered on subscription or usage record, Rounding is based on unit of measure’s rounding rule
quantity displayed on invoices
Tax rounding rules
• When tax information is displayed on invoices, rounding applies to tax calculation. For details and examples, see this article.
• With tax-inclusive mode, rounding can happen to the tax amount or the net amount. For details and examples, see this article.
Rounding and "As Is"
Rounding can occur in the following cases:
• Before a value is stored.
• Before a value is used in calculations.
• Before a value is displayed or returned via the API.
• After values of the same type are aggregated (added) together.
If numbers are not rounded, they are referred to "as is."
Data types rounded "As Is"
The following data types are rounded as is:
Data Type Prior to Storage Prior to Use After Aggregation Prior to Display/API Example
Price As Is As Is N/A As Is $3.479/Gallon
Amount Rounded Rounded Rounded Rounded Extended Price of $36.67, Total Tax of $18.29
Metrics As Is As Is As Is Rounded MRR = $1533.333333333
Tax As Is As Is N/A (aggregated tax items are an amount) As Is Tax item of $12.1275
Quantity (Transaction) Rounded Rounded Rounded Rounded 7.65 GB, 8 users
Quantity (Tier) Rounded Rounded N/A Rounded Volume Pricing for 1-100 Units
Quantity (Other) As Is Rounded Rounded As Is Usage Quantity Entered
Examples of rounding
For example, a fictitious SaaS company sells seat licenses to its software and charges for any storage used. It offers the following monthly rate plan for its product:
│ Recurring Fee │ Usage Fee │
│ $59.99/seat license per month │ $1/GB per month │
It has defined the following units of measure:
│ Unit of measure │ Number of decimal places │ Rounding rule │
│ Seat License │ 0 │ Round down │
│ Gigabytes │ 2 │ Round down │
The company also charges sales tax at the rate of 7.775%. A subscription is then created and a quantity of 4.6 is mistakenly entered for the number of seat licenses. Since seat licenses have 0
decimal places and transaction quantities are rounded prior to being stored, 4.6 is rounded to 4 based on its round down rounding rule, and then stored. Any query via the API or display of this value
in the UI will return a value of 4.
The usage is subsequently uploaded into the system with a value of 12.31245 GB. Since usage quantity is stored as is, the value 12.31245 is then stored. Any display in the UI or query via the API
will return the value of 12.31245. When the invoice is generated, charges are calculated as defined in the following table:
│ Amount │ Calculation │ Total │
│ Recurring Charge │ Unit price * transaction quantity = $59.99/seat license * 4 seat licenses │ $239.96 │
│ Usage Charge │ Unit price * usage quantity = $1/GB * Round(12.31245 GB)= $1/GB * 12.32 GB ) │ $12.32 │
│ Tax item on recurring charge │ Charge * tax rate = $239.96 * 0.0775 │ $18.5969 │
│ Tax item on usage charge │ UCharge * tax rate = $12.32 * 0.0775 │ $0.9548 │
│ Invoice Total │ Recurring charge + usage charge + total tax = $239.96 + $12.32 + $19.55 │ $271.83 │
Usage quantities (like GB) are not rounded when stored, while transaction quantities (like seat licenses) are rounded when stored. For example, 12.3124 GB is not rounded, while 4.6 seat licenses are
stored as 4 seat licenses. Individual tax items are not rounded. When they are aggregated (added together) for a total tax they are then rounded.
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Verifiable Privacy-Preserving Computing: Conclusion & References
This paper is available on arxiv under CC 4.0 license.
(1) Tariq Bontekoe, University of Groningen;
(2) Dimka Karastoyanova, University of Groningen;
(3) Fatih Turkmen, University of Groningen.
Table of Links
Zero-knowledge proofs and verifiable computing
Privacy-preserving computations
Requirements: an application’s perspective
Verifiable, privacy-preserving computing
Open challenges and future directions
8. Conclusion
We presented a systematic overview of solutions for verifiable privacy-preserving computations, applicable to scenarios with distributed data. Specific attention was paid to solutions that offer the
stronger notion of public verifiability.
First, we established the relevant background knowledge on decentralized computations, verifiable computing and ZKPs. This was followed by a summary of predominant approaches in privacy-preserving
computations: MPC and HE, and a discussion of the need for verifiability. Second, we analyzed the use of verifiable PETs in different application domains, and discussed the required properties.
Third, a systematic overview on existing solutions for verifiable privacy-preserving computations for (distributed) private data was given. Solutions were categorized based on the underlying PET and
verifiability approach that was used. Moreover, we compared the solution approaches based on efficiency and the earlier identified properties.
Based on our analysis, we note a number of challenges and open research questions. First, we recommend to investigate the combination of verifiability with input authentication and/or reusability, to
improve security and efficiency in practice, where data is not simply a oneuse object. Also, we see a need to design post-quantum secure schemes and derive efficient public verifiability from
standard assumptions. Finally, we see a lack of efficient publicly verifiable HE schemes, whereas HE itself is already in the first stages of practical adoption.
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Zhang, Eds. Cham: Springer International Publishing, 2018, pp. 100–115.
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Pfitzmann, Ed. Berlin, Heidelberg: Springer, 2001, pp. 280–300.
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Curtain Rails
This came up before. I have, it turns out, 100mm of empty space between the platser, plaster boards, insulation and just empty space it seems in the living room. This is all super dooper until one
tries to hang a curtin rail. The fittings won't take more than a 6mm screw/hammer in fitting. But the longest 6mm screw and plug one seems to be able to get is only 100mm. So I have a fitting that
takes a screw that doesn't even reach concrete block, never mind actually penetrate it 30mm+.
So other than me getting a length of wood and using 10mm X 135mm screws to mount the wood and then screwing the fittings to the wood itself, can anyone come up with a suggestion that would allow me
to mount the curtain direct without using an ugly length of wood.
Legend, Try and find the wooden studs behind the plaster board, I heard they should be no more than 17" apart. You could be lucky and find they fall at the right place to hang the curtain correctly,
a stud finder gadget would do this no bother.
Have you heard of/considered butterfly bolts? You should, with a bit of patience, be able to get a size suitable for the curtain rail mounts, and once in place and tightened, they will hold a
surprising amount of weight.
Know the ones you mean Sunny. The ones that have the spring toggle in them I take it?
Only problem is, the only way you'd find out if they would hold the curtain is to put everything up. An dyou'd know they wouldn't hold it whren the thing crashed down on you cracking the plasterboard
as it did it!
Re: ..
these things can support the weight of a telly so i dont think curtains wold be much of an issue
We are talkling about the mettle ones that have a V in them. You drill quite a large, circa 12-14 mm hole, squueze the V together, push it through the hole and then tighten the screw into it, such
that the back of the v is held against the plaster board....but doesn't that mean that the weight of the curtains is basically being held by the plasterboard???
Re: ..
We are talkling about the mettle ones that have a V in them. You drill quite a large, circa 12-14 mm hole, squueze the V together, push it through the hole and then tighten the screw into it,
such that the back of the v is held against the plaster board
Hi legend,
that's exactly the ones I'm talking about. The first time I used them, it was to hold a c.50kg "box" to a stud wall, with just 4 of them. I was s***ing it that it would fall, but, alas, it's still
there, and rock solid! Maybe just use a few more than if it were conventional screws.
Now I think about it though, won't the 12-14mm hole be a problem? Would it be feasible to fix the curtain rail to the ceiling rather than to the wall?
Presume you mean my problem might be that the hole would be so wide as to be visible because the fitting would not cover it entirely?. We'd rise above that...could fill the visible semi-circle with
painters mate etc...if thats what you mean.
Can't fix to ceiling. 9 and a half feet ceilings....(Selling point of house in the fecking brouchre!!!). These curtains are for sliding door, so no way in the world the curtains would be 9.5 feet
One more thing though, if the plaster board has the 1 inch foam aeroboard on the back, and I know it does, would that stop m yspring opening fully....can't recall how much recoil is on those springs?
Re: ..
Presume you mean my problem might be that the hole would be so wide as to be visible because the fitting would not cover it entirely?.
Yeah, not just that, but the fact the bracket for the rail would actually go through the hole. But ok, fill the hole, with something which sets hard.
Good high ceilings!
As regards the foam back, I don't think that will hinder you too much, and the "wings" should just bite through that when you tighten them up.
No, the bracket would defo not punch through. of course, becaude the holes on the bracket are close together, the max scress I could put in would be 2. Its one of those white rails, with strings. The
part that screws up is about 2.5 inches by maybe 1-1.5 inches. The 2.5 inches is the horizontal part, the 1.5 inches is across. it is drilled with 6 holes. Obviously you wouldn;t use all 6, I had
used 2 of them in the upstairs rooms. Upsatris its only about 65mm before you hit concrete....
Thanks for the ideas, I'l lwork on it over the weekend!
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WORKING WITH ANALYTICAL FUNCTIONS - Sayantan's Blog On OracleWORKING WITH ANALYTICAL FUNCTIONS - Sayantan's Blog On Oracle
• Analytical functions compute an aggregate value based on a group of rows. This group of rows are called Window of rows.
• The Window determines the no of rows to be used to perform the calculation of the current row.
• Analytical function is differ from normal aggregate function as normal aggregate function returns a single row for a group of rows, while the analytical function returns multiple rows for a group
of rows.
• Analytical functions are the last set of operation in a query except the last order by clause.
• Joining and all other clauses like WHERE, GROUP BY, HAVING clauses are completed before the analytical functions are processed.
• Analytical functions can appear only in SELECT list or ORDER BY clause.
• Analytical functions can take 0 to 3 arguments.
• The datatype of the arguments can be any numeric data type or any data type that can be implicitly converted into numeric data types.
• The OVER analytic clause indicate that the function operates on a group of values.
• OVER clause is computed after the FROM, WHERE, GROUP BY or HAVING clauses.
• PARTITION BY clause is used to partition the query result set into multiple groups based on one or more columns provided in PARTITION BY clause.
• ORDER BY clause is used to specify how data is sorted within a partition. Multiple columns can be used in ORDER BY clause.
• ASC|DESC
• NULLS FIRST|NULLS LAST
• ROW|RANGE
• BETWEEN|AND
• UNBOUNDED PRECEDING
• UNBOUNDED FOLLOWING
• CURRENT ROW
Specify the ordering sequence as ascending or descending. Default is ASC.
Specify in ordering sequence whether the rows containing NULL should appear first or last.
• ROWS specifies each row of a window which is a physical set rows and that is used for calculating the function result.
• RANGE specifies each row of a window which is a logical set rows and that is used for calculating the function result.
• These clauses cannot be used unless you have specified the ORDER BY clause.
Few example of Analytical Functions for which Window clause is applicable:
• AVG
• SUM
• COUNT
• MIN
• MAX
• FIRST_VALUE
• LAST_VALUE
• NTH_VALUE
• STDDEV
EAXMPLE 1:
ROUND(SUM(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS 1 PRECEDING)) CUM_SUM_SAL_BW,
ROUND(SUM(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 0 PRECEDING AND 1 FOLLOWING)) CUM_SUM_SAL_FW,
ROUND(SUM(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) CUM_SUM_SAL_BW_FW
In the first cumulative salary column CUM_SUM_SAL_BW, for department_id 90, in the 1st row since there is no previous row, its value shown same as salary column, i.e. 2640. In the second row salary
value is 1870, so cumulative column 1870 is added with previous row value 2640 and total is visible as (1870 + 2640) = 4510. And in 3rd row salary value is 1870 and cumulative value is 1870 + prev.
row value 1870) = 3740.
In the second cumulative salary column CUM_SUM_SAL_FW, for department_id 90, since every row added with the value of next row, in the 1st row its value shown as (2640 + 1870) = 4510. In the second
row salary value is 1870, so in cumulative column 1870 is added with next row salary value 1870 and total is visible as (1870 + 1870) = 3740. And the 3rd row salary value is 1870 and since its the
last row of department_id 90, cumulative value is visible as 1870.
3rd cumulative salary column CUM_SUM_SAL_BW_FW, for department_id 90, since every row added with the value of previous row as well as next row, in the 1st row its value shown as (2640 + 1870) = 4510,
since there is no previous row. In the second row salary value is 1870, previous row value is 2640 & next row value is 1870, so in cumulative column total is visible as (2640 + 1870 + 1870) = 6380.
In the 3rd row salary value is 1870, previous row value is 1870 and since its the last row of department_id 90, cumulative value is visible as (1870 + 1870) = 3740.
EAXMPLE 2:
ROUND(AVG(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) AVG_SAL
In the above example,
• 1st avg_sal -> for employee_id 100 and 101 avg sal = ROUND((2640 + 1870)/2) = 2255.
• 2nd avg_sal -> for employee_id 100, 101, 102 avg_sal = ROUND((2640+1870+1870)/3) = 2127………so on..
EAXMPLE 3:
ROUND(AVG(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) AVG_SAL,
ROUND(SUM(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) SUM_SAL,
ROUND(MAX(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) MAX_SAL,
ROUND(MIN(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) MIN_SAL,
ROUND(FIRST_VALUE(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) FIRST_SAL,
ROUND(LAST_VALUE(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING)) LAST_SAL
All possible functions are used in above example. Impact is same like Average and Sum functions.
EXAMPLE 4:
ROUND(SUM(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY SALARY RANGE BETWEEN 50 PRECEDING AND 100 FOLLOWING)) CUM_SUM_SAL_LR,
ROUND(SUM(SALARY) OVER(PARTITION BY DEPARTMENT_ID ORDER BY SALARY ROWS BETWEEN 20 PRECEDING AND 50 FOLLOWING)) CUM_SUM_SAL_SM
WHERE DEPARTMENT_ID IN (30, 50)
In the above example,
• in CUM_SUM_SAL_LR column, 1st row salary value is 1210 and function is set for range between 50 and 100. So Cumulative Sum will add any value between 50 less than 1210 and any value 100 greater
than 1210. Since there is no such value exists in department 30, its 1st row total is equal to the 1st row salary value 1210.
• In the 2nd row salary value is 341. There is no value exists in the range, greater than 100 of 341, but there are values in the range, 50 less than 341, i.e. (341 – 50) = 291. Such values are:
308, 319. So Cumulative total of 2nd row is (308 + 319 + 341) = 968.
• Applying the same formula in 3rd row, there is one value in the range 100 greater than 319 in department 30, i.e. 341. Also there are values in the range. 50 less than 319 i.e. (319 – 50) = 269.
Such values are: 275, 286, 308, 319. So Cumulative Sum is (275 + 286 + 308 + 319 + 341) = 1529. So on…
EXAMPLE 5: (DATES)
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING) MAX_HIREDATE,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING) MIN_HIREDATE,
FIRST_VALUE(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING) FIRST_HIREDATE,
LAST_VALUE(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY EMPLOYEE_ID ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING) LAST_HIREDATE
In The above example,
• max_hiredate showing us the maximum hiredate among previous row, current row and the next row. So in the 1st row its showing the value of '21-09-05' which is the maximum of current row
('17-06-03') and next row ('21-09-05').
• Similarly for 2nd row hiredate is '21-09-05', previous row hiredate is '17-06-03' and next row hiredate is '13-01-01'. So the maximum among the 3 dates is '21-09-05'..so on.
• Min_hiredate is working similar way as max_hiredate but minimum among the dates will be considered.
• First_value will be considered only the first value among the dates. In the example the first row against First_hiredate showing the value '17-06-03' which is the first value among the first row
('17-06-03') and next row ('21-09-05').
• In the second row of First_hiredate is showing the value again as '17-06-03' as this is the first value among current row ('21-09-05'), previous row ('17-06-03') and next row ('13-01-01').
• Last value will consider only the last value among the dates.
EXAMPLE 6:
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN 0 PRECEDING AND 30 FOLLOWING) MAX_HD_30_FW,
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE 30 PRECEDING) MAX_HD_30_BW,
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN 30 PRECEDING AND 30 FOLLOWING) MAX_HD_30_BW_30_FW,
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN 0 PRECEDING AND 365 FOLLOWING) MAX_HD_1YR_FW,
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE 365 PRECEDING) MAX_HD_1YR_BKW,
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN 365 PRECEDING AND 365 FOLLOWING) MAX_HD_1YR_BKW_1YR_FW
In example 6:
• hire_date is showing the value as '17-06-03'. The The MX_HD_30_FW column showing the maximum of hire_date among all dates whose values are in the range between '17-06-03' and '17-06-03' + 30 i.e.
'17-07-03'. But in hire_date column there are no such dates so that its showing '17-06-03'.
• Similarly, MAX_HD_30_BW column is showing '17-06-03' as thee is no such dates between '17-06-03'and '17-06-03' – 30 i.e. '18-05-03'. But in hire_date column there are no such dates so that its
showing '17-06-03'.
• In the MAX_HD_30_BW_30_FW column the value is showing again as '17-06-03' as there are no such hire_dates range between '17-06-03' – 30 and '17-06-03' + 30.
• Same formula will be applicable for remaining fields as well.
• Its indicates that the Window starts at the first row of the partition.
• Its the start point specification and cannot be used as end point of specification.
EXAMPLE 1:
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS UNBOUNDED PRECEDING) MAX_HD_UB_PR,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS UNBOUNDED PRECEDING) MIN_HD_UB_PR
WHERE DEPARTMENT_ID IN (30, 60, 90)
In the above example,
• The MAX_HD_UB_PR column shows the maximum of hire_date between first row and preceding rows where values is less than that of current row. In this case every row will be the maximum of all the
preceding rows whose values are less than that of current row.
• The MIN_HD_UB_PR column shows the minimum of hire_dates between current row and the preceding rows whose values are less than that of current row. In this case minimum hire_date of any department
will be shown against each row.
EXAMPLE 2:
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW) MAX_HD_UB_PR_CR,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW) MIN_HD_UB_PR_CR
WHERE DEPARTMENT_ID IN (30, 60, 90)
In case of Example 2 has the same output as Example 1 for the same reason.
EXAMPLE 3:
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS BETWEEN UNBOUNDED PRECEDING AND 1 FOLLOWING) MAX_HD_UB_PR_1_FO,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS BETWEEN UNBOUNDED PRECEDING AND 1 FOLLOWING) MIN_HD_UB_PR_1_FO
WHERE DEPARTMENT_ID IN (30, 60, 90)
In the Example 3,
• the current row of MAX_HD_UB_PR_1_FO will always return the next higher value within a department except its the maximum hire_date of that department. For maximum hire_date it will return the
same value as the hire_date.
• In the 1st row its showing the value as '21-09-05' as in hire_date column, following next row has the value '21-09-05' and its greater than the 1st row hire_date value '17-06-03'.
• For 2nd row the hire_date and MAX_HD_UB_PR_1_FO has the same value as its the maximum value of hire_date in department 90.
• For the 3rd row the hire_date has the value '13-01-01' and next higher value in department 90 is '17-06-03', so in MAX_HD_UB_PR_1_FO column the value is showing as '17-06-03'.
EXAMPLE 4:
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN UNBOUNDED PRECEDING AND 1 FOLLOWING) MAX_HD_UB_PR_1_FO,
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN UNBOUNDED PRECEDING AND 1000 FOLLOWING) MAX_HD_UB_PR_1000_FO,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN UNBOUNDED PRECEDING AND 1 FOLLOWING) MIN_HD_UB_PR_1_FO,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE RANGE BETWEEN UNBOUNDED PRECEDING AND 1000 FOLLOWING) MIN_HD_UB_PR_1000_FO
WHERE DEPARTMENT_ID IN (30, 60, 90)
In Example 4,
• The MAX_HD_UB_PR_1_FO column is showing the same value as that of hire_date column. Its logically searching a value one day larger than the current row.
• In First row hire_date is '17-06-03' and there is no value in department 90 where hire_date is '18-06-05'. So its showing the value as '17-06-03'. Same is repeated for 2nd and 3rd row of
department 90.
• In MAX_HD_UB_PR_1000_FO column, the value is showing '21-09-05' as its logically searching a value 1000 days higher than hire_date which is '17-06-03'. So the searching range will be between
'17-06-03' and ('17-06-03' + 1000) = '13-03-06'. There is only one value in department 90 in that range and i.e. '21-09-05'. In second row the hire_date and MAX_HD_UB_PR_1000_FO is showing the
same value as its the maximum hire_date in department 90.
• In both MIN_HD_UB_PR_1_FO and MIN_HD_UB_PR_1000_FO the same value is showing in all rows and columns and its the minimum hire_date in department 90. With MIN function logical range will always
returns the minimum in the given condition.
MAX(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS BETWEEN CURRENT ROW AND UNBOUNDED FOLLOWING) MAX_HD_UB_FO,
MIN(HIRE_DATE) OVER(PARTITION BY DEPARTMENT_ID ORDER BY HIRE_DATE ROWS BETWEEN CURRENT ROW AND UNBOUNDED FOLLOWING) MIN_HD_UB_FO
WHERE DEPARTMENT_ID IN (30, 60, 90)
Query to display salary with next highest salary of employees table in descending order:
LEAD(SALARY, 1, 0) OVER(ORDER BY SALARY DESC) NEXT_HIGHEST_SALARY,
(SALARY - LEAD(SALARY, 1, 0) OVER(ORDER BY SALARY DESC)) SALARY_DIFF
The query to display salary with next lowest salary of employees table:
LAG(SALARY, 1, 0) OVER(ORDER BY SALARY) PREVIOUS_LOWEST_SALARY,
(SALARY - LAG(SALARY, 1, 0) OVER(ORDER BY SALARY)) SALARY_DIFF
Displaying employees hire_date in ascending order with its next lowest hire_date and the gap days between two hire_dates:
LAST_NAME, EMAIL,
LEAD(HIRE_DATE, 1, HIRE_DATE) OVER(ORDER BY HIRE_DATE) NEXT_HIRE_DATE,
NVL((LEAD(HIRE_DATE, 1, HIRE_DATE) OVER(ORDER BY HIRE_DATE) - HIRE_DATE), 0) GAP_DAYS,
ORDER BY HIRE_DATE;
Query to display employees hire_date in descending order with its previous hire_date and the gap days between two hire_dates:
LAST_NAME, EMAIL,
LEAD(HIRE_DATE, 1, HIRE_DATE) OVER(ORDER BY HIRE_DATE DESC) PREVIOUS_HIRE_DATE,
NVL((HIRE_DATE - LEAD(HIRE_DATE, 1, HIRE_DATE) OVER(ORDER BY HIRE_DATE DESC)), 0) GAP_DAYS,
ORDER BY HIRE_DATE DESC;
1 thought on “WORKING WITH ANALYTICAL FUNCTIONS”
Nice blog here! Also your website loads up very fast! Kai Ludvig Lucine
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Quantum Gravity
[a theoretical progression from my previous post] "What On Earth Is Wrong With Gravity?"(I have written an extensive thesis on this but I thought I'd better not post it all here!!! However, I'll try
and condense the theory so I can get some input from you all.)
Gravity is everywhere. Or is it?
To us, on Earth, and to cosmology in general, gravity is king. It explains everything from the orbital motion of planets and galaxies to the very laws of Newtonian motion themselves.
Deep within the atomic nucleus, at the quantum level, it is the Strong Nuclear Force that is by far the most abundantly powerful force – binding together protons despite the naturally repulsive
electromagnetic forces of the positively charged particles.
This Strong Nuclear Force is the most powerful fundamental force in the Universe (a trillion, trillion, trillion times stronger that gravity on Earth), but it is only effective at quantum distances
(smaller than a trillionth of a millimetre).
To maintain equilibrium, I assert that as the Strong Nuclear Force depletes with increasing distances so must the forces of electromagnetism and gravity correspondingly strengthen; electromagnetism
with increasing wavelength and gravity comparable with an increase in mass.
As Max Planck detailed the “Planck Constant” (length) of h (6.626x10^-34) below with length electromagnetic waves could never shrink, so too should there be a “
Le Marcs Mass
” – being the amount of mass at which gravity gains dominion (either via Newtonian attraction or Einsteinian warping of space-time), but below which level of mass renders gravity insignificant in the
sub-atomic world and entirely submissive to the other fundamental forces.
Arguably there could also exist a measure of constant length relating to the Strong Nuclear Force, being the corresponding distance required between particles/objects once achieved “Le Marcs Mass” to
overcome the Strong Nuclear Force, but in line with quantum theory this may be comparable to Planck’s length. However, as far as coining a name for this theoretical quantum distance I would suggest
the “
Peake Distance
” or maintaining the ITLADian tradition of dreadful puns………….the “Peake District”
*English groans*
An enormous amount of time and money is presently being spent by leading scientific researchers into finding the role of gravity at the sub-atomic level. Here I postulate that it has NO role at some
quantum levels in an inverse way that the Strong Nuclear Force has no effect in the macro-scale universe. The two are kings in their own dominion ONLY. And the search for the elusive Graviton (the
assumed particle of gravity) seems futile to me as we have never even recorded Einsteinian Gravity Waves at the Cosmological level. So trying to find a single Graviton when we can't even see any
evidence of the wave seems proposterous. Indeed can there even occur a gravitational wave collapse at observation if Gravitational waves themselves are not proven?
To summarise: I postulate that the “
Le Marcs Mass
” is the quantity of mass required for any material object/particle to obtain before it can possibly exert any gravitational force. Below this mass level it possesses neither no gravitational pull
nor sufficient space-time warpage and consequently, at such small mass levels and deep within the quantum field, gravity has no effect in sub-atomic quantum physics.
The “
Peake Distance
” is the distance required between sub-atomic particles (particularly protons) for the Strong Nuclear Force to weaken and electromagnetism and gravity to become effective.
This, in as simplistic form as I can manage, is my theory of Quantum Gravity.
Discuss !
(any input is valuable as I develop this theory so your thoughts are welcomed either in the comments section or, given the complexity of this subject, via direct email to me.)
A Dark PhilosopherKarl L Le Marcs
24 comments:
Karl Le Marcs said...
*some hard science for those interested*
At the Cosmological level (C)
Mass of Sun = 1.98892 x 10^30 kg
Mass of Earth = 5.9742 x 10^24 kg
Distance between the two = 150 x 10^9m
At the Quantum level (Q)
Mass of Proton = 1.6726 x 10^-27 kg
Mass of Electron = 9.10938 x 10^-31 kg
Mass of Neutron = approx 1,839 x Electron Mass
Distance or Bohr Radius = 5.3nm (5.3 x 10^-9 m)
Now, the ratio of Cosmological (C) to Quantum (Q) calculates to roughly 1:10^73
(which is a considerable number)
Given that F = G M1 M2
Where F = Force; G = Gravitational Constant; M1 is the Mass of 1 object and M2 is the mass of the other object; and r is the distance between them – if M1 M2 is viewed as the ratio (C) to (Q)
∑ F = G (1:10^73)
(Q) -----------
“Le Marcs Mass” would be a constant above which G makes F a positive result and below which leaves F at zero.
“Peake Distance” would be r where the Strong Nuclear Force effect is depleted so G again takes a positive dominant F.
(Writing mathematical equations on blogger is chuffing difficult !!!!)
I hope this makes (some) sense, but comments or direct emails to me are welcomed as I try and work on this thesis in the long insomniacal nights (and it gives my hypergraphia something to do).
This comment has been removed by the author.
*looks around the room*
Has no-one got anything to say?
Ken, surely!! Tony will, I know, chuck in his lot when he returns from the SMN but I hope some more of you will help me along.
Karl, I need more time to assimilate this AND your third-person comments to my post. My Pentium II is chugging along full bore and I'm paging in and out of my 512MB of memory like ... like ...
like ... like ... an oyster shucker ... at ... a ...
Sorry, segmentation fault. Where was I?
I agree that "finding the role of gravity at the sub-atomic level" seems useless if you are considering the force between sub-atomic particles. But, what is interesting to me is how gravity
interacts with matter and, more specifically, how does it do this on the sub-atomic level? The earth runs around the sun but how does it "know" that it should do this? Is this the role of the
graviton -- to interact with matter and transfer this "knowledge"?
Whether or not the Graviton is measurable, I cannot give an informed opinion. But something must be telling the matter in the earth that it needs to move in such a way as it does.
As far as the "Le Marcs Mass," I would suggest that you consider defining it as the amount of mass at which gravity equals the strong force, or something like that. "Gains dominion" seems a
rather subjective measurement but "becomes equal to" is more objective. But isn't distance also necessary? You could have a tiny black hole with enough mass to be significant but it would still
need to be appropriately close. Maybe you need a "Le Marcs Constant" with dimensions mass/length?
The problem is that I don't have a feel for how the strong force depends on mass and distance. Does it decrease smoothly or is there a sharp transition as distance increases? Gravity, presumable,
would be a smooth function as mass and distance decrease even into the sub-atomic range. Where the two profile cross would seem to be the most interesting point.
After a little bit of rumination...
I think that considering the Newtonian gravitational force (i.e. F=GMM/r2) between sub-atomic particles is not relevant. It is far too small compared to the strong force. I totally agree with you
that on the sub-atomic scales, gravity is not important and on larger scales, the strong force is not important.
I'm also not convinced that comparing the gravitational force on the cosmological and sub-atomic scales is meaningful. What is special about the sun-earth system that it would be the standard?
Why not the gravitational force between our sun and alpha centauri or the earth and the moon? And why should gravity on the small scale necessarily be comparable to gravity on the cosmological
scale? The force between me and the earth as I step off my second story balcony is some10^20 times less than between the earth and sun but it's still important.
I must admit I did not understand (probably due to problematic formatting due to blogger) your last equation and the "above which G makes F a positive result and below which leaves F at zero."
Gravity goes like the product of the two masses under consideration so it will never be identically zero.
From some VERY brief readings about quantum gravity, the point is not the gravitational force between protons and electrons but the mediation of gravity by a particle called the graviton. It is
the anticipated merging of gravity with the other three forces (and about here my current comprehension starts to fade).
Karl, you and I have touched upon this issue when we last met and I am even more interested in this idea now - particularly after reading a couple of books discussing the concept of 'quantum
fields'. In my opinion you are suggesting something very powerful and I am hoping that your concept of the Le Marcs Mass (otherwise known as the 'Dark Mass' - 'Dark Matter' maybe?) will be
published elsewhere and commented upon by specialists far more qualified than myself (although I suspect that Ken has clear specialism in this field - indeed I just stand back in awe at the
vlevel of your debate). Honoured by your 'Peake Distance' by the way. Lets hope that the followers of Abraham Maslow do not sue you - wouldn't that be an 'experience'.
Ken, thank you as ever for your input, it is all greatly appreciated.
You asked: "something must be telling the matter in the earth that it needs to move in such a way as it does." which reminded me of the famous quote by MIT professor Alan Guth - "Space tells
matter how to move. Matter tells space how to curve."
I agree with the semantics of "gains dominion" so that will be amended. And you said: "But isn't distance also necessary?" which is where the "Peake Distance" or puntastically the "Peake
District" comes into my equations.
Again Ken, thanks for your help.
Looking forward to discussing Quantum Fields with you in the pub shortly then. *smile*
I'm not sure of your connection between my "Le Marcs Mass" and "Dark Mass" or "Dark Matter", I think some more explanation maybe required from me over a soothing ale or two.
And your standing "in awe" is deeply humbling especially as you should remember that many of my theories are as a direct consequence, or certainly majorly inspired by both your book and yourself.
I rather like the "Peake District" myself by the way - It's great when you can make yourself laugh !!
And congratulations at throwing a name at me that I have no knowledge of.
*runs off to revise on Abraham Maslow*
Karl-- I feel that I'm missing something in your ideas (which would not be surprising to me). Are you, indeed, talking about the gravitational force between sub-atomic particles and is this what
scientists are, indeed, investigating?
Anthony-- I am a scientist/engineer by education but I have not speciality in physics or quantum theory or anything.
Ken, don't worry, I'm missing somethings in my ideas as well *smile*
Yes I am theorising on the sub-atomic gravitational effects (or non-effects as I postulate). The elusive Graviton that is being sought is the hypothetical particle of gravity. However, I argue 2
points. 1, That as the Gravitational Waves that Einstein predicted have not even been discovered at the Cosmological level how likely is it that we will discover such a small fragment when the
bigger force is not evident. And 2, I wonder if it is even possible to collapse the wave function of gravity to produce a graviton anyway.
And to answer your question on how the SNF depletes either uniformly or with sharp decline I've been doing some calculations and I would think that as between nucleons, neutrons and protons, the
strong force is mediated by pions it would simply decrease with increasing distance by the inverse-square law and an exponential term something like exp(-x/d) where x is distance and d = hc/
(2.Pi.E) from the uncertainty principle.
But I may be getting in way-hey-hey too deep here.
*tries pulling self out of mental black hole*
KARL and ANTHONY-- Have you heard of the Penrose Interpretation? In this one, gravity causes the wave function collapse eliminating the need for both Many Worlds and conscious observation. Found
it on wikipedia
Ken, problem being the word "gravity". In my theory of Quantum Gravity there is NO gravitational effect, either Newtonian or Einsteinian at the Quantum Level therefore the Penrose Interpretation
collapses as the hypothetical wave as "Gravity" cannot be a force to collapse the wave at such a quantum distance (until both "Le Marcs Mass" and "Peake Distance" have been achieved - see
original post for definitions of these "Constants")
However, by very nature of MWI we all become observers, not just in our own universe but also in those in which we have an entaglement (Quantum or otherwise).
I feel you and I Ken (and also Tony) ideally should discuss much of this over a plentiful supply of Malt Whiskey.
Make it a nice bourbon (preferably something 16 years or older) and you're on!
Course there is the slight problem of that puddle between our domiciles. Surely there is one of my Many Worlds in which I reside in the UK. Just need to swap places with that ken. Wonder if you
can carry a bottle with you on that trip? That way I could bring my own nectar.
KARL-- But gravity has infinite range (unlike the SNF which seems to reach only a few fm before going to zero) and there's no competition with the SNF in Penrose's theory (at least that I see).
I did a quick calculation and the gravitational attraction between two dust specks each being the Planck mass in size and separated by 1 micrometer is equal to an acceleration on one of the dust
specks of almost 1.5 micrometer/s^2. This is far from negligible.
I used the Planck mass for the calculation because that's the scale at which Penrose suggests that standard quantum mechanics will fail. So, he's saying gravity is a factor for things much larger
than electrons.
Which, I guess is kind of off-topic for a post on Quantum Gravity.
I'd suggest a nice 16yr old Lagavulin Malt. I'll see if I can carry one on an Astral travel??!!!
I don't agree that "gravity has infinite range", this is the basis of my theory of Quantum Gravity below the "Le Marcs Mass" and at distances within the "Peake Distance". Elementary particles,
and even Quarks and Leptons do not have sufficient mass, nor energy to generate what we "know" as gravity, neither in the Newtonian "pull" variety nor the Einsteinian "warping of space-time".
I agree with Penrose regarding gravity at scales "much larger than electrons" which is the case I'm trying to present. It is when mass of any particle falls below my "Le Marcs Mass" that any
gravitation effect become negligible or if the "Peake Distance" has also been reached, that the SNF becomes the dominant force.
KARL-- I feel like I'm coming across as argumentative, and I don't mean to be. At some point you'll just have to tell me to shut up until you share your theory with me.
So, that being said . . .
1) I don't think comparing the Cosmological level to the Quantum level of the gravitational force makes sense. From what I've found online, the Strong Force potential energy is on the order of
2-50 MeV. I calculated the potential energy of the earth-sun system (i.e. the energy required to remove the earth from it's current location to infinity) as 3.3x10^46 MeV.
2) The SNF goes to zero for distances of only a few femtometers. For a distance of 2 fm, to get a 2 MeV potential via gravity would require particles of mass 3.1x10^-9 (only one order of
magnitude away from the Planck Mass, interestingly enough) But this mass is 10^18 bigger than a proton which would make a very dense particle, indeed! So, I don't see any feasible, realistic mass
or distance for which the SNF still has a non-zero value and gravity is comparable.
Ken, you're not argumentative at all. This type of dialectical arguement is what I need.
I have no time today but I will get your comments answered soon.
Ok, in answer to your points:
1) I'm NOT comparing Cosmological Forces to Quantum Forces, merely pointing out the vastly differing laws of physics within each. At a ratio of 1:10^73 the difference between the two is awesomely
inconceivable. The SNF "potential energy" is problematic as its existence, as you rightly state in point 2 is only felt over a few femtometers BUT it is SO powerful that it overcomes all the
other elemental forces combined to a trillion times. And I'm rather lost as to why you mention PE in cosmology to "remove the earth from it's current location". *scratched head and looks
comedically confused in the now legendary Stan Laurel manner*
2) You actually confirm my point in point 2 while dialectically positing your argument in an interestingly paradoxical way.
You mention this hypothetical particle just a magnitude away from Planck Mass which is 10^18 bigger than a proton. This still makes the mass much larger than that which I argue about in my
Quantum Gravity post. At protonic level, even are quark and lepton level, atomic physicists are currently seeking the elusive "Graviton" the particle of gravity to learn the role of gravity at
the atomic level and also back to the role of gravity within the singularity (which of course is a different matter than we are discussing here as a singularity would have infinite mass at the
atomic level, but anyway.......) At the atomic level of "nature" the SNF fades to zero effect over a few femtometers after which the forces of electromagneticism and ultimately gravity take over.
If the MeV potential via gravity requires a specific mass then this would relate to two constants - namely the "Le Marcs Mass" and "Peake Distance" of elementary particles.
*lies down*
1) I computed the PE of the earth-sun system to have a comparison with the values I was taking from plots of the SNF. I thought, from way back at the beginning of this post (your first comment,
to be precise), that you were comparing Cosmological to Quantum forces of gravity and defining your "Le Marcs Mass" in terms of that ratio. I was trying to demonstrate why I don't think you
should do that. But you're not. So a moot point.
2) I try to be dialectically interesting in a paradoxical way at least once a week. "Use it or lose it" and all that. In any case, and despite the fact that I don't remember what I was positing
(I try not to posit but once a fortnight) I think I was going somewhere along the lines of where does gravity overwhelm the SNF and trying to show that the SNF goes to zero long before gravity
has a chance because gravity would require particle masses so massive but so small (so as to fit into a femtometer or so) that they would be extremely dense and so impossible.
In short, I think I have demonstrated that I am ignorant enough of your theory to offer much in the way of constructive comment yet knowledgeable enough to do a few computations and report my
results. So, until I am better informed, I think I shall have to refrain from further comment.
You make me smile and challenge me at the same time which if you were female would have some interesting alternative potential.
Your comments are ALWAYS welcomed, and as I'm always telling Tony, I put my theories on here specifically to get an alternate view. Theories based on a one-sided blinkered view are weak, but
those built via good diallectical discussions are strong.
So thank you Ken, and I do hope that's not the last I hear of your thoughts on this matter.
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Overview Package Class Use Tree Deprecated Index Help
PREV CLASS NEXT CLASS FRAMES NO FRAMES
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Class UnsignedInts
public final class UnsignedInts
extends Object
Static utility methods pertaining to int primitives that interpret values as unsigned (that is, any negative value x is treated as the positive value 2^32 + x). The methods for which signedness is
not an issue are in Ints, as well as signed versions of methods for which signedness is an issue.
In addition, this class provides several static methods for converting an int to a String and a String to an int that treat the int as an unsigned number.
Users of these utilities must be extremely careful not to mix up signed and unsigned int values. When possible, it is recommended that the UnsignedInteger wrapper class be used, at a small efficiency
penalty, to enforce the distinction in the type system.
Louis Wasserman
│ Method Summary │
│ static int │ compare(int a, int b) │
│ │ Compares the two specified int values, treating them as unsigned values between 0 and 2^32 - 1 inclusive. │
│ static int │ divide(int dividend, int divisor) │
│ │ Returns dividend / divisor, where the dividend and divisor are treated as unsigned 32-bit quantities. │
│ static String │ join(String separator, int... array) │
│ │ Returns a string containing the supplied unsigned int values separated by separator. │
│ static Comparator<int[]> │ lexicographicalComparator() │
│ │ Returns a comparator that compares two arrays of unsigned int values lexicographically. │
│ static int │ max(int... array) │
│ │ Returns the greatest value present in array, treating values as unsigned. │
│ static int │ min(int... array) │
│ │ Returns the least value present in array, treating values as unsigned. │
│ static int │ parseUnsignedInt(String s) │
│ │ Returns the unsigned int value represented by the given decimal string. │
│ static int │ parseUnsignedInt(String string, int radix) │
│ │ Returns the unsigned int value represented by a string with the given radix. │
│ static int │ remainder(int dividend, int divisor) │
│ │ Returns dividend % divisor, where the dividend and divisor are treated as unsigned 32-bit quantities. │
│ static long │ toLong(int value) │
│ │ Returns the value of the given int as a long, when treated as unsigned. │
│ static String │ toString(int x) │
│ │ Returns a string representation of x, where x is treated as unsigned. │
│ static String │ toString(int x, int radix) │
│ │ Returns a string representation of x for the given radix, where x is treated as unsigned. │
│ Methods inherited from class java.lang.Object │
│ clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait │
public static int compare(int a,
int b)
Compares the two specified int values, treating them as unsigned values between 0 and 2^32 - 1 inclusive.
a - the first unsigned int to compare
b - the second unsigned int to compare
a negative value if a is less than b; a positive value if a is greater than b; or zero if they are equal
public static long toLong(int value)
Returns the value of the given int as a long, when treated as unsigned.
public static int min(int... array)
Returns the least value present in array, treating values as unsigned.
array - a nonempty array of unsigned int values
the value present in array that is less than or equal to every other value in the array according to compare(int, int)
IllegalArgumentException - if array is empty
public static int max(int... array)
Returns the greatest value present in array, treating values as unsigned.
array - a nonempty array of unsigned int values
the value present in array that is greater than or equal to every other value in the array according to compare(int, int)
IllegalArgumentException - if array is empty
public static String join(String separator,
int... array)
Returns a string containing the supplied unsigned int values separated by separator. For example, join("-", 1, 2, 3) returns the string "1-2-3".
separator - the text that should appear between consecutive values in the resulting string (but not at the start or end)
array - an array of unsigned int values, possibly empty
public static Comparator<int[]> lexicographicalComparator()
Returns a comparator that compares two arrays of unsigned int values lexicographically. That is, it compares, using compare(int, int)), the first pair of values that follow any common prefix, or
when one array is a prefix of the other, treats the shorter array as the lesser. For example, [] < [1] < [1, 2] < [2] < [1 << 31].
The returned comparator is inconsistent with Object.equals(Object) (since arrays support only identity equality), but it is consistent with Arrays.equals(int[], int[]).
public static int divide(int dividend,
int divisor)
Returns dividend / divisor, where the dividend and divisor are treated as unsigned 32-bit quantities.
dividend - the dividend (numerator)
divisor - the divisor (denominator)
ArithmeticException - if divisor is 0
public static int remainder(int dividend,
int divisor)
Returns dividend % divisor, where the dividend and divisor are treated as unsigned 32-bit quantities.
dividend - the dividend (numerator)
divisor - the divisor (denominator)
ArithmeticException - if divisor is 0
public static int parseUnsignedInt(String s)
Returns the unsigned int value represented by the given decimal string.
NumberFormatException - if the string does not contain a valid unsigned integer, or if the value represented is too large to fit in an unsigned int.
NullPointerException - if s is null
public static int parseUnsignedInt(String string,
int radix)
Returns the unsigned int value represented by a string with the given radix.
string - the string containing the unsigned integer representation to be parsed.
radix - the radix to use while parsing s; must be between Character.MIN_RADIX and Character.MAX_RADIX.
NumberFormatException - if the string does not contain a valid unsigned int, or if supplied radix is invalid.
public static String toString(int x)
Returns a string representation of x, where x is treated as unsigned.
public static String toString(int x,
int radix)
Returns a string representation of x for the given radix, where x is treated as unsigned.
x - the value to convert to a string.
radix - the radix to use while working with x
IllegalArgumentException - if radix is not between Character.MIN_RADIX and Character.MAX_RADIX.
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How do you graph y < x^3+ x^2? | Socratic
How do you graph #y < x^3+ x^2#?
1 Answer
This is a cubic polynomial ...
Set up a table for x and y values, then plot the points. Finally, sketch the line and shade the area below the line (not including the line).
hope that helped
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Volatilidad: VIX S&P Dow Jones Indices - Forex World
» Blog » Volatilidad: VIX S&P Dow Jones Indices
Volatilidad: VIX S&P Dow Jones Indices
Also known as the “fear index,” the VIX can thus be a gauge of market sentiment, with higher values indicating greater volatility and greater fear among investors. While variance captures the
dispersion of returns around the mean of an asset in general, volatility is a measure of that variance bounded by a specific period of time. Thus, we can report daily volatility, weekly, monthly, or
annualized volatility. It is, therefore, useful to think of volatility as the annualized standard deviation.
Though volatility isn’t the same as risk, volatile assets are often considered riskier because their performance is less predictable. With investments, volatility refers to changes in an asset’s or
market’s price — especially as measured against its usual behavior or a benchmark. Other works have agreed, but claim critics failed to correctly implement the more complicated models. Some
practitioners and portfolio managers seem to completely ignore or dismiss volatility forecasting models.
Volatility origin
Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and
behavioral finance. Adam received his master’s in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as
well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem. And volatility is a
useful factor when considering how to mitigate risk.
Instead, they have to estimate the potential of the option in the market. There are several ways to measure volatility, including beta coefficients, option pricing models, and standard deviations of
returns. Volatility is a statistical measure of the dispersion of returns for a given security or market index.
Some investors can also use volatility as an opportunity to add to their portfolios by buying the dips, when prices are relatively cheap. Market volatility can also be seen through the VIX or
Volatility Index, a numeric measure of broad market volatility. Also referred to as statistical volatility, historical volatility gauges the fluctuations of underlying securities by measuring price
changes over predetermined periods of time. It is the less prevalent metric compared to implied volatility because it isn’t forward-looking. Unlike historical volatility, implied volatility comes
from the price of an option itself and represents volatility expectations for the future. Because it is implied, traders cannot use past performance as an indicator of future performance.
Volatility: Meaning In Finance and How it Works with Stocks
When applied to the financial markets, the definition isn’t much different — just a bit more technical. Glosten and Milgrom shows that at least one source of volatility can be explained by the
liquidity provision process. When market makers infer the possibility of adverse selection, they adjust their trading ranges, which in turn increases the band of price oscillation. The Greeks, in the
financial markets, are the variables used to assess risk in the options market. This means adding each value and then dividing it by the number of values. If we add, $1, plus $2, plus $3, all the way
to up to $10, we get $55.
For example, a stock with a beta value of 1.1 has historically moved 110% for every 100% move in the benchmark, based on price level. To annualize this, you can use the “rule of 16”, that is,
multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year .
Whether volatility is a good or bad thing depends on what kind of trader you are and what your risk appetite is. For long-term investors, volatility can spell trouble, but for day traders and options
traders, volatility often equals trading opportunities. This is a measure of risk and shows how values are spread out around the average price. It gives traders an idea of how far the price may
deviate from the average. HV and IV are both expressed in the form of percentages, and as standard deviations (+/-). If you say XYZ stock has a standard deviation of 10%, that means it has the
potential to either gain or lose 10% of its total value.
But conflating the two could severely inhibit the earning capabilities of your portfolio. Assessing the risk of any given path — and mapping out its more hair-raising switchbacks — is how we evaluate
and measure volatility. Many or all of the offers on this site are from companies from which Insider receives compensation . Advertising considerations may impact how and where products appear on
this site but do not affect any editorial decisions, such as which products we write about and how we evaluate them. There has been increasing volatility in international dairy markets in recent
Market volatility is defined as a statistical measure of a stock’s (or other asset’s) deviations from a set benchmark or its own average performance. Loosely translated, that means how likely there
is to be a sudden swing or big change in the price of a stock or other financial asset. Performance of VIX compared to past volatility as 30-day volatility predictors, for the period of Jan 1990-Sep
2009. Volatility is measured as the standard deviation of S&P500 one-day returns over a month’s period. The blue lines indicate linear regressions, resulting in the correlation coefficients r shown.
Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical.
Actual current volatility of a financial instrument for a specified period , based on historical prices over the specified period with the last observation the most recent price. One way to measure
an asset’s variation is to quantify the daily returns of the asset. Historical volatility is based on historical prices and represents the degree of variability in the returns of an asset. Volatility
often refers to the amount of uncertainty or risk related to the size of changes in a security’s value. A higher volatility means that a security’s value can potentially be spread out over a larger
range of values. This means that the price of the security can change dramatically over a short time period in either direction.
Investopedia does not include all offers available in the marketplace. XYZ, Inc. has a beta coefficient of 1.45, making it significantly more volatile than the S&P 500 index. ABC Corp. has a beta
coefficient of .78, which makes it slightly less volatile than the S&P 500 index. Pete Rathburn is a copy editor and fact-checker with expertise in economics and personal finance and over twenty
years of experience in the classroom.
The Cboe Volatility Index, or VIX, is an index created by Cboe Global Markets, which shows the market’s expectation of 30-day volatility. Volatility is often used to describe risk, but this is
necessarily always the case. Risk involves the chances of experiencing a loss, while volatility describes how large and quickly prices move. If those increased price movements also increase the
chance of losses, then risk is likewise increased. It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout.
Implied volatility , also known as projected volatility, is one of the most important metrics for options traders. As the name suggests, it allows them to make a determination of just how volatile
the market will be going forward. One important point to note is that it shouldn’t be considered science, so it doesn’t provide a forecast of how the market will move in the future. Volatile assets
are often considered riskier than less volatile assets because the price is expected to be less predictable. That said, the implied volatility for the average stock is around 15%.
If prices are randomly sampled from a normal distribution, then about 68% of all data values will fall within one standard deviation. Ninety-five percent of data values will fall within two standard
deviations (2 x 2.87 in our example), and 99.7% of all values will fall within three standard deviations (3 x 2.87). In this case, the values of $1 to $10 are not randomly distributed on a bell
curve; rather.
This is divided by 10 because we have 10 numbers in our data set. And there’s always the potential for unpredictable volatility events like the 1987 stock market crash, when the Dow Jones Industrial
Average plummeted by 22.6% in a single day. The second issue is the reduction in the volatility of food markets, especially by increasing stocks.
Understanding Volatility
VIX does that by looking at put and call option prices within the S&P 500, a benchmark index often used to represent the market at large. Those numbers are then weighted, averaged, and run through a
formula that expresses a prediction not only about what might lie ahead but how confident investors are feeling. The greater the volatility, the higher the market price of options contracts across
the board. Much research has been devoted to modeling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place.
Variance is a measurement of the spread between numbers in a data set.
You then back-solve for implied volatility, a measure of how much the value of that stock is predicted to fluctuate in the future. Volatility does not measure the direction of price changes, merely
their dispersion. This is because when calculating standard deviation , all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with
different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time. The VIX is the CBOE volatility
index, a measure of the short-term volatility in the broader market, measured by the implied volatility of 30-day S&P 500 options contracts. The VIX generally rises when stocks fall, and declines
when stocks rise.
What Is Volatility, Mathematically?
Volatility is a key variable in options pricing models, estimating the extent to which the return of the underlying asset will fluctuate between now and the option’s expiration. Volatility, as
expressed as a percentage coefficient within option-pricing formulas, arises from daily trading activities. How volatility is measured will affect the value of the coefficient used. Using a
simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value
near 10,000, has moved about 100 points a day, on average, for many days. Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not
reflect current volatility.
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S. Mangiarotti, F. Le Jean, M. Huc & C. Letellier
Global modeling of aggregated and associated chaotic dynamics
Chaos, Solitons & Fractals, 83, 82-96, 2016. Online
Abstract Spatially distributed systems are rather difficult to investigate due to two distinct problems which can be sometimes combined. First, the spatial extension is taken into account by
monitoring the system evolution at different locations. Second, the dynamics cannot always be continuously tracked in time, and segments of data – sometimes recorded at different places – are
only available. When the dynamics underlying a single marker is under consideration – as for instance the normalized difference vegetation index which can be used for assessing the vegetation
canopy of a given area – a global model can be obtained from a single scalar time series built by aggregating the available time series recorded at different places and/or associating the
segments of data recorded at different times (and possibly at different locations). We investigated how these two data preprocessing – common in environmental studies – may affect the model
dynamics by using a system of spatially distributed Rössler systems which are phase synchronized or not.
L. A. Aguirre & Letellier
Controllability and synchronizability : Are they related ?
Chaos, Solitons & Fractals, 83, 242-251, 2016. Online
Abstract In the two last decades the concept of observability has been formally linked to that of embedding in the context of nonlinear dynamics. Such a concept has been shown to play an
important role in global modeling, data analysis and filtering, to mention a few examples. Preliminary results suggested that observability, at least in some cases, has some influence in
synchronization problems. Could the dual concept of controllability also be important in such problems ? In the context of synchronization, in general, the role played by controllability
properties may not be as relevant as observability is for data analysis. In this work we compute controllability coefficients analogous to the observability ones, now established in the
literature, and evaluate their importance in synchronization problems. Two benchmarks have been used in the simulations : the Rössler and the cord systems. The following schemes were investigated
: synchronization to external sinusoidal force, complete replacement, uni- and bi-directional coupling of identical oscillators. The results discussed in this work show that controllability and
synchronizability are not related in general.
A. Kerfourn, B. Lamia, J.-F. Muir & C. Letellier
A dynamical model for heart remodeling during the two phases of pulmonary arterial hypertension
EPJ Nonlinear Biomedical Physics, 4, 1, 2016. Online
Abtract Background. Pulmonary arterial hypertension is a rare and lethal disease affecting small diameter pulmonary arteries and leading to a progressive increase of the right vascular
resistances. Patients with such a disease have no specific symptom, a feature which delays the diagnosis by 18 months to 2 years in average. In most cases, pulmonary arterial hypertension is
diagnosed when the cardiac output is already reduced, inevitably leading to death when the disease is not efficiently treated. During the evolution of the disease, the right ventricle is clearly
affected in two different ways : first the thickness of its walls increases (compensation) and second the ventricle inflates (decompensation). The latter phase remained unexplained.
Methods. We developed a dynamical model of the cardiovascular system in order to explain such a feature by regulation mechanisms. Since pulmonary arterial hypertension is a slowly evolving
pathology, we took into account long-term regulation mechanisms as the myocardial development (muscular heart development) ; we only considered the heart rate variations among the short-term
regulation mechanisms.
Results. Using a static model, we showed that the two phases observed during the evolution of pulmonary arterial hypertension can be explained by the interplay between the right and left
ventricles. We then showed that our dynamical model can reproduce the heart remodeling during these two phases (compensation and decompensation of the right ventricle). After the compensation
phase, the right ventricle can no longer maintain the cardiac output without the help of the left ventricle, which produces a part of the required work with the side effect of inflating the right
Conclusion. By taking into account slow regulation mechanisms, the cardiac remodeling during pulmonary arterial hypertension was reproduced. We thus showed that the two phases observed during the
increase in the pulmonary arterial resistance result from an interplay between the left and right ventricles.
F. Denis & C. Letellier,
Is high cancer rate in human due to a weakness in biology resulting from the rapid increase in lifetime expectancy ?
Bulletin du Cancer, 103 (3), 224-226, 2016. Online
Introduction Keane et al. recently showed that specific genes related to cell cycle and DNA repair of bowhead whale may explain the very low lifespan cancer rate measured in this cetacean [1], a
particularity selected during million years of evolution. In the other hand, Tomasetti et al. reported that variations in cancer risk among tissues can be correlated to the number of stem cell
divisions during the average lifetime of humans : longer one lives, greater his cancer risk [2]. There are thus two puzzling aspects in carcinogenesis that are i) the lack of correlation between
body size (the number of cells in an organism) and the incidence of cancer (Peto’s paradox), and ii) the lifetime cancer rate significantly greater in humans than in any other wild mammals.
C. Letellier & J.-M. Malasoma,
Architecture of chaotic attractors for flows in the absence of any singular point
Chaos, 26, 063115, 2016. Online
Abstract Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space
remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the
flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2x2 matrix describing the stability of flow in a subspace transverse to it. We will show
that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such
one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.
E. Fresnel, J.-F. Muir & C. Letellier
Performances of domiciliary ventilators compared by using a parametric procedure
EPJ Nonlinear Biomedical Physics, 4, 6, 2016. Online
Background Noninvasive mechanical ventilation is sufficiently widely used to motivate bench studies for evaluating and comparing performances of the domiciliary ventilators. In most (if not in
all) of the previous studies, ventilators were tested in a single (or a very few) conditions, chosen to avoid asynchrony events. Such a practice does not reflect how the ventilator is able to
answer the demand from a large cohort of patients with their inherent inter-patient variability. We thus developed a new procedure according which each ventilator was tested with more than 1200
“simulated” patients.
Methods Three lung mechanics (obstructive, restrictive and normal) were simulated using a mechanical lung (ASL 5000) driven by a realistic muscular pressure. 420 different dynamics for each of
these three lung mechanics were considered by varying the breathing frequency and the mouth occlusion pressure. For each of the nine ventilators tested, five different parameter settings were
investigated. The results are synthesized in colored maps where each color represents the ventilator (in)ability to synchronize with a given muscular pressure dynamics. A synchronizability ε is
then computed for each map.
Results The lung model, the breathing frequency and the mouth occlusion pressure strongly affect the synchronizability of ventilators. The Vivo 50 (Breas) and the SomnoVENT autoST (Weinmann) are
well synchronized with the restrictive model (ε=86 and 78 %, respectively), whereas the Elisée 150 (ResMed), the BiPAP A40 and the Trilogy 100 (Philips Respironics) better fit with an obstructive
lung mechanics (ε=87, 86 and 86 %, respectively). Triggering and pressurization performances of the nine ventilators present heterogeneities due to their different settings and operating
Conclusion Performances of domiciliary ventilators strongly depend not only on the breathing dynamics but also on the ventilator strategy. One given ventilator may be more adequate than another
one for a given patient.
I. Sendińa-Nadal, S. Boccaletti, & C. Letellier
Observability coefficients for predicting the class of synchronizability from the algebraic structure of the local oscillators
Physical Review E, 94, 042205, 2016. online
Understanding the conditions under which a collective dynamics emerges in a complex network is still an open problem. A useful approach is the master stability function --- and its related
classes of synchronization --- which offers a necessary condition to assess when a network successfully synchronizes. Observability coefficients, on the other hand, quantify how well the original
state space of a system can be observed given only the access to a measured variable. The question is therefore pertinent : given a generic dynamical system (represented by a state variable x)
and given a generic measure on it h(x) (which may be either an observation of an external agent, or an output function through which the units of a network interact), are classes of
synchronization and observability actually related to each other ? We explicitly address this issue, and show a series of non trivial relationships for networks of different popular chaotic
systems (Rössler, Lorenz and Hindmarsh-Rose oscillators). Our results suggest that specific dynamical properties can be evoked for explaining the classes of synchronizability.
L. Viger, F. Denis, C. Draghi, T. Ménard & C. Letellier
Spatial avascular growth of tumor in a homogeneous environment,
Journal of Theoretical Biology, 416, 99-112, 2016. online
Describing tumor growth is a key issue in oncology for correctly understanding the underlying mechanisms leading to deleterious cancers. In order to take into account the micro-environment in
tumor growth, we used a model describing — at the tissue level — the interactions between host (non malignant), effector immune and tumor cells to simulate the evolution of cancer. The spatial
growth is described by a Laplacian operator for the diffusion of tumor cells. We investigated how the evolution of the tumor diameter is related to the dynamics (periodic or chaotic oscillations,
stable singular points) underlying the interactions between the different populations of cells in proliferation sites. The sensitivity of this evolution to the key parameter responsible for the
immuno-evasion, namely the growth rate of effector immune cells and their inhibition rate by tumor cells, is also investigated.
[1] M. Keane, J. Semeiks, B. Thomsen, J. P. de Magalhaes, Insights into the Evolution of longevity from the Bowhead Whale Genome, Cell Reports, 10, 1-11, 2015.
[2] C. Tomasetti & B. Vogelstein, Variation in cancer incidence among tissues can be explained by the number of stem cell divisions, Science, 347, 78, 2015.
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What Does Out of Mean in Math?
Taking a course in math can be a entertaining activity that all students get pleasure from.
On the other hand, there are lots of topics that students have problems with when they 1st enroll in the course, and what will help them comprehend what calculus is all about.
Students who buy essays cheap discover that their math homework is too tricky, have taken a final exam which will not be approved for their class, or have questions regarding the idea behind
calculus, may possibly take into consideration taking an excursion in modern mathematics. A enjoyable excursion in math may aid a student find out additional concerning the topic.
The very first notion to grasp about what out of signifies in math is that it has to complete with multiplication. The word out of in math will be the prefix out- which suggests “out of” and – by
implies of which it signifies “by a similar approach.” As a result, out of in math suggests the addition of numbers.
Then, out of suggests “to be performed,” or to become completed. http://xroads.virginia.edu/%7EHYPER2/thoreau/civil.html Also, out of suggests “to start or start up.” As an example, d’Out=d ‘out ou’
“a D’s apple”
What does out of imply in math? When we add two numbers collectively, it can be accomplished by way of out of-multiplication. That is because the number right after the addition will normally be a
number lower than the a single prior to it. As a result, the only time it does not occur is when the second number includes a larger number than the initial one particular.
The following is from the Wikipedia.org internet site around the topic of mathematics. This definition of out of is additionally:
Math refers to a variety of distinct terms. A few of these are the mathematical operations involving numbers, mathematical reasoning, the study of number systems, the use of symbols and equations to
explain the properties of mathematical objects, as well as the abstraction of mathematical difficulties. The terms are vital mainly because they’re utilised to express or study something regarding
the factors called numbers and also other sorts of objects.
What does out of imply in math? https://www.buyessay.net/ It implies the addition of numbers, after which the multiplication of them.
What does out of mean in math? It suggests what out suggests in English and French.
What does out of mean in math? It implies that an addition has to do with multiplying by a number of the same steps as multiplying a number by its aspect. One example is, moreover, adding two numbers
requires either adding their things together or adding them separately.
What does out of imply in math? It means that the quantity before the addition will probably be a number lower than the one ahead of it.
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Basic College Mathematics (10th Edition) Chapter 5 - Ratio and Proportion - Review Exercises - Page 373 26
Work Step by Step
We can take the cross products of the given proportion to determine whether or not the proportion is true. If the cross products are equal, then we know that the proportion is true. $\frac{3\frac{1}
{2}}{2\frac{1}{3}}=\frac{\frac{7}{2}}{\frac{7}{3}}=\frac{6}{4}$ $\frac{7}{2}\times4=\frac{7}{3}\times6$ Left side: $\frac{28}{2}=\frac{28\div2}{2\div2}=14$ Right side: $\frac{42}{3}=\frac{42\div3}{3\
div3}=14$ Therefore, the proportion is True.
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What are the basic building blocks of Nature? What forces hold them together? These questions are at the heart of elementary particle physics. Theoretical physicists study and explore how to describe
elementary particles and their interactions. There are several concrete hints that tell us that we currently know only part of the puzzle.
In order to explore what pieces are missing, and to get hints for completing the picture, it is paramount to get clues from experimental observations. The scientific approach is to compare
experimental data to theoretical predictions. Obtaining the latter requires understanding of quantum field theory, which underpins the model for the known elementary particles and their interactions.
Scientists from the Quantum Field Theory department (Photo: Axel Griesch/MPP)
The Amplitudes research field focuses on understanding and computing probabilities of scattering processes in quantum field theory. The latter are the essential building blocks for cross-sections
that can be measured at particle colliders, such as the LHC at CERN, Geneva.
Apart from their phenomenological interest, scattering amplitudes also have intriguing mathematical properties. They satisfy a number of physical conditions, and have obvious and hidden symmetries.
In some situations, the amplitudes can be uniquely determined from their symmetry and analytic properties. Many modern methods of computing them are based on such structural insights.
Key words for specific research directions of the group:
• Scattering amplitudes of elementary particles
• Feynman integrals, differential equations, and special functions
• Infrared divergences in quantum field theory
• Conformal symmetry and supersymmetry
• AdS/CFT correspondence, N=4 super Yang-Mills
Quantum field theory at the MPP
E-mail address: e-mail@mpp.mpg.de
Phone number: +49 89 32354-extension
name function e-mail extension office
Carrôlo, Sérgio PhD Student scarrolo 201 A.3.73
Ditsch, Sara PhD Student ditsch 293 A.3.52
Hegde, Subramanya, Dr. Postdoc subbu 596 A.3.73
Henn, Johannes, Prof. Dr. Director henn 420 A.3.51
Kuo, Chia-Kai, Dr. Postdoc ckkuai 200 A.3.54
Lagares, Martín PhD Student lagares 608 A.3.66
Lim, Jungwon PhD Student wonlim 317 A.3.52
López-Falcón, Diana, Dr. Scientific Coordinator dlf 427 A.3.77
Ma, Rourou PhD Student marr 200 A.3.89
Mazzucchelli, Elia PhD Student eliam 200 A.3.54
Raman, Prashanth, Dr. Postdoc praman 317 A.3.75
Redl, Julia Secretary redl 334 A.3.47
Salvatori, Giulio, Dr. Postdoc giulios 334 A.3.54
Scholtes, Sorana Communication & Outreach scholtes 434 A.3.09
Vazão, Francisco PhD Student fvvazao 200 A.3.89
Willen, Aaron Student willen 334 A.3.66
Yang, Qinglin Postdoc qlyang 544 A.3.75
Zhang, Shun-Qing, Dr. Postdoc sqzhang 405 A.3.87
Zhu, Yujiao Postdoc yzhu 427 A.3.50
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Research Profile
I work in differential geometry and dynamical systems. These two fields have many connections. The geodesic flow in particular introduces dynamical techniques to study geometric problems. For
example, finding closed geodesics is equivalent to finding periodic orbits of the geodesic flow. Does every compact Riemannian manifold have infinitely many closed geodesics? For the case of the
two-dimensional sphere, this age old problem was resolved using ideas from low-dimensional dynamics in the late 90s.
Most of my work concerns rigidity properties of both Riemannian manifolds and dynamical systems. This area is has developed from the classical theorems due to Mostow and Margulis in the 1960s and
70s. The Strong Rigidity Theorem by Mostow is the paradigm: In dimension at least 3, the fundamental group of a closed manifold of constant negative curvature determines its isometry type. Ideas and
techniques from differential geometry, group theory and dynamical systems were all essential to the proof. Margulis greatly strenghtened this result in his superrigidity and arithmeticity theorems
under the much more stringent higher rank assumption that the manifolds in questions have lots of totally geodesic flat subspaces.
These results have inspired many works. In geometry, there are various characterizations of symmetric spaces in terms of simple geometric properties. For example, non-positively curved closed
manifolds have to be locally symmetric if every geodesic is contained in a totally geodesic flat subspace, much in the spirit of Margulis’ theorem. The latter also was generalized to certain
symmetric spaces with negative curvature, and some non-Riemannian spaces of higher rank.
In dynamics, these results inspired the investigation of actions of higher rank abelian and semi-simple groups. Surprisingly, suitable such actions are locally rigid, i.e. cannot be perturbed, they
do not have time changes and few invariant probability measures. Some of these actions incorporate phenomena from number theory where the scarcity of invariant measures has given rise to important
If you want to get a better feeling for this area of research, I will be glad to talk with you and/or direct you to some introductory reading, e.g. my recent survey An Invitation to Rigidity Theory.
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Pseudo almost periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays and leakage delays on time scales
[1] B. Xu, X. Liu, X. Liao, Global asymptonic stability of high-order Hopfield neural networks with time delays, Comput. Math. Appl., 45 (2003), 1729-1737. doi: 10.1016/S0898-1221(03)00151-2
[2] X. Lou, B. Cui, Novel global stability criteria for high-order Hopfield-type neural networks with time-varying delays, J. Math. Anal. Appl., 330 (2007), 144-158. doi: 10.1016/j.jmaa.2006.07.058
[3] C. Ou, Anti-periodic solutions for high-order Hopfield neural networks, Comput. Math. Appl., 56 (2008), 1838-1844. doi: 10.1016/j.camwa.2008.04.029
[4] Y. Yu, M. Cai, Existence and exponential stability of almost-periodic solutions for high-order Hopfield neural networks, Math. Comput. Model., 47 (2008), 943-951. doi: 10.1016/j.mcm.2007.06.014
[5] B. Xiao, H. Meng, Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks, Appl. Math. Model., 33 (2009), 532-542. doi: 10.1016/
[6] L. Duan, L. Huang, Z. Guo, Stability and almost periodicity for delayed high-order Hopfield neural networks with discontinuous activations, Nonlinear Dynam., 77 (2014), 1469-1484. doi: 10.1007/
[7] C. Xu, P. Li, Pseudo almost periodic solutions for high-order Hopfield neural networks with time-varying leakage delays, Neural Process. Lett., 46 (2017), 41-58. doi: 10.1007/s11063-016-9573-3
[8] C. Xu, P. Li, Y. Pang, Global exponential stability for interval general bidirectional associative memory (BAM) neural networks with proportional delays, Math. Method. Appl. Sci., 39 (2016),
5720-5731. doi: 10.1002/mma.3957
[9] F. Kong, Q. Zhu, K Wang, J. J. Nieto, Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator, J. Franklin. I., 356
(2019), 11605-11637. doi: 10.1016/j.jfranklin.2019.09.030
[10] F. Kong, R. Rajan, Finite-time and fixed-time synchronization control of discontinuous fuzzy Cohen-Grossberg neural networks with uncertain external perturbations and mixed time, Fuzzy Set.
Syst., 411 (2021), 105-135. doi: 10.1016/j.fss.2020.07.009
[11] W. Shen, X. Zhang, Y. Wang, Stability analysis of high order neural networks with proportional delays, Neurocomputing, 372 (2020), 33-39. doi: 10.1016/j.neucom.2019.09.019
[12] Z. Dong, X. Zhang, X. Wang, State estimation for discrete-time high-order neural networks with time-varying delays, Neurocomputing, 411 (2020), 282-290. doi: 10.1016/j.neucom.2020.06.047
[13] K. Gopalsamy, Kondalsamy, Leakage delays in BAM, J. Math. Anal. Appl., 325 (2007), 1117-1132. doi: 10.1016/j.jmaa.2006.02.039
[14] P. Balasubramaniam, V. Vembarasan, R. Rakkiyappan, Leakage delays in T-S fuzzy cellular neural networks, Neural Process. Lett., 33 (2011), 111-136. doi: 10.1007/s11063-010-9168-3
[15] R. Sakthivel, P. Vadivel, K. Mathiyalagan, A. Arunkumar, M. Sivachitra, Design of state estimator for bidirectional associative memory neural networks with leakage delays, Inform. Sciences, 296
(2015), 263-274. doi: 10.1016/j.ins.2014.10.063
[16] C. Xu, L. Chen, P. Li, Effect of proportional delays and continuously distributed leakage delays on global exponential convergence of CNNs, Asian J. Control, 21 (2019), 2476-2483. doi: 10.1002/
[17] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56. doi: 10.1007/BF03323153
[18] Y. Li, C. Wang, Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstr. Appl. Anal., 2011 (2011), 341520.
[19] Y. Li, L. Yang, Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales, Appl. Math. Comput., 242 (2014), 679-693.
[20] W. Yang, W. Yu, J. Cao, F. Alsaadi, T. Hayat, Almost automorphic solution for neutral type high-order Hopfield BAM neural networks with time-varying leakage delays on time scales, Neurocomputing
, 267 (2017), 241-260. doi: 10.1016/j.neucom.2017.05.089
[21] Y. Li, X. Meng, L. Xiong, Pseudo almost periodic solutions for neutral type high-order Hopfield neural networks with mixed time-varying delays and leakage delays on time scales, Int. J. Mach.
Learn. Cyb., 8 (2017), 1915-1927. doi: 10.1007/s13042-016-0570-7
[22] T. Isokawa, T. Kusakabe, N. Matsui, F. Peper, Quaternion neural network and its application, Berlin, Springer, 2003.
[23] N. Matsui, T. Isokawa, H. Kusamichi, F. Peper, H. Nishimura, Quaternion neural network with geometrical operators, J. Intell. Fuzzy Syst., 15 (2004), 149-164.
[24] M. Yoshida, Y. Kuroe, T. Mori, Models of hopfield-type quaternion neural networks and their energy functions, Int. J. Neural Syst., 15 (2005), 129-135. doi: 10.1142/S012906570500013X
[25] Y. Li, J. Qin, B. Li, Existence and global exponential stability of anti-periodic solutions for delayed quaternion-valued cellular neural networks with impulsive effects, Math. Method. Appl.
Sci., 42 (2019), 5-23. doi: 10.1002/mma.5318
[26] N. Huo, B. Li, Y. Li, Existence and exponential stability of anti-periodic solutions for inertial quaternion-valued high-order Hopfield neural networks with state-dependent delays, IEEE Access,
7 (2019), 60010-60019. doi: 10.1109/ACCESS.2019.2915935
[27] Y. Li, H. Wang, X. Meng, Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays, IMA J. Math. Control I., 36 (2019),
983-1013. doi: 10.1093/imamci/dny015
[28] Y. Li, J. Xiang, Existence and global exponential stability of almost periodic solution for quaternion-valued high-order Hopfield neural networks with delays via a direct method, Math. Method.
Appl. Sci., 43 (2020), 6165-6180. doi: 10.1002/mma.6363
[29] H. Wang, G. Wei, S. Wen, T, Huang, Impulsive disturbance on stability analysis of delayed quaternion-valued neural networks, Appl. Math. Comput., 390 (2021), 125680.
[30] C. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl., 151 (1994), 62-76.
[31] T. Diagana, Pseudo almost periodic solutions to some differential equations, Nonlinear Anal. Theor., 60 (2005), 1277-1286. doi: 10.1016/j.na.2004.11.002
[32] F. Kong, X. Fang, Pseudo almost periodic solutions of discrete-time neutral-type neural networks with delays, Appl. Intell., 48 (2018), 3332-3345. doi: 10.1007/s10489-018-1146-x
[33] A. Zhang, Almost periodic solutions for SICNNs with neutral type proportional delays and D operators, Neural Process. Lett., 47 (2018), 57-70. doi: 10.1007/s11063-017-9631-5
[34] A. Zhang, Pseudo almost periodic high-order cellular neural networks with complex deviating arguments, Int. J. Mach. Learn. Cyb., 10 (2019), 301-309. doi: 10.1007/s13042-017-0715-3
[35] Y. Li, C. Wang, Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales, Adv. Differ. Equ., 2012 (2012), 77. doi: 10.1186/1687-1847-2012-77
[36] A. Arbi, J. Cao, Pseudo-almost periodic solution on time-space scales for a novel class of competitive neutral-type neural networks with mixed time-varying delays and leakage delays, Neural
Process. Lett., 46 (2017), 719-745. doi: 10.1007/s11063-017-9620-8
[37] A. Zhang, Pseudo almost periodic solutions for neutral type SICNNs with D operator, J. Exp. Theor. Artif. In., 29 (2017), 795-807. doi: 10.1080/0952813X.2016.1259268
[38] M. Bohner, A. Peterson, Dynamic equations on time scales: An introuduction with applications, Springer Science & Business Media, 2001.
[39] M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Springer Science & Business Media, 2002.
Reader Comments
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Missing Angles In Triangles And Quadrilaterals Worksheet Tes - TraingleWorksheets.com
Angles In Quadrilaterals And Triangles Worksheet – Triangles are among the most fundamental patterns in geometry. Understanding the triangle is essential to learning more advanced geometric terms. In
this blog this post, we’ll go over the different types of triangles that are triangle angles. We will also explain how to calculate the size and perimeter of a triangle, and give the examples for
each. Types of Triangles There are three types to triangles: the … Read more
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Cyflex User Portal
Set filter coefficient for engine acceleration (dN/dt) measurement.
The set_dndt_k command sets or changes the value of the filter coefficient on the engine acceleration (dN/dt) measurement channel. The coefficient is used in an optional recursive filter on the
analog input signal for the acceleration measurement to “smooth” the transient response. Filter values near 0.0 and 1.0 produce small and large filtering effects, respectively.
The command rejects any value for the filter coefficient that is less than 0.0 (no filter) or is greater than or equal to 1.0 (total filter).
set_dndt_k [filter_coefficient]
set_dndt_k [frequency_cutoff in hz]
set_dndt_k [time_constant]
filter_coefficient Specify the coefficient for the recursive filter on the engine acceleration measurement channel. Valid values are 0.0-1.0
frequency_cutoff Specify the 1st order filter frequency.
time_constant Specify the 1st order filter time constant.
set_dndt_k 0.10
specifies recursive filter coeff
set_dndt_k .5[ hz ]
specifies cutoff frequency
set_dndt_k .1[ sec ]
specifies time constant
The recursive filter with coefficient k computes a filtered value Y(n) that is the contribution of the previous filtered value Y(n-1) and the current raw measurement X(n):
Y(n) = k * Y(n-1) + (1-k) * X(n)
See Also:
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Photoelectric Effect
Photons: The Quanta of Light
According to the Planck hypothesis, all electromagnetic radiation is quantized and occurs in finite "bundles" of energy which we call photons. The quantum of energy for a photon is not Planck's
constant h itself, but the product of h and the frequency. The quantization implies that a photon of blue light of given frequency or wavelength will always have the same size quantum of energy. For
example, a photon of blue light of wavelength 450 nm will always have 2.76 eV of energy. It occurs in quantized chunks of 2.76 eV, and you can't have half a photon of blue light - it always occurs in
precisely the same sized energy chunks.
But the frequency available is continuous and has no upper or lower bound, so there is no finite lower limit or upper limit on the possible energy of a photon. On the upper side, there are practical
limits because you have limited mechanisms for creating really high energy photons. Low energy photons abound, but when you get below radio frequencies, the photon energies are so tiny compared to
room temperature thermal energy that you really never see them as distinct quantized entities - they are swamped in the background. Another way to say it is that in the low frequency limits, things
just blend in with the classical treatment of things and a quantum treatment is not necessary.
│ │ Interaction of radiation with matter │ Electromagnetic spectrum │ Basic quantum processes │
Photoelectric effect
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ASHIGIFT - editorial
Author: Pankaj Jindal, Piyush Kumar
Tester: Sergey Kulik
Editorialist: Aman jain
Binary search
Poor Chef now wants to go to Byteland. On the way to Byteland he found B tasty dishes, each dish requires B_i people to complete, after eating the dish one can’t move forward. He can ask support from
C tribal clans, found on the way, but on one condition by each clan that is, “they will join only if he approaches them with a group of size atleast q_i”. Now chef wants to know with what minimum
number of people including him he should start his journey to Byteland.
Binary search over the range of group size Chef can start with and check whether it is possible to reach Byteland with the group size.
Its easy to observe that if there are no tribal clans in the path, then Chef has to start with x = 1+\sum{B_i} people to reach Byteland, first sub-task can be solved using this. But how can he
minimize number of people on start when help from tribal clans is available?
Basic idea that might struck to your mind would be to try all possible values in range and check what should be minimum size to reach Byteland, one can easily see that the ans would be in range [1,x]
, since in the worst case no clan members joins you. Complexity of this solution would be O(x*(B+C)), where B is number of dishes and C is number of tribal clans, that’s about 10^{13} computations
which do not fit in our time limit.
If Chef start with x people (including him) and can reach Byteland, then if he starts with more than x people, then also he can reach Byteland.
Let f(n): checks whether its possible to reach Byteland with n people or not. You can do binary search over range [1,x] to find out minimum n such that f(n) is true, as answer would lie in range
[1,x] only. For binary searching in a range, the idea is to check value of function in mid point of current range and decide to go to one half depending on its value. This will ensure that in log(x)
steps we will reach the optimal value.
low = 1, high = x, ans = x
while(low <= high){
m = (low+high)/2
ans = min(ans,m);
high = m-1;
else low = m+1;
// store dishes and clans in a vector, in increasing order of their distance from Chef's town
// pi,qi,ri are same convention described in problem
for each item in vector:
if item is a dish:
else if v >= qi: v+= ri
return (v>0)
Binary search takes O(log(x))
Possible function takes O(B+C)
Total complexity should be O(log(x)*(B+C))
AUTHOR’S, TESTER’S and Editorialist’s SOLUTIONS:
4 Likes
i had a different solution by dynamic programming:
The original problem was to find the minimum number of people such that at least one man can reach X. I added a dummy dish at the final point X with y=1 so that now 0 men need to reach the end (after
Now, i stored all the locations of dishes and tribes in an array and sorted them according to the distance from origin. Let min[i] be the minimum number of people required to reach the end, when
starting from ith tribal location. I found this value for all tribal locations starting from the farthest tribe from the origin to the one nearest to the origin.
EDIT: there is no need for sorting, since it is given that the elements will be given in the sorted order. So, Complexity : O(B+C)
[1]: http://www.codechef.com/viewsolution/6593138
8 Likes
hmm, its a good technique to solve the problem. First, finding the upper limit, then using binary search to get the exact solution. Really enjoyed after solving
1 Like
My Binary-search based solution gave WA for one testcase in both categories (30 and 70).
Finally the DP worked!
Nice editorial… very good explanation… thank you very much
can any body tell me why this is giving wrong answer
Can you please explain how will the answer be in range [1,X]? X is the distance of byteland from chef’s town. In worst case, shouldn’t the answer be sum of all yi(no. of people req. to finish a
And can anyone tell me how to solve this problem using dp?
can anyone tell y am i getting wrong answer. Thanks in advance
your code would not work for following test,
actual ans = 2
Moreover greedy approach is wrong.
I am continuously getting WA for 2nd subtask. Can anyone provide me a test case to get it AC?
1 Like
I did it using dynamic programming. Very similar to dungeon princess problem.
Wat was the reason behind your assumption that ans would be in range[1,x] ?
This x is not the same as the x defined in the question. In this, it’s the number of people required without clan tribes
I am getting TLE. do not know why. can someone help?
I am getting TLE on subtask 2. Can anyone check and tell why?
Thanks in advance!
I am getting WA in last case.Can anyone check and tell why?
I used greedy approach and it worked. I traversed all the points (dishes and clans together) backward and at each clan, decided whether its support was required or not.
Here’s my passing solution. Here’s a passing solution from another user following the same idea.
What if BOTH Positions (Dish && Tribals ) are same ?
@ad It is stated in the problem that they are distinct.
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What is Safety factor / Factor of safety ? - ExtruDesign
In the design of machine members, it is necessary to understand the Safety factor (SF). It is also known as Factor of safety (FoS). It describes that the load carrying capacity of any designed
member. How much stronger the system than the expected or intended. Make sense? Let’s get into the topic thoroughly.
The Safety Factor
It is defined as the ratio of the Maximum Stress to the applicable Working stress.
Safety Factor(SF) = Maximum stress/Working stress
Further, we can describe the factor of safety between the ductile materials and the brittle materials.
The safety Factor based on material property
For Ductile materials, the yield point is defined so the factor of safety can be defined as the ratio of the Yield point stress to the Working stress.
The factor of safety (FoS) = Yield point stress/ Working stress.
For Brittle materials, the yield point is cannot be defined well. so the factor of safety can be defined as the ratio of the Ultimate stress to the Working stress.
The factor of safety (FoS) = Ultimate stress/ Working stress.
Have you heard Service factor?
And the term Service factor is not to be confused with the Safety factor.
Service Factor vs Safety Factor
Service factor is the ratio of the capacity to the duty. Here we are talking about the oversizing the life of a product.
Service Factor = Capacity/ Duty
If Service factor is 1.50 that means the device can be operated at 50% more load than its rated capacity without breakdown.
Whereas the Safety factor will be considered in the designing methodologies to determine at which point it is going to fail.
Let’s take an example of a bearing needed to be designed for a shaft that is operating at 3600 RPM. We have a bearing in the market that is capable to run at 10000 RPM. Then the factor of safety is
defined as
Service Factor = Capacity/ Duty
Service factor = 10000/3600 = 2.7778
2.78 is the Service factor.
And not only with the speed, there are so many other factors like stress, torque, etc that we can calculate the Service Factor. Generally, torque is the most significant factor to calculate service
The safety factor for any machine or machine part can be defined as the Capacity divided by the Duty or operating condition of the machine part.
1. Sunil K says
It is So helpful for mechanical students.
THANK U SO MUCH SIR
□ Sundar Dannana says
Thank you Sunil. Glad you like it.
Leave a Reply Cancel reply
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De Nederlandse Vereniging voor Logica &
Wijsbegeerte der Exacte Wetenschappen
Symposium `Proofs and Problems' (plus ALV)
The Vereniging voor Logica (VvL) organizes a symposium
Proofs and Problems
on Friday November 25, 2005, in the Jaarbeurs in Utrecht. Participation is free, registration not necessary. The programme is as follows:
│12.30 - 13.30│lunch en algemene ledenvergadering │
│13.30 - 14.15│prof. dr. Jan van Leeuwen, Utrecht University │
│ │Great Open Problems in Computational Complexity │
│14.15 - 14.30│tea and coffee │
│14.30 - 15.15│dr. Lev Beklemishev, Utrecht University │
│ │Problems in the Logic of Provability │
│15.15 - 15.30│tea and coffee │
│15.30 - 16.15│prof. dr. Henk Barendregt, Radboud University │
│ │State of the Art of Computer Mathematics │
│16.15 - 17.00│drinks │
Abstract of the talk by Jan van Leeuwen:
Ever since it was listed among the seven Millennium Prize problems by the Clay Mathematics Institute in 2000, the P-versus-NP problem is one of the most widely known open problems of our field. The
problem was `discovered' some thirty-four years ago and is known to capture the computational difficulty of many questions in efficient algorithm design, ranging from combinatorial optimization to
logic and further. For known attempts to solve P-versus-NP (either way) see here. However, P-versus-NP is only one of the key open problems in complexity theory. What do we know about the problem,
and what are typical open problems in other computational contexts? We survey the salient features of P-versus-NP and related problems and of some of the historic approaches and results known today
such as the highly non-trivial PCP-theorem.
Abstract of the talk by Lev Beklemishev:
Provability logic emerged in the 1970s as a study of formal provability as a "mathematical" kind of modality. The field experienced a period of rapid development in the 80s and 90s. We shall discuss
some long standing open problems in this area as well as possible directions for future research.
Abstract of the talk by Henk Barendregt:
Progress in the foundations of mathematics has made it possible to formulate all thinkable mathematical concepts, algorithms and proofs in one language and in an impeccable way. This not in spite of,
but partially based on the famous results of Goedel and Turing. In this way statements are about mathematical objects and algorithms, proofs show the correctness of statements and computations, and
computations are dealing with objects and proofs. Interactive computer systems for a full integration of defining, computing and proving are based on this. The human defines concepts, constructs
algorithms and provides proofs, while the machine checks that the definitions are well-formed and the proofs and computations are correct. Results formalized so far demonstrate the feasibility of
computer mathematics. There are very good applications. The challenge is to make the systems more mathematician-friendly, by building libraries and tools. The eventual goal is to help humans to
learn, develop, communicate, referee and apply mathematics.
Address and how to get there
Jaarbeurs Utrecht
Jaarbeursplein 6, Beatrixgebouw, zaal 412
3521 AL Utrecht
Telefoon 030 - 295 59 11
only 10 minutes walk from Utrecht Centraal Station
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DukeSpace :: Browsing by Author "Zavlanos, Michael M"
Browsing by Author "Zavlanos, Michael M"
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Zeroth-Order (ZO) methods have been applied to solve black-box or simulation-based optimization prroblems. These problems arise in many important applications nowa- days, e.g., generating
adversarial attacks on machine learning systems, learning to control the system with complicated physics structure or human in the loop. In these problem settings, the objective function to
optimize does not have an explicit math- ematical form and therefore its gradient cannot be obtained. This invalidates all gradient-based optimization approaches. On the other hand, ZO methods
approxi- mate the gradient by using the objective function values. Many existing ZO methods adopt the two-point feedback scheme to approximate the unknown gradient due to its low estimation
variance and fast convergence speed. Specifically, two-point ZO method estimates the gradient at the current iterate of the algorithm by querying the objective function value twice at two
distinct neighbor points around the current iterate. Such scheme becomes infeasible or difficult to implement when the objective function is time-varying, or when multiple agents collaboratively
optimize a global objective function that depends on all agents’ decisions, because the value of the objective function can be queried only once at a single decision point. However, con-
ventional ZO method based on one-point feedback is subject to large variance of the gradient estimation and therefore slows down the convergence.
In this dissertation, we propose a novel one-point ZO method based on the residual feedback. Specifically, the residual feedback scheme estimates the gradient using the residual between the
values of the objective function at two consecutive iterates of the algorithm. When optimizing a deterministic Lipschitz function, we show that the query complexity of ZO with the proposed
one-point residual feedback matches that of ZO with the existing two-point schemes. Moreover, the query complexity of the proposed algorithm can be improved when the objective function has
Lipschitz gradient. Then, for stochastic bandit optimization problems, we show that ZO with one-point residual feedback achieves the same convergence rate as that of two-point scheme with
uncontrollable data samples.
Next, we apply the proposed one-point residual-feedback gradient estimator to solve online optimizaiton problems, where the objective function varies over time. In the online setting, since each
objective function can only be evaluated once at a single decision point, existing two-point ZO methods are not feasible and only one-point ZO methods can be used. We develop regret bounds for ZO
with the proposed one- point residual feedback scheme for both convex and nonconvex online optimization problems. Specifically, for both deterministic and stochastic problems and for both
Lipschitz and smooth objective functions, we show that using residual feedback can produce gradient estimates with much smaller variance compared to conventional one- point feedback methods. As a
result, our regret bounds are much tighter compared to existing regret bounds for ZO with conventional one-point feedback, which suggests that ZO with residual feedback can better track the
optimizer of online optimization problems. Additionally, our regret bounds rely on weaker assumptions than those used in conventional one-point feedback methods.
The proposed residual-feedback scheme is next decentralized to conduct dis- tributed policy optimization in the multi-agent reinforcement learning (MARL) prob- lems. Existing MARL algorithms
often assume that every agent can observe the states and actions of all the other agents in the network. This can be impractical in large-scale problems, where sharing the state and action
information with multi-hop neighbors may incur significant communication overhead. The advantage of the pro- posed zeroth-order policy optimization method is that it allows the agents to compute
the local policy gradients needed to update their local policy functions using local estimates of the global accumulated rewards that depend on partial state and action information only and can
be obtained using consensus. Specifically, the local ZO policy gradients relying on one-point residual-feedback significantly reduces the vari- ance of the local policy gradient estimates
compared to the conventional one-point policy gradient estimates, improving, in this way, the learning performance. We show that the proposed distributed zeroth-order policy optimization method
with constant stepsize converges to a neighborhood of the global optimal policy that depends on the number of consensus steps used to calculate the local estimates of the global accumulated
rewards.Another challenge in the distributed ZO optimization problems is that the agents may conduct local updates in an asynchronous fashion when they do not have ac- cess to a global clock. To
deal with this challenge, we propose an asynchronous zeroth-order distributed optimization method that relies on the proposed one-point residual feedback gradient estimator. We show that this
estimator is unbiased under asynchronous updating, and theoretically analyze its convergence. We demonstrate the effectiveness of all proposed algorithms via extensive numerical experiments.
• Adaptive Planning in Changing Policies and Environments
(2023) Sivakumar, Kavinayan Pillaiar
Being able to adapt to different tasks is a staple of learning, as agents aim to generalize across different situations. Specifically, it is important for agents to adapt to the policies of other
agents around them. In swarm settings, multi-agent sports settings, or other team-based environments, agents learning from one another can save time and reduce errors in performance. As a result,
traditional transfer reinforcement learning proposes ways to decrease the time it takes for an agent to learn from an expert agent. However, the problem of transferring knowledge across agents
that operate in different action spaces and are therefore heterogeneous poses new challenges. Mainly, it is difficult to translate between heterogeneous agents whose action spaces are not
guaranteed to intersect.
We propose a transfer reinforcement learning algorithm between heterogeneous agents based on a subgoal trajectory mapping algorithm. We learn a mapping between expert and learner trajectories
that are expressed through subgoals. We do so by training a recurrent neural network on trajectories in a training set. Then, given a new task, we input the expert's trajectory of subgoals into
the trained model to predict the optimal trajectory of subgoals for the learner agent. We show that the learner agent is able to learn an optimal policy faster with this predicted trajectory of
It is equally important for agents to adapt to the intentions of agents around them. To this end, we propose an inverse reinforcement learning algorithm to estimate the reward function of an
agent as it updates its policy over time. Previous work in this field assume the reward function is approximated by a set of linear feature functions. Choosing an expressive enough set of feature
functions can be challenging, and failure to do so can skew the learned reward function. Instead, we propose an algorithm to estimate the policy parameters of the agent as it learns, bundling
adjacent trajectories together in a new form of behavior cloning we call bundle behavior cloning. Our complexity analysis shows that using bundle behavior cloning, we can attain a tighter bound
on the difference between the distribution of the cloned policy and that of the true policy than the same bound achieved in standard behavior cloning. We show experiments where our method
achieves the same overall reward using the estimated reward function as that learnt from the initial trajectories, as well as testing the feasibility of bundle behavior cloning with different
neural network structures and empirically testing the effect of the bundle choice on performance.
Finally, due to the need for agents to adapt to environments that are prone to change due to damage or detection, we propose the design of a robotic sensing agent to detect damage. In such
dangerous environments, it may be unsafe for human operators to manually take measurements. Current literature in structural health monitoring proposes sequential sensing algorithms to optimize
the number of locations measurements need to be taken at before locating sources of damage. As a result, the robotic sensing agent we designed is mobile, semi-autonomous, and precise in measuring
a location on the model structure we built. We detail the components of our robotic sensing agent, as well as show measurement data taken from our agent at two locations on the structure
displaying little to no noise in the measurement.
• Decentralized State Estimation using Robotic Sensor Networks
(2016) Freundlich, Charles
This dissertation proposes three control algorithms for active sensing with one or several autonomous robots.
The algorithms all rely on models of the information content of the sensor measurement with respect to the relative poses between sensors and subjects.
The approaches each predict how new information may impact the uncertainty in the subjects, controlling sensors to new locations or trajectories from where these uncertainties will me minimized.
The first algorithm deals with the Next-Best-View (NBV) problem for a single robot, where the goal is to control a mobile camera so that the next image of a set of possibly mobile targets will be
as informative as possible.
The NBV controller is designed for a rig that hosts two cameras in a fronto-parallel arrangement, commonly known as stereo vision.
Assuming that the objects, landmarks, or targets being estimated are visible by both cameras in the rig and that these observations are corrupted by zero-mean Gaussian errors, the control
algorithm moves the rig through pose space in order to reduce the expected Kalman-filtered uncertainty in the next location point-estimate.
This is done by differentiating the KF output error covariance matrix with respect to the sensor pose, which results in a nonlinear control problem.
The controller is decomposed so that first the robot computes the NBV in coordinates relative to the body-frame of the stereo rig, and then it moves in pose space to realize this view.
When an image is acquired, a switching signal changes the goal of pose control, giving rise to a stable hybrid system.
Experiments of on a real robot localizing targets in a laboratory setting are presented.
The second algorithm addresses the problem of estimating a finite set of hidden state vectors using a mobile robotic sensor network.
For every hidden state that needs to be estimated, a local Dynamic Program (DP) in the joint state-space of robot positions and state uncertainties determines robot paths and associated sequences
of state observations that collectively minimize the estimation uncertainty.
It divides the collection of hidden states into clusters based on a prior belief of their geographic locations and, for each cluster, defines a second DP that determines how far along the local
optimal trajectories the robot should travel before transitioning to estimating the next hidden state within the cluster.
Finally, a distributed assignment algorithm dynamically allocates controllers to the robot team from the set of optimal control policies at every cluster.
Assuming Gaussian priors on the hidden state vectors, the distributed state estimation method scales gracefully to large teams of mobile robots and hidden vectors and provide extensive
simulations and real-world experiments using stereoscopic vision sensors to illustrate the approach.
The third chapter addresses the problem of controlling a network of mobile sensors so that a set of hidden states are estimated up to a user-specified accuracy. The sensors take measurements and
fuse them online using an Information Consensus Filter (ICF). At the same time, the local estimates guide the sensors to their next best configuration. This leads to an LMI-constrained
optimization problem that we solve by means of a new distributed random approximate projections method. The new method is robust to the state disagreement errors that exist among the robots as
the ICF fuses the collected measurements. Assuming that the noise corrupting the measurements is zero-mean and Gaussian and that the robots are self localized in the environment, the integrated
system converges to the next best positions from where new observations will be taken. This process is repeated with the robots taking a sequence of observations until the hidden states are
estimated up to the desired user-specified accuracy. It presents simulations of sparse landmark localization, where the robotic team is achieves the desired estimation tolerances while exhibiting
interesting emergent behavior.
Experiments of the first two algorithms are also presented.
• Deep Reinforcement Learning with Temporal Logic Specifications
In this thesis, we propose a model-free reinforcement learning method to synthesize control policies for mobile robots modeled by Markov Decision Process (MDP) with unknown transition
probabilities that satisfy Linear Temporal Logic (LTL) specifications. The key idea is to employ Deep Q-Learning techniques that rely on Neural Networks (NN) to approximate the state-action
values of the MDP and design a reward function that depends on the accepting condition of the Deterministic Rabin Automaton (DRA) that captures the LTL specification. Unlike relevant works, our
method does not require learning the transition probabilities in the MDP, constructing a product MDP, or computing Accepting Maximal End Components (AMECs). This significantly reduces the
computational cost and also renders our method applicable to planning problems where AMECs do not exist. In this case, the resulting control policies minimize the frequency with which the system
enters bad states in the DRA that violate the task specifications. To the best of our knowledge, this is the first model-free deep reinforcement learning algorithm that can synthesize policies
that maximize the probability of satisfying an LTL specification even if AMECs do not exist. We validate our method through numerical experiments.
• Distributed Intermittent Connectivity Control of Mobile Robot Networks
Wireless communication is known to play a pivotal role in enabling teams of robots to successfully accomplish global coordinated tasks. In fact, network connectivity is an underlying assumption
in every distributed control and optimization algorithm. For this reason, in recent years, there is growing research in designing controllers that ensure point-to-point or end-to-end network
connectivity for all time. Nevertheless, all these methods severely restrict the robots from accomplishing their tasks, as motion planning is always restricted by connectivity constraints on the
network. Instead, a much preferred solution is to enable robots to communicate in an intermittent fashion, and operate in disconnect mode the rest of the time giving rise to an intermittently
connected communication network. While in disconnect mode, the robots can accomplish their tasks free of communication constraints. The goal of this dissertation is to design a distributed
intermittent connectivity framework that (i) ensures that the communication network is connected over time, infinitely often (ii) is flexible enough to account for arbitrary dynamic tasks, and
(iii) can be applied to large-scale networks.
The great challenge in developing intermittent connectivity protocols for networks of mobile robots is to decide (i) which robots talk to which, (ii) where, and (iii) when, so that the
communication network is connected over time infinitely often. To address these challenges, we decompose the network into small groups of robots, also called teams, so that every robot belongs to
at least one team and that there is a path, i.e., a sequence of teams, where consecutive teams have non-empty intersections, connecting every two teams of robots, so that information can
propagate in the network. First, given such fixed teams, we design infinite sequences of communication events for all robots, also called communication schedules, independent of the tasks
assigned to the robots, that determine when every team should communicate, so that the communication network is connected over time infinitely often. The designed communication schedules ensure
that all teams communicate infinitely often, i.e., that the communication network is connected over time infinitely often. Between communication events the robots can move in the workspace free
of communication constraints to accomplish their assigned tasks. Theoretical guarantees and numerical experiments corroborate the proposed framework. This is the first distributed intermittent
connectivity framework that can be applied to large-scale networks and is flexible enough to account for arbitrary dynamic robot tasks.
Next, given user-specified fixed teams, we integrate the respective communication schedules with task planning. Specifically, we consider high-level complex tasks captured by temporal logic
formulas, state-estimation tasks, and time-critical dynamic tasks. The proposed distributed integrated path planning and intermittent connectivity frameworks determine both where and when every
team should communicate so that the assigned task is accomplished, the communication network is connected over time infinitely often, and a user-specified metric, such as total traveled distance
or consumed energy, is minimized. We show that employing the proposed intermittent connectivity framework for such tasks results in significant performance gains compared to the existing
solutions in the literature that maintain connectivity for all time. Theoretical guarantees, numerical and experimental studies support the proposed distributed control algorithms.
Finally, we propose a fully autonomous intermittent connectivity framework that can handle arbitrary dynamic tasks and also allows the robots to locally and online update the structure of the
teams and the communication schedules, effectively allowing them to decide who they should talk to, so that they can better accomplish newly assigned tasks. The structure of the teams, the
associated communication locations, and the time instants when communication within teams will occur are integrated online with task planning giving rise to paths, i.e., sequences of waypoints,
that ensure that the assigned task is accomplished, the communication network is connected over time infinitely often, and a user specified metric is minimized. This is the first fully
autonomous, distributed, and
online intermittent connectivity framework that can handle arbitrary dynamic tasks and also controls the topology of the intermittently connected robot network to better accomplish these tasks.
At the same time, the proposed framework scales well with the size of the robot network. Theoretical guarantees and numerical experiments corroborate the proposed distributed control scheme.
• Distributed Optimization Algorithms for Networked Systems
(2015) Chatzipanagiotis, Nikolaos
Distributed optimization methods allow us to decompose an optimization problem
into smaller, more manageable subproblems that are solved in parallel. For this
reason, they are widely used to solve large-scale problems arising in areas as diverse
as wireless communications, optimal control, machine learning, artificial intelligence,
computational biology, finance and statistics, to name a few. Moreover, distributed
algorithms avoid the cost and fragility associated with centralized coordination, and
provide better privacy for the autonomous decision makers. These are desirable
properties, especially in applications involving networked robotics, communication
or sensor networks, and power distribution systems.
In this thesis we propose the Accelerated Distributed Augmented Lagrangians
(ADAL) algorithm, a novel decomposition method for convex optimization prob-
lems with certain separability structure. The method is based on the augmented
Lagrangian framework and addresses problems that involve multiple agents optimiz-
ing a separable convex objective function subject to convex local constraints and
linear coupling constraints. We establish the convergence of ADAL and also show
that it has a worst-case O(1/k) convergence rate, where k denotes the number of
Moreover, we show that ADAL converges to a local minimum of the problem
for cases with non-convex objective functions. This is the first published work that
formally establishes the convergence of a distributed augmented Lagrangian method
ivfor non-convex optimization problems. An alternative way to select the stepsizes
used in the algorithm is also discussed. These two contributions are independent
from each other, meaning that convergence of the non-convex ADAL method can
still be shown using the stepsizes from the convex case, and, similarly, convergence
of the convex ADAL method can be shown using the stepsizes proposed in the non-
convex proof.
Furthermore, we consider cases where the distributed algorithm needs to operate
in the presence of uncertainty and noise and show that the generated sequences of
primal and dual variables converge to their respective optimal sets almost surely. In
particular, we are concerned with scenarios where: i) the local computation steps
are inexact or are performed in the presence of uncertainty, and ii) the message
exchanges between agents are corrupted by noise. In this case, the proposed scheme
can be classified as a distributed stochastic approximation method. Compared to
existing literature in this area, our work is the first that utilizes the augmented
Lagrangian framework. Moreover, the method allows us to solve a richer class of
problems as compared to existing methods on distributed stochastic approximation
that consider only consensus constraints.
Extensive numerical experiments have been carried out in an effort to validate
the novelty and effectiveness of the proposed method in all the areas of the afore-
mentioned theoretical contributions. We examine problems in convex, non-convex,
and stochastic settings where uncertainties and noise affect the execution of the al-
gorithm. For the convex cases, we present applications of ADAL to certain popular
network optimization problems, as well as to a two-stage stochastic optimization
problem. The simulation results suggest that the proposed method outperforms
the state-of-the-art distributed augmented Lagrangian methods that are known in
the literature. For the non-convex cases, we perform simulations on certain simple
non-convex problems to establish that ADAL indeed converges to non-trivial local
vsolutions of the problems; in comparison, the straightforward implementation of the
other distributed augmented Lagrangian methods on the same problems does not
lead to convergence. For the stochastic setting, we present simulation results of
ADAL applied on network optimization problems and examine the effect that noise
and uncertainties have in the convergence behavior of the method.
As an extended and more involved application, we also consider the problem
of relay cooperative beamforming in wireless communications systems. Specifically,
we study the scenario of a multi-cluster network, in which each cluster contains
multiple single-antenna source destination pairs that communicate simultaneously
over the same channel. The communications are supported by cooperating amplify-
and-forward relays, which perform beamforming. Since the emerging problem is non-
convex, we propose an approximate convex reformulation. Based on ADAL, we also
discuss two different ways to obtain a distributed solution that allows for autonomous
computation of the optimal beamforming decisions by each cluster, while taking into
account intra- and inter-cluster interference effects.
Our goal in this thesis is to advance the state-of-the-art in distributed optimization by proposing methods that combine fast convergence, wide applicability, ease
of implementation, low computational complexity, and are robust with respect to
delays, uncertainty in the problem parameters, noise corruption in the message ex-
changes, and inexact computations.
• Formal Verification of Stochastic ReLU Neural Network Control System
In this work, we address the problem of formal safety verification for stochastic cyber-physical systems (CPS) equipped with ReLU neural network (NN) controllers. Our goal is to find the set of
initial states from where, with a predetermined confidence, the system will not reach an unsafe configuration within a specified time horizon. Specifically, we consider discrete-time LTI systems
with Gaussian noise, which we abstract by a suitable graph. Then, we formulate a Satisfiability Modulo Convex (SMC) problem to estimate upper bounds on the transition probabilities between nodes
in the graph. Using this abstraction, we propose a method to compute tight bounds on the safety probabilities of nodes in this graph, despite possible over-approximations of the transition
probabilities between these nodes. Additionally, using the proposed SMC formula, we devise a heuristic method to refine the abstraction of the system in order to further improve the estimated
safety bounds. Finally, we corroborate the efficacy of the proposed method with a robot navigation example and present comparative results with commonly employed verification schemes.
• Human-in-the-Loop Robot Planning with Non-Contextual Bandit Feedback
In this paper, we consider robot navigation problems in environments populated by humans. The goal is to determine collision-free and dynamically feasible trajectories that also maximize human
satisfaction, by ensuring that robots are available to assist humans with their work as needed and avoid actions that cause discomfort. In practice, human satisfaction is subjective and hard to
describe mathematically. As a result, the planning problem we consider in this paper may lack important contextual information. To address this challenge, we propose a semi-supervised Bayesian
Optimization (BO) method to design globally optimal robot trajectories using bandit human feedback, in the form of complaints or satisfaction ratings, that expresses how desirable a trajectory
is. Since trajectory planning is typically a high-dimensional optimization problem in the space of waypoints that need to be decided, BO may require prohibitively many queries for human feedback
to return a good solution. To this end, we use an autoencoder to reduce the high-dimensional space into a low dimensional latent space, which we update using human feedback. Moreover, we improve
the exploration efficiency of BO by biasing the search for new trajectories towards dynamically feasible and collision-free trajectories obtained using off-the-shelf motion planners. We
demonstrate the efficiency of our proposed trajectory planning method in a scenario with humans that have diversified and unknown demands.
(2019) Khodayi-mehr, Reza
Mathematical models that describe different processes and phenomena are of paramount importance in many robotics applications. Nevertheless, utilization of high-fidelity models, particularly
Partial Differential Equations (PDEs), has been hindered for many years due to the lack of adequate computational resources onboard mobile robots. One such problem of interest for the
roboticists, that can hugely benefit from more descriptive models, is Chemical Plume Tracing (CPT). In the CPT problem, one or multiple mobile robots are equipped with chemical concentration and
flow sensors and attempt to localize chemical sources in an environment of interest. This problem has important applications ranging from environmental monitoring and protection to search and
rescue missions. The transport of a chemical in a fluid medium is mathematically modeled by the Advection-Diffusion (AD) Partial Differential Equation (PDE). Despite versatility, rigorous
derivation, and powerful descriptive nature, the AD-PDE has seldom been used in its general form for the solution of the CPT problem due to high computational cost. Instead, often simplified
scenarios that render closed-form solutions for the AD-PDE or various heuristics are used in the robotics literature.
Using the AD-PDE to model the transport phenomenon enables generalization of the CPT problem to estimate other properties of the sources, e.g., their intensity, in addition to their locations. We
refer to this problem as Source Identification (SI) which we define as the problem of estimating the properties of the sources using concentration measurements that are generated under the action
of those sources. We can also put one step further and consider the problem of controlling a set of sources, carried by a team of mobile robots, to generate and maintain desired concentration
levels in select regions of the environment with the objective of cloaking those regions from external environmental conditions; we refer to this problem as the AD-PDE control problem that has
important applications in search and rescue missions.
Both SI and AD-PDE control problems can be formulated as PDE-constrained optimization problems. Solving such optimization problems onboard mobile robots is challenging due to the following
reasons: (i) the computational cost of solving the AD-PDE using traditional numerical discretization schemes, e.g., the Finite Element (FE) method, is prohibitively high, (ii) obtaining accurate
knowledge of the environment and Boundary and Initial Conditions (BICs), required to solve the AD-PDE, is difficult and prone to error and finally, (iii) obtaining accurate estimates of the
velocity and diffusivity fields is challenging since for typical transport mediums like air even in very small velocities, the flow is turbulent. In addition, we need to plan the actions of the
mobile robots, e.g., measurement collection for SI or release rates for the AD-PDE control problem, to ensure that they accomplish their tasks optimally. This can be done by formulating a
planning problem that often is solved online to take into account the latest information that becomes available to robots. Solving this planning problem by itself is a challenging task that has
been the subject of heavy research in the robotics literature. The reason is that (i) the objective is often nonlinear, (ii) the planning is preferred to be done for more than the immediate
action to avoid myopic, suboptimal plans, and (iii) the environment that the robots operate in is often non-convex and cluttered with obstacles.
In order to address the computational challenges that rise due to the use of numerical schemes, we propose using multiple mobile robots that decompose the high-dimensional optimization variables
among themselves or using nonlinear representations of the sources. In addition we utilize Model Order Reduction (MOR) approaches that facilitate the evaluation of the AD-DPE at the expense of
accuracy. In order to alleviate the loss of accuracy, we also propose a novel MOR method using Neural Networks that can straight-forwardly replace the traditional MOR methods in our formulations.
To deal with uncertainty in the the PDE input-data, i.e., the geometry of environment, BICs, and the velocity and diffusivity fields, we formulate a stochastic version of the SI problem that
provides posterior probabilities over all possible values of these uncertain parameters. Finally, to obtain the velocity and corresponding diffusivity fields that are required for the solution of
the AD-PDE, we rely on Bayesian inference to incorporate empirical measurements, collected and analyzed by mobile robots, into the numerical solutions obtained from computational fluid dynamics
In order to demonstrate the applicability of our proposed model-based approaches, we have devised and constructed an experimental setup and designed a mobile robot equipped with concentration and
flow sensors. To the best of our knowledge, this dissertation is the first work to use the AD-PDE, in its general form, to solve realistic problems onboard mobile robots. Note that although here
we focus on the AD-PDE and particularly chemical propagation, many other transport phenomena including heat and acoustic transport can be considered and the same principles apply. Our results are
a proof of concept that we hope will convince many roboticists to use more general mathematical models in their solutions.
• Passive Acoustic Localization and Tracking with Mobile Robots
(2021) Calkins, William Lucas
Acoustic sensing has received a lot of attention in the underwater domain as this is usually the only form of sensing available. As robotic platforms have been ever increasing in terms of
computational capabilities, there now exists the ability to autonomously make decisions and navigate without human intervention. This dissertation proposes and demonstrates acoustic sensing
onboard mobile robotic platforms in passive bearing-only tracking of surface vessels in the water and in detecting nearby obstacles in aerial systems.
First, we consider the problem of target tracking with a bearing-only sensor in the presence of merged measurements. Assuming the number of targets in the domain is known, we incorporated a
merged measurement model into a nonlinear joint probabilistic data association filter (JDPAF). We demonstrate the ability to track multiple targets through merging events. Furthermore, we propose
a novel planning algorithm that incorporates the merged measurement model into the planning process. The result is a planning trajectory biased away from regions where targets will merged in the
measurement space, as this leads to higher uncertainty in the target state estimates. We present experimental results with unmanned ground vehicles equipped with camera sensors acting as a
surrogate for a bearing-only passive sonar sensor.
Next, we consider the problem of bearing-only tracking of multiple targets using a port-starboard ambiguous sensor. This is the type of sensor used onboard our Autonomous Underwater vehicles
(AUVs). We address the problem of resolving the ambiguity by using a likelihood ratio detection and tracking (LRDT) method. The LRDT serves as a front end detector to initialize tracks and pass
off to a tracking algorithm. We show that as long as the ambiguity is resolved, the JPDAF algorithm can track targets even with ambiguous measurements. We run our detector-tracker system on a
dataset taken in Boston Harbor in August 2018. We show effective functioning of the detector tracker system and provide a discussion for improvements that we are currently working on at the time
of writing this dissertation.
We also explore acoustic sensing in aerial vehicles using the self-generated noise caused by the vehicles normal operation. We first propose an algorithm to actively control the distance between
a motor propeller system (MPS) and large obstacle using data from a single microphone. By first recording and storing the free-field response of the MPS, we show that by subtracting the power
spectrum of the free-field response from the power spectrum when a wall is present, we can reveal a broadband interference pattern. The dominant oscillating frequency of this interference pattern
is linearly related to the distance from the microphone to the wall. By performing a fast Fourier transform on the difference between the spectra, we show that we can extract this distance and
actively control it in real time. We present a test rig demonstrating the algorithm experimentally.
Finally, we offer an improvement to the aerial acoustic sensing system by adding an additional microphone. We develop a novel cross-correlation processing algorithm that is able to extract the
distance from the microphones to the wall. This method does not rely on computing the free-field response of the MPS. We demonstrate the algorithm in experiment by controlling the altitude of a
blimp-like vehicle using only the self-generated noise and two microphones placed on the bottom of the vehicle.
• Scalable Control Synthesis for Multi-Robot Systems under Temporal Logic Specifications
The study of high-level complex tasks for robotics, captured by temporal logics, e.g., Linear Temporal Logic (LTL), has gained significant research interest in the last decade, which extends the
traditional point-to-point navigation by incorporating temporal goals. This dissertation proposes and evaluates scalable path planning and control synthesis methods for robotic systems under
temporal logic specifications. The scalability is measured by the number of robots, the size of the environment, and the complexity of temporal logic specifications.
First, we consider the optimal control synthesis to satisfy the task specified by temporal logic specifications. Given the same discrete workspace, rather than solving a new formula from scratch,
we propose a method that exploits experience from solving similar LTL tasks before. The key idea is to decompose complex LTL tasks into simpler subtasks appropriately and define sets of skills,
or plans, needed to solve these subtasks. These skills can be stored in a library of reusable skills and can be used to quickly synthesize plans for new tasks that have not been encountered
before, meanwhile expanding the library with new skills. We present numerical experiments that show that our approach generally outperforms these methods in terms of time to generate feasible
plans. We also show that our proposed algorithm is probabilistically complete and asymptotically optimal.
Next, we consider the problem of optimally allocating tasks, expressed as global LTL specifications to teams of heterogeneous mobile robots. The robots are classified in different types that
capture their different capabilities in accomplishing tasks, and each task may require robots of multiple types. The specific robots assigned to each task are immaterial, as long as they are of
the desired type. Given a discrete workspace, our goal is to design paths, i.e., sequences of discrete states, for the robots so that the LTL specification is satisfied. To obtain a scalable
solution to this complex assignment problem, we propose a hierarchical approach that first allocates specific robots to tasks using the information about tasks provided by the Nondeterministic
Büchi Automaton (NBA) that captures the LTL specification, and then designs low-level executable plans for the robots that respect the high-level assignment. We provide theoretical results
showing completeness and soundness of our proposed method and present numerical simulations demonstrating that our method can generate robot paths with lower cost, considerably faster than
existing methods.
The majority of existing LTL planning methods rely on the construction of a discrete product automaton that combines a discrete abstraction of robot mobility and the NBA corresponding to the LTL
specification. However, constructing expressive discrete abstractions makes the synthesis problem computationally intractable. Finally, we propose a new sampling-based LTL planning algorithm that
does not require any discrete abstraction of robot mobility. Instead, it incrementally builds trees that explore the product state-space, until a maximum number of iterations is reached or a
feasible plan is found. To accelerate the construction of feasible plans, we introduce bias in the sampling process which is guided by transitions in the Büchi automaton that belong to the
shortest path to the accepting states. We show that our planning algorithm, with and without bias, is probabilistically complete and asymptotically optimal. Finally, we present numerical
experiments showing that our method outperforms relevant temporal logic planning methods.
• Sensor Planning for Bayesian Nonparametric Target Modeling
Bayesian nonparametric models, such as the Gaussian process and the Dirichlet process, have been extensively applied for target kinematics modeling in various applications including environmental
monitoring, traffic planning, endangered species tracking, dynamic scene analysis, autonomous robot navigation, and human motion modeling. As shown by these successful applications, Bayesian
nonparametric models are able to adjust their complexities adaptively from data as necessary, and are resistant to overfitting or underfitting. However, most existing works assume that the sensor
measurements used to learn the Bayesian nonparametric target kinematics models are obtained a priori or that the target kinematics can be measured by the sensor at any given time throughout the
task. Little work has been done for controlling the sensor with bounded field of view to obtain measurements of mobile targets that are most informative for reducing the uncertainty of the
Bayesian nonparametric models. To present the systematic sensor planning approach to leaning Bayesian nonparametric models, the Gaussian process target kinematics model is introduced at first,
which is capable of describing time-invariant spatial phenomena, such as ocean currents, temperature distributions and wind velocity fields. The Dirichlet process-Gaussian process target
kinematics model is subsequently discussed for modeling mixture of mobile targets, such as pedestrian motion patterns.
Novel information theoretic functions are developed for these introduced Bayesian nonparametric target kinematics models to represent the expected utility of measurements as a function of sensor
control inputs and random environmental variables. A Gaussian process expected Kullback Leibler divergence is developed as the expectation of the KL divergence between the current (prior) and
posterior Gaussian process target kinematics models with respect to the future measurements. Then, this approach is extended to develop a new information value function that can be used to
estimate target kinematics described by a Dirichlet process-Gaussian process mixture model. A theorem is proposed that shows the novel information theoretic functions are bounded. Based on this
theorem, efficient estimators of the new information theoretic functions are designed, which are proved to be unbiased with the variance of the resultant approximation error decreasing linearly
as the number of samples increases. Computational complexities for optimizing the novel information theoretic functions under sensor dynamics constraints are studied, and are proved to be
NP-hard. A cumulative lower bound is then proposed to reduce the computational complexity to polynomial time.
Three sensor planning algorithms are developed according to the assumptions on the target kinematics and the sensor dynamics. For problems where the control space of the sensor is discrete, a
greedy algorithm is proposed. The efficiency of the greedy algorithm is demonstrated by a numerical experiment with data of ocean currents obtained by moored buoys. A sweep line algorithm is
developed for applications where the sensor control space is continuous and unconstrained. Synthetic simulations as well as physical experiments with ground robots and a surveillance camera are
conducted to evaluate the performance of the sweep line algorithm. Moreover, a lexicographic algorithm is designed based on the cumulative lower bound of the novel information theoretic
functions, for the scenario where the sensor dynamics are constrained. Numerical experiments with real data collected from indoor pedestrians by a commercial pan-tilt camera are performed to
examine the lexicographic algorithm. Results from both the numerical simulations and the physical experiments show that the three sensor planning algorithms proposed in this dissertation based on
the novel information theoretic functions are superior at learning the target kinematics with
little or no prior knowledge
• Transfer learning in continuous RL under unobservable contextual information
In this paper, we consider a transfer Reinforcement Learning (RL) problem in continuous stateand action spaces, under unobserved contextual information. The context here can represent a specific
unique mental view of the world that an expert agent has formed through past interactions with this world. We assume that this context is not accessible to a learner agent who can only observe
the expert data and does not know how they were generated. Then, our goal is to use the context-aware continuous expert data to learn an optimal context-unaware policy for the learner using only
a few new data samples. To this date, such problems are typically solved using imitation learning that assumes that both the expert and learner agents have access to the same information.
However, if the learner does not know the expert context, using the expert data alone will result in a biased learner policy and will require many new data samples to improve. To address this
challenge, in this paper, we formulate the learning problem that the learner agent solves as a causal bound-constrained Multi-Armed-Bandit (MAB) problem. The arms of this MAB correspond to a set
of basis policy functions that can be initialized in an unsupervised way using the expert data and represent the different expert behaviors affected by the unobserved context. On the other hand,
the MAB constraints correspond to causal bounds on the accumulated rewards of these basis policy functions that we also compute from the expert data. The solution to this MAB allows the learner
agent to select the best basis policy and improve it online. And the use of causal bounds reduces the exploration variance and, therefore, improves the learning rate. We provide numerical
experiments on an autonomous driving example that show that our proposed transfer RL method improves the learner’s policy faster compared to imitation learning methods and enjoys much lower
variance during training
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replaced Divided by zero message with the 100%
I'm calculating the difference in a number of cases between various years like in the screenshot, but when for a particular year there is 0 cases, i get Divided by zero message.
Here is the example of the formula: =([2022]@row - [2021]@row) / [2021]@row.
I'm just replacing the years depending on which year I need to compare.
I understand why this error mathematically, but can i replace the error message with "100%" to show an increase or "0" if both years have 0 in them and apply this formula to all the years even if the
cases are not 0 - to keep the formula unified?
thank you
Best Answer
• You would use an IFERROR function.
=IFERROR(original formula, output when original formula errors)
• You would use an IFERROR function.
=IFERROR(original formula, output when original formula errors)
Help Article Resources
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Hans Lewy - Wikipedia
Hans Lewy (20 October 1904 – 23 August 1988) was an American mathematician, known for his work on partial differential equations and on the theory of functions of several complex variables.^[3]
Lewy was born to a Jewish family in Breslau, Silesia, on October 20, 1904. He began his studies at the University of Göttingen in 1922, after being advised to avoid the more local University of
Breslau because it was too old-fashioned,^[4]^[5] supporting himself during the Weimar hyperinflation by a side job doing railroad track maintenance.^[5] At Göttingen, he studied both mathematics and
physics; his teachers there included Max Born, Richard Courant, James Franck, David Hilbert, Edmund Landau, Emmy Noether, and Alexander Ostrowski. He earned his doctorate in 1926, at which time he
and his friend Kurt Otto Friedrichs both became assistants to Courant and privatdozents at Göttingen. The famous Courant-Friedrichs-Lewy condition originated from that time in 1928.^[4]^[5]
At the recommendation of Courant, Lewy was granted a Rockefeller Fellowship, which he used in 1929 to travel to Rome and study algebraic geometry with Tullio Levi-Civita and Federigo Enriques, and
then in 1930 to travel to Paris, where he attended the seminar of Jacques Hadamard. After Hitler's election as chancellor in 1933, Lewy was advised by Herbert Busemann to leave Germany again. He was
offered a position in Madrid, but declined it, fearing for the future there under Francisco Franco. He revisited Italy and France, but then at the invitation of the Emergency Committee in Aid of
Displaced Foreign Scholars and with the assistance of Hadamard found a two-year position in America at Brown University. At the end of that term, in 1935, he moved to the University of California,
During World War II, Lewy obtained a pilot's license, but then worked at the Aberdeen Proving Ground. He married Helen Crosby in 1947.^[5]
In 1950, Lewy was fired from Berkeley for refusing to sign a loyalty oath.^[5]^[6]^[7] He taught at Harvard University and Stanford University in 1952 and 1953^[5] before being reinstated by the
California Supreme Court case Tolman v. Underhill.^[6]^[7]
He retired from Berkeley in 1972, and in 1973 became one of two Ordway Professors of Mathematics at the University of Minnesota. He died on August 23, 1988, in Berkeley.^[5]^[6]^[8]
Lewy is known for his contributions to partial differential equations. In 1957, his famous example of a second-order linear partial differential equation was so stunning and unexpected that the whole
field steered in a new direction, as well as shaping modern analysis in a significant way. Based on this example, Louis Nirenberg, Lars Hörmander and others have outlined some important changes to
the theory and structure of the field. This was adopted by many analysts and mathematicians as a major development.
He also worked on several complex variables in relation to nonlinear hyperbolic equations and elliptic equations, well-posedness for initial value problems of wave fronts (now commonly called Sobolev
spaces) in the early 1930s, solutions of the classical problems of Hermann Weyl and Hermann Minkowski for analytical data (the original problem was solved by Louis Nirenberg in 1949 as part of his
PhD thesis), the extendibility of minimal surfaces on and analytical nature of its boundaries which is fully free or in part, free boundary problems of water wave fronts in hydrodynamics, and the
proof of quadratic reciprocity theorem in number theory from 'hydrodynamical' perspective.
Lewy was elected to the National Academy of Sciences in 1964, and was also a member of the American Academy of Arts and Sciences.^[6] He became a foreign member of the Accademia dei Lincei in 1972.^[
5] He was awarded a Leroy P. Steele Prize in 1979,^[5] and a Wolf Prize in Mathematics in 1986 for his work on partial differential equations.^[1] In 1986, the University of Bonn gave him an honorary
A selection of his work, edited by David Kinderlehrer and including his most important works, was published as the two volume work (Kinderlehrer 2002a) and (Kinderlehrer 2002b)
• Kinderlehrer, David, ed. (2002a), Hans Lewy Selecta. Volume 1, Contemporary Mathematicians, Boston-Basel-Stuttgart: Birkhäuser Verlag, pp. lxvi+357, ISBN 0-8176-3523-8, Zbl 1132.01312. With
biographical essays by Helen Lewy and Constance Reid, and commentaries on Lewy's work by Erhard Heinz, Peter D. Lax, Jean Leray, Richard MacCamy, Louis Nirenberg and François Treves.
• Kinderlehrer, David, ed. (2002b), Hans Lewy Selecta. Volume 2, Contemporary Mathematicians, Boston-Basel-Stuttgart: Birkhäuser Verlag, pp. xviii, 446, ISBN 0-8176-3524-6, Zbl 1147.01335.
The following works are included in his "Selecta" in their original language or translated form.
• Courant, R.; Friedrichs, K.; Lewy, H. (1928), "Über die partiellen Differenzengleichungen der mathematischen Physik", Mathematische Annalen (in German), 100 (1): 32–74, Bibcode:
1928MatAn.100...32C, doi:10.1007/BF01448839, JFM 54.0486.01, MR 1512478, S2CID 120760331. There are also two English translations of the 1928 German original paper: the first one is a translation
from the German by Phyllis Fox, circulated as a research report: Courant, R.; Friedrichs, K.; Lewy, H. (September 1956) [1928], On the partial difference equations of mathematical physics, AEC
Research and Development Report, vol. NYO-7689, New York: AEC Computing and Applied Mathematics Centre – Courant Institute of Mathematical Sciences, pp. V + 76, archived from the original on
October 23, 2008. The second one is a typographical improvement of the first, published by IBM as: Courant, R.; Friedrichs, K.; Lewy, H. (March 1967) [1928], "On the partial difference equations
of mathematical physics", IBM Journal of Research and Development, 11 (2): 215–234, Bibcode:1967IBMJ...11..215C, doi:10.1147/rd.112.0215, MR 0213764, Zbl 0145.40402, archived from the original on
2017-01-25, retrieved 2011-07-26. A freely downloadable version of this one can be found here
• Lewy, Hans (1957), "An example of a smooth linear partial differential equation without solution", Annals of Mathematics, 66 (1): 155–158, doi:10.2307/1970121, JSTOR 1970121, MR 0088629, Zbl
• Lewy, Hans (1977), On the boundary behavior of holomorphic mappings (Lezione tenuta il 3 maggio 1976) (Lecture given on May 3, 1976), Contributi del Centro Linceo Interdisciplinare di Scienze
Matematiche e Loro Applicazioni, vol. 35, Rome: Accademia Nazionale dei Lincei, p. 8.
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Concatenate Double Quotation Marks in Excel
Excel is a great tool for presenting numbers, figures, and data. It is probably not the best option to manipulate the text, as there are programs that do it better.
However, this does not mean that Excel cannot manipulate text in a great way. One of the options that we have at our disposal for this is CONCATENATE, and it serves as a great tool for grouping the
text. There are some other options as well, and we will show them in the text above.
Concatenate Double Quotation Marks
For our example, let’s that we have a short list of books that were published in 2023, with their respective authors:
It is clear that the book’s titles are not in quotation marks, and we will say that we need it to be. For the first book, we will use the simplest formula for concatenation (in cell C2):
When we insert it, the result will be the one that we need:
This formula connects the two double quotation marks at the beginning and at the end of the cell value in cell B2. The fact that we have four double quotation marks tells us that we actually have a
pair of quotation marks within our string.
For the second book, we can use the CONCATENATE formula. We will insert the following formula in cell B3:
1 =CONCATENATE("""",B3,"""")
We use the same logic as in the previous example, with the difference being that we use a proper CONCATENATE formula. We will get the same result:
We can also use the CHAR formula to get double quotation marks. CHAR formula, by its description, returns the character by the number from the list of characters set on our computer. The CHAR number
for quotation marks is number 34, so our formula in cell C4 will be:
1 =CHAR(34) & B4 & CHAR(34)
This will lead us to the same result as before:
Another way to get quotation marks is to use the REPT formula in combination with the CHAR formula. REPT function does what its name suggests- it repeats a certain character a specified number of
times. We will simply add it to the previously shown CHAR formula, and will insert the following formula in cell C5:
1 =REPT(CHAR(34),1) & B5 & REPT(CHAR(34),1)
And we will get:
For this example, we did not even need to add the REPT formula, but it is useful to see what it can be used for. If we have chosen to add another number after number 34 in our formula, for example,
number 2, we would end up with two quotation marks.
For the last book, we can choose to repeat any of the options we used before.
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MSCS Faculty & Staff
Curtis J Balz
Visiting Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: Mon 12:55-1:50 pm; Wed 12:55-1:50 pm; Thur 1-2 pm; Fri 12:55-1:50 pm; or by appointment: balz1
Adam H Berliner (he, him, his)
Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 509
Office Hours: M 2-3, T 11:30-12:30, W 10:30-11:30, Th 8:15-9:15 (CAGE), F 2:30-3:30
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Laura Boehm Vock (she, her, hers)
Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 400
Office Hours: Fall 2024 Mon 1145-1245 | Wed 2-3 | Thurs 1-2 | Fri 9-10
Debra Czarneski
Adjunct Associate Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 600
Jaime Davila (he, him, his)
Assistant Professor of Mathematics, Statistics, & Computer Science
Regents Hall (Math) 401
Office Hours: Mon:2PM-3PM;Tue:2:30PM-4:30PM;Wed:12:30PM-2PM;Friday:2PM-3PM or set up by email
Maria Davis (she, her, hers)
Term Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 501
Jill Dietz (she, her, hers)
Professor of Mathematics, Statistics, and Computer Science; CURI Director
Regents Hall (Math) 500
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Erin Ellefsen (she, her, hers)
Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 505
Francesca Gandini
Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: MWF: 11 am – 12 pm; MW: 4:30 – 5:30 pm; or by appointment: gandin1
Kristina Garrett
Professor of Mathematics, Statistics, and Computer Science; North American Director of Budapest Semesters in Mathematics; Associate Dean of Natural Sciences and Mathematics
Regents Hall (Natural Science) 302A
Olaf Hall-Holt
Associate Professor of Computer Science
Regents Hall (Math) 309
Office Hours: Mon 1-2 pm; Tues 3:30-4:30 pm; Wed 2-3 pm; Fri 3-4 pm or by appointment, email: olaf – Zoom Link
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Alex Hanhart
Visiting Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: Mon 4:30-6 pm; Wed 1-2 pm & 3-4 pm; Thurs 12:45-2:15 pm; and by appointment email: hanhar1
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Visiting Assistant Professor of Mathematics, Statistics, and Computer Scienc
Barbara Johnson
Visiting Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 600
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Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: Mondays 1:30 – 03:00 pm; Wednesdays 1:30- 03:00 pm; Thursdays 12:55-01:55 pm; or by choosing from Available slots
Anna Kraut (she, her, hers)
Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 403
Office Hours: Mon 10-11 am; Wed 3-4:30 pm; Thu 1-2 pm; Fri 10-11 am; or by appointment via email
Will Leeson (he, him, his)
Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 304
Office Hours: M,Th 9-10am, W,F 3-4pm
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Melissa Lynn
Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: Mondays, Wednesdays & Fridays 9:30 – 11 am; or by appointment, email: lynn5
Kim Mandery (she, her, hers)
Visiting Instructor of Mathematics, Statistics, and Computer Science
Office Hours: W 12-2p, Th 2-3p, F 12-1p, or virtual by appt
Ryota Matsuura
Professor of Mathematics, Statistics, and Computer Science; North American Director of Budapest Semesters in Mathematics Education; Department Chair of MSCS; Director of Computer Science
Regents Hall (Math) 507
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Paula Mercurio (she, her, hers)
Visiting Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: RMS 406: Mon/Fri 12:30-2 pm; Wed 3:00-4:00 pm, or by appointment, email mercur1
Rachael Norton (she, her, hers)
Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: Mon 10:30-11:30am, Wed 1:00-2:00pm, Thurs 1:30-2:30pm, Fri 10:00-11:00am, or by appointment (email norton10)
Paul Roback
Professor of Mathematics, Statistics, and Computer Science
Joe Roith (he, him, his)
Associate Professor of Practice in Mathematics, Statistics, and Computer Science
Regents Hall (Math) 408
Office Hours: M 10:30-12, T 12-1, W 10:30-12, Th 1-2, or by appointment, email: roith1
Melissa Schori (she, her, hers)
Academic Administrative Assistant – Team South
Regents Hall (Math) 307
Office Hours: Monday-Friday 7:30am-4:00pm
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Visiting Assistant Professor of Mathematics, Statistics, and Computer Science
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David P Walmsley (he, him, his)
Assistant Professor of Mathematics, Statistics, and Computer Science
Regents Hall (Math) 404
Office Hours: Mon 1-2:25 pm; Tues 2:30-3:30 pm; Wed 10:30 – Noon; Thurs 1:30-2:30 pm; or by appointment, email walmsl1
Matthew Wright
Associate Professor of Mathematics, Statistics, and Computer Science; Director of Mathematics
Kathryn Ziegler-Graham (she, her, hers)
Professor of Statistics; Director of Statistics and Data Science, Co-Director of CILA
Regents Hall (Math) 308
Office Hours: Monday 2-3pm, Tuesday: 1:30-3pm, Wednesday 1-2pm, or by appointment.
Martha Zillig (she, her, hers)
Visiting Assistant Professor of Mathematics, Statistics, and Computer Science
Office Hours: M/F: 1:00-2:30, T: By appt only, W: 2:30-3:30, Th: 9:00-10:00am
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Lesson 4
Dive Back Into Division
Warm-up: Notice and Wonder: Blank Spaces (10 minutes)
The purpose of this warm-up is to elicit the relationship between multiplication and division, which will be useful when students identify missing dividends and divisors in a later activity. While
students may notice and wonder many things about this equation, possible numbers to fill in the blanks are the important discussion points.
• Groups of 2
• Display the equation.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.
Student Facing
What do you notice? What do you wonder?
\(\underline{\hspace{1 cm}} \div \underline{\hspace{1 cm}} = 136\)
Activity Synthesis
• “What numbers could go in the blanks? How do you know?” (I could put in 136 and 1 or 272 and 2. The first number has to be 136 times the second number.)
Activity 1: Reasonable Estimates (15 minutes)
The purpose of this activity is for students to estimate quotients of multi-digit numbers and reason about multiplication expressions that are helpful when dividing. In previous units, students
learned a partial quotients algorithm to divide multi-digit whole numbers. This lesson prepares them to revisit this algorithm in the next lesson.
• 5–10 minutes: independent work time
• 5 minutes: partner discussion
Student Facing
1. Circle the most reasonable estimate. Show your reasoning.
2. Find the value of each quotient.
Activity Synthesis
• “How did multiplication help you with your estimates?” (Multiplying by tens is easy to do in my head so I could find the product that was closest to the dividend.)
• “How did your estimates help you find the quotient?” (My estimates gave me a good idea what multiple of the divisor to subtract to begin my calculation. They helped me check that my answer was
Activity 2: Missing Dividends and Divisors (20 minutes)
The purpose of this activity is for students to use the relationship between multiplication and division to determine possible dividends and divisors that have a given value for their quotient (MP7).
Monitor for students who:
• multiply the quotient by friendly numbers such as 1, 2, or 5.
• multiply the quotient by powers of ten and use place value understanding.
• can explain the relationship between multiplication and division.
This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy to determine which whole number makes the equation true before they begin. Students can speak
quietly to themselves or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language
• 3–5 minutes: independent work time
• “Share your responses with your partner. If you have any equations that are the same, write a new equation that is different. Together, work to find 10 different equations.”
• 3–5 minutes: partner discussion
• Give each group tools for creating a visual display.
MLR7 Compare and Connect
• “Work with your partner to create a visual display that shows your thinking about problems 1 and 2.”
• 2–5 minutes: independent or group work
• 3–5 minutes: gallery walk
Student Facing
1. Write different numbers in the blanks that make the equations true.
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700\)
\(\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78\)
2. What strategy did you use to choose numbers to write in the blanks?
Activity Synthesis
• Display: \(\underline{\hspace{1cm}} \div \underline{\hspace{1cm}} = 700\)
• Invite students to share the numbers they used to make the equation true.
• “How did you find numbers that make the equation true?” (I multiplied 700 by different numbers. I chose simple quotients, like dividing by 1 or 2 or 10.)
• Display: \(\underline{\hspace{1cm}} \div \underline{\hspace{1cm}} = 78\)
• “Did the same strategies work to find numbers that make this equation true?” (Yes, but 78 is a more difficult number to multiply by in my head. So, 1 worked and 2 and 10, but it was harder to
find a variety of numbers.)
Lesson Synthesis
Display or write for all to see.
\(\underline{\hspace{1cm}} \div \underline{\hspace{1cm}} = 25\)
“How can you use multiplication to find numbers that make the equation true?” (I know the number in the first blank has to be 25 times the number in the second blank.)
“What are some examples of numbers that make the equation true?” (25 and 1, 50 and 2, 100 and 4, 250 and 10)
“How did you choose the numbers?” (I looked for numbers that are easy to calculate in my head. I know lots of multiples of 25 that I can find in my head.)
Cool-down: Estimate and Evaluate (5 minutes)
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The implementation may regroup operators according to the usual mathematical rules only where the operators really are associative or commutative
For example, in the following fragment
int a, b;
/* ... */
a = a + 32760 + b + 5;
the expression statement behaves exactly the same as
a = (((a + 32760) + b) + 5);
due to the associativity and precedence of these operators
Thus, the result of the sum
(a + 32760)
is next added to
, and that result is then added to 5 which results in the value assigned to
On a machine in which overflows produce an exception and in which the range of values representable by an
[-32768, +32767]
, the implementation cannot rewrite this expression as
a = ((a + b) + 32765);
since if the values for
were, respectively, -32754 and -15, the sum
a + b
would produce an exception while the original expression would not; nor can the expression be rewritten as either
a = ((a + 32765) + b);
a = (a + (b + 32765));
since the values for
might have been, respectively, 4 and -8 or -17 and 12
However on a machine in which overflows do not produce an exception and in which the results of overflows are reversible, the above expression statement can be rewritten by the implementation in any
of the above ways because the same result will occur
end note
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EPR Paradox, a Thought, Myth, or, a Reality? - Physics In My View
EPR Paradox, a Thought, Myth, or, a Reality?
EPR paradox – a thought experiment, that simply asks a question that Can Quantum-Mechanical Description of Physical Reality be Considered Complete? In other words, Albert Einstein – one of the
authors of the EPR paper, refused to accept the principle of quantum indeterminacy.
In fact, in Einstein’s view, everything, whether quantum or classical should be deterministic in nature. Of course, not to mention, the whole of classical physics comes within the realm of
Therefore, with this in mind, Albert Einstein and two of his younger colleagues, Boris Podolsky and Nathan Rosen asked some fundamental questions regarding our understanding of reality.
In this article, I will try to give you a brief flashback on the Einstein Podolsky Rosen Paradox. In order to get a clear picture, stick with me till the end. Let’s start this journey to understand
whether the EPR paradox is a thought, a myth, or simply a reality?
EPR Paradox – a Thought?
Credit: The Economist
What is the EPR paradox anyway? Well, it’s a quantum paradox that directly questions the fundamental fabrics of quantum mechanical theory. A fabric – in the quantum mechanical world is known as the
foundational interpretation of quantum mechanics. Not to mention, there are so many different interpretations of quantum mechanics.
Out of all, the Copenhagen interpretation of quantum mechanics is the widely accepted one. According to the Copenhagen interpretation, the concept of measuring position or momentum at the same time
has no true physical meaning in nature.
An interpretation, now famously known as the Heisenberg Uncertainty principle. In other words, Quantum mechanics is indeterministic, or, you can say probabilistic in nature.
Check out, Wave-Particle Duality – Physical Reality of Quantum Physics
With this in mind, the thought of the EPR paradox surfaced in the conscience of Einstein. Simply because the idea that quantum mechanics is probabilistic, not deterministic, deeply bothered Einstein.
Copenhagen Interpretation of Quantum Mechanics
Credit: Linkedin
The formulation of quantum physics is based on two principal points. First, until the moment of measurement, a particle could be in multiple places at one time. Second, as soon as we measure the
position of particle A; with simple mathematics, the position of particle B could be easily predicted. (Same goes for the momentum of a particle).
When you combine these two key points, what you get is the Copenhagen interpretation of quantum mechanics. Or in other words, you get the Heisenberg uncertainty principle.
Well, what basically I want to tell you is that according to the Copenhagen interpretation, the observation creates reality. This is where Einstein had issues.
In Einstein’s view, observation does not create reality, reality is itself fundamental. As once Max Planck said, and I quote,
I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing,
postulates consciousness.
Principle of Locality and Realism
Observation does not create reality, reality is fundamental, meaning the moon is always there, doesn’t matter we see it or not/Credit: Wallpaperflare
According to the Copenhagen interpretation of quantum mechanics, as soon as we measure the position or momentum of particle A, the position or momentum of particle B could be easily predicted with
Therefore, for that to happen, the information has to be transferred faster than the speed of light, which is prohibited by the theory of relativity. Einstein termed this bizarre phenomenon as
Spukhafte Fernwirkung or Spooky Action At a Distance.
In other words, not only the transfer of information will contradict the idea or principle of locality. It will also oppose the idea of realism. The idea of locality means that no physical effect
could take place faster than the speed of the light.
On the other hand, the idea of realism means the observation does not create reality, reality is fundamental. In fact, the principle of locality has its roots in the classical world. In classical
mechanics, the idea of locality is known as action at a distance.
And, when you combine these two, what you get is local realism. Or, in the words of Einstein Podolsky Rosen, you get the EPR paradox.
God Does Not Play Dice
Bohr and Einstein having a deep conversation, maybe regarding the fundamental nature of reality/Credit: Wikimedia Commons
As, the information is being able to transfer faster than the speed of light, therefore, as a result, quantum theory is non-local. Einstein suggested that there have to be some hidden elements which
would make quantum theory local again. That’s why he theorized that quantum theory is incomplete.
In fact, Neil Bohr – The proprietor of Copenhagen’s interpretation, never challenged the fact that quantum physics is indeed showing non-locality. He just can’t accept that quantum theory has to be
deterministic in nature, rather than being a probabilistic one.
And, with that, started the series of publicly heated debates between Albert Einstein and Neil Bohr, regarding the fundamental nature of reality.
In other to justify that quantum physics indeed complete, therefore, does not require local hidden variables; Bohr even wrote a paper with the same title just like the EPR paper.
After Bohr’s response, Einstein famously said that “God does not play dice”. Well, don’t think Bohr would lay down his guards. He responded to Einstein by advising him to “stop telling God what to
Quantum mechanics is very worthy of regard. But an inner voice tells me that this is not yet the right track. The theory yields much, but it hardly brings us closer to the Old One’s secrets. I,
in any case, am convinced that He does not play dice – Albert Einstein.
Concluding Remarks
Well, my concluding remark would be that the EPR paradox is an awesome thought experiment. I mean, even Neil Bohr acknowledged it. Okay, maybe partially, but he did. So, we have reached the point to
check out the next goal of this article.
EPR Paradox – a Myth?
As a matter of fact, the way Einstein’s spooky theory was presented, was a bit confusing. I mean, even professional theoretical physicists were hesitant to even take a look.
Years later, American physicist and philosopher David Bohm developed an alternative approach to the EPR experiment.
EPR Paradox in Terms of Bohemian Mechanics
Unlike the position-momentum correlation calculation; according to bohemian mechanics or Bohm formulation, EPR Paradox can be explained in terms of electron-positron pairs. Not to mention, the way
Bohm interpreted the EPR experiment was quite easy to understand.
Bohemian Way
Let’s take an example of an electron-positron pair in terms of spin-singlet. Suppose a decaying atom emits electron-positron pair in opposite directions.
The EPR thought experiment, performed with electron-positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to Bob (right), who can perform
spin measurements/Credit: Wikimedia Commons
As the momentum is conserved, the electron goes towards destination A, where there is an observer sitting named Alice. (See the above figure for proper understanding). Similarly, the positron goes
towards destination B, where an observer is sitting named Bob.
Don’t forget both went in the opposite direction. Moreover, like the uncertainty principle or so-called Copenhagen interpretation of quantum mechanics states, “it is impossible to determine both the
position and momentum of electron or positron precisely.
Nevertheless, it also states that if the exact momentum of the electron is known (of course, after the collapse of the wave function). Therefore, with the proper mathematical calculation, we can for
sure determine the exact momentum of the positron.
Until the time of measurement, both electron and position can have multiple states. Simply because they are in entangled states or in a quantum superposition.
Case 1
As the electron went towards destination A, where Alice is taking the measurement. She will get the spin of an electron as (+x). Therefore, as a result, when bob will take the measurement, he will
surely get the spin of the positron as (-x). Or, vice versa.
Case 2
Similarly, as the electron went towards destination A (this time in the perpendicular direction), Alice will get the spin of an electron as (+z). Therefore, as a result of the collapse of the wave
function, Bob will definitely get the spin of the positron as (-z). Or, vice versa.
Case 3 (take away point)
Again, the electron went towards destination A and the positron went towards destination B. If Alice and Bob are taking measurements along X-axis, they will definitely get the result as expected in
Case 1. Likewise, if Alice and Bob are taking measurements along Z-axis, they will unquestionably get the result as expected as Case 2.
Now suppose, if Alice is taking measurement along X-axis, therefore get the result as (+x) or (-x). Here comes the catch. What if Bob takes his measurement along Z-axis, rather than taking
measurements along X-axis?
Uncertainty Fails
Credit: YouTube
Yup, you heard me right. According to the quantum indeterminacy or uncertainty principle, the position-momentum complementarity only applies to the measurement taken in opposite direction.
In other words, if both Alice and Bob are taking measurements along the same axis but in opposite direction, they both will get their answer in terms of the Uncertainty principle.
On the other hand, if Alice is taking measurement along X-axis and Bob is taking measurement along Z-axis. It doesn’t matter what spin value (in terms of x) Alice got during her measurement. When Bob
will take his measurement along Z-axis, he will have 50-50 chances of having either (+z) or (-z).
Recommended, Top 6 game-changing facts about quantum mechanics you never knew
To put it differently, Bob’s positron will have a definite spin value, only and only if, when measured along the same axis as Alice’s electron. On the contrary, when measured along a perpendicular
axis i.e Z-axis, Bob’s Positron spin will be uniformly random.
Concluding Remarks
As you can see from the above case study (especially case 3), in order to determine the spin of the positron (Bob), information (from Alice) has to be transferred faster than the speed of light.
Therefore, there is a definite need for a hidden variable to account for this Non-locality to make Quantum physics local again.
As far as Einstein’s spooky theory or EPR Paradox is concerned, you can say that Bohm’s formulation is nothing but the extension of the EPR Paradox. I mean, you can see that Case 1 and 2 are almost
the same as the EPR example.
Therefore, my concluding remarks will be that EPR Paradox is definitely not a myth. Simply because the Bohemian mechanics or Bohm formulation is one of the best examples of hidden variable theory.
Okay, not a local hidden variable theory. But it still has some deterministic approach.
EPR Paradox – a Reality?
Well, frankly speaking, Bohm’s hidden variable theory solidified the fact that Einstein was somehow correct regarding the spooky action at a distance. Hence, the EPR riddle came into the limelight
that how information could instantaneously transfer from one location to another.
In the year 1964, an Irish physicist john Stewart bell published a research paper On the Einstein Polosky Rosen Paradox. In today’s world, we know this paper as Bell’s theorem.
Bell’s Inequalities
Bell extended the research done by Einstein Podolsky Rosen, and later by David Bohm. Not to mention, he was totally aware of the fact that how Einstein and the company argued about the errors in the
Copenhagen Interpretation.
Therefore, designated that quantum mechanics show non-locality (spooky action at a distance), and suggested some hidden elements to make quantum theory local again.
On the other hand, he was also aware that how bohemian mechanics or Bohm formulation explained the first successful hidden variable theory.
Scheme of a “two-channel” Bell test, The source S produces pairs of “photons”, sent in opposite directions. Each photon encounters a two-channel polariser whose orientation (a or b) can be set by the
experimenter. Emerging signals from each channel will detect and coincidences of four types (++, −−, +− and −+) counted by the coincidence monitor/Credit: Wikimedia Commons
In fact, Bell was himself so baffled in the thought that how can information be transferred instantaneously. Hence, did some experiments, which in today’s world is known as Bell’s inequalities. In
Bell’s view, there can only be two ways to explain this phenomenon.
In order to understand it properly, let’s go back to the above-mentioned example of electron-positron singlet pairs. But, this time, we will talk in terms of Photons, rather than orthodox
electron-positron pairs. (See the above figure for proper understanding).
If the second photon is affected immediately by the measurement performed on the first photon, this means they are co-related with each other. In other words, information from the first photon is
immediately transferring to the second photon.
Therefore, violating the principle of locality and showing non-locality. The above-mentioned cases of Bohemian formulation hold true in the first way.
What if we take measurement along with the other angles (say 45 degrees), rather than the horizontal or vertical axis? The correlation showed by the Bohm formulation will become null and void.
To put it differently, if the information does not get transferred immediately from the first photon to the second (45 degrees case), therefore, the outcome of the second photon is completely random
or simply not correlated to the outcome of the first photon.
Hence, violating the bell’s inequalities and proving that quantum physics is indeterministic and show non-locality. In other words, the hidden variables theory (Bohm formulation) can only represent
the correlations only in terms of the horizontal and vertical axis.
As soon as, the other angles between the horizontal and vertical axis (say 45 degrees) are taken into consideration; Bohm formulation becomes impotent to replicate the correlation between two
photons. This difference is essentially known as Bell’s inequalities.
If a hidden-variable theory is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local – John Stewart Bell
Concluding Remarks
Well, my concluding remarks for this section would be that EPR Paradox is indeed a reality. I mean, if there were no hypothesis of Einstein’s spooky theory, then, we won’t be having this amazing
theory of Bell’s Inequalities. Moreover, just because of the idea of the EPR experiment, we got the quantum entanglement theory.
Final Words On EPR Paradox
One can say that Bell’s theorem proved that Einstein was indeed wrong. I mean, experimental pieces of evidence suggest that quantum theory is indeed complete. Well, maybe Einstein was wrong, but you
can’t deny that the EPR Paradox is an awesome thought, therefore definitely not a myth.
Plus, no one can deny that with the help of Bell’s inequality, Einstein’s spooky theory of quantum entanglement became reality. Let us try to understand where Einstein went wrong.
• According to EPR Paradox, quantum mechanics is incomplete and shows its non-local behavior. Therefore, needs some local hidden variables to complete it.
• According to Bell’s Theorem, quantum mechanics is indeed complete. And, yes, it shows non-locality. Therefore, does not need local hidden variables to complete it.
One More Thing
At the very last leg of this Article, let me clear some air regarding quantum entanglement or spooky action at a distance. Maybe, quantum entanglement seems to be violating the fundamentals of
relativity. But in reality, it doesn’t.
Okay, see, the quantum non-locality only accounts for some specific cases, such as electron-positron pairs or the pairs of photons. In another word, there is no transfer of information at a speed
faster than the speed of light.
What I mean to say is that maybe two particles are linked through entanglement, that doesn’t mean this will allow communication faster than the speed of light. Simply speaking, there is no quantum
entanglement between humans or atoms, just with special cases.
What’s your take on this? I mean, is it possible to have quantum entanglement between humans? That’s it for this post. If you like this article, share it if you like it, like it if you share it. You
can also find us on Mix, Twitter, Pinterest, and Facebook.
12 thoughts on “EPR Paradox, a Thought, Myth, or, a Reality?”
1. Great post about this paradox. A lot of people have a hard time understanding it but gathered everything in one place so it works.
□ I’m happy that you liked it. Thank you!
2. I don’t know why The Copenhagen interpretation got struck in my head while reading this. For some reason, it was like reading in Sheldon’s voice. I’m going to be honest, I’m not the best when it
comes to quantum mechanics but this article is pretty detailed. I can’t even begin to process it. Haha!
□ Well, it’s a probabilistic approach of quantum mechanics towards the reality. Personally, I believe quantum world should be deterministic. I mean, the world will become more easy to
3. Wonderful article, in fact I was glued to it. EPR Paradox – I was not aware of that much details like Bohm example and all, interesting post.
□ Well, yeah, the way he extended epr experiment to the electron-positron pair is outstanding. Not to mention, with that, he opened the way for bell’s theorem too..
4. A very interesting read for sure. I’ll send it to my brother who is a theoretical physics.
□ Thank you
5. Hello 🙂 Your post is very brilliant and fascinated, I like the idea and conception. I retargeting the main address for all friends… 🙂 Thanks!
□ I am quite happy that you liked my work. Keep visiting us!!!
6. Great job 🙂
□ thanx. Keep visiting us!!!
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In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with
homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori
error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.
Turbulent flow simulation methods based on finite differences are attractive for their simplicity, flexibility and efficiency, but not always for accuracy or stability. This paper demonstrates that a
good compromise is possible with the advected grid explicit (AGE) method. Starting from the same initial field as a previous spectral DNS, AGE method simulations of a planar turbulent wake were
carried out as DNS, and then at three levels of reduced resolution. The latter cases were in a sense large‐eddy simulations (LES), although no specific sub‐grid‐scale model was used. Results for the
two DNS methods, including variances and power spectra, were very similar, but the AGE simulation required much less computational effort. Small‐scale information was lost in the reduced resolution
runs, but large‐scale mean and instantaneous properties were reproduced quite well, with further large reductions in computational effort. Quality of results becomes more sensitive to the value
chosen for one of the AGE method parameters as resolution is reduced, from which it is inferred that the numerical stability procedure controlled by the parameter is acting in part as a
sub‐grid‐scale model. Copyright © 2002 John Wiley & Sons, Ltd.
It is proved that the variety of relevant disjunction lattices has the finite embeddability property. It follows that Avron's relevance logic RMI
has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron's result that RMI
is decidable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
A finite difference/boundary integral procedure to determine the acoustic reflected pressure from a fluid-loaded bi-laminated plate is described. The bi-laminate is composed of a piezoelectric layer
and an elastic layer in contact with the fluid, and is held by an acoustically hard baffle. In the numerical model, the fluid pressure at fluid/solid interface is replaced by a continuum of point
sources weighted by the normal acceleration of the elastic plate, and the governing equation system is solved in the solid domain. With the normal acceleration found, the reflected pressure in the
fluid is determined by an integral expression involving the Green's function. It is demonstrated that an appropriate applied voltage potential across the piezoelectric layer has the effect of
cancelling either the reflected or scattered pressure of the plate at any chosen field points in the fluid. Project supported by the National Natural Science Foundation of China (No. 10172039).
Developement of numerical methods for obtaining approximate solutions to the three dimensional diffusion equation with an integral condition will be carried out. The numerical techniques discussed
are based on the fully explicit (1,7) finite difference technique and the fully implicit (7,1) finite difference method and the (7,7) Crank‐Nicolson type finite difference formula. The new developed
methods are tested on a problem. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithms. The results of numerical testing show that the numerical
methods based on the finite difference techniques discussed in the present article produce good results. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 193–202, 2002; DOI
The turbulent flow in a compound meandering channel with a rectangular cross section is one of the most complicated turbulent flows, because the flow behaviour is influenced by several kinds of
forces, including centrifugal forces, pressure‐driven forces and shear stresses generated by momentum transfer between the main channel and the flood plain. Numerical analysis has been performed for
the fully developed turbulent flow in a compound meandering open‐channel flow using an algebraic Reynolds stress model. The boundary‐fitted coordinate system is introduced as a method for coordinate
transformation in order to set the boundary conditions along the complicated shape of the meandering open channel. The turbulence model consists of transport equations for turbulent energy and
dissipation, in conjunction with an algebraic stress model based on the Reynolds stress transport equations. With reference to the pressure–strain term, we have made use of a modified pressure–strain
term. The boundary condition of the fluctuating vertical velocity is set to zero not only for the free surface, but also for computational grid points next to the free surface, because experimental
results have shown that the fluctuating vertical velocity approaches zero near the free surface. In order to examine the validity of the present numerical method and the turbulent model, the
calculated results are compared with experimental data measured by laser Doppler anemometer. In addition, the compound meandering open channel is clarified somewhat based on the calculated results.
As a result of the analysis, the present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent flow in a compound meandering open channel. Copyright © 2005 John
Wiley & Sons, Ltd.
The widely used locally adaptive Cartesian grid methods involve a series of abruptly refined interfaces. In this paper we consider the influence of the refined interfaces on the steady state errors
for second‐order three‐point difference approximations of flow equations. Since the various characteristic components of the Euler equations should behave similarly on such grids with regard to
refinement‐induced errors, it is sufficient enough to conduct the analysis on a scalar model problem. The error we consider is a global error, different to local truncation error, and reflects the
interaction between multiple interfaces. The steady state error will be compared to the errors on smooth refinement grids and on uniform grids. The conclusion seems to support the numerical findings
of Yamaleev and Carpenter (
J. Comput. Phys.
2002; 181: 280–316) that refinement does not necessarily reduce the numerical error. Copyright © 2005 John Wiley & Sons, Ltd.
A time discrete scheme is used to approximate the solution toa phase field system of Penrose Fife type with a non-conservedorder parameter. An
a posteriori
error estimate is presentedthat allows the estimation of the difference between continuousand semidiscrete solutions by quantities that can be calculatedfrom the approximation and given data.
Trabecular bone fracture is closely related to the trabecular architecture, microdamage accumulation, and bone tissue properties. Primary constituents of trabecular tissue are hydroxyapatite (HA)
mineralized type-I collagen fibers. In this research, dynamic fracture in two dimensional (2-D) micrographs of ovine (sheep) trabecular bone is modeled using the mesoscale cohesive finite element
method (CFEM). The bone tissue fracture properties are obtained based on the atomistic strength analyses of a type-I collagen + HA interfacial arrangement using molecular dynamics (MD). Analyses show
that the presented framework is capable of analyzing the architecture dependent fracture in 2-D micrographs of trabecular bone.
The paper addresses the problem of calculation of the local stress field and effective elastic properties of a unidirectional fiber reinforced composite with anisotropic constituents. For this aim,
the representative unit cell approach has been utilized. The micro geometry of the composite is modeled by a periodic structure with a unit cell containing multiple circular fibers. The number of
fibers is sufficient to account for the micro structure statistics of composite. A new method based on the multipole expansion technique is developed to obtain the exact series solution for the micro
stress field. The method combines the principle of superposition, technique of complex potentials and some new results in the theory of special functions. A proper choice of potentials and new
results for their series expansions allow one to reduce the boundary-value problem for the multiple-connected domain to an ordinary, well-posed set of linear algebraic equations. This reduction
provides high numerical efficiency of the developed method. Exact expressions for the components of the effective stiffness tensor have been obtained by analytical averaging of the strain and stress
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IPA: kwˈɑdrʌtʃɝ
• The process of making something square; squaring.
• (mathematics) The act or process of constructing a square that has the same area as a given plane figure, or of computing that area.
• (mathematics, numerical analysis) The calculation of a definite integral by numerical means.
• (mathematics, mathematical analysis) The act or process of solving an indefinite integral by symbolic means.
• (astronomy) A situation in which the directions of two celestial bodies (or a celestial body and the Sun) form a right angle from the perspective of the observer.
• (physics) The condition in which the phase angle between two alternating quantities is 90°.
• (art) A painting painted on a wooden panel.
Examples of "quadrature" in Sentences
• Quadrature amplitude modulation.
• Quadrature division multiple access.
• See 'The Quadrature of the Parabola'.
• Squaring or quadrature as integration.
• Archimedes' quadrature of the parabola.
• Quadrature of the circle in ancient Egypt.
• I'm not familiar with quadrature as a term.
• The quadrature of the circle was also interesting.
• They are often referred to as quadrature carriers .
• The number pi is as fictitious as the quadrature of circle.
• More curious than his quadrature is his name; what are we to make of it?
• (A quadrature is the inverse problem, that of determining the fluents when the fluxions are given.)
• La quadrature notes that the names appearing in the petition are not artistic names but real names of artists.
• The error by Archimedes is ultimately comparable to the use of that fallacy of the notion of quadrature which was already implicit in the Aristotelean presumption expressed by
• Before Newton, quadrature or integration had rested ultimately “on some process through which elemental triangles or rectangles were added together”, that is, on the method of indivisibles.
• Further more, La quadrature du net has investigated the petition… Many of the names are fake, a lot of these “artists” dont exist, are members of the french IFPI, or their names have been used
without their knowledge… Oh yes nice petition indeed…
• Further more, La quadrature du net has investigated the petition … Many of the names are fake, a lot of these “artists” dont exist, are members of the french IFPI, or their names have been used
without their knowledge … Oh yes nice petition indeed …
• “Further more, La quadrature du net has investigated the petitionâ ¦ Many of the names are fake, a lot of these â artistsâ dont exist, are members of the french IFPI, or their names have been
used without their knowledgeâ ¦ Oh yes nice petition indeedâ ¦”
• “Further more, La quadrature du net has investigated the petitionâ ¦ Many of the names are fake, a lot of these â artistsâ dont exist, are members of the french IFPI, or their names have been
used without their knowledgeâ ¦ Oh yes nice petition indeedâ ¦”
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sci.physics Frequently Asked Questions (Part 4 of 4)
Posted-By: auto-faq 3.1.1.2
Archive-name: physics-faq/part4
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FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 4/4
Item 24. Special Relativistic Paradoxes - part (a)
The Barn and the Pole updated 4-AUG-1992 by SIC
--------------------- original by Robert Firth
These are the props. You own a barn, 40m long, with automatic
doors at either end, that can be opened and closed simultaneously by a
switch. You also have a pole, 80m long, which of course won't fit in the
Now someone takes the pole and tries to run (at nearly the speed of
light) through the barn with the pole horizontal. Special Relativity (SR)
says that a moving object is contracted in the direction of motion: this is
called the Lorentz Contraction. So, if the pole is set in motion
lengthwise, then it will contract in the reference frame of a stationary
You are that observer, sitting on the barn roof. You see the pole
coming towards you, and it has contracted to a bit less than 40m. So, as
the pole passes through the barn, there is an instant when it is completely
within the barn. At that instant, you close both doors. Of course, you
open them again pretty quickly, but at least momentarily you had the
contracted pole shut up in your barn. The runner emerges from the far door
But consider the problem from the point of view of the runner. She
will regard the pole as stationary, and the barn as approaching at high
speed. In this reference frame, the pole is still 80m long, and the barn
is less than 20 meters long. Surely the runner is in trouble if the doors
close while she is inside. The pole is sure to get caught.
Well does the pole get caught in the door or doesn't it? You can't
have it both ways. This is the "Barn-pole paradox." The answer is buried
in the misuse of the word "simultaneously" back in the first sentence of
the story. In SR, that events separated in space that appear simultaneous
in one frame of reference need not appear simultaneous in another frame of
reference. The closing doors are two such separate events.
SR explains that the two doors are never closed at the same time in
the runner's frame of reference. So there is always room for the pole. In
fact, the Lorentz transformation for time is t'=(t-v*x/c^2)/sqrt(1-v^2/c^2).
It's the v*x term in the numerator that causes the mischief here. In the
runner's frame the further event (larger x) happens earlier. The far door
is closed first. It opens before she gets there, and the near door closes
behind her. Safe again - either way you look at it, provided you remember
that simultaneity is not a constant of physics.
References: Taylor and Wheeler's _Spacetime Physics_ is the classic.
Feynman's _Lectures_ are interesting as well.
Item 24. Special Relativistic Paradoxes - part (b)
The Twin Paradox updated 04-MAR-1994 by SIC
---------------- original by Kurt Sonnenmoser
A Short Story about Space Travel:
Two twins, conveniently named A and B, both know the rules of
Special Relativity. One of them, B, decides to travel out into space with
a velocity near the speed of light for a time T, after which she returns to
Earth. Meanwhile, her boring sister A sits at home posting to Usenet all
day. When B finally comes home, what do the two sisters find? Special
Relativity (SR) tells A that time was slowed down for the relativistic
sister, B, so that upon her return to Earth, she knows that B will be
younger than she is, which she suspects was the the ulterior motive of the
trip from the start.
But B sees things differently. She took the trip just to get away
>from the conspiracy theorists on Usenet, knowing full well that from her
point of view, sitting in the spaceship, it would be her sister, A, who
was travelling ultrarelativistically for the whole time, so that she would
arrive home to find that A was much younger than she was. Unfortunate, but
worth it just to get away for a while.
What are we to conclude? Which twin is really younger? How can SR
give two answers to the same question? How do we avoid this apparent
paradox? Maybe twinning is not allowed in SR? Read on.
Paradox Resolved:
Much of the confusion surrounding the so-called Twin Paradox
originates from the attempts to put the two twins into different frames ---
without the useful concept of the proper time of a moving body.
SR offers a conceptually very clear treatment of this problem.
First chose _one_ specific inertial frame of reference; let's call it S.
Second define the paths that A and B take, their so-called world lines. As
an example, take (ct,0,0,0) as representing the world line of A, and
(ct,f(t),0,0) as representing the world line of B (assuming that the the
rest frame of the Earth was inertial). The meaning of the above notation is
that at time t, A is at the spatial location (x1,x2,x3)=(0,0,0) and B is at
(x1,x2,x3)=(f(t),0,0) --- always with respect to S.
Let us now assume that A and B are at the same place at the time t1
and again at a later time t2, and that they both carry high-quality clocks
which indicate zero at time t1. High quality in this context means that the
precision of the clock is independent of acceleration. [In principle, a
bunch of muons provides such a device (unit of time: half-life of their
The correct expression for the time T such a clock will indicate at
time t2 is the following [the second form is slightly less general than the
first, but it's the good one for actual calculations]:
t2 t2 _______________
/ / / 2 |
T = | d\tau = | dt \/ 1 - [v(t)/c] (1)
/ /
t1 t1
where d\tau is the so-called proper-time interval, defined by
(c d\tau) = (c dt) - dx1 - dx2 - dx3 .
d d
v(t) = -- (x1(t), x2(t), x3(t)) = -- x(t)
dt dt
is the velocity vector of the moving object. The physical interpretation
of the proper-time interval, namely that it is the amount the clock time
will advance if the clock moves by dx during dt, arises from considering
the inertial frame in which the clock is at rest at time t --- its
so-called momentary rest frame (see the literature cited below). [Notice
that this argument is only of heuristic value, since one has to assume
that the absolute value of the acceleration has no effect. The ultimate
justification of this interpretation must come from experiment.]
The integral in (1) can be difficult to evaluate, but certain
important facts are immediately obvious. If the object is at rest with
respect to S, one trivially obtains T = t2-t1. In all other cases, T must
be strictly smaller than t2-t1, since the integrand is always less than or
equal to unity. Conclusion: the traveling twin is younger. Furthermore, if
she moves with constant velocity v most of the time (periods of
acceleration short compared to the duration of the whole trip), T will
approximately be given by ____________
/ 2 |
(t2-t1) \/ 1 - [v/c] . (2)
The last expression is exact for a round trip (e.g. a circle) with constant
velocity v. [At the times t1 and t2, twin B flies past twin A and they
compare their clocks.]
Now the big deal with SR, in the present context, is that T (or
d\tau, respectively) is a so-called Lorentz scalar. In other words, its
value does not depend on the choice of S. If we Lorentz transform the
coordinates of the world lines of the twins to another inertial frame S',
we will get the same result for T in S' as in S. This is a mathematical
fact. It shows that the situation of the traveling twins cannot possibly
lead to a paradox _within_ the framework of SR. It could at most be in
conflict with experimental results, which is also not the case.
Of course the situation of the two twins is not symmetric, although
one might be tempted by expression (2) to think the opposite. Twin A is
at rest in one and the same inertial frame for all times, whereas twin B
is not. [Formula (1) does not hold in an accelerated frame.] This breaks
the apparent symmetry of the two situations, and provides the clearest
nonmathematical hint that one twin will in fact be younger than the other
at the end of the trip. To figure out *which* twin is the younger one, use
the formulae above in a frame in which they are valid, and you will find
that B is in fact younger, despite her expectations.
It is sometimes claimed that one has to resort to General
Relativity in order to "resolve" the Twin "Paradox". This is not true. In
flat, or nearly flat, space-time (no strong gravity), SR is completely
sufficient, and it has also no problem with world lines corresponding to
accelerated motion.
Taylor and Wheeler, _Spacetime Physics_ (An *excellent* discussion)
Goldstein, _Classical Mechanics_, 2nd edition, Chap.7 (for a good
general discussion of Lorentz transformations and other SR basics.)
Item 24. Special Relativistic Paradoxes - part (c)
The Superluminal Scissors updated 31-MAR-1993
------------------------- original by Scott I.Chase
A Gedankenexperiment:
Imagine a huge pair of scissors, with blades one light-year long.
The handle is only about two feet long, creating a huge lever arm,
initially open by a few degrees. Then you suddenly close the scissors.
This action takes about a tenth of a second. Doesn't the contact point
where the two blades touch move down the blades *much* faster than the
speed of light? After all, the scissors close in a tenth of a second, but
the blades are a light-year long. That seems to mean that the contact
point has moved down the blades at the remarkable speed of 10 light-years
per second. This is more than 10^8 times the speed of light! But this
seems to violate the most important rule of Special Relativity - no signal
can travel faster than the speed of light. What's going on here?
We have mistakenly assumed that the scissors do in fact close when
you close the handle. But, in fact, according to Special Relativity, this
is not at all what happens. What *does* happen is that the blades of the
scissors flex. No matter what material you use for the scissors, SR sets a
theoretical upper limit to the rigidity of the material. In short, when
you close the scissors, they bend.
The point at which the blades bend propagates down the blade at
some speed less than the speed of light. On the near side of this point,
the scissors are closed. On the far side of this point, the scissors
remain open. You have, in fact, sent a kind of wave down the scissors,
carrying the information that the scissors have been closed. But this wave
does not travel faster than the speed of light. It will take at least one
year for the tips of the blades, at the far end of the scissors, to feel
any force whatsoever, and, ultimately, to come together to completely close
the scissors.
As a practical matter, this theoretical upper limit to the rigidity
of the metal in the scissors is *far* higher than the rigidity of any real
material, so it would, in practice, take much much longer to close a real
pair of metal scissors with blades as long as these.
One can analyze this problem microscopically as well. The
electromagnetic force which binds the atoms of the scissors together
propagates at the speeds of light. So if you displace some set of atoms in
the scissor (such as the entire handles), the force will not propagate down
the scissor instantaneously, This means that a scissor this big *must*
cease to act as a rigid body. You can move parts of it without other parts
moving at the same time. It takes some finite time for the changing forces
on the scissor to propagate from atom to atom, letting the far tip of the
blades "know" that the scissors have been closed.
The contact point where the two blades meet is not a physical
object. So there is no fundamental reason why it could not move faster
than the speed of light, provided that you arrange your experiment correctly.
In fact it can be done with scissors provided that your scissors are short
enough and wide open to start, very different conditions than those spelled
out in the gedankenexperiment above. In this case it will take you quite
a while to bring the blades together - more than enough time for light to
travel to the tips of the scissors. When the blades finally come together,
if they have the right shape, the contact point can indeed move faster
than light.
Think about the simpler case of two rulers pinned together at an
edge point at the ends. Slam the two rulers together and the contact point
will move infinitely fast to the far end of the rulers at the instant
they touch. So long as the rulers are short enough that contact does not
happen until the signal propagates to the far ends of the rulers, the
rulers will indeed be straight when they meet. Only if the rulers are
too long will they be bent like our very long scissors, above, when they
touch. The contact point can move faster than the speed of light, but
the energy (or signal) of the closing force can not.
An analogy, equivalent in terms of information content, is, say, a
line of strobe lights. You want to light them up one at a time, so that
the `bright' spot travels faster than light. To do so, you can send a
_luminal_ signal down the line, telling each strobe light to wait a
little while before flashing. If you decrease the wait time with
each successive strobe light, the apparent bright spot will travel faster
than light, since the strobes on the end didn't wait as long after getting
the go-ahead, as did the ones at the beginning. But the bright spot
can't pass the original signal, because then the strobe lights wouldn't
know to flash.
Item 25. Can You See the Lorentz-Fitzgerald Contraction? 12-Oct-1995
Or: Penrose-Terrell Rotation by Michael Weiss
People sometimes argue over whether the Lorentz-Fitzgerald contraction is
"real" or not. That's a topic for another FAQ entry, but here's a short
answer: the contraction can be measured, but the measurement is
frame-dependent. Whether that makes it "real" or not has more to do with your
choice of words than the physics.
Here we ask a subtly different question. If you take a snapshot of a rapidly
moving object, will it *look* flattened when you develop the film? What is the
difference between measuring and photographing? Isn't seeing believing? Not
always! When you take a snapshot, you capture the light-rays that hit the
*film* at one instant (in the reference frame of the film). These rays may
have left the *object* at different instants; if the object is moving with
respect to the film, then the photograph may give a distorted picture.
(Strictly speaking snapshots aren't instantaneous, but we're idealizing.)
Oddly enough, though Einstein published his famous relativity paper in
1905, and Fitzgerald proposed his contraction several years earlier,
no one seems to have asked this question until the late '50s. Then
Roger Penrose and James Terrell independently discovered that the
object will *not* appear flattened [1,2]. People sometimes say that
the object appears rotated, so this effect is called the
Penrose-Terrell rotation.
Calling it a rotation can be a bit confusing though. Rotating an object brings
its backside into view, but it's hard to see how a contraction could do that.
Among other things, this entry will try to explain in just what sense
the Penrose-Terrell effect is a "rotation".
It will clarify matters to imagine *two* snapshots of the same object, taken by
two cameras moving uniformly with respect to each other. We'll call them *his*
camera and *her* camera. The cameras pass through each other at the origin at
t=0, when they take their two snapshots. Say that the object is at rest with
respect to his camera, and moving with respect to hers. By analysing the
process of taking a snapshot, the meaning of "rotation" will become clearer.
How should we think of a snapshot? Here's one way: consider a pinhole camera.
(Just one camera, for the moment.) The pinhole is located at the origin, and
the film occupies a patch on a sphere surrounding the origin. We'll ignore all
technical difficulties(!), and pretend that the camera takes full spherical
pictures: the film occupies the entire sphere.
We need more than just a pinhole and film, though: we also need a shutter. At
t=0, the shutter snaps open for an instant to let the light-rays through the
pinhole; these spread out in all directions, and at t=1 (in the rest-frame of
the camera) paint a picture on the spherical film.
Let's call points in the snapshot *pixels*. Each pixel gets its color due to
an event, namely a light-ray hitting the sphere at t=1. Now let's consider his
& her cameras, as we said before. We'll use t for his time, and t' for hers.
At t=t'=0, the two pinholes coincide at the origin, the two shutters snap
simultaneously, and the light rays spread out. At t=1 for *his* camera, they
paint *his* pixels; at t'=1 for *her* camera, they paint *hers*. So the
definition of a snapshot is frame-dependent. But you already knew that. (Pop
quiz: what shape does *he* think *her* film has? Not spherical!) (More
technical difficulties: the rays have to pass right through one film to hit the
So there's a one-one correspondence between pixels in the two snapshots. Two
pixels correspond if they are painted by the same light-ray. You can see now
that her snapshot is just a distortion of his (and vice versa). You could take
his snapshot, scan it into a computer, run an algorithm to move the pixels
around, and print out hers.
So what does the pixel mapping look like? Simple: if we put the usual
latitude/longitude grid on the spheres, chosen so that the relative motion is
along the north-south axis, then each pixel slides up towards the north pole
along a line of longitude. (Or down towards the south pole, depending on
various choices I haven't specified.) This should ring a bell if you know
about the aberration of light: if our snapshots portray the night-sky, then the
stars are white pixels, and aberration changes their apparent positions.
Now let's consider the object--- let's say a galaxy. In passing from his
snapshot to hers, the image of the galaxy slides up the sphere, keeping the
same face to us. In this sense, it has rotated. Its apparent size will also
change, but not its shape (to a first approximation).
The mathematical details are beautiful, but best left to the textbooks [3,4].
Just to entice you if you have the background: if we regard the two spheres as
Riemann spheres, then the pixel mapping is given by a fractional linear
transformation. Well-known facts from complex analysis now tell us two things.
First, circles go to circles under the pixel mapping, so a sphere will *always*
photograph as a sphere. Second, shapes of objects are preserved in the
infinitesimally small limit. (If you know about the double-covering of SL(2),
that also comes into play. [3] is a good reference.)
References: [1] and [2] are the original articles. [3] and [4] are textbook
treatments. [5] has beautiful computer-generated pictures of the
Penrose-Terrell rotation. The authors of [5] later made a video [6] of this
and other effects of "SR photography".
[1] Penrose, R.,"The Apparent Shape of a Relativistically Moving Sphere",
Proc. Camb. Phil. Soc., vol 55 Jul 1958.
[2] Terrell, J., "Invisibility of the Lorentz Contraction",
Phys. Rev. vol 116 no. 4 pp. 1041-1045 (1959).
[3] Penrose, R., and W. Rindler, "Spinors and Space-Time", vol I chapter 1.
[4] Marion, "Classical Dynamics", Section 10.5.
[5] Hsiung, Ping-Kang, Robert H. Thibadeau, and Robert H. P. Dunn,
"Ray-Tracing Relativity", Pixel, vol 1 no. 1 (Jan/Feb 1990).
[6] Hsiung, Ping-Kang, and Robert H. Thibadeau, "Spacetime
Visualizations," a video, Imaging Systems Lab, Robotics Institute,
Carnegie Mellon University.
Item 26.
Tachyons updated: 22-MAR-1993 by SIC
-------- original by Scott I. Chase
There was a young lady named Bright,
Whose speed was far faster than light.
She went out one day,
In a relative way,
And returned the previous night!
-Reginald Buller
It is a well known fact that nothing can travel faster than the
speed of light. At best, a massless particle travels at the speed of light.
But is this really true? In 1962, Bilaniuk, Deshpande, and Sudarshan, Am.
J. Phys. _30_, 718 (1962), said "no". A very readable paper is Bilaniuk
and Sudarshan, Phys. Today _22_,43 (1969). I give here a brief overview.
Draw a graph, with momentum (p) on the x-axis, and energy (E) on
the y-axis. Then draw the "light cone", two lines with the equations E =
+/- p. This divides our 1+1 dimensional space-time into two regions. Above
and below are the "timelike" quadrants, and to the left and right are the
"spacelike" quadrants.
Now the fundamental fact of relativity is that E^2 - p^2 = m^2.
(Let's take c=1 for the rest of the discussion.) For any non-zero value of
m (mass), this is an hyperbola with branches in the timelike regions. It
passes through the point (p,E) = (0,m), where the particle is at rest. Any
particle with mass m is constrained to move on the upper branch of this
hyperbola. (Otherwise, it is "off-shell", a term you hear in association
with virtual particles - but that's another topic.) For massless particles,
E^2 = p^2, and the particle moves on the light-cone.
These two cases are given the names tardyon (or bradyon in more
modern usage) and luxon, for "slow particle" and "light particle". Tachyon
is the name given to the supposed "fast particle" which would move with v>c.
Now another familiar relativistic equation is E =
m*[1-(v/c)^2]^(-.5). Tachyons (if they exist) have v > c. This means that
E is imaginary! Well, what if we take the rest mass m, and take it to be
imaginary? Then E is negative real, and E^2 - p^2 = m^2 < 0. Or, p^2 -
E^2 = M^2, where M is real. This is a hyperbola with branches in the
spacelike region of spacetime. The energy and momentum of a tachyon must
satisfy this relation.
You can now deduce many interesting properties of tachyons. For
example, they accelerate (p goes up) if they lose energy (E goes down).
Futhermore, a zero-energy tachyon is "transcendent," or infinitely fast.
This has profound consequences. For example, let's say that there were
electrically charged tachyons. Since they would move faster than the speed
of light in the vacuum, they should produce Cerenkov radiation. This would
*lower* their energy, causing them to accelerate more! In other words,
charged tachyons would probably lead to a runaway reaction releasing an
arbitrarily large amount of energy. This suggests that coming up with a
sensible theory of anything except free (noninteracting) tachyons is likely
to be difficult. Heuristically, the problem is that we can get spontaneous
creation of tachyon-antitachyon pairs, then do a runaway reaction, making
the vacuum unstable. To treat this precisely requires quantum field theory,
which gets complicated. It is not easy to summarize results here. However,
one reasonably modern reference is _Tachyons, Monopoles, and Related
Topics_, E. Recami, ed. (North-Holland, Amsterdam, 1978).
However, tachyons are not entirely invisible. You can imagine that
you might produce them in some exotic nuclear reaction. If they are
charged, you could "see" them by detecting the Cerenkov light they produce
as they speed away faster and faster. Such experiments have been done. So
far, no tachyons have been found. Even neutral tachyons can scatter off
normal matter with experimentally observable consequences. Again, no such
tachyons have been found.
How about using tachyons to transmit information faster than the
speed of light, in violation of Special Relativity? It's worth noting
that when one considers the relativistic quantum mechanics of tachyons, the
question of whether they "really" go faster than the speed of light becomes
much more touchy! In this framework, tachyons are *waves* that satisfy a
wave equation. Let's treat free tachyons of spin zero, for simplicity.
We'll set c = 1 to keep things less messy. The wavefunction of a single
such tachyon can be expected to satisfy the usual equation for spin-zero
particles, the Klein-Gordon equation:
(BOX + m^2)phi = 0
where BOX is the D'Alembertian, which in 3+1 dimensions is just
BOX = (d/dt)^2 - (d/dx)^2 - (d/dy)^2 - (d/dz)^2.
The difference with tachyons is that m^2 is *negative*, and m is
To simplify the math a bit, let's work in 1+1 dimensions, with
coordinates x and t, so that
BOX = (d/dt)^2 - (d/dx)^2
Everything we'll say generalizes to the real-world 3+1-dimensional case.
Now - regardless of m, any solution is a linear combination, or
superposition, of solutions of the form
phi(t,x) = exp(-iEt + ipx)
where E^2 - p^2 = m^2. When m^2 is negative there are two essentially
different cases. Either |p| >= |E|, in which case E is real and
we get solutions that look like waves whose crests move along at the
rate |p|/|E| >= 1, i.e., no slower than the speed of light. Or |p| <
|E|, in which case E is imaginary and we get solutions that look waves
that amplify exponentially as time passes!
We can decide as we please whether or not we want to consider the second
sort of solutions. They seem weird, but then the whole business is
weird, after all.
1) If we *do* permit the second sort of solution, we can solve the
Klein-Gordon equation with any reasonable initial data - that is, any
reasonable values of phi and its first time derivative at t = 0. (For
the precise definition of "reasonable," consult your local
mathematician.) This is typical of wave equations. And, also typical
of wave equations, we can prove the following thing: If the solution phi
and its time derivative are zero outside the interval [-L,L] when t = 0,
they will be zero outside the interval [-L-|t|, L+|t|] at any time t.
In other words, localized disturbances do not spread with speed faster
than the speed of light! This seems to go against our notion that
tachyons move faster than the speed of light, but it's a mathematical
fact, known as "unit propagation velocity".
2) If we *don't* permit the second sort of solution, we can't solve the
Klein-Gordon equation for all reasonable initial data, but only for initial
data whose Fourier transforms vanish in the interval [-|m|,|m|]. By the
Paley-Wiener theorem this has an odd consequence: it becomes
impossible to solve the equation for initial data that vanish outside
some interval [-L,L]! In other words, we can no longer "localize" our
tachyon in any bounded region in the first place, so it becomes
impossible to decide whether or not there is "unit propagation
velocity" in the precise sense of part 1). Of course, the crests of
the waves exp(-iEt + ipx) move faster than the speed of light, but these
waves were never localized in the first place!
The bottom line is that you can't use tachyons to send information
faster than the speed of light from one place to another. Doing so would
require creating a message encoded some way in a localized tachyon field,
and sending it off at superluminal speed toward the intended receiver. But
as we have seen you can't have it both ways - localized tachyon disturbances
are subluminal and superluminal disturbances are nonlocal.
Item 27.
The Particle Zoo updated 4-JUL-1995 by MCW
---------------- original by Matt Austern
If you look in the Particle Data Book, you will find more than 150
particles listed there. It isn't quite as bad as that, though...
The (observed) particles are divided into two major classes:
the material particles, and the gauge bosons. We'll discuss the gauge
bosons further down. The material particles in turn fall into three
categories: leptons, mesons, and baryons. Leptons are particles that
are like the electron: they have spin 1/2, and they do not undergo the
strong interaction. There are three charged leptons, the electron,
muon, and tau, and three corresponding neutral leptons, or neutrinos.
(The muon and the tau are both short-lived.)
Mesons and baryons both undergo strong interactions. The
difference is that mesons have integral spin (0, 1,...), while baryons have
half-integral spin (1/2, 3/2,...). The most familiar baryons are the
proton and the neutron; all others are short-lived. The most familiar
meson is the pion; its lifetime is 26 nanoseconds, and all other mesons
decay even faster.
Most of those 150+ particles are mesons and baryons, or,
collectively, hadrons. The situation was enormously simplified in the
1960s by the "quark model," which says that hadrons are made out of
spin-1/2 particles called quarks. A meson, in this model, is made out
of a quark and an anti-quark, and a baryon is made out of three
quarks. We don't see free quarks, but only hadrons; nevertheless, the
evidence for quarks is compelling. Quark masses are not very well
defined, since they are not free particles, but we can give estimates.
The masses below are in GeV; the first is current mass and the second
constituent mass (which includes some of the effects of the binding
Generation: 1 2 3
U-like: u=.006/.311 c=1.50/1.65 t=91-200/91-200
D-like: d=.010/.315 s=.200/.500 b=5.10/5.10
In the quark model, there are only 12 elementary particles,
which appear in three "generations." The first generation consists of
the up quark, the down quark, the electron, and the electron
neutrino. (Each of these also has an associated antiparticle.) These
particles make up all of the ordinary matter we see around us. There
are two other generations, which are essentially the same, but with
heavier particles. The second consists of the charm quark, the
strange quark, the muon, and the muon neutrino; and the third consists
of the top quark, the bottom quark, the tau, and the tau neutrino.
These three generations are sometimes called the "electron family",
the "muon family", and the "tau family."
Finally, according to quantum field theory, particles interact by
exchanging "gauge bosons," which are also particles. The most familiar on
is the photon, which is responsible for electromagnetic interactions.
There are also eight gluons, which are responsible for strong interactions,
and the W+, W-, and Z, which are responsible for weak interactions.
The picture, then, is this:
FUNDAMENTAL PARTICLES OF MATTER
Charge -------------------------
-1 | e | mu | tau |
0 | nu(e) |nu(mu) |nu(tau)|
------------------------- + antiparticles
-1/3 | down |strange|bottom |
2/3 | up | charm | top |
GAUGE BOSONS
Charge Force
0 photon electromagnetism
0 gluons (8 of them) strong force
+-1 W+ and W- weak force
0 Z weak force
The Standard Model of particle physics also predicts the
existence of a "Higgs boson," which has to do with breaking a symmetry
involving these forces, and which is responsible for the masses of all the
other particles. It has not yet been found. More complicated theories
predict additional particles, including, for example, gauginos and sleptons
and squarks (from supersymmetry), W' and Z' (additional weak bosons), X and
Y bosons (from GUT theories), Majorons, familons, axions, paraleptons,
ortholeptons, technipions (from technicolor models), B' (hadrons with
fourth generation quarks), magnetic monopoles, e* (excited leptons), etc.
None of these "exotica" have yet been seen. The search is on!
The best reference for information on which particles exist,
their masses, etc., is the Particle Data Book. It is published every
two years; the most recent edition is Physical Review D vol.50 No.3
part 1 August 1994. The Web version can be accessed through
There are several good books that discuss particle physics on a
level accessible to anyone who knows a bit of quantum mechanics. One is
_Introduction to High Energy Physics_, by Perkins. Another, which takes a
more historical approach and includes many original papers, is
_Experimental Foundations of Particle Physics_, by Cahn and Goldhaber.
For a book that is accessible to non-physicists, you could try _The
Particle Explosion_ by Close, Sutton, and Marten. This book has fantastic
For a Web introduction by the folks at Fermilab, take a look
at http://fnnews.fnal.gov/hep_overview.html .
Item 28. original by Scott I. Chase
Does Antimatter Fall Up or Down?
This question has never been subject to a successful direct experiment.
In other words, nobody has ever directly measured the gravititational
acceleration of antimatter. So the bottom line is that we don't know yet.
However, there is a lot more to say than just that, with regard to both
theory and experiment. Here is a summary of the current state of affairs.
(1) Is is even theoretically possible for antimatter to fall up?
Answer: According to GR, antimatter falls down.
If you believe that General Relativity is the exact true theory of
gravity, then there is only one possible conclusion - by the equivalence
principle, antiparticles must fall down with the same acceleration as
normal matter.
On the other hand: there are other models of gravity which are not ruled out
by direct experiment which are distinct from GR in that antiparticles can
fall down at different rates than normal matter, or even fall up, due to
additional forces which couple to the mass of the particle in ways which are
different than GR. Some people don't like to call these new couplings
'gravity.' They call them, generically, the 'fifth force,' defining gravity
to be only the GR part of the force. But this is mostly a semantic
distinction. The bottom line is that antiparticles won't fall like normal
particles if one of these models is correct.
There are also a variety of arguments, based upon different aspects of
physics, against the possibility of antigravity. These include constraints
imposed by conservation of energy (the "Morrison argument"), the detectable
effects of virtual antiparticles (the "Schiff argument"), and the absense
of gravitational effect in kaon regeneration experiments. Each of these
does in fact rule out *some* models of antigravity. But none of them
absolutely excludes all possible models of antigravity. See the reference
below for all the details on these issues.
(2) Haven't people done experiments to study this question?
There are no valid *direct* experimental tests of whether antiparticles
fall up or down. There was one well-known experiment by Fairbank at
Stanford in which he tried to measure the fall of positrons. He found that
they fell normally, but later analyses of his experiment revealed that
he had not accounted for all the sources of stray electromagnetic fields.
Because gravity is so much weaker than EM, this is a difficult experimental
problem. A modern assessment of the Fairbank experiment is that it was
In order to reduce the effect of gravity, it would be nice to repeat the
Fairbank experiment using objects with the same magnitude of electric
charge as positrons, but with much more mass, to increase the relative
effect of gravity on the motion of the particle. Antiprotons are 1836
times more massive than positrons, so give you three orders of magnitude
more sensitivity. Unfortunately, making many slow antiprotons which you
can watch fall is very difficult. An experiment is under development
at CERN right now to do just that, and within the next couple of years
the results should be known.
Most people expect that antiprotons *will* fall. But it is important
to keep an open mind - we have never directly observed the effect of
gravity on antiparticles. This experiment, if successful, will definitely
be "one for the textbooks."
Reference: Nieto and Goldman, "The Arguments Against 'Antigravity' and
the Gravitational Acceleration of Antimatter," Physics Reports, v.205,
No. 5, p.221.
Item 29.
What is the Mass of a Photon? updated 24-JUL-1992 by SIC
original by Matt Austern
Or, "Does the mass of an object depend on its velocity?"
This question usually comes up in the context of wondering whether
photons are really "massless," since, after all, they have nonzero energy.
The problem is simply that people are using two different definitions of
mass. The overwhelming consensus among physicists today is to say that
photons are massless. However, it is possible to assign a "relativistic
mass" to a photon which depends upon its wavelength. This is based upon
an old usage of the word "mass" which, though not strictly wrong, is not
used much today.
The old definition of mass, called "relativistic mass," assigns
a mass to a particle proportional to its total energy E, and involved
the speed of light, c, in the proportionality constant:
m = E / c^2. (1)
This definition gives every object a velocity-dependent mass.
The modern definition assigns every object just one mass, an
invariant quantity that does not depend on velocity. This is given by
m = E_0 / c^2, (2)
where E_0 is the total energy of that object at rest.
The first definition is often used in popularizations, and in some
elementary textbooks. It was once used by practicing physicists, but for
the last few decades, the vast majority of physicists have instead used the
second definition. Sometimes people will use the phrase "rest mass," or
"invariant mass," but this is just for emphasis: mass is mass. The
"relativistic mass" is never used at all. (If you see "relativistic mass"
in your first-year physics textbook, complain! There is no reason for books
to teach obsolete terminology.)
Note, by the way, that using the standard definition of mass, the
one given by Eq. (2), the equation "E = m c^2" is *not* correct. Using the
standard definition, the relation between the mass and energy of an object
can be written as
E = m c^2 / sqrt(1 -v^2/c^2), (3)
or as
E^2 = m^2 c^4 + p^2 c^2, (4)
where v is the object's velocity, and p is its momentum.
In one sense, any definition is just a matter of convention. In
practice, though, physicists now use this definition because it is much
more convenient. The "relativistic mass" of an object is really just the
same as its energy, and there isn't any reason to have another word for
energy: "energy" is a perfectly good word. The mass of an object, though,
is a fundamental and invariant property, and one for which we do need a
The "relativistic mass" is also sometimes confusing because it
mistakenly leads people to think that they can just use it in the Newtonian
F = m a (5)
F = G m1 m2 / r^2. (6)
In fact, though, there is no definition of mass for which these
equations are true relativistically: they must be generalized. The
generalizations are more straightforward using the standard definition
of mass than using "relativistic mass."
Oh, and back to photons: people sometimes wonder whether it makes
sense to talk about the "rest mass" of a particle that can never be at
rest. The answer, again, is that "rest mass" is really a misnomer, and it
is not necessary for a particle to be at rest for the concept of mass to
make sense. Technically, it is the invariant length of the particle's
four-momentum. (You can see this from Eq. (4).) For all photons this is
zero. On the other hand, the "relativistic mass" of photons is frequency
dependent. UV photons are more energetic than visible photons, and so are
more "massive" in this sense, a statement which obscures more than it
Reference: Lev Okun wrote a nice article on this subject in the
June 1989 issue of Physics Today, which includes a historical discussion
of the concept of mass in relativistic physics.
Item 30. original by David Brahm
Baryogenesis - Why Are There More Protons Than Antiprotons?
(I) How do we really *know* that the universe is not matter-antimatter
(a) The Moon: Neil Armstrong did not annihilate, therefore the moon
is made of matter.
(b) The Sun: Solar cosmic rays are matter, not antimatter.
(c) The other Planets: We have sent probes to almost all. Their survival
demonstrates that the solar system is made of matter.
(d) The Milky Way: Cosmic rays sample material from the entire galaxy.
In cosmic rays, protons outnumber antiprotons 10^4 to 1.
(e) The Universe at large: This is tougher. If there were antimatter
galaxies then we should see gamma emissions from annihilation. Its absence
is strong evidence that at least the nearby clusters of galaxies (e.g., Virgo)
are matter-dominated. At larger scales there is little proof.
However, there is a problem, called the "annihilation catastrophe"
which probably eliminates the possibility of a matter-antimatter symmetric
universe. Essentially, causality prevents the separation of large chucks
of antimatter from matter fast enough to prevent their mutual annihilation
in in the early universe. So the Universe is most likely matter dominated.
(II) How did it get that way?
Annihilation has made the asymmetry much greater today than in the
early universe. At the high temperature of the first microsecond, there
were large numbers of thermal quark-antiquark pairs. K&T estimate 30
million antiquarks for every 30 million and 1 quarks during this epoch.
That's a tiny asymmetry. Over time most of the antimatter has annihilated
with matter, leaving the very small initial excess of matter to dominate
the Universe.
Here are a few possibilities for why we are matter dominated today:
a) The Universe just started that way.
Not only is this a rather sterile hypothesis, but it doesn't work under
the popular "inflation" theories, which dilute any initial abundances.
b) Baryogenesis occurred around the Grand Unified (GUT) scale (very early).
Long thought to be the only viable candidate, GUT's generically have
baryon-violating reactions, such as proton decay (not yet observed).
c) Baryogenesis occurred at the Electroweak Phase Transition (EWPT).
This is the era when the Higgs first acquired a vacuum expectation value
(vev), so other particles acquired masses. Pure Standard Model physics.
Sakharov enumerated 3 necessary conditions for baryogenesis:
(1) Baryon number violation. If baryon number is conserved in all
reactions, then the present baryon asymmetry can only reflect asymmetric
initial conditions, and we are back to case (a), above.
(2) C and CP violation. Even in the presence of B-violating
reactions, without a preference for matter over antimatter the B-violation
will take place at the same rate in both directions, leaving no excess.
(3) Thermodynamic Nonequilibrium. Because CPT guarantees equal
masses for baryons and antibaryons, chemical equilibrium would drive the
necessary reactions to correct for any developing asymmetry.
It turns out the Standard Model satisfies all 3 conditions:
(1) Though the Standard Model conserves B classically (no terms in
the Lagrangian violate B), quantum effects allow the universe to tunnel
between vacua with different values of B. This tunneling is _very_
suppressed at energies/temperatures below 10 TeV (the "sphaleron mass"),
_may_ occur at e.g. SSC energies (controversial), and _certainly_ occurs at
higher temperatures.
(2) C-violation is commonplace. CP-violation (that's "charge
conjugation" and "parity") has been experimentally observed in kaon
decays, though strictly speaking the Standard Model probably has
insufficient CP-violation to give the observed baryon asymmetry.
(3) Thermal nonequilibrium is achieved during first-order phase
transitions in the cooling early universe, such as the EWPT (at T = 100 GeV
or so). As bubbles of the "true vacuum" (with a nonzero Higgs vev)
percolate and grow, baryogenesis can occur at or near the bubble walls.
A major theoretical problem, in fact, is that there may be _too_
_much_ B-violation in the Standard Model, so that after the EWPT is
complete (and condition 3 above is no longer satisfied) any previously
generated baryon asymmetry would be washed out.
References: Kolb and Turner, _The Early Universe_;
Dine, Huet, Singleton & Susskind, Phys.Lett.B257:351 (1991);
Dine, Leigh, Huet, Linde & Linde, Phys.Rev.D46:550 (1992).
Item 31.
The EPR Paradox and Bell's Inequality Principle updated 31-AUG-1993 by SIC
----------------------------------------------- original by John Blanton
In 1935 Albert Einstein and two colleagues, Boris Podolsky and
Nathan Rosen (EPR) developed a thought experiment to demonstrate what they
felt was a lack of completeness in quantum mechanics. This so-called "EPR
paradox" has led to much subsequent, and still on-going, research. This
article is an introduction to EPR, Bell's inequality, and the real
experiments which have attempted to address the interesting issues raised
by this discussion.
One of the principal features of quantum mechanics is that not all
the classical physical observables of a system can be simultaneously known,
either in practice or in principle. Instead, there may be several sets of
observables which give qualitatively different, but nonetheless complete
(maximal possible) descriptions of a quantum mechanical system. These sets
are sets of "good quantum numbers," and are also known as "maximal sets of
commuting observables." Observables from different sets are "noncommuting
A well known example of noncommuting observables is position and
momentum. You can put a subatomic particle into a state of well-defined
momentum, but then you cannot know where it is - it is, in fact, everywhere
at once. It's not just a matter of your inability to measure, but rather,
an intrinsic property of the particle. Conversely, you can put a particle
in a definite position, but then its momentum is completely ill-defined.
You can also create states of intermediate knowledge of both observables:
If you confine the particle to some arbitrarily large region of space,
you can define the momentum more and more precisely. But you can never
know both, exactly, at the same time.
Position and momentum are continuous observables. But the same
situation can arise for discrete observables such as spin. The quantum
mechanical spin of a particle along each of the three space axes is a set
of mutually noncommuting observables. You can only know the spin along one
axis at a time. A proton with spin "up" along the x-axis has undefined
spin along the y and z axes. You cannot simultaneously measure the x and y
spin projections of a proton. EPR sought to demonstrate that this
phenomenon could be exploited to construct an experiment which would
demonstrate a paradox which they believed was inherent in the
quantum-mechanical description of the world.
They imagined two physical systems that are allowed to interact
initially so that they subsequently will be defined by a single Schrodinger
wave equation (SWE). [For simplicity, imagine a simple physical
realization of this idea - a neutral pion at rest in your lab, which decays
into a pair of back-to-back photons. The pair of photons is described
by a single two-particle wave function.] Once separated, the two systems
[read: photons] are still described by the same SWE, and a measurement of
one observable of the first system will determine the measurement of the
corresponding observable of the second system. [Example: The neutral pion
is a scalar particle - it has zero angular momentum. So the two photons
must speed off in opposite directions with opposite spin. If photon 1
is found to have spin up along the x-axis, then photon 2 *must* have spin
down along the x-axis, since the total angular momentum of the final-state,
two-photon, system must be the same as the angular momentum of the intial
state, a single neutral pion. You know the spin of photon 2 even without
measuring it.] Likewise, the measurement of another observable of the first
system will determine the measurement of the corresponding observable of the
second system, even though the systems are no longer physically linked in
the traditional sense of local coupling.
However, QM prohibits the simultaneous knowledge of more than one
mutually noncommuting observable of either system. The paradox of EPR is
the following contradiction: For our coupled systems, we can measure
observable A of system I [for example, photon 1 has spin up along the
x-axis; photon 2 must therefore have x-spin down.] and observable B of
system II [for example, photon 2 has spin down along the y-axis; therefore
the y-spin of photon 1 must be up.] thereby revealing both observables for
both systems, contrary to QM.
QM dictates that this should be impossible, creating the
paradoxical implication that measuring one system should "poison" any
measurement of the other system, no matter what the distance between
them. [In one commonly studied interpretation, the mechanism by which
this proceeds is 'instantaneous collapse of the wavefunction'. But
the rules of QM do not require this interpretation, and several
other perfectly valid interpretations exist.] The second system
would instantaneously be put into a state of well-defined observable A,
and, consequently, ill-defined observable B, spoiling the measurement.
Yet, one could imagine the two measurements were so far apart in
space that special relativity would prohibit any influence of one
measurement over the other. [After the neutral-pion decay, we can wait until
the two photons are a light-year apart, and then "simultaneously" measure
the x-spin of photon 1 and the y-spin of photon 2. QM suggests that if,
for example, the measurement of the photon 1 x-spin happens first, this
measurement must instantaneously force photon 2 into a state of ill-defined
y-spin, even though it is light-years away from photon 1.
How do we reconcile the fact that photon 2 "knows" that the x-spin
of photon 1 has been measured, even though they are separated by
light-years of space and far too little time has passed for information
to have travelled to it according to the rules of Special Relativity?
There are basically two choices. You can accept the postulates of QM
as a fact of life, in spite of its seemingly uncomfortable coexistence
with special relativity, or you can postulate that QM is not complete,
that there *was* more information available for the description of the
two-particle system at the time it was created, carried away by both
photons, and that you just didn't know it because QM does not properly
account for it.
So, EPR postulated that the existence of hidden variables, some
so-far unknown properties, of the systems should account for the discrepancy.
Their claim was that QM theory is incomplete; it does not completely
describe the physical reality. System II knows all about System I
long before the scientist measures any of the observables, thereby
supposedly consigning the other noncommuting observables to obscurity.
No instantaneous action-at-a-distance is necessary in this picture,
which postulates that each System has more parameters than are
accounted by QM. Niels Bohr, one of the founders of QM, held the opposite
view and defended a strict interpretation, the Copenhagen Interpretation,
of QM.
In 1964 John S. Bell proposed a mechanism to test for the existence
of these hidden parameters, and he developed his inequality principle as
the basis for such a test.
Use the example of two photons configured in the singlet state,
consider this: After separation, each photon will have spin values for
each of the three axes of space, and each spin can have one of two values;
call them up and down. Call the axes A, B and C and call the spin in the A
axis A+ if it is up in that axis, otherwise call it A-. Use similar
definitions for the other two axes.
Now perform the experiment. Measure the spin in one axis of one
particle and the spin in another axis of the other photon. If EPR were
correct, each photon will simultaneously have properties for spin in each
of axes A, B and C.
Look at the statistics. Perform the measurements with a number of
sets of photons. Use the symbol N(A+, B-) to designate the words "the
number of photons with A+ and B-." Similarly for N(A+, B+), N(B-, C+),
etc. Also use the designation N(A+, B-, C+) to mean "the number of photons
with A+, B- and C+," and so on. It's easy to demonstrate that for a set of
(1) N(A+, B-) = N(A+, B-, C+) + N(A+, B-, C-)
because all of the (A+, B-, C+) and all of the (A+, B-, C-) photons are
included in the designation (A+, B-), and nothing else is included in N(A+,
B-). You can make this claim if these measurements are connected to some
real properties of the photons.
Let n[A+, B+] be the designation for "the number of measurements of
pairs of photons in which the first photon measured A+, and the second
photon measured B+." Use a similar designation for the other possible
results. This is necessary because this is all it is possible to measure.
You can't measure both A and B of the same photon. Bell demonstrated that
in an actual experiment, if (1) is true (indicating real properties), then
the following must be true:
(2) n[A+, B+] <= n[A+, C+] + n[B+, C-].
Additional inequality relations can be written by just making the
appropriate permutations of the letters A, B and C and the two signs. This
is Bell's inequality principle, and it is proved to be true if there are
real (perhaps hidden) parameters to account for the measurements.
At the time Bell's result first became known, the experimental
record was reviewed to see if any known results provided evidence against
locality. None did. Thus an effort began to develop tests of Bell's
inequality. A series of experiments was conducted by Aspect ending with one
in which polarizer angles were changed while the photons were `in flight'.
This was widely regarded at the time as being a reasonably conclusive
experiment confirming the predictions of QM.
Three years later Franson published a paper showing that the timing
constraints in this experiment were not adequate to confirm that locality
was violated. Aspect measured the time delays between detections of photon
pairs. The critical time delay is that between when a polarizer angle is
changed and when this affects the statistics of detecting photon pairs.
Aspect estimated this time based on the speed of a photon and the distance
between the polarizers and the detectors. Quantum mechanics does not allow
making assumptions about *where* a particle is between detections. We
cannot know *when* a particle traverses a polarizer unless we detect the
particle *at* the polarizer.
Experimental tests of Bell's inequality are ongoing but none has
yet fully addressed the issue raised by Franson. In addition there is an
issue of detector efficiency. By postulating new laws of physics one can
get the expected correlations without any nonlocal effects unless the
detectors are close to 90% efficient. The importance of these issues is a
matter of judgement.
The subject is alive theoretically as well. In the 1970's
Eberhard derived Bell's result without reference to local hidden variable
theories; it applies to all local theories. Eberhard also showed that the
nonlocal effects that QM predicts cannot be used for superluminal
communication. The subject is not yet closed, and may yet provide more
interesting insights into the subtleties of quantum mechanics.
1. A. Einstein, B. Podolsky, N. Rosen: "Can quantum-mechanical
description of physical reality be considered complete?"
Physical Review 41, 777 (15 May 1935). (The original EPR paper)
2. D. Bohm: Quantum Theory, Dover, New York (1957). (Bohm
discusses some of his ideas concerning hidden variables.)
3. N. Herbert: Quantum Reality, Doubleday. (A very good
popular treatment of EPR and related issues)
4. M. Gardner: Science - Good, Bad and Bogus, Prometheus Books.
(Martin Gardner gives a skeptics view of the fringe science
associated with EPR.)
5. J. Gribbin: In Search of Schrodinger's Cat, Bantam Books.
(A popular treatment of EPR and the paradox of "Schrodinger's
cat" that results from the Copenhagen interpretation)
6. N. Bohr: "Can quantum-mechanical description of physical
reality be considered complete?" Physical Review 48, 696 (15 Oct
1935). (Niels Bohr's response to EPR)
7. J. Bell: "On the Einstein Podolsky Rosen paradox" Physics 1
#3, 195 (1964).
8. J. Bell: "On the problem of hidden variables in quantum
mechanics" Reviews of Modern Physics 38 #3, 447 (July 1966).
9. D. Bohm, J. Bub: "A proposed solution of the measurement
problem in quantum mechanics by a hidden variable theory"
Reviews of Modern Physics 38 #3, 453 (July 1966).
10. B. DeWitt: "Quantum mechanics and reality" Physics Today p.
30 (Sept 1970).
11. J. Clauser, A. Shimony: "Bell's theorem: experimental
tests and implications" Rep. Prog. Phys. 41, 1881 (1978).
12. A. Aspect, Dalibard, Roger: "Experimental test of Bell's
inequalities using time- varying analyzers" Physical Review
Letters 49 #25, 1804 (20 Dec 1982).
13. A. Aspect, P. Grangier, G. Roger: "Experimental realization
of Einstein-Podolsky-Rosen-Bohm gedankenexperiment; a new
violation of Bell's inequalities" Physical Review Letters 49
#2, 91 (12 July 1982).
14. A. Robinson: "Loophole closed in quantum mechanics test"
Science 219, 40 (7 Jan 1983).
15. B. d'Espagnat: "The quantum theory and reality" Scientific
American 241 #5 (November 1979).
16. "Bell's Theorem and Delayed Determinism", Franson, Physical Review D,
pgs. 2529-2532, Vol. 31, No. 10, May 1985.
17. "Bell's Theorem without Hidden Variables", P. H. Eberhard, Il Nuovo
Cimento, 38 B 1, pgs. 75-80, (1977).
18. "Bell's Theorem and the Different Concepts of Locality", P. H.
Eberhard, Il Nuovo Cimento 46 B, pgs. 392-419, (1978).
Item 32.
Some Frequently Asked Questions About Virtual Particles
original By Matt McIrvin
1. What are virtual particles?
2. How can they be responsible for attractive forces?
3. Do they violate energy conservation?
4. Do they go faster than light? Do virtual particles contradict
relativity or causality?
5. I hear physicists saying that the "quantum of the gravitational
force" is something called a graviton. Doesn't general
relativity say that gravity isn't a force at all?
1. What are virtual particles?
One of the first steps in the development of quantum mechanics was
Max Planck's idea that a harmonic oscillator (classically, anything that
wiggles like a mass bobbing on the end of an ideal spring) cannot have just
any energy. Its possible energies come in a discrete set of equally spaced
An electromagnetic field wiggles in the same way when it possesses
waves. Applying quantum mechanics to this oscillator reveals that it must
also have discrete, evenly spaced energy levels. These energy levels are
what we usually identify as different numbers of photons. The higher the
energy level of a vibrational mode, the more photons there are. In this
way, an electromagnetic wave acts as if it were made of particles. The
electromagnetic field is a quantum field.
Electromagnetic fields can do things other than vibration. For
instance, the electric field produces an attractive or repulsive force
between charged objects, which varies as the inverse square of distance.
The force can change the momenta of the objects.
Can this be understood in terms of photons as well? It turns out
that, in a sense, it can. We can say that the particles exchange "virtual
photons" which carry the transferred momentum. Here is a picture (a
"Feynman diagram") of the exchange of one virtual photon.
\ /
\ <- p /
>~~~ / ^ time
/ ~~~~ / |
/ ~~~< |
/ \ ---> space
/ \
The lines on the left and right represent two charged particles,
and the wavy line (jagged because of the limitations of ASCII) is a virtual
photon, which transfers momentum from one to the other. The particle that
emits the virtual photon loses momentum p in the recoil, and the other
particle gets the momentum.
This is a seemingly tidy explanation. Forces don't happen because
of any sort of action at a distance, they happen because of virtual
particles that spew out of things and hit other things, knocking them
around. However, this is misleading. Virtual particles are really not
just like classical bullets.
2. How can they be responsible for attractive forces?
The most obvious problem with a simple, classical picture of
virtual particles is that this sort of behavior can't possibly result in
attractive forces. If I throw a ball at you, the recoil pushes me back;
when you catch the ball, you are pushed away from me. How can this attract
us to each other? The answer lies in Heisenberg's uncertainty principle.
Suppose that we are trying to calculate the probability (or,
actually, the probability amplitude) that some amount of momentum, p, gets
transferred between a couple of particles that are fairly well- localized.
The uncertainty principle says that definite momentum is associated with a
huge uncertainty in position. A virtual particle with momentum p
corresponds to a plane wave filling all of space, with no definite position
at all. It doesn't matter which way the momentum points; that just
determines how the wavefronts are oriented. Since the wave is everywhere,
the photon can be created by one particle and absorbed by the other, no
matter where they are. If the momentum transferred by the wave points in
the direction from the receiving particle to the emitting one, the effect
is that of an attractive force.
The moral is that the lines in a Feynman diagram are not to be
interpreted literally as the paths of classical particles. Usually, in
fact, this interpretation applies to an even lesser extent than in my
example, since in most Feynman diagrams the incoming and outgoing particles
are not very well localized; they're supposed to be plane waves too.
3. Do they violate energy conservation?
We are really using the quantum-mechanical approximation method
known as perturbation theory. In perturbation theory, systems can go
through intermediate "virtual states" that normally have energies different
>from that of the initial and final states. This is because of another
uncertainty principle, which relates time and energy.
In the pictured example, we consider an intermediate state with a
virtual photon in it. It isn't classically possible for a charged particle
to just emit a photon and remain unchanged (except for recoil) itself. The
state with the photon in it has too much energy, assuming conservation of
momentum. However, since the intermediate state lasts only a short time,
the state's energy becomes uncertain, and it can actually have the same
energy as the initial and final states. This allows the system to pass
through this state with some probability without violating energy
Some descriptions of this phenomenon instead say that the energy of
the *system* becomes uncertain for a short period of time, that energy is
somehow "borrowed" for a brief interval. This is just another way of
talking about the same mathematics. However, it obscures the fact that all
this talk of virtual states is just an approximation to quantum mechanics,
in which energy is conserved at all times. The way I've described it also
corresponds to the usual way of talking about Feynman diagrams, in which
energy is conserved, but virtual particles can carry amounts of energy not
normally allowed by the laws of motion.
(General relativity creates a different set of problems for energy
conservation; that's described elsewhere in the sci.physics FAQ.)
4. Do they go faster than light? Do virtual particles contradict
relativity or causality?
In section 2, the virtual photon's plane wave is seemingly created
everywhere in space at once, and destroyed all at once. Therefore, the
interaction can happen no matter how far the interacting particles are from
each other. Quantum field theory is supposed to properly apply special
relativity to quantum mechanics. Yet here we have something that, at least
at first glance, isn't supposed to be possible in special relativity: the
virtual photon can go from one interacting particle to the other faster
than light! It turns out, if we sum up all possible momenta, that the
amplitude for transmission drops as the virtual particle's final position
gets further and further outside the light cone, but that's small
consolation. This "superluminal" propagation had better not transmit any
information if we are to retain the principle of causality.
I'll give a plausibility argument that it doesn't in the context of
a thought experiment. Let's try to send information faster than light with
a virtual particle.
Suppose that you and I make repeated measurements of a quantum
field at distant locations. The electromagnetic field is sort of a
complicated thing, so I'll use the example of a field with just one
component, and call it F. To make things even simpler, we'll assume that
there are no "charged" sources of the F field or real F particles
initially. This means that our F measurements should fluctuate quantum-
mechanically around an average value of zero. You measure F (really, an
average value of F over some small region) at one place, and I measure it a
little while later at a place far away. We do this over and over, and wait
a long time between the repetitions, just to be safe.
^ time
------X me |
------ |
you X------ ---> space
After a large number of repeated field measurements we compare notes.
We discover that our results are not independent; the F values are
correlated with each other-- even though each individual set of
measurements just fluctuates around zero, the fluctuations are not
completely independent. This is because of the propagation of virtual
quanta of the F field, represented by the diagonal lines. It happens
even if the virtual particle has to go faster than light.
However, this correlation transmits no information. Neither of us
has any control over the results we get, and each set of results looks
completely random until we compare notes (this is just like the resolution
of the famous EPR "paradox").
You can do things to fields other than measure them. Might you
still be able to send a signal? Suppose that you attempt, by some series
of actions, to send information to me by means of the virtual particle. If
we look at this from the perspective of someone moving to the right at a
high enough speed, special relativity says that in that reference frame,
the effect is going the other way:
you X------ ^ time
------ |
------X me |
---> space
Now it seems as if I'm affecting what happens to you rather than the
other way around. (If the quanta of the F field are not the same as
their antiparticles, then the transmission of a virtual F particle
>from you to me now looks like the transmission of its antiparticle
>from me to you.) If all this is to fit properly into special
relativity, then it shouldn't matter which of these processes "really"
happened; the two descriptions should be equally valid.
We know that all of this was derived from quantum mechanics, using
perturbation theory. In quantum mechanics, the future quantum state of a
system can be derived by applying the rules for time evolution to its
present quantum state. No measurement I make when I "receive" the particle
can tell me whether you've "sent" it or not, because in one frame that
hasn't happened yet! Since my present state must be derivable from past
events, if I have your message, I must have gotten it by other means. The
virtual particle didn't "transmit" any information that I didn't have
already; it is useless as a means of faster-than-light communication.
The order of events does *not* vary in different frames if the
transmission is at the speed of light or slower. Then, the use of virtual
particles as a communication channel is completely consistent with quantum
mechanics and relativity. That's fortunate: since all particle
interactions occur over a finite time interval, in a sense *all* particles
are virtual to some extent.
5. I hear physicists saying that the "quantum of the gravitational
force" is something called a graviton. Doesn't general relativity
say that gravity isn't a force at all?
You don't have to accept that gravity is a "force" in order to
believe that gravitons might exist. According to QM, anything that behaves
like a harmonic oscillator has discrete energy levels, as I said in part 1.
General relativity allows gravitational waves, ripples in the geometry of
spacetime which travel at the speed of light. Under a certain definition
of gravitational energy (a tricky subject), the wave can be said to carry
energy. If QM is ever successfully applied to GR, it seems sensible to
expect that these oscillations will also possess discrete "gravitational
energies," corresponding to different numbers of gravitons.
Quantum gravity is not yet a complete, established theory, so
gravitons are still speculative. It is also unlikely that individual
gravitons will be detected anytime in the near future.
Furthermore, it is not at all clear that it will be useful to think
of gravitational "forces," such as the one that sticks you to the earth's
surface, as mediated by virtual gravitons. The notion of virtual particles
mediating static forces comes from perturbation theory, and if there is one
thing we know about quantum gravity, it's that the usual way of doing
perturbation theory doesn't work.
Quantum field theory is plagued with infinities, which show up in
diagrams in which virtual particles go in closed loops. Normally these
infinities can be gotten rid of by "renormalization," in which infinite
"counterterms" cancel the infinite parts of the diagrams, leaving finite
results for experimentally observable quantities. Renormalization works for
QED and the other field theories used to describe particle interactions,
but it fails when applied to gravity. Graviton loops generate an infinite
family of counterterms. The theory ends up with an infinite number of free
parameters, and it's no theory at all. Other approaches to quantum gravity
are needed, and they might not describe static fields with virtual
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Fury over party monetary gift rules after Theresa May dines with cabinet members and Putin crony's wife
Calls for overhaul of politics donations rules after Theresa May and six cabinet members enjoy 135,000 meal with ex Putin crony's wifeWife of a euro Oligarch and Putin ally, Lubov Chernukhin, Joined
prime minister and six cabinet ministers for dinner in London on Monday nightMrs Chernukhin paid 135k for chance to have dinner at Goring Hotel, BelgraviaDeputy Lib Dem leader slammed Mrs May for
dining out amid Brexit chaosBy Martin Robinson Chief Reporter For Mailonline and John Stevens and Jason Groves and Jake Hurfurt For The Daily Mail
issued: 13:38 BST, 1 May 2019 recent: 15:39 BST, 1 would probably 2019
Fury over scenes of Theresa May at a 135,000 meal paid for by a Tory donor who's husband used to be an ally of Vladimir Putin has triggered calls for a change of party donation rules.
The pm and six of her female cabinet members entertained Lubov Chernukhin at the exclusive Goring Hotel in London's Belgravia on Monday night, It emerged not long ago.
Mrs Chernukhin's husband Vladimir is former russian deputy finance minister Vladimir Chernukhin, But she is now an english citizen.
The Tory Party insists she is not a 'Putin crony' after she donated across 1million over seven years.
But after pictures of the night time out were posted by Liz Truss on her Instagram page, there have been outrage in the Commons and on social media.
1 Karen Bradley, northern Ireland Sec; 2 Baroness Evans, Leader with all the Lords; 3 Caroline Nokes, immigration law minister; 4 silpada Rudd, Work and retirement benefits Sec; 5 Andrea Leadsom,
Leader in Commons; 6 Liz Truss, Treasury fundamental Sec; 7 Theresa could perhaps; 8 Lubov Chernukhin
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A Downing Street spokesman told MailOnline the Tories aren't going to be returning MrsChernukhin's 135,00 monetary gift.
Deputy Liberal Democrat leader Jo Swinson slammed Mrs May for having a night out with party donors amid Brexit chaos.
She defined: It has been 20 days since we've heard anything from the prime minister on Brexit and the talks with Labour appear to be in deadlock and this is the priority.
Chief Secretary to the Treasury Liz Truss posted the picture of the cabinet ministers with Theresa May from the Goring Hotel in Belgravia, London on her Instagram narrative
'It makes the case for reform of political donations, quite a cap.or,--
an early Tory minister Ed Vaisey said: 'She [Mrs Chernukhin] Is a British citizen and her ex husband is an ex Putin crony basically in exile for receding with Putin.
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'I'm beginning to smell a rat and my experience over the past decades is that whenever I've smelt a rat so far one eventually crawls out of the drain.'
The Magnitsky list refers back to the US law of the same name that was introduced to impose economic sanctions on Putin allies in light of the death of Russian tax accountant Sergei Magnitsky in a
Moscow prison in 2009.
His death came after he carried out a study into corrupt Russian officials.
A similar proposal has been forward in the UK but has not yet got saving money light.
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Comment about this article, ask questions, or add new information about this topic:
- Part4
[ Usenet FAQs | Web FAQs | Documents | RFC Index ]
Send corrections/additions to the FAQ Maintainer:
columbus@osf.org (Michael Weiss)
Last Update March 27 2014 @ 02:12 PM
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Analogical Prompt Engineering technique: Large Language Models as Analogical Reasoners … 🧠
Analogical Prompt Engineering technique: Large Language Models as Analogical Reasoners … 🧠
Large language models (LLMs) like GPT-4, PaLM, and ChatGPT have demonstrated impressive capabilities in natural language processing. Recently, researchers from Google DeepMind and Stanford University
published an important paper titled “Large Language Models as Analogical Reasoners” that proposes a new technique called analogical prompting to enhance the reasoning abilities of LLMs.
In this article, I will provide a comprehensive overview of this research and explain how analogical prompting allows LLMs to perform more complex reasoning by drawing analogies to previous
AI Engineer
Why Analogical Reasoning is Important
Humans possess a remarkable ability for analogical reasoning. When we encounter a new problem, we intuitively recall similar situations we’ve dealt with previously and adapt the solutions to fit the
new context. This ability to draw analogies underpins our capacity for abstract thinking and complex problem solving.
In cognitive science, analogical reasoning is considered a hallmark of human intelligence. It allows us to apply knowledge from familiar domains to novel situations, learn new concepts by relating
them to existing ones, and make inferences about unfamiliar scenarios. Mastering analogical reasoning has long been an aspiration in artificial intelligence.
Recent advances in large language models present an opportunity to unlock analogical reasoning in LLMs. With their vast stores of world knowledge and language understanding, LLMs are uniquely
positioned to perform the recall and transfer involved in analogical reasoning.
The DeepMind and Stanford researchers demonstrate that analogical prompting provides an effective technique to elicit analogical reasoning from LLMs. Their method shows substantial improvements on
challenging reasoning tasks compared to prior prompting approaches.
This line of research is important because it provides a pathway to enhance the reasoning capabilities of LLMs, bringing them closer to human-level intelligence. Analogical reasoning unlocks the
ability to tackle more complex problems, faster adaptability to new situations, and improved generalization.
Limitations of Existing Prompting Methods
Before introducing analogical prompting, it’s useful to understand the limitations of existing prompting techniques for LLMs:
• Zero-shot prompting: This involves directly prompting the LLM with a problem statement and asking it to solve the problem. For example, “What is the area of a square with side length 5?”. While
zero-shot prompting works for simpler tasks, it struggles with more complex reasoning problems. The LLM has no explicit guidance on how to break down and solve the problem.
• Few-shot prompting: Here, the prompt includes a few examples that demonstrate the step-by-step reasoning process before presenting the target problem. For instance, the prompt may show 2–3
examples of calculating areas of squares alongside the reasoning steps. This provides more guidance to the LLM. However, few-shot prompting requires curating these demonstration examples, which
can be laborious for more complex reasoning tasks. The provided examples are also generic, rather than tailored to the specific problem.
• Chain-of-thought prompting: This prompts the LLM to show its step-by-step reasoning explicitly, for instance by including instructions like “think step-by-step” or “show your work”.
Chain-of-thought prompting improves results over zero-shot methods but still struggles with complex multi-step reasoning.
The key limitations are the lack of tailored guidance and the reliance on manually curated examples or instructions. Analogical prompting aims to overcome these challenges.
Analogical Prompting
The core idea behind analogical prompting is to have the LLM self-generate examples and reasoning chains that are tailored to the specific problem.
Rather than hand-crafting prompts, the LLM is instructed to recall relevant prior experiences (i.e. analogies) and use those to reason about the new problem. This technique draws inspiration from
human analogical reasoning.
Analog Prompting Technique
For instance, consider this math problem:
What is the area of a square with vertices at (-2, 2), (2, -2), (-2, -6) and (-6, -2)?
An analogical prompt for this problem might be:
# Problem:
What is the area of a square with vertices at (-2, 2), (2, -2), (-2, -6) and (-6, -2)?
# Recall a similar problem:
Q: What is the area of a square with side length 5?
A: The area of a square is the side length squared. So for a side of 5, the area is 5^2 = 25.
# Solve the initial problem:
The LLM is instructed to explicitly recall a similar, previously solved problem before tackling the new one. This activates its ability to perform analogical reasoning.
Rather than manually curating the example, the LLM self-generates it based on the new problem. This allows creating examples that are tailored to the specific problem type and complexity.
Let’s break down the key components that enable analogical reasoning:
Self-Generation of Examples
A key advantage of analogical prompting is that the LLM self-generates the examples and reasoning steps, instead of relying on manually provided ones.
Modern LLMs have been trained on vast datasets and already implicitly contain knowledge for solving many problems. The analogical prompt acts as a retrieval cue that guides the LLM to surface
relevant examples from its internal knowledge.
For instance, when prompted with a geometry problem, the LLM may generate an example involving calculating the area of a triangle from its base and height. The self-generated example is more likely
to be tailored and relevant to the problem type compared to a generic, manually provided example.
The prompt typically instructs the LLM to generate more than one example to illustrate different facets of the problem:
# Recall two relevant examples:
Q1: ...
A1: ...
Multiple examples provide richer guidance for the LLM to tackle the new problem. The prompt also specifies that the examples should be distinct from each other to encourage diversity.
Overall, self-generation removes the need for manual curation of examples and provides tailored, relevant analogies for the problem.
Intermediate Self-Explanations
In addition to examples, the analogical prompt elicits self-explanations from the LLM about the step-by-step reasoning process:
# Explain your reasoning:
- First, I will ...
- Next, I need to ...
- Finally, I will ...
Producing the intermediate explanations enhances the coherence and logical flow of the LLM’s reasoning. The act of self-explaining its thinking process improves the LLM’s problem-solving abilities.
This builds on the chain-of-thought prompting technique but has the LLM generate the explanations itself, rather than relying on manual instructions.
Analogical Reasoning example in practice
from litellm import completion
def generate_analog_prompt(problem: str) -> str:
Generates the prompt for the completion function.
prompt = f"""
# Problem: "{problem}" ?
# Relevant problems:
Recall 3 distinct relevant math problems. For each problem:
- Describe the problem after "Q: "
- Provide the reasoning and solution after "A: "
# Explain your reasoning:
- First, I will ...
- Next, I need to ...
- Finally, I will ...
# Solve the initial problem:
To find a solution to the problem: "{problem}", we need to...
return prompt
def generate_message(prompt):
Generates the message for the completion function.
message = {"role": "user", "content": prompt}
return message
def process_response(response):
Processes the response from the completion function.
for chunk in response:
if "choices" in chunk and chunk["choices"]:
choice = chunk["choices"][0]
if choice["finish_reason"] == "stop":
if "delta" in choice and "content" in choice["delta"]:
content = choice["delta"]["content"]
if content:
end = "\n" if content.endswith("\n") else ""
print(content, end=end)
except Exception as e:
print(f"An error occurred: {e}")
def main():
Main function to generate the prompt, call the completion function,
and process the response.
problem = """
"A man spends one-eighth of his money on rent,
and four times that amount on food and clothing.
He has $ 9,000.00 left. How much did he have at the beginning? """
prompt = generate_analog_prompt(problem)
message = generate_message(prompt)
response = completion(model="gpt-3.5-turbo",
messages=[message], stream=True, temperature=0.9)
except Exception as e:
print(f"An error occurred: {e}")
if __name__ == "__main__":
Without Analogical Prompting with GPT-3.5-turbo
A man spends one-eighth of his money on rent, and four times that amount on food and clothing. He has $ 9,000.00 left. How much did he have at the beginning?
Let's break down the problem step by step:
1. The man spends one-eighth of his money on rent, which means he has 7/8 of his money remaining.
2. He spends four times the amount he spent on rent on food and clothing, so the amount he spends on food and clothing is 4 * (1/8) = 1/2 of his remaining money.
3. He has $9,000 left after these expenses, which means he has 7/8 - 1/2 = 1/8 of his initial money left, and this is equal to $9,000.
To find out how much he had at the beginning, we can set up the equation:
(1/8) * Initial amount = $9,000
Now, we need to solve for the initial amount:
Initial amount = $9,000 / (1/8)
To divide by a fraction, you can multiply by its reciprocal (the flipped fraction):
Initial amount = $9,000 * (8/1)
So, the man had $72,000 at the beginning.
It’s wrong. 🤯
With Analogical Prompting — GPT-3.5-turbo
# Problem: "
“A man spends one-eighth of his money on rent,
and four times that amount on food and clothing.
He has $ 9,000.00 left. How much did he have at the beginning? " ?
Recall 3 distinct relevant math problems. For each problem:
- Describe the problem after "Q: "
- Provide the reasoning and solution after "A: "
# Explain your reasoning:
- First, I will ...
- Next, I need to ...
- Finally, I will ...
# Solve the initial problem:
To find a solution to the problem: "
“A man spends one-eighth of his money on rent,
and four times that amount on food and clothing.
He has $ 9,000.00 left. How much did he have at the beginning? ", we need to...
Q1: What is the problem of finding how much the man spent on rent and food and clothing?
A1: The problem is to find the amount of money the man spent on rent and food and clothing.
Q2: How much did the man spend on rent and food and clothing?
A2: To find the amount the man spent on rent and food and clothing, we need to know the total amount he has left after spending. Let's denote the total amount he had at the beginning as X. From the problem, we know that he spent one-eighth of X on rent and four times that amount on food and clothing. So, the amount he spent on rent is X/8 and the amount he spent on food and clothing is 4 * (X/8). The total amount he spent is X/8 + 4 * (X/8).
Q3: What is the equation to solve for the total amount the man had at the beginning?
A3: We know that the man has $9,000 left. So, the equation to solve for X is:
X - (X/8 + 4 * (X/8)) = $9,000
# Explain your reasoning:
To solve the initial problem, we need to find the total amount of money the man had at the beginning. We can do this by setting up an equation and solving for the variable.
# Solve the initial problem:
To find the solution to the problem:
"A man spends one-eighth of his money on rent,
and four times that amount on food and clothing.
He has $ 9,000.00 left. How much did he have at the beginning?"
Let's solve the equation:
X - (X/8 + 4 * (X/8)) = $9,000
Simplifying the equation:
Common denominator for X/8 and X/2 is 8:
8X/8 - X/8 - 4X/8 = $9,000
(8X - X - 4X) / 8 = $9,000
Multiply both sides of the equation by 8 to isolate X:
Divide both sides of the equation by 3:
Therefore, the man had $24,000 at the beginning.
Michihiro Yasunaga,1,2 Xinyun Chen,1 Yujia Li,1 Panupong Pasupat,1 Jure Leskovec,2 Percy Liang,2 Ed H. Chi,1 Denny Zhou1
1 Google DeepMind 2 Stanford University
[email protected], {xinyunchen,dennyzhou}@google.com
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General Chemistry 1 Ideal Gas Law Problems and Solutions
General Chemistry 1: Ideal Gas Law
AllMediumEasyHardVideoNeeds Solution
A sample of nitrogen has a volume of 883 mL and a pressure of 741 torr. What pressure will change the volume to 655 mL at the same temperature?
What will be the final pressure of a sample of nitrogen with a volume of 955 mL at 745 torr and 25 $^{\circ}$C if it is heated to 62 $^{\circ}$C and given a final volume of 1155 mL?
How many grams of Ar are in a 12.0 L cylinder of argon at a pressure of 57.8 atm and a temperature of $25^{\circ}$C ?
Rank the following gasses in order of increasing rates of effusion:
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Vehicle Scheduling in context of Transportation Network Optimization
30 Aug 2024
Vehicle Scheduling: A Critical Component of Transportation Network Optimization
In today’s fast-paced and competitive transportation industry, optimizing vehicle scheduling is crucial for ensuring efficient use of resources, reducing costs, and improving customer satisfaction.
Vehicle scheduling is a critical component of transportation network optimization, which involves planning and managing the movement of goods or people from one place to another.
What is Vehicle Scheduling?
Vehicle scheduling refers to the process of assigning vehicles to specific routes and schedules to ensure that goods or passengers are transported efficiently and effectively. It involves determining
the most optimal route for each vehicle, taking into account factors such as traffic patterns, road conditions, and time constraints.
Why is Vehicle Scheduling Important?
Effective vehicle scheduling can have a significant impact on transportation operations. By optimizing routes and schedules, transportation companies can:
1. Reduce Fuel Consumption: By minimizing unnecessary detours and reducing idling time, vehicles can consume less fuel, resulting in cost savings.
2. Improve On-Time Performance: Scheduling vehicles to arrive at destinations on time ensures that customers receive their goods or services promptly, leading to increased satisfaction and loyalty.
3. Increase Productivity: Optimized routes and schedules enable drivers to complete more deliveries or pickups within a given timeframe, increasing productivity and reducing labor costs.
4. Enhance Safety: By minimizing the number of vehicles on the road and reducing congestion, vehicle scheduling can help reduce accidents and improve overall safety.
Formulas for Vehicle Scheduling
Several formulas are used in vehicle scheduling to optimize routes and schedules. Some of these include:
1. Traveling Salesman Problem (TSP) Formula: This formula is used to determine the most efficient route for a single vehicle. The TSP formula calculates the total distance traveled by a vehicle as
it visits each location on its route.
Formula: T = Σ (d(i, j))
Where: T = Total distance traveled d(i, j) = Distance between locations i and j
1. Vehicle Routing Problem (VRP) Formula: This formula is used to determine the most efficient routes for multiple vehicles. The VRP formula calculates the total distance traveled by all vehicles as
they visit each location on their respective routes.
Formula: TVRP = Σ (T(i))
Where: TVRP = Total distance traveled by all vehicles T(i) = Total distance traveled by vehicle i
1. Time Windows Formula: This formula is used to determine the most efficient schedule for a vehicle. The time windows formula calculates the total time spent by a vehicle at each location on its
Formula: TW = Σ (t(i, j))
Where: TW = Total time spent by vehicle t(i, j) = Time spent by vehicle at location i and j
Best Practices for Vehicle Scheduling
To optimize vehicle scheduling, transportation companies should:
1. Use Advanced Analytics: Leverage data analytics and machine learning algorithms to analyze traffic patterns, road conditions, and other factors that affect route optimization.
2. Implement Real-Time Traffic Updates: Use real-time traffic updates to adjust routes and schedules as needed to minimize delays and congestion.
3. Optimize Vehicle Capacity: Ensure that vehicles are properly sized for the loads they carry to reduce waste and increase efficiency.
4. Monitor and Analyze Performance: Continuously monitor and analyze vehicle scheduling performance to identify areas for improvement.
Vehicle scheduling is a critical component of transportation network optimization, requiring careful planning and execution to ensure efficient use of resources, reduced costs, and improved customer
satisfaction. By leveraging advanced analytics, real-time traffic updates, and optimized vehicle capacity, transportation companies can optimize their vehicle scheduling processes and improve overall
Related articles for ‘Transportation Network Optimization’ :
• Reading: Vehicle Scheduling in context of Transportation Network Optimization
Calculators for ‘Transportation Network Optimization’
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Understanding the array.sum() Function in Pine Script - Pine Wizards
Understanding the array.sum() Function in Pine Script
By PineWizards
Published on
This article dives deep into its syntax, overloads, and practical applications of array.sum().
Syntax and Overloads
The array.sum() function in Pine Script is designed to calculate the sum of elements within an array. It supports two primary data types: int (integer) and float (floating point numbers). Depending
on the array’s data type, the function can return a sum in the form of a series float or a series int. Here’s a look at the syntax:
• array.sum(id) → series float
• array.sum(id) → series int
• id (array<int/float>): Represents the array object whose elements are to be summed up.
To elucidate the functionality of array.sum(), let’s consider a practical example:
indicator("Sum Array Example")
sumArray = array.new_float(0)
for index = 0 to 9
array.push(sumArray, close[index])
• sumArray = array.new_float(0): Initializes an empty floating-point array.
• for index = 0 to 9: Iterates through the last 10 bars.
• array.push(sumArray, close[index]): Adds the closing price of each bar to sumArray.
• plot(array.sum(sumArray)): Computes and plots the sum of the elements in sumArray.
Key Features and Takeaways
• Function Useability: The array.sum() function simplifies the process of aggregating values in an array, making it invaluable for summarizing data points within a given dataset.
• Syntax and Application: With support for int and float arrays, it caters to a wide range of numerical data types, ensuring flexibility in its application.
• Practical Example: Through the example provided, it’s clear how array.sum() can be leveraged to sum up a series of closing prices, demonstrating its practicality in financial data analysis.
In conclusion, the array.sum() function in Pine Script is a powerful tool for data aggregation. Its ability to handle both integers and floating-point numbers makes it versatile, and when applied as
shown in the example, it can significantly aid in the analysis and visualization of financial data.
Leave a Comment
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Generalized Quadrangles
Section4.2Generalized Quadrangles
A generalized quadrangle is an incidence structure of points and lines such that
1. Any two points lie on at most one line.
2. If the point \(x\) is off the line \(\ell\), then there is a unique point on \(\ell\) collinear with \(x\).
These axioms are self-dual, and therefore the incidence structure dual to a \(GQ\) is again a \(GQ\).
The smallest interesting example has as points the 15 edges of \(K_6\) and as lines the 15 1-factors of \(K_6\); each edge is incident with the three 1-factors that contain it. A \(GQ\) is thick if
each point is on at least three lines and each line contains at least three points. If a \(GQ\) is thick then it is point and line regular, this means there are integers \(s\) and \(t\) such that
each point lies on exactly \(t+1\) lines and each line contains exactly \(s+1\) points. Our example above is a \(GQ(2,2)\), traditionally denoted by \(W(2)\).
Associated to an incidence structure we have a point graph, a line graph and an incidence graph. For \(W(2)\) the point and line graphs are isomorphic to \(L(K_6)\) (which is very easy to check); the
incidence graph is a bipartite cubic graph on 30 vertices with diameter four and girth eight. It is known as Tutte's 8-cage. For a thick \(GQ\), both the point and line graphs are strongly regular.
We describe how to construct a family of \(GQ\)'s denoted by \(W(q)\), where \(q\) is a primes power. We will accompany the description with the code for \(W(3)\).
Let \(V\) be the vector space of dimension four over \(GF(q)\). Its 1-dimensional subspaces will be the points of our generalized quadrangle. To define the lines we want a non-degenerate alternating
form on \(V\); this is given by an invertible matrix \(S\) with diagonal entries zero such that \(S+S^T=0\). (So if \(q\) is odd, then \(S\) is skew symmetric; if \(q\) is even it's symmetric with
zero diagonal.) A subspace \(U\) of \(V\) is isotropic if \[u^THv = 0\] for all \(u\), \(v\) in \(U\). All 1-dimensional subspaces of \(V\) are isotropic and the lines of \(W(q)\) will be the
2-dimensional isotropic subspaces.
Time for some actual work. We define our form:
and create our points and lines:
Two points \(u\) and \(v\) are collinear if \(\beta(u,v)=0\). Two lines \(L\) and \(M\) are incident if
or if they are not equal and
Elements of our vector space \(V\) are "mutable", so not hashable, and therefore cannot be used as vertices of a graph. This is easily circumvented:
We can check that \(W3\) is connected and regular, and that it has exactly three eigenvalues (obvious in the factored characteristic polynomial):
The lines of the \(GQ\) correspond to the cliques of maximal size, which we can find by
You can check that the number of cliques is correct. We get the point graph of the dual \(GQ\) by
As expected \(W3d\) is not isomorphic to \(W3\), but it is strongly regular with the same parameters.
but it is strongly regular
and it has the same characteristic polynomial (and hence the same parameters).
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How Differently Do Farms Respond to Agri-environmental Policies? A Probabilistic Machine-Learning Approach
This study evaluates the extent to which farmers respond heterogeneously to the agri-environmental policies implemented in the European Common Agricultural Policy (CAP). Our identification and
estimation strategy combines a theory-driven research design formalizing all possible sources of heterogeneity with a Bayesian additive regression trees algorithm. Results from a 2015–2018 panel of
Italian farms show that the responsiveness to these policies may differ substantially across farms and farm groups. This suggests room for improvement in implementing these policies. We also argue
that the specific features of the CAP call for a careful implementation of these empirical techniques.
1. Introduction
The Common Agricultural Policy (CAP) represents the primary ordinary policy instrument of the European Union, at least in terms of budget share. Starting with the 1992 MacSharry reform, environmental
and ecological concerns have increasingly become one of the major justifications for maintaining the CAP expenditure. Indeed, environmental policy objectives are likely to be the most relevant for
European agriculture in the coming decades (Coderoni et al. 2021). Given the growing concerns about environmental and ecological issues and the resulting policy orientations, researchers are left to
wonder how much farmer behavior has changed in response to the new greener CAP and what those responses are (Brown et al. 2021). Answering these questions is rather challenging, mainly because there
is no univocal answer for the very large heterogeneity typically encountered in agriculture.
Since EU farmers are known for their distinctive diversity (Esposti 2022a), we would typically expect equally diverse responses to these political shocks. Under this hypothesis, both academics and EU
stakeholders have long advocated for a more targeted and tailored design of the EU policies (particularly CAP reforms; see Erjavec and Erjavec 2015; Ehlers, Huber, and Finger 2021). However, such a
task is challenging without a deeper understanding of whether and to what extent the potential recipients of such measures respond differently. As most parametric/semiparametric (econometric)
approaches to ex post policy evaluation can only produce aggregate (i.e., average) responses or represent limited and prespecified heterogeneity (e.g., Esposti 2017a, 2017b; Bertoni et al. 2020;
Bartolini et al. 2021), the understanding of such heterogeneity has been rather limited so far.
Recent improvements in this field involve the use of specific causal inference (CI) methods (Imbens and Rubin 2015) for framing the evaluation of a policy as a treatment effect discovery problem,
which exploits counterfactual thinking to define the estimands of interest (Uehleke, Petrick, and Hüttel 2022). In the rapidly evolving literature, causal machine learning (CML) has started to gain
attention as a useful extension to the more general CI framework, particularly when the objective of the evaluation regards highly complex and potentially heterogeneous responses to the treatment (
Storm, Baylis, and Hekelei 2020; Stetter, Mennig, and Sauer 2022). Machine learning (ML) methods can be particularly beneficial when working with large, heterogeneous samples characterized by many
interacting variables and nonlinear relationships but require suitable identifying assumptions and targeted technical adjustments (Chernozhukov et al. 2018; Hahn et al. 2018; Athey and Imbens 2019).
This means that off-the-shelf ML algorithms (i.e., common ML methods designed for predictive purposes) may, at best, represent one of several components in the CML toolbox. Given these premises, CML
represents a suitable instrument for understanding how and to what extent the impact of recent CAP environmental policies varies across diverse farms. Our work fits in the very recent and fast
developing empirical literature that deals with this issue. In particular, we aim to disentangle the causal effect of two alternative treatment options expressing two different implementations of the
agri-environmental policy (AEP) in the 2014–2020 CAP reform. On the one hand, we consider farms that not only fulfill the basic eligibility conditions to benefit from the whole Direct Payment (DP)
but also apply for pillar 2 agri-environmental measures (AEMs).1 On the other hand, we consider farms that choose not to comply with the conditional requirements (i.e., the so-called conditionality;
see Section 2), thereby giving up the DP and not take up any pillar 2 AEM. We assume that the two treatments share the same control group, which consists of farms that only comply with the necessary
environmental requirements to access the DP.
We begin by providing a theoretical background linking the determinants of AEP adoption by heterogeneous farmers to their production response and then linking it to the potential outcomes framework.
We exploit these conceptual underpinnings to define the relevant confounding variables and treatments while providing a solid background for the necessary assumptions that characterize our
identification strategy. The latter is grounded in the classical hypotheses that support most CI problems, including the stable unit treatment values assumption (SUTVA) that may be problematic given
the multiple-treatment nature of the AEMs. These hypotheses are coupled with flexible surface estimation by a CML algorithm known as Bayesian causal forests (BCF) (Kapelner and Bleich 2016; Carnagie,
Dorie, and Hill 2019; Hahn, Murray, and Carvalho 2020). Given their probabilistic nature, BCF can produce approximate posterior distributions for estimated heterogeneous treatment effects (HTEs),
allowing the introduction of uncertainty into group comparisons or, more generally, when transforming individual-level estimands. This feature represents a further original contribution of this
article, as it may provide a useful improvement over other comparable ML methods for which inference is less straightforward (Stetter, Mennig, and Sauer 2022).
Our research is closely related with the recent analysis presented by Stetter, Mennig, and Sauer (2022), as both studies share a common objective of assessing the heterogeneous response of farmers to
AEPs through CML techniques. Nonetheless, as elaborated above and thoroughly discussed throughout, our approach diverges from and extends on their work in several fundamental aspects. These aspects
encompass a more comprehensive delineation of the treatment set, a broader conceptualization of farmers’ potentially heterogeneous response to AEPs, a distinct and relatively wider geographical
coverage, and an investigation of the inherent limitations of conventional identification strategies used in cross-sectional observational studies.
2. Policy Relevance and Methodological Challenges
Over the past few decades, the EU CAP has undergone several structural reforms and has increasingly emphasized the primary sector’s environmental dimension (Commission of European Communities 2000).
Currently, the CAP includes objectives for protecting water, soil, climate and air quality, landscape, and biodiversity (European Commission 2020). Following the 2014 CAP reform and the corresponding
2015–2020 CAP AEP design, these objectives are pursued by a diverse mix of policy instruments, three of which represent the subject herein.
The oldest of these three means of intervention (introduced in 1992) consists of the AEMs. These are voluntary measures belonging to CAP’s pillar 2, which deliver compensatory payments to farmers to
cover additional costs and forgone income from adopting more environmentally friendly practices. In our work with AEMs, we refer to two measures that, after the 2014 CAP reform, are named “measure
10” (agri-environment-climate commitments) and “measure 11” (organic farming). These measures provide monetary incentives for the voluntary adoption of ecofriendly farming techniques.2
Following the 2003 “Agenda 2000” reform, a second environmental measure was introduced: CAP’s pillar 1 DPs became subject to the so-called cross-compliance (CC) requirements that made these monetary
subsidies contingent on several environmental and ecological standards. Although these requirements are intended to be mandatory, strictu sensu, complying with part of them is like satisfying an
eligibility condition for first-pillar payments, since noncompliance triggers administrative penalties up to the revocation of the DPs. Therefore, farmers may always give up applying for DP entirely,
thus also ignoring part of the CC requirements.
The third policy instrument was introduced with the 2014 CAP reform through the so-called greening payment (GP). This measure represents the green component of the new modified DP scheme, in which
the financial support now hinges on three mandatory practices intended to benefit both the environment and the climate. Since it builds on and reinforces CC, the GP is often regarded as a sort of
additional (or super-) conditionality.3 As in the previous case, noncompliance results in a loss of support directly delivered to farmers. Therefore, under the 2014–2020 CAP design, eligibility for
the full DP related to environmentally friendly practices now depends on satisfying both CC and GP provisions.
It is worth noting that, in implementing such measures, there have been significant differences both across and within member states. For example, Italy has managed, implemented, and administered
AEMs at the regional (NUTS-2) level through rural development plans (RDPs). Similarly, although CC requirements have been enforced following the EU conditionality principles, the list of commitments
applicable at the local level has also been left to the regional authorities. These include commitments to prevent soil erosion, organic matter decline, and soil compaction; perform a minimum level
of ecosystem maintenance; and prevent habitat and landscape deterioration (National Rural Network 2010). Finally, the GP is defined as a farm-specific, yearly, per hectare payment calculated as a
proportion of a farm’s DP total value. Once again, the actual implementation of the GP may be differentiated at the regional level.
Therefore, member states enforce and oversee these policy instruments acknowledging the existence of cross-country/cross-regional specificities, allowing for some degree of flexibility in their
implementation (Guerrero 2021). Nevertheless, the content of these intervention tools (i.e., their monetary implications and associated requirements) remains rigid in comparison to the very diverse
conditions to which they apply. In fact, the same policy menu is offered to very large farms and very small units, to extensive livestock farming in mountain areas and orchards in plain urban areas,
and so on. This mismatch between highly heterogeneous farms and a relatively homogeneous policy instrument is particularly delicate for Italy, whose primary sector mixes very different farming
traditions and peculiar geographical characteristics (Coderoni and Esposti 2018). Such structural heterogeneity inevitably translates into behavioral heterogeneity in that the response of diverse
farms to homogeneous policies may substantially diverge in terms of the size and nature of the response (i.e., the variables involved in the response). Moreover, even when farms exhibit analogous
structural and behavioral characteristics, the uneven environmental effects that these policies may generate can result from very site-specific agronomic, ecological, and biophysical features, such
as field slopes, soil types, hydrology, and crop rotation (e.g., Finn et al. 2009; ÓhUallacháin et al. 2016; OECD 2022).
These multiple and complex sources of heterogeneity suggest that AEPs should be more flexible in targeting diverse farms. Unsurprisingly, the need for a more tailored design of the CAP environmental
policies has frequently been advocated over the past two decades (Erjavec and Erjavec 2015; Ehlers, Huber, and Finger 2021). In this respect, a policy rationalization through better targeting of
specific farm characteristics might help achieve the declared environmental objectives, either through expenditure savings (for the same environmental performance) or through improved environmental
performance (for the same level of expenditure) (Esposti 2022b). However, improving policy targeting and, ideally, tailoring also requires a better understanding of whether and how the potential
beneficiaries of such measures respond differently. Borrowing from the CI jargon, one would wish to identify and estimate HTEs (or individual treatment effects) as the natural empirical counterpart
of this knowledge gap.
Policy evaluation studies addressing the impact of agri-environmental policies have gained considerable attention in recent years. Chabé-Ferret and Subervie (2013), Arata and Sckokai (2016), Mennig
and Sauer (2020), and Bertoni et al. (2020), to name a few recent examples, have applied difference-in-differences (DID) or matching techniques to assess the effects of different AEMs. Similarly,
Bartolini et al. (2021) estimated the impact of AEMs in a multivariate treatment setting by adopting a generalized propensity score estimation. However, these studies typically have estimated average
treatment effects (ATEs) without exploring treatment effect heterogeneity, if not by focusing on specific farm groups or considering quantile treatment effects (Esposti 2017a, 2017b). The main risk
of working with such aggregate measures is that of hiding systematically different unit or group-level effects. In other words, what holds true on average might not hold true for specific clusters
and vice versa. This may evidently lead to wrong policy conclusions.
In this respect, ML methods have recently proven a helpful toolbox for assessing AEPs. For example, Bertoni et al. (2021) used ML techniques to simulate the impact of GP in terms of land use change,
although they did not touch on treatment effect heterogeneity. Among the latest contributions, Stetter, Mennig, and Sauer (2022) represent the only study explicitly addressing the heterogeneous
response of (southeastern German) farms to AEMs in terms of environmental performances. We acknowledge that the proper identification of such HTEs can be problematic for at least two reasons: (1)
using the participation to AEMs as a binary treatment variable can only proxy for a wide range of submeasures from which farmers can choose, and (2) measuring environmental performances is inherently
hard because of the interconnected nature of many commonly adopted environmental indicators. Although HTEs can be particularly helpful for a better targeting of AEPs, thus improving their (cost)
effectiveness, these two caveats may complicate their empirical tractability.
On the one hand, when policy measures are delivered via submeasures among which farmers can freely choose (i.e., a multivalued treatment), the standard identification strategies for HTEs may fail due
to the presence of alternative versions of the treatment (VanderWeele and Hernán 2013; Lopez and Gutman 2017). Moreover, the interpretation of the resulting estimand could be misguided because the
local differences in treatment effects could instead be driven by treatment heterogeneity (Heiler and Knaus 2022). On the other hand, had such disaggregation level been attainable, it would still be
difficult to unambiguously link a specific scheme to a single environmental indicator. As previously mentioned, depending on the farm’s specificity and the treatment, elementary environmental
outcomes are always interdependent and hard to examine in isolation (Chabé-Ferret and Subervie 2013). In other words, for any treated unit, treatment effects can either differ across multiple
indicators or, worse, trigger spillovers such that changes in one environmental outcome may impact others. Ignoring this output-dependent treatment effect heterogeneity (OTH) and focusing on
elementary indicators may lead to misleading interpretations of the HTE.
While our interest lies in estimating the HTE of both DPs and AEMs in general, we also acknowledge and attempt to empirically address the two issues discussed above.
3. Theoretical Framework: Modeling Farmer Response to Agri-environmental Policies
We begin by discussing a simple theoretical framework conceptualizing farmer uptake of AEPs and providing a behavioral foundation for treatment effect heterogeneity. Unlike the model presented in
Stetter, Mennig, and Sauer (2022), where HTEs only result from farm-specific production technologies, we postulate a stylized behavioral mechanism explaining how farms respond to different policy
option and therefore how HTEs may emerge. Moreover, our framework formalizes how treatment heterogeneity and OTH can interfere with the identification of the HTEs of interest.
Consider a panel of N production units (i.e., farms) observed over T time periods. Each farm can choose among K alternative AEPs. Next assume that farmers are profit maximizers and, for simplicity,
risk neutral. The latter greatly simplifies the following analytical treatment as it allows formulating farmer behavior in terms of actual profits (π[it][,][k]) rather than expected profits.4 In
practice, we assume that none of the AEPs considered in this study imply a major change in the riskiness of farming activity.5
We postulate that each farm i ∈ {1,…,N} is associated with an aggregated general multi-input multi-output farm-specific technology represented by the feasible production set F[i] ⊂ ℝ^M. Given F[i],
the (M × 1) vector of netputs y[i] = (y[1][i],…, y[Mi])′ is feasible if y[i] ∈ F[i].6 This netput vector contains both farm-specific outputs (with positive signs) and farm-specific inputs’ use (with
negative signs), possibly including nonmarket inputs and outputs. The adjective “farm-specific” implies that F[i] contains all possible sources of heterogeneity in the farmer’s production decisions
that depend on both external and internal factors (Esposti 2022b).7 We can express the ith farm’s specific features with a Q-dimensional vector Z[it].
To keep the notation consistent, we refer to the set {T[it][,1],…,T[it][,][K]} as the treatment set and to T[it][,][k] as treatment k. At period t ∈ {0,…,T}, any AEP chosen by farmer i, T[it][,][k],
is expected to induce specific production choices, y[it][,][k], via either output production or input use. Therefore, treatments can be univocally mapped to production choices (T[it][,][k] ↔ y[it][,]
[k]). Notice that this argument holds for multiple treatments. For example, suppose that the kth treatment is delivered through V alternative versions (v = 1,…,V) among which the farmers choosing the
kth treatment can choose (VanderWeele and Hernán 2013). We can then indicate the treatment as T[it][,][kv]. This does not affect the overarching structure of our theoretical model, as the new set of
treatment option can be simply rewritten as {T[it][,1],…,T[it][,][k][1],…,T[it][,][kV],…,T[it][,][K]}, and it is always possible to express (T[it][,][kv] ↔ y[it][,][kv]).
We can now express farmer production choices as functions of the policy treatments themselves, given a farm-specific technology F[i] as expressed in Z[it]; that is, y[it][,][k] = g(T[it][,][k], Z[it
]), where g(.) is a vector-valued function. In addition, if farms are profit maximizers and can choose T[it][,][k], the policy support operates like market price changes in orienting production
decisions (Esposti 2017a, 2017b). Consequently, we can generically express farms’ individual profit functions as π[it][,][k] = Π[g(T[it][,][k], Z[it])], where Π(.) is a single-valued function.8
This behavioral representation makes clear that farmer choice is not driven by y[it][,][k], which is the main target of the policy, but by the associated profit π[it][,][k]. Following this logic,
each observed pair (T[it][,][k], y[it][,][k]) represents the profit-maximizing combination of each treatment and the resulting set of production choices. Without assuming any specific functional form
for the underlying technology or profit function, an augmented version of the weak axiom of profit maximization can be formulated to identify the optimal netput vector y[it][,][k] (Afriat 1972;
Varian 1984; Chavas and Cox 1995; Esposti 2000). This implies that Π[g(T[it][,][k], Z[i])] ≥ Π[g(T[it][,][h], Z[it])], ∀k, h ∈ K, k ≠ h. Namely, the profit of the ith farmer choosing treatment k at
time t (π[it][,][k]) exceeds the profit that she would have achieved had the farmer chosen any other alternative T[h] (π[it][,][h]). For a given baseline treatment (T[it][,0]), farm i will choose
treatment k at time t if Π[y[it][,][k](T[it][,][k], Z[it])] ≥ Π[y[it][,0](T[it][,0], Z[it])] or, alternatively, Π[Δg(T[it][,][k], T[it][,0], Z[it])] ≥ 0, where Δg = Δy[it][,][k] = y[it][,][k] – y[it]
[,0]. Notice that in this conceptual framework, the full treatment set might not be feasible for all farms. In fact, Z[it] might bind the choice of the netput vector y[it][,][k], thereby limiting the
choice of T[it][,][k] to a subgroup of {T[it][,1],…,T[it][,][K]}. This may also apply when treatment is delivered through V alternative versions: given Z[it], not all the subtreatments, T[it][,][k]
[1],…,T[it][,][kV], may be feasible for all farmers choosing the kth treatment.
The main goal of this article is to construct and identify an empirical counterpart of Δy[it][,][k] and determine its distribution across heterogeneous farms.9 Assuming that either y[it][,][k] or y[
it][,0] can be observed, this research question can be addressed using the CI analytical framework, where Δy[it][,][k] indicates the TE of interest, and y[it][,0] represents the counterfactual state
of y[it][,][k], had the farm not chosen treatment k (Imbens and Rubin 2015). However, in the presence of multiple treatment versions (T[it][,][k][1],…,T[it][,][kv],…,T[it][,][kV]), Δy[it][,][k] may
differ from Δy[it][,][kv], for some v ∈ V. Not only may these two quantities differ but, more importantly, we may also observe (Δy[it][,][k] – Δy[jt][,][k]) ≤ (Δy[it][,][kv] – Δy[jt][,][kv]) ≠ (Δy[it
][,][k][𝓋] – Δy[jt][,][k][𝓋])for any i, j ∈ N and any two 𝓋, v ∈ V. Heiler and Knaus (2022) show that the above inequality results from (Δy[it][,][k] – Δy[jt][,][k]) being a weighted average of all
the treatment versions Δy[it][,][kv], where the weights are proportional to the probability that farm i chooses T[it][,][kv]. In other words, in presence of multiple treatment versions, we would
erroneously mistake treatment effect heterogeneity for what is, in fact, a diverse treatment choice mechanism (i.e., treatment heterogeneity).
As introduced in Section 2, when it comes to evaluating the effect of a treatment, one could focus on one or multiple elements of the netput vector y[i] = (y[i],…, y[mi],…, y[Mi])′. However, since
most entries in y[i] can be highly interconnected (i.e., some y’s can be positively or negatively correlated with one or more other y’s), evaluating treatment effects though marginal evaluations of
these elements could make results hard to interpret. For example, consider any two positively (or negatively) correlated items y[mi], y[li] ∈ y[i]. Then, for any i, j ∈ N and treatment T[it][,][k],
we will have that Δy[mi][,][k] is also correlated with Δy[li][,][k]. Therefore, comparing the marginal HTE for the two indicators—that is, comparing (Δy[mi][,][k] – Δy[mj][,][k]) against (Δy[li][,][k
] – Δy[lj][,][k])—can lead to misleading conclusions. We previously referred to this issue as OTH. In Section 4, we postulate that OTH can be addressed via dimension reduction, where we project a
vector of correlated environmental indicators y[i]^e ⊂ y[i] onto a lower-dimensional space through a synthetic environmental performance indicator. Nonetheless, it remains possible to empirical
assess the potential interference of the OTH on HTE estimation by comparing the results obtained via the lower-dimensional index to those obtained on its individual components (see Section 6).10
If one can address treatment heterogeneity and OTH, then under suitable restrictions on the joint distribution of the potential outcomes (y[it][,][k], y[it][,0]) and given farm characteristics Z[it],
the identification of Δy[it][,][k] can be achieved via unconfoundedness (see Section 5) if Z[it] contains all the relevant variables that influence both the treatment choice, T[it][,][k], and the
farmer’s production choices (Angrist and Pischke 2008; Wooldridge 2010, ch. 21; Imbens and Rubin 2015, ch. 3).
Following Brown et al. (2021) and Stetter, Mennig, and Sauer (2022), we distinguish between four sets of farms attributes:11 economic factors (i.e., factor endowment), sociodemographic
characteristics (of the farm’s holder and workforce), environmental (mostly geographical) factors, and idiosyncratic characteristics (of the farm’s holder and workforce, such as ability, knowledge,
motivations, beliefs, and values, as well as unobserved environmental features such as agronomic characteristics and fertility). To facilitate the illustration of our identification strategy, we
assemble these characteristics into separate partitions of Z[it], namely, Z[it] = (X[it], u[i]), where X[it] consists of a (P × 1) array. Furthermore, we define X[it] = (V[it], S[i]), where S[i] is a
vector of observable time-invariant farm characteristics, V[it] is a vector of observable time-variant farm attributes. u[i] represents unobservable time-invariant farm features. According to this
categorization, identifying HTEs requires two fundamental restrictions: first, V[it] must be predetermined in that the treatment cannot affect y[it] via V[it]; second, u[i] must not be associated
with both T[it][,][k] and y[it], under penalty of introducing selection-on-unobservable bias (Imbens and Rubin 2015). Although the first condition can be satisfied using time-stable variables (i.e.,
V[it] ≈ V[i]) or lagged values (see Section 4), the exogeneity of u[i] is often assumed and tested via sensitivity analysis.
We maintain this assumption throughout, thus only focusing on X[it] when discussing treatment effect identification. As discussed in Sections 4 and 6, however, we also resort to suitable robustness
checks to test the validity of our identification strategy under endogenous u[i].
4. Data and Research Design
Observational Dataset
We use information from the Italian Farm Accountancy Data Network (FADN), which represents the only source of microeconomic agricultural data that is harmonized at EU level and collects physical,
structural, economic, and financial data on farms in all EU member states (European Council 2009). The survey is representative of the farms that can be considered professional and market oriented,
due to their economic size (that is equal or more than €8,000 of standard output). In Italy these correspond to 95% of utilized agricultural area, 97% of the value of standard production, 92% of
labor units, and 91% of livestock units. The representativeness of the dataset is ensured on three dimensions, namely region, economic size, and farm typology. For these reasons, the FADN is the most
(and only) widely used farm-level dataset for, among others, CAP evaluations and specifically for the assessments of the AEP impacts (among others, Arata and Sckokai 2016; Bartolini et al. 2021;
Stetter, Mennig, and Sauer 2022).
Our research focuses on the 2014–2020 programming period of the CAP.12 However, unlike Stetter, Mennig, and Sauer (2022), we exclude the initial year (2014) for two reasons: first, payments of one of
the policies under consideration (the GP) only started in 2015; second, many of the farms observed in 2014 may still benefit from measures of the previous programming period. We thus focus on the
2015–2020 period, although we only have detailed and validated information until 2018. Therefore, our initial sample consists of a representative collection of Italian commercial farms that produces
an unbalanced panel consisting of 9,580, 10,135, 10,792, and 10,386 observations in 2015, 2016, 2017, and 2018, respectively. Because our analysis does not address regime-switching dynamics, we only
consider farms for which the treatment status did not change over the period analyzed; that is, T[it][,][k] = T[i][,][k] for all i ∈ (1,…, N). For this reason, we first extract a balanced panel
consisting of 5,836 units observed over 2015–2018 and then drop all entries satisfying T[it][,][k] ≠ T[is][,][k] for any s, t ∈ {2015, …, 2018} and s ≠ t.13 The resulting dataset consists of 4,001
farms repeated over four years, for a total of 16,004 observations. Compared with other related works (Bertoni et al. 2020; Stetter, Mennig, and Sauer 2022), our study provides wide coverage of the
agricultural sector by focusing on the entire national area instead of a single region. Furthermore, since the treatments presented in Section 4 are likely to affect the agri-environment over several
years, our outcome variable uses information from the last two years in the series to account for potential accumulation effects (see Section 4 for details).
Definition of Treatments
As mentioned in Section 2, the 2015–2020 CAP AEP design is based primarily on two main policy instruments that belong to CAP’s pillar 1, pillar 2, or both. On the one hand, we observe pillar 1
subsidies that are conditional on a set of compulsory requirements (i.e., CC and the GP) with which farmers must comply to preserve the DP. On the other hand, we have voluntary measures aimed at
compensating farmers for income losses or increased costs resulting from the voluntary adoption of more sustainable farming practices (i.e., the AEM of pillar 2). Consequently, farms are subscribed
to—in fact, they voluntarily choose—one of three possible policy alternatives, which effectively reflect the interplay between the two pillars of the CAP: (1) farms failing to meet all the CC and GP
requirements, that is, farms receiving neither pillar 1 nor pillar 2 payments; (2) farmers receiving both pillar 1 (DP and GP) and pillar 2 (AEM) payments; and (3) farms complying with the CC and GP
requirements but not adopting any AEM.
Table 1 indicates how the farms in our sample are distributed across the three policy categories. The third cohort is the largest group, which includes approximately 71% of the observed farms (2,841
units). Using the terminology introduced in Section 3, we consider the corresponding policy option as the baseline treatment, T[i][,0], associated with the netput vector y[it][,0]. Next, all farms
choosing not to benefit from pillar 1 and pillar 2 payments (i.e., the first cohort, corresponding to approximately 13% of the sample) take up the first treatment, T[i][,][k][=1], which implies
giving up both pillar 1 and pillar 2 resources. We assume that this decision follows the behavioral model stylized in Section 3, according to which, conditional on X[it], T[i][,1] produces higher
profits than T[i][,0]. Similarly, farms applying for pillar 2 AEM supports (i.e., the second cohort, corresponding to approximately 16% of the sample) choose treatment T[i][,][k][=2] through the same
profit-maximizing mechanism. In this respect, our work extends the analysis in Stetter, Mennig, and Sauer (2022) by distinguishing between the two different AEPs (i.e., the AEMs and the pillar 1
environmental requirements).
We postulate that treatments T[i][,1] and T[i][,2] belong to two nonoverlapping choice sets; in other words, we rule out a multiple treatment setup by positing treatment T[i][,1] as infeasible for
farms choosing T[i][,2] and vice versa. Although this assumption is quite strong, it is necessary to identify the treatment effects of interest. Given that T[i][,1] and T[i][,2] represent two ends of
a rather wide spectrum of policy options, it is plausible that both treatments may appeal to (i.e., are feasible for) farms with very distinctive characteristics. Conversely, our setup implies that
both T[i][,1] and T[i][,2] are feasible alternatives to the baseline treatment T[i][,0]. This presupposes that farms in the control group are characterized by features X[it] that overlap with the
characteristic of the units in T[i][,1] or T[i][,2]. That is, we can always find comparable farms in either of the two groups in different strata of X[it], that is, 0 < Pr(T[i][,][k] = 1 | X[it] = x[
it]) < 1. This restriction is also commonly known as common support (or positivity), and as we discuss in Section 5 and Appendix E, it limits extrapolation issues, thus preventing unreliable
treatment effects.
One caveat in our setup is that unlike T[i][,1] farms choosing T[i][,2] may in fact opt for one among four treatment versions. As outlined in Section 2, T[i][,2] aggregates measure 10 and 11 which in
turn can be decomposed in two submeasures: agri-environment-climate commitments (10.1); conservation and sustainable use and development of genetic resources in agriculture (10.2); payment to convert
to organic farming practices and methods (11.1); and payment to maintain organic farming practices and methods (11.2). While measure 10.2 only concerns a small share of farms (roughly 3% of our
sample) and can be thus excluded or safely merged into measure 10.1 (our current choice), submeasures 11.1 and 11.2 are substantially equivalent in terms of farmer behavior, the only difference being
the amount of support granted. For this reason, we de facto consider submeasure 11.1 and 11.2 as a unique measure (i.e., measure 11). As put forward in Sections 2 and 3, disregarding such
distinctions may greatly affect the interpretation of the HTEs via treatment heterogeneity.
It is also worth mentioning that in principle, the submeasures could be further disaggregated into specific actions (using the RDP jargon). Unfortunately, the Italian FADN data do not provide enough
information on AEM actions. In fact, to our knowledge, there are no high-quality representative datasets that can provide more detail on AEMs (e.g., Stetter, Mennig, and Sauer 2022, who use the
German version of our dataset). Had this level of disaggregation been observable, it would imply a very large number of actions (i.e., treatment versions), as evidenced by the 21 RDPs implemented in
Italy.14 Clearly, expanding the treatment options well beyond the four submeasures would greatly affect the sample size of each subgroup and challenge the estimation of any HTE under the standard
conditions discussed in Section 5 (Heiler and Knaus 2022). Finally, focusing on more specific measures does not necessarily imply a more refined outcome variable (see Section 4 for further
Since organic farming (measure 11) is homogeneous across the RDPs and involves a reasonable number of farms (271), we repeat our analysis by redefining treatment T[2] as a two-versions treatment T[2]
= (T[2][o], T[2][n]), where o = organic and n = nonorganic. Given our initial definition of the treatments, T[2][n] coincides with measure 10 which, unlike measure 11, is not entirely homogeneous
across RDPs and could be exposed to further treatment heterogeneity. We therefore estimate the HTE of T[2] under two different setups: (1) we analyze the HTE of participating to AEMs as in Stetter,
Mennig, and Sauer (2022); (2) we break down the treatment in setup 1 into T[2][o] and T[2][n] and obtain the corresponding HTE; and (3) we compare the results from setups 1 and 2 and discuss their
implications for the interpretation of the HTE of interest (see Section 6).
Outcome Variable
The theoretical framework presented in Section 3 expresses the farm response to the treatment as Δy[it][,][k], that is, a vector whose nonzero elements represent all of the farmer’s production
choices associated with the treatment in terms of both input and output.16 These elements may consist of a long list of the farmer’s specific production decisions, ranging from crop and livestock
management practices to water and nutrient use (Burton and Schwarz 2013; Guerrero 2021, 11). One way to reduce the dimensionality of Δy[it][,][k] consists of identifying and extracting the elementary
indicators expressing the change in farming practices toward extensification or environmentally friendly practices. However, as discussed in Sections 2 and 3, focusing on elementary indicators might
cause ambiguity when interpreting treatment effect heterogeneity because of the OTH problem. Given the potential correlation among the components of Δy[it][,][k], one way to retain all the
information in the netput vector while avoiding multiple marginal evaluations is to perform dimension reduction (Chipman and Gu 2005) to obtain composite dimensional indices (Bartolini et al. 2021).
This strategy not only provides an insulation against OTH but also resonates the need for a comprehensive evaluation of complex policy instruments such as the AEM discussed in Sections 2 and 4. As
also argued by Stetter, Mennig, and Sauer (2022, 727), despite the articulation of AEMs in specific submeasures, the goal of the AEPs remains more general, aiming to improve the overall environmental
performance of the agricultural sector. Although many studies have tried to evaluate the effectiveness of distinct AEPs with respect to specific policy targets (e.g., the impact on biodiversity), the
integrated assessment of multifaceted goals involving, for example, soil and water protection and the curbing of greenhouse gas (GHG) emissions have received relatively little attention until
recently (Hudec et al. 2007; Zhen et al. 2022). However, the literature has long suggested that the intricate and ecosystemic nature of the agri-environment requires that any assessment should be
based on a comprehensive integration of indicators across many environmental dimensions (Wascher 2003; Purvis et al. 2009).
In this respect, Purvis et al. (2009) propose an interesting, harmonized approach to evaluating AEMs: the so-called agri-environmental footprint index (AFI). The AFI expresses a multidimensional
assessment as a univariate index that can be flexibly adapted to diverse contexts. We use the AFI framework as adapted by Westbury et al. (2011) with the FADN data. We refer to this methodology as
FADN-AFI, as the resulting index uses elementary information included in the FADN dataset. We extend the FADN-AFI to evaluate whether and to what extent the implementation of the CC requirements,
GPs, and AEMs meet the CAP 2015–2020 environmental objectives.17
Table 2 presents the elementary components of our FADN-AFI (see Appendix Table B2). The land use diversity indicator (the Shannon index) is detailed in Appendix A. Appendix B discusses the definition
of a farm-level GHG emissions indicator using farm-level information. This measure should provide a reliable proxy of the contribution of a farm’s practices to climate change mitigation (Dabkiene,
Balezentis, and Streimikiene 2021). The FADN-AFI’s elementary components are then standardized to obtain dimensionless z-scores that we eventually aggregate using the weights indicated in the last
column of Table 2 (i.e., giving a positive or negative sign for positive or negative environmental externalities, respectively).18 The resulting FADN-AFI is monotonic in farms’ environmental
performance in that higher FADN-AFI scores correspond to “better” environmental performance. Since the range of the FADN-AFI is not bounded, the index might be difficult to interpret per se. However,
since HTEs are defined through pairwise differences, these can easily be understood comparatively. Finally, we average the FADN-AFI in 2017–2018 to provide more stable values for the outcome
Confounding Variables
As discussed in Section 3, the choice of covariates entering the X[it] vector becomes crucial for identifying the HTEs of interest. These should encompass farm heterogeneity as extensively as
possible, thereby allowing fair comparisons between treated and untreated units. Selecting all the relevant confounders such that the assumptions outlined in Section 5 are satisfied may follow
multiple routes. On the one hand, one may construct a very large collection of internal farm characteristics and external socioeconomic indicators that might explain the individual decision of
adopting one of the treatments. In this case, we would let ML algorithm choose which feature contributes the most to predict farmer behavior through a regularization mechanism. However, as recently
outlined by Hünermund, Louw, and Caspi (2023), this strategy may lead to severely biased treatment effects if the covariate set includes potentially endogenous confounding variables. Ultimately, the
authors advocate that when the goal is conducting CI, researchers need to justify the controls they want to include and, more importantly, make sure that these are exogenous (i.e., pretreatment).
For these reasons, we begin by defining the confounders in S[i] and V[it] through an extensive literature review covering several empirical studies addressing farmer participation in AEPs and the
impact of AEPs on farms’ economic and environmental performance. The results of this survey are displayed in Table 3, where the list of covariates resulting from this desk research is classified
using the taxonomy elaborated by Brown et al. (2021) and discussed in Section 3. We invite the reader to refer to the individual studies for a throughout explanation of how these regressors are
relevant for the research questions. The abundance of controls compiled in this long list might suggest some form of preliminary selection to avoid redundancy and achieve a more parsimonious set of
variables. Nevertheless, unlike most parametric econometric tools, forest-based ML algorithms can easily accommodate multiple overlapping information sources and use them to either create
intermediate features or discard redundant ones through regularization. Therefore, our empirical analysis makes use of all the covariates in Table 3.20
To satisfy the identifying conditions anticipated in Section 3, the time-varying controls, V[it], must be exogenous with respect to the treatment (i.e., predetermined). In theory, this would preclude
the use of certain direct measures of farm physical and economic size, such as utilized arable land, profit, revenue, costs, and total workforce. To circumvent this issue, some authors suggest using
covariates measured before the introduction of the treatment (for studies assessing AEMs, see, e.g., Bertoni et al. 2020; Uehleke, Petrick, and Hüttel 2022; Stetter, Mennig, and Sauer 2022). However,
this strategy is sometimes infeasible, as such measurements may not be available if the policies under investigation were introduced several years before the outcome is measured. When this happens,
going back in time may imply a major loss of observations. This concern is particularly relevant for our application, as the rotating structure of the Italian FADN panel shows that 582 farms
(approximately 15% of the sample) included in the 2015–2018 dataset are not present in the 2014 data. Therefore, our choice is to follow the strategy of Arata and Sckokai (2016) and Pufhal and Weiss
(2009), which consists of using the first year since the introduction of the policy as the pretreatment period (2015, in this case).21 Notice that since our outcome variable is calculated using the
years 2017 and 2018, V[it] contains lagged (by two years) elementary components of the FADN-AFI. Moreover, since farms usually sign up for participating in certain AEMs over several years (Bertoni et
al. 2020; Uehleke, Petrick, and Hüttel 2022), we also include information on previous participation to such programs in V[it] (Chabé-Ferret and Subervie 2013). Appendix Tables C1 and C2 report
descriptive statistics for the outcome variable and all the control variables discussed above.
Unobservable Characteristics
The theoretical derivation in Section 3 provides the behavioral foundation of the farmer’s treatment choice and response to the treatment. This behavior depends on some observable characteristics but
also on unobservable farm characteristics, u[i]. The conditional independence between any of the treatments and the corresponding potential outcomes also hinges on the last component of the
conditioning vector Z[it], namely, the unobservable farm characteristics, u[i]. If these latent features influence the choice between T[i][,1] and T[i][,2] and the corresponding potential outcomes,
the identification of the HTE becomes challenging because of the violation of unconfoundedness. Even though X[it] can be extended to collect as many observable farm characteristics as possible, this
strategy may be insufficient to insulate against selection-on-unobservable. Policy conclusions drawn from the HTE estimation could be problematic and even erroneous if the relevance of these
unobservables and their possible association with the observable characteristics are not properly investigated and understood.
In these situations, ML methods (including BCFs) can help in identifying automatically creating nonlinearities and complex interactions among the variables in X[it], generating artificial strata that
allow more precise comparisons between treated/untreated units and their counterfactuals. These “synthetic traits” not only greatly expand the initial set of confounders but also correlate with the
unobservable characteristics, thereby making the unconfoundedness assumption more credible. This argument is also put forward by Stetter, Mennig, and Sauer (2022, 738–39, 744), who provide a nice
example of how this property of ML techniques may help to control for farmer attitudes toward environmental issues.22 Since this is not directly testable, we check the robustness of the above
propositions through several sensitivity analysis tests. As illustrated in Appendix H, we probe the stability of our results in the presence of omitted variable bias from unobserved endogenous
heterogeneity by introducing synthetically generated u[i] into the covariate set. See Section 6 for more details and caveats of this approach.
5. Methodology
Research on the estimation of HTE has flourished recently, stimulated by an increasing interest in the development of ML methods able to provide theoretically sound inferences in such research
settings (Athey and Imbens 2019; Athey, Tibshirani, and Wager 2019; Hahn, Murray, and Carvalho 2020; Knaus, Lechner, and Strittmatter 2021, 2022). Recent studies have proposed two ways ML can be used
to estimate HTE. First, off-the-shelf ML algorithms can be tweaked to address some of the relevant identification issues of CI directly (Imai and Ratkovic 2013; Athey and Imbens 2016;; Wager and
Athey 2018; Hahn, Murray, and Carvalho 2020).23 Second, direct modifications of the loss functions and data-splitting techniques can also help address one challenging problem of traditional ML
techniques in causal settings: regularization-induced confounding (RIC) (Chernozhukov et al. 2018, and references therein; Hahn et al. 2018; Hahn, Murray, and Carvalho 2020; Nie and Wager 2021). We
broadly refer to all these methods as CML.
Among the diverse approaches proposed in the literature, BART-based algorithms (Chipman, George, and McCulloch 2010; Hill 2011; Hill, Linero, and Murray 2020) stand out as promising additions to the
CML toolbox. These methods not only exhibit encouraging performance in terms of unbiasedness and coverage rates (Carvalho et al. 2019; Dorie et al. 2019; Hahn, Murray, and Carvalho 2020; Lee,
Bargagli-Stoffi, and Dominici 2020) but also take advantage of a fully probabilistic (i.e., Bayesian) inferential approach, which enables the introduction of uncertainty measures when comparing
groups of individuals (an aspect that currently limits the extent of other comparable ML methods; Stetter, Mennig, and Sauer 2022) and facilitates investigating the extent of overlap between treated
and untreated groups (see Appendix E for details; Hill and Su 2013; Li, Ding, and Mealli 2022). The latter is particularly important when it comes to treatment T[1], as the farms associated with this
group are likely to exhibit very specific characteristics (see Appendix E; Esposti 2017a, 2017b). Both traits hinge on the full posterior distributions of, on the one hand, the estimated HTE and, on
the other hand, the fitted individual-level conditional expectations.
As with many other tree-based methods, BART can flexibly fit complex response surfaces by creating regularized ensembles of shallow Bayesian regression trees (Chipman, George, and McCulloch 1998),
making it possible to perform predictive inference using the resulting posterior distributions (Chipman, George, and McCulloch 2010). This flexibility is achieved via recursive partitioning of the
covariate space at the tree level, a procedure that is adept at defining nonlinearities and interactions between the observed covariates without the need to prespecify them (Hill 2011). However,
since the original BART was not purposely designed for CI, a naive application of such methods for the estimation of HTE might potentially introduce RIC. For this reason, Hahn, Murray, and Carvalho
(2020) recently proposed an extension of the original algorithm, which they refer to as BCFs.24 In addition to exploiting the estimated propensity score (PS) to deal with potential distortions
attributable to RIC (see Appendix D), the BCF algorithm also provides for a more flexible structure that separates the prognostic component from the heterogeneous treatment effect, thereby enabling
direct control over the latter to avoid overfitting.
Estimating Treatment Effects via BCF
The estimation of HTEs using the BCF algorithm requires the usual assumptions of unconfoundedness and SUTVA, which can be expressed as follows: 1 where Y[i] represents the FADN-AFI defined in Section
4, X[i] indicates the vector of confounders defined in Section 4, while Y[i](1) and Y[i](0) indicate potential outcomes for individuals in a treatment group (T[i][,][k] = 1) or control group (T[i][,]
[k] = 0), respectively (Imbens and Rubin 2015, ch. 1). Notice that SUTVA implies no hidden variations of the treatment. As discussed in Sections 2 and 3, binarized multiple-versions treatments can
lead to violations of this assumption unless one imposes stringent restrictions on the treatment assignment mechanism. For example, in case any individual i with characteristics X[i] can only choose
one of the hidden treatments, SUTVA is still a credible assumption (VanderWeele and Hernán 2013; Lopez and Gutman 2017). As previously discussed, we make this assumption for the treatments defined in
Section 4, except for the distinction between organic and nonorganic farming. We therefore set k to k ∈ {1,2} such that T[i][,][k] = 1 indicates either T[i][,1] = 1 or T[i][,2] = 1, while T[i][,][k]
= 0 always refers to farms in the control group. We discuss the implication for disaggregating T[i][,2] into T[i][,2][o] and T[i][,2][n] in Section 6. For notational convenience, we drop the
subscript k. Of these elements, we only observe the potential outcome that corresponds to the realized T[i], namely, Y[i] = T[i]Y[i](1) + (1 – T[i])Y[i](0). Equation [1] postulates independence
between the potential outcomes and the treatment, conditional on the set of exogenous variables, X[i].
Combining unconfoundedness, SUTVA, and overlap (as discussed in Section 4) allows the estimation of causal effects via strong ignorability; that is, E[Y[i](t) | X[i] = x[i]] = E[Y[i] | T[i] = t[i], X
[i] = x[i]], with t[i] ∈ {0,1}. The latter implies that the estimand of interest is simply the difference between two conditional expectation functions: 2 where τ(x[i]) is typically referred to as a
conditional average treatment effect (CATE). Since one can use μ[T](x[i]) to impute conditional treatment effects at the individual level, equation [2] is sometimes referred to as individualized
average treatment effect (IATE) (Lechner 2018; Knaus, Lechner, and Strittmatter 2021, 2022). This estimand represents the most disaggregated form of HTE.
Often researchers may be interested in subgroups or intermediate aggregation levels of the exogenous covariates, leading to the definition of group average treatment effects (GATEs): 3 where ϕ(.)
represents a generic probability density of mass function, G[i] denotes the collection of possible groups, and g[i] denotes one such group. GATEs have recently gained considerable attention in the
applied literature as treatment effect heterogeneity is often better understood for subsets of the population (Lechner 2018; Lee, Bargagli-Stoffi, and Dominici, 2020). ATEs can also be obtained by
averaging the IATEs over the full distribution of X[i]: 4 To estimate the IATEs (and then the GATEs and ATEs), we assume that the data-generating process for follows a stochastic process defined as
follows: 5 where f indicates an arbitrarily complex function25 and ε[i] represents an additive idiosyncratic error term ε[i] ~ N (0,σ^2), independently distributed.
In this context, E[Y[i] | T[i] = t[i], X[i] = x[i]] = f (x[i], t[i]) therefore, at least in principle, τ(x[i]) can be estimated by the simple difference f (x[i], t[i] = 1) – f (x[i], t[i] = 0) = μ[1]
(x[i]) – μ[0](x[i]), as illustrated above. However, as discussed by Künzel et al. (2019) and Nie and Wager (2021), training two separate conditional mean functions and taking their difference may
produce highly unstable estimates. For this reason, Hahn, Murray, and Carvalho (2020) proposed a slightly different approach, wherein the expected value of the outcome of interest has two components:
a prognostic function, 𝓂(x[i]) plus an additive heterogeneous treatment effect, τ(x[i]): 6 where both 𝓂(.) and τ(.) represent stochastic functions with BART priors, namely, 𝓂 ~ BART(θ | PS(x[i]), x[i
]) and τ ~ BART(ϑ | x[i]), and PS(x[i]) indicates the estimated PS. The two vectors θ and ϑ collect the hyperparameters regulating the number of trees in the BART ensembles, their depth, and the
splitting rule associated with each single tree (see Appendix F for details). As previously mentioned, the specification in equation [6] allows regularizing τ(x[i]) directly and independently,
thereby reducing the noisiness of the IATEs with respect to the same estimates obtained from simple differences in conditional mean functions. Furthermore, the additive nature of equation [6] ensures
that the prior on f (x[i], t[i]) is also a BART (Chipman, George, and McCulloch 2010; Hill, Linero, and Murphy 2020). Finally, notice that the model presented in equation [6] also appears in Nie and
Wager (2021), who propose a frequentist approach to estimating τ(x[i]). In contrast to the setup discussed above, however, the authors propose a residuals-on-residuals reparameterization of equation
[6] which is then used to obtain (regularized) consistent estimates of τ(x[i]) via a two-stage optimization procedure.
The full Bayesian model requires the definition of a likelihood function for the outcome variable (Gelman et al. 2013; McElreath 2020). Consistent with equation [5] and Chipman, George, and McCullogh
(2010), Hill (2011), and Hahn, Murray, and Carvalho (2020), we employ a normal model for Y[i], along with a semiconjugate inverse chi square prior for its variance: 7 where ω is set following
Chipman, George, and McCullogh (2010) (see Appendix F for further details). Samples from the posterior distribution of τ(x[i]) are obtained via Markov chain Monte Carlo sampling, as implemented in
the R package bcf. We indicate posterior draws from ϕ(τ(x[i]) | x[i], t[i], y[i],…, y[N]) as {τ ^s(x[i])}^S[s][=1], where S indicates the number of Markov chain Monte Carlo simulations.
Subgroup Search via Shallow Regression Trees
The approximated posterior {τ ^s(x[i])}^S[s][=1] is a multivariate probability distribution over a complex P-dimensional function, and as such, it might be difficult to interpret directly. One way to
compress such information consists of obtaining marginal distributions of the IATEs for one covariate of interest and plotting them against the full range of that variable. A similar approach was
adopted by Stetter, Mennig, and Sauer (2022), who used Shapley values (Shapley 1953) to identify the marginal contributions of several treatment effect drivers and used these indicators to construct
partial dependence plots. Another sensible approach to investigating IATE heterogeneity consists of comparing farm subgroups obtained by projecting the full posterior distribution onto a
lower-dimensional covariate space. In this respect, we follow the work of Yeager et al. (2019), Hahn, Murray, and Carvalho (2020), Woody, Carvalho, and Murray (2021) and (and partially Lee,
Bargagli-Stoffi, and Dominici 2020), who suggest eliciting the relvant subgroups by partitioning the IATE maximum a posteriori (MAP) estimates, τ[i] = S ^–1∑^S[s][=1] τ ^s[q](x[i]), using shallow
regression trees (CART) (Breiman 1984). Specifically, the authors propose to split τ[i] along w[i], where w[i] ⊆ x[i] indicates a vector of policy-relevant variables and setting w[i] ⊂ x[i] implies
using domain knowledge to enforce an initial regularization of the resulting tree. We restrict our attention to a subset of simple and understandable characteristics that policy makers might find
helpful to improve the targeting of AEMs (see Section 6). Once farm subgroups have been identified, GATEs can be obtained as weighted averages of the IATEs that fall into each cluster. This approach
to calculating GATEs is also consistent with Lechner (2018) in that group-level effects are obtained as convex combinations of the IATEs. In our application, however, weighting is automatically
performed when fitting a tree to τ[i].
Finally, for some potential effect moderator x[p] ∈ x, the comparison between pointwise estimates (or intervals) computed at different levels of x[p] ignores any potential correlation between IATEs
along other variables x[l], for all p, l ∈ {1,…, P}. In other words, the marginal distribution of τ(x[p][,][i]) disregards the information encoded in the correlation between τ(x[l][,][i]) and τ(x[l]
[,–][i]) when x[l][,][i] and x[l][,–][i] are close. This might lead to misleading comparisons along x[p] and, consequently, unreliable policy implications. Therefore, once the relevant subgroups have
been identified, one can obtain the full posterior distribution of each pairwise difference as: ϕ[g][1][,][g][2] = ϕ(τ[i|i][∈][g][1] – τ[i|i][∈][g][2]), where g[1] and g[2] indicate any two subsets
of τ[i].
6. Results
The two graphs numbered “1” in Figure 1 display the MAP; that is, the average over the S samples from the posterior distribution of τ(x[i]) estimates and corresponding 95% confidence intervals (CrI)
of the IATEs over the two treatment comparisons. These are ordered across the respective samples from the lowest to the highest individual value. We start our discussion by presenting the results for
T[2], the treatment that is more frequently addressed by the literature. First, it is worth noting that overall, the modal direction of the responses to the treatment (T[2]) is fully consistent with
theoretical expectations: adding the AEM to the environmental standards implied by the CC and the GP (Figure 1a [1]) induces an improvement in the FADN-AFI, that is, in the farm-level environmental
performance. The opposite response is observed when the environmental standards implied by the CC and GP are dropped (i.e., treatment T[1]) (Figure 1b [1]). Whereas in the first case, most estimated
IATEs exhibit CrI not including zero (black dots), the converse applies to the second comparison group, for which a large proportion of farms have inconclusive individual-level TEs (light gray dots).
The first graph in Figure 1b also indicates that some farms might even exhibit opposite responses, although the corresponding IATEs appear quite noisy. This evidence is presented in greater detail in
Table 4, which provides descriptive summaries of our main results.
The two graphs numbered “2” in Figure 1 show the IATE’s MAP frequency distribution for the two cases. These plots highlight the variability of the responses, with few cases showing a treatment effect
direction that conflicts with the expected direction (despite exhibiting CrI including positive and negative values). Apart from these rare extreme cases, however, our MAP estimates range between
roughly 0.1 and 1.0 for treatment T[2] and between approximately −3 and 1.5 for treatment T[1]. The nature and determinants of these different patterns can be further investigated by estimating
GATEs, as addressed in the next section.
The irregularity of farms’ responses to the treatments is a clear sign of heterogeneity, one that would be lost by the mere inspection of ATEs (see the two graphs numbered “3” in Figure 1). Whereas
these latter aggregated estimands provide clear indications of policy effectiveness (as both show an effect in the expected direction), the inspection of the IATEs tells a different and more subtle
story. This is especially true for the treatment T[1], whereas the responses seem more homogeneous when studying the treatment effect of implementing CC and GP requirements together with AEMs
(treatment T[2]).
Finally, for each individualized treatment effect, we calculate the posterior probability that the corresponding IATE is either greater than zero or lower than zero for the T[2] and T[1],
respectively. Our results show that when comparing farms implementing CC and GP requirements plus AEMs with the control group, most of the IATEs’ posterior distributions lie above zero. For example,
the proportions of IATEs with at least 60%, 75%, and 90% positive posterior are 100%, 88.5%, and 5%, respectively. Conversely, when comparing the control group to farms with no adherence to CC or GP,
the posterior distributions of their IATEs are largely negative. In this case, the proportions of IATEs with at least 60%, 75%, and 90% negative posterior are 83%, 15%, and 0%, respectively.
Notice that all the results discussed thus far are based on observations satisfying the common support as defined by rule I in Appendix E. Under such a restriction to the range of X[i], however, our
dataset does not suffer drops. The sensitivity of these figures to different exclusion rules is discussed in Section 6 (robustness check), in which the selection method we used based on the estimated
PS is also discussed.
We partition the posterior distribution of τ(x[i]) using a set of policy-relevant measures w[i] covering the most relevant dimensions of heterogeneity, as evidenced by the measures of feature
importance produced by the BCF. Our characterization of w[i] involves (1) examining the variable importance metrics generated as a by-product of the fitting model [7],26 and (2) choosing the 10 most
predictive dimensions that policy makers might target to improve the effectiveness of AEMs. We fit a CART algorithm to τ[i] using the attributes selected using the procedure illustrated above:
latitude, longitude, altitude (geographical location); total arable land, share of rented land, revenue (physical or economic size); farm specialization (relative importance of the first and second
crop, farms specialized in livestock, crop and livestock farms, farms specialized in annual crops, and farm specialized in perennial crops). The results for the two treatments are shown in Figures 4
and 5, wherein, for the sake of interpretability, we do not allow the trees to split more than three times.
When we consider the adoption an AEM in addition to CC and GP requirements (treatment T[2]) (Figure 2b [1]), we find that TE heterogeneity is mostly associated with five variables: latitude, physical
farm size, altitude, crop specialization (share of the second crop in the crop mix), and livestock intensity. These covariates trace out eight subgroups with different levels of treatment effects.
For example, subgroup g[8] exhibits the lowest treatment effect and consists of farms in southern Italy with less than 85 ha arable land. On the opposite end of the spectrum, we find subgroup g[15],
which comprises crop-specialized farms in northern Italy with low livestock intensity. One can then obtain the full posterior distribution of g[15] – g[8] with 95% CrI between −0.27 and 0.49 (Figure
2b [2]), which indicates that the difference between the two subgroups is in fact small, if not zero. Interestingly, if we repeat this exercise across all the leaves defined by the tree in Figure 2b
[1], no group differences emerge (see Appendix G). These results are consistent with our discussion in Section 6 (i.e., our preliminary findings suggested limited treatment effect heterogeneity for
treatment T[2]).
In the case of treatment T[1] (Figure 2a [1]), we see that the shallow tree picks up four moderating variables: specialization in perennial crops, latitude, altitude, and livestock intensity. In this
case, the subgroup with the strongest TE is g[8], which consists of farms specialized in perennial crops in Italy’s southmost regions. Subgroup g[15] includes observations from farms in the Po Valley
that are not specialized in perennial crops. The difference in TE between these subgroups lies approximately between −2.2 and −0.41 (95% CrI; Figure 2a [2]), indicating the presence of treatment
effect heterogeneity. Repeating this exercise across all the terminal nodes, we find that unlike treatment T[2], when the treatment consists of dropping both CC and GP requirements, many groups
exhibit diversified responses. These further details are provided in Appendix G, where we also provide a deeper tree to gain further insights into these HTEs and a graphic representation of the
geographical distribution of the IATEs.
It is finally worth stressing that although our main goal is to explore which observable farm characteristics exhibit a greater heterogeneity of response, some of these features might not be easily
addressed by AEPs due to cost constraints or infeasibility or because they could potentially lead to discriminatory outcomes. From a policy perspective, it would be more useful to evaluate the level
of heterogeneity associated with covariates that can be targeted more easily and effectively through policy measures. Most of the geographical features considered in our study, along with variables
indicating long-term farm production specialization, appear particularly suitable for this purpose. In this respect, our results confirm that most of these geographical features significantly
contribute to the observe heterogeneity of response. Similarly, the presence of perennial crops, crop specialization, and livestock density, all of which relate to distinct and consistent farming
practices, pinpoint to patterns of strong heterogeneity. This suggests that AEPs could significantly enhance their effectiveness by specifically targeting these features. For a more detailed
discussion on this matter, please refer to Appendix G.
Robustness Checks
We check the consistency of our results to the assumptions formulated in Sections 4 and 5. Our first robustness check concerns the common support condition. As anticipated in Section 5 and further
detailed in Appendix E, we use both the posterior distribution of the BART algorithm and a PS-based algorithm to investigate common support. Our tests show that the results presented in Section 6 are
robust to these different methods to achieve overlap (see Appendix Tables H1 and H2).
We perform a battery of tests that largely encompass those discussed by Stetter, Mennig, and Sauer (2022) in that we reestimate our BCF multiple times, each time manipulating different model
features. We begin by probing unconfoundedness through a recursive procedure in which we fit model [7] after dropping: (1) the most important feature in terms of relative frequency in the forest, (2)
the three most important features, and (3) the five most important features. As detailed in Appendix Figures H1–H3), this exercise yields the first indication that the BCF in equation [7] is
fundamentally resilient against unobserved heterogeneity as long as this is associated with the set of observed confounders. Put differently, the complex interactions and nonlinearities generated by
the tree ensemble seem to work as additional synthetic controls associated with the left-out covariates, thus compensating for their absence in the model. However, this line of reasoning hinges on
the (strong) assumption that the most predictive features are also associated with both Y and T[k]. In case this assumption fails, the procedure discussed above cannot be interpreted as a robustness
check for unconfoundedness. For this reason, we build on these preliminary results and devise an additional test targeting endogenous unobserved heterogeneity directly. Our strategy consists of
generating a random variable correlated with both Y and T[k], forming the vector Z[it] as described in Section 3, and rerunning the model. As shown in Appendix Figure H4, our results do not change
substantially, even under a strong imposed association between the unobserved variable and (Y, T[k]). This stability could result from the properties of the BART ensemble in that when the forest is
dense, the marginal contribution of each covariate becomes increasingly small (Chipman, George, and McCulloch 2010). Alternatively, it could be that the correlation between the nonlinear interactions
generated by the BCF and the new confounder is strong enough to prevent distortions in the IATEs. In either case, it is worth warning that treatment effect estimates might deteriorate quickly when
unobserved heterogeneity is more abundant and complex. This test is in fact only restricted to a single unobserved factor, which we model as linearly associated with Y and T[k] (i.e., through
correlations, which do not necessarily imply a direct effect of the synthetic u[i] on the outcome or the treatment). We thus expect that in presence of multiple endogenous latent confounders,
possibly related to the treatment and the outcome (or other elements of X[i] in a nonlinear fashion, our estimates might tun out sensibly different. Although the literature offers other methods to
perform sensitivity analysis with respect to omitted confounders (Dorie et al. 2016; VanderWeele and Ding 2018), we believe that they either do not overcome the limitations discussed above or they
remain difficult to implement in HTE estimation. Therefore, despite the promising results presented so far, we stress that these only hold if several important restrictions are met.
The following robustness check consists of creating both a placebo treatment and a placebo outcome, replacing their observed counterparts in equation [7], and fitting the model two more times. If the
model is correctly specified, the IATEs resulting from these “fake” variables should be uncorrelated with τ(x[i]). As Appendix Figures H5 and H6 show, the new results obtained through placebo
treatments and outcomes not only have no correlation with our estimated IATEs but also produce zero ATE with minimal treatment effect heterogeneity.
Finally, we assess the robustness of the estimated IATEs with respect to the OTH problem discussed in previous sections. We proceed by replacing the FADN-AFI with its elementary components and
reestimating model [7] as discussed. Appendix Figure H7 suggests that focusing on marginal indicators produces TEs whose individual directions are essentially in line with those presented in Section
6. For example, implementing AEMs seems to yield lower GHGs, higher crop diversity, lower fertilizer expenditure, and more woodland areas. Nonetheless, a noteworthy difference emerges in terms of
treatment effect heterogeneity. Whereas adopting the FADN-AFI points to a limited diversity across farms, using marginal measurements would suggest that treatment T[2] is environmentally beneficial
only when the treatment effect is large. For this reason, our results invite to caution when it comes to choosing the dependent variable of model [7]. Although addressing individual indicators may
appear more attractive and interpretable, it is worth stressing that missing out on the potential correlation or interdependence among them can affect the TE estimates in a nontrivial way.
Role of Heterogeneous Treatments
As discussed in Sections 3 and 4, one potential limitation of our results (as well as other works investigating HTE of aggregated treatments) is that part of the estimated treatment effect
heterogeneity in T[2] might be a statistical artifact. This would result from the fact that T[2] is a multiple-versions treatment as it aggregates two distinct measures which admit, in turn, several
submeasures (see Section 4). As introduced in Section 5, the presence of treatment heterogeneity may affect our results by violating SUTVA (Heiler and Knaus 2022). Since in this case, the resulting
interpretation of τ(x) would be misleading, we reestimate model [7] replacing T[2] with the two respective measures (measure 11 and measure 10) and approach the problem from a multiple-versions
treatment perspective as discussed in Lopez and Gutman (2017).
To assess the possible bias in HTE estimation due to treatment heterogeneity, we compare the posterior distribution of the IATEs presented in Section 6 with the posterior density of the IATEs
estimated using either T[2][o], τ[2][o](x[i]), or T[2][n], τ[2][n](x[i]). Figure 3 shows the 95% CrI for the differences τ[2][o](x[i]) – τ[2](x[i]) and τ[2][n](x[i]) – τ[2](x[i]), respectively, where
τ[2](x[i]) indicates the IATE for individual i under treatment T[2]. As we can see from these plots, the difference between our initial estimates and those obtained by substituting T[2] with T[2][o]
are minimal. Indeed, although τ[2][o](x[i]) is on average (black line in the left graph in Figure 3) slightly smaller than τ[2](x[i]) for all i ∈ N[o], where N[o] indicates the number of units
choosing T[2][o], all the CrI include both positive and negative values. At the same time, when focusing on T[2][n], we see that τ[2][n](x[i]) – τ[2](x[i]) are on average higher than zero for all i ∈
N[n], where N[n] indicates the units choosing T[2][n]. However, the CrI once again includes zero for all such comparisons, although they are all moderately skewed toward positive values. Moreover, as
mentioned in Section 4, T[2][n] could still entail some degree of treatment heterogeneity, which recommends caution when interpreting the corresponding estimates. Overall, examining the two measures
separately highlights that the posterior distribution of the IATEs does not seem to change markedly when the aggregated (T[2]) or the disaggregated (T[2][o], T[2][n]) treatment is considered. This
would suggest a limited impact of treatment heterogeneity on our interpretation of the HTEs discussed above. Nonetheless, further research effort remains desirable to better clarify the possible role
of multiple versions in the correct identification and estimation of the HTE.
7. Concluding Remarks
Giving the CAP a more explicit environmental orientation and justification has been at the core of all its recent reforms. This necessarily means shifting the support from undifferentiated and
unconditional payments to more tailored and target measures. The efficiency and effectiveness of AEPs in this respect critically depend on how farmers respond to these measures. This response, in
turn, largely depends on the individual characteristics of supported farms. This makes the response itself highly heterogeneous and, consequently, suggests that there is still room for substantial
improvement through better policy targeting.
In this article, we present a CML approach to assessing the heterogeneous response of farmers to different AEPs implemented through the 2015–2020 CAP reform. Building on the existing literature, this
study’s main contribution is twofold. First, we explicitly conceptualize and investigate the different sources of heterogeneity that we expect influence farms’ environmental performances under such
policies. Second, we take advantage of the most recent developments in Bayesian nonparametrics and conduct the analysis using a relatively unexplored algorithm called Bayesian causal forest. This
method allows using the posterior distribution of the individualized treatment effect (the IATEs) to draw inferences about arbitrary transformations of these highly disaggregated estimands. We
leverage this property, particularly when discussing group-level treatment effects and testing the robustness of our results against identification assumptions.
More generally, estimating IATEs can prove insightful in that some beneficiaries of an AEP may exhibit limited or unsatisfactory responses, thereby calling for an intensification of the support,
while others may show responses that are well beyond the policy target, suggesting a reduction of support. Our results illustrate how informative the approach can be in detecting the extent, nature,
and source of this heterogeneous response. For instance, we demonstrate that contrasting different farm subgroups can provide additional information on the nature of the heterogeneous response.
Specifically, we highlighted that the treatment effect from implementing pillar 2 agri-environmental measures and fulfilling pillar 1 conditionality requirements seems more homogeneous than the
response to adopting none of the above.
The primary policy implication of our results concerns the need for a better targeting of AEPs. In this respect, caution is necessary, as not all farm characteristics considered can be easily
targeted due to practical or political constraints. Nonetheless, our analysis suggests that significant heterogeneity in treatment effects is concentrated in farm subgroups that can be feasibly
targeted. These subgroups often involve geographical features and specific production specializations. Therefore, delivering some CAP measures at a local scale and tailoring them to specific
production orientations, along with broader adoption of results-based payment schemes, may represent a sensible initial step toward better targeting. The new CAP acknowledges greater flexibility for
member states through the new delivery model, allowing them to address the environmental aspects of pillar 1 (the reinforced CC and the eco-schemes replacing the GP) and the AEMs in pillar 2 more
effectively. In principle, this flexibility seems to go along with the goal of improved targeting for these AEPs.
Although our empirical results provide valuable insights, our work also contributes to the constructive discussion on the potential and limitations of these relatively new policy assessment methods.
How useful is CML and the analysis of heterogeneous treatment effects in informing policy improvements related to the CAP? Our conceptual framework and empirical investigation suggest that they can
be useful. However, as with all emerging econometric approaches, several issues require careful consideration.
Because standard causal ML methods cannot be used for policy analysis without additional identifying restrictions and assumptions, selecting appropriate confounders and ensuring overlapping/
treatment-stable units necessitates a solid theoretical understanding of treatment selection mechanisms. Developing these conceptual foundations also facilitates result interpretation, as the complex
output of these estimation methods can be challenging to put into perspective. Among the standard assumptions presented herein, unconfoundedness and the stable unit treatment value are often regarded
as restrictive. Although the former can be corroborated via robustness checks and the use of ML algorithms, the latter finds little practical help from flexible estimation techniques and thus remains
debatable. In this respect, specifying the correct treatment variable(s) is quintessential for an unbiased interpretation of the resulting treatment effect, an aspect that is still relatively
underdiscussed in the literature.
More generally, investigating the effectiveness of CAP’s agri-environmental policies in a binary-treatment logic may prove limiting when the analysis targets heterogeneous causal effects. The risk is
that the elicited estimates do not entirely reflect farms’ heterogeneous responses to a treatment but encapsulate the heterogeneity of the treatment itself. Besides the prototypical case of
multiple-versions treatments (whether hidden or observable), problems can also arise when a policy measure is not only adopted (i.e., a discrete choice) but also exhibits different intensity levels
in different cohorts of farms. In such cases, binary treatments should be extended to incorporate dosage information. How to define the treatment intensity (i.e., the “dose”) of different
agri-environmental policies is an ambitious empirical question that we leave to future research.
• ↵1 We consider AEMs as a subset of whole menu of AEPs.
• ↵2 Measure 10 supports (among other things) integrated production, manure management, increasing soil organic matter, sustainable management of extensive grassland, and management of buffer
strips against nitrates. Measure 11 supports conversion to and maintenance of organic practices and methods. It is worth noticing that Stetter, Mennig, and Sauer (2022, 732) do not consider the
organic farming measure “due to [the] distinctly different farming approach compared to conventional farms.” As clarified in Section 4, we include this measure in the analysis to compare the
results obtained on the whole sample.
• ↵3 At the member state level, the total amount of GP must correspond to 30% of the total DPs. In several EU countries (including Italy), this condition is satisfied by automatically assigning to
eligible farms 30% of total DP as the GP.
• ↵4 Since production decisions must be taken ex ante, their consequences are evidently subject to some degree of uncertainty. Consequently, farmers actually maximize E{Π[g(T[it][,][k],X[i])]} and,
more importantly, the condition E{Π[g(T[it][,][k],X[i])]} ≥ E{Π[g(T[it][,][h],X[i])]}, ∀k, h ∈ K, k ≠ h remains valid only if we are willing to assume farmer’s risk neutrality. Otherwise, the
variance of π[it][,][k] and π[it][,][h], and the possible impact of T[it][,][k] on them, would also matter.
• ↵5 It can be argued that under risk aversion, farmers are expected to be more prudent and conservative; therefore, ceteris paribus, the participation in the treatment and the observed response, Δ
y should be smaller. At the same time, the monetary support granted to participant farmers may represent a guaranteed income, making participation in the measure a less risky situation. Also
notice that under risk aversion, risk can be interpreted as an additional source of costs and/or forgone income that the AEP is expected to compensate. Therefore, as noted in previous studies (
Esposti 2017a, 2017b), it is difficult to model and predict the differential impact of these support measures between risk-neutral and risk-averse farmers.
• ↵6 Unlike the other vectors of model variables, the netput vector is here indicated with a small letter, y[it], to avoid confusion with the conventional notation of potential outcomes, Y[i](0)
and Y[i](1) (see Section 5).
• ↵7 As will be clarified in Section 4, examples of internal factors are the farm size and the farmer’s age and education. Examples of external factors are latitude and farm’s location in a
disadvantaged area.
• ↵8 Following the conventional terminology of production theory, this should be a direct profit function as opposed to the more frequently used indirect profit function, where profit is a function
of only output and input prices. In fact, in addition to netput quantities, the direct profit function includes the respective prices expressed as Π[𝒗′[it]g(T[it][,][k], X[i])], where 𝒗′[it] is
the (M × 1) vector of netput prices. For nonmarket netputs, there are no prices, but these elements in 𝒗′[it] can still be interpreted as shadow prices. Nonetheless, prices have been excluded
from the present notation under the assumption, maintained that the prices are constant or, more precisely, unaffected by the policy regime.
• ↵9 The heterogeneity among farms is the core of this theoretical framework. With homogeneous farms, we would have π[it][,][k] = π[jt][,][k] = π[t], ∀i ≠ j, ∀k and ∀t, so all farmers would opt for
the same policy, and we would observe only one treatment. A policy response would thus be only conjectural but not actually observable if not by comparing farms before and after the treatment.
• ↵10 Notice that this assessment applies to both single treatment and multiple-treatments versions.
• ↵11 See Zimmerman and Britz (2016), Dessart, Barreiro-Hurlé, and van Bavel (2019), Brown et al. (2021) for recent and extensive reviews of structural and behavioral factors underlying farmer’s
• ↵12 The programming period has been subsequently extended to 2022, also because of the COVID-19 pandemic. Validated data from 2021 and 2022 have still to be released.
• ↵13 It is worth noticing that extracting the balanced sample from the unbalanced one does not imply a relevant loss in terms of representativeness of the sample; see Baldoni, Coderoni, and
Esposti (2021) for a detailed explanation.
• ↵14 More specifically, from a survey carried out at national level, it emerged that there are 65 different versions of measure 10 that can be applied at regional programming level, corresponding
to a total of 100 commitment categories for the whole 21 RDPs; see https://www.reterurale.it/flex/cm/pages/ServeBLOB.php/L/IT/IDPagina/23816.
• ↵15 The support for organic farming is exemplary in this respect. The nature of the response may vary largely across different farming types, even under such a very specific measure. The same
argument applies to CC requirements, where each element and constraint becomes applicable to the farm depending on the characteristics of the farmland or the agricultural activities carried out.
• ↵16 For elements of y[it][,][k] that are only marginally (or not at all) affected by the policy treatment under consideration, we have Δ y[it][,][k] ≈ 0. Therefore, we may restrict the analysis
only to input and output decisions that are related to the environmental measures, all the rest being orthogonal by assumption.
• ↵17 These goals are related to (1) the mandatory practices devised to benefit the environment (soil and biodiversity in particular) and climate (with the GP of pillar 1), and (2) the new RDP
priority areas specifically addressing the environment and climate change (pillar 2). The latter are aimed at restoring, preserving and enhancing ecosystems dependent on agriculture and forestry
(priority 4) and promoting resource efficiency and supporting the shift toward a low-carbon and climate-resilient economy in the agriculture, food and forestry sectors (priority 5).
• ↵18 Following Purvis et al. (2009), all the indicators and assessment criteria in the FADN-AFI receive a subjectively equal weighting.
• ↵19 Averaging only over the last two years reduces the risk of integrating out potential accumulation effects by smoothing over a longer period (i.e., the cumulative benefit of environmentally
friendly practices).
• ↵20 This explains the presence of insurance expenditure among covariates. This variable might seem contradictory to the risk neutrality assumed in deriving the theoretical framework (Section 3).
However, it is worth remembering that in most cases, farms incur these costs not because of their risk aversion but because taking out an insurance contract is mandatory to receive public or
private investment support. For this reason, this variable was considered in previous studies and thus in the present study.
• ↵21 This requires assuming no anticipation and no instantaneous impact of either T[1] or T[2] on V[it]. With no anticipation, we refer to the assumption that farmers have not changed their
characteristics V[it][–1] in response to the foreseen implementation of the policy at time t.
• ↵22 In short, the authors discuss how a construct resulting from the interaction between farm type, farm size, farmer’s age, farm capital intensity, and proxies for risk behavior is conceivably
strongly correlated with the unobservable trait, thereby contributing to deconfounding the treatment effect.
• ↵23 For an inventory of these methods, see Nie and Wager (2021).
• ↵24 Notice that although the terminology “causal forests” resembles that used in Wager and Athey (2018), BCF differs substantially from the frequentist counterpart in their definition,
functioning, and in how inference is performed.
• ↵25 f (X[i], T[i]) could be specified. as a fully parametric function, although this would inevitably constraint the cross-farm technological and behavioral heterogeneity. Admitting an
arbitrarily complex function is thus more consistent with the assumption of a farm-specific production set F[i].
• ↵26 The importance metric is obtained from a BART that includes the PS (PS-BART). Unlike the algorithm in equation [7], the PS-BART does not distinguish the prognostic from the treatment effect
component. However, in terms of variable importance, the difference between the two techniques is negligible.
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101. Symmetric Tree 🚀
Solution Developed In:
Explaining The Question
So, we're given a Binary Tree and we have to check if it's a symmetric tree. Meaning that from the tree, it should be that of a mirror. We don't need to check the given root node, because that won't
ever have a mirror. But it's left and right children need to be mirrored.
Mirrored meaning, that on the left side of the tree, the right side of the tree should have the same values but with their children swapped. It sounds confusing and it is a little. Visually speaking
it's easy to imagine but to get this down into code, it's a little more complicated.
Recommended Knowledge
What do we know?
1. We have a Binary Tree and we need to check if it's a symmetric tree. Meaning that from the tree should be that of a mirror. We don't need to check the given root node, because that won't ever
have a mirror. But it's left and right children need to be mirrored.
2. We know that at any given node, it's children should have the same value but with their children swapped. Meaning that on one side, it's on the left and on the other side it's on the right.
How we're going to do it:
We're going to perform post-order traversal on both sides of the tree at the same time. Meaning, that on the left side, we're going to go as far left as possible, and on the right side, we're going
to go as far right as possible. Compare the values of the nodes on both sides. If they're the same, then we're good. If not, we know it's a bad tree.
We're going to do this recursively.
1. Firstly, we need to treat every node as if they're all the same. Meaning we subject every node to the same process. So firstly we ask the nodes from both the left and right trees if they're
empty? If this is the case, it mean's we reached the end of the binary tree without error. Meaning it's a symmetric tree.
2. We then ask 'Are the left and right trees both not empty?'. Meaning that one of them isn't a leaf node while the other is. If this is the case, then we know it's a bad tree. So we return false
3. We then compare values of both trees. 'Is the current left value the same as the current right value?' If this is the case, we have symmetrically nodes. If not, we return false
4. We then continue our search of the right and lefts trees until we have exhausted the tree. Meaning we have visited every node we can.
5. Once all nodes are visited, we ask if the trees was symmetric. Meaning that we have visited every node and we have checked if they're symmetric. If they were, then we return true. If not, we
return false. If we got a false anywhere, it would have bubbled up through the stack to let us know that we had a bad node and thus this is a bad tree. If they were all true, then we return true.
Big O Notation:
• Time Complexity: O(n) | Where n is the number of nodes the tree has | As we will always be traversing the entire tree
• Space Complexity: O(h) | As we will be using the call Stack to store the nodes | Although, it can be argued that the Space Complexity is **O(n)** as in the worst case, the call stack contains as
many nodes n as the tree has.
' Could this be improved? ' Yes! Morris Traversal could do this problem in O(1) space complexity. But Morris Traversal is tricky and tough to read. For the sake of simplicity, I don't use it here.
Leetcode Results:
See Submission Link:
• Runtime: 63 ms faster than 96.46% of JavaScript online submissions for Symmetric Tree
• Memory Usage: 55.6 MB, less than 44.80% of JavaScript online submissions for Symmetric Tree
The Solution
var isSymmetric = function (left_tree, right_tree = left_tree) {
/* -------------------------------------------------------------------------- */
/* 101. Symmetric Tree */
/* -------------------------------------------------------------------------- */
* @author Samuel Hinchliffe
* @see {@link linkedin.com/in/samuel-hinchliffe-🚀-2bb5801a5/ | Author's Linkedin }
* @see {@link github.com/Samuel-Hinchliffe}
// https://leetcode.com/submissions/detail/684314946/
// Both are trees are the same
if (!left_tree && !right_tree) {
return true;
// One exists without another?
if (!left_tree || !right_tree) {
return false;
// Are left and right of same value?
// If not return false
if (left_tree.val != right_tree.val) {
return false;
// Do all the left trees and right trees
let outer_tree = isSymmetric(left_tree.left, right_tree.right);
let inner_tree = isSymmetric(left_tree.right, right_tree.left);
// Are both trees the same?
return outer_tree && inner_tree;
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Post-Quantum Cryptography
There's been a bunch of talk in the news this week around a group from China claiming to break RSA encryption with a D-Wave quantum computer.
In discussing this, first we need to distinguish between the two types of cryptography that we use on the web today: symmetric encryption (where the same key is used for encryption and decryption)
and asymmetric encryption (where a public key is used for encryption and a private key is used for decryption).
The current standard for symmetric encryption: AES-GCM 256 is thought to be quantum safe at this point in time. That means that we don't know of an algorithm that you can use on a quantum computer
that will allow you to easily decrypt files that use this type of symmetric encryption.
Asymmetric Encryption is an entirely different story: Shor's Alogrithm is widely accepted as a way to recover asymmetric encryption keys rapidly on a sufficiently large quantum computer (on the order
of millions qbits). And in this case a small quantum computer (50 qbits) was able to factor a 22bit RSA key. Note that most people that use RSA for SSH would use a 2048bit or 4096bit key. So this
development isn't really a threat.
However, it is widely expected that we will have quantum computers with millions of qbits in the next 10 years. If someone was to save data transmitted today and use one of these computers in 10
years they would be able to decrypt anything that uses Asymmetric Encryption. In order to prevent this several groups have been working on algorithms to replace what we currently use and this summer
the NSA approved several of them for use (and another is coming).
Below I've compiled a list of the current approved standards along with one standard under development. For more information I recommend reading:
ML-KEM aka CRYSTALS-Kyber
Intended to be used for key exchange (like ECDH) it replaces algorithms used to share keys for symmetric encryption.
There are three parameter sets defined by the standard:
• Kyber-512, offering security equivalent to AES-128
• Kyber-768, offering security equivalent to AES-192
• Kyber-1024, offering security equivalent to AES-256
ML-DSA aka CRYSTALS-Dilithium
Intended to be used for cryptographic signing it is intended to replace ECDSA and RSA.
There are three parameter sets defined by the standard:
• ML-DSA-44, offering security equivalent to SHA3-256
• ML-DSA-65, offering security equivalent to AES-192
• ML-DSA-87, offering security equivalent to AES-256
SLH-DSA aka SPHINCS+
Intended to be used for calculating hashes. It doesn't use the "lattice-based" approach that Kyber uses.
There are twelve parameter sets defined by the standard, for example:
• SLH-DSA-SHA2-128s, equivalent to AES-128.
• SLH-DSA-SHA2-192s, equivalent to AES-192.
• SLH-DSA-SHA2-256s, equivalent to AES-256.
SHA2 and SHAKE are both used.
FN-DSA aka FALCON
Being developed as an alternative to ML-DSA. Several standards are being developed incase a flaw is discovered in one of them.
A draft standard for FFT (fast-Fourier transform) over NTRU-Lattice-Based Digital Signature Algorithm (FN-DSA) is expected soon.
Member discussion
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Dynamical phase transition in large-deviation statistics of the Kardar-Parisi-Zhang equation
We study the short-time behavior of the probability distribution P(H,t) of the surface height h(x=0,t)=H in the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension. The process starts from a
stationary interface: h(x,t=0) is given by a realization of two-sided Brownian motion constrained by h(0,0)=0. We find a singularity of the large deviation function of H at a critical value H=Hc. The
singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x↔-x of optimal paths h(x,t) predicted by the weak-noise theory of the
KPZ equation. At |H| |Hc| the corresponding tail of P(H) scales as -lnP∼|H|3/2/t1/2 and agrees, at any t>0, with the proper tail of the Baik-Rains distribution, previously observed only at long
times. The other tail of P scales as -lnP∼|H|5/2/t1/2 and coincides with the corresponding tail for the sharp-wedge initial condition.
Bibliographical note
Publisher Copyright:
© 2016 American Physical Society.
Dive into the research topics of 'Dynamical phase transition in large-deviation statistics of the Kardar-Parisi-Zhang equation'. Together they form a unique fingerprint.
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How To Calculate Time And A Half And Double Time | Artinice
How To Calculate Time And A Half And Double Time
How to Calculate Time and a Half and Double Time
Time and a half and double time are terms usually heard in a working environment. These terms refer to the computation of an employee’s rate of pay when they work overtime, beyond the regular hours
of work.
Depending on the employer, an employee may be entitled to receive additional compensation when they work outside their regular working hours. To avoid confusion, it is best to know how to calculate
time and a half and double time so that you can understand how your earnings for overtime work are computed.
In this article, we will guide you on how to calculate time and a half and double time. We will also provide helpful information on these two payment schemes, including their differences and
similarities. Furthermore, we have included a list of FAQs that will answer many questions you might have about these payment schemes.
What is Time and a Half?
When an employee works hours beyond their regular shift, they are entitled to receive a rate of pay that is higher than their regular pay rate – that is time and a half. Time and a half refer to one
and a half times an employee’s regular hourly wage.
To calculate for time and a half, use this formula:
Time and a half = hourly rate x 1.5
For example, let’s say that you earn $20 per hour during your regular shift. You work on a public holiday, making it an overtime shift. Your usual shift is eight hours but you had to work for 12
hours that day. Here’s how to calculate your pay for overtime:
Regular working hours: 8 hours x $20 = $160
Overtime: 4 hours x $30 ($20 x 1.5) = $120
Total pay for the day: Regular pay + overtime pay = $280
Therefore, if you work with a rate of $20 per hour and worked an overtime shift, your rate of pay will be $30 per hour ($20 x 1.5).
What is Double Time?
Double time, on the other hand, refers to employees’ payment that is twice their hourly rate, received for working overtime beyond a specified regular working hours or in special cases such as
extreme weather conditions or emergencies.
To calculate double time, use this formula:
Double time = hourly rate x 2
Using the same example above, let’s say you are paid $20 per hour and are required to work on a weekend, making it an overtime shift. Double time applies in this situation. Here’s how to calculate
your pay:
Regular working hours: 8 hours x $20 = $160
Overtime: 4 hours x $40 = $160 ($20 x 2)
Total pay for the day: Regular pay + overtime pay = $320
Therefore, if you work with a rate of $20 per hour and worked on an overtime shift where double time applies, your rate of pay will be $40 per hour ($20 x 2).
FAQs about Time and a Half and Double Time
1. What are regular hours of work?
Regular hours of work refer to the set hours or shift that an employee is expected to work per day/week. These hours are agreed upon in the employment contract and are also specified in the labor
code of each country. An employee’s hourly rate of pay is usually based on their regular hours of work.
2. When does time and a half apply?
Time and a half applies when an employee works beyond their regular shift but not more than a certain number of hours specified in the labor code. For instance, in the United States, the Fair Labor
Standards Act (FLSA) indicates that employees must receive time and a half for working more than 40 hours in a week. However, the rules vary depending on the jurisdiction.
3. When does double time apply?
Double time applies in cases of extreme weather conditions, natural disasters or other emergencies that may require employees to work regardless of their regular schedules. Double time may also apply
when a shift is considered to be “unsociable hours” or outside of the regular daytime or work hours.
4. Do all employees qualify for time and a half and double time?
No, not all employees qualify for time and a half and double time. These payment schemes typically apply to non-exempt employees. Exempt employees, such as managers or executives, may not qualify for
overtime pay as they are not covered by the same labor laws.
5. Can an employer choose to pay more than time and a half or double time?
Yes, an employer can choose to pay more than time and a half or double time. This is at the discretion of the employer and is usually part of the employment contract.
6. Can an employer choose not to pay anything extra for overtime work?
No, if an employee works overtime, they are entitled to receive an additional payment for their extra hours of work.
7. Are there countries where time and a half and double time are not legally required?
Yes, there are countries where there are no legal requirements for employers to provide overtime pay or pay a rate of time and a half or double time. However, many countries have labor laws designed
to protect the rights of employees when it comes to working beyond the regular hours.
Now that you know how time and a half and double time are computed, you can better understand how your employer arrives at your pay rate for overtime work. Remember that you must be an eligible
employee to receive overtime pay, and that only certain circumstances qualify you for double time. It is always best to check your employment contract or consult with your employer to clarify their
rules on overtime pay.
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IJEECS Paper Template - P.PDFKUL.COM
(IJEECS) INTERNATIONAL JOURNAL OF ELECTRICAL, ELECTRONICS AND COMPUTER SYSTEMS. VOLUME: 2 ISSUE: 1, MAY 2011
Efficient Key Pre-Distribution Approach for Wireless Sensor Networks Saurabh Kumar Singh#1, Dr Harsh K Verma*2, Ravindra Kumar Singh#3 #
Department Of Computer Science and Engineering, National Institute of Technology Jalandhar Punjab India 1
[email protected]
[email protected]
[email protected]
Abstract—Wireless sensor network is highly vulnerable to attacks because it consists of various resource-constrained sensor nodes which communicate among themselves via wireless links. Securely
distributing keys among sensor nodes is a fundamental challenge for providing security services in WSNs. In the random key pre-distribution approach is suitable for low power and resource constrained
sensor nodes, a shared key between a pair of nodes is not guaranteed and thus they may not be able to communicate with each other. Matrix based scheme for key predistribution essentially use LU
decomposition of matrix which can provide keys between any pair of nodes but are quite vulnerable to attack. This paper proposes an improved and efficient key pre-distribution scheme based on LU
composition of matrices. Our scheme, we use integer as elements of symmetric matrices. The existing approach use decomposition of matrices which is compute intensive but our proposed scheme uses
composition of matrices. The proposed scheme allows almost 100 % connectivity indifferent of the number of keys and provides resilience against node capture.
Keywords—Wireless sensor network; key pre-distribution; LU matrix; security; matrix decomposition; symmetric matrix.
I. INTRODUCTION Wireless Sensor Networks (WSNs) have potential to provide economical solutions to many problems of practical importance. Some general purpose applications where Sensor Networks can be
used are: Emergency Response System, Energy Management, Battlefield Management, Health Monitoring, and Inventory management etc [1]. For example, power load that should be carried over an electrical
line depends on the temperature of the wire and the environmental conditions. If the parameters are monitored by remote sensors and transmitted to a base station, it would be possible to meet load
requirements optimally. Sensor Networks consist of various resource-constrained devices. Each sensor node has low battery power, less memory and very less computational capability. Same battery is
used throughout the life time of a sensor node [1, 8]. However, there are still a lot of unresolved issue in WSN of which security is one of the hottest research issue [1,3,12]. Sensor Networks are
deployed in hostile environments. Environmental conditions along with resource-constraints give rise to many type of security threats or attacks. Adversary can physically capture and get the
information contained in the sensor node, eavesdrop and inject new messages, modify messages.
Hence there must be some sort of mechanism for node to node data transmission. The message is sent encrypted with a key that is shared by sender and receiver. Keys play a central role in realizing
security services like: authenticity, integrity, confidentiality etc. Keys need to be distributed securely among sensor nodes. For the distribution of keys, many ordinary security mechanisms such as
public key-based authentication and corresponding key management scheme are impractical and infeasible for WSN. In this paper, we present a novel scheme for key distribution scheme that is based on
polynomial over finite field. This scheme has advantage of both probabilistic and deterministic approaches. It is based on polynomial for a node it guarantees the establishment of key with every node
[2,13]. The rest of this paper is organized as follows. Section II describes the existing key distribution scheme for wireless sensor network. Section III gives an overview of the polynomial-based
key pre-distribution scheme approach in detail. Section IV presents the key pre distribution scheme with matrix decomposition. Section V deals with the detailed performance analysis. Finally, Section
VI concludes this paper. II. RELATED WORKS Eschenauer and Gligor [4] propose a probabilistic key predistribution scheme for pair wise key establishment. For each sensor node, a set of keys are chosen
from a big pool of keys and given to each node before deployment. In order to establish a pair wise key, two sensor nodes only need to identify the common keys they share. Thus every pair of nodes in
the network shares a key with certain probability. Since keys are randomly chosen from the key pool, they are not related. Hence it is not possible to calculate other keys by knowing some of the keys
from the key pool. Chan et al.[9,10] further extended this idea and developed two key predistribution techniques: q-composite key pre-distribution and random pair wise scheme[9]. q-composite key
pre-distribution also uses a key pool but requires two sensors to compute a pair wise key from at least q pre-distributed keys they share. In the random pair wise keys scheme, random pairs of sensors
are picked and assigned a unique random key. In both the schemes, resilience is improved because probability that a link is compromised, when a sensor node is captured, decreases. But, probability of
key sharing also decreases because a pair of sensor nodes has to share q keys instead of one. This scheme achieves good security under small scale attacks, while being vulnerable to large scale
(IJEECS) INTERNATIONAL JOURNAL OF ELECTRICAL, ELECTRONICS AND COMPUTER SYSTEMS. VOLUME: 2 ISSUE: 1, MAY 2011
III. THE POLYNOMIAL-BASED KEY PREDISTRIBUTION SCHEME To pre-distribution pair-wise key, the key pre-distribution server first randomly generate a bivariate t-degree polynomial over a finite field Fq
( , )=
a x
Definition 4: A Circular shift function (CS) is a function that defines an operation of rearranging the entries in a tuple, by moving the final entry to the first position. CS ( tuple , n) indicates
that n circular shifts are applied to a given tuple. Definition 5: A Reverse function(R) is a function that rearranges the entries of a tuple in a reverse order. B. The proposed key pre-distribution
Where q is a prime number that is large enough to accommodate a cryptographic key, such that it has the property of f (x, y) = f (y, x). Then for each node i , the setup server compute a polynomial
share of f (x, y) , that is f (i, y) and store it in sensor node i. For any two sensor node i , j , node i can compute the pair-wise key f (i, j) by evaluating f (i, y) at point j, and node j can
compute the pair-wise key f (j, i) by evaluating f (j, y) at point i . From the property of symmetry of f (x, y), f (i, j) = f (j, i) . So the pair-wise key between node i and j can be established.
In this scheme each sensor node needs to store a bivariate tdegree polynomial’s coefficients, which would occupy (t+1)log2 q storage space. The security proof [3] in ensure that this scheme is
unconditionally secure and t-collision resistant. In other words the coalition of no more than t compromised sensor node know nothing about the pair-wise key between any two non compromised sensor
node. IV. THE PROPOSED SCHEME In this section we briefly describe how the proposed key predistribution scheme works in detail. The basic idea can be based on E-G scheme, to remove the drawback of the
existing schemes, we proposed a new key pre-distribution scheme with LU matrix for wireless sensor network. The following procedure is executed by base station in order to construct L, U and U′
matrices. Which provides full connectivity and better resilience? A. Preliminaries We start with a brief description of various concepts and definitions used in this paper. Definition 1: If a square
matrix M has the property MT = M, where transpose of matrix M is denoted by MT, we say that M is a symmetric matrix. M is a symmetric matrix means MIj = Mji, where Mij is the element in the ith row
and jth column of matrix M . Definition 2: LU matrix decomposition of an n × n matrix M decomposes it into two matrices L and U such that M = LU , where L is an n × n lower triangular matrix and U is
an n × n upper triangular matrix, respectively. Definition 3: Let M is a square matrix with M = LU, and the pivots on the diagonal of U are all nonzero. By dividing ith (1 ≤ I ≤ n) row of U by the
nonzero pivot di , the matrix U is decomposed into a diagonal matrix D whose diagonals are just the pivots d1,d2,...,dn and a new upper triangular matrix, denoted by U’ , whose diagonal elements are
all 1. Then M = LDU’ .
The proposed key pre-distribution scheme consists of five step Step 1: Generate a large pool of key and setup server generate randomly a large pool of bivariate t-degree polynomial over the finite
field. Step 2: Construct a lower triangular matrix using the randomly selected elements from the key pool. The first condition for selecting elements from the large pool is that all elements present
in a column should be multiple of the diagonal element of the same column, some elements should be zero, some elements should be same as diagonal element and all the selected elements should be
Step 3: Forming an upper triangular matrix using lower triangular matrix: Upper Triangular matrix is formed by taking the simple transpose of lower triangular matrix, i.e., U = LT U=
Step 4: Forming a diagonal D matrix using U matrix: Diagonal matrix D is constructed by choosing diagonal elements from matrix U. ⎡1 ⎤ ⎥ D= 0 U′ = ⎢⎢ ⎥ 0 1 ⎢ ⎥ 0 0 ⎣0 0 1 ⎦ Step 5: After computing L,
U, D, and U′ matrices, the base station selects one row from lower triangular matrix L, i.e., L and one column from upper triangular matrix U , i.e., U for each node and sends both the tuples along
with diagonal matrix D to each node in the network separately. This is done by using the condition that the row number and column number selected for a particular node should be equal. 0
IV. PERFORMANCE ANALYSIS AND COMPARISON In this section, we present the evaluation of the performances of our schemes, and compare the scheme with Eschenauer and Gligor scheme [4]. Our focus are on
analysis of the network connectivity, analysis of resilience against nodes capture and analysis memory usage by each node in the network. A. Analysis of authentication The proposed scheme allows node
to node mutual
(IJEECS) INTERNATIONAL JOURNAL OF ELECTRICAL, ELECTRONICS AND COMPUTER SYSTEMS. VOLUME: 2 ISSUE: 1, MAY 2011
authentication by following the below process: Step 1: Initially Node A applies Reverse function (R) on the selected column elements U′ci and sends it to the Node B. Step 2: Node B applies reverse
function on the data received from Node A and then applies reverse function to calculate U′ci. Then Node B computes the cross product U = D × U′ci. After computing this cross product, Node B′
generates a key Kji by multiplying Lrj with Uci , and apply hash function on key K ji , i.e., H (Kji ). Step 3: Now Node B applies the same process as done by Node A in step 1 own column U′cj and
send this value with the generated hash key H(Kji) to Node A . Fig. 1 Analysis of networks connectivity
Step 4: After receiving data from Node B, Node A calculates Kij similarly as Node B calculated in step 2 and apply the hash function H on K ij. Step 5: If H(Kij) and H(Kji ) are equal then Node A Yes
message along with H(Kij) to Node B otherwise sends errmsg to Node B. If the response is yes then Node B verifies H(Kij ) with H (Kji ) to establish a secure channel. B. Analysis of the network
connectivity In this section , we compare the proposed scheme with E & G scheme. Network connectivity, probability p in the proposed scheme sharing at least between any two sensor nodes. We define an
event in which a pair of node that does not have a common any one key an[Eve] and pr[eve] is the probability of such event. In proposed scheme, network connectivity p as following.
Connectivity without concern for the number of key per node. In addition key in our scheme occupy less memory space in sensor nodes. C . Analysis of resilience against node capture In this section we
analyzed resilience to node capture. In wireless sensor networks an adversary can easily calculate the information of compromised nodes, intentionally provide misleading information to the entire
network, and break the whole network security. In this section we evaluated that the proposed scheme improves WSN resilience by calculating the fraction of compromised nodes among non-compromised
nodes. In addition, we plan to compare our scheme with E & G schemes based on performance. In E & G schemes, the probability of compromising the shared keys between any two non-compromised nodes is
following: Pcompromised = (1 – (1 –k/S)m)
P = 1- Pr[Eve] =1-(1-K/S)2S-2K+1/(1-2K/S)S-2K+1/2
In the proposed scheme, rows from lower triangular matrix L, column from upper triangular matrix U′ , diagonal matrix D, Where S is the total number of node in network and are deleted after the
Establishment of the keys. Polynomials K is the number of key in each node which are pre-distributed to each node are randomly selected from the lower triangular matrix L and its degree will be left
Our scheme show that any two sensor node can always find a in each node. For the purpose of addition of new no des in shared key between themselves using LU matrix network. When m nodes have been
compromised, the decomposition. In other words, the probability of not sharing probability of compromising the shared keys between any two a common key between any two network sensor nodes is zero.
non-compromised nodes is equal to the probability of Figure. 1compares network connectivity p of the proposed compromising the shared polynomials between any two nonscheme with E & G scheme. In the
performance analysis, we compromised nodes But in our scheme if adversary did not get assume that the size of key pool for each node is 1000, 2000, any information from compromised no des about
non5000 and 10000. The result shows that the proposed has 100%. compromised nodes, we can say that m is equal to zero. So for our scheme, the probability of compromising the shared keys between any
two non-compromised node is following: Pcompromised = (1 – (1 –k/S)0) Pcompromised = (1 – 1) =0 C. Memory usage analysis In this subsection, any two sensor nodes establish the shared key by using
polynomial with LU matrix decomposition based key pre-distribution scheme. The network-wide memory usage of our scheme is mostly of polynomial cost. We have ©IJEECS
(IJEECS) INTERNATIONAL JOURNAL OF ELECTRICAL, ELECTRONICS AND COMPUTER SYSTEMS. VOLUME: 2 ISSUE: 1, MAY 2011
proposed an effcient method to store the row and column information of L, and U matrices. Our scheme only needs to store each element in the non-zero-element part and one value specifying the maximum
number of following zeros in zeroelement part of L and U matrices.
[6] Fig. 2 Resilience comparison E & G scheme and our scheme
This technique is specially suitable for large wireless senor networks. Some notations given below which are used to estimate the storage efficiency: VI. CONCLUSION In recent years key distribution
has been one of the hot issues in security research. In This paper, we have proposed a new key pre-distribution scheme based on symmetric matrix with LU decomposition. This scheme guaranteed that any
pair of nodes can find a common key between themselves and also it allows more security enhancement on node to node pair-wise key establishment and very good resilience to data exchanged between the
nodes and also took very less time to establishment of key between the nodes. This proposed scheme has very less computational overhead to calculate the key in the network. ACKNOWLEDGMENT The author
would like to thanks the anonymous referess for their valuable comments, which greatly improved the readability of the paper.
REFERENCE A. Perrig, R. Szewczyk, V. Wen et al.,”SPIN:security protocols for sensor network,” Wireless Network,Vol.8.,No.5,pp. 521-534,2002. Carlo Blundo, Alfredo De Santis, Amir Herzberg, Shay
Kutten, Ugo Vaccaro, and Moti Yung. Perfectly-secure key distribution for dynamic conferences. In CRYPTO, pages 471–486, 1992. A. K. Pathan, H. W. Lee, and C. S. Hong, “ Security in wireless sensor
network: issues and challenges,” In proceeding of the 8th ICACT 06,Volume 2, Phoenix Park, Korea, pp. 1043-1048,February, 2006 L. Eschenauer, V. D. Gligor, ”A key –managements cheme for distributed
sensor network,” In proceeding of the 9th ACM conference on Computer and Communication,Washington, DC, USA, pp. 41-47, Nov. 2002. Taejo on Park and Kang G. Shin. Secure routing based on distributed
key sharing in large-scale sensor networks. ACM Trans. Embed. Comput. Syst., 7(2):1– 28, 2008. Hangyang Dai and Hongbing Xu, “ Key Predistribution Approach in Wireless Sensor Network Using LU
Matrix”, IEEE Sensors Journal, Vol, 10. No.8. August 2010. Ni Chen,Jian Bo Yao and Gang Jun Wen, “An Improved LU Matrix Key Pre-distribution Scheme for Wireless Sensor Networks”, International
Conference on Advanced Computer Theory and Engineering 2008 Al-Sakib Khan Pathan, Tran Thanh Dai and Choong Seon Hong, “An Efficient LU Decomposition-based Key Pre-distribution Scheme for Ensuring
Security in Wireless Sensor Network”, Proceeding of The Sixth IEEE International Conference on Computer and Information Technology 2006. Wenliang Du, Jing Deng, Yunghsiang S. Han, Pramod K. Varshney,
Jonathan Katz, and Aram Khalili. A pairwise key predistribution scheme for wireless sensor networks. ACM Trans. Inf. Syst. Secur., 8(2):228–258, 2003. Chang-Won Park, Sung Jin Choi, and Hee Yong
Youn. A noble key pre-distribution scheme with lu matrix for secure wireless sensor networks. In CIS (2), pages 494– 499, 2005. Sung Jin Choi and Hee Yong Youn. Mkps: A multilevel key
pre-distribution scheme for secure wireless sensor networks. In HCI (2), pages 808–817, 2007. Karlof, C. and Wagner, D. “Secure routing in wireless sensor network: Attack and countermeasure”,
Elsevier’s Ad Hoc Network Journal, Special Issue on Sensor Network Application and Protocol, September 2003,pp.293-315. Sencun Zhu, Shouhuai Xu, Sanjeev Setia, and Sushil Jajodia. Establishing
pairwise keys for secure communication in ad hoc networks: A probabilistic approach. In ICNP, pages 326– 335, 2003.
(IJEECS) INTERNATIONAL JOURNAL OF ELECTRICAL, ELECTRONICS AND COMPUTER SYSTEMS. VOLUME: 2 ISSUE: 1, MAY 2011
Saurabh Kumar Singh He received B.Tech degree in computer science and engineering in 2009 from Institute of Engineering and Technology Jhansi, India. Currently he is pursuing M.Tech degree,in
computer science and engineering from National Institute of Technology Jalandhar India in 2009 and 2011 respectively. His current research include in wireless sensor network, computer network and
information security. Ravindra Kumar Singh received B.Tech degree in Information Technology in 2009 from Ajay Kumar Garg Engineering College Ghazibad Utter Pradesh Technical University, India.
Currently he is pursuing M.Tech degree,in computer science and engineering from National Institute of Technology Jalandhar India in 2009 and 2011 respectively. His current research include in search
engine optimization, wireless sensor network and information security.
Dr. Harsh K Verma He has completed Phd in Numerical Computing from Punjab Technical University Punjab, India. He is currently a professor and Head of Department of computer science and engineering in
National Institute of Technology Jalandhar India. His research includes in numerical computing, information security and computer networks. He has published various paper in national and
international journal and conferences. He has attended various national and international workshop training schools and other technical activity during his academic carrier.
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Pedal Circle
The pedal Circle of a point in a Triangle is the Circle through the feet of the perpendiculars from to the sides of the Triangle (the Circumcircle about the Pedal Triangle). When is on a side of the
Triangle, the line between the two perpendiculars is called the Pedal Line. Given four points, no three of which are Collinear, then the four Pedal Circles of each point for the Triangle formed by
the other three have a common point through which the Nine-Point Circles of the four Triangles pass. The radius of the pedal circle of a point is
(Johnson 1929, p. 141).
See also Miquel Point, Nine-Point Circle, Pedal Triangle
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.
© 1996-9 Eric W. Weisstein
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Best Player Combinations For
URH vs SSS
How it Works?
This guide is super useful for identifying player combinations that are likely to score big. Use the filters to identify the best player combinations according to specific match scenarios.
Player correlate positively
Both players generally perform similarly when picked together. They either score high together or score low together. Maximise FPts by picking players that correlate positively with top players.
Player correlate negatively
Both players generally perform in an opposite manner. If one player scores high, the other usually scores low. Avoid picking players that correlate negatively with top player
Show Player Combinations on the basis of:
Best URH Player Combinations
Best URH Player Combinations
S Shinwari AR URH
G Singh Mann BAT URH
S Binny AR URH
S Shukla BAT URH
S Jakati BOW URH
J Malhotra WK URH
B Sharma BOW URH
P Trego AR URH
S Tyagi BOW URH
M Panesar BOW URH
R Jangid AR URH
Best SSS Player Combinations
Best SSS Player Combinations
H Hassan BOW SSS
P Negi AR SSS
S Goswami WK SSS
J Karia AR SSS
S Bhati BOW SSS
M Guptill BAT SSS
H Masakadza BAT SSS
C Gandhi BAT SSS
J Mendis AR SSS
E Chigumbura BAT SSS
K Jadhav BAT SSS
Best URH Player Combinations
Best URH Player Combinations
S Shinwari AR URH
G Singh Mann BAT URH
S Binny AR URH
S Shukla BAT URH
S Jakati BOW URH
J Malhotra WK URH
B Sharma BOW URH
P Trego AR URH
S Tyagi BOW URH
M Panesar BOW URH
R Jangid AR URH
Pick S Shinwari and J Malhotra together
Positive Correlation
Pick S Shukla and S Tyagi together
Positive Correlation
Pick S Jakati and J Malhotra together
Positive Correlation
Pick J Malhotra and P Trego together
Positive Correlation
Best SSS Player Combinations
Best SSS Player Combinations
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Boosting Your Test-Taking Skills - Getting into the Act: An Overview of ACT Math Basics - ACT Math For Dummies
ACT Math For Dummies (2011)
Part I. Getting into the Act: An Overview of ACT Math Basics
Chapter 2. Boosting Your Test-Taking Skills
In This Chapter
Some folks say that the only thing tests measure is how well you take a test. Such news would be great, because then you could skip all the math stuff in this book and head directly to tips such as
“always guess Choice (D).” But guess what? That tactic isn’t gonna help!
So before you take any test, you need to know a few strategies that relate specifically to that test. In this chapter, I focus on some basic facts about ACT math and show you how best to approach the
test to take advantage of these facts.
To begin, I discuss the strategy of tackling the math section in two passes — the first to handle the easy questions and the second to focus on the more difficult ones. I also discuss a few
strategies about guessing on the ACT and provide you with a few ways to approach multiple-choice math questions. Finally, I end the chapter with some advice on selecting a calculator, including some
tips for using the one you select.
Two-Timing the Test: Taking Two Passes to Answer Questions
On the ACT math test, you have 60 minutes to answer 60 questions. So you have roughly 1 minute per question. Every question you answer correctly is worth 1 point toward your raw score.
But not all ACT math questions are created equal. Generally speaking, the questions increase in difficulty as you proceed from Question 1 to Question 60. Here’s the general breakdown of difficulty:
Easy: Questions 1 through 20
Medium: Questions 21 through 40
Hard: Questions 41 through 60
You should obviously start the test at Question 1 and proceed in order as much as you can. However, every student is different, so you may find some questions along the way that are more difficult
for you than later questions. So if you read a question and don’t have a clue how to answer it, feel free to jump over it. The next question may be easier for you to answer.
I suggest you use the tried-and-true strategy of taking two passes over the ACT:
Pass #1: Start with Question 1 and work your way forward, answering questions that look relatively quick and easy and jumping over those that look difficult or time-consuming.
Pass #2: After you’ve answered all the quick and easy questions, circle back to the first question you skipped over and work your way forward to the end again.
This strategy maximizes the number of questions you can answer with confidence. It also helps you save time for the tough questions, which usually take more than 1 minute to solve.
begging for you to skip over them. For example, you may consider passing over questions that
Of course, not every problem with the preceding characteristics is as difficult as it looks. But as you run across problems like this, feel free to jump over them — even on Pass #2. If you have time
at the end of the test, you can always try to pick off a few of these questions. But if you’re going to skip questions, you may as well skip these hairy beasts.
To Guess or Not to Guess
On the ACT, you don’t lose points from your raw score when you fill in a wrong answer. So strategically you should fill in every answer, even if you have to make a wild guess.
Of course, you don’t want to guess on questions that you may be able to answer correctly — especially among the earlier questions, which tend to be easier. And keep in mind that an educated guess is
always better than a wild guess. So whenever possible, rule out answers that you know are wrong. Keep track of these wrong answers by crossing them out in your test booklet.
Don’t guess at any answers while you’re still on the first pass (see the previous section, “Two-Timing the Test: Taking Two Passes to Answer Questions,” where I discuss tackling the test in two
separate passes). Instead, begin guessing on your second pass. At this point, if you can confidently rule out a couple of answers but don’t know how to proceed with a question, you can save time by
guessing at the answer and moving on to the next question.
Finally, keep close track of your time. When your 60 minutes are almost up, take a moment to guess at all the remaining answers — don’t leave any blank. With a bit of luck, you may pick up a few
additional points on some of these questions.
not return to that part, even to fill in guesses. If you get caught doing this, you could be expelled from the test with no score and no refund. ’Nuff said!
ACT versus SAT: When do I guess?
On the ACT, you don’t lose points when you fill in a wrong answer. This rule is different on the SAT, where a quarter of a point is taken off for each wrong answer. This small distinction makes a big
difference in your strategy if you’re planning to take both tests.
When you take the ACT, be sure to fill in every answer, even if you have to guess. To do this, keep a sharp eye on the time. Then when you’re down to the last minute or two, make sure that every
answer is filled in. In contrast, if you also take the SAT, don’t fill in any answer for which you can’t make an educated guess. That is, don’t answer unless you can confidently rule out at least one
choice as wrong.
Answering Multiple-Choice Questions
The math section of the ACT comprises 60 multiple-choice questions. Each question provides five possible answers. You likely have been taking standardized tests for most of your life as a student, so
you’re probably already familiar with this type of question. However, math teachers generally don’t use multiple-choice questions when assigning homework or testing. So in this section, I provide a
few strategies for approaching multiple-choice math questions.
Considering the five answer choices
Every multiple-choice question gives you a little extra information, because you know the correct answer must be one of the five choices given. Always take a moment to notice these answer choices,
because they may guide you as you work on solving the problem. The following example shows you how you can rely on answer choices to correctly solve a problem.
j^2 – 14j + 48 = 0, which of the following shows all of the possible values of j?
(A) –6
(B) 8
(C) 6, 8
(D) –6, 8
(E) –6, –8
You can solve the equation j^2 – 14j + 48 = 0 by factoring. (I show you the details of factoring in Chapter 5 and explain how to apply this technique to solving quadratic equations in Chapter 7.) In
this case, every value in each of the five answers includes either 6 or 8 (give or take a minus sign), so you have a head start on the factoring:
j^2 – 14j + 48 = 0
(j – 6)(j – 8) = 0
At this point, you only need to fill in the signs (+ or –) inside the parentheses. Because 48 in the original equation is positive, the two signs must be the same (either both + or both –). And
because –14 is negative, at least one of the signs is negative. Therefore, both signs are negative:
(j – 6)(j – 8) = 0
Now you can solve this equation by breaking it into two separate equations:
j – 6 = 0j – 8 = 0
j = 6j = 8
Thus, the correct answer is Choice (C).
Plugging and playing
Multiple-choice questions give you an opportunity to arrive at the correct answer by plugging in the answer choices and solving. Please note that plugging in answers can be a little time-consuming,
so if you can find a better way to solve the problem, go for it. But when you get stuck, this tactic gives you a chance at answering questions that you really aren’t sure how to solve. Consider the
following example.
x =
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
You may or may not know how to solve this type of equation (flip to Chapter 5 to see how it’s done). And in any case, solving it may be time-consuming. So you can try to plug in each possible answer
for x to see which one works. Start with Choice (A) and plug in 4 for x:
This answer choice is obviously wrong, because 21 isn’t a square number. Therefore, x + 1 must be a square number. Try testing Choices (B) through (E) in this way, keeping in mind that you’re looking
for a value of x that makes 5x + 1 a square number:
(B): 5(5) + 1 = 25 + 1 = 26not a square number
(C): 5(6) + 1 = 30 + 1 = 31not a square number
(D): 5(7) + 1 = 35 + 1 = 36a square number!
(E): 5(8) + 1 = 40 + 1 = 41not a square number
Only one value produces a square number, so the correct answer is Choice (D). You can verify this by plugging in 7 for x:
Some questions ask you for the greatest or least number that has a certain property. These questions provide a great opportunity to test answers individually until you find the correct one. Consider
the following strategies:
lowest or least value, begin with the lowest number and work your way up.
greatest or highest value, begin with the greatest number and work your way down.
The following example illustrates this strategy.
(F) 60
(G) 120
(H) 144
(J) 240
(K) 288
Because you’re looking for the least common denominator, you can find the correct answer by testing numbers and ruling out wrong answers, starting with the lowest number.
Begin by testing to see whether 60 is divisible by 6, 9, and 16:
60 ÷ 6 = 10
60 ÷ 9 = 6 r 6not divisible
So Choice (F) is wrong. Now test 120:
120 ÷ 6 = 20
120 ÷ 9 = 13 r 3not divisible
So Choice (G) also is wrong. Next, test 144:
144 ÷ 6 = 24
144 ÷ 9 = 16
144 ÷ 16 = 9
So Choice (H) is the correct answer. By the way, before moving on, notice that 288 is also divisible by all three denominators. However, Choice (K) is wrong because the question asks for the least
common denominator, which is why you started plugging in the lowest numbers first.
Calculating Your Way to Success: Calculators and the ACT
If you hate doing long division as much as I do, you’re probably glad to hear that calculators are allowed on the ACT. In this section, I answer a few basic questions about calculators.
When should I use a calculator?
A calculator is a great tool for solving problems more quickly than you can either in your head or using a pencil and scratch paper. At the same time, however, you want to avoid overusing it for
calculations that you can easily and accurately do in your head.
When you take a practice test, notice how you use your calculator. Do you almost forget about it? If so, here are a few tips to train yourself to use the calculator to your advantage:
On the other hand, do you use your calculator for just about every problem? If so, consider these tips for backing off a bit:
correctly without a calculator but then doubt the answer and check it to be sure — you probably need to convince yourself that you’re on track. Start to notice how often you check simple math and
find that your answer was already correct. If you’re almost always correct, you’re spending time using your calculator that could be used in a better way.
What kind of calculators can I use?
The calculators that are allowed on the ACT are divided into these categories:
Basic calculators: You can buy this type of calculator for less than $10 in almost any store that sells stationery. Basic calculators are perfect for balancing your checkbook or keeping a tally of
your groceries, but they’re simply not adequate for the ACT. If you’re currently using this kind of a calculator, I strongly suggest that you consider upgrading to a scientific or graphing
Scientific calculators: This type of calculator typically costs more than $10, but you get a lot of functionality not found on a basic calculator. Depending on the model, a scientific calculator
usually includes exponents, square roots, logarithms, trig functions, a reciprocal function, and lots of other stuff that may come in handy on the ACT.
Graphing calculators: A graphing calculator has all the bells and whistles of a scientific calculator, plus a larger screen for visual display of graphs and tables. If you’re thinking of upgrading to
a graphing calculator from either of the other two types, consider this: The main advantage you gain is directly related to your proficiency with these visual elements. So plan to spend at least four
or five hours practicing with your new toy, creating input-output tables for functions, graphing lines and parabolas, and exploring other related visual options. If you’re not convinced you’re really
going to practice, you may as well save your money. Stick with a scientific calculator, which should serve you well enough.
What kind of calculators can’t I use?
not use a calculator that includes any of the following features:
Texting and Internet access: Sure, your iPhone (or iPad or laptop) may have a calculator function, but this function doesn’t make it a calculator. It also has lots of other fancy capabilities that
aren’t allowed on the ACT. Obviously the elders don’t want you texting your genius Uncle Roy at MIT or looking up answers in Wikipedia if you get stuck on a question.
Talking or other weird noises: If your calculator makes noise and disturbs people, the monitors may separate you from it for the duration of the test. Of course, that separation wouldn’t be good for
your test score.
Electrical access: I can’t guarantee that your testing site will have a place to plug in a calculator that requires power. Even though you may get lucky, your best bet is to bring a battery-powered
calculator (along with a fresh set of batteries).
How do I use my calculator?
After you purchase your calculator, don’t let it sit in its impenetrable plastic packaging until the night before the ACT. Use it for at least one practice test so you can get the feel of where the
important keys are. At a minimum, make sure you know how to enter the following:
Negative numbers: Scientific calculators almost always have a special key to enter a negative number. This key usually is distinct from the minus sign used for subtraction. Find this key and test it
by calculating a few things, such as –1 – 5 = –6
Parentheses: When you enter complicated calculations, you may need parentheses.
The most common example of this is when you enter a fraction such as
this fraction as (2 + 7)/(4 – 1) and make sure you get the answer 3.
Pi (π): On most scientific calculators, the π function is simple to use, requiring only one or two key strokes. Locate it and calculate 10π — the answer should be about 31.4.
Square roots: On many popular scientific calculators, the square root function doesn’t have its own key. Instead, it’s often the 2nd function on the x^2 key used for squaring a number. This
manipulation isn’t complicated, but you want to know it cold before you take your ACT. Make sure you can calculate some square roots, such as
Trig functions: Truthfully, you may never need to use the sin, cos, or tan keys. Even so, be prepared. Locate them and calculate cos 0 = 1.
Also very important: Scientific calculators accept trig inputs in either degrees or radians, depending on which mode you choose. (Here’s a quick test: Enter sin π and see what you get. If the answer
is 0, you’re in radian mode; if it’s something weird, you’re in degree mode.) Both modes work equally well, so decide whether you like working with degrees or radians best and make sure your
calculator is set for this mode.
If you have a fancy graphing calculator, here are some of the useful features that are worth checking out:
Input-output tables: This feature allows you to enter a function such as y = 2x – 3, and then the calculator builds a table showing the resulting y-values (given x = 1, x = 2, and so on).
Solving equations: You won’t want to miss this time-saving feature. Be sure to enter equations carefully, using parentheses as needed, especially for complicated fractions.
Graphs: This feature allows you to enter a function and view the resulting graph. More advanced features allow you to solve equations — including quadratic equations — using a graph.
TI-83 Plus Graphing Calculator For Dummies (Wiley) and TI-89 Graphing Calculator For Dummies (Wiley), which are both written by C. C. Edwards. Finally — and I know that in the fun department, this
ranks somewhere between comparison shopping for snow tires and cleaning the lint filter on your dryer — you can learn a lot about your calculator from that little manual that comes along with it.
Don’t just toss it aside!
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A department conducted a survey and the results are as follows:-Turito
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A department conducted a survey and the results are as follows:
What is the total measure of painter’s and programmer’s block?
A. 15^o
B. 225^o
C. Both a and b
D. None of the above
We will add the angles subtended by the painter's and the programmer's arc to the center.
The correct answer is: 225^o
The total measure of Painter's and programmer's blocks
See and understand the problem, then solve it.
Get an Expert Advice From Turito.
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A consistent pricing process (CPP) is any representation of (frictionless) "prices" of assets in a market. It is a stochastic process in a filtered probability space ${\displaystyle (\Omega ,{\
mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}$ such that at time ${\displaystyle t}$ the ${\displaystyle i^{th}}$ component can be thought of as a price for the ${\displaystyle i^{th}}$ asset.
Mathematically, a CPP ${\displaystyle Z=(Z_{t})_{t=0}^{T}}$ in a market with d-assets is an adapted process in ${\displaystyle \mathbb {R} ^{d}}$ if Z is a martingale with respect to the physical
probability measure ${\displaystyle P}$, and if ${\displaystyle Z_{t}\in K_{t}^{+}\backslash \{0\}}$ at all times ${\displaystyle t}$ such that ${\displaystyle K_{t}}$ is the solvency cone for the
market at time ${\displaystyle t}$.^[1]^[2]
The CPP plays the role of an equivalent martingale measure in markets with transaction costs.^[3] In particular, there exists a 1-to-1 correspondence between the CPP ${\displaystyle Z}$ and the EMM $
{\displaystyle Q}$.
1. ^ Schachermayer, Walter (November 15, 2002). "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time".{{cite journal}}: Cite journal requires |
journal= (help)
2. ^ Yuri M. Kabanov; Mher Safarian (2010). Markets with Transaction Costs: Mathematical Theory. Springer. p. 114. ISBN 978-3-540-68120-5.
3. ^ Jacka, Saul; Berkaoui, Abdelkarem; Warren, Jon (2008). "No arbitrage and closure results for trading cones with transaction costs". Finance and Stochastics. 12 (4): 583–600. arXiv:math/0602178.
doi:10.1007/s00780-008-0075-7. S2CID 17136711.
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class hyppo.independence.HHG(compute_distance='euclidean', **kwargs)¶
Heller Heller Gorfine (HHG) test statistic and p-value.
This is a powerful test for independence based on calculating pairwise Euclidean distances and associations between these distance matrices. The test statistic is a function of ranks of these
distances, and is consistent against similar tests 1. It can also operate on multiple dimensions 1.
☆ compute_distance (str, callable, or None, default: "euclidean") -- A function that computes the distance among the samples within each data matrix. Valid strings for compute_distance are,
as defined in sklearn.metrics.pairwise_distances,
■ From scikit-learn: ["euclidean", "cityblock", "cosine", "l1", "l2", "manhattan"] See the documentation for scipy.spatial.distance for details on these metrics.
■ From scipy.spatial.distance: ["braycurtis", "canberra", "chebyshev", "correlation", "dice", "hamming", "jaccard", "kulsinski", "mahalanobis", "minkowski", "rogerstanimoto",
"russellrao", "seuclidean", "sokalmichener", "sokalsneath", "sqeuclidean", "yule"] See the documentation for scipy.spatial.distance for details on these metrics.
Set to None or "precomputed" if x and y are already distance matrices. To call a custom function, either create the distance matrix before-hand or create a function of the form metric(x,
**kwargs) where x is the data matrix for which pairwise distances are calculated and **kwargs are extra arguments to send to your custom function.
☆ **kwargs -- Arbitrary keyword arguments for compute_distance.
The statistic can be derived as follows 1:
Let \(x\) and \(y\) be \((n, p)\) samples of random variables \(X\) and \(Y\). For every sample \(j \neq i\), calculate the pairwise distances in \(x\) and \(y\) and denote this as \(d_x(x_i,
x_j)\) and \(d_y(y_i, y_j)\). The indicator function is denoted as \(\mathbb{1} \{ \cdot \}\). The cross-classification between these two random variables can be calculated as
\[A_{11} = \sum_{k=1, k \neq i,j}^n \mathbb{1} \{ d_x(x_i, x_k) \leq d_x(x_i, x_j) \} \mathbb{1} \{ d_y(y_i, y_k) \leq d_y(y_i, y_j) \}\]
and \(A_{12}\), \(A_{21}\), and \(A_{22}\) are defined similarly. This is organized within the following table:
\(d_x(x_i, \cdot) \leq d_x(x_i, x_j)\) \(d_x(x_i, \cdot) \leq d_x(x_i, x_j)\)
\(d_x(x_i, \cdot) \leq d_x(x_i, x_j)\) \(A_{11} (i,j)\) \(A_{12} (i,j)\) \(A_{1 \cdot} (i,j)\)
\(d_x(x_i, \cdot) > d_x(x_i, x_j)\) \(A_{21} (i,j)\) \(A_{22} (i,j)\) \(A_{2 \cdot} (i,j)\)
\(A_{\cdot 1} (i,j)\) \(A_{\cdot 2} (i,j)\) \(n - 2\)
Here, \(A_{\cdot 1}\) and \(A_{\cdot 2}\) are the column sums, \(A_{1 \cdot}\) and \(A_{2 \cdot}\) are the row sums, and \(n - 2\) is the number of degrees of freedom. From this table, we can
calculate the Pearson's chi squared test statistic using,
\[S(i, j) = \frac{(n-2) (A_{12} A_{21} - A_{11} A_{22})^2} {A_{1 \cdot} A_{2 \cdot} A_{\cdot 1} A_{\cdot 2}}\]
and the HHG test statistic is then,
\[\mathrm{HHG}_n (x, y) = \sum_{i=1}^n \sum_{j=1, j \neq i}^n S(i, j)\]
The p-value returned is calculated using a permutation test using \(hyppo.tools.perm_test\).
The fast version of this test performs a multivariate independence test based on univariate test statistics 2. The univariate test statistic used is Hoeffding's independence test, derived as
follows 3:
Let \(x\) and \(y\) be \((n, p)\) samples of random variables \(X\) and \(Y\). A center point - the center of mass of points in 'X' and 'Y' - is chosen. For every sample \(i\), calculate the
distances from the center point in \(x\) and \(y\) and denote this as \(d_x(x_i)\) and \(d_y(y_i)\). This will create a 1D collection of distances for each sample group.
From these distances, we can calculate the Hoeffding's dependence score between the two groups - denoted as \(D\) - using,
\[ \begin{aligned}D &= \frac{(n-2) (n-3) D_{1} + D_{2} - 2(n-2) D_{3}} {n (n-1) (n-2) (n-3) (n-4)}\\D_{1} &= \sum_{i} (Q_{i}-1) (Q_{i}-2)\\D_{2} &= \sum_{i} (R_{i} - 1) (R_{i} - 2) (S_{i} - 1)
(S_{i} - 2)\\D_{3} &= \sum_{i} {R_{i} - 2} (S_{i} - 2) (Q_{i}-1)\end{aligned} \]
where \(R_{i}\) is the rank of \(x_{i}\), \(D_{i}\) is the rank of \(y_{i}\), \(Q_{i}\) is the bivariate rank = 1 plus the number of points with both x and y values less than the \(i\)-th point.
\(D\) is notably sensitive to ties and gets smaller the more pairs of variables with identical values. If there are no ties in the data,D ranges between -0.5 and 1, with 1 indicating complete
dependence. 3
The p-value returned is calculated using a permutation test using hyppo.tools.perm_test.
Ruth Heller, Yair Heller, and Malka Gorfine. A consistent multivariate test of association based on ranks of distances. Biometrika, 100(2):503–510, 2013.
Ruth Heller and Yair Heller. Multivariate tests of association based on univariate tests. In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information
Processing Systems, volume 29. Curran Associates, Inc., 2016. URL: https://proceedings.neurips.cc/paper/2016/file/7ef605fc8dba5425d6965fbd4c8fbe1f-Paper.pdf.
SAS. Hoeffding dependence coefficient. https://support.sas.com/documentation/cdl/en/procstat/63104/HTML/default/viewer.htm#procstat_corr_sect016.htm. Accessed: 2021-12-17.
Methods Summary
HHG.statistic(x, y) Helper function that calculates the HHG test statistic.
HHG.test(x, y[, reps, workers, auto, ...]) Calculates the HHG test statistic and p-value.
HHG.statistic(x, y)¶
Helper function that calculates the HHG test statistic.
x,y (ndarray of float) -- Input data matrices. x and y must have the same number of samples. That is, the shapes must be (n, p) and (n, q) where n is the number of samples and p and q are the
number of dimensions. Alternatively, x and y can be distance matrices, where the shapes must both be (n, n). For fast version, x and y can be 1D collections of distances from a chosen center
point, where the shapes must be (n,1) or (n-1,1) depending on choice of center point.
stat (float) -- The computed HHG statistic.
HHG.test(x, y, reps=1000, workers=1, auto=False, random_state=None)¶
Calculates the HHG test statistic and p-value.
☆ stat (float) -- The computed HHG statistic.
☆ pvalue (float) -- The computed HHG p-value.
>>> import numpy as np
>>> from hyppo.independence import HHG
>>> x = np.arange(7)
>>> y = x
>>> stat, pvalue = HHG().test(x, y)
>>> '%.1f, %.2f' % (stat, pvalue)
'160.0, 0.00'
In addition, the inputs can be distance matrices. Using this is the, same as before, except the compute_distance parameter must be set to None.
>>> import numpy as np
>>> from hyppo.independence import HHG
>>> x = np.ones((10, 10)) - np.identity(10)
>>> y = 2 * x
>>> hhg = HHG(compute_distance=None)
>>> stat, pvalue = hhg.test(x, y)
>>> '%.1f, %.2f' % (stat, pvalue)
'0.0, 1.00'
Examples using hyppo.independence.HHG¶
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8.9 Use Direct and Inverse Variation - Elementary Algebra 2e | OpenStax
By the end of this section, you will be able to:
• Solve direct variation problems
• Solve inverse variation problems
Before you get started, take this readiness quiz.
If you miss a problem, go back to the section listed and review the material.
When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss
direct variation and inverse variation in this section.
Solve Direct Variation Problems
Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation
Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if
one is the product of a constant and the other.
For any two variables x and y, y varies directly with x if
The constant k is called the constant of variation.
In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find
values of y for other values of x.
How to Solve Direct Variation Problems
If y varies directly with x and $y=20y=20$ when $x=8x=8$, find the equation that relates x and y.
If $yy$ varies directly as $xx$ and $y=3,whenx=10.y=3,whenx=10.$ find the equation that relates x and y.
If $yy$ varies directly as $xx$ and $y=12whenx=4y=12whenx=4$ find the equation that relates x and y.
We’ll list the steps below.
Solve direct variation problems.
1. Step 1. Write the formula for direct variation.
2. Step 2. Substitute the given values for the variables.
3. Step 3. Solve for the constant of variation.
4. Step 4. Write the equation that relates x and y.
Now we’ll solve a few applications of direct variation.
When Raoul runs on the treadmill at the gym, the number of calories, c, he burns varies directly with the number of minutes, m, he uses the treadmill. He burned 315 calories when he used the
treadmill for 18 minutes.
1. ⓐ Write the equation that relates c and m.
2. ⓑ How many calories would he burn if he ran on the treadmill for 25 minutes?
The number of calories, $cc$, varies directly with
the number of minutes, $mm$, on the treadmill,
and $c =315c=315$ when $m=18m=18$.
Write the formula for direct variation.
We will use $cc$ in place of $yy$ and $mm$ in place of $xx$.
Substitute the given values for the variables.
Solve for the constant of variation.
Write the equation that relates $cc$ and $mm$.
Substitute in the constant of variation.
Find $c c$ when $m =25m =25$.
Write the equation that relates $cc$ and$mm$.
Substitute the given value for $mm$.
Raoul would burn 437.5 calories if he used the
treadmill for 25 minutes.
The number of calories, c, burned varies directly with the amount of time, t, spent exercising. Arnold burned 312 calories in 65 minutes exercising.
1. ⓐ Write the equation that relates c and t.
2. ⓑ How many calories would he burn if he exercises for 90 minutes?
The distance a moving body travels, d, varies directly with time, t, it moves. A train travels 100 miles in 2 hours
ⓐ Write the equation that relates d and t. ⓑ How many miles would it travel in 5 hours?
In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do
in most applied problems.
The number of gallons of gas Eunice’s car uses varies directly with the number of miles she drives. Last week she drove 469.8 miles and used 14.5 gallons of gas.
1. ⓐ Write the equation that relates the number of gallons of gas used to the number of miles driven.
2. ⓑ How many gallons of gas would Eunice’s car use if she drove 1000 miles?
The number of gallons of gas varies directly with the number of miles driven.
First we will name the variables. Let $g=g=$ number of gallons of gas.
$m=m=$ number of miles driven
Write the formula for direct variation.
We will use $gg$ in place of $yy$ and $mm$ in place of $xx$.
Substitute the given values for the variables.
Solve for the constant of variation.
We will round to the nearest thousandth.
Write the equation that relates $gg$ and $mm$.
Substitute in the constant of variation.
$Findgwhenm=1000.Write the equation that relatesgandm.g=0.031mSubstitute the given value form.g=0.031(1000)Simplify.g=31Eunice’s car would use 31 gallons of gas if she drove it 1,000 miles.Findgwhenm
=1000.Write the equation that relatesgandm.g=0.031mSubstitute the given value form.g=0.031(1000)Simplify.g=31Eunice’s car would use 31 gallons of gas if she drove it 1,000 miles.$
Notice that in this example, the units on the constant of variation are gallons/mile. In everyday life, we usually talk about miles/gallon.
The distance that Brad travels varies directly with the time spent traveling. Brad travelled 660 miles in 12 hours,
1. ⓐ Write the equation that relates the number of miles travelled to the time.
2. ⓑ How many miles could Brad travel in 4 hours?
The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons.
1. ⓐ Write the equation that relates the weight to the volume.
2. ⓑ If a liquid has volume 13 gallons, what is its weight?
In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is $y=kx2y=kx2$. We solve these applications just as we did
the previous ones, by substituting the given values into the equation to solve for k.
The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 4” will support a maximum load of 75 pounds.
1. ⓐ Write the equation that relates the maximum load to the cross-section.
2. ⓑ What is the maximum load that can be supported by a beam with diagonal 8”?
The maximum load varies directly with the square of the diagonal of the cross-section.
Name the variables. Let $L=L=$ maximum load.
$c=c=$ the diagonal of the cross-section
Write the formula for direct variation, where $yy$ varies directly with the square of $xx$.
We will use $LL$ in place of $yy$ and $cc$ in place of $xx$.
Substitute the given values for the variables.
Solve for the constant of variation.
Write the equation that relates $LL$ and $cc$.
Substitute in the constant of variation.
$FindLwhenc=8.Write the equation that relatesLandc.L=4.6875c2Substitute the given value forc.L=4.6875(8)2Simplify.L=300A beam with diagonal 8” could supporta maximum load of 300 pounds.FindLwhenc=
8.Write the equation that relatesLandc.L=4.6875c2Substitute the given value forc.L=4.6875(8)2Simplify.L=300A beam with diagonal 8” could supporta maximum load of 300 pounds.$
The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds.
1. ⓐ Write the equation that relates the distance to the time.
2. ⓑ How far will an object fall in 4 seconds?
The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.
1. ⓐ Write the equation that relates the area to the radius.
2. ⓑ What is the area of a pizza with a radius of 9 inches?
Solve Inverse Variation Problems
Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is $y=kxy=kx$.
For any two variables x and y, y varies inversely with x if
The constant k is called the constant of variation.
The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is $1x1x$.
We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change
‘direct’ to ‘inverse’.
Solve inverse variation problems.
1. Step 1. Write the formula for inverse variation.
2. Step 2. Substitute the given values for the variables.
3. Step 3. Solve for the constant of variation.
4. Step 4. Write the equation that relates x and y.
If y varies inversely with $xx$ and $y=20y=20$ when $x=8x=8$, find the equation that relates x and y.
Write the formula for inverse variation.
Substitute the given values for the variables.
Solve for the constant of variation.
Write the equation that relates $xx$ and $yy$.
Substitute in the constant of variation.
If $pp$ varies inversely with $qq$ and $p=30p=30$ when $q=12q=12$ find the equation that relates $pp$ and $q.q.$
If $yy$ varies inversely with $xx$ and $y=8y=8$ when $x=2x=2$ find the equation that relates $xx$ and $yy$.
The fuel consumption (mpg) of a car varies inversely with its weight. A car that weighs 3100 pounds gets 26 mpg on the highway.
1. ⓐ Write the equation of variation.
2. ⓑ What would be the fuel consumption of a car that weighs 4030 pounds?
The fuel consumption varies inversely with the weight.
First we will name the variables. Let $f=f=$ fuel consumption.
$w=w=$ weight
Write the formula for inverse variation.
We will use $ff$ in place of $yy$ and $ww$ in place of $xx$.
Substitute the given values for the variables.
Solve for the constant of variation.
Write the equation that relates $ff$ and $ww$.
Substitute in the constant of variation.
Write the equation that relates f and w. $f=80,600wf=80,600w$
Substitute the given value for w. $f=80,6004030f=80,6004030$
Simplify. A car that weighs 4030 pounds would
have fuel consumption of 20 mpg.
A car’s value varies inversely with its age. Elena bought a two-year-old car for $20,000.
ⓐ Write the equation of variation. ⓑ What will be the value of Elena’s car when it is 5 years old?
The time required to empty a pool varies inversely as the rate of pumping. It took Lucy 2.5 hours to empty her pool using a pump that was rated at 400 gpm (gallons per minute).
1. ⓐ Write the equation of variation.
2. ⓑ How long will it take her to empty the pool using a pump rated at 500 gpm?
The frequency of a guitar string varies inversely with its length. A 26” long string has a frequency of 440 vibrations per second.
1. ⓐ Write the equation of variation.
2. ⓑ How many vibrations per second will there be if the string’s length is reduced to 20” by putting a finger on a fret?
The frequency varies inversely with the length.
Name the variables. Let $f=f=$ frequency.
$L=L=$ length
Write the formula for inverse variation.
We will use $ff$ in place of $yy$ and $LL$ in place of $xx$.
Substitute the given values for the variables.
Solve for the constant of variation.
Write the equation that relates $ff$ and $LL$.
Substitute in the constant of variation.
Write the equation that relates f and L. $f=11,440Lf=11,440L$
Substitute the given value for L. $f=11,44020f=11,44020$
Simplify. A 20” guitar string has frequency
572 vibrations per second.
The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees.
1. ⓐ Write the equation of variation.
2. ⓑ How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?
The force needed to break a board varies inversely with its length. Richard uses 24 pounds of pressure to break a 2-foot long board.
1. ⓐ Write the equation of variation.
2. ⓑ How many pounds of pressure is needed to break a 5-foot long board?
Section 8.9 Exercises
Practice Makes Perfect
Solve Direct Variation Problems
In the following exercises, solve.
If $yy$ varies directly as $xx$ and $y=14,whenx=3y=14,whenx=3$, find the equation that relates $xandyxandy$.
If $pp$ varies directly as $qq$ and $p=5,whenq=2p=5,whenq=2$, find the equation that relates $pandqpandq$.
If $vv$ varies directly as $ww$ and $v=24,whenw=8v=24,whenw=8$, find the equation that relates $vandw.vandw.$
If $aa$ varies directly as $bb$ and $a=16,whenb=4a=16,whenb=4$, find the equation that relates $aandb.aandb.$
If $pp$ varies directly as $qq$ and $p=9.6,whenq=3p=9.6,whenq=3$, find the equation that relates $pandq.pandq.$
If $yy$ varies directly as $xx$ and $y=12.4,whenx=4,y=12.4,whenx=4,$ find the equation that relates $xandyxandy$
If $aa$ varies directly as $bb$ and $a=6,whenb=13a=6,whenb=13$, find the equation that relates $aandb.aandb.$
If $vv$ varies directly as $ww$ and $v=8,whenw=12v=8,whenw=12$, find the equation that relates $vandw.vandw.$
The amount of money Sally earns, P, varies directly with the number, n, of necklaces she sells. When Sally sells 15 necklaces she earns $150.
1. ⓐ Write the equation that relates P and n.
2. ⓑ How much money would she earn if she sold 4 necklaces?
The price, P, that Eric pays for gas varies directly with the number of gallons, g, he buys. It costs him $50 to buy 20 gallons of gas.
1. ⓐ Write the equation that relates P and g.
2. ⓑ How much would 33 gallons cost Eric?
Terri needs to make some pies for a fundraiser. The number of apples, a, varies directly with number of pies, p. It takes nine apples to make two pies.
1. ⓐ Write the equation that relates a and p.
2. ⓑ How many apples would Terri need for six pies?
Joseph is traveling on a road trip. The distance, d, he travels before stopping for lunch varies directly with the speed, v, he travels. He can travel 120 miles at a speed of 60 mph.
1. ⓐ Write the equation that relates d and v.
2. ⓑ How far would he travel before stopping for lunch at a rate of 65 mph?
The price of gas that Jesse purchased varies directly to how many gallons he purchased. He purchased 10 gallons of gas for $39.80.
1. ⓐ Write the equation that relates the price to the number of gallons.
2. ⓑ How much will it cost Jesse for 15 gallons of gas?
The distance that Sarah travels varies directly to how long she drives. She travels 440 miles in 8 hours.
1. ⓐ Write the equation that relates the distance to the number of hours.
2. ⓑ How far can Sally travel in 6 hours?
The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.
1. ⓐ Write the equation that relates the mass to the volume.
2. ⓑ What is the volume of this liquid if its mass is 128 kilograms?
The length that a spring stretches varies directly with a weight placed at the end of the spring. When Sarah placed a 10 pound watermelon on a hanging scale, the spring stretched 5 inches.
1. ⓐ Write the equation that relates the length of the spring to the weight.
2. ⓑ What weight of watermelon would stretch the spring 6 inches?
The distance an object falls varies directly to the square of the time it falls. A ball falls 45 feet in 3 seconds.
1. ⓐ Write the equation that relates the distance to the time.
2. ⓑ How far will the ball fall in 7 seconds?
The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds.
1. ⓐ Write the equation that relates the load to the diagonal of the cross-section.
2. ⓑ What load will a beam with a 10 inch diagonal support?
The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.
1. ⓐ Write the equation that relates the area to the radius.
2. ⓑ What is the area of a personal pizza with a radius 4 inches?
The distance an object falls varies directly to the square of the time it falls. A ball falls 72 feet in 3 seconds,
1. ⓐ Write the equation that relates the distance to the time.
2. ⓑ How far will the ball have fallen in 8 seconds?
Solve Inverse Variation Problems
In the following exercises, solve.
If $yy$ varies inversely with $xx$ and $y=5y=5$ when $x=4x=4$ find the equation that relates $xx$ and $y.y.$
If $pp$ varies inversely with $qq$ and $p=2p=2$ when $q=1q=1$ find the equation that relates $pp$ and $q.q.$
If $vv$ varies inversely with $ww$ and $v=6v=6$ when $w=12w=12$ find the equation that relates $vv$ and $w.w.$
If $aa$ varies inversely with $bb$ and $a=12a=12$ when $b=13b=13$ find the equation that relates $aa$ and $b.b.$
Write an inverse variation equation to solve the following problems.
The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds and gets 33 mpg on the highway.
1. ⓐ Write the equation that relates the mpg to the car’s weight.
2. ⓑ What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?
A car’s value varies inversely with its age. Jackie bought a 10 year old car for $2,400.
1. ⓐ Write the equation that relates the car’s value to its age.
2. ⓑ What will be the value of Jackie’s car when it is 15 years old ?
The time required to empty a tank varies inversely as the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute),
1. ⓐ Write the equation that relates the number of hours to the pump rate.
2. ⓑ How long would it take Janet to pump her basement if she used a pump rated at 400 gpm?
The volume of a gas in a container varies inversely as pressure on the gas. A container of helium has a volume of 370 cubic inches under a pressure of 15 psi.
1. ⓐ Write the equation that relates the volume to the pressure.
2. ⓑ What would be the volume of this gas if the pressure was increased to 20 psi?
On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second.
1. ⓐ Write the equation that relates the string length to its frequency.
2. ⓑ What is the frequency of a 10-inch string?
Paul, a dentist, determined that the number of cavities that develops in his patient’s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had 4
cavities when brushing her teeth 30 seconds (0.5 minutes) each night.
1. ⓐ Write the equation that relates the number of cavities to the time spent brushing.
2. ⓑ How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?
The number of tickets for a sports fundraiser varies inversely to the price of each ticket. Brianna can buy 25 tickets at $5each.
1. ⓐ Write the equation that relates the number of tickets to the price of each ticket.
2. ⓑ How many tickets could Brianna buy if the price of each ticket was $2.50?
Boyle’s Law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a
pressure of 220 psi.
1. ⓐ Write the equation that relates pressure to volume.
2. ⓑ If the pressure increases to 330 psi, how much air can Braydon’s tank hold?
Mixed Practice
If $yy$ varies directly as $xx$ and $y=5,whenx=3.y=5,whenx=3.$, find the equation that relates $xandyxandy$.
If $vv$ varies directly as $ww$ and $v=21,whenw=8.v=21,whenw=8.$ find the equation that relates $vandw.vandw.$
If $pp$ varies inversely with $qq$ and $p=5p=5$ when $q=6q=6$, find the equation that relates $pp$ and $q.q.$
If $yy$ varies inversely with $xx$ and $y=11y=11$ when $x=3x=3$ find the equation that relates $xx$ and $y.y.$
If $pp$ varies directly as $qq$ and $p=10,whenq=2.p=10,whenq=2.$ find the equation that relates $pandqpandq$.
If $vv$ varies inversely with $ww$ and $v=18v=18$ when $w=13w=13$ find the equation that relates $vv$ and $w.w.$
The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot long board, how many pounds of pressure would he need to use to break a 6
foot long board?
The number of hours it takes for ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours when the temperature is 54 degrees. How long would it take for the same block
of ice to melt if the temperature was 45 degrees?
The length a spring stretches varies directly with a weight placed at the end of the spring. When Meredith placed a 6-pound cantaloupe on a hanging scale, the spring stretched 2 inches. How far would
the spring stretch if the cantaloupe weighed 9 pounds?
The amount that June gets paid varies directly the number of hours she works. When she worked 15 hours, she got paid $111. How much will she be paid for working 18 hours?
The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that
weighs 5,500 pounds? Round to the nearest tenth.
The volume of a gas in a container varies inversely as the pressure on the gas. If a container of argon has a volume of 336 cubic inches under a pressure of 2,500 psi, what will be its volume if the
pressure is decreased to 2,000 psi?
The distance an object falls varies directly to the square of the time it falls. If an object falls 52.8 feet in 4 seconds, how far will it fall in 9 seconds?
The area of the face of a Ferris wheel varies directly with the square of its radius. If the area of one face of a Ferris wheel with diameter 150 feet is 70,650 square feet, what is the area of one
face of a Ferris wheel with diameter of 16 feet?
Everyday Math
Ride Service It costs $35 for a ride from the city center to the airport, 14 miles away.
1. ⓐ Write the equation that relates the cost, c, with the number of miles, m.
2. ⓑ What would it cost to travel 22 miles with this service?
Road Trip The number of hours it takes Jack to drive from Boston to Bangor is inversely proportional to his average driving speed. When he drives at an average speed of 40 miles per hour, it takes
him 6 hours for the trip.
1. ⓐ Write the equation that relates the number of hours, h, with the speed, s.
2. ⓑ How long would the trip take if his average speed was 75 miles per hour?
Writing Exercises
In your own words, explain the difference between direct variation and inverse variation.
Make up an example from your life experience of inverse variation.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?
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QUBO solver limitations
Dear Sirs, I have a problem formulated as a QUBO and want to use GUROBI to try to optimize it.
I am completely new to GUROBI, so I want to ask you about the limitations of the solver as the size of the problem and maybe some "best practices" on how to use/tune solver for that type of problems.
Note, for the toy problem I have about 20k variables while the actual size can be up to 1,000,000 variables - so available memory is the only limitation?
Because I understand that even 100,000 x 100,000 matrix of float64 values in NumPy will consume 80,000,000,000 bytes of memory, which is approx 74.51 gigabytes.. and it's a rough estimate..
Appreciate for links and examples.
• Hi,
A QUBO is essentially an instance of binary quadratic programming and you can in theory solve an optimization model with Gurobi as long as the sum of the number of variables and constraints is
less than ~2 billions. Of course, this is just in theory and the actual size of a model that can be solved by Gurobi can be much smaller depending on the model structure.
You are correct that as the size of the model increases, the machine should have enough memory to store the model and carry the optimization.
Is your QUBO fully connected or sparse? Solving a fully connected QUBO with 1 million variables would be very challenging especially if the goal is to prove optimality.
You can experiment with parameters below when solving a QUBO:
1. NoRelHeurTime to force running the no-relaxation heuristic algorithm before solving the root relaxation with the aim of reaching a high-quality solution as quickly as possible.
2. PreQLinearize=1|2 to force the linearization of quadratic terms in the objective function and solve the problem as an instance of binary linear programming
If your problem is an instance of constrained optimization problem where the constraints are penalized in the objective function to transfer the model into a QUBO, we highly recommend using
Gurobi to solve the constrained problem directly.
Best regards,
If your problem is an instance of constrained optimization problem where the constraints are penalized in the objective function to transfer the model into a QUBO, we highly recommend using
Gurobi to solve the constrained problem directly.
What do you mean by "solve the constrained problem directly"? - without PreQLinearize?
What do you mean by "solve the constrained problem directly"? - without PreQLinearize?
The optimization problems are not naturally QUBOs because they are usually constrained. To reformulate the problem as a QUBO, the constraints are penalized in the objective function. My point was
that instead of reformulating the problem as a QUBO, solve the problem as an instance of constrained optimization with Gurobi. For example, consider the constrained optimization problem as \(\min
\{c^Tx| Ax = b\}\). To solve this problem as a QUBO, you need to reformulate it as \(\min\{c^Tx + P(Ax-b)^2\}\) with \(P\) being penalty multipliers. My point was to solve the problem as \(\min\
{c^Tx| Ax = b\}\) with Gurobi.
Best regards,
Please sign in to leave a comment.
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Data::Float - details of the floating point data type
use Data::Float qw(have_signed_zero);
if(have_signed_zero) { ...
# and many other constants; see text
use Data::Float qw(
float_class float_is_normal float_is_subnormal
float_is_nzfinite float_is_zero float_is_finite
float_is_infinite float_is_nan);
$class = float_class($value);
if(float_is_normal($value)) { ...
if(float_is_subnormal($value)) { ...
if(float_is_nzfinite($value)) { ...
if(float_is_zero($value)) { ...
if(float_is_finite($value)) { ...
if(float_is_infinite($value)) { ...
if(float_is_nan($value)) { ...
use Data::Float qw(float_sign signbit float_parts);
$sign = float_sign($value);
$sign_bit = signbit($value);
($sign, $exponent, $significand) = float_parts($value);
use Data::Float qw(float_hex hex_float);
print float_hex($value);
$value = hex_float($string);
use Data::Float qw(float_id_cmp totalorder);
@sorted_floats = sort { float_id_cmp($a, $b) } @floats;
if(totalorder($a, $b)) { ...
use Data::Float qw(
pow2 mult_pow2 copysign nextup nextdown nextafter);
$x = pow2($exp);
$x = mult_pow2($value, $exp);
$x = copysign($magnitude, $sign_from);
$x = nextup($x);
$x = nextdown($x);
$x = nextafter($x, $direction);
This module is about the native floating point numerical data type. A floating point number is one of the types of datum that can appear in the numeric part of a Perl scalar. This module supplies
constants describing the native floating point type, classification functions, and functions to manipulate floating point values at a low level.
Floating point values are divided into five subtypes:
The value is made up of a sign bit (making the value positive or negative), a significand, and exponent. The significand is a number in the range [1, 2), expressed as a binary fraction of a
certain fixed length. (Significands requiring a longer binary fraction, or lacking a terminating binary representation, cannot be obtained.) The exponent is an integer in a certain fixed range.
The magnitude of the value represented is the product of the significand and two to the power of the exponent.
The value is made up of a sign bit, significand, and exponent, as for normalised values. However, the exponent is fixed at the minimum possible for a normalised value, and the significand is in
the range (0, 1). The length of the significand is the same as for normalised values. This is essentially a fixed-point format, used to provide gradual underflow. Not all floating point formats
support this subtype. Where it is not supported, underflow is sudden, and the difference between two minimum-exponent normalised values cannot be exactly represented.
Depending on the floating point type, there may be either one or two zero values: zeroes may carry a sign bit. Where zeroes are signed, it is primarily in order to indicate the direction from
which a value underflowed (was rounded) to zero. Positive and negative zero compare as numerically equal, and they give identical results in most arithmetic operations. They are on opposite sides
of some branch cuts in complex arithmetic.
Some floating point formats include special infinite values. These are generated by overflow, and by some arithmetic cases that mathematically generate infinities. There are two infinite values:
positive infinity and negative infinity.
Perl does not always generate infinite values when normal floating point behaviour calls for it. For example, the division 1.0/0.0 causes an exception rather than returning an infinity.
This type of value exists in some floating point formats to indicate error conditions. Mathematically undefined operations may generate NaNs, and NaNs propagate through all arithmetic operations.
A NaN has the distinctive property of comparing numerically unequal to all floating point values, including itself.
Perl does not always generate NaNs when normal floating point behaviour calls for it. For example, the division 0.0/0.0 causes an exception rather than returning a NaN.
Perl has only (at most) one NaN value, even if the underlying system supports different NaNs. (IEEE 754 arithmetic has NaNs which carry a quiet/signal bit, a sign bit (yes, a sign on a
not-number), and many bits of implementation-defined data.)
Mixing floating point and integer values
Perl does not draw a strong type distinction between native integer (see Data::Integer) and native floating point values. Both types of value can be stored in the numeric part of a plain (string)
scalar. No distinction is made between the integer representation and the floating point representation where they encode identical values. Thus, for floating point arithmetic, native integer values
that can be represented exactly in floating point may be freely used as floating point values.
Native integer arithmetic has exactly one zero value, which has no sign. If the floating point type does not have signed zeroes then the floating point and integer zeroes are exactly equivalent. If
the floating point type does have signed zeroes then the integer zero can still be used in floating point arithmetic, and it behaves as an unsigned floating point zero. On such systems there are
therefore three types of zero available. There is a bug in Perl which sometimes causes floating point zeroes to change into integer zeroes; see "BUGS" for details.
Where a native integer value is used that is too large to exactly represent in floating point, it will be rounded as necessary to a floating point value. This rounding will occur whenever an
operation has to be performed in floating point because the result could not be exactly represented as an integer. This may be confusing to functions that expect a floating point argument.
Similarly, some operations on floating point numbers will actually be performed in integer arithmetic, and may result in values that cannot be exactly represented in floating point. This happens
whenever the arguments have integer values that fit into the native integer type and the mathematical result can be exactly represented as a native integer. This may be confusing in cases where
floating point semantics are expected.
See perlnumber(1) for discussion of Perl's numeric semantics.
Truth value indicating whether floating point zeroes carry a sign. If yes, then there are two floating point zero values: +0.0 and -0.0. (Perl scalars can nevertheless also hold an integer zero,
which is unsigned.) If no, then there is only one zero value, which is unsigned.
Truth value indicating whether there are subnormal floating point values.
Truth value indicating whether there are infinite floating point values.
Truth value indicating whether there are NaN floating point values.
It is difficult to reliably generate a NaN in Perl, so in some unlikely circumstances it is possible that there might be NaNs that this module failed to detect. In that case this constant would
be false but a NaN might still turn up somewhere. What this constant reliably indicates is the availability of the nan constant below.
The number of fractional bits in the significand of finite floating point values. The significand also has an implicit integer bit, not counted in this constant; the integer bit is always 1 for
normalised values and always 0 for subnormal values.
The difference between adjacent representable values in the range [1, 2] (where the exponent is zero). This is equal to 2^-significand_bits.
The maximum exponent permitted for finite floating point values.
The maximum representable power of two. This is 2^max_finite_exp.
The maximum representable finite value. This is 2^(max_finite_exp+1) - 2^(max_finite_exp-significand_bits).
The maximum representable number. This is positive infinity if there are infinite values, or max_finite if there are not.
The maximum integral value for which all integers from zero to that value inclusive are representable. Equivalently: the minimum positive integral value N for which the value N+1 is not
representable. This is 2^(significand_bits+1). The name is somewhat misleading.
The minimum exponent permitted for normalised floating point values.
The minimum positive value representable as a normalised floating point value. This is 2^min_normal_exp.
The base two logarithm of the minimum representable positive finite value. If there are subnormals then this is min_normal_exp - significand_bits. If there are no subnormals then this is
The minimum representable positive finite value. This is 2^min_finite_exp.
Special Values
The positive zero value. (Exists only if zeroes are signed, as indicated by the have_signed_zero constant.)
If Perl is at risk of transforming floating point zeroes into integer zeroes (see "BUGS"), then this is actually a non-constant function that always returns a fresh floating point zero. Thus the
return value is always a true floating point zero, regardless of what happened to zeroes previously returned.
The negative zero value. (Exists only if zeroes are signed, as indicated by the have_signed_zero constant.)
If Perl is at risk of transforming floating point zeroes into integer zeroes (see "BUGS"), then this is actually a non-constant function that always returns a fresh floating point zero. Thus the
return value is always a true floating point zero, regardless of what happened to zeroes previously returned.
The positive infinite value. (Exists only if there are infinite values, as indicated by the have_infinite constant.)
The negative infinite value. (Exists only if there are infinite values, as indicated by the have_infinite constant.)
Not-a-number. (Exists only if NaN values were detected, as indicated by the have_nan constant.)
Each "float_" function takes a floating point argument to operate on. The argument must be a native floating point value, or a native integer with a value that can be represented in floating point.
Giving a non-numeric argument will cause mayhem. See "is_number" in Params::Classify for a way to check for numericness. Only the numeric value of the scalar is used; the string value is completely
ignored, so dualvars are not a problem.
Each "float_is_" function returns a simple truth value result.
Determines which of the five classes described above VALUE falls into. Returns "NORMAL", "SUBNORMAL", "ZERO", "INFINITE", or "NAN" accordingly.
Returns true iff VALUE is a normalised floating point value.
Returns true iff VALUE is a subnormal floating point value.
Returns true iff VALUE is a non-zero finite value (either normal or subnormal; not zero, infinite, or NaN).
Returns true iff VALUE is a zero. If zeroes are signed then the sign is irrelevant.
Returns true iff VALUE is a finite value (either normal, subnormal, or zero; not infinite or NaN).
Returns true iff VALUE is an infinity (either positive infinity or negative infinity).
Returns true iff VALUE is a NaN.
Returns "+" or "-" to indicate the sign of VALUE. An unsigned zero returns the sign "+". dies if VALUE is a NaN.
VALUE must be a floating point value. Returns the sign bit of VALUE: 0 if VALUE is positive or a positive or unsigned zero, or 1 if VALUE is negative or a negative zero. Returns an unpredictable
value if VALUE is a NaN.
This is an IEEE 754 standard function. According to the standard NaNs have a well-behaved sign bit, but Perl can't see that bit.
Divides up a non-zero finite floating point value into sign, exponent, and significand, returning these as a three-element list in that order. The significand is returned as a floating point
value, in the range [1, 2) for normalised values, and in the range (0, 1) for subnormals. dies if VALUE is not finite and non-zero.
String conversion
Encodes the exact value of VALUE as a hexadecimal fraction, returning the fraction as a string. Specifically, for finite values the output is of the form "s0xm.mmmmmpeee", where "s" is the sign,
"m.mmmm" is the significand in hexadecimal, and "eee" is the exponent in decimal with a sign.
The details of the output format are very configurable. If OPTIONS is supplied, it must be a reference to a hash, in which these keys may be present:
The number of digits of exponent to show, unless this is modified by exp_digits_range_mod or more are required to show the exponent exactly. (The exponent is always shown in full.) Default 0,
so the minimum possible number of digits is used.
Modifies the number of exponent digits to show, based on the number of digits required to show the full range of exponents for normalised and subnormal values. If "IGNORE" then nothing is
done. If "ATLEAST" then at least this many digits are shown. Default "IGNORE".
The string that is prepended to a negative exponent. Default "-".
The string that is prepended to a non-negative exponent. Default "+". Make it the empty string to suppress the positive sign.
The number of fractional digits to show, unless this is modified by frac_digits_bits_mod or frac_digits_value_mod. Default 0, but by default this gets modified.
Modifies the number of fractional digits to show, based on the length of the significand. There is a certain number of digits that is the minimum required to explicitly state every bit that
is stored, and the number of digits to show might get set to that number depending on this option. If "IGNORE" then nothing is done. If "ATLEAST" then at least this many digits are shown. If
"ATMOST" then at most this many digits are shown. If "EXACTLY" then exactly this many digits are shown. Default "ATLEAST".
Modifies the number of fractional digits to show, based on the number of digits required to show the actual value exactly. Works the same way as frac_digits_bits_mod. Default "ATLEAST".
The string that is prefixed to hexadecimal digits. Default "0x". Make it the empty string to suppress the prefix.
The string that is returned for an infinite magnitude. Default "inf".
The string that is returned for a NaN value. Default "nan".
The string that is prepended to a negative value (including negative zero). Default "-".
The string that is prepended to a positive value (including positive or unsigned zero). Default "+". Make it the empty string to suppress the positive sign.
The manner in which subnormal values are displayed. If "SUBNORMAL", they are shown with the minimum exponent for normalised values and a significand in the range (0, 1). This matches how they
are stored internally. If "NORMAL", they are shown with a significand in the range [1, 2) and a lower exponent, as if they were normalised. This gives a consistent appearance for magnitudes
regardless of normalisation. Default "SUBNORMAL".
The manner in which zero values are displayed. If "STRING=str", the string str is used, preceded by a sign. If "SUBNORMAL", it is shown with significand zero and the minimum normalised
exponent. If "EXPONENT=exp", it is shown with significand zero and exponent exp. Default "STRING=0.0". An unsigned zero is treated as having a positive sign.
Generates and returns a floating point value from a string encoding it in hexadecimal. The standard input form is "[s][0x]m[.mmmmm][peee]", where "s" is the sign, "m[.mmmm]" is a (fractional)
hexadecimal number, and "eee" an optionally-signed exponent in decimal. If present, the exponent identifies a power of two (not sixteen) by which the given fraction will be multiplied.
If the value given in the string cannot be exactly represented in the floating point type because it has too many fraction bits, the nearest representable value is returned, with ties broken in
favour of the value with a zero low-order bit. If the value given is too large to exactly represent then an infinity is returned, or the largest finite value if there are no infinities.
Additional input formats are accepted for special values. "[s]inf[inity]" returns an infinity, or dies if there are no infinities. "[s][s]nan" returns a NaN, or dies if there are no NaNs
All input formats are understood case insensitively. The function correctly interprets all possible outputs from float_hex with default settings.
This is a comparison function supplying a total ordering of floating point values. A and B must both be floating point values. Returns -1, 0, or +1, indicating whether A is to be sorted before,
the same as, or after B.
The ordering is of the identities of floating point values, not their numerical values. If zeroes are signed, then the two types are considered to be distinct. NaNs compare equal to each other,
but different from all numeric values. The exact ordering provided is mostly numerical order: NaNs come first, followed by negative infinity, then negative finite values, then negative zero, then
positive (or unsigned) zero, then positive finite values, then positive infinity.
In addition to sorting, this function can be useful to check for a zero of a particular sign.
This is a comparison function supplying a total ordering of floating point values. A and B must both be floating point values. Returns a truth value indicating whether A is to be sorted
before-or-the-same-as B. That is, it is a <= predicate on the total ordering. The ordering is the same as that provided by float_id_cmp: NaNs come first, followed by negative infinity, then
negative finite values, then negative zero, then positive (or unsigned) zero, then positive finite values, then positive infinity.
This is an IEEE 754r standard function. According to the standard it is meant to distinguish different kinds of NaNs, based on their sign bit, quietness, and payload, but this function (like the
rest of Perl) perceives only one NaN.
EXP must be an integer. Returns the value two the the power EXP. dies if that value cannot be represented exactly as a floating point value. The return value may be either normalised or
EXP must be an integer, and VALUE a floating point value. Multiplies VALUE by two to the power EXP. This gives exact results, except in cases of underflow and overflow. The range of EXP is not
constrained. All normal floating point multiplication behaviour applies.
VALUE and SIGN_FROM must both be floating point values. Returns a floating point value with the magnitude of VALUE and the sign of SIGN_FROM. If SIGN_FROM is an unsigned zero then it is treated
as positive. If VALUE is an unsigned zero then it is returned unchanged. If VALUE is a NaN then it is returned unchanged. If SIGN_FROM is a NaN then the sign copied to VALUE is unpredictable.
This is an IEEE 754 standard function. According to the standard NaNs have a well-behaved sign bit, which can be read and modified by this function, but Perl only perceives one NaN and can't see
its sign bit, so behaviour on NaNs is not standard-conforming.
VALUE must be a floating point value. Returns the next representable floating point value adjacent to VALUE with a numerical value that is strictly greater than VALUE, or returns VALUE unchanged
if there is no such value. Infinite values are regarded as being adjacent to the largest representable finite values. Zero counts as one value, even if it is signed, and it is adjacent to the
smallest representable positive and negative finite values. If a zero is returned, because VALUE is the smallest representable negative value, and zeroes are signed, it is a negative zero that is
returned. Returns NaN if VALUE is a NaN.
This is an IEEE 754r standard function.
VALUE must be a floating point value. Returns the next representable floating point value adjacent to VALUE with a numerical value that is strictly less than VALUE, or returns VALUE unchanged if
there is no such value. Infinite values are regarded as being adjacent to the largest representable finite values. Zero counts as one value, even if it is signed, and it is adjacent to the
smallest representable positive and negative finite values. If a zero is returned, because VALUE is the smallest representable positive value, and zeroes are signed, it is a positive zero that is
returned. Returns NaN if VALUE is a NaN.
This is an IEEE 754r standard function.
VALUE and DIRECTION must both be floating point values. Returns the next representable floating point value adjacent to VALUE in the direction of DIRECTION, or returns DIRECTION if it is
numerically equal to VALUE. Infinite values are regarded as being adjacent to the largest representable finite values. Zero counts as one value, even if it is signed, and it is adjacent to the
positive and negative smallest representable finite values. If a zero is returned and zeroes are signed then it has the same sign as VALUE. Returns NaN if either argument is a NaN.
This is an IEEE 754 standard function.
As of Perl 5.8.7 floating point zeroes will be partially transformed into integer zeroes if used in almost any arithmetic, including numerical comparisons. Such a transformed zero appears as a
floating point zero (with its original sign) for some purposes, but behaves as an integer zero for other purposes. Where this happens to a positive zero the result is indistinguishable from a true
integer zero. Where it happens to a negative zero the result is a fourth type of zero, the existence of which is a bug in Perl. This fourth type of zero will give confusing results, and in particular
will elicit inconsistent behaviour from the functions in this module.
Because of this transforming behaviour, it is best to avoid relying on the sign of zeroes. If you require signed-zero semantics then take special care to maintain signedness. Avoid using a zero
directly in arithmetic and handle it as a special case. Any flavour of zero can be accurately copied from one scalar to another without affecting the original. The functions in this module all avoid
modifying their arguments, and where they are meant to return signed zeroes they always return a pristine one.
As of Perl 5.8.7 stringification of a floating point zero does not preserve its signedness. The number-to-string-to-number round trip turns a positive floating point zero into an integer zero, but
accurately maintains negative and integer zeroes. If a negative zero gets partially transformed into an integer zero, as described above, the stringification that it gets is based on its state at the
first occasion on which the scalar was stringified.
NaN handling is generally not well defined in Perl. Arithmetic with a mathematically undefined result may either die or generate a NaN. Avoid relying on any particular behaviour for such operations,
even if your hardware's behaviour is known.
As of Perl 5.8.7 the % operator truncates its arguments to integers, if the divisor is within the range of the native integer type. It therefore operates correctly on non-integer values only when the
divisor is very large.
Data::Integer, Scalar::Number, perlnumber(1)
Andrew Main (Zefram) <zefram@fysh.org>
Copyright (C) 2006, 2007, 2008, 2010, 2012, 2017 Andrew Main (Zefram) <zefram@fysh.org>
This module is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
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Applying Artificial Neural Networks (ANNs) for Linear Regression: Yay or Nay?
Applying Artificial Neural Networks (ANNs) for Linear Regression: Yay or Nay?
Did you know that the term “Regression” was first coined by ‘Francis Galton’ in the 19^th Century for describing a biological phenomenon?
Regression models have been around for decades.
They have turned out to be extremely useful for providing valuable predictions and modeling real-world problems.
Linear regression, one of the most common and simplest regression models, is useful for determining the relationship between one or more independent variables and a dependent variable.
(Watch our Youtube video on Linear regression by Dr. Ry and get to know all necessary basics).
Whereas, on the other side, artificial neural networks, also referred to as connectionist systems, is a computational model that’s capable of learning new things and making decisions, just like
It’s a component of Artificial Intelligence and has been designed for stimulating the functioning of the human brain.
Regular computers are programmed to behave as if they are interconnected brain cells in order to achieve the goal.
That’s what brings us to our next question,
“Why Should Artificial Neural Networks Be Used For Linear Regression?”
And that’s what we will be taking a look at here.
In this blog post, not only will we take a brief look at each of the concepts, but we will even shift our focus towards the benefits of using artificial neural network for regression as well as some
real-life examples.
Well, then!
Let’s dive in.
Fundamentals of Artificial Neural Networks
As we took a look above, an artificial neural network is a piece of a computing system that’s designed for stimulating the manner in which a human brain analyses as well as processes information.
Similar to a human brain, artificial neural networks (ANNs) are built with interconnected nodes, just like a web.
As you might already know, a single human brain consists of innumerable cells referred to as neurons.
Each neuron, made up of a cell body, processes information by carrying it away and towards your brain.
Also called as processing units, an ANN comprises of hundreds or thousands of artificial neurons. Each of these processing units is made up of input as well as output units.
The input ones are subject to receiving different structures and forms of information that are based on an internal weighting system.
In order to produce an output report, the neural network tries to learn about the information.
The set of learning rules followed by ANNs is called as backpropagation.
In the training phase, the ANN tries to recognize the different patterns in the data fed to it. In the supervised phase, the ANN compares the actual output with the desired one in order to detect any
differences. These differences are adjusted, as a result of backpropagation.
What this means is that the network operates backward. During both of these stages, the ANN is trained on what exactly it should be looking for as well as what the output looks like. And that’s what
makes the network intelligent, which we also call “Deep Learning.”
What Is Linear Regression Exactly?
Linear regression is a commonly-used and basic kind of predictive analysis.
It determines the relationship between one or more independent variables and a dependent variable by simply fitting a linear equation to the observed data.
In simpler terms, this model can help you predict the dependent variable, from one or more independent variables.
Based on your income, a linear regression model can help you predict your expenses. This model can even help you predict Facebook’s revenue, based on its userbase.
Believe me, when I tell you, it’s the simplest model out there that you can easily try out on any data.
Let’s take a look at how you can apply the linear regression model to your own set of data:
• Plot your dependent variable on Y-Axis, against your independent variable on X-Axis
• Plot a straight line
• Measure correlation
• Change the straight line’s direction, until you get the perfect correlation
• Find out the new values on Y-Axis, based on this line
Here’s its equation:
• Where Y is a dependent variable
• X is an independent variable
• m is the slope of the line
• C is the intercept
Let’s take into consideration a simple example.
For each value of x, there’s a response y.
x: Feature vector
y: Response vector
Number of observations (n): 10
Here’s what the scatter plot for the above dataset looks like:
Now, we need to find a line that fits the above scatter-plot perfectly. This line is called the “Regression line.”
The equation for this line is:
In order to create the model, we need to estimate or learn the values of m and C. Once we do, we will be able to predict a response.
Here, we will be using the “Least Squares Technique.”
Listed below are the values we calculated:
Scatter plot:
Sounds amazing, right?
But the downside to applying the linear regression model is that it’s limited to linear relationships.
This model simply assumes that there’s a linear relationship between the dependent and independent variables. It doesn’t take into consideration other factors.
For instance, the relationship between age and income is curved. During the beginning of your career, you might find your income to below. As you gain experience, your income will begin rising and
soon, it will flatten out.
And if we predict your income after you retire, according to the linear regression model, it should be at an all-time high. But that’s not the case. People usually witness a major decline in the
income, upon retirement. And that’s the reason you can’t reply to linear regression, in every scenario.
Remember ANNs?
What if I told you that you can use artificial neural network for linear regression?
That’s correct.
Let’s take a look.
Benefits Of Using ANN For Linear Regression
Let’s dive into neural network linear regression basics.
Neural networks can be reduced to regression models.
Well, not exactly “reduced.” But, a neural network can easily “pretend” to act as any kind of regression model.
Let’s take a look at why you should use ANN for linear regression.
Ability To Learn & Model Linear & Non-Linear Relationships
One of the downsides of the Linear regression model that we took a look at above is that it assumes that there’s always a linear relationship between the dependent as well as independent variables.
What if it’s curved?
What if this model showed you a huge boost in your income after you retire? That’d be pretty disappointing, right?
And one of the biggest benefits of using ANNs for linear regression is that it takes into account the real-life relationships as well.
It’s able to adapt to linear as well as non-linear models, depending on how it has been trained.
It’s a flexible model that can easily adapt to the shape of the data. And if the results aren’t accurate enough, you can easily add further hidden neuron layers in order to improve its prediction
capabilities and turn it more complex.
What if one or multiple cells of the ANN have been corrupted?
Will it stop working entirely?
That’d be a disaster, right?
But the thing is an artificial neural network is fault-tolerant. No matter what the situation is, it will always provide you with output.
Will Perform Gradient Descent
ANNs will never stop learning. After the training process, ANNs will be performing gradient descent in order to find even better-fitting coefficients that fit the data accurately.
“Gradient descent is a mathematical technique that plays a role in modifying a function’s parameters to decline from a higher-value to a lower value of a function. It does so by analyzing the
derivatives of a function, on the basis of each of the parameters, and by learning about the next best step for minimizing the function. Applying it to the error function will result in the discovery
of the weights with lower error values, thereby playing a role in turning the entire model perfect.”
ANNs will continue doing so until the model arrives at the most favorable linear regression coefficients.
Let’s learn how you can apply ANNs to Linear Regression problems.
3 Examples Of How To Apply Artificial Neural Network For Linear Regression
Listed below are the neural network regression examples we’ll be taking a look at:
• Linear Regression Using Tensorflow
• Linear Regression Using PyTorch
• Linear Regression Using Theano
Linear Regression Using Tensorflow
Let’s look at Tensorflow linear regression example. You need to complete a few steps.
• Import all the 3 libraries.
• For making the random numbers unpredictable, we’ll be defining fixed seeds for Tensorflow as well as Numpy.
• The above-mentioned steps will help you generate random data.
• Now, we will be visualizing the training data.
• Define the placeholders for x & y
• Define 2 trainable Tensorflow variables for bias and weights. np.random.randn() will initialize them randomly.
• Define model’s hyperparameters: Number of Epochs & the Learning Rate
• Build the cost optimizer, cost function, and the hypothesis. We are not going to implement the Gradient Descent Optimizer manually. That’s because it’s in-built. Now, it’s time to initialize the
• Let’s begin with training the model.
• Let us analyze the result of linear regression using Tensorflow
• Remember that both bias and weight are scalars. Let’s plot the result.
Ladies & Gentlemen, that’s how linear regression at Tensorflow works.
Now, let’s take at another example using PyTorch.
Linear Regression Using PyTorch
This time let’s see a linear regression PyTorch example.
• First of all, if you don’t have PyTorch installed, it’s time you headed over to pytorch.org.
• Once you do, begin by importing the required objects and library functions
• Define some data. Assign them to x_data and y_data variables
• x_data -> Independent Variable & y_data -> dependent variable. Now, it’s time to define the model. Listed below are the two most vital steps:
□ Initializing the model
□ Declaring forward pass
• Here’s the class we’ll be using:
• The model class is the subclass of torch.nn.module. We’ve 1 input as well as 1 output. Let’s now create an object.
• Now, select the loss criteria and optimizer. We’ll be using Mean Squared Error as loss function & stochastic gradient descent as the optimizer. Let’s fix the learning rate as 0.01
• Now, it’s time to train the model
• Once the training phase is completed successfully, we’ll be testing whether the results are correct or not.
• If all of the above-mentioned steps were correctly performed, you’d get the value somewhere close to 8 for input 4.0. Here’s what we got:
Ladies & Gentlemen, that’s how you can use PyTorch linear regression.
Linear Regression Using Theano
The last example is neural networks for regression using Theano.
• Import the mandatory library functions and the training data
• Now, we will randomly initialize the coefficients, i.e. m & c as shared variables. That’s because they will be extensively used during the entire process.
• Now let’s define x and y as symbolic variables. These differ from the ones we are used to in Python. These help Theano achieve outstanding computation speeds.
• Define cost function: It’s the cumulative error that will be predicted by the model for the current coefficients. During the entire process, the model will be trying to minimize the cost function
to better fit the training data. Here, we will be considering cumulative MSE of the predicted value as well as the true value as a cost parameter
• Now, we will be multiplying the input values by slope m and adding intercept c in order to get the predicted value. Once you calculate the cost factor, it’s time, to begin with, the optimization
process. Here the learning rate will be 0.01 and the training frequency will be 10000.
• Here’s the final predicted model that you’ll get:
Congratulations! You have successfully uncovered the secret of using ANNs for linear regression. Also you’ve got familiar with neural network regression examples. Good job!
In order to run neural network for regression, you will have to utilize one of the frameworks we mentioned above. There are various other. But, these 3 are my personal favorite. I’ve rarely seen a
regression equation perfectly fitting all of the expected datasets.
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Math Symbols For Number Sets: Explanations!
Math Symbols for Number Sets: Explanations!
Mathematical symbols for number sets are essential tools for identifying and categorizing various groups of numbers.
They encompass a wide range of numerical categories from the simplest natural numbers to the more complex real and imaginary numbers. Recognizing these symbols is key to understanding and working
with different types of numbers in mathematics.
Number sets in mathematics have specific symbols that signify the entire set of numbers within that category.
For instance:
Natural Numbers (ℕ): These include all positive integers starting from 1. Example: 1, 2, 3…
Integers (ℤ): This set includes all whole numbers and their negative counterparts. Example: -3, -2, -1, 0, 1, 2, 3…
Whole Numbers: Natural numbers plus zero.
Rational Numbers (ℚ): Numbers that can be expressed as a fraction of two integers. Example: 1/2, 3/4, 5…
Real Numbers (ℝ): All numbers on the number line, including both rational and irrational numbers.
Complex Numbers (ℂ): Numbers that have a real and an imaginary part. Example: 3 + 4i, where i is the imaginary unit.
Imaginary Numbers: Numbers that are multiples of the imaginary unit i, where i^2 = -1.
Recognizing number set symbols is a fundamental skill in mathematics, aiding in the precise expression of mathematical ideas.
Key Takeaway
The symbol ℝ represents the set of real numbers.
The symbol ℂ represents the set of complex numbers.
The symbol ℤ represents the set of integers.
The symbol ℚ represents the set of rational numbers.
Natural Numbers
The set of natural numbers, denoted by the symbol N, encompasses all positive integers starting from 1 and extending to infinity. Natural numbers are used to count objects and are fundamental in
various mathematical concepts.
They form the building blocks for other number sets and mathematical operations. Natural numbers cannot be negative or fractions, and they are represented as {1, 2, 3, 4, …}. In set notation, N = {1,
2, 3, …}.
Understanding natural numbers is crucial for comprehending the basic principles of arithmetic and number theory.
The concept of integers encompasses negative whole numbers, including zero, and positive whole numbers. Integers do not include fractional or decimal numbers.
Understanding the properties and operations of integers is fundamental in various mathematical disciplines, including algebra and number theory.
Negative Whole Numbers
Negative whole numbers, also known as integers, are an important part of the number set in mathematics. They are represented as the set of numbers {…, -3, -2, -1}. Integers include all the natural
numbers (1, 2, 3, …) as well as their additive inverses (−1, −2, −3, …).
They are crucial in various mathematical operations and real-life applications, such as in accounting (for representing debts), temperature measurements (below zero), and in many other contexts.
Integers are essential in various mathematical operations and real-life applications. Understanding and working with negative whole numbers are fundamental in mathematics and other fields where
quantitative analysis is required.
No Fractional or Decimal
Fractions and decimals are not included in the set of integers, which solely consists of whole numbers and their additive inverses.
• Understanding the exclusion of fractions and decimals may feel limiting, but it provides a clear and distinct set of numbers to work with.
• While it may seem restrictive, the concept of integers allows for focused study and problem-solving.
• Recognizing the boundary between integers and other number sets can evoke a sense of clarity and precision in mathematical operations.
• The absence of fractions and decimals in the set of integers can evoke a sense of purity and simplicity, making it easier to comprehend and work with these numbers.
This distinction sets the stage for the subsequent section about ‘rational numbers’, where we will explore the inclusion of fractions and decimals within the number set framework.
Rational Numbers
Rational numbers can be expressed as the quotient of two integers. They can be written in the form p/q, where p and q are integers and q is not equal to 0. Rational numbers include fractions and
integers since every integer can be expressed as a fraction with a denominator of 1.
Rational numbers can be positive, negative, or zero. They can also be recurring or terminating decimals.
When represented on a number line, rational numbers occupy specific, finite positions. The set of rational numbers is denoted by the symbol Q.
Understanding rational numbers is fundamental in various mathematical concepts, including algebra, calculus, and number theory.
Their properties and operations play a crucial role in solving practical problems and theoretical proofs in diverse fields of mathematics.
Real Numbers
Real numbers encompass a wide range of mathematical entities, including both rational and irrational numbers. They are characterized by their ability to be represented on the number line as points,
each corresponding to a unique position.
The properties of real numbers play a fundamental role in various mathematical operations and provide a foundation for advanced mathematical concepts.
Real Numbers Definition
Real numbers are defined as the set of all numbers that can be represented on the number line. This includes both rational numbers, which can be expressed as a fraction, and irrational numbers, which
cannot be expressed as a fraction and have non-repeating decimal expansions.
The concept of real numbers encompasses a wide range of values, from negative infinity to positive infinity, and includes familiar numbers such as integers, fractions, and decimals.
Understanding the nature of real numbers is fundamental in various mathematical fields, such as calculus, analysis, and geometry.
It provides a foundation for comprehending the continuous nature of quantities and the infinite precision required in many scientific and mathematical applications.
Real numbers form the basis for modeling and understanding the physical world, making them a cornerstone of mathematical study.
Properties of Real Numbers
The properties of numbers on the number line provide fundamental principles for mathematical operations and relationships. Real numbers exhibit several key properties, including closure,
commutativity, associativity, distributivity, identity, and inverses.
The closure property states that the sum or product of any two real numbers is also a real number. Commutativity indicates that the order of addition or multiplication does not affect the result.
Associativity implies that the grouping of numbers in addition or multiplication does not change the outcome. Distributivity governs the relationship between addition and multiplication.
The existence of additive and multiplicative identities means that for any real number, there exists an element such that the sum or product with the identity yields the original number.
Furthermore, every real number has an additive and multiplicative inverse, except for zero. Understanding these properties is essential for comprehensive mathematical reasoning and problem-solving.
Real Numbers Examples
Examples of real numbers can be found in various mathematical contexts, illustrating the practical application of the properties discussed earlier.
Real numbers include familiar numbers like 3, 0.5, and -1, as well as less obvious ones like √2 and π.
These numbers are used in everyday measurements, such as the length of a bookshelf (3 feet), the weight of an apple (0.5 pounds), or the temperature outside (-1 degree Celsius).
Real numbers also appear in more complex mathematical concepts, such as the distance between two points on a coordinate plane or the solutions to a quadratic equation.
Understanding real numbers is essential for grasping the fundamental principles of mathematics, providing a foundation for solving real-world problems and unlocking the mysteries of the universe.
Complex Numbers
Complex numbers are a fundamental concept in mathematics, representing numbers in the form a + bi, where ‘a’ and ‘b’ are real numbers and ‘i’ is the imaginary unit.
One way to represent complex numbers is through a table, as shown below:
Type Representation Example
Real a + 0i 5 + 0i
Imaginary 0 + bi 0 + 3i
Complex a + bi 2 + 3i
Conjugate a – bi 2 – 3i
complex numbers is through a table
In this table, ‘a’ and ‘b’ stand for real numbers, and ‘i’ represents the imaginary unit. The table illustrates the different types of complex numbers and provides examples for each type, offering a
comprehensive understanding of their representations.
Imaginary Numbers
Discussing the representation of imaginary numbers involves understanding their role in mathematical operations and applications.
Imaginary numbers, denoted as a multiple of the imaginary unit “i,” are essential in various mathematical contexts, including electrical engineering, quantum mechanics, and signal processing.
To evoke emotion in the audience, consider the following points:
• Imaginary numbers can initially seem perplexing but offer a fascinating insight into the depth and beauty of mathematical abstraction.
• The concept of imaginary numbers challenges conventional understanding, igniting a sense of curiosity and wonder about the boundless nature of mathematics.
• Despite initial skepticism, grasping the significance of imaginary numbers can evoke a sense of intellectual achievement and appreciation for the elegance of mathematical concepts.
Understanding imaginary numbers enriches one’s perception of the interconnectedness and intricacy of mathematical systems, fostering a profound appreciation for the discipline’s complexity.
The mathematical symbols for number sets serve as the foundation for understanding the abstract and interconnected nature of mathematics.
Each set represents a unique realm of numbers, with its own properties and relationships. Just as different ecosystems in nature support diverse forms of life, these number sets provide the structure
and framework for mathematical concepts to thrive and evolve.
Understanding and mastering these symbols is essential for navigating the complex terrain of mathematics.
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Trading myths in forex and financial markets - Forex Wink
Trading myths in forex and financial markets
When I first started trading, I discovered that there was so much information out there about the financial markets, forex and stock exchanges that it was difficult to say what was true. I spent a
lot of time learning through trial and error in hopes of finding something that was truly useful and working. It's only after years of trading that I realize that simplicity is best and what may seem
obvious may not actually be so.
I hope that by dismantling popular trading myths, I will help you avoid the unnecessary disappointments and failures that I once encountered. So, what are the most common trading myths?
I will succeed if I work hard
I know that everyone has watched films like: “Rocky”, “Never Back Down”, etc. and so on. Their plot is simple: the main character, through hard work, achieves his cherished goal and takes all the
cream from the top of Olympus.
I have to disappoint you, in Forex trading everything is not so simple. You will not get good results by looking at charts 24/7 or studying hundreds of trading systems. The market doesn't just
require hard work, it requires hard work with intelligence.
To become a good trader, you need to develop your skills and discipline. This means that you should work on the things that you can control. Start improving your trading by maintaining and analyzing
a trading journal.
All I need is discipline
I won’t argue, discipline is one of the most important qualities required for trading in the market. But if you find yourself getting one losing trade after another, it may be worth reconsidering
your trading system rather than working on your discipline.
Traders who have lost their deposits are not necessarily undisciplined. They may have put little effort and time into identifying the trading style that suits them best.
Emotions are the main enemy of a trader
Of course, emotions play an important role in trading, but for some traders they are limited to the phrase: “Let profits run and cut losses.”
Don't forget that profitable trading is a combination of skill, talent, passion, discipline, and not just your emotional state.
It is worth remembering that the more legs a chair has, the more stable it stands.
The entry point is most important
When I first started learning, I knew there was no holy grail in trading. But I was obsessed with finding the best entry point. I thought that if I had a good entry point, then the price would
immediately go in my favor and I could make money.
However, there are many other factors in trading that are just as important to consider: position sizing, stop losses, trade management, etc. And a good entry point is just part of the equation for
profitable trading.
Now the entry point for me just means the frequency of my trades. The more entry points I get, the more often I open new positions.
Did you know that even with random entry points, you can still remain profitable if you think about stop losses, position sizing, and trade management?
Fixed income
Trading is a marathon, not a sprint. We strive for constant profitability over a long period of time. But how can you determine your ongoing profitability?
Constant returns will differ for different traders on the selected timeframe. For scalpers, consistent returns mean making a profit every quarter. For traders who trade on daily charts, consistent
returns will mean a year's worth of profits.
As you can see, consistency correlates with the time frames you trade on. The lower the time frame, the higher the frequency of your trading, with the law of large numbers working in your favor over
a shorter period of time. Likewise for a trader who trades on higher time frames. He needs a longer period of time for the law of large numbers to start working in his favor.
You may be wondering what the consistent returns are for high frequency trading? If trades are completed in milliseconds, the law of large numbers can work even throughout the day. It is therefore
not surprising that high frequency trading can generate profits every day.
Please note: before you want to determine consistent returns in your trading, you must first decide which time frame you will use.
Risk no more than 2% of your capital per trade
The 2% risk is the standard in trading that everyone talks about. In the long term, this level of risk helps protect your deposit from a complete loss of all funds. However, it does not take into
account the trading timeframe, the profitability of the trading strategy, and your personal risk tolerance.
Managing risk using a fixed percentage of capital works if you are a day trader, swing trader or position trader. You will have enough time to calculate your stop losses and position size. But if
you're a scalper who's been in the market for a few minutes, the 2% rule won't make much sense to you because you won't have enough time to calculate the position size every time you open a new trade
without yet knowing where Your stop loss will be placed.
A return of 80% with a risk to reward ratio of 1 to 1 is very good for any trading strategy. The probability of making three losing trades in a row will be less than 1%. If you are risking 2% of your
deposit on every trade, you are being too conservative and not making optimal use of all your options. Even if you risk 10%, your risk of ruin will be 0 after 1000 trades.
Each trader has his own risk tolerance. There are traders who set a loss limit for a certain period of time. Let's say you are willing to tolerate a 25% drawdown on your trading account, and you have
a trading strategy that is profitable 50% of the time with a risk to reward ratio of 1 to 2. If you risk 2%, you will have a 0.56% chance of making 25 % drawdown to the trading account. If you risk
1% of your deposit, you will have zero chance of experiencing a 25% drawdown. In this case, a 2% risk may no longer be acceptable to you.
There is no universal amount of risk for every trader. You always need to consider your trading style, the effectiveness of your trading strategy, and your own risk management strategy.
Trading is a risk
Everything in life comes with some risk, and your job is to minimize all possible risks.
When you cross the road, you risk getting hit by a car. When you learn to drive, you risk the safety of others. When you enter a trade without a stop loss, you risk getting a margin call.
When you cross the road, you go to the green traffic light, look left and right. The chances of you getting hit by a car are greatly reduced.
After 30 driving lessons with an instructor, you are a much better driver. The risk of an accident is reduced.
Similarly, in trading, by learning how to trade properly and knowing your entry point, stop loss and risk level, you reduce the likelihood of losing your deposit.
Risk arises from lack of knowledge and practice. If you devote yourself to studying trading for a long time and adhere to the rules in your trading, your risks will be significantly reduced.
The greater the leverage, the greater the income
If you Google “forex trading” you will see promotions from hundreds of brokers who will offer you crazy leverage for your trading. Brokers will tell you that you can easily earn 100% of your deposit,
but forget to mention that you can also quickly lose all your money.
Let's say you have a trading account of $1,000 and you buy 50 shares of Apple at a price of $20. The stock price then rises to $25. You make a profit of $250. But if Apple shares fall to $15 per
share, you will lose $250.
Let's say the broker allows you to use 5 times the capital you have. This allows you to buy 250 shares of Apple for $1,000. If the stock price rises to $25 per share, you make a profit of $1,250. But
if the stock price drops to $15, your loss will also be $1,250!
Always remember that leverage is a double-edged sword. It can magnify both your gains and your losses.
Risk to reward ratio
One of the most talked about topics in trading is none other than the risk to reward ratio. Often we see traders trying to maintain a risk to reward ratio of at least 1 to 2 before placing a trade,
and that anything less than that would mean a bad trade. In short, the quality of trading depends entirely on the risk to reward ratio.
Risk-to-reward ratio is interpreted as risk (the pips you stand to lose) to reward (the pips you earn), which pretty much tells us nothing because it doesn't take into account the probability factor.
Instead, it would be more correct to use the following relationship:
Risk (number of points you can lose * probability of making a loss) / Profit (number of points you can earn * probability of making a profit).
Risk-to-reward ratio and probability of profit are two sides of the same coin. Thus, it is not worth talking separately about the risk-to-reward ratio without knowing the probability of a successful
Does correlation increase your chances of profit?
We can talk about correlation in trading when two or more technical analysis factors point to the same market direction, thereby increasing our chances of making a profitable trade. For example, a
bullish engulfing pattern is placed at a support level equal to the 61.8% Fibonacci level.
However, there are two features of the correlation that are not mentioned anywhere.
Firstly, correlation has a lot to do with the number of patterns, setups and indicators you use in your trading. The more elements you use, the more correlations you will find. A trader using ten
different patterns, setups and indicators should have more correlations in his trading compared to someone using just a few. Does this mean that a trader with 10 elements has a higher probability of
trading profitably?
This cannot be proven statistically, because there are many different trading setups, patterns and indicators, as well as various combinations of them that can be used in trading.
Second, let's assume that correlation increases our chances of success, but by how much?
Let's assume that trading strategy A has a risk to reward ratio of 1 in 5 and is profitable 80% of the time. However, it requires the correlation of several elements, so this strategy only generates
trading signals 5 times per year. Now take trading strategy B, which has a risk to reward ratio of 1 in 5 and is successful 50% of the time, generating 100 signals per year. Which trading system is
more profitable?
With a risk of $100:
Trading strategy A would have (0.8 * 500) – (0.2 * 100) * 5 = $1900 after 1 year of trading.
Trading strategy B would have (0.5 * 200) – (0.5 * 100) * 100 = $5000 after 1 year of trading.
Although trading system A has a higher profitability than B, the low frequency of trades makes it less profitable.
Thus, if you are looking for trading opportunities on a chart with a high probability of profit, you may lose to a trader who takes advantage of less profitable but high-frequency trading
You can make $100,000 out of $1000
Beginners are lured into trading with promises of quick riches and easy money. You see advertisements that promise 300% profits within a few months. But is this possible?
Let's take a long-term view of everything. On average, hedge funds will earn 20% annual returns. Let’s assume that you manage to earn 20% on an ongoing basis. It's easy to calculate that it will take
you 26 years for $1,000 to turn into $100,000. If you have $20,000, then it will take you 9 years to turn that amount into $100,000.
In trading, you need to have a lot of money to earn a lot. If you have a small initial capital, then you will not be able to earn a million dollars in the near future.
You need to know exactly where the market is going
When I first started trading, I thought that those who make money always know where the market is going to go. In fact, this is not true.
If you think you need to know when you can make money in the market, then the casino needs to know if the next hand will win. But casinos make billions of dollars every year without knowing whether
each hand will win or not.
Likewise, in trading, you don't need to know when the market will give you an opportunity to make money. In fact, you can still remain profitable if you are wrong most of the time. How is this
Let's play a simple game according to the rules of which an incorrect prediction will cost you $100, and a correct one will cost you $300. You make a total of 20 predictions, of which 14 are wrong
and 6 are correct. You lose $1400 for your incorrect predictions and earn $1800 for your correct ones. As a result, you earn $400.
This becomes possible if you lose little when you are wrong and win a lot when you are right.
Only new traders need to know where the market will go. Professionals use a trading strategy and trading plan, as well as competent risk management.
The broker is hunting our feet
Traders often complain when their stops are repeatedly triggered. It happens that the price hits the stop loss for several pips before moving in the opposite direction. And in this situation, brokers
often seem to be to blame. But is your broker really hunting your stops?
My answer is no, and here's why:
If you consistently see a price from your broker that is different from other traders trading with the same broker, you can take a screenshot and post it on most trading forums. Because of this, the
broker may lose its reputation and lose trust.
Forex brokers make money by taking the spread from their clients' trades. Therefore, it is in their best interest for traders to trade consistently over a long period of time. However, if they are
trying to make money by taking out clients' stops just to earn a few extra pips, it is a matter of time before their clients find out about it and move on to more reputable brokers.
There is a wealth of information about trading in financial markets. Some of this is fact, but most of it is trading myths without concrete evidence. Be smart and don't fall for common misconceptions
and mistakes.
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[WIP] Introduction to Statistics & Data Analysis — Notes
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[WIP] Introduction to Statistics & Data Analysis — Notes
— notes — 2 min read
Chapter 1
1. Statistics is the scientific discipline that provides methods to help us make sense of data.
2. Statistical methods are used to organize, summarize, and draw conclusions from data.
3. The entire collection of individuals or objects about which information is desired is called the population of interest.
4. A sample is a subset of the population, selected for study in some prescribed manner.
5. Descriptive statistics is the branch of statistics that includes methods for organizing and summarizing data.
6. Inferential statistics is the branch of statistics that involves generalizing from a sample to the population from which it was selected and assessing the reliability of such generalizations.
7. The following questions should be addressed as part of a study evaluation:
□ Understanding the nature of the problem: What were the researchers trying to learn? What questions motivated their research?
□ Deciding what to measure and how to measure it: Was relevant information collected? Were the right things measured?
□ Data collection: Were the data collected in a sensible way?
□ Data summarization and preliminary analysis: Were the data summarized in an appropriate way?
□ Formal data analysis: Was an appropriate method of analysis used, given the type of data and how the data were collected?
□ Interpretation of results: Are the conclusions drawn by the researchers supported by the data analysis?
8. A variable is any characteristic whose value may change from one individual or object to another.
9. Data result from making observations on variable(s).
10. A data set consisting of observations on a single variable is a univariate data set.
11. A data set consisting of observations on multiple variable is a multivariate data set. A bivariate data set is a special case of a multivariate data set consisting of only two variables.
12. A univariate data set is categorical (or qualitative) if the individual observations are categorical responses.
13. A univariate data set is numerical (or quantitative) if each observation is a number.
14. A numerical variable results in discrete data if the possible values of the variable correspond to isolated points on the number line. We can place an interval that is small enough that no other
possible value is included in the interval. Discrete data usually arise when each observation is determined by counting.
15. A numerical variable results in continuous data if the set of possible values forms an entire interval on the number line. In general, data are continuous when observations involve making
16. A frequency distribution for categorical data is a table that displays the possible categories along with the associated frequencies and/or relative frequencies.
17. The frequency for a particular category is the number of times the category appears in the data set.
18. The relative frequency for a particular category is the fraction or proportion of the observations resulting in the category. It is calculated as:$relative~frequency = \frac{frequency}
19. If the table includes relative frequencies, it is sometimes referred to as a relative frequency distribution.
20. A bar chart is a graph of the frequency distribution of categorical data. Each category in the frequency distribution is represented by a bar or rectangle of equal width, and the picture is
constructed in such a way that the height of each bar is proportional to the corresponding frequency or relative frequency.
21. A dot-plot is a simple way to display numerical data when the data set is reasonably small. Each observation is represented by a dot above the location corresponding to its value on a horizontal
measurement scale. When a value occurs more than once, there is a dot for each occurrence and these dots are stacked vertically.
22. Dotplots convey information about:
□ A representative or typical value in the data set.
□ The extent to which the data values spread out.
□ The nature of the distribution of values along the number line.
□ The presence of unusual values in the data set.
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lca.model: fits latent class models for phenotypic measurements in... in abureau/LCAextend: Latent Class Analysis (LCA) with familial dependence in extended pedigrees
This is the main function for fitting latent class models. It performs some checks of the pedigrees (it exits if an individual has only one parent in the pedigree, if no children is in the pedigree
or if there are not enough individuals for parameters estimation) and of the initial values (positivity of probabilites and their summation to one). For models with familial dependence, the child
latent class depends on his parents classes via triplet-transition probabilities. In the case of models without familial dependence, it performs the classical Latent Class Analysis (LCA) where all
individuals are supposed independent and the pedigree structure is meaningless. The EM algorithm stops when the difference between log-likelihood is smaller then tol that is fixed by the user.
1 lca.model(ped, probs, param, optim.param, fit = TRUE,
2 optim.probs.indic = c(TRUE, TRUE, TRUE, TRUE), tol = 0.001,
3 x = NULL, var.list = NULL, famdep = TRUE, modify.init = NULL)
lca.model(ped, probs, param, optim.param, fit = TRUE, optim.probs.indic = c(TRUE, TRUE, TRUE, TRUE), tol = 0.001, x = NULL, var.list = NULL, famdep = TRUE, modify.init = NULL)
ped a matrix representing pedigrees and measurements: ped[,1] family ID, ped[,2] subjects ID, ped[,3] dad ID, ped[,4] mom ID, ped[,5] sex, ped[,6] symptom status (2: symptomatic, 1:
without symptoms, 0: missing), ped[,7:ncol(ped)] measurements, each column corresponds to a phenotypic measurement,
probs a list of initial probability parameters (see below for more details). The function init.p.trans can be used to compute an initial value of the component p.trans of probs,
param a list of initial measurement distribution parameters (see below for more details). The function init.ordi can be used to compute an initial value of param in the case of discrete
or ordinal data (product multinomial distribution) and init.norm in the case of continous data (mutivariate normal distribution),
optim.param a variable indicating how measurement distribution parameter optimization is performed (see below for more details),
fit a logical variable, if TRUE, the EM algorithm is performed, if FALSE, only computation of weights and log-likelihood are performed with the initial parameter values without
log-likelihood maximization,
optim.probs.indic a vector of logical values indicating which probability parameters to estimate,
tol a small number governing the stopping rule of the EM algorithm. Default is 0.001,
x a matrix of covariates (optional), default is NULL,
var.list a list of integers indicating the columns of x containing the covariates to use for a given phenotypic measurement, default is NULL,
famdep a logical variable indicating if familial dependence model is used or not. Default is TRUE. In models without familial dependence, individuals are treated as independent and
pedigree structure is meaningless. In models with familial dependence, a child class depends in his parents classes via a triplet-transition probability,
modify.init a function to modify initial values of the EM algorithm, or NULL, default is NULL.
a matrix representing pedigrees and measurements: ped[,1] family ID, ped[,2] subjects ID, ped[,3] dad ID, ped[,4] mom ID, ped[,5] sex, ped[,6] symptom status (2: symptomatic, 1: without symptoms, 0:
missing), ped[,7:ncol(ped)] measurements, each column corresponds to a phenotypic measurement,
a list of initial probability parameters (see below for more details). The function init.p.trans can be used to compute an initial value of the component p.trans of probs,
a list of initial measurement distribution parameters (see below for more details). The function init.ordi can be used to compute an initial value of param in the case of discrete or ordinal data
(product multinomial distribution) and init.norm in the case of continous data (mutivariate normal distribution),
a variable indicating how measurement distribution parameter optimization is performed (see below for more details),
a logical variable, if TRUE, the EM algorithm is performed, if FALSE, only computation of weights and log-likelihood are performed with the initial parameter values without log-likelihood
a vector of logical values indicating which probability parameters to estimate,
a small number governing the stopping rule of the EM algorithm. Default is 0.001,
a list of integers indicating the columns of x containing the covariates to use for a given phenotypic measurement, default is NULL,
a logical variable indicating if familial dependence model is used or not. Default is TRUE. In models without familial dependence, individuals are treated as independent and pedigree structure is
meaningless. In models with familial dependence, a child class depends in his parents classes via a triplet-transition probability,
a function to modify initial values of the EM algorithm, or NULL, default is NULL.
The symptom status vector (column 6 of ped) takes value 1 for subjects that have been examined and show no symptoms (i.e. completely unaffected subjects). When applying the LCA to measurements
available on all subjects, the status vector must take the value of 2 for every individual with measurements.
a probability vector, each p[c] is the probability that an symptomatic founder is in class c for c>=1,
the probability that a founder without symptoms is in class 0,
an array of dimension K times K+1 times K+1, where K is the number of latent classes of the model, and is such that p.trans[c_i,c_1,c_2] is the conditional probability that a symptomatic individual i
is in class c_i given that his parents are in classes c_1 and c_2,
a vector of length K, where p0connect[c] is the probability that a connector without symptoms is in class 0, given that one of his parents is in class c>=1 and the other in class 0,
a probability vector, each p[c] is the probability that an symptomatic individual is in class c for c>=1,
the probability that an individual without symptoms is in class 0,
param is a list of measurement distribution parameters: the coefficients alpha (cumulative logistic coefficients see alpha.compute) in the case of discrete or ordinal data, and means mu and
variances-covariances matrices sigma in the case of continuous data,
optim.param is a variable indicating how the measurement distribution parameter estimation of the M step is performed. Two possibilities, optim.noconst.ordi and optim.const.ordi, are now available in
the case of discrete or ordinal measurements, and four possibilities optim.indep.norm (measurements are independent, diagonal variance-covariance matrix), optim.diff.norm (general variance-covariance
matrix but equal for all classes), optim.equal.norm (variance-covariance matrices are different for each class but equal variance and equal covariance for a class) and optim.gene.norm (general
variance-covariance matrices for all classes), are now available in the case of continuous measurements, One of the allowed values of optim.param must be entered without quotes.
optim.probs.indic is a vector of logical values of length 4 for models with familial dependence and 2 for models without familial dependence.
All defaults are TRUE. If the dataset contains only nuclear families, there is no information to estimate p0connect and p.connect, and these parameters will not be estimated, irrespective of the
indicator value.
param the Maximum Likelihood Estimator (MLE) of the measurement distribution parameters if fit=TRUE or the input param if fit=FALSE,
probs the MLE of probability parameters if fit=TRUE or the input probs if fit=FALSE,
the Maximum Likelihood Estimator (MLE) of the measurement distribution parameters if fit=TRUE or the input param if fit=FALSE,
the MLE of probability parameters if fit=TRUE or the input probs if fit=FALSE,
When measurements are available on all subjects, the probability parameters p0 and p0connect are degenerated to 0 and p.afound, p.child and p.aff to 1 in the output.
weight an array of dimension n (the number of individuals) times 2 times K+1 (K being the number of latent classes in the selected model and the K+1th class being the unaffected class) giving the
individual posterior probabilities. weight[i,s,c] is the posterior probability that individual i belongs to class c when his symptom status is s, where s takes two values: 1 for symptomatic
and 2 for without symptom. In particular, all weight[,2,] are 0 for symptomatic individuals and all weight[,1,] are 0 for individuals without symptoms. For missing individuals (unkown symptom
status), both weight[,1,] and weight[,2,] may be greater than 0.
ll the maximum log-likelihood value (log-ML) if fit=TRUE or the log-likelihood computed with the input values of param and probs if fit=FALSE,
an array of dimension n (the number of individuals) times 2 times K+1 (K being the number of latent classes in the selected model and the K+1th class being the unaffected class) giving the individual
posterior probabilities. weight[i,s,c] is the posterior probability that individual i belongs to class c when his symptom status is s, where s takes two values: 1 for symptomatic and 2 for without
symptom. In particular, all weight[,2,] are 0 for symptomatic individuals and all weight[,1,] are 0 for individuals without symptoms. For missing individuals (unkown symptom status), both weight[,1,]
and weight[,2,] may be greater than 0.
the maximum log-likelihood value (log-ML) if fit=TRUE or the log-likelihood computed with the input values of param and probs if fit=FALSE,
TAYEB, A. LABBE, A., BUREAU, A. and MERETTE, C. (2011) Solving Genetic Heterogeneity in Extended Families by Identifying Sub-types of Complex Diseases. Computational Statistics, 26(3): 539-560. DOI:
LABBE, A., BUREAU, A. et MERETTE, C. (2009) Integration of Genetic Familial Dependence Structure in Latent Class Models. The International Journal of Biostatistics, 5(1): Article 6.
1 #data
2 data(ped.ordi)
3 fam <- ped.ordi[,1]
4 #probs and param
5 data(param.ordi)
6 data(probs)
7 #the function applied only to two first families of ped.ordi
8 lca.model(ped.ordi[fam%in%1:2,],probs,param.ordi,optim.noconst.ordi,
9 fit=TRUE,optim.probs.indic=c(TRUE,TRUE,TRUE,TRUE),tol=0.001,x=NULL,
10 var.list=NULL,famdep=TRUE,modify.init=NULL)
#data data(ped.ordi) fam <- ped.ordi[,1] #probs and param data(param.ordi) data(probs) #the function applied only to two first families of ped.ordi lca.model(ped.ordi
[fam%in%1:2,],probs,param.ordi,optim.noconst.ordi, fit=TRUE,optim.probs.indic=c(TRUE,TRUE,TRUE,TRUE),tol=0.001,x=NULL, var.list=NULL,famdep=TRUE,modify.init=NULL)
For more information on customizing the embed code, read Embedding Snippets.
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Methods implemented in NeuroElf
Whenever a program is used for data analysis, it is important for the community at large to understand what algorithms were used in the analysis. And while NeuroElf is mostly written to make
algorithms accessible (user friendliness aspect), it is equally relevant to ascertain that the methods implemented in any program have been accepted by the scientific community as “useful and
reliable” (to achieve the intended goal) and are (as much as possible) free of errors, both when it comes to potential flaws in the algorithm as well as its specific implementation in the given
As an example, initially when using the alphasim button (NeuroElf GUI access to the alphasim.m function) the GUI would demand user input for the estimated smoothness of the data, and as a default
value 6mm was presented to the user. This choice (of default value) was motivated by the fact that, at the lab where I work, the smoothing operation during the preprocessing stage would be configured
with a 6mm Gaussian kernel. However, the correct number to use ought to be an estimate of the spatial smoothness of the residual, because that determines how likely it is that, by chance, a cluster
of a given size will be encountered in a statistical map (at any given uncorrected threshold), and this issue has since been addressed!
List of methods (overview)
The following list gives an overview on what methods of analysis and parameter estimation are implemented in NeuroElf (as far as they exceed basic operations, such as for example plain averaging
across a dimension, or auxiliary functions that are used for string manipulation, file in-/output, or extended array operations, etc.):
Cluster size threshold estimation (alphasim)
Cluster size threshold estimation is a method that can be used to account for the fact that a regular whole-brain map is made up of multiple (partially) independent tests. One common way is to simply
adapt the statistical threshold by dividing the desired false-positive rate (i.e. typically 5 per cent = 0.05) by (an estimate of) the number of independent tests. However, this can be too stringent
in some cases where larger swaths of cortex (neurocomputational network nodes) respond to an experimental manipulation below the then required detection threshold. Instead of ensuring significance of
results solely by applying a voxel-wise corrected statistical threshold it is possible to estimate how large clusters are, given the smoothness of the residual, that appear in a given search space at
random. I.e. the alpha-rate (false positives among performed tests) can be estimated by simulating statistical maps of the desired kind and then selecting the appropriate cluster size threshold to
ensure that at most 5 per cent of maps (with the residual exhibiting the same smoothness) would show a false positive cluster. The resulting pair of uncorrected statistical threshold and cluster size
threshold together then correct a whole-brain map to a family-wise-error corrected threshold of desired strength (again usually 0.05). This algorithm is
• accepts a mask (sub-space specification)
• can be applied to surface statistics (given the mesh vertices and topology, as well as an estimate of the smoothness)
• allows to estimate the cluster size threshold for fully independent components of a conjunction analysis
• as a still experimental feature allows to apply a shift in the Z-distribution to account for shifts in the observed distribution of a statistical map (e.g. by-chance global “signal” in a
covariate regression)
Cluster table generation
Cluster tables are often presented in publications describing analyses where whole-brain mapping was performed, i.e. the attempt in localizing the spatial nodes within cortex that subserve a specific
function. This function is
□ whereas the M-file provides a command-line interface with rich options for output formatting, converting coordinates, thresholding volumes, etc.
□ and the C/MEX-file provides the actual clustering of the binary (thresholded and masked) volume into separate spatial nodes
Once a (thresholded) map has been segregated into separate volumes (such that voxels of different clusters do not “touch” voxels of another cluster), clusters of considerable size (e.g. more than 100
voxels) sometimes exhibit “local maxima”, i.e. the spatial gradient becomes positive again from the overall maximum outwards after being negative in the beginning. To detect this, a 3D watershed
algorithm has been implemented in the function splitclustercoords.m.
Conjunction analysis (minimum t-statistic)
A conjunction analysis can be informative when, across the brain, the overlap of two statistical tests is of interest. The most stringent test that can be applied is that of requiring that, in each
considered voxel, both tests must be significant at the desired level. This functionality is
• implemented in the function
for statistics of the same kind and with the same D.F. parameter, i.e. higher value means greater significance
• implemented in the function
for p-values (and possibly other statistics for which lower values mean greater significance; also accepts negative values)
Mediation analysis
Mediation analysis as a whole can be described as the estimation (and test) of separate path coefficients, a and b, as well as their product, a*b, such that the “transmission” of an existing effect
between an indepedent/explanatory variable, X, and an outcome variable, Y, is accomplished via one or several mediators, Mi. The analysis includes a test for significance of the a*b product term (as
well as the individual path coefficients), and also allows to specify covariates. It is
• implemented in function
, where the pset indicates that the function returns path coefficients (p), standard errors (se), and t-statistics (t)
• options are: a*b product testing via bootstrapping or Sobel test, and robust regression
• supports multi-dimensionaging) data for X, M, and Y
An example would be, on the level of a between-subject effect, that a randomly assigned condition (X, e.g. strategy to apply to stimuli) has an effect on outcome (Y, e.g. appetite to a specific type
of stimulus or difference in appetite to two kinds of stimuli) via a specific brain region (or network of regions) that work/s as a mediator/s (Mi, e.g. pre-frontal control regions). For a
within-subjects design, a test could be whether, on any given trial, the response in pre-frontal cortex during an instructional cue (strategy stimulus) has an effect on outcome (self-reported craving
for depicted food) via another brain region. In that case, either X (which brain regions has an influence on the “craving center” of the brain) or M (which brain region is influenced by the “control
region” of the brain) could be “searched for”…
Multi-level kernel density analysis (MKDA / meta analysis)
Multi-level kernel density analysis is trying to determine whether reported “peak coordinates” in previously published papers (given a selection criterion, such as publications concerned with a
specific psychological construct, e.g. fear or working memory) occur in specific spatial locations (spatial specificity) significantly more often than warranted by chance, as a means to pool several
publications to reduce the influence of a single publication on the “knowledge” of spatial distributions of activation patterns. It is
Ordinary least-squares (OLS) regression
Ordinary least-squares (OLS) regression is the most generic way of applying the General Linear Model (GLM) so as to estimate “effect sizes”. Given the different applications, there are several
functions implementing forms of this regression:
• the most general implementation is done in the
□ to assess the significance of the regression (single beta or computed contrasts), the
function must be used
An additional small number of function files also perform some flavor of linear regression, but those are not applied to functional imaging data (e.g. the function regress_coords.m can be used to
determine the transformation required to minimize the error between two sets of coordinates after a rigid-body transform).
Robust regression
Robust regression, in NeuroElf, is the estimation of regression parameters using an iteratively-reweighted-least-squares approach where outliers are “detected” using the bi-square weighting function.
It is
neuroelf_methods.txt · Last modified: 2013/02/02 00:38 by jochen
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Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and Compound Channels
National Research University “Moscow Power Engineering Institute”, Krasnokazarmennaya 14, 111250 Moscow, Russia
Author to whom correspondence should be addressed.
Submission received: 15 November 2022 / Revised: 14 December 2022 / Accepted: 20 December 2022 / Published: 23 December 2022
At present, software products for numerical simulation of fluid dynamics problems (ANSYS Fluent, Ansys CFX, Star CCM, Comsol, etc.) problems are widely used. These software products are mainly based
on the numerical solution of the Navier–Stokes equations, the most common and computationally easy method of solving, which is Reynolds averaging (RANS), and further closing the system using
semi-empirical turbulence models. Currently, there are many modeling methods and turbulence models; however, there are no generalized recommendations for setting up grid models for modeling flows,
while for practical use both the correct mathematical models and the setting of the computational grid are important. In particular, there are no generalized recommendations on the choice of scale of
global elements of grid models for typical single channels. This work is devoted to the development and study of relations for a priori estimation of the parameters of a grid model in relation to
solving hydrodynamic problems with fluid flow in channels. The paper proposes the introduction of a generalized grid convergence criterion for single channels at high Reynolds numbers. As single
channels, a channel with a sudden expansion, a channel with a sudden contraction, and diffuser channels with different opening angles are considered. Based on the results of variant calculations of
typical single channels at various Reynolds numbers and various geometric parameters, generalized criterion correlations were obtained to find dimensionless linear scales of grid elements relative to
the hydrodynamic characteristics of the flow in the channel. Variant calculations of the compound channel were investigated, which showed the adequacy of correlations proposed.
1. Introduction
At present, the modeling of fluid dynamics processes in the flow in channels is an urgent task associated with the development and design of complex science-intensive products in power engineering,
aviation and other industries. The main difficulty in modeling hydro-gas-dynamic processes lies in the correct modeling of the turbulence phenomenon, which is characterized by:
• Non-stationarity;
• Irregularity, lack of strict order in time;
• Randomness;
• Three-dimensionality;
• Viscous and vortex nature of the flow;
• Coherence of large vortex structures.
Usually, the flow regime becomes turbulent at Reynolds number (Re) more than a critical values, which is different for various types of flow. The Re by definition is:
• $D H$—characteristic size, m.
• $u$—characteristic velocity, m/s;
• $ν$—viscosity, m^2/s. There are currently several main approaches to modeling turbulence:
□ Direct numerical simulation (DNS) [
] calculates the Navier–Stokes equations for eddies of all scales up to the Kolmogorov scale. With an increase in the size of the computational cell, the system of equations for DNS, as a
rule, does not converge. At present, the DNS is used mainly for research purposes and in modeling low-Re currents.
□ Large eddy modeling (LES) [
] uses the separation of eddies by scale: large eddies are directly resolved, small eddies are modeled using subgrid models. LES methods require a grid of the order of the scale of large
eddies. Currently, LES is used in scientific research and technical applications as a highly accurate method.
□ Modeling of Reynolds-averaged Navier-Stokes equations or unsteady Reynolds-averaged Navier-Stokes equations (RANS, URANS) [
]. Modeling is done by closing the equations through Boussinesq hypothesis and semi-empirical turbulence models over the entire energy spectrum.
At present, this method is most widely used in technology in the development of new equipment, scientific research, etc.
Figure 1
shows a comparative diagram of DNS, LES and RANS/URANS approaches for modeling turbulent flows. The hybrid methods as a detached eddy simulation (DES) [
] are between LES and RANS/URANS in terms of physicality and volume of calculations.
There are a large number of turbulence models for the RANS method, the most widely used of which are the two-parameter models of the k-ε [
] and k-ω [
] groups. The main difference between the turbulence models of the k-ε group is the type of differential equations that allow, together with the averaged system of Navier–Stokes equations, to
determine the kinetic energy of turbulence and the rate of its dissipation. Additional differential equations for k and ε are transport equations of the given quantities. The k-ω and Shear Stress
Transport (SST) turbulence models are traditionally referred to as the low Reynolds group of turbulence models. The turbulence model k-ω is similar to the turbulence models from the high-Reynolds k-ε
family, only in this case, to determine the turbulent viscosity, differential equations are additionally solved for the specific dissipation rate of kinetic energy ω. One of the most versatile
turbulence models is the k-ω SST hybrid model. The group of equations of the low Reynolds SST [
] turbulence model includes equations for the high Reynolds k-ε turbulence model solved for the flow core and equations for the low Reynolds k-ω turbulence model solved for the boundary layer region.
Since RANS methods, unlike LES and DNS methods, do not directly calculate vortex structures, but model them using semi-empirical models, there are no unambiguous criteria for constructing grid models
] related to the scales of resolvable vortex structures that can be built for these methods for DNS and LES methods [
]. At the same time, the formation of recommendations for the construction of computational grids that ensure high calculation accuracy for specific geometric and regime characteristics is an
important applied task, since it allows for advanced verification of flow modeling, which in turn reduces the number of numerical and physical experiments.
The paper [
] summarizes recommendations for numerical modeling of flow processes in the interblade channels of turbines. Based on the results of the review, the authors recommend the use of the k-ω SST model,
the use of inflation in the vicinity of the wall with the thickness of the first near-wall element corresponding to y
< 1, as well as recommendations for choosing the size of the global mesh element, corresponding to the number of elements in the range between 500,000 and 1,500,000. The y
is the dimensionless universal flow coordinate, which is by definition:
• y—wall distance, m.
• $u τ$—shear velocity, m/s.
• $ν$—viscosity, m^2/s.
Using the example of modeling thermal–hydraulic processes occurring in the core of the CANDU (Canada Deuterium Uranium) reactor, it was shown in [
] that a good approximation at a Reynolds number of 2.3 · 10
is provided with a number of elements of the grid model of 84 million, while the size of the element in the zones of a high-velocity gradient should not exceed 0.4 mm. It is also noted in the work
that during modeling, a much greater sensitivity is manifested to a change in the size of the global element than to a change in the size of the near-wall cells.
The main problem of using the data of the above works is the lack of generalized recommendations for constructing grid models for the sizes of global elements, while for near-grid cells there are
recommendations for choosing sizes (y
< 5 for low Reynolds models and y
> 30 for high Reynolds ones) [
For a global element of a grid model, an important characteristic is the grid convergence point [
] when conducting a grid-independence study. The grid convergence point is the value of the element size, at which a further decrease in the element size does not lead to a noticeable increase in the
accuracy of calculations; therefore, this value is of great importance, since it is the optimal element size that provides the required modeling accuracy at relatively low computational costs.
The paper [
] presents the results of the study of grid convergence as applied to modeling the external flow around a wind turbine with a Darrieus rotor. The paper proposes a methodology and correlations for
estimating element sizes in relation to URANS modeling using the SST model, using a combination of the Courant number (Co) and the newly introduced GRV (Grid-Reduced Vorticity) criterion. The
advantages of the described approach include a high degree of physical validity, as well as high accuracy. The disadvantages are the need to construct the vorticity field, which excludes the
possibility of a priori estimation of the size of the grid model.
The purpose of this work is to form correlations for estimating the sizes of global elements of grid models that provide grid convergence in modeling fluid dynamics processes in turbulent flow in
typical single channels, which are considered channels with sudden expansion, sudden contraction, as well as diffuser channels. In this case, the ratios must be designed for application for use in a
wide range of operating characteristics and meet the following requirements:
• The size of the grid element in correlations should be associated with characteristic hydrodynamic quantities that have a length scale and characterize the flow regime.
• The nature of the quantities used should allow one to estimate the scale of the element a priori, before conducting numerical studies.
• Correlations should take into account the results of empirical and analytical studies of turbulent flows in channels.
The main novelty of this paper is the formulation of new grid convergence criteria, which has a physical sense and is related to the characteristic flow length scale. In the previous study [
] were estimated some recommendations for turbine blade channel modelling, but there are no generalized criteria, which can help to expand recommendations for more cases. On the contrary, in this
study, a step is being taken towards the formulation of generalized criteria, which can be expanded on another geometry in the same class of cases in further studies. The formulation of generalized
recommendations on the optimal mesh settings will allow the future to form a methodology for advanced verification of numerical simulation results.
2. Research Object
The objects of study in this paper are channels with a sudden expansion, with a sudden contraction, as well as diffuser channels. These types of channels are typical in relation to the problems of
power engineering, in particular, channels with a sudden expansion and diffuser channels are characteristic geometries for combustion chambers of gas turbine plants, and channels with a sudden
contraction are often found in shut-off and control valves, which, in particular, include faucets and valves.
To build a methodology for advanced verification, it is necessary to simulate the processes of fluid dynamics in single circular channels in a wide range of geometric and regime characteristics.
Figure 2
shows the sketches of the single circular channels considered in this work;
Table 1
shows their geometric and operating characteristics.
Characteristic Re for interblade channels of gas turbines ~10
] are considered as modes.
For the cases F[0], l[0], D[0] are the cross-sectional area, length and diameter, respectively, of the channel until the diameter changes and F[1], l[1], D[1] are these ones after changing. The $α$
is the diffuser opening angle and Re[0] is the inlet Reynolds number.
In this paper, the compound channel is considered a control geometry for checking the correlations obtained during the study. The compound channel includes a diffuser section, a sudden contraction
and a sudden expansion. The geometric and operating characteristics of the compound channel are shown in
Table 2
. For compound channel
is the length and diameter, respectively, of channel until diffuser expansion and,
are the ones after the diffuser region. The α is the diffuser opening angle and
is the inlet Reynolds number. The
and the
is the diameter and length of the region after sudden contraction and after sudden expansion, respectively.
3. Research Method
Studying the problem for grid convergence is an important step in solving any problem; however, in the case of complex problems, the need to calculate several configurations leads to a significant
increase in the overall computational complexity of the problem. In the case of forward verification, if the value of the optimal size of the global element is known, there is no need to conduct a
large grid convergence study from the biggest scales of the grid element.
In the numerical simulation of hydrodynamic problems, the main source of simulation error is the low resolution of the grid model in areas with high velocity and pressure gradients. As applied to the
flow in channels, the main velocity gradient is contained in the near-wall boundary layer, while in the core of the flow, the mean velocity practically does not change. The biggest local pressure
gradients are contained near vortex generation sites as sharp corners or sudden flow direction change.
Figure 3
shows the velocity profile for a turbulent fluid flow in a channel in natural coordinates. This profile, when translated into dimensionless quantities in logarithmic coordinates, corresponds to the
standard logarithmic turbulent profile.
Figure 3
shows that the largest velocity gradient during fluid flow is concentrated in the viscous sublayer, for which there are current recommendations for choosing the heights of prismatic layers [
]. In a turbulent logarithmic boundary layer, the change in velocity is slower, but the gradient is also significant.
If we talk about the section of the logarithmic turbulent sublayer, then, as a rule, this section corresponds to the area of global grid elements, for which there are currently no practical
generalized a priori recommendations for choosing linear dimensions. It is also important to note that a change in the size of the global element also affects the density of the grid in the direction
of the fluid flow, which in turn affects the correctness of the calculation of the pressure field, which changes during a steady flow, mainly along the length of the channel.
Taking into account the structure of the flow, it is assumed that in order to ensure the required accuracy, the optimal sizes of global elements should be related by correlation dependences with
characteristic scales characterizing the scale of turbulence in the flow. The optimal scale, in this case, would be the characteristic size of the vortex structures, but it cannot be estimated a
priori, and therefore it is not suitable as a scale for constructing a priori correlation dependences.
On the other hand, using the relations for the logarithmic Prandtl velocity profile, one can construct correlations in which the thickness of the turbulent logarithmic sublayer, approximately
corresponding to the dimensionless coordinate y
= 200, is taken as the characteristic linear dimension. Taking into account the expression for near-wall layer height [
], it is possible to convert dimensionless coordinates into dimensional ones.
It is assumed that for all considered regimes within the framework of one considered turbulence model, the correlation:
—dimensionless value of the optimal value of the length of the control element, related to near-wall layer height, which is estimated by equations, presented in [
$K o = Δ o p t y + D 74 R e ′ − 13 / 14 ,$
• $D$—channel characteristic size, m;
• $Δ o p t$—optimal linear size of the global element of the grid model, m;
• $y +$
—dimensionless distance from the wall corresponding to the transition to the flow core,
$y + = 200$
• $R e ′$—the Reynolds number in the characteristic cross section of the channel, $R e ′ = u ′ D ν$; where $u ′$ is the characteristic flow velocity.
The Ko is the new criteria number of grid quality related to the characteristic length scale in the flow, proposed in this study. The physical meaning of Ko is the grid length scale divided by a
near-wall layer height, which is estimated by $y + D 74 R e ′ − 13 / 14$. With increasing of Ko the grid quality will decrease because there will be less control volumes per wall layer, and on the
contrary with decreasing of Ko grid quality will increase.
In this paper, the optimal size of a grid model element is considered to be the size that ensures grid convergence. This decision is due to the fact that in this size the numerical solution is quite
accurate and further decrease in grid element size does not significantly effect. Wherein the decrease in cell size leads to an increased number of element, which require larger computing resources.
As the characteristic size of the channel to determine Re’ and Ko in the case of channels with a varying cross-section, the size of the channel in the region of the greatest pressure losses is taken.
For example, in the case of a canal with a sudden expansion, this is the diameter of the long wide part, in the case of a canal with a sudden contraction this is the diameter of tight part, for the
diffuser this is the mean diameter between wide part and the tight part. The characteristic velocity to determine Re’ is the velocity, which matched the characteristic diameter of the channel.
The physical meaning of the hypothesis lies in the dependence of the element value and the height of the turbulent boundary layer on the flow regime in order to provide the necessary discreteness in
the velocity gradient zone, which is the turbulent boundary layer.
The general research methodology includes:
• Conducting research on grid convergence for single channels with various regime and geometric characteristics.
• Revealing transition points to grid convergence using approximation power expressions.
• Reducing the values of the size of the element corresponding to the transition to the grid convergence to the dimensionless form Ko by dividing by the thickness of the turbulent boundary layer.
• Formation of correlations Ko(Re’) for individual channels with a test of statistical significance.
• Formation of the overall correlation Ko(Re’) with a test of statistical significance.
• Verification of the obtained general correlation dependence on the compound channel.
Modeling is planned to be carried out in an axisymmetric 2D RANS formulation, which provides fast calculation, which is valuable for practical engineering calculations, k-ω SST Menters Model [
] is used as a turbulence model. This model was chosen because of one of the best predictions for a flow with separation. Equations for the SST turbulence model for an incompressible fluid:
$∂ k ∂ t + u j ∂ k ∂ x j = P k − β * k ω + ∂ ∂ x j [ ( μ ρ + σ * μ t ρ ) ∂ k ∂ x j ] ∂ ω ∂ t + u j ∂ ω ∂ x j = γ P k ρ μ t − β ω 2 + ∂ ∂ x j [ ( μ ρ + σ * μ t ρ ) ∂ ω ∂ x j ] + 2 ( 1 − F 1 ) σ ω 2 ∇
k · ∇ ω ω P k = min ( μ t ρ S 2 , 10 β * k ω ) F 1 = tan h ( a r g 1 4 ) a r g 1 = min [ m a x ( k 0.09 ω d w , 500 ν d w 2 ω ) , 4 ρ σ ω 2 k C D k ω d w 2 ] C D k ω = max ( 2 ρ σ ω 2 1 ω ∂ k ∂ x j ∂
ω ∂ x j , 10 − 10 ) F 2 = tanh ( a r g 2 2 ) a r g 2 = max ( 2 k 0.09 ω d w , 500 ν d w 2 ω ) σ k = F 1 σ k 1 + ( 1 − F 1 ) σ k 2 σ ω = F 1 σ ω 1 + ( 1 − F 1 ) σ ω 2 β = F 1 β 1 + ( 1 − F 1 ) β 2 γ =
β β * − σ ω κ 2 β * μ t = ρ a 1 k max ( a 1 ω , S F 2 ) S = 2 S i j S i j S i j = 1 2 ( ∂ u i ∂ x j − ∂ u j ∂ x i )$
$k$—turbulent kinetic energy;
$ω$—turbulent kinetic energy specific dissipation rate;
$ρ$—fluid density;
$μ$—fluid dynamic viscosity;
$μ t$—fluid eddy viscosity;
$d w$—distance to the wall;
$x j$—coordinates by space;
$F 1$, $F 2$—blending functions;
$S$—strain rate;
$u$—fluid velocity;
$C D k ω$—cross-diffusion term.
In this paper, there are no discussions about closing empirical coefficients, which are standard for the k-ω SST model in ANSYS Fluent 19.2:
$β * = 0.09 , = 0.41 , a 1 = 0.31 , σ k 1 = 0.85 , σ k 2 = 1.0 , σ ω 1 = 0.5 , σ ω 2 = 0.856 , β 1 = 0.075 , β 2 = 0.0828 .$
The Ansys Fluent 19.2 software package was used for numerical simulation. The mesh and solver settings used in the simulation are shown in
Table 3
. An example mesh is shown in
Figure 4
. The boundary condition on the inlet is the velocity which is vary depending on Re and in the outlet is the Gauge pressure, which is equal to zero. To provide better accuracy there are prismatic
pipe regions: 10D
before the inlet and 10D
after the outlet.
As a control parameter characterizing the accuracy of the simulation, the coefficient of hydraulic resistance was used, calculated by the formula [
$ζ A N S Y S = 2 · P 1 − P 0 ρ w 0 2 ,$
$ζ A N S Y S$—coefficient of hydraulic resistance, obtained by numerical simulation;
$P 1$—mass-averaged total inlet pressure, Pa;
$P 0$—mass-averaged total outlet pressure, Pa.
The modeling error was estimated by the formula:
$δ = 100 · | ζ A N S Y S − ζ t h | ζ t h ,$
$ζ t h$
—the value of the coefficient of hydraulic losses according to the literature data.
$ζ t h = ∑ ζ p i p e , i · ( D 0 D p i p e , i ) 4 + ∑ ζ s p e c , i · ( D 0 D s p e c , i ) 4 ,$
$ζ p i p e , i$—coefficient of hydraulic losses of pipes;
$ζ s p e c , i$
—coefficient of local losses, determined from the data [
$D p i p e , i$—pipe’s diameter, m;
Coefficient of hydraulic losses of pipes calculated by the formula:
$ζ p i p e , i = 0.3164 R e 0.25 · l p i p e , i D p i p e , i ,$
$l p i p e , i$—pipe section length, m.
To test the hypothesis of the presence of a correlation, the Pearson test [
] is used. The condition for the presence of a correlation:
$r p$—Pearson’s test statistics;
$r k r i t$—critical value of the Pearson criterion;
The Pearson test statistic is determined by the formula [
$c p$—Pearson’s linear correlation coefficient.
The critical value of the statistics of the Pearson test is determined by the formula:
$r k r i t = T 0.95 , 2 n − 2 ,$
$T 0.95 , 2 n − 2$—quantile of Student’s distribution of 0.95 level with 2n − 2 degrees of freedom;
$n$—number of points.
4. Results and Discussion
As a result of research on grid convergence for all considered geometries, it was possible to reach an acceptable error level of 10%. Graphs illustrating grid convergence are shown in
Figure 5
Figure 6
Figure 7
. As can be seen from
Figure 5
Figure 6
Figure 7
, the error achieved during the transition to grid convergence mostly increases with an increase in the degree of flow turbulence with an increase in the average Re along the channel length.
According to the results of research on grid convergence for each of the channels under consideration for the considered modes, the sizes of global elements were found that ensure grid convergence,
after which these sizes were reduced to a dimensionless form to obtain the value of the parameter Ko. The results of data processing are shown in
Table 4
. In some cases, the smaller grid leads to a local decrease in accuracy. It can be related to the changing of the aspect ratio of the global element and near-wall element, and also, it can be related
to local errors within the convergence process. This effect should be considered in detail in further studies.
The correlations obtained during the correlation analysis are shown in
Figure 8
Table 5
. As can be seen from
Figure 8
, the correlation for each type of channel has its own slope, while all correlations are statistically significant. Correlations at low Re have a small slope angle, from which we can conclude that in
this zone, approximately the same number of layers is needed to resolve the logarithmic boundary layer. At high values of Re’, which corresponds to the transition to developed turbulence, the ratio
of the size of the optimal element to the thickness of the boundary layer increases at a faster rate, which indicates that in this zone a smaller number of elements is required to resolve the
logarithmic boundary layer, which can be explained on average the smaller contribution of the wall to the flow pattern under these regimes. The overall correlation that unites the points for all the
considered geometries also has statistical significance and is quite strong.
In the overall correlation, the power at Re′ is 0.762, while the thickness of the boundary layer is proportional to Re′^−13/14, which indicates that the element size increases more slowly than the
thickness of the boundary layers, so that higher Re corresponds to a smaller element size, which is a physical result.
To check the adequacy of the overall correlation, parametric calculations of the compound channel were carried out at different Reynolds numbers at the input. During validation, for each Re, Re’ was
found to correspond to the average channel diameter, after which, using correlations, the number Ko was found, from which the linear size of the global grid element was expressed. Using the element
size estimated a priori from correlations, a grid is constructed similarly to the considered channels. According to the results of calculations, the values of the control parameter corresponding to
the literature data with an acceptable accuracy of 10%. The calculation results are shown in
Figure 9
Thus, based on the results of the calculations of the compound channel, it can be concluded that the proposed ratios are generally adequate in relation to channels containing diffuser sections,
sudden expansion and the contraction, since when modeling the compound channel, the correlations predict with acceptable accuracy the value of the element size that provides the necessary calculation
accuracy. If the correlations do not provide the required accuracy, the values obtained from them can be used as a first approximation in the study of grid convergence.
5. Conclusions
Based on the results of the simulation of turbulent flows in typical channels, as well as the analysis of the calculated data, the following conclusions can be drawn:
• There are regularities that relate the size of the grid model element, which ensures convergence along the grid, with the regime and geometric parameters of the flow in the channel;
• As a dimensionless similarity criterion, one can introduce the coefficient Ko, the ratio of the size of the grid model element that ensures grid convergence to the thickness of the turbulent
boundary layer;
• There are statistically significant correlations Ko(Re’) for channels with sudden expansion, sudden contraction and diffusers, and there is also an overall statistically significant correlation
• This correlation makes it possible to a priori estimate the required size of the grid model element, including for compound channels, the simulation results using the obtained grid settings are
within acceptable limits compared to the literature data.
Author Contributions
Conceptualization, S.O. and I.K.; methodology, P.B. and N.R.; software, P.B.; validation, P.B. and S.O.; formal analysis, A.R.; investigation, A.R.; resources, N.R. and I.K.; data curation, N.R.;
writing—original draft preparation, P.B.; writing—review and editing, A.R.; visualization, P.B.; supervision, S.O.; project administration, I.K.; funding acquisition, N.R. All authors have read and
agreed to the published version of the manuscript.
This study conducted by the Moscow Power Engineering Institute was financially supported by the Ministry of Science and Higher Education of the Russian Federation (project no. FSWF-2020-0020).
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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246–261. [Google Scholar] [CrossRef]
14. Ben-Mansour, R.; Al-Hadhrami, L. Effect of Reynolds Number and Property Variation on Fluid Flow and Heat Transfer in the Entrance Region of a Turbine Blade Internal-Cooling Channel. Int. J.
Rotating Mach. 2005, 2005, 36–44. [Google Scholar] [CrossRef]
15. Komarov, I.; Osipov, S.; Vegera, A.; Kharlamova, D.; Zonov, A. Verification of Computer Flow Simulation in Confuser and Diffuser Channels. In Proceedings of the International Symposium on
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Figure 2. Sketches of the considered geometries: (a)—channel with a sudden expansion; (b)—channel with a sudden contraction; (c)—diffuser channel; (d)—compound channel.
Figure 5. Plots δ(1/Δ) for channel with sudden expansion: (a) F[0]/F[1] = 0.1; (b) F[0]/F[1] = 0.3; (c) F[0]/F[1] = 0.5.
Figure 6. Plots δ(1/Δ) for channel with sudden contraction: (a) F[0]/F[1] = 0.1; (b) F[0]/F[1] = 0.3; (c) F[0]/F[1] = 0.5.
Figure 9. Graph of the coefficient of hydraulic resistance of the compound channel from the Reynolds number.
Channel with a Sudden Expansion
$F 0 F 1$ 0.1 0.3 0.5
$D 0$, mm 84 84 84
$D 1$, mm 265.6 153.4 118.8
$l 0$, mm 50 50 50
$l 1$, mm 1400 700 700
$R e 0$ 20,000; 60,000; 100,000 20,000; 60,000; 100,000 20,000; 60,000; 100,000
Channel with a sudden contraction
$F 1 F 0$ 0.1 0.3 0.5
$D 0$, mm 84 84 84
$D 1$, mm 48.1 83.3 107.5
$l 0$, mm 50 50 50
$l 1$, mm 700 700 700
20,000 20,000 20,000
$R e 0$ 60,000 60,000 60,000
100,000 100,000 100,000
Diffuser channel
$α , °$ 10 15 20
$D 0$, mm 84 84 84
$D 1$, mm 220.4 289.32 359
$l 0$, mm 168 168 168
$l 1$, mm 700 700 1078
$R e 0$ 20,000; 60,000; 100,000 20,000; 60,000; 100,000 20,000; 60,000; 100,000
$D 0 , mm$ $l 0 , mm$ $D 1 , mm$ $l 1 , mm$ $D 2 , m m$ $l 2 , mm$ $D 3 , mm$ $l 3 , mm$
$α , °$ 14
$R e 0$ 60,000
General Steady State RANS, 2D Axisymmetric Turbulence Model k-ω SST
Velocity inlet, m/s 10.434 Gauge pressure outlet, Pa 0
Fluid $Air at 25 ℃$ First near-wall prismatic layer y[+] 1
$ρ$, kg/m^3 1.225 Number of prismatic layers 10
$ν$, M^2/c 1.46 ·10^−5 Growth coefficient 1.1
Meshing method Unstructured, triangles Global element size, mm 0.2–40
Sudden Expansion
Re 20,000 60,000 100,000 20,000 60,000 100,000 20,000 60,000 100,000
$F 0 F 1$ 0.1 0.1 0.1 0.3 0.1 0.1 0.5 0.5 0.5
Re′ 6324 18,973 31,622 10,959 32,879 54,799 14,142 42,426 70,710
$Δ o p t$, mm 14.48 5.4 3.1 6.8 2.67 1.28 3.37 2.02 1.09
Ko 0.107 0.111 0.103 0.146 0.158 0.122 0.117 0.196 0.169
Sudden contraction
Re 20,000 60,000 100,000 20,000 60,000 100,000 20,000 60,000 100,000
$F 1 F 0$ 0.1 0.1 0.1 0.3 0.1 0.1 0.5 0.5 0.5
Re′ 61,632 184,704 308,160 106,735 319,872 533,675 137,743 412,800 688,716
$Δ o p t$, MM 0.63 0.40 0.85 0.74 1.01 0.88 1.16 0.75 0.89
Ko 0.213 0.591 1.294 0.087 0.329 0.462 0.083 0.149 0.414
Re 20,000 60,000 100,000 20,000 60,000 100,000 20,000 60,000 100,000
Re′ 15,245 45,735 76,225 11,613 34,840 58,067 9,359 28,077 46,796
$Δ o p t$, mm 0.4 0.3 0.25 1 0.5 0.3 2.5 1 0.8
Ko 0.016 0.033 0.045 0.023 0.033 0.031 0.039 0.043 0.056
Geometry Correlation c[p] $| r p |$ $r k r i t$
Sudden expansion $K o = 0.031 R e ′ 0.1424$ 0.497 2.29 1.74
Sudden contraction $K o = 8 · 10 − 7 R e ′ 1.108$ 0.97 16.06 1.74
Diffuser $K o = 0.0027 R e ′ 0.2456$ 0.492 2.26 1.74
General $K o = 8 · 10 − 5 R e ′ 0.7062$ 0.652 3.44 1.70
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MDPI and ACS Style
Bryzgunov, P.; Osipov, S.; Komarov, I.; Rogalev, A.; Rogalev, N. Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and
Compound Channels. Inventions 2023, 8, 4. https://doi.org/10.3390/inventions8010004
AMA Style
Bryzgunov P, Osipov S, Komarov I, Rogalev A, Rogalev N. Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow Simulation in Single and Compound
Channels. Inventions. 2023; 8(1):4. https://doi.org/10.3390/inventions8010004
Chicago/Turabian Style
Bryzgunov, Pavel, Sergey Osipov, Ivan Komarov, Andrey Rogalev, and Nikolay Rogalev. 2023. "Research and Development of Criterial Correlations for the Optimal Grid Element Size Used for RANS Flow
Simulation in Single and Compound Channels" Inventions 8, no. 1: 4. https://doi.org/10.3390/inventions8010004
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How to Estimate EAMCET 2013 Rank ? How to Calculate AP EAMCET 2013 Rank ? Free Eamcet 2013 Rank Estimators Free eamcet 2013 rank CalculatorsHow to Estimate EAMCET 2013 Rank ? How to Calculate AP EAMCET 2013 Rank ? Free Eamcet 2013 Rank Estimators Free eamcet 2013 rank Calculators
The AP Eamcet 2013 Marks are published shortly All the Students of Eamcet 2013 are eargly waiting for their Eamcet 2013 Marks and Ransk. Finally Ranks are declared by the JNTU Hyderabad Shortly .
This time we providing fast accessing links for EAMCET 2013 Ranks in this website . The EAMCET Ranks will be available fastly.
• Eamcet 2013 Marks
• Eamcet 2013 Ranks
• EAMCET 2013 Free Mock Counselling
• Free EAMCET 2013 Mock Counselling
Post a Comment
1. thank you sir for your Eamcet estimators , i count my rank in it and i get good rank from your calculator , but i have some doubt sir ,i snd a mail to plz read and reply me to my mail id
rupa_rupa@gmail.com or contact to my mobile number which is mail to u
2. saicharan patelSunday, June 20, 2010
Sir my marks in eamcet are 102 and my group marks are 594 what is my expected rank?
3. VENUGOPAL KRISHNASunday, June 20, 2010
My marks in eamcet 2010 is 86. I got 538 marks out of 600 in groups of Inter exam. What rank can I expected in EAMCET 2010 ?
4. MY MARKS IN EAMCET 2010 IS 42 AND I GOT IN MY OPTIONAL SUBJECTS SCORE 327,WHAT IS THE MY EXPECTED RANK IN EAMCET 2010?PLEASE GIVE ME ANSWER NOW.
5. my marks in eamcet 2010 is 42 and I got intermediate optional subjects score 327 so what is the my expected rank in eamcet 2010?
6. my marks in eamcet are 105 i have got 571/600 i belong to oc please tell my approximate rank sir
7. sir my marks in eamcet are 102 and my group marks are 420/600 please can u help me estimate my rank........
8. sir..i checked your rank estimator n it showed 5000+ in medical stream.. i am a girl n belong to bc-e category..so i d like to know if this is the final rank which i will be gettin..because the
the reservation category is not included in it..or will my rank change if bc-e category is included..plz help me sir .. my id.
9. sir plz reply d above question to my id
regarding 5000+ marks n bc-e reservation.. i shall really be grateful to u..thanks in advance..
10. i g.alekhya got 78 in eamcet and 570\600 in group in inter please estimate my rank thanking you sir
11. sir my marks in eamcet are 50 and my marks in ipe are 811/1000 so plese estimate my rank
12. VAISHNAVISAID I GOT IN IPE 551 AND IN EAMCET 73 MARKS FOR THAT WHICH RANK I EXPECTED
13. I GOT IN EAMCET 73 MARKS AND IN IPE 551 WHAT RANK I EXPECT
14. i got in eamcet 59&ipe marks in groups r 420 what rank i accept
15. sir I GOT IN EAMCET 50 MARKS AND IN IPE 421 WHAT RANK I EXPECT
16. D.Anil kumar
caste oc
eamcet 44
ip 515/600
17. ANIL said
sir I GOT IN EAMCET 44 MARKS AND IN IPE 515/600 WHAT RANK I EXPECT
18. i got 81 marks and 572
19. 0
m,mv i have got109marks in eamscet and589 out
of 600 in intermedjiate public
examinations . please estimate my rank.
20. bittu said igot 74 marks n 556 marks .please estimate my rank.
21. bittu,i got 74 marks and 556 out of 600 marks. please estimate my rank.
22. Mounika said.,.,I have got 103 marks in Eamcet & 594/600.,.,So please check my estimate rank.,Mail me or call me on .,.,
23. divya said i got 76 marks in eamcet.in ipe 316 marks out of 600.pls estimate my rank.
24. i scored 45 in EAMCET and 329 in optionals. what will be my rank in EAMCET.
In which college i'll be getting free seat in Ranga Reddy dist.
25. sir,i got 51 marks in eamcet&326 marks in groups what is my rank
26. i scored 81 marks in eamcet and 559 marks in ipe please estimate my rank
27. i got 67 in eamcet&557 in groups.please tell my rank
28. I am mukesh, i got 44 marks in eamcet& 388/600 in groups pls estimate my rank & which college i'll be free seat in RR DIST, MED DIST, HYD
29. My name is mukesh i got 44 in eamcet & 388/600 in groups what is my rank
30. i got 66 marks in eamcet and 537 marks in groups tell me my rank caste:oc
31. sir i got 61 marks in eamcet.521 in ipe out of 600.pls estimate my rank
32. i got 136 marks in EAMCET and 591 marks out of 600 in inter. my expected rank ?
33. sir i got 73 marks in eamcet and 540 in groups. pls tell me my rank.
34. Sir my EAMCET Marks is 57 My INTER Marks is 702 What is my expected
35. sir i got 86 in eamcet and 491/600 in ipe.i am a bipc student what rank can i expect
36. sir i got 65 marks in eamcet and 517 in groups.so,please tell me my rank
In which college i'll be getting free seat in andhra pradesh
manikanta from vijayawada
37. This is manikanta from vijayawada
i got 65 marks in eamcet and 517 in groups. so,please tell me my rank.
38. i got 49 marks in eamcet and 694 in mpc pls tell me my
rank in eamcet
39. divya:i got eamcet marks 96and ipe 498.what rank could i get,please tell me sir.
40. i got 48 marks in eamcet and 523 marks in groups please tell me my rank
41. i got 48 marks in eamcet and 523 marks in groups please tell me my rank
42. i got 52 in eamcet and 857 in Inter. Please tell my rank. i am Bi.P.C student.
43. i got 88 marks in eamcet and 557/600 in intermediate please send my estimated rank to my mail : nenu_naadesam@yahoo.com
44. i got 40 marks in eamcet and 150/300 in intermediate estimate the rank to my email: nenu_naadesam@yahoo.com
45. I got 101 marks in EAMCET and 577/600 in optionals what is my expected rank in EAMCET 2010
46. hai, i am sailaja. i got 98 marks in eamcet and 418 marks in groups. what is my expected rank in eamcet 2010?piease tell me now...
47. i got 43 in mpc total 556 what is my rank in eamcet
48. i got 63 marks in eamcet mpc and 519 in group what is my rank
49. hhi dnhi i am Prashanth . i want to know my rank in eamcet and which college i will get aseat . my marks are 47 in eamcet and 503/600 in ipe. i belong to sc caste
50. i am Prashanth igot 47 in eamcet .can u tell me which rank i will get. and which college i would get seat as i belong to sc cast
51. hi! i m kavya.i want to know my rank in eamcet engineering stream and in which colleges i will get seats.my eamcet marks are 61 and 540/600 in ipe.i belong to oc caste.
52. Iam rakesh i got 58 in eamcet and i got 544 in groups i belong to bc-a caste
53. G.KISHORE (MALE)EAM-99 (MED) IPE (AVG)-16 AU BC-B
54. i m.ram mohan got 97 in eamcet 578in ipe i belongs to bc-b caste
55. i am midhun i got 79 marks in eamcet and got 512 in IPE i belong to oc caste tell which rank i will get
56. hi
57. E-44,I-730.CASTE-BC A,WHICH RANK I WILL GET
58. hi, my name is B.LAVANYA. my group b.p.c. and my eamcet marks i secured is 47. my caste sc(c). i am interested in doing MBBS/ BPHARMACY.. Could you kindly estimate my rank and which place.. my
brother email id is . and contact no:
59. sir,I got 61 marks in EAMCET, in IPE i got 566 out of 600 and i belong to BC-b, so can u pleace calculate my rank and sent to my id sir i.e.,
60. My EAMCET marks are 52.Group marks are 314,I belongs to ST_girl ,SVU local,Kindly let me know the chances of getting MBBS,BDS,B.Pharmacy,Ag Bsc
or any other courses.
D Priyanka
61. hai iam sitamaha lakshmi i got 62 marks in eamcet and 89% in inter i got 22%out of 25 do i get the medical seat in collage
62. hai iam shaik saleem of hnk my marks are 43 and my ipe marks are 336 out of 600 how much rank can i expect in eamcet
63. sir i got 84 in eamdet and i belong to oc i scored 578 out of 600 in groups how much rank can i expect in eamcet
64. my name is manikanta i got 68 marks in eamcet and a total of 537 in groups so estimate my rank
65. sir i got 85 marks in EAMCET.568 marks in IPE.could you please tell me what rank i will get?
66. I got 53marks in EAMCET and 543/600 in groups will u pls estimate my rank
67. sir i got 53 marks in eamcet and 543 in groups my caste is oc will u pls estimate my rank my email id is
68. i got 41 in eamcet and 284 my self s.t pis estimat my rank
69. sir i got 41 in eamcet and 284 in groups m.p.c my cast is st pls estimat my rank
70. sir i got 41 in eamcet and 284 in groups my caste is st pls estimate my rank
71. sir i got 51 in eamcet and 778/1000 in my ipe,please estimate my rank
72. i got 78 eamcet 568 in groups
73. swapnika
i got 83 in eamcet and 574/600 in groups of ipe.please tell me my estimated rank and in which college of hyd i may get my seat.im local to rayalaseema
74. i(female) scored 71 in eamcet(engineering) and 572/600 in groups. wat could b my estimated rank?i belong to bc-b category and OU region.
75. hi sir iam ashok i got marks in mcet 51 and ipe 518/600 i belongs to bc-b category so pls estimate my rank
76. eamcet 2010 ranks released time on 20-07-2010
77. puja singhTuesday, July 13, 2010
eamcet 2010 ranks released on what time
78. hi sir iam ashok i got marks in mcet 51 and ipe 518/600 i belongs to bc-b category so pls estimate my rank
79. hi sir iam ashok i got marks in mcet 51 and ipe 518/600 i belongs to bc-b category so pls estimate my rank
80. M.Balachander raoTuesday, July 13, 2010
sir i secured 521/600 in groups and 54 marks in eamcet in b.p.c i belongs to sc-c category. please estimate my rank and say that am i eligible to AGRICULTURE B.sc
81. SONY HYDWednesday, July 14, 2010
when will eamcet 2010 ranks be released ?
82. my name is harish and iam an mpc student and my marks in eamcet is 72 and i got 960/1000 marks in intermediate which rank can i expect?
83. hi siriam nos studant.eamct 46 bipc 160/300.BC-B.PLEASE ESTIMATE MY RANK.
84. my name is sai i got 46 in eamcet in mpc i got 416/600 which rank can i expect?
85. Anonymous said...
my name is lachakoti i got 30 in eamcet in mpc i got 336/600 which rank can i expect?
86. vijaya lakshmiSaturday, July 17, 2010
i got 72 marks in eamcet,my ipe marks are548/600.
i belogs to {caste}bc-b.gender-girl
87. what is the date of eamcet ranks 2010
88. sir i got 585 in ipe and 110 in eamcet may i know what rank i can get (oc)
89. sir my eamcet score is 110 and ipe marks are 585 may i know my probabale rank plz send a mail to harihara.pulipaka@gmail.com
90. sir,my eamcet marks is 68 and in ipe 510 and i belong to BC-B. I want to know my eamcet rank.pls reply soon
91. when are the eamcet final rankings of 2010
92. when will ap eamcet 2010 ranks willcome
93. timings of eamcet ranks
94. final rank of eamcet 2010
95. vinay; sir i got 54 marks in eamcet and in optionals 524 so how much rank i will get plz send 2 my mail ID Vinayrockstar@YAHOOMAIL.COM
96. hai
97. timing
98. my name is roja i got 42 marks in eamcet and in optionals 492 may i know my probable rank
99. You wouldn¡¯t believe it but I have wasted all day digging for some articles about this. You are a lifesaver, it was an excellent read and has helped me out to no end. Cheers,
100. HI igot 77marks in eamcet &545 marks in groups can you tell me my rank
101. hi my name is roja I have got 42 marks in eamcet and 549marks in optionals can you tell me my rank
102. my marks 114 in eamcet,956 total,groups=587,send to mail
103. g.venkatesh
i got emcet rank 227116
which college&gruop i getin khammam&warangal distrit
i intrested in CIVIL
my catogiry Oc
104. my eamcet rank is 2948 in medicon which medical college will i get a seat if my 4% reservation is considered please email it to me at adilunplugged_ar007@yahoo.com or send an sms to 8801571927
thank you
105. i am sathish i got47 in eamcetand443inI.P.E can u estimate my rank
106. i am wajeed igot 40 marks &my rank is 205828
with collage &group i get in khammam (or) hyd
i interrested in eee
my catagiry is bc.e
plese send for my life is right
107. sir
this is vivek. please update the eamcet councling help line numbrs.....
throuch sms alerts......
my mobile number
108. IAM SIVA MY RANK IS 75059 I WANT EEE BRANCH IN BEST COLLEGE ARROUND SVC REGION PLEASE SENT MY MOBLIE NUMBERS SMS 812578332
109. i got 1,31097 in eamcet what will be my muslim minority local rank
110. how can i get 150 marks in eamcet exam
111. i want eamcet 2010 rank for seat in ou ?
112. wat is the max rank to be obtained in 2011 to get into ou,jntu,cbit
113. hi...........sir i got 485 marks out of 600 and 50 marks out of 160 what are my marks and weightage
114. sir i have got 56 marks in eamcet and ipe marks of about 580/600 what would be my rank in eamcet
115. sir my total marks in ipe is 832/1000.in groups i got 474/600 marks what would be my weightage marks in eamcet
116. I got 140 marks in Eamcet and 585 marks in IPE . What would my approximate rank be?
117. i got 45 in eamcet and 525 in IPE what would be my rank?
118. chunduru sai kishoreWednesday, May 25, 2011
my name kishor from kadapa i got 55 marks 280 ipe marks.what would my approximent rank be?
119. sir iam from guntur my name is sandeep igot ipe % 80 and eamcet marks 64 tell me near rank send sms to this number 8985221961
120. sir i got 53 marks in eamcet and 338 marks in ipe what may be my rank in eamcet msg me at 9700783764
121. sir i got 56in eamcet and 867/1000.how much rank can i expect
122. sir my eamcet marks is 78 and my IPE optional marks is 585 my cast is bc-e what rank i can expect?
123. sir i got 47 marks in eamcet and 73% in IPE tell me near rank send sms to this number 9160143670
124. sir i got 73 marks in eamcet and 369 marks in IPE what may be my rank in eamcet message me at 7893907690
125. sir i got 38 marks in eamcet and 70% in intermediate. what may be my rank in eamcet message me 9642055426
126. sir i have got 62 marks in eamcet and ipe marks of about 528/600 what would be my rank in eamcet
127. sir, i am varshini i have go 67 marks in eamcent and ipe marks 560/600 what would be my rank in engineering
128. estimate eamcet rank in m.p.c
129. sir i got 69/159 in eamcet
n 520/600
what is ma rank
130. my eamcet marks are 75 and inter 585/600 what is my rank
131. sir i got 83 in eamcet and igot 24%out of 25% in i.p.e and my caste is o.c .what rank shall i expect
132. sir i got 45 marks in eamcet and ipe 72% what may be rank in eamcet sead me 8885643542
133. Sir I got 44 in eamcet and ipe 823 what may be the rank I can expect
134. sir i got 44 in eamcet and my percentage is 82% which rank i will get sir plz mail to me lalu.pooji@gmail.com
135. sir i got 44 in eamcet and 82% is my Inter what rank i can expect sir plz send the mail
136. sir i got 85 marks ineamcet ipe548 oc
can i get aseat in bds
137. sir my friend got 53 marks. . .na group percentage is nearly 20%. .plzz tell me that how much rank na wich college she can get.. ...
138. i got eamcet 45 marks and ipe of 19.26% .can u please guess my rank.which college can i get
139. sir i got 75 marks in EAMCET and i have 536/600 for this how much rank i can expect????in EAMCET2012
140. sir i got 75/160 and i have 536/600
how much rank i can expect for dis
141. i have got 44 marks in eamcet and 630 in inter what can be my rank in 2012 eamcet
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Is the Birthday Paradox Really a Paradox?
Posted by Juniorsbook on Oct 13, 2016 in TellMeWhy |
Is the Birthday Paradox Really a Paradox?
Is the Birthday Paradox Really a Paradox? How many people would you have to gather together in a room to ensure at least a 50% chance that two people in the room share a birthday?
Some people might think you’d need 183 people, since that’s half of 366. But they would be wrong! Would you believe you only need 23 people? It seems impossible, but it’s true! This interesting
mathematical oddity is known as the birthday paradox.
Of course, it’s not a true logical paradox, because it’s not self-contradictory. It’s just much unexpected and surprises most people, so it seems like a paradox.
How does the math work? Before we get started on that, let’s assume from here on out that there are only 365 possible birthdays and that every birthday is equally likely. While those assumptions
aren’t completely accurate, they make the math easier and don’t affect the results in any meaningful way.
The birthday paradox is so surprising because we usually tend to view such problems from our own perspective. For example, if you walk into a room with 22 other people, the chances are pretty good
that no one else will have the same birthday as you. With only 22 of the possible 365 days taken up, that leaves 343 out of 365 chances that your birthday will be unique.
Only considering things from our own perspective, however, limits our expectations. Instead of making 22 comparisons (our own birthday versus the other 22 people in the room), we have to compare each
person’s birthday to every other person’s birthday in the room.
The first person compares with 22 other people. The second person compares with 21 other people (subtracting one since the first person already compared with the second).
The third person makes 20 comparisons and so on, down to the second-to-last person only comparing with one other person, the last person. Adding up all these comparisons among 23 people (22 + 21 +
20…+ 1) gives us a total of 253 possible pairs, which makes it much more likely that we’ll find a pair with matching birthdays.
Without diving too deeply into complex probability calculations, let’s take a look at the probability that, in a room of 23 people, no one has the same birthday as another person. Experts say that’s
the easier calculation to make.
The probability that person 1 has a unique birthday is 365/365 since every date is available. For person 2, the probability drops to 364/365, since one date is taken by person 1.
That trend continues until we get to person 23, whose probability of having a unique birthday is 343/365. We must multiply all 23 separate probabilities to find out the probability of everyone having
unique birthdays. Doing the math, we would end up with a probability of 0.491.
Logic tells us that subtracting our result from 1 will give us the probability that at least two people out of the 23 share a birthday. That means that 1 – 0.491, or 0.509 or 50.9%, is the
probability that at least two people in the group of 23 share a birthday.
Adding people to the room will increase the probability that at least one pair of people share a birthday. For example, in a classroom of 30 students, you’d have a 70% chance of two classmates
sharing a birthday. If you increase the number of people in the room to 70, there’s a 99.9% chance that a pair of people will have the same birthday!
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An alternative way to visualize stocks’ correlations
In the universe of Asset Management, and more precisely in the field of portfolio construction, two of the most used mathematical elements are covariance and correlation matrices. These elements are
used to suggest a risk estimation on assets given a specific investment universe. They allow to bring two informations, one relates to the estimation of a standalone risk for each assets (a.k.a the
volatility), the second tries to estimate how all the assets behave together (a.k.a the correlation). Here we will focus on the use of correlation matrix.
Before starting to elaborate a portfolio construction process to invest in a specific universe, one should investigate how the universe “looks like”. In other words, we would like to investigate if
there are some groups of stocks that tend to behave similarly, that have similar features (a.k.a clusters).
Let’s take an example on major US equities, i.e. on a universe of around 600 stocks.
The basic visualization of a correlation matrix is simply a heatmap of all stocks correlation.
Chart 1 – Correlation matrix of US stocks
We can’t see much information, as it is not very readable.
Remark: By the way, here correlations are nearly all positive as we are looking in “absolute” terms on stocks, i.e. keeping their beta market component. The purpose of this article is simply to show
how to generate a good visualization of stocks’ correlation structure, and not about the matrix generation which alone is a wide subject.
One improvement would be to cluster data thanks to a “linkage” function.
Chart 2 – Clustered correlation matrix of US stocks
This is a bit better, as we can spot some clusters. But still, with a lot of data, we can’t visually analyze it properly.
Now, the method we are going to study is part of the Graph Theory field.
Graphs are used to model pairwise relations between objects of a specified universe.
Explained briefly, graphs are mathematical structures that possess two components, nodes and edges. Nodes represent the objects we are studying (here stocks), edges represent the links between all
nodes (here correlations, or more precisely distances) and the strength of the links are reflected by a weight feature (here the correlation/distance coefficient).
Like for the clustered heatmap, we apply a linkage function (to compute a distance measure from the correlation matrix) and map data into a graph. Then we apply a minimization algorithm to chart what
is called a “minimum spanning tree” (i.e. we minimize the total distance, i.e. the sum of edges’ weights, between all stocks to generate a tree).
Chart 3 – Minimum Spanning Tree on US stocks
Here each node represents a stock, and one “edge” (a link) relates each stock to at least another one which exhibits the most “similarity”. The method is quite simple, starting from the correlation
matrix, we compute a distance^(1) matrix, which simply reflects the intensity of the correlation between stocks. When much correlated the distance is quite small and vice versa. Nodes are colored by
their sectoral affiliation and size of nodes are dependent of the market capitalization of the stock.
Now let’s dig into the code to produce this chart.
We will be using the python library NetworkX which allows to generate and analyze networks, and what is our interest here, a minimum spanning tree.
First, let’s import all the librairies we will use.
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import networkx as nx
Then let’s define our data. We load our correlation matrix as well as our stocks data (i.e. name, sector, market cap), inside pandas dataframes.
correl_matrix = pd.read_excel('correlation_data.xlsx')
data_labels = pd.read_excel('label_data.xlsx')
Now that we have our data, we should create a graph object G from our correlation matrix.
G = nx.from_numpy_matrix(np.asmatrix(correl_matrix)) # create a graph G from a numpy matrix
G = nx.relabel_nodes(G,lambda x: correl_matrix.index[x]) # relabel nodes with our correlation labels
From our graph G, we ask to generate the minimum spanning tree^(2) (MST) T.
T = nx.minimum_spanning_tree(G)
In order to chart our tree, we need to choose a “layout”, i.e. to position graphically our nodes. We will be using Fruchterman-Reingold force-directed algorithm^(3) to create the nodes positions.
pos = nx.fruchterman_reingold_layout(T)
Now we focus on aggregating our graph data into dataframes, the primary key being “Asset ID”.
nodes = pd.DataFrame(sorted(T.nodes),columns=['Asset ID']).set_index('Asset ID')
nodes = pd.merge(nodes,data_labels,how='left',left_index=True,right_index=True)
edges = pd.DataFrame(sorted(T.edges(data=True)),columns=['Source','Target','Weight'])
As we would like to use sectors to define colors, we transform the “GICS Sector” serie to a pandas Categorical type and store the corresponding sectors codes.
nodes_cat = nodes.copy()
nodes_cat['GICS Sector Cat']=pd.Categorical(nodes_cat['GICS Sector'])
nodes_cat['GICS Sector Cat Code'] = nodes_cat['GICS Sector Cat'].cat.codes
nodes_cat = nodes_cat.reindex(T.nodes()) # keep the right index order
Thanks to the sector codes, we can map a “colormap” to the sectors.
# we create our color pallet
color_df = pd.DataFrame(data=[plt.cm.Paired.colors]).transpose().rename(columns={0:'color'})
# we map the color for each stock accordingly to their sector affiliation
nodes_cat = pd.merge(nodes_cat,color_df, left_on="GICS Sector Cat Code", right_index=True)
nodes_cat = nodes_cat.reindex(T.nodes()) # keep the right index order
For the sake of a better visualization, we will only display the names of stocks that are the biggest in terms of market cap (here a weight in index > 1%). So we create a “labels_to_draw” vector that
will store only the label name we want to display and an empty string otherwise.
labels_to_draw = {
n: (nodes[nodes.index == n]['Asset Name'].values[0]
if nodes[nodes.index == n]['Active Weight (%)'].values[0]*100 > 1.0
else '')
for n in T.nodes
Same for the size of nodes, we create a vector that will store the size for each node, depending on its index weight (i.e. market cap).
node_size_list = {
n: ((nodes[nodes.index == n]['Active Weight (%)'].values[0]100+1)500)
for n in T.nodes
As we have all our attributes ready, we can create our plot, with the nx.draw() function. To display a legend, we use empty scatter plots (as there is no such feature in networkx).
plt.figure(figsize=(30, 30))
nx.draw(T, pos, with_labels=True,
edge_color = "grey",
width = .1,
node_color = nodes_cat['color'])
for v in range(len(nodes_cat['GICS Sector Cat'].cat.categories)):
plt.scatter([],[], color=plt.cm.Paired.colors[v], label=nodes_cat['GICS Sector Cat'].cat.categories[v],s=10)
plt.legend(loc=1,markerscale=14., labelspacing=1, borderpad=3, fontsize=30)
To sum up, in this article we were able to produce an alternative way to visualize the correlation structure of an equity market universe. With this methodology, we can visually analyze more
precisely on the existence of clusters and link it to some stocks characteristics (market cap, sectors, etc.).
• (1) The distance measure usually used for correlations is the Euclidian distance:
d = \sqrt {2 \times(1- \rho)}
• (2) The MST algorithm used here is Kruskal’s algorithm. Another widely used algorithm is PRIM’s algorithm.
• (3) An alternative layout, that fits well in our use case, is the Force Atlas algorithm, but unfortunately it is not directly available on the networkx lib.
• To manually produce MSTs in a more advanced way, Gephi software is really useful and well equiped.
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The Stacks project
Lemma 14.32.3. Let $A$, $B$ be sets, and that $f : A \to B$ is a map. Consider the simplicial set $U$ with $n$-simplices
\[ A \times _ B A \times _ B \ldots \times _ B A\ (n + 1 \text{ factors)}. \]
see Example 14.3.5. If $f$ is surjective, the morphism $U \to B$ where $B$ indicates the constant simplicial set with value $B$ is a trivial Kan fibration.
Comments (2)
Comment #6779 by Bogdan on
Shouldn't A and B be interchanged in the diagram?
Comment #6935 by Johan on
Thanks and fixed here.
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Why is outdoor maths important?
Why is outdoor maths important?
Exploring mathematics outdoors is beneficial as it will: encourage a mathematical disposition and enable children to make useful links in their learning. Be memorable in that it offers mathematical
pursuits in a context not normally associated with mathematical learning. encourage exploration and risk taking.
How do you teach math outside?
Check out our favourite outdoor Maths activities to enjoy with your primary school class; most of these are entirely adaptable for any year from Reception upwards.
1. Find a Maths connection.
2. Playground problem questions.
3. Get the chalk out!
4. Go on an angle hunt.
5. Surveys.
6. Plan and run Summer Fete activities.
7. Fun and games.
What is mathematical development EYFS?
In the EYFS, mathematics is also subdivided into two aspects: numbers and shape, space and measures. Development Matters in the Early Years Foundation Stage is guidance material that provides
practitioners with examples of mathematical development across different ages and stages.
How do you create a math environment in early years?
7 simple ways to add mathematics to your environment
1. Incorporating Mathematics into your learning environment.
2. Role Play.
3. Sand and Water Play.
4. Nursery Rhymes.
5. Book Area.
6. Creative Area.
7. Investigation.
8. Outside environment.
How can the environment support mathematics?
Young children actively construct mathematical knowledge through everyday interactions with their environment. It should provide access to objects and materials that encourage children to experiment
and learn about key mathematical concepts through everyday play. …
How is mathematics evident in children’s everyday lives?
Math is Everywhere! Mathematics plays a major role in a child’s development and helps children makes sense of the world around them. Math play is evident in pretend play, block play, literacy play,
outdoor play and science play, for example.
How do I teach outside area?
19 Creative Ways to Teach Area and Perimeter
1. Make an area and perimeter anchor chart. Start with an anchor chart!
2. Decorate your classroom.
3. Snack while you learn.
4. Pull out the pattern blocks.
5. Draw a Perimeter Person.
6. Make a math mosaic.
7. Explore area and perimeter with LEGO bricks.
8. Write block letter names.
How can you use math to teach nature?
25 Math Activities Using Natural Materials
1. Finding Symmetry in Nature at Buggy and Buddy.
2. Nature Numbers Made by Children at Everyday Smilestones.
3. Pine Cone Math Station at Nurturestore.
4. Autumn Estimating Jars at Science Spark.
5. Rock Number Sums at Creative Family Fun.
What are the areas of mathematics in early years?
As articulated in professional standards, researchers have identified number and operations, geometry, and measurement as areas particularly important for 3- to 6-year- olds [19]. These play an
especially significant role in building the foundation for mathemat- ics learning [47].
How do you scaffold children’s mathematical development?
Asking questions, putting students in groups and using reading strategies can provide a scaffold. Ask questions: Asking questions isn’t only a scaffolding technique, you probably also do it during
math talk. Good questions get students to analyze concepts you presented and think about how they come to an answer.
How do you support a child’s mathematical development?
Group activities – fun group learning games such as lotto or dominoes games. These encourage other skills as well promote mathematical development. Child led – Having the environment set up in a
maths-friendly manner, so the children have lots of opportunities to discover and learn about maths for themselves.
Why incorporate maths into early years outdoor provision?
This is why incorporating mathematics into early years outdoor provision is so important. If activities are presented to children in the right way they should not even be aware that they are building
up mathematical skills and knowledge. You may after the activity has completed recap and talk to the children about what new skills they have learnt.
What age group are the outdoor maths activities for?
I have grouped theses outdoor maths activities & ideas based on different areas of learning for EYFS (Early Years Foundation Stage). They are primarily for children ages 2-5, but they can be adapted
for older children as well. *Please note that this post contains affiliate links to help with the running cost of this website.
How can we support children’s maths learning outdoors?
In short, supporting children’s maths learning outdoors often means offering them mathematical opportunities that are bigger, noisier and messier, and using the natural and built environment in ways
that are less easy indoors.
How can I promote children’s mathematical development?
Children’s mathematical development can be promoted at any age in the early years through the outdoors environment. You may choose to add some images to your outdoor provision which promote
mathematical development also so that the environment offers a range of supporting material.
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Topological insulator of bosons in 3d via dyon condensation and the statistical Witten effect
In this talk, I will construct a symmetry protected topological phase of bosons in 3d with particle number conservation and time reversal symmetries, which is the direct bosonic analogue of the
familiar electron topological insulator. The construction employs a parton decomposition of bosons, followed by condensation of parton-monopole composites. The surface of the resulting state supports
a gapped symmetry respecting phase with intrinsic toric code topological order where both e and m anyons carry charge 1/2.
It is well-known that one signature of the 3d electron topological insulator is the Witten effect: if the system is coupled to a compact electromagnetic gauge eld, a monopole in the bulk acquires a
half-odd-integer polarization charge. I will discuss the corresponding phenomenon for the constructed topological insulator of bosons: a monopole can remain electrically neutral, but its statistics
are transmuted from bosonic to fermionic. This "statistical Witten effect" guarantees that the surface is either gapless, symmetry broken or carries an intrinsic topological order.
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EPSRC Reference: EP/C014014/1
Title: Coalgebras, Modal Logic, Stone Duality
Principal Investigator: Kurz, Professor A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Computer Science
Organisation: University of Leicester
Scheme: First Grant Scheme Pre-FEC
Starts: 01 February 2006 Ends: 31 May 2008 Value (£): 118,516
EPSRC Research Topic Classifications: Fundamentals of Computing Logic & Combinatorics
EPSRC Industrial Sector Classifications: No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
I.One of the central problems of programming computers is that it isvery difficult to write correct programs or to convince yourself ofthe correctness of some program. One way to tackle this problem
is theuse of logic.Let us first take a brief look at logic. We can use logic to (1) make statements about the world, (2) define when a statement holds or does not hold in the world, (3) deduce new
statements from given ones using rules of reasoning.`World' can mean the world we live in and logic was originally indeeddeveloped to reason about everyday problems. In mathematics, the worldone
reasons about is the world of mathematical objects. Themathematical world is rich enough to model different notions ofcomputation. Mathematical logic thus allows us to devise differentlogics for
different models of computation. The logics relevant forthis proposal are known as modal logics.The upshot of this effort should be to make reasoning aboutcomputations completely precise and thus to
eliminate the errorshumans tend to make when reasoning about programs.II.In my project I will look at particular models of computation whichare called transition systems. Transition systems consist
of statesand relations between states. The idea is that each state representsa given moment of the computation and the relations describe how thecomputation proceeds from on state to another.The
project aims at a general theory of logics for transitionsystems. It will establish the relationship between logics andtransition systems via the following detour that allows us to use acertain
mathematical theory known as Stone duality. Recent developments suggest using co-algebras to represent transitionsystems. Coalgebras are in a special relationship---calledduality---to algebras. In a
similar way as known from solvingequations in school, algebra can be used to formulate reasoningprinciples.In particular, the aims of this proposal are the following. Toassociate to any type of
transition system an appropriate logic. Toshow how these logics can be applied to the verification of statementsabout programs. To investigate how certain concepts and tools ofmathematical logic can
be adapted to coalgebras and their logics.III.The project will contribute to the theory of coalgebras as a generaltheory of transition systems as developed in the 1990s by manyresearchers. It will
also be an important contribution to the recentworks on the connections between (modal) logic andcoalgebras. Coalgebras and modal logic have received attention fromresearchers in different areas of
mathematics and computer science andthis research will bring to light new connections them.In a wider context, the project concerns the fundamental relationshipunderlying models of computation on the
one hand and logic on theother hand. The development of the theory of coalgebras opens up thepossibility of integrating existing insights and to explore newdirections.
Key Findings
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Potential use in non-academic contexts
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Description This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Date Materialised
Sectors submitted by the Researcher
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Project URL:
Further Information:
Organisation Website: http://www.le.ac.uk
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On the degree of groups of polynomial subgroup growth
Let G be a finitely generated residually finite group and let a[n](G) denote the number of index n subgroups of G. If a[n](G) ≤ n^a for some a and for all n, then G is said to have polynomial
subgroup growth (PSG, for short). The degree of G is then defined by deg(G) = lim sup log an(G)/log n Very little seems to be known about the relation between deg(G) and the algebraic structure of G.
We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if H ≤ G is a finite
index subgroup, then deg(G) ≤ deg(H) + 1. A large part of the paper is devoted to the structure of groups of small degree. We show that a(G) is bounded above by a linear function of n if and only if
G is virtually cyclic. We then determine all groups of degree less than 3/2, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely
generated group cannot lie in the open interval (1,3/2). Our methods are largely number-theoretic, and density theorems àla Chebotarev play essential role in the proofs. Most of the results also rely
implicitly on the Classification of Finite Simple Groups.
Dive into the research topics of 'On the degree of groups of polynomial subgroup growth'. Together they form a unique fingerprint.
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Bachelor"s Programme in Mathematics at HSE University in Nizhny Novgorod
Abrashkin, Anatoly A.
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Mathematics: Professor
Best Teacher – 2017, 2016
Багаев Андрей Владимирович
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor
Barinova, Marina
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor; HSE Campus in Nizhny
Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Laboratory of Topological Methods in Dynamics: Senior Research Fellow; HSE Campus in Nizhny Novgorod;
Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Senior Research Fellow
Best Teacher – 2023
Bobrovskii, Andrei
HSE Campus in Nizhny Novgorod; Postgraduate Programme (Aspirantura): Doctoral Student; HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod);
Department of Fundamental Mathematics: Lecturer; HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Laboratory of Topological Methods in
Dynamics: Research Assistant; HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and
Applications : Research Assistant
Bozhenova, Valeria
Faculty of Law; School of Digital Law and Bio-Law: Lecturer; Faculty of Computer Science; Big Data and Information Retrieval School: Senior Lecturer; Faculty of Computer Science; Digital Skills
Development Unit: Curriculum Coordinator
Galkin, Oleg
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor; HSE Campus in Nizhny
Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Senior Research Fellow
Best Teacher – 2022
Galkina, Svetlana
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor; HSE Campus in Nizhny
Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Senior Research Fellow
Best Teacher – 2024
Grishakova, Marina
HSE Campus in Nizhny Novgorod; Department of Physical Training: Department Head, Senior Lecturer
Gurevich, Elena
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor; HSE Campus in Nizhny
Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Senior Research Fellow; HSE Campus in
Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Научно-учебная группа «Регулярная динамика и хаос»: Associate Professor
Zhuzhoma, Evgeny V.
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Professor; HSE Campus in Nizhny Novgorod;
Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Laboratory of Topological Methods in Dynamics: Leading Research Fellow; HSE Campus in Nizhny Novgorod; Faculty of
Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Chief Research Fellow
Best Teacher – 2022
Klinshov, Vladimir
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Professor; HSE Campus in Nizhny Novgorod;
Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Chief Research Fellow
Kocherov, Sergey
HSE Campus in Nizhny Novgorod; Faculty of Humanities (Nizhny Novgorod); School of Literature and Intercultural Communication: Professor
Kuzenkov, Pavel
Faculty of Humanities; School of History: Associate Professor
Medvedev, Timur V.
HSE Campus in Nizhny Novgorod; Laboratory of Algorithms and Technologies for Networks Analysis (Nizhny Novgorod): Deputy Laboratory Head; HSE Campus in Nizhny Novgorod; Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor
Best Teacher – 2024, 2020, 2019, 2018
Melnikov, I. E.
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Lecturer; HSE Campus in Nizhny Novgorod; Faculty
of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Research Assistant
Mulyar, Olga
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor
Nozdrinova, Elena
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor; HSE Campus in Nizhny
Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Senior Research Fellow
Best Teacher – 2024, 2023
Polotovskiy, Grigory
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Associate Professor; HSE Campus in Nizhny
Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Senior Research Fellow
Best Teacher – 2024, 2023, 2022, 2021
Chilina, Ekaterina
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Lecturer; HSE Campus in Nizhny Novgorod; Faculty
of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Laboratory of Topological Methods in Dynamics: Research Assistant; HSE Campus in Nizhny Novgorod; Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Junior Research Fellow
Shubin, Danila
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Lecturer; HSE Campus in Nizhny Novgorod; Faculty
of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Laboratory of Topological Methods in Dynamics: Research Assistant; HSE Campus in Nizhny Novgorod; Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Research Fellow
Yakovlev, Evgeniy
HSE Campus in Nizhny Novgorod; Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); Department of Fundamental Mathematics: Professor; HSE Campus in Nizhny Novgorod;
Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod); International Laboratory of Dynamical Systems and Applications : Leading Research Fellow, Senior Research Fellow
Best Teacher – 2021
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Tired worn out and aching this wekk.
First full week of work since before Christmas and really suffering this week. So washed out that i don't want to get up in the morning and now getting the old pains back which had subsided during
the school hols.
I am on new drugs for another condition and am wondering if they are causing the tiredness...
I am not sure whether it is the stress of the job that increases the aches and pains etc or rushing around more than i do when i am off. Does anyone else have this problem?
• Hi Lupin,
Just now ll the time..... I am lucky and self employed but its taking me hours to do hat used to take mins and I am so tired as well.
Hopefully as you get more used to it it will get better? Fingers crossed you have a better day tomorrow. Cris x
• I am constantly tired. I think it's the PA, the continual pain and the meds. One's body can only take so much. Never mind, eh? Onwards and downwards! DD
Have you got the despatches? No, I always walk like this. Eddie Braben
• Hello, I've been really tired since Christmas. I think its been the worry of hospital visits and waiting for test results as well as the pain, which has increased especially in my neck.
Since getting the 'more or less all clear' from the hospital Gyneocologist, I've felt better, but still tired! These horrible dark dreary days don't help either. If you've started back to work, I
expect you are doing a lot more and have a lot more stress, all these things add up and together with the pain and illness, make things feel worse. Sorry this sounds depressing reading it back,
but I don't mean to depress you, only to say its not uncommon and you'r not alone with the way you feel.
• [These horrible dark dreary days don't help either.
Reckon we are programmed to hibernate?!
Serioulsy Lupin rest as much as poss over the weekend and then see how it goes.
If it doesn't settle then you could talk to the doc aboujt it next visit?
toni xx
• Hi Lupin
I think it's not surprising you feel so tired and achey... going back to work really does take it out of your bones. :roll: ol' arther though knows how to make us feel 'wished' too!!
hope you get a chance to rest over the weekend and recharge the batteries.
Iris xxx
• Hello lupin,
Ive just done my first proper week since before Xmas too and iam shattered. Struggling to keep up with not only work but house etc too. I went shopping last night at 8 and what should have taked
me half hour tops took an hour and a half! Pain always seems to eat away more with me when iam tired too. I should have text my partner but was just too physically tired to get phone out of
pocket before i set off for home.
Try to rest although i know thats so easily said.
Hope next week is better for you
all the best
• Hi Lupin
I am sorry you are suffering, hopefully work will get easier for you, I am lucky in a way that I dont work...but I am bringing up my two granddaughters...well me and OH but he does work...the
thing is since Christmas I cant get going, so I have an idea of what you mean.
All I can suggest is rest when you can, I do hope you feel better very soon...sending you loads of hugs (((((()))))
Barbara xx
• Thankyou folks...
Feeling worse today just think i have picked something up from work...feel sick and for good measure both hips are aching like mad as well as my knees burning...(thats a new one). Hands are now
playing up but that is because i decided to do some knitting.
Going to try and take it easy tomorrow. Roll on the warmer weather..
• Hi Lupes, really hope you can have a restful day tomorrow, sounds like you need it, I get so tired sometimes, never sure if it's the disease or the meds, combination of both, plus I haven't been
eating right for over two months now so not only do I not have the energy reserves but the meds are more or less being taken on an empty stomach.. :roll: :roll:
One good thing, I have lost about a stone and a half :shock:
Be well, make some time for yourself
Ra-1996 -2013 RIP...
Cleo - 1996 to 2011. RIP
• 7 hours !!!!! spent trailing around a shopping center yesterday with hubby and two wingey children have taken their toll and today i have felt so fatigued i could weep. Dark, wet and damp
mornings certainly dont help. Hubby is now also arranging a weekend away going straight after i have done an early shift at work - I feel sometimes I talk to the wall about how awful i feel. wish
i could wave a wand for us all and banish this away xxx
• Hi Lupin
as many of us I too have same problem, work really does wipe me out as i am up and down the stairs all day. poor diet and lack of sleep are not helping so cant only blame RA
Study work and cleaning do not mix with RA and it's usually the cleaning that suffers but u just have to do what u can.
try not to push yourself too much and hope u can get some rest soon.
• Hi Lupin,
You really must try and have a rest tomorrow. Err that means no knitting either. Seriously it is so hard when we go back to work and then try and keep up with everything.
I work part time now and I have to have a sleep after tea every night. I wouldn’t mind but I like many of us find it difficult to sleep at night so it becomes a vicious circle doesn’t it.
There are such a lot of bugs about at the moment that it makes it difficult to avoid catching things. A nurse told me to wear gloves when out and to use wipes whenever possible to keep some of
the germs at bay.
I hope you feel better soon,
Lv, Ix
• oppps....i wish i would listen to my body but i am a foul at times. Decided to start a tidy the attic....Up and down the ladder. Yes you.ve guessed it my hips are giving me grief alone with my
knees. Keep thinking i am still wonder woman and then my body lets me down.
Ah well now slowed down thanks god i only work morning. As for the houswork it quite often does not get done. Well i am going to tire and get some sleep. Nite nite.
• You should've listened to ya Uncle Tony Lupes..... :roll:
Make sure you rest as much as you can tomorrow... Or else..
Best wishes...
Ra-1996 -2013 RIP...
Cleo - 1996 to 2011. RIP
• Hi lupin15,
You remind me of myself, thinking you can do things and then let down once agian my our bodies. You say a weekend away being organised? mayb you can relax there? At least dont try tidying thier
attic they might take offence!
Best wishes
• Gosh i would love to go away at the moment but no can do....boooooooo. Hope that in Feb we can get away for a few days to see my sister and hubby. That is weather permitting...as it always snows
in feb.....Sitting and relaxing at the moment, had my lunch and a pudding (heheeh) well someone had to eat the jersey cream in the fridge. Now all i have got to do is NOT fall asleep.
• spoil yourself a bit Lupes... As long as you are taking it easy that's the main thing.. No more going on flash in the attic!! Lol
Ra-1996 -2013 RIP...
Cleo - 1996 to 2011. RIP
• Well spoke to soon the **** hit the fan.....not a quite afternoon. Foster lad playing up then various other bits...Must admit my hips are not so bad today. I think it was going up and down the
ladder which did them in. Now eating an dip dab then off to try and get some sleep...
• Hope you sleep well Lupes....
Goodnight hun.
Ra-1996 -2013 RIP...
Cleo - 1996 to 2011. RIP
• 21 Welcome
• 11.9K Our Community
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Predicate Simplification - Analytics Database - Teradata Vantage
Predicate simplification is a part of the query rewrite system that transforms a query into an equivalent form, but one that is simpler and more amenable to query optimization. Several of the key
categories of predicate simplification are outlined in this section.
Satisfiability and Transitive Closure
The SAT-TC (SATisfiability-Transitive Closure) rewrite analyzes a set of predicates to determine if either of the following can be exploited to rewrite a request:
• A contradiction.
A simple example is a condition like a=1 AND a=0.
• Inferring new predicates from the existing predicate set by using transitive closure.
For example, consider a request that specifies the predicate a=1 AND a=b. This implies that b=1, which might be useful information for rewriting or otherwise optimizing the request.
Other examples of deriving new conditions by using transitive closure include the following samples:
A=B AND A=C --> B=C
A=5 AND A=B --> B=5
A=5 AND A IS NULL --> FALSE
A=5 AND A IS NOT NULL --> A=5
X > 1 AND Y > X --> Y >= 3
X IN (1,2,3) AND Y=X --> Y IN (1,2,3)
Transitive closure is also applied in the context of extract predicates. For example, for the set of conditions {o_orderdate=‘1999-05-01’ AND EXTRACT(MONTH FROM o_orderdate)>2}, the system adds
EXTRACT(MONTH FROM o_orderdate)=5 based on o_orderdate=‘1999-05-01’, which can then be used to simplify the existing EXTRACT predicate in the query.
For some categories of DML requests specified with BEGIN END constraints on a column with a Period data type, the query rewrite system adds the implied constraint BEGIN(column_1) <END(column_1) to
help to identify unsatisfiable conditions. When these unsatisfiable conditions occur, EXPLAIN text for the query identifies them using the EXPLAIN text phrase unsatisfiable (see EXPLAIN Request
Modifier Phrase Terminology).
As an example, consider the following table definition:
CREATE TABLE t1 (a INTEGER b PERIOD (DATE));
Suppose that you use the following query:
SELECT *
FROM t1
WHERE BEGIN(b) > DATE ‘2010-02-03’ AND
END(b) < DATE ‘2010-02-03’;
The query rewrite system adds an implied constraint BEGIN(b) < END(b), and is able to derive unsatisfiability in conjunction with the query predicates.
The query rewrite system can also apply transitive closure across query blocks, meaning transitive closure between outer and inner query blocks. This allows conditions to be pushed into and out of
subqueries. The basic approach is to combine the query block conditions before computing the transitive closure. The IN and NOT IN operators are treated as = and ≠ operators, respectively. Derived
conditions are added, as appropriate, to each query block.
Consider a simple SQL example. The following SELECT request implies that x<1.
SELECT * FROM t1 WHERE x IN (SELECT y FROM t2 WHERE y<1);
Similarly, the following SELECT request implies that x < 3 and y is in (1,4).
SELECT *
FROM t1
WHERE EXISTS (SELECT *
FROM t2
WHERE y<3
AND x=y)
AND x IN (1,4);
In the current context, the conditions under analysis are referred to as connecting conditions. A connecting condition is one that connects an outer query with a subquery. See Connecting Predicates
for further information about connecting conditions.
Applications of Transitive Closure
Transitive Closure (TC) can optimize date ranges and IN clauses. The following request illustrates one of these cases:
SELECT l_shipmode, SUM (CASE
WHEN o_orderpriority = '1URGENT'
OR o_orderpriority = '2-HIGH'
THEN 1
ELSE 0
FROM lineitem
WHERE l_commitdate < l_receiptdate
AND l_shipdate < l_commitdate
AND l_receiptdate >= '1994-01-01'
AND l_receiptdate < ('1994-06-06')
GROUP BY l_shipmode;
The new set of constraints that can be derived is as follows:
(l_shipdate < l_receiptdate AND
l_commitdate <= '1994-06-04' AND
l_shipdate <= '1994-06-03')
If lineitem or one of its covering indexes is either value-ordered or row-partitioned on l_shipdate, the new constraint l_shipdate<='1994-06-03' enables Vantage to access only a portion of the table
instead of doing a full-table scan.
You might notice performance improvements for some queries because of the extra predicates TC adds. The extra predicates can also be seen in the EXPLAIN reports for requests that use them.
SAT-TC and Query Rewrite
One important aspect of using transitive closure for query rewrite is its application across ON and WHERE clauses. For example, suppose you have the following request:
SELECT product_name, sum(amount*quantity) AS qty
FROM product LEFT OUTER JOIN sales1
ON product_key=sales_product_key
WHERE product_key=10
GROUP BY product_key, product_name ;
For this request, transitive closure adds the inferred predicate sales_product_key=10 to the ON clause. This is particularly effective when the predicate added to the ON clause is a constraint on the
primary index of the inner table in a join.
Another important property of transitive closure is its ability to infer new predicates across the ON clauses of consecutive inner joins. For example, consider the following request.
SELECT product_key, product_name, SUM(s1.amount * s1.quantity+s2.amount * s2.quantity)
AS total
FROM product LEFT OUTER JOIN ((sales1 AS s1 INNER JOIN store
ON s1.sales_store_key=store_key)
INNER JOIN sales2 AS s2
ON s2.sales_store_key=store_key
AND s2.sales_store_key=10)
ON product_key=s1.sales_product_key
AND product_key=s2.sales_product_key
GROUP BY product_key, product_name;
To see this application of transitive closure, consider the consecutive inner joins between s1, s2, and store. The predicates in consecutive inner joins can be treated collectively by transitive
closure as if they were specified in a single WHERE clause. In this example transitive closure processes these predicates as if they appeared as the following compound predicate.
WHERE s1.sales_store_key = store_key
AND s2.sales_store_key = store_key
AND s2.sales_store_key = 10
By grouping the predicates logically like this, transitive closure can derive the new predicates store_key=10 and s1.sales_store_key=10 and then place them in the ON clause of the uppermost inner
join in the set of consecutive inner joins. In this example, that is the ON clause joining to s2.
If a condition is false mathematically, it is said to be contradictory or unsatisfiable. In this context, the opposite of contradictory is satisfiable, regardless of the data. An example might be
specifying the conditions a=1 and a=2 in the same request. If Query Rewrite discovers such an unsatisfiable condition, it simplifies and optimizes the condition in such a way that all joins and
retrievals can be done on a single AMP basis.
One way to take advantage of a contradictory condition is to add CHECK constraints to tables, enabling Query Rewrite to eliminate unnecessary conditions, thus permitting the Optimizer to later
construct better execution plans (see Applications of Transitive Closure).
For example, assume that you want to list all orders made in the first three months of the fiscal year. Assuming that you have access only to the ordertbl view that is a UNION ALL of orders1,
orders2,…,orders12, and the query looks like this:
SELECT *
FROM order_tbl
WHERE EXTRACT(MONTH FROM o_orderdate)<= 3;
Without CHECK constraint, the system must access all of the tables orders1, orders2, … orders12 using the constraint EXTRACT(MONTH FROM o_orderdate)<=3, even though it only needs to access orders1,
orders2, and orders3 to satisfy the request. The only way Query Rewrite knows to filter out the other nine tables is to add CHECK constraints for every table, and then to determine the contradiction
between the CHECK constraints and the query constraint.
For example, if the CHECK constraint on order4 is added, Query Rewrite sees the following compound predicate, which is a contradiction.
EXTRACT(MONTH FROM o_orderdate)<= 3 -- from query
EXTRACT(MONTH FROM o_orderdate)=4 -- from CHECK constraint
For this particular case, Query Rewrite can simply eliminate this step. In general, Query Rewrite needs to know if a set of conditions is satisfiable.
Of course, you would rarely submit a contradictory query that does not return results regardless of the data like a=1 AND a=2. The example in the previous paragraph indicates the need for such
checks. As previously stated, this issue is referred to as the satisfiability problem. Any solution to satisfiability generates a set of conditions and either declares them to be FALSE to denote that
they are contradictory, or TRUE, which means that for some specific data, the set of conditions is satisfiable.
You should be aware of the cost of enforcing CHECK constraints if you decide to use them for horizontal table partitioning. This type of table partitioning is not identical to the row partitioning
that you can specify with a partitioning expression in a PARTITION BY clause for a table or join index, though it is similar conceptually.
SAT-TC and Query Optimization
Query optimization introduces the following additional applications of satisfiability in the usage and maintenance of join indexes:
• Determining whether a join index must be updated to keep it synchronized with an update operation on one or more of its base tables. The term update operation here signifies an insert, update, or
delete operation against the base table in question.
• Determining whether a join index partly or fully covers a query.
The join index update problem can be solved by using the satisfiability check for the conjunction of the join index conditions and the condition applied in the base table maintenance.
Assume the following sparse join index definitions:
CREATE JOIN INDEX j1 AS
SELECT *
FROM lineitem
WHERE EXTRACT (MONTH FROM l_shipdate)<=6;
CREATE JOIN INDEX j2 AS
SELECT *
FROM lineitem
WHERE EXTRACT(MONTH FROM l_shipdate)>=7;
Now consider the following delete operation on the lineitem table for example:
DELETE lineitem
WHERE EXTRACT(MONTH FROM l_shipdate)=12;
This implies that there is a need to update j2, but not j1. The system can make this decision because the Satisfiability check returns TRUE for the following predicate:
EXTRACT(MONTH FROM l_shipdate)=12
EXTRACT(MONTH FROM l_shipdate)>=7
but FALSE for this predicate.
EXTRACT(MONTH FROM l_shipdate)=12
EXTRACT(MONTH FROM l_shipdate)<=6
The problem of determining whether a join index completely or partially covers a query is solved as a set of satisfiability problems. Note that the use of satisfiability in this problem becomes more
important when more complex conditions, like constants in WHERE clause predicates, are specified in the join index definition. This happens, for example, when you create a sparse join index. See the
information about hash and join indexes in Teradata Vantage™ - Database Design, B035-1094 and the documentation for CREATE JOIN INDEX in Teradata Vantage™ - SQL Data Definition Language Syntax and
Examples, B035-1144 for more information about sparse join indexes.
Constant Predicate Evaluation
Constant predicate evaluation evaluates predicates that contain only constants (those having no column references) at the time a request is parsed. For example, consider the following complex
OR ‘a’ IN (‘b’,‘c’)
The system identifies and evaluates constant predicates and then replaces the predicate with the result if it is either TRUE or FALSE.
The IN predicate in this example can be evaluated to FALSE when the database parses the request, so Query Rewrite transforms it as follows:
Domain-based Simplification
Domain-based simplification uses the domain range of the underlying column to simplify a predicate. For example, consider the following table definition:
create table t1 (a1 smallint)
With this table definition, the predicate a1 = 64000 leads to unsatisfiability, as the constant 64000 is not within the smallint range of values:
a1 = 64000 => FALSE
Similarly, consider the following predicate:
a1 in (32800, 80000, 1, 2, 3)
This predicate can be simplified as follows, as the constants 32800 and 80000 are not within the smallint range of values:
a1 in (1, 2, 3)
Check Constraint-based Simplification
Check constraint-based simplification is similar to domain-based simplification, but the range of the underlying column is based on check constraints specified on the column. For example, consider
the following table definition:
create table t1 (a1 int NOT NULL check (a1 < 10))
With this table definition, the following predicate leads to unsatisfiability, as the predicate is outside the allowed range of values based on the check constraint:
a1 > 30 => FALSE
Consolidating Single-Table Predicates
Query rewrite evaluates single-table predicates and rewrites them to eliminate overlapping or redundant conditions. For example, consider the following set of predicates:
a1 NOT IN (1,3) AND
a1 <4 AND
a1 >= 2
If a1 is an INTEGER column these predicates can be simplified as follows:
Similarly, consider the following predicate:
a1 IN (1, 3, 5, 7, 9) AND
a1 > 4
where a1 is an INTEGER column.
The predicate can be simplified as follows:
a1 IN (5,7,9)
Consolidation of single-table predicates enables the Optimizer to calculate more accurate selectivity estimates while at the same time eliminating the execution of redundant predicates.
Query Rewrite also supports simplification of single-table predicates for the BEGIN and END bound functions on Period columns.
For example, consider the following predicate based on the BEGIN bound function:
BEGIN(b)>DATE ‘2005-02-03’ AND
BEGIN(b)>DATE ‘2010-02-03’
Query Rewrite simplifies this to the following predicate:
BEGIN(b)>=DATE ‘2010-02-04’
Similarly, consider the following predicate based on the END bound function:
END(b)< DATE‘2005-02-03’ AND
END(b)< DATE‘2010-02-03’
Query Rewrite simplifies this to the following predicate:
END(b)<= DATE‘2005-02-02’
Consolidation of single table predicate can lead to unsatisfiability as follows:
a1 NOT IN (1, 2, 3) AND
a1>=1 AND
If a1 is an INTEGER column, the conditions are non-overlapping and the predicate is rewritten as follows:
0=1 evaluates to FALSE.
In other cases, the predicate can be simplified to a predicate that does not constrain the set of possible values, as shown:
a1>1 OR a1<2
The predicate can be rewritten as follows:
a1 IS NOT NULL
For a more complex example, consider the following predicate:
(a1 >= 1 AND a1 <= 3) OR
(a1 >= 4 AND a1 <= 10)
If a1 is an INTEGER column, the predicate can be rewritten as follows:
a1 >= 1 AND a1 <= 10
Constant Movearound
The constant movearound rewrite can enable better selectivity estimates and wider use of indexes by the Optimizer. Constant movearound attempts to move constants from one side of a boolean comparison
operator to the other side in order to rewrite predicates of the following form:
<column> <±> <constant_1> <comparison_operator><constant_2>
Constant movearound rewrites predicates of the preceding form to predicates of the following form:
<column> <comparison_operator> <constant_3>
The method does this by removing the plus or minus operation from the column side of the predicate and then adding its negation to the constant-only side of the predicate. The constant expression is
then folded. Any errors that occur during folding, such as overflow or underflow, cause the rewrite to be rejected in favor of using the original expression.
This transformation is only done for the plus and minus arithmetic operators.
For a simple example, consider a1 + 1 > 4, which can be rewritten as a1 > 3.
The constant expression that is moved can also be an INTERVAL expression, as in DATE operations.
For example, consider the following predicate:
date_col + INTERVAL '3' MONTH <= '2007-03-31'
Using constant movearound, this predicate can be rewritten as follows:
date_col <= '2006-12-31'
Constant Substitution
The constant substitution rewrite substitutes the values of columns, if they are available, and attempts to derive unsatisfiability or to simplify the predicate, if possible. Predicates of the
following form are simplified by substituting the values for column_1 and column_2 whenever possible:
<column_1> <±> <constant_1> <comparison_operator>
<column_2> <±> <constant_2>
As an example, consider the following predicate:
a = 10 AND
b = 20 AND
a + 2 = b + 1
Substituting the value 10 for a and 20 for b, the predicate can be simplified to 12 = 21, which is FALSE.
For additional examples, consider a > a + 1, which simplifies to FALSE and a >= a - 1, which is rewritten as a IS NOT NULL.
Simplification by Distribution
This simplification by distribution rewrite attempts to simplify complex AND/OR predicates by using distributive properties.
Consider the following predicate:
a > 5 AND b < 6 AND (a < 2 OR b > 9)
By distributing the AND into the OR branch, the predicate can be simplified as follows:
a > 5 AND b < 6 AND ((a > 5 AND a < 2) OR (b < 6 AND b > 9)) =>
a > 5 AND b < 6 AND ( FALSE OR FALSE) =>
a > 5 AND b < 6 AND FALSE =>
Factoring Common Predicates
The factoring of common predicates is another rewrite based on distributive properties. If a complex AND/OR predicate has some common predicates across the AND/OR branches, these common predicates
can be factored out to simplify the predicate.
Consider the following predicate:
(a1 = 1 AND b1 = 1 AND c1 = 1) OR
(a1 = 1 AND b1 = 1 AND d1 = 1) OR
(a1 = 1 AND e1 = 1)
Note that (a1 = b1 AND b1 = 1) is common to the first two branches of the overall OR predicate. In the first step of factoring, (a1 = b1 AND b1 = 1) can be factored out, resulting in the following
equivalent predicate:
(a1 =1 AND b1 = 1 AND (c1 = 1 OR d1 = 1)) OR
(a1 = 1 AND e1 = 1)
After the first step, note that (a1 = 1) is common to the remaining two branches of the OR predicate. In the second step of factoring, (a1 = 1) can be factored out, resulting in the following
equivalent predicate:
a1 = 1 AND ((b1 = 1 AND (c1 = 1 OR d1 = 1)) OR e1 = 1)
Simplification Based on Containment
Simplification based on containment attempts to simplify complex AND/OR predicates by using the set containment properties of set algebra. There are two kinds of containment rewrite rules: the OR
containment rule and the AND containment rule.
The OR containment rule is as follows:
If A contains A',
A OR (A' AND ..) => A
For example, a > 5 OR (a > 7 AND b < 6) can be simplified to a > 5.
The AND containment rule is as follows:
If A' contains A,
A AND (A' OR ..) => A
For example, a < 10 AND (a < 12 OR b < 6) can be simplified to a < 10.
Duplicate Predicate Removal
Duplicate Predicate Removal eliminates identical conjuncts, disjuncts, or both from AND and OR predicates. The maximum number of duplicate conjuncts and disjuncts compared per predicate is 100.
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How I Teach Algebra to Earn College Credit
As far as exams go, the College Algebra CLEP exam seems pretty straight forward. Learn algebra, pass exam in sha Allah (God willing), done. Still, it requires a little advanced planning.
In our 9th Grade curriculum, we’ve chosen to stick with Saxon, using their Algebra 2 program (3rd edition, specifically). Saxon isn’t the prettiest curriculum around, but it holds water when looking
for a rigorous curriculum to get you ready for exam time. Knowing that Care Bear’s plans are for college, we wanted something well known and respected.
I think there are a lot of great math curriculums out there, and Saxon just seemed to fit the way I was taught (making it easy to step in and help when needed). At the high school level, the parents
aren’t always at the student’s side helping each day. I’ve found when we used other curriculum, I couldn’t just jump in and help when she had a question because they were teaching in a completely
different way than I was as a student. Since Saxon makes sense to Care Bear, and it makes sense to me, we stick with it.
So, what does our weekly work look like?
Care Bear is expected to work on math every school day, 5 days a week on a typical week. It’s pretty straight forward with a “do the next thing” plan. Simply, she works on one lesson or completes a
test each day. Normally she does all her work independently. Test days are admittedly very easy, and aid in making for a shorter work day. In years past we would use the test as the practice set,
rather than the problems in the lesson she just completed, but this year we’ve chosen to make test day its own day. It works for us, and it takes the full school year rather than ending early,
narrowing the summer gap between math programs.
This year has been relatively easy for Care Bear, with her needing very little extra explanation. When she does need assistance, I’m ready and waiting, though she dislikes asking me for help. Why? I
get too excited! Haha! Math was always my favorite subject, with Algebra and Trigonometry at the very top of my list. Care Bear doesn’t share in the math love, so she gets annoyed at my excitement.
Instead, Khan Academy fills in for me (and they are more monotone than I, so my teen is happy). I’ve offered to buy the Saxon Teacher CD if she is having troubles on a regular basis, but since it is
pretty expensive, we hold off until it’s necessary.
CLEP College Algebra Prep
I have to remember, college algebra is harder than high school algebra. The concepts are the same, but you will be faced with more challenging problems than you might see in a high school test.
Knowing that, we use some test prep to fill in that potential gap (though, I do think this is being quite prudent on our part, and probably unnecessary). Since we are aiming for the CLEP College
Algebra exam, we will be purchasing the REA CLEP College Algebra Review book, and the fact that it is one edition back shouldn’t be any trouble. You save money, can get the Kindle version, and math
doesn’t really change, so yay!
That’s about it! Math is right or wrong, black or white. At this level, there aren’t many manipulatives, so it isn’t as fun and pretty as the earlier years, but it’s just as important!
This post is part of iHomeschool Network’s January Hopscotch. Please visit other great bloggers as they explain how they teach various subjects as well!
The rest of the series to check out!
1. How I Teach Writing to Earn College Credit {Lesson Plan Included} (January 13, 2014)
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Understanding Weighted Average Cost of Capital (WACC) - richriddles.com
Overview of WACC
The Weighted Average Cost of Capital (WACC) serves as a comprehensive measure of a company’s cost of capital, encompassing all sources. It is a pivotal benchmark in financial analyses, often
determining the viability of investment opportunities.
Definition of WACC
WACC represents the average rate of return a company needs to compensate all its different investors. The formula for calculating WACC is a proportional average of each component of capital, which
includes equity, debt, preferred stocks, and any other securities that comprise a company’s capital structure. The cost of each type of capital is accordingly weighted by its relative size in the
total capital pool.
Importance of WACC in Financial Decisions
The clear-cut importance of WACC lies in its role as a tool for assessment and decision-making. A company’s WACC serves as a discount rate for the appraisal of potential investment opportunities.
Applying WACC in valuation practices enables financiers to determine the net present value (NPV) of future cash flows. When a project’s rate of return exceeds the WACC, it generally indicates that
the project will add value to the company, making it a potentially worthwhile investment.
Calculating WACC
When assessing a company’s cost of capital, understanding the Weighted Average Cost of Capital (WACC) is crucial. It reflects the average after-tax cost of a company’s various capital sources,
including stock and debt.
WACC Formula Components
WACC is the weighted mean of the cost of equity and the cost of debt, proportionally weighted by their usage in a company’s capital structure. The general WACC formula is:
WACC = (E/V) * Re + (D/V) * Rd * (1-T)
• E = Market value of equity
• D = Market value of debt
• V = E + D (Total market value of the firm’s financing)
• Re = Cost of equity
• Rd = Cost of debt
• T = Corporate tax rate
Cost of Equity
The cost of equity represents the return that investors require for their investment in the firm. One common method to calculate this is the Capital Asset Pricing Model (CAPM) which formula is:
Re = Risk-free rate + Beta * (Equity risk premium)
• The risk-free rate typically refers to the yield on government bonds.
• Beta measures the volatility, or systematic risk, of a security in comparison to the market.
• The equity risk premium reflects the extra returns investors expect for the additional risk of equity over risk-free investments.
Cost of Debt
The cost of debt is based on the average rate the company pays on its debt. This can be estimated using the yield to maturity on existing debt or the interest rate on new debt. It’s important because
debt provides a tax shield due to the deductibility of interest expenses, hence why the cost of debt is factored into the WACC formula as Rd * (1-T).
Market Value Metrics
For calculating WACC accurately, it’s imperative to determine the market value of equity (E) and debt (D), rather than their book values. The market value of a company’s equity can be found by
multiplying its current stock price by the number of outstanding shares. The market value of debt requires calculating the present value of all future cash flows from debt using the market’s required
rate of return for the firm’s debt.
Factors Influencing WACC
Weighted Average Cost of Capital (WACC) calculations are directly impacted by several factors, each contributing to a company’s cost of financing. Understanding these variables is essential for
accurate computation and effective financial decision-making.
Capital Structure
The capital structure significantly dictates the WACC as it represents the proportion of debt to equity a company utilizes to finance its operations. A higher proportion of debt in the capital
structure can reduce WACC due to the tax-deductible nature of interest expense, but it also increases financial risk, which may affect a firm’s credit rating. Conversely, relying more heavily on
equity can lower financial risk but may result in a higher WACC since equity typically demands a higher return than debt.
Tax Considerations
Tax considerations play a pivotal role in influencing WACC. The corporate tax rate directly affects the cost of debt because interest payments on loans are tax-deductible. This means that the
after-tax cost of debt is the interest rate multiplied by (1 – tax rate), thereby decreasing the WACC as the tax rate increases. However, changes in the corporate tax rate can have complex
implications for WACC as they may also shift investor expectations and market valuations.
Market Conditions
Lastly, market conditions reflect the overall health of the economy and the status of the capital markets, encompassing factors such as inflation, perceived riskiness, and availability of capital.
High inflation rates generally lead to higher interest rates, thus raising a company’s cost of borrowing. The perceived riskiness of a company can influence investor demand, affecting both equity and
debt costs. Strong capital markets may provide more favorable conditions for a company to raise finance, potentially lowering WACC.
Applications of WACC
The Weighted Average Cost of Capital (WACC) is pivotal in finance, serving as a crucial benchmark for various strategic evaluations. It provides businesses and investors with a comprehensive metric
to assess opportunity costs and make informed decisions.
Investment Valuation
In investment valuation, particularly with Discounted Cash Flow (DCF) analysis, WACC is used as the discount rate to determine the present value of future cash flows. Analyzing an investment’s value
requires estimating the cash flows it may generate and then discounting them back to their present value using WACC. This process is essential in DCF valuations to calculate an asset’s equity value.
The formula applied is often rooted in forecast expectations and the potential investment opportunities that an asset might present.
Corporate Finance Decisions
Corporate finance decisions rely on WACC for gauging the feasibility of projects. WACC serves as a hurdle rate; if the internal rate of return (IRR) on a project exceeds the WACC, the project is
likely to add value to the company. This benchmark helps in comparing projects with different risk profiles and selecting the one that is most beneficial for the shareholders and the company’s growth
trajectory. When considering raising capital, companies assess WACC to understand the least costly way to finance their operations or growth, thereby directly influencing share price indirectly.
In financial modeling and valuation, WACC is frequently employed to test different scenarios and understand the sensitivity of the company’s value to changes in the capital structure. This analysis
is crucial for making robust financing decisions that align with the firm’s overall strategy and market conditions.
WACC’s Impact on Corporate Strategy
Weighted Average Cost of Capital (WACC) serves as a foundational benchmark in strategic financial decision-making. It influences how a corporation aligns its objectives regarding investment
opportunities and shareholder returns.
Mergers and Acquisitions
WACC plays a critical role in mergers and acquisitions strategy by helping to assess whether an investment will yield returns above the company’s cost of capital. Identifying the blended cost of
capital aids in determining the value creation potential of a merger or acquisition. A company with a lower WACC may pursue more aggressive acquisitions as their financing costs are less, while a
company with higher WACC must be more cautious, as additional risk can drive the WACC up further, potentially diminishing value for the shareholders.
Determining Growth and Dividend Policy
The corporate strategy for growth and dividend policy relies heavily on WACC for its direction. A firm’s growth plan must account for projects and investments that exceed the WACC to ensure they are
adding value. If the growth rate in earnings is expected to outpace the WACC, reinvestment in the business may be favored over paying out dividends. Conversely, if projects do not promise returns
over the WACC, it may be financially strategic to increase dividends paid to shareholders. Additionally, deciding on issuing common stock or preferred stock involves evaluating levered beta and
unlevered beta, voting rights, and flotation costs, all while maintaining an optimal WACC to satisfy investor expectations and secure cost-effective financing.
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At a given instant there are 25% undecayed radioactive nuclei i... | Filo
At a given instant there are undecayed radioactive nuclei in a sample. After the number of undecayed nuclei reduces to . Calculate
the time in which the number of undecayed nuclei will further reduce to of the reduced number.
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From initial to reduction till , it takes four half lives.
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At a given instant there are undecayed radioactive nuclei in a sample. After the number of undecayed nuclei reduces to . Calculate
Question Text the time in which the number of undecayed nuclei will further reduce to of the reduced number.
Topic Nuclei
Subject Physics
Class Class 12
Answer Type Text solution:1
Upvotes 127
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BHASKARA Biography - Craftmen, artisans and people from other Occupations
Leelavati looked entranced at the water-clock her father had brought home. Its movements were fascinating. She had a slight feeling of guilt, for her father had told her never to enter that room. But
because it was forbidden territory, its exploration gave her a sense of adventure. And she continued looking at the clock.
Then came disaster, though she was never to know about it. A tiny pearl slipped out of her nose-ring and fell into the clock. She was so alarmed that she fled. And in the excitement of the
arrangements being made for her wedding the next day she forget all about the clock and the pearl. Which was not surprising for she was only six years old.
Leelavati was married, but a week later her husband fell off a cliff and died. This was what her father, Bhaskara, a great mathematician and astrologer, had feared. Astrological calculations had
shown Bhaskara that, if the marriage of his daughter was not performed at a particular hour on that particular day, she would become a widow. And he had bought the water-clock to ensure that he would
know the right time. He did not know that the pearl in it had made the clock inexact. And, going by that clock, he had made an error. Bhaskara thought that it was his astrological calculations that
had gone wrong and blamed himself for the tragedy.
In those days, widowed girls were not allowed to marry again. Bhaskara, therefore, began to try to arouse her interest in mathematics, so that she would forget her grief. It is not known how good a
mathematician she turned out to be, but he made her immortal in the history of mathematics in India by titling after his daughter a chapter of the book Siddhantasiromani that he wrote when he was
only 30 years old. At one time there was even a popular saying: “Whosoever is well-versed with Leelavati can tell the exact number of leaves on a tree.”
The part of the book titled Leelavati dealt essentially with arithmetic. The other three parts were on different aspects of mathematics: Bijaganita dealt with algebra, Coladhyaya with spheres and
Grahaganita with planetary mathematics. Basically the book was a text-book, a collection of the works of some eminent scholars like Brahmagupta, Mahavira and Sridhara, after they had been simplified
to help students. The book contained problems presented in such a way as to stimulate the student’s interest. It was so popular and authoritative that four to five centuries later it was translated
twice into Persian.
Bhaskara was an original thinker, too. He was the first mathematician to declare confidently that any term and infinity is infinity.
In algebra, Bhaskara considered Brahmagupta his guru and mostly extended Brahmagupta’s work. But his introduction of Chakrawal, or the cyclic method, to solve algebraic equation is a remarkable
contribution. It was only after six centuries that European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it “inverse cyclic". Determination of the area and
volume of a sphere in a rough integral calculus manner was also mentioned for the first time in his book. It contained some important formulas and theorems in trigonometry and permutation and
Bhaskara can also be called the founder of differential calculus. He had conceived it several centuries before Isaac Newton and Gottfried Leibniz, who are considered in the West to be the founders of
this subject. He had even given an example of what is now called “differential coefficient” and the basic idea of what is now known as “Rolle’s theorem". Although Bhaskara attained such excellence in
calculus, no one in the land took any notice of it.
As an astronomer Bhaskara is renowned for his concept of Tatkalikagati, which means instantaneous motion. This enables astronomers to determine the motion of the planets accurately.
Bhaskara was born in 1114 at Bijjada Bida (Bijapur, Karnataka) in the Sahyadri Hills. He learnt mathematics from his saintly father. Later, the works of Brahmagupta inspired him so much that he
devoted himself entirely to mathematics. At the age of 69 he wrote his second book, Karanakutuhala, a manual of astronomical calculations. Though it is not as well known as his other book, it is
still referred to in making calendars.
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Mediation in PLS-SEM
Mediation occurs when a third mediator variable intervenes between two other related constructs. More precisely, a change in the exogenous construct causes a change in the mediator variable, which,
in turn, results in a change in the endogenous construct in the PLS path model. Thereby, a mediator variable governs the nature (i.e., the underlying mechanism or process) of the relationship between
two constructs.
Analyzing the strength of the mediator variable’s relationships with the other constructs allows substantiating the mechanisms that underlie the cause-effect relationship between an exogenous
construct and an endogenous construct. In the simplest form, the analysis considers only one mediator variable, but the path model can include a multitude of mediator variables simultaneously (i.e.,
multiple mediator analysis).
Hair et al. (2017) describe the systematic mediator analysis process in PLS-SEM in more detail; also see Nitzl et al. (2016) and Cepeda et al. (2017).
The following figure shows the example of a simple mediator model, whereby ${p_{3}}$ is the direct effect, ${p_{1}·p_{2}}$ is the indirect effect, and the direct effect ${(p_{3})}$ + the indirect
effect ${(p_{1}·p_{2})}$ = the total effect:
To analyze a mediator model, Zhao et al. (2010) suggest a model, as shown in the following figure, which Hair et al. (2017) also propose to use for PLS-SEM:
As a result, the researcher decides with regards to the indirect effect, if mediation and what kind of mediation occurs.
Researchers also can apply the model to situations with multiple mediators as shown in the following figure:
In this figure, constructs $M_1$ and $M_2$ mediate in parallel the relationship between constructs $Y_1$ and $Y_2$. In case of an additional relationship from construct $M_1$ to $M_2$ in that figure,
we would describe the situation as serial mediation (i.e, mediator $M_2$ follows $M_1$).
When considering multiple mediators, the researcher shall analyze the model that includes all relevant mediators at the same time (as, for example, shown in the above figure). For such a mediator
model, one can use the before described analysis procedure as suggested by Hair et al. (2022) and Zhao et al. (2010). It also allows in a multi mediator model to analyze the total indirect effect $
({p_{1}·p_{2}}+{p_{4}·p_{5}})$ for the total mediation via both mediators $M_1$ and $M_2$. Alternatively, the researcher can use the procedure to analyze the specific indirect effects per mediator
variable (i.e., ${p_{1}·p_{2}}$ for the $M_1$ mediator and ${p_{4}·p_{5}}$ for the $M_2$ mediator).
Mediation in SmartPLS
SmartPLS supports to model and analyze mediators. The following model shows the corporate reputation model example. CUSA mediates the relationships between COMP and CUSL as well as LIKE and CUSL.
In SmartPLS, the results of the PLS-SEM algorithm and the bootstrap procedure include the direct, the total indirect effect, the specific indirect effects, and the total effect. These outcomes, which
are available in the SmartPLS results reports, permit conducting a mediator analysis (e.g., as suggested by Hair et al. 2017). Note that the SmartPLS results allow analyzing both single and multiple
mediation models (i.e., parallel and serial mediation).
Cite correctly
Please always cite the use of SmartPLS!
Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. Bönningstedt: SmartPLS. Retrieved from https://www.smartpls.com
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high energy planetary ball mill
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Term Test 2 sample P3
Find all singular points of
z^3 \tan ( \pi z ) \cot^2 (\pi z ^2) $$
and determine their types (removable, pole (in which case what is it's order), essential singularity, not isolated singularity, branching point).
In particular, determine singularity at $\infty$ (what kind of singularity we get at $w=0$ for $g(w)=f(1/w)$?).
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A Math Major’s Take on JobZones and Matter of Wissen, Inc., by David Froman
As previewed in my previous post, this essay will explore the mysterious realm of SVP (specific vocational preparation) codes as implemented by the O*NET JobZone classification system. Business
Immigration practitioners have long understood the critical importance of SVP codes in establishing employment-based second preference (EB2) advanced degree education and experience qualifications
normal to the occupation–particularly with the chronic waits experienced in skilled worker and professionals (EB3) category. Unfortunately, the chief authority analyzing JobZones to date, Matter of
Wissen, Inc., 2009-PER-00405 (BALCA 2010), suffers from incomplete analysis, faulty logic, and self-contradiction.
Having come to the Law via mathematics has colored my approach to analyzing legal questions: do all parts “add up”? Does any part contradict any other part? Have the proper principles and
“theorems” been applied? Does the result “make sense”? Four years ago, I had to analyze the meaning of O*NET JobZone Four for an EB2 labor certification application. Since then I have reused the
explanation on multiple occasions with minor modifications. Then in April of 2010 came the Board of Alien Labor Certification Appeals (BALCA) decision in Matter of Wissen, Inc., supra, which clashed
with my analysis by essentially deleting the SVP 8 category from JobZone Four. In an effort to raise awareness concerning the logical errors in the Wissen decision, I offer a post-Wissen version of
my JobZone analysis (with single quotation marks from electronic filing format in portions).
Before turning to the analysis, however, let’s review SVP codes[1] and O*NET JobZones:
Specific Vocational Preparation is defined as the amount of lapsed time required by a typical worker to learn the techniques, acquire the information, and develop the facility needed for
average performance in a specific job-worker situation.
This training may be acquired in a school, work, military, instructional, or vocational environment. It does not include the orientation time required of fully qualified worker to become
accustomed to the special conditions of any new job. Specific vocational training includes: vocational education, apprenticeship training, in-plant training, on-the-job training, and essential
experience in other jobs.
. . . .
The following is an explanation of the various level of specific vocational preparation:
SVP 1 - Short demonstration only
SVP 2 - Anything beyond short demonstration up to and including 1 month
SVP 3 - Over 1 month up to and including 3 months
SVP 4 - Over 3 months up to and including 6 months
SVP 5 - Over 6 months up to and including 1 year
SVP 6 - Over 1 year up to and including 2 years
SVP 7 - Over 2 years up to and including 4 years
SVP 8 - Over 4 years up to and including 10 years
SVP 9 - Over 10 years
The JobZone categories[2] equate to SVP codes as follows:
JobZone One: Little or No Preparation Needed . . . SVP Range: Below 4.0
JobZone Two: Some Preparation Needed . . . SVP Range: 4.0 to < 6.0
JobZone Three: Medium Preparation Needed . . . SVP Range: 6.0 < 7.0
JobZone Four: Considerable Preparation Needed . . .SVP Range: 7.0 < 8.0
JobZone Five: Extensive Preparation Needed . . . SVP Range: 8.0 and above . . . .
The matter involved an occupation that fell under O*NET JobZone Four, requiring ‘considerable preparation’ and listing the ‘SVP Range: 7.0 < 8.0.’ The equivalent occupation in the Dictionary of
Occupational Titles (DOT) carried an SVP code of 8, that is, a minimum of four years, up to ten years, of work-related skill, knowledge, or experience. This range includes a Master's degree plus up
to 6 years of experience. Based on the ‘SVP Range: 7.0 < 8.0’ designation contained in the new JobZone rating system (literally meaning greater than level 7, up to level 8), this would include a
range from a Bachelor's degree at the low end of SVP 7 through a Master's Degree plus up to, but not including, six years of experience at the SVP 8 level. The proposed minimum education and
experience level of a Master’s degree and 5 years of experience for the position in question fell within this zone and was therefore within the job opportunity’s requirements normal for the
occupation. How did I arrive at this conclusion?
First, a few general observations: All five JobZone categories cover multiple SVP codes. Each of these multiple SVP code groups overlaps with its adjacent JobZone groups. This suggests that the
JobZone scheme is flexible and overlapping, rather than rigid and monolithic. In fact overlapping education and experience descriptions pervade the JobZone scheme, as will be seen. However, the
effect of the Wissen holding, if applied to all JobZones, would be to strip them of their flexibility. The Wissen world of JobZones appears as follows:
JobZone One: Little or No Preparation Needed . . . SVP Range: 1-3
JobZone Two: Some Preparation Needed . . . SVP Range: 4 & 5
JobZone Three: Medium Preparation Needed . . . SVP Range: 6
JobZone Four: Considerable Preparation Needed . . .SVP Range: 7
JobZone Five: Extensive Preparation Needed . . . SVP Range: 8 & 9
No overlap. No flexibility. No mathematical or analytical basis. So let’s see what analysis shows.
The meaning of JobZone Four
The O*NET explanation of JobZone Four indicates ‘SVP Range: 7.0 < 8.0.’ At first glance, the meaning of the specified SVP Range of ‘7.0 < 8.0’ is unclear. The decimal notation is superfluous; SVP
designations appear in whole integers from one to nine. The connecting symbol borrowed from mathematics means less than, that is, ‘seven is less than eight.’ Obviously, this notation was intended
to convey something more than simply normal counting order. The word ‘Range’ implies more than a single point; thus, neither SVP 7 nor SVP 8 alone could be correct. Without both involved, there
would be no range. Indeed, if the O*NET drafters intended only one SVP level for JobZone Four, inclusion of the other would serve no purpose, except to cause confusion. Thus a simple ‘less than’
symbol analysis–as relied on in Wissen–proves inadequate. However, when restated in algebraic terms, with x equal to the acceptable range of years of specific vocational preparation, the JobZone
Four SVP Range formula takes on a plausible meaning:
7 < x < 8.
That is, x is greater than 7, and x is less than 8. But 7 and 8 are not just single mathematical values. Rather, they are symbols, each of which includes its own range of time values. So we are
dealing with a numerical range of time ranges.[3] Borrowing further from algebra by substituting the SVP time ranges represented by the symbols 7 and 8 into this restated formula, we achieve the
following expression:
(over 2 years up to and including 4 years) < x < (over 4 years up to and including 10 years)
In strict mathematical terms, this formula describes a total continuum, where x, if it exists at all, is ever so tightly wedged somewhere between ‘4 years’ and ‘over 4 years.’ (4.00000001 years has
already gone past x, the putative JobZone Four SVP value.) This yields a nonsensical result. But observing that the high end of the first element is continuous with the low end of the second
element suggests that the JobZone ‘range’ in question should focus not on the adjacent ends of the two elements, but instead on the remote ends. This yields the following formula:
(over 2 years) < x < (up to and including 10 years)
Refining this expression to give effect to the inequality symbols, the formula for JobZone Four in terms of years of specific vocational preparation may be restated as follows:
2 years < x < 10 years
This means that the acceptable range of specific vocational preparation specified in JobZone Four encompasses from more than 2 years up to, but not including, 10 years.
From this specific algebraic inspection we may tentatively conclude that a range of ranges joined by an inequality function should encompass all of the values between the end points. Algebraic
analysis shows that the normal value of x (the required level of specific vocational preparation) for any given JobZone Four occupation may fall between the lowest level of SVP 7 and the highest
level of SVP 8, like the two ends of a football goalpost. The ‘ball’ (the required preparation for a given position) may pass anywhere between the ends of the goalpost (the low end of 7 and the high
end of 8) and still ‘score.’ No other explanation of this official governmental notation makes sense. The range includes from more than two years to less than 10 years of specific vocational
preparation–which encompasses a Master’s degree and 5 years, a total of 9 years of specific vocational preparation, within the ‘job opportunity’s requirements normal for the occupation.’ Therefore,
the above analysis shows that the requirements for the job opportunity that prompted this inquiry lie within the normal range for this occupation, ‘SVP Range: 7.0 < 8.0,’ – or, more precisely, SVP
Range = x, where SVP codes 7 < x < 8.
Confirmation by comparing other JobZones
Inspection of other SVP ranges appearing in the JobZones confirms this analysis and shows that the conclusion of the Certifying Officer ratified by the BALCA in Wissen, supra, erred in using the low
end rather than the high end of SVP 8 as the top of the JobZone range. Analysis of similarly stated SVP ranges for JobZone Two (‘SVP Range: 4.0 to < 6.0’) and JobZone Three (‘SVP Range 6.0 < 7.0’)
show both of these descriptions include experience, education, and training requirements that stretch from the lowest level of the lowest SVP number to the highest level of the highest SVP number
within the specified range. JobZone Two includes a few months (SVP 4) to one year (SVP 5) of job training but ‘in some cases, an associate’s or bachelor’s degree (SVP 6) could be needed.’ This
verbal description clearly indicates a range that encompasses all three listed SVP codes.
Likewise, the description of JobZone Three discusses three or four years of apprenticeship (SVP 7) or several years of vocational training (SVP 6), a bachelor’s degree (SVP 6), followed by one or two
years of job training (SVP 7). Again, both SVP codes are necessary to cover the written description. This shows a pattern of intent on the part of the drafters of the O*Net Job Zones to utilize the
entire specified range of SVP codes for each JobZone.
JobZone Four lists both a ‘four-year bachelor’s degree’ and a ‘minimum of two to four years of work-related experience. Wissen held that education and experience should be combined when determining
the applicable SVP level. Wissen, slip op. at 6. If this is so, two years of specific vocational preparation for a bachelor’s degree plus four years of work-related experience, a total of six years
of specific vocational preparation, falls within SVP code 8. Thus, even the internal logic of Wissen in determining the appropriate SVP level–that education and experience must be
combined–contradicts its own conclusion that JobZone Four does not include SVP code 8. The Board in Wissen focused in error on the low end of SVP 8 as the ‘ceiling’ for JobZone Four and came to the
erroneous conclusion: ‘Stated another way, the top end of the total preparation required must be less than an SVP of 8.0 (over 4 years and up to and including 10 years).’ Instead, they should have
been looking at the upper end of SVP 8.
Therefore, paraphrasing Wissen, the proper description of JobZone Four should be: ‘Stated another way, the top end of the total preparation required must be less than the top end of SVP 8 (over 4
years and up to but not including 10 years).’ Wissen, slip op. at 6, with deletions and additions, emphasis added; compare original statement, infra, note 3.
Mathematical logic concludes that the JobZone Four SVP description follows the goalpost model: from, but not including, the low end of SVP 7, to, but not including, the high end of SVP 8. The
structure and descriptions of the JobZone system of ranges confirms this conclusion. Finally, the previously assigned SVP code for the equivalent occupation listed in the Dictionary of Occupational
Titles supports the conclusion–contrary to Wissen–that SVP 8 remains an equal partner with SVP 7 in O*NET JobZone Four.
Therefore, BALCA or the Department of Labor needs to step beyond Wissen to restore SVP 8 from the limbo of surplusage and to affirm a uniform, coherent application of the JobZone SVP categories that
will withstand critical analysis and foster sensible results. D.F.
© David Froman 2012
1. http://www.flcdatacenter.com/svp.aspx, last visited 3/27/2012.
2. http://www.flcdatacenter.com/JobZone.aspx#4, last visited 3/27/2012.
3. This is where the Wissen analysis jumped the tracks. The Board relied on the Certifying Officer (CO) for the meaning of the inequality sign without taking into account the nature of the
symbols it joined:
"Thus, when the SVP range is stated as “7.0 to < 8.0,” it means that O*Net determined the SVP level of preparation to be anywhere from two years up to and including four years.11 Stated
another way, the top end of the total preparation required must be less than an SVP of 8.0 (over 4 years and up to and including 10 years).
11 As the CO asserted in his appellate brief, “<” is the universal symbol for “less than” and the Employer’s position that this symbol actually means “less than or equal to” is
unsubstantiated." Wissen, supra, Slip Op. at 6 (emphasis in original). The Board’s logic makes JobZone Four equal to SVP ‘7.0’ only–and SVP ‘8.0’ irrelevant surplusage.
4. The Board in Wissen assumed that SVP 8 represented a single value. Accordingly, they failed to inquire, ‘Which part of SVP 8 is indicated here?’ Instead–following the advice of the CO–they
automatically excluded all of SVP 8 by treating the low end of its time range as synonymous with its numerical designation. As the analysis shows, the low end of SVP 8 is contiguous with the
high end of SVP 7, with no gaps. This continuous property admits no ‘foothold’ for the inequality symbol to function: there is no room for ‘x.’ To give the inequality and ‘x’ room to function,
there must be a gap for them to operate in. Excluding SVP 8 from JobZone Four eliminates any gap. Only using the ‘goalpost’ end points of SVP 7 and 8 gives meaning to the inequality notation as
a ‘range.’
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Trigonometric Ratios (sin, cos, tan only) Mazes - All Things Algebra®
Students will practice identifying the sine, cosine, or tangent ratio of an angle in a right triangle with this set of two mazes.
Mazes Included:
• Maze 1: Pythagorean Theorem Triples Only
• Maze 2: Non-Pythagorean Triples (requires simplifying radicals and rationalizing denominators)
Note: Only two side lengths of the triangle are given for both versions. Therefore, Pythagorean Theorem is required on both versions to find the third side length in order to identify the
trigonometric ratio.
The solutions are used to navigate students through the maze. Mazes work well as warm-ups, alternative homework assignments, classwork, or quick assessments.
This resource is included in the following bundle(s):
Geometry Curriculum (with Activities)
Geometry Activities Bundle
License Terms:
This purchase includes a single non-transferable license, meaning it is for one teacher only for personal use in their classroom and can not be passed from one teacher to another. No part of this
resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. A transferable license is not available for this
Copyright Terms:
No part of this resource may be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by
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[Bug?] Arbitrary precision not working in Coloring Algorithm
posted Jun 14 at 9:08 pm
I recently picked UF.5 up again to do some math experiments. I wrote a coloring algorithm, but calculations became very slow at a zoom depth of 1e24. I read UF.6 was a lot faster and maybe
perturbation could also help (as I'm using Mandelbrot).
Opening the fractal in UF.6, the coloring algorithm doesn't seem to work anymore. I tried loading a non-perturbation version of Mandel, but it still doesn't work. The symptoms I'm seeing are equal to
those when using Extended precision, i.e. large quantization of input values
gives the same result as:
0.44444444444486132 and
and the next number giving a different result is:
This is tell-tale of limited precision calculations.
Furthermore, increasing the precision from 70 to 100 or to 1000 doesn't change the result, or the calculation time.
Copy7OfExt3rnalRays-anim {
mjd_private.ucl:ExternalRayFinder(OUTSIDE) {
mjd_private.ucl:DistanceEstimator(OUTSIDE) {
; Distance-estimator coloring algorithm for Mandelbrot and
; other z^n fractal types (Phoenix, Julia). This coloring
; algorithm estimates the distance to the boundary of the
; fractal (for example the Mandelbrot set) and colors points
; accordingly.
; Written by Damien M. Jones
Incorrect UF.6 result:
Correct UF.5 result:
[edit] Placed parameter in code block
I recently picked UF.5 up again to do some math experiments. I wrote a coloring algorithm, but calculations became very slow at a zoom depth of 1e24. I read UF.6 was a lot faster and maybe
perturbation could also help (as I'm using Mandelbrot). Opening the fractal in UF.6, the coloring algorithm doesn't seem to work anymore. I tried loading a non-perturbation version of
Mandel, but it still doesn't work. The symptoms I'm seeing are equal to those when using Extended precision, i.e. large quantization of input values e.g. Entering:
0.444444444444861296182757053786418265553146200444003440041363028157654069228480050281532280718162049115995 gives the same result as: 0.4444444444448613 0.44444444444486131 0.44444444444486132 and
0.44444444444486133 and the next number giving a different result is: 0.44444444444486134 This is tell-tale of limited precision calculations. Furthermore, increasing the precision from 70 to 100 or
to 1000 doesn't change the result, or the calculation time. Parameter: ```` Copy7OfExt3rnalRays-anim { ::BFWD1gn2tTVTPSNOQ07j08fwKcZ5w0TZH/5gs0ItMcYl4CzVkQmEnu9OJxROuZme/1vlT6G6
dBkQASch+QU7ylr3zue1rL5ayu+bu8CCJHy9eb1fGnOQ+D1zJxOi/pcda01TSuDzX5GDDVkH Dt5dWuGI78ht7yWGHI9uD+0slVqTXyN4x/LkkupZrEKBbS+2Qe2W9Pxhre/LkXT5XzAG/Fv2 leg8X+UEL++5z24/lY1lXsAyCVbcT5Qc0Wd3TZ/
JCWRiTumQ+glCAZwn26HiteLiMuKvL2aH 23nDTu55l7byNOP5S+xs9gfmMl8Nh5SZVIlHcTThxtrwhp4T2qrobUSTtmRNahwo0g4aYDAc Wt0Q5UtCUatRf7zg7vvksiJxslctS+2ShoUMw6ZkMJ1IFamwwYKuRxv9ZMgDQVJxB3W82xoS
MP1G+SBNbApBRqumLY1cOjLNGhgpN1CFjqYKqBUCkaatnyudBx1aeXFxNut0dvCJJeJ40jc8 jr+E2z5kP3szWtmzp9waM/g/RbFsEG+0R6iJ8h1dzKxfKUerwWAQm8pmd+mHsxuOSXo3PiKD
b19Z3YrL1uZfHqnw320Bb1rcz5Xjx99LkY6dzZXKbhrxy8u37C9x9ZL1TxWTYcO06v5j9wOE vx4o/yLwc+8t6DjeXik8TeXGT8MmM83tYbP8BX2vZfzKyH5zJl1bcHeVApV6ItWfHhN8z+pl
UmRS1MlQBialGfRx2uQIqpcJDFIc8lqu8hTrX6LyagppClUwLdVmGHoAxSsacFooaKeSuhSF YXeFbUs+A9XI2s/L2CJvI+oGGgfFUQKpKDvmiLLAjrROYq5Y1M81JApWrMSmgyYCgjEQBgWw p8aFjJxhLDHwJJgZAuSh4ua3g4i8Yby1Gw+zS/
deIGxdKTupY2t4HQJlW1TWg0E7joKUqwZQ q4Y4aj54GoK6oZx51C1GnO/JrEmQ8FNeeZYO/1Mc6L2i+xvFTnf7z8T1nJF3vd3Xzp5sh7zM b+J4006x0QVA/H1zpIqcjN+7mzhBXOmKsbK+IWQ2G47S/TL9W4HU//vAbb1P8D== }
mjd_private.ucl:ExternalRayFinder(OUTSIDE) { ::XdgTbgn29J5SzNNMQc87dm+dYJ5SSGMoVvFDlpcAOzNOr2YnqBXZPOq86TPrtcidIpxHk1+6 v+trUIGSf42bAoquxngU5+0nj7qLh7g799b6jFiJ4l42yumfW2RRYTF4HTmcV300OIVoae2j
pf8EeqM5pC8Jfkva5fXDvHWxhNwy2w6bvJn6BVzJXMh1phz/LguG68WNYtOnBhQO4HvDKu/X htpnyBgTakCM7tMuNU9fV+JaC8qFeedzncT73A8rF+A6H9Mg/oqVhoveYgeofxC/D7z95G76 zG1v54oe2tyMIX+YTdTP9d7eIrybh5/IF3WW5fpO/iIFSDis4L/
OV2RsAd+/AfN0feL6To13 5fG8T3LP6bThm4J1MEeRO+o69g9O5sPrG5ONa5GlhpEGrWSGalSJQpmzYUOMmofRiCtgxtoy oVSm2x5WplxUD+EkFzgWkKS6QU5cqxRwAsTUTr/APj6vReDxdw+2mEgXnalWytOB64MaVhMt
GNOpAJzekJbienQSw4kat1acauC5cFTSobYMrSiShhz1anwJZoEZcHTaMvO08rDN/CQzusaz ecPTtv379SduK775Q8Si+PwRETy0 } mjd_private.ucl:DistanceEstimator(OUTSIDE) { ; ; Distance-estimator coloring algorithm for
Mandelbrot and ; other z^n fractal types (Phoenix, Julia). This coloring ; algorithm estimates the distance to the boundary of the ; fractal (for example the Mandelbrot set) and colors points ;
accordingly. ; ; Written by Damien M. Jones ; ::DfjUign29AVsONMMQ09Il/hTJLplSClRkQiBKiF2gNUlcjvUbJHfGbHRb/65sdLWewv79u39 Ort64T1VAMSzODeCkXgnhutbgHWVXZIylZzVfxR/ieYN0eZfXBc/2VMmZvD2WXNptCTWfr2K
ZzYnK6WbojdjiDhu2LrStc7NMUeK5y772OUknUE+xH7anFHtcQk4kYxUSaUHNIbdzr6QUYHR YXIqnFRy3k4VoxNpLS+aad/oa+/6RypHTEvHnNDjkh8a7xhkPShXOIvaJezxeFLM3uT4FzQO eJI/
jJcRNZTut7kjsoN2UYumWm5x+HKlUabC38pSHggiWMS4ACBMCRC4RNqgoCB8qTANlxfX a/2Zi8zLGBcmWAhHhlAH/e4NyDfwrAaO4p4GuTeK8dJsIMmzcMyJDty8WUX9Hc8yY6I= } ```` Incorrect UF.6 result:  Correct UF.5 result: [edit] Placed parameter in code block
edited Jun 15 at 11:25 pm
try turning off perturbation, I assume that is what is causing the problem because whichever coloring algo you are using is relying on intermediate results that are not available when using
perturbation. Some CAs require the entire orbit to be explicitly calculated. I hope this helps.
Hello, try turning off perturbation, I assume that is what is causing the problem because whichever coloring algo you are using is relying on intermediate results that are not available when using
perturbation. Some CAs require the entire orbit to be explicitly calculated. I hope this helps.
If you read my first post, I've already tried without perturbation.
If you read my first post, I've already tried without perturbation.
I detected something strange by testing. arbitrary seams to have problems with big numbers, there where the red color is, the numbers get too big and the numbers NaN. not in extended, but in double
and arbitrary. Don't know if this is related, or i made something wrong. I only changed the number in "Additional Precision".
I detected something strange by testing. arbitrary seams to have problems with big numbers, there where the red color is, the numbers get too big and the numbers NaN. not in extended, but in double
and arbitrary. Don't know if this is related, or i made something wrong. I only changed the number in "Additional Precision". double  extended arbitrary
edited Jun 20 at 12:36 am
Sorry, I missed that. What happens if you manually increase precision?
//edit: I just tried your parameters and the formula does not have perturbation settings. Also, if I swap from Mandelbrot (built-in) to just Mandelbrot, the result changes completely:
Sorry, I missed that. What happens if you manually increase precision? //edit: I just tried your parameters and the formula does not have perturbation settings. Also, if I swap from Mandelbrot
(built-in) to just Mandelbrot, the result changes completely:
edited Jun 20 at 2:35 pm
Set bailout value to 1e10 and it changes back to how it's supposed to look
Set bailout value to 1e10 and it changes back to how it's supposed to look
edited Jun 20 at 10:33 pm
Ah, will try this later. But to be honest, your correct result looks like it's bugged and vice versa. Did you accidentally swap the images in your opening post?
Ah, will try this later. But to be honest, your correct result looks like it's bugged and vice versa. Did you accidentally swap the images in your opening post?
In UF6, coloring algorithms always use double or extended precision, never arbitrary, as a general optimization. If your coloring algorithm requires arbitrary precision internally, it will give
different results in UF6, so I think that explains it.
In UF6, coloring algorithms always use double or extended precision, never arbitrary, as a general optimization. If your coloring algorithm requires arbitrary precision internally, it will give
different results in UF6, so I think that explains it.
Ultra Fractal author
Wait, did they use arbitrary in UF5?
Wait, did they use arbitrary in UF5?
Ah so it IS a bug, just not the one we expected. The input fields allow for an arbitrary amount of decimals, giving the impression that the numbers are actually used as such. That's the bug. (And
maybe also a documentation issue, as I read through the changelog and manual and didn't find anything about this)
Is there any workaround for this?
Ah so it IS a bug, just not the one we expected. The input fields allow for an arbitrary amount of decimals, giving the impression that the numbers are actually used as such. That's the bug.
(And maybe also a documentation issue, as I read through the changelog and manual and didn't find anything about this) Is there any workaround for this?
edited Jun 21 at 1:49 pm
OK this explains my bug i found. in this case it would be good if UF not uses double but extended as a fallback in case arbitrary is requested. And a option to use arbitrary also in coloring
algorithms could be useful, maybe as a switch in "default:" section. it maybe solves Mark Jeronimus bug too?
(i tried to use a tool in a coloring algorithm to find Misiurewicz points in high precision, and this not worked because of the same reason.)
OK this explains my bug i found. in this case it would be good if UF not uses double but extended as a fallback in case arbitrary is requested. And a option to use arbitrary also in coloring
algorithms could be useful, maybe as a switch in "default:" section. it maybe solves Mark Jeronimus bug too? (i tried to use a tool in a coloring algorithm to find Misiurewicz points in
high precision, and this not worked because of the same reason.)
I just found another reason why removing arbitrary precision from coloring was a bad idea.
The plain old Distance Estimator breaks down at magnifications above 1e76 on standard Mandelbrot.
My Better Distance Estimator (which adjusts the Color Density automatically with #magn) still breaks down at magnifications above 1e140 and high bailout values. Lowering the bailout makes it work for
up to 1e149 (bailout 4) but obviously Distance Estimator will show artifacts at this bailout, and it's literally the end of zooming.
The next parameter is with the default Distance Estimator and a Color Density of 3e38. Try increasing it to 4e38 and it turns completely black.
Copy10 {
I just found another reason why removing arbitrary precision from coloring was a bad idea. The plain old Distance Estimator breaks down at magnifications above 1e76 on standard Mandelbrot. My Better
Distance Estimator (which adjusts the Color Density automatically with #magn) still breaks down at magnifications above 1e140 and high bailout values. Lowering the bailout makes it work for up to
1e149 (bailout 4) but obviously Distance Estimator will show artifacts at this bailout, and it's literally the end of zooming. The next parameter is with the default Distance Estimator and a
Color Density of 3e38. Try increasing it to 4e38 and it turns completely black. ```` Copy10 { ::Stz93gn2tJVPvtNMQ0dD4/DEaqdI287PiBnazSR7UXLQAtElFbkIFIpRs7v+eS2pJDVaQk39
u39Odv+srt6Gfc7GEqGqjebzXSzXRfig/MK1j8Xqsc0NiyurlHcxwUD61QXdwK5Y0gPcaoaZ Y8S9jur+cxSQ9Z3kvYpC5tjWFH1PXsyVYtZfXoWsN/wlfB9NfOBseucQunh3TxU+h/kme44B
ze9yVWz2NrMvqxW3cNki2mvvECRaQpZXboe1SwY0kvOk6sTnHrhZXpsdzkbeOEPdrWfs6zW8 OOnpISlWwoUuSz0YJb9Llw1aiQoFYJ2QJEmgyNcO1IIYiEy8rFi4GpgoJGMXykaKla4UukCR
pCx+Hw7YKKmDsToamRZAWhXqQJ0KORbgOZWJSzFYBWZkcDhZ0UBEgQxGjUSB9BkYMAVCQZMD Gr0alUsU4k7U0y2pxGimpeSp2upPlh5294t0XCLjqSghH0sP3O4bfxmTnPNgcd/+cpav6Lo+ woPusha+Z1F7c5udn7naW4A+Zlvuskid+xj5UtBN/
cp6yVLePw5zzpXhWQXPf0FGTnrWinA7 4QsE6838UZXs0D4ipof7GAz/S15BYwij5Z63xNGieXGl9zeXFq6/Kx2xPKxvGAZFb9PVqhJX NlbeXd7A5cK76CA21uWmSJw9uM8wM5WNTEUAGyLWMqNNmyWwboUUi4eYmxcPBQ2d72H5CE5b 1/mZE8Bb38XAhVB/IC==
} ````
edited Jun 30 at 11:33 am
Interesting. This change seemed like a good idea to me while implementing perturbation calculations, because if you still need to do a bunch of arbitrary precision math during each iteration (for the
loop section of a coloring algorithm), there is hardly any speedup at all. From my testing, it didn't seem to make a difference but I can see why distance estimation would break down.
I'll investigate and see if a different approach is needed. Of course it would be possible to go back to UF5 behavior and do all coloring algorithm calculations with arbitrary precision, but that
would really slow down a fractal that uses perturbation calculations in combination with e.g. an orbit trap coloring.
Another possibility would be to only use double/extended if perturbation is used and allow to turn off perturbation in the Formula tab.
Interesting. This change seemed like a good idea to me while implementing perturbation calculations, because if you still need to do a bunch of arbitrary precision math during each iteration (for the
loop section of a coloring algorithm), there is hardly any speedup at all. From my testing, it didn't seem to make a difference but I can see why distance estimation would break down. I'll
investigate and see if a different approach is needed. Of course it would be possible to go back to UF5 behavior and do all coloring algorithm calculations with arbitrary precision, but that would
really slow down a fractal that uses perturbation calculations in combination with e.g. an orbit trap coloring. Another possibility would be to only use double/extended if perturbation is used and
allow to turn off perturbation in the Formula tab.
Ultra Fractal author
I think the most people witch investigate time to create there own formulas, know at least do a degree what they do. And so it would make sense if they have the chance to make there decisions and
have all tools for it. And in case is a simple info in the Help text useful for example that arbitrary precision can cause much slower calculations. And if the forum is floated whit always the same
questions, the software could give a hint for optimization if the calculations are very slow.
(I usually spent way much more time for solutions and workarounds, if there is something missing, an way less if i just have all tools to make better decisions. I for example spend much time to
search the error in my code, when actually the software made something unlogical and unexpected.)
(i have not checked jet if only the coloring algo not uses arbitrary precision, or the codes in libraries too, even when used in formulas...)
I think the most people witch investigate time to create there own formulas, know at least do a degree what they do. And so it would make sense if they have the chance to make there decisions and
have all tools for it. And in case is a simple info in the Help text useful for example that arbitrary precision can cause much slower calculations. And if the forum is floated whit always the same
questions, the software could give a hint for optimization if the calculations are very slow. (I usually spent way much more time for solutions and workarounds, if there is something missing, an way
less if i just have all tools to make better decisions. I for example spend much time to search the error in my code, when actually the software made something unlogical and unexpected.) (i have not
checked jet if only the coloring algo not uses arbitrary precision, or the codes in libraries too, even when used in formulas...)
(Something about the perturbation, it should be a visible option. Because it uses a shortcut for the calculations, i would expect that some of my coloring algos would not work anymore. For example,
some add up all results in every calculation step.)
(Something about the perturbation, it should be a visible option. Because it uses a shortcut for the calculations, i would expect that some of my coloring algos would not work anymore. For example,
some add up all results in every calculation step.)
Exactly, it would be good to have the option to force it anyway and just decide to wait it out.
Exactly, it would be good to have the option to force it anyway and just decide to wait it out.
posted Jun 17 at 8:51 am
posted Jun 19 at 9:12 pm
posted Jun 20 at 12:36 am
posted Jun 20 at 2:16 pm
posted Jun 20 at 7:56 pm
posted Jun 21 at 8:45 am
posted Jun 21 at 1:30 pm
posted Jun 21 at 1:47 pm
posted Jun 21 at 3:09 pm
posted Jun 30 at 11:14 am
posted Jul 3 at 6:42 pm
posted Jul 3 at 6:51 pm
posted Jul 7 at 6:40 pm
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Question: REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has b… | Assignment Writing Service: EssayNICE
REM (rapid eye movement) sleep is sleep during which most dreams
occur. Each night a person has both REM and non-REM sleep. However,
it is thought that children have more REM sleep than adults†.
Assume that REM sleep time is normally distributed for both
children and adults. A random sample of n[1] = 10
children (9 years old) showed that they had an average REM sleep
time of x[1] = 2.6 hours per night. From previous
studies, it is known that σ[1] = 0.6 hour.
Another random sample of n[2] = 10 adults showed
that they had an average REM sleep time of x[2] =
1.90 hours per night. Previous studies show that
σ[2] = 0.7 hour. Do these data indicate that, on
average, children tend to have more REM sleep than adults? Use a 1%
level of significance.
(a) What is the level of significance?
State the null and alternate hypotheses.
H[0]: μ[1] =
μ[2]; H[1]:
μ[1] >
μ[1] = μ[2];
H[1]: μ[1] â‰
μ[2] H[0]:
μ[1] < μ[2];
H[1]: μ[1] =
μ[1] = μ[2];
H[1]: μ[1] <
(b) What sampling distribution will you use? What assumptions are
you making?
The Student’s t. We assume that both population
distributions are approximately normal with known standard
deviations.The standard normal. We assume that both population
distributions are approximately normal with known standard
deviations. The standard normal. We
assume that both population distributions are approximately normal
with unknown standard deviations.The Student’s t. We
assume that both population distributions are approximately normal
with unknown standard deviations.
What is the value of the sample test statistic? (Test the
difference μ[1] − μ[2]. Round
your answer to two decimal places.)
(c) Find (or estimate) the P-value. (Round your answer to
four decimal places.)
Sketch the sampling distribution and show the area corresponding to
the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis? Are the data statistically
significant at level α?
At the α = 0.01 level, we reject the null hypothesis
and conclude the data are statistically significant.At the
α = 0.01 level, we fail to reject the null hypothesis and
conclude the data are not statistically
significant. At the α = 0.01
level, we reject the null hypothesis and conclude the data are not
statistically significant.At the α = 0.01 level, we fail
to reject the null hypothesis and conclude the data are
statistically significant.
(e) Interpret your conclusion in the context of the
Fail to reject the null hypothesis, there is sufficient evidence
that the mean REM sleep time for children is more than for
adults.Reject the null hypothesis, there is insufficient evidence
that the mean REM sleep time for children is more than for
adults. Reject the null hypothesis,
there is sufficient evidence that the mean REM sleep time for
children is more than for adults.Fail to reject the null
hypothesis, there is insufficient evidence that the mean REM sleep
time for children is more than for adults.
|
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|
Concrete Floor Load Calculator
You will find a range of different strategies you are able to use to concrete flooring to produce gorgeous decorative floors ideal for both the office of yours and your home. In the event that you
desire to get a tank of it, when poured, concrete can easily bring it. With acid stained concrete floor, you have alternatives readily available to meet the design must have of yours.
Images about Concrete Floor Load Calculator
If you are contemplating a polished concrete floor then your very first choice is going to be if you stain the flooring. Additionally, floors made up of concrete are actually durable, costs cheaper
thanks to reduced fee of labor and also easy to clean compared to other flooring sorts. Treated concrete floors are actually several of the world's most durable.
Structure Load Calculation Load Calculation Of Building
With regards to floor surfaces, polished concrete floors are great materials, which are a lot more commonly used. The point is: concrete comes pre-installed in the majority of homes, since most
houses are crafted on concrete slabs. The period it takes to complete the assembly of a concrete floor is actually depending on how big or small the floor area is actually.
Concrete Calculator How To Calculate Concrete
How to Load Calculation on Column, Beam, Wall u0026 Slab Column
Load On Column, Beam u0026 Slab Column Design Calculations Pdf How To Calculate Column Size For Building Slab Load Calculation
How to calculate self weight of slab dead load of slab – Civil Sir
How Do We Calculate The Dead Load In The Slab? – Lceted -lceted
How to Calculate Load on Column, Beam, Slab u0026 Wall
How to Load Transfer from Slab to Beam Formulas with Example
Concrete Calculator How To Calculate Concrete
Concrete Slab on Grade Thickness Analysis
Design Concrete Slab NZ Standards Spreadsheet Calculator
slab – weight
Load Calculation on Column – Load Calculation of Column, Beam
Related Posts:
|
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|
Power Steering Fluid
I was replacing the power steering fluid in my car when I stumbled upon some exponential decay math. First, some background: There is no drain plug on a power steering system. You need to siphon
out fluid from the reservoir and replace it with new fluid. This new fluid then mixes into the rest of the system to create a slightly cleaner mixture. This idea is that if you repeat this a few
times, you’ll replace most of the old fluid with new fluid.
So, as you can see, there exists some sort of formula that can determine exactly how many times you need to extract and replace to reach X% replacement.
My car’s entire power steering system holds about 1 liter, and the reservoir itself holds .4 liters of that. So, 40% of the steering fluid is replaced each time I drain and refill the reservoir, and
60% of the old fluid remains elsewhere in the system. I can do this repeatedly, each time replacing 40% of the “mixed” fluid with brand new fluid.
Let p = Percentage of the system replaced each time you empty the reservoir.
Let n = number of times you empty/fill the reservoir (“flush”)
Let FD = Percentage of dirty fluid in the system.
Let FN = Percentage of new fluid in the system.
If p = percentage of new fluid introduced by a flush, then (1-p) is percentage of old fluid remaining. (eg: if \(p = .40, then (1-.40) = .60)\)
\(FD_0 = 1\) (initial proportion that is dirty)
\(FD_1 = (1-p)\)
\(FD_2 = (FD_1)(1-p) = (1-p)(1-p)\)
\(FD_3 = (FD_2)(1-p) = (1-p)(1-p)(1-p)\)
\(FD_n = (FD_{n-1})(1-p) = (1-p)^n\)
\(FD = (1-p)^n\)
\(FN = 1 – FD\)
For my car, p=.40 so \(FD = (1-.40)^n\)
How many times do I need to empty and fill the reservoir to get to 80% clean?
Just set \(FN = .8\) and solve for n:
\(.80 = 1- (1-.40)^n\)
\(.80 = 1- (.60)^n\)
\(.6^n = .2\)
\(log(.6)^n = log(.2)\)
\(n*log(.6) = log(.2)\)
\(n = \frac{log(.2)}{log(.6)}\)
\(n = 3.15\)
With a reservoir that holds 40% of capacity, I need to empty and replace it about 3 times to get 80% of the old fluid replaced.
You can also use the formula to figure out what percentage of the system contains old vs. new fluid, based on the number of flushes you’ve done. For this, you just plug in n and calculate \(FD\)
eg: If you’ve done 5 flushes, \(FD = (1-.4)^5 = .08\) So, after 7 refills, 8% is dirty, and 92% is new.
RE: Birthday Dinner
Yes, life does go by fast. Strangely, the older you get, the faster it goes. I do not know why this is.
Ever get an email like this? Well, as your age varies, the percentage of your life that a single calendar year represents also varies. As you get older, a year is a smaller percentage of your
overall life. In other words, 1 year represents 50% of a 2 year old’s life. However, it is only 2% of a 50 year old’s life. So, perhaps that is why each year seems to go by faster.
Want to see the percentage for every age from 0 to 80? Let’s make a formula and graph it. The percentage of your life that a single year represents is just a function of your age: \(f(age) = \frac
{1}{age}\) If you graph this on a spreadsheet, you’ll get the following:
How would you interpret this graph? You’ll notice that once you pass the inflection point, the percentage seems to flatten out. So, at what point can a person legitimately start saying “Wow, this
year really flew by?” Based on the graph, teenagers might feel this almost as much as middle aged people.
Lastly, do you notice how scaling of the y-axis makes the difference between age 15 and 50 look trivial? In order to properly display percentage changes, I will scale the y-axis logarithmically.
Here is the result:
With this scaling, you can see a year in the life of a teenager (~6%) is quite different than a year in the life of someone in their 50s (~2%)
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