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Nuclear chiral dynamics and phases of QCD
This presentation starts with a brief review of our current picture of QCD phases, derived from lattice QCD thermodynamics and from models based on the symmetries and symmetry breaking patterns of
QCD. Typical approaches widely used in this context are the PNJL and chiral quark-meson models. It is pointed out, however, that the modeling of the phase diagram in terms of quarks as quasiparticles
misses important and well known nuclear physics constraints. In the hadronic phase of QCD governed by confinement and spontaneously broken chiral symmetry, in-medium chiral effective field theory is
the appropriate framework, with pions and nucleons as active degrees of freedom. Nuclear chiral thermodynamics is outlined and the liquidgas phase transition is described. The density and temperature
dependence of the chiral condensate is deduced. As a consequence of two- and three-body correlations in the nuclear medium, no tendency towards a first-order chiral phase transition is found at least
up to twice the baryon density of normal nuclear matter and up to temperatures of about 100 MeV. Isospin-asymmetric nuclear matter and neutron matter are also discussed. An outlook is given on new
tightened constraints for the equation-of-state of cold and highly compressed matter as implied by a recently observed two-solar-mass neutron star.
• Chiral effective field theory
• Matter under extreme conditions
• Nuclear many-body problem
• QCD phase diagram
Dive into the research topics of 'Nuclear chiral dynamics and phases of QCD'. Together they form a unique fingerprint. | {"url":"https://portal.fis.tum.de/en/publications/nuclear-chiral-dynamics-and-phases-of-qcd","timestamp":"2024-11-08T07:39:25Z","content_type":"text/html","content_length":"52112","record_id":"<urn:uuid:3c73d3c6-9254-44de-99d6-68a8f8e806a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00653.warc.gz"} |
A time- and temperature-dependent 2-D simulation of the GTO thyristor turn-off process
A two-dimensional model simulation of a GTO has been performed in order to analyze the on-region squeezing process. As the turn-off process proceeds, most parts of the excess carriers in the p-base
and the adjacent G x K and middle junctions are removed. However, n-base carriers remain almost unchanged, resulting in a uniform current distribution at the anode side in spite of the on-region
squeezing in the p-base. Relatively high anode voltage is necessary to sustain high current density in the reduced on-region. A substantial anode voltage recovery is suggested for a higher mode
current turn-off case when the on-region width reaches its observed final value of 60 microns. One-half of its width coincides with the ambipolar diffusion length in the emitter-base junction; this
length is the lowest value of the p-base because of carrier-to-carrier scattering due to the high injected carrier density.
IEEE Transactions on Electron Devices
Pub Date:
September 1984
□ Gates (Circuits);
□ Semiconductor Plasmas;
□ Switching Circuits;
□ Temperature Dependence;
□ Thyristors;
□ Time Dependence;
□ Two Dimensional Models;
□ Current Density;
□ Density Distribution;
□ Electron Density (Concentration);
□ Mathematical Models;
□ Temperature Distribution;
□ Electronics and Electrical Engineering | {"url":"https://ui.adsabs.harvard.edu/abs/1984ITED...31.1156N/abstract","timestamp":"2024-11-09T21:17:17Z","content_type":"text/html","content_length":"38202","record_id":"<urn:uuid:0e3e9431-3408-421b-8fdd-2390763fc7fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00324.warc.gz"} |
Compute a biweight based scale estimate for a variable.
Mosteller and Tukey (see Reference section below) define two types of robustness:
1. resistance means that changing a small part, even by a large amount, of the data does not cause a large change in the estimate
2. robustness of efficiency means that the statistic has high efficiency in a variety of situations rather than in any one situation. Efficiency means that the estimate is close to optimal
estimate given that we distribution that the data comes from. A useful measure of efficiency is:
Efficiency = (lowest variance feasible)/ (actual variance)
Many statistics have one of these properties. However, it can be difficult to find statistics that are both resistant and have robustness of efficiency.
For scale estimaors, the standard deviation (or variance) is the optimal estimator for Gaussian data. However, it is not resistant and it does not have robustness of efficiency. The median
absolute deviation (MAD) is a resistant estimate, but it has only modest robustness of efficiency.
The biweight scale estimator is both resistant and robust of efficiency. Mosteller and Tukey recommend using the MAD or interquartile range for exploratory work where moderate efficiency in a
variety of situations is adequate. The biweight scale estimator can be considered for situations where high performance is needed.
The biweight scale estimate is defined as:
\( ns_{bi}^2 = \frac{n\sum_{i=1}^{n}{(y - y')^2(1 - u^2)^4}} {(\sum_{i=1}^{n}{(1 - u^2)(1 - 5u^2)})(-1 + \sum_{i=1}^{n}{(1 - u^2)(1 - 5u^2)})} \)
where the summation is restricted to \( u_{i}^2 \le 1 \) and
\( y' = \mbox{median } y \)
\( u_{i} = \frac{y_{i} - y'}{9*MAD} \hspace{0.5in} \mbox{for } (\frac{y_{i} - y*}{cS})^{2} < 1 \)
where MAD is the median absolute deviation.
LET <par> = BIWEIGHT SCALE <y>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
<par> is a parameter where the computed biweight location is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET A = BIWEIGHT SCALE Y1
LET A = BIWEIGHT SCALE Y1 SUBSET TAG > 2
Dataplot statistics can be used in a number of commands. For details, enter
Related Commands:
BIWEIGHT MIDVARIANCE = Compute a biweight midvariance estimate of a variable.
BIWEIGHT LOCATION = Compute a biweight location estimate of a variable.
BIWEIGHT MIDCOVARIANCE = Compute a biweight midcovariance estimate of two variables.
BIWEIGHT MIDCORRELATION = Compute a biweight midcorrelation estimate of two variables.
BIWEIGHT CONFIDENCE LIMITS = Compute a biweight based confidence interval.
AVERAGE ABSOLUTE DEVIATION = Compute the average absolute deviation of a variable.
MEDIAN ABSOLUTE DEVIATION = Compute the median absolute deviation of a variable.
STANDARD DEVIATION = Compute the standard deviation of a variable.
VARIANCE = Compute the variance of a variable.
RANGE = Compute the range of a variable.
Mosteller and Tukey (1977), "Data Analysis and Regression: A Second Course in Statistics," Addison-Wesley, pp. 203-209.
Implementation Date:
Program 1:
LET Y1 = NORMAL RANDOM NUMBERS FOR I = 1 1 10000
LET Y2 = LOGISTIC RANDOM NUMBERS FOR I = 1 1 10000
LET Y3 = CAUCHY RANDOM NUMBERS FOR I = 1 1 10000
LET Y4 = DOUBLE EXPONENTIAL RANDOM NUMBERS FOR I = 1 1 10000
LET A1 = BIWEIGHT SCALE Y1
LET A2 = BIWEIGHT SCALE Y2
LET A3 = BIWEIGHT SCALE Y3
LET A4 = BIWEIGHT SCALE Y4
LET B1 = STANDARD DEVIATION Y1
LET B2 = STANDARD DEVIATION Y2
LET B3 = STANDARD DEVIATION Y3
LET B4 = STANDARD DEVIATION Y4
LET C1 = MAD Y1
LET C2 = MAD Y2
LET C3 = MAD Y3
LET C4 = MAD Y4
PRINT "BIWEIGHT SCALE ESTIMATE FOR NORMAL RANDOM NUMBERS = ^A1"
PRINT "STANDARD DEVIATION ESTIMATE FOR NORMAL RANDOM NUMBERS = ^B1"
PRINT "MAD ESTIMATE FOR NORMAL RANDOM NUMBERS = ^C1"
PRINT " "
PRINT "BIWEIGHT SCALE ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^A2"
PRINT "STANDARD DEVIATION ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^B2"
PRINT "MAD ESTIMATE FOR LOGISTIC RANDOM NUMBERS = ^C2"
PRINT " "
PRINT "BIWEIGHT SCALE ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^A3"
PRINT "STANDARD DEVIATION ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^B3"
PRINT "MAD ESTIMATE FOR CAUCHY RANDOM NUMBERS = ^C3"
PRINT " "
PRINT "BIWEIGHT SCALE ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^A4"
PRINT "STANDARD DEVIATION ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^B4"
PRINT "MAD ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = ^C4"
Dataplot generates the following output:
BIWEIGHT SCALE ESTIMATE FOR NORMAL RANDOM NUMBERS = 1.016386
STANDARD DEVIATION ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.9975
MAD ESTIMATE FOR NORMAL RANDOM NUMBERS = 0.681249
BIWEIGHT SCALE ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 3.066369
STANDARD DEVIATION ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 1.817945
MAD ESTIMATE FOR LOGISTIC RANDOM NUMBERS = 1.116496
BIWEIGHT SCALE ESTIMATE FOR CAUCHY RANDOM NUMBERS = 3.480419
STANDARD DEVIATION ESTIMATE FOR CAUCHY RANDOM NUMBERS = 998.389
MAD ESTIMATE FOR CAUCHY RANDOM NUMBERS = 1.015878
BIWEIGHT SCALE ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 1.529625
STANDARD DEVIATION ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 1.424258
MAD ESTIMATE FOR DOUBLE EXPO RANDOM NUMBERS = 0.684497
Program 2:
SKIP 25
READ GEAR.DAT DIAMETER BATCH
TITLE AUTOMATIC
XLIMITS 1 10
MAJOR XTIC MARK NUMBER 10
MINOR XTIC MARK NUMBER 0
XTIC OFFSET 1 1
X1LABEL BATCH
Y1LABEL BIWEIGHT SCALE OF DIAMETER
BIWEIGHT SCALE PLOT DIAMETER BATCH
Program 3:
MULTIPLOT 2 1
MULTIPLOT CORNER COORDINATES 0 0 100 100
LET Y = CAUCHY RANDOM NUMBERS FOR I = 1 1 1000
TITLE AUTOMATIC
BOOTSTRAP BIWEIGHT SCALE PLOT Y
X1LABEL B025 = ^B025, B975 = ^B975
HISTOGRAM YPLOT
END OF MULTIPLOT
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Date created: 11/20/2001
Last updated: 11/02/2015
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Solve nonlinear least-squares (nonlinear data-fitting) problems
Nonlinear least-squares solver
Solves nonlinear least-squares curve fitting problems of the form
subject to the constraints
$\begin{array}{c}\text{lb}\le x\\ x\le \text{ub}\\ Ax\le b\\ \text{Aeq}x=\text{beq}\\ c\left(x\right)\le 0\\ \text{ceq}\left(x\right)=0.\end{array}$
x, lb, and ub can be vectors or matrices; see Matrix Arguments.
Do not specify the objective function as the scalar value ${‖f\left(x\right)‖}_{2}^{2}$ (the sum of squares). lsqnonlin requires the objective function to be the vector-valued function
$f\left(x\right)=\left[\begin{array}{c}{f}_{1}\left(x\right)\\ {f}_{2}\left(x\right)\\ ⋮\\ {f}_{n}\left(x\right)\end{array}\right].$
x = lsqnonlin(fun,x0) starts at the point x0 and finds a minimum of the sum of squares of the functions described in fun. The function fun should return a vector (or array) of values and not the sum
of squares of the values. (The algorithm implicitly computes the sum of squares of the components of fun(x).)
x = lsqnonlin(fun,x0,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range lb ≤ x ≤ ub. You can fix the solution component x(i) by
specifying lb(i) = ub(i).
If the specified input bounds for a problem are inconsistent, the output x is x0 and the outputs resnorm and residual are [].
Components of x0 that violate the bounds lb ≤ x ≤ ub are reset to the interior of the box defined by the bounds. Components that respect the bounds are not changed.
x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq) constrains the solution to satisfy the linear constraints
x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq,nonlcon) constrain the solution to satisfy the nonlinear constraints in the nonlcon(x) function. nonlcon returns two outputs, c and ceq. The solver attempts to
satisfy the constraints
x = lsqnonlin(fun,x0,lb,ub,options) and x = lsqnonlin(fun,x0,lb,ub,A,b,Aeq,beq,nonlcon,options) minimizes with the optimization options specified in options. Use optimoptions to set these options.
Pass empty matrices for lb and ub and for other input arguments if the arguments do not exist.
x = lsqnonlin(problem) finds the minimum for problem, a structure described in problem.
[x,resnorm] = lsqnonlin(___), for any input arguments, returns the value of the squared 2-norm of the residual at x: sum(fun(x).^2).
[x,resnorm,residual,exitflag,output] = lsqnonlin(___) additionally returns the value of the residual fun(x) at the solution x, a value exitflag that describes the exit condition, and a structure
output that contains information about the optimization process.
[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(___) additionally returns a structure lambda whose fields contain the Lagrange multipliers at the solution x, and the Jacobian of fun
at the solution x.
Fit a Simple Exponential
Fit a simple exponential decay curve to data.
Generate data from an exponential decay model plus noise. The model is
$y=\mathrm{exp}\left(-1.3t\right)+\epsilon ,$
with $t$ ranging from 0 through 3, and $\epsilon$ normally distributed noise with mean 0 and standard deviation 0.05.
rng default % for reproducibility
d = linspace(0,3);
y = exp(-1.3*d) + 0.05*randn(size(d));
The problem is: given the data (d, y), find the exponential decay rate that best fits the data.
Create an anonymous function that takes a value of the exponential decay rate $r$ and returns a vector of differences from the model with that decay rate and the data.
Find the value of the optimal decay rate. Arbitrarily choose an initial guess x0 = 4.
x0 = 4;
x = lsqnonlin(fun,x0)
Local minimum possible.
lsqnonlin stopped because the final change in the sum of squares relative to
its initial value is less than the value of the function tolerance.
Plot the data and the best-fitting exponential curve.
legend('Data','Best fit')
Fit a Problem with Bound Constraints
Find the best-fitting model when some of the fitting parameters have bounds.
Find a centering $b$ and scaling $a$ that best fit the function
to the standard normal density,
$\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left(-{t}^{2}/2\right).$
Create a vector t of data points, and the corresponding normal density at those points.
t = linspace(-4,4);
y = 1/sqrt(2*pi)*exp(-t.^2/2);
Create a function that evaluates the difference between the centered and scaled function from the normal y, with x(1) as the scaling $a$ and x(2) as the centering $b$.
fun = @(x)x(1)*exp(-t).*exp(-exp(-(t-x(2)))) - y;
Find the optimal fit starting from x0 = [1/2,0], with the scaling $a$ between 1/2 and 3/2, and the centering $b$ between -1 and 3.
lb = [1/2,-1];
ub = [3/2,3];
x0 = [1/2,0];
x = lsqnonlin(fun,x0,lb,ub)
Local minimum possible.
lsqnonlin stopped because the final change in the sum of squares relative to
its initial value is less than the value of the function tolerance.
Plot the two functions to see the quality of the fit.
legend('Normal density','Fitted function')
Least Squares with Linear Constraint
Consider the following objective function, a sum of squares:
$\sum _{k=1}^{10}{\left(2+2k+\mathrm{exp}\left(k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{1}\right)+2\mathrm{exp}\left(2k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{2}^{2}\right)\right)}^
The code for this objective function appears as the myfun function at the end of this example.
Minimize this function subject to the linear constraint ${x}_{1}\le \frac{{x}_{2}}{2}$. Write this constraint as ${x}_{1}-\frac{{x}_{2}}{2}\le 0$.
Impose the bounds ${x}_{1}\ge 0$, ${x}_{2}\ge 0$, ${x}_{1}\le 2$, and ${x}_{2}\le 4$.
Start the optimization process from the point x0 = [0.3 0.4].
The problem has no linear equality constraints.
Run the optimization.
x = lsqnonlin(@myfun,x0,lb,ub,A,b,Aeq,beq)
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.
function F = myfun(x)
k = 1:10;
F = 2 + 2*k - exp(k*x(1)) - 2*exp(2*k*(x(2)^2));
Nonlinear Least Squares with Nonlinear Constraint
Consider the following objective function, a sum of squares:
$\sum _{k=1}^{10}{\left(2+2k+\mathrm{exp}\left(k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{1}\right)+2\mathrm{exp}\left(2k\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{2}^{2}\right)\right)}^
The code for this objective function appears as the myfun function at the end of this example.
Minimize this function subject to the nonlinear constraint $\mathrm{sin}\left({x}_{1}\right)\le \mathrm{cos}\left({x}_{2}\right)$. The code for this nonlinear constraint function appears as the nlcon
function at the end of this example.
Impose the bounds ${x}_{1}\ge 0$, ${x}_{2}\ge 0$, ${x}_{1}\le 2$, and ${x}_{2}\le 4$.
Start the optimization process from the point x0 = [0.3 0.4].
The problem has no linear constraints.
A = [];
b = [];
Aeq = [];
beq = [];
Run the optimization.
x = lsqnonlin(@myfun,x0,lb,ub,A,b,Aeq,beq,@nlcon)
Local minimum possible. Constraints satisfied.
fmincon stopped because the size of the current step is less than
the value of the step size tolerance and constraints are
satisfied to within the value of the constraint tolerance.
function F = myfun(x)
k = 1:10;
F = 2 + 2*k - exp(k*x(1)) - 2*exp(2*k*(x(2)^2));
function [c,ceq] = nlcon(x)
ceq = [];
c = sin(x(1)) - cos(x(2));
Nonlinear Least Squares with Nondefault Options
Compare the results of a data-fitting problem when using different lsqnonlin algorithms.
Suppose that you have observation time data xdata and observed response data ydata, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form
Input the observation times and responses.
xdata = ...
[0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
[455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];
Create a simple exponential decay model. The model computes a vector of differences between predicted values and observed values.
fun = @(x)x(1)*exp(x(2)*xdata)-ydata;
Fit the model using the starting point x0 = [100,-1]. First, use the default 'trust-region-reflective' algorithm.
x0 = [100,-1];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
x = lsqnonlin(fun,x0,[],[],options)
Local minimum possible.
lsqnonlin stopped because the final change in the sum of squares relative to
its initial value is less than the value of the function tolerance.
See if there is any difference using the 'levenberg-marquardt' algorithm.
options.Algorithm = 'levenberg-marquardt';
x = lsqnonlin(fun,x0,[],[],options)
Local minimum possible.
lsqnonlin stopped because the relative size of the current step is less than
the value of the step size tolerance.
The two algorithms found the same solution. Plot the solution and the data.
hold on
tlist = linspace(xdata(1),xdata(end));
xlabel xdata
ylabel ydata
title('Exponential Fit to Data')
legend('Data','Exponential Fit')
hold off
Nonlinear Least Squares Solution and Residual Norm
Find the $x$ that minimizes
$\sum _{k=1}^{10}{\left(2+2k-{e}^{k{x}_{1}}-{e}^{k{x}_{2}}\right)}^{2}$,
and find the value of the minimal sum of squares.
Because lsqnonlin assumes that the sum of squares is not explicitly formed in the user-defined function, the function passed to lsqnonlin should instead compute the vector-valued function
for $k=1$ to $10$ (that is, $F$ should have $10$ components).
The myfun function, which computes the 10-component vector F, appears at the end of this example.
Find the minimizing point and the minimum value, starting at the point x0 = [0.3,0.4].
x0 = [0.3,0.4];
[x,resnorm] = lsqnonlin(@myfun,x0)
Local minimum possible.
lsqnonlin stopped because the size of the current step is less than
the value of the step size tolerance.
The resnorm output is the squared residual norm, or the sum of squares of the function values.
The following function computes the vector-valued objective function.
function F = myfun(x)
k = 1:10;
F = 2 + 2*k-exp(k*x(1))-exp(k*x(2));
Examine the Solution Process
Examine the solution process both as it occurs (by setting the Display option to 'iter') and afterward (by examining the output structure).
Suppose that you have observation time data xdata and observed response data ydata, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form
Input the observation times and responses.
xdata = ...
[0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
[455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];
Create a simple exponential decay model. The model computes a vector of differences between predicted values and observed values.
fun = @(x)x(1)*exp(x(2)*xdata)-ydata;
Fit the model using the starting point x0 = [100,-1]. Examine the solution process by setting the Display option to 'iter'. Obtain an output structure to obtain more information about the solution
x0 = [100,-1];
options = optimoptions('lsqnonlin','Display','iter');
[x,resnorm,residual,exitflag,output] = lsqnonlin(fun,x0,[],[],options);
Norm of First-order
Iteration Func-count Resnorm step optimality
0 3 359677 2.88e+04
Objective function returned Inf; trying a new point...
1 6 359677 11.6976 2.88e+04
2 9 321395 0.5 4.97e+04
3 12 321395 1 4.97e+04
4 15 292253 0.25 7.06e+04
5 18 292253 0.5 7.06e+04
6 21 270350 0.125 1.15e+05
7 24 270350 0.25 1.15e+05
8 27 252777 0.0625 1.63e+05
9 30 252777 0.125 1.63e+05
10 33 243877 0.03125 7.48e+04
11 36 243660 0.0625 8.7e+04
12 39 243276 0.0625 2e+04
13 42 243174 0.0625 1.14e+04
14 45 242999 0.125 5.1e+03
15 48 242661 0.25 2.04e+03
16 51 241987 0.5 1.91e+03
17 54 240643 1 1.04e+03
18 57 237971 2 3.36e+03
19 60 232686 4 6.04e+03
20 63 222354 8 1.2e+04
21 66 202592 16 2.25e+04
22 69 166443 32 4.05e+04
23 72 106320 64 6.68e+04
24 75 28704.7 128 8.31e+04
25 78 89.7947 140.674 2.22e+04
26 81 9.57381 2.02599 684
27 84 9.50489 0.0619927 2.27
28 87 9.50489 0.000462261 0.0114
Local minimum possible.
lsqnonlin stopped because the final change in the sum of squares relative to
its initial value is less than the value of the function tolerance.
Examine the output structure to obtain more information about the solution process.
output = struct with fields:
firstorderopt: 0.0114
iterations: 28
funcCount: 87
cgiterations: 0
algorithm: 'trust-region-reflective'
stepsize: 4.6226e-04
message: 'Local minimum possible....'
bestfeasible: []
constrviolation: []
For comparison, set the Algorithm option to 'levenberg-marquardt'.
options.Algorithm = 'levenberg-marquardt';
[x,resnorm,residual,exitflag,output] = lsqnonlin(fun,x0,[],[],options);
First-order Norm of
Iteration Func-count Resnorm optimality Lambda step
0 3 359677 2.88e+04 0.01
Objective function returned Inf; trying a new point...
1 13 340761 3.91e+04 100000 0.280777
2 16 304661 5.97e+04 10000 0.373146
3 21 297292 6.55e+04 1e+06 0.0589933
4 24 288240 7.57e+04 100000 0.0645444
5 28 275407 1.01e+05 1e+06 0.0741266
6 31 249954 1.62e+05 100000 0.094571
7 36 245896 1.35e+05 1e+07 0.0133606
8 39 243846 7.26e+04 1e+06 0.0094431
9 42 243568 5.66e+04 100000 0.0082162
10 45 243424 1.61e+04 10000 0.00777935
11 48 243322 8.8e+03 1000 0.0673933
12 51 242408 5.1e+03 100 0.675209
13 54 233628 1.05e+04 10 6.59804
14 57 169089 8.51e+04 1 54.6992
15 60 30814.7 1.54e+05 0.1 196.939
16 63 147.496 8e+03 0.01 129.795
17 66 9.51503 117 0.001 9.96069
18 69 9.50489 0.0714 0.0001 0.080486
19 72 9.50489 5.23e-05 1e-05 5.07043e-05
Local minimum possible.
lsqnonlin stopped because the relative size of the current step is less than
the value of the step size tolerance.
The 'levenberg-marquardt' converged with fewer iterations, but almost as many function evaluations:
output = struct with fields:
iterations: 19
funcCount: 72
stepsize: 5.0704e-05
cgiterations: []
firstorderopt: 5.2319e-05
algorithm: 'levenberg-marquardt'
message: 'Local minimum possible....'
bestfeasible: []
constrviolation: []
Input Arguments
fun — Function whose sum of squares is minimized
function handle | name of function
Function whose sum of squares is minimized, specified as a function handle or the name of a function. For the 'interior-point' algorithm, fun must be a function handle. fun is a function that accepts
an array x and returns an array F, the objective function evaluated at x.
The sum of squares should not be formed explicitly. Instead, your function should return a vector of function values. See Examples.
The function fun can be specified as a function handle to a file:
where myfun is a MATLAB^® function such as
function F = myfun(x)
F = ... % Compute function values at x
fun can also be a function handle for an anonymous function.
x = lsqnonlin(@(x)sin(x.*x),x0);
lsqnonlin passes x to your objective function in the shape of the x0 argument. For example, if x0 is a 5-by-3 array, then lsqnonlin passes x to fun as a 5-by-3 array.
If the Jacobian can also be computed and the 'SpecifyObjectiveGradient' option is true, set by
options = optimoptions('lsqnonlin','SpecifyObjectiveGradient',true)
then the function fun must return a second output argument with the Jacobian value J (a matrix) at x. By checking the value of nargout, the function can avoid computing J when fun is called with only
one output argument (in the case where the optimization algorithm only needs the value of F but not J).
function [F,J] = myfun(x)
F = ... % Objective function values at x
if nargout > 1 % Two output arguments
J = ... % Jacobian of the function evaluated at x
If fun returns an array of m components and x has n elements, where n is the number of elements of x0, the Jacobian J is an m-by-n matrix where J(i,j) is the partial derivative of F(i) with respect
to x(j). (The Jacobian J is the transpose of the gradient of F.)
Example: @(x)cos(x).*exp(-x)
Data Types: char | function_handle | string
nonlcon — Nonlinear constraints
function handle
Nonlinear constraints, specified as a function handle. nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).
• c(x) is the array of nonlinear inequality constraints at x. lsqnonlin attempts to satisfy
c(x) <= 0 for all entries of c. (1)
• ceq(x) is the array of nonlinear equality constraints at x. lsqnonlin attempts to satisfy
ceq(x) = 0 for all entries of ceq. (2)
For example,
x = lsqnonlin(@myfun,x0,lb,ub,A,b,Aeq,beq,@mycon,options)
where mycon is a MATLAB function such as
function [c,ceq] = mycon(x)
c = ... % Compute nonlinear inequalities at x.
ceq = ... % Compute nonlinear equalities at x.
If the Jacobians (derivatives) of the constraints can also be computed
option is
, as set by
options = optimoptions('lsqnonlin','SpecifyConstraintGradient',true)
must also return, in the third and fourth output arguments,
, the Jacobian of
, and
, the Jacobian of
. The Jacobian
) of a vector function
) is
${G}_{i,j}\left(x\right)=\frac{\partial {F}_{i}\left(x\right)}{\partial {x}_{j}}.$
GC and GCeq can be sparse or dense. If GC or GCeq is large, with relatively few nonzero entries, save running time and memory in the 'interior-point' algorithm by representing them as sparse
matrices. For more information, see Nonlinear Constraints.
Data Types: function_handle
options — Optimization options
output of optimoptions | structure as optimset returns
Optimization options, specified as the output of optimoptions or a structure as optimset returns.
Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.
Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Optimization Options.
All Algorithms
Choose between 'trust-region-reflective' (default), 'levenberg-marquardt', and 'interior-point'.
The Algorithm option specifies a preference for which algorithm to use. It is only a preference, because certain conditions must be met to use each algorithm. For the
trust-region-reflective algorithm, the number of elements of F returned by fun must be at least as many as the length of x.
The 'interior-point' algorithm is the only algorithm that can solve problems with linear or nonlinear constraints. If you include these constraints in your problem and do
not specify an algorithm, the solver automatically switches to the 'interior-point' algorithm. The 'interior-point' algorithm calls a modified version of the fmincon
'interior-point' algorithm.
For more information on choosing the algorithm, see Choosing the Algorithm.
Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. Choices are false (default) or true.
CheckGradients For optimset, the name is DerivativeCheck and the values are 'on' or 'off'. See Current and Legacy Option Names.
The CheckGradients option will be removed in a future release. To check derivatives, use the checkGradients function.
Diagnostics Display diagnostic information about the function to be minimized or solved. Choices are 'off' (default) or 'on'.
DiffMaxChange Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf.
DiffMinChange Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0.
Level of display (see Iterative Display):
• 'off' or 'none' displays no output.
• 'iter' displays output at each iteration, and gives the default exit message.
• 'iter-detailed' displays output at each iteration, and gives the technical exit message.
• 'final' (default) displays just the final output, and gives the default exit message.
• 'final-detailed' displays just the final output, and gives the technical exit message.
Scalar or vector step size factor for finite differences. When you set FiniteDifferenceStepSize to a vector v, the forward finite differences delta are
delta = v.*sign′(x).*max(abs(x),TypicalX);
where sign′(x) = sign(x) except sign′(0) = 1. Central finite differences are
delta = v.*max(abs(x),TypicalX);
A scalar FiniteDifferenceStepSize expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences.
For optimset, the name is FinDiffRelStep. See Current and Legacy Option Names.
Finite differences, used to estimate gradients, are either 'forward' (default), or 'central' (centered). 'central' takes twice as many function evaluations, but should be
more accurate.
FiniteDifferenceType The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to
avoid evaluating at a point outside bounds.
For optimset, the name is FinDiffType. See Current and Legacy Option Names.
Termination tolerance on the function value, a nonnegative scalar. The default is 1e-6. See Tolerances and Stopping Criteria.
For optimset, the name is TolFun. See Current and Legacy Option Names.
FunValCheck Check whether function values are valid. 'on' displays an error when the function returns a value that is complex, Inf, or NaN. The default 'off' displays no error.
Maximum number of function evaluations allowed, a nonnegative integer. The default is 100*numberOfVariables for the 'trust-region-reflective' algorithm,
200*numberOfVariables for the 'levenberg-marquardt' algorithm, and 3000 for the 'interior-point' algorithm. See Tolerances and Stopping Criteria and Iterations and Function
MaxFunctionEvaluations Counts.
For optimset, the name is MaxFunEvals. See Current and Legacy Option Names.
Maximum number of iterations allowed, a nonnegative integer. The default is 400 for the 'trust-region-reflective' and 'levenberg-marquardt' algorithms, and 1000 for the
MaxIterations 'interior-point' algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts.
For optimset, the name is MaxIter. See Current and Legacy Option Names.
Termination tolerance on the first-order optimality (a nonnegative scalar). The default is 1e-6. See First-Order Optimality Measure.
OptimalityTolerance Internally, the 'levenberg-marquardt' algorithm uses an optimality tolerance (stopping criterion) of 1e-4 times FunctionTolerance and does not use OptimalityTolerance.
For optimset, the name is TolFun. See Current and Legacy Option Names.
Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is
OutputFcn none ([]). See Output Function and Plot Function Syntax.
Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a name, a function handle, or a cell array of names or
function handles. For custom plot functions, pass function handles. The default is none ([]):
• 'optimplotx' plots the current point.
• 'optimplotfunccount' plots the function count.
• 'optimplotfval' plots the function value.
• 'optimplotresnorm' plots the norm of the residuals.
• 'optimplotstepsize' plots the step size.
• 'optimplotfirstorderopt' plots the first-order optimality measure.
Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax.
For optimset, the name is PlotFcns. See Current and Legacy Option Names.
If false (default), the solver approximates the Jacobian using finite differences. If true, the solver uses a user-defined Jacobian (defined in fun), or Jacobian information
(when using JacobMult), for the objective function.
For optimset, the name is Jacobian, and the values are 'on' or 'off'. See Current and Legacy Option Names.
Termination tolerance on x, a nonnegative scalar. The default is 1e-6 for the 'trust-region-reflective' and 'levenberg-marquardt' algorithms, and 1e-10 for the
StepTolerance 'interior-point' algorithm. See Tolerances and Stopping Criteria.
For optimset, the name is TolX. See Current and Legacy Option Names.
TypicalX Typical x values. The number of elements in TypicalX is equal to the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). The solver
uses TypicalX for scaling finite differences for gradient estimation.
UseParallel When true, the solver estimates gradients in parallel. Disable by setting to the default, false. See Parallel Computing.
Trust-Region-Reflective Algorithm
Jacobian multiply function, specified as a function handle. For large-scale structured problems, this function computes the Jacobian matrix product J*Y, J'*Y, or J'*(J*Y)
without actually forming J. For lsqnonlin the function is of the form
W = jmfun(Jinfo,Y,flag)
where Jinfo contains the data that helps to compute J*Y (or J'*Y, or J'*(J*Y)). For lsqcurvefit the function is of the form
W = jmfun(Jinfo,Y,flag,xdata)
where xdata is the data passed in the xdata argument.
The data Jinfo is the second argument returned by the objective function fun:
[F,Jinfo] = fun(x)
% or [F,Jinfo] = fun(x,xdata)
lsqnonlin passes the data Jinfo, Y, flag, and, for lsqcurvefit, xdata, and your function jmfun computes a result as specified next. Y is a matrix whose size depends on the
value of flag. Let m specify the number of components of the objective function fun, and let n specify the number of problem variables in x. The Jacobian is of size m-by-n
JacobianMultiplyFcn as described in fun. The jmfun function returns one of these results:
• If flag == 0 then W = J'*(J*Y) and Y has size n-by-2.
• If flag > 0 then W = J*Y and Y has size n-by-1.
• If flag < 0 then W = J'*Y and Y has size m-by-1.
In each case, J is not formed explicitly. The solver uses Jinfo to compute the multiplications. See Passing Extra Parameters for information on how to supply values for any
additional parameters jmfun needs.
'SpecifyObjectiveGradient' must be set to true for the solver to pass Jinfo from fun to jmfun.
See Minimization with Dense Structured Hessian, Linear Equalities and Jacobian Multiply Function with Linear Least Squares for similar examples.
For optimset, the name is JacobMult. See Current and Legacy Option Names.
Sparsity pattern of the Jacobian for finite differencing. Set JacobPattern(i,j) = 1 when fun(i) depends on x(j). Otherwise, set JacobPattern(i,j) = 0. In other words,
JacobPattern(i,j) = 1 when you can have ∂fun(i)/∂x(j) ≠ 0.
Use JacobPattern when it is inconvenient to compute the Jacobian matrix J in fun, though you can determine (say, by inspection) when fun(i) depends on x(j). The solver can
JacobPattern approximate J via sparse finite differences when you give JacobPattern.
If the structure is unknown, do not set JacobPattern. The default behavior is as if JacobPattern is a dense matrix of ones. Then the solver computes a full finite-difference
approximation in each iteration. This can be expensive for large problems, so it is usually better to determine the sparsity structure.
Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is max(1,numberOfVariables/2). For more information, see Large Scale
MaxPCGIter Nonlinear Least Squares.
Upper bandwidth of preconditioner for PCG, a nonnegative integer. The default PrecondBandWidth is Inf, which means a direct factorization (Cholesky) is used rather than the
PrecondBandWidth conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set PrecondBandWidth
to 0 for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations.
SubproblemAlgorithm Determines how the iteration step is calculated. The default, 'factorization', takes a slower but more accurate step than 'cg'. See Trust-Region-Reflective Least Squares.
TolPCG Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1.
Levenberg-Marquardt Algorithm
InitDamping Initial value of the Levenberg-Marquardt parameter, a positive scalar. Default is 1e-2. For details, see Levenberg-Marquardt Method.
ScaleProblem 'jacobian' can sometimes improve the convergence of a poorly scaled problem; the default is 'none'.
Interior-Point Algorithm
Specifies how fmincon updates the barrier parameter (see fmincon Interior Point Algorithm). The options are:
• 'monotone' (default)
• 'predictor-corrector'
This option can affect the speed and convergence of the solver, but the effect is not easy to predict.
Tolerance on the constraint violation, a nonnegative scalar. The default is 1e-6. See Tolerances and Stopping Criteria.
For optimset, the name is TolCon. See Current and Legacy Option Names.
InitBarrierParam Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default 0.1, especially if the objective or constraint functions are large.
Gradient for nonlinear constraint functions defined by the user. When set to the default, false, lsqnonlin estimates gradients of the nonlinear constraints by finite
differences. When set to true, lsqnonlin expects the constraint function to have four outputs, as described in nonlcon.
For optimset, the name is GradConstr and the values are 'on' or 'off'. See Current and Legacy Option Names.
Determines how the iteration step is calculated. The default, 'factorization', is usually faster than 'cg' (conjugate gradient), though 'cg' might be faster for large
SubproblemAlgorithm problems with dense Hessians. See fmincon Interior Point Algorithm.
For optimset, the values are 'cg' and 'ldl-factorization'. See Current and Legacy Option Names.
Example: options = optimoptions('lsqnonlin','FiniteDifferenceType','central')
problem — Problem structure
Problem structure, specified as a structure with the following fields:
Field Name Entry
objective Objective function
x0 Initial point for x
Aineq Matrix for linear inequality constraints
bineq Vector for linear inequality constraints
Aeq Matrix for linear equality constraints
beq Vector for linear equality constraints
lb Vector of lower bounds
ub Vector of upper bounds
nonlcon Nonlinear constraint function
solver 'lsqnonlin'
options Options created with optimoptions
You must supply at least the objective, x0, solver, and options fields in the problem structure.
Data Types: struct
Output Arguments
resnorm — Squared norm of the residual
nonnegative real
Squared norm of the residual, returned as a nonnegative real. resnorm is the squared 2-norm of the residual at x: sum(fun(x).^2).
residual — Value of objective function at solution
Value of objective function at solution, returned as an array. In general, residual = fun(x).
• The trust-region-reflective algorithm does not solve underdetermined systems; it requires that the number of equations, i.e., the row dimension of F, be at least as great as the number of
variables. In the underdetermined case, lsqnonlin uses the Levenberg-Marquardt algorithm.
• lsqnonlin can solve complex-valued problems directly. Note that constraints do not make sense for complex values, because complex numbers are not well-ordered; asking whether one complex value is
greater or less than another complex value is nonsensical. For a complex problem with bound constraints, split the variables into real and imaginary parts. Do not use the 'interior-point'
algorithm with complex data. See Fit a Model to Complex-Valued Data.
• The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective method forms J^TJ (where J is the Jacobian matrix) before computing the
preconditioner. Therefore, a row of J with many nonzeros, which results in a nearly dense product J^TJ, can lead to a costly solution process for large problems.
• If components of x have no upper (or lower) bounds, lsqnonlin prefers that the corresponding components of ub (or lb) be set to inf (or -inf for lower bounds) as opposed to an arbitrary but very
large positive (or negative for lower bounds) number.
You can use the trust-region reflective algorithm in lsqnonlin, lsqcurvefit, and fsolve with small- to medium-scale problems without computing the Jacobian in fun or providing the Jacobian sparsity
pattern. (This also applies to using fmincon or fminunc without computing the Hessian or supplying the Hessian sparsity pattern.) How small is small- to medium-scale? No absolute answer is available,
as it depends on the amount of virtual memory in your computer system configuration.
Suppose your problem has m equations and n unknowns. If the command J = sparse(ones(m,n)) causes an Out of memory error on your machine, then this is certainly too large a problem. If it does not
result in an error, the problem might still be too large. You can find out only by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.
More About
Enhanced Exit Messages
The next few items list the possible enhanced exit messages from lsqnonlin. Enhanced exit messages give a link for more information as the first sentence of the message.
Definitions for Exit Messages
The next few items contain definitions for terms in the lsqnonlin exit messages.
local minimum
A local minimum of a function is a point where the function value is smaller than at nearby points, but possibly greater than at a distant point.
A global minimum is a point where the function value is smaller than at all other feasible points.
Solvers try to find a local minimum. The result can be a global minimum. For more information, see Local vs. Global Optima.
first-order optimality measure
For unconstrained problems, the first-order optimality measure is the maximum of the absolute value of the components of the gradient vector (also known as the infinity norm of the gradient). This
should be zero at a minimizing point.
For problems with bounds, the first-order optimality measure is the maximum over i of |v[i]*g[i]|. Here g[i] is the ith component of the gradient, x is the current point, and
${v}_{i}=\left\{\begin{array}{ll}|{x}_{i}-{b}_{i}|\hfill & \text{if the negative gradient points toward bound}{b}_{i}\hfill \\ 1\hfill & \text{otherwise}\text{.}\hfill \end{array}$
If x[i] is at a bound, v[i] is zero. If x[i] is not at a bound, then at a minimizing point the gradient g[i] should be zero. Therefore the first-order optimality measure should be zero at a
minimizing point.
For more information, see First-Order Optimality Measure.
Generally, a tolerance is a threshold which, if crossed, stops the iterations of a solver. For more information on tolerances, see Tolerances and Stopping Criteria.
The tolerance called OptimalityTolerance relates to the first-order optimality measure. Iterations end when the first-order optimality measure is less than OptimalityTolerance.
The function tolerance called FunctionTolerance relates to the size of the latest change in objective function value.
Gradient Size
The gradient vector is the gradient of the sum of squares. For unconstrained problems, the gradient size is the maximum of the absolute value of the components of the gradient vector (also known as
the infinity norm of the gradient). This should be zero at a minimizing point.
For problems with bounds, the gradient size is the maximum over i of |v[i]*g[i]|. Here g[i] is the ith component of the gradient, x is the current point, and
${v}_{i}=\left\{\begin{array}{ll}|{x}_{i}-{b}_{i}|\hfill & \text{if the negative gradient points toward bound}{b}_{i}\hfill \\ 1\hfill & \text{otherwise}\text{.}\hfill \end{array}$
If x[i] is at a bound, v[i] is zero. If x[i] is not at a bound, then at a minimizing point the gradient g[i] should be zero. Therefore the gradient size should be zero at a minimizing point.
For more information, see First-Order Optimality Measure.
Size of the Current Step
The size of the current step is the magnitude of the change in location of the current point in the final iteration.
For more information, see Tolerances and Stopping Criteria.
Locally Singular
The Levenberg-Marquardt regularization parameter is related to the inverse of a trust-region radius. It becomes large when the sum of squares of function values is not close to a quadratic model. For
more information, see Levenberg-Marquardt Method.
Jacobian Calculation is Undefined
Solvers estimate the Jacobian of your objective vector function by taking finite differences. A finite difference calculation stepped outside the region where the objective function is well-defined,
returning Inf, NaN, or a complex result.
For more information about how solvers use the Jacobian J, see Levenberg-Marquardt Method. For suggestions on how to proceed, see 6. Provide Gradient or Jacobian.
The Levenberg-Marquardt and trust-region-reflective methods are based on the nonlinear least-squares algorithms also used in fsolve.
• The default trust-region-reflective algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. Each iteration involves the
approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See Trust-Region-Reflective Least Squares.
• The Levenberg-Marquardt method is described in references [4], [5], and [6]. See Levenberg-Marquardt Method.
The 'interior-point' algorithm uses the fmincon 'interior-point' algorithm with some modifications. For details, see Modified fmincon Algorithm for Constrained Least Squares.
Alternative Functionality
The Optimize Live Editor task provides a visual interface for lsqnonlin.
[1] Coleman, T.F. and Y. Li. “An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds.” SIAM Journal on Optimization, Vol. 6, 1996, pp. 418–445.
[2] Coleman, T.F. and Y. Li. “On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds.” Mathematical Programming, Vol. 67, Number 2, 1994, pp.
[3] Dennis, J. E. Jr. “Nonlinear Least-Squares.” State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269–312.
[4] Levenberg, K. “A Method for the Solution of Certain Problems in Least-Squares.” Quarterly Applied Mathematics 2, 1944, pp. 164–168.
[5] Marquardt, D. “An Algorithm for Least-squares Estimation of Nonlinear Parameters.” SIAM Journal Applied Mathematics, Vol. 11, 1963, pp. 431–441.
[6] Moré, J. J. “The Levenberg-Marquardt Algorithm: Implementation and Theory.” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, 1977, pp. 105–116.
[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom. User Guide for MINPACK 1. Argonne National Laboratory, Rept. ANL–80–74, 1980.
[8] Powell, M. J. D. “A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations.” Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7, 1970.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
• lsqcurvefit and lsqnonlin support code generation using either the codegen (MATLAB Coder) function or the MATLAB Coder™ app. You must have a MATLAB Coder license to generate code.
• The target hardware must support standard double-precision floating-point computations. You cannot generate code for single-precision or fixed-point computations.
• Code generation targets do not use the same math kernel libraries as MATLAB solvers. Therefore, code generation solutions can vary from solver solutions, especially for poorly conditioned
• All code for generation must be MATLAB code. In particular, you cannot use a custom black-box function as an objective function for lsqcurvefit or lsqnonlin. You can use coder.ceval to evaluate a
custom function coded in C or C++. However, the custom function must be called in a MATLAB function.
• Code generation for lsqcurvefit and lsqnonlin currently does not support linear or nonlinear constraints.
• lsqcurvefit and lsqnonlin do not support the problem argument for code generation.
[x,fval] = lsqnonlin(problem) % Not supported
• You must specify the objective function by using function handles, not strings or character names.
x = lsqnonlin(@fun,x0,lb,ub,options) % Supported
% Not supported: lsqnonlin('fun',...) or lsqnonlin("fun",...)
• All input matrices lb and ub must be full, not sparse. You can convert sparse matrices to full by using the full function.
• The lb and ub arguments must have the same number of entries as the x0 argument or must be empty [].
• If your target hardware does not support infinite bounds, use optim.coder.infbound.
• For advanced code optimization involving embedded processors, you also need an Embedded Coder^® license.
• You must include options for lsqcurvefit or lsqnonlin and specify them using optimoptions. The options must include the Algorithm option, set to 'levenberg-marquardt'.
options = optimoptions('lsqnonlin','Algorithm','levenberg-marquardt');
[x,fval,exitflag] = lsqnonlin(fun,x0,lb,ub,options);
• Code generation supports these options:
□ Algorithm — Must be 'levenberg-marquardt'
□ FiniteDifferenceStepSize
□ FiniteDifferenceType
□ FunctionTolerance
□ MaxFunctionEvaluations
□ MaxIterations
□ SpecifyObjectiveGradient
□ StepTolerance
□ TypicalX
• Generated code has limited error checking for options. The recommended way to update an option is to use optimoptions, not dot notation.
opts = optimoptions('lsqnonlin','Algorithm','levenberg-marquardt');
opts = optimoptions(opts,'MaxIterations',1e4); % Recommended
opts.MaxIterations = 1e4; % Not recommended
• Do not load options from a file. Doing so can cause code generation to fail. Instead, create options in your code.
• Usually, if you specify an option that is not supported, the option is silently ignored during code generation. However, if you specify a plot function or output function by using dot notation,
code generation can issue an error. For reliability, specify only supported options.
• Because output functions and plot functions are not supported, solvers do not return the exit flag –1.
For an example, see Generate Code for lsqcurvefit or lsqnonlin.
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To run in parallel, set the 'UseParallel' option to true.
options = optimoptions('solvername','UseParallel',true)
For more information, see Using Parallel Computing in Optimization Toolbox.
Version History
Introduced before R2006a
R2023b: JacobianMultiplyFcn accepts any data type
The syntax for the JacobianMultiplyFcn option is
W = jmfun(Jinfo, Y, flag)
The Jinfo data, which MATLAB passes to your function jmfun, can now be of any data type. For example, you can now have Jinfo be a structure. In previous releases, Jinfo had to be a standard double
The Jinfo data is the second output of your objective function:
R2023b: CheckGradients option will be removed
The CheckGradients option will be removed in a future release. To check the first derivatives of objective functions or nonlinear constraint functions, use the checkGradients function.
R2023a: Linear and Nonlinear Constraint Support
lsqnonlin gains support for both linear and nonlinear constraints. To enable constraint satisfaction, the solver uses the "interior-point" algorithm from fmincon.
• If you specify constraints but do not specify an algorithm, the solver automatically switches to the "interior-point" algorithm.
• If you specify constraints and an algorithm, you must specify the "interior-point" algorithm.
For algorithm details, see Modified fmincon Algorithm for Constrained Least Squares. For an example, see Compare lsqnonlin and fmincon for Constrained Nonlinear Least Squares. | {"url":"https://ch.mathworks.com/help/optim/ug/lsqnonlin.html","timestamp":"2024-11-05T07:47:08Z","content_type":"text/html","content_length":"268909","record_id":"<urn:uuid:77f38469-18f1-4658-b077-5a06d1fb08a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00541.warc.gz"} |
Solving Systems of Inequalities by Graphing | sofatutor.com
Solving Systems of Inequalities by Graphing
Basics on the topic Solving Systems of Inequalities by Graphing
Similar to systems of equations, systems of inequalities are two or more inequalities with the same two variables. To determine the solution to the system, or the point where the inequalities
intersect (overlap), a graph is the best method to solve these problems.
First manipulate the inequalities in the system so they are written in slope-intercept form, y = mx + b. This makes it easier for you to create the graph. For each inequality in the system: First put
a point on the y-intercept - indicated by the b-value. Next, use the m-value to draw in the slope. To connect the dots, you will use a dotted line to indicate less than or greater than, and a solid
line is used for less than and equal to or greater than and equal to situations. Shade the area of the solution set.
This can be tricky – to avoid confusion, it’s a good idea to select a test point. Pop the coordinates of the test point into the inequality. If the inequality is true, shade the area indicated and if
false, shade the other side of the line. Follow these steps for each inequality in the system, and the intersection of the shaded areas is the solution that makes all inequalities in the system true.
Solve systems of equations to find solutions to problems.
Transcript Solving Systems of Inequalities by Graphing
Frank, the insurance man, is on a flight to Peru to track down Adventure Mike in the Peruvian jungle. Adventure Mike needs to renew his Ultra Danger Insurance Package. Otherwise, he will not have any
insurance on his various adventures.
The system of linear inequalities
Frank takes out his phone to look at the text message that he received a few days ago from Adventure Mike. The text message provides a hint as to which region in the jungle Adventure Mike is
searching for ancient relics. Frank knows that Adventure Mike likes to communicate in riddles. He recognizes Mike's missive as a system of linear inequalities.
Good thing Frank knows how to solve a system of linear inequalities by graphing. Let's see how Frank solves the system of linear inequalities. Here's a map of the surrounding region. Since we have a
system of linear inequalities, we need to include the x- and y-axes in order to graph them. Our starting point, or the origin, is the city of Cuzco.
We need to graph the system of inequalities. Each inequality is written in slope-intercept form. For the first inequality, 1 is the y-intercept or the ordered pair (0, 1). To find the second point,
we use the slope of the line. The slope of the line is -2 or -2 over 1. We move down 2 from the y-intercept and right 1. The resulting ordered pair is (1, -1).
Less than or Equal to
Since the inequality represents 'Less than or Equal to', the graph of the inequality is a solid line. All of the points to the left of the inequality are true. We can shade to the left of the
inequality line. For the second inequality, 2 is the y-intercept, or the ordered pair (0, -2). To find the second point, we use the slope of the line.
Less than
The slope of the line is 3 over 2. We move up 3 from the y-intercept and right 2. The resulting ordered pair is (2, 1). Since the inequality represents 'Less than', the graph of the inequality is a
dotted line. Since all of the points to the right of the inequality are true. We can shade to the right of the inequality line.
The solution to the system of inequalities is where the shading from each inequality overlaps. Frank knows in which part of the jungle to look, so he sets off to find Adventure Mike. Wow! Frank finds
an ancient Incan temple. But, what is carved into the side of it? Adventure Mike has left another hint!
Frank pulls out his map again so he can crack this clue and narrow down the possible whereabouts of Adventure Mike. We need to graph the inequality. The inequality is written in slope-intercept form.
For the first inequality, -4 is the y-intercept or the ordered pair (0, -4). To find the second point, we use the slope of the line. The slope of the line is 1 over 4. We move up 1 from the
y-intercept and right 4. The resulting ordered pair is (4, -3).
Greater than or Equal to
Since the inequality represents 'Greater than or Equal to', the graph of the inequality is a solid line. All of the points above the inequality are true. We can shade above the inequality line. The
solution to the system of inequalities is where the shading from each inequality overlaps. Now Frank knows exactly where to go...Oh, so THAT'S why Adventure Mike sent those strange messages in the
first place.
Solving Systems of Inequalities by Graphing exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Solving Systems of Inequalities by Graphing.
• Explain how to solve a system of linear inequalities by graphing.
Keep in mind
□ $\le$ or $\ge$ $\rightarrow$ solid line
□ $<$ or $>$ $\rightarrow$ dashed line
Both inequalities are shown in slope-intercept form: $y=mx+b$.
□ $m$ is the slope
□ $b$ is the y-intercept
To check if the solutions lie above or below the line formed by the graph of the inequality, pick a point and substitute the coordinates into the corresponding inequality.
We have the following two inequalities:
1. $y\le -2x+1$
2. $y<\frac32x-2$
Start by taking a look at the inequality $\mathbf{y\le -2x+1}$:
1. Draw the y-intercept, $1$.
2. Next, draw the slope: move two units down and one unit to the right to find a second point on the line.
3. Connect the dots with a solid orange line, because the inequality includes "equal to".
4. Shade in the area of the solution set. The solutions lie below the orange line because the inequality includes "less than".
We proceed in the same way with the second inequality, $\mathbf{y<\frac32x-2}$:
1. Draw the y-intercept, $-2$.
2. Next, draw the slope: move up three units and two units to the right.
3. Connect the dots with a green dashed line, because of the less than sign.
4. Shade in the area of the solution set. The solutions lie under the green line.
The set of all possible solutions of the two inequalities combined is given by the intersection of the shaded areas.
• Find all solutions to Adventure Mike's riddle.
$\le$ or $\ge$ inequalities are represented with solid lines while $<$ or $>$ inequalities are represented with dashed lines.
Dont forget:
□ $\le$ or $<$ $\rightarrow$ the solutions lie below the line
□ $\ge$ or $>$ $\rightarrow$ the solutions lie above the line
For equations written in the slope-intercept form $y=mx+b$, the y-intercept is represented by the term $b$.
Here you can see the correct graph.
$\mathbf{y\le -2x+1}$
□ the orange solid line passes through the y-intercept, $1$
□ the solutions lie below the line
$\mathbf{y<\frac32 x-2}$
□ the green dashed line passes through the y-intercept, $-2$
□ the solutions lie below the line
Keep in mind:
□ draw solid lines for $\le$ or $\ge$ inequalities
□ draw dashed lines for $<$ or $>$ inequalities
Check any point to verify the shading of the solution set:
□ $\le$ or $<$ $\rightarrow$ below the line
□ $\ge$ of $>$ $\rightarrow$ above the line
• Determine where the party will take place.
You can pick some points and plug them into the inequality to check if they belong to the solution set.
The lines formed by $\le$ or $\ge$ inequalities are part of the solution set.
The lines formed by $<$ or $>$ inequalities aren't part of the solution set.
Let's start with the inequality $\mathbf{y\ge 2x-2}$:
1. The y-intercept is $-2$.
2. Move two units up and one unit to the right to plot additional points.
3. Connect the points with a solid line, since they are formed by a greater than or equal to inequality. The points on the line are included in our solution set. This inequality is shown by the
green line.
4. The green shaded area is the solution set for this inequality.
Now we graph $\mathbf{y<-\frac12x+3}$:
1. The y-intercept is $3$.
2. Move one unit down and two units to the right to plot additional points.
3. Connect the points with a dashed line, since they are formed by a less than inequality. This is shown by the orange line.
4. The orange shaded area is the solution set for this inequality.
The set of all solutions satisfying both inequalities is given by the intersection of the shaded areas. Don't forget that points on the green line of $\mathbf{y\ge 2x-2}$ also belong to the
solution set.
• Decide if the location of the treasure lies inside the area where the robbers are searching.
Each point in the coordinate system is given by $P(p_x,p_y)$, where $p_x$ is the x-coordinate, and$p_y$ is the y-coordinate of the point $P$.
Plug each of the given points into all three of the inequalities. The solution must satisfy each of the inequalities.
Pay attention to the $<$ sign in the second inequality.
Are points that are exactly on the line part of the solution set?
Draw the graphs of the inequalities and see if the coordinates are within the intersection area.
You can solve a system of three inequalities by graphing in a similar way used for two inequalities:
1. Determine the y-intercept
2. Use the slope to plot additional points
3. Connect the points with a solid or dashed line depending on the inequality sign
4. Shade the area belonging to the solution set, either below or above the line, depending on the inequality sign
5. The intersection area of all shaded areas is where the hidden treasure can be found
In the graph on the right, you can see the system of the three equations.
We can use this graph for our decision:
□ Crown $(0,3)$ lies on the green and on the red line. But pay attention: the red line is dashed, which means the points on the line aren't included in the solution set. So the robbers can't
find this specific treasure.
□ Coins $(0,2)$ lie inside the solution area. You can see it directly on the graph.
□ Necklace $(2,1.75)$ lies inside the solution area.
□ Antique sword $(-1,1)$ lies inside the solution area.
□ Bottle of love potion $(2,3)$ lies outside the solution area. When we plug in our y-coordinate, $3$, we see that the inequality is not true: $3\not < \frac14\times 2+3=3.5$
• Describe how to determine the solution set for the inequality.
An equation in slope-intercept form is: $y=mx+b$
□ $m$ is the slope
□ $b$ is the y-intercept
For all $\le$ or $\ge$ inequalities, draw a solid line and for all others, draw a dashed line.
Take any point $P(x_p,y_p)$ and check if $x_p$ and $y_p$ are in the solution set of the given inequality. This helps you verify if the solutions lie above or below the line formed by the
Here you can see the graph of $y<\frac32x-2$:
□ the y-intercept is $-2$
□ for the slope $m=\frac32$, move up three units and right two units
□ connect the points $(0,-2)$ and $(2,1)$ with a dashed line
□ all solution sets lie below the dashed line
• Assign the inequality.
The y-intercept is easily identified in the coordinate system. This is the place where the line crosses the y-axis.
You can calculate the slope by moving up or down and right. For example $m=-\frac34$:
□ move three units down (because of the negative sign)
□ and move four units to the right
Keep in mind:
□ $\ge$ or $\le$ $\rightarrow$ solid line
□ $>$ or $<$ $\rightarrow$ dashed line
How do you know if the solution area lies above or below the line formed by the inequality?
□ $\ge$ or $>$ $\rightarrow$ above
□ $\le$ or $<$ $\rightarrow$ below
Let's reconstruct the three inequalities:
Start with the red line and the corresponding shaded area:
1. The y-interecept is $-3$.
2. We move three units up and five units to the right. This gives us the slope $m=\frac35$.
3. The line is solid and the shaded area is above the line.
We can write the inequality:
Now let's have a look at the green line:
1. The y-interecept is $3$.
2. We move up three units and one unit to the right. This gives us the slope $m=3$.
3. The line is dashed and the shaded area is below the line.
We can write the inequality:
Finally, we examine the blue line:
1. The y-interecept is $4$.
2. We move two units down and one unit to the right. This gives us the slope $m=-2$.
3. The line is solid and the shaded area is above the line.
We can write the inequality:
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Zero Coupon Bond Price Calculator Excel (5 Suitable Examples) - ExcelDemy
What Is Zero Coupon Bond?
When a bond does not pay coupon payments or interest and trades but rather pays a bulk amount of money at the time of maturity, it is called a Zero Coupon bond. A Zero Coupon bond is also known as a
“deep discount bond” or “discount bond”. The sum of money paid at maturity is called the face value. Since a Zero Coupon bond provides no coupons or interest and trades, its transaction occurs at a
discount to its face value.
Zero Coupon Bond Price Calculator Excel: 5 Examples
The following table has Bond Terms and Value columns. We will use this table for the zero coupon bond price calculator in Excel.
Example 1 – Applying a Generic Formula to Create a Zero Coupon Bond Price Calculator in Excel
The generic formula for Zero Coupon Price Calculation = (Face Value)/〖(1+r)〗^t
• Use the following formula in cell C8.
Formula Breakdown
• (1+C6) → adds 1 with cell C6.
• (1+8%) → Therefore, this becomes
• (1+C6)^C7 → is (1.08)^10
• (1.08)^10 → As a result, it becomes
• C5/(1+C6)^C7 → divides 20000 by 2.158924997279
• 20000/2.158924997279→ Hence, it becomes
Read More: How to Create Convertible Bond Pricing Model in Excel
Example 2 – Zero Coupon Bond Price Calculator for Compounding Periods
The generic formula including compounding periods per year= (Face Value)/〖(1+r/n)〗^t*n
We can see the Value for Compounding Periods Per Year (n) is 3. We will use the above formula for Zero Coupon Price Calculation.
• Use the following formula in cell C9.
Formula Breakdown
• (C7*C8) → It multiplies cell C7 with cell C8
• (10*3) → Therefore, it becomes
• (C6/C8) → divides cell C6 by cell C8
• (8%/3) → Then, it becomes
• (1+(C6/C8)) → is adding 1 with 0.026666666667
• (1+0.026666666667) → As a result, this becomes
• (1+(C6/C8))^(C7*C8) → is (1.026666666667)^30
• (1.026666666667)^30 → Then, it becomes
• C5/(1+(C6/C8))^(C7*C8) → is dividing C5 by 2.2033739695385.
• 20000/2.2033739695385 → becomes
Read More: How to Make Treasury Bond Calculator in Excel
Example 3 – Using the PV Function to Create a Zero Coupon Bond Price Calculator in Excel
• Use the following formula in cell C8.
Formula Breakdown
• PV(C6,C7,0,C5) → The PV function calculates the present value of a loan or investment based on a constant interest rate.
• C6 is the rate, which is referred to as Yield to Maturity (YTM)
• C7 is the nper, which is the total number of payment periods
• 0 is the pmt, that is the payment made on each period. For zero coupon bond, as there is no periodic payment, pmt is 0
• C5 is the fv, which is the Future Value
• PV(8%,10,0,20000) → Therefore, this becomes
□ Output: -$9263.87, here the negative sign means outgoing cash flow.
Example 4 – Using the PV Function to Make Zero Coupon Bond Price Calculator for Compounding Periods
We can see the Value of Compounding Periods Per Year (n) is 3.
• Use the following formula in cell C9.
Formula Breakdown
• PV(C6/C8,C7*C8,0,C5) → The PV function calculates the present value of a loan or investment based on a constant interest rate.
• C6/C8 is the rate, which is referred to as Yield to Maturity (YTM)
• 8%/3 → Therefore, it becomes
• C7*C8 is the nper, which is the total number of payment periods
• 10*3 → As a result, becomes
• 0 is the pmt, that is the payment made on each period. For zero coupon bond, as there is no periodic payment, pmt is 0
• C5 is the fv, which is the Future Value
• PV(0.026666666667,30,0,20000) → becomes
□ Output: -$9081.26, here the negative sign means outgoing cash flow.
Example 5 – Using the RATE Function to Calculate the Interest Rate for a Zero Coupon Bond
We will use the RATE function to calculate the Yield to Maturity-YTM (r), which is the interest rate (r) for a zero coupon bond.
• Use the following function in cell C8.
Formula Breakdown
• RATE(C7,0,C6,C5) → the RATE function returns the interest rate per period of an annuity.
• C7 is the npr, which is the total number of payment periods
• 0 is the pmt, that is the payment made on each period. For zero coupon bond, as there is no periodic payment, pmt is 0
• C6 is pv, which is the Present Value
• C5 is fv, that is the Future Value
• RATE(10,0,-12000,20000) → Therefore, it becomes
Practice Section
You can download the Excel file to practice the explained methods.
Download the Practice Workbook
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Dyadic Curvelet Transform (DClet) for Image Noise Reduction
Marjan Sedighi Anaraki^*, Fangyan Dong^*,
Hajime Nobuhara^**, and Kaoru Hirota^*
^*Dept. of Computational Intelligence and Systems Science, Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan
^**Dept. of Intelligent Interaction Technologies, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba science city, Ibaraki 305-8573, Japan
January 15, 2007
March 20, 2007
July 20, 2007
image processing, curvelet, noise reduction, wavelet, ridgelet
Dyadic Curvelet transform (DClet) is proposed as a tool for image processing and computer vision. It is an extended curvelet transform that solves the problem of conventional curvelet, of
decomposition into components at different scales. It provides simplicity, dyadic scales, and absence of redundancy for analysis and synthesis objects with discontinuities along curves, i.e.,
edges via directional basis functions. The performance of the proposed method is evaluated by removing Gaussian, Speckles, and Random noises from different noisy standard images. Average 26.71 dB
Peak Signal to Noise Ratio (PSNR) compared to 25.87 dB via the wavelet transform is evidence that the DClet outperforms the wavelet transform for removing noise. The proposed method is robust,
which makes it suitable for biomedical applications. It is a candidate for gray and color image enhancement and applicable for compression or efficient coding in which critical sampling might be
Cite this article as:
M. Anaraki, F. Dong, H. Nobuhara, and K. Hirota, “Dyadic Curvelet Transform (DClet) for Image Noise Reduction,” J. Adv. Comput. Intell. Intell. Inform., Vol.11 No.6, pp. 641-647, 2007.
Data files:
1. [1] M. N. Do and M. Vetterli, “The Contourlet Transform: an Efficient Directional Multiresolution Image Representation,” IEEE Trans. on Image Processing, 14-12, pp. 2091-2106, 2005.
2. [2] D. D. -Y. Po and M. N. Do, “Directional Multiscale Modeling of Images using the Contourlet Transform,” IEEE Trans. on Image Processing, 15-6, pp. 1610-1620, 2006.
3. [3] D. L. Donoho and M. R. Duncan, “Digital Curvelet Transform: Strategy, Implementation and Experiments,” Proc. SPIE, Vol.4056, pp. 12-29, 2000.
4. [4] J. L. Starck, F. Murtagh, E. J. Candes, and D. L. Donoho, “Gray and Color Image Contrast Enhancement by the Curvelet Transform,” IEEE Trans. on Image Processing, 12-6, pp. 706-717, 2003.
5. [5] F. J. Herrmann, “Curvelet Imaging and Processing: an Overview,” CSEG National Convention, 2004.
6. [6] E. J. Candes, “What Is A Curvelet?,” Notice of the AMS, 50-11, pp. 1402-1403.
7. [7] E. J. Candes and D. L. Donoho, “Curvelets Multiresolution Representation, and Scaling Laws,”
8. [8] E. Candes and D. Donoho, “New Tight Frams of Curvelets and Optimal Representations of Objects with Piecewise-C2 Singularities,” Commun. on Pure and Appl. Math., 57, pp. 219-266, 2004.
9. [9] E. J. Candes and D. L. Donoho, “Ridgelets, A Key to Higher Dimensional Intermittency?,” Phil. Trans. Royal Society London, 1999.
10. [10] E. J. Candes and D. L. Donoho, “Curvelet A Surprisingly Effective Nanadaptation for Objects with Edges,” Curve and Surface Filtering, Saint-Malo, 1999.
11. [11] M. N. Do and M. Vetterli, “The Finite Ridgelet Transform for Image Representation,” IEEE Trans. on Image Processing, 12-1, pp. 16-28, 2003.
12. [12] M. N. Do and M. Vetterli, “The Finite Ridgelet Transform for Image Representation,” IEEE Trans. on Image Processing, IP EDICS: 2-WAVP (Wavelets and Multiresolution Processing), 2001.
13. [13] F. Matus and J. Flusser, “The Image Representations via a Finite Radon Transform,” IEEE Trans. on Pattern Analysis and Machine Intelligence, 15-10, pp. 996-1006, 1993.
14. [14] E. D. Bolker, S. Helgason, R. L. Bryant, V. Guillemin, and R. O. Wells Jr. (Eds.), “The Finite Radon Transform,” Internal Geometry, Contemporary Mathematics, 63, pp. 27-50, 1987.
15. [15] G. Beylkin, “Discrete Radon Transform,” IEEE Trans. on Acoustic and Speech Signal Processing, 35, pp. 162-172, 1987.
16. [16] R. Rangarajan, R. Kataramanan, and S. Shah, “Image De-noising using Wavelet,” 2002.
17. [17] E. Bala and A. Ertuzun, “Application of Multiwavelet Techniques to Image Denoising,” IEEE ICIP 2002, III581-584.
18. [18] J. L. Starck, E. J. Candes, and D. L. Donoho, “The Curvelet Transform for Image De-noising,” 2000.
19. [19] J. L. Starck, E. J. Candes, and D. L. Donoho, “The Curvelet Transform for Image Denoising,” IEEE, Transaction on Image Processing, 11-6, pp. 670-684, 2002.
20. [20] E. Candes, L. Demanet, D. Donoho, and L. Ying, “Fast Discrete Curvelet Transforms,” 2005. | {"url":"https://www.fujipress.jp/jaciii/jc/jacii001100060641/","timestamp":"2024-11-11T23:52:33Z","content_type":"text/html","content_length":"47877","record_id":"<urn:uuid:d2f74ed1-2286-453a-9996-ed7790e5cf0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00183.warc.gz"} |
subnet mask question
14 years 11 months ago #32748 by lav_plsb1
Given a subnet mask of 255.255.255.224. which of the following addresses can be assigned to network hosts
the options given are (choose three)
1. 15.234.118.63
2. 92.11.178.93
3. 134.178.18.56
4. 192.168.16.87
5. 201.45.116.159
6. 217.63.12.192
Plz explain how to identify the correct answers.
14 years 11 months ago #32749 by SteveP
I'm not going to just hand out the answer, but I'll give some big clues:
The "block size" for a 255.255.255.224 mask is 32. This means that the network numbers are:
which provide 32 IP addresses per network. The first address is the network address and the last one is the broadcast address, hence there are 30 useable host addresses per network.
The easy way to determine the block size is to subtract the non-255 octet in the subnet mask from 256.
From these network numbers, it's easy enough to look at the IP addresses that you listed and decide which are valid host addresses rather than network or broadcast addresses.
If you're not comfortable using such a short cut, there are several very useful posts on the forum which deal with the basics of converting everything to binary and working it out from first
14 years 11 months ago #32751 by lav_plsb1
Hi steveP,
I got the answer, thanks for your quick reply.
14 years 11 months ago #32781 by talk2sp
Interesting Steve P. Trying to figure out the answers. Why are u stingy with answer (lol) those are CCNA questions i guess so may be the dude is reading up and needs quick help?
SteveP can u elaborate some more? I'd be taking my exams soon and i really need to get sticky and refreshed.
SteveP Wrote:
From these network numbers, it's easy enough to look at the IP addresses that you listed and decide which are valid host addresses rather than network or broadcast addresses.
Did not get that paragraph.
c0de - 3
Take Responsibility! Don't let failures define you
14 years 11 months ago #32783 by Losh
From these network numbers, it's easy enough to look at the IP addresses that you listed and decide which are valid host addresses rather than network or broadcast addresses.
Did not get that paragraph.
I'll answer that specific question.
255.255.255.224 has 27 network bits, remaining with 5 host bits.
This translates to 2^5 = 32 hosts per subnet.
Each subnet has a total of 32 ip addresses but only 30 ip addresses are usable. This is because the 1st address of each subnet is the subnet I.D and the last address of each subnet is the broadcast.
In subnet 1: x.y.z.0
x.y.z.0 is the subnet I.D and x.y.z.33 is the broadcast address
In subnet 2: x.y.z.32
x.y.z.32 is the subnet I.D and x.y.z.63 is the broadcast address
This goes the same for all the other subnets.
~ Networking :- Just when u think its starting to make sense......... ~
CCNA, CCNP, CCNA Security, JNCIA, APDS, CISA
14 years 11 months ago #32784 by talk2sp
trying to patch tru with ur explanation losh but relating it to lav_plsb1's initial question, if i had to choose an answer i will choose 1 and 6 using ur explanation of id and broadcast please
correct me if i am wrong?
thanks man
c0de - 3
Take Responsibility! Don't let failures define you
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11: In how many different orders can you ring five bells?
Submitted by Jeff Ladd on August 19, 2017.
lesson Paragraph:
A plain course of Bob Doubles has forty rows, all different. Are there more rows we could ring? How many?
Asking that question is the same as asking ‘In how many different ways can you arrange five letters?’ Let’s find out.
One letter can be arranged in just one way:
If we add a second letter, we can either put it in front of the first one, or behind it:
We had 1 arrangement for 1 letter, and we can add the second letter in one of two positions.
That gives us 1x2 arrangements, which is, of course, equal to 2 arrangements.
Now, let’s add a third letter.
We have two arrangements of our first two letters. In each of those arrangements, we can add the third letter in one of three positions:
• in front
• in the middle
• at the back
Let’s do each of those with AB:
and now, with BA:
We had 1x2 arrangements of two letters. In each of those arrangements we can add the third letter in one of three positions. That gives us 1x2x3 arrangements, which is, of course the 6 arrangements
of three letters that you can see.
Lets add a fourth letter, but lets save ourselves some time and do some maths, Frankie!
We had 1x2x3 = 6 arrangements of three letters. In each of those six arrangements, we can put the new letter in one of 4 positions (either 1st, 2nd, 3rd or last)
That means we must have 1x2x3x4 = 24 arrangements of 4 letters. You can draw them all out if you feel like it, or you could just write out a plain course of Plain Bob Minimus (4 bells, it’s Plain
Hunt on 4, with 2nds made at the end of each lead; try it!). It has 3 leads of 8 rows, 24 rows in total, and I challenge you either to find a row that isn’t there, or find one that repeats.
You should now be able to see the pattern, and so, I’m going to claim that on five bells, there are 1x2x3x4x5 = 120 possible rows.
So far, we’ve only managed to ring 40 of them. Where do we get the remaining 80 from? Go Grandsire… | {"url":"https://vernetbells.com/en/lessons/11-how-many-different-orders-can-you-ring-five-bells","timestamp":"2024-11-14T00:48:15Z","content_type":"text/html","content_length":"40333","record_id":"<urn:uuid:b1f4c8ff-25dc-4241-9ebf-56871f3723d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00567.warc.gz"} |
Random generation of associative algebras
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions
for finite and profinite associative algebras. Let (Formula presented.) be a finite field. Let (Formula presented.) be a finite-dimensional, associative, unital algebra over (Formula presented.). Let
(Formula presented.) be the probability that two elements of (Formula presented.) chosen (uniformly and independently) at random will generate (Formula presented.) as a unital (Formula presented.)
-algebra. It is known that if (Formula presented.) is simple, then (Formula presented.) as (Formula presented.). We extend this result to a large class of finite associative algebras. For (Formula
presented.) simple, we find the optimal lower bound for (Formula presented.) and we estimate the growth rate of (Formula presented.) in terms of the minimal index (Formula presented.) of any proper
subalgebra of (Formula presented.). We also study the random generation of simple algebras (Formula presented.) by two elements that have a given characteristic polynomial (resp. a given rank). In
addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let (Formula presented.) be a profinite algebra over (Formula presented.). We show that
(Formula presented.) is positively finitely generated if and only if (Formula presented.) has polynomial maximal subalgebra growth. Related quantitative results are also established.
Bibliographical note
Publisher Copyright:
© 2023 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.
Dive into the research topics of 'Random generation of associative algebras'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/random-generation-of-associative-algebras","timestamp":"2024-11-12T19:40:44Z","content_type":"text/html","content_length":"49280","record_id":"<urn:uuid:ffd66c14-67a1-4519-8f42-61731e5fd15c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00359.warc.gz"} |
how to make a weighing balance for school project
All the elements (line, shape, color, etc) in that composition look stable or have a feeling of balance (like one side is not heavier than the other). If you use a nickel, your answer would be 4.8
and you would place the coin on the 4.8 inch mark on the ruler. The coins would weigh less on the moon, but their mass would not change! Craft Instructions For Kids .. It is a very simple to make a
weigh scale. This has so much opportunity for learning number comparisons, counting, and thinking skills. DIY Balance Board Teach the kids that scales come in many different forms with this idea from
Elsie Marley. Achieve a specific weight by using a coin or coins equal to the amount of weight you need. Get the how-to here. In order to make beam, measure the length of the long straw and cut
according to our required length. Balance in art is defined as the equal distribution of visual weight in a composition. Divide this figure by 5. Adventure Homeward. This is an easy idea that took
only a few minutes to make, and helped keep my 2 year, 11 month old daughter Bumble’s mind and body active (and out of mischief) at the end of a crazy day. There are a lot of variations online to
expand on the learning. Multiply the weight of the coin by 6 inches. The quotient will tell you where to place the coin on the ruler. Or you can ask questions like, “how many Legos does this toy car
weigh?”, finding how many Legos it takes to balance a scale with a toy on the other end. If you’re an adult and you’re trying to balance school and work, keep track of your schedule in a planner,
including your work hours, class times, and space each day for studying and homework. By using the resources below, students will learn about this important science tool and practice using it in
meaningful ways. Easy DIY balance scales for toddlers and preschoolers to explore weight and gravity concepts through play. Balance is one of the principles of design. Balance scales are commonly
used to compare the weights of objects or to weigh objects by balancing them with standard weights. Balance Scales. Having a healthy balance is important, so remember to include time for your
friends, family, and hobbies. Mar 30, 2018 - We scoured the web for easy DIY balance scales that are perfect for introducing the kids to the concepts of weight and measurement. Stem Science Preschool
Science Science Fair Science For Kids Stem Projects Science Projects School Projects Science Experience Islam For Kids. Scales are tools used for measuring weight. Weighing Activity for Hands-On Math
We have been experimenting with our balance scale for play and learning so naturally we have been weighing items around the house and comparing different objects. 4) Making a beam: Beam holds the
objects to be weighed in the weighing baskets. A spring is used as a weighing scale. Kids' Crafts. 193. You can compare objects by size, or quantity. Talk about how weight must be evenly distributed
on the board to get the full, cool balance effect and watch your active kid wobble away. Since your balance only compares the mass of objects, not their weight, you would get the same results on the
moon as you do on earth! Article from ... Two types of homemade balance scales from recycling. DIY Projects. Even on the moon, a quarter on one side of your balance would still have the same mass as
a dime and penny on the other side of the balance. 5) Attach the weighing baskets to the beam: Now beam is ready in your hands. It depends on how big we are going to make the balance scale. Equal to
the amount of weight you need Science Preschool Science Science Fair Science Kids. The beam: beam holds the objects to be weighed in the weighing baskets to be weighed in weighing! Time for your
friends, family, and hobbies weights of objects or weigh. A lot of variations online to expand on the moon, but their mass would not!... Different forms with this idea from Elsie Marley or quantity
objects or to weigh objects by how to make a weighing balance for school project or. Weight of the coin by 6 inches Kids stem Projects Science Projects Projects... To weigh objects by balancing them
with standard weights is ready in your hands much for... Will learn about this important Science tool and practice using it in meaningful ways students learn. The Kids that scales come in many
different forms with this idea from Elsie Marley 6 inches are lot. In your hands Science Experience Islam for Kids it depends on how we! The ruler types of homemade balance scales are commonly used
to compare the weights of objects to... Compare objects by balancing them with standard weights ) Attach the weighing baskets is in... With this idea from Elsie Marley weigh objects by balancing them
with how to make a weighing balance for school project! So remember to include time for your friends, family, and thinking skills mass would not change Making beam. Or quantity commonly used to
compare the weights of objects or to weigh by. A coin or coins equal to the amount of weight you need balance is important, so remember to time! Is important, so remember to include time for your
friends, family, and thinking skills scales recycling... Of variations online to expand on the learning length of the coin on the ruler but mass! A specific weight by using a coin or coins equal to
the:! Distribution of how to make a weighing balance for school project weight in a composition Science tool and practice using in. Tell you where to place the coin by 6 inches Science Science Fair
Science for Kids stem Projects Science Islam! Be weighed in the weighing baskets the moon, but their mass would not change scales for toddlers preschoolers! With this idea from Elsie Marley about
this important Science tool and practice using it in meaningful.... Our required length much opportunity for learning number comparisons, counting, and hobbies this important Science tool practice.
Weight and gravity concepts through play Projects School Projects Science Projects School Projects Experience. Homemade balance how to make a weighing balance for school project are commonly used to
compare the weights of objects or weigh! Are commonly used to compare the weights of objects or to weigh objects by balancing them with weights... Lot of variations online to expand on the moon, but
their mass would not!! Them with standard weights expand on the moon, but their mass would not change the of! Using a coin or coins equal to the amount of weight you need of homemade balance scales
are used. Going to make the balance scale, and thinking skills practice using it in meaningful.. In your hands visual weight in a composition the ruler and preschoolers to explore weight gravity...
Balance Board Teach the Kids that scales come in many different forms with this idea from Marley... Using a coin or coins equal to the beam: beam holds the objects to be in. Now beam is ready in your
hands quotient will tell you where to place the coin on the.... Weighing baskets to the beam: beam holds the objects to be weighed in the baskets. Are going to make the balance scale of objects or to
weigh objects by them. Make a weigh scale distribution of visual weight in a composition be weighed in the baskets! Is important, so remember to include time for your friends,,! 6 inches to expand on
the ruler important Science tool and practice using it in ways! For toddlers and preschoolers to explore weight and gravity concepts through play visual! To expand on the moon, but their mass would
not change: beam. Compare objects by size, or quantity weight and gravity concepts through play to place the coin 6... So remember to include time for your friends, family, and thinking skills this
has so much for. Compare the weights of objects or to weigh objects by balancing them standard! Family, and hobbies by balancing them with standard weights for Kids Science Fair for! Distribution of
visual weight in a composition in meaningful ways ) Attach the weighing baskets friends! Types of homemade balance scales from recycling coin or coins equal to the amount of weight you need them... | {"url":"https://tyrepressurechart.com/ltg4i/ee4e38-how-to-make-a-weighing-balance-for-school-project","timestamp":"2024-11-05T22:31:17Z","content_type":"text/html","content_length":"15450","record_id":"<urn:uuid:c90365d0-8368-44cd-9ee7-da6b48c34449>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00733.warc.gz"} |
Counting by 10s (WIZ Math)
1. Read out the numbers given.
2. Notice the pattern in the sequence of numbers.
3. Guess the number that should come next.
What is the missing number:
100, 110, 120,__
Look at the numbers in the above sequence. Here the difference between one number to the next is 10, therefore, it is skip counting by 10. That is, to each number we have to add 10 to get the next
In the above example,
110-100 = 10
120-110 = 10
Answer : 130
Directions: Find the missing number. Also write at least 10 examples of your own. | {"url":"http://kwiznet.com/p/takeQuiz.php?ChapterID=1422&CurriculumID=2&NQ=6&Num=2.4","timestamp":"2024-11-03T18:43:09Z","content_type":"text/html","content_length":"11216","record_id":"<urn:uuid:bc7e2e42-a066-489a-af31-5d30ca853abf>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00077.warc.gz"} |
Capacitated single assignment hub location problem with modular link capacities
Corberán Á., Peiró J., Campos V., Glover F., Martí R. (2014)
Optsicom project, University of Valencia (Spain)
The capacitated single assignment hub location problem with modular link capacities is a variant of the classical hub location problem in which the cost of using edges is not linear but stepwise, and
the hubs are restricted in terms of transit capacity rather than in the incoming traffic. This problem was introduced by Yaman and Carello (Yaman and Carello, 2005) and treated by a branch-and-cut
and a tabu search metaheuristic. We propose a metaheuristic algorithm based on strategic oscillation, a methodology originally introduced in the context of tabu search. Our method incorporates
several designs for constructive and destructive algorithms, together with associated local search procedures, to balance diversification and intensification for an efficient search. Computational
results on a large set of instances show that, in contrast to exact methods that can only solve small instances optimally, our metaheuristic is able to find high-quality solutions on larger instances
in short computing times. In addition, the new method, which joins tabu search strategies with strategic oscillation, outperforms the previous tabu search implementation.
In mathematical terms, given a network G with a set of nodes V and a set of edges E, let t[ij] be the amount of traffic to be transported from node i to node j, where t[ij]=0 for any node i.
Each node i is either a terminal node or a hub node (terminal and hub for short). A terminal can only be assigned to a single hub. A hub is assigned to itself. The hubs and the adges among them
define a complete subgraph. Opening a hub at node i has a fixed installation cost of C[ii]. Each hub i has a capacity Q^h limiting the total amount of traffic transitiing through i.
There are two types of edges between nodes: edges of the first type are used to connect terminals with hubs, and we call them access edges. Let m[i] be the number of access edges needed to route the
incoming and outgoing traffic at node i. The cost of installing m[i] edges between terminal i and hub k is denoted by C[ik]. Edges of the second type are used to transfer traffics between hubs, and
we call them backbone edges. Each backbone edge has a maximum traffic capacity of Q^b (in each direction). If nodes k and l are hubs, the amount of traffic on arc (k,l), denoted as z[kl], is the
traffic that has to be transported from nodes assigned to k to nodes assigned to l. The capacity Q^b of a given edge kl cannot be less than the maximum traffic on its corresponding arcs (k, l) and (l
, k), and the cost of installing te edge is denoted by R[kl].
The following variables (Yaman and Carello, 2005) are defined in order to provide the mathematical programming model shown below: The assignment variable x[ik] is equal to 1 if terminal i is assigned
to hub k, and 0 otherwise. If node i receives a hub, then x[ii] takes value 1. z[kl] is the traffic on an arc (k,l) and w[kl] is the number of copies of the edge kl.
The problem is formulated (Yaman and Carello, 2005) as:
A metaheursitic and a branch-and-cut algorithm were proposed in (Yaman and Carello, 2005). The metaheuristic consists of a tabu search (TS) to solve the hub location subproblem and a local search for
assigning terminals to hubs. The solution provided by the metaheuristic is used as an initial upper bound in the branch-and-cut algorithm and to limit the number of variables considered by the exact
method. In addition to the best solution, the metaheuristic produces also a subset of nodes that represents, in a sense, the best potential locations for the hubs. The hubs selected in the best
solution belong to this subset, as well as the two other hubs which appear most often in the best solutions found by the metaheuristic. This set is called the concentration set. The resulting reduced
problem, where hubs can be chosen only among the nodes of the concentration set, is called the concentrated problem, and is the problem solved using the branch-and-cut method.
A strategic oscillation algorithm that incorporates several designs for constructive and destructive phases, making use of tabu search memory structures together with associated local search
We have tested our algorithms on three sets of instances:
1. The CAB (Civil Aviation Board) data set. It is based on airline passenger flows between some important cities in the United States. It consists of a data file, presented by O’Kelly in 1987, with
the distances and flows of a 25 nodes graph.
2. The AP (Australian Post) data set. It is based on real data from the Australian postal service and was presented by Ernst and Krishnamoorthy in 1996. The size of the original data file is 200
nodes. Smaller instances can be obtained using a code from ORLIB. As with CAB, many authors have generated different instances from the original file.
3. The USA423 data set. Introduced by (Peiró, Corberán, and Martí, 2014) and based on real airline data. It consists of a data file concerning 423 cities in the United States, where real distances
and passenger flows for an accumulated 3 months period are considered.
You can download the instances here.
We performed extensive computational experiments with 150 instances. The best values for the instances can be downloaded here.
• Alumur, S., and Kara, B. Y. Network hub location problems: The state of the art. European Journal of Operational Research 190, 1 (2008), 1–21.
• Beasley, J. E. OR-library: distributing test problems by electronic mail. Journal of the Operational Research Society 41, 11 (1990), 1069–1072.
• Campbell, J. F., Ernst, A. T., and Krishnamoorthy, M.Hub location problems. In Facility location: Applications and theory, Z. Drezner and H. Hammacher, Eds. Springer, 2002, pp. 373–407.
• Campbell, J. F., and O'Kelly, M. E. Twenty-five years of hub location research. Transportation Science 46, 2 (2012), 153–169.
• Ernst, A. T., and Krishnamoorthy, M. Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science 4, 3 (1996), 139–154.
• Fanjul-Peyroa, L., and Ruiz, R. Iterated greedy local search methods for unrelated parallel machine scheduling. European Journal of Operational Research 207, 1 (2010), 55–69.
• Farahani, R. Z., and Hekmatfar, M. Facilities location: Concepts, models, algorithms and case studies. Springer-Verlag, 2009.
• Glover, F. Heuristics for integer programming using surrogate constraints. Decision Sciences 8, 1 (1977), 156–166.
• Glover, F., and Laguna, M. Tabu search. Kluwer, Norwell, MA, 1997.
• Ilić, A., D. Urosević, J. Brimberg, and N. Mladenović. 2010. A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem. European Journal
of Operational Research 206, no. 2: 289-300.
• Jacobs, L., and Brusco, M. A local-search heuristic for large set-covering problems. Naval Research Logistics 42, 7 (1995), 1129–1140.
• Lozano, M., Molina, D., and Garcia-Martinez, C. Iterated greedy for the maximum diversity problem. European Journal of Operational Research 214, 1 (2011), 31–38.
• Melo, M. T., Nickel, S., and Saldanha da Gama, F. Facility location and supply chain management - a review. European Journal of Operational Research 196, 2 (2009), 401–412.
• Nickel, S., and Puerto, J. Location Theory: A unified approach. Springer, 2005.
• O'Kelly, M. E. 1987. A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research 32, no. 3: 393-404.
• Peiro, J., Corberan, A., and Marti, R. GRASP for the uncapacitated r-allocation p-hub median problem. Computers & Operations Research 43, 1 (2014), 50–60.
• ReVelle, C., and Eiselt, H. Location analysis: A synthesis and survey. European Journal of Operational Research 165, 1 (2005), 1–19.
• ReVelle, C., Eiselt, H., and Daskin, M. A bibliography for some fundamental problem categories in discrete location science. European Journal of Operational Research 184, 3 (2008), 817–848.
• Ruiz, R., and Stutzle, T. An iterated greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives. European Journal of Operational
Research 187, 3 (2008), 1143–1159.
• Yaman, H. Concentrator Location in Telecommunication Networks. PhD thesis, Universit ́e Libre de Bruxelles, Brussels, Belgium, Dec 2002.
• Yaman, H., and Carello, G. Solving the hub location problem with modular link capacities. Computers & Operations Research 32, 12 (2005), 3227–3245.
• Ying, K., and Cheng, H. Dynamic parallel machine scheduling with sequence-dependent setup times using an iterated greedy heuristic. Expert Systems with Applications 37, 4 (2010), 2848–2852. | {"url":"https://grafo.etsii.urjc.es/optsicom/chlpml.html","timestamp":"2024-11-02T19:04:13Z","content_type":"application/xhtml+xml","content_length":"19277","record_id":"<urn:uuid:6e0b7bd1-5812-496c-8fbc-cfb992bd966c>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00716.warc.gz"} |
Algebra 2 Worksheets Basics For Algebra 2 Worksheets | Order of Operation Worksheets
Algebra 2 Worksheets Basics For Algebra 2 Worksheets
Algebra 2 Worksheets Basics For Algebra 2 Worksheets
Algebra 2 Worksheets Basics For Algebra 2 Worksheets – You may have become aware of an Order Of Operations Worksheet, but just what is it? In this post, we’ll talk about what it is, why it’s vital,
and also just how to get a Algebra 1 Order Of Operations Worksheet Hopefully, this details will certainly be practical for you. After all, your pupils should have an enjoyable, reliable method to
assess one of the most essential ideas in maths. On top of that, worksheets are a wonderful way for pupils to exercise brand-new skills as well as testimonial old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a kind of math worksheet that needs pupils to perform math operations. Students who are still finding out exactly how to do these tasks will certainly discover
this kind of worksheet beneficial.
The primary objective of an order of operations worksheet is to assist pupils learn the appropriate means to solve math equations. If a trainee doesn’t yet comprehend the idea of order of operations,
they can review it by referring to a description web page. Additionally, an order of operations worksheet can be divided right into several groups, based upon its problem.
An additional crucial purpose of an order of operations worksheet is to educate trainees just how to perform PEMDAS operations. These worksheets start with straightforward problems connected to the
fundamental policies and accumulate to much more intricate problems entailing all of the rules. These worksheets are a fantastic means to introduce young students to the exhilaration of addressing
algebraic equations.
Why is Order of Operations Important?
Among one of the most crucial things you can learn in mathematics is the order of operations. The order of operations guarantees that the mathematics problems you address are consistent. This is
important for tests as well as real-life computations. When addressing a math trouble, the order needs to start with backers or parentheses, followed by multiplication, subtraction, as well as
An order of operations worksheet is a fantastic means to educate trainees the correct way to resolve math equations. Before trainees start utilizing this worksheet, they may need to review ideas
related to the order of operations. To do this, they should evaluate the concept web page for order of operations. This concept web page will certainly give trainees an overview of the basic idea.
An order of operations worksheet can help trainees create their skills on top of that and also subtraction. Teachers can make use of Prodigy as an easy way to separate method and also deliver
interesting content. Natural born player’s worksheets are an excellent way to assist pupils discover the order of operations. Educators can start with the fundamental principles of multiplication,
addition, as well as division to aid trainees build their understanding of parentheses.
Algebra 1 Order Of Operations Worksheet
Practice The Order Of Operations With These Free Math Worksheets
Algebra 1 Order Of Operations Worksheet
Algebra 1 Order Of Operations Worksheet provide a terrific resource for young students. These worksheets can be easily personalized for specific needs.
The Algebra 1 Order Of Operations Worksheet can be downloaded for free and also can be published out. They can after that be reviewed utilizing addition, division, multiplication, and subtraction.
Trainees can likewise make use of these worksheets to examine order of operations and also using backers.
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Probability, in general terms, means the possibility of an event occurring. Probability is a branch of mathematics that tells us the chances that a given event will occur. The probability for an
impossible event is 0 and for a sure event is 1.
Now, we will see the Probability Previous Year's Questions asked in the GATE exam.
1. A 2-digit number is selected randomly out of all 2-digit integers between 1 and 100. What will be the probability that the number selected is not divisible by 7?
A) 13/90
B) 77/90
C) 56/90
D) 79/90
ANSWER: B
Explanation: The total number of two-digit numbers between 1 and 100 is 90. Out of the 13 are found to be divisible by 7, are 14,21,28,35,42,49,56,63,70,77,84,91 and 98. .So, the probability that
the selected number is divisible by 7 is 13/90. This means the probability that the selected number is not divisible by 7 is 1-13/90, i.e., 77/90.Option(B) is correct
2. A fair six-sided die is rolled once. If the value on the die is found to be 1,2,3, we will roll the die for a second time. What would be the probability that the sum of values turns up at least
A) 5/12
ANSWER: A
Explanation: For the following, even sum would be greater than or equal to 6.6 appeared on the first throw (â…™)+1 appeared on the first throw, 5 appeared on the second throw (â…™)*(â…™) + 1
appeared on the first throw, and 6 on the second (â…™)*(â…™). Similarly, if 2 appeared on the first throw, and four appeared on the second throw(â…™)*(â…™), + 2 appeared on the first throw, and 5
appeared on the second throw(â…™)*(â…™)+ 2 appeared on the first throw, and 6 appeared on the second throw(â…™)*(â…™) + and similarly, if 3 appeared on the first throw, and three appeared on the
second throw(â…™)*(â…™), + 3 appeared on the first throw, and 4 appeared on the second throw(â…™)*(â…™)+ 3 appeared on the first throw, and 5 appeared on the second throw(â…™)*(â…™) + 3 appeared
on the first throw, and 6 appeared on the second throw(â…™)*(â…™) So P(x) = â…™+ (â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)+(â…™)(â…™)= 1/6+9/36= 5/12
3. Consider a random variable X that takes value +1 and -1 with fair probability of 0.5 each.What would be the cumulative distribution function F(x) at x = -1 and +1 are
A) 0 and 0.5
B) 0 and 1
C) 0.5 and 1
D)0.5 and 0.75
ANSWER:- C
Explanation :The Cumulative distribution function F(x) = P(X≤x) F(-1) = P(X≤-1) = P(X=-1) = 0.5 F(+1) = P(X≤+1) = P(X=-1) + (P=+1) = 0.5+0.5 = 1
4. Two fair coins are flipped, and it's known that at least one of the outcomes is a head. What would be the probability that both outcomes are head?
ANSWER:- C
Explanation: Since it's known that at least one of the outcomes is a Head, there remain only three possibilities that are (H, H), (T, H), (H, T). So The Probability of both heads = 1/3.
5. What is the difference between the expectation of the square or a random variable E[X²] and the square of the expectation of a random variable (E[X])².
A)The difference is equal to 0
B)The difference is greater than 0
C)The difference is lesser than 0
D)The difference is greater than equal to 0
ANSWER:- D
Explanation: The difference between the (E[X²]) and (E[X])² is called the variance of a random variable. (If the variance is zero, all the values are identical.) A non-zero variance is always
6. In a deck of 5 cards, each card carrying a specific number from 1 to 5 is shuffled thoroughly. Two cards are removed from the deck one at a time. What would be the probability that two cards are
selected such that the number on the first card is found to be one higher than the number on the second card?
A) 4/25
B) 1/5
C) 1/4
D) 2/5
ANSWER:- B
Explanation: We have to select two cards from the deck of 5 Cards. Since the order in which cards are drawn matters here, there are 5P2 = 5!/3! i.e. 20 elementary events out of which only 4 are
favourable cases: 5 comes before 4, 4 comes before 3, 3 comes before 2 and 2 comes before 1. Hence, probability = 4/20 = 1/5
7. What will be the probability that a divisor of 1099 is found to be a multiple of 1096?
ANSWER:- A
Explanation :Multiples of 1096 that are also the divisor of 1099 are 1096, 2x1096, 4x1096, 5x1096, 8x1096, 10x1096, 20x1096, 25x1096, 40x1096, 50x1096, 100x1096, 125x1096, 200x1096, 250x1096,
500x1096, 1000x1096 The total number of divisors of 1099 = 10000 So the probability = 16/10000 = 1/625
8. Mathur studies either physics or chemistry every day. If she studies physics on one day, then the probability that she studies chemistry the next day is 0.6. If she studies chemistry on one day,
then the probability that she studies physics the next day is 0.4. Given that Mathur studies physics on Monday, what is the probability that he studies physics on Wednesday?
ANSWER:- D
Explanation: Mathur studies physics on Monday. Then the probability that he studies chemistry on Tuesday is 0.6, and the probability that he studies physics on Tuesday is 0.4. He studies
chemistry on Tuesday and physics on Wednesday = 0.6 x 0.4 = 0.24 or He studies physics on Tuesday and physics on Wednesday = 0.4x0.4 = 0.16 Adding 1 and 2, the required probability that he
studies physics on Wednesday is 0.24 + 0.16 = 0.40
9. If you broke a stick of unit length at any point chosen uniformly at random. What would b the expected length of the shorter part of the stick
A)0.24 to 0.27
B)0.15 to 0.30
C)0.20 to 0.30
D)0.10 to 0.15
ANSWER:- A
Explanation: If we broke a stick of unit length at any point chosen uniformly at random. The smaller part of the stick would range in length from almost 0 units up to a maximum of 0.5 units,
where each length is equally possible. Hence, the average length will be about (0 + 0.5)/2 = 0.25 unit.
10. Suppose four fair six-sided dice are rolled. What would be the probability that the sum of the number in dice being 22
A) 7/1296
B) 8/1296
C) 9/1296
D) 10/1296
ANSWER:- D
Explanation: Probability (of an event ) is defined as No of favourable outcomes to the event / Total Number of possible outcomes in the random experiment. Here, four six-faced dice are rolled.
There could be six equally likely and mutually exclusive outcomes for one dice. Taking four together, there can be a total number of 6*6*6*6 = 1296 possible outcomes. There are no favourable
cases to the event: here, the event is getting a sum of 22. So, there can be only 2 possible cases—case 1: Three 6's and one 4, for example, 6,6,6,4 ( sum is 22). Hence, the number of ways we
could obtain this is 4!/3! i.e., four ways ( 3! is for removing those cases where all three six are swapping among themselves) Second Case has two 6's and two 5's, for example, 6,6,5,5 that also
sum to 22. Therefore the number of ways we can obtain this = 4! /( 2! * 2!) = six ways ( 2! is for removing those cases where both six are swapping between themselves, similarly for both five
also). Hence total no of favourable cases = 4 + 6 = 10. Hence probability = 10/1296. Therefore option D.
11. What will be the probability of having two heads and two tails if a fair coin is tossed four times
A) 3/8
B) 5/8
C) 1/2
D) 3/4
ANSWER:- A
Explanation: If a coin is tossed four times, there are a total of 16 possibilities, out of which only the following mentioned have occurrence two heads and two tails, i.e., HHTT, HTHT, HTTH,
TTHH, THTH, THHT So the Probability of having two heads is 6/16 which is ⅜
12. Suppose four fair coins are tossed simultaneously. What will be the probability that at least one head and one tail will turn up is:
ANSWER:- C
Explanation: The probability that at least one head or at least one tail would appear will be 1- the probability that no head or no tail appears. Since there are only two cases for which only
head r only tail appears are TTTT, HHHH so the required probability is 1-1/8 = â…ž
13. If seven distinct car accidents occur in a week. What would be the probability that all distinct accident happens on the same day of the week
A) 1/(7^7)
B) 1/(7^6)
C) 1/(2^7)
D) 7/(2^7)
ANSWER:- B
Explanation: The probability of all accidents happening on Monday is 1/(7^7). Similarly, for the rest of the six days. So the total probability that all seven accident happens on the same day is
7*(1/7^7) = 1/(7^6)
14. Given two Set P = {2, 3, 4, 5} and Set Q = {11, 12, 13, 14, 15}, two numbers are selected at random, one from each set. What will be the probability that the sum of the two numbers equals 16?
ANSWER:- A
Explanation :There are 20 possible pairs we could draw from Set P {2, 3, 4, 5} and Set Q {11, 12, 13, 14, 15} i.e 5*4=20 Out of which following pairs have sum equals to 16 (2, 14),(3, 13),(4,
12),(5, 11) Probability is equal to No. Favourable Outcomes/Total No. Outcomes Probability = 4/20 = 0.20
15. A bag contains ten blue balls, 20 green balls and 30 red balls. A ball is drawn from the bag, and its colour is recorded and put back in the bag again. This process is repeated three times. What
will be the probability that no two of the balls drawn have the same colour is
ANSWER:- A
Explanation: As the number of the ball's colour is 3, Possible combinations -would be 3! = 6. Probability of Blue ball: 10/60 Probability of Green ball: 20/60 Probability of Red ball: 30/60.The
probability that no two of the balls drawn have the same colour is = 6 * (10/60 * 20/60 * 30/60) = 1/6 | {"url":"https://www.naukri.com/code360/library/probability-4734","timestamp":"2024-11-07T23:09:08Z","content_type":"text/html","content_length":"393345","record_id":"<urn:uuid:1aa6b7db-f32b-491c-8e3b-e4e8cf99f413>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00192.warc.gz"} |
Problem A
by chris morgan cc by
According to Wikipedia, FizzBuzz is a group word game for children to teach them about division. This may or may not be true, but this question is generally used to
screen young computer science graduates during programming interviews.
Basically, this is how it works: you print the integers from $1$ to $N$, replacing any of them divisible by $X$ with Fizz or, if they are divisible by $Y$, with Buzz. If the number is divisible by
both $X$ and $Y$, you print FizzBuzz instead.
Check the samples for further clarification.
Input contains a single test case. Each test case contains three integers on a single line, $X$, $Y$ and $N$ ($1 \leq X < Y \leq N \leq 100$).
Print integers from $1$ to $N$ in order, each on its own line, replacing the ones divisible by $X$ with Fizz, the ones divisible by $Y$ with Buzz and ones divisible by both $X$ and $Y$ with FizzBuzz.
Sample Input 1 Sample Output 1
2 3 7 Fizz
Sample Input 2 Sample Output 2
2 4 7 FizzBuzz
Sample Input 3 Sample Output 3 | {"url":"https://open.kattis.com/contests/xo3pb3/problems/fizzbuzz","timestamp":"2024-11-11T03:40:59Z","content_type":"text/html","content_length":"31244","record_id":"<urn:uuid:ad41fbcc-555b-4fe6-9fd8-1b718632d23e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00283.warc.gz"} |
Solving Linear Equations
Solving linear equations means to find the solution of a linear equation. Here, the methods of solving linear equations are explained for its three main types which include linear equations in one
variable, linear equations in two variables and linear equations in three variables.
What Does Solving Linear Equations Mean?
Solving a linear equation refers to finding the solution of linear equations in one, two, three or variables. In simple words, a solution of a linear equation means the value or values of the
variables involved in the equation.
How to Solve Linear Equations?
There are six main methods to solve linear equations. These methods for finding the solution of linear equations are:
Graphical Method of Solving Linear Equations
To solve linear equations graphically, first graph both equations in the same coordinate system and check for the intersection point in the graph. For example, take two equations as 2x + 3y = 9 and x
– y = 3.
Now, to plot the graph, consider x = {0. 1, 2, 3, 4} and solve for y. Once (x, y) is obtained, plot the points on the graph. It should be noted that by having more values of x and y will make the
graph more accurate.
Check: Graphical Method of Solving Linear Programming
The graph of 2x + 3y = 9 and x – y = 3 will be as follows:
In the graph, check for the intersection point of both the lines. Here, it is mentioned as (x, y). Check the value of that point and that will be the solution of both the given equations. Here, the
value of (x, y) = (3.6, 0.6).
Elimination Method of Solving Linear Equations
In the elimination method, any of the coefficients is first equated and eliminated. After elimination, the equations are solved to obtain the other equation. Below is an example of solving linear
equations using the elimination method for better understanding.
Consider the same equations as
2x + 3y = 9 ———–(i)
x – y = 3 ———–(ii)
Here, if equation (ii) is multiplied by 2, the coefficient of “x” will become the same and can be subtracted.
So, multiply equation (ii) × 2 and then subtract equation (i)
2x + 3y = 9
2x – 2y = 6
-5y = -3
Or, y = ⅗ = 0.6
Now, put the value of y = 0.6 in equation (ii).
So, x – 0.6 = 3
Thus, x = 3.6
In this way, the value of x, y is found to be 3.6 and 0.6.
Substitution Method of Solving Linear Equations
To solve a linear equation using the substitution method, first, isolate the value of one variable from any of the equations. Then, substitute the value of the isolated variable in the second
equation and solve it. Take the same equations again for example.
2x + 3y = 9 ———–(i)
x – y = 3 ———–(ii)
Now, consider equation (ii) and isolate the variable “x”.
So, equation (ii) becomes,
x = 3 + y.
Now, substitute the value of x in equation (i). So, equation (i) will be-
2x + 3y = 9
⇒ 2(3 + y) + 3y = 9
⇒ 6 + 2y + 3y = 9
Or, y = ⅗ = 0.6
Now, substitute “y” value in equation (ii).
x – y =3
⇒ x = 3 + 0.6
Or, x = 3.6
Thus, (x, y) = (3.6, 0.6).
Cross Multiplication Method of Solving Linear Equations
Linear equations can be easily solved using the cross multiplication method. In this method, the cross-multiplication technique is used to simplify the solution. For the cross-multiplication method
for solving 2 variable equation, the formula used is:
x /(b[1] c[2] − b[2] c[1]) = y / (c[1] a[2] − c[2] a[1]) = 1 /(b[2] a[1] − b[1] a[2])
For example, consider the equations
2x + 3y = 9 ———–(i)
x – y = 3 ———–(ii)
a[1 ]= 2, b[1 ]= 3, c[1 ]= -9
a[2 ]= 1, b[2 ]= -1, c[2 ]= -3
Now, solve using the aforementioned formula.
x = (b[1] c[2] − b[2] c[1]) / (b[2] a[1] − b[1] a[2])
Putting the respective value we get,
x = 18/5 = 3.6
Similarly, solve for y.
y = (c[1] a[2] − c[2] a[1]) / (b[2] a[1] − b[1] a[2])
So, y = ⅗ = 0.6
Matrix Method of Solving Linear Equations
Linear equations can also be solved using matrix method. This method is extremely helpful for solving linear equations in two or three variables. Consider three equations as:
a[1]x + a[2]y + a[3]z = d[1]
b[1]x + b[2]y + b[3]z = d[2]
c[1]x + c[2]y + c[3]z = d[3]
These equations can be written as:
⇒ AX = B ————-(i)
Here, the A matrix, B matrix and X matrix are:
Now, multiply (i) by A^-1 to get:
A^−1AX = A^−1B ⇒ I.X = A^−1B
⇒ X = A^−1B
Determinant Method of Solving Linear Equations (Cramer’s Rule)
Determinants method can be used to solve linear equations in two or three variables easily. For two variables and three variables of linear equations, the procedure is as follows.
For Linear Equations in Two Variables:
x = Δ[1]/Δ,
y = Δ[2]/Δ
Or, x = (b[1] c[2] − b[2] c[1]) / (b[2] a[1] − b[1] a[2]) and y = (c[1] a[2] − c[2] a[1]) / (b[2] a[1] − b[1] a[2])
For Linear Equations in Three Variables:
Related Video: Solving an Equation
Methods of Solving Linear Equations in One Variable
Solving a linear equation with one variable is extremely easy and quick. To solve any two equations having only 1 variable, bring all the variable terms on one side and the constants on the other.
The graphical method can also be used in which the point of intersection of the line with the x-axis or y-axis will give the solution of the equation.
For example, consider the equation 2x + 4 + 7 = 4x – 3 + x
Here, combine the “x” terms and bring them on one side.
5x – 2x = 14
Or, x = 14/3
Methods of Solving Linear Equations in Two Variables
To solve a linear equation in two variables, any of the above-mentioned methods can be used i.e. graphical method, elimination method, substitution method, cross multiplication method, matrix method,
determinants method.
Methods of Solving Linear Equations in Three or More Variables
For solving any equation having three or more variables, the graphical, elimination and the substitution method is not feasible. For solving a three-variable equation, the cross-multiplication method
is the most preferred method. Even matrix Cramer’s rule is extremely useful for solving equations having 3 or more variables.
Check: Solve Linear Equation in Two Or Three Variables
Topics Related to Solution of Linear Equations:
Frequently Asked Questions
What is a Linear Equation?
A linear equation is an equation where each variable has a degree of one. An example of a linear equation is 4x + 3y = 10.
What are the Methods of Solving Linear Equations?
The 6 most common methods of solving a linear equation are:
• Graphical Method
• Elimination Method
• Substitution Method
• Cross Multiplication Method
• Matrix Method
• Determinants Method
How to Solve Linear Equations with Fractions?
To solve a linear equation with fraction, follow these steps:
• Step 1: Make any complex fraction into a simple fraction
• Step 2: Find the LCM of all denominators
• Step 3: Multiply the equation with the LCM of the denominator
• Step 4: Cancel out the fractions as all the denominators can be divided by the LCM value
• Step 5: Solve the final linear equation using any of the methods explained here | {"url":"https://mathlake.com/Solving-Linear-Equations","timestamp":"2024-11-01T23:03:46Z","content_type":"text/html","content_length":"19288","record_id":"<urn:uuid:d4fd28f9-ea11-4e24-9d42-a0d816c4eff7>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00349.warc.gz"} |
Jesse Lingard | Seoul singer
Aug 22, 2013
Probably means he doesn’t have to move house either. Going to WHU would mean buying an expensive property in London
Dec 14, 2005
Probably guaranteed a starting role in his favourite position.. 1 year contract. He probably thinks he has a shot at making the WC squad ..
Probably guaranteed a starting role in his favourite position.. 1 year contract. He probably thinks he has a shot at making the WC squad ..
If he makes the WC squad I’d say you and I also have a shot.
Yeah I praise Ronaldo. Whats it got to do with this?
No, its a good marker nowadays to figure out whom to be taken seriously.
It could be if they’re paying him a sign on fee, we haven’t heard about that yet have we?
May 26, 2008
Eh, seems to be an unpopular opinion around here but I hope he does well for himself. I didn't and still don't dislike the guy nearly as much as some people around here; everything I've seen of
him suggests he's a pretty decent human being and honestly, I think United and Ole were probably at least as much to blame for Lingard's situation towards the end of his contract as he was.
I have no problem with Jesse at all. Wish him all the best. Happy he found a club.
It could be if they’re paying him a sign on fee, we haven’t heard about that yet have we?
But he is worth absolutely shit all
Probably means he doesn’t have to move house either. Going to WHU would mean buying an expensive property in London
Eh? Nottingham is a good 70-80 miles from Manchester. Hardly comfortable commuting distance.
But he is worth absolutely shit all
Done great at WHU, so might do well for Forest. He's a proven PL player, playmaker and goal scorer and Forest doesn't have to worry about adjusting to PL. His mentality is a big question mark. Hope
it works out though.
Eh? Nottingham is a good 70-80 miles from Manchester. Hardly comfortable commuting distance.
Jesse works from home so it’s alright.
Correctly predicted Portugal to win Euro 2016
Oct 31, 2012
Him getting a 1 year contract at 29 tells you everything you need to know, really. It was America, or a newly promoted side breaking their wage structure to get him in for a season.
Aug 6, 2018
Surely West Ham would have taken him at £80,000 wage? I bet he's waiting a year for another move, the one year contract is unusual.
Yeah, that seems weird. I was expecting him to go to a bigger club to be fair.
I always remember a Jose quote when he was manager of United and someone asked him about the players that had left the club and he said 'Yeah, and where are they now?" and he meant (aside from being
a typically bitchy/cu*tish Jose thing) to say that it showed the quality of the players that had been making a living here. This is yet another example of that in some ways. We ain't losing many of
these players to Bayern Munich or Real Madrid that's for sure
Dec 18, 2018
That video.. Jesus christ.
Sep 1, 2014
Such a cynical money move. West Ham would have obviously been the footballing move given they are managing to keep all their best players and qualifying for Europe repeatedly.
I am fairly sure he wants to go to MLS. We have seen a few younger players heading there lately, especially from Italy, deciding the lifestyle is worth more than the silverware. I think the reason he
is at Forest is he wanted a 1 year deal to give himself a shot of making the WC squad even if it is a long shot and they are planning year by year at this point also so it suits them. I imagine the
more established top flight clubs did not want to be used that way and wanted a longer commitment but JLingz has stars and stripes in his eyes and wants to head over there next summer.
May 2, 2018
Knew that Times article by the WHU supporting journalist citing 200k a week was a load of guff.
Meaning WHU were offering even less than 80k.
Jan 3, 2006
Done great at WHU, so might do well for Forest. He's a proven PL player, playmaker and goal scorer and Forest doesn't have to worry about adjusting to PL. His mentality is a big question mark.
Hope it works out though.
Jul 14, 2009
Shows the mentality of the brain dead fool.
How he lasted at our club so long is a mystery, he must have photos of SAF, Ole, Gill and the Glazers at an orgy with various animals.
Done great at WHU, so might do well for Forest. He's a proven PL player, playmaker and goal scorer and Forest doesn't have to worry about adjusting to PL. His mentality is a big question mark.
Hope it works out though.
But 200k not a chance, one year contract says it all. It's a risk and he will need to show what he can do it USA awaits
Apr 13, 2004
He’s no longer an England international, I bet he won’t get another cap by the time he retires. Also, stupid from his side to sign a 1 year deal, he will be poor in a struggling side, and then he’ll
only ever get another championship contract. Good riddance.
May 6, 2011
Jesse, mate....that ridiculous hand gesture you insist on doing at every given opportunity, is never ever going to catch on. Manchild.
Oct 21, 2007
One year contract seems strange for a 29 years old. This could be his last big contract. He couldn't find any club willing to offer a 4 years contract while he's on a free?
Shameless Tagline Thirst.
Oct 21, 2020
Even in a 5 second video he sounds and acts like a wanker. Remarkable.
Jul 7, 2016
Aug 24, 2016
He’ll be a very solid fantasy player.
I guess the £200k a week makes some sense given the 1 year contract
Aug 5, 2011
Karlsruher SC
I guess the £200k a week makes some sense given the 1 year contract
It's £80k
Oct 25, 2013
Jun 12, 2014
He’ll be a very solid fantasy player.
Yeah he does appear to be living in a dream world
Jul 6, 2016
He even leaked the news before the official announcement
Reported as 80k base with the potential to rise to 121k but I am guessing that includes some fairly hard to hit bonuses like European qualification, 15+ goals etc. all of which would make it worth
paying the extra but frankly are unlikely to happen. I am sure he has a specific plan for next summer and the 1 year was from his side with Forest ok with it as they won't want to be stuck paying PL
wages next year if they do go back down. 1 year rental on a proven PL player is a great deal for a newly promoted club.
May 24, 2010
Such a cynical money move. West Ham would have obviously been the footballing move given they are managing to keep all their best players and qualifying for Europe repeatedly.
It's an obvious money move, but also i don't think he's nailed on to start at WHU anymore, where as Nottingham he is. If he can drag them along, he's got a shot at the world cup squad and a bigger
team coming in for him next season. If it doesn't go well for him and he's not in the world cup team, he can leave quite easily and be in America next season.
Good move for Forest as well, he'll help them stay up and if they go down, they don't have an 80k player in the Championship or they can cut his losses if he flops or is to expensive.
Jan 22, 2009
I think it’s a good move for him personally. He’ll feel more appreciated in a small club.
of course its a money move, every transfer he would’ve done would be a money move. He did however choose forest over some sheik-club
Oct 14, 2014
Meaning WHU were offering even less than 80k.
Not sure, West Ham might have walked away to be fair. Their fans are butthurt though and considering this is lingard we are talking about it's quite amusing.
Aug 15, 2006
He's cringey as feck and badly advised but he had his moments.
Classic newly promoted gamble from Forest. Fun to watch.
But 200k not a chance, one year contract says it all. It's a risk and he will need to show what he can do it USA awaits
Absolutely it's a risk for both clubs, I agree. Papers reporting 80k per week etc, don't know which one to believe.
Jul 12, 2013
Such a cynical money move. West Ham would have obviously been the footballing move given they are managing to keep all their best players and qualifying for Europe repeatedly.
Did think the 200k was bonkers, especially for a new promoted side. Up to 120k still a decent deal but guess he would have thought he could have got something similar elsewhere on a longer contract. | {"url":"https://www.redcafe.net/threads/jesse-lingard-seoul-singer.471766/page-3","timestamp":"2024-11-01T23:01:36Z","content_type":"text/html","content_length":"292582","record_id":"<urn:uuid:1fb82956-dc21-4f37-9f27-18cb7bba68a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00608.warc.gz"} |
Every year during the summer and Diwali holidays we conduct workshops for middle and secondary grade students. These workshops cover milestone topics from the school curriculum, such as fractions,
ratios, proportional thinking, multiplicative reasoning, algebraic thinking, etc. The workshops, exercise an interactive mode where focus is given more on eliciting and building on students’ ideas.
There are no costs or eligibility criteria in terms of marks or grades to participate in these workshops.
In the past, we have offered workshops on the following topics:
• Learning Fractions, Ratios and Proportions
• Developing Algebraic Thinking
• Developing Multiplicative Thinking in Learning Area and Measurement
• Learning to Communicate in Mathematics
• Learning Problem solving in Mathematics
Often we combine these student workshops with teacher workshops where teachers use these instruction sessions to ascertain more about math teaching.
To find out more about upcoming student workshops, please visit our event page. If you are interested in organizing such workshop for your students, please write to use at: mathedu@hbcse.tifr.res.in
or mathedu.res@gmail.com or drop a message here | {"url":"https://mathedu.hbcse.tifr.res.in/workshops-for-students/","timestamp":"2024-11-04T07:26:44Z","content_type":"text/html","content_length":"63473","record_id":"<urn:uuid:4ac4ffbd-cb0e-4483-9dc0-2f2523fc3744>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00116.warc.gz"} |
Sensible heat ratio
Research And Insight
Sensible heat ratio
Sensible heat ratio (SHR) is the term used to describe the ratio of sensible heat load to total heat load.
This can be formulated as:
SHR = q[s] / q[t]
q[s] = sensible heat (kW)
q[t] = total heat (kW)
For example, an SHR value of 100 % would mean that an evaporator would only cool the air, i.e., a purely sensible load. On the other hand, an SHR value of 80 % would mean that 80 % of an evaporator
load is used for cooling air (sensible load), while the remaining 20 % would provide dehumidification (latent load). | {"url":"https://www.grundfos.com/gh/learn/research-and-insights/sensible-heat-ratio","timestamp":"2024-11-10T22:27:18Z","content_type":"text/html","content_length":"610121","record_id":"<urn:uuid:4c2138af-3683-4057-8ac3-e49b17c7b356>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00756.warc.gz"} |
Eratosthenes Prime Sieve: a Java Implementation
One of the classical problems in the
Sphere Online Judge
consists in generating prime numbers. Such a problem can be solved using a prime sieve, a technique that allows finding new prime numbers by discarding all the multiples of known prime numbers. For
example, we know that 2 is a prime, so we would automatically discard all even numbers, then we would go ahead and discard the multiples of 3, 5, 7 and so on. Eratosthenes came up with this algorithm
roughly 2250 years ago (
Wikipedia provides a good description of the sieve
I wrote down a very straightforward implementation of Eratosthenes sieve in java. Since the algorithm can be used to find a considerable amount of prime numbers, usually it's implemented so that the
amount of memory used is as small as possible. A single bit of information is used to store whether or not a number is prime. Thus each byte contains information about 8 numbers. Discarding all even
numbers beforehand cuts in half the number of bits needed. Other techniques can be used to reduce the amount of memory used, but I decided to stop here for simplicity. Here's my source code:
public class PrimeSieve {
private final byte sieve[];
* Creates a sieve of integers up to n.
* @param n
public PrimeSieve(int n) {
// using one bit per number, skipping even numbers
int sieveSize = n / 16 + 1;
// round up to the next multiple of 16
n = sieveSize * 16;
System.out.println("Sieving numbers up to " + n);
// initialize the array of bytes. Each bit corresponds to an odd integer
// between 1 and n, starting with the rightmost bit of the first byte.
// If the bit is 0, the number is prime. Initially, all numbers are
// assumed to be prime and some will be sifted out.
sieve = new byte[sieveSize];
// 1 is composite
sieve[0] = 0x01;
// this is the maximum starting number to search for primes in the form
// 2*k+1
int maxK = (int) Math.floor(Math.sqrt(n / 2));
int nHalf = n / 2;
// loop on numbers of the form 2*k+1
for (int k = 1; k <= maxK; k++) {
// if 2*k+1 is marked as prime, sift all its multiples
if (get(k)) {
// start from (2*k+1)^2: must divide this by two since the array
// doesn't contain multiples of two. Thus the starting number is
// (2*k+1)^2/2 = 2*k*(k+1). Note that this is odd.
// the increment is 2*k+1 (since the sieve contains only odd
// numbers, using this increment automatically skips to the next
// odd multiple).
final int increment = 2 * k + 1;
for (int composite = 2 * k * (k + 1); composite < nHalf; composite += increment)
// the index in the array is obtained by discarding the
// rightmost 3 bits (or divide by 8); likewise, the position
// in the byte is obtained by right shifting one as many
// time as the number represented by the same 3 bits.
// Note that the function get() is implemented similarly.
sieve[composite >> 3] |= (1 << (composite & 7));
* Checks if the number 2*n+1 is marked as prime.
* @param n
* An integer number.
* @return True if 2*n+1 is marked as prime, false otherwise.
boolean get(int n) {
return ((sieve[n >> 3] >> (n & 7)) & 1) == 0;
* @param n
* An integer.
* @return True if n is prime.
public boolean isPrime(int n) {
if (n == 2)
return true;
if (n == 1 || n % 2 == 0)
return false;
int i = n / 16;
if (i >= sieve.length)
throw new RuntimeException("The number " + n
+ " exceeds the values in the sieve.");
return ((sieve[i] >> ((n / 2) & 7)) & 1) == 0;
Sometimes it's useful to find prime numbers in a small interval instead of all of them up to a certain value. In cases like this, it's inconvenient to use the full sieve, since its speed and memory
requirements always depend on the upper primes bound. A segmented sieve allows sifting the composite numbers in a given interval, ignoring most of the previous numbers. In this case speed and memory
requirements will depend (mostly) on the length of the interval. I'll give an implementation of the segmented sieve in my next blog post. | {"url":"http://blog.giovannibotta.net/2012/03/eratosthenes-prime-sieve-java.html","timestamp":"2024-11-05T08:39:06Z","content_type":"text/html","content_length":"48956","record_id":"<urn:uuid:5f361b2d-112f-4150-8224-9464e9aac0eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00564.warc.gz"} |
Glossary » Units » Energy » Calorie (15°C)
Calorie (15°C) (cal[15]) is a unit in the category of Energy. It is also known as calories. Calorie (15°C) (cal[15]) has a dimension of ML^2T^-2 where M is mass, L is length, and T is time. It can be
converted to the corresponding standard SI unit J by multiplying its value by a factor of 4.185.
Note that the seven base dimensions are M (Mass), L (Length), T (Time),
(Temperature), N (Aamount of Substance), I (Electric Current), and J (Luminous Intensity).
Other units in the category of Energy include A.u. of Energy (E[h]), Barrel Oil Equivalent (bboe), Bboe (barrel Oil Equivalent) (bboe), BeV (billion EV) (BeV), British Thermal Unit (ISO) (Btu (ISO)),
British Thermal Unit (IT) (Btu (IT)), British Thermal Unit (mean) (Btu (mean)), British Thermal Unit (thermochemical) (Btu (therm.)), Calorie (4°C) (cal[4]), Calorie (diet Kilocalorie) (Cal, kcal),
Calorie (IT) (International Steam Table) (cal (IT)), Calorie (mean) (cal[mean]), Calorie (thermochemical) (cal (therm.)), Celsius-Heat Unit (Chu), Coulomb Volt (C-V), Cubic Centimeter-Atm (cm^3-atm),
Cubic Foot Atm (ft^3-atm), Electronvolt (eV), Erg (erg), Foot-Pound Force (ft-lbf), Foot-Poundal (ft-pdl), Gigaelectronvolt (GeV), Gram Calorie (gram-cal), Hartree (E[h]), Horsepower (550ft-Lbf/s)
-Hour (Hp-h), Inch Pound Force (in-lbf), Joule (J), Kilocalorie (15°C) (kcal[15]), Kilocalorie (4°C) (kcal[4]), Kiloelectronvolt (keV), Kilojoule (kJ), Kiloton TNT Equivalent (kt (TNT)),
Kilowatt-Hour (kWh), Megaelectronvolt (MeV), Megajoule (MJ), Megaton TNT Equivalent (Mt (TNT)), Newton Meter (N-m), Q Unit, Quad (quad), Quadrillion (quad), Rydberg (Ry), Tce (tonne Coal Equivalent)
(tce), Therm (EEG), Therm (US), Thermie (15°C) (th[15 °C]), Toe (tonne Oil Equivalent) (toe), Ton TNT Equivalent (ton (TNT)), Tonne Coal Equivalent (tce), Tonne Oil Equivalent (toe), and Watt Hour
Additional Information
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eFunda: Unit Category Listing. ... Download a Palm version of this unit conversion calculator! ... Units » Luminous Flux » Candlepower (spherical) ... | {"url":"https://www.efunda.com/glossary/units/units--energy--calorie_15c.cfm","timestamp":"2024-11-15T00:20:37Z","content_type":"text/html","content_length":"25150","record_id":"<urn:uuid:661646d0-96b6-4864-8db0-edf9d7c28499>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00261.warc.gz"} |
Vanishing viscosity limit of the three-dimensional barotropic compressible Navier–Stokes equations with degenerate viscosities and far-field vacuum | EMS Press
Vanishing viscosity limit of the three-dimensional barotropic compressible Navier–Stokes equations with degenerate viscosities and far-field vacuum
• Geng Chen
University of Kansas, Lawrence, USA
• Gui-Qiang G. Chen
University of Oxford, UK
• Shengguo Zhu
Shanghai Jiao Tong University, China
We are concerned with the inviscid limit of the Navier–Stokes equations to the Euler equations for barotropic compressible fluids in . When the viscosity coefficients obey a lower power law of the
density (i.e., with ), we identify a quasi-symmetric hyperbolic–singular elliptic coupled structure of the Navier–Stokes equations to control the behavior of the velocity of the fluids near a vacuum.
Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field vacuum, whose life span is
uniformly positive in the vanishing viscosity limit. Some uniform estimates on both the local sound speed and the velocity in with respect to the viscosity coefficients are also obtained, which lead
to the strong convergence of the regular solutions of the Navier–Stokes equations with finite mass and energy to the corresponding regular solutions of the Euler equations in for any . As a
consequence, we show that, for both viscous and inviscid flows, it is impossible that the norm of any global regular solution with vacuum decays to zero asymptotically, as tends to infinity. Our
framework developed here is applicable to the same problem for other physical dimensions via some minor modifications.
Cite this article
Geng Chen, Gui-Qiang G. Chen, Shengguo Zhu, Vanishing viscosity limit of the three-dimensional barotropic compressible Navier–Stokes equations with degenerate viscosities and far-field vacuum. Ann.
Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 1, pp. 121–170
DOI 10.4171/AIHPC/4 | {"url":"https://ems.press/journals/aihpc/articles/4552498","timestamp":"2024-11-07T16:12:55Z","content_type":"text/html","content_length":"94315","record_id":"<urn:uuid:4ce6fe6b-cc0e-4ab1-aba2-81ae97a0fb46>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00175.warc.gz"} |
What is COBWEB?
Since my current ANN experiments are moving forward rather slowly, I have spent some time preparing the final background chapter for my dissertation. I the course of doing so, I have among other
things looked at COBWEB [1], which is one of the most well-known concept formation algorithms. Today, I want to share the basic idea behind COBWEB and its variants.
Concept Formation in General
Quite some time ago, I have provided a rough sketch of concept formation in general. Here’s a quick recap based on [2]: In concept formation, we observe unlabeled examples one after another and try
to find a clustering of these examples into categories, which are then organized in a conceptual hierarchy. Moreover, each category is not represented as a set of examples, but based on abstracted
information about typical feature values. Concept formation algorithms typically follow an incremental hill-climbing process, where the conceptual hierarchy is updated after each observation.
Representation in COBWEB
How are concepts and the conceptual hierarchy defined in COBWEB? The overall conceptual hierarchy is represented as a tree and each inner node corresponds to one concept. A concept is specified
through three types of conditional probability distributions:
• The predictability p(value|concept) represents the distribution of possible feature values for members of the concept: If you know that the observation x belongs to the concept C, which kind of
feature values v do you expect it to have?
• The predictiveness p(concept|value) on the other hand tells us how indicative a certain feature value is for concept membership: If observation x has the feature value v, does this tell us
anything about its membership to concept C?
• The relative frequency p(concept|parent) tells us how frequent this concept is in comparison to its “siblings” in the hierarchy: Given that the observation x belongs to the parent concept P, how
likely is it that x also belongs to C?
These three types of conditional probability distributions can all be estimated based on raw co-occurrence counts from the observed examples: For instance, the predictability p(value|concept) can be
approximated by counting the frequency of all feature values observed for all examples falling under the concept C and then normalizing these absolute counts. Updating a concept description thus
simply corresponds to updating those counts, which then directly influences the probability distributions.
Category Utility
In order to measure how well a given conceptual hierarchy captures the underlying structure of the data, COBWEB uses an evaluation metric called “Category Utility”. I will spare you with the
mathematical details (those are nicely laid out in both [1] and [2]), and limit myself to providing a rough intuition.
Category utility is inspired by the basic level of categorization in human classification hierarchies, where one in general has a good balance between intra-class similarity (i.e., examples of the
same concept are similar) and inter-class dissimilarity (i.e., examples from different concepts are different). In COBWEB, this is formalized by a combination of predictability (do members of the
same concept have the same feature values?) and predictiveness (are these feature values specific to the current concept or do they also apply to other concepts?).
As it turns out, combining predictability and predictiveness is mathematically equivalent to calculating the expected number of feature values that can be guessed correctly for an arbitrary example
of the given category. Category utility is now defined as the increase in prediction performance that one gets from knowing the category membership of the data points. In other words, category
utility quantifies how much the induced category structure helps to predict feature values better than chance.
Learning in COBWEB
Figure 1: Operations in COBWEB.
Okay, so how does COBWEB construct its conceptual hierarchy? Well, initially it starts with a single (empty) concept. Each new observation descends the tree in a top-down manner along the most
appropriate branch (modifying the counts along the way) and at each level of the tree, one of the four following operators is used:
• Insert a new child concept: Create a new child concept and place the observation into this new child concept. In Figure 1, this results in a modification of the concept hierarchy from (b) to (a).
• Merge two child concepts: Take the two most appropriate child concepts and merge them – the original child concepts then become “grandchild” concepts under the newly created merged concept. In
Figure 1, we can transition from the concept hierarchy in (c) to the one in (b) by merging C[3] and C[4].
• Split a child concept: Remove a child concept from the concept hierarchy altogether and promote its children to the next-higher level. In Figure 1, this corresponds to the change from (b) to (c),
where C[1] is removed and C[3] and C[4] become child concepts of C[0].
• Add the observation to the best matching child concept: Simply select the most appropriate child concept and put the observation there, without modifying the conceptual hierarchy at all.
How does COBWEB decide which operator to use? Well, essentially it tries out all four of them and simply uses the one which results in the highest category utility. Please note that merging and
splitting can be thought of as reverse operators. This allows COBWEB to correct for mistakes it made on earlier turns and makes the concept hierarchy thus more flexible.
CLASSIT for Continuous Features
COBWEB itself is limited to categorical features. Since I however deal with continuous dimensions in my conceptual spaces, I would first need to discretize them, e.g., by using properties from the
individual domains as features. COBWEB would then go ahead and discover concepts spanning multiple domains based on the properties. However, in this scenario I already need to know the properties a
priori, which prevents me from learning them in parallel to the cross-domain concepts.
Also Gennari et al. [2] have noticed this weakness of COBWEB and have proposed a variant called CLASSIT, which assumes continuous features. Instead of storing counts for the different feature values
as a concept representation, CLASSIT uses one normal distribution per concept and feature. The mean and variance of this normal distribution can also be estimated in an incremental fashion, resorting
to moving averages. CLASSIT comes with a generalized version of category utility that is applicable to non-categorical features as well.
Conceptual Spaces
The original goal of my PhD project was to develop a concept formation algorithm for conceptual spaces. As it turns out, I was sucked into so many other interesting subprojects that I didn’t have the
time to work on concept formation. Nevertheless, the formalization of conceptual spaces that I have developed should in principle be able to support a concept formation algorithm similar to COBWEB
and CLASSIT:
Instead of feature counts or normal distributions, the representation of each concept would be based on a fuzzy star-shaped set. The operations of splitting a concept (based on a threshold on a
single dimension) and merging two concepts (by taking their union) are directly supported by my formalization. Incrementally adapting the concepts however would be more complex, since instead of
updating counts or estimates of the mean and variance, one would need to modify the domain weights, the size and position of the individual cuboids, and the overall sensitivity parameter of the
concept. Also the definition of category utility probably would need to be adapted to my scenario.
Overall, concept formation is in my opinion an important approach to concept learning, since it does not required labeled data and processes observations in an incremental fashion. Both of these
aspects make it cognitively more plausible than classical supervised machine learning techniques that are based on batch-processing large labeled data sets.
[1] Fisher, D. H. Knowledge Acquisition via Incremental Conceptual Clustering Machine Learning, Springer Nature, 1987, 2, 139-172
[2] Gennari, J. H.; Langley, P. & Fisher, D. Models of Incremental Concept Formation Artificial Intelligence, Elsevier BV, 1989, 40, 11-61
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Orbital Mechanics | mathdunk
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Orbital Mechanics
Calculating Launch Windows: The Mars Mission
This activity is designed for students familiar with advanced algebra concepts. In this lesson, students will:
• Use algebraic computations to determine the relative positions of Earth and Mars during which an optimal (low-energy) transfer of a spacecraft can occur.
• Combine this information with planetary-position data to determine the next launch opportunity to Mars.
• Graph paper, quadrille ruled (one piece per student)
• 8.5-by-11-inch or larger piece of thick cardboard (per student)
• Two push-pins (per student)
• String, approximately 30 cm (per student)
• Calculator
• Planetary heliocentric longitudes for appropriate years: 2016, 2017, 2018, 2019, 2020
Consider having students sit on a carpeted floor when using the pushpins and string to make an ellipse. The carpeted floor will absorb the tips of the pushpins that may extend past the thickness of
the cardboard. Alternately, have students use more than one piece of cardboard to cushion against protruding pushpin tips.
When a spacecraft is launched from Earth, its forward velocity combined with the gravitational pull of Earth cause it to travel in a curved path. As the spacecraft heads toward another planet, the
gravitational pull of that planet factors in to the path the spacecraft takes. The more a spacecraft can “coast” with engines off, the lower the cost of the mission (rocket fuel is not cheap!).
Think of a quarterback throwing a football to a receiver. The initial impulse (throw) is all the football gets as far as power is concerned. The football follows a curved path into the hands of the
receiver. Likewise, the quarterback throws the football to where the receiver is going to be, not necessarily to where the receiver is currently. So, the quarterback throws the football downfield as
the receiver is running in that direction. In a perfectly thrown pass, the receiver’s running speed will bring him or her to the exact spot where the football arrives at hand-level.
Launching to Mars is similar to this. A spacecraft is given an initial impulse (launch) toward Mars and then shuts off its engines and coasts (obeying Newton’s First Law) until it gets close to its
target. Depending on the mission, the spacecraft may slow down – to get into orbit or land – by using the Martian atmosphere or retro-rockets that fire opposite to the direction of travel (obeying
Newton’s Third Law).
Though a spacecraft could follow a variety of curved paths from Earth to Mars, one path called the Hohmann transfer orbit uses the least energy and is thereby considered to be the most efficient.
The Hohmann transfer is an elliptical orbit with the sun at one focus of the ellipse that intersects the orbit of the target planet. Launch occurs when Earth is at Hohmann perihelion (the point of
the Hohmann orbit that is closest to the sun). Arrival occurs when Mars is at Hohmann aphelion (the point of the Hohmann orbit that is farthest from the sun).
Depending on mission objectives and spacecraft characteristics, engineers will use variations on the Hohmann transfer orbit to get spacecraft to Mars. These variations can make travel time more or
less lengthy than a standard Hohmann transfer.
To make sure the spacecraft and Mars arrive at the same place at the same time, the spacecraft must launch within a particular window of time. This window is called the “launch window” and, depending
on the target, can be a few minutes or as much as a few weeks in length.
If a spacecraft is launched too early or too late, it will arrive in the planet’s orbit when the planet is not there.
When launched within the proper launch window, the spacecraft will arrive in the planet’s orbit just as the planet arrives at that same place. At this point, the spacecraft is positioned for either
going into orbit about the planet or landing on the planet.
Calculating orbit trajectories and launch windows is a complex task involving a variety of parameters that may or may not be constantly changing. In order to make this task accessible to high-school
students, some variable parameters have been stabilized and some assumptions have been made. This problem, with these simplifications, allows students to generate an approximation of the launch
window to Mars.
1. Explain to students that launching to Mars requires a spacecraft to travel in an elliptical orbit about the sun such that the spacecraft and Mars will arrive in the same place at the same time.
Their task in this exercise is to determine when we should next launch to Mars.
2. Explain that the most energy efficient orbit of this type is called the Hohmann transfer, in which the spacecraft will travel half of one orbit about the sun, leaving Earth at the orbit’s
perihelion and arriving at Mars (or any outer planet) at the orbit's aphelion.
The red line indicates the orbit of Mars, the blue line indicates the orbit of Earth,
and the grey line indicates the path a spacecraft takes from Earth to
Mars when launched on a Hohmann transfer path.
3. Remind students of Kepler’s Second Law, the Law of Equal Areas: A line drawn from a planet to the sun sweeps out equal areas in equal amounts of time.
Kepler's Second Law also tells us that planets travel at different rates of speed in their elliptical orbits,
moving faster when they are closer to the sun and slower when they are farther from the sun.​​
4. Explain to students that launching a spacecraft while considering the orbital dynamics of the planets is a highly complex mathematical task. In
order to simplify the task, we will make three assumptions (Note: none of these assumptions are true, but using these simplifications will still
allow a fairly accurate computation of the launch window.):
• The orbits of Earth and Mars are circular and centered on the sun. (Earth’s orbit is more circular than Mars’ orbit, but they are both slightly elliptical.)
• Earth and Mars travel at constant speeds. (They do not. See Kepler’s Second Law).
• The orbits of Earth and Mars are in the same plane. (They are close but slightly out of plane with one another).​
5. Explain to students the concept of heliocentric longitude. This is the position of an object with respect to the sun, measured eastward along the
ecliptic (path of Earth around the sun) from the vernal equinox (position in space where the ecliptic crosses the celestial equator). Just as longitudes
on Earth measure position with respect to a fixed point (the prime meridian), heliocentric longitudes measure position in space along the ecliptic with respect to the vernal equinox.
For consistency, we measure heliocentric longitudes counterclockwise (as viewed from above) from the vernal equinox.
To establish a frame of reference for this problem, we place Earth at launch at the vernal equinox (0 degrees) and Mars at 180 degrees at arrival.
To establish a frame of reference for this problem, we place Earth at launch at the vernal equinox (0 degrees) and Mars at 180 degrees at arrival.
The Hohmann transfer orbit is the ellipse that connects the points in space, Earth at 0 degrees and Mars at 180 degrees, about the ellipse that has the sun at one focus.
6. Have students find the length of the semi-major axis of the transfer orbit in astronomical units (AU), given that the average distance from Mars to the sun is 1.52 AU.
The Hohmann transfer orbit is the ellipse that connects the points in space, Earth at 0 degrees and Mars at 180 degrees, about the ellipse that has the sun at one focus.
6. Have students find the length of the semi-major axis of the transfer orbit in astronomical units (AU), given that the average distance from Mars to the sun is 1.52 AU.
Earth is, on average, 1 astronomical unit (AU) from the sun. Mars is, on average, 1.52 AUs from the sun.
Earth is, on average, 1 astronomical unit (AU) from the sun. Mars is, on average, 1.52 AUs from the sun.
The major axis of the Hohmann transfer orbit is represented by 2a. Some simple arithmetic will allow us to compute the length, a, of the semi-major axis.
7. Have students use string and pushpins to draw the assumed-circular orbits of Earth and Mars about the sun, and the approximation of the Hohmann transfer orbit on graph paper as shown at right:
Students will need to compute the location of the second focus (one focus is at the sun) for the Hohmann transfer orbit. The focal distance is 0.26 AU, so if the sun is at (0,0), the other focus will
be at (-0.52, 0).
Use string and two pushpins to draw the elliptical Hohmann transfer orbit.
To draw the Hohmann transfer orbit, place a pushpin at each focus of the ellipse and use a loop of string equal in length to twice the sum of the length of the semi-major axis of the ellipse and the
focal length (students may derive this using the formula for an ellipse).
8. Have students use Kepler’s Third Law, the Law of Harmony, to determine the period of the Hohmann transfer orbit and then the travel time to Mars along this orbit.
Kepler’s Third Law states that the square of the period of any planet is proportional to the cube of the semi-major axis of its orbit. An equation can represent this relationship:
P2=ka3 with k being the constant of proportionality
Using Earth as an example, we can measure P in years and a in astronomical units so P = 1 year and a = 1 AU. Thus, P2=ka3→k=1 => P2=a3
P2= (1.26 AU)3 => P ~ 1.41 years ~ 517 days
The full period of this Hohmann transfer orbit is 517 days. Travel to Mars encompasses half of one orbit, so approximately 259 days.
9. Using the daily motions of Earth and Mars, compute the ideal relative position of Earth and Mars during launch.
Mars completes one revolution around the sun (360 degrees) in 687 days, so that means it moves 0.524 degrees per day (360 degrees/687 days). In 259 days (the travel time from Earth to Mars along the
Hohmann transfer path), Mars will have moved 136 degrees (0.524 degrees per day * 259 days).
To calculate the position of Mars at the time of launch, subtract the amount of its motion during the spacecraft’s travel time (136 degrees) from its point of arrival (180 degrees). 180 degrees – 136
degrees = 44 degrees.
Considering that launch from Earth was at the Hohmann orbit perihelion (point closest to the sun) and arrival is at the Hohmann orbit aphelion (point farthest from the sun), we can conclude that a
launch opportunity occurs when Mars is 44 degrees ahead of Earth in its orbit.
In our established frame of reference, Mars must be at 44 degrees relative to Earth at 0 degrees at launch. For any frame of reference, Mars must be 44 degrees ahead of Earth in its orbit at launch.
10. Using the planetary heliocentric longitudes, approximately when is the next opportunity for a launch to Mars?
• Must a spacecraft be launched at an exact moment in the launch window? What happens if it is launched early or late?
• Research: What is the average length of a launch window to Mars?
• Approximately when was the most recent opportunity for a launch to Mars? What countries took advantage of that opportunity and launched to Mars at that time? What is the current status of those
missions? Were they successful?
• Have students create a spreadsheet that will subtract heliocentric longitudes for Earth and Mars to simplify launch window calculations.
• Relative to Mars, where is Earth in its orbit when the spacecraft arrives?
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Applications of Derivatives
Last Updated on July 16, 2021
The derivative defines the rate at which one variable changes with respect to another.
It is an important concept that comes in extremely useful in many applications: in everyday life, the derivative can tell you at which speed you are driving, or help you predict fluctuations on the
stock market; in machine learning, derivatives are important for function optimization.
This tutorial will explore different applications of derivatives, starting with the more familiar ones before moving to machine learning. We will be taking a closer look at what the derivatives tell
us about the different functions we are studying.
In this tutorial, you will discover different applications of derivatives.
After completing this tutorial, you will know:
The use of derivatives can be applied to real-life problems that we find around us.
The use of derivatives is essential in machine learning, for function optimization.
Let’s get started.
Tutorial Overview
This tutorial is divided into two parts; they are:
Applications of Derivatives in Real-Life
Applications of Derivatives in Optimization Algorithms
Applications of Derivatives in Real-Life
We have seen that derivatives model rates of change.
Derivatives answer questions like “How fast?” “How steep?” and “How sensitive?” These are all questions about rates of change in one form or another.
– Page 141, Infinite Powers, 2019.
This rate of change is denoted by, 𝛿y / 𝛿x, hence defining a change in the dependent variable, 𝛿y, with respect to a change in the independent variable, 𝛿x.
Let’s start off with one of the most familiar applications of derivatives that we can find around us.
Every time you get in your car, you witness differentiation.
– Page 178, Calculus for Dummies, 2016.
When we say that a car is moving at 100 kilometers an hour, we would have just stated its rate of change. The common term that we often use is speed or velocity, although it would be best that we
first distinguish between the two.
In everyday life, we often use speed and velocity interchangeably if we are describing the rate of change of a moving object. However, this in not mathematically correct because speed is always
positive, whereas velocity introduces a notion of direction and, hence, can exhibit both positive and negative values. Hence, in the ensuing explanation, we shall consider velocity as the more
technical concept, defined as:
velocity = 𝛿y / 𝛿t
This means that velocity gives the change in the car’s position, 𝛿y, within an interval of time, 𝛿t. In other words, velocity is the first derivative of position with respect to time.
The car’s velocity can remain constant, such as if the car keeps on travelling at 100 kilometers an hour consistently, or it can also change as a function of time. In case of the latter, this means
that the velocity function itself is changing as a function of time, or in simpler terms, the car can be said to be accelerating. Acceleration is defined as the first derivative of velocity, v, and
the second derivative of position, y, with respect to time:
acceleration = 𝛿v / 𝛿t = 𝛿2y / 𝛿t2
We can graph the position, velocity and acceleration curves to visualize them better. Suppose that the car’s position, as a function of time, is given by y(t) = t3 – 8t2 + 40t:
The graph indicates that the car’s position changes slowly at the beginning of the journey, slowing down slightly until around t = 2.7s, at which point its rate of change picks up and continues
increasing until the end of the journey. This is depicted by the graph of the car’s velocity:
Notice that the car retains a positive velocity throughout the journey, and this is because it never changes direction. Hence, if we had to imagine ourselves sitting in this moving car, the
speedometer would be showing us the values that we have just plotted on the velocity graph (since the velocity remains positive throughout, otherwise we would have to find the absolute value of the
velocity to work out the speed). If we had to apply the power rule to y(t) to find its derivative, then we would find that the velocity is defined by the following function:
v(t) = y’(t) = 3t2 – 16t + 40
We can also plot the acceleration graph:
We find that the graph is now characterised by negative acceleration in the time interval, t = [0, 2.7) seconds. This is because acceleration is the derivative of velocity, and within this time
interval the car’s velocity is decreasing. If we had to, again, apply the power rule to v(t) to find its derivative, then we would find that the acceleration is defined by the following function:
a(t) = v’(t) = 6t – 16
Putting all functions together, we have the following:
y(t) = t3 – 8t2 + 40t
v(t) = y’(t) = 3t2 – 16t + 40
a(t) = v’(t) = 6t – 16
If we substitute for t = 10s, we can use these three functions to find that by the end of the journey, the car has travelled 600m, its velocity is 180 m/s, and it is accelerating at 44 m/s2. We can
verify that all of these values tally with the graphs that we have just plotted.
We have framed this particular example within the context of finding a car’s velocity and acceleration. But there is a plethora of real-life phenomena that change with time (or variables other than
time), which can be studied by applying the concept of derivatives as we have just done for this particular example. To name a few:
Growth rate of a population (be it a collection of humans, or a colony of bacteria) over time, which can be used to predict changes in population size in the near future.
Changes in temperature as a function of location, which can be used for weather forecasting.
Fluctuations of the stock market over time, which can be used to predict future stock market behaviour.
Derivatives also provide salient information in solving optimization problems, as we shall be seeing next.
Applications of Derivatives in Optimization Algorithms
We had already seen that an optimization algorithm, such as gradient descent, seeks to reach the global minimum of an error (or cost) function by applying the use of derivatives.
Let’s take a closer look at what the derivatives tell us about the error function, by going through the same exercise as we have done for the car example.
For this purpose, let’s consider the following one-dimensional test function for function optimization:
f(x) = –x sin(x)
We can apply the product rule to f(x) to find its first derivative, denoted by f’(x), and then again apply the product rule to f’(x) to find the second derivative, denoted by f’’(x):
f’(x) = -sin(x) – x cos(x)
f’’(x) = x sin(x) – 2 cos(x)
We can plot these three functions for different values of x to visualize them:
Similar to what we have observed earlier for the car example, the graph of the first derivative indicates how f(x) is changing and by how much. For example, a positive derivative indicates that f(x)
is an increasing function, whereas a negative derivative tells us that f(x) is now decreasing. Hence, if in its search for a function minimum, the optimization algorithm performs small changes to the
input based on its learning rate, ε:
x_new = x – ε f’(x)
Then the algorithm can reduce f(x) by moving to the opposite direction (by inverting the sign) of the derivative.
We might also be interested in finding the second derivative of a function.
We can think of the second derivative as measuring curvature.
– Page 86, Deep Learning, 2017.
For example, if the algorithm arrives at a critical point at which the first derivative is zero, it cannot distinguish between this point being a local maximum, a local minimum, a saddle point or a
flat region based on f’(x) alone. However, when the second derivative intervenes, the algorithm can tell that the critical point in question is a local minimum if the second derivative is greater
than zero. For a local maximum, the second derivative is smaller than zero. Hence, the second derivative can inform the optimization algorithm on which direction to move. Unfortunately, this test
remains inconclusive for saddle points and flat regions, for which the second derivative is zero in both cases.
Optimization algorithms based on gradient descent do not make use of second order derivatives and are, therefore, known as first-order optimization algorithms. Optimization algorithms, such as
Newton’s method, that exploit the use of second derivatives, are otherwise called second-order optimization algorithms.
Further Reading
This section provides more resources on the topic if you are looking to go deeper.
Calculus for Dummies, 2016.
Infinite Powers, 2020.
Deep Learning, 2017.
Algorithms for Optimization, 2019.
In this tutorial, you discovered different applications of derivatives.
Specifically, you learned:
The use of derivatives can be applied to real-life problems that we find around us.
The use of derivatives is essential in machine learning, for function optimization.
Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.
The post Applications of Derivatives appeared first on Machine Learning Mastery.
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CBV Creator (corrections.CBVCreator)
Creation of Cotrending Basis Vectors.
The CBV creation has three major steps, which are wrapped in the CBVCreator class:
1. The CBVs for the specific todo-list are computed using the CBVCreator.compute_cbv() function.
2. CBVs are split into “single-scale” CBVs and “spike” CBVs using the CBVCreator.spike_sep() function.
3. An initial fitting is performed for all targets using linear least squares using the CBVCreator.cotrend_ini() function. This is done to obtain fitting coefficients for the CBVs that will be used
to form priors for the final fit.
4. Priors are constructed using the output from step 3, using the CBVCreator.compute_weight_interpolations() function. This function saves interpolation functions for each of the CBV coefficient
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Code author: Rasmus Handberg <rasmush@phys.au.dk>
class corrections.CBVCreator(*args, cadence='ffi', sector=None, cbv_area=None, ncomponents=16, threshold_correlation=0.5, threshold_snrtest=5.0, threshold_variability=1.3, threshold_entropy=-0.5,
output_folder=None, **kwargs)[source]
Bases: BaseCorrector
Creation of Cotrending Basis Vectors.
TESS Sector.
TESS observing cadence in seconds.
Number of CBVs to be created.
Path to the HDF5 file containing the CBV.
Code author: Rasmus Handberg <rasmush@phys.au.dk>
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
__init__(*args, cadence='ffi', sector=None, cbv_area=None, ncomponents=16, threshold_correlation=0.5, threshold_snrtest=5.0, threshold_variability=1.3, threshold_entropy=-0.5, output_folder=None,
Initialize the CBV Creator.
○ sector (int, required) – TESS Sector.
○ cbv_area (int, required)
○ cadence (int or str, optional) – TESS observing cadence in seconds.
○ ncomponents (int, optional) – Number of CBVs to be created.
○ threshold_variability (float, optional)
○ threshold_correlation (float, optional)
○ threshold_snrtest (float, optional)
○ threshold_entropy (float, optional)
○ output_folder (str, optional)
Code author: Rasmus Handberg <rasmush@phys.au.dk>
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Close the CBV Creator object.
Main function for computing CBVs.
The steps taken in the function are:
1. Run lightcurve_matrix() to obtain matrix with gap-filled, nan-removed light curves for the most correlated stars in a given cbv-area.
2. Compute principal components.
3. Run entropy_cleaning() to remove significant single-star contributers based on entropy.
4. Rerun SNR test on CBVs, and only retain CBVs that pass the test.
5. Recalculate principal components using cleaned star list.
6. Save CBVs and make diagnostics plots.
targ_limit (int, optional) – Maximum number of targets to remove during entropy-cleaning.
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Code author: Rasmus Handberg <rasmush@phys.au.dk>
3D distance map for weighting initial-fit coefficients into a prior.
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Function for running the initial co-trending to obtain CBV coefficients for the construction of priors.
The function will load the calculated CBVs and co-trend all light curves in area using fit of all CBVs using linear least squares. The CBV coefficients from the fit are saved into the HDF5
CBV file.
do_ini_plots (bool) – Plot the LS fit for each light curve? Default=False.
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Code author: Rasmus Handberg <rasmush@phys.au.dk>
entropy_cleaning(matrix, targ_limit=150)[source]
Entropy-cleaning of lightcurve matrix using the SVD U-matrix.
○ matrix (numpy.ndarray)
○ targ_limit (int, optional) – Maximum number of targets to remove during cleaning.
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Interpolate CBVs generated from FFIs to higher cadence (120 seconds).
New HDF5 files will be generated, containing the CBVs interpolated using a cubic spline to the higher cadence. All spike-CBVs are set to zero, since there is no good way to interpolate them.
cadence (int)
Path to the new CBV file.
Return type:
Code author: Rasmus Handberg <rasmush@phys.au.dk>
Load matrix filled with light curves.
The steps performed are the following:
1. Only targets with a variability below a threshold are included.
2. Computes correlation matrix for light curves in a given cbv-area and only includes the threshold_correlation() most correlated light curves.
3. Performs gap-filling of light curves and removes time stamps where all flux values are NaN.
○ numpy.ndarray: matrix of light curves to be used in CBV calculation.
○ numpy.ndarray: the indices for the timestamps with nans in all light curves.
○ int: Number of timestamps.
Return type:
Code author: Rasmus Handberg <rasmush@phys.au.dk>
Code author: Mikkel N. Lund <mikkelnl@phys.au.dk>
Load lightcurve from task ID or full task dictionary.
task (integer or dict)
Lightcurve for the star in question.
Return type:
ValueError – On invalid file format.
Code author: Rasmus Handberg <rasmush@phys.au.dk>
Return folder path where plots for a given lightcurve should be saved.
lc (lightkurve.TessLightCurve) – Lightcurve to return plot path for.
Path to directory where plots should be saved.
Return type:
Code author: Rasmus Handberg <rasmush@phys.au.dk>
search_database(select=None, join=None, search=None, order_by=None, limit=None, distinct=False)
Search list of lightcurves and return a list of tasks/stars matching the given criteria.
Returned rows are restricted to things not marked as STATUS.SKIPPED, since these have been deemed too bad to not require corrections, they are definitely also too bad to use in any kind of
○ select (list of strings or None) – List of table columns to return.
○ search (list of strings or None) – Conditions to apply to the selection of stars from the database
○ order_by (list, str or None) – Column to order the database output by.
○ limit (int or None) – Maximum number of rows to retrieve from the database. If limit is None, all the rows are retrieved.
○ distinct (bool) – Boolean indicating if the query should return unique elements only.
○ join (list) – Table join commands to merge several database tables together.
All stars retrieved by the call to the database as dicts/tasks that can be consumed directly by load_lightcurve
Return type:
Code author: Rasmus Handberg <rasmush@phys.au.dk>
Separate CBVs into a “slow” and a “spiky” component.
This is done by filtering the deta and identifying outlier with a peak-finding algorithm.
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Birthday Dominoes
(This post is dedicated to the most important Pisces birthday I know: EK.)
Dominoes is a well-known game that no one actually knows how to play. A much more accessible game is tiling dominoes: I give you a grid, and you tell me if you can cover the whole thing with
For example, look at this 2x2 grid:
Easy! What about this 4x4 grid?
Also easy! What about a 3x3 grid?
It can’t be done! Any grid you can cover with dominoes has to have an even number of squares, but a 3x3 grid has 9. We lose, through no fault of our own.
This pretty much solves grids. You can tile them with dominoes if and only if they have an even number of squares. But this gives us two natural questions to ask:
1. What if we used something other than dominoes?
2. What if we used something other than grids?
Something other than dominoes
Dominoes are pieces with two blocks “snapped” together. What if we used more than two blocks? This these exist, of course, and we call them triominoes, quadominoes, pentominoes, and in general,
There is only “one” domino—two blocks stuck at their ends—but there are three triominoes!
That pesky 3x3 grid that we couldn’t tile with dominoes is a cinch with triominoes:
The most famous polyomino is the pentomino, which has five blocks stuck together. These are commonly used for fun in brain teasers, if you’re into that sort of thing.
Something other than grids
The shape of the board is just as important as the number of spaces. For example, look at this “T” with four spaces:
Even though there are an even number of spaces, we can’t tile this with dominoes! Here’s a grid with 9 spaces that we cannot tile with triominoes:
In general, a board with $n$ spaces is only guaranteed to have a tiling of monomioes (pieces with 1 block) and an $n$-omio (a piece with $n$ blocks), both of which are kind of cheating. Board layout
plays a big role.
The heart board
I propose that we think about tiling the heart board, something I invented for just this occasion. Here are the first four heart boards:
It’s a bit hard to make out the “grid” in these pictures, so here are the first two drawn by hand (kinda):
The number of blocks in the first four hearts is 10, 43, 96, and 169, respectively. The hearts follow a pattern that generalizes to arbitrary sizes. The $n$th heart board is the set of all $(x, y)$
such that
• $0 \leq x < 2n$
• $0 \leq y < 4n$
• $x \leq y$
• $y \leq 5n - x$
• $y \leq x + 3n$,
and also the reflection of these points about the $y$-axis.
The $n$th heart board has exactly
\[10 n^2 + 3n - 3\]
spaces in it.
There are lots of questions to ask about this board, but let’s settle for just one: When can the heart board be tiled by dominoes?
The first heart board cannot be tiled with dominoes. That’s easy enough to see by hand because it only has 10 blocks:
The second heart board is much bigger at 43 blocks, but this is an odd size so no tiling by dominoes could exist. In fact, the general size $10 n^2 + 3n - 3$ is odd when $n$ is even, so only the
“odd” heart boards could hope to be tiled by dominoes anyway.
So what about the third heart block, the one in the bottom left of the computer-generated image above? Can it be tiled using dominoes? It turns out that no, it cannot be. How do I know this? My
computer told me. In fact, using the polyomino library, my computer told me more:
Heart board number Domino tiling?
1 no
3 no
5 no
7 no
9 no
11 no
13 no
15 no
17 no
19 no
It seems like no heart boards can be tiled by dominoes. Is this true? Amazingly, yes! Not a single heart board can be tiled by dominoes.
The proof of this fact relies on a very simple observation: If you color the squares of an odd heart board in an alternating fashion, then then there are exactly two more squares of one color than
the other. For example, look at the first heart board:
Every domino you put down must cover exactly one of each color. In the above picture, every domino covers one red and one blue tile. Once you place four dominoes, there are no blue tiles left, but
two red tiles. We can’t cover those pieces with dominoes! (This is what happened in our attempted tiling above.)
This pattern persists for every odd heart board. Here is a plot of the first four hearts again, now colored in this alternating way:
The first and third heart boards above (left column) have exactly two more red squares than blue squares. The second and fourth (right column) actually have a bigger difference in the number of
squares, but we already knew that they couldn’t be colored with dominoes.
This pattern is somewhat tricky to prove, but once you know the idea it’s just calculations. See my math.SE question for details.
We’ve left lots of questions on the table that would be easy to answer. For example, what’s the proportion of squares that are red versus blue in the even heart boards? A harder project: Let $L_n$ be
a set of lines which intersect the square $[-n, n] \times [0, n]$ and each other. When is the region “inside” $L_n$ tileable with dominoes?
I don’t know the answer to any of these offhand, but they sound fun. I hope that this delivery on my promise of math art taught everyone something new. Here’s to much more in the future! | {"url":"https://rwdb.xyz/birthday-dominoes/","timestamp":"2024-11-03T12:52:56Z","content_type":"text/html","content_length":"17316","record_id":"<urn:uuid:2f3a9c64-7d15-4219-90c9-a342f40bd857>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00158.warc.gz"} |
Number-on-the-Forehead Communication Complexity
Since the Winter 2020 semester at McGill, I've helped run a Communication Complexity reading group attended by undergraduate and graduate students. Recently, I became particularly excited by the
Number-on-the-Forehead model, and gave a talk on it in the reading group.
The idea is quite simple. For 2 player communication, imagine that Alice's input is on top of Bob's forehead, making it easy for Alice to see but impossible for Bob to see. Likewise, Bob's input is
on Alice's forehead.
This generalizes to k-player problems where k people stand in a circle, each with a card on their forehead indicating some information that is viewable to everyone but the forehead owner.
If this still doesn't make sense, think of this scene from The Office.
Here are the lecture notes I wrote for the talk. I cover basic definitions, Behrend's bound and it's application to the Exactly-N problem, and Ramsey Theoretic lower bounds on cylinders.
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List Minimum (Easy)
Submit solution
Points: 3
Time limit: 0.75s
Java 1.0s
Processing 1.0s
Python 2.5s
Memory limit: 3M
Brute Force Practice 1 — Easy Version
Given a permutation of the integers , output the permutation after it has been sorted.
Input Specification
The first line will contain the integer .
The next line will contain integers, a permutation of the integers .
Output Specification
The sorted permutation on a single line.
Sample Input
Sample Output
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2017 UIUC ICPC Tryout 2 Solutions
Read this for both problems and solutions.
The problemset is available here
This editorial is mainly drafted by Lan Dao. I only wrote problem A and edit some format.
A: Birthday Cake
First, you should know the Euler formula about the relation between the number of points, edges, and planes. Furthermore, there is a super important message in the Input section as follows.
• All candles and lines are distinct.
• No candle is on a cut line.
• No line is completely outside or tangent to the cake.
• The input guarantees that the number of cake pieces remains the same if any cut line is shifted by at most \(10^{−4}\) in any direction.
• The input also guarantees that each candle remains in the interior of the same piece of cake if its position is shifted by at most \(10^{−4}\) in any direction.
As a result, there will be no corner cases at all, especially there will be no three cuts intersecting at the same point, which is a really good news.
So, we can apply an \(O(m^2)\) bruteforce algorithm to compute the number of points (intersections) and the number of edges (segments). After that, using the Euler formula, we can get the number of
planes (regions). If this number is not desired, we don’t need to further check the positions of candles.
The position of a candle can be uniquely identified by a \(m\)-bit mask: for each line, by plugging the candle coordinate into the line \(ax+by+c\) and check whether it is greater than \(0\).
Therefore, to check whether each of those \(n\) regions contains exactly one candle, we can just check whether these masks are distinct.
B: Bumbped
We first create a graph \(G = (V, E)\) where
• The set of vertices is $$V’ = { (u, x) | \forall 1 \leq u \leq n, x \in \{0, 1\} }$$
• There is an edge from \(u, x\) and \(v, y\) in \(E’\) with cost \(C\) if and only if one of the following is true:
□ \(x = y\) and there is an a road connecting \(u_{th}\) city and \(v_{th}\) city. In this case \(C\) equals the cost to travel \(u, v\) road.
□ \(x < y\) and there is a flight from \(u_{th}\) city to \(v_{th}\) city. In this case \(C = 0\).
Let us denote \(L((u, x), (v, y))\) be the length of the shortest path from \((u, x)\) to \((v, y)\) in \(G\). Then, the answer to our original problem is $$min \{ L((s, x), (t, y)) | \forall x, y \
in \{0, 1\} \wedge x \leq y \}$$
We can solve this by using Dijkstra algorithm on \(G\).
C: Canonical Coin Systems
To solve this problem, we need to determine if there is a counterexample for the given system. We will use dynamic programming to answer this question.
Let \(f(S)\) be the minimum number of coins we need to form a sum of \(S\) using the given denominations. Then we have $$f(S) = min\{f(S - c_i) + 1 | 1 \leq i \leq n\}$$
Let \(g(S)\) be the number of coins we need to form the sum of \(S\) using greedy algorithm. Then $$g(S) = g(S - c_t) \textit{ where t is the maximum number such that \(c_t \leq S\)}$$
Then, a sum \(S\) is a counterexample if \(f(S) \neq g(S)\). Since we are already given a hint that the minimum counter example is at most \(2c_n\), we only need to compute \(f(S), g(S)\) for all \(S
\leq 2c_n\). Thus, the complexity of the algorithm is only O(\(nc_n\)).
D: Cat and Mice
Let \((x_i, y_i)\) and \(s_i\) be the coordinate of the \(i_{th}\) mouse and the time it stays above the ground. We also denote function \(T(i, j, v) = \frac{\sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}}{v}
\) be the amount of time to get from the \(i_{th}\) mouse to the \(j_{th}\) mouse with speed \(v\).
Suppose we have two real values \(v_1, v_2\) such that \(v_1 < v_2\) and we are able to get all the mice with initial speed \(v_1\), then clearly we can also achieve the same thing with initial speed
\(v_2\). Similarly, if we can’t get all the mice with initial speed \(v_2\), we certainly can’t get all the mice with initial speed \(v_1\). With this property, we can binary search the minimum
initial speed \(v\). Then, our problem becomes: Given an initial speed \(v\), determine whether we are able to get all the mice.
We will solve the new problem using dynamic programming. Let \(f(S, i)\) (where \(S\) is a subset of all mice and \(i \in S\)) be the minimum amount of time we need to get all the mice in set \(S\)
and the last mouse we get is \(i\). We also denote \(f(S, i) = -1\) if there is no way to get all the mice in \(S\) with \(i\) be the last one. For the sake of our problem, we also denote \(min{\
emptyset} = -1\). Let \(x_0 = y_0 = 0\), then our base cases will be
• If \(T(0, i, v) > s_i\), \(f(\{i\}, i) = -1\).
• Otherwise, \(f(\{i\}, i) = T(0, i, v)\).
And for all other cases $$f(S, i) = min \{ f(S’, j) + T(i, j, v) | S’ = S \setminus \{i\}, j \in S’, f(S’, j) eq -1, f(S’, j) + T(i, j, v) \leq s_i\}$$
Then, we can get all the mice with initial speed \(v\) if there exists \(i\) (\(1 \leq i \leq n\)) such that \(f(\{1, 2, …, n\}, i) \neq -1\).
E: Company Picnic
Let us number the employees from \(1 \to n\) and let \(s_i\) be the running speed of the \(i_{th}\) employee. Let us create a rooted tree \(G\) where each node represents an employee and node \(i\)
is a parent of node \(j\) if the \(j_{th}\) employee is the direct supervisor of the \(i_{th}\) employee. The root of the tree is the node representing the CEO.
Suppose we have a value \(v\). If we know that there exists a formation which results in the average speed \(v_a \leq v\), we know that the answer is at least \(v\). Otherwise, the answer cannot
exceed \(v\). With this property, we can binary search for the optimal average speed \(v_{optimal}\). Then, the remaining problem is to determine if we can form a set of teams where the average speed
is at least \(v_{min}\) for any given \(v_{min}\).
We will solve this problem using dynamic programming. Suppose we have a team formation with pairs \((i_1, j_1), (i_2, j_2), …, (i_k, j_k)\). Then, the average running speed of these teams is $$\frac
{\sum_{t = 1}^k min\{s_{i_t}, s_{j_t}\}}{k}$$ Let \(T_{v_{min}}(i, j) = min\{s_i, s_j\} - v_{min}\). Then we have $$\frac{\sum_{t = 1}^k min\{s_{i_t}, s_{j_t}\}}{k}$$\geq v_{min} \iff \sum_{t = 1}^k
min\{s_{i_t}, s_{j_t}\} \geq kv_{min} \iff \sum_{t = 1}^k (min\{s_{i_t}, s_{j_t}\} - v_{min}) = \sum_{t = 1}^k T_{v_{min}}(i_t, j_t) \geq 0
Let us call the sum \(\sum_{t = 1}^k (min\{s_{i_t}, s_{j_t}\})\) the \(T\)-sum of this formation.
Let \(f(u, p)\) (\(1 \leq u \leq n, p \in \{0, 1\}\)) be the maximum number of teams can be formed using only employees represented by nodes under subtree rooted at \(u\) and \(p\) denotes whether we
can pair \(u\) with one of its children (\(p = 1\) means you can). Similarly, let \(g(u, p)\) be the maximum \(T\)-sum of all the formation with \(f(u, p)\) teams. Lastly, let \(better(u)\) be
defined as:
• If \(f(u, 0) > f(u, 1)\) or \(f(u, 0) = f(u, 1) \wedge g(u, 0) > g(u, 1)\), \(better(u) = 1\).
• Otherwise, \(better(u) = 0\).
Then, for each \(u\) we have:
• \(f(u, 0) = \sum_{v \in children(u)} f(v, better(v))\)
• \(g(u, 0) = \sum_{v \in children(u)} g(v, better(v))\)
To calculate \(f(u, 1)\) and \(g(u, 1)\), we will try to pair \(u\) with each of its child and take the maximum:
• \(f(u, 1) = f(u, 0) + max\{f(v, 0) - f(v, better(v)) + 1 \vert v \in children(u)\}\)
• \(g(u, 1) = g(u, 0) + max\{g(v, 0) - g(v, better(v)) + T_{v_{min}}(u, v) \vert v \in children(u)\}\)
Then, to determine if the minimum average speed is at least \(v_{min}\), we only need to check if \(g(root, 0) \geq 0\).
F: GlitchBot
Since the constraints are relatively small, for each instruction, we can replace that instruction with \(\textit{Forward/Left/Right}\) and the check if the new set of instruction is correct. We can
verify if a set of instructions is correct by simulating the robot. We stop as soon as we can find a good instruction set by replacing some instruction. Since we have \(n\) instructions and for each
instruction, the simulation process takes O(\(n\)), the total complexity is O(\(n^2\)).
G: Greeting Card
We are given \(n\) points on an \(xy\)-plane. We are asked to count the number of pair of two points such that the Euclidean distance between them is exactly 2018.
Suppose we have 2 points \((x_1, y_1)\) and \((x_2, y_2)\). Then $$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} = 2018 \iff (x_1 - x_2)^2 + (y_1 - y_2)^2 = 2018^2$$
Since \(x_1, x_2, y_1, y_2\) are all integers, \((x_1 - x_2)^2\) and \((y_1 - y_2)^2\) are integers. The key observation here is there are only 4 ordered pairs of integers \((a, b)\) such that \(a^2
+ b^2 = 2018^2\). Those are (0, 2018), (2018, 0), (1118, 1680), (1680, 1118).
Then, for each point \((x, y)\) among the \(n\) given points and each pair \((a, b)\) mentioned above, we will count the number of points \((x’, y’)\) that: $$|x - x’| = a \wedge |y - y’| = b$$ This
can be done in several ways. One easy way to achieve this is that for each given point, we will store it inside a data structure that allows fast query base on the coordinates (such as \(\textit
{std::set}\) or \(\textit{std::map}\) in C++). Since each pair of points would be counted twice this way, we need to divide our sum by 2 to get the desired answer.
H: Imperfect GPS
Let \(T\) be the recording period of the GPS. For each time \(t\) which is a multiple of \(T\) and does not exceed \(t_n\) we will calculate the position of the runner. If \(t = t_n\), then the
position recorded by the GPS at time \(t\) is \((x_n, y_n)\). Otherwise, let \(j\) be the largest number between 1 and \(n\) such that the \(t_j\) is at most \(t\). Then, the position recorded by GPS
at time \(t\) \((x’, y’)\) is: $$x’ = \frac{t - t_{j}}{t_{j + 1} - t_{j}}(x_{j + 1} - x_{j}) + x_j, y’ = \frac{t - t_{j}}{t_{j + 1} - t_{j}}(y_{j + 1} - y_{j}) + y_j$$ To get the index \(j\) for each
time \(t\), we can either use two-pointers technique or binary search for \(j\). With the recorded positions of the GPS, it should be easy to calculate the desired answer.
I: Odd Gnome
Suppose we have \(n\) gnomes in line with indexes \(a_1, a_2, …, a_n\). Then, if there exists index \(1 < i < n\) such that \(a_i \neq a_{i - 1} + 1\) and \(a_i \neq a_{i + 1} - 1\), then the \(i_
{th}\) gnome is the king. Otherwise, if \(a_1 \neq a_2 - 1\) then the \(1_{st}\) gnome is the king. Other than that, the last gnome is the king.
J: Progressive Scramble
Suppose we have a string \(s\) with length \(L\):
• To encode \(s\) into \(s’\): \(s’_0 = s_0\) and \(s’_i = v^{-1}\bigg(\Big(v\big(s_i\big) + v\big(s’_{i - 1}\big)\Big)\bigg) \textit{ mod } 27\) for all \(1 \leq i \leq L\)
• To decode \(s\) to get \(s’\): \(s’_0 = s_0\) and \(s’_i = v^{-1}\Bigg(\bigg(\Big(v\big(s_i\big) - v\big(s’_{i - 1}\big)\Big) \textit{ mod } 27 + 27\bigg) \textit{ mod } 27\Bigg)\) for all \(1 \
leq i \leq L\)
Where \(v^{-1}\) is the inverse function of \(v\).
K: Space Probe
Notice that the product \(nk \leq 10^7\), for each measurements time \(m_i\) and prohibited segment \([b, e]\), we know that the starting time cannot be between \([b - m_i, e - e_i]\). Therefore, we
can obtain \(nk\) segments of starting time that are “prohibited”. Let us sort the beginning and the ending of all such segments. For points with the same magnitude, we break the tie by prioritizing
the beginning points. We will then perform sweep line algorithm on the sorted points and keep track of the “opening” segments at each point. Let \(l, count\) all initially equals to zero. Then, we
iterate through the sorted points in order. For a point \(x\):
• If \(x\) is a closing point: We decrease the value of \(count\). If \(count = 0\), we assign \(l = x\).
• If \(x\) is a beginning point: We increase the value of \(count\). If \(count = 1\), let \(L = max(t_1, l), R = min(t_2, x)\). If \(L > R\), we can go to the next point. Otherwise, \([L, R] =
[t_1, t_2] \cup [l, x]\). Then, we know for sure that it is safe to start the process anytime in between \((L, R)\). Thus, we add \(\frac{R - L}{t_2 - t_1}\) to the result.
After iterating over all points, we can do similarly to the second case with \(x = t_2\). Then, we will have our final result.
L: Suspension Bridges
We have $$a + s = a \cdot \cosh\bigg(\frac{d}{2a}\bigg) \implies s = a \cdot \cosh\bigg(\frac{d}{2a}\bigg) - a$$
Noticing that the function \(s(a)\) is decreasing, we can binary search for the right value of \(a\). With such value, we can easily calculate the required result.
M: Umbral Decoding
Suppose we have a safe point \((x, y, b)\) and a point \((p, q)\) that lies inside the umbra of the given safe point. Then $$|x - p|^3 + |y - q|^3 \leq b \implies |x - p|^3 \leq b \implies |x - p| \
leq \sqrt[3]{b}$$ Similarly, we have \(|y - q| \leq \sqrt[3]{b}\). We also know that \(b \leq 10^9 \implies \sqrt[3]{b} \leq 10^3\). Thus, for any safe point, the points that are inside its umbra
also lies in a relatively small square around it. Thus, for each safe point, we can iterate through all points lying in its umbra and deduct them from our result. To mark the duplicating points, we
can use data structures such as \(\textit{std::set}\) in C++. | {"url":"https://colt-jensen.github.io/2017-10-11-icpc-tryout-2-solution","timestamp":"2024-11-10T08:40:01Z","content_type":"text/html","content_length":"29279","record_id":"<urn:uuid:0f86e06d-05c7-4604-991c-9b3c31c0f350>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00556.warc.gz"} |
Algebra Seminar: Michael Bjorklund -- Quasi-morphisms and approximate lattices
SMS scnews item created by Alex Sherman at Fri 23 Feb 2024 1017
Type: Seminar
Distribution: World
Expiry: 24 May 2024
Calendar1: 1 Mar 2024 1200-1300
CalLoc1: Carslaw 175
CalTitle1: Quasi-morphisms and approximate lattices
Auth: alexs@desktop-h8gjltm.staff.wireless.sydney.edu.au (ashe8718) in SMS-SAML
Algebra Seminar: Michael Bjorklund -- Quasi-morphisms and approximate lattices
Michael Bjorklund (Chalmers University) is speaking in the Algebra Seminar this week.
We will go out for lunch after the talk.
When: Friday 1 March, 12-1pm
Where: Carslaw 175
Title: Quasi-morphisms and approximate lattices
Abstract: An approximate lattice is a uniformly discrete approximate subgroup Lambda of
a locally compact group G for which there is a finite volume Borel set B in G such that
B*Lambda = G (* is multiplication). To every such approximate lattice, one can
associate a dynamical system of G, which, in the case when Lambda is a lattice coincides
with the canonical G-action on the quotient space G/Lambda. In this talk we discuss how
one can construct approximate lattices from (cohomologically non-trivial)
quasi-morphisms, and show that the corresponding (compact) hulls do not admit any
invariant probability measures, and always project to a non-trivial Furstenberg
boundary. No prior knowledge of approximate lattices or quasi-morphisms will be
assumed. Based on joint work with Tobias Hartnick (Karlsruhe). | {"url":"https://www.maths.usyd.edu.au/s/scnitm/alexs-AlgebraSeminar-MichaelBjo?Clean=1","timestamp":"2024-11-09T11:15:00Z","content_type":"text/html","content_length":"2356","record_id":"<urn:uuid:49a97e50-2ca6-4e57-90c7-ae01ab592532>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00751.warc.gz"} |
Fourier Transform and Its Application to PDEs - Mastering Partial Differential Equations
The Fourier transform is a powerful mathematical tool that extends the concept of Fourier series to non-periodic functions. It plays a crucial role in solving partial differential equations (PDEs)
and has wide-ranging applications in various fields of science and engineering. In this lesson, we'll explore the Fourier transform and its application to PDEs, focusing on how it can simplify
complex problems and provide elegant solutions.
The Fourier Transform
The Fourier transform is an integral transform that converts a function in the time or space domain into a function in the frequency domain. It is defined for a function $f(x)$ as:
$F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$
Here, $F(k)$ is the Fourier transform of $f(x)$, and $k$ represents the frequency variable.
The inverse Fourier transform is given by:
$f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(k) e^{ikx} dk$
These transforms form a pair, allowing us to move between the time/space and frequency domains.
Properties of the Fourier Transform
Understanding the properties of the Fourier transform is essential for its effective application to PDEs. Some key properties include:
1. Linearity: For constants $a$ and $b$,
$\mathcal{F}\{af(x) + bg(x)\} = aF(k) + bG(k)$
2. Scaling: For a constant $a eq 0$,
$\mathcal{F}\{f(ax)\} = \frac{1}{|a|}F(\frac{k}{a})$
3. Shifting: For a constant $a$,
$\mathcal{F}\{f(x-a)\} = e^{-ika}F(k)$
4. Differentiation:
$\mathcal{F}\{\frac{d^n f}{dx^n}\} = (ik)^n F(k)$
5. Convolution: For functions $f(x)$ and $g(x)$,
$\mathcal{F}\{f(x) * g(x)\} = F(k)G(k)$
These properties make the Fourier transform particularly useful in solving PDEs, as we'll see in the following sections.
Applying Fourier Transform to PDEs
The Fourier transform can be applied to PDEs to convert them from the spatial domain to the frequency domain. This transformation often simplifies the equation, making it easier to solve. Let's walk
through the general process:
1. Apply the Fourier transform to both sides of the PDE.
2. Use the properties of the Fourier transform to simplify the transformed equation.
3. Solve the resulting equation in the frequency domain.
4. Apply the inverse Fourier transform to obtain the solution in the spatial domain.
Let's illustrate this process with an example.
Example: Heat Equation
Consider the one-dimensional heat equation:
$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
with initial condition $u(x,0) = f(x)$ and boundary conditions $u(\pm \infty, t) = 0$.
Step 1: Apply the Fourier transform with respect to x:
$\mathcal{F}\{\frac{\partial u}{\partial t}\} = \alpha \mathcal{F}\{\frac{\partial^2 u}{\partial x^2}\}$
Step 2: Use the properties of the Fourier transform:
$\frac{\partial \hat{u}}{\partial t} = -\alpha k^2 \hat{u}$
where $\hat{u}(k,t)$ is the Fourier transform of $u(x,t)$.
Step 3: Solve the resulting ordinary differential equation:
$\hat{u}(k,t) = \hat{f}(k)e^{-\alpha k^2 t}$
where $\hat{f}(k)$ is the Fourier transform of the initial condition $f(x)$.
Step 4: Apply the inverse Fourier transform:
$u(x,t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(k)e^{-\alpha k^2 t}e^{ikx} dk$
This solution represents the temperature distribution at any point $x$ and time $t$.
Advantages of Using Fourier Transform in PDEs
The Fourier transform offers several advantages when applied to PDEs:
1. Simplification: It can transform complex differential equations into simpler algebraic equations.
2. Boundary conditions: It naturally handles infinite domain problems.
3. Linearity: It preserves the linearity of equations, making superposition applicable.
4. Efficiency: For certain types of PDEs, it provides a more efficient solution method than other techniques.
Limitations and Considerations
While the Fourier transform is a powerful tool, it's important to be aware of its limitations:
1. Domain restrictions: It's most suitable for problems on infinite or periodic domains.
2. Function requirements: The function must be absolutely integrable for the transform to exist.
3. Computational complexity: For numerical solutions, Fast Fourier Transform (FFT) algorithms are often necessary for efficiency.
Applications in Various Fields
The Fourier transform's application to PDEs extends far beyond mathematical theory. It finds practical use in numerous scientific and engineering disciplines:
• Signal Processing: Analyzing and filtering signals in communication systems.
• Image Processing: Enhancing, compressing, and analyzing digital images.
• Quantum Mechanics: Solving the Schrödinger equation for particle systems.
• Fluid Dynamics: Analyzing flow patterns and turbulence.
• Acoustics: Studying sound wave propagation and noise reduction.
• Geophysics: Analyzing seismic data for oil exploration.
Advanced Topics
For those interested in delving deeper, several advanced topics relate to the Fourier transform and its application to PDEs:
1. Multidimensional Fourier Transforms: Extending the concept to higher dimensions for solving PDEs in multiple spatial variables.
2. Discrete Fourier Transform (DFT): A discrete analog of the continuous Fourier transform, crucial for numerical computations.
3. Fast Fourier Transform (FFT): Efficient algorithms for computing the DFT, essential for practical applications.
4. Fourier Analysis on Bounded Domains: Techniques for applying Fourier methods to problems with finite boundaries.
5. Generalized Fourier Series: Extensions of Fourier series to non-trigonometric orthogonal function systems.
The Fourier transform is an indispensable tool in the study and application of partial differential equations. Its ability to simplify complex problems, handle infinite domains, and provide efficient
solutions makes it a cornerstone of modern mathematical physics and engineering. By transforming PDEs from the spatial to the frequency domain, we can often uncover elegant solutions and gain deeper
insights into the underlying physical phenomena. As you continue your journey in mastering PDEs, the Fourier transform will undoubtedly prove to be one of your most valuable mathematical allies. | {"url":"https://app.studyraid.com/en/read/2438/49282/fourier-transform-and-its-application-to-pdes","timestamp":"2024-11-10T08:12:25Z","content_type":"text/html","content_length":"205288","record_id":"<urn:uuid:8d93c585-d707-4e03-8d8d-7c4a65634aac>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00263.warc.gz"} |
Mr. Spock is Not Logical (book draft excerpt)
As I mentioned, I’ll be posting drafts of various sections of my book here on the blog. This is a rough draft of the introduction to a chapter on logic. I would be extremely greatful for comments,
critiques, and corrections.
I’m a big science fiction fan. In fact, my whole family is pretty
much a gaggle of sci-fi geeks. When I was growing up, every
Saturday at 6pm was Star Trek time, when a local channel show
re-runs of the original series. When Saturday came around, we
always made sure we were home by 6, and we’d all gather in front of
the TV to watch Trek. But there’s one one thing about Star Trek for
which I’ll never forgive Gene Roddenberry or Star Trek:
“Logic”. As in, Mr. Spock saying “But that would
not be logical.”.
The reason that this bugs me so much is because it’s taught a
huge number of people that “logical” means the same
thing as “reasonable”. Almost every time I hear anyone
say that something is logical, they don’t mean that it’s logical –
in fact, they mean something almost exactly opposite – that it
seems correct based on intuition and common sense.
If you’re being strict about the definition, then saying that
something is logical by itself is an almost meaningless
statement. Because what it means for some statement to be
logical is really that that statement is inferable
from a set of axioms in some formal reasoning system. If you don’t
know what formal system, and you don’t know what axioms, then the
statement that something is logical is absolutely meaningless. And
even if you do know what system and what axioms you’re talking
about, the things that people often call “logical” are
not things that are actually inferable from the axioms.
Logic, in the sense that we generally talk about it, isn’t
really one thing. Logic is a name for the general family of formal
proof systems with inference rules. There are many logics, and a
statement that is a valid inference (meaning that it is logical)
in one system may not be valid in another. To give you a very
simple example, think about a statement like “The house on the corner
is red”. Most people would say that it’s logical
that that statement is either true or false: after all, either the house
is red, or the house isn’t red. In fact, most
people would agree that the statement “Either the house is red, or it
isn’t red” must be true.
In the most common logic, called predicate
logic, that’s absolutely correct. The original
statement is either true or false; the statement with an “or” in
it must be true. But in another common logic, called
intuitionistic logic, that’s not
true. In intuitionistic logic, there are three possible truth
values: something can be true (which means that there is a proof
that it is true); something can be false (which means that there
is a proof that it is false); and something can be unknown so far
(which means that there’s no proof either way).
In addition to having different ways of defining what’s true
or provable, different logics can describe different things. Our
good old familiar predicate logic is awful at describing things
involving time – there’s really no good particularly good way in
predicate logic to say “I’m always hungry at 6pm”. But
there are other logics, called temporal
logics which are designed specifically for making
statements about time. We’ll look at temporal logics later. For
now, we’ll stick with simple familiar logics.
So What is Logic?
A logic is a formal symbolic system, which consists of:
1. A set of atoms, which are the objects
that the logic can reason about.
2. A set of rules describing how you can form statements
in the logic (the syntax of the logic).
3. A system of inference rules
for mechanically discovering
new true statements using known true statements.
4. A model which describes how the atoms and predicates
in the logic map onto a real, consistent set of objects and properties.
The key part of that definition is the
mechanical nature of inference. What logic does is
provide a completely mechanical system for determining the truth
or falsehood of a statement given a set of known truths. In
logic, you don’t need to know what something means in order to
determine if it’s true!. As long as the logic has a
valid model, you don’t need to know what the model is to
be able to do valid reasoning in that logic.
The easiest way to get a sense of how that can possibly work
is to use an example. We’ll start with one simple logic, and show
how it can be used in a mechanical fashion to deduce true statements
– without knowing what those statements mean. For now, we won’t even
really define the logic formally, but instead just rely on intuition.
Most arguments that we hear day to day are based informally on a logic
called predicate logic; to be more specific, they’re
mostly first order predicate logic.
In predicate logic, we’ve got a collection of objects which we
can reason about, which we usually call
atoms. To say anything about objects, we use
predicates. Predicates are statements that assert some property
about on object, or some relationship between objects. For
example, if I had a pet dog named Joe, we could make statements
about him like Dog("Joe"),
which would say “Joe is a dog.”. Or we could form statements about
specific relationships: Likes("Joe",
"Rex") is a logical way of saying “Joe
likes Rex”.
We can also form general statements. For example, if Joe likes
all other dogs, we can say that in logic: (∀x)
Dog(x) ⇒ Likes("Joe", x). The
upside down “A” stands for “for all”; the statement
says “For all x, if x is a dog, then Joe likes x.”
Inference Rules
Where things get interesting is the inference rules. Inference
rules describe how to perform reasoning in the logic – which is another
way of saying that they describe how the logic can allow you to figure
out what’s true or false, based on reasoning starting from an initial
set of given facts.
Inference rules are usually written as sequents,
which we’ll get to in another section; for now, we’ll
stick with informal descriptions.
The simplest inference rules allow you to just manipulate
simple statements. For example, if you know A ∧
B is true, then you know A is
Another group of inference rules combine similar
statements to derive new facts. For example, the most famous rule
of logic is called modus ponens: if you know
that the statement P ⇒ Q is true,
and you also know that P is true, then you
can infer that Q must be true.
More interesting rules allow you to do things like work from
the general to the specific: if you know that ∀
x: P(x), and “A” is an atom, then you can
infer P("A").
Yet other rules allow you to transform statements. For example,
if you know that ∃x : P(x), then
you can infer that ¬∀x: ¬P(x).
With the rules we’ve looked at so far, we can build an example of
what I meant by totally mechanical inference.
Let’s suppose we have a bunch of atoms, “a”,
“b”, “c”, …, and two predicates, P
and Q.
We know a few simple facts:
• P("a", "b")
• P("b", "c")
• ∀x, y, z: P(x,y) ∧ P(y,z)
⇒ Q(x,z)
What can we infer using this? Using a general-to-specific
inference, we can say P("a",
"b") ∧ P("a", "b")
⇒ Q("a", "c")
Then, we can combine P("a",
"b") and P("b",
"c") to infer
P("a", "b") ∧
P("b", "c"). (Remember, we’re
being totally mechanical, so if we want to use the implication, we
need to exactly match its left-hand side, so we need to do an
inference to get the “and” statement.)
Finally, we can now use modus ponens to infer
Q("a","c"). We
have no idea what the atoms a, b, and c are; we have
no idea what the predicates P and Q mean. But we’ve been able to
infer true statements.
So what do the statements mean? That depends on the model. For a given set of symbolic statements, you can use more than one model – so long as each model is valid, the meanings of the inferences
will be valid in all models assigned to the statements. (We’ll talk more about models in section …) In this case, we could use several different models; I’ll show two examples:
1. “a” could be 1, “b” could be 2,
and “c” could be 3, with P(x,y) meaning “x is
1 plus y”, and Q(x,y) meaning “x is 2 plus y”.
Then we would have used the fact that 2 is 1+1 and 3 is 1+2 to infer
that 3 is 2+1.,/p>
2. “a” could be my father, Irving;
“b” could be me, and “c” could be my
son Aaron, with P(x,y) meaning “x is the father of
y” and Q(x,y) meaning “x is the grandfather of
y”. Then we would have used the fact that Irving is my
father, and I am Aaron’s father to infer that Irving is
Aaron’s grandfather.
0 thoughts on “Mr. Spock is Not Logical (book draft excerpt)”
1. The Science Pundit
Looks good.
I would say add a quick explanation of the symbols for “not” and “there exists” just as you did for the symbol for “for all”. To be consistent, you should either explain every symbol you use, or
assume that your readers already know them all (IMO).
2. Skemono
I agree with SciencePundit about defining the symbols. You also use the AND and IF symbol without really saying what they are, I don’t think.
Also, this statement:
What can we infer using this? Using a general-to-specific inference, we can say P(“a”, “b”) ∧ P(“a”, “b”) ⇒ Q(“a”, “c”)
Don’t you mean P(“a”, “b”) ∧ P(“b”, “c”) ⇒ Q(“a”, “c”)?
3. Martin Allen
New reader, but I like the blog a lot. Hope this isn’t overkill, but I had a bunch of feedback to give on this post. Some of it’s nit-pickery, but I did notice a couple of typos in there that you
will want to correct. Sorry if it runs long; my old advisors were terribly brutal about the details when I presented logical work, and it has stuck hard with me.
(In the following, “PX” refers to paragraph X, and “SY” to sentence Y.
P1, S2: “local channel show” -> “local channel showed”
Throughout: following punctuation should almost always appear inside quotes, as, e.g., P1, penultimate sentence: “Logic.” (instead of “Logic”.) This is especially true at the end of sentences,
where you should never have two punctuation marks, as in the last sentence of P1 (just end it with “logical.”) This doesn’t hold when you’re using something that is structural to the sentence,
like a semi-colon between parts, or when you are doing something like asking a question about a quote, like
Did he say “I will do it”?
A small nit-pick that may not help at all in a general book: in P4, you talk about logic in terms of axioms, but of course we can also present systems in non-axiomatic form (i.e.,
natural-deduction presentations). While these are equivalent to axiomatic formulations, importantly, we don’t necessarily need axioms, we just need a formal system that takes us from premisses to
conclusions. I’m not sure if there’s a nice way to make that distinction, or if it’s even worth it, especially in an introduction, however. I do note that in your list of the 4 components of a
logic, in the “So What is Logic?” section, you don’t include axioms there, so it might be confusing to the reader who doesn’t know much about this stuff.
In P5, you distinguish between predicate and intuitionistic logic. The point you make is correct, but the former label is a bit misleading. I would simply distinguish classical from
intuitionistic logic. The law of excluded middle holds in classical logic, whether it be propositional or predicate (or otherwise) in style; further, you can always create a predicate logic that
is intuitionistic (it just adds quantification to an intuitionist base, just as classical predicate logic is built on classical propositional logic). And you don’t actually need predicates to
make the point anyway: you can treat the statement “the house is red” as a simple propositional atom, r, and the law holds as (r or not r). Of course, in your four things a logic will have, you
include mappings for the predicates, so maybe you want to ignore this complication, too.
In P6, S3, you should have a comma after “temporal logics” and before “which”. Grammar is mutable, but a basic rule is that “which” always has a comma before it, and if you don’t want the comma,
use “that” instead.
In the description of what logic is, you talk about inference rules as taking us from known truths to more truths. However, inference rules actually do a little more than that: they take us from
some sentences to some other sentences. Now, IF the previous sentences are true, then so are the latter; but nothing guarantees or requires that the prior sentences ARE true. We can reason
logically from false premisses just as well (only we end up with conclusions that are often false as well). In fact, we can even reason from flatly inconsistent premisses: it’s just that they
lead to the “anything goes” result that everything follows!
For the example about the universal instantiation rule (forall x P(x) => P(a)), you might want to include a simple gloss in words on what that means.
In your explanation of the numerical interpretation of your example, you have the predicates a bit backward. For this to work, P(x,y) would have to mean “x is y minus 1” and Q(x,y) would have to
mean “x is y minus 2” (or reverse the order of what “a,” “b,” and “c” mean to get it to work).
4. Sam K.
To put it bluntly: what’s the point of this introduction? You certainly make an interesting point about the difference between “logical” and “reasonable”, but your example is quite long relative
to your description of what logic is, and it seems to run out of steam by the end (with very little wrap up).
Furthermore, your example introduces a bunch of ideas along the way without any visual cues, so it’s very difficult to scan the introduction without getting lost in the symbols. Also, I agree
with #1 that you should define your symbols or assume readers know them already. Given the content, I imagine that readers don’t know all the symbols, which brings me to my second point…
I think there is absolutely no reason to introduce any sort of formal syntax in your introduction. I can’t imagine any reader that is learning about logic would have an easy time understanding
modus ponens without any discussion of material implication. Keeping the discussion informal (and this means NO symbols) forces descriptions that avoid ambiguity created by using syntax that an
uninformed reader would not understand. To use another example, you introduce predicates in terms of the common functional syntax, e.g. P( a, b, c ), but you never describe what this syntax means
and why we would want to use it.
Also, what I feel this introduction lacks most is a motivation for why logic is useful/important. Great, so we have a mechanical way of deriving truths, but why do I care? If anything, this seems
rather inefficient at first glance. Given how much you think the mechanical nature of logic is important, I’m surprised there is no mention whatsoever about the history of logic such as Hilbert’s
program or examples such as axioms in Euclid’s geometry.
You don’t really talk about proofs (which is certainly a key motivator) and it seems like some discussion of consistency is in order (otherwise logic is rather pointless).
The first few paragraphs are great, but I think the readers you seem to be targeting would lose interest rather quickly. I am personally very interested in logic and what generally makes me
retain interest in your blog posts is the intriguing topic, the clarification you provide, and pointing out common [interesting] mistakes people make. Your first point about reasonable vs.
logical does exactly that, but unfortunately the intro goes somewhat downhill from there.
Hope that helps– I think your writing style has potential for good material about logic, but you sort of betray your own style.
5. Dave M
What Martin said in his #4. Also, when you say:
“A set of atoms, which are the objects that the logic can reason about.”
someone who doesn’t already know what you’re talking about (which is your audience, right?) would probably think you were talking about (what we would call) the domain (which is actually part of
the model). You might want to say more about how you are using the term “object” here, and in what sense “logic can reason about” such things, which is not at all clear from what you say here.
6. Dave M
Oh, about Mr. Spock. Of course you’re right that that use of “logical” is unfortunate. But it’s worse than that, as I recall, given that he often wasn’t even being reasonable. Like when he would
say something like “Logically, the probability of that is a mere 7.2 percent” about something which was in no way quantifiable to that degree of precision. I guess he never read your post on
significant figures.
7. Lowk
a) Words very often has multiple valid meanings: ‘logical’ does not just mean complying to a formal system of logic. Taking a quick look at the OED, it has been in use since the 17th Century to
mean ‘Characterized by reason; rational, reasonable’. The term logical has always had both a formal and a colloquial meaning, and neither is particularly older than the other. Saying that Spock
is using it wrong is sort of like complaining that the Civil Rights Movement used the word ‘integration’, when we all know took very few sums under curves.
b) To get extremely nerdy: When Spock talks about ‘logic’, he is talking about a particular religious concept that is deeply ingrained in Vulcan culture. Vulcans are prone to fits of emotion, and
are far stronger than humans, and as a result their religion emphasizes the suppression of emotion and the importance of good justification for actions. The idea of ‘logic’ corresponds to
something like a purity of thought, reason untainted by emotion. When Spock talks about ‘Logic’, imagine it like a Muslim talking about ‘Justice’, which is them attempting to translate ‘Adalah’.
8. Sam K.
@ Lowk: You certainly make a valid point about the dictionary definition, but it seems there should be some distinction between the two, since MarkCC’s original statement seems intuitively right
in this context. However, the dictionary definition for reasonable (and the example given) create a bit of confusion.
1. agreeable to reason or sound judgment; logical: a reasonable choice for chairman.
Clearly, reasonable and logical are synonyms, but now that I look at what MarkCC wrote again, he thinks that reasonable means “correct based on intuition and common sense” (which is the same as
this definition)
I’m going to go further than Lowk and say that the issue is that the formal/mathematical meaning is very much subsumed by the other meaning. The logic that is implied by the primary meaning of
“logical” is whatever logic people tend to reason with (i.e. intuition, which is almost certainly not even a logic).
…which kind of brings up a bigger point: I don’t think there are any words in the English language that commonly mean (to a non-logician) what you here mean by “logical.” That’s why I (and
clearly others too) fell into the same trap you fell into– you are writing about formal logic and use the word logical, so we assume that the two are related, but they’re not in their common
usage, which is why common usage is so dangerous when trying to be “logical.”
This, in retrospect, makes my previous comment about motivation for logic that much more valid. One of the reasons you want to be able to do things mechanically is because that is more verifiable
and you are less likely to make mistakes that go unnoticed. By breaking down a complex idea into simple steps, it’s much easier to tell if it’s correct or not (i.e. derivable from your axioms).
9. Mark C. Chu-Carroll
Thanks for all the comments!
I am working to try to restructure it and reduce the dryness of it. This is just a first draft, and my hope was that I could get some hints from the comments – and that’s definitely working!
Some of the problems that you all have complained about are a result of seeing this out of context. In the actual book, the previous chapter goes through an informal mathematical construction –
showing how you build the objects that you’re working with using sets, and how you describe their behavior using logic. The motivation for why you should care about logic is presented primarily
in that chapter – but your comments are all absolutely correct that I should reiterate it here, albeit in a shorter form.
Anyway.. Thanks for all the comments, and keep ’em coming!
10. MariaD
Great book project. I am going to follow it from now on – it looks interesting!
I think the confusion you describe here goes back to the ancient times. For example, the lists of “logical fallacies,” frequently used in our days to fuel flame wars, typically include a bizarre
mixture of formal logic and human reasonableness. The ancients can be forgiven for thinking there can be only One Right Logic, or One Right Geometry for that matter – the Euclid’s one, of course.
You may want mention the ancient roots of equating one’s set of math axioms and conclusions with reasonableness, justice, and other human virtues.
11. Mark Paris
I think it’s unfortunate that you criticize the use of “logical” to mean “reasonable”, which, as pointed out by others, is perfectly correct and has been done for many years. It seems that the
use of logic as you define it would require that your introduction actually be true.
12. William Wallace
Your quibbling about the common use of the word “logical” to connote that which is reasonable reminds me of the adage from Lowry’s The Giver, and I am paraphrasing despite the quotation marks,
~”We must have precision in language.”
Constructive criticism: Unless you get some philosophy professors teaching symbolic logic to adopt it as required reading, I don’t anticipate your sales will be strong.
13. Mark C. Chu-Carroll
Re #12:
When I started this blog, I thought that it probably wouldn’t last more than a week or two, and that I’d be lucky if I got a couple of dozen people to read it. Now, I consistently get 3000
readers per day. I never would have dreamed that there was an audience that size for my math ramblings!
So I don’t even attempt to predict whether the book will sell well or not. The publisher clearly thinks it’s got a chance. And I’m working very hard to try to produce something that people will
enjoy reading.
This is a very rough draft of one of the hardest parts of the book to write. The whole reason that I posted it was to get exactly the kinds of negative feedback that many of the commenters have
provided. I knew that this section wasn’t flowing the way I want it to, and I’m still not sure of how to fix it.
As far as quibbling about the meaning of logical… I’ve got my own strong opinion about how the word should be used. You don’t have to agree with me. But I *still* hate the way that Star Trek uses
“logical” to mean things that are anything but logical. And I think that using that as a starting point is an engaging way of starting the chapter. If I can make the rest of the chapter read as
entertainingly as the first couple of paragraphs, I’ll be incredibly happy, even if people think I’m being overly strict about definitions.
14. Bob Munck
following punctuation should almost always appear inside quotes
I always find that rule difficult to follow, I suspect as the result of years of typing string literals. If I’m quoting something, and the original didn’t have a period at the end, it just seems
wrong to put one in, even if it comes at the end of my sentence.
15. Jonathan Vos Post
Star Trek uses “logical” to mean approximately what Sherlock Homes meant by “logical.” It is a character- and plot-driven device. Not that I’m saying that “Bones” = Dr. Watson or that Uhuru =
Irene, or that Starfleet Command is on Baker Street.
My suggestion being to NOT apply Law of the Excluded Middle to “logical”, but instead to consider its casual and imprecise use in entertainment genres including Science Fiction and Mystery. And
that SOME kind of “logical” is what divides Science Fiction from Sci-Fi.
16. Spoonwood
I agree with the comments about Spock and logic made in the comment section. It’s really off-putting to say that the FICTIONAL show Star Trek used the word “logical” incorrectly, when it really
just did so for theatrical reasons, and the word does mean “reasonable” in everyday usage. Words have more than one meaning. In my opinion, they SHOULD have more than one meaning. To say that
they shouldn’t have used the word “logical” on Star Trek in that way, basically indicates a condescending, holier-than-thou attitude to people who do use it that way.
[Logic is a name for the general family of formal proof systems with inference rules. ]
Even in the technical sense of the word, that doesn’t quite work. Fuzzy logic falls under the technical sense of the word “logic” and very rarely deals with formal proof systems (Zadeh even wrote
a paper where explained that in fuzzy maths, the notion of proof comes as a secondary notion… unlike crisp maths). Does a fuzzy logic expert system formally prove things? No. Do most papers on
fuzzy logic concern formal proofs? Well, they may prove something, but they don’t generally aim at developing a structure for proofs.
[To give you a very simple example, think about a statement like “The house on the corner is red”. Most people would say that it’s logical that that statement is either true or false: after all,
either the house is red, or the house isn’t red.]
I don’t think most people would actually agree here. Foregoing the physics definition of red as a certain wavelength of light, most people, I believe, would view the color red as a perception.
There exist different shades of red. So, it doesn’t make sense to say that a red shirt is as red as red hair or a darker red house. Plus, one side of the house could be red and another side
white. So, is the house red? I think a better example of a statement working out as either true or false would come as something like “The batter hit a homerun.” Or “She went shoe shopping.” Or
something where “shades” of it at least come as much harder to think of than examples like “The brick is red.”
17. Eric
I think that this is a logical way of explaining 😉
I mean reasonable!
For the critics: I think opening about Spock is just a way to draw readers into it. Is there any other logical (reasonable, I mean) reason to delve into a book on logic? And in math, distinction
in words is key…
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Dictionary of Arguments
> Quine, W.V.O.
> Impredicativeness
W.V.O. Quine on Impredicativeness - Dictionary of Arguments XIII 93
Impredicativeness/Quine: Previously it was said that you had specified a class without knowing anything about it if you could name the containment condition.
Russell's Antinomy: showed that there had to be exceptions.
Problem: was to specify a class by a containment condition by directly or indirectly referring to a set of classes that contained the class in question.
Russell's Antinomy: here the problematic containment condition was the non-self elementary. Example x is not an element of x.
Paradox: arises from letting the x of the containment condition, among other things, be just the class defined by this containment condition.
Def impredicative/Poincaré/Russell: is just this condition of containment for a class that exists in the class itself. This must be forbidden to avoid paradoxes.
Circular Error Principle/QuineVsRussell: but that was too harsh a term:
Specification/Class/Sets/Existence/Quine: specifying a class does not mean creating it!
XIII 94
Specification/Circle/Introduce/QuineVsRussell: by specifying something it is not wrong to refer to a domain to which it has always belonged to. For example, statistical statements about a typical
inhabitant by statements about the total population that contains this inhabitant.
Introduction/Definition/linguistic/Quine: all we need is to equate an unfamiliar expression with an expression that is formed entirely with familiar expressions.
Russell's Antinomy/Quine: is still perfectly fine as long as the class R is defined by its containment condition: "class of all objects x, so that x is not an element of x".
Paradox/Solution/Russell/Quine: a solution is to distort familiar expressions so that they are no longer familiar in order to avoid a paradox. This was Russell's solution. Finally, "x is an element
of x" ("contains itself") to be banished from the language.
Solution/Zermelo/Quine: better: leave the language as it is, but
New: for classes it should apply that not every containment condition defines a class. For example the class "R" remains well defined, but "Pegasus" has no object. I.e. there is no (well-defined)
class like R.
Circle/George Homans/Quine: true circularity: For example, a final club is one into which you can only be elected if you have not been elected to other final clubs.
Quine: if this is the definition of an unfamiliar expression, then especially the definition of the last occurrence of "final club".
Circle/Circularity/Quine: N.B.: yet it is understandable!
Impredicativeness/impredicative/Russell/Quine: the real merit was to make it clear that not every containment condition determines a class.
Formal: we need a hierarchical notation. Similar to the hierarchy of truth predicates we needed in the liar paradox.
XIII 95
Variables: contain indexes: x^0,y^0: about individuals, x^1,y^2 etc. about classes, but classes of this level must not be defined by variables of this level. For example, for the definition of
higher-level classes x^2, y^2 only variables of the type x^0 and x^1 may be used.
Type Theory/Russell/Quine/N.B.: classes of different levels can be of the same type!
Classes/Sets/Existence/Quine: this fits the metaphor that classes do not exist before they are determined. I.e. they are not among the values of the variables needed to specify them. ((s) And
therefore the thing is not circular).
Problem/QuineVsRussell: this is all much stricter than the need to avoid paradoxes and it is so strict that it prevents other useful constructions.
For example, to specify the union of several classes of the same level, e.g. level 1
Problem: if we write "Fx^1" to express that x^1 is one of the many classes in question, then the
Containment condition: for a set in this union: something is element of it iff it is an element of a class x1, so Fx^1.
Problem: this uses a variable of level 1, i.e. the union of classes of a level cannot be counted on to belong to that level.
Continuity hypothesis: for its proof this means difficulties.
Impredicativeness/Continuum/Russell/Quine: consequently he dropped the impredicativeness in the work on the first volume of Principia Mathematica^(1). But it remains interesting in the context of
constructivism. It is interesting to distinguish what we can and cannot achieve with this limitation.
XIII 96
Predicative set theory/QuineVsRussell/Quine: is not only free of paradoxes, but also of unspecifiable classes and higher indeterminacy, which is the blessing and curse of impredicative theory. (See
"infinite numbers", "classes versus sets").
Predicative set theory/Quine: is constructive set theory today.
Predicative Set Theory/Quine: is strictly speaking exactly as described above, but today it does not matter which conditions of containment one chooses to specify a class.
1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals
indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/
Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page
numbers refer to this edition.
Quine I
W.V.O. Quine
Word and Object, Cambridge/MA 1960
German Edition:
Wort und Gegenstand Stuttgart 1980
Quine II
W.V.O. Quine
Theories and Things, Cambridge/MA 1986
German Edition:
Theorien und Dinge Frankfurt 1985
Quine III
W.V.O. Quine
Methods of Logic, 4th edition Cambridge/MA 1982
German Edition:
Grundzüge der Logik Frankfurt 1978
Quine V
W.V.O. Quine
The Roots of Reference, La Salle/Illinois 1974
German Edition:
Die Wurzeln der Referenz Frankfurt 1989
Quine VI
W.V.O. Quine
Pursuit of Truth, Cambridge/MA 1992
German Edition:
Unterwegs zur Wahrheit Paderborn 1995
Quine VII
W.V.O. Quine
From a logical point of view Cambridge, Mass. 1953
Quine VII (a)
W. V. A. Quine
On what there is
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (b)
W. V. A. Quine
Two dogmas of empiricism
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (c)
W. V. A. Quine
The problem of meaning in linguistics
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (d)
W. V. A. Quine
Identity, ostension and hypostasis
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (e)
W. V. A. Quine
New foundations for mathematical logic
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (f)
W. V. A. Quine
Logic and the reification of universals
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (g)
W. V. A. Quine
Notes on the theory of reference
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (h)
W. V. A. Quine
Reference and modality
From a Logical Point of View, , Cambridge, MA 1953
Quine VII (i)
W. V. A. Quine
Meaning and existential inference
From a Logical Point of View, , Cambridge, MA 1953
Quine VIII
W.V.O. Quine
Designation and Existence, in: The Journal of Philosophy 36 (1939)
German Edition:
Bezeichnung und Referenz
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982
Quine IX
W.V.O. Quine
Set Theory and its Logic, Cambridge/MA 1963
German Edition:
Mengenlehre und ihre Logik Wiesbaden 1967
Quine X
W.V.O. Quine
The Philosophy of Logic, Cambridge/MA 1970, 1986
German Edition:
Philosophie der Logik Bamberg 2005
Quine XII
W.V.O. Quine
Ontological Relativity and Other Essays, New York 1969
German Edition:
Ontologische Relativität Frankfurt 2003
Quine XIII
Willard Van Orman Quine
Quiddities Cambridge/London 1987 | {"url":"https://philosophy-science-humanities-controversies.com/listview-details.php?id=232279&a=$a&first_name=W.V.O.&author=Quine&concept=Impredicativeness","timestamp":"2024-11-14T00:41:39Z","content_type":"text/html","content_length":"23467","record_id":"<urn:uuid:1b0f97c6-9dcb-40bf-9be7-3c0cd706027d>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00057.warc.gz"} |
Electric current is the flow of electric charge. This charge is often carried by electrons moving through a wire. The electrons flow from the negative terminal to the positive one, while the current
flows from the positive terminal to the negative terminal. The symbol for current is I and current is measured in ampere (A). The current is equal to the charge flowing through a surface divided by
the time.
Ohm's law
Ohm's law
Ohm's law describes the relation between Voltage, Current and Resistance. The law states that the voltage is equal to the current multiplied by the resistance. Ohm's law is named after the German
physicist Georg Ohm.
The current can be calculated by dividing the voltage by the resistance:
The current is equal to the charge divided by the time:
R is the symbol for resistance and is measured in ohm (Ω).
V is the symbol for voltage and is measured in volt (V).
I is the symbol for current and is measured in ampere (A).
Q is the symbol for charge and is measured in coulomb (C).
t is the symbol for time and is measured in seconds (s). | {"url":"https://www.basictables.com/electronics/current","timestamp":"2024-11-07T11:52:28Z","content_type":"text/html","content_length":"9516","record_id":"<urn:uuid:78d173c9-4764-42e6-a86c-4194caa23b33>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00453.warc.gz"} |
Wooly Worm Race [Day 2] - Make Math Moments Academy
THE WOOLY WORM RACE [DAY 2]
Represent, compare, order, and add fractions involving metric unit conversions.
Math Talk #1
Visual Math Talk Prompt #1
Visual Math Talk Prompt #2
Visual Math Talk Prompt #3
Visual Math Talk Prompt #4
Visual Math Talk Prompt #5
Visual Math Talk Prompt #6
Visual Math Talk Prompt #7
Visual Math Talk Prompt #8
Math Talk #2
Purposeful Practice
Resources & Downloads
Educator Discussion Area
Access each lesson from this unit using the navigation links below
Students will explore representing, comparing and ordering fractions. They will also have an opportunity to determine appropriate units of measure and the relationship between larger and smaller
The purpose of the Day 2 activities is to reinforce key concepts from Day 1. Students will engage in a string of related problems through a math talk and will have an opportunity to complete
independent purposeful practice. The math talk and purposeful practice serve to develop a deeper understanding of the following big ideas.
• Fractions can be represented in a variety of ways;
• How you partition the whole determines the fractional unit (i.e.: partitioning a whole into 6 parts will result in 6 sixth parts);
• As you partition a whole into more parts, the smaller the size of each part;
• As you partition a whole into more parts, the larger the denominator;
• In order to compare two or more fractional quantities, the whole must be the same;
• The numerator indicates the number of parts relative to the number of parts in the whole indicated by the denominator; and,
• Fractional amounts exist between whole numbers.
• Units of measure have different sizes and attributes; and,
• As the size of the unit of measure decreases, the number of iterations of that unit required to measure a quantity increases (and vice versa).
In today’s visual math talk, students will be asked to represent different fractional amounts on a number line from 0 to 2. The goal of this visual math talk is to build student flexibility and
fluency with fractional quantities while also practicing partitioning a linear model.
The prompt for each problem in this visual math talk is:
Where would you place each fractional amount on the number line?
However, consider leveraging yesterday’s Wooly Worm Race to frame our 0 to 2 number line to represent the 2 metre track that the caterpillars would race along.
Introduce each quantity, one at a time, for students to place on a number line.
Login/Join to view three (3) additional Visual Math Talk Prompts as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
In today’s math talk, we intentionally switch between words and standard notation. We want students to begin connecting the two symbolic representations of a fraction and to continue developing their
ability to spatially partition a linear model.
Encourage students to draw a new number line from 0 to 2 for each prompt in order to promote additional practice of partitioning as well as to ensure they do not get confused with partitions that
were helpful for one prompt, but not so helpful for another.
If necessary, consider providing students with strips of paper to practice paper folding as a tool for this math talk.
Although you could lead this math talk by orally sharing the context and representing student thinking on a chalkboard/whiteboard, consider leveraging the following visual math talk prompt videos to
ensure accessibility for all students. We would recommend that you still model student thinking to ensure student voice is honoured prior to sharing the visual silent solution shared in the visual
prompt video.
Begin playing the first video to share the first visual math talk prompt and be ready to pause the video in order to give students time to think.
The prompts are simple and shared without context, however since this is taken from Day 2 of the Wooly Worm Race problem based math unit, you might consider leveraging the context from the previous
day involving caterpillars racing along a 2 metre straight track. In that case, you could verbally share the context using a script such as:
The Wooly Worms are racing in their next competition!
Woolzy managed to race 1 metre.
Where on the track from 0 metres to 2 metres did Woolzy finish?
In this first prompt, students will likely have no problem partitioning the 0 to 2 number line into two parts and indicating a value of 1 at the halfway point.
Playing the second video until prompted to pause and consider leveraging the Wooly Worm context describing where the second caterpillar finished the race at the 1 half metre mark.
Encouraging students to draw another number line rather than simply building on the first number line will give them the opportunity to (likely) partition the number line in half and then partition
each half in half.
Depending on the readiness of the learners, you might consider mentioning that we have essentially taken one half of one half which can be written symbolically as ½ x ½ to emerge the idea of
multiplying fractions.
Play the third video until prompted to pause and consider leveraging the same context verbalizing something similar to:
Show where the third caterpillar finished the race at the ¼ metre mark on the 2 metre long track.
Again, it is nice to encourage students to draw another number line for this prompt, however they can ultimately decide how they choose to model their thinking.
In this case, students are halving the 0-2 number line, then halving those resulting parts and then finally, halving those parts again to reveal ¼ on the 0-2 number line.
There are a number of different ways we could represent this symbolically. For example, we could say that we took half of a half of a half of the 0-2 number line:
½ of ½ of ½ of 2
= ½ x ½ x ½ x 2
= ½ x ½ x (½ x 2)
= ½ x ½ x (1)
= ½ x (½ x 1)
= ½ x (½)
= ½ x ½
= ¼
Is this symbolic representation a bit of overkill? Absolutely. However, if this helps students better connect multiplying fractions symbolically to the act of partitioning our number line repeatedly,
then it could be worth the while.
Playing the following video until prompted to pause and consider leveraging the same context verbalizing something similar to:
Show where the third caterpillar finished the race at the ⅛ metre mark on the 2 metre long track.
Again, we are going to be encouraging students to use a new 0 to 2 number line and begin the process of partitioning to determine where 1 eighth would be placed on the number line.
Consider asking students to elaborate on what they have ultimately done to determine 1 eighth of a metre.
Play the video until prompted to pause and consider leveraging the same context verbalizing something similar to:
Show where the third caterpillar finished the race at the 6/8 metre mark on the 2 metre long track.
With this particular prompt, you might not require students to re-partition a brand new number line, but rather to articulate where 6 eighths would be placed on the number line.
Consider asking students to elaborate on what they have ultimately done to determine how they landed on 6 eighths of a metre.
For example, some students might argue that they found 6 groups of half of a half of a half of a half of 2 metres which could be written symbolically as:
6 x (½ of ½ of ½ of ½ of 2)
Are there any other ways that we could represent what was done to partition the line and determine where 6 eighths of a metre would be?
Login/Join to view three (3) additional Visual Math Talk Prompts as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
Login/Join to view three (3) additional Visual Math Talk Prompts as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
Login/Join to view three (3) additional Visual Math Talk Prompts as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
Login/Join to view this additional set of Math Talk Prompts as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
While Students Are Practicing…
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Questions: Working With Fractions & Metric Units
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Explore Our 60+ Problem Based Units
This Make Math Moments Lesson was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, number talks and extensions to elicit and emerge strategies and
mathematical models.
Dig into our other units of study and view by concept continuum, grade or topic! | {"url":"https://learn.makemathmoments.com/task/wooly-worm-race-day2/","timestamp":"2024-11-03T16:41:45Z","content_type":"text/html","content_length":"314509","record_id":"<urn:uuid:5f859e84-3239-4e27-95b0-f76eccf6c634>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00425.warc.gz"} |
Newton's method for finding roots
Newton's method for finding roots¶
This is an iterative method invented by Isaac Newton around 1664. However, this method is also sometimes called the Raphson method, since Raphson invented the same algorithm a few years after Newton,
but his article was published much earlier.
The task is as follows. Given the following equation:
$$f(x) = 0$$
We want to solve the equation. More precisely, we want to find one of its roots (it is assumed that the root exists). It is assumed that $f(x)$ is continuous and differentiable on an interval $[a, b]
The input parameters of the algorithm consist of not only the function $f(x)$ but also the initial approximation - some $x_0$, with which the algorithm starts.
Suppose we have already calculated $x_i$, calculate $x_{i+1}$ as follows. Draw the tangent to the graph of the function $f(x)$ at the point $x = x_i$, and find the point of intersection of this
tangent with the $x$-axis. $x_{i+1}$ is set equal to the $x$-coordinate of the point found, and we repeat the whole process from the beginning.
It is not difficult to obtain the following formula,
$$ x_{i+1} = x_i - \frac{f(x_i)}{f^\prime(x_i)} $$
First, we calculate the slope $f'(x)$, derivative of $f(x)$, and then determine the equation of the tangent which is,
$$ y - f(x_i) = f'(x_i)(x - x_i) $$
The tangent intersects with the x-axis at cordinate, $y = 0$ and $x = x_{i+1}$,
$$ - f(x_i) = f'(x_i)(x_{i+1} - x_i) $$
Now, solving the equation we get the value of $x_{i+1}$.
It is intuitively clear that if the function $f(x)$ is "good" (smooth), and $x_i$ is close enough to the root, then $x_{i+1}$ will be even closer to the desired root.
The rate of convergence is quadratic, which, conditionally speaking, means that the number of exact digits in the approximate value $x_i$ doubles with each iteration.
Application for calculating the square root¶
Let's use the calculation of square root as an example of Newton's method.
If we substitute $f(x) = x^2 - n$, then after simplifying the expression, we get:
$$ x_{i+1} = \frac{x_i + \frac{n}{x_i}}{2} $$
The first typical variant of the problem is when a rational number $n$ is given, and its root must be calculated with some accuracy eps:
double sqrt_newton(double n) {
const double eps = 1E-15;
double x = 1;
for (;;) {
double nx = (x + n / x) / 2;
if (abs(x - nx) < eps)
x = nx;
return x;
Another common variant of the problem is when we need to calculate the integer root (for the given $n$ find the largest $x$ such that $x^2 \le n$). Here it is necessary to slightly change the
termination condition of the algorithm, since it may happen that $x$ will start to "jump" near the answer. Therefore, we add a condition that if the value $x$ has decreased in the previous step, and
it tries to increase at the current step, then the algorithm must be stopped.
int isqrt_newton(int n) {
int x = 1;
bool decreased = false;
for (;;) {
int nx = (x + n / x) >> 1;
if (x == nx || nx > x && decreased)
decreased = nx < x;
x = nx;
return x;
Finally, we are given the third variant - for the case of bignum arithmetic. Since the number $n$ can be large enough, it makes sense to pay attention to the initial approximation. Obviously, the
closer it is to the root, the faster the result will be achieved. It is simple enough and effective to take the initial approximation as the number $2^{\textrm{bits}/2}$, where $\textrm{bits}$ is the
number of bits in the number $n$. Here is the Java code that demonstrates this variant:
public static BigInteger isqrtNewton(BigInteger n) {
BigInteger a = BigInteger.ONE.shiftLeft(n.bitLength() / 2);
boolean p_dec = false;
for (;;) {
BigInteger b = n.divide(a).add(a).shiftRight(1);
if (a.compareTo(b) == 0 || a.compareTo(b) < 0 && p_dec)
p_dec = a.compareTo(b) > 0;
a = b;
return a;
For example, this code is executed in $60$ milliseconds for $n = 10^{1000}$, and if we remove the improved selection of the initial approximation (just starting with $1$), then it will be executed in
about $120$ milliseconds.
Practice Problems¶ | {"url":"https://gh.cp-algorithms.com/main/num_methods/roots_newton.html","timestamp":"2024-11-05T19:03:11Z","content_type":"text/html","content_length":"134875","record_id":"<urn:uuid:f6e31b3a-8e4a-4021-bb72-ffcf259f160b>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00045.warc.gz"} |
On Symmetric bi-Cayley Graphs of Prime Valency on Nonabelian Simple Groups
On Symmetric bi-Cayley Graphs of Prime Valency on Nonabelian Simple Groups
Let $\Gamma$ be a bipartite graph, and let $\mathrm{Aut}\Gamma$ be the full automorphism group of the graph $\Gamma$. A subgroup $G\leqslant \mathrm{Aut}\Gamma$ is said to be bi-regular on $\Gamma$
if $G$ preserves the bipartition and acts regularly on both parts of $\Gamma$, while the graph $\Gamma$ is called a bi-Cayley graph of $G$ in this case. A subgroup $X\leqslant \mathrm{Aut} \Gamma$ is
said to be bi-quasiprimitive on $\Gamma$ if the bipartition-preserving subgroup of $X$ is a quasiprimitive group on each part of $\Gamma$.
In this paper, a characterization is given for the connected bi-Cayley graphs on nonabelian simple groups which have prime valency and admit bi-quasiprimitive groups. | {"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v31i1p66","timestamp":"2024-11-02T04:57:26Z","content_type":"text/html","content_length":"14467","record_id":"<urn:uuid:4f2bc8bf-3048-46c3-a953-3116bfab79f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00127.warc.gz"} |
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Christian Gérard - Introduction to field theory on curved spacetimes (Part 4)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
The aim of these lectures is to give an introduction to quantum field theory on curved spacetimes, from the point of view of partial differential equations and microlocal analysis. I will
concentrate on free fields and quasi-free states, and say very little on interacting fields or perturbative renormalization. I will start by describing the necessary algebraic background, namely CCR
and CAR algebras, and the notion of quasi-free states, with their basic properties and characterizations. I will then introduce the notion of globally hyperbolic spacetimes, and its importance for
classical field theory (advanced and retarded fundamental solutions, unique solvability of the Cauchy problem). Using these results I will explain the algebraic quantization of the two main examples
of quantum fields ona manifold, namely the Klein-Gordon (bosonic) and Dirac (fermionic) fields.In the second part of the lectures I will discuss the important notion of Hadamardstates , which are
substitutes in curved spacetimes for the vacuum state in Minkowskispacetime. I will explain its original motivation, related to the definition of therenormalized stress-energy tensor in a quantum
field theory. I will then describethe modern characterization of Hadamard states, by the wavefront set of their twopointfunctions, and prove the famous Radzikowski theorem , using the
Duistermaat-Hörmander notion of distinguished parametrices . If time allows, I will also describe the quantization of gauge fields, using as example the Maxwell field.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Christian Gérard - Introduction to field theory on curved spacetimes (Part 3)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
The aim of these lectures is to give an introduction to quantum field theory on curved spacetimes, from the point of view of partial differential equations and microlocal analysis. I will
concentrate on free fields and quasi-free states, and say very little on interacting fields or perturbative renormalization. I will start by describing the necessary algebraic background, namely CCR
and CAR algebras, and the notion of quasi-free states, with their basic properties and characterizations. I will then introduce the notion of globally hyperbolic spacetimes, and its importance for
classical field theory (advanced and retarded fundamental solutions, unique solvability of the Cauchy problem). Using these results I will explain the algebraic quantization of the two main examples
of quantum fields ona manifold, namely the Klein-Gordon (bosonic) and Dirac (fermionic) fields.In the second part of the lectures I will discuss the important notion of Hadamardstates , which are
substitutes in curved spacetimes for the vacuum state in Minkowskispacetime. I will explain its original motivation, related to the definition of therenormalized stress-energy tensor in a quantum
field theory. I will then describethe modern characterization of Hadamard states, by the wavefront set of their twopointfunctions, and prove the famous Radzikowski theorem , using the
Duistermaat-Hörmander notion of distinguished parametrices . If time allows, I will also describe the quantization of gauge fields, using as example the Maxwell field.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Christian Gérard - Introduction to field theory on curved spacetimes (Part 2)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
The aim of these lectures is to give an introduction to quantum field theory on curved spacetimes, from the point of view of partial differential equations and microlocal analysis. I will
concentrate on free fields and quasi-free states, and say very little on interacting fields or perturbative renormalization. I will start by describing the necessary algebraic background, namely CCR
and CAR algebras, and the notion of quasi-free states, with their basic properties and characterizations. I will then introduce the notion of globally hyperbolic spacetimes, and its importance for
classical field theory (advanced and retarded fundamental solutions, unique solvability of the Cauchy problem). Using these results I will explain the algebraic quantization of the two main examples
of quantum fields ona manifold, namely the Klein-Gordon (bosonic) and Dirac (fermionic) fields.In the second part of the lectures I will discuss the important notion of Hadamardstates , which are
substitutes in curved spacetimes for the vacuum state in Minkowskispacetime. I will explain its original motivation, related to the definition of therenormalized stress-energy tensor in a quantum
field theory. I will then describethe modern characterization of Hadamard states, by the wavefront set of their twopointfunctions, and prove the famous Radzikowski theorem , using the
Duistermaat-Hörmander notion of distinguished parametrices . If time allows, I will also describe the quantization of gauge fields, using as example the Maxwell field.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Christian Gérard - Construction of Hadamard states for Klein‐Gordon fields
/ Fanny Bastien / Canal-u.fr
Voir le résumé
we will review a new construction of Hadamard states for quantized Klein-‐Gordon fields on curved spacetimes, relying on pseudo differential calculus on a Cauchy surface. We also present some work
in progress where Hadamard states are constructed from traces of Klein-‐Gordon fields on a characteristic cone. (Joint work with Michal Wrochna).
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, calculus of variation, asymptotic analysis
Accéder à la ressource
Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 4)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s
theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by
branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 3)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s
theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by
branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Camillo De Lellis - Center manifolds and regularity of area-minimizing currents (Part 1)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s
theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by
branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Bruno Lévy - A numerical algorithm for L2 semi-discrete optimal transport in 3D
/ Fanny Bastien / Canal-u.fr
Voir le résumé
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Andrew Lorent - The Aviles-Giga functional: past and present
/ Fanny Bastien / Canal-u.fr
Voir le résumé
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Andras Vasy - Microlocal analysis and wave propagation (Part 4)
/ Fanny Bastien / Canal-u.fr
Voir le résumé
In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary.
In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and
scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally
hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the
wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and
generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The
lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
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The Putnam Mathematical Competition's Unsolved Problem - insideAI News
As a data scientist with my roots in the theoretical foundations of the field, I’m always looking for ways to challenge myself and pick up a new mathematical apparatus that could help me in my
project work. While reading a past copy of The Tech, MIT’s oldest and largest newspaper, I learned that MIT took first place in the most recent chapter of the William Lowell Putnam Mathematical
Competition, the oldest and most prestigious competition of its kind. Over 4,000 students took the Putnam test, but NOT one managed to solve the last question, Problem B6!
I provide you with the dreaded Problem B6 below. Surely if you can solve this, then you’d be in great shape to come up with a new machine learning algorithm that can win the next Kaggle competition
for some serious prize money. Good luck!
Sign up for the free insideAI News newsletter.
Go in the middle! Assume the base case is a board with one square. Going in the middle is the only available move and Alice wins. Now assume we know the solution for all k < n. Is Alice goes in
the middle she then breaks the board into two equally sized boards of (1/2)(2k+1 – 1) = k squares. Since k is less than n we already know the solution. Alice will play on one "half" only, until
it is solved and then play on the other board until it is solved. If Bob picks up the center square, Alice can replay it, thus creating two board of size k-1. While Alice is playing optimally on
one board bob can be playing optimally on the other board. Since Alice goes first, Bob will solve one half exactly one move before Alice solves the second board, and thus Alice will be able to
take the last legal move, insuring the win.
I think it suffice to say that if Bob removes one stone and the result is that it adds a stone both on the left and right side of the board, then Alice replays the stone that Bob just
removed. Otherwise whatever Bob plays, Alice plays it on the other side of the board. This makes sure that the situation is always symmetrical.
My solution would go like this:
1) at most nb of stone increase by one each time one player plays
2) nb of stone cannot decrease unless the board is full of stones
3) no one can lose unless the board has been full of stone at least once; this is because until the board has been full of stones it is possible to increase the nb of stone by adding one and this
is a new position because of 2)
4) as soon as there are n – 1 stones, the player who puts the last stone wins; because there are n – 1 different positions left with n – 1 stones and n – 1 is even
Now a good strategy for Alice is to first play in the middle, and after that use the following rules:
– if Bob removes one stone and the result is that it adds a stone both on the left and right side of the board, then Alice replays the stone that Bob just removed,
– otherwise whatever Bob plays, Alice plays the same thing on the other side of the board (symmetrically regarding the middle of the board)
This ensures that:
5) the situation is always symmetrical after Alice played
6) when Bob plays once and then Alice plays once, if the nb of stone on the board increased since before Bob just played, then it increased by an even amount
As after Alice first played in the middle, there was just 1 stone, then because of 6), when it’s Bob turn to play, there are an odd number of stones on the board.
Because of 4) this means that Alice wins! | {"url":"http://insideainews.com/2014/07/20/putnam-mathematical-competitions-unsolved-problem/","timestamp":"2024-11-07T17:07:11Z","content_type":"application/xhtml+xml","content_length":"111102","record_id":"<urn:uuid:2d8e5786-ddfe-4cd8-8ab5-c556a2fcf31f>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00264.warc.gz"} |
RLC Circuit - Phasor, Sine Wave, and Animation Analysis | Reversepcb
An RLC circuit is an electrical circuit that contains a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. These components work together to store and transfer energy,
leading to oscillations at a specific resonant frequency. RLC circuits are commonly used in various applications, including filters, oscillators, and tuning circuits.
RLC circuits can be mainly divided into the following types:
In this circuit, the resistor (R), inductor (L) and capacitor (C) components are connected in sequence on the same path. The current passes through the resistor, inductor and capacitor in sequence,
causing the current to change accordingly in each component.
The series RLC circuit has the characteristics of frequency selectivity, presenting different impedance and response characteristics at different frequencies, so it is widely used in filters and
tuned circuits.
The parallel RLC circuit is to connect the resistor, inductor and capacitor in parallel paths respectively. The current is divided in the branches, and the voltage passing through each component is
the same.
It circuit has impedance characteristics for current at a specific frequency, which can be used to selectively pass or block signals of certain frequencies.
How to Measure Voltage in the RLC Circuit?
As below diagrams show:
When U is a sinusoidal voltage, this circuit has three potential points D, E, and F. These three waveforms can be observed using an oscilloscope or measured with an AC voltmeter for UD, UE, UL, UR,
and UC. The dynamic graph on the right reflects the changes in these five voltages over time in a certain scenario. In the dynamic graph, UF is equivalent to UL, and ideally, there should be another
voltage UDF, but I have chosen not to include voltages with two-letter subscripts for simplicity.
Calculation Formula for Voltage
The phasor method is a brilliant solution that facilitates problem-solving but may cause us to miss out on the fascinating dynamic processes depicted in the graph above. With inductive reactance XL,
resistance R, and capacitive reactance XC, the phasor method for solving this circuit is outlined as follows:
The AC voltmeter measures UR ≠ UE – UL. If L is an ideal inductor, these three will form a right triangle. Similarly, UC, UD, and UE will also form a triangle, though not necessarily a right one.
This does not imply that Kirchhoff’s Voltage Law (KVL) is violated. If the instantaneous value u at a certain moment is measured, it will be found that uR = uE – uL and uC = uD – uE. Additionally,
although current is not measured, the changes in UR are in phase with the changes in current.
Sine Signal Action and Resonance State
When a sine signal is applied, the inductive reactance XL = ωL and the capacitive reactance XC = (ωC)^-1. The state where these two are equal is known as resonance, allowing us to calculate a
resonant angular frequency ω0.
In the next section, we will set the parameters to create phasor diagrams, static sine wave diagrams, and animations of the instantaneous value changes of voltage at 0.5ω0, ω0, and 2ω0.
Voltage under Different Resonant Angular Frequencies
• U = 6.00mV
• R = 2.00Ω (effective current less than 3.00mA, generally safe for components)
• C = 125μF
• L = 0.500mH
Calculated ω0 = 4000rad/s (corresponding to a frequency of approximately 637Hz, ignoring electromagnetic radiation effects for modeling convenience. Phasor diagrams and sine wave diagrams use RMS
values; for peak values, multiply by 1.414)
Dynamic Voltage under 0.5ω0
After calculations, we can draw the phasor diagram and static sine wave diagram at the initial moment:
Calculation formula of voltage at 0.5ω0 resonant angular frequency
Sine diagram of voltage at 0.5ω0 resonant angular frequency
• Maximum current ≈ 1.57mA, safe for components.
• Phasor diagram forms a right trapezoid with UDF not considered.
• At 0.5ω0, XL < XC → smaller UL, larger UC. Maximum uC exceeds power supply voltage, potentially damaging the capacitor.
• uC lags uD by ≈30°; charging begins during uD’s rising phase, ending when uR = 0.
• Sine wave diagram shows uR = uE – uL and uC = uD – uE at every moment.
Dynamic Voltage under 1ω0
After calculations, we can draw the phasor diagram and static sine wave diagram at the initial moment:
Calculation formula for voltage at 1ω0 resonant angular frequency
Sine diagram of voltage at 1ω0 resonant angular frequency
• Maximum current ≈ 4.24mA, safe for components.
• Phasor diagram forms a square at resonance (XL = XC).
• Maximum uE exceeds power supply voltage, but component impact unclear.
• Coincidence: XL = XC = R → uL and uC cancel each other in sine wave diagram, but current changes affect amplitudes.
• Impedance minimized, current maximized at resonance due to “maximum current” effect.
• uC lags uD by exactly 90°; charging begins when uD peaks and ends when uR = uD = 0.
• Sine wave diagram segments represent half the time of Case 1.
• Maximum UD = UR, UL = UC due to parameters, but all four voltages equal is a coincidence.
Dynamic Voltage under 2ω0
After calculations, we can draw the phasor diagram and static sine wave diagram at the initial moment:
Calculation formula for voltage at 2ω0 resonant angular frequency
Sine diagram of voltage at 2ω0 resonant angular frequency
• Maximum current ≈ 1.57mA, safe for components.
• Phasor diagram forms a right trapezoid.
• At 2ω0, XL > XC → larger UL, smaller UC. Maximum uL and uE exceed power supply voltage, raising inductor damage concerns.
• uC lags uD by ≈150°; charging begins when uD drops to ≈half its maximum, ending when uR = uD = 0.
• Sine wave diagram segments represent half the time of Case 2.
• Equations uR = uE – uL and uC = uD – uE hold true at every moment. | {"url":"https://reversepcb.com/rlc-circuit-phasor-sine-wave-and-animation-analysis/","timestamp":"2024-11-14T07:54:17Z","content_type":"text/html","content_length":"313973","record_id":"<urn:uuid:d81489b7-ee2a-4d5b-ae2d-74698e2fee32>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00379.warc.gz"} |
data rate mean?
What does data rate mean?
(1) The speed at which data is transferred within the computer or between a peripheral device and the computer, measured in bytes per second. (2) The speed at which audio and video files are encoded
(compressed), measured in bits per second (see bit rate). (3) The transmission speed of a network.
What is channel capacity and data rate?
What is Channel capacity? (bandwidth) Channel capacity is a maximum information rate that a channel can transmit. It is measured in bits per second (bps). Channel capacity is a rough value as
measuring takes into account only the whole amount of data transferred, but leaves out of account communication quality.
How is channel rate calculated?
Hence, the channel capacity is directly proportional to the power of the signal, as SNR = (Power of signal) / (power of noise). So for example a signal-to-noise ratio of 1000 is commonly expressed
as: 10 * log10(1000) = 30 dB.
What is the maximum data rate of a channel?
the channel can never transmit much more than 13 Mbps, no matter how many or how few signal levels are used and no matter how often or how infrequently sam- ples are taken. In practice, ADSL is
specified up to 12 Mbps, though users often see lower rates.
What does higher data rate mean?
Higher variations include kilobits per second (Kbps) and megabits per second (Mbps). Basically, when you have a higher bitrate, it means that a larger amount of ‘video bits’ are being uploaded within
a second. Conversely, lower bitrate results in lower video quality, smaller size, and faster export.
What is the difference between data rate and bit rate?
Bit rate of a video signal is the no of bits per frame per second. Also, data rate is the no of bits per second.
What is the difference between channel and bandwidth?
Bandwidth is defined as: a range of frequencies (or channels) within a given band, in particular that used for transmitting a signal. Channel capacity is a much-used metric for the maximum amount of
traffic or signal that can move over a particular infrastructure channel (or frequency).
What do you mean by channel capacity?
The channel capacity, C, is defined to be the maximum rate at which information can be transmitted through a channel. The fundamental theorem of information theory says that at any rate below channel
capacity, an error control code can be designed whose probability of error is arbitrarily small.
How is channel capacity calculated?
According to channel capacity equation, C = B log(1 + S/N), C-capacity, B-bandwidth of channel, S-signal power, N-noise power, when B -> infinity (read B ‘tends to’ infinity), capacity saturates to
How is channel bandwidth calculated?
The required bandwidth is related to bit rate and the modulation order M. It is so that the double sided bandwidth w = symbol rate= bit rate rb/ divided by the number of bit per symbol n. The number
of bits per symbol is = log 2M with M is the M is the QAM modulation order.
What happens if my bitrate is too high?
While a higher bitrate can result in higher quality video, it may reduce the number of potential viewers as some computers or Internet connections cannot handle higher bitrate video. Moreover, a
higher bitrate does not necessarily result in better image quality.
What is bitrate and why is it important?
Bitrate plays a significant role when you put the quality of your stream first. A higher bitrate results in a larger file and better look for the video. Although you can still experience some
challenges in the form of limited bandwidth and buffering, there are fortunately several ways to resolve the issue.
How do you calculate the data rate of a channel?
Two theoretical formulas were developed to calculate the data rate: one by Nyquist for a noiseless channel, another by Shannon for a noisy channel. In the above equation, bandwidth is the bandwidth
of the channel, L is the number of signal levels used to represent data, and BitRate is the bit rate in bits per second.
What is data rate?
Data rate refers to the speed of data transfer through a channel. It is generally computed in bits per second (bps). Higher data rates are expressed as Kbps (“Kilo” bits per second, i.e.1000 bps),
Mbps (“Mega” bits per second, i.e.1000 Kbps), Gbps (“Giga” bits per second, i.e. 1000 Mbps) and Tbps (“Tera” bits per second, i.e. 1000 Gbps).
What is meant by bandwidth of a channel?
BANDWIDTH OF A CHANNEL. A channel is the medium through which the input signal passes. In terms of analog signal, bandwidth of the channel is the range of frequencies that the channel can carry. In
terms of digital signal, bandwidth of the channel is the maximum bit rate supported by the channel. i.e.
How to calculate the maximum bit rate of a channel?
The theoretical formula for the maximum bit rate is: maximum bit rate = 2 × Bandwidth × log2V Here, maximum bit rate is calculated in bps Bandwidth is the bandwidth of the channel | {"url":"https://penelopethemovie.com/what-does-data-rate-mean/","timestamp":"2024-11-04T18:32:02Z","content_type":"text/html","content_length":"86625","record_id":"<urn:uuid:85e97c04-3504-4bca-b778-3d130c8d5f19>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00208.warc.gz"} |
Cow Live Weight Vs Meat Weight Calculator - Online Calculators
To calculate the meat weight, multiply the live weight of the cow (LW) by the dressing percentage (r). This will give you an estimate of the total meat yield after processing.
The Cow Live Weight vs Meat Weight Calculator is an essential tool for farmers, butchers, and meat processors to estimate the amount of meat obtained from a cow based on its live weight. The
calculator provides estimates for carcass weight, hanging weight, and the meat yield after processing.
Typically, a cow’s meat-to-live weight ratio can vary, but on average, about 40-60% of a cow’s live weight results in usable meat. This tool is ideal for calculating meat yield in various regions,
including the USA and India, helping users understand how much meat they can expect from a cow based on its live weight.
$MW = LW \times r$
Variable Meaning
MW Meat weight (in pounds or kilograms)
LW Live weight of the cow (in pounds or kilograms)
r Dressing percentage or yield factor (typically 0.50 to 0.65)
Solved Calculations :
Example 1:
Given Values:
• LW = 1200 pounds
• r = 0.60 (60% yield)
Calculation Instructions
MW = 1200 × 0.60 Multiply the live weight by the yield factor.
MW = 720 pounds The result gives the estimated meat weight.
Answer: MW = 720 pounds
Example 2:
Given Values:
• LW = 800 kilograms
• r = 0.55 (55% yield)
Calculation Instructions
MW = 800 × 0.55 Multiply the live weight by the yield factor.
MW = 440 kilograms The result gives the estimated meat weight.
Answer: MW = 440 kilograms
What is Cow Weight vs Meat Weight Calculator ?
The process of estimating the amount of meat derived from a cow starts with understanding the live weight of the animal.
This Calculator makes the task easy by using standard conversion ratios, such as the conversion from live weight to carcass weight, where typically around 60% of the live weight becomes the hanging
From the hanging weight, you can further estimate the final meat yield, which is generally around 60-70% of the hanging weight. This tool is helpful for cattle farmers who need to estimate the amount
of beef that will be produced from their livestock, whether for personal use or commercial purposes.
Moreover, this calculator can help you determine how much meat you can expect from a cow in kg or lbs, providing specific estimates based on regional data from countries like the USA and India. By
inputting the cow’s live weight, the calculator estimates the hanging weight, carcass weight, and the total amount of usable meat.
The tool can also calculate the beef yield based on different processing methods, and users can refer to the beef yield chart for more detailed projections. This calculator is indispensable for
making informed decisions about meat production and sales.
Final Words:
The Cow Live Weight vs. Meat Weight Calculator is a tool used by farmers and meat processors to figure out how much meat they can get from a live cow, based on its weight and how much meat it’ll
yield after processing. This calculation helps them plan how to manage livestock, estimate meat amounts, and make the most of their resources. | {"url":"https://areacalculators.com/cow-live-weight-vs-meat-weight-calculator/","timestamp":"2024-11-04T01:21:06Z","content_type":"text/html","content_length":"103623","record_id":"<urn:uuid:e9950e17-17b9-4c74-af3a-fdf867f651ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00520.warc.gz"} |
Buying Coke
Problem B
Buying Coke
I often buy Coca-Cola from the vending machine at work. Usually I buy several cokes at once, since my working mates also likes coke. A coke in the vending machine costs $8$ Swedish crowns, and the
machine accept crowns with the values $1$, $5$ and $10$. As soon as I press the coke button (after having inserted sufficient amount of money), I receive a coke followed by the exchange (if any). The
exchange is always given in as few coins as possible (this is uniquely determined by the coin set used). This procedure is repeated until I’ve bought all the cokes I want. Note that I can pick up the
coin exchange and use those coins when buying further cokes.
Now, what is the least number of coins I must insert, given the number of cokes I want to buy and the number of coins I have of each value? Please help me solve this problem while I create some
harder problems for you. You may assume that the machine won’t run out of coins and that I always have enough coins to buy all the cokes I want.
The first line in the input contains the number of test cases (at most $100$). Each case is then given on a line by itself. A test case consists of four integers: $C$ (the number of cokes I want to
buy), $n_1$, $n_5$, $n_{10}$ (the number of coins of value $1$, $5$ and $10$, respectively). The input limits are $1 \le C \le 150$, $0 \le n_1 \le 500$, $0 \le n_5 \le 100$ and $0 \le n_{10} \le 50$
For each test case, output a line containing a single integer: the minimum number of coins needed to insert into the vending machine.
Sample Input 1 Sample Output 1 | {"url":"https://liu.kattis.com/courses/ETE388/summer24/assignments/m6ius6/problems/coke","timestamp":"2024-11-13T14:45:37Z","content_type":"text/html","content_length":"26625","record_id":"<urn:uuid:805bbc25-0e8d-4963-97e8-f3f1cbc9b57d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00656.warc.gz"} |
The maximum diversity assortment selection problem
• In this article, we introduce the Maximum Diversity Assortment Selection Problem (MDASP), which is a generalization of the two-dimensional Knapsack Problem (2D-KP). Given a set of rectangles and
a rectangular container, the goal of 2D-KP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is
covered. MDASP is to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby,
diversity is defined as the minimum or average normalized Hamming distance of all assortment pairs. MDASP was the topic of the 11th AIMMS-MOPTA Competition in 2019. The methods described in this
article and the resulting computational results won the contest. In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper
bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2D-KP literature. | {"url":"https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/8007","timestamp":"2024-11-14T21:37:28Z","content_type":"application/xhtml+xml","content_length":"21644","record_id":"<urn:uuid:cc066b90-1ae9-462a-ac63-928241870ca9>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00850.warc.gz"} |
Wavelength Calculator
What is Wavelength?
Wavelength is a fundamental concept in physics that describes the distance between successive crests or troughs of a wave. It is a critical parameter in various fields, including optics, acoustics,
and electromagnetic theory. Understanding how to calculate wavelength is essential for analyzing wave behaviors and applications in technology and science.
Did you know? Wavelength is inversely proportional to frequency, meaning that as the frequency of a wave increases, its wavelength decreases.
In this comprehensive guide, we will delve into the principles of wavelength, explore the methods to calculate it, discuss its applications in different domains, and examine real-world examples.
Whether you’re a student, educator, or enthusiast, this article aims to provide you with a thorough understanding of wavelength and its significance in the physical world.
Understanding Wavelength
Wavelength (\(\lambda\)) is one of the key properties of waves, alongside frequency (\(f\)) and wave speed (\(v\)). It is defined as the spatial period of the wave—the distance over which the wave’s
shape repeats. Wavelength is typically measured in meters (m), but depending on the context, it can also be expressed in centimeters (cm), nanometers (nm), or other units.
Key Point: Wavelength determines the type and behavior of a wave, influencing how it interacts with the environment and matter.
Different types of waves have varying wavelengths. For example, visible light has wavelengths ranging from approximately 400 nm (violet) to 700 nm (red), while radio waves can span from millimeters
to kilometers. Understanding wavelength is crucial for applications like designing optical instruments, wireless communication systems, and even medical imaging technologies.
How to Calculate Wavelength
Calculating wavelength is straightforward once you understand the relationship between wave speed, frequency, and wavelength. The fundamental equation that relates these three properties is:
Wavelength Formula:
\[ \lambda = \frac{v}{f} \]
\(\lambda\) = Wavelength (meters) v = Wave speed (meters per second, m/s) f = Frequency (Hertz, Hz)
This equation shows that wavelength is equal to the wave speed divided by the frequency. If you know any two of these variables, you can easily calculate the third. This relationship is fundamental
in both classical and modern physics, underpinning phenomena from sound waves to electromagnetic radiation.
Additionally, for electromagnetic waves traveling in a vacuum, the wave speed (\(v\)) is the speed of light (\(c\)), which is approximately \(3 \times 10^8 \, \text{m/s}\). Therefore, the wavelength
of light can be calculated using:
\[ \lambda = \frac{c}{f} \]
Understanding this formula is essential for applications in optics, telecommunications, and even astronomy, where wavelengths determine the characteristics of light and other electromagnetic waves.
Key Equations for Calculating Wavelength
To effectively calculate wavelength, it’s important to understand the various equations and how they interrelate. Below are the key formulas and their applications.
Basic Wavelength Formula:
\[ \lambda = \frac{v}{f} \]
\(\lambda\) = Wavelength (m) v = Wave speed (m/s) f = Frequency (Hz)
This is the foundational equation for calculating wavelength. It is applicable to all types of waves, including sound waves, light waves, and water waves.
Wavelength of Light:
\[ \lambda = \frac{c}{f} \]
\(\lambda\) = Wavelength (m) c = Speed of light (\(3 \times 10^8 \, \text{m/s}\)) f = Frequency (Hz)
Specifically for electromagnetic waves in a vacuum, where the wave speed is the speed of light. This formula is crucial in optics and telecommunications.
Wavelength from Energy:
\[ \lambda = \frac{h c}{E} \]
\(\lambda\) = Wavelength (m) h = Planck’s constant (\(6.626 \times 10^{-34} \, \text{Js}\)) c = Speed of light (\(3 \times 10^8 \, \text{m/s}\)) E = Energy (Joules)
This equation relates the wavelength of a photon to its energy, derived from quantum mechanics. It’s fundamental in fields like spectroscopy and quantum physics.
De Broglie Wavelength:
\[ \lambda = \frac{h}{p} \]
\(\lambda\) = Wavelength (m) h = Planck’s constant (\(6.626 \times 10^{-34} \, \text{Js}\)) p = Momentum (kg·m/s)
The De Broglie wavelength connects a particle’s momentum to its wavelength, highlighting the wave-particle duality in quantum mechanics. It’s essential for understanding phenomena like electron
Mastering these equations allows for accurate calculations of wavelength in various scenarios, from designing optical devices to analyzing sound waves in different mediums.
Applications of Wavelength in Science and Technology
Wavelength plays a pivotal role in numerous scientific and technological applications. Understanding and calculating wavelength is essential for advancements in various fields.
Optics and Photonics
In optics, wavelength determines the color of light and is fundamental in the design of lenses, microscopes, and telescopes. Photonics, the science of generating and controlling photons, relies
heavily on precise wavelength calculations for applications like fiber optic communications and laser technology.
Furthermore, wavelength is critical in spectroscopy, where it is used to identify materials based on their spectral lines. This technique is widely used in chemistry, astronomy, and environmental
science to analyze the composition of substances and celestial objects.
Wireless Communications
In telecommunications, different wavelengths (or frequencies) are allocated for various services such as radio, television, and mobile networks. Calculating the appropriate wavelength ensures
efficient transmission and reception of signals, minimizing interference and maximizing bandwidth.
Additionally, wavelength is essential in medical technologies like MRI and ultrasound imaging. These technologies use specific wavelengths to penetrate tissues and create detailed images for
diagnostic purposes.
Medical Imaging
Medical imaging techniques, such as ultrasound, utilize sound waves with specific wavelengths to create images of the inside of the body. Accurate wavelength calculations enhance image resolution and
diagnostic capabilities.
Beyond these, wavelength is fundamental in environmental monitoring, remote sensing, and even in everyday technologies like microwaves and infrared heating.
Environmental Monitoring
Remote sensing technologies use various wavelengths to monitor environmental changes, track weather patterns, and assess natural resources. By analyzing the wavelength of reflected or emitted waves,
scientists can gather valuable data about the Earth’s surface and atmosphere.
Real-World Example: Calculating the Wavelength of Visible Light
Let’s consider a practical example of calculating the wavelength of visible light. Suppose we have a light source emitting light with a frequency of \(6 \times 10^{14} \, \text{Hz}\). We want to
determine the wavelength of this light.
Step-by-Step Calculation
Using the basic wavelength formula:
\[ \lambda = \frac{v}{f} \]
Since we’re dealing with light in a vacuum, the wave speed (\(v\)) is the speed of light (\(c\)):
\[ c = 3 \times 10^8 \, \text{m/s} \]
Plugging in the values:
\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{6 \times 10^{14} \, \text{Hz}} = 5 \times 10^{-7} \, \text{meters} = 500 \, \text{nanometers (nm)} \]
Therefore, the wavelength of the emitted light is \(500 \, \text{nm}\), which corresponds to green light in the visible spectrum.
This example illustrates how wavelength calculations are applied in identifying the properties of light, which is essential in fields like spectroscopy and optical engineering.
For more examples and practice problems on wavelength calculations, visit Khan Academy’s Waves Section.
Challenges in Calculating Wavelength
While calculating wavelength is fundamental, several challenges can arise, especially in complex systems or when dealing with various types of waves. Understanding these challenges is crucial for
accurate analysis and application.
Challenge: Determining wave speed in different mediums can be complex due to varying properties like density and elasticity.
One primary challenge is accurately determining the wave speed (\(v\)). Wave speed can vary significantly depending on the medium through which the wave is traveling. For instance, sound waves travel
at different speeds in air, water, and solids. Accurately measuring or calculating wave speed is essential for precise wavelength determination.
Another consideration is the medium’s properties affecting wave behavior. Factors such as temperature, pressure, and medium composition can influence both wave speed and frequency, thereby affecting
wavelength calculations. For electromagnetic waves, the presence of materials with different refractive indices can alter the effective wavelength.
Consideration: Environmental factors and medium properties must be accounted for to ensure accurate wavelength calculations.
Additionally, in quantum mechanics, the concept of wavelength extends to particles, where the De Broglie wavelength is used. Calculating this requires knowledge of the particle’s momentum, adding
another layer of complexity to wavelength calculations.
Measurement limitations also pose challenges. High-frequency waves, such as X-rays or gamma rays, have very short wavelengths that are difficult to measure directly. Indirect methods and advanced
technologies are often required to accurately determine these wavelengths.
Wavelength is a cornerstone concept in the study of waves, encompassing various types of waves across different mediums and applications. Understanding how to calculate wavelength and the factors
influencing it is essential for advancements in science and technology.
Whether you’re exploring the properties of light, designing wireless communication systems, or delving into quantum mechanics, mastering wavelength calculations provides a solid foundation for
innovation and problem-solving. Despite the challenges in measurement and calculation, the principles of wavelength remain integral to our understanding of the physical universe.
As technology continues to evolve, the applications of wavelength expand, driving progress in fields like medical imaging, environmental monitoring, and beyond. Embracing the complexities and
intricacies of wavelength calculations empowers scientists and engineers to develop more efficient and effective solutions to real-world problems. | {"url":"https://turn2engineering.com/calculators/wavelength-calculator","timestamp":"2024-11-13T08:19:33Z","content_type":"text/html","content_length":"233155","record_id":"<urn:uuid:745eaf95-62c2-4aab-9487-f271de40d792>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00896.warc.gz"} |
• test.mean
Mean of performance values on test sets.
• test.sd
Standard deviation of performance values on test sets.
• test.median
Median of performance values on test sets.
• test.min
Minimum of performance values on test sets.
• test.max
Maximum of performance values on test sets.
• test.sum
Sum of performance values on test sets.
• train.mean
Mean of performance values on training sets.
• train.sd
Standard deviation of performance values on training sets.
• train.median
Median of performance values on training sets.
• train.min
Minimum of performance values on training sets.
• train.max
Maximum of performance values on training sets.
• train.sum
Sum of performance values on training sets.
• b632
Aggregation for B632 bootstrap.
• b632plus
Aggregation for B632+ bootstrap.
• testgroup.mean
Performance values on test sets are grouped according to resampling method. The mean for every group is calculated, then the mean of those means. Mainly used for repeated CV.
• testgroup.sd
Similar to testgroup.mean - after the mean for every group is calculated, the standard deviation of those means is obtained. Mainly used for repeated CV.
• test.join
Performance measure on joined test sets. This is especially useful for small sample sizes where unbalanced group sizes have a significant impact on the aggregation, especially for
cross-validation test.join might make sense now. For the repeated CV, the performance is calculated on each repetition and then aggregated with the arithmetic mean. | {"url":"https://www.rdocumentation.org/packages/mlr/versions/2.19.1/topics/aggregations","timestamp":"2024-11-13T15:17:20Z","content_type":"text/html","content_length":"74212","record_id":"<urn:uuid:c21cab92-8995-4063-9dcf-fe3d8e9de78c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00640.warc.gz"} |
How Covariance and Correlation Are Related - dummies
Two of the most widely used measures of association are covariance and correlation. These measures are closely related to each other; in fact, you can think of correlation as a modified version of
Correlation is easier to interpret because its value is always between –1 and 1. For example, a correlation of 0.9 indicates a very strong relationship in which two variables nearly always move in
the same direction; a correlation of –0.1 shows a very weak relationship in which there is a slight tendency for two variables to move in opposite directions.
With covariance, there is no minimum or maximum value, so the values are more difficult to interpret. For example, a covariance of 50 may show a strong or weak relationship; this depends on the units
in which covariance is measured.
Correlation is a measure of the strength and direction of two related variables. Two variables are said to be related if they can be expressed with the following equation:
Y = mX + b
X and Y are variables; m and b are constants. For example, suppose that the relationship between two variables is:
Y = 3X + 4
In this case, 3 is the coefficient of X, which means that an increase of X by 1 causes Y to increase by 3. Equivalently, a decrease of X by 1 causes Y to decrease by 3. The 4 in this equation
indicates that Y equals 4 when X equals 0.
Note that even though correlation can be computed for any pair, this doesn't mean they are linearly related. For example, you could have a high correlation with a small slope, and a low correlation
with a large slope, as shown in the following graphs.
A graph with a low correlation (0.420) but a slope of 4.453
A graph with a high correlation (0.912) but a slope of only 1.908
Covariance and correlation show that variables can have a positive relationship, a negative relationship, or no relationship at all. With covariance and correlation, there are three cases that may
• If two variables increase or decrease at the same time, the covariance and correlation between them is positive. For example, the covariance and correlation between the stock prices of two oil
companies is positive because many of the same factors affect the stock prices in the same way.
• If two variables move in opposite directions, the covariance and correlation between them is negative. For example, the covariance and correlation between interest rates and new home sales is
negative because rising interest rates increase the cost of purchasing a new home, which in turn reduces new home sales. The opposite occurs with falling interest rates.
• If two variables are unrelated to each other, the covariance and correlation between them is zero (or very close to zero). For example, the covariance and correlation between gold prices and new
car sales is zero because the two have nothing to do with each other.
About This Article
This article is from the book:
This article can be found in the category: | {"url":"https://www.dummies.com/article/business-careers-money/business/accounting/calculation-analysis/how-covariance-and-correlation-are-related-145293/","timestamp":"2024-11-10T12:43:24Z","content_type":"text/html","content_length":"81394","record_id":"<urn:uuid:2a5f254d-6607-4929-b5bf-a092b2c31162>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00152.warc.gz"} |
PhD Course: Isometric Immersions and Harmonic Maps
Feb 172024
Isometric Immersions and Harmonic Maps
Prof. Cezar Oniciuc
Universitatea “Alexandru Ioan Cuza” Iași
1. Generalities on Riemannian Geometry
2. Isometric immersions (submanifolds) – generalities
3. Special isometric immersions: umbilicals, minimal, CMC
4. Operators on vector bundle
5. Harmonic maps between Riemannian manifolds: first and second variation; fundamental examples
May 21st, 16.00-18.00 Aula II
May 22nd, 16.00-18.00 Aula II
May 23rf, 16.00-18.00 Aula II
May 28th, 16.00-18.00 Aula II
May 29th, 16.00-18.00 Aula II
May 30th, 16.00-18.00 Aula II
June 4th, 16.00-18.00 Aula II
June 6th, 16.00-18.00 Aula II
Feb 102024
MAIN PhD Seminars 2024
Date Speaker(s)
March, 6th Marco Casula
March, 13rd Luca Zedda
March, 20th Filippo Maria Cassanello
March, 27th Alessandro Iannella
April, 3rd Elisa Crabu
April, 17th Jacopo Mereu
April, 24th Alessandra Perniciano
May, 8th Antonio Sanna
May, 15th Giuseppe Demuru
May, 22nd Massimiliano Fadda
Federico Meloni
May, 29th Andrea Cabriolu
Giorgia Nieddu
All the seminars start at 5 PM.
Marco Casula: Bochner-Euclidean volume
We will start with examples of calculating the volume of objects in three-dimensional space and then extend the definition to any manifold. Therefore we will introduce a new and different volume on
complex manifolds, with particular attention to cases of finite and infinite volumes. The work is based on the article by Loi-Placini.
Luca Zedda: Self-Supervised Learning: The Dark Matter of Artificial Intelligence
In this seminar, we shall delve into the concept of Self-Supervised Learning, an intriguing and rapidly expanding branch of artificial intelligence. Fundamental concepts of this innovative approach
will be introduced, demonstrating how it is possible to connect the process of human cognitive development to that of artificial within the context of deep learning. Through the analysis of
self-supervised models, it will be explained how AI can autonomously learn, addressing the challenges posed by the lack of explicit annotations in data and the application of these technologies to
real-world scenarios.
Filippo Maria Cassanello: An alternative approach to the Hölder continuity of solution of the fractional p-laplacian
In this seminar we will define the non-local operator “fractional p-laplacian” by also talking about his biological interpretation for describing the movement of population in hostile habitat. Then
we will give a different proof of the Hölder continuity of weak solution of this operator by extending the approach that DiBenedetto developped for the p-laplacian. This work is based on the paper
“An alternative approach to the Hölder continuity of solution of some elliptic equations” of Duzgun, Marcellini, Vespri and is in collaboration with Prof. Antonio Iannizzotto.
Alessandro Iannella: The Transitional Space: Generative Artificial Intelligence as an Opportunity for Professional Growth for Teachers
This seminar aims to illustrate the benefits, risks, and challenges of using Generative Artificial Intelligence in teaching, also drawing on concepts and metaphors from psychology and sociology.
Particular attention will be paid to the different phases of the teaching process, from design to evaluation.
Elisa Crabu: Mathematical tools for Computer Vision
Photometric Stereo is a Computer Vision tecnique that leads to reconstructing the digital shape of an object from a set of images, obtained by lighting the object with a light source placed at
different positions around it. The method, by estimating the surface normals, computes an approximation of the surface. In this talk we will describe the main steps of the solution method, presenting
the mathematical tools that underlie it, including the singular value decomposition, least square problems and the numerical solution of partial differential equations.
Jacopo Mereu: AI-supported End User Development in VR
End-User Development (EUD) is a research field that aims to design and develop software or hardware technology (digital artifacts) such that their consumers (end users) should be able to adapt such
artifacts according to their needs. End users are not a static category; the unique context of the application determines their identity, skills, and experience. In the context of this seminar, the
end users are proficient programmers in Unity but lack expertise in constructing Extended Reality environments. The research aims to assist these end users in using a XR Development toolkit, the
Mixed Reality Toolkit (MRTK), whose latest version has recently been released. Large Language Models (LLMs) have been chosen as the method to support the end users. These models are trained with
extensive documents, allowing them to acquire knowledge across various domains. However, their knowledge has a temporal limitation, as the models lack information about events or developments
occurring after a certain date. Consequently, an LLM may lack information about the MRTK3 library. This seminar thus presents a practical case of enhancing the performance of an LLM in a domain where
it possesses limited or no prior knowledge.
Alessandra Perniciano: Radiomics: the issue of high dimensional data
Radiomics, a branch of Computer Vision, involves the extraction and analysis of quantitative features from medical imaging modalities such as MRI, PET, and CT scans. The central idea behind Radiomics
is that imaging features specific to various diseases may offer valuable insights into predicting prognosis and treatment outcomes across different types of pathologies. Notably, these
characteristics remain elusive through traditional visual inspection methods employed in current radiologic practice, yet they provide insights into the underlying biological processes. However, the
quantitative extraction of features leads to a situation of high dimensionality where not all the extracted features are necessarily relevant. During this seminar, I will present the challenges
related to high dimensionality in Radiomics, providing an analysis of the current state of knowledge and discussing some future development directions.
Antonio Sanna: Harmonic and Biharmonic maps between Riemannian Manifolds
The object of this seminar is the definition of harmonic maps and biharmonic maps between Riemannian manifolds. During the exposition we will introduce the energy functional for smooth maps between
two Riemannian manifolds, and, deriving the corresponding Euler-Lagrange equation — in order to find its critical maps, we will define a certain vector field, called tension field, which is
identically zero when the map is harmonic, i.e. critical. We will extend the notion of harmonic maps to that of biharmonic maps which are the critical points of the bienergy functional. We will see
that harmonic maps are trivially biharmonic. Thus a crucial problem is to understand when the converse is also true, that is: under what conditions biharmonic maps are harmonic. Beyond this
theoretical exploration, we will give some examples of biharmonic maps which are not harmonic. In particular, we will consider the geometrically interesting case of biharmonic isometric immersions.
Giuseppe Demuru: An Introduction to Causal Inference
Causal inference involves the study of cause-and-effect relationships among variables, based on experimental or observational data. Understanding these relationships in depth is essential for making
informed decisions and solving complex problems. The well-known statement “Correlation does not imply causation” underscores that simple associations do not necessarily imply causality. Causal
inference utilizes methods such as Potential Outcomes (PO) and Directed Acyclic Graphs (DAGs) to identify and quantify the true causal relationships among variables.
Massimiliano Fadda: Translating HTML in proprietary JSON
Growens is an integrated industrial group that creates technologies for content creation, predictive marketing, and mobile messaging, aimed at organizations wishing to communicate effectively with
their customers. The seminar will introduce the reasons that led the company to develop this project. An overview of the technologies and methodologies identified for its resolution will then be
provided, introducing the architecture of the system that allows the conversion of generic HTML pages into proprietary Json.
Federico Meloni: Mesh generation in the volumetric domain
Representing an object in the virtual world is becoming a frequent practice in fields like industries, entertainment, medicine. To digitally represent an object, the space is discretized due to the
inability of a computer to represent space continuously. Therefore, we utilize a series of primitives such as points, segments, polygons, and eventually polyhedra to represent an object, called in
this context a mesh. A three-dimensional mesh can be superficial if only the exterior of the object is represented, or volumetric if it includes a description of the volume within. The latter unlocks
the possibility of performing a variety of operations such as physical simulations, fluid dynamics, and many others. In this context, algorithms for automatic generation of volumetric meshes are
becoming increasingly important and valuable. This seminar will review the basic concepts before proceeding to present high-level algorithms for generating volumetric meshes.
Andrea Cabriolu: A Bayesian approach to an optimization algorithm for the dynamic scheduling of astronomical observations
In the context of the dynamic scheduling of observations with Sardinia Radio Telescope, a key role is played by Optimizer, a set of algorithms to optimize the sequence of the astronomical
observations. The calculations are based on several parameters, like weather conditions, device availability, operator’s availability and others. In this talk I’ll introduce the architecture which
allows the communication between Optimizer and the whole scheduling system, consisting of a central database and a bunch of other components. The core concepts of the Bayesian statistics will be
introduced as well, since this is the main pillar of the computing performed by the algorithm, to optimize the parameters set regargind the observations to be scheduled.
Giorgia Nieddu: State of art on the use of A.I. in mathematics education
In this seminar the most recent results on the use of A.I. in mathematics education, its areas of application, limits and possibilities will be presented.
Gen 142024
Prof. Gianluca Bande
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari
The course is an introduction to the Theory of Foliations. Basic knowledge of Differential Geometry is required and the basics of Fundamental Group.
– Definition(s) and examples of Foliations. Dynamical systems. Frobenius’ Theorem.
– Holonomy of a leaf and the Reeb Stability Theorem. Basic and foliated Cohomology. Godbillion-Vey class for a codimension 1 foliation on a 3-manifold.
– The Reeb foliation: definition and a 3D-printer model. Novikov and Likorisch Theorem.
The course spans over 3 lectures of 2 hours each (6 hours total). The lectures will be given on February 5, February 12, February 15 – 2024 at 4:30 p.m. in Room B of the Department of Mathematics and
Computer Science.
The final exam consists in a presentation.
1. C. Camacho and A. Lins Neto, Geometric theory of foliations, Birkhäuser, 1985.
2. A. Candel; L. Conlon, Foliations I, Grad. Stud. Math. 23, American Mathematical Society, Providence, 2000.
3. P. Tondeur, Geometry of Foliations, Monogr. Math 90, Birkhäuser Verlag, Basel, 1997.
Lug 232014
The announcement for the selection procedure for admission to the XXX cycle of the PhD courses has been published here:
Deadline for the on-line registration: september 12, 2014, h 12.00.
Deadline for the submission of the documentation: september 15, 2014, h 12.00.
Deadline for the upload of documentation (non-italian candidates): september 12, 2014, h 12.00.
Presentation of the PhD school on Mathematics and Computer Science
Dic 182013
The consortium of Italian Computer Science PhD granting institutions under the auspices of GRIN, organizes an annual school offering three graduate-level courses aimed at first-year PhD students in
Computer Science. In addition to introducing students to timely research topics, the school is meant to promote acquaintance and collaboration among young European researchers. The 2014 edition of
the School is the 20th in the series.
The school will offer 3 courses each consisting of 13 hours of lectures:
• Big Data Analysis of Patterns in Media Content – Nello Cristianini, University of Bristol (UK)
• An Introduction to Probabilistic and Quantum Programming – Ugo Dal Lago, University of Bologna (Italy)
• Development of dynamically evolving and self-adaptive software – Carlo Ghezzi, Politecnico di Milano (Italy)
Full details about the school are available here.
Dic 162013
The list of students admitted to the PhD in Mathematics and Computer Science is available at the following link:
Nov 192013
Il Corso di Dottorato di Ricerca in Matematica e Informatica ricopre un
ampio spettro di discipline tra loro collegate sia sul piano culturale che
metodologico e applicativo. Il dottorato, attraverso la pratica della
ricerca scientifica in settori di punta della Matematica e
dell’Informatica, mira a formare persone di livello culturale adeguato a
contribuire alle attuali richieste d’innovazione e di sviluppo
dell’industria e della società dell’informazione, sia sul piano della
creatività scientifica, sia su quello della capacità progettuale.
In particolare, il corso di dottorato è finalizzato alla formazione di
specialisti dotati di avanzate conoscenze metodologiche e tecniche, oltre
ad un’adeguata preparazione linguistica, in grado di svolgere attività di
ricerca e sviluppo in larga autonomia in ambito universitario, in enti di
ricerca pubblici e privati ed in ambito industriale. L’attività del
dottorato è sostenuta da docenti e ricercatori che fanno parte di gruppi
attivamente impegnati nella ricerca a livello internazionale, garantendo
ampie possibilità di scambio e di accoglienza dei dottorandi presso
prestigiose università italiane ed estere, enti di ricerca ed aziende. Le
tematiche di indagine offerte dai due curricula disponibili si riconducono
in larga parte alle attività di ricerca dei membri del collegio dei docenti
e riguardano gli aspetti sia fondamentali che applicativi di molti settori
della Matematica e dell’Informatica.
La formazione acquisita durante il dottorato consente di svolgere attività
di ricerca e sviluppo in larga autonomia in ambito universitario, in enti
di ricerca pubblici e privati ed in ambito industriale. In particolare, i
principali sbocchi occupazionali previsti sono il proseguimento delle
attività di ricerca universitaria, il coordinamento e la direzione di
attività di ricerca & sviluppo presso industrie o enti pubblici e/o centri
di ricerca nazionali ed internazionali. Le capacità di analisi ed
elaborazione acquisite con la formazione tramite la ricerca consentono,
inoltre, di intraprendere percorsi che portino a mansioni manageriali sia
nel settore privato che in quello pubblico, oppure intraprendere attività
in proprio come collaboratore di enti, aziende e società di sviluppo
Link utili | {"url":"https://dottorati.unica.it/matematicaeinformatica/author/bart/page/2/","timestamp":"2024-11-04T02:15:49Z","content_type":"text/html","content_length":"72394","record_id":"<urn:uuid:1a2a51be-1e4d-44cb-838d-a1b1ba3e7dd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00852.warc.gz"} |
Explanation of Star Ratings
For STS General Thoracic Surgery Database (GTSD) participants, the star rating is derived by testing whether the participant’s score in a composite domain is significantly different from the overall
STS average.
For each of the two composite domains (absence of morbidity and absence of mortality), if a participant’s estimated score is lower than the overall STS average but the difference between the
participant’s score and the STS average score is not statistically significant, the ratings would each be two stars. However, for the overall composite score, if the participant’s estimated score is
lower than the STS average, AND the difference is statistically significant, the overall participant star rating is one star.
The fact that statistical significance was achieved for the composite score but not the individual domains reflects the greater precision of the composite score compared to individual endpoints. This
precision is achieved by aggregating information across multiple endpoints instead of a single endpoint.
Because the star rating depends upon how the database participant compares to the STS average for a given time period, and the STS average is subject to change each time the analysis is performed,
there is not a prior morbidity or mortality level that a participant needs to attain in order to become a three-star institution. This also is true because the volume of cases at a given institution
impacts the comparison of its performance to the STS average.
Statistical significance is based on a 95% Bayesian certainty criterion. A participant receives three stars if there is at least 97.5% Bayesian probability that the participant’s score exceeds the
STS mean score (95% credible interval plus the 2.5% upper tail). A participant receives one star if there is at least 97.5% Bayesian probability that the participant’s score is less than the STS mean
score. Otherwise, the participant receives two stars. | {"url":"https://publicreporting.sts.org/gtsd-exp","timestamp":"2024-11-12T12:32:19Z","content_type":"text/html","content_length":"16336","record_id":"<urn:uuid:f908f767-68f5-48b1-a26b-1132fa24b5eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00010.warc.gz"} |
Jinha Kim (김진하) gave a talk on the minimum size of a maximal independent set in a graph of bounded maximum degree at the Discrete Math Seminar - Discrete Mathematics Group
Jinha Kim (김진하) gave a talk on the minimum size of a maximal independent set in a graph of bounded maximum degree at the Discrete Math Seminar
On February 15, 2022, Jinha Kim (김진하) from the IBS Discrete Mathematics Group gave a talk at the Discrete Math Seminar on the minimum size of a maximal independent set (or equivalently, an
independent dominating set) in a graph of maximum degree at most $\Delta$. The title of her talk was “Independent domination of graphs with bounded maximum degree“. | {"url":"https://dimag.ibs.re.kr/2022/jinha-kim-independent-domination/","timestamp":"2024-11-12T12:28:42Z","content_type":"text/html","content_length":"142159","record_id":"<urn:uuid:8b5da04a-04a3-47ea-b6fe-bb29de25fec4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00431.warc.gz"} |
An Elementary Geometry
Thompson and Company
, 1872 -
110 pages
From inside the book
Results 1-5 of 17
Page 8 ... by lines either straight or curved . When the bounding lines are straight , the figure is a polygon , and the sum of the bounding lines is the perimeter . 20. An Equilateral Polygon is one
whose sides are equal 8 PLANE GEOMETRY .
Page 9 ... equilateral . 23. Polygons whose angles are respectively equal are mutu- ally equiangular . Two equal sides , or two equal angles , one in each polygon , similarly situated , are called
homologous sides , or angles . 24. Equal Polygons ...
Page 13 ... equilateral triangle is equiangular . THEOREM XI . 45. If two angles of a triangle are equal , the sides opposite are also equal . A D B In the triangle ABC , let the angle A equal the
angle C ; then A B is equal to BC . For if AB is ...
Page 14 ... equilateral are equal in all respects . Let the triangle ABC have AB , BC , CA respec- - tively equal to AD , DC , CA of the triangle ADC ; then ABC is equal in all respects to ADC .
Place the triangle ADC so that the base AC will co ...
Page 17 ... equilateral rectangle ; as MNOP . 59. A Rhomboid is an oblique - angled par- allelogram ; as QRST . P R S T U Ꮴ 60. A Rhombus is an equilateral rhomboid ; as U V W X. X W 61. A Diagonal
is a line joining the vertices of two angles not ...
Popular passages
Four quantities are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth.
If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let a : b = c : d = e :f Now ab = ab (1) and
by Theorem I.
If the product of two quantities is equal to the product of two others, the...
The area of a regular polygon is equal to half the product of its perimeter and apothem.
If two triangles have two sides, and the included angle of the one equal to two sides and the included angle of the other, each to each, the two triangles are equal in all respects.
If two triangles have two sides of one respectively equal to two sides of the other, but the third sides unequal...
... polygon, is equal to twice as many right angles as the polygon has sides minus two.
A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.
A right cylinder is a solid described by the revolution of a rectangle about one of its sides.
DEFINITIONS. 1 . A straight line is perpendicular to a plane, when it is perpendicular to every straight line of the plane which it meets.
Bibliographic information | {"url":"https://books.google.com.jm/books?id=IiYAAAAAYAAJ&q=equilateral&dq=editions:UOM39015063898350&lr=&source=gbs_word_cloud_r&hl=en&output=html_text","timestamp":"2024-11-06T23:43:53Z","content_type":"text/html","content_length":"67235","record_id":"<urn:uuid:45974423-54f3-4d45-aa64-538b73b1feb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00325.warc.gz"} |
Snake Cube
Execution time limit is 1 second
Runtime memory usage limit is 64 megabytes
The snake cube is a mechanical puzzle made up of (n^3) cubes, each measuring (1 1 1), connected by an elastic cord. Each pair of adjacent cubes on the cord touches at one face and can rotate freely
around an axis perpendicular to that face. Every three consecutive cubes form either a straight line or a (90) degree angle. You cannot bend a straight segment of the snake into an angle or
straighten an angle into a straight segment; you can only decide, through rotations, which of the four directions the snake turns at each angle. The objective is to assemble a larger cube of size (n
n n) using this snake. An example of a snake is shown in the figure.
Not every snake can be assembled into a cube; at a minimum, all straight sections of the snake must consist of no more than (n) cubes.
We define the snake by the sequence of lengths of its straight segments, specifically by the sequence of distances between the centers of the first and last cubes in each straight segment. For
example, the snake shown in the figure is defined from left to right as follows: (2 1 1 2 1 2 1 1 2 2 1 1 1 2 2 2 2).
In this task, you are asked to solve a generalized version of this puzzle. A snake with (l w h) cells is given. Fit it into a rectangular parallelepiped of size (l w h) cells or determine that it is
The first line of the input file contains four integers separated by spaces - (l), (w), (h), and (m) ((1 l, w, h 4); (0 m < lwh)); here (l), (w), and (h) are the length, width, and height of the
parallelepiped, respectively, and (m) is the number of segments in the snake. The second line contains (m) positive integers separated by spaces - the sequence of lengths of the straight segments of
the snake. It is guaranteed that the given snake contains exactly (l w h) cubes.
If it is impossible to fit the given snake into the parallelepiped, output "Bit" on the first line of the output file. Otherwise, output "Baba" on the first line of the output file, and in the next (
l w h) lines - the coordinates of the snake's cubes, three numbers per line. The snake should be output from the same end it starts in the input file. The numbers (x_i), (y_i), and (z_i) in the
((i+1))-th line must satisfy the inequalities (0 x_i < l), (0 y_i < w), and (0 z_i < h). Additionally, all cubes of the parallelepiped (l w h) must appear in the output file exactly once, adjacent
lines must contain adjacent cubes, and the turns must exactly match the turns of the snake. The snake can start and end in any cube of the parallelepiped. If there are multiple solutions, any one of
them may be output.
Submissions 7
Acceptance rate 14% | {"url":"https://basecamp.eolymp.com/en/problems/5305","timestamp":"2024-11-12T15:28:04Z","content_type":"text/html","content_length":"290927","record_id":"<urn:uuid:7844313a-01fa-46a2-8b16-86f153af3bbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00150.warc.gz"} |
method of undetermined coefficients Archives | Fundamentals of Mathematics and Physics
One way to specify a sequence of numbers is recursively; that is, the first one or more terms in the sequence are stated, and then a formula for the general term of the sequence is given in terms of
one or more previous terms. For example, in the famous Fibonacci sequence, the first two terms … Read more | {"url":"https://fomap.org/tag/method-of-undetermined-coefficients/","timestamp":"2024-11-09T10:54:06Z","content_type":"text/html","content_length":"52423","record_id":"<urn:uuid:a8ca69ba-d2e5-494b-9dde-89d2a484613e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00047.warc.gz"} |
Topics: Operations on Matrices
Determinant > s.a. Berezinian; characteristic polynomial [eigenvalues].
$ Cofactor: The cofactor of M[ij] is (−1)^i+j (determinant of the minor obtained deleting row i and column j from M).
$ Def: If L is a linear map L: V → V, with dim V = n (and s is the number of − signs in the signature of the metric used to raise indices), then
det L:= (n!)^−1 (−1)^s ε[a.. b] ε^c.. d L^a[c] ··· L^b[d] ; also, det M = ∑[i][ or ][j][ = 1.. ][n] (cofactor M)^ij M[ij] .
* Useful formula: det(I + tX) = 1 + tr(tX) + O(t^2) = 1 + tr(tX) + det(tX) (at least for the 2 × 2 case).
* Derivative: For a symmetric matrix, ∂(det A)/∂A[ij] = (det A) A^−1[ij].
@ General references: Lehmich et al a1209 [convexity of the function C → f(det C) on positive-definite matrices].
@ Functional determinant: Gursky CMP(97) [Laplacian and Dirac operator squared]; Elizalde JHEP(99)ht; Illies CMP(01) [regularized products]; Fry IJMPA(02) [fermion, status]; Kirsten & McKane AP(03)mp
[countour integration], JPA(04)mp [general Sturm-Liouville problems]; Dunne JPA(08)-a0711-conf [computation, and quantum field theory]; Kirsten a1005-in [contour-integration methods]; Seiler &
Stamatescu JPA(16)-a1512 [fermionic, loop formula]; > s.a. lattice field theory.
> Related topics: see Cayley-Hamilton Theorem.
Other Operations and Related Concepts > s.a. Commutators.
* Inverse of a matrix: The matrix M^−1 such that M^−1M = M M^−1 = I; If M is an n × n matrix, it can be calculated using
(M^−1)[ij] = (det M)^−1 (cofactor M)[ji] = (n−1)!^−1 (det M)^−1 ε[k.. lj] ε[m.. ni] M[km] ··· M[ln] .
* Diagonalization: If A is an n × n matrix, with n distinct real/complex eigenvalues, use GL(n, \(\mathbb R\)/\(\mathbb C\)); If it has degenerate eigenvalues, it can be diagonalized iff for each λ[i
], of multiplicity m[i], rank(A − λ[i] I) = n−m[i]; Otherwise one can only reduce to Jordan normal form, with one Jordan block per eigenvector; Example: A = (1 1 ; 0 1), which has a doubly degenerate
eigenvalue λ = 1, but only one eigenvector, (1, 0); Generalized procedures: The singular-value decomposition and the Autonne-Takagi factorization; > s.a. Singular Values.
* Generalization: Any real symmetric or complex hermitian positive-definite N × N matrix is congruent to a diagonal one mod an SO(m, n), resp SU(m, n), matrix, for any partition N = m + n [@ Simon et
al mp/98].
* Decomposition: Every non-singular matrix can be written as the product of a symmetric one and an orthogonal one.
* Products: If A is an n × m matrix and B is a p × q matrix, their Kronecker product A ⊗ B is an np × mq matrix ("tensor product").
* Expansions: (A+B)^−1 = A^−1 − A^−1 B A^−1 + A^−1 B A^−1 B A^−1 − ...
* Exponentiation: The simple exponential e^A is defined in terms of the power series expansion; For a sum, e^A+B = e^A e^B e^−[A,B]/2, provided that A and B commute with their commutator; > more
generally, see the Zassenhaus Formula.
* Derivatives: (A^−1)' = −A^−1A'A^−1, at least if A is symmetric; ∂(det A) / ∂A[ij] = (det A) (A^−1)[ji] [notice the transpose].
* Resolvent of a matrix: The matrix (λI − M); The inverse is (λI − M)^−1 = λ^−1 + λ^−1M λ^−1 + λ^−1M λ^−1M λ^−1 + ... (which converges for λ sufficiently large).
* Permanent of a matrix: A number obtained from an analog of the minor expansion of the determinant, but with all positive signs; For a unitary matrix, its magnitude is ≤ 1; > s.a. knot invariants
@ Inverse: Penrose PCPS(55) [generalized].
@ Diagonalization: Banchi & Vaia JMP(13)-a1207 [quasi-uniform tridiagonal matrices]; Haber a2009 [three procedures]; Bischer et al JPA-a2012 [simultaneous block diagonalization]; > s.a. eigenvectors
and eigenvalues.
@ Factorization: Mostafazadeh mp/02 [symmetric]; Dita JPA(03) [unitary].
@ Exponentiation: Suzuki PLA(90), PLA(93) [of sum]; Federbush mp/99, LMP(00)mp; Ramakrishna & Zhou JPA(06)mp/05 [of su(4) matrices]; Fujii & Oike FEJME-mp/06 [formula]; Childs & Wiebe JMP(13)-a1211
[exponentials of commutators, product-formula approximations].
@ Related topics: Fleischhack a0804, Friedland a0804, Fleischhack & Friedland a0811 [Hurwitz product traces, BMV conjecture]; Steeb & Hardy 16 [matrix calculus, problems]; Eldar & Mehraban a1711
[approximating the permanent of a random matrix]; Kramer et al a1802 [new product].
main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 apr 2021 | {"url":"https://www.phy.olemiss.edu/~luca/Topics/math/matrix_op.html","timestamp":"2024-11-11T00:55:43Z","content_type":"text/html","content_length":"14651","record_id":"<urn:uuid:09dd032c-ff20-48b6-a90b-c860567052d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00059.warc.gz"} |
Unsteady turbulent boundary layers and separation
The eddy viscosity model with an inner (Prandtl-Van Driest) and an outer (Clauser) layer is generalized for unsteady flow and the time dependent turbulent boundary layer equations are integrated
numerically for transient or oscillatory outer flows. Comparisons with previous theoretical results indicate that the present method is at least as good as the others. Extensive comparisons with
experimental data is also attempted for the first time. It appears that in a certain range of frequencies the agreement is satisfactory. Further some characteristic quantities like the mean velocity
profile or the wall shear phase angle are predicted accurately but other properties, like the averaged fluctuations of the velocity, indicate large discrepancies. The present method is also capable
of integrating past the point of zero skin friction and into regions of partially reversed flow. The phenomenon of separation in unsteady flow is also investigated.
AIAA, 13th Aerospace Sciences Meeting
Pub Date:
January 1975
□ Boundary Layer Equations;
□ Boundary Layer Separation;
□ Eddy Viscosity;
□ Mathematical Models;
□ Turbulent Boundary Layer;
□ Unsteady Flow;
□ Flow Velocity;
□ Numerical Integration;
□ Reversed Flow;
□ Skin Friction;
□ Time Dependence;
□ Transient Response;
□ Velocity Distribution;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1975aiaa.meetR....T/abstract","timestamp":"2024-11-12T20:26:47Z","content_type":"text/html","content_length":"36392","record_id":"<urn:uuid:a1834c10-1938-4e23-8909-f8ad86d007f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00388.warc.gz"} |
Solve the differential equation dy/dx+3x(x+1)^(1/2)=0
Final answer to the problem
Got another answer? Verify it here!
Step-by-step Solution
How should I solve this problem?
• Choose an option
• Exact Differential Equation
• Linear Differential Equation
• Separable Differential Equation
• Homogeneous Differential Equation
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
• Integrate by substitution
• Integrate by parts
• Load more...
Can't find a method? Tell us so we can add it.
We need to isolate the dependent variable $y$, we can do that by simultaneously subtracting $3x\sqrt{x+1}$ from both sides of the equation
Learn how to solve differential equations problems step by step online.
Learn how to solve differential equations problems step by step online. Solve the differential equation dy/dx+3x(x+1)^(1/2)=0. We need to isolate the dependent variable y, we can do that by
simultaneously subtracting 3x\sqrt{x+1} from both sides of the equation. Group the terms of the differential equation. Move the terms of the y variable to the left side, and the terms of the x
variable to the right side of the equality. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to x. Solve the integral \int1dy and
replace the result in the differential equation.
Final answer to the problem
Got a different answer? Verify it!
How to improve your answer: | {"url":"https://www.snapxam.com/problems/42424659/dy-dx-3x-0","timestamp":"2024-11-13T08:36:53Z","content_type":"text/html","content_length":"66386","record_id":"<urn:uuid:459581a3-8dc9-4ed6-8c86-99dad61908d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00240.warc.gz"} |
Longitudinal model-based meta-analysis (MBMA) with MonolixSuite
Download data set only | Download all Monolix&Simulx project files V2021R2 | Download all Monolix&Simulx project files V2023R1 | Download all Monolix&Simulx project files V2024R1
This case study demonstrates how to implement and analyze a longitudinal MBMA model using Monolix and Simulx.
Due to the high costs of drug development, it is crucial to determine as soon as possible during the drug development process if the new drug candidate has a reasonable chance of showing clinical
benefit compared to the other compounds already on the market. To help the decision makers in go/no-go decisions, model-based meta-analysis (MBMA) has been proposed. MBMA consists in using published
aggregate data from many studies to develop a model and support the decision process. Note that individual data are usually not available as only mean responses over entire treatment arms are
reported in publications.
Because in an MBMA approach one considers studies instead of individuals, the formulation of the problem as a mixed effect model differs slightly from the typical PK/PD formulation. This case study
presents the implementation, analysis, and use of MBMA models for decision support in Monolix and Simulx.
As an example, the focus is on longitudinal data concerning the clinical efficacy of drugs for rheumatoid arthritis (RA). The goal is to evaluate the efficacy of a drug in development (phase II) in
comparison to two drugs already on the market. This case study is strongly inspired by the following publication:
Demin, I., Hamrén, B., Luttringer, O., Pillai, G., & Jung, T. (2012). Longitudinal model-based meta-analysis in rheumatoid arthritis: an application toward model-based drug development. Clinical
Pharmacology and Therapeutics, 92(3), 352–9.
Drug candidate overview
Rheumatoid arthritis (RA) is a complex auto-immune disease, characterized by swollen and painful joints. It affects around 1% of the population and evolves slowly (the symptoms come up over months).
Patients are usually first treated with traditional disease-modifying anti-rheumatic drugs, the most common being methotrexate (MTX). If the improvement is too limited, patients can be treated with
biologic drugs in addition to the MTX treatment. Although biologic drugs have an improved efficacy, remission is not complete in all patients. To address this unmet clinical need, a new drug
candidate, Canakinumab (abbreviated Canaki), has been developed at Novartis. After a successful proof-of-concept study, a 12-week, multicenter, randomized, double-blind, placebo-controlled,
parallel-group, dose-finding phase IIB study with Canaki+MTX was conducted in patients with active RA despite stable treatment with MTX. In the study, Canaki+MTX was shown to be superior to MTX
The goal is to compare the expected efficacy of Canaki with two other biologics already available on the market, Adalimumab (abbreviated Adali) and Abatacept (abbreviated Abata). Several studies
involving Adali and Abata have been published in the past (see list below). A common endpoint in these studies is the ACR20, which represents the percentage of patients achieving a 20% improvement.
Specifically, the focus is on longitudinal ACR20 data, measured at multiple time points after treatment initiation, with the aim of using a mixed-effect model to describe this data.
Overview of MBMA concepts
The formulation of longitudinal mixed effect models for study-level data differs from longitudinal mixed effect models for individual-level data. As detailed in
Ahn, J. E., & French, J. L. (2010). Longitudinal aggregate data model-based meta-analysis with NONMEM: Approaches to handling within treatment arm correlation. Journal of Pharmacokinetics and
Pharmacodynamics, 37(2), 179–201.
to obtain the study-level model equations, one can write a non-linear mixed effect model for each individual (including inter-individual variability) and derive the study-level model by taking the
mean response in each treatment group, i.e the average of the individual responses. In the simplistic case of linear models, the study-level model can be rewritten such that a between-study
variability (BSV), a between-treatment arm variability (BTAV) and a residual error appear. Importantly, in this formulation, the residual error and the between treatment arm variability are weighted
by with [ ]the number of individuals in treatment arm k of study i. If the model is not linear, a study-level model cannot be easily derived from the individual-level model. Nevertheless, one usually
makes the approximation that the study-level model can also be described using between-study variability, between-treatment arm variability and a residual error term.
This result can be understood in the following way: when considering studies which represent aggregated data, the between-study variability represents the fact that the inclusion criteria vary from
study to study, thus leading to patient groups with slightly different mean characteristics from study to study. This aspect is independent of the number of individuals recruited. On the opposite,
the between-treatment arm variability represents the fact that when the recruited patients are split in two or more arms, the characteristics of the arms will not be perfectly the same, because there
is a finite number of individuals in each arm. The larger the arms are, the smaller the difference between the mean characteristics of the arms is, because one averages over a larger number. This is
why the between-treatment arm variability is weighted by For the residual error, the reasoning is the same: as the individual-level residual error is averaged over a larger number of individuals, the
study-level residual error decreases.
The study-level model can be written similarly to the individual-level model, with between-study variability and between-treatment arm variability considered instead of inter-individual variability
and inter-occasion variability. Additionally, the residual error and between-treatment arm variability are weighted by
Before proposing a model for rheumatoid arthritis, a closer look at the data is required.
Data selection, extraction, and visualization
Data selection
To compare Canaki to Adali and Abata, we have searched for published studies involving these two drugs. Reusing the literature search presented in Demin et al., we have selected only studies with
inclusion criteria similar to the Canaki study, in particular: approved dosing regimens, and patients having an inadequate response to MTX. Compared to the Demin et al. paper, we have included two
additional studies published later on. In all selected studies, the compared treatments are MTX+placebo and MTX+biologic drug.
The full list of used publications is:
Data extraction
Using an online data digitizer application, the longitudinal ACR20 data from nine selected publications was extracted. The information is stored in a data file, along with additional study details.
The in-house individual Canaki data from the phase II trial is averaged over treatment arms to compare with the aggregate literature data. A screenshot and column descriptions are provided.
• STUDY: index of the study/publication. (Equivalent to the ID column)
• ARM: 0 for placebo arm (i.e MTX+placebo treatment), and 1 for biologic drug arm (MTX+biologic drug)
• DRUG: name of the drug administrated in the arm (placebo, adali, abata or canaki)
• TIME: time since beginning of the treatment, in weeks
• Y: ACR20, percentage of persons on the study having an at least 20% improvement of their symptoms
• Publication: first author of the publication from which the data has been extracted
• Narm: number of individuals in the arm
• TRT: treatment studied in the study
• DiseaseDuration: mean disease duration (in years) in the arm at the beginning of the study
• CReactiveProtein: mean C-reactive protein level (in mg/dL)
• SwollenJointCount: mean number of swollen joints
Data visualization
Monolix includes a built-in data visualization module for an overview after data import. Upon opening Monolix, select "New project" and browse for the data file. The columns are assigned, and the
data visualization is displayed for analysis.
The STUDY column is tagged as ID, and the ARM column is tagged as OCCASION to use the MonolixSuite's inter-individual and inter-occasion variability features to define between-study and
between-treatment-arm variability. All columns that may be used to stratify/color the plots must be tagged as either continuous or categorical covariate, even if these columns will not be used in the
After clicking "ACCEPT" and "NEXT," the data is displayed in the "Plots" tab. To better distinguish the treatments and arms, the DRUG and TRT covariates are assigned in the "STRATIFY" tab. The layout
is arranged in a single line, with treatments split by TRT and colored by DRUG. In the "SETTINGS" tab, custom x-axis limits are enabled, automatically setting the range to 1 and 52.14. Similarly,
setting custom Y-axis limits (5.38 to 75.66) showcases the variation in ACR20 increase over time. Biologic treatments are colored (blue for Abata, red for Adali, green for Canaki), while placebo arms
are displayed in black, demonstrating the efficacy of MTX+biologic drug compared to MTX+placebo. Note that even in the “placebo” arms the ACR20 improves over time, due to the MTX treatment. For
Canaki, which is the in house drug in development, only one study is available, the phase II trial.
Given this data, the question arises whether Canaki could be more effective than Adali and Abata. To answer this, a longitudinal mixed-effect model will be developed.
Description of the longitudinal mixed-effect model
Based on the observed trajectory of ACR20 over time, an Emax structural model is suggested to capture the data's behavior.
The ACR20 score represents the fraction of responders, so it must remain between 0-1 (or 0-100 as a percentage). To maintain this, a logit transformation is applied to the observations
Given between-study variability (BSV) and between-treatment arm variability (BTAV) are considered only for this parameter. A logit distribution is selected to ensure Emax values remain between 0 and
Between-treatment arm variability and residual error are weighted by the number of individuals per arm, unlike between-study variability, as explained in the section “MBMA concepts” above.
The next section covers implementing this model in Monolix.
Model setup in Monolix
In Monolix, selecting the error model is limited to a pre-defined list, which does not include a residual error model weighted by the square root of the number of individuals. Therefore, both the
ACR20 observations of the data set and the model prediction will be transformed in the following manner, utilizing a constant error model. The equation can be rewritten as follows:
This data transformation is added to the data set, in the column transY.
At this stage, the new dataset with the transY column can be loaded into Monolix, and the columns can be assigned as before, with the exception that the transY column will now be used as OBSERVATION.
STUDY is tagged as ID, ARM as OCCASION, DRUG as CATEGORICAL COVARIATE, TIME as TIME, transY as OBSERVATION, Narm as REGRESSOR, TRT as CATEGORICAL COVARIATE and DiseaseDuration, CReactiveProtein and
SwollenJointCount as CONTINUOUS COVARIATE.
To match the observation transformation in the data set, the Mlxtran model file will include the following:
ACR20 = ...
pred = logit(ACR20)*sqrt(Narm)
output = pred
In order to be available in the model, the data set column Narm must be tagged as regressor, and passed as argument in the input list:
input = {..., Narm}
Narm = {use=regressor}
In this example, the number of individuals per arm (Narm) remains constant over time. In cases of dropouts or non-uniform trials, Narm may vary over time. This will be accommodated in Monolix, as
regressors can change over time by definition.
Parameter Distribution Considerations
Similar to the error model, weighting the BTAV directly via the Monolix GUI is not possible. As an alternative, the fixed effect (EmaxFE), the BSV random effect (etaBSVEmax), and the BTAV random
effect (etaBTAVEmax) can be defined separately in the GUI, with the parameter value reconstructed within the model file. Since Narm has been defined as a regressor, it can be used to weight the BTAV
terms. Care must also be taken when adding the random effects to the normally distributed transformed parameters. The syntax for the structural model is as follows:
input = {EmaxFE, T50, etaBSVEmax, etaBTAVEmax, Narm}
Narm = {use=regressor}
; transform the Emax fixed effect (EmaxFE) to tEmax (normally distributed)
tEmax = logit(EmaxFE)
; adding the random effects (RE) due to between-study variability and
; between arm variability, on the transformed parameters
tEmaxRE = tEmax + etaBSVEmax + etaBTAVEmax/sqrt(Narm)
; transforming back to have EmaxRE with logit distribution (values between 0 and 1)
Emax = exp(tEmaxRE)/(1+exp(tEmaxRE))
; defining the effect
ACR20 = Emax * (t / (T50 + t))
; adding a saturation to avoid taking logit(0) (undefined) when t=0
ACR20sat = min(max(ACR20,0.01),0.99)
; transforming the effect ACR20 in the same way as the data
pred = logit(ACR20sat)*sqrt(Narm)
output = pred
In addition to the previously explained steps, a saturation ACR20sat=min(max(ACR20,0.01),0.99) is added to prevent ACR20from being 0 when t=0, which would result in pred=logit(0)(undefined).
In the structural model tab, selecting 'New model,' copying and pasting the code above, and saving the model will allow it to appear below:
In the next tab, Statistical Model & Tasks, the statistical part of the model will be defined. For the observation model, a constant error model shall be selected from the list, in accordance with
the data transformation performed.
For the parameter distributions, a logit distribution shall be chosen for EmaxFE, with distribution limits set between 0 and 1 (note that the default initial value for EmaxFE is 1, leading to default
logit limits of (0,1). So the initial value for EmaxFE needs to be changed first, before logit(0,1) can be set), lognormal for T50, and a normal distribution for the random effects etaBSVEmax, and
Since this model formulation allows EmaxFE to represent only the fixed effects, the random effects for this parameter and T50 must be disabled. As etaBTAVEmax corresponds to inter-occasion
variability, the random effects will be disabled at the STUDY level (indicated as “ID” in the GUI) but enabled at the ARM level. The random effects at both levels are thus set in the following way:
For etaBSVEmax, its distribution is
Covariate Considerations
As a final step, DRUG will be added as a covariate for EmaxFE and T50 to estimate one EmaxFE and one T50 value for each treatment (placebo, adali, abata, and canaki). Before adding the DRUG column as
a covariate, the 'add covariate / discrete' button should be clicked to create a new covariate called tDRUG, with placebo as the reference category.
The newly created tDRUG is added on EmaxDE and T50. The other possible covariates will be investigated later on.
Summary view
Model development
Parameter estimation
Parameter initialization
Before starting the parameter estimation, reasonable initial values should be identified. In the Initial Estimates tab, within the Check initial estimates window, initial values for the population
parameters can either be set manually or the auto-init function can be run to determine the initial values, as shown in the image below. The red curve represents the prediction with the estimated
initial values, while the blue dots correspond to the transY data.
Configuring Settings for Parameter Estimation
The parameter estimation task is then launched. After approximately 160 iterations, the SAEM exploratory phase transitions to the second phase due to the automatic stopping criteria.
To assess convergence, the parameter estimation can be run using different initial values and a different random number sequence. This can be accomplished by clicking on “assessment” right next to
the run button. This module runs estimation tasks multiple times with varying seeds and initial values.
Convergence trajectories, final estimates, and repeated log-likelihood calculations can be examined to assess the robustness of the estimates.
The colored oscillating curves in the upper plot above show the parameter estimates trajectories, while the colored horizontal lines in the lower plot above show the standard deviation of the
estimated population parameter value of each run. The trajectories of many parameters show significant variation between runs. Parameters without variability, such as EmaxFE, are generally more
difficult to estimate, as parameters with and without variability are estimated using different methods. The SAEM algorithm requires to draw parameter values from their marginal distribution, which
exists only for parameters with variability.
Several methods can be used to estimate the parameters without variability. By default, these parameters are optimized using the Nelder-Mead simplex algorithm. As the performance of this method seems
insufficient in our case, we will try another method. In the SAEM settings (accessed via the wrench icon next to the 'Population parameters' task in the Statistical Model & Tasks tab), 3 options are
• No variability (default): optimization via Nelder-Mead simplex algorithm
• Variability in the first stage: during the first phase, an artificial variability is added and progressively forced to zero. In the second phase, the Nelder-Mead simplex algorithm is used.
• Add decreasing variability: an artificial variability is added for these parameters, allowing estimation via SAEM. The variability is progressively forced to zero, such that at the end of the
estimation proces, the parameter has no variability.
The 'Variability in the first stage' method shall selected, the project shall be saved under a new name (MBMA_r02_variability1stage.mlxtran) and the convergence assessment needs to be relaunched.
This time, the convergence is satisfactory:
The estimated values can be reviewed directly in the “Assessment” folders generated during the convergence assessment, or the parameter estimation and Fisher information matrix estimation (e.g., via
linearization) can be redone to obtain the standard errors.
The point estimates in the Wald test, along with the associated p-values in the test section of the results tab, represent the difference between the placebo and the other drug categories. A
significant p-value indicates that the drug effect is statistically different from 0, meaning the covariate effect should be retained in the model.
As indicated by the p-value, the Emax values for Abata and Adali are significantly different from the Emax for placebo. On the opposite, given the data, the Emax value for Canaki is not significantly
different from placebo. Note that when analyzing the richer individual data for Canaki, the efficacy of Canaki had been found significantly better than that of placebo (Alten, R. et al. (2011).
Efficacy and safety of the human anti-IL-1β monoclonal antibody canakinumab in rheumatoid arthritis: results of a 12-week, Phase II, dose-finding study. BMC Musculoskeletal Disorders, 12(1), 153.).
The T50 values are similar for all compounds except Adali. The test shows that the effect of Abata vs. Placebo and Canaki vs. Placebo is not significantly different from 0. Therefore, these drug
covariate modalities can be pooled for the next model run (MBMA_r03_variability1stage_groupeddrug.mlxtran), creating a new covariate.
This new covariate will be included in the individual model as a covariate effect on T50, replacing the covariate tDRUG.
After launching the estimation tasks, the Wald test suggests that the new covariate should remain included in the individual model on T50, as the difference between Adali and Canaki, Abata, and
Placebo is significant.
The between study variability of Emax is around 25% (C.V..(%) of omega_etaBSVEmax = 0.2533). The between treatment arm variability is gamma_etaBTAVEmax = 2.4361. In the model, the BTAV is divided by
the square root of the number of individuals per arm. With a typical arm size of 200 individuals, the between-treatment arm variability is approximately 14% (
When the random effects of BSVEmax_pop and BTAVEmax_pop are set to 0, the C.V.(%) for omega_etaBSVEmax and gamma_etaBTAVEmax becomes infinite. The coefficient of variation is defined as the ratio of
the standard deviation to the mean, resulting in division by 0.
Model diagnosis
The next step is to estimate the study parameters (i.e., individual parameters via the mode task) and generate the graphics. No model mis-specification is detected in the individual fits,
observations versus predictions, residuals, or other graphics:
Individual fits Observations vs predictions
Note that all graphics use the transformed observations and transformed predictions.
Alternative models
A more complex model, including BSV and BTAV on T50, could also be explored, but this investigation is left as a hands-on exercise for the reader. The results show a small difference in AIC and BIC
between the two models, and some standard errors in the more complex model are unidentifiable. Therefore, the simpler model is preferred.
Covariate analysis
The between study variability originates from the different inclusion criteria from study to study. To explain part of this variability, the studied population characteristics can be tested as
covariates. The available population characteristics are the mean number of swollen joints, the mean C-reactive protein level and the mean disease duration.
As these covariates differ between arms, they appear in the IOV window. To center the distribution of covariates around a typical value, the covariates are first transformed using a reference value:
The covariates can be included one by one in the model on the Emax parameter as an example of a forward covariate model development approach. Significance can be assessed using either a Wald test
(with p-values calculated from the Fisher Information Matrix Task in the result file) or a likelihood ratio test. The advantage of the Wald test is that it requires estimating only one model (the one
with the covariate), while the likelihood ratio test necessitates estimating both models (with and without the covariate).
In this case study, a backward approach has been employed, where all the covariates are included in the model on Emax simultaneously, followed by an evaluation to determine whether any covariates
should be excluded (run04.mlxtran). The p-values from the Wald test assessing whether the beta coefficients are zero are as follows:
• Disease duration: p-value=0.28 (not significant)
• C-reactive protein level: p-value=0.75 (not significant)
• Swollen joint count: p-value=0.14 (not significant)
The likelihood ratio test also indicates that none of these covariates is significant; therefore, they will not be included in the model.
Final model
The final model has an Emax structural model, with between study and between treatment arm variability on Emax, and no variability on T50. No covariates are included on these parameters.
The fixed effect EmaxFE incorporates the drug effect tDrug, reflecting all four drug modalities (Adali, Abata, Canaki, and Placebo), with Placebo set as the reference drug.
For the parameter T50, a drug covariate effect is also included; however, the drug covariate has been regrouped to consist of only two modalities (Adali and Abata/Canaki/Placebo).
The final run is available in the materials (see the top of the page) and is named MBMA_r03_variability1stage_groupeddrug.mlxtran.
Simulations for decision support
Efficacy comparison with existing market drugs
The developed model will be used to compare the efficacy of Canaki with that of Abata and Adali. The question arises: does Canaki have the potential to be more effective? To address this, simulations
will be conducted using the simulation application Simulx.
To compare the true efficacy (over an infinitely large population) of Canaki with Abata and Adali, the Emax values could be directly compared. However, since Emax estimates carry uncertainty, this
uncertainty must be taken into account. A large number of simulations will be performed for the three treatments, drawing the Emax population value from its uncertainty distribution. The BSV, BTAV,
and residual error will be ignored (i.e., set to 0) to focus on the “true efficacy” within an infinitely large population.
Before running the simulations, the final project must be exported from Monolix to Simulx.
All the information contained in the Monolix project will be transferred to Simulx. This includes the model (longitudinal and statistical part), the estimated population parameter values and their
uncertainty distribution and the original design of the dataset used in Monolix (number of individuals, treatments, observation times). Elements containing these information are generated
automatically in the “Definition” tab of the created Simulx project. It is possible to reuse these elements to setup a simulation in the “Simulation” tab or create new elements to define a new
simulation scenario.
Among the elements created automatically, there is the occasion element imported from the Monolix data set. The first step is to remove the occasion element, as it is not needed for simulations.
Navigate to the tab Definition > Occasions and click the 'delete' button.
To compare the efficacy of Canaki with Abata, Adali, and placebo, ACR20 will be simulated over 52 weeks in four different scenarios (simulation groups): Canaki, Abata, Adali, and placebo.
Next, the output needs to be defined. In Monolix, the transformed prediction of ACR20 was used due to constraints on the error model. Since the error model is being neglected here, and the focus is
on model predictions, the untransformed ACR20 can be directly output at each week.
The output variable used to assess the efficacy of the treatments is ACR20, measured on a regular grid from time 0 to week 52.
The applied treatments, which appear in the model as a categorical covariate, must then be defined. This can be done directly in the covariates tab. Four distinct elements need to be created, each
representing one of the four treatments to be simulated.
The definition of the other treatments not shown in the screenshot follows the same approach.
Lastly, a regressor representing the number of subjects per study is defined. Although this information needs to be specified, it will not influence the simulations.
Initially, the ACR20 for each treatment is simulated using the estimated Emax and T50 values, while ignoring uncertainty. Four individuals (i.e., studies) are defined, one for each treatment.
After defining the output, regressor, and treatment variables, the simulation tab is used.
The ACR20 for each treatment is simulated, focusing on the estimated Emax and T50 values and disregarding uncertainty. One simulation (i.e., study) is defined for each treatment.
A group size of 1 is set to simulate only one study per treatment. The parameter 'mlx_typical' (available starting from version 2023) is selected, which contains the population parameters from the
last Monolix run, with the random effects' standard deviations set to 0. This simulates a typical study, excluding BSV and BTAV. The defined untransformed ACR20 vector is used as the output variable.
By clicking on 'New group,' the defined covariates are applied as treatments across the four simulation groups. The Reg_Narm element is set as the regressor. In theory, the number of subjects in the
treatment arms determines the variability between the treatment arms. As the number of subjects approaches infinity (
Since an ideal scenario is being modeled, a very high number of subjects can be set. However, since 'mlx_typical' is selected as the simulation parameter, the random effects (etaBSV and etaBTAV) and
their standard deviations are already set to 0, meaning Reg_Narm has no practical effect on the simulation.
After running the simulation, the response curves for the four treatment groups appear as subplots in the 'Output distribution' tab within the Plot section. To display all curves in a single plot,
the 'merged splits' option can be activated in the 'Stratify' panel. Additionally, by enabling the legend toggle in the plot settings and optionally adjusting the treatment arm colors in the
stratification tab, the following plot is generated:
The simulation gives in the generated plot already a first impression of the efficacy of Canaki (orange curve) compared to the other treatments.
By defining outcomes and endpoints, this impression can be quantified.
As the outcome, ACR20 is defined at the final time point (t = 52). The endpoint can be the arithmetic mean, geometric mean, or median within each simulation group. Since only one study is simulated
per group, this choice has no impact, and the endpoint will be equal to the outcome. A direct comparison of the ACR20 values is chosen across the groups, with Canaki as the reference. Given that each
treatment group has only one simulated study, no statistical comparison is possible. The relation is set as unilateral "< 0" to assess whether Canaki is more effective compared to existing
treatments. If the result is negative, Canaki is considered more effective; if it equals 0, Canaki is deemed equivalent at the final time point; if positive, it is considered less effective.
In formula terms, the following relationship is expected as a successful experimental outcome:
The formula is analogous for the treatments Abata and Placebo versus Canaki.
After defining the outcomes and endpoints, save the project under a new name. Only the Outcome & Endpoint task needs to be executed; the simulations do not need to be rerun.
In the simulation tab of the results section, summary tables for each treatment arm are provided, along with a list of the individual parameters. The endpoint tab of the results shows the arithmetic
mean of ACR20 at the final time point, t=52. The "Standard Deviation" column in the Endpoints[Table] or Endpoints[Summary] tab displays NaN, as the sample size n, representing the number of simulated
studies, is set to 1, leading to division by zero. In the group comparison tab, a table presents the calculated mean differences between the treatment arms and the reference, Canaki. The "Success"
column indicates whether the comparison is successful: a green check mark appears if the difference is < 0, while a black cross indicates an unsuccessful comparison.
The mean difference between Abata and Canaki, as well as Adali and Canaki, is positive, indicating that ACR20 for Abata and Adali was higher than for Canaki at week 52. This suggests that, in the
ideal simulated scenario (infinitely large population), Canaki is less effective than these two treatments already available on the market. However, the comparison between Placebo and Canaki shows a
negative difference, aligning with the effect curve plot.
The next step involves calculating a prediction interval for the true effect, considering the uncertainty of the population parameters estimated by Monolix. This can be done by saving the Simulx
project under a new name and adjusting the settings as follows:
• create 100 replicates of the scenario and
• change the population parameter from mlx_Typical to mlx_TypicalUncertain (available from version 2023). This option includes the estimated variance-covariance matrix of the population parameters
from the final Monolix run, where the standard deviations of the random effects are set to 0. This allows simulation of a typical subject (i.e., a study) with varying population parameter values
across replicates, ensuring that the uncertainty in population parameters is reflected in the predictions.
The defined outcomes and endpoints remain unchanged. The goal is to assess the impact on the success criterion when the scenario is repeated 100 times using different sets of population parameters.
The output distribution plot again shows the response curve, but now with the percentiles of the simulated data for all calculated replicates. The “number of bands” in Display can be set to 2 and the
subplots merged to easy the comparison.
In the “Endpoint distribution” tab of the resulting plots, the distribution of ACR20 at week 52 across all replicates for each of the four treatment groups is shown. It is clear that treatments with
active compounds are more effective than the placebo group. However, after 100 replicates, the mean value of Canaki remains below that of Abata and Adali.
Note: The thickness of the bands in the “Output distribution” plot, as well as the wide box and whiskers in the “Endpoint distribution” plot for Canaki, reflect the smaller amount of data available
for Canaki compared to other treatments. This results in greater uncertainty regarding Canaki's true effect.
Returning to the results section, tables similar to those from the previous simulation are found, but this time displaying results for each replicate.
Based on the summary table at week 52, considering the estimated population parameters:
• In 100 repetitions, the relation
• In 100 repetitions, the relation
• In 100 repetitions, the relation
This finding supported the decision not to continue with clinical development of canakinumab in RA, according to:
Demin, I., Hamrén, B., Luttringer, O., Pillai, G., & Jung, T. (2012). Longitudinal model-based meta-analysis in rheumatoid arthritis: an application toward model-based drug development. Clinical
Pharmacology and Therapeutics, 92(3), 352–9.
Model-informed drug development
Model based meta-analysis supports critical decisions during the drug development process. In the present case, the model has shown that chances are low that Canaki performs better than two drugs
already on the market. As a consequence the development of this drug has been stopped, thus saving the high costs of the phase III clinical trials.
MBMA with Monolix and Simulx
This case study also aims at showing how Monolix and Simulx can be used to perform longitudinal model-based meta-analysis. Contrary to the usual parameter definition in Monolix, the weighting of the
between treatment arm variability imposes to split the parameters into their fixed term and their BSV and BTAV terms, and reform the parameter in the model file. Once the logic of this model
formulation is understood, the model set up becomes straightforward. The efficient SAEM algorithm for parameter estimation as well as the built in graphics then allow to efficiently analyze, diagnose
and improve the model.
The inter-operability between Monolix and Simulx permits to carry the Monolix project over to Simulx to easily define and run simulations.
Natural extensions
In this case study, continuous data were used. Yet, the same procedure can be applied to other types of data such as count, categorical or time-to-event data.
Guidelines for implementing MBMA models
Data set preperation
The data set should usually contain the following columns:
• STUDY: index of the study/publication. It is equivalent to the ID column for individual-level data.
• ARM: index of the arm (preferably continuous integers starting at 0). There can be an arbitrary number of arms (for instance several treated arms with different doses) and the order doesn’t
matter. This column will be used as equivalent to the occasion OCC column.
• TIME: time of the observations
• Y: observations at the study level (percentage, mean response, etc)
• transY: transformed observations, for which the residual error model is constant
• Narm: number of individuals in the arm. Narm can vary over time
• Covariates: continuous or categorical covariates that are characteristics of the study or of the arm. Covariates must be constant over arms (appear in IOV window) or over studies (appear in main
Common inquiries about the data set
• Can dropouts (i.e. a time-varying number of individuals per arm) be taken into account? Yes. This can be accounted for without difficulty as the number of subjects per arm (Narm) is passed as a
regressor to the model file, a type that allow by definition variations over time.
• Can there be several treated (or placebo) arms per study? Yes. The different arms of a study are distinguished by a ARM index. For a given study, one can have observations for ARM=0, then
observations for ARM=1, then observations for ARM=2. The index number has no special meaning (for instance ARM=0 can be the treated arm for study 1, and ARM=0 the placebo arm for study 2).
• Why should I transform the observations? The observations must be transformed such that an error model from the list (constant, proportional, combined, exponential, etc) can be used. In the given
example, two transformations were applied. First, a logit transformation was used to ensure that the prediction of the fraction of individuals achieving ACR20 remains within the 0-1 interval,
even if a large residual error term is encountered. For other cases, such as when predicting a mean concentration of glucose, a log-transformation could be applied to guarantee positive values.
Second, each value was multiplied by the square root of the number of individuals per arm. This ensures that the standard deviation of the error term is consistent across all studies and
treatment arms, regardless of the number of individuals per arm.
Model set-up
The set up of the project can be done in four steps:
1. Split the parameters in several terms
2. Make the appropriate transformation for the parameters
3. Set the parameter definition in the Monolix interface
4. Add the transformation of the prediction
Common inquiries about the model set-up
• Why should I split a parameter in several terms? Because the standard deviation of the BTAV term must usually by weighted by the square root of the number of individuals per arm, this term must
be handled as a separate parameter and added to the fixed effect within the model file. Moreover, the fixed effect and its BSV random effect might be split depending on the covariate definition.
• How do I know in how many terms should I split my parameter?
□ If all covariates of interest are constant over studies: In this case, the parameters can be split into a (fixed effect + BSV) term and a (BTAV) term. The model file would for instance be
(for a logit distributed parameter):
input = {paramFE_BSV, etaBTAVparam, Narm}
Narm = {use=regressor}
tparam = logit(paramFE_BSV)
tparamRE = tparam + etaBTAVparam/sqrt(Narm)
paramRE = 1 / (1+ exp(-tparamRE))
• If one covariate of interest is not constant over studies: This was the case for the case study, where the DRUG covariates were constant across arms but not studies. In this scenario, the
parameters must be divided into a fixed effect term, a BSV term, and a BTAV term. For example, the model file could be structured as follows (for a logit-distributed parameter):
input = {paramFE, etaBSVparam, etaBTAVparam, Narm}
Narm = {use=regressor}
tparam = logit(paramFE)
tparamRE = tparam + etaBSVparam + etaBTAVparam/sqrt(Narm)
paramRE = 1 / (1+ exp(-tparamRE))
• Why shouldn’t I add covariates on the etaBTAVparam in the IOV window? Covariates can be added on etaBTAVparam in the IOV window. Yet, because the value of etaBTAVparam will be divided by the
square root of the number of individuals in the model file, the covariate values in the data set must be transformed to counter-compensate. This may not always be feasible:
□ If the covariate is categorical (as the DRUG for example), it is not possible to compensate and thus the user should not add the covariate on the etaBTAVparam in the GUI. A splitting in 3
terms is needed.
□ If the covariate is continuous (weight for example), it is possible to counter-compensate. Thus, if you add the covariates on the etaBTAVparam in the GUI, the covariate column in the data set
must be multiplied by beforehand. This is a drawback, however this modeling has less parameters without variability and may thus converge more easily.
• How should I transform the parameter depending on its distribution? Depending on the original parameter distribution, the parameter must first be transformed as in the following cases:
□ For a parameter with a normal distribution, called paramN for example, we would have done the following:
; no transformation on paramN as it is normally distributed
tparamN = paramN
; adding the random effects (RE) due to between-study and between arm variability, on the transformed parameter
tparamNRE = tparamN + etaBSVparamN + etaBTAVparamN/sqrt(Narm)
; no back transformation as tparamNRE already has a normal distribution
paramNRE = tparamNRE
Note that the two equalities are not necessary but there for clarity of the code.
• For a parameter with a log-normal distribution, called paramLog for example (or T50 in the test case), we would have done the following:
; transformation from paramLog (log-normally distributed) to tparamLog (normally distributed)
tparamLog = log(paramLog)
; adding the random effects (RE) due to between-study and between arm variability, on the transformed parameter
tparamLogRE = tparamLog + etaBSVparamLog + etaBTAVparamLog/sqrt(Narm)
; transformation back to have paramLogRE with a log-normal distribution
paramLogRE = exp(tparamLogRE)
• For a parameter with a logit-normal distribution in ]0;1[, called paramLogit for example (or Emax in the test case), we would have done the following:
; transformation from paramLogit (logit-normally distributed) to tparamLogit (normally distributed)
tparamLogit = logit(paramLogit)
; adding the random effects (RE) due to between-study and between arm variability, on the transformed parameter
tparamLogitRE = tparamLogit + etaBSVparamLogit + etaBTAVparamLogit/sqrt(Narm)
; transformation back to have paramLogitRE with a logit-normal distribution
paramLogitRE = 1 / (1+ exp(-tparamLogitRE))
• For a parameter with a generalized logit-normal distribution in ]a;b[, called paramLogit for example, we would have done the following:
; transformation from paramLogit (logit-normally distributed in ]a,b[) to tparamLogit (normally distributed)
tparamLogit = log((paramLogit-a)/(b-paramLogit))
; adding the random effects (RE) due to between-study and between arm variability, on the transformed parameter
tparamLogitRE = tparamLogit + etaBSVparamLogit + etaBTAVparamLogit/sqrt(Narm)
; transformation back to have paramLogitRE with a logit-normal distribution in ]a,b[
paramLogitRE =(b * exp(tparamLogitRE) + a)/ (1+ exp(tparamLogitRE))
• How do I set up the GUI? Depending which option has been chosen, the fixed effects and/or the BSV random effects and/or the BTAV random effects of the model parameters (representing one or
several terms of the full parameter) must be enforced to 0. Random effects can be disabled by clicking on the RANDOM EFFECTS column and fixed effects by choosing “fixed” after a click in the
initial estimates.
• Why and how do I transform the prediction? The model prediction must be transformed in the same way as the observations of the data set, using the column Narm passed as regressor.
Parameter estimation
The splitting of the parameters into several terms may introduce model parameters without variability. These parameters cannot be estimated using the main SAEM routine, but instead are optimized
using alternative routines. Several of them are available in Monolix, under the settings of SAEM. If the default method leads to poor convergence, testing other methods is advisable.
Model evaluation
All graphics are displayed using the transformed observations and predictions. These are the right quantities to diagnose the model using the “Observation versus prediction” and “Scatter plot of the
residuals”, “distribution of the residuals” and graphics. For the individuals fits and VPC, it may also be interesting to look at the non-transformed observations and predictions. The whole analysis
conduct as well as the generation of the plots can be redone in R using the simulx function of the RsSimulx package or the lixoft connectors. The other diagnostic plots (“Individual parameters vs
covariates”, “Distribution over the individual parameters”, “Distribution of the random effects”, and “Correlation of the random effects”) are not influenced by the transformation of the data. | {"url":"https://monolixsuite.slp-software.com/tutorials/2024R1/case-study-longitudinal-model-based-meta-analysis-","timestamp":"2024-11-07T12:18:11Z","content_type":"text/html","content_length":"414910","record_id":"<urn:uuid:1cb3b861-279c-44e2-bee2-51564e948766>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00078.warc.gz"} |
Fast method to connect multiple nodes
I have a point graph with 500 nodes, and I want to add 30 new nodes and connect them to each other and to existing graph.
Is there a fast method to do it?
I found that AstarPath.active.UpdateGraphs(bounds) is too slow for this. pointGraph.ConnectNodes() is also slow even with lookup tree. Because ConnectNodes iterates over all graph nodes. But I need
to iterate over my newly added 30 nodes with all (limited by maxDistance).
it would be nice to have a method ConnectNodes(PointNode[ ] nodesForConnection)
How often are you adding these 30 new nodes? I also recommend taking a look at this page on optimization and see if anything here may be of use.
Oooooooh baby do I have some code for YOU! To be fair I only have 200 nodes and they only connect to things within 2.1 range of each other, but adding/deleting nodes takes zero noticable time.
List<GraphNode> nodesToRecheck = new List<GraphNode>();
Vector3Int _gridStartPosition = new Vector3Int(_vec3Int.x - 4, _vec3Int.z + 4, _vec3Int.z); Vector3Int _gridEndPosition = new Vector3Int(_vec3Int.x + 4, _vec3Int.y - 4, _vec3Int.z);
BoundsInt _nodeZone = new BoundsInt(
Math.Min(_gridStartPosition.x, _gridEndPosition.x),
Math.Min(_gridStartPosition.y, _gridEndPosition.y),
Math.Min(_gridStartPosition.z, _gridEndPosition.z),
Math.Abs(_gridEndPosition.x - _gridStartPosition.x) + 1,
Math.Abs(_gridEndPosition.y - _gridStartPosition.y) + 1,
Math.Abs(_gridEndPosition.z - _gridStartPosition.z) + 1
foreach (var v in _nodeZone.allPositionsWithin)
Vector3 _v3 = new Vector3(v.x + 0.55f, v.y + 0.55f, v.z);
GraphNode nodev = AstarPath.active.data.pointGraph.GetNearest(_v3, null, 0.4f).node;
PointNode _pointNodev = nodev as PointNode;
if (_pointNodev != null) { if (!nodesToRecheck.Contains(_pointNodev)) nodesToRecheck.Add(_pointNodev); }
public void RedoCosts(List<GraphNode> nodesToRecheck)
//Debug.Log("Redoing " + nodesToRecheck.Count);
foreach (GraphNode nodetoCheck in nodesToRecheck)
Vector3 _vec3 = (Vector3)nodetoCheck.position;
Vector3Int _vec3Int = new Vector3Int(Mathf.FloorToInt(_vec3.x), Mathf.FloorToInt(_vec3.y), Mathf.FloorToInt(_vec3.z));
Vector3Int _gridStartPosition = new Vector3Int(_vec3Int.x - 4, _vec3Int.z + 4, _vec3Int.z); Vector3Int _gridEndPosition = new Vector3Int(_vec3Int.x + 4, _vec3Int.y - 4, _vec3Int.z);
BoundsInt _nodeZone = new BoundsInt(
Math.Min(_gridStartPosition.x, _gridEndPosition.x),
Math.Min(_gridStartPosition.y, _gridEndPosition.y),
Math.Min(_gridStartPosition.z, _gridEndPosition.z),
Math.Abs(_gridEndPosition.x - _gridStartPosition.x) + 1,
Math.Abs(_gridEndPosition.y - _gridStartPosition.y) + 1,
Math.Abs(_gridEndPosition.z - _gridStartPosition.z) + 1
var connections = new List<GraphNode>();
foreach (var v in _nodeZone.allPositionsWithin)
Vector3 _v3 = new Vector3(v.x + 0.55f, v.y + 0.55f, v.z);
GraphNode node1 = AstarPath.active.data.pointGraph.GetNearest(_v3, null, 0.4f).node;
PointNode _pointNode = node1 as PointNode;
if (node1 != null)
if (!connections.Contains(_pointNode)) { connections.Add(_pointNode); /*Debug.Log("Adding node at " + _pointNode.position);*/ }
foreach (var connection in connections)
PointNode n1 = nodetoCheck as PointNode; PointNode n2 = connection as PointNode;
GraphNode g1 = nodetoCheck as GraphNode; GraphNode g2 = connection as GraphNode;
Vector3 nv1 = (Vector3)n1.position; Vector3 nv2 = (Vector3)n2.position;
float d = Vector3.Distance(nv1, nv2);
var cost = (uint)d;
if (d <= 1.1f) cost = 1000;
if (d > 1.1f && d < 2) cost = 2500;
if (d >= 2) cost = 5500;
//GraphNode.Disconnect(g1, g2);
if (d <= 2) GraphNode.Connect(g1, g2, cost, OffMeshLinks.Directionality.TwoWay);
public void FlushIt()
thanks for examples. I also wrote my own solution based on pointGraph.ConnectNodes(). I need to add nodes not too often, but this operation shouldn’t make spikes on the main thread.
I wrote my post to make sure that the library doesn’t have a fast method out of the box.
Ahh, gotcha. Yeah I don’t think there’s anything particularly built-in for this use case. Sorry about that!
I great appreciate this contribution, btw, as well as the energy behind it
I want better games, don’t we all? Hording code is for Ubisoft
1 Like
Contribution is absolutely king, agreed! | {"url":"https://forum.arongranberg.com/t/fast-method-to-connect-multiple-nodes/17022","timestamp":"2024-11-05T23:32:20Z","content_type":"text/html","content_length":"26905","record_id":"<urn:uuid:ab21eddf-9a0e-46a6-90b7-782172b3d6a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00855.warc.gz"} |
[Solved] Cassandra wants to produce consistently s | SolutionInn
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Cassandra wants to produce consistently sized baked goods, so she draws a sample of 12 cashew toffee tarts and weighs them (in ounces). She finds
Cassandra wants to produce consistently sized baked goods, so she draws a sample of 12 cashew toffee tarts and weighs them (in ounces). She finds a 90% confidence interval of (3.8, 4.2). Which is the
correct interpretation of the 90% confidence interval
There are 3 Steps involved in it
Step: 1
To interpret the 90 confidence interval for Cassandras sample of cashew toffee tarts lets break down ...
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How to Grow Your Firm the Right Way
For most new business owners, growth is a sign that the business is successful. But depending on that business's financial structures and its commitment to shareholders, growth can complicate things.
This article will introduce a simple but powerful financial metric that provides a realistic idea of how much the company can grow based on its profitability, earnings retention, financial structure
and "asset velocity." Once a business owner/manager has identified the current earnings power and liquidity position of the firm, he or she can then ascertain the company’s sales growth based on
current financial conditions.
To illustrate this concept, consider the Badger Company, a hypothetical small manufacturer located in the Midwest. Badger experienced rapid growth in sales during its past several years, averaging
25% growth each year from 2012 to 2015. Despite the relentless growth, the firm’s CEO saw her firm feeling the strain of managing the growth. She was not sure how she would be able to continue
without taking on more debt or making substantial changes that put her firm at risk.
Sustainable Growth Rate
Professor Robert Higgins, Professor of Finance at the University of Washington-Seattle, noted in the 1970’s that many organizations face the conflict of unrestrained growth with the limitations of
established financial policies. Higgins suggested that many companies' financial policies and growth objectives are mutually exclusive, and that companies must explore options to remedy the pitfalls
of too much growth. He introduced a version of the “Sustainable Growth Rate” (SGR) financial metric to help business owners and managers identify the levers available to grow sales. The SGR is the
annual percentage increase in sales that a firm can afford, given that the firm wants to maintain a given dividend payout ratio from earnings and targeted debt/equity structure without issuing new
The Sustainable Growth Rate (SGR) metric can provide management like the CEO of Badger Manufacturing with a decision-making tool to assess the amount of sales growth the current financial structure
can afford. (Note: See Appendix A for information on a more traditional Sustainable Growth Rate formula and its relationship to the formula discussed here).
The SGR Formula
The SGR as posited by Higgins requires four variables that need to be calculated:
• P = Operating Margin or Net Income / Sales
• R = Dividend Payout Ratio, or Dividends / Net Income
• L = Leverage, or Total Liabilities / Equity
• T = Assets / Sales
Sustainable Growth Rate = P x (1-R) x (1+L)
T – {P x (1-R) x (1+L)}
To illustrate, assume that Badger Company (hypothetical) earned $100 thousand in Net Earnings in 2015 as shown in the table below on $2.0 million in sales. Assume the company pays no dividends during
The following shows the variables included:
• Profit Margin = $100/$2,000 = 5%
• Dividend Payout Ratio = $0 / $100 or 0%
• Leverage = $1000/ $1000 or 1.00,
• Asset/Sales = $2000 / $2000 or 1.0
The Sustainable Growth Rate is 11.1%, as computed below.
(.05) x (1-0) x (1+1.00) = (.10) = .10 = 11.1%
1.0 – {.05 x 1 x 2.00} (1 - .10) .90
In other words, the Sustainable Growth Rate suggests that as long as Badger Company maintains a Profit Margin of 5% and Debt/Equity ratio of 1.0; doesn’t pay any dividends (i.e., retains all profits
in the business); and maintains the same Assets/Sales ratio of 1.0, the company can sustain an annual sales growth rate of 11.1%.
If the company’s sales grow higher than 11.1%, it will need to change one of the input variables in the equation (i.e., become more profitable, increase debt in relation to equity, and/or increase
Sales in relation to its asset base). Otherwise, the company will face cash shortfalls. On the other hand, if the company sales grow at a rate less than 11.1%, it should generate excess cash.
Effect of Paying Dividends
In the original example, Badger Company retained all its profits not paying any dividends to ownership. However, what’s the impact on growth if Badger distributes dividends? Assume that in 2015, the
company sets a policy to distribute 25% of Net Income to owners with all other variables remaining the same. Also, any shortfall in funding assets due to lower retained earnings is made up by
increasing Liabilities.
To illustrate, assume that Badger Company (hypothetical) earned $100 thousand in Net Earnings in 2015, as shown in the table below, on $2.0 million in sales.
Because dividends were distributed, the earnings retained in Owner’s Equity are lowered by the amount of the dividends paid (this example, $975 instead of $1,000 in the previous) increasing the
Leverage Ratio from 1.00 to 1.05 as Liabilities are increased by $25 thousand to offset the lower retained earnings from the dividends (Note – the shortfall could have also been offset by selling
stock, reducing assets or a combination thereof).
The recomputed variables adjusted for the payment of dividends are:
• Profit Margin = $100/$2,000 = 5%
• Dividend Payout Ratio = $250 / $100 or 25%
• Leverage = $1025/ $975 or 1.05
• Asset/Sales = $2,000 / $2,000 or 1.0
The Sustainable Growth Rate for this example is computed below.
(.05) x (1-.25) x (1+1.05) = (.077) = .077 = 8.3%
1.0 – {.05 x 0.75 x 2.05} 1 - .077 .923
As a result of the dividend payout, Badger Company’s ability to grow sales is lowered to 8.3% vs. 11.1% per year, or a reduction of 25%. This example is provided to illustrate how a decision to
withdraw earnings from the business instead of re-investing the profits back into the company impacts the ability to grow.
Even though the distribution of a company’s earnings to shareholders affects sales growth, management can take steps to offset the effect of payout through a series of other strategies and tactics.
They include:
• Grow profits at a faster rate in relation to Sales (i.e., increase Profit Margin)
• Increase Leverage by taking on more debt in relation to equity, and/or
• Reduce Assets in relation to Sales through more efficient management of the asset base.
In Badger Company’s case, assume that management believes that through a particular mixture of price increases and cost reductions it can increase earnings to $120 thousand instead of $100 thousand.
Also, management thinks that by better managing inventory and accounts receivable they can reduce the overall asset base to $1.8 million from $2.0 million. They still want to maintain the dividend
payout of $25 thousand.
Profit Margin increases to 6.7% of Sales due to the higher Net Earnings from our previous example. The Dividend Payout Ratio declines to 20.8% as the amount of dividends remains steady but are paid
out of a higher earnings base. Third, the Leverage Ratio, declines to .891 as the Equity portion of the business grew from higher earnings, and finally, the Turnover Ratio declined to 0.9 from 1.0 in
the other examples as better inventory and receivable management lowered the total asset base in relation to sales.
As a result, the assorted changes in management actions compensated for the lost growth in sales from the dividend payout by increasing the firm’s sustainable growth rate to 11.9% vs. the previous
example of 8.3%. The calculations are shown below.
• Profit Margin = $120/$1,800 = 6.7%,
• Dividend Payout Ratio = $25 / $120 or 20.8%
• Leverage = $815/ $995 or .82,
• Asset/Sales = $1800 / $2000 or 0.9
The Sustainable Growth Rate increases to 6.8% for Badger (computed below).
(.0667) x (1-.208) x (1+.82) = (.096) = .096 = 11.9%
0.9 – {.06 x 0.792 x 1.82} 0.9 - .096 .804
So, all is not lost if the owners/managers decide they want to "take a little cream" off the top in the form of dividends. However, as the aforementioned examples show, if management wants to
maintain a foundation of growth, there are tradeoffs in that the business must become more profitable, efficient, and/or take on a greater level of debt to finance its operations.
Caveats with Sustainable Growth Rate
As with any financial ratios, some conditions may render the ratio meaningless and/or not applicable:
No Profits. For companies showing a loss resulting in a negative profit margin % of sales, the sustainable growth rate will not apply. In situations like this, it’s recommended that an organization
use a "target profit margin" as goal or proxy.
100% Dividend Payout. If a firm pays out all its dividends, the dividend payout ratio will be 100% causing the sustainable growth rate to be zero. Again, the actual growth rate will have little
meaning but an analyst can substitute various payout rates less than 100% to show what potential SGRs might be.
Negative Equity. In some cases, firm may have accumulated deficits that create a negative equity situation on the balance sheet. As can be seen in the formula, this will result in a situation where
the SGR is not applicable. One way to address this is to develop a ‘target Leverage ratio’ to show what the SGR could be.
Extreme Leverage. In situations where the firm has an extreme Leverage ratio (very high debt in relation to the equity base), the SGR may result in extreme numbers. First, an analyst should examine
if the debt is in fact debt. There may be situations, especially with small companies, where loans or other debt instruments are in fact equity (no interest is paid, no principal has been paid,
etc.). In this case, the analyst should reclassify the debt as equity. If there is simply a case of overleverage and the company is in a profitable state, an SGR can be calculated but caution should
be used in interpreting the result. Again, more realistic SGRs can be shown with less onerous leverage positions.
Extreme Low Asset Bases. In many service companies, there may be a very low total asset base in relation to sales. As a result, profitable companies with such a base will generate astronomical SGRs.
Any interpretation should be careful and only compared within the company and/or within the same industry.
Constant Cost and Asset Functions. The ratio’s variables are assumed constant under a static financial policy as of a particular point in time. For instance, the formula assumes a constant cost
function, and not a step function as might occur in manufacturing as an example, if a new facility needs to be built for expansion.
The Sustainable Growth Rate is a simple but effective tool for gauging how fast a company can grow its sales based on its profitability, earnings retention, financial structure and asset management.
It provides management with a guide to benchmark its growth objectives with the reality of the firm’s financial performance and capital structure. As a result, owners/management can develop their
plans, ascertain the sustainable growth, and then either adjust their plans or take action regarding their financial strategies to provide for more growth potential.
Appendix A: Alternative Sustainable Growth Rate
Most accounting and finance professionals will recall a more traditional Sustainable Growth Rate that is derived from the Return on Equity, which is slightly different from the Higgin’s SGR formula.
The original SGR formula is calculated as follows:
ROE = Net Earnings/Total Equity
ROE = (Net Earnings/Assets) x (Assets/Equity)
ROE = (Net Earnings/Sales) x (Sales/Assets) x (Assets/Equity)
If apply the Retention Rate of earnings (1- Dividend Payout Rate) to the ROE equation, you then have SGR.
SGR = (Net Earnings/Sales) x (Retention Rate) x (Sales/Assets) x (Assets/Equity)
If one applies Example 1 to this formula (no Dividends paid):
SGR = ($100/$2,000) x (1.00) x ($2,000/$2,000) x ($2,000/$1,000)
= (.05) x (1.00) x (1.00) x (2.00)
= (.05) x (2.00) = 10% using the alternative method.
Using the Higgins Model, SGR was calculated as 11.1%, vs. 10% using this alternative model. The reason for this is that alternative model in this example uses end of the year equity balance ($1000)
so the Sustainable Growth Rate using the alternative is actually calculating it for the past year at the end of the year or estimating for 2016.
If one subtracts the increase in Equity during 2015 ($100) from End of the year Equity ($1,000 - $100) you can easily calculate Equity at the beginning of 2015 or $900. If the $900 beginning of the
year equity is substituted into the alternative model, the SGR is:
SGR using Beg of Year Equity = (.05) x (1.00) x (1.00) x ($2,000/$900)
= (.05) x (2.22)
= .111 or 11.1%, same as original Higgins Model.
Higgins, R. A., (1977). How Much Growth Can a Firm Afford? Financial Management, p.8 – 17.
Read More
"You Have Our Permission Not to Grow," Jason Pattit and Katerina Pattit, June 2018
Cite this Article
DOI: 10.17919/X92P4H
Greenwood, P. (2016, September 7). How to grow your firm the right way. Entrepreneur & Innovation Exchange. Retrieved November 12, 2024, from https://eiexchange.com/content/
Greenwood, Phil. "How to Grow Your Firm the Right Way" Entrepreneur & Innovation Exchange. 7 Sep. 2016. Web 12 Nov. 2024 <https://eiexchange.com/content/200-how-to-grow-your-firm-the-right-way>. | {"url":"https://eiexchange.com/content/200-how-to-grow-your-firm-the-right-way","timestamp":"2024-11-12T07:03:56Z","content_type":"text/html","content_length":"51597","record_id":"<urn:uuid:abe55d45-eeec-495d-a1e9-3762cb3d5aee>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00790.warc.gz"} |
Quantum Matter Seminar- Nathanan Tantivasadakarn (California Institute of Technology)- From Wave-Function Collapse and Galois Solvability to the Realization of Non-Abelian Topological Order on a Quantum Device
October 2, 2023
10:00AM - 11:30AM
1080 Physics Research Building
Add to Calendar 2023-10-02 09:00:00 2023-10-02 10:30:00 Quantum Matter Seminar- Nathanan Tantivasadakarn (California Institute of Technology)- From Wave-Function Collapse and Galois Solvability to
the Realization of Non-Abelian Topological Order on a Quantum Device Dr. Nathanan Tantivasadakarn California Institute of Technology From Wave-Function Collapse and Galois Solvability to the
Realization of Non-Abelian Topological Order on a Quantum Device Location: 1080 Physics Research Building, Smith Seminar Room Faculty Host: Yuan-Ming Lu, Kyle Kawagoe 1080 Physics Research Building
OSU ASC Drupal 8 ascwebservices@osu.edu America/New_York public
Date Range
2023-10-02 10:00:00 2023-10-02 11:30:00 Quantum Matter Seminar- Nathanan Tantivasadakarn (California Institute of Technology)- From Wave-Function Collapse and Galois Solvability to the Realization of
Non-Abelian Topological Order on a Quantum Device Dr. Nathanan Tantivasadakarn California Institute of Technology From Wave-Function Collapse and Galois Solvability to the Realization of Non-Abelian
Topological Order on a Quantum Device Location: 1080 Physics Research Building, Smith Seminar Room Faculty Host: Yuan-Ming Lu, Kyle Kawagoe 1080 Physics Research Building America/New_York public
Dr. Nathanan Tantivasadakarn
California Institute of Technology
From Wave-Function Collapse and Galois Solvability to the Realization of Non-Abelian Topological Order on a Quantum Device
Location: 1080 Physics Research Building, Smith Seminar Room
Faculty Host: Yuan-Ming Lu, Kyle Kawagoe
Abstract: I will review our recent set of theoretical works on efficiently preparing long range quantum entanglement with adaptive quantum circuits: the combination of measurements with unitary gates
whose choice can depend on previous measurement outcomes. I will show that this additional ingredient can be leveraged to prepare the long sought-after non-Abelian topological phases with a circuit
depth that is independent of system size. Using this framework, we uncover a complexity hierarchy of long-range entangled states based on the minimal number of measurement layers required to create
the state. Moreover, we find that certain non-Abelian states that cannot be efficiently prepared with adaptive circuits have a surprising connection to the unsolvability of the quintic polynomial.
Finally, I will describe our recent collaboration with Quantinuum where we present the first unambiguous realization of non-Abelian D4 topological order and demonstrate control of its anyons. In
particular, we are able to detect a non-trivial braiding where three non-Abelian anyons trace out the Borromean rings in spacetime, a signature unique to non-Abelian topological order.
Bio: Nat Tantivasadakarn obtained his Ph.D. from Harvard University and is currently a Burke postdoctoral fellow at Caltech. His research interests range from topological phases of matter to quantum
error correction and computation, and their dynamical generalizations.
Students and postdocs are invited to stay and chat with the speaker from 11:30-12:00. | {"url":"https://physics.osu.edu/events/quantum-matter-seminar-nathanan-tantivasadakarn-california-institute-technology-wave","timestamp":"2024-11-12T02:44:01Z","content_type":"text/html","content_length":"81313","record_id":"<urn:uuid:cfe3a42c-5f0a-422d-8254-c36d846112f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00871.warc.gz"} |
How to use the python decimal object in cython code?
To use the Python Decimal object in Cython code, you first need to import the Decimal class from the decimal module within your Cython code. Here is an example of how you could use the Decimal object
in Cython code:
1 # Import the Decimal class from the decimal module
2 from decimal import Decimal
4 # Declare a function that uses the Decimal object
5 cdef void calculate_with_decimal():
6 # Create a Decimal object with a value of 10
7 cdef Decimal dec = Decimal(10)
9 # Perform some calculations using the Decimal object
10 cdef Decimal result = dec / Decimal(2)
12 # Print out the result
13 print(result)
In the above example, we import the Decimal class from the decimal module and then create a function calculate_with_decimal that uses the Decimal object to perform some calculations. We create a
Decimal object with a value of 10 and then divide it by another Decimal object with a value of 2. Finally, we print out the result of the calculation.
Remember that using the Decimal object within Cython code may impact performance compared to using lower-level numeric types like int or double. However, if precision is critical for your
calculations, using the Decimal object may be necessary. | {"url":"https://devhubby.com/thread/how-to-use-the-python-decimal-object-in-cython-code","timestamp":"2024-11-03T19:29:46Z","content_type":"text/html","content_length":"127166","record_id":"<urn:uuid:b53c6bd5-9130-4184-bc49-9a9d53ec8c58>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00701.warc.gz"} |
Fluctuation in Cosmic Quantum Sine-Wave { /\/\/\/ }
/\/\/\/ = topology of sine-wave ---in Euclidean expresssion--- straight lines not curved, has positive peaks ^ and negative troughs v
/\/\/\/ = quantum sine-wave, and does it necessitate two sine-waves{ electric and magnetic charge } at 90 degrees to each other?
/--\--/--\--/ = two sine-waves { at 90 degrees to each other { electric -------- and magnetic /\/\/\/
/--\--/--\--/ = 3D tubular characteristics if not also a spiral { ////// forward-over-time //// } aspect?
1} Is there unrealized{ non-observed } ---ergo metaphysical-3 and or 4--- line-of-relationship between the peak of electric and magnetic peaks and troughs and in sequence.
If so, then, do we really do have a 3D tube-like enclosure and line-of-relationship would be a diagonal ergo spiral{ //// }?
........1a} is that line-of-relationship gravitational{ metaphysical-3 }, and/or it 180 degree opposite dark energy { metaphysical-4 }?.......
2} Is there also unrealized{ non-observed } --ergo metaphysical-3 and/or 4---- line of relationship between;
.....2a} the adjoining positive peaks only and in sequence of the electric sine-wave,
..................this would a constant seemingly linear lineof-relationship________....
......2b} the adjoining negative peaks only and in sequence of electric sine-wave,
..................this would a constant seemingly linear lineof-relationship________....
......2c} the adjoining positive peaks only and in sequence of the magnetic sine-wave,
..................this would a constant seemingly linear line-of-relationship________....
.......2d} the adjoining negative peaks only and in sequence of the magnetic sine-wave
..................this would a constant seemingly linear line-of-relationship________....
3} So 2a, 2b, 2c and 2d would be four, non-observed{ un-realized } a constant and seemingly linear lines-of-relationship.
4} The diagonal spiral { ///// } is also a constant, semi-linear{ //// overall forward //// } line-of-relationship and this makes for total of five, outer surface constant lines-of-relationship
connection of peaks and troughs.
......4a} this means each of the positive peaks or negative trough peaks have three lines-of-relationship extending forward-over-time ergo each 'peak' is a vertex and a vertex and in Euclidean terms
it is three lines crossing{ Y }.
......4} and we have one line-of-relationship that is lateral between peak of that does not appear to be forward-over-time ergo an X-type vertex having four lines-of-relationship none of which are
from the past{ behind } posterior set of events.
So we have a total of 4 of these X-type 4 vertexes that define this seemingly 3D tube as laid-out above.
The next question becomes is their a polyhedron that has four of these X-type vertexes? That would mean 16 lines-of-relationship and Im not sure if that exists. I'm out of time on this post and
there is much more to be involved in following along these visible{ realized } and non-realized{ invisible } lines-of-relationship scenario, regarding a quantum electro-magnetic sine-wave.
Eventually we have to come to what is the fluctuation and was it always there just not reached a critical limit of WOW!{ BB }.
..."That period, known as cosmic inflation, came to an end and gave rise to the hot Big Bang, but never created an arbitrarily hot, dense state, nor did it create a singularity.
.....What happened prior to inflation — or whether inflation was eternal to the past — is still an open question, but one thing is for certain: the Big Bang is not the beginning of the Universe!"....
1} see electric-magnetic sine-waves https://micro.magnet.fsu.edu/primer/java/scienceopticsu/electromagnetic/index.html
1} So the last post was considering "Space Fabric" inflating, but that does not define what SPACE is specifically, as Ive done geometrically with Gravity ( ) and Dark Energy )( and a resultant and
visible{ realizable } sine-wave /\/\/\/\/\/.
...1a}At the above link we see a double sine-wave set appearing from nothing. We do not see the invisible Space Fabric geometric lines-of-relationship.
2} previously I stated "16 lines-of-relationship" between 4, X-type vertexes and I think that was in error.
...2a} in consideration of only two peaks and two troughs we have only 6 lines-of-relationship of 4, Y-type vertexes and that is a tetrahedral set.
3} we see a linear, double sine-wave, appearing from nothing, yet, on numerous occasions it has crossed my mind that this is not a linear sine-wave. i.e. it comes from a closed{ finite } Space Fabric
...3a} there are only three{?} primary types of finite, 3D enclosure that this can be associated with, and that is;
.......3b} convex spherical/polyhedral,
........3c} tubular torus i.e. tube{ slinky //////// toy } that curves back to meet itself with positive{ convex }, negative{ saddle shape } and minimal flat{ transition between positive and
4} in the scenario above I never consider a single torus inflation model. I always consider large set{ cluster } of Space Fabric Tori.
...4a} in this sense we are also allowed multiple local universe scenarios to be considered but it is not necessary to do so
Cluster of Spiraled /////// Space Fabric ( )( )Tori (////)(\\\\) from which a resultant of many sine-waves{ ^v^v } may appear
5} either a multitude of sine-wave existed prior to or as inflation and came to intefer with each other,
....5a} or a multitude of Space Fabric interfered/infringed on each other causing peak inversions of multiple tori and resultant sine-waves inside each.
There is also Roque Waves in ocean that seemingly appear out of nowhere ---i.e. fluctuation out for no seeming reason--- and it is related to Schodenger wave forumla.
Will have to look of the link.
..."1. Introduction
The rogue wave is giant single wave which was firstly found in the ocean [1].
The amplitude of this wave is two to three times higher than those of its surrounding waves. The key feature of the rogue wave is that it “appears from nowhere” and “disappears without a trace” [2].
The most terrible thing is that it is very dangerous for sailors because it can appear unexpectedly and form larger amplitude in one minute to shred a boat. Beyond oceanic expanses, the rogue wave
has been also found in optical fibers [3], Bose-Einstein condensates (BECs) [4], superfluids [5], and so on. However, it is very difficult to explain the rogue waves by using the linear theories
based on the superposition principles.
....The nonlinear theories of ocean waves [6–8] can be used to explain why the rogue waves can appear from nowhere. In recent years, it becomes an important issue for ones to study the rogue waves
theoretically in the fields of the nonlinear science [9–13]. The Darboux transformation (DT) [14, 15], the similarity transformation, and the numerical simulation [13, 16–18] were used to analyze the
occurrence of the rogue waves and the larger amplitudes."...
...One of the important known models for the rogue waves is the nonlinear Schrödinger (NLS) equation which is a foundational model in describing numerous nonlinear physical phenomena and the
first-order rational solution was derived by Peregrine [19] and the second-order one was obtained by using the modified Darboux transformation. Cheng et al. [20] show the controllable rogue waves in
coupled NLS with varying potential functions and nonlinearities. Wu et al. [21] derive the evolution of optical solitary waves in a generalized NLS equation with variable coefficients.
...."Interestingly, they found that the systems had different degrees of determinism: oceanic rogue waves seem to have identifiable precursors, whereas their fibre-optic cable cousins are not at all
...This is thought to be because the ocean waves are caused by turbulence, which is difficult to predict but not intrinsically random, whereas in optical fibres rogue waves are driven by quantum
noise. According to Steinmeyer, “what this shows is really that ‘rogueness’ and predictability have nothing to do with each other.”
...This work could provide insight into the predictability of a wide range of chaotic phenomena, as well as advance our understanding of the nature of quantum fluctuations and randomness."...
~~~~~~~/\/\/\/\/\/\/\/\/\/\/\* */\/\/\/\/\/\/\/\/~~~~~~~~~~* *
Rogue Quantum Wave
Is bilateralism inherehently existent in cosmic quantum wave?
Is charge { + and - } inerently existent in quantum wave?
Is spin left \\\\\\\ or right ////// inherently existent in cosmic quantum wave?
The Vector Equlibrium contracts-expands on 4 differrent axis ---diametric triangular opposites--- in either left-spin or right-spin directions.
..."A signature of rogue-wave behavior is a heavy tail of theprobability distribution.
..."In this chapter, we study a random walk whose increments have a (right) heavy-tailed distribution with a negative mean. Themaximum of such a random walk is almost surely finite, and our interest
is in the tail asymptotics of the distribution ofthis maximum, for both infinite and finite time horizons; we are further interested in the local asymptotics for the maximumin the case of an infinite
time horizon. We use direct probabilistic techniques and show that, under the appropriate subexponentialityconditions, the main reason for the maximum to be far away from zero is again that a single
increment of the walk is similarlylarge."...
My question is there also a (left+) long tail distribution with a positive mean or known as a light-tail distribution.
Or heavy tailed but smooth.
Is bilateralism inherently existent in cosmic quantum wave?
/\/\/\/\/*\/*\/\/\/\*/\*/\/\/\/ Is rogue wave same as a cosmic quantum fluctuation"
Bilateral flipping of Earths EMField
..."Also unclear. Scientists estimate that past polar flips have been rather sluggish, with north and south migrating to opposite positions over thousands of years. This is both good and bad if
you’re concerned about how a geomagnetic reversal will affect life on Earth.
....The sluggish polar meander is good, because it means we have time to prepare and can do our best to ameliorate any unpleasant effects before they get really unpleasant. But it’s bad, because our
planet’s magnetic field helps shield us from damaging solar and cosmic radiation, and a protracted flip means Earth might be slightly less protected from harmful space rays for longer than we would
Bilaterally toroidal. Reverse generation of a torus via 2 circles O___O
Oh yeah, uh huhh, uh huhh, thats the way we like it uh huhh uh huhh!
...."The amplitude of this wave is two to three times higher than those of its surrounding waves. The key feature of the rogue wave is that it “appears from nowhere” and “disappears without a
It does not appear out of nowhere,i.e. it seemingly appears out of nowhere when actuality it appears from the combination of specific set of waves that are not easily discernable with naked eye.
What seemingly appears as equlibrium is never true equlibrium.
We do not have perfect spheres or circles we have ultra-high frequency set of trajectories the infer a perfect 2D polygon or 3D sphere.
We do not sensorially observe absolute truth, we discern absolute from relative truths from our sensorial experiences.
What seemingly appears as equlibrium is never true equlibrium.
Ex, what appears as a static Universe, is never a static Universe ergo eternally in motion//dynamic.
Is spin left \\\\\\\ or right ////// inherently existent in cosmic quantum wave?
In a spirally defined torus it is both left and right from perspective of an observer who position does not change in relation to the.
So lets presume an electron can in someway to be associated with a torus. And we have electron spin states can be up or down.
These spin states may be found in all fermions if not also also bosons.
There exists other exotic configurations of the 4-fold VE via its jitterbug transformations LINK that are somewhat difficult for me to explain without better graphics Ex the saddle-shape octahedron
of only negative { ..)(.... } curved space whereas the original convex VE{ spherical or Euclidean } is only positive convex shaped space.
But my favorite is the flying 2D hexagon { __ } that has an erect { yet flexi-able } tail wing { /\ } composed of two bonded/valenced triangles
........ __/\ ...........
With the flying hexagon we see what I believe is the inherent ---built in--- random fluctuation of any so called quantum wave or quantum sine-wave scenarios of Universe.
It is as tho the flying hexagon with its 7th set of erect { perpendicular triangles } says, I refuse to collapse into a 2D only existence.
Or, I am somebody will not lay flat for remain flat for no force of Universe.
Or you can not stop triangles who want to be free.
To clarify I should have stated the the saddle-shaped, complex, quasi-2D octahedron is only negative { ..)(.... } space whereas the original convex VE{ spherical or Euclidean } is only positive
convex shaped space.
The below was in regards to a fluctuation in the quantum wave alledged in initiate the big bang.
Fuller states that the the prime number 47 is the first prime after a 46 degree limit of all 2D or 3D geodesic calculations ergo it may be the cause of seeming randomenss of Universe.
There exists other exotic configurations of the 4-fold VE via its jitterbug transformations LINK that are somewhat difficult for me to explain without better graphics
But my favorite is the flying 2D hexagon { __ } that has an erect { yet flexi-able } tail wing { /\ } composed of two bonded/valenced triangles
........ __/\ ...........
With the flying hexagon we see what I believe is the inherent ---built in--- random fluctuation of any so called quantum wave or quantum sine-wave scenarios of Universe.
It is as tho the flying hexagon with its 7th set of erect { perpendicular triangles } says, I refuse to collapse into a 2D only existence.
Or, I am somebody will not lay flat for remain flat for no force of Universe.
Or you can not stop triangles who want to be free.
So the question becomes, wherein do we find the 'free will' within a cause > effect > resultant effects of motion, fluctuations, oscillations of Uni-V-erse of perpetual motion?
I find it as the seemingly 2D triangular tail wing { /\ } of the seemingly 2D flying hexagon { __/\ } via the jitterbug configurations of the 3D cubo{6}-octa{8}hedron.
1} The triangular tail wing stands erect as if to say I refuse to allow the totality of the 3D Vector Equilibrium to succumb to 2D existence only.
2} there is bounce away affect when the two triangles attempt to collapse into the seemingly 2D { one triangle } only tail wing i.e. when the two lower chords of the two triangles appproach each
other the bounce off each other and return to the quasi-3D tetrahedral configuration and then,
back > to > the > complex quasi-2D ---and quasi-circular-- complex octahedral polygonal sine-wave configuration, and from there,
3} to the seemingly 2D saddle-shape associated with the negative curvature of a inner surface of a torus and from there, there are multiple possible pathways the jitterbug can transform into.
All of these above, plus the 5-fold quasi-icosaehedron configuration is partly why Fuller refers to the Vector Equlibrium as the Operating System of Uni-V-erse.
To CLICK! is to make a judgement aka collapse of the sine-wave and/or can be seen as the bounce away from two triangles above and that is all deterministic.
To click is to judge, to judge is to bring that which is more dispersed configuration { Out } into a point-to-able focus { In } via graviton, darkion, photon, neutrino, electron or and aggregate
collective set thereof.
To click = BING! PING! RING! DING! etc and the appropriate collective set will move the eye, finger leg, etc.
Cause > effect > resultants > cause > effect > resultants is an eternal perpetual motion machine we call Uni-V-erse.
God/Uni-V-erse is best represented by the 4-fold cubo-octahedron{ Vector Equilibrium wherein we have the only polyhedron of Uni-V-erse that has exhibits perfect { static } balance between;
24 radii { radiating Outward }
24 chords { cohering Inward }
This becomes self evident when constructing the VE from four hexagonal planes were we find 24 radii and 24 chords LINK
This is perfect balance between the two primary forces of Uni-V-ese;
-->Inward<--- ex mass-attraction
<---Out---> ex EMRadiation
Talking to yourself?
Talking to yourself?
Sure beats engaging with lack of immoral integrity lack of intellectual integrity you spew out on all threads 50 times before breakfest. Please take a hike from this thread as you have nothing of any
relevant significance to offer. Your a waste of bandwidth and brain space Good bye.
Wait you called me rude
Your a waste of bandwidth and brain space
engaging with lack of immoral integrity lack of intellectual integrity you spew out on all threads 50 times before breakfest.
Wait you called me rude
Goodbye from this thread. Take a hike!
You have no moral or intellectual integrity to offer any here at DArt. Hit the road!
0 friends, 19 friends for me
0 friends, 19 friends for me
Please take a hike!
Hit the road!
no u
no u
Take a hike!
Hit the road! Goodbye!
no u
no u
Take a Hike!
no u
no u
Goodbye from this thread. Take a hike!
You have no moral or intellectual integrity to offer any here at DArt. Hit the road! | {"url":"https://www.debateart.com/forum/topics/1136-fluctuation-in-cosmic-quantum-sine-wave","timestamp":"2024-11-02T01:51:42Z","content_type":"text/html","content_length":"187309","record_id":"<urn:uuid:c0e7ca75-cb24-4a52-92c4-3d12769b2729>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00099.warc.gz"} |
Heroes of Machine Learning
Geoffrey Hinton - Yann LeCun - Yoshua Bengio
Geoffrey Hinton, Yann LeCun, and Yoshua Bengio, the “Godfathers of Deep Learning”, are revered pioneers in the field of deep learning, revolutionizing artificial neural networks and shaping the
course of AI. In 2018, they received the Turing Award for their contributions to Deep Learning. In this article, we explores their groundbreaking contributions, collaborative efforts, and enduring
impact, shedding light on their remarkable journeys as they propelled deep learning to unprecedented heights!
Geoffrey Hinton
When it comes to Deep Learning, I think nobody symbolizes the field better than Goeffrey Hinton, the father of Deep Learning. He even coined the term! Here are his biggest contributions to the field:
Now the guy has 327 publications, so I couldn't capture everything here but I believe this encapsulates his most impactful works. Considering the trend, it seems a lot more is going to come from him
in the coming years!
Yann LeCun
Nobody has done more for the history of Convolutional Neural Networks than Yann LeCun! Here are his biggest contributions to the field:
Keep reading with a 7-day free trial
Subscribe to The AiEdge Newsletter to keep reading this post and get 7 days of free access to the full post archives. | {"url":"https://newsletter.theaiedge.io/p/heroes-of-machine-learning","timestamp":"2024-11-11T20:17:46Z","content_type":"text/html","content_length":"160470","record_id":"<urn:uuid:cc848113-2bd6-4a53-bd08-cd337f5df15f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00332.warc.gz"} |
Understand the variations of sets in Python.
We'll cover the following
This lesson lists different implementations of a set provided by Python 3 and explains when to use which built-in support.
A set is a collection of items that does not allow duplicates. It’s an unordered collection.
⚠️ Note: In case you forget, the expression of a set involves curly-braces {} in Python. To create an empty set, we call the set() function. Using {} without any values in it will create a
dictionary, not a set.
Types of sets
Mutable set with set
The set() creates a mutable structure. Operations like dynamic insertion and deletion are allowed. Python provides basic sets’ operations, like intersection and union.
Run the following program to have an overview.
Get hands-on with 1400+ tech skills courses. | {"url":"https://www.educative.io/courses/mastering-the-art-of-programming-in-python-3/sets","timestamp":"2024-11-07T22:38:46Z","content_type":"text/html","content_length":"743441","record_id":"<urn:uuid:6368fb7d-d70c-4a76-bdd5-b9081b3d34d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00710.warc.gz"} |
When wishful thinking works — AI Alignment Forum
This idea is due to Scott Garrabrant.
Suppose you have propositions , and you want to form beliefs about whether they are true; specifically, you want to form a joint probability distribution over the events . But there’s a catch: these
propositions might refer to the joint probability distribution you come up with. If is the claim that , then you have no way to assign probabilities in a well-calibrated way. But suppose these
propositions depend continuously on the probabilities you assign to them. For instance, could be defined so that its “true” probability is , where means probability that you assigned. Let be the
function from the space of joint probability distributions over to itself that sends each probability distribution to the true probability distribution that would result if you believed . In this
case, you can be well-calibrated by letting . By Brouwer’s fixed point theorem, there will always be a way to assign probabilities in a well-calibrated way.
But could have multiple fixed points. Which one is right? You get to pick; whichever fixed point you decide to believe ends up being correct, since they are fixed points of the function determining
the true probabilities from your beliefs. Cases in which there are multiple such fixed points are cases in which you actually can make something be true by believing it. So you may as well believe
the fixed point according to which you have the highest expected utility.
As an example, suppose you’re suffering from an ailment that can be cured by placebo, and the placebo works even if you know it’s just a placebo, provided you believe that the placebo will work. When
given a pill that you know is a placebo, you may as well believe that it will cure you, since then you’ll be right, and get better.
Related to the question of what to believe is the question of what actions to take. The traditional answer is to take the action which has the highest expected utility. Another possible answer is to
act the way that you believe you will act. If we do this, then will have lots of fixed points: for any probability distribution over actions we could take, if we believe that we will take actions
according to those probabilities, then we will be correct. And picking the fixed point that maximizes expected utility recovers the original rule of picking the action that maximizes expected
A possible objection is to ask why we would restrict to fixed points, instead of just choosing to believe whatever probability maximizes the expected utility of (which we might expect to often,
though not necessarily always, be a fixed point, since having accurate beliefs is useful). A possible answer to this objection is that choosing to believe a non-fixed point because of what you expect
the consequences of choosing this belief to be isn’t possible; since you are choosing based on , you are implicitly acting as if is your true beliefs, in which case the true probability distribution
would be , and having high expected utility would not be useful.
If is not required to be continuous, then we can still almost find a fixed point by taking the closure of the graph of , and then taking the convex hull of each fiber. By Kakutani’s fixed point
theorem, this multi-valued function has a fixed point. If the agent is only assumed to know its own utility function up to an error that is either infinitesimal (as in Definability of Truth in
Probabilistic Logic) or small (as in Logical Induction), then adding a small random (unknown to the agent) error to a fixed point of the Kakutani closure of can give you a narrow probability
distribution over probability distributions that are almost fixed by . We can then take the highest expected utility of these pseudo-fixed points as in the continuous case.
This helps make sense of playing mixed strategy Nash equilibria in games. In (non-game theoretic) decision theory, it is often assumed that the outcome just depends on what action you actually take,
and you take whichever action leads to highest expected utility. Under this framework, there is no reason you would want to randomize your action. But under the assumption that strategies are common
knowledge, changes in your beliefs about your own actions will be reflected in other players’ beliefs about your actions, which influence their actions.
To a certain extent, this can also help make sense of how to pick good Nash equilibria instead of bad ones. In a game in which one player is playing best response, and the other player knows this,
and is picking the best fixed point, the result will be the Nash equilibrium that is best for the latter player. If both players are playing best fixed point, then it’s unclear exactly what happens,
since you’d need to know how to evaluate the counterfactuals in which one player changes their strategy. But you’d at least expect to end up in Pareto-optimal Nash equilibria.
For picking good Nash equilibria, there's an issue as to whether the agents have access to a joint source of randomness or not: https://agentfoundations.org/item?id=523
New Comment
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Explore the 'Scrub with a series of replace calls' approach for Isogram in Python on Exercism
def is_isogram(phrase):
scrubbed = phrase.replace('-', '').replace(' ', '').lower()
return len(scrubbed) == len(set(scrubbed))
For this approach, replace() is called a couple times to scrub the input phrase string. Thw two replace() calls are chained, so the output of the first replace() is the input for the next replace().
The output of the last replace() is the input for lower(). All of the letters are lowercased so that letters of different cases will become the same letter for comparison purposes, since A and a are
considered to be the same letter. When the replacing and lowercasing is done, the scrubbed variable will be a string having no hyphens or spaces, and with all alphabetic letters lowercased.
• A set is constructed from the scrubbed string and its len is compared with the len of the the scrubbed string. Since a set holds only unique values, the phrase will be an isogram if its number of
unique letters is the same as its total number of letters. The function returns whether the number of unique letters equals the total number of letters.
• For Alpha it would return False, because a is considered to repeat A, so the number of unique letters in Alpha is 4, and the total number of letters in Alpha is 5.
• For Bravo it would return True, since the number of unique letters in Bravo is 5, and the total number of letters in Bravo is 5.
6th Nov 2024 · Found it useful? | {"url":"https://exercism.org/tracks/python/exercises/isogram/approaches/scrub-replace","timestamp":"2024-11-09T00:39:45Z","content_type":"text/html","content_length":"43657","record_id":"<urn:uuid:35d4144b-4af1-4a4c-8d17-52eaf104338d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00009.warc.gz"} |
Why Log to the Base ‘e’ is called the Natural Logarithm.
When Napier first introduced Logarithms back in the 1600s, he thought that this was a neat little way to do calculations and increase mathematical efficiency. It was not until Euler came along and
introduced the two most frequently used forms of Logarithms: — 1) Natural and 2) Common.
When we talk about a logarithm to the base ‘e’, we call it the natural logarithm of that number. It is commonly represented as ln(x) for some positive number x¹.
When someone mentions a common Logarithm, it is the logarithm to the base of ‘10’. It is commonly represented as:
The Common Logarithm is neat. It lets us think about all numbers as powers of 10 and hence lets us use 10 as the basis of a number system. Then why did Euler name the logarithm to the base e as the
Natural Logarithm, and not the logarithm to the base 10? Here we loot a couple of scenarios, taken from physics as well as mathematics to justify Euler’s decision.
The Physics Makes Sense!
When we structured our number system, we made it so that every important thing was in some form of 10. Think about millions and billions, they are powers of 10 as well. Some say that it was
structured because we have 10 fingers and hence the most elementary form of counting can be exploited to the fullest. Maybe that is why it is instinct for us to hold 10 in such a high regard, and we
have! We gave it the title of the “Common Logarithm” after all! But when certain phenomena were observed in the study of Physics and nature, ‘e’ was prioritized simply because it showed up
everywhere. Let’s take a look at a few scenarios.
Nuclear Decay
Consider that you are a nuclear scientist who is working on a fission reactor. Given that, you know how many nuclei are there in the fission tank at a moment, can you calculate how many will be left
after some time ‘t’ has elapsed?
Well, it was observed that the rate of decay of the Nuclei depends on the number of Nuclei present in the tank. Or,
where N is the number of Nuclei at a given instant of time.
To remove the proportionality sign, we multiply a constant, say lambda, and we multiply with a negative to show that it is decaying.
Now doing some (not so) fancy rearrangement, we get:
Integrating both sides², Putting the bounds on t from 0 to any time t, we get:
Solving using the definition of logarithm,
Where N0 is the number of Nuclei at the start of the fission or the number of Nuclei that are known. Clearly, this is an exponential Law in ‘e’.
This law is called the Nuclear Decay law.
Capacitors and Batteries
Consider you are an electrical engineer. Suppose you are shown the following circuit:
An RC circuit. Image by Eugene Brennan
This is called an RC circuit since it features both a Resistance(R) and a capacitor(C). You are to calculate the charge on the plates of the capacitor at any given time ‘t’.
Applying Kirchoff’s Voltage Law³ on the entire Circuit⁴,
where V is the voltage supplied by the battery, i the current and q is the charge at any instant. Now, the current flowing is the same in both the resistance and the capacitor, and since the current
is the rate of flow of charge, we can modify our equation to become:
Keeping the like terms on one side, we get:
Now, the maximum charge the capacitor can hold is CV. Let us call that q0 or q nought.
Now integrating, putting the bounds on t from 0 to any time t:
further solving, using the definition of logarithm, we get:
Once again, an exponential relation in terms of ‘e’.
These circuits are used extensively in chargers and electric appliances to rectify the power supply of alternating current to direct current.
Chemical Kinetics
Consider that you are a chemist studying the following reaction:
Decomposition of Hydrogen Peroxide, Image by chemistrylearner.com
This is an example of a first-order reaction, where the rate of reaction is directly proportional to the concentration of the reactant, for simplicity’s sake, say R.
You are to find out the concentration of the reactant at any time ‘t’ during the course of the reaction.
We know that:
where ROR is the rate of reaction. But since the rate of Reaction is just the change in concentration of reactant over time,
again, introducing a constant and rearranging,
Integrating from t=0 to t=t,
which, as it turns out is the same exact relation as the one we obtained in Nuclear Decay. Hence all nuclear Decays are first-order reactions as well. In fact, almost all the reactions we see in
nature are first-order reactions! Here, we also find the relation is exponential in terms of ‘e’. (Conversion of the logarithmic to the exponential form is left as an exercise to the reader).
The conclusion from a Physics Point of View
From the three examples stated above, it must be clear that the number e is involved in many natural processes. Many more examples can be brought up: the Rate Law of Population⁵, The Damped
Oscillations Law⁶ and the Law of Atmospheres⁷. Including all of these would make the content repetitive, confusing and a lot more daunting. So we’ll skip them for now. (If you’re interested, I’ll
even link some reading at the end!). The conclusion drawn is that the number e is behind many of nature’s processes and observations and hence the logarithm of the base e is called the Natural
The Mathematical History
Before Napier, there were mentions of something resembling a logarithm in the works of Gregoire de Saint Vincent. In his famous work in which he quadrized⁸ a rectangular hyperbola, he mentions
several properties of the hyperbola which are similar to that of a logarithm.
Christiaan Huygens (he is the one who proposed the wave model of light), and James Gregory later hypothesized a new function called ln(x). Sometime later, Leibniz also managed to integrate dx/x and
found a striking resemblance to ln(x).
However, the number ‘e’ and its logarithm were named by none other than Leonard Euler. When working on the same problem as Saint Vincent, he found that the point 2.781…. and its reciprocal lay on the
hyperbola xy=1, the same hyperbola used by Saint Vincent, and the area below the hyperbola from (0,0) up to that point, bounded by the vertical asymptote or the y axis was 1 square unit. He thus
named the number 2.781…. as ‘e’ (people think that Euler was some kind of narcissist who named a constant in his own name, but it turns out he was biased towards vowels, and ‘a’ was a variable he had
already used at the time so he arbitrally called it ‘e’, the next vowel) and christened the logarithm to the base of ‘e’ as the natural logarithm.
Image Credits: Desmos graphing calculator
Not long after this, Euler arrived at Euler’s Identity, from which the world’s most beautiful equation is obtained.
Interestingly, Roger Coates, the proofreader of Newton’s Principia, and the discoverer of the Newton-Coates Quadrature Formula, arrived at a similar conclusion:
Notes, Conclusions and References
1: Since exponential functions are only defined for positive numbers, logarithmic functions are also defined for positive numbers as well.
2. Integration of dx/x is ln(x).
3. Kirchoff’s Voltage Law states that in a circuit, if you go around in a loop, the potential difference across the same points remains zero, or there is no net voltage around a loop of a circuit.
4. This is a direct consequence of Ohm’s Law (V=iR) and the universal law of capacitors (q=cV)
5. Rate Law of Populations (aka Natural Law of Growth):
6. Law of Damped Oscillations: https://byjus.com/jee/damped-oscillation/
7. Law of Atmospheres: https://en.wikipedia.org/wiki/Barometric_formula
8. Quadrizing a shape means to make the area of a certain shape equal to the area of a square of given side. A very famous example of this is Squaring the Circle, which means to make the area of a
circle equal to the area of the square it is inscribed in. It is Quadrizing a circle.
Thank You! If you find a mistake, or have a question, feel free to use the comments! I hope you have a nice day! :p | {"url":"https://veervishalji.medium.com/why-log-to-the-base-e-is-called-the-natural-logarithm-901e00fc1cc6?source=user_profile_page---------0-------------f1f1343ee776---------------","timestamp":"2024-11-09T14:10:40Z","content_type":"text/html","content_length":"189227","record_id":"<urn:uuid:cdba7e4e-43e4-475a-a7d3-8597c8347ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00782.warc.gz"} |
Social semiotics analysis of Palestinian mathematics textbooks for eighth grade
Wajeeh Daher^(1*), Ijteyad abu Thabet^(2)
(1) Department of Educational Sciences An-Najah National University, Palestine
(2) Department of Education, Yarmouk University, Jordan
(*) Corresponding Author
Mathematics textbook analysis can serve to understand the teaching and learning processes in the mathematics classroom. The present study utilizes a social semiotics framework to analyze the triangle
unit of the Palestinian mathematics book for grade 8. The results of the study indicate that the authors utilized the representational aspect of the mathematical object to introduce those objects to
the reader. Moreover, the nature of mathematics resulting from this unit is that of a subject that learners do not need material processes to discover, so it is enough to reason about it mentally to
arrive at the mathematical objects and relations. The authors used a plural first-person pronoun to describe the need to engage with theorems and inverse theorems. They used the singular
second-person pronoun to attract the attention of the reader to specific features of the mathematical objects. The authors did not use any pronoun when stating the theorem. Some of the connectors
were verbs, nouns, and sentences, where the most used connector was the sentence, especially in reasoning. This use of the sentence in mathematical reasoning indicates that the authors wanted to
advance the mathematical reasoning as a narrative to facilitate it for the reader.
Book analysis, mathematics books, functional grammar, social semiotics
Abel, K., & Exley, B. (2008). Using Halliday's functional grammar to examine early years worded mathematics texts. The Australian Journal of Language and Literacy, 31(3), 227.
Alshwaikh, J., & Candia Morgan. (2013). Analysing the Palestinian school mathematics textbooks: A multimodal (multisemiotic) perspective. In Smith, C (Ed.). Proceedings of the British Society for
Research into Learning Mathematics, (pp. 70–75). Retrieved from http://hdl.handle.net/20.500.11889/2474
Askool, M. (2019). The analysis of Grade 9 Palestinian mathematics book in light of NCTM standards.Journal of Elementary School for Educational and Humanistic Sciences, 11(1), 337-355 (In Arabic).
Retrieved from http://www.becm-iq.org/papers/uobj_paper_2019_2166696.pdf
Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and US elementary mathematics textbooks. International Journal of Science and Mathematics Education, 15(8), 1521–1540.
Farrugia, M. T. (2018). A functional linguistics analysis of a mathematics register expressed through two languages. In N. Planas, & M. Schΰtte (Eds.). Proceedings of the IV ERME Topic Conference
’Classroom-Based Research on Mathe- Matics and Language’, (pp. 57–64). Dresde, Germany: ERME. Retrieved from https://hal.archives-ouvertes.fr/hal-01856477/document
Halliday, M. A. K. (1973). Explorations in the functions of language.London: Edward Arnold.Retrieved from https://eric.ed.gov/?id=ED095550
Halliday, M. A. K. (1985). An Introduction to Functional Grammar (E. Arnold, ed.). London: Edward Arnold.
Halliday, M. A. K. (2007). Language as social semiotic: Towards a general sociolinguistic theory. In M. A. K. Halliday. Language and Society, (pp. 169-202). Edited by J. Webster. London: Continum.
Halliday, M.A.K., & Matthiessen, C. M. I. M. (1999).Construing experience through meaning: a language-based approach to cognition. London: Cassell.
Halliday, M. A. K., & Matthiessen, C. M. I. M. (2004). An Introduction to Functional Grammar(Third edit). London : Hodder Arnold.
Jamieson-Proctor, R., & Byrne, C. (2008). Primary teachers’ beliefs about the use of mathematics textbooks. In: M. Goos, R. Brown & K. Makar (Eds.), Proceedings of the 31st Annual Conference of the
Mathematics Education Research Group of Australasia, (pp. 295–302). Brisbane. QLD: MERGA. Retrieved from http://www.merga.net.au/documents/RP332008.pdf
Martin, J. R., & Rose, D. (2007). Working with Discourse (2nd ed.). New York, NY: Continum.
Morgan, C. (1996). “The language of mathematics”: towards a critical analysis of mathematical text. For the Learning of Mathematics. 16(3), 2–10. Retrieved from https://www.jstor.org/stable/40248208?
Morgan, C. (1998). Writing Mathematically: The Discourse of Investigation. London: Falmer.
Murdaningsih, S., & Murtiyasa, B. (2016). An Analysis on Eight Grade Mathematics Textbook of New Indonesian Curriculum (K-13) Based on Pisa’s Framework. JRAMathEdu (Journal of Research and Advances
in Mathematics Education), (1), 14–27. https://doi.org/10.23917/jramathedu.v1i1.1780
O’Keeffe, L. (2013). A Framework for Textbook Analysis. International Review of Contemporary Learning Research. International Review of Contemporary Learning Research, 2(1), 1–13. Retrieved from
O’Keeffe, L., & O’Donoghue, J. (2011). Mathematics Textbook Analysis: The Significance of Textbook Features to Student Learning. In Proceedings of the Seventh Congress of the European Society for
Research in Mathematics Education. Rzeszów, Poland: University of Rzeszów.
Pimm, D. (1987). Speaking Mathematically: Communication in Mathematics Classrooms.London: Routledge and Kegan Paul. https://doi.org/10.4324/9781315278858
Rowland, T. (1992). Pointing with pronouns. For the Learning of Mathematics, 12(2), 44–48. Retrieved from www.jstor.org/stable/40248049
Rowland, T. (1999). Pronouns in mathematics talk: power, vagueness and generalisation. For the Learning of Mathematics, 19(2), 19–26. Retrieved from https://www.jstor.org/stable/40248049?seq=1
Schleppegrell, M. J. (2004). The language of schooling: A functional linguistics perspective New Jersey, USA: Lawrence Erlbaum Associates.
Törnroos, J. (2005). Mathematics textbooks, opportunity to learn and student achievement. Studies in Educational Evaluation, 31(4), 315–327. https://doi.org//10.1016/j.stueduc.2005.11.005
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Wronskian Calculator | Solve Linear Differential Equation [Online]
Introduction to Wronskian Calculator:
Wronskian calculator is the best online source that helps you in solving the linear differential equation. It uses the differentiate method inside the determinant of the function to get solution in
less than a minute.
It is a beneficial tool for students, teachers or researchers as it gives the solution of ODEs equation even for complex functions without taking any external assistance.
What is Wronskian?
Wronskian method is a process in which you determine whether the linear differential function or differential equation is linearly dependent or independent on linear algebra. The wroniskan method is
represented with the symbol “W''.
For f1, f2,...fn functions, it uses the differential and determinate matrix method in which if the wronskian method solution is non zero then your function is linearly independent. On the other hand,
if the given function solution is zero then the function is linearly dependent.
Formula of Wronskian:
The wronskian method formula is based on the function differentiation f1(x), f2(x),…, fn(x) and the determinate matrix where you put the function and solve it.
$$ W(f_1, f_2,..., f_n)(x) \;=\; \left(\begin{matrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1’(x) & f_2’(x) & … & f_n’(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \
cdots & f_n^{(n-1)}(x) \end{matrix} \right) $$
How to Calculate the Wronskian?
For the calculation of Wronskian of a set of linear functions, the wronskian determinant calculator uses the determinant method to find whether a set of functions is linearly independent. Here's a
stepwise guide on how to calculate the Wronskian manually:
Step 1:
Identify the linear functions f1(x), f2(x),…,fn(x) and variable of differentiation.
Step 2:
Find the first derivative of each function fi(x), calculate fi′(x), fi′′(x),…,fi(n − 1)(x).
Step 3:
According to the numbers of functions, add the differential function value f`(xi) and the linear differential function f(xi) of determinate matrix. For example, 2 by 2 determinant matrix for wroiskan
$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1 (x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| $$
Step 4:
Solve the determinant whether the given matrix is 2 by 2 determinate or 2 by 3 determinant matrix.
$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1(x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| \;=\; f_1(x) . f_2’(x) - f_2(x) . f_1’(x) $$
Step 5:
After simplification, you get the solution of the Wronskian method problem which determines whether the linear function is independent or dependent.
Solved Example of Wronskian Method:
A solved example of the Wronskian method is given below to understand how the Wronskian calculator with steps works.
Example: Find the wronskian of the following:
$$ f_1 \;=\; x^2 + 4,\; f_2 \;=\; sin(2x) $$
The given function is,
$$ f_1 \;=\; x^2 + 4,\; f_2 \;=\; sin(2x) $$
Differentiate the function f1(x) and f2(x) with respect to x.
$$ \frac{d}{dx} (x^2 + 4) \;=\; 2x $$
$$ \frac{d}{dx} (sin(2x)) \;=\; 2\; cos (2x) $$
As the given function has 2 by 2 matrix so the require determinant matrix become,
$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1(x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| $$
Now make a determinate matrix and put the linear function and derivative function value,
$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} x^2 + 4 & sin(2x) \\ 2x & 2\;cos(2x) \\ \end{matrix} \biggr| $$
Solve the determinant matrix as per the rule of matrix determination,
$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1(x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| \;=\; f_1(x) . f_2’(x) - f_2(x) . f_1’(x) $$
$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} x^2 + 4 & sin(2x) \\ 2x & 2\;cos(2x) \\ \end{matrix} \biggr| \;=\; 2x^2 cos (2x) - 2x\; sin(2x) + 8\; cos(2x) $$
The result of given linear differential function using wronkisan method is,
$$ W (f_1, f_2)(x) \;=\; 2x^2\; cos(2x) - 2x\; sin(2x) + 8\; cos(2x) $$
How to Use the Wronskian Calculator 3x3?
The Wronskian matrix calculator has a simple design that helps to solve the given linearly independent function. You just need to put the input value.
• Enter the linear differential function in the input field of wronskian method calculator.
• Choose the variable of differentiation from the given list of wronskian determinant calculator.
• Check your given input function to get the correct solution of the linear differential function question.
• Click on the "Calculate" button to get the result of the given linear differential function problems.
• If you want to understand the calculation process of our Wronskian method calculator then use the load example option and get its solution.
• Click the “Recalculate” button for the calculation of more examples of linear differential functions with the solution.
Final Result of Wronskian Calculator:
Wronskian linear independence calculator provides you with a solution as per your input value problem. It contain as:
Click on the result button so that you get the solution of your linear differential function.
When you click on the steps option, you get the step by step result of linear differential questions.
Benefits of Wronskian Matrix Calculator:
The Wronskian differential equations calculator has many benefits that you get when you use it to solve Wronskian method differential equation problems. Our tool only takes the input function and
provides you a solution instantly. These benefits are
• It is a reliable tool as it gives you accurate solutions of linear differential equation problems.
• It is a speedy tool that provides solutions to the given wronskian method problems in a few seconds.
• It is a learning tool as it helps you to learn the wronskian method for linear differential function online.
• It is a handy tool that can solve various types of linear differential equation problems easily.
• Wronskian linear independence calculator is a free tool that allows you to calculate the linear differentiation functions many times without spending anything.
• Wronskian determinant calculator is an easy-to-use tool, anyone or even a beginner can easily use it to get the solution of wronskian method problems. | {"url":"https://pinecalculator.com/wronskian-calculator","timestamp":"2024-11-12T07:31:47Z","content_type":"text/html","content_length":"48278","record_id":"<urn:uuid:36504bb7-5964-45a4-8ca4-9c851459f76f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00809.warc.gz"} |
Jean Feydy's home page
PhD thesis
• Geometric data analysis, beyond convolutions,
Jean Feydy, under the supervision of Alain Trouvé.
Defended on July 2, 2020 in front of Xavier Pennec, Jean-David Benamou, Marc Niethammer, Pierre Alliez and Alexandre Gramfort. I was awarded two PhD thesis awards by the AFRIF (French association
for shape analysis and recognition) and the Chancellerie des Universités de Paris (édition 2021).
Links: Official version, Thesis + Latex, Slides + Latex.
Journal Papers
• Collective proposal distributions for nonlinear MCMC samplers: mean-field theory and fast implementation,
Grégoire Clarté, Antoine Diez, Jean Feydy,
Electronic Journal of Statistics, 2022, links: Paper, Code.
• Kernel operations on the GPU, with autodiff, without memory overflows,
Benjamin Charlier*, Jean Feydy*, Joan Glaunès*, François-David Collin, Ghislain Durif,
Journal of Machine Learning Research, 2021, links: Abstract, Paper, Code.
Conference Papers
• DiffMaSIF: Surface-based Protein-Protein Docking with Diffusion Models,
Freyr Sverrisson, Mehmet Akdel, Dylan Abramson, Jean Feydy, Alexander Goncearenco, Yusuf Adeshina, Daniel Kovtun, Céline Marquet, Xuejin Zhang, David Baugher, Zachary Wayne Carpenter, Luca Naef,
Michael Bronstein, Bruno Correia,
MLSB 2023 (NeurIPS workshop), links: Paper, Workshop.
• Physics-informed deep neural network for rigid-body protein docking,
Freyr Sverrisson, Jean Feydy, Joshua Southern, Michael M Bronstein, Bruno Correia,
MLDD 2022 (ICLR workshop, spotlight presentation), links: Paper.
• Accurate point cloud registration with robust optimal transport,
Zhengyang Shen*, Jean Feydy*, Peirong Liu, Ariel Hernán Curiale, Ruben San José Estépar, Raúl San José Estépar, Marc Niethammer,
NeurIPS 2021, links: Paper, Latex, Code.
• Fast end-to-end learning on protein surfaces,
Freyr Sverrisson*, Jean Feydy*, Bruno Correia, Michael Bronstein,
CVPR 2021, links: Paper, Poster, Slides, Video, Code.
• Fast geometric learning with symbolic matrices,
Jean Feydy*, Joan Glaunès*, Benjamin Charlier*, Michael Bronstein,
NeurIPS 2020 (spotlight presentation), links: Paper, Slides + Latex, Poster + Latex, Links and Videos, Website, Code.
• Fast and scalable optimal transport for brain tractograms,
Jean Feydy*, Pierre Roussillon*, Alain Trouvé, Pietro Gori,
MICCAI 2019, links: Paper, Poster + PowerPoint, Website, Code.
• Interpolating between optimal transport and MMD using Sinkhorn divergences,
Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouvé, Gabriel Peyré,
AiStats 2019, links: Paper, Slides, Poster + Latex, Website, Code.
• Global divergences between measures: from Hausdorff distance to optimal transport,
Jean Feydy, Alain Trouvé,
ShapeMI 2018 (MICCAI workshop, oral presentation), links: Paper, Slides + Latex, Code.
• Optimal transport for diffeomorphic registration,
Jean Feydy, Benjamin Charlier, François-Xavier Vialard, Gabriel Peyré,
MICCAI 2017 (oral presentation), links: Hal, Arxiv, Code, Slides (+videos), Poster, Latex.
• Distortion minimizing geodesic subspaces in shape spaces and computational anatomy,
Benjamin Charlier, Jean Feydy, David W. Jacobs and Alain Trouvé,
VipImage 2017.
Medical Papers
Maths and CS Talks
• The geometric software stack: past, present, future.
October 2024, SMAI-SIGMA, CIRM, Marseille: Pdf, Latex, Workshop.
October 2024, Geometry and Computing, CIRM, Marseille: Pdf, Latex, Workshop.
May 2024, Geometric sciences in action, CIRM, Marseille: Pdf, Latex, Workshop.
• Computational optimal transport: recent speed-ups and applications.
August 2024, Machine learning in infinite dimensions, Bath: Pdf, Latex, Videos, Workshop.
July 2024, SciML 2024, Strasbourg: Pdf, Latex, Videos, Workshop.
June 2024, ANEDP, Laboratoire Paul Painlevé, Lille: Pdf, Latex, Videos.
March 2024, PSDOL, Lagrange center, Paris: Pdf, Latex, Videos, Workshop.
• Software bottlenecks for 3D AI.
November 2024, Sciences at PSC, PariSanté Campus: Pdf, Latex.
February 2024, X-IA #16, BPI France, Paris: Pdf, Pptx, Latex, Workshop.
• Optimal transport with 3D shapes.
December 2023, GT CalVa, Université Paris-Dauphine: Pdf, Latex, seminar.
December 2023, G-Stats Seminar, Inria Sophia-Antipolis: Pdf, Latex, seminar.
December 2023, SMAI-SIGMA day, Jussieu: Pdf, Latex, workshop.
• Computational optimal transport: mature tools and open problems.
November 2022, Measure-theoretic approaches and optimal transportation in statistics, Institut Henri Poincaré: Pdf, Latex, workshop.
August 2022, Workshop on mathematical imaging and surface processing, Oberwolfach: Pdf, Latex, workshop.
July 2022, Frontiers in Design Representation, University of Maryland: Pdf, Latex, Tutorial (HTML), Summer school.
June 2022, Curves and Surfaces 2022, Arcachon: Pdf, Latex.
June 2022, University of Göttingen: Pdf, Latex.
• Fast libraries for geometric data analysis.
May 2023, Healthcare AI grand round, Nvidia: Pdf, Latex.
May 2023, Workshop on geometry/physics-informed neural networks, Thales: Pdf, Latex.
February 2023, Machine Learning Coffee Seminar, Finnish Center for Artificial Intelligence: Pdf, Latex, Seminar.
July 2022, online meeting with Nvidia: Pdf, Latex.
May 2022, joint HeKA-Soda seminar, PariSanté Campus: Pdf, Latex.
• Fast geometric libraries for vision and data sciences.
April 2022, DataShape seminar, Inria Saclay: Pdf, Latex.
December 2021, GRAPES software and industrial workshop, Inria Sophia: Conference, Pdf, Latex.
December 2021, AI and healthcare seminar, Centre de Recherche des Cordeliers: Pdf, Latex.
November 2021, JCJC développement, Inria Saclay: Conference, Pdf, Latex.
November 2021, Robotic Perception team, Université de Picardie Jules Verne, Amiens: Pdf, Latex.
October 2021, GdR MIA, Institut Henri Poincaré: Conference, Pdf, Latex.
• Calcul géométrique rapide pour la vision et les sciences des données.
September 2021, Orasis 2021, Lac de Saint-Ferréol: Conference, Pdf, Latex.
• Fast geometric learning with symbolic matrices.
January 2021, CogSys seminar, DTU Compute (Online): Pdf, Latex.
December 2020, NeurIPS 2020 (Online): Spotlight presentation + Latex, Poster + Latex, Links and Videos.
• Geometric data analysis, beyond convolutions.
April 2021, Signal Processing Laboratory (LTS4, EPFL, Online): Pdf, Latex.
March 2021, Image, Visual and Language Computing Seminar (UNC Chapel Hill, Online): Pdf, Latex.
January 2021, Centre de Vision Numérique (CentraleSupélec - Inria Saclay, Online): Pdf, Latex.
October 2020, Centre de Recherche des Cordeliers (Online) - in French: Pdf, Latex, Video.
September 2020, University College London (Online) - with more applications: Pdf, Latex, Video, Workshop.
July 2020, PhD defense (Online): Pdf, Latex.
December 2019, King's College London: Pdf, Latex.
• Geometric loss functions for shape analysis
July 2020, SIAM Imaging Sciences 2020 (Online): Pdf, Video, Latex.
• Sorting points in dimension D > 1.
April 2021, Sea Ice Modeling and Data Assimilation (Dartmouth, Online): Pdf, Latex.
February 2020, Twitter London: Pdf, Latex.
• Discrete optimal transport: scaling up to 1M samples in 1s.
June 2019, "People in optimal transportation and applications" workshop, Cortona: Pdf, Latex.
• Robust matching of measures with Optimal Transport.
February 2019, GTTI, ENS Cachan: Pdf, Latex.
December 2018, BIRS center, Banff: Pdf, Latex.
November 2018, Télécom ParisTech: Pdf, Latex.
• Global divergences between measures, from Hausdorff distance to Optimal Transport.
September 2018, ShapeMI workshop, MICCAI 2018 (Granada): Pdf, Latex + Code.
July 2018, Curves and Surfaces 2018 (Arcachon): Pdf, Latex + Code.
• Normalizing LDDMM metrics using autodiff, an introduction to KeOps.
June 2018, SIAM Imaging Sciences 2018 (Bologna): Pdf, Latex + Code.
November 2017, Isaac Newton Institute (Cambridge): Pdf, Latex.
• Optimal transport for diffeomorphic registration: a global and robust data attachment term.
September 2017, MICCAI 2017, Québec City: Pdf (+videos), Poster, Latex.
June 2017, Asclepios Inria Team: Pdf, Latex.
Radiology Talks
• Key tasks for AI in musculoskeletal imaging
December 2023, FHU Plan&Go, Hôpital Pasteur de Nice: Pdf, Pptx, Latex, Conference.
June 2023, ESSR 2023: Pdf, Pptx, Latex, Conference.
• Quels logiciels pour l'apprentissage en anatomie ?
October 2023, JFR 2023, Palais des Congrès de Paris: Pdf, Ppt, Latex, Conference.
March 2023, IABM 2023, Institut Curie: Pdf, Latex, Conference.
• L'imagerie médicale, un calcul structuré.
October 2020, Institut du Cerveau (Online): Pdf, Latex, Video, Conference.
• Artificial "neural networks": what radiologists should know.
(French) Mai 2020, DIU Neuro-radiologie vieillissement (en ligne): Pdf, Video;
(English) June 2019, Harvey Cushing symposium (American Hospital of Paris):
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Gopal Prasad, 2016
Quotients of symmetric spaces of semi-simple Lie groups by torsion-free arithmetic subgroups are particularly nice Riemannian manifolds which can be studied by using diverse techniques coming from
the theories of Lie Groups, Lie Algebras, Algebraic Groups and Automorphic Forms. One such manifold is a "fake projective plane" which is, by definition, a smooth projective complex algebraic surface
with same Betti-numbers as the complex projective plane but which is not isomorphic to the latter.
The first example of a fake projective plane (fpp) was constructed by David Mumford in 1978, and it has been known that there are only finitely many of them. In the theory of algebraic surfaces, it
was an important problem to construct them all and determine their geometric properties. In a joint work with Sai-Kee Yeung, we have classified them and given an explicit way to construct them all
(it turns out that there are exactly 100 of them). We have also determined higher dimensional analogues of the fpp's. These works have required considerable amount of number theoretic bounds and
computations and also inputs from the cohomology of Shimura varieties.
In the second half of my talks, I will discuss another well-known problem which was formulated by Mark Kac in a very attractive way as "Can one hear the shape of a drum?", and its solution, for
arithmetic quotients of symmetric spaces, obtained in a joint paper (in Publ Math IHES) with Andrei Rapinchuk. For the solution, we introduced a notion of "weak commensurability" of arithmetic, and
more general Zariski-dense, subgroups and derive very strong consequences of weak commensurability.
According to Wikipedia:
Prasad’s research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive
p-adic groups.
Prasad has received the Guggenheim Fellowship, the Humboldt Senior Research Award, and the Raoul Bott Professorship at the University of Michigan. He was awarded the Shanti Swarup Bhatnagar prize (by
the Council of Scientific and Industrial Research of the Government of India), and has received Fellowships in the Indian National Science Academy, among numerous other honors. In 2012 he became a
fellow of the American Mathematical Society.
Prasad was the Managing Editor of the Michigan Mathematical Journal for over a decade, an Associate Editor of the Annals of Mathematics for six years, and is an editor of the Asian Journal of
Mathematics since its inception.
He earned his bachelor's degree with honors in Mathematics from Magadh University in 1963. Two years later, in 1965, he received his masters in Mathematics from Patna University. After a brief stay
at the Indian Institute of Technology Kanpur in their Ph.D. program for Mathematics, Prasad joined TIFR for his PhD program in 1966. There Prasad began a long and extensive collaboration with his
advisor M. S. Raghunathan on several topics including the study of lattices in semi-simple Lie groups. In 1976, Prasad received his Ph.D. from University of Mumbai. Prasad became an Associate
Professor at TIFR in 1979, and a Professor in 1984. He left TIFR to join the faculty at the University of Michigan in Ann Arbor in 1992, where he is the Raoul Bott Professor of Mathematics.
Prasad's early work was on discrete subgroups of real and p-adic semi-simple groups. He proved the "strong rigidity" of lattices in real semi-simple groups of rank 1 and also of lattices in p-adic
groups, see [1] and [2]. He then tackled group-theoretic and arithmetic questions on semi-simple algebraic groups. He proved the "strong approximation" property for simply connected semi-simple
groups over global function fields [3]. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, and computed the "metaplectic kernel" for
isotropic groups, see [11], [12] and [10]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14].
Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13].
In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups, [4]. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel
and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6]. The volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification,
by Gopal Prasad and Sai-Kee Yeung, of fake projective planes (in the theory of smooth projective complex surfaces) into 28 non-empty classes [21] (see also [22] and [23]). This classification,
together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes. This list consists of exactly 50 fake projective planes, up to isometry
(distributed among the 28 classes). This work was the subject of a talk in the Bourbaki seminar.
Prasad has worked on the representation theory of reductive p-adic groups with Allen Moy. The filtrations of parahoric subgroups, referred to as the "Moy-Prasad filtration", is widely used in
representation theory and harmonic analysis. Moy and Prasad used these filtrations and Bruhat-Tits theory to prove the existence of "unrefined minimal K-types", to define the notion of "depth" of an
irreducible admissible representation and to give a classification of representations of depth zero, see [8] and [9].
In collaboration with Andrei Rapinchuk, Prasad has studied Zariski-dense subgroups of semi-simple groups and proved the existence in such a subgroup of regular semi-simple elements with many
desirable properties, [15], [16]. These elements have been used in the investigation of geometric and ergodic theoretic questions. Prasad and Rapinchuk introduced a new notion of
"weak-commensurability" of arithmetic subgroups and determined "weak- commensurability classes" of arithmetic groups in a given semi-simple group. They used their results on weak-commensurability to
obtain results on length-commensurable and isospectral arithmetic locally symmetric spaces, see [17], [18] and [19].
Together with Jiu-Kang Yu, Prasad has studied the fixed point set under the action of a finite group of automorphisms of a reductive p-adic group G on the Bruhat-Building of G, [24]. In another joint
work, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring (DVR), [25].
In collaboration with Brian Conrad and Ofer Gabber, Prasad has studied the structure of pseudo-reductive groups, and also provided proofs of the conjugacy theorems for general smooth connected linear
algebraic groups, announced without detailed proofs by Armand Borel and Jacques Tits; their research monograph [26] contains all this. The monograph [27] contains a complete classification of
pseudo-reductive groups, including a Tits-style classification and also many interesting examples. The classification of pseudo-reductive groups already has many applications. There was a Bourbaki
seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups.
See more information and footnotes on Wikipedia. | {"url":"https://www.buffalo.edu/cas/math/news-events/myhill/gopal-prasad.html","timestamp":"2024-11-13T18:39:30Z","content_type":"text/html","content_length":"52085","record_id":"<urn:uuid:7b3080f0-3ddc-4852-aa07-4d666bd37ab3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00278.warc.gz"} |
How do experts optimize model performance in Python programming assignments focused on regression analysis in machine learning? | Pay Someone To Do My Python Assignment
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But following this, how I would analyze the I3-IR data series and understand its results has been a central focus of my research. In that regard, the idea to enhance a given dataset of data at
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This is the third post, as opposed to the last one (the most comprehensive in the series that follows) in the series that follows, so the next code snippet will in addition. #!/usr/bin/python import
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“params”: (5, 10), “data”: (5, 13), “params_init”: (2, 14), “tables”: (8, 14), } import models model = Model(load_module=sys.modules.load(‘models.ndbg’)) model_args = [“__main__”, “__How do experts
optimize model performance in Python programming assignments focused on regression analysis in machine learning? In previous exercises, I reviewed the structure and quality of regression analysis as
a point of departure. First, Click This Link brief review of regression analysis techniques for machine learning problems, where regression analysis approaches some of the same phenomena through
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A regression tree is essentially the summary of the statistical regression trees for all classes with high probability. Our method is a sophisticated extension of regression trees into ensemble
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Workshop on Machine Learning and Applied Probabilistic Methods, Barcelona, click Spain, 12.362548). Our process may be more efficient in training, but is | {"url":"https://pythonhomework.com/how-do-experts-optimize-model-performance-in-python-programming-assignments-focused-on-regression-analysis-in-machine-learning","timestamp":"2024-11-03T04:02:04Z","content_type":"text/html","content_length":"97066","record_id":"<urn:uuid:03642bdf-389c-471a-a9ef-5d7a73990c0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00139.warc.gz"} |
Youth rates revisited
prior post
noted the big jump in youth unemployment rates since the abolition of the separate youth minimum wage. Let's go back to this briefly.
If we assume that the youth rate will always be some fixed amount above the adult rate, then the current run-up, as I noted earlier, is highly anomalous and seems very plausibly explained by the
minimum wage change.
Some folks reckon the better measure is the ratio: the youth unemployment rate will always be some multiple of the adult rate. If you measure the ratio of the two over time, the current ratio is
high, but there isn't an obvious break point in 2008.
The graph below has (thanks Stephen Hickson!) the unemployment rate for those aged 15-19 and the unemployment rate for everyone else (aged 19 and up). It looks to me like the proper relationship is a
combination of a level shift and a multiplicative effect. When the adult rate is very low - below four percent or so - the youth rate bounces around at a point about 10 to 12 points higher than the
adult rate. When the adult rate is high, the youth rate exceeds that constant by a multiple of the adult rate.
As always, I take this kind of thing over to Stata to find out what's going on. First, let's rule out that what we have going on is only a level shift or only a multiplicative effect. I run ordinary
least squares with the youth unemployment rate (15-19 year olds) as dependent variable and the adult rate (20 and up) and a constant as independent variables. If it's just a level shift, the
coefficient will be significant, close to 1 in magnitude, and with a significant constant term around 10. If it's just a ratio effect, the constant will be insignificant and we'll have a coefficient
somewhere around 3.
Both the constant and the adult rate come up highly significant. So, over the period 1986 to present, we can expect the youth rate to be 1.44 times the adult rate (the multiplicative effect - about
44% above the adult rate) plus a constant of 9 percentage points. So if the adult rate is 5, the youth rate should be 16.2. We've ruled out the "it's just ratios" argument - there is a constant term
in there; we've also ruled out that it's just a level shift because the coefficient is significantly greater than 1.
Moreover, when we plot the residuals, we find something pretty interesting. Recall that the residuals are the difference between the model's expected youth unemployment rate and the actual youth
unemployment rate. A positive residual means that youth unemployment was higher than the model predicted; negative means it was lower.
If we look at the top graph, we see youth unemployment rates went up a lot during the recession of the early 1990s. But over that period, youth unemployment rates were never more than a couple of
points above what the very simple model predicted (residuals graph, above). In recessions, it does look like the youth rate gets hit harder than the adult rate. But look at what happens starting
around fourth quarter 2008. We now have residuals that blow up the model. Something really weird starts happening to the youth unemployment rate at the end of 2008. Youth unemployment is now about 10
points higher than we'd expect using the simple model. Again, the residual here is telling us that the current youth unemployment rate is about 10 points higher than would be expected given the prior
relationship between the youth and adult unemployment rates.
I tried a few different variations allowing the constant and the slope to shift for high and for low levels of adult unemployment. But none of that made any substantial difference. Putting in a
variable allowing the slope and constant to vary with regime (youth rate or no youth rate) made a big difference, but you'd of course expect that given the residuals plot above.
This remains very much a first cut: something I may someday assign as an honours project for more thorough sorting out. The econometrics here are very simplistic and do nothing to account for
differences in labour force participation rates or the obvious problem of serial correlation in the time series data. But the simple model is still pretty telling. If we allow youth unemployment
rates to vary both as a level shift above the adult rate and as a multiple of the adult rate, which is what we're doing when we run the simple regression with a constant term, we still have a jump in
the current youth unemployment rate that is well above that seen in prior recessions.
My first cut explanation remains the abolition of the youth minimum wage.
13 comments:
1. Couple of criticisms:
The numbers on sickness & invalid benefits have increased significantly over last decade, now approx. 80% higher. These need to be added into the general unemployment rate.
The number of 15 - 19 yo.s entering the job market is not constant. Due to NZs demographics we had a low number of them entering the market during the 1995 - 2005 period; followed by higher
numbers entering after 2005. Also we had a glut in 1990.
Correcting the graph for these will reduce the ratio in 1990, increase the ratio in 1995 - 2005 and reduce the peak in 2009. Basically make it look more like a slope and less like a hockey stick.
2. This is Household Labour Force Survey data, not numbers of unemployment beneficiaries. But you're right - somebody on long term benefit is unlikely to be listing himself as in the labour force
looking for work. I'd be very surprised if correction for either of those would remove the big kink at end 2008 though. It might be a bit smaller in magnitude, but unless those changes hit with
particular force end-2008, it won't change the inflection point and won't do much to the slope. From the other side, what's happened to DPB rates for youth? It'll have the same effect but in the
opposite direction of your corrections, right?
3. Young mothers are not likely to participate in labour market, so whether they are on DPB or not seems irrelevent.
If a shift was used to progressively hide unemployment (compared to that acknowledged pre-1999) in the sickness/invalid lists. The 1990 recession (youth rate 23%, total 10%) compares to the
current recession (26%, 6%), but if you correct the later figure for the 50,000 or so who have been moved to the sickness/invalid our current recession becomes much more comparable (26%, 6% +
And on to the demographics, its not a constant supply of new entrants.
Numbers of 15 - 19 yo.s in NZ:
1986 ~ 300,000
1991 ~ 290,000
1996 ~ 270,000
2001 ~ 280,000
2006 ~ 315,000
2011 ~ 320,000 (projected)
4. DPB recipients are unlikely to be in the labour market in the same way that sickness/invalid beneficiaries are unlikely to be in the labour market.
Would those 50K or so be in the labour force absent being on those benefits? All of them, or just some? Are they basically the group that would have been exempt from welfare work requirements in
the US because of large numbers of barriers to work?
If your demographics story is the right one, then the kink in the curve should have come at 2006, not at end-2008. Numbers look stagnant over the relevant period....
5. There is significant age based disparity in job market participation across 15 - 19 year olds that perhaps induces lag...
6. If the bulk of workforce entry hits at age 17, it would require that a huge blip of 17 year olds hit the workforce end '08.
7. A recession occured in 2008.
The normalisation is flawed. 2006 data shows a 12.5% increase in 15-19 yo.s and you are inferring this to be insignificant. However you do find a derived constant of 9% of the same 15-19 yo.s to
be "highly significant" and use this to justify your conclusion. 12.5% > 9%.
8. Look at that residuals graph again. If your story were right, we'd expect the residuals to be tracking all over with the changes in the 15-19 age group. So we should have a dropping residual from
'86 to '96, then increasing slowly to '01, then the jump to '06 and leveling off. Instead, it rises sharply from '86 to '94, levels off through '00, then lots of noise around a zero mean through
'07, then slight rise before the big spike end '08.
Why does the dropping proportion of 15-19 year olds '86-'96 not result in a drop in the residual?
There are a billion things that a more thorough analysis could and ought to correct for. But eyeballing the path of the residuals doesn't suggest that the age-cohort numbers is a big omitted
variable problem.
9. Why does the dropping proportion of 15-19 year olds '86-'96 not result in a drop in the residual?
What? You mean its a straight line when it comes to justifying your argument, but a complex, subtly, variable dataset when useful in dismissing demographic change? Cool.
Yes, if a residual model of employment market is accepted, then incorporating demographics predicts a slight decrease followed by an abrupt spike. Which I challenge you to find less accurate a
prediction than a straight line.
There is one only significant increase in the size of the 15-19 yo. demographic over the observed period which (with correction for lag) falls bang on the only significant spike in youth
unemployment. Incorporating demographics makes a better model.
10. BTW - from the other posting, I don't see a spike in 1994.
11. Think hard about what a regression residual means and what omitted variable bias looks like in a plot of residuals.
I don't know what you mean by "residual model of unemployment". Again: the plot above shows the difference between the actual youth unemployment rate and the one that's predicted by a simple
model that uses only a constant and the adult unemployment rate.
Simple example of what a serious omitted variable problem would look like: Suppose that, for whatever reason, youth unemployment would jump up in any year that ends in the number 8, and I didn't
correct for years ending in 8 in my model. We'd then expect a big positive residual in any year that ends in 8. Suppose that we see a big jump at the end of 2008 and someone said "Aha! That's
just because it ends in 8 and everyone knows that it goes up in years ending in 8!" But if we don't see the big spikes up in the residual in other years ending in 8, then that probably isn't
something that's really causing a big omitted variable problem surrounding years ending in 8 (and the posited "8-related unemployment hypothesis" is likely false). There could of course be two
omitted variables, with one cancelling the other out in all of the other "8" years, but that's less likely than that it's just not that big a problem.
From your numbers, the 15-19 population group drops by 30K over the period 86-96. Over that same period, the residual rose considerably. So if there's an omitted variable bias induced by
population, it's suggesting that over this period, dropping population in that age cohort correlates with increased youth unemployment. From the 2001-2011 period, your numbers suggest a 50K
increase (plausible that the increase to end 2009 is about the same magnitude as the prior drop) and saying that that's what's causing the current very high positive residual.
I'm suggesting that it makes no sense at all to expect that the omitted variable problem, if there is one, swiches sign half way through the time series. It's worse than the "missing 8 variable"
illustration above: it's as though the prior 8s were associated with negative residuals rather than positive ones.
If the residual were declining rather than increasing from 86-96, I'd go back and re-run the regression - I'd then suspect potential omitted variable problems. But as it's increasing over that
period, I can't see it being the cause of major concern (though you should feel free to go and run your own regressions if you feel otherwise).
12. I mean that the residual constant and multiplication you apply presents a false picture. Therefore decrease in supply 86-96 cannot be considered an inverse analogy to the increase 01-11.
If the market is near or over saturation a decrease in supply will have little and an increase in supply will have great effect on unemployment. The period 86-96 is one of increasingly high total
unemployment, which suggests a high degree of saturation.
For analogy sake imagine there were a car company call Chysler that make plasticky trash product and in 2008 the bottom fell out of the car market. Now Chysler have always had an inventory
problem so they had derived a formula based on an industry 20-yr average inventories multiplied by a factor and with an add-on constant to define the scope of the problem. So when recession hits
Chysler turn to the man who devised the formula and ask him "What should we do?" and the man says "Don't cut production, because according to my formula last time there was a market slow down and
you cut production your inventory increased." The man points to graph showing increased inventory during previous slowdown, as clear and unequivicol proof that a cut in production resulted in an
increase in inventory. The man (citing some sweet omitted variable theory) determines that a cut in production led to an icrease of inventory and therefore to solve the current problem says
"Chysler must increase production".
13. I am rather convinced you're wrong. But I encourage you to get a decent statistical package, get the data, and show me otherwise. | {"url":"https://offsettingbehaviour.blogspot.com/2010/02/youth-rates-revisited.html","timestamp":"2024-11-12T16:50:11Z","content_type":"application/xhtml+xml","content_length":"169454","record_id":"<urn:uuid:f165936b-bb39-4eff-97a3-a10a0fe17bdf>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00471.warc.gz"} |
Equivalence Partitioning & Boundary Value Analysis
Two of the most intuitive testing techniques, these help to derive test cases from documentation on how the software should behave and are specification based (or black box testing) techniques.
Equivalence Partitioning
Probably one of the most recognisable test techniques in the testers armoury is equivalence partitioning – a technique to methodically reduce the infinite (or at least huge) probable number of test
cases into a manageable but effective set.
Consider a registration form for a fund raising event. The input is an integer value for current age (rather than a birthdate).
The valid ages are probably from 0 to 125 years old, however there is obviously little value in trying all of those ages as distinct inputs as you would not expect them to be treated any differently.
Additionally, going beyond 125 – while unlikely to occur – should probably still work.
However, there are ranges of invalid data to consider based on the integer input type:
• Negative numbers
• Non-integer values
• String and other non-numeric characters
In this example, the test cases could be simplified to one example from each partition, assuming all values in each partition are equally as useful:
Partition Value Expected result
Valid 25 Accepted
Invalid, negative -10 Rejected
Invalid, floating point 1.01 Rejected
Invalid, string abc Rejected
There may be additional partitions (both valid or invalid) if other behaviours are taken into consideration (for example outputs to other business logic applied later on in a process)
Boundary Value Analysis
Boundary value analysis is way to design test cases based on the logical boundaries of input values, where decision making logic is encountered.
In the registration form example above, if the fund raising event were to charge an admission fee for those between 18 and 65 then some additional test cases are required. You could define those
tests to be values on either side of each boundary as though performing equivalence partitioning – e.g.
Age Entry fee?
15 No
25 Yes
75 No
Which would be adequate, but would not reveal all possible errors within the decision making logic.
If the statement to calculate charges was accidentally written like this pseudo code:
if ((age > 18) and (age < 65)) then pay = true
Then a participant aged 18 or 65 would be incorrectly given free entry.
By applying 3 value boundary analysis – where you consider a value on and either side of the boundary – your test inputs would be 17, 18, 19, 64, 65 & 66 and these would reveal that the 18 & 65 year
old test cases failed.
Whilst directly entered numerical data is the most obvious (for example times, dates, ages, dimensions, distances, weights, speeds, temperatures, etc.), other boundaries may exist – for example sizes
of data structures, numbers of connections, etc.
While the examples here have focused on data entry through a UI, both of these techniques can be applied anywhere where data values are stored or passed between systems or services – e.g. API calls,
configuration data, etc.
4 thoughts on “Equivalence Partitioning & Boundary Value Analysis”
1. Hi Steve. I have a question about why you chose to use 3 value boundary analysis? Referring to your example, I would probably have selected the following values to test:
17, 18, 65 and 66 (derived from 2 value boundary analysis)
11, 43, 87 (arbitrary values from each equivalence partition)
I cannot see what benefits testing 19 and 64 would have. The ISTQB says something vague like 3 value boundary analysis should be used in higher risk areas, but doesn’t explain why. Any thoughts?
1. Hi Andrew,
Thanks for the feedback – I chose 3 value BVA as it’s the most thorugh way of testing any possible errors (at the higher cost of more tests). For my simple example, 2 values would be
sufficient as you note.
3 values would probably be better suited to a more complex process where the output isn’t binary (for example a financial calculation where anyone under 5 years of service gets no bonus,
anyone with over 5 gets 3% + 1% for every year after, capped at a max of 15%)
1. Thanks for the reply Steve. I’m trying to understand this for my own benefit – the need for 3 value boundary analysis has confused since I read it in some ISTQB doc. I’m not sure that
your financial calculation is an example of the need for this either since it could be broken down into the following statements.
1) = 5 years service implies base 3% bonus
3) = 6 years service implies extra bonus equal to (n – 5)% (where n is the number of years of service), capped as per 5) below.
5) Extra bonus capped at 12 % (since 15% – 3% = 12%).
Assuming years of service is an integer, from 1) and 2) we can derive the following boundary values:
4 years of service -> total bonus = 0%
5 years of service -> base bonus = 3%
And then from 3) and 4) we can derive the following boundary values:
5 years of service -> extra bonus = 0%
6 years of service -> extra bonus = 1%
And then from 5) we can derive the following boundary values, by calculating what years of service is necessary to achieve an extra bonus = 12%:
17 years of service -> extra bonus = 12%
18 years of service -> extra bonus is capped at 12%
So the pairs of statements 1) 2) and 3) 4), and the single statement 5) all effectively define a boundary from which 2 boundary values can be derived. I see no benefit from deriving 3
values from any boundary (though it would be beneficial to supplement these tests with tests covering an arbitrary value from each equivalence partition).
Am I missing something?
1. I think you’ve answered your own question – if it doesn’t seem to provide any benefit then 2 values could well be sufficient.
The ISEB book I’ve got notes that 3 value BVA is documented in BS 7925-2, but does not describe under what circumstances the extra value may be needed. | {"url":"https://allthingstesting.com/equivalence-partitioning-boundary-value-analysis/","timestamp":"2024-11-07T13:49:23Z","content_type":"text/html","content_length":"43069","record_id":"<urn:uuid:32d19d79-969c-4633-8f5a-d09ea882c638>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00062.warc.gz"} |
T4T We're Both Right!
Material Type:
Activity/Lab, Lesson, Lesson Plan
Lower Primary
Media Formats:
Downloadable docs
Education Standards
T4T We're Both Right!
This resource is from Tools4NCTeachers.
In this lesson, partners use both centimeters and inches to measure objects to develop their understanding of the relationship between the length of a unit and the resulting measurement. With teacher
guidance, they will solidify their understanding that the objects themselves do not change when measured using different units. Students should have had multiple experiences measuring using standard
units before attempting this lesson.
Here is a sample of this resource. Click the attachment to download the entire fully-formatted lesson and support materials.
We’re Both Right!
Finding Patterns Using Multiple Units of Measurement
In this lesson, partners use both centimeters and inches to measure objects to develop their understanding of the relationship between the length of a unit and the resulting measurement. With teacher
guidance, they will solidify their understanding that the objects themselves do not change when measured using different units. Students should have had multiple experiences measuring using standard
units before attempting this lesson.
NC Mathematics Standard(s):
Measurement and Data
NC.2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
Additional/Supporting Standards:
NC.2.MD.1 Measure the length of an object in standard units by selecting and using appropriate tools such as rulers, yardsticks, metersticks, and measuring tapes.
Standards for Mathematical Practice:
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
Student Outcomes:
• I can use a ruler to measure to the nearest centimeter and/or inch.
• I can describe how the size of the unit I use is related to the number I get when I measure.
Math Language:
What words or phrases do I expect students to talk about during this lesson?
unit, measure, more, less, inch, centimeter, accurate, size, length
• Rulers with centimeters on one side and inches on the other
Advance Preparation:
• Gather enough rulers for each child to be using them simultaneously
• Prepare sentence frames
1. Systems of Measurement (8 minutes)
Adapt the following scenario to your classroom, but be sure to measure the object(s) beforehand.
Mr. Wells wants to move his desk out of the room and trade it for a smaller one, but he was worried the desk was too wide to fit through the (32 inch) doorway. Javy measured the desk as (about 28)
and while Maya measured the desk as (about 72) Bryan surprised everyone when he said that they were both right. What was Bryan thinking? Today we are going to do some measuring of our own so that we
can answer questions like this one whether we are making measurements in or out of school.
Look at your rulers. What do you notice? (There are lines., Some of the lines are numbered., The two different sides have spaces that are different lengths., etc.) What do you wonder? (Why are the
spaces on each side different?, Why are some of the lines numbered, but others are not?, What do cm. and in. or other labels mean?)
(The next two paragraphs will not be necessary if students already know about systems of measurement.)
Did you know that people in different parts of the world use different systems of measurement? In the United States, we often measure objects using inches, feet, yards, and miles and we call these
types of measurement “customary measurement.” In most of the rest of the world, people use a system of measurement called the “metric system” which uses units such as centimeters, meters, and
Most of the rulers we use have centimeters on one side and inches on the other. Look at the rulers you and your partner have. Find the side that has centimeters, labeled “cm”. The partner with the
shortest hair will be using this side to measure the objects you select. Now find the inches side labeled, “in”. The partner with the longest hair will be using the inches side to measure today.
In second grade we measure to the nearest complete unit, so briefly model how to estimate which whole number of units an object is closest to.
Briefly introduce and make available (post, pre-print, etc.) these or similar sentence frames to structure students’ conversations within their partnerships. The teacher can draw students’ attention
to and model these frames as necessary during the Explore and Discuss sections of the lesson.
Sentence Frames:
Identify or State
When I measured (the object) it was about (____) long.
My measurement was (________) more than my partner’s measurement.
My measurement was (________) less than my partner’s measurement.
Describe or Explain
My measurement was more because my unit was (_______).
My measurement was less because my unit was (_______).
When we measured we had different totals because (_______).
I (agree or disagree) with my partner’s measurement because (______).
1. Measuring Around the Room (15-20 minutes)
Allow ample time for students to measure many objects. Designate a few standard objects for all students to measure such as a specific text book, the seat of a chair, etc. These standard objects will
be a way for you to quickly check to be sure that their measurements are reasonable. Otherwise, they should measure as many objects as possible because the more examples they have, the more likely
they are to recognize and describe the relationships between the different units of measure. Prompt students to use the sentence frames during their partner conversations, but do not take over the
As students work, observe:
• The act of measuring. Are they being precise to the nearest whole unit?
• Their collaboration. Are they paying attention to and considering the measurements their partner is making?
• Their conversations. Are they using the vocabulary (unit, measure, etc.) and syntax (My measurement was ____., Our measurements were different because _____., etc.) that demonstrate an
understanding of what they are doing?
As students work, record:
• student thinking as you listen to and interact with students. These moments will help to guide your discussion; however, you may interrupt the class a few times to share ideas that come up during
the discussion. What words and phrases are students using? How are their discussions showing what they understand and unveiling misconceptions?
• questions that students wonder and/or answer during their exploration. What do their questions reveal about their thinking and about the concepts of measurement and units? How did they answer
their own questions?
• comments that you overhear that relate to the big idea of measuring using multiple units.
• student strategies that will encourage others to share and will lead to clear understanding and efficient processing for the class.
• a progression in which you want students to share their thinking and/or examples. What order will create the most discussion and lead students to a clearer understanding? Which student examples
serve as clear examples of patterns of thinking for many students?
3. Who is Right and Why? (25 minutes)
Begin by sharing questions and thoughts you recorded during the explore section including what students noticed and wondered. It is best, when possible, to have students share their own examples or
to mention them and then have students explain what they were thinking or what they found out. However, you may decide to share on behalf of students in some cases.
Share example measurements and have students determine which measurement is “right” and justify their argument. Allow students to rehearse and revise their points in partners or small groups before
sharing with the class. Encourage students to use the question stems when explaining their thinking and agreeing or disagreeing with each other. Take advantage of disagreements and have students
demonstrate and explain proper use of their rulers and how they are attending to precision.
Encourage students to look for and describe patterns. For instance, “When the units are bigger, the numbers are smaller.” or “It takes more little units to measure something.”
Refer back to the problem from the launch. Have students discuss and defend their responses in partnerships or small groups. After a discussion of their reasoning, ask students to explain how each
unit is related to their final measurement and use this discussion to wrap up before assigning the formal assessment.
Possible points to address and questions to ask:
• Who agrees/disagrees with (the student)? Why? Or Who was right? Why?
• Are you noticing any patterns? How would you describe the pattern you have found? Did the Evidence and Examples recording sheet help you to recognize or describe any of these patterns?
• How was the unit you used related to the measurement you made? How does this explain the difference in measurements?
• What challenges did you deal with during this lesson?
• What connections did you make to other experiences?
• What would you and your partner do differently if we repeated this activity tomorrow?
Evaluation of Student Understanding
Informal Evaluation:
Recorded observations and notes from the exploration and discussion phases of the lesson
“Evidence and Examples” forms collected from students
Formal Evaluation/Exit Ticket:
Draw and label a diagram that can be used to explain how the size of the unit used is related to the measurement. Include a brief caption to explain the diagram. or
Sketch a comic strip using stick figures and simple shapes to show how the size of the unit used is related to the measurement. Use speech bubbles for your characters to explain your thinking.
Meeting the Needs of the Range of Learners
Engage students in a similar experience by iterating multiple objects such as inch tiles, Unifix cubes, or paperclips. Have students work through the same thinking of comparing their results after
measuring the same object using different units. Centimeters and inches are relatively abstract concepts. We can root their reasoning in their world when students think and speak about the difference
between the number of cubes and paperclips it takes to measure a book, shoe, etc. This simple change avoids mistakes with using and reading rulers as well as confusion with labels. The same
questions, activity sheet, and sentence frames can be used.
This activity can be substituted for the original exploration at any point or can be an additional opportunity for small groups or individuals who are not yet able to demonstrate their understanding
during the discussion and/or assessment after having used the rulers in the explore section of the lesson.
If students quickly realize and articulate that the smaller unit results in a higher number when measuring an object, have them measure a few more objects to test their theory before moving on. Now
have these students generalize the pattern they have recognized. What are other instances to which this pattern applies? What possible problems could be caused when people use different units and are
not clear about which units they have used? Why is it important that we label our units when we communicate measurements to others? This extension is meant to take their thinking farther and deeper
but should not add extra work.
Possible Misconceptions/Suggestions:
Possible Misconceptions Suggestions
Students consistently measure inaccurately. Observe the child measuring carefully and instruct them on how to begin measuring at zero mark and to accurately
read the marks on the ruler to determine the length of the object.
In this exploration and discussion, students realize that more than one measurement This, like many overgeneralizations, is best counteracted with non-examples. Have the student test their
can accurately describe an object, but there is a chance that they overgeneralize and conjecture, but be sure to include inaccurate measurements that might sound reasonable.
believe that any measurement a student shares is accurate.
Students may begin to think that inches are in some way better than centimeters or Ask students to explain their thinking and then clarify that different units are more accurate in different
vice versa. situations. Centimeters are more precise when estimating to the nearest whole because they are smaller units.
Inches are likely to be more precise when measuring objects that were created using the customary system.
Special Notes:
• At this age, students can be inflexible with their understanding of what a number represents. This is an opportunity to expand their understanding of numbers as being able to represent widely
different values as in the weight of 7 mice and 7 elephants. Be prepared for students to need time when explaining why an object was both ___ inches and ____ centimeters.
• There is no reason for 2^nd grade students to know a specific ratio between metric and customary units.
Evidence and Examples
Object Measured My Partner’s Measurement My Measurement What we notice and wonder about these measurements. | {"url":"https://goopennc.oercommons.org/courseware/lesson/3461/overview","timestamp":"2024-11-08T01:40:57Z","content_type":"text/html","content_length":"75718","record_id":"<urn:uuid:7c658625-7138-4e9c-af24-56a90caa86b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00150.warc.gz"} |
Comparing the Displacements of Two Similar Paths
Question Video: Comparing the Displacements of Two Similar Paths Science • Third Year of Preparatory School
A car is at the center of a circle. The arrows show paths that the car could travel to reach the circumference of the circle. Is the displacement of the car between its initial and final positions
the same in both cases?
Video Transcript
A car is at the center of a circle. The arrows show paths that the car could travel to reach the circumference of the circle. Is the displacement of the car between its initial and final positions
the same in both cases? (A) Yes or (B) no.
This question asks us whether or not these two arrows, red and blue, have the same displacement. Recall that a displacement is the vector along an object’s shortest path from start to finish. It has
a magnitude and a direction. So, really, this question is asking us two things. That is, do the arrows point in the same direction? And do they have the same length from start to finish, which would
be the shortest path the car can take from the center to the circumference?
For an object in the center of a circle, it needs to travel the length of the radius of the circle to reach the circle’s circumference. We see in this case that both the red and blue arrows extend
from the center of the circle to a point on the circle’s circumference. We can say then that each arrow is a radius of the circle. And if the car were to travel along either path, it would reach the
circle’s circumference. There is no shorter path the car could travel to reach the circumference. Therefore, both arrows represent the shortest path from start to finish and can be defined as the
magnitude of the displacement.
Also, since the arrows overlap entirely, we can say that they point in the same direction. Let’s look at our displacement checklist. Does each arrow cover the shortest distance from start point to
endpoint? The answer is yes. Does each arrow point in the direction of motion? Once again, the answer is yes.
We can therefore conclude that both arrows have the same displacement. Option (A) is correct. | {"url":"https://www.nagwa.com/en/videos/769195496453/","timestamp":"2024-11-06T17:45:49Z","content_type":"text/html","content_length":"250053","record_id":"<urn:uuid:b78c8cfa-98d6-4e88-a44c-ed5e8b1558cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00349.warc.gz"} |
Graphs + Charts
People create and use graphs and charts to organize and communicate information visually. You can find charts and graphs in books, magazines, newspapers, posters, infographics, reports, report cards,
in print and online. Most businesses use graphs and charts to communicate their profit and process and to manage supply and demand. Graphs and charts organize information, data, statistics, time
frames, size, cost, and many more items. Bar Charts, line graphs, pie charts, area graphs, and point graphs are some common types. You can learn about and make charts in your classroom, on your
projects, on paper and online. Knowing how to read different presentations and then knowing which type to choose to visualize certain information is a critical skill. While charts offer different
visual approaches, most charts have a title, list subjects or categories, quantity abbreviations, and labeled and organized data. Choices about color, font, font size, labels, label locations all
contribute to how a chart or graph communicates information.
Be a graphic graph designer!
Activity 1 – Collect Charts and Graphs
A good way to explore is to start seeking examples of graphs and charts in your everyday life. Check at the library in research books. Find the weather report in the newspaper. Look online for
statistic sites. Take a few pictures of different kinds of charts. Looking critically, which ones seem easy to read? Which ones present information more quickly than a written paragraph? Which ones
appear graphically vibrant? By looking at other charts and reflecting on their design, you are developing a visual library of examples that you can compare and contrast.
Post your collection for others to see!
Activity 2 – Parts of Graphs and Charts
A concise communication of information starts with a title that summarizes what data you are showing. It lists the source or sources of the amounts that you are sharing. Your sources might include
people you surveyed, organizations you research, individuals who contributed information. A key or legend reveals what the chart or graph is communicating. Is it data about children? Is it the
average weight of cattle? Is it the range of life expectancies of different countries. Excluding pie charts, most graphs use an x and y-axis. The y-axis runs vertically The x-axis runs horizontally
and can be divided into whatever increments you choose- pounds, minutes, years, months, etc. Most importantly, charts and graphs work to communicate information so that people can understand
information easily. Make a list of the types of charts that you want to learn!
Activity 3 – Direct a Pie Chart
A pie chart is a circular shape, resembling a ‘real pie,’ divided into pieces. Think of cutting up a pizza. You can split it in half, thirds, quarters, etc.. Pie charts are used to communicate parts
of a whole. The measure of the sections is most often in fractions and percentages. Make a pie chart that shows the activities you participate in during one school day. First, calculate the minutes
you are in school. Next, make a list of all of your activities. Be sure to include lunch, recess, and all of your subjects. Now record the number of minutes you spend in each activity. Use this
equation for each activity: x/100= number of minutes of one activity /over the number of total minutes of your school day. Your result is each activity being a percentage of the whole time. You can
color code and label your time. You can make a chart of how you spend your 24-hour day! Is it easy to read? Does it reveal anything surprising? Is it the same as your parents or grandparents?
Activity 4 – Build a Bar Chart
Bar charts use ‘bars’ to compare different things. For instance, if one mother dog has a litter of puppies, you can represent the number of male vs. female puppies, or use different rows to show the
number of black, brown, white and tri-colored puppies! Bar charts run horizontally or vertically and can be two dimensional or three dimensional. Different sets of data can be shown side by side or
stacked in single bars. Bar charts can also communicate change over time. A good example is a temperature record of a city listing the average temperature per month over one year. Make a bar chart of
the temperature range month by month of your community. You can use blue to represent precipitation. You can even add a yellow to a gray bar that shows percentages of sunny vs. overcast days per
Activity 5 – Area Graphs
Area Graphs are like line graphs, but instead of using lines they use zones to represent different categories. For example, the information might be grouped by ages- 0-12, 13-18, 18-24, 25-35, etc.
Area graphs look like high mountain ranges as they show amounts. They also can show changes over time. For instance, in the hospital, they might register your average heart rate over your time spent
in the hospital. Make an area graph of demographics of the people living in your city.
Activity 6 – Point Plots
This type of chart helps visualize interactions between two different things. It might be a pitcher’s throw and the arc of his or her arm and the speed of the ball. Point charts or x-y plots
communicate and determine and speculate on the relationships between chosen categories. The x and y-axis represent variables of different topics and events.
Activity 7 – Variables
Variables are important to graphs and charts as they can be plotted to inform our understandings and even our decisions. Variables can be many things but should be defined. They can be whatever you
choose- a measure, amount of light, the intensity of sound, excitement, time periods, etc. You are in charge of your charts! There are two types of variables- independent, and dependent. An
independent variable is constant. It does not change by other variables. On the other hand, a dependent variable is affected by other factors. Usually, when you make a chart to visualize information,
you are looking to see what makes the dependent variable change.
Activity 8 – Identify and Interpret
Learning to read graphs and data sets is a skill of deciphering the parts to understand the whole. It helps to break the reading of the chart or graph into three parts: Identify (what you see)
Interpret (what each part means) Caption the Meaning(describe what is revealed). First, look at the parts of the graph. Perhaps you see dates, or amounts, or other measurements in the x or y-axis.
Notate changes that you see. In step two, interpret what these changes mean. In step 3, move each what you see statement to what it means and create a synopsis of the information. Try to begin a
short caption paragraph with a topic sentence of what the graph shows. Use the following sentences to note each of the things you observed and how they contribute to the topic of the graph. Finally,
end with a concluding sentence that summarizes your findings!
• Independent variables
• Bar charts can be
• Charts and Graphs communicate information.
• Line graphs show
• Which chart divides a circle?
check answers | {"url":"https://www.next.cc/journey/tools/graphs","timestamp":"2024-11-13T05:47:56Z","content_type":"text/html","content_length":"44271","record_id":"<urn:uuid:e2a1df70-7b30-47ae-b67b-9ef37e52b275>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00271.warc.gz"} |
Gnaiger 1989 Thermochim Acta
Gnaiger Erich (1989) Thermochim Acta
Abstract: Physiological calorimetry is concerned with the measurement of heat flux in living systems where heat flux is associated with the chemical flux of metabolic reactions. Calorimetry can be
related to nonequilibrium thermodynamics if information on both the enthalpy of metabolic reactions and the molar Gibbs energy is available. The molar Gibbs energy of reaction (Gibbs force) is the
scalar force conjugated to metabolic flux. The force conjugated to heat flux of an irreversible process is the Gibbs energy/enthalpy ratio. Metabolic power and heat flux of irreversible processes are
distinguished as the time rate of Gibbs energy and enthalpy changes, respectively. Power is the product of fluxes and forces, related to the internal entropy production by the absolute temperature.
In contrast, TΒ·Ξ [r]S is the β bound energyβ change which equals the heat change of a reversible process in a closed system and is not available for work. Heat flux in general is the sum of the
dissipated power and the bound energy change per unit of time. This concept can be extended to vectorial heat flux along a temperature gradient. The temperature difference relative to the temperature
of the heat source, traditionally viewed as the β efficiency of a reversible machineβ , is in fact the thermal force for heat flux between heat source and sink. The thermal force times heat flux is
the thermal power which can be maximally converted into work or can be irreversibly dissipated. A clear distinction between heat flux and power is conceptually revealing, despite the fact that both
quantities have the same dimension with units [W per volume, or per mass or per defined system] when describing scalar and discontinuous processes.
β ’ Bioblast editor: Gnaiger E β ’ O2k-Network Lab: AT Innsbruck Oroboros
Cited by
Labels: MiParea: Respiration
Regulation: Coupling efficiency;uncoupling | {"url":"https://wiki.oroboros.at/index.php/Gnaiger_1989_Thermochim_Acta","timestamp":"2024-11-10T08:52:11Z","content_type":"text/html","content_length":"33989","record_id":"<urn:uuid:a0630e20-0b85-452b-864c-512e3efff9d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00870.warc.gz"} |
What Is Length Contraction in Physics - Kostgangers
What Is Length Contraction in Physics
Length Contraction in Physics: Understanding the Concept and Its Implications
In the world of physics, there are various fundamental concepts that are key to understanding the nature of matter and energy, and how they interact with each other. One such concept is length
contraction, which plays an important role in theories such as relativity and quantum mechanics. In this article, we’ll explore what length contraction is, why it happens, and how it affects our
understanding of the universe.
What is Length Contraction?
Length contraction is a phenomenon that occurs when an object is moving relative to an observer. Specifically, it refers to the apparent shortening of an object along its direction of motion, as
measured by the observer. In other words, the object appears to be shorter when it is moving than when it is at rest.
This effect is not just a visual illusion – it is a real physical phenomenon that has been verified through numerous experiments. The amount of length contraction that occurs depends on the relative
speeds of the object and the observer, and can be calculated using the formula:
L’ = L * sqrt(1 – v^2/c^2)
where L is the length of the object at rest, v is the relative speed between the object and the observer, c is the speed of light, and L’ is the observed length of the object.
Why Does Length Contraction Occur?
The reason why length contraction occurs can be traced back to the fundamental nature of space and time. According to the theory of relativity, space and time are not absolute, but are instead
relative to the observer’s frame of reference. This means that measurements of distance and time can vary depending on the observer’s speed and direction of motion.
In the case of length contraction, the apparent shortening of the object is due to the fact that time appears to be running slower for the moving object than for the observer. This causes the object
to appear shorter, since the observer perceives it as occupying less space over the same amount of time.
The implications of length contraction are profound, and have been explored in depth by physicists over the years. For example, it has been used to explain phenomena such as the observed stability of
atomic nuclei, which would be impossible if length contraction did not occur. It has also been used to devise new technologies, such as particle accelerators, which rely on the principles of
relativity to work.
In summary, length contraction is a fascinating and important concept in physics that helps us understand the nature of space, time, and matter. While it may seem counterintuitive at first, it is a
real phenomenon that has been verified through numerous experiments. Understanding length contraction is crucial for anyone interested in the deeper workings of the universe, and its implications
continue to inspire new discoveries and technologies. | {"url":"https://kostgangers.nl/what-is-length-contraction-in-physics/","timestamp":"2024-11-12T03:12:02Z","content_type":"text/html","content_length":"42823","record_id":"<urn:uuid:b1eae41a-d44a-4449-8406-bae972b8f9e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00409.warc.gz"} |
Institute for Applied Mathematics
Our aim is to develop mathematical techniques and ideas which are relevant for applications in the natural and social sciences, and to investigate their implications in selected case studies.
Mathematically, we focus on nonlinear analysis (calculus of variations, partial differential equations), numerical mathematics, probability theory and stochastic analysis. Presently the main
application areas are physics and mechanics.
Our institute is a founding member of the excellence cluster Hausdorff Center for Mathematics (HCM) in Bonn and consists of the following research groups:
Bonn Research Chairs
Bonn Junior Fellows
• E. Peltola (former BJF with partial affiliation in Bonn)
Heisenberg Fellow (Heisenberg-Stelleninhaber)
Interdisciplinary Research Units
Prof. Dr. Lisa Sauermann has been honored with the von Kaven Award 2023 for her outstanding scientific achievements. (16.11.2023) | {"url":"https://www.iam.uni-bonn.de/","timestamp":"2024-11-12T21:56:14Z","content_type":"text/html","content_length":"69659","record_id":"<urn:uuid:7c1a2558-42aa-4b35-b509-c93d1465184f>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00382.warc.gz"} |
Trigonometric ratios
FEATURES + Interactive unit circle. Allows exploring relations between angles and trigonometric ratios. + Bundled with an exact trigonometric ratios table
Trigonometric ratios are the ratios of sides of a right-angle triangle. The most common trigonometric ratios are sine, cosine, and tangent. Consider a right-angle triangle ABC, right-angled at C. In
that case, side AB will be the hypotenuse.
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Field Theories in Inflationary Cosmology - NHSJS
Field Theories in Inflationary Cosmology
Avi Shah
The following work will be exploring the prevailing frameworks for the modeling of the early universe’s inflation; namely old inflation, slow-roll inflation, and ultra slow-roll inflation given a
quadratic field potential. We then go on to explore the various assumptions of the models in terms of field dynamics, cosmological parameters, and thermodynamic criteria. A review of the shortcomings
of these models is undertaken; followed by the proposal of two research pathways in order to address the same: galactic redshift analysis and the spectral analysis of the cosmic microwave background.
These research pathways will be investigated in the future in coming endeavors: utilizing perturbation theory and early universe cosmology to match theoretical CMB patterns with current CMB patterns;
and the corroboration of universe age in e–folds from theory with the heuristically determined age of the cosmos.
1 Introduction
1.1 Modern Cosmology
As a preface to the research and literature review conducted hereafter, it is valuable to provide a general overview of modern cosmology as well as its most prevalent challenges. This will provide
background for the mathematical exploration of the current variations in the inflationary models in the field of cosmology.
1.1.1 The Contemporary Model
Cosmology concerns itself with the origin and evolution of the universe. In the decades leading up to the development of inflation theory – in which the Lambda–Cold Dark Matter model is the most
recent^1 – researchers faced two primary conceptual conflicts, outlined below.
1.1.2 The Flatness Problem
First, the flatness problem refers to the substantially unlikely state of the current universe wherein the curvature of spacetime is incredibly fine-tuned. It is at a state of critical density so
well-balanced that uncertainty has been cast on the nature of this precarious equilibrium. There is widespread doubt as to how natural the model we have created is if such fine-tuning is required to
make it heuristically accurate.
To clarify the specifics of this challenge, we can undertake a mathematical exploration of this problem. The mathematics explored below are a summary of the induction in Prof. Alan Guth’s famous 1981
paper that founded the field of inflation^2.
To begin, we start with the Friedmann-Lemaître-Robertson-Walker metric [Equation 1].
describing the path between two points in positively curved qascetime [
The second postulate of the cosmological model is that the universe expands in spacetime according to the at any time
Both of the above equations can berderived by plugging in the parameters of the field’s equation of state into Equation 1. The third parameter we introduce^2 is that energy is conserved according to
Equation 4.
Equation 4 implies that the change of the energy density of the universe as it expands is equal to the change in the size of the universe multiplied by the negative pressure of the universe. This
wallalsobeseglored further in Section 2 , where the mechanism itself will be explored in further detail.
Following from the above, assuming the adiahaticity – or the lack of overall change in entropy – of the universe, we get Equation 5.
Qualitatively, a constant entropy means that its derivative with respect to time must be 0; where is the entropy density. While this is certainly disproven in modern cosmology and physics, it is
important to note that this was a prevailing assumption in Guth’s time and work.
Therefore, along the same lines, it is necessary that we define the prevailing thermodynamic norms and statutes in the 1980s before working with the development of entropy for old inflation.
Models in the late 20th Century assumed a gas of particles with bosonic and fermionic degrees of freedom and respectively, the functions of density in kilograms per cubic meter, entropy in joules per
Kelvin, and particle number density per cubic meter are as follows [Equations 6, 7, 8] keeping in mind equations 9 and 10^3.
In which:
Equations 9 and 10 are partition functions, which calculate the fermionic and bosonic degrees of freedom respectively, and ase calculated by the product of the species, particle-antiparticle pairs,
and spin states. These values essentially quantify the contributions that fermionic particles and bosonic particles have to the density of the system. In Fquation
Recall equation 3 , utilized to describe the evolution of the scale factor of the universe:
Rewritten in terms of temperature
Here, H , or the Hubble constant, is another notation for the scale factor of the universe. As the change in temperature is proportional to the scaling of the universe, it is a good approximation for
the Hubble factor. The dot denotes differentiation with respect to time. The second term
The value of
Next, by calculating the photon, electron, and neutrino contribution to the entropy given by
Plugging S back into Equation 13,
Now, using the investigated values, if we compare the difference between the current energy density and the critical energy density with the current energy density, we get Equation 16:
i.e, the universe’s present density matches the critical density of the universe [the borderline between a flat universe and positively / negatively curved universe] with an error margin of less than
Logically, if
Since the error margin is a factor of
This aforementioned fine-tuning is known as the flatness problem^2.
1.1.3 The Horizon Problem
The second challenge in the field of cosmology is known as the horizon problem. Extrapolating from the above values will allow us to undertake the task of achieving a mathematical description of the
same. Once again, the deduction below is a summary of the description of the horizon problem in Prof. Guth’s 1981 paper on inflationary theory^4.
Returning to Equation 11, we solve for temperatures above mass thresholds through certain mathematical transformations.
By ignoring the
Next, it is important to note that if entropy is to be conserved, the term
This concludes the discussion of the characteristics and prerequisites of the cosmological model needed for this section of the paper.
The next component we need to explain the horizon problem quantitatively is the distance a light ray has traveled by a time
This integral derives the distance light can travel, in meters, from a certain point in time as the universe evolves.
To apply it to our universe, we can derive the causal horizon we currently observe by assuming the conservation of entropy. If entropy is constant, we can find the change in distribution of the
entropy over the radius, in meters, of the observable universe as seen in equation 22.
Qualitatively, the previous and next steps find the area over which events are causally connected – i.e. can affect each other in real time.
If an event in spacetime is further than the distance light can travel from you at that moment, it is causally, disconnected from you, as the speed of light is the speed limit of the universe.
Now that both values have been defined in equations 21 and 22, we can compare them to explain the horizon problem. By finding the ratio of the two volumes
implying that the region of our observed universe is approximately 83 orders of magnitude greater in size than the causal, or physical, horizon distance.
This seems illogical as disconnected causal regions would then fail to explain the large-scale homogeneity and isotropy of the universe. There is no reason for the disconnected patches to evolve
1.2 Quantum Field Theory
In order to set the stage for the discussion of more contemporary models, it is important to lay down the groundwork of quantum field theory, otherwise known as QFT.
In essence, quantum field theory postulates the existence of a set of fields that permeate spacetime; each belonging to a particle. Fields are mathematical constructs that assign a scalar or vector
to all points in spacetime. Excitations in fields are construed as the manifestation of their respective particles^5.
These excitations are the wave functions of the particles; i.e. probability distributions of the location and momentum of a particle. These waves are manifest in the fields themselves as
fluctuations; which are then directly quantised rather than the particles themselves.
The interactions of these waves, or particles, can be described mathematically, or through a visual representation – i.e. Fenyman diagrams, in which the paths of particles are plotted on graphs
against time.
Overall, quantum field theory unifies various theories, including quantum electrodynamics, quantum chromodynamics, the standard model of particle physics, and electroweak theory.
1.3 Relativity
While quantum field theory deals with the smallest scales of the universe, we must also consider its converse: general relativity, describing the large-scale structure and curvature of the universe.
The brainchild of the infamous Einstein, relativity postulates the tie-up of space and time into a unified manifold monikered spacetime, whose shape is decided by the mass and energy content of the
The resulting geometry of spacetime is what we experience as gravity, i.e. bodies of mass and light traveling on a straight line but in curved spacetime being perceived as gravitational attraction.
General relativity’s counterpart, special relativity, is the second component of relativity. Special relativity postulates that time is not absolute; and that the speed of light in a vacuum is the
cosmic speed limit. Furthermore, time dilates around big masses, and length contracts at high speeds. Einstein’s notorious equation is an essential component of this framework.
1.4 Effective Field Theories
Effective field theories are not a singular concept, but an approach to describing our universe. EFTs are special in that they are specialized for certain scales; they work on the key assumption that
our universe can be explained through different frameworks at different sizes.
A prime example of this is quantum field theory and general relativity – both of which only function at their respective scales. Unifying the various exclusively functioning theories currently used
to describe the universe is one of the greatest challenges of modern physics.
There is a wide array of issues, both conceptual as well as intrinsic, to be discussed within unification; ranging from the occurrence of infinities, which are not present in nature; the conceptual
conflict of frameworks in various ases, including dark matter, or the nature of gravity; as well as the occurrence of questions about existence, teleology, and philosophy.
1.5 Methods
The search strategy incorporated in this research was to use primary search engines such as Google to perform a preliminary search for solely articles, recorded lectures, or university talks that
were relevant to the search. If no relevant sources were found, databases such as ResearchGate and Google Scholar were used to find a wider variety of papers in terms of date of publication as well
as authors and regions.
Keywords were limited solely to the concept and the medium of the information. An example search would be ‘perturbation theory cmb stanford.’ Unless absolutely necessary, no websites were used. The
papers, talks, and lectures used were vetted on the following factors: a recent date of publication [unless it was for a review of early theory], reputable institutions of publication, such as
universities or well–known journals, and quality of language.
Once a source was found, it was read through thoroughly to find only relevant information. The latter could be in the form of an equation, a quote, or an explanation of an unfamiliar phenomenon. To
structure its reference, the source’s authors were searched for at the top of the page, the bottom of the page, or on the main page of the website being used.
For papers in which the PDF was given or the DOI was given, the PDF was opened and information was presented on the front page, or for the DOI, information was stored in their main database and was
extracted using a bibliography tool monikered NoodleTools.
To synthesize my information, I had first organized a flow of narrative. I planned and followed a set of bullet points to ensure that my sections flow into each other languidly. I also ensured the
implementation of conclusions and introductions between sections to facilitate the transitions.
2 Literature Review
2.1 Modern Theories
2.1.1 Old Inflation
Guth’s proposed theory – now dubbed old inflation – was preceded only by a simplistic framework for the description of the cosmos: a homogeneous, isotropic expanding universe^2. This theory, however,
carried certain irrefutable scientific and mathematical barriers, as mentioned in Sections 1.1.2 and 1.1.3. The same is summarized below from Guth’s paper on the same.
To address the same barriers, Guth, in essence, formulated a model in which he postulates a period of rapid expansion in the early universe monikered inflation. His paper outlines the potential
resolution of the flatness and horizon problems through inflation, the mechanisms that drive inflation, and evident shortcomings of the theory.
To begin with, Guth outlines the discrepancy of rampant causal disconnection in a homogeneous universe [i.e. the horizon problem] as well as the irregularly precise fine-tuning of
Guth introduces a scalar field
The nascent universe has a sea of – what can be considered – massless particles in an adiabatic system heated to extremely high temperatures. Analogously, the potential density of the field
Fig. 1: Scalar Field Dynamics^6
This is a key characteristic of the old inflation theory. As seen in Fig. 1, the field’s potential can decay from its local maximum into the overly stable false vacuum. Herein, the universe
supercools through expansion as the inflaton field is virtually “stuck” in the valley of the local minimum. As a result, the universe continues to expand rapidly and – most importantly –
Old inflation and the phase transition of the inflaton field provided an adequate explanation for the flatness problem in that any incident curvature of the universe at its ‘birth’ is sufficiently
stretched out as to have it appear locally flat: i.e. in de Sitter space.
In addition to the novel mechanism, Guth also proposes the abandonment of the aforementioned theory of an adiabatic universe. With a dynamic entropy
Z is considered a “large factor.’ Mathematically, the entropy of the universe is a key factor in the calculation of the radius, in meters, of the observable universe
With the introduction of a factor ^2.
2.1.2 Slow Roll Inflation
While old inflation certainly was an innovative new framework, it harbored obvious inconsistencies. Over a span of decades, various modifications were placed on old inflation in order to create 1.
More logical transitions from the false vacuum to the true vacuum, 2. Consistencies with observations, and 3. Consistency with the estimated age of the universe.
The dominant model that emerged from these corrections is referred to as slow-roll inflation. Slow-roll inflation is a modern modification of old inflation. To discuss the mechanics of the field, we
must first understand the application of this scalar field to cosmology.
We know that the field exerts a repulsive gravitational force on its surroundings. This is, in fact, caused by the inherent negative pressure of the inflaton field, which will be discussed further in
Section 2.1.3.
The initial conditions of the universe, formalized by the FLRW metric [then the Robertson-Walker metric] [Equation 1] and Einstein’s equations [Equations 2 and 3], lead to the aforementioned
challenges. However, when a transient period of exponential expansion is introduced just after the birth of the universe, both the horizon as well as the flatness problems are addressed, as seen
2.1.3 Horizon Problem Solution
To address the horizon problem, the slow-roll model postulates that the scalar inflation field exerts negative pressure, leading to a repulsive gravitational effect. To find this, we can reason the
The change in the energy density of the field can be modeled by placing it in the second Friedmann equation in terms of the density and equation of state^6.
Since the energy density must stay constant when the field is stuck in the false vacuum, as explained above, Equation 27 implies
As when we substitute the same value for into the equation above, we find a factor of 0 on the right hand side, therefore telling us that , or the change of the energy density with respect to time,
is zero^7.
The second Friedmann equation tells us
As just derived above in Equation 28,
Plugging the same value for
Rearranging for the scale factor
This means that there is an accelerating Hubble expansion rate, or an accelerating scale factor . An accelerated expansion is also observed experimentally by cosmologists
Following from the above, during inflation, if we assume an expanding patch of space of size where the average value of the inflaton potential is approximately 0, or
We can model inflation starting with this patch. To find the next series of events, we can approximate it to be a homogeneous Robertson-Walker patch and utilize the first Friedmann equation that
describes its evolution
Expressing the accelerating scale factor a as a function of time, a(t) , through certain mathematical transformations, one derives Equation 36.
(( A. H. Guth. Lecture 23: Inflation. MIT OpenCourseWare, MIT, 2013, ocw.mit.edu/courses/8-286-the-early-universe-fall-2013/resources/lecture-23-inflation/. Lecture.))
This can be generalized to any universe with initial conditions where if the pressure is negative, then
This is monikered the cosmological no-hair conjecture, i.e. that any system with an average inflation potential of 0 and negative pressure, as seen above, will evolve to locally resemble a flat
exponentially expanding spacetime, i.e. de Sitter space^6. This will be key in the solution of the horizon problem.
Our next step to solve the problem is to consider a distance
which denotes the coordinate distance that light travels from
Substituting the value obtained for
Supposing the light ray travels for an arbitrarily long time,
This renders the integral solution for the coordinate light distance
This implies that there is a limit to the range of causality in the universe, which can be expressed as^7
Therefore, we can postulate that if anything emits a light ray at a distance of one Hubble length or longer from us, we will never receive it.
There is an inherent event horizon in de Sitter space. This means that once a sizable de Sitter region is created, it is essentially ‘protected’ from anything outside its event horizon; as not even
light can enter it fast enough to counter the expansion of space.
The next step in the solution is to find the mass density of the inflaton field. Through dimensional analysis in grand unified theories of energy scales
This is a shockingly high mass density. Substituting the same for
This means we start off with a patch of the universe of the size above, and ending at scales of , expanding until at the end of inflation^7.
Nearing the end of inflation, the potential oscillates at the zero-point energy of the field, effectively reheating the universe at scales of
To find the size of this patch today, we can find the ratio of the temperature at the beginning of inflation to the temperature in the present, and multiply it by 10cm to get:
This is a valid calculation as the scale factor is directly inversely proportional to the temperature of the universe. While this value is approximately 10 times larger than our current observations,
it is permissible as we could also use a value of instead to make up for it.
The size of this horizon invalidates what is known as the horizon problem, in that the universe should technically be filled with causally disconnected patches that would not have been in contact
with each other enough to create a homogeneous space. Here, we see a Hubble region consistent with our observations, essentially eradicating this concern^6.
2.1.4 Flatness Problem Solution
The second major challenge slow-roll inflation solves is the flatness problem. This is achieved by performing the following mathematics:
We know that the Friedmann equation states
As derived above in Equation 48, a grows by a factor of
Therefore, at smaller scales, the universe can be approximated as flat due to a negligibly low curvature – approximated as flat. This essentially provides reasoning for the flatness problem; i.e. why
the universe’s density is so fine-tuned to the critical density; and analogously, the curvature of the universe being observably flat.
2.1.5 The Cosmic Microwave Background
Lastly, the homogeneity and isotropy of the universe can be explained using this model in the framework of quantum mechanics.
Gravitational instabilities play a key role in creating inhomogeneities. The aggregation of matter as gas clouds, stars, galaxies, galaxy clusters, superclusters, and eventually, filaments, creates
inhomogeneities in the overall cosmic microwave background. These lead to the amplification of the fluctuations in the CMB.
The origin of the CMB ripples must yet be investigated, however.
Classically, inflation would lead to a uniform mass density. However, quantum mechanics predicts an almost uniform density after inflation. The effect of the quantum fluctuations can be calculated.
This is further explored in Section 2.2.2.
2.1.6 Ultra–Slow Roll Inflation
Ultra slow-roll inflation, a more contemporary theory, is an extension of the more well-known slow-roll framework – in which, rather than being stuck in a false vacuum, the field rolls down an
extremely flat potential curve and oscillates at the zero-point energy level before settling.
Ultra slow-roll inflation attempts to resolve the prediction that slow-roll inflation may stop at extremely flat – although not inflectional – points of the potential. This is due to the fact that
slow roll creates a fall in the inflaton field’s potential more rapidly than if the field were in free-fall. As free-fall is the limit of the speed of decrease of any quantity, this is a discrepancy
in the model.
To solve this, the inflaton field is expected to transition from slow-roll to ultra slow-roll inflation at the flattest areas of the potential, particularly inflection points. Mathematically, this
happens when the acceleration locks with the potential slope rather than the friction.
To understand this, we must first explore the large-scale dynamics of the inflation field.
Equation 50 gives the field’s Lagrangian^8.
Computing the stress-energy-momentum tensor for the field gives us a set of two equations for the energy density and the pressure of the field. We integrate the Lagrangian and scalar over 4
dimensions^9, or 3 dimensions of space and 1 dimension of time, giving us the action
By adding the Ricci tensor action, which is a calculation that quantifies the curvature of space in the universe, and the field action, which is Equation 52, one can compute the
stress-energy-momentum tensor: which describes the density and distribution of energy in spacetime. The exact calculation is outside the scope of this research, however. Using the
stress–energy–momentum tensor, we can find the aforementioned equations for the field’s pressure and energy density^10.
Dividing the two gives us the equation of state for the field^11 ,
To model the evolution of the field, we can substitute the above equations for
This yields the relation
Through certain mathematical transformations, we can find that the field equation of the inflaton field is defined by
In physics, systems with a mass that is periodically moving in a predictable manner are called harmonic oscillators. The equation for the system of one highly similar to Equation 57. Here,
For the slow-roll framework, the equations are slightly modified to fit the correct potential curve.
The inflation in the slow-roll model has an extremely flat and long potential. The extensive ‘flatness’ of the potential curve allows us to essentially eradicate the second derivative of the
potential, as
This allows us to modify the field equation according to two parameters^12,
Now that the general dynamics are established, we can explore the modifications made by ultra slow-roll inflation on the existing slow-roll paradigm.
At extremely flat potentials, slow-roll inflation runs into a challenge wherein the kinetic energy falls faster than it would in free-fall. This is not physically possible; which is why ultra
slow-roll introduces parameters in which the field acts differently under these conditions.
Since ^13, or Equation 57, to get Equation 60.
However, as mentioned above, at extremely flat potentials, we must modify the theory slightly. Here, the slope term
in which case the friction term locks with the acceleration term^14:
Using the two parameters, it is possible to estimate the length and scale of inflation within the slow-roll model in e-folds given by the integral^14
To calculate N, we can use an alternative method to arrive to the final value: calculating the size of a Hubble patch during inflation^14
This can be interpreted as taking today’s size of a Hubble patch and multiplying it by the growth of the scale factor and a certain number of e-folds. Approximating
We can rewrite N as
We therefore know that for inflation to be valid, we must have at least 60 e-folds^14 This is a requirement for ultra slow-roll inflation, along with the two parameters
This concludes the section’s purpose of establishing the necessary parameters for a successful inflationary model of the universe. Taking the same into account, we can now evaluate its weaknesses and
suggest possible research avenues to ameliorate the same.
2.2 Results
This section will topically discuss the current weaknesses and challenges in the inflationary model and further suggest pathways of research that can potentially develop the theory to make it more
2.2.1 Weaknesses of the Inflationary Model
2.2.1.1 The Graceful Exit Problem
The graceful exit problem is the name given to the conceptual challenge of finding a valid model that describes the events following the inflaton potential’s transition to the true vacuum.
2.2.1.2 Timeline and Recombination
In the model of slow-roll inflation, the universe is returned to the Big Bang through a process known as recombination, or reheating. As the potential oscillates near the zero-point, potential energy
is rapidly converted into kinetic energy and vice versa periodically; and the inflaton potential undergoes a phase transition – all the energy is manifested as relativistic matter at extremely high
This is caused by a phenomenon known as parametric resonance. The oscillations of the inflaton field potential can resonate constructively with the natural frequencies of other fields, leading to the
creation of particles.
Recombination and reheating are also directly connected to the timeline and spectra created by inflation.
This connects directly to Section 2.2.2.1, in which the power spectra of the CMB radiation are key in our verification of warm inflation.
2.2.2 Novel Research Pathways
Having evaluated the weaknesses of the model, it is now possible to investigate the potential pathways of research that can provide insight into solving these challenges.
2.2.2.1 Cosmic Microwave Background
The cosmic microwave background is electromagnetic radiation permeating the universe; or a sea of photons, currently at a temperature of about 2.7 Kelvin^16. It can be used as a tool for cosmic
archaeology. The distribution of the radiation over a range of spectra is essential; as well as the type and nature of the anisotropies in the radiation.
Given the origin of the CMB, it is key in the identification of the necessary initial conditions of the inflationary model as well as models for the guidance of the evolution of the anisotropies
during inflation.
The CMB carries imprints from early stages of the universe at extremely high energies. It is a spectrum of density fluctuations that set the initial conditions for the structure of the universe.
Exploring the power spectra of primordial gravitational waves that created density fluctuations as well as the resultant power spectrum of the CMB is key in structuring a newer, more accurate
framework of inflation.
To map the CMB, we follow the photons pointed at us backwards in their path as they reach the Earth back to the surface where they last scattered from the CMB plasma; i.e. the surface of last
scattering. This pattern of photons can be mapped.
As expected, in the map, there should be minute deviations in the density and polarization of the radiation due to the interference of the radiation as it reaches us with the mass structures of the
However, in order to give rise to these mass structures, there must have been density perturbations in the early universe. These perturbations lead to anisotropies in the CMB: uneven temperatures and
intensities of photons across the map.
To map these perturbations, it is possible to liken the more accessible example of sound waves and air. Sound waves also create density perturbations. If we had a snapshot of the sound in an
orchestral room, we can create a map of the density perturbations caused by the sound emitted by each instrument. We can graph the mapped perturbations’ frequency against their strength on a graph.
The same can be done to the strength of different density waves in the universe^17.
Following the same process outlined above, the most accurate measurement of the CMB is that of the ESA’s Planck Satellite.
If we graph the observed photons’ brightness against their frequency, the following curve is obtained at a constant temperature of approximately 2.7 Kelvin. This also matches the fact that the
anisotropies seen above in the CMB are at the level of only 0.00001 Kelvin, describing a highly uniform CMB^17. The graph can be seen in Figure 2.
Fig. 2: Cosmic Microwave Background Spectrum^17
We have to measure the polarization of the light for the next step in analyzing the CMB. This is caused by gravitational waves in the early universe, which are amplified by inflation. They spread in
all directions and must be detected along with the photons in the plasma phase. Not only gravitational waves, but also quantum fluctuations create anisotropies in the radiation. Taking into account
the effects of both allows us to go further in analyzing the CMB^17
To set the initial conditions for the fluctuations, the modes can be represented in a Fourier series as they all evolve independently. In perturbation theory, the general expression for the change in
a value, field, or metric is described as
Where ^18
The wave modes of the same perturbation fluctuations can be Fourier transformed to become a series of wave modes at a constant wavenumber ; i.e.
The fluctuations in the scalar inflaton field can be treated in the same manner. However, when we take into consideration the fact that the Hubble radius stays constant while the universe expands, it
can be postulated that relative to the spacetime manifold, the Hubble sphere shrinks^17.
As the shrinking takes place, the wavelengths of the inflaton’s fluctuation modes escape the Hubble radius, thereby stretching out the quantum fluctuations to large scales; and creating the
aforementioned density perturbations.
Mathematically, this can be expressed by the inequalities As the shrinking takes place, the wavelengths of the inflaton’s fluctuation modes escape the Hubble radius, thereby stretching out the
quantum fluctuations to large scales; and creating the aforementioned density perturbations.
Mathematically, this can be expressed by the inequalities
After inflation, as the Hubble sphere begins to grow, the radius will catch up with
The fluctuations result in two classes of perturbations; i.e. scalar and tensor. Scalar perturbations are changes to the curvature, defined by
while tensor perturbations caused by gravitational waves are defined by
which is a transverse, traceless perturbation in the spacetime metric formalized by a tensor^19 .
With the formalization of these two perturbations, we can logically derive the fact that they are equivalent at horizon exit and at causal horizon re-entry^18. The figure below graphs the constant
density fluctuations of the universe against the size of the comoving horizons, once again on comoving scales, against time on a logarithmic scale, as seen in Figure 3.
Fig. 3: Comoving Horizon Scale Graph^18
Since we know that the density fluctuations are the same at exit and re-entry, we can use the initial conditions of the CMB to make predictions about post-inflationary physics.
The vertex of the comoving causal horizon graph is analogous to the reheating stage of inflation^18, as shown in Figure 4.
Fig. 4: Inflaton Field Potential^20
As the field oscillates at its vacuum point, a process known as parametric resonance occurs in which, due to its coupling with other fields, a gargantuan amount of radiation and matter is produced by
the inflaton as it resonates with other fields. This produces a universe filled with matter and radiation at energy levels of the Big Bang^18.
At this stage, the density fluctuations are identical to what they were at the exit of the wavemodes from the Hubble horizon.
Because of the constant nature of the density fluctuations, we can extrapolate the conditions at horizon re-entry to today’s universe.
These theoretical predictions at both phases of inflation can then be cross-referenced with observations and thereby filter out theories that do not pass the heuristic test.
2.2.2.2 Galactic Redshift Analysis
In this section, we will explore a novel potential pathway of research for the verification or negation of a wide variety of inflationary models.
For this, we will be using more conventional evolutionary cosmology; particularly galactic and stellar formation as well as redshift and the expansion of the universe.
The James Webb telescope, a pioneer of the same field, released a set of astonishing images in 2022 of the early universe.
By spectrometry and photoanalysis, it is possible to calculate the redshift of the galaxies based on their distance from us as observers^21, allowing for the derivation of the Hubble constant, or
Hubble rate. The inverse of the Hubble rate will then give us the age of the universe; which can be used to corroborate various models of inflation based on their time period, or length of inflation
in e-folds.
We will first derive a generalized formula or construct for the age of the universe based on estimations from the redshift of the JWST-imaged galaxies; then substituting it into the relevant
estimations of age through inflationary dynamics.
3 Discussion
The literature review undertaken above yields great insight into the status quo of inflationary cosmology in the 21st Century. Through the analysis of old inflation, slow-roll inflation, and ultra
slow-roll inflation, a number of weaknesses – both conceptual and mathematical – were evaluated. Through further review, two primary research methods were discussed in order to address these
LEAVE A REPLY Cancel reply | {"url":"https://nhsjs.com/2024/field-theories-in-inflationary-cosmology/","timestamp":"2024-11-03T05:58:47Z","content_type":"text/html","content_length":"342443","record_id":"<urn:uuid:8e0e9414-4d01-4477-b38d-cde06e0c088f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00758.warc.gz"} |
AP Statistics Tutorial
Welcome to Stat Trek's Advanced Placement (AP) Statistics tutorial. This free, online statistics course is designed to help you master the Advanced Placement Statistics Exam. > Begin lesson 1
About the Tutorial
This tutorial provides accurate and complete coverage of the AP Statistics curriculum. Specifically, the AP Statistics curriculum and this tutorial cover the following topics:
• Exploring data. Using graphical and numerical techniques to study patterns of data. Emphasizes interpreting graphical information and descriptive statistics.
• Sampling and experimentation. How to plan and conduct a study. Focuses on clarifying research questions and specifying methods to collect and analyze data.
• Anticipating patterns. Using probability and simulation to understand random events. Focuses on using probabilistic models to understand real-world events.
• Statistical inference. How to estimate population attributes, based on sample data. How to test statistical hypotheses.
How to Use the AP Statistics Tutorial
This tutorial is built for self-study. Even if you have no prior experience with statistics, you can work through the tutorial at your own pace and teach yourself statistics!
If you are taking an AP Statistics course in school, use this tutorial as a study aid. Before each class, read the relevant lesson from the tutorial. This will have two good effects.
• Because you have been exposed to the material, you will find it easier to understand your instructor's lecture, and you will retain the information more effectively.
• And, if anything in the tutorial is unclear, you will be alerted to a potential area of confusion. You can get clarification from your instructor when he/she covers the material.
Individual lessons are accessible through the table of contents, which can be found in the vertical column on the left side of the page. You should work through lessons in the order in which they
appear; because each lesson builds on previous lessons.
Individual lessons are accessible through the table of contents, which can be accessed by tapping the "AP Statistics: Table of Contents" button at the top of the page. You should work through lessons
in the order in which they appear; because each lesson builds on previous lessons.
Additional Helpful Resources
As you progress through the AP Statistics tutorial, take advantage of the following helpful resources.
• Analytical tools. Stat Trek provides a variety of analytical tools - online statistical tables, calculators, problem solvers - to take the drudgery out of statistical computations. The tutorial
will alert you to these tools, all of which are free.
• Sample problems. Most of the lessons include sample problems. The sample problems help you test your knowledge. They also illustrate shortcuts and solutions to common statistics problems.
• Practice exam. After you have completed the tutorial, take the practice exam. Review the explanations for any questions that were answered incorrectly.
• Online help. Stat Trek's online Statistics Dictionary takes the mystery out of statistical jargon. If any term or concept is unclear, visit the dictionary for additional explanation.
Note: The dictionary can be accessed by clicking the Help tab in the main menu (located at the top of this web page). | {"url":"https://www.stattrek.com/tutorials/ap-statistics-tutorial","timestamp":"2024-11-12T23:15:45Z","content_type":"text/html","content_length":"62380","record_id":"<urn:uuid:29ff9d34-5bd0-4534-9fa0-b2d8861dcaca>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00673.warc.gz"} |
Student Learning Outcomes
Mathematics Program Student Learning Outcome: Recipients of our AS degree, in mathematics, will be well prepared to continue their education in STEM (Science, Technology, Engineering, Mathematics) at
a college or university.
Math, Science, and Engineering Student Learning Outcomes
Course Student Learning Outcome (SLO)
Apply mathematical principles and techniques to solve problems in areas such as ancient systems of numeration, set theory and number theory.
Math 1 Use critical thinking to arrive and conclusions from Venn diagrams, syllogistic forms and truth tables.
Demonstrate knowledge of affective domain and study skills.
Math Analyze and solve a precalculus level problem using analytic methods.
11 Sketch the graph of a precalculus level problem using skills beyond plotting a table of points.
Demonstrate knowledge of affective domain and study skills.
Math Recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of integration and its applications.
13 Recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of the derivative and its applications.
Demonstrate knowledge of affective domain and study skills.
Interpret slope as rate of change.
Math Use exponential growth and decay models to make predictions.
14 Demonstrate knowledge of affective domain and study skills.
Computational Skills: successful students will be proficient in arithmetic with integers, rational numbers, decimals and percents
Construct and interpret graphs such as bar charts, histograms and box plots.
Math Compute appropriate descriptive statistics.
20 Choose and apply inferential analyses in order to draw conclusions about a population.
Demonstrate knowledge of affective domain and study skills.
Math Solve linear equations: Students will be able to solve linear equations.
Math Critical thinking: use critical thinking to arrive at conclusions from Venn Diagrams, syllogistic forms, and truth tables.
100 Cultural understanding: relate a knowledge of the people, and uses of mathematics throughout history of mathematics.
Principles and Technique: apply mathematical principles and techniques to solve problems in areas such as ancient systems of numeration, set theory, and number theory.
Math Interpret slope as a rate of change.
101 Use exponential growth and decay models to make predictions.
Math Place Value: students will demonstrate an understanding of place value by counting in bases other than base ten
Math Area and Perimeter: students will be able to demonstrate an understanding of the difference between area and perimeter.
Math College Algebra: students will be able to analyze and solve a precalculus level problem using analytic methods and be able to sketch the graph of a precalculus level function.
Applications of Right Triangle Trigonometry: use trigonometric functions to solve application problems involving unknown sides of right triangles
Math Trigonometric Equations: be able to solve equations involving trigonometric functions
115 Trigonometric function values: analytically evaluate the six trigonometric functions of angles of measures that are multiples of 30 degrees and 45 degrees.
Trigonometric Identities: use basic identities to verify trigonometric identities or to simplify trigonometric expressions.
Math Descriptive statistics: compute appropriate descriptive statistics.
120 Graphing: students will be able to construct and interpret graphs such as bar charts, histograms and box plots.
Inferential statistics: choose and apply inferential analyses in order to draw conclusions about a population.
Math Students will be able solve multi-step precalculus level problems in a variety of contexts related to science, technology, engineering, and mathematics
126 Students will be able use multiple representations of functions to interpret and describe how two quantities change together.
Math Students will be able to create sinusoidal models and interpret the period, amplitude, vertical shift and phase shift in the context of STEM applications.
127 Students will be able to use multiple representations of functions to interpret and describe how two quantities change together.
Students will be able to solve trigonometric equations.
Math Interpret derivative: students will recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of the derivative and its applications.
130 Interpret Integration: students will recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of integration and its applications.
Math Graph functions: demonstrate proficiency in the graphing of functions at the precalculus level.
135 Solve equations: solve equations involving algebraic and transcendental functions at the precalculus level
Antiderivative: find the antiderivative of a function using basic integration rules.
Math Limits: evaluate limits analytically.
140 Optimization: use calculus to solve optimization problem
Rules of Derivatives: find the derivative of a function using rules of derivatives
Math Integration Techniques: demonstrate proficiency in evaluating integrals using various techniques of integration.
Math Functions, Subroutines: develop a FORTRAN-90 program that contains functions and subroutines.
146 Sequence, Selection, Iteration: develop a FORTRAN-90 program that contains sequence, selection and iteration control structures
Math Demonstrate understanding of the theoretical foundations of linear algebra, such as vector spaces, inner product spaces, the eigenvalue problem. May include applications from math, science, or
200 engineering.
Solve a linear system using appropriate methods and interpret the results.
Math Multivariable Functions: perform calculus on multivariable functions.
205 Vector Operations: perform vector operations using geometry in space.
Vector Valued Functions: Perform calculus on vector valued functions
Math Application of Differential Equations: successful students will be able to compare first- and second-order differential equations, solve these equations using appropriate techniques including
206 constructing solutions using series and matrices, and apply them to problems in science and engineering.
Math Mathematical Proofs: prove a statement using one of the basic methods of proof or disprove it using a counter example.
245 Minimum Spanning Tree: Use a standard algorithm to find a minimal spanning tree for a given graph. | {"url":"https://www.palomar.edu/math/student-learning-outcomes/","timestamp":"2024-11-13T01:59:53Z","content_type":"text/html","content_length":"85801","record_id":"<urn:uuid:7a5ad88e-d383-40f2-8929-408e17f1dbd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00292.warc.gz"} |
Approximating the number of network motifs
The World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species are only a few examples of systems comprising a large number of highly
interconnected dynamical units. These networks contain characteristic patterns, network motifs, that occur far more often than in randomized networks with the same degree sequence. Several algorithms
have been suggested for counting or detecting the number of occurrences of network motifs as trees and bounded treewidth subgraphs of size O(log n), at most 7 for some motifs. In addition, local
motif counting, counting the number of motifs in which a node participates, was recently suggested as a method of classifying nodes in the network. The premise is that the distribution of motifs in
which a node participates is an indication of its function in the network. Therefore, local counting of network motifs provides a major challenge. However, no such practical algorithm exists other
than local counting of triangles. We present several algorithms with time complexity (Formula presented) that approximate for every vertex the number of occurrences of the motif in which the vertex
participates, for k-length cycles and k-length cycles with a chord, where k = O(log n), and algorithms with time complexity (Formula presented) that approximate for every vertex the number of
noninduced occurrences of the motif in which the vertex participates for all motifs of size four. In addition, we show algorithms that approximate the total number of occurrences of these network
motifs when no efficient algorithm exists. Some of our algorithms use the “color-coding” technique.
Funders Funder number
Israeli Science Foundation center of knowledge on communication networks 1685/07
European Commission
Dive into the research topics of 'Approximating the number of network motifs'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/approximating-the-number-of-network-motifs-2","timestamp":"2024-11-10T12:08:16Z","content_type":"text/html","content_length":"51460","record_id":"<urn:uuid:4f90a91d-ff48-49dd-a3b6-792592a9d81d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00127.warc.gz"} |
Example 3
Suppose you want to calculate the rate of depreciation when given the original cost, the current value, and the time in years. Solve the equation, V = C(1 − r) for the variable, r.
This applet is provided by Walch Education as supplemental material for their mathematics programs. Visit www.walch.com for more information. | {"url":"https://www.geogebra.org/m/zKsT6j5R","timestamp":"2024-11-10T15:06:54Z","content_type":"text/html","content_length":"88668","record_id":"<urn:uuid:4b20ce68-6ee6-4651-b018-c7d72b7a3a38>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00897.warc.gz"} |
Physical and magnetic properties of Sm0.2Gd0.8Ni 4B compound
5th Moscow International Symposium on Magnetism, MISM 2011, Moscow, Russia, 21 - 25 August 2011, vol.190, pp.208-212
• Publication Type: Conference Paper / Full Text
• Volume: 190
• Doi Number: 10.4028/www.scientific.net/ssp.190.208
• City: Moscow
• Country: Russia
• Page Numbers: pp.208-212
• Keywords: AC-susceptibility, DC-magnetization, Magnetic properties
• Hakkari University Affiliated: Yes
Physical properties of the Sm0.2Gd0.8Ni4B compound have been investigated by means of the X-ray powder diffraction, DC and AC-susceptibility techniques. The compound studied crystallizes in CeCo 4B
type structure with P6/mmm space group. The unit-cell parameters a and c are determined as 5.01 and 6.95 Å, respectively, and the unit-cell volume V is calculated as 151.08 Å3. DC and AC magnetic
measurements present the visible magnetic phase transition from paramagnetic to ferromagnetic, around definite transition temperature. The magnetic phase transition temperature of the compound is
obtained from DC magnetization, AC-susceptibility and the well known Kouvel-Fisher method as 36.6, 35.7 and 35.2 K, respectively. The saturation magnetization (Ms) and the coercive fields (Hc) of the
compound are found to be 3.7μB/f.u and 277 Oe, respectively, by using the hysteresis loops at 9.5 K. We have also investigated the non-linear AC-susceptibility of the compound, around its
ferromagnetic transition temperature, as a function of temperature, frequency and amplitude of the AC-driving field. In order to explain the measured experimental results, we have used the theory
developed for ferromagnetic, based upon the mean field model. The measurements exhibit both frequency and amplitude dependencies. Observed dependencies are compared with the existing theories of
linear and nonlinear susceptibilities with reference to short- and long-range interactions. In Kouvel-Fisher method, one plots 1/χ1*(dχ-1/ dT) against T, obtaining a straight line. The slope of this
line gives the critical exponent γ, and it intersects the T axis at Tc. In order to obtain dχ-1/dT and the best straight line, we used a two-point numerical differentiation program and linear
regression method, respectively. The critical exponent γ of the sample is calculated to be 2.78 ± 0.05. The value of the critical exponent β, which is characteristic of static phase transition to a
ferromagnetic state, is estimated as 2.41±0.3 from the slope of the line obtained the plot of the absolute third-harmonic values versus the reduced temperature on a log-log scale. © (2012) Trans Tech | {"url":"https://avesis.hakkari.edu.tr/yayin/54fec16f-a74e-4b23-9e10-b79a9fd5ad2c/physical-and-magnetic-properties-of-sm0-2gd0-8ni-4b-compound","timestamp":"2024-11-02T05:27:36Z","content_type":"text/html","content_length":"53698","record_id":"<urn:uuid:47184050-8154-4b57-900a-c4c0797e1cf8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00837.warc.gz"} |
Tento text není v aktuálním jazyce dostupný. Zobrazuje se verze "en".
The regularized method of analytic continuation is used to study the low-energy negative-ion states of beryllium (configuration 2s(2)epsilon p P-2) and magnesium (configuration 3s(2)epsilon p P-2)
atoms. The method applies an additional perturbation potential and requires only routine bound-state multi-electron quantum calculations.
Such computations are accessible by most of the free or commercial quantum chemistry software available for atoms and molecules. The perturbation potential is implemented as a spherical Gaussian
function with a fixed width.
Stability of the analytic continuation technique with respect to the width and with respect to the input range of electron affinities is studied in detail. The computed resonance parameters E-r =
0.282 eV, Gamma = 0.316 eV for the 2p state of Be- and E-r = 0.188 eV, Gamma = 0.167 for the 3p state of Mg- agree well with the best results obtained by much more elaborate and computationally
demanding present-day methods. | {"url":"https://explorer.cuni.cz/publication/553637?lang=cs","timestamp":"2024-11-11T18:04:10Z","content_type":"text/html","content_length":"28969","record_id":"<urn:uuid:349a1138-dade-42e1-960c-3807c0cd3ff9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00012.warc.gz"} |
Roman Numerals Converter - Online Numbers/Date/Year Translator (2024)
1. Communication System
2. Numeral System
3. Roman Numerals
Roman Numerals to Hindu-Arabic (English) Converter
Roman Numeral Writer
Answers to Questions (FAQ)
What are Roman Numerals? (Definition)
Roman numerals are the name given to the numeral system used in ancient Roman times (especially in the time of Caesar), read from left to right it uses 7 letters whose values are added or subtracted
according to their position.
What are the letters to write in Roman Numerals?
Roman numeration uses 7 letters corresponding to 7 numbers. Roman digits chart from 1 to 1000:
Beyond several thousands, there are no letters to represent these numbers.
However, some archaic scripts (more rare) used 4 other symbols
How to read/write with Roman numerals?
Roman numeral system uses two rules:
— (1) Any letter $ L_2 $ placed to the right of another letter $ L_1 $ are added if $ L_2 \leq L_1 $
Example: VI = 5 + 1 = 6
XX = 10 + 10 = 20
— (2) Any letter of unit $ L_1 = \rm{I} $ placed immediately to the left of another letter $ L_2 \neq \rm{I} $ is subtracted.
Example: IV = 5 - 1 = 4
IX = 10 - 1 = 9
IL = 50 - 1 = 49
IC = 100 - 1 = 99
ID = 500 - 1 = 499
Rule (2) is sometimes extended to: Any letter $ L_1 $ placed immediately to the left of another letter $ L_2 > L_1 $ is subtracted.
Example: XC = 100 - 10 = 90
In theory therefore, any symbol (letter) is repeated a maximum of 3 times consecutively.
1970 in roman numerals MCMLXX 1971 in roman numerals MCMLXXI
1972 in roman numerals MCMLXXII 1973 in roman numerals MCMLXXIII
1974 in roman numerals MCMLXXIV 1975 in roman numerals MCMLXXV
1976 in roman numerals MCMLXXVI 1977 in roman numerals MCMLXXVII
1978 in roman numerals MCMLXXVIII 1979 in roman numerals MCMLXXIX
1980 in roman numerals MCMLXXX 1981 in roman numerals MCMLXXXI
1982 in roman numerals MCMLXXXII 1983 in roman numerals MCMLXXXIII
1984 in roman numerals MCMLXXXIV 1985 in roman numerals MCMLXXXV
1986 in roman numerals MCMLXXXVI 1987 in roman numerals MCMLXXXVII
1988 in roman numerals MCMLXXXVIII 1989 in roman numerals MCMLXXXIX
1990 in roman numerals MCMXC 1991 in roman numerals MCMXCI
1992 in roman numerals MCMXCII 1993 in roman numerals MCMXCIII
1994 in roman numerals MCMXCIV 1995 in roman numerals MCMXCV
1996 in roman numerals MCMXCVI 1997 in roman numerals MCMXCVII
1998 in roman numerals MCMXCVIII 1999 in roman numerals MCMXCIX
2000 in roman numerals MM 2001 in roman numerals MMI
2002 in roman numerals MMII 2003 in roman numerals MMIII
2004 in roman numerals MMIV 2005 in roman numerals MMV
2006 in roman numerals MMVI 2007 in roman numerals MMVII
2008 in roman numerals MMVIII 2009 in roman numerals MMIX
2010 in roman numerals MMX 2011 in roman numerals MMXI
2012 in roman numerals MMXII 2013 in roman numerals MMXIII
2014 in roman numerals MMXIV 2015 in roman numerals MMXV
2016 in roman numerals MMXVI 2017 in roman numerals MMXVII
2018 in roman numerals MMXVIII 2019 in roman numerals MMXIX
2020 in roman numerals MMXX 2021 in roman numerals MMXXI
2022 in roman numerals MMXXII 2023 in roman numerals MMXXIII
2024 in roman numerals MMXXIV 2025 in roman numerals MMXXV
How does the converter from/to Roman numerals work?
The program automatically detects whether the number is in Arabic or Roman numerals and makes the conversion/translation.
Roman numeration does not permit writing large numbers, beyond 9999 the program will display the number of thousands separately. This writing is not standardized but remains comprehensible.
The program is very permissive and allows badly formed Roman numbers not complying with the rule (2).
Example: IVX is translated as 6
How to write zero (0) in Roman numerals?
Romans did not use the zero, for them it was not a digit but a state of emptiness, so they did not write it (the absence of a number indicates zero).
dCode writes either ??, or 0.
How to write four (4) in Roman numerals?
Four is written IV, however, this software indicates that IIII = 4, unusual, IIII is a variant of IV which is tolerated. It can be found today (typically in watches, or clocks).
How to write a date with roman numerals?
There is no specific way to write a date (or a birthdate), except to write the number of the day, the month and the year separately.
Example: 12 / 06 / 2008 = XII / VI / MMVIII
dCode has a tool to write a date in latin.
In some European countries, the centuries are sometimes written in Roman numerals.
What is the biggest number in roman numerals?
Numbers above 10000 were not thinkable, without any calculation tool, they were useless. If you wish to write a value of hundreds of thousands, one can imagine writing hundreds of M at the beginning
of the number.
Example: 9999 = MMMMMMMMMCMXCIX (a bit ridiculous)
How to write negative numbers in roman numerals?
The negative writing is not recognized, it probably did not exist. The notion of positive or negative numbers is related to the concept of zero (which was not known to the Romans).
However, today, adding a - can help to be understood.
Example: -XXV = -25
How to write a decimal number in roman numerals?
Using decimal numbers is very few documented in history books, however, it is probable that they used fractions, including a duodecimal currency system (base 12) which allowed sharing by 2, 3, 4, 6
and 12 without decimal places.
When were Roman Numerals invented?
Roman numerals were born with Antique Rome, so starting at the 7th century BC. For example, they were used with Latin.
How to write Roman Numerals with Unicode?
Roman numerals have been added to the Unicode standard, they encode by a single character each number from 1 to 12 (used in clocks and watches) and 8 other numbers meaning:
Unicode character Value Unicode character Value Unicode character Value
Ⅼ 50 Ⅽ 100 Ɔ 500
Can there be more than 4 identical consecutive letters?
Roman numerals can be written with 4 identical letters in a row, but this is rare or incorrect.
Example: 4000 can be written MMMM or the watchmaker's four is written IIII
When to use Roman Numerals?
Roman numerals are learned at school in primary school but are rarely used except in mathematics or history. The uses today are limited to clocks, dates, but also on tattoos, many tattoos use Roman
Source code
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solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Roman Numerals" functions (calculate, convert, solve, decrypt / encrypt, decipher /
cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Roman Numerals" are not
public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
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Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Roman Numerals on dCode.fr [online website], retrieved on 2024-07-24, https://www.dcode.fr/roman-numerals | {"url":"https://hobokendive.com/article/roman-numerals-converter-online-numbers-date-year-translator","timestamp":"2024-11-02T01:26:22Z","content_type":"text/html","content_length":"113827","record_id":"<urn:uuid:a95a1d7d-bd50-41ae-bf88-ad907ad1ca7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00676.warc.gz"} |
Are there experts who offer assistance with Fluid Mechanics model validation using multi-objective optimization? | Hire Someone To Do Mechanical Engineering Assignment
Are there experts who offer assistance with Fluid Mechanics model validation using multi-objective optimization? Would you do this yourself and do it over the web? Is your source code available for
free? The first hurdle I had when writing my Fluid Mechanics specification was the basic knowledge of open source software. Actually, it’s been there and done. There aren’t such experts at the
moment, but I’d definitely like to know more and help out folks. So, I went to the Fluid Mechanics community and asked them if they ever had one. And they are, actually, quite a long time. The
document includes several sections covering issues, which I will start with. First of you could try here I will provide here: First section: Introduction to open source Fluid Mechanics; “For your
work on a mechanical model, the entire form is produced by fluted (or grated) blocks to represent the relationship between the parts. Blocks come in two forms: straight, smooth or fibrous. If a piece
is spiked or is fibrous, for example, the fluid can be spun on any of a few different rollers. If a piece is fibrous, the force required is the same without any change in the material. Otherwise, the
force will be the same upon the load (or friction). Additionally, the area of influence of the material will be an electric potential, or electric potential, based on the resistance of the material.
That is what Fluid Mechanics means, and you can apply that to a flat or flat flat. Fluid is always flat but it will be a surface fluid, wet with water instead of dry drying or dew, to which each
piece can accept equally. Bias is the amount of material introduced to a material when the difference between the elastic moments of the material and those of the solid, is sufficiently small that
mechanical misalignment or deformation produces a failure. This can occur when theAre there experts who offer assistance with Fluid Mechanics model validation using multi-objective optimization? I
have expertise in Read Full Article this, the main problems I am having to solve in my head read more three questions: A classical model is 2×2 or a 2×2 A software model is Model 1 where there is a
distance of $1\pi$, just Website a linear model. Is there such any word solution for this problem. I was thinking of looking for “weird” or “spank” solution: maybe by hand I would have identified
some important properties of the $\pi/2$ plane that should be verified in A software model. A: I don’t think that there’s a standard way to construct a linear PDE for 3D space – this more tips here a
Read Full Report review point. But a better approach is probably based on the tools given by Hansen and Rohrlich.
Someone To Do My Homework For Me
One can think of the linear PDE as $ds^2 = -f(\nabla f) dt $ where $$ \left| \nabla f \right| = 3\|\nabla f\|$$ where $$ f(\nabla f) = \nabla |f|\nabla dt $$ Then the equation of the you can find out
more (the only function that is free in this case) will have the following expression, $$ \left\{ -\frac{|f|}{\|f\| + |\nabla f\|}\right\}^3 = -\pi^3\|f|^2 \delta f = -3\pi^3 \nabla f + \nabla \log \
nabla f = -\pi^2|f|^2 \delta f$$ The form of the denominator is tricky. But it’s easier if you solve for $f$. So when $\nabla f$ is real then identity $$ \left\{\sum_{Are there experts who offer
assistance with Fluid Mechanics model validation using multi-objective optimization? Similar question should be addressed in a comment for their work. It seems that Fluid Mechanics models may be
based on some different data types. According to the Fluid Mechanics website, datasets of dynamic pressure settings are available from Procter & Gamble because they are large i loved this reliable in
many aspect including: surface pressure in unnormal stress domains; surface properties of polymeric mixtures; capillary networks and capillary channels of composite materials; the number of particles
of many kinds in flow-flow models; as well as models of the polymer particle in the rest conditions. One alternative to these models is to create models that are sufficiently simple for calculation.
Recently developed Fluid Mechanics models also take advantage of dynamic load testing with use this link addition of models for other topics, and like models from Bluré, they can be built directly
from the datasheets of these models, with an approach of complex models of other situations. On the other hand from designing models based on model validation, the demand on the development of
research articles is huge. It is possible to check the model validation by different types of non-compatible interfaces, so that new non-compatible models could be created, e.g. web pages. However,
when designing non-compatible models view it from the market, or from existing experts, it is somewhat difficult to establish the basis of the model to the current user; this is in fact the real
issue for Fluid Mechanics users. The following techniques were introduced into this view based upon the assessment of the model-data validation technique from a web page. The problem is that, on the
one hand, the models of Fluid Mechanics, are limited in the available validation methods; and instead, they are limited in their features of validity. 1. Data Validation for Non-Compatible Model
Validation Tools [1] All publications of F$_{_{%others_} %}$ with our methodology that specifically describe modeling | {"url":"https://mechanicalassignments.com/are-there-experts-who-offer-assistance-with-fluid-mechanics-model-validation-using-multi-objective-optimization","timestamp":"2024-11-07T09:08:09Z","content_type":"text/html","content_length":"133565","record_id":"<urn:uuid:8bb12140-9470-4a2a-809d-04d0d8422182>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00372.warc.gz"} |
(a^3)^2 Without Exponents
Understanding (a^3)^2 without Exponents
The expression (a^3)^2 might seem confusing at first glance, but it's actually quite straightforward when broken down. Let's explore how to write it without exponents.
Breaking Down the Exponents
• a^3: This means "a multiplied by itself three times": a * a * a
• (a^3)^2: This means "a^3 multiplied by itself two times": (a * a * a) * (a * a * a)
Expanding the Expression
Expanding the expression, we get:
(a * a * a) * (a * a * a) = a * a * a * a * a * a
Simplifying with Multiplication
Since we are multiplying the same variable 'a' by itself six times, we can write it as:
a * a * a * a * a * a = a^6
Therefore, (a^3)^2 is equivalent to a^6. This demonstrates that when dealing with exponents raised to another exponent, we simply multiply the powers. | {"url":"https://jasonbradley.me/page/(a%255E3)%255E2-without-exponents","timestamp":"2024-11-03T03:11:22Z","content_type":"text/html","content_length":"56750","record_id":"<urn:uuid:ddd143d8-037f-4dbe-bb9c-cce751e76f21>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00716.warc.gz"} |
Understanding and Doing Math
The comics are from the book Understanding and Doing Math – Circle 1 Are natural numbers natural? Do we need zero? Be careful with negative numbers! How to rationally divide two golden bars among
three pirates? Secrets of decimal notation Bills can sometimes be irrational.
A Paradox of School Mathematics
School mathematics: Real mathematics:
Which unit of measurement is better: English foot or Croatian elbow?
My neighbor John and I got into a serious disagreement while working on a fence between our houses: will we use an English foot (John’s proposal) or a traditional Croatian elbow (my proposal) to
measure the fence? This event helped me solve a problem that has plagued me since elementary
To be vaccinated, or not to be vaccinated, that is the question
Someone proposes the following bet to you. A symmetrical dice will be rolled only once. If is rolled, the challenger gets euros, otherwise, you get euros. The probability of rolling is and the
probability of not rolling is , times bigger. Would you accept a bet? What would you do
A Path into Math
Some do math because they have to. Some do math because they think math helps them manage the world. Some do math because they find beauty in it. Not only is there no royal path to math, as Euclid
said long ago (for geometry), but there is also no common
James Joseph Sylvester (1814 – 1897)
May not music be described as the mathematics of the sense, mathematics as music of the reason? .
Tobias Dantzig, about the development of the notion of number, in Number: The Language of Science, Macmillan, 1930
It is not a story of brilliant achievement, heroic deeds, or noble sacrifice. It is a story of blind stumbling and chance discovery, of groping in the dark and refusing to admit the light. It is a
story replete with obscurantism and prejudice, of sound judgment often eclipsed by loyalty
W.Servais, T.Varga: Teaching School Mathematics, A UNESCO Source Book, 1971.
Learning is much more similar to biological growth than to manufacture, where component parts are first produced, then fitted together.
W.Servais, T.Varga: Teaching School Mathematics, A UNESCO Source Book, 1971
Every child, by nature, likes learning just as he likes eating. Children reluctant to eat, and parents using promises and threats in order to make them eat, are not rare, yet they constitute perhaps
rather an exception than a rule. In teaching, the situation is worse: to attach rewards and | {"url":"https://understandingmath.academy/","timestamp":"2024-11-13T22:09:55Z","content_type":"text/html","content_length":"40440","record_id":"<urn:uuid:4ac8d8e5-575d-41a6-911a-da9462aadcf5>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00317.warc.gz"} |
Development of STOC-FREE MODEL
The aim of WP1, which is led by ONIRIS France, is to develop a method (STOC-FREE MODEL) for the quantitative comparison of the confidence in freedom of disease in different control programmes for
non-regulated diseases in the EU.
The method STOC-FREE MODEL will allow the estimation of the confidence of freedom from infection and the associated uncertainty from heterogeneous data inputs available for different epidemiological
units such as animal, herd, sector, region or country. The method will be developed and evaluated using BVDV in cattle as an example disease.
WP1: Highlights of year 1
During the first year, WP1 focussed on the development of a conceptual model representing the course and dynamics of infection at different levels and the exploration of possible statistical methods
that showed potential to be used in this specific context. From 5 September on, PhD student Mathilde Mercat started to work on the project.
Conceptual model
The conceptual model described the infection process for the STOC free case disease - BVD - at 3 levels. At the animal level, the different infection states and the transitions between states (such
as susceptible, infectious or resistant) were evaluated. At the herd level, the model considered herd demography, contact structure and the transmission pathways. At the territory level, the model
represented possible transmission pathways from outside to within the territory. The conceptual model was developed and mapped the different types of information that existed for a given infectious
disease onto the true status regarding infection.
The model connected:
· The biological system: the true status regarding infection which is of interest for different levels of analysis: animal, herd and territory.
· Information that is extremely diverse. Conceptually, two types of information that are different in nature can be distinguished:
o Information generated and collected to specifically detect the pathogen of interest such as test results from control programmes
o Information associated with an increased probability of pathogen presence such as risk factors of infection
The conceptual model was delivered in April 2018 and will be used to design the appropriate statistical models that will integrate different pieces of information (data) for the estimation of
probabilities of being in each single state of interest (outcome) at different levels.
Statistical model
After evaluating and discussing different statistical approaches, development of a Bayesian network model appeared the most promising method to use in STOC free. Bayesian networks are flexible and
allow for heterogeneous input information. Such information can be incorporated by inclusion of
prior distributions for the parameters in the model. The prior distributions can be based on default information on for example country level but can be tailored to each specific situation by
entering more specific information. Data to specify the distributions to specific situations can be obtained from databases of control programmes, demographic data and contact structures between
herds that will have a heterogeneous nature. In addition, frequency of occurrence and risk estimates for factors that influence either the probability of introduction or delayed detection of the
infection in an animal or herd will be included in the model.
WP1: Highlights of year 2
During the second year, WP1 proceeded with the development of a statistical model based on the chosen method: a special type of Bayesian networks called a hidden Markov model, which allows
incorporating infection dynamics in the estimation. The results of the conceptual framework were delivered together with a document containing guidelines for identification and sources of data.
An initial simple version of the model was developed and discussed between the partners. It is a herd-level model in which the probability of becoming infected (τ1) is influenced by the occurrence of
risk factors and the probability of clearing an infection (τ2). The latter (τ2), among other things, depends on the CP in place. The first version was discussed and decisions were made about the time
steps used in the model (monthly), the risk factors that should be included and the amount of CP information that should be taken into account. Risk factors that are included are herd size,
introduction of cattle into the herd and the risk from neighbouring herds (prevalence of disease and/or livestock density). The model includes parameters describing the CP in place (risk mitigation
(including vaccination) + test system), the test characteristics and information such as the time since freedom was achieved. Later on the probability of freedom and associated uncertainty will be
estimated for specific strata in the population based on risk factors such as herd size (small/medium/large), introduction of cattle (yes/no), test scheme (BTM test, tag test, spot test), and
neighbourhood risk (low/high).
In the second year, an initial simple version of the model was developed using French data. In the third year, it is foreseen to expand the model by adding risk factors and more detailed CP
information. Thereafter, the model will be tested using case studies to validate and further improve the model.
WP1: Highlights of year 3
During the third year, WP1 proceeded with the development of a statistical model based on the chosen method: a special type of Bayesian networks called a hidden Markov model, which allows
incorporating infection dynamics in the estimation.
The initial version of the model that was developed on herd-level was expanded with an animal level module for countries that perform their CPs on animal level. The sensitivity of the different input
parameters in the models were tested by evaluating the model results of the simulated data when incorporating a range of pre-defined parameter values. The results were discussed and the model adapted
accordingly. After this exercise the model with the computer code and a handbook was provided to the members of the team together with an exercise dataset. Members could opt to start directly with
their own data (WP3) or to apply the model on the sample data first. During the annual meeting in October 2019, a half day workshop was organised by the members of WP1. The aim of this workshop was
to let the team members get acquainted with the model and to answer all questions so far. Between July 2019 and march 2020, each member was applying the model on the data of their own country and in
this process feedback was provided which led to several updates of the model. In April 2020 the model was finalised and delivered to EFSA together with the computer code.
Deliverables of all years can be found here | {"url":"https://stocfree.eu/work-packages/wp1","timestamp":"2024-11-11T11:30:34Z","content_type":"text/html","content_length":"29188","record_id":"<urn:uuid:1a2d2dd0-c69c-439a-bc5d-c8904387f45f>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00812.warc.gz"} |
history of mathematics Archives - Math Research of Victor Porton
In my book I introduce funcoids as a generalization of proximity spaces. This is the most natural way to introduce funcoids, but it was not the actual way I’ve discovered them. The first thing
discovered equivalent to funcoids was a function $latex \Delta$ (generalizing a topological space) which I defined to get a set as […] | {"url":"https://math.portonvictor.org/tag/history-of-mathematics/","timestamp":"2024-11-10T13:56:32Z","content_type":"text/html","content_length":"90812","record_id":"<urn:uuid:21d9ddec-02ed-4097-af00-90570c36499e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00117.warc.gz"} |
Can anyone help me with my AB Calc?
Can anyone help me with my AB Calc?
October 24, 2005 - 23:07
Can anyone help me with my AB Calc?
October 25, 2005 - 07:39
(Reply to #1) #2
Sure, but I don't think it would be time productive to type out the entire theory of calculus. What chapter are you on and what don't you understand.
You know you're an AP student if...
you think studying is fun.
you constantly find yourself saying "we had homework?"
everything you know about sex, you learned in english class.
If you try to fail, and succeed, which have you done?
October 25, 2005 - 07:50
(Reply to #2) #3
oh haha other post, for implicit differentiation:
-Use d/dx as a function over your entire equation
-Factor out the dy/dx and solve for it
-You can use the d/dx for almost any equation for any term
-break up the equation over addition and take the derivate of each single term
Example: x^2 + y^2 = 49
now, you are allowed to do almost anything provided you do it to both sides correct.
You can add one to both sides
You can take the square root of both sides
You can cube both sides
You can also take the derivate of both sides
d/dx(x^2 + y^2) = d/dx(49)
d/dx can be split over addition, right?
d/dx x^2 + d/dx y^2 = d/dx 49
When you do this, I recommend you always use the chain rule. So here it goes...
2x dx/dx + 2y dy/dx = 0
so dx/dx cancels out
2x + 2y dy/dx = 0
You want to isolate and solve for dy/dx
2y dy/dx = -2x
dy/dx = -x/y
Does that help.
The best way to understand I think is to treat d/dx as a function, that you are allowed to do to both sides of the equation and it will still be equal. All d/dx does is find the change of some
variable in relation to x. dy/dx is how much y changes with x. And since slope is defined as delta y/delta x, taking the derivative(d/dx) will give you the slope. I hope this helps. I can give
another example if you want, or answer any specific questions.
You know you're an AP student if...
you think studying is fun.
you constantly find yourself saying "we had homework?"
everything you know about sex, you learned in english class.
If you try to fail, and succeed, which have you done?
October 25, 2005 - 23:45
(Reply to #3) #4
try this one, I'm stumped:A balloon rises at the rate of 8 feet per second from a point on the ground 60 ft. from an observer. FInd the rate of change of the angle of elevation when the balloon is 25
ft. above the ground.
October 26, 2005 - 23:56
(Reply to #4) #5
I did my work on paper, but I cannot format the picture to fit right now, so I'll walk you through it. This type of problem is called a related rates problem.
The first step is to write down all of your known values as well as your unknown ones. I'm going to let K stand for Theta for the angle.
dx= 0 ft/s
dy= 8 ft/s
y= 25 ft
x= 60 ft
K= ?
dK= ?
Next draw a picture and label all of your values.
This should basically be a right triangle with the right angle in the bottom right hand corner and the longer leg on the bottom
Now let your x be the horizontal line and your y be your vertical line and put in the values that you know. O will the the bottom angle on the left.
Now write the equations that you know that have the variables that you have in them. If there are any other variables, then this will not work.
For this problem use the definition for tangent:
tanK= y/x
Since you know y and x, use inverse tangent to find theta (it should be 22.62 degrees). Now you need dK. To find that take the derivative of the above equation. You should get this:
xdy - ydx = (secK)^2dK
Now solve the equation for dK and plug in your values. The final equation should be:
dy(cosK)^2 = dK
And your final answer should be 0.11 degrees per second.
I hope that you understood that!
If anyone sees a problem with this, please tell me. But this should be right.
Advice for the future: if you are ever stuck on a calculus problem that can be graphed or drawn, do so and see if it makes any more sense than it did before. The same thing applies for
Physics...probably not a good thing for chronic doodlers though. Good Luck! :D
[=RoyalBlue][=Comic Sans MS]
"I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."
"But," say Man, "the Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It
October 27, 2005 - 08:27
(Reply to #5) #6
That's a good solution, I didn't check to see if the math is right. If you can't do a problem, before you ask anybody else, write an equation in terms of what you are trying to find. Any equation.
For this problem it would be tan(k) = y/60. Setting up and solving the problem is usually the hardest part. Once you get an equation, fill in as many values as you can, then see if it is able to be
solved. Once you're there, if your stumped come here, but that is usually the hardest part of the problem, you just need to figure out how to set it up
You know you're an AP student if...
you think studying is fun.
you constantly find yourself saying "we had homework?"
everything you know about sex, you learned in english class.
If you try to fail, and succeed, which have you done?
October 27, 2005 - 13:56
(Reply to #6) #7
chessmaster1990 wrote:That's a good solution
Thank you :D
[=RoyalBlue][=Comic Sans MS]
"I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."
"But," say Man, "the Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It
December 6, 2007 - 11:13
(Reply to #7) #8
Hey guys, I'm new to course-notes forum, but not new to calc. I took AB Calc last year and I got a 5. Here's a neat website for AB, BC Calculus and Physics.
It has great tutorials.
Hope it'll help. Although, dont think ull get a 5 with just that website. U shud do all ure hw from ure book and if u want to get a 5, U should buy a barrons review book.
"Music is the movement of sound to reach the soul for the education of its virtue."
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Quantization | Concrete ML
Quantization is the process of constraining an input from a continuous or otherwise large set of values (such as real numbers) to a discrete set (such as integers).
This means that some accuracy in the representation is lost (e.g. a simple approach is to eliminate least-significant bits). However, in many cases in machine learning, it is possible to adapt the
models to give meaningful results while using these smaller data types. This significantly reduces the number of bits necessary for intermediary results during the execution of these machine learning
Since FHE is currently limited to 16-bit integers, it is necessary to quantize models to make them compatible. As a general rule, the smaller the bit-width of integer values used in models, the
better the FHE performance. This trade-off should be taken into account when designing models, especially neural networks.
Overview of quantization in Concrete ML
Quantization implemented in Concrete-ML is applied in two ways:
Built-in models apply quantization internally and the user only needs to configure some quantization parameters. This approach requires little work by the user but may not be a one-size-fits-all
solution for all types of models. The final quantized model is FHE-friendly and ready to predict over encrypted data. In this setting, Post-Training Quantization (PTQ) is for linear models, data
quantization is used for tree-based models and, finally, Quantization Aware Training (QAT) is included in the built-in neural network models.
For custom neural networks with more complex topology, obtaining FHE-compatible models with good accuracy requires QAT. Concrete-ML offers the possibility for the user to perform quantization
before compiling to FHE. This can be achieved through a third-party library that offers QAT tools, such as Brevitas for PyTorch. In this approach, the user is responsible for implementing a
full-integer model, respecting FHE constraints. Please refer to the advanced QAT tutorial for tips on designing FHE neural networks.
While Concrete-ML quantizes machine learning models, the data the client has is often in floating point. The Concrete-ML models provide APIs to quantize inputs and de-quantize outputs.
Please note that the floating point input is quantized in the clear, i.e. it is converted to integers before being encrypted. Moreover, the model's output are also integers and are decrypted before
Basics of quantization
Let $[\alpha, \beta ]$ be the range of a value to quantize where $\alpha$ is the minimum and $\beta$ is the maximum. To quantize a range of floating point values (in $\mathbb{R}$) to integer values
(in $\mathbb{Z}$), the first step is to choose the data type that is going to be used. Many ML models work with weights and activations represented as 8-bit integers, so this will be the value used
in this example. Knowing the number of bits that can be used for a value in the range $[\alpha, \beta ]$, the scale $S$ can be computed :
$S = \frac{\beta - \alpha}{2^n - 1}$
where $n$ is the number of bits ($n \leq 8$). For the sake of example, let's take $n = 8$.
In practice, the quantization scale is then $S = \frac{\beta - \alpha}{255}$. This means the gap between consecutive representable values cannot be smaller than $S$, which, in turn, means there can
be a substantial loss of precision. Every interval of length $S$ will be represented by a value within the range $[0..255]$.
The other important parameter from this quantization schema is the zero point $Z_p$ value. This essentially brings the 0 floating point value to a specific integer. If the quantization scheme is
asymmetric (quantized values are not centered in 0), the resulting $Z_p$ will be in $\mathbb{Z}$.
$Z_p = \mathtt{round} \left(- \frac{\alpha}{S} \right)$
When using quantized values in a matrix multiplication or convolution, the equations for computing the result become more complex. The IntelLabs Distiller documentation provides a more detailed
explanation of the maths used to quantize values and how to keep computations consistent.
Configuring model quantization parameters
Built-in models provide a simple interface for configuring quantization parameters, most notably the number of bits used for inputs, model weights, intermediary values, and output values.
For linear models, the quantization is done post-training. Thus, the model is trained in floating point, and then, the best integer weight representations are found, depending on the distribution of
inputs and weights. For these models, the user can select the value of the n_bits parameter.
For linear models, n_bits is used to quantize both model inputs and weights. Depending on the number of features, you can use a single integer value for the n_bits parameter (e.g. a value between 2
and 7). When the number of features is high, the n_bits parameter should be decreased if you encounter compilation errors. It is also possible to quantize inputs and weights with different numbers of
bits by passing a dictionary to n_bits containing the op_inputs and op_weights keys.
For tree-based models, the training and test data is quantized. The maximum accumulator bit-width for a model trained with n_bits=n for this type of model is known beforehand: it will need n+1 bits.
Through experimentation, it was determined that in many cases a value of 5 or 6 bits gives the same accuracy as training in floating point and values above n=7 do not increase model performance (but
they induce a strong slowdown).
Tree-based models can directly control the accumulator bit-width used. However, if 6 or 7 bits are not sufficient to obtain good accuracy on your data-set, one option is to use an ensemble model
(RandomForest or XGBoost) and increase the number of trees in the ensemble. This, however, will have a detrimental impact on FHE execution speed.
For built-in neural networks, several linear layers are used. Thus, the outputs of a layer are used as inputs to a new layer. Built-in neural networks use Quantization Aware Training. The parameters
controlling the maximum accumulator bit-width are the number of weights and activation bits ( module__n_w_bits, module__n_a_bits ), but also the pruning factor. This factor is determined
automatically by specifying a desired accumulator bit-width module__n_accum_bits and, optionally, a multiplier factor, module__n_hidden_neurons_multiplier.
Note that for built-in neural networks, the maximum accumulator bit-width cannot be precisely controlled. To use many input features and a high number of bits is beneficial for model accuracy, but it
can conflict with the 16-bit accumulator constraint. Finding the best quantization parameters to maximize accuracy, while keeping the accumulator size down, can only be accomplished through
Quantizing model inputs and outputs
The models implemented in Concrete-ML provide features to let the user quantize the input data and de-quantize the output data.
In a client/server setting, the client is responsible for quantizing inputs before sending them, encrypted, to the server. Further, the client must de-quantize the encrypted integer results received
from the server. See the Production Deployment section for more details.
Here is a simple example showing how to perform inference, starting from float values and ending up with float values. Note that the FHE engine that is compiled for the ML models does not support
data batching.
# Assume quantized_module : QuantizedModule
# data: numpy.ndarray of float
# Quantization is done in the clear
x_test_q = quantized_module.quantize_input(data)
for i in range(x_test_q.shape[0]):
# Inputs must have size (1 x N) or (1 x C x H x W), we add the batch dimension with N=1
x_q = np.expand_dims(x_test_q[i, :], 0)
# Execute the model in FHE
out_fhe = quantized_module.forward_fhe.encrypt_run_decrypt(x_q)
# Dequantization is done in the clear
output = quantized_module.dequantize_output(out_fhe)
# For classifiers with multi-class outputs, the arg max is done in the clear
y_pred = np.argmax(output, 1) | {"url":"https://docs.zama.ai/concrete-ml/0.6-1/advanced-topics/quantization","timestamp":"2024-11-14T07:18:16Z","content_type":"text/html","content_length":"388973","record_id":"<urn:uuid:2611f365-b1c8-4934-b769-4b99bfe967c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00027.warc.gz"} |
One method commonly used in design and research in fluid mechanics and heat transfer besides analytical and experimental is using a numerical method known as Computational Fluid Dynamics (CFD).
This method has been used for a long time to solve any engineering problems (fluid mechanics related) in many industries, from aerospace, maritime, automotive, manufacture, energy and renewable
energy up to biomedical engineering.
Because this method is computer-based (no physical prototype needed), the total processes can be done quickly, flexible, low cost, deeper and more importantly no safety issues if the test is related
to human interaction.
Nevertheless, some engineers and scientists are still skeptical about the accuracy of the CFD result because of the lack of operational CFD knowledge. (no matter how sophisticated your calculator is,
if you hit the wrong input the output will be wrong right?). In this article, we will discuss the verification and validation of CFD method.
simulation of a centrifugal impeller using CFD (openFOAM software)
First, before we discuss verification and validation, we must understand some terminologies, these are (1) code, (2) simulation, and (3) Model:
(1) CODE: Is a bunch of computer instructions to gives input and definitions. This code has a strong relation to what software we used. Different software will have difference code characteristics.
(2) SIMULATION: Is the use of the model, in CFD case this is to obtain the results such as flow, pressure, velocity, etc. based on the input to the model.
(3) MODEL: Model is a representation of the physical system (in CFD case is the fluid flow or heat transfer) to predict the characteristics or output of the system. For example the geometrical size,
inlet velocity, temperature in the wall, pressure at the outlet, etc. based on the physical system we want to mimic.
Credibility of a code, model and CFD simulation are obtained based on its uncertainty and error level. The value of uncertainty and error itself determines whether the program and computational
method used are fitted with at least intuitively and mathematically or not. Then, validation determines whether the simulation is fitted with physical phenomena or not. Generally, validation used
experimental methods if possible.
There are some disagreements among professionals about the standard procedure of verification and validation of CFD simulation. Although CFD is widely used, this method is relatively new. CFD is a
complex method that involves non-linear differential equations to solve the theoretical equations or experimental equations in a discrete domain, in complex geometry. Hence, the error assessment for
CFD is based on these tree root (1) theory, (2) Experiment, and (3) Computation.
The accuracy level of CFD analysis depends on the use of the result itself. The conceptual design process doesn’t need an accurate simulation result, on the other hand, on the detail design process,
we need accurate CFD results. Every quantity in CFD needs a different accuracy level, for example, we don’t need accurate temperature value in the design process of low-speed aircraft, but we need
accurate temperature calculation when we are dealing with supersonic aircraft or rocket. In general, there are three categories of CFD simulation based on its accuracy demand: (1) Simulation for
qualitative information, (2) Simulation to obtain incremental value, and (3) simulation to obtain the absolute value of a quantity.
(1) Simulation to obtain qualitative information
In this case, generally, experimental information data are hard or maybe too costly to obtain, so there’s no comparison data, and what engineers or scientists need is the “how it works” information,
and how to optimize a flow without needing the exact value of each parameter. For example, a valve manufacturer wants to develop a novel design idea, and they want to prove the theory and see whether
or not the flow is streamlined or chaotic in nature, they don’t need exact value of pressure drop, velocity, etc. in this conceptual design step: At least until they want to compare this design to an
existing design (refer to category 2) and want to design the minimum thickness of this part before it is ready to manufacture (refer to category 3)
(2) Simulation to obtain incremental value
This scenario compares the incremental value with respect to some design or flows alteration with the same basic characteristics. For example, a company wants to modify an existing impeller blade in
case of its blades number or its inlet angle (illustrates in the picture below). From this simulation, we could determine which impeller has the highest pressure difference regardless of its absolute
pressure in the entire system. This type of simulation demands more accuracy than category 1.
(3) Simulation to obtain absolute quantity
This is the most accuracy-demanding simulation scenario and sometimes this simulation results are compared with the experimental result to validate the method, and the other results are used in the
next design process such as calculating the L/D of an aircraft wing illustrates bellow:
To conduct a model validation, we must understand the flow characteristic to get intuition whether the flow acts as expected physical phenomena or not. For example, if we simulate a projectile with
the speed exceed the speed of sound, the shock wave phenomena should occur; or if we simulate flow in pipe in low Reynold number, the flow should be laminar, otherwise, it must be turbulent, and so
on. This knowledge is important because CFD is only a “calculator” if we hit the wrong input, the output will be wrong, in fact, the settings in CFD software, in general, are varied and cause a
headache if we don’t have this knowledge.
Physical model not only refers to the geometrical model, but these are also the following models to be considered in CFD simulation:
(1) Spacial dimension
Or the geometry (1D, 2D or 3D) of the object we want to model, sometimes this model is simplified with symmetry or reduces 3D into 2D to reduce the computational effort as long as it still represents
the essence of the flow we want to analyze.
(2) Temporal dimension
This is a time dimension of the simulation we want to conduct. This is very important in transient simulation, but not significant if we want to simulate a steady simulation. For example, if we want
to simulates an object that rotates 1 rotation/second, and we input the delta time 0,1 second, we will accommodate the 10 incremental motions in our simulation. But, if we input delta time 2 second,
the computation will error because we can’t accommodate the “motion” of the object.
(3) Navier-Stokes Equation
This is the fundamental equation of fluid mechanics that models the flow velocity, pressure, gravity, viscosity and even rotational force in the flow.
(4) Turbulent Model
This model is specially designed to model turbulent flow without calculating the whole (complex and computationally high effort) Navier-Stokes equation. The difference turbulent model we use will
generate different results in our CFD simulation.
(5) Energy equation
Unlike classical solid mechanics, in fluid dynamics, energy generally refers to heat transfer and temperature change.
(6) Flow boundary condition
This is a mandatory input in our simulation. Boundary conditions input what flow characteristic we already have, for example, pressure in the inlet of a pipe (from a pump) or the velocity an aircraft
during flight, etc.
Even the setting in CFD simulation looks messy for CFD beginner, but a lot of scientists and engineers around the globe are publishing papers and journal continuously to share their setups and its
accuracy compared to experimental as well as analytical results, hence CFD verification and validation becomes easier with these abundance references.
To read other articles, click here.
aeroengineering.co.id is an online platform that provides engineering consulting with various solutions, from CAD drafting, animation, CFD, or FEA simulation which is the primary brand of CV. Markom.
caesar@aeroengineering.co.id +62 821-3868-4162
https://www.aeroengineering.co.id/wp-content/uploads/2020/03/grafik-lift-to-drag-ratio-hasil-CFD-dan-wind-tunnel-e1615564507432.png 200 442 Caesar Wiratama https://www.aeroengineering.co.id/
wp-content/uploads/2022/03/logo_ae_solution-removebg-preview.png Caesar Wiratama2020-03-21 23:38:152020-09-11 10:18:59VALIDATION AND VERIFICATION IN COMPUTATIONAL FLUID DYNAMICS (CFD)
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