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The actual three-body problem and chaos theoryThe actual three-body problem and chaos theory
Since the excellent The Three-Body Problem book trilogy and the similarly great Netflix adaptation, I have been wondering why exactly is the name-giving physics/mathematics problem called the
three-body problem considered unsolvable. Frustratingly, searching on Google didn’t let me find any good answers to my biggest question: given that the universe (erm, at least on a macro scale) is
deterministic, isn’t it just a matter of refining our understanding of the rules (of physics)? Like, if we understood the exact rules, couldn’t we just build a computer simulation of a solar system
with 3 suns and run it to get perfect predictions?
Finally, all these months later, I’ve stumbled upon this video from Up and Atom that has answered all my questions and more:
The gist of it is that yes, in principle if you understand all the rules, and you know the exact position, speed, direction, mass, rotation, etc. of every object in a system, then you can build a
simulation and get perfect predictions. Unfortunately, in reality, for all practical purposes it is not possible to get exact measurements of all these properties, so we rely on approximations. For
many things, approximations are good enough, and we will end up with a simulation or prediction that is going to be extremely close to the reality. This allowed us to get this far as a technological
Some things are not like that, and even a small inaccuracy in our measurement or approximation will yield a prediction dramatically different from the reality. This is called extreme sensitivity to
initial conditions, and is one of the core ideas in chaos theory:
Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical
systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique
evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This
behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:
Chaos: When the present determines the future but the approximate present does not approximately determine the future.
The three-body problem happens to be an example of this.
I find these limits of knowability equally fascinating and horrifying. There are just things that we will never be able to do, no matter the knowledge, the understanding, or the resources we
accumulate. Humans can thus never become gods.
Some additional examples are:
• The uncertainty principle, related to the three-body problem and chaos theory
• Gödel’s incompleteness theorems, which proves fundamental limits to any model or system
• Then of course there’s the entire field of quantum mechanics, with the central idea that at the heart of the universe it’s all just probabilities, which idea I just find viscerally disturbing.
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ACM Other ConferencesOn Petri Nets with Hierarchical Special Arcs
We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer
arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability
status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that
deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem.
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Breadth-first tree traversal
Breadth-first tree traversal
blogentry, programming, breadthfirst, traversal
Featured Image - "Trees" by RichardBH, used under CC BY 2.0
In the previous article, Depth-First Tree Traversal, I wrote about how to traverse trees with Depth-first approaches;In-Order, Pre-Order, and Post-Order traversals.
In this article, I will talk about the algorithm and an implementation for a breadth-first version.
As a spoiler, you don't have a use a recursion and it needs a familiar data structure.
I am going to use the same tree structure I used in the previous article.
Breadth-first traverse means going through each node from the root of the tree, then next level down and down until you reach the maximum height the tree.
When traversing a tree above, an algorithm needs to go from 4 2 6 1 3 5 7 from top down, left to right;4 is the root, next level down on the left, it's 2, then the node on the same depth, 6, and so
As you pass over each level, you need to keep a track of all the nodes sharing the same depth.
From the description, it looks like we need to process whichever node we encountered first in each depth. This is where a queue comes into play.
The algorithm is fairly simple.
Add the root to the queue
While the queue is not empty,
Dequeue a node
Process the node
Queue left node of the node for further processing
Queue right node of the node for further processing
Here is the implementation in C#.
1private static void TraverseBreadthFirst(TreeNode<int> root, List<int> list)2{3if (root == null) return;4
5 Queue<TreeNode<int>> queue = new Queue<TreeNode<int>>();6 queue.Enqueue(root);7
8 while (queue.Count > 0)9 {10 var node = queue.Dequeue();11 list.Add(node.Value);12
13 if (node.Left != null)14 queue.Enqueue(node.Left);15
16 if (node.Right != null)17 queue.Enqueue(node.Right);18 }19}
The implementation follows the algorithm almost word by word except for the simple validation guard clause in the first line.
After the method exits, the list will contain 4 2 6 1 3 5 7.
The working source is available on GitHub.
I've covered bothDepth-first and Breadth-first traversals in two entries.
Breadth-first traversal is less intuitive to implement than depth-first traversals but still easy to do so using a queue abstract data structure.
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Linear Momentum: Example Problems
Try to solve these problems before watching the solutions in the screencasts.
The aluminum elbow is bolted to a pipe, which is not shown. The bolts provide a force that anchors the elbow in place. What are the x- and y-components of this anchoring force? The diameters of the
entrance and exit are D[1] = 10 cm and D[2] = 5 cm. The mass of the elbow is m[e] = 2.4 kg. The mass of the water inside of it is m[w] = 3 kg. The volumetric flow rate of the water is Q = 15 L/s. The
gage pressure of the water entering the elbow is P[1] = 30 kPa. The water exits the elbow as a free jet.
How much reverse thrust is developed? The diameter of the fan is 2 m. The engine is drawing in a mass flow rate of air of 500 kg/s. Air flows through the engine and is deflected at an angle of 60
degrees from the horizontal. The velocity exiting through the thrust reversers (V[2]) is 3 times the magnitude of the velocity entering the engine (V[1]).
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Guess The Correct Numbers
Daily Math Puzzle Challenge
Guess the randomly generated daily math puzzle within 6 tries.
The goal of the challenge is to input the correct sum that equals the displayed answer. Additionally, a hint indicating the number of arithmetic signs used in the sum will be shown.
Once you have made your attempt, the next math puzzle will be available in 6 hours, which is up to 4 attempts in 24 hours for each of the 3 levels. (Easy, Medium, Hard)
You can track your statistics to monitor your performance over time and strive to improve your results with each attempt.
Begin by entering your initial guess for the math equation. In the given example, the sum should equal 61, using only one arithmetic sign.
If a number or sign is present in the equation but in the wrong position, it will be highlighted in ORANGE.
If it is in the correct position, it will be highlighted in GREEN.
If the number or sign is not present in the equation at all, the colour will be GREY.
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Research Guides: SPSS: Tests
Cropper, C. 1977. "Recovery of Patients from Stroke." OzDASL – Australasian Data and Story Library. Accessed from http://www.statsci.org/data/oz/stroke.html on Dec. 15, 2015.
Essenberg, C. J., R. A. Easter, R. A. Simmons, and D. R. Papaj. 2015. The value of information in floral cues: bumblebee learning of floral size cues. Unpublished raw data.
Hanley, J. A., and Shapiro, S. H. 1994. Sexual Activity and the Lifespan of Male Fruitflies: A Dataset That Gets Attention. Journal of Statistics Education 2(1). Accessed from http://www.amstat.org/
Rasmussen, Marianne. "Activities of Dolphin Groups." OzDASL – Australasian Data and Story Library. Accessed from http://www.statsci.org/data/general/dolpacti.html on December 15, 2015.
Wilson, Richard J. "Pulse Rate Before and After Exercise." OzDASL – Australasian Data and Story Library. Accessed from http://www.statsci.org/data/oz/ms212.html on December 15, 2015.
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ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 8 Ratio and Proportion Ex 8.1
ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 8 Ratio and Proportion Ex 8.1
Question 1.
Express the following ratios in simplest form:
(i) 20 : 40
(ii) 40 : 20
(iii) 81 : 108
(iv) 98 : 63
Question 2.
Fill in the missing numbers in the following equivalent ratios:
Question 3.
Find the ratio of each of the following in simplest form :
(i) 2.1 m to 1.2 m
(ii) 91 cm to 1.04m
(iii) 3.5 kg to 250gm
(iv) 60 paise to 4 rupees
(v) 1 minute to 15 seconds
(vi) 15 mm to 2 cm
Question 4.
The length and the breadth of a rectangular park are 125 m and 60 m respectively. What is the ratio of the length to the breadth of the park?
Question 5.
The population of village is 4800. If the numbers of females is 2160, find the ratio of males to that of females.
Question 6.
In a class, there are 30 boys and 25 girls. Find the ratio of the numbers of
(i) boys to that of girls.
(ii) girls to that of total number of students.
(iii) boys to that of total numbers of students.
Question 7.
In a year, Reena earns ₹ 1,50,000 and saves ₹ 50,000. Find the ratio of
(i) money she earns to the money she saves.
(ii) money that she saves to the money she spends.
Question 8.
The monthly expenses of a student have increased from ₹350 to ₹500. Find the ratio of
(i) increase in expenses and original expenses.
(ii) original expenses to increased expenses.
(iii) increased expenses to increased in expenses.
Question 9.
Mr Mahajan and his wife are both school teachers and earn ₹20900 and ₹ 18700 per month respectively. Find the ratio of
(i) Mr Mahajan’s income to his wife’s income.
(ii) Mrs Mahajan’s income to the total income of both.
Question 10.
Out of 30 students in a class, 6 like football, 12 like cricket and remaining like tennis. Find the ratio of
(a) Number of students liking football to number of students liking tennis.
(b) Number of students liking cricket to total number of students.
Question 11.
Divide ₹ 560 between Ramu and Munni in the ratio 3 : 2.
Question 12.
Two people invested ₹ 15000 and ₹25000 respectively to start a business. They decided to share the profits in the ratio of their investments. If their profit is ₹ 12000, how much does each get?
Question 13.
The ratio of Ankur’s money to Roma’s money is 9 : 11. if Ankur has ₹540, how much money does Roma have?
Question 14.
The ratio of weights of tin and zinc in on alloy is 2 : 5. How much zinc is there in 31.5g of alloy?
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A Statistical Appraisal Of Bond And Equity Behaviour
Bonds and equities exhibit fundamentally different patterns
I have just updated a dataset of monthly real returns from US stocks - equities - and US bonds going back to January 1850, then rerun some old analyses on the data as well as some new ones. This post
simply presents the results of the analyses. In another post I will set out what I think are the implications of them. Suffice to say, much of the conventional wisdom about these two main asset
classes is flawed.
First off, all returns have been adjusted for inflation. Nominal returns are utterly irrelevant and always will be. What use is a 10% nominal return if goods and services prices have risen 20%?
Also, to be clear, bonds are long maturity - as opposed to bills - and are high grade/have very low default risk - as opposed to credit.
Charts 1 and 2 below are simply of the inflation adjusted equity and bond indices on semilog scales.
What I find extraordinary about US equities is that despite underlying economic growth having gradually declined as the US economy matured, the trend real growth in US equities has not wavered. If
economic growth gradually fell, corporate revenue growth should also have fallen, in which case one might have expected equity trend growth to fall. It didn't.
One can derive the trend growth rate from the trend line formula on the chart. Since the data is monthly, the exponent constant of 0.005189 means that trend monthly growth is 0.5189pct which equates
to 6.4pct annualised.
Trend real growth in bonds can be derived from the trend line formula in Chart 2. Trend monthly growth of 0.1778pct equates to 2.2pct annualised.
Equities have gone up far more than bonds over the 172 years so to compare like with like I have put them together in Chart 3.
Chart 1:
Source: Credit Suisse/Yahoo
Chart 2:
Source: Credit Suisse/Yahoo
Chart 3 clearly shows that equities have performed much better than bonds over the 170 years - the difference between 6.4pct and 2.2pct per annum. It should also be noted that the bonds' line is less
jagged than equities', and also that it contains three distinct cycles that equities' line does not.
Chart 3:
Source: Credit Suisse/Yahoo
Charts 4 and 5 are normalised indices, derived by dividing the actual index by the trend line - another term for normalised is de-trended. Thus a value of 2 means that the index is 100% above trend
and 0.5 means 50% below. The chart is semilog because doubling - 2.00/100% - is equivalent to having - 0.50/50%. If you double something - +100pct - then halve it - -50pct - you get back to where you
started, not to 100pct-50pct = +50pct.
Note that the jaggedness/cycles are now even more apparent in the two charts.
Chart 4:
Source: Credit Suisse/Yahoo
Chart 5:
Source: Credit Suisse/Yahoo
Charts 4 and 5 are overlain in Chart 6 below. You would be forgiven for thinking that there might just be some correlation between the two. They both reach lows in around 1860/70, 1920, and 1980, as
well as highs around 1900 and, perhaps more tenuously, 2020. In fact the correlation coefficient between them, R, is 0.16 which is low - Chart 7.
It is also instructive that the range for both series is around 0.50 to 2.00, though each has one excursion towards 0.25, i.e. towards 75% below trend.
Chart 6:
Source: Credit Suisse/Yahoo
While the correlation coefficient, R, between bonds and stocks over the entire 170 years is a low 0.16, there are fairly long periods - 20 years - when there is a high degree of correlation, whether
positive or negative. For example, 1900-1920 saw both real stocks and bonds falling - a high, positive correlation coefficient of 0.90 - while 1948-1968 saw stocks falling and bonds rising - a
"high", negative correlation coefficient of -0.89.
Periods can generally be classified into five groups: both are rising; both are falling; one is rising and the other falling; one is falling and the other rising; and, finally, no obvious
relationship. Which of these is prevalent depends largely on the underlying economic/inflation regime.
Chart 7:
Source: Credit Suisse/Yahoo
The behaviour of financial assets, whether bonds or equities, individual names or markets, is generally defined by two parameters: average return, and standard deviation of returns. If you plot a
histogram of returns you will get something close to a bell curve, with the average return in the centre.
If the financial asset follows a random walk, 68pct of the returns will be within one standard deviation on the mean, 95% within two, 99.7pct within three, etc.
In the case of equities, we already know that the average monthly return over the 172 years is 0.52pct. To calculate the standard deviation, one must first take logarithms to remove skew - think of a
wonky bell curve - then convert back.
Once you have a mean and standard deviation, you can create simulations of time series of financial asset prices, ones that follow a random walk.
Charts 8 below depicts 100 randomly selected actual 30 year equity market periods. Chart 9 depicts 100 simulations based on mean and standard deviation based on actual monthly returns. The red lines
represent +1 and -1 standard deviations.
As you would expect from the random walks in Chart 9, many of the series lie outside +/-1 standard deviation. In Chart 8 however, this is not the case. The difference between the two charts is stark
- remember that the vertical scales are the same. A number of the actual series in the early years do veer outside of +/- 1 standard deviation but then they are drawn back inside the red lines.
Charts 10 and 11 are exactly the same but for bonds. Again, there is a difference between the two - the actual series veer farther from the mean than the simulations. This is the opposite to the
pattern seen in equities.
Chart 8:
Source: Credit Suisse/Yahoo
Chart 9:
Source: Credit Suisse/Yahoo
Chart 10:
Source: Credit Suisse/Yahoo
Chart 11:
Source: Credit Suisse/Yahoo
Just by eyeballing Charts 8 and 10 it is clear that neither bonds nor equities follow a random walk. However, there is a mathematical test called the variance test that can quantify the extent of the
non-randomness. Essentially, if a series follows a random walk, the variance of 4 month returns should be double the variance of 2 month returns, 10 month double 5 month, etc. Dividing the variance
over, say, 9 months by 9, then dividing it by the variance over 1 month, should give you a value close to 1 if over 9 month periods the time series is following a random walk.
Charts 12 and 13 below show actual/simulated variance ratios for, respectively, equities and bonds. As expected, the variance ratios for both simulated series are close to 1, indicating that they are
following random walks. Not so with the actual equities and bonds.
In the case of equities, they start off being more volatile than would be expected, then over longer periods get progressively less so. In the case of bonds, they are immediately more volatile over
short periods, then become even more so over longer ones.
The two actual series are plotted together in Chart 14. The difference I think is extraordinary, and demonstrates the almost fundamentally different nature of these two main asset classes.
Variance is simply the square of standard deviation, so one can create series of standard deviation of actual returns over a particular number of months, then plot them on a chart. This is what is
depicted in Chart 15, the final chart.
One would naturally expect the volatility of, say, 10 year returns to be higher than that of, say, 6 month returns - in other words, both lines on the whole are moving up. Equities get more volatile
in relation to bonds over shorter periods, but then around 3 years the gap starts to narrow. This gap continues to narrow until the ten year mark, at which point equities become less volatile than
bonds. Thereafter, the gap between bond volatility and equity volatility continues to widen. In other words, over longer periods bond returns are much more volatile than those from equities.
As I will expand upon in another post this week, this is a fascinating and counterintuitive finding. Patterns in markets mean that there are opportunities to make predictions and thus to outperform.
The difference in patterns between equities and bonds highlighted in this post provide further knowledge about these two key asset classes which can provide further scope for outperformance.
Chart 12:
Source: Credit Suisse/Yahoo
Chart 13:
Source: Credit Suisse/Yahoo
Chart 14:
Source: Credit Suisse/Yahoo
Chart 15:
Source: Credit Suisse/Yahoo
The views expressed in this communication are those of Peter Elston at the time of writing and are subject to change without notice. They do not constitute investment advice and whilst all reasonable
efforts have been used to ensure the accuracy of the information contained in this communication, the reliability, completeness or accuracy of the content cannot be guaranteed. This communication
provides information for professional use only and should not be relied upon by retail investors as the sole basis for investment.
Originally Posted on chimpinvestor.com
Sound investments
don't happen alone
Find your crew, build teams, compete in VS MODE, and identify investment trends in our evergrowing investment ecosystem. You aren't on an island anymore, and our community is here to help you make
informed decisions in a complex world.
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PRBS Input Signals
A pseudorandom binary sequence (PRBS) is a periodic, deterministic signal with white-noise-like properties that shifts between two values. A PRBS signal is inherently periodic with a maximum period
length of 2^n–1, where n is the PRBS order.
You can use a PRBS input signal for frequency-response estimation at the command line or in Model Linearizer. The estimation algorithm injects the PRBS signal at the input analysis point you specify
for estimation, and measures the response at the output analysis point. PRBS signals are useful for estimating frequency responses for communications and power electronics applications.
Using PRBS input signals, you can:
• Reduce total estimation time compared to using sinestream input signals, while producing comparable estimation results.
• Obtain faster frequency response estimation with a higher frequency resolution than using chirp input signals.
When you create your PRBS input signal, specify the following parameters.
• Signal amplitude — The peak-to-peak range of the signal.
• Sample time — Set the sample time to match the sample time at the signals that correspond to the input and output linear analysis points.
• Signal order — The maximum length of the PRBS signal is 2^n–1, where n is the signal order.
• Number of periods — Number of periods N[p] in the PRBS signal.
When specifying your PRBS signal parameters, consider the following:
• Set the amplitude such that the system is properly excited for your application. If the input amplitude is too large, the signal can deviate too far from the model operating point. If the input
amplitude is too small, the PRBS signal is indistinguishable from noise and ripples in your model.
• For a given sample time, to increase the resolution over the low-frequency range, increase the order of the PRBS signal.
• For most frequency response estimation applications, use a single period. Doing so produces a flat frequency response across the frequency range of the signal.
• The frequency range of the generated PRBS signal is [F[min],F[max]], where ${F}_{\mathrm{min}}=\left({F}_{N}/{N}_{p}\right)\cdot \left(2/{2}^{n}-1\right)$ and ${F}_{\mathrm{max}}={F}_{N}$. F[N]
is the Nyquist frequency of the signal.
You can also create a PRBS signal with parameters based on the dynamics of a linear system, sys. For instance, if you have an exact linearization of your system, you can use it to initialize the
When you set the PRBS parameters using a linear system, the amplitude of the signal is 0.05 and the number of periods is 1. To set the sample time and order of the signal, the software first selects
a signal frequency range, [F[min],F[max]], based on the dynamics of sys.
If sys is a discrete-time system, then:
• The sample time of the PRBS is equal to the sample time of sys.
• The order of the PRBS is as follows, where ⌈.⌉ is the ceiling operator.
$\text{Order}=⌈\frac{\mathrm{log}\left(\frac{2\pi }{\text{Ts}\cdot \text{Fmin}}\right)}{\mathrm{log}\left(2\right)}⌉$
If sys is a continuous-time system, then:
• The sample time of the PRBS is
$\text{Ts}=\frac{2\pi }{5\cdot \text{Fmax}}$
• The order of the PRBS is as follows, where ⌊.⌋ is the floor operator.
$\text{Order}=⌊\frac{\mathrm{log}\left(\frac{2\pi }{\text{Ts}\cdot \text{Fmin}}\right)}{\mathrm{log}\left(2\right)}⌋$
Create PRBS Signals Using Model Linearizer
In the Model Linearizer, to use a PRBS input signal for estimation, on the Estimation tab, select Input Signal > PRBS Pseudorandom Binary Sequence.
In the Create PRBS input dialog box, specify the name of the PRBS signal object in Variable Name. You can then specify the parameters of your PRBS input signal using the following fields.
• Amplitude — Signal amplitude
• Sample time — Sample time
• Number of periods — Number of periods
• Signal order — Signal order
You can also automatically determine the parameters Number of periods and Signal order based on a frequency range of interest. Automatic parameter determination helps create an input signal that
leads to an accurate frequency response over a specified frequency range.
To determine the parameters automatically, first set the Sample time parameter to match the sample time at the point of signal injection. Next, specify the frequency range of interest in rad/s using
the Min and Max parameters, and then click Compute parameters.
Additionally, you can:
• Use the One sample per clock period parameter to specify whether the signal remains constant for one sample per clock period or multiple samples per clock period. Use this parameter if you have
Number of periods > 1. By default, this option is enabled and the generated signal is constant over one sample. When you disable this option, the generated signal is constant for the specified
number of samples.
• Use the Perform window-based filtering to improve estimation results parameter to apply Hann window-based filtering, which produces a smoother frequency response estimation result.
Create PRBS Signals Using MATLAB Code
To create a PRBS signal for estimation at the command line with frestimate, use a frest.PRBS object.
Create PRBS Signals in Simulink
Since R2024a
To create a PRBS signal for estimation in Simulink^®, use the PRBS Signal Generator block. This block is helpful when you to generate perturbation signals to inject in your plant models in desktop
simulation or on hardware through code generation. You can then collect the plant response data to the perturbation signal and perform custom processing to identify plant characteristics.
Improve Frequency Response Result at Low Frequencies
To improve the frequency response estimation result at low frequencies, you can use a different sample time other than the sample time in the original model. To do so, modify your model to use a
Constant block at the input analysis point and a Rate Transition block at the output analysis point.
For both the Constant block and the Rate Transition block, specify a new sample time for your PRBS signal that is larger than the original sample time of the model.
The ability to change the sample time of the PRBS input signal provides an additional degree of freedom in the frequency response estimation process. By using a larger sample time than in the
original model, you can obtain a higher resolution frequency response estimation result over the low-frequency range. Additionally, running estimation at lower sampling rate reduces processing
requirements when deploying to hardware.
See Also
frest.PRBS | frestimate
Related Topics
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The Basics of Quantum Computing .. Quantum Superposition - Development.ie
The Basics of Quantum Computing .. Quantum Superposition
I am the first to admit that a deep understanding of quantum physics is not something I have, and my goal ( or your goal ) of becoming a quantum developer does not necessarily need it. No more do you
need to know the inner workings of transistors or microchips in order to be a classical developer using java, the same applies for quantum. However, let us delve into some basics, that will help us
with the nomenclature of the software libraries we will use.
The Nobel prize-winning physicist Richard Feynman is attributed to the quote “If you think you understand quantum mechanics, then you don’t understand quantum mechanics”, and he was the leading
physicist in the area, so let’s not get too caught up if we don’t fully understand everything. Try and develop a sense of meaning, as if you were going to try and explain it to someone else.
(Quantum) Superposition
The noun superposition is defined as “the action of placing one thing on or above another, especially so that they coincide.” Quantum superposition means that any two quantum states can be added
together (superposed) and a valid quantum state will result. Or, that any quantum state can be defined by one of more quantum states.
Now … using this idea of superposition , let us take the example of sets of coins. We write out their states ( H – heads, T – tails) first…
2 coins – HH, TT, HT , TD – 4 results
3 coins – HHH, HHT, HTH, HTT, THH, THT, TTH, TTT – 8 results
4 coins – … – 16 results
5 coins – … – 32 results
So, as we add a coin each time the number of results that we can have are 2^n .
Whereas the n (classical) coins are in only one of the 2^n possible results, n qubits can be in a superposition of all 2^n possible results. ( we will dig into qubits later )
The Probability Difference
In the case of a set of coins, the state that they can be in is a 2^n space. And they can only hold that one particular state, even if we don’t know what that is. As we explained above on
superposition , quantum computers can hold superpositions of 2^n distinct logical states, which means they can solve problems potentially exponentially faster. These values can be positive, negative
or complex numbers unlike probabilities which are positive or zero.
Quantum Circuits
Where does superposition come into developing quantum algorithms. If you take sound waves for example, one is noise and the other is a cancellation tone to remove noise like in noise cancelling
headphones, then the principle of superposition and interference is used to result in cancelled noise.
In the quantum circuit below, which we will develop, the same principles apply. We start with a superposition and then we apply an algorithm by creating a quantum circuit to apply interference on the
superposition to result in our solution.
When we are talking about quantum development using qiskit.org for example we are talking about developing these quantum circuits. Next up, entanglement.
My Quantum challenge ?
It’s time to become a quantum developer. And yes I will update my linkedin profile to say that! 😉 I will endeavour to learn everything I can in the area of quantum development using IBM’s resources
and its software libraries. Where possible I will share all the links out and you can follow along. My “beginners mind” is set and ready to go ( Shoshin) .
Shoshin : It’s the open-minded attitude of being ready to learn; without preconceived notions, judgement or bias.
To follow along then keep an eye out on my blog or follow me on linkedin https://www.linkedin.com/in/andrewpenrose/ where I will share the blog post links and updates.
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The ABC ConjectureHigh School
The ABC conjecture is a powerful statement about positive integers which, if proved, would imply a large number of other significant results, among them Fermat's Last Theorem. As of 2012, a proof by
the Japanese mathematician Shinichi Mochizuki is being checked by members of the mathematical community. But we don't have to wait for the check to explore some ramifications of the conjecture. Read
about its exact statement, about how it implies the truth of Fermat's Last Theorem, and about several other results it implies.
2012 Pan-African Mathematical Olympiad ProblemsHigh School
The 2012 Pan-African Mathematical Olympiad was held in Tunis, Tunisia in September. These are two of the problems set by the jury for the students participating from 12 African countries. For the
first time, a US team participated (unofficially). Three of the team members performed on the level of a medalist (one gold, one silver, and one bronze). The solutions provided here are intended to
show students not just a mathematically correct solution, but to indicate at least one path to the solution for students unused to solving Olympiad problems.
AAS Olympiad Training Program ProblemsHigh School
In the summer of 2012, the AAAS (supported by a generous grant from the Alfred P. Sloan Foundation) gathered 20 students, mostly from backgrounds under-represented in the mathematical sciences, for a
training session in Washington, D.C. The students practiced solving Olympiad-level mathematics problems. Four of them were then selected to represent the US in the Pan-African Mathematical Olympiad.
Four more were selected to attend the Mexican Mathematical Olympiad. These problems are typical of those the students worked on. We give formal solutions, but also hints and discussions about how to
find a pathway to the solution.
Classroom Worksheet: IMO 2012High School
This file contains a guide to a solution to IMO 2012 problem 1, a geometry problem. The Olympiad-level problem is broken down into fourteen easier problems. An average (successful) student of the
first year of geometry will be able to work most of these problems. The student will need to know how to measure angles inscribed in circles. For the very last stage of the solution, the student
needs to know that opposite angles of a cyclic quadrilateral are supplementary. But no more advanced knowledge is needed. In working the problem this way, students will begin to understand how a
mathematical palace can be built out toothpicks--tiny slivers of information.
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Questions & CorrectionsQuestions & Corrections
I have worked really hard – alongside my editor, fact-checker, and lawyers in the US and UK – to make sure everything in the book is accurate. If there are any errors left in the text, I’d be
grateful for your help in correcting them for future editions and for the record. If you spot any mistakes, please email me at the address above. I’ll post corrections on the same date, and give you
a shout-out for spotting them.
If there was anything the book left you wondering, please do message me at chasingthescream@gmail.com and I’ll be happy to try to figure out the answer.
• Corrections IX – posted 7th March 2015
On p51, I refer to the World Series baseball game rigged by Arnold Rothstein in 1919, and say “50 million people were listening in.” In fact, the first live radio broadcast of the World Series
was in 1921 – I will change this in future editions to say “50 million people were following the result.” Thanks to the reader who emailed about this – I haven’t heard back from him yet about
whether I can use his name but I’ll post it if he gives permission.
On p120, I refer to the videogame Rosalio Reta used to play as ‘The Mask of Zelda.’ In fact, it is called ‘The Legend of Zelda.’ Thanks to Mark Whitfield for emailing me about this.
On p251, there’s a page reference that is given incorrectly in the footnotes. I refer to how HIV transmission among drug users has fallen dramatically in Portugal since drugs were decriminalized,
and the footnote says the evidence for this can be found on page 36 of Arthur Domaslawski’s research; in fact, it is on page 40. Thanks to Stuart Rodger for pointing this out.
• Corrections VIII – posted February 27th
Alison Wrbik emailed to point out two typos:
On page 276, I wrote: “The cops demanded to know: Where do you buy your marijuana? What suppliers to you know?” The ‘to’ in the last sentence should of course be ‘do.’
On page 293, where it says “I looked him just now,” it should say “I looked at him just now”.
And Ron Dodd emailed to point out some more typos.
On page 51, where it says “he said was broke” – it should say “he said he was broke.”
On page 70, where it says “And yet sometimes Chino went looking for Deborah, in the park, on the benches, or on the corner where should could be looking for business, because Chino wanted her.”
It should say: “And yet sometimes Chino went looking for Deborah, in the park, on the benches, or on the corner where she would be looking for business, because Chino wanted her.”
On page 76, where it says “sent to back to prison,” the first ‘to’ should be cut.
On page 191, where it says “if I can just stay enough long enough”, the first ‘enough’ is a mistake.
On page 296, when it says “When I learned from Chino and Bud is…” it should say “What I learned from Chino and Bud is…”
Also, there is an error in Chapter Six that was picked up in my fact-checking conversations with Leigh Maddox and was supposed to be fixed but – due to an editing mistake – appeared
incorrectly in the final edition. On page 86, it refers to Leigh’s dad as having been in the US Army. He was in fact in the US Navy.
In addition, Stuart Rodger spotted that I forgot to post one of the audio clips from the book. It is of Chino Hardin saying an act “actually made us look weaker” – I have found that audio clip
and it will be posted soon.
Seth Mnookin, writing in the New York Times, suggested an attribution in the book should have been clearer and taken out of the footnotes and inserted into the main text. On reflection, I have
concluded he is right. On page 213 I write: “Research published in the Proceedings of the Royal College of Physicians of Edinburgh compared Widnes, which had a heroin clinic, to the very similar
Liverpool borough of Bootle, which didn’t.” This sentence should be clearer, and in future will read: “Research published by John in the Proceedings of the Royal College of Physicians of
Edinburgh compared Widnes, which had a heroin clinic, to the very similar Liverpool borough of Bootle, which didn’t.”
Thanks to Alison, Ron, Stuart and Seth.
If you are reading the book and you spot any other errors please do email me – chasingthescream gmail.com - because it is important to me to make sure everything about the book is entirely
• Corrections VII – posted February 17th 2015
Clare Barlett and Charles Cairns both emailed to point out a typo on page 231 – where it refers to the “patents” of Dr Hal Vorse, it should say “patients.” Thanks to both of you for spotting
• Corrections VI – posted 8th Feb 2015
There is a typo on page 268. It says: "When you ban a drug, it's very risk to transport it - so dealers will always choose the drug that packs the strongest possible kick into the smallest
possible space." It should of course say 'risky', not risk. Thank you to Erin Klassen for pointing this out.
• Corrections V – posted February 8th 2015
On page 183, I state that menthol cigarettes are less addictive than tobacco cigarettes. Ben Richards got in touch to let me know this is based on outdated science, and this claim is now strongly
scientifically contested and may well be wrong - indeed,
the US Food and Drug Administration says research suggests they are significantly more addictive.
I will remove this line from future editions. Thanks to Ben for pointing this out.
• Corrections IV – posted 31st January 2015
On some computers, when you click to look at the corrections, it is only displaying the first few hundred words of each new entry, and appears to cut off abruptly in mid-sentence. If your
computer is displaying in that way, just click on the headline for the post - for example, where it says 'Corrections IV' - and it will display the full text.
Patrick Riesterer emailed to point out a mistake. In the book, there are three places where I have changed somebody's name to protect their identity. Each time I do so, I explain in the text that
I am doing it. They are: 'Dee', the stripper Chino has a relationship with in prison, who was raped by a prison guard (because I felt I shouldn't disclose her sexual assault to people who might
know she was in prison in Riker's at that time - it would be a violation of her privacy); 'Hannah', one of Liz Evans' clients, who has subsequently died (because Liz asked me to preserve her
client's anonymity); and 'Jean', one of the addicts who is prescribed heroin in the clinic in Switzerland, and who described to me his past smuggling drugs (an offence for which he could still be
prosecuted if he was identifiable through the book).
The decision to change the name of 'Dee' was suggested late in the editing process by one of the lawyers who worked on 'Chasing The Scream.' By that time, I had already written the 'note on
narrative technique' that appears at the end of the book. In that note as it currently stands, I say that I have altered the names of two people in the book. It should say three people, and I
should have updated that reference. I'll do so in all future editions. Thanks to Patrick for spotting this - I really appreciate it. If you spot any mistakes in the book, please do email me -
chasingthescream -at- gmail.com.
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Reynolds Number
Have anyone thought about calculating Reynolds Number for each bar on any TF ?
By this we could define whether flow of orders was turbulent(chaotic) or laminar. It would be important piece of information.
Interesting topic. I am not so familiar with physics.
First of all, why exactly do you claim that this would be an important piece of information? How would it look and how would you interpretate this number?
What would you use and different parameters in the formula for the Reynolds number?
I did a quick search and found one interesting application. Chapter 10.4 and further.
Attached File(s)
Financial Market Risk - Measurement and Analysis.pdf 3.5 MB | 3,524 downloads
Interesting topic. I am not so familiar with physics. First of all, why exactly do you claim that this would be an important piece of information? How would it look and how would you interpretate
this number? What would you use and different parameters in the formula for the Reynolds number? I did a quick search and found one interesting application. Chapter 10.4 and further. {image}
Great sharing. Thanks.
Interesting topic. I am not so familiar with physics. First of all, why exactly do you claim that this would be an important piece of information? How would it look and how would you interpretate
this number? What would you use and different parameters in the formula for the Reynolds number? I did a quick search and found one interesting application. Chapter 10.4 and further. {image}
Thanks for your valuable reply.
Once laminar process changes into chaotic we may assume that some important actions are taking place, like selling/buying by lets say big money. It it is like a footprint left by important investors.
What I have seen till now is that turbulance occur during and after news release, but not only. The problem is I cannot detect them by myself due to my lack of math background.
I do not know what to put in the given equation. It is mystery to be solved.
Also the book, you gave, includes some essential info for understending how markets work. I believe deterministic chaos theory is an answer and the final one for the price forecasting.
{quote} Thanks for your valuable reply. Once laminar process changes into chaotic we may assume that some important actions are taking place, like selling/buying by lets say big money. It it is
like a footprint left by important investors. What I have seen till now is that turbulance occur during and after news release, but not only. The problem is I cannot detect them by myself due to
my lack of math background. I do not know what to put in the given equation. It is mystery to be solved. Also the book, you gave, includes some essential info for...
I think the footprint you are referring to is basically Price Action. Big money does leave a trail and it is up to us traders to find the trail and follow it. I find it an interesting topic but I
gotta say I don't believe there is some magic number that will solve your problems. There is no holy grail
For more reading on the holy grail I would refer you to Van Tharp.
{quote} I think the footprint you are referring to is basically Price Action. Big money does leave a trail and it is up to us traders to find the trail and follow it. I find it an interesting
topic but I gotta say I don't believe there is some magic number that will solve your problems. There is no holy grail For more reading on the holy grail I would refer you to Van Tharp. Best,
PA is not a trail, certainly you cannot tell anything by just looking at the candles.
Magic is not tha case here, but turbulance is a basic information about chaotic system. It is just a step for further research. Also to ne remembered, it is objective number that tells us some fact.
It not probabilitie of something.
Sure it is not a cash machine, but still better than stochastics(sic!), linear tools and indicators that are derivatievs of price.
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The Mathematical Foundations of Post-Quantum Cryptography
The security of lattice-based post-quantum cryptography relies on the computational hardness of the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) in lattices, which are
equivalent to sphere packing and sphere covering problems, and can be formulated as arithmetic problems of positive definite quadratic forms.
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zlarz: applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right - Linux Manuals (l)
zlarz (l) - Linux Manuals
zlarz: applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right
ZLARZ - applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right
SIDE, M, N, L, V, INCV, TAU, C, LDC, WORK )
CHARACTER SIDE
INTEGER INCV, L, LDC, M, N
COMPLEX*16 TAU
COMPLEX*16 C( LDC, * ), V( * ), WORK( * )
ZLARZ applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right. H is represented in the form
= I - tau * v * vaq
where tau is a complex scalar and v is a complex vector.
If tau = 0, then H is taken to be the unit matrix.
To apply Haq (the conjugate transpose of H), supply conjg(tau) instead tau.
H is a product of k elementary reflectors as returned by ZTZRZF.
SIDE (input) CHARACTER*1
= aqLaq: form H * C
= aqRaq: form C * H
M (input) INTEGER
The number of rows of the matrix C.
N (input) INTEGER
The number of columns of the matrix C.
L (input) INTEGER
The number of entries of the vector V containing the meaningful part of the Householder vectors. If SIDE = aqLaq, M >= L >= 0, if SIDE = aqRaq, N >= L >= 0.
V (input) COMPLEX*16 array, dimension (1+(L-1)*abs(INCV))
The vector v in the representation of H as returned by ZTZRZF. V is not used if TAU = 0.
INCV (input) INTEGER
The increment between elements of v. INCV <> 0.
TAU (input) COMPLEX*16
The value tau in the representation of H.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the M-by-N matrix C. On exit, C is overwritten by the matrix H * C if SIDE = aqLaq, or C * H if SIDE = aqRaq.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK (workspace) COMPLEX*16 array, dimension
(N) if SIDE = aqLaq or (M) if SIDE = aqRaq
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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Introduction to Statistics: Goals, Phobias, and Descriptive & Inferential Statistics - Pro | Study notes Psychology | Docsity
Download Introduction to Statistics: Goals, Phobias, and Descriptive & Inferential Statistics - Pro and more Study notes Psychology in PDF only on Docsity! 1 Introduction to Statistics 1. Goals of
the Course 2. The What and Why of Statistics 3. The Syllabus 2 Introduction to Statistics The Course Syllabus Available on Oncourse Note: syllabus may be subject to change, including exam dates.
Changes will be announced in class and via email Syllabus is approximate only! 5 Introduction to Statistics A Word About Phobias 1. All statistical procedures were developed to serve a purpose 2. If
you understand why a new procedure is needed, you will find it much easier to learn the procedure After each class/chapter, try to read the “Preview” section at the beginning of each book chapter. 6
Introduction to Statistics Why Statistics? 2. Science is based on observation - statistics allows us to organize, summarize and interpret empirical data 1. Statistics is all around us 7 Introduction
to Statistics Examples 1. In a psychological experiment, we need to determine a human subject’s reaction time. The measurements we obtain vary a great deal from one trial to the next. What can we do
to get a reliable estimate? 2. A drug company has developed a new substance that, they claim, reduces blood pressure. How do we test this claim? 3. The mayor of a large city must decide whether to
build an extension of the downtown highway system or not. The mayor is concerned about voter support. How does the mayor find out what people think? 10 Introduction to Statistics Fundamental Logic of
Statistical Reasoning samplinginference 11 Introduction to Statistics Populations and Samples Population of individuals population of scores Sample of individuals sample of scores A parameter
describes a population A statistic describes a sample Observation / measurement = datum / score / raw score For instance, the mean of the population And the mean of the sample 12 Introduction to
Statistics Descriptive Statistics Descriptive statistics are used to summarize, organize and simplify data some examples... Descriptive Statistics: Tornadoes a0 ao 70 Average Number &0 of Tornadoes
Per Hour of o the Day a.m. a.m. a.m. p.m. p.m. p.m. 1997 Gklahoma Climatological Survey. All rights reserved. 15 16 Introduction to Statistics Inferential Statistics Inferential statistics study
samples and allow generalizations (inferences) about the population from which the sample was obtained (assuming the sample was representative). For example: I want to use the data from 100 students
to make conclusions about all of the incoming students of IU By the end of the course, you should have an understanding of why this works (and its limitations) 17 Introduction to Statistics
Inferential Statistics Sampling error is the discrepancy between a sample statistic and the corresponding population parameter Keep the sampling error small: Use large samples Use random sampling 20
Introduction to Statistics Inferential Statistics Sampling error is the discrepancy between a sample statistic and the corresponding population parameter Keep the sampling error small: Use large
samples Use random sampling 21 Introduction to Statistics Fundamental Logic of Statistical Reasoning: Let’s try an example! samplinginference 22 Introduction to Statistics Population: K300 Mean =
66.9 Student # Sample 1: Mean = 68.5 Sample 1: Mean = 65.8 25 Introduction to Statistics Correlational Method Correlational method: Two variables are observed and checked to see if a relationship
exists 26 Introduction to Statistics Correlational Method Correlational method: Two variables are observed and checked to see if a relationship exists Does early wake-up time cause better academic
performance? Correlation does not imply causation • Experimental method: Goal is to establish causal relationships between variables • Requires manipulation and control conditions • The independent
variable is the one that is manipulated • The dependent variable is the one that is observed 27 Introduction to Statistics Experimental Method 30 Introduction to Statistics Discrete and Continuous
Variables A discrete variable consists of separate, indivisible categories (e.g., number of male children in family). A continuous variable is divisible into an infinite number of fractional parts
(e.g., weight of male children in family). 31 Introduction to Statistics Scales of Measurement Different kinds of scales: nominal ordinal interval ratio When collecting data we need to make
measurements. How do we measure things? By putting them into categories: Qualitative By using numbers: Quantitative • Nominal – Set of categories that have different names – “more than” or “less
than” not defined – Major = {Math, Stats, Physicis} • Ordinal – Organized in an ordered sequence – You can determine the direction of difference (i.e., order) – Groups = {lower, middle, upper
socioeconomic class} 32 Introduction to Statistics Scales of Measurement 35 From Jaccard and Becker (5th ed., Fig. 1.1) Example 36 Introduction to Statistics Statistical Notation One score: X Two
scores: X, Y Number of scores (sample): n Number of scores (population): N Summation: X (“sum of X”) (think of summing over a column in a spreadsheet) Note: (X-5) X-5 X2 (X)2 Example: X =
{3, 1, 7, 4} X = 3 + 1 + 7 + 4 = 15 X2 = 9 + 1 + 49 + 16 = 75 37 Introduction to Statistics Statistical Notation Note: (X-5) X-5 X2 (X)2 X = {3, 1, 7, 4} X = 15 (X-5) = (-2) + (-4) + 2 +
(-1) = -5 X-5 = (X)-5 = 15 – 5 – 10 (X)2 = (X) x (X) = 15 x 15 = 225 X2 = 75
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To Catch Terrorists, Think Quantum Mechanically?
An interesting paper in PNAS from a few years back that I missed, “Strong profiling is not mathematically optimal for discovering rare malfeasors” by William H. Press.
Suppose you have a large population of people and a single bad guy who you want to catch. You could look through the entire population to find the bad guy, or you could set up checkpoints (err I
mean airline security screening areas) to look for people, sampling only some small fraction of the population that goes through the checkpoint. Now, if you don’t know anything about the population
you’re sampling, you might as well just sample them randomly until you find your baddie, since you don’t have any other information that could help you.
But suppose that you are able to come up with some prior probabilities for different people to be the bad guy. I.e. you’ve developed a model for what makes someone more likely to be a bad guy, and
further assume that this model is really pretty accurate. To each person you can assign a probability $p_i$ that the person is indeed the bad guy. Now you could continue to sample uniformly, but
you have this great model that you want to use to help catch the bad guy. What do you do?
It turns out that the wrong thing to do is to sample in proportion to the probability $p_i$. To figure out the correct strategy, suppose that you sample from the population with probability $q_i$.
Then if $k$ is your man, the probability that you get him in one sample is $q_k$. Or, another way to say it is that the mean number of screenings you’ll need to find the baddie is $1/q_k$.
Assuming your model is correct, the mean number of people you will have to sample is $sum_i p_i/q_i$. So now to calculate the optimum we need to minimize this expression for $q_i$ subject to the
constraint that $sum_i q_i =1$.
To calculate this optimum, you use a Lagrange multiplier
$${partial over partial q_j} left[ sum_i {p_i over q_i} + lambda (sum_i q_i -1)right]=0$$
$$ – {p_j over q_j^2} + lambda = 0$$
Which, in order to satisfy our contraints (and also positive probabilities) gives us the answer for the optimum of
$$ q_j = { sqrt{p_j} over sum_i sqrt{p_i}}$$
Or, in other words, you should sample proportional to the square root of the probabilities. Pretty cool, a nice easy, yet surprising answer.
Even more awesome is that we got some square roots of probabilities in there. Quantum probability amplitudes are, of course, like square roots of probabilities. Now if only we could massage this
into insight into quantum theory. Do it. Or the terrorist win.
2 Replies to “To Catch Terrorists, Think Quantum Mechanically?”
1. There’s a more direct geometric intuition for this result. There’s a natural map that takes points in the simplex to points on the positive orthant of the sphere (one dimension higher) by taking
the square root of each coordinate. Now, the resulting value of the q corresponds to taking the ray from the origin of the sphere through the mapped point for p, and seeing where it cuts the
simplex. Essentially, you’re projecting from the sphere back down to the simplex, finding the “nearest” point. This sort of makes sense in the context of information geometry.
Which is not to say that there isn’t a quantum interpretation.
2. Thanks Suresh. I think the quantum connection is actually to some old work of Bill Wootters, but I can’t seem to find the relevant paper right now.
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What is a solution to the differential equation (2+x)y'=2y?
| HIX Tutor
What is a solution to the differential equation #(2+x)y'=2y#?
Answer 1
$y = \beta {\left(2 + x\right)}^{2}$
#1/y y'=2/(2+x) #
#int \ 1/y \ dy=int \ 2/(2+x) \ dx #
#ln y = 2 ln (2+x) + alpha#
#ln y = 2 ln (2+x) + ln beta#
#ln y = ln beta(2+x)^2#
# y = beta (2+x)^2#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities Exercise 8.1
NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities (EX 8.1) Exercise 8.1
The Central Board of Secondary Education introduced the National Council of Educational Research and Training. It was established to support a unified educational system with a national character for
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areas related to school education. NCERT discusses all significant topics and offers information about the globe. Students are taken off the traditional path of study. They can easily understand its
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The variety of study tools available to students makes deciding which to use as they prepare for their examinations difficult. Many of them usually spend time looking for the best study source in
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NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities (EX 8.1) Exercise 8.1
“Comparing Quantities ” refers to the quantitative relationship between two quantities that represents their respective quantities. It is simply a tool for comparing data.
The concepts of ratio and proportion are taught to students in Class 7th 8.1 Maths on Comparing Quantities. Additionally, they will learn about the Unitary Method, Percentages, And Simple Interest
concepts and how to use them in real-world situations. This chapter also discusses the Conversion Of Fractions, Decimals, and Percentages into one another. If students had access to the NCERT
Solutions Class 7 Maths Chapter 8 Exercise 8.1, learning would be simpler. Many sites offer the NCERT Solutions Class 7 Maths Chapter 8 Exercise 8.1, and it gets tough for them to understand which
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Students should refer to the Extramarks website for the NCERT Solutions Class 7 Maths Chapter 8 Exercise 8.1 as it is an authenticated website that provides reliable solutions to the students who are
registered with it. The NCERT Solutions for Class 7 Maths Chapter 8 Exercise 8.1, Comparing Quantities, expands on this concept by giving comparisons of precise names in Mathematics, which are
Ratios, Proportions, and Percentages. The Chapter deals with real-world problems that will assist students in comprehending how to compare quantities. It also explains the concepts of profit, loss,
and basic interest. In addition to teaching whole numbers, NCERT Solutions Class 7 Maths Chapter 8 Exercise 8.1 teaches how to convert fractions and decimals into ratios and percentages.Students will
learn some crucial information, such as the necessity of using the same unit when comparing any two quantities. The four quantities in question are said to be in proportion if any two ratios are
Access NCERT Solutions for Class 7 Mathematics Chapter 8- Comparing Quantities
The best option for CBSE students to practise the chapter titled Comparing Quantities in terms of exam preparation is choosing the NCERT Solutions Class 7 Maths Chapter 8 Exercise 8.1. There are
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website or mobile application, or they can download it as needed. Extramarks’ internal subject-matter experts, carefully and in accordance with all CBSE regulations, solved the problems and questions
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Any student in Class 7 who is familiar with all the concepts in the Mathematics textbook and is well-versed in all the exercises in it can easily achieve the highest scores on the final examination.
Students may readily comprehend the types of problems that may be given in the examination from this chapter. They can also learn the chapter weightage in terms of overall grade by using the NCERT
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Exercise 8.1
The NCERT Solutions Class 7 Maths Chapter 8 Exercise 8.1 are available on the Extramarks website so that students may fully understand the material. The best professionals in India who are
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Students can efficiently prepare for the board examination by practising the key questions for each chapter of the Class 7 Mathematics course. Complete solutions will be given to them. Students can
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NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities Exercise 8.1
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$\begin{array}{l}\mathrm{Find}\mathrm{the}\mathrm{ratio}\mathrm{of}:\\ \left(\mathrm{a}\right)\mathrm{Rs}5\mathrm{to}50\mathrm{paise}\left(\mathrm{b}\right)15\mathrm{kg}\mathrm{to}210\mathrm{g}\\ \
$\begin{array}{l}\left(\text{a}\right)\text{Rs 5 to 5}0\text{paise}\\ \text{}1\text{ruppe}=\text{100 paise}\\ \text{So,}\text{\hspace{0.17em}}\text{5 rupee}=\text{500 paise}\\ \text{So,}\frac{\text
{Rs}5}{50\text{paise}}=\frac{500}{50}=\frac{10}{1}\\ \text{Thus, the required ratio is}\overline{)\text{10:1}}.\\ \left(\text{b}\right)\text{15 kg to 21}0\text{g}\\ \text{1 kg}=\text{1000 g}\\ \text
{So, 15 kg}=\text{15000 g}\\ \text{So,}\frac{15\text{kg}}{210\text{g}}=\frac{15000}{210}=\frac{500}{7}\\ \text{Thus, the required ratio is}\overline{)\text{500:7}}.\\ \left(\text{c}\right)\text{9 m
to 27 cm}\\ \text{1 m}=\text{100 cm}\\ \text{So, 9 m}=\text{900 cm}\\ \text{So,}\frac{9\text{m}}{27\text{cm}}=\frac{900}{27}=\frac{100}{3}\\ \text{Thus, the required ratio is}\overline{)\text
{100:3}}.\\ \left(\text{d}\right)\text{3}0\text{days to 36 hour}\\ \text{1 day}=\text{24 hours}\\ \text{So, 30 days}=\text{720 hours}\\ \text{So,}\frac{30\text{days}}{36\text{hour}}=\frac{720}{36}=\
frac{20}{1}\\ \text{Thus, the required ratio is}\overline{)\text{20:1}}.\end{array}$
$\begin{array}{l}\mathrm{In}\mathrm{a}\mathrm{computer}\mathrm{lab},\mathrm{there}\mathrm{are}3\mathrm{computers}\mathrm{for}\mathrm{every}6\\ \mathrm{students}.\mathrm{How}\mathrm{many}\mathrm
{computers}\mathrm{will}\mathrm{be}\mathrm{needed}\mathrm{for}24\\ \mathrm{students}?\end{array}$
$\begin{array}{l}\text{For 6 students, number of computers required}=\text{3}\\ \text{So, for 1 student, number of computers required}=\frac{3}{6}=\frac{1}{2}\\ \text{Thus, for 24 students, number of
computers required}=\frac{1}{2}×24\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}=\overline{)12}\\ \text{Therefore, 12 computers are needed for 24 students}\text{.}\end{array}$
$\begin{array}{l}\mathrm{Population}\mathrm{of}\mathrm{Rajasthan}=570\mathrm{lakhs}\mathrm{and}\\ \mathrm{population}\mathrm{of}\mathrm{UP}=1660\mathrm{lakh}\\ \mathrm{Area}\mathrm{of}\mathrm
{Rajasthan}= 3\mathrm{lakh}{\mathrm{km}}^{\mathrm{2}}\mathrm{and}\\ \mathrm{area}\mathrm{of}\mathrm{UP}= 2\mathrm{lakh}{\mathrm{km}}^{\mathrm{2}}\mathrm{.}\\ \left(\mathrm{i}\right)\mathrm{How}\
mathrm{many}\mathrm{people}\mathrm{are}\mathrm{there}\mathrm{per}{\mathrm{km}}^{\mathrm{2}}\mathrm{in}\mathrm{both}\mathrm{these}\mathrm{States}?\\ \left(\mathrm{ii}\right)\mathrm{Which}\mathrm
$\begin{array}{l}{\text{(i) Population of Rajasthan in 3 lakh km}}^{\text{2}}=\text{570 lakhs}\\ {\text{Population of Rajasthan in 1 lakh km}}^{\text{2}}\text{}=\frac{\text{570}}{3}\\ \text{}\text{}\
text{}\text{}\text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}=\overline{)190\text{}}\\ {\text{Population of UP in 2 lakh km}}^{\text{2}}\text{}\text{}\text{}=\text{1660
lakhs}\\ {\text{Population of UP in 1 lakh km}}^{\text{2}}\text{}\text{}\text{}=\frac{1660}{2}\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}=\overline{)830\text{\hspace{0.17em}}}\
\ \text{(ii)}\\ \text{From above data, clearly Rajasthan is less populated}\text{.}\end{array}$
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Understanding Quantum Optics: A Comprehensive Overview
Welcome to our comprehensive overview of quantum optics, one of the most fascinating and cutting-edge fields of physics research today. From its origins in the late 19th century to its current
applications in quantum computing and communication, quantum optics has revolutionized our understanding of light and its interactions with matter. In this article, we will delve into the fundamental
principles of quantum optics, explore its various applications, and discuss the latest advancements in this ever-evolving field. Whether you are a seasoned researcher or just starting to explore the
world of optics, this article will provide you with a solid foundation of knowledge on quantum optics.
So, let's dive in and unlock the mysteries of quantum optics together. Quantum Optics is a fascinating branch of physics that delves into the nature of light and its interaction with matter. From its
basic principles to its applications in modern technology, this article aims to provide a comprehensive overview of Quantum Optics for readers interested in physics. The fundamental concepts of
Quantum Optics are essential to understanding this field. One of the key concepts is wave-particle duality, which states that light can behave both as a wave and a particle. This concept was first
proposed by Albert Einstein in his theory of photoelectric effect, and it revolutionized our understanding of light. Another important concept in Quantum Optics is the quantization of energy, which
states that energy is not continuous but exists in discrete packets known as quanta.
This concept was introduced by Max Planck in his studies on blackbody radiation and is crucial to understanding the behavior of light at the atomic level. The uncertainty principle, proposed by
Werner Heisenberg, states that it is impossible to simultaneously know the exact position and momentum of a particle. This principle has significant implications in Quantum Optics experiments and is
crucial to understanding the behavior of particles at the quantum level. To aid in understanding these concepts, this article will use examples and diagrams to illustrate their applications in
Quantum Optics. These visual aids will make it easier for readers to grasp these complex concepts. One of the most exciting aspects of Quantum Optics is the various experiments that are conducted in
this field. Some examples include photon counting experiments, which involve detecting individual photons, and entanglement experiments, where two or more particles are linked in such a way that
their properties are correlated even when separated by large distances. This article will also cover some of the key formulas used in Quantum Optics.
These include Planck's constant, which relates the energy of a photon to its frequency, and the Schrödinger equation, which describes the behavior of particles at the quantum level. For those
interested in furthering their understanding and conducting their own experiments, this article will provide resources and tutorials to help them get started. These resources will include books,
online courses, and simulation software. For those considering a career in physics, Quantum Optics offers a wide range of opportunities. This article will provide information on the various career
paths available in this field, including research positions in academia and industry. Finally, this article will discuss the latest research and advancements in Quantum Optics. This field is
constantly evolving, with new discoveries and breakthroughs being made every day.
Staying up to date with the latest developments is essential for anyone interested in Quantum Optics.
Exploring the World of Quantum Optics
Quantum Optics is a fascinating field of study that delves into the fundamental concepts of light and its interaction with matter. It has led to groundbreaking discoveries and has revolutionized
modern technology. In this section, we will delve deeper into the world of Quantum Optics and explore its core principles.
The Latest Research and Advancements
In a field as rapidly evolving as Quantum Optics, staying up-to-date on the latest research and advancements is crucial. As scientists continue to push the boundaries of our understanding of light
and matter, new discoveries and breakthroughs are being made on a regular basis. One way to stay informed on the latest developments in Quantum Optics is to regularly read scientific journals and
These sources provide in-depth analyses of current research and experiments, allowing readers to stay informed on the cutting-edge of the field. Attending conferences and workshops is another great
way to stay updated on the latest advancements in Quantum Optics. These events bring together experts from around the world to share their findings and discuss future directions for research.
Finally, keeping an eye on funding opportunities and collaborations can also provide valuable insight into the current state of Quantum Optics research. By staying connected with other researchers
and institutions, scientists can stay informed on new projects and initiatives that are shaping the field.
Resources and Tutorials for Further Learning
Expanding your knowledge on Quantum Optics can be an exciting journey that opens up endless possibilities. To help you delve deeper into this subject, we have compiled a list of resources and
tutorials that cover various aspects of Quantum Optics. For those new to the field, introductory textbooks such as 'Introduction to Quantum Optics' by Grynberg, Aspect, and Fabre provide a solid
For a more in-depth understanding, 'Quantum Optics' by Scully and Zubairy offers a comprehensive overview of the mathematical framework and experimental techniques used in Quantum Optics. Online
resources such as video lectures and tutorials from prestigious universities like MIT and Caltech are also great options for expanding your knowledge. These resources cover a wide range of topics
from basic principles to advanced concepts in Quantum Optics. Additionally, attending conferences and workshops on Quantum Optics can provide a valuable opportunity to learn from experts in the field
and network with fellow researchers.
Careers in Quantum Optics
Quantum Optics is a rapidly growing field with endless possibilities for those interested in pursuing a career in physics. This branch of physics combines the principles of quantum mechanics and
optics to study the behavior of light and its interaction with matter at a fundamental level. As technology continues to advance, the demand for professionals in Quantum Optics has increased. This
has led to a wide range of
career opportunities
in various industries such as telecommunications, healthcare, and defense. Some common career paths in Quantum Optics include research scientists, optical engineers, and laser physicists.
These professionals work on cutting-edge projects, developing new technologies and techniques that have the potential to revolutionize the way we use light in our daily lives. If you are passionate
about physics and have a strong understanding of the principles of Quantum Optics, then pursuing a career in this field can be an exciting and rewarding experience. With the right education and
training, you can become a part of this dynamic and innovative field and contribute to groundbreaking discoveries and advancements. Whether you choose to work in academia or in the industry, a career
in Quantum Optics offers endless possibilities for growth and development. So if you are ready to embark on a path of exploration and discovery, then Quantum Optics may just be the perfect field for
Experiments and Formulas in Quantum Optics
Applying Theory to PracticeQuantum Optics is a field that not only explores the theoretical aspects of light and matter, but also puts these theories into practice through experiments and formulas.
These experiments and formulas play a crucial role in advancing our understanding of Quantum Optics and have led to many groundbreaking discoveries. One such experiment is the famous double-slit
experiment, which demonstrates the wave-particle duality of light.
This experiment showed that light behaves both as a wave and a particle, depending on how it is observed. This discovery has had a major impact on our understanding of quantum mechanics and has paved
the way for many applications in modern technology. Other important experiments in Quantum Optics include the photoelectric effect, which led to the development of the photon concept, and the Bell
test, which confirmed the existence of quantum entanglement. These experiments have provided evidence for some of the most fundamental principles in Quantum Optics. Formulas also play a crucial role
in Quantum Optics, allowing us to make precise calculations and predictions about the behavior of light and matter. Some of the most well-known formulas in this field include Planck's law, which
describes the spectral energy density of blackbody radiation, and Schrödinger's wave equation, which describes the evolution of quantum states. By applying theory to practice through experiments and
formulas, we are able to gain a deeper understanding of Quantum Optics and its practical applications.
As technology continues to advance, we can only imagine the new experiments and formulas that will be developed in this fascinating field. In conclusion, Quantum Optics is a fascinating field of
study that offers endless possibilities for exploration and discovery. From understanding the basics of light and energy to applying theories in experiments, this field has something to offer for
everyone interested in physics. With the resources and opportunities available, pursuing a career in Quantum Optics is an exciting prospect. Stay updated on the latest research and advancements in
this field to continue expanding your knowledge and understanding of the world around us.
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We introduce the Distributed Post Office, an idea for routing messages in a decentralized mesh network. The Distributed Post Office is a method for creating an instant hierarchical structure from a
mesh network.
We describe the basic form of the Distributed Post Office, and mention two improvements to its structure.
Next, we run some experiments [github]. Our results show that the Distributed Post Office in its current form does not scale well as a solution for routing messages in large networks.
In this text we do not discuss security or reliability topics related to the Distributed Post office.
The distributed post office question
What if there there were no post offices in the world, and you wanted to send a package to some far friend?
Assume that you meet some people you trust on a regular basis. More generally, assume that every person $x$ in the world meets a few people he trusts on a regular basis. We also call those trusted
regulars of $x$ the neighbours of $x$. If $y$ is a neighbour of $x$, we also say that $x$ is a neighbour of $y$. The neighbours relationships could be used to transfer packages.
For the feasibility of things, let's also add in the assumption that this system is connected. This means: We can get from any person to any other person using a sequence of neighbours. Or, in other
words: If we draw a graph of all the people in the world (As nodes), and put edges between every person and his neighbours, then we get a connected graph.
In the picture: A connected network of people. There is a path of neighbours that connects $x$ and $y$.
Would these assumptions be enough to let us send a message to anyone in the world?
A few initial thoughts:
• Person $a$ can send a package to person $B$ only if there is a "path" of neighbours between $a$ and $b$. That means: $a$ has some neighbour $a_1$, and $a_1$ has some neighbour $a_2$ and so on,
until finally some $a_k$ has $b$ as a neighbour.
• The lack of any centralized post office might create the urge to create one. This artificial post office might be some person that everyone knows. However, picking such a person that is agreed
upon all the participants might be difficult task.
• What kind of addressing are we going to use for sending packages, if there is no central post office? (What are we going to write on the package on the "TO" field?) And what if the network layout
changes? Will the address remain valid?
Basic Distributed Post office
Finding the highest person
One way to solve this is finding some special person. This person will be used as a reference point for sending messages. All our messages will pass through this person. This centralized point of
view is a bit different from our usual decentralized point of view, but it will present some idea that we might later be able to utilize in a decentralized way, so bear with me here.
Finding a special person could be achieved as follows: Every person $x$ will maintain a link (Maybe through a few neighbours) to the highest person he knows of (We assume that there are no two people
of the exact same height). By maintaining a link we mean that $x$ will remember some person $y$, his height and a path from $x$ to $y$ that goes through neighbours.
Initially, every person $x$ will look at his neighbours (And himself), and find the highest person among them. Then $x$ will inform all his neighbours of the highest person he knows (He will also
send his path to that person). Next, $x$ will look at all the highest known people he received from his neighbours. He will update his highest known person accordingly.
The algorithm could be described simply as follows: In every iteration, every person $x$ sends to all his neighbours the highest person $y$ that he knows, and also the shortest path he knows from $x$
to $y$. (Iteration could happen every few seconds, for example).
After enough iterations, we expect that every person $x$ will find the highest person in the world $t$. In addition, $x$ will know a shortest path possible from $x$ to $t$.
Let's explain this result: Why do we expect that the highest person $t$ will always be found by all people, and also that a shortest path will be found?
Assume that $x$ is some person, and $t$ is the highest person in the world. There is some shortest path of neighbours between $x$ and $t$ (However $x$ doesn't know that path yet). Let's assume that
this path of neighbours have the people: $(x,x_1,x_2,\dots,x_{k-1},x_k,t)$ in this order.
In the picture: A shortest path between $x$ and $t$. Note that this is not the only shortest path between $x$ and $t$.
After the first iteration, $x_k$ will have $t$ as the highest person in the world. $x_k$ will also have a shortest path to $t$. In the next iteration, $x_{k-1}$ will know $t$ as the highest person in
the world (Because $x_k$ has told him about it). $x_{k-1}$ will also know a shortest path to $t$: The path: $(x_{k-1},x_{k},t)$. What if this is not the shortest path? Then there must be some path of
length two, as follows: $x_{k-1},t$. In that case, there is also a shorter path between $x$ and $t$: $(x,x_1,x_2,\dots,x_{k-1},t)$. But this will contradict our assumption that $x,x_1,x_2,\dots,x_
{k-1},x_k$ is a shortest path between $x$ and $t$.
We can continue this argument until we get to $x$. (The formal way to do it is using mathematical induction). Then we conclude that after $k+1$ iterations, $x$ will have $t$ as the highest person in
the world, and $x$ will also have a shortest path from $x$ to $t$.
Another way to think about it is that the amount of iterations needed until every person finds the highest person in the world is not more than the diameter of the neighbours graph.
We don't really deal with security here, but I want to mention this question: Whenever a person sends to all his neighbours the highest person he knows, how can we know he doesn't lie? We will deal
with this later.
Addressing and Drift
So far we got some special person $t$ that every person can reach: Every person $x$ knows a path to $t$, and thus $x$ can send a package to $t$.
How could $x$ send a message to some arbitrary person $y$? $x$ already knows the path from $x$ to $t$. If $x$ knew a path from $t$ to $y$, he could first deliver his package to $t$, and then ask $t$
to send his package to $y$.
This idea leads us to choose the address of an arbitrary node $y$ as a shortest path between $t$ and $y$. (This is the reversed path between $y$ and $t$). We will mark this as the address of $y$, or
Given $A(y)$, $x$ can send a package to $y$ by sending the package first to $t$, with the address $A(y)$ written on the package. $t$ will then use the path $A(y)$ to deliver the message all the way
to $y$.
We ignore the centrality issues of this idea for a moment ($y$ has to deal with all the packages sent in the world!!!), and try to think about the addressing idea.
$y$'s address is induced from the structure of the network of neighbours. Certain changes could invalidate $y$'s address. In other words: There is a drift in the addresses, as the network of
neighbours changes.
One way to deal with this issue is to refresh the addresses from time to time: If $x$ is in contact with $y$, then $y$ will send his current address to $x$ every few seconds. $x$ will also send his
current address to $y$ every few seconds. This might not be very reliable, but it's an idea.
Changing people into nodes
Formally we solved the problem of delivering packages, however the solution is not very satisfying. All the packages has to go through some special person. This is not good for us because of security
reasons (Could we trust the special person), and also because of load issues (Just because he is the highest person in the world, he has to deal with all the packages?)
Before we start proposing more ideas, it is probably a good time to change our terminology to networks and nodes. We get the following question: Assume that we are given a mesh network, where every
node has a few neighbours. How can we deliver messages between every two arbitrary nodes?
Note that we already tried to solve this question using flooding, $sqrt{n}$ routing and Virtual DHT routing.
Instead of checking the height of people, we can use some other properties of nodes. We will use some public key cryptography. Every node $y$ will create a key pair $prv_x,pub_x$ (Private and
Public). We assume that every node $x$ knows the public keys of all his neighbours.
Next, we choose some cryptographic hash function $h$. Instead of person $x$'s height, we will take a look at $h(pub_x)$. We will call this value $x$'s height. The highest person in the world will
turn into the node $t$ that maximizes the value $h(pub_t)$. In that case we will also say that $t$ is the "highest" node in the network with respect to the cryptographic hash function $h$.
While it was difficult to verify the height of a person from a distance, we could verify the public key of a node from a distance. If node $x$ is informed by node $y$ about some remote node $t$ that
has a certain $h(pub_t)$ value, $x$ can verify it himself. $x$ can send some challenge all the way to $t$, and $t$ will send back a response that proves he owns the public key $pub_t$.
In the picture: $x$ wants to verify that $t$ owns the public key $pub_t$. Therefore $x$ sends a challenge to $t$. If $t$ returns a correct response, $x$ will believe that $t$ owns the public key
Note however that this challenge response idea is not a magic cure to all the security problems in this model. It just helps a bit.
Adding hierarchy
Highest node in some radius
One approach to make things better is to create some kind of hierarchy. Earlier, every node $x$ maintained contact to the "highest" node in the network (with respect to some cryptographic hash
function $h$) through some path of nodes.
This time, instead of remembering just the "highest" node in the network, every node $x$ will remember a few special nodes. Every node will be the "highest" in some certain area around $x$:
• $t_x^1$: The "highest" node of maximum distance $1$.
• $t_x^2$: The "highest" node of maximum distance $2$.
• $t_x^3$: The "highest" node of maximum distance $3$.
• $t_x^d$ The "highest" node in the network.
In the picture: Distance rings around $x$. The first ring is distance $0$ from $x$. It contains only $x$ itself. The next ring contains all nodes of distance exactly $1$ from $x$. The next ring
contains all nodes of distance exactly $2$, and so on. Note that by definition, A node from a certain ring can only be connected to nodes of adjacent rings, or nodes from the same ring.
We mark by $t_x^j$ The "highest" node of maximum distance $j$ from $x$. In our example, $t_x^3 = t_x^4$.
Side question: How can we choose a good value for $d$? One suggestion is to keep increasing the distance until we stop getting new highest nodes. Another suggestion would be to just assume that the
graph diameter won't be more than some constant number.
First let's assume that somehow we managed to have the above information for every node in the network, and see what we can do with it. (Note that we didn't yet describe how to get this information.
It will be described soon later).
We define $x$'s address to be $A(x) = (p_x^1,p_x^2,\dots,p_x^d)$, where $p_x^j$ is the path from $t_x^j$ to $x$. This definition of $A(x)$ is an extension of our previous definition of $A(x)$, where
we only had the $t_x^d = t$. Also note that looking at some $p_x^j$, one can conclude $t_x^j$ (It is just the first node on the path).
Looking at two different nodes: $x,y$, the first thing to note is that $t_x^d$ and $t_y^d$ are the same, assuming that $d$ is large enough. Why? Because $t_x^d = t_y^d = t$, the highest node in the
network. For other distances, the nodes $x$ and $y$ have chosen might differ. For example, $t_x^1$ and $t_y^1$ are likely to be different.
How to deliver messages using the address information? Assume that $x$ has the address of $y$: $A(y)$, as described above. $x$ will compare his own address: $A(x)$ with $A(y)$. $x$ will try to find
the smallest $j$ such that $t_x^j = t_y^j$. $x$ knows a shortest path from $x$ to $t_x^j$. $x$ also knows $A(y)$, So $x$ knows $p_y^j$, which is a shortest path from $t_y^j = t_x^j$ to $y$. Finally,
$x$ can create a full path from $x$ to $y$ that goes through $t_y^j = t_x^j$. This path could be used to send messages.
In the picture: The node $t_x^4 = t_y^4$ is a mediator between $x$ and $y$. $x$ and $y$ can route messages through $t_x^4 = t_y^4$.
Addresses should not be too large if we want them to be practical to use. Let's estimate the size of a typical address, as defined above. For some node $x$, $x$'s address is $(p_x^1,p_x^2,\dots,p_x^
d)$. Every such $p_x^j$ is a path. Assuming that the diameter of the network graph is $d$, we get that each path is of length no more than $d$. Therefore we get at most $d^{2} q$, where $q$ is the
size of a typical public key. This could become more than a few kilobytes if the public key size and the network diameter are big. (Much more than an IP address, unfortunately).
Obtaining "highest" nodes
We now explain how a node $x$ can obtain contact to the nodes $t_x^1,\dots,t_x^d$. (And also a shortest path to each of those nodes).
In every iteration, the node $x$ will ask every neighbour $y$ for the value $t_y^j$ for every $1 \leq j \leq d$. Then $x$ will update his values of $t_x^j$ accordingly: The value $t_y^j$ from $y$ is
a candidate for $t_x^{j+1}$. If $t_y^j$ is "higher" than $t_x^{j+1}$, then $x$ will replace it with $t_y^j$.
Simply speaking: In iteration number $k$, a node $x$ is aware of all the "highest" nodes in the network until distance $k$.
Formally, Using mathematical induction (Over the amount of iterations) it can be shown that after $k$ network iterations, Every node $x$ knows the correct value for $t_x^j$ for every $1 \leq j \leq
k$, and also a shortest path to $t_x^j$.
Hierarchy benefits?
Why would we want to have hierarchy from the first place? Earlier we complained that all the messages will go through one special node $t$, and $t$ won't be able to deal with the load. Maybe the
hierarchy we have added can help a bit.
How often will a message go through the "highest" node $t$?
Assume that $x$ wants to send a message to some node $y$. $x$ checks $y$'s address, and tries to find the first $j$ such that $t_x^j = t_y^j$. (As described above). If any small enough such $j$ is
found, the message will be routed through $t_x^j = t_y^j$, and not through the "highest" node $t$. We can think of $t$, the "highest" node, as some kind of backup. If nothing better was found, we can
always route through $t$.
But how do we know if $x$ will route his message to $y$ through $t$, or through some lower level node $t_x^j = t_y^j$? Generally speaking, we expect that the more $x$ and $y$ are close, the more
their addresses $A(x),A(y)$ are similar, and the more likely it is to route the message using a "high" node that is not the "highest" node in the network.
The "highest" node is still overloaded
We expect that messages between close nodes are routed using a local "high" node, while messages between very far nodes are routed using a globally "high" node. Therefore idealy we expect that the
"highest" node in the network will not be so loaded, because the lower level "high" nodes will take part of the load.
[DEL:This somehow resembles the way physical post offices work. You have the global post office which handles messages between countries, and smaller post offices that handle messages between cities,
and so on.:DEL] EDIT: Thanks to rubygeek I now know that post offices work differently in some countries :)
However, this is just the ideal. It is true that the local "high" nodes in the network take part of the load from the "highest" node in the network, but usually they only take a small part of it.
Intuitively, this happens for a few reasons:
• In physical mail system people tend to send packages and letters to people close to them geographically, so the structure of global post offices and local post offices makes sense. However, this
is not the case with mesh networks: In a mesh network, any two arbitrary nodes $x$ and $y$ might want to communicate. With high probability $x$ and $y$ are far away from each other (With respect
to the network graph), and so their messages will be routed through the "highest" node in the network.
• In a grid style graph (Or any planar graphs), close nodes are expected to have many "high" nodes in common. However, for other types of graphs, close nodes might only have the "highest" node in
I want to discuss the second reason with a bit more detail. (Though with a bit of hand waving). You can skip directly to the code experiments results below.
For some node $x$ in the network, we denote by $R_j(x)$ the set of nodes of distance no more than $j$ from $x$. You can think about it as a ball around $x$ of radius $j$.
Consider two nodes $x$ and $y$ in the network. We observe the sets $R_i(x) \cap R_j(y)$ and $R_i(x) \cup R_j(y)$.
A schematic picture of the sets $R_i(x)$ and $R_j(y)$ intersecting. $R_i(x)$ contains all the nodes of distance at most $i$ from $x$. $R_j(y)$ contains all the nodes of distance at most $j$ from $y$.
Let $w$ be the "highest node in $R_i(x) \cup R_j(y)$. If $w$ is inside $R_i(x) \cap R_j(y)$ then $t_x^i = w = t_y^j$. (In other words: $w$ is the "highest" node in distance $i$ from $x$, and the
"highest" node in distance $j$ from $y$). In that case, $x$ and $y$ could route messages through $w$.
What are the odds that such $w$ exists? As the "highest" node in $R_i(x) \cup R_j(y)$ could be any node in that set, the odds are: [\frac{\left|R_i(x) \cap R_j(y)\right|}{\left|R_i(x) \cup R_j(y)\
I don't know to calculate those odds for every type of graph, but let me leave you with my intuition about it. For every type of graphs, the odds for the existence of $w$ decrease as the distance
between $x$ and $y$ increases. However, for some graphs the odds decrease slowly, and for other graphs, the odds decrease quickly.
The odds decrease slowly (Quadratically) when we consider a planar graph, like a grid. However, for random graphs (Like the Erdos-Renyi model) the odds decrease quickly (exponentially).
This might be related to the fact that intersection between higher dimensional spheres becomes smaller with respect to their union, as we increase the dimension.
Experiments results
I wrote some Python3 code to check the load over the "high" nodes in the network. It can be found here [github]
To run this code, you will need the python package networkx. I could be installed as follows:
pip install networkx
If you want to change any parameter in the code, check out the go() function. All the parameters are there.
The code creates a network $G$ of nodes with random identity numbers. Using iterations as described above (In "Obtaining highest nodes"), every node $x$ finds the highest node in distance $j$, for
every distance $0 < j \leq diameter(G)$.
After creating the network and finding the "high" nodes, some large amount of pairs of nodes are chosen randomly. Between every pair of nodes $x,y$ the best mediator node is found. A mediator node is
some "high" node that both $x$ and $y$ know. A best mediator is a mediator that minimizes the sum of distances from $x$ and $y$.
For every mediator ever chosen, we count the amount of messages that were routed through that mediator. We sort the mediators list by the amount of messages they have routed, and show as output the
mediators that routed the largest amount of messages. Those are the mediators that had the highest load.
These are the results for a two dimensional grid graph of about $2^{12}$ nodes, and simulation of $2^{16}$ messages.
||| i = 12
||| num_hashes = 1
||| ident_bits = 32
Generating graph...
Generating Network...
Calculating specials...
Simulating 65536 messages delivery...
most common mediators:
mediator index | ratio | messages routed
565 | 0.507568 | 33264
3251 | 0.210510 | 13796
3661 | 0.078995 | 5177
1573 | 0.058914 | 3861
3724 | 0.031265 | 2049
1806 | 0.022171 | 1453
3333 | 0.018906 | 1239
1341 | 0.006180 | 405
159 | 0.006027 | 395
1884 | 0.005585 | 366
2047 | 0.005035 | 330
978 | 0.003891 | 255
3109 | 0.003662 | 240
377 | 0.002518 | 165
26 | 0.002228 | 146
2269 | 0.001862 | 122
How to read this table? Mediator index is a unique number that identifies the node used as a mediator. In our code, every node has a unique number. This number doesn't really matter to you. (If you
really care, it is the index number inside a python list)
The ratio is the amount of messages routed through a given node, divided by the total amount of messages delivered. In our results the first ratio is $0.507$. This ratio could be calculated as $33264
/65536$. The last column shows the amount of messages routed through a specific node.
It can be seen from the table that the first node (565) routes most of the messages (about half of the messages). It is probably the "highest" node in the network.
Next, let's look at the results for an Erdos-Renyi Random graph with $2^{12}$ nodes, and $p = (2\cdot 12)/{2^{12}}$ (This is the probability for every edge in the graph to exist). Again we simulate
the delivery of $2^{16}$ messages.
||| i = 12
||| num_hashes = 1
||| ident_bits = 32
Generating graph...
Generating Network...
Calculating specials...
Simulating 65536 messages delivery...
most common mediators:
mediator index | ratio | messages routed
3425 | 0.918594 | 60201
2300 | 0.029877 | 1958
3935 | 0.012985 | 851
3232 | 0.006516 | 427
2453 | 0.005585 | 366
767 | 0.004410 | 289
2281 | 0.003174 | 208
943 | 0.002869 | 188
457 | 0.002625 | 172
2189 | 0.002319 | 152
3682 | 0.001694 | 111
3215 | 0.001648 | 108
641 | 0.001144 | 75
1049 | 0.000565 | 37
3469 | 0.000534 | 35
782 | 0.000519 | 34
It can be seen that there is a main difference between the results of the Erdos-Renyi model and the two dimensional grid. In the Erdos-Renyi model the ratios are more condensed to the upper part of
the table. The most common mediator routes 0.91 of the messages. (Compared to about half in the grid case). Also note that the rest of the ratios decrease much faster in the Erdos-Renyi model,
compared to the two dimensional grid.
Adding hash functions
Another idea to take off the load from the "highest" node in the network would be to add more cryptographic hash functions.
Recall that the "highest" node in the network, $t$, is a node that maximizes the value $h(pub_t)$, for some cryptographic hash function $h$. We could add a few more cryptographic hash functions, to
end up with a few "highest" nodes, one for each hash function. The process of obtaining "highest" nodes for different distances will be invoked independently for each of the hash functions.
With more hash functions, every two nodes $x,y$ are likely to have more "high" nodes in common with respect to a few different hash functions. On the other hand, having $k$ hash functions means
having an address that is $k$ times bigger. (Because it contains paths to highest nodes for $k$ different hashes). It also means that every node has to maintain contact with $k$ times more nodes.
You might be wondering where will we get all those cryptographic hash functions from. If you have one cryptographic hash function $h$, you can get more for free by appending something to the input.
For example, given a function $h$ we can define $h_i(x) = h(i . x)$, where $.$ is string concatenation.
This is a pretty simple example of how to do it, and it could be not very secure in some cases, so be careful. We are just experimenting here, so we don't really care.
Let's look at some run results with more than one hash function. We show here the results for an Erdos-Renyi network with $2^{12}$ nodes, and $p = (2\cdot 12)/2^{12}$. We use $4$ hash functions:
||| i = 12
||| num_hashes = 4
||| ident_bits = 32
Generating graph...
Generating Network...
Calculating specials...
Simulating 65536 messages delivery...
most common mediators:
mediator index | ratio | messages routed
3685 | 0.189270 | 12404
2935 | 0.189026 | 12388
2010 | 0.187103 | 12262
3136 | 0.186569 | 12227
2466 | 0.025085 | 1644
3546 | 0.021988 | 1441
3886 | 0.021057 | 1380
1330 | 0.015961 | 1046
2039 | 0.010941 | 717
400 | 0.010269 | 673
761 | 0.009781 | 641
1057 | 0.008957 | 587
502 | 0.006882 | 451
2890 | 0.006760 | 443
2487 | 0.005554 | 364
3204 | 0.005341 | 350
We can see from the results that the first $4$ most common mediator nodes route about the same amount of messages: About 0.18 of all the messages sent. The rest of the nodes route much less messages.
(The next one routes 0.02 messages out of the total amount).
It seems that having $k$ cryptographic hash functions approximately divides the amount of work the "highest" node has to do by $k$. It's an improvement, but it seems like each of the "highest" nodes
still has to route a constant fraction of all the message sent in the network, which is unacceptable for a symmetric mesh network. (In other words: The average computer out there can not deal with
this amount of traffic).
Final simplification
We began with a simple idea of the "highest" node in the network, and later added two improvements: First, every node $x$ had to remember the "highest" node of distance $j$: $t_x^j$, for every $0 < j
\leq diameter(G)$. Next, we added some more "height" properties. In other words: We added more hash functions.
We noticed that the first improvement: remembering local "high" nodes didn't help much. It took some of the traffic from the "highest" node in the network to some other local "high" nodes, but still
most of the traffic was routed by the "highest" node in the network.
The second improvement: Changing to $k$ hash functions instead of one made a bigger difference. Instead of having just one "highest" node in the network, now there are $k$ of them, and the task of
routing the network messages is divided somewhat equally between those $k$ "highest" node.
We could discard the first improvement (Remembering local "high" nodes), and stay only with the second improvement (Adding more hash functions). That means: Every node will have to remember one
"highest" node $l_j$ for every one of $k$ hash function. Note that $l_j$ is the "highest" node in the network with respect to hash function number $j$. All the messges sent in the network will then
be routed by the $k$ "highest" nodes $l_1,\dots,l_k$. (One "highest" node for every cryptographic hash function).
Using this simpler method we are going to have results comparable to the more complex method described above, and at the same time have much shorter addresses for nodes. We call the $k$ "highest"
nodes in the graph the $k$ landmarks.
In the picture: The nodes $l_j$ for $1 \leq j \leq 5$ are the landmarks used for routing messages. $l_j$ is the "highest" node in the network with respect to cryptographic hash function number
$j$.Each node in the network maintains a shortest path to each of the landmarks.
Note that each message sent in the network will be routed through one of the landmarks, therefore each landmarks is going to route about $1/5$ of all the messages in the network.
In the example: Both $x$ and $y$ keep contact with each of the $5$ landmarks, so messages between $x$ and $y$ could be routed using any of those $5$ landmarks.
We end up with a routing method that is not very practical for large networks, but it gives us some new ideas about routing. This method will serve as a starting point for some of our next ideas.
The Distributed Post Office is an algorithm for creating a hierarchical structure in a decentralized mesh network.
Our results show that the Distributed Post Office works, but not very efficiently: A few nodes (The "highest" node with respect to every hash function) have the responsibility of routing messages for
the whole network.
In our last simplified solution, every node has to remember exactly $k$ landmarks: The "highest" nodes with respect to $k$ different cryptographic hash functions. In this solution we get that every
landmark has to route $1/k$ of the messages sent in the network.
A question to think about: Is there a way to route the messages in the network using the knowledge about the $k$ landmarks, without actually routing the messages through the landmarks themselves?
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Crank-Nicolson method
From Encyclopedia of Mathematics
One of the most popular methods for the numerical integration (cf. Integration, numerical) of diffusion problems, introduced by J. Crank and P. Nicolson [a1] in 1947. They considered an implicit
finite difference scheme to approximate the solution of a non-linear differential system of the type which arises in problems of heat flow.
In order to illustrate the main properties of the Crank–Nicolson method, consider the following initial-boundary value problem for the heat equation
To approximate the solution
and numerical initial condition
Whenever the theoretical solution Taylor series expansions that there exists a positive constant
The stability study ([a2], [a3]) in the discrete
[a3]. Denoting
An important question is to establish a maximum principle for the approximations obtained with the Crank–Nicolson method, similar to the one satisfied by the solutions of the heat equation. Related
topics are monotonicity properties and, in particular, the non-negativity (or non-positivity) of the numerical approximations. Maximum-principle and monotonicity arguments can be used to derive
convergence in the [a2], [a3]) that a choice of the step sizes such that
where [a4]), and also it holds without any restriction between the step sizes but then a value [a3], [a5]).
From a computational point of view, the Crank–Nicolson method involves a tridiagonal linear system to be solved at each time step. This can be carried out efficiently by Gaussian elimination
techniques [a2]. Because of that and its accuracy and stability properties, the Crank–Nicolson method is a competitive algorithm for the numerical solution of one-dimensional problems for the heat
The Crank–Nicolson method can be used for multi-dimensional problems as well. For example, in the integration of an homogeneous Dirichlet problem in a rectangle for the heat equation, the scheme is
still unconditionally stable and second-order accurate. Also, the system to be solved at each time step has a large and sparse matrix, but it does not have a tridiagonal form, so it is usually solved
by iterative methods. The amount of work required to solve such a system is sufficiently large, so other numerical schemes are also taken into account here, such as alternating-direction implicit
methods [a6] or fractional-steps methods [a7]. On the other hand, it should be noted that, for multi-dimensional problems in general domains, the finite-element method is better suited for the
spatial discretization than the finite-difference method is.
The Crank–Nicolson method can be considered for the numerical solution of a wide variety of time-dependent partial differential equations. Consider the abstract Cauchy problem
where Differential equation, partial; Differential operator) which differentiates the unknown function
For the spatial discretization one can use finite differences, finite elements, spectral techniques, etc., and then a system of ordinary differential equations is obtained, which can be written as
The phrase "Crank–Nicolson method" is used to express that the time integration is carried out in a particular way. However, there is no agreement in the literature as to what time integrator is
called the Crank–Nicolson method, and the phrase sometimes means the trapezoidal rule [a8] or the implicit midpoint method [a6]. Let
while when the implicit midpoint method is considered, one obtains
where [a9], while the trapezoidal rule does not satisfy this property.
The Crank–Nicolson method applied to several problems can be found in [a8], [a10], [a11], [a12], [a13] and [a14].
[a1] J. Crank, P. Nicolson, "A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type" Proc. Cambridge Philos. Soc. , 43 (1947) pp.
[a2] K.W. Morton, D.F. Mayers, "Numerical solution of partial differential equations" , Cambridge Univ. Press (1994)
[a3] V. Thomee, "Finite difference methods for linear parabolic equations" P.G. Ciarlet (ed.) J.L. Lions (ed.) , Handbook of Numerical Analysis , 1 , Elsevier (1990) pp. 9–196
[a4] J.F.B.M. Kraaijevanger, "Maximum norm contractivity of discretization schemes for the heat equation" Appl. Numer. Math. , 9 (1992) pp. 475–496
[a5] C. Palencia, "A stability result for sectorial operators in Banach spaces" SIAM J. Numer. Anal. , 30 (1993) pp. 1373–1384
[a6] A.R. Mitchell, D.F. Griffiths, "The finite difference method in partial differential equations" , Wiley (1980)
[a7] N.N. Yanenko, "The method of fractional steps" , Springer (1971)
[a8] J.G. Verwer, J.M. Sanz-Serna, "Convergence of method of lines approximations to partial differential equations" Computing , 33 (1984) pp. 297–313
[a9] J.M. Sanz-Serna, M.P. Calvo, "Numerical Hamiltonian problems" , Chapman&Hall (1994)
[a10] Y. Tourigny, "Optimal IMA J. Numer. Anal. , 11 (1991) pp. 509–523
[a11] G.D. Akrivis, "Finite difference discretization of the Kuramoto–Sivashinsky equation" Numer. Math. , 63 (1992) pp. 1–11
[a12] S.K. Chung, S.N. Ha, "Finite element Galerkin solutions for the Rosenau equation" Appl. Anal. , 54 (1994) pp. 39–56
[a13] I.M. Kuria, P.E. Raad, "An implicit multidomain spectral collocation method for stiff highly nonlinear fluid dynamics problems" Comput. Meth. Appl. Mech. Eng. , 120 (1995) pp. 163–182
[a14] G. Fairweather, J.C. Lopez-Marcos, "Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions" Adv. Comput. Math. , 6 (1996) pp. 243–262
How to Cite This Entry:
Crank-Nicolson method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Crank-Nicolson_method&oldid=13787
This article was adapted from an original article by J.C. Lopez-Marcos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Quadratic Approximations to Pi, or What if Archimedes Had Had Mathematica?
Mark Bollman
Associate Professor
Department of Mathematics and Computer Science
Albion College
Archimedes (c. 287 BCE--c. 212 BCE) used polygons inscribed within and circumscribed about a circle to approximate pi. In this talk, we will extend his work by approximating the areas of circular
sectors. This is done by adjoining parabolic segments to triangular subregions of his inscribed regular polygons. While much of the mathematics would have been familiar to Archimedes, the
calculations involved quickly outstrip the computational power of ancient Greece, and so Mathematica is used to facilitate calculations. The method allows us to derive recurrence relations that can
be used to approximate pi more accurately.
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Balancing Equations
Combining Like Terms
Balancing equations: ax + b = cx + d
Combining like terms: Move variables to one side, constants to the other
Properties of Equality (Addition, Subtraction, Multiplication, Division)
Suitable Grade Level
Grades 7-9
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Ultimate Guide to the AP Calculus AB Exam 2024 (2024)
What’s Covered:
• What Does the AP Calculus AB Exam Cover?
• Sample AP Calculus AB Exam Questions
• AP Calculus AB Exam Score Distribution, Average Score, and Passing Rate
• Tips for Preparing for the Exam
• How Will Your AP Scores Affect Your College Chances?
AP Calculus is a popular course for high schoolers, particularly those planning to pursue a college education. Hundreds of thousands of high schoolers each year study to obtain a passing/high score
on the AP Calculus AB Exam so that they can “test out” of math distribution requirements in the early university years and save time and money.
In 2022, over 250,000 of the 1.2 million students taking AP exams took the AP Calculus AB exam. This places it among the top 4 most popular AP exams. If you are interested in taking the AP Calculus
AB exam, whether you have taken the class or are planning to self-study, read on for a breakdown of the test and advice on how to best prepare for it.
What Does the AP Calculus AB Exam Cover?
The purpose of the AP Calculus AB exam is to test your knowledge of specific “big concepts” that you have learned either through taking the AP Calculus AB course or through self-study.
The “big concepts” of AB Calculus, as defined by College Board, are:
• Limits
• Derivatives
• Integrals and the Fundamental Theorem of Calculus
With regard to limits, students should be comfortable with computing limits, including one-sided limits, limits at infinity, the limit of a sequence, and infinite limits. The exam will also test each
student’s ability to estimate the limit of a function at a point and apply limits to understand the behavior of a function near a point.
With regard to derivatives, students should be comfortable with finding the slope of a tangent line to a graph at a point and using a graph to determine whether a function is increasing or
decreasing. Students should also be able to find concavity and find extreme values. Additionally, the exam will require students to solve problems involving rectilinear motion.
Finally, with regard to integrals, students should be able to use various techniques and methods to approximate an integral. Students should also be familiar with area, volume, and motion
applications of integrals, as well as with the use of the definite integral as an accumulation function.
How Long Is the AP Calculus AB Exam? What Is the Format?
The AP Calculus AB exam is one of the longest AP exams, clocking in at three hours and 15 minutes. It has two sections. The first section contains 45 multiple choice questions, spans one hour and 45
minutes, and accounts for 50% of your total score. The second section consists of six free response questions, spans one hour and 30 minutes, and accounts for the remaining 50% of your score.
Each section is divided into two parts, Part A and Part B. Students are permitted to use calculators during one part and not allowed to use them during the other.
Section Skill Assessed Types of Question Number of Scoring
Questions Weight
Section I, Part A Multiple Choice without Graphing Algebraic, exponential, logarithmic, trigonometric, and general types of functions 30 33.3%
Section I, Part B Multiple Choice with Graphing Calculator Analytical, graphical, tabular, and verbal types of representations 15 16.7%
Section II, Part Free Response with Graphing Calculator Various types of functions and function representations and a roughly equal mix of procedural and 2 16.7%
A conceptual tasks
Section II, Part Free Response without Graphing Questions that incorporate a real-world context or scenario into the question 4 33.3%
B Calculator
Can I Use a Calculator?
While taking the AP Calculus AB exam, you may use a scientific calculator on Part B of the multiple choice section and on Part A of the free response section. Your calculator should be able to plot
the graph of a function within an arbitrary viewing window, find the zeros of functions, numerically calculate the derivative of a function, and numerically calculate the value of a definite
integral. More information and a list of acceptable calculator models can be found in the official Calculator Policy.
Sample AP Calculus AB Exam Questions
Multiple Choice: Section I, Part A
Note: A calculator may not be used on questions on this part of the exam.
1. The graphs of the functions f and g are shown above. The value of is
(A) 1
(B) 2
(C) 3
(D) nonexistent
(A) 6
(B) 2
(C) 1
(D) 0
Multiple Choice: Section I, Part B
Note: A graphing calculator is required for some questions on this part of the exam.
1. The derivative of the function f is given by
At what values of x does f have a relative minimum on the interval 0 < x < 3?
(A) 1.094 and 2.608
(B) 1.798
(C) 2.372
(D) 2.493
2. The second derivative of a function g is given by
For -5 < x < 5, on what open intervals is the graph of g concave up?
(A) -5 < x < -1.016 only
(B) -1.016 < x < 5 only
(C) 0.463 < x < 2.100 only
(D) -5 < x < 0.463 and 2.100 < x < 5
Free Response: Section II, Part A
Note: A graphing calculator is required for problems on this part of the exam.
1. Let R be the region in the first quadrant bounded by the graph of g, and let S be the region in the first quadrant between the graphs of f and g, as shown in the figure above. The region in the
first quadrant bounded by the graph of f and the coordinate axes has area 12.142. The function g is given by and the function f is not explicitly given. The graphs of f and g intersect at the point
(A) Find the area of S.
(B) A solid is generated when S is revolved about the horizontal line y = 5. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
(C) Region R is the base of an art sculpture. At all points in R at a distance x from the y-axis, the height of the sculpture is given by h(x) = 4 – x. Find the volume of the art sculpture.
Free Response: Section II, Part B
Rochelle rode a stationary bicycle. The number of rotations per minute of the wheel of the stationary bicycle at time t minutes during Rochelle’s ride is modeled by a differentiable function r for 0
≤ t ≤ 9 minutes. Values of r(t) for selected values of t are shown in the table above.
(A) Estimate r’(4). Show the computations that lead to your answer. Indicate units of measure.
(B) Is there a time t, for 3 ≤ t ≤ 5 at which r(t) is 106 rotations per minute? Justify your answer.
(C) Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate Using correct units, explain the meaning of in the context of the problem.
(D) Sarah also rode a stationary bicycle. The number of rotations per minute of the wheel of the stationary bicycle at time t minutes during Sarah’s ride is modeled by the function s, defined by for
0 ≤ t ≤ 9 minutes. Find the average number of rotations per minute of the wheel of the stationary bicycle for 0 ≤ t ≤ 9 minutes.
AP Calculus AB Exam Score Distribution, Average Score, and Passing Rate
While many exam distributions fall along a bell curve, with the majority of students receiving a score of 3, the AP Calculus AB exam shows a flatter distribution. Simply put, many students do well
and many students do poorly. In 2022:
• 20.4% of test-takers received a 5
• 16.1% of test-takers received a 4
• 19.1% of test-takers received a 3
• 22.6% of test-takers received a 2
• 21.7% of test-takers received a 1
This means that 55.6% of students who took the exam received a 3 or higher (typically considered passing).
Note: The credit you will receive for AP exam scores varies widely from school to school. For example, prestigious schools (and even prestigious programs at schools) might accept only a 4 or a 5 to
receive course credit. Though a score of 3 is typically considered passing, it’s not always enough. You can use this search tool to see what scores will allow you to receive credit at a specific
college or university.
Tips for Preparing for the Exam
Step 1: Assess Your Skills
Take a practice test to assess your initial knowledge of the material. It’s important to know where you are, so that you know how far you need to go.
Keep in mind that Calculus is an age-old study, so you can use practice tests from before you were even born and you’ll be assessing/learning just the same!
There are a couple of options for taking practice tests:
• Print a practice exam and self-proctor
• Use a diagnostic test offered through a commercial study guide
• Talk to your teacher about proctoring a practice exam after school or over the span of a few lunch periods
The 2012 exam has been openly published by the College Board and might be a good place to start. The College Board also has free response questions from the last few decades published online, though
you should note that these are not complete assessments.
Once you have taken your formative assessment, score it to identify the areas you already understand and those in need of improvement.
Note: When grading the free response portion of the exam, make sure you grade yourself based on the rubric! Act like you are an AP scorer, scrutinizing and nitpicking every portion of your answer.
The little points add up, and your area of improvement could very well be “needing to show my work.”
Step 2: Study the Material
After taking your assessment, you should be able to identify areas that need improvement. These areas could be related to content—not knowing which technique should be used to approximate which type
of integral or not understanding the relationship between concavity and limits on a graph. Alternatively, the areas you struggle with might have more to do with form—like struggling to read graphs or
conceptualize tables.
Identify your areas for improvement, write them down, and focus on one area during each study session. Look over your mistakes and put in the work to understand them. Watch videos online about
specific concepts, read sections in books about them, and talk to your friends and classmates about them.
Then do it again and again and again until the areas you struggle with become areas you excel in.
Some students choose to use commercial study guides when studying. This can be extremely beneficial, depending on your learning style. That said, if you choose to use a commercial study guide, use it
in conjunction with your initial assessment. Study books are divided into sections organized around both big and small concepts. Don’t get stuck reading a guide front to back and don’t waste time on
content that you have already mastered!
Lastly, you might consider looking into the free resources that are available online! For decades, AP teachers have been publicly posting complete study guides, review sheets, and test questions. Use
these for your benefit.
Step 3: Practice Multiple Choice Questions
Once you feel like you’ve mastered the concepts you initially struggled with, put them into action by answering some multiple choice practice questions.
The College Board provides a set of sample questions with scoring explanations. Additionally, the College Board Course Description includes many practice multiple choice questions along with
explanations of their answers. As you go through these, try to keep track of which areas are still tripping you up, and go over those concepts again until you have a better grasp on them. Focus on
understanding what each question is asking and keep a running list of any vocabulary that is still unfamiliar to you.
Step 4: Practice Free Response Questions
When you score your own formative assessment, you will notice that every step you take to arrive at a solution to a free response question must be clearly notated for the exam reader. Even if you use
your calculator to solve an equation, compute a numerical derivative, or find a definite integral, write the equation, derivative, or integral first. Otherwise, you can lose little points—and little
points add up!
The free response portion of the AP Calculus AB exam tests your ability to solve problems using an extended chain of reasoning. In most cases, an answer without supporting work will receive no
credit. This means that, as you answer practice free response questions, you are not just practicing getting the right answer, but getting the right answer in the right way!
You can get a better understanding of the free response section’s scoring by reading scoring commentary from the Development Committee and authentic examples of student responses and their scoring
explanations from previous exam administrations.
Step 5: Take Another Practice Test
Every couple of weeks, when you are feeling confident or when you just want to see your progress, we recommend that you take another complete practice test. This will allow you to see which areas
have improved the most and which areas still need improvement. Taking new practice tests at some interval will serve as a progress report of sorts.
Step 6: Exam Day Specifics
In 2024, the AP Calculus AB Exam will be administered on Monday, May 13 at 8 AM local time. The day before, make sure you have everything you need, and then focus on getting a good night’s sleep.
Studies show that being well-rested is far more likely to lead to improved performance than last-minute cramming!
How Will Your AP Scores Affect Your College Chances?
WhileAP scores themselves don’t play a major role in the college admissions process, having AP classes on your transcript can be a crucial part of your application, especially at highly selective
institutions. College admissions officers want to see that you enjoy challenging yourself intellectually, and that you’re capable of handling college-level coursework, and taking AP classes
demonstrates both of those qualities.
The main benefit of scoring high on AP exams comes once you land at your dream school, as high scores can allow you to “test out” of entry-level requirements, often called GE requirements or
distribution requirements. This will save you time and money.
If you’re starting to think about what schools you should apply to, we recommend that you use CollegeVine’s free chancing engine. This tool will consider your test scores, GPA, extracurriculars, and
more, to calculate your chances of acceptance at various schools and to help you decide where to apply. Itcan also give you suggestions for how to boost your chances of acceptance—for example, by
taking more AP classes in your junior or senior year.
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The great problem in the traditional analysis of curved manifolds was the fact that the only originally available geometry was Euclidian geometry.
This meant that the mathematics of any other geometry, such as that of curved surfaces, had to be constructed out of Euclidean mathematics, and the result was that a Euclidian geometry of a flat
embedding space had to be used in order to describe the geometry of a curved surface within it. This allowed the construction of the well known First and Second Fundamental Forms, to describe the
properties of the curved surface.
Attempts were made by Gauss and Riemann in the direction of constructing a geometry wholly intrinsic to a curved surface, that is, without reference to any embedding space. I haven't been able to
find a work that makes clear the degree to which this was actually successful.
Riemann, for example, used the property that a
vector parallel transported around a closed loop, upon a curved surface, from a point, P, back to the same point, P, changes direction in a manner determined by the curvature of the surface. This
process itself is wholly intrinsic to the surface. Nevertheless, Riemann's curvature tensor derived from this process uses the Christoffel symbols, and I do not think that it has been made clear
whether or not these can be derived entirely from within the surface itself. They are usually derived either from the two fundamental forms, which involve the surface embedding space, or by some
other means, also using the embedding space.
The question, therefore, that I ask, is, could a geometry of a curved surface be discovered from a perspective entirely confined within the surface itself? In what follows I will consider how this
question might be given a clear, unambiguous answer.
PART 1 - THE GEOMETRIC ARGUMENT:
I shall first set up a scenario that will establish what we are to mean by a perspective confined to being wholly within, and thus intrinsic to, a curved surface. To do this I introduce the concept
of so-called 'flatlanders', that is, two dimensional beings confined to the two dimensional arena of a curved two dimensional surface. The light they see is only that which travels within the
surface, and they can see only two dimensional objects that exist within the two dimensional surface. This is what will be meant by the flatlanders 'perspective', also called an 'intrinsic
perspective', to mean that it is confined to their two dimensional surface.
This scenario is somewhat problematic to visualise, as two dimensional beings can see only the one dimensional edges of two dimensional objects within the surface. We, in a three dimensional world,
likewise see only in two dimensions, one dimension less than the world itself. Stereo vision is a contrivance that allows us to create a three dimensional interpretation of what we can see
In order to facilitate visualisation of the discussion, I will introduce a modification of the flatlanders experience by allowing them a very small thickness in a third dimension. This is to be
considered such as will allow at least some of the very thin two dimensional objects to slide over one another in the very thin third dimension, and allow the flatlanders to see the two dimensional
objects from slightly above them, just sufficiently to gain an impression of their two dimensional shapes. This will not, however, allow arbitrary movements of any kind in the third dimension, or
release any objects from their conformity to the curvature of the surface. That is, this contrivance does not allow the possibility of three dimensional measurements of any kind; but it will help us
to visualise the still two dimensional experience of the flatlanders from our three dimensional perspective by making them similar to us in a very restricted way. The flatlanders themselves will not
be aware that this involves a third dimension, and will interpret such objects as being made of a material having a peculiar and unexplained kind of transparency.
We must now begin to determine what the flatlanders can discover, experimentally, about the geometry of the surface they inhabit. The usual analysis of curved surfaces, using an embedding space,
makes it clear that, at any point, P, on a smooth, 2-dimensional, curved surface, the surface in the immediate vicinity of the point, P, approximates to its tangent plane at that point. The smaller
region considered the better is this approximation. That is to say, the smaller the region that is being examined the more accurately will measurements made within it approximate to corresponding
measurements made within the tangent plane around the same point. This means that a surface geometry confined within a small region around the point, P, will approximate to the Euclidian geometry of
the tangent plane, or will contain measurements that depart from it by only very small amounts. If the flatlanders have rulers less accurate than would be required to detect the errors caused by the
curvature of the surface itself, they will be able to discover, experimentally, within such a small region, an apparently strictly Euclidian geometry. If they succeed in doing this, they will find,
experimentally, however, that any investigations carried out within a larger region will not give geometric results that correspond to those within the much smaller region. This will be, for them, a
puzzle they will have to solve, which they will do by investigating whether or not the departure of the larger scale geometry from that of the small region can be formally determined in some
consistent way.
However, in order to experimentally discover a Euclidian geometry within a small region of the curved surface, the flatlanders will have to have discovered the local equivalent of the straight line.
They will have to do this in spite of the fact that the line is curved and not really straight in the strict sense. So we must ask the question: how might they be able to do this? How, indeed, might
the strictly correct straight line have been discovered initially, even in a Euclidian space?
One might answer immediately that it can be defined as the path involving the shortest distance between two points. This, however, appears tautological, since 'distance' is a concept defined in terms
of the straight line, and depends on it, and cannot, therefore, be used to define the straight line before the straight line exists. That is to say, if the straight line doesn't yet exist, the paths
connecting any two points whatsoever are all arbitrary, and an arbitrary path cannot be used to define a non-arbitrary concept of 'distance'. Thus, a comprehensive analysis of our problem leads us to
the necessity to define how a straight line might come to be defined or discovered before any geometrical concepts whatsoever yet exist. This will be the task of the next section.
At first, as mentioned, the flatlanders initially have no geometrical concepts, no straight line, and thus no formal concept of 'distance' or 'direction'. They do, however, have primitive cutting
tools. They have discovered, also, that, by sliding certain two dimensional objects over one another (which they can do in the very thin third dimension), and working around the edges with their
primitive cutting tools, they can make identical copies of two dimensional objects. So one day, in this way, for amusement, a flatlander makes a large number of identical copies of some arbitrarily
shaped two dimensional object. Having nothing else better to do with them, he lays them end to end between two fixed points in his two dimensional environment, along an arbitrary path and also
randomly oriented relative to one another, and finds that, by varying the path, he can connect the two fixed points using different numbers of the arbitrary shapes.
He suddenly gains an intuition that there is some peculiar kind of meaning hidden somewhere in this discovery, which leads him to experiment more extensively with this phenomenon. Eventually he
discovers that there is a unique path, and a particular relative orientation of the shaped objects, which allows him to use a minimum number of the shapes to connect together the two fixed points. He
further makes the remarkable observation that this path is the same as that which the flatlanders would use to walk from the one fixed location to the other, in the particular case in which they walk
directly towards the light coming to them from the location towards which they are walking.
Over a period of time, continuing to amuse himself experimenting with these observations, he discovers that the same path will be identified by the minimum number of shapes necessary to connect the
one fixed point to the other, no matter what might be the actual size of the identical shapes used to carry out the experiment. He also discovers that the number of steps a flatlander needs to take
to walk from one point to the other is also roughly a minimum along that same path. He further eventually discovers that the same results obtain for any other pairs of fixed points in his world.
It is to be noted, here, that the flatlanders have no conception of the geometrical significance of a local geometry compared to a large scale geometry, within their curved two dimensional world, and
our experimenter makes no attempt to limit what we would call the distance apart of the various pairs of fixed points which he uses in his experiments. Nevertheless, he always gets the same results,
whether the locations are what we would describe as close together or far apart.
I shall now show that the flatlander's experiments can be described in our mathematical terms, if the terms are initially given entirely non-mathematical meanings.
An arbitrary shape I shall label with the label ds.
This label has nothing to do with any concept of an element of 'distance'. It is merely the name of one of the arbitrary shapes the flatlander used for his experiments. Now I shall call the result of
his process of counting the shapes Σds. His process of removing shapes, in changing from one path to another, thus causing him to count a reduced total number shapes, I shall refer to as δΣds, to
mean the number of shapes he has removed in this operation. When he reaches the minimum number of shapes connecting the two locations, and finds he cannot remove any more shapes, if he wants the two
locations to remain connected together, I shall refer to the removal of no shapes as δΣds = 0.
We now have the remarkable result that the non-mathematical flatlander's experiments (apart from the process of counting) can be described by a virtually non-mathematical version of Einstein's
equation δΣds = 0, and that this description applies independently of whether the flatlanders two dimensional world is curved or not. This equation is therefore pre-mathematical and pre-geometrical
and is the fundamental basis of all geometry, on which all geometry depends, including even the definition of the straight line.
If I call the unique path discovered by the flatlander the geodesic connecting his two locations, I can conclude that the geodesic is the fundamental and immediate intrinsic geometrical property of
all spaces, curved or flat, and the straight line is only a special case of this, and not prior to, or more immediate than, any other geodesic.
This is established only by the experimental scenario described so that, if its validity is questioned, it can be confirmed by repeating experiments on a curved two-dimensional surface, equivalent to
those carried out by the flatlander.
The consequence of all this is that the geodesics of a curved surface are no less immediate than straight lines, and can be assumed equally with them, and do not have to be derived from them. That
is, we can begin the analysis of a curved manifold by assuming its geodesics in the same way we can begin the analysis of a flat space by assuming its straight lines. This is because the experiments
can show that geodesics are no less immediately discoverable experimentally in a curved surface than are straight lines in a flat surface or space.
I shall introduce, here, for simplicity, a limitation within the flatlander's experiments, which occurs accidentally, and which he does not deliberately choose. That is, the regions within which he
carries out his experiments are never large enough for him to discover more than one geodesic connecting any two points, such as, for example, the two geodesics connecting any two points on a great
circle of a spherical surface.
Having established the geodesic as the pre-mathematical fundamental basis of all geometry, we can proceed to connect the pre-mathematical version of Einstein's geodesic equation with its normal
mathematical version. I shall not continue to follow the flatlanders gradual experimental
development of a strictly mathematical geometry, which would involve a long story, including possibly a good deal of irrelevant collateral diversions. Instead, relying on what we already know,
mathematically, I shall leap ahead to simply examine the results the flatlanders would eventually discover.
The flatlanders have established for us the initial non-mathematical version of the geodesic equation:
δΣds = 0
Now, if their experiment is carried out with a third, fourth, or more set of shapes, reducing the number joining P and Q, as before, each time. We know that a practical experiment would produce the
same minimum number of shapes, n, each time and, furthermore, each final set of shapes will always perfectly coincide with the shapes in the path formed by those of all of the other experiments. This
establishes the unique nature of the geodesic path joining P and Q,
To make this even clearer, let us set up a point Q’, very close to Q, such that the same number of shapes, n, with the same points of contact between the individual shapes, will only just reach Q‘.
It will be found that, as with Q, no other path formed by the shapes, and no other rotational orientation of the individual shapes, will allow P to be joined to Q’. Furthermore, it will be found that
if all the shapes joining P to Q are used to join P to Q’, by moving them from the path PQ to the path PQ’, every one of the shapes will have to be moved onto the new path PQ’, and not only some of
them. That is, the paths nowhere perfectly share any portion of their paths in common, which is a concept not contradicted by the fact that the shapes will overlap. The point being made is that none
of the shapes forming one path will perfectly coincide with any of the shapes forming the other. This will be true of all geodesic paths from P to any other point in the space that is not on the path
between P and Q, or that does not incorporate the whole of the path PQ
We can thus declare two fundamental properties of
geodesic paths, created from any point, P:
1) A geodesic path is a unique path defined by a unique arrangement of the minimum number of shapes necessary to join any point P with any other point Q, and
2) Any geodesic path originating at a point, P, is unique and distinct in its entirety, and in all its parts, from every other geodesic path originating at P.
These are the fundamental properties of geodesics, immediately observed, or discovered by experiment, and are not derived from any more fundamental principle. Since we have as yet no concept of
length or distance, and no concept of any kind of geometry, the two properties of geodesics are prior to all measurement and all geometric truths, apart from themselves, and are not dependent on them
or derived from them. On the contrary, the whole body of geometric truths of all kinds, including the whole of Euclidian geometry, originate in these two properties of geodesics, which apply equally
and without distinction to all geodesics. I therefore repeat, and reemphasise, that this means that a geodesic of a curved manifold is as immediate and fundamental a connection between one point an
another as is the straight line in Euclidian space, and is not generated by, or derived from, anything prior to it.
The two fundamental properties of geodesics can be used to immediately generate useful concepts, but first let us refine the geodesic thus far constructed to make it more suitable to support such
concepts with greater clarity.
Let the shapes represented by ds now be made so small that they are almost invisible, but still sufficient to carry out the procedure involved in creating a geodesic. The appearance of the set of
shapes joining P and Q will now have a character approximating what we recognise as a very thin, or one dimensional, line joining P and Q.
Now let ds be taken to represent a small subset of the set of shapes, instead of only a single shape, and let it contain m shapes, where m << n, n being the total number of shapes along the whole
geodesic PQ, as before. It will be seen immediately that the shape referred to by ds is very much like the shape of the line joining P and Q as a whole, i.e. it has the
appearance of a small line segment. Furthermore, since it coincides with the geodesic path joining P and Q, and forms part of it, ds must itself contain the minimum number of shapes that can form its
own part of the larger path; that is to say ds, as a path segment, is itself a geodesic. Thus, an important feature of the geodesic PQ, and consequently of any geodesic, is that the path by which it
connects P with any other point, O, along it between P and Q also contains a minimum number of shapes, and is also a geodesic connecting P with O. By a similar argument, any intermediate portion, ON,
of a geodesic path, joining points O and N on it, is also a geodesic connecting O to N.
The above argument establishes that any portion, ON, of a geodesic path is also a geodesic joining O and N. Let the segment, ON, be now taken to be so small that it exists within a region of the
surface around the point O which approximates to the surface tangent plane at O. That is to say,
within this region, the surface geometry would approximate to Euclidian geometry and therefore the geodesic segment, ON, as a portion of the large scale surface geodesic, approximates to a straight
line within the surface.
Let us now arrange that a second set of very small, almost-invisible shapes form another line joining P and Q, and let it also be a geodesic, which will therefore lie alongside the first and in
contact with it all along its length. This assumes that both lines are so narrow that together they can effectively be taken to form the same single geodesic path, such that the distinction between
the two lines is, in reality, only a construct of the mind. Now we may mentally transfer ds to the second line and, in addition, we may successively redefine the location of ds so as to make it slide
along the new path. In this case it will at all times be in contact in all its constituent shapes
with corresponding constituent shapes in the first path, since both paths are effectively identical. That is, the shapes will perfectly coincide with one another. Let us call this principle of
keeping the segment ds in contact with the geodesic path as keeping it ’parallel’ to the path. By means of this procedure, the motion of ds along the geodesic can be referred to as a 'parallel
transport' along it, since it remains parallel at all times. This procedure can be used for the dynamic generation of geodesics, as described further on. But first we must consider some useful
consequences that flow from the two fundamental properties of geodesics.
Consequence of the 1st property of geodesics - distance and length
If ds is composed of only a single shape, and is parallel transported along different paths from P to Q, the number of shapes it will have to pass in order to get to Q is arbitrary, and does not give
rise to any special concept. Since, however, the geodesic path identifies a unique, minimum number of shapes ds must pass in order to get to Q, this can easily be given a convenient name, as the
‘distance’ from P to Q, which always implies that it refers to the geodesic path only. Since ds itself, if it is composed of m shapes, is also a geodesic segment, we can, by similar reasoning, call m
the ‘length’ of ds, to refer to the number of shapes contained within ds when it is a geodesic
Consequence of the 2nd property of geodesics - direction
Since the geodesic path from P to Q is unique in its entirety, and in all its parts, a parallel transport of ds along this path from P must get to Q. Any geodesic transport from P of a geodesic
segment ds, where the transport is not parallel to the path from P to Q, cannot get to Q, as no other geodesic path can connect P to Q. This can be conveniently expressed by saying that the geodesic
path from P to Q is the ‘direction’ of Q from P, which also introduces and defines the concept of 'direction'. In order, therefore, to get to Q, from P, along a geodesic path, it can be said simply
that ds must travel from P in the direction of Q. It is to be remembered, here, that the concept of ‘parallel’ is restricted to what we have defined it to be, since any broader notion of parallel, as
we usually think of it, does not yet exist. Since the concept of 'direction' also signifies that travel along a geodesic, from P, in any other direction, will not get to Q, this also establishes the
significance of the concept of relatively different directions existing at the point P.
Let us now return to the definition of 'parallel transport', which involved transporting a segment, ds, along a second geodesic that was effectively identical to the first, and thus 'parallel' with
it. If ds is very small, which is to say, if its 'length', which we have now defined, is very small, ds will occupy a local region of the surface, within which the surface geometry is effectively
Euclidian, so that ds will effectively be a straight line segment. It will therefore be parallel transported along local, straight line geodesic segments of the global geodesic, since every segment
of a geodesic, however small, is itself necessarily also a geodesic, as argued before. The transport of ds locally, as a straight line segment, will therefore correspond to the notion of parallel as
it exists within a Euclidian geometry. This inevitably means that, if a straight line segment is parallel transported locally, in a Euclidian sense, within a
curved surface, and this process is continued beyond the initial local region, it will enter another local region that overlaps with it, within which it will again continue to be parallel transported
in a Euclidian sense, even though the local regions involved are not locally Euclidian when taken together. In other words, if a small, straight line geodesic segment is parallel transported in a
Euclidian sense, it will progressively and dynamically generate a global, non-Euclidian geodesic across a curved surface. Our flatlanders could, in fact, experimentally use this method to dynamically
generate the geodesics of the surface, without having to use only the geodesic equation as the basis of doing so. This justifies regarding curved geodesics, in general, as auto-parallel, in the same
way that a straight line is auto-parallel within a flat surface.
Up to this point the fundamental geodesic equation has not yet acquired its usual mathematical form. Since, however, we are now able to identify ds as a segment having the significance of a
measurement of ‘length’, or distance, and our geodesic equation, which originated merely as a counting of a number of shapes, has now acquired its usual mathematical significance. It now appears, in
terms of the summation of segments of the measurement of distance, to give an overall measurement of distance, as a specification, and definition, of the ‘shortest’ path joining P and Q. We remember,
of course, that our experiment has excluded cases such as that in which we can reach Q via geodesic paths in opposite directions along the great circle of a sphere.
With ds being used to signify line segments of infinitesimal length, the equation δΣds = 0 can be written in the usual form
This mathematical version of the equation is thus the fundamental mathematical equation of all geometry, and is prior to all further geometrical equations, concepts, and relationships of every kind.
It applies equally to flat or curved spaces and is equally fundamental to them all.
The whole of geometry likewise depends on the two properties of geodesics, and could not exist without them. They must be seen to have a fundamentally immediate, experimental basis, for the reason
that they could originally have had no other. Thus, any geometrical conclusions, in any manifold, that are derived from an assumption of the foregoing properties of geodesics will, if they are true,
be confirmed by any experimental test of them.
We have now reached the point where we must consider the question whether or not the flatlanders will be able to discover and measure the curvature of their world. If they cannot, it will have to be
concluded that a completely intrinsic analysis of a curved manifold is not possible.
We have already said enough to recognise that the flatlanders will eventually be able to discover the locally Euclidian geometry of any small region of their two dimensional world. They will thus be
able to draw arcs of circles at different radii, and will have discovered the equation X = rθ, where X is the arc length, drawn through the angle θ, in radians, at a radial distance r. We may suppose
that the flatlanders will also have established, by experiment, that, if the radius used to draw the arc becomes too large, this equation will cease to be true, and that the arc length X will become
progressively less than the equation would predict, as the radius is increased.
Let us suppose that eventually one of the flatlanders will try to discover whether or not this occurs in any regular and predictable manner. Let us say that he rewrites the equation in the form X =
krθ, with k being some fractional value, as a function of the arc radius, to make the equation correspond to experimental results.
We can suppose that the flatlanders, who now possess Euclidian geometry, will have Sine and Cosine tables, etc. I will now make the supposition that, for some reason I don't know, some flatlander
also created a table of values of (Sinθ)/θ, and that our flatlander, experimenting with circular arcs, discovers that the values of k in his equation correspond to those of the (Sinθ)/θ table.
He now feels certain that he is on the verge of discovering some property of his world which causes the arcs of circles to be modified in some specific way. He eventually notices that his discovery
relates to the relative values of the circular arc to the partial chord shown below, so that the actual arc within his world is being drawn as if the radius is really rSinθ
This suggests that the surface radius used to draw the arc is not the 'real' radius that is actually drawing the arc, as shown below.
Our flatlander puzzles over this for a considerable time, in an attempt to discover where the 'real' radius of the arc might actually be, and where the fulcrum of rotation of the real radius, shown
as P', might be. He cannot conceive any possible location for it, within his visible world, that makes any sense in terms of providing a meaningful explanation as to why the value of k has to have
the form Sinθ/θ. This is made worse by the fact that the value of θ does not appear to be related to the angle through which the radius of the arc is drawn on the surface. That is, if the surface
radius of the arc, r, draws an arc through an angle θ, k will adopt values of another angle Sinφ/φ, which will depend on r, but will be unaffected by, and unrelated to, the value of the angle θ. Our
flatlander thus also cannot make sense of the angle φ, or discover where it might be located.
But suddenly, one day, his failures lead him to a flash of intuition, that the 'real' radius that is drawing the arc may, in fact, not be within his world at all but, rather, somehow outside of it
altogether. He immediately senses the significance of this consideration and, knowing that he inhabits a two dimensional world, is led to suppose that, in fact, he may have to conclude that he is
inhabiting a three dimensional world, from within which he can see in only two dimensions, with the third being inaccessible to him. He then gradually realises that, in this three dimensional
reality, his world must be actually curved, and that the fulcrum, P', of the real radius that draws the arc is located somewhere
outside his world, in the manner shown below. He thus also realises that the angle φ must be related to a radius R, completely separate from the surface radius r, which is actually curved, with R
being the radius of its curvature.
Our flatlander then concludes that the complete explanation of his experimental observations is accounted for by a curved two dimensional surface segment, in the manner shown below, although he has
as much difficulty visualising this in three dimensions as we have in attempting to visualise what a four dimensional object might look like.
He now knows why the value of k has to have the form Sinφ/φ and is, moreover, able to determine the curvature of the surface, since both r and θ are measured on the surface, and φ can be calculated
from his tables. He can thus obtain the value of R from the equation r = Rφ
This result, of course, depends on a supposition that the value of R does not change along the length of r. We must therefore ask whether or not the flatlanders can determine whether or not the value
of R is constant?
If they examine the total differential of X, in which both r and R are allowed to change, they will obtain
dX = (δX/δr)dr + (δX/δR)dR
If the radius R does not change as the length of r is increased, then
dX = (δX/δr)dr, since dR is zero
= θ(Sinφ/φ)dr
If the total value, dX, begins to depart from being directly proportional to dr, we know that the curvature is beginning to change with increasing values of r.
If R is not constant, the flatlanders can try dividing the length r into two segments of value r/2 each, and separately examine the curvatures of the two halves, as shown below (where I still call
each half r, for convenience)
In this way they should be able to determine the change in the radius of curvature R along the geodesic r
If the value of the angle θ is small, the flatlanders will be able to calculate the radius of curvature of their world in roughly one direction, and, in a similar manner, can go on to build up a
profile of the curvature of their world in all other directions.
Since the flatlanders have been able to obtain the value of the curvature of their two dimensional world, they have direct theoretical access to its three dimensional Euclidian embedding space. They
will thus be able to determine the first and second fundamental forms applicable to their surface. They can also construct a geodesic curvilinear coordinate system on their surface, which is the
natural coordinate system of a curved space, just as it is in a Euclidian space. We would not construct a non-geodesic coordinate system in a Euclidian space, unless for a very special, extraordinary
The flatlanders do not require a Riemannian analysis of their world, and do not need an Affine connection. The geodesic is the immediate, fundamental connection in any space, and does not have to be
derived or
determined by any equation.
It is not my intention to proceed with a further analysis of the geodesic geometry of curved spaces, since I have achieved the purpose of establishing the fundamental nature of the geodesic, the
fundamental equality of all geodesics, and the fact that the geodesics of a curved space do not have to be derived. There are any number of mathematicians who can easily develop the consequences of
this fundamental starting point. Instead, it will be much more fruitful to examine (in part II) the way in which the fundamental geodesic principle implies that Nature, and the laws of Physics must
relate to, and must make use of, geodesics in any space, flat or curved.
© Alen, March 2007; update Dec 2010.
June 2019 - re-engineered to display properly on Chrome and Firefox.
Material on this page may be reproduced
for personal use only.
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The Binary Adder:
Andy Long
Spring, 2020 - Year of Covid-19
Having constructed a
working example of a finite state machine
(FSM), from Gersting's 7th edition (p. 730, Example 29), I decided to create a more useful one -- a binary adder (p. 732). It works!
Subject to these rules:
1. Your two binary numbers should start off the same length -- pad with zeros if necessary. Call this length L.
2. Now pad your two binary numbers with three extra 0s at the end; this lets the binary-to-decimal conversion execute.
3. numbers are entered from ones place (left to right).
4. In Settings, choose "simulation start" as 1, your "simulation length" as L+2 -- two more than the length of your initial input number vectors. (I wish that the Settings issues could be set
without having to explicitly change it each time -- maybe it can, but I don't know how.)
Be attentive to order -- start with 1s place, 2s place, 4s, place, etc., and your output answer will be read in the same order.
To understand why we need three additional inputs of 0s:
1. For the useless first piece of output -- so n -> n+1
2. For the possibility of adding two binary numbers and ending up with an additional place we need to force out: 111 + 111 = 0 1 1 1
3. For the delay in computing the decimal number: it reads the preceding output to compute the decimal value.
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hdu-5738 Eureka (combination count + polar order)
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What are the factors of 12x^3+12x^2+3x? | Socratic
What are the factors of #12x^3+12x^2+3x#?
1 Answer
Your problem is $12 {x}^{3} + 12 {x}^{2} + 3 x$ and you are trying to find its factors. Try factoring out 3x: $3 x \left(4 {x}^{2} + 4 x + 1\right)$ does the trick to decrease the size of the numbers
and the powers. Next, you should look to see if the trinomial that is inside the parentheses can be factored further. $3 x \left(2 x + 1\right) \left(2 x + 1\right)$ breaks the quadratic polynomial
down into two linear factors, which is another goal of factoring. Since the 2x + 1 repeats as a factor, we usually write it with an exponent: $3 x {\left(2 x + 1\right)}^{2}$.
Sometimes, factoring is a way to solve an equation like yours if it was set = 0. Factoring allows you to use the Zero Product Property to find those solutions. Set each factor = 0 and solve: $3 x =
0$ so x = 0 or $\left(2 x + 1\right) = 0$ so 2x = -1 and then x = $- \frac{1}{2}$.
Other times, the factoring can help us to graph the function y = $12 {x}^{3} + 12 {x}^{2} + 3 x$ by again helping to find the zeros or x-intercepts. They would be (0,0) and $\left(- \frac{1}{2} , 0\
right)$. That can be helpful information to start to graph this function!
Impact of this question
15053 views around the world
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Concept information
alpha-ocimene mass concentration
• Mass concentration means mass per unit volume and is used in the construction mass_concentration_of_X_in_Y, where X is a material constituent of Y. A chemical species denoted by X may be
described by a single term such as 'nitrogen' or a phrase such as 'nox_expressed_as_nitrogen'. The chemical formula for alpha-ocimene is C10H16. Alpha-ocimene is a member of the group of
monoterpenoids. The IUPAC name for alpha-ocimene is (3E)-3,7-dimethylocta-1,3,7-triene.
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Area of a Triangle SAS Calculator
Last updated:
Area of a Triangle SAS Calculator
Our area of a triangle SAS calculator can determine a triangle's area from any of its two sides and the corresponding inscribed angle. Note that the abbreviation SAS stands for Side-Angle-Side.
In this article, we shall briefly discuss the following:
• How to find the area of a triangle given 2 sides and an angle.
• What is a triangle's SAS area formula.
• Some FAQs.
SAS area formula of a triangle
You might be familiar with the formula of a triangle's area given its base and height:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
A triangle whose two sides and the inscribed angle are known.
In the triangle above, we know only its two sides, $a$ and $b$, and the angle $\gamma$ between them. If we consider the side $b$ as the triangle's base, using trigonometry, we obtain its height as:
$h = a \cdot \sin(\gamma)$
Therefore, the SAS area formula for a triangle is given by:
$\text{Area} = \frac{1}{2} \times a × b ×\sin(\gamma)$
We can use this formula to calculate the triangle area with 2 sides and an angle.
How do you find a triangle's area given two sides and an angle?
To find the area of a triangle given its two sides a and b, and the inscribed angle γ, follow these simple steps:
1. Multiply the lengths of the two sides together to get a × b.
2. Multiply this value with the sine of the angle γ, to get a × b × sin(γ).
3. Divide this value by half to get the triangle area as A = (a × b × sin(γ))/2.
4. Verify using our area of a triangle SAS calculator.
Other relevant calculators
We have put together a collection of similar calculators that you might find useful:
How to use this area of a triangle SAS calculator
Our calculator for the area of a triangle given 2 sides and an angle is simple and easy to use:
1. Enter the two sides you know.
2. Provide the value of the inscribed angle. The calculator will automatically find the area.
And just like that, you can find the triangle area with 2 sides and an angle. Note that this area of a triangle SAS calculator can also work backward! Play around with it providing different inputs
in any order, and enjoy the results!
How do you find the missing side of a triangle from its two sides and angle?
The formula to calculate the missing side c of a triangle from its two sides a and b and the inscribed angle γ is:
c = √(a^2 + b^2 - 2abcos(γ))
What is the triangle area with two sides 3 and 4 which subtend 90°?
6 units. To find this answer yourself, follow these steps:
1. Multiply the lengths of the two sides together to get 3 × 4 = 12.
2. Multiply this value with the sine of the angle 90°, to get 12 × sin(90°) = 12 × 1 = 12.
3. Divide this value by half to get the triangle area as A = 12/2 = 6.
4. Verify using our area of a triangle SAS calculator.
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Simulatable Commitments and Efficient Concurrent
Simulatable Commitments and Efficient Concurrent Zero-Knowledge
Authors: Daniele Micciancio and Erez Petrank
Advances in Cryptology - Eurocrypt 2003. Warsaw, Poland, May 2004. LNCS 2656, Springer, pp. 614-629.
[BibTeX] [Postscript] [PDF]
Abstract: We show how to efficiently transform any public coin honest verifier zero knowledge proof system into a proof system that is concurrent zero-knowledge with respect to any (possibly
cheating) verifier via black box simulation. By efficient we mean that our transformation incurs only an additive overhead, both in terms of the number of rounds and the computational and
communication complexity of each round, independently of the complexity of the original protocol. Moreover, the transformation preserves (up to negligible additive terms) the soundness and
completeness error probabilities. The new proof system is proved secure based on the Decisional Diffie-Hellman (DDH) assumption, in the standard model of computation, i.e., no random oracles, shared
random strings, or public key infrastructure is assumed. In addition to the introduction of a practical protocol, this construction provides yet another example of ideas in plausibility results that
turn into ideas in the construction of practical protocols.
We prove our main result by developing a mechanism for simulatable commitments that may be of independent interest. In particular, it allows a weaker result that is interesting as well. We present an
efficient transformation of any honest verifier public-coin computational zero-knowledge proof into a (public coin) computational zero-knowledge proof secure against any verifier. The overhead of
this second transformation is minimal: we only increase the number of rounds by 3, and increase the computational cost by 2 public key operations for each round of the original protocol. The cost of
the more general transformation leading to concurrent zero knowledge is also close to optimal (for black box simulation), requiring only omega(log n) additional rounds (where n is a security
parameter and omega(log n) can be any superlogarithmic function of n (e.g., log(n)log^*(n)), and omega(log n) additional public key operations for each round of the original protocol.
Preliminary version: [IACR ePrint 2002-090] or [ECCC TR 02-045]
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Identify quaternion group as a subgroup of the general linear group of dimension 2 over complex numbers - Solutions to Linear Algebra Done Right
Identify quaternion group as a subgroup of the general linear group of dimension 2 over complex numbers
Let $i$ and $j$ be the generators of $Q_8 = \langle i, j \ |\ i^4 = j^4 = 1, i^2 = j^2, ji = i^3 j \rangle$. Prove that the map $\varphi : Q_8 \rightarrow GL_2(\mathbb{C})$ defined on generators by $
\varphi(i) = A = \left[ {\sqrt{-1} \atop 0} {0 \atop -\sqrt{-1}} \right]$ and $\varphi(j) = B = \left[ {0 \atop 1} {-1 \atop 0} \right]$ extends to a homomorphism. Prove that $\varphi$ is injective.
Solution: First we prove a lemma.
Lemma. Let $S = \{ i,j \}$ and $G$ be a group. If $\overline{\varphi} : S \rightarrow G$ such that $\overline{\varphi}(i)^4 = \overline{\varphi}(j)^4 = 1$, $\overline{\varphi}(i)^2 = \overline{\
varphi}(j)^2$, and $\overline{\varphi}(j) \overline{\varphi}(i) = \overline{\varphi}(i)^3 \overline{\varphi}(j)$, then $\overline{\varphi}$ extends to a homomorphism $\varphi : Q_8 \rightarrow G$.
Moreover, if $\overline{\varphi}(j)$ is not a power of $\overline{\varphi}(i)$ and the powers $\overline{\varphi}(i)^k$ are distinct for $0 \leq k < 4$, then $\varphi$ is injective.
Proof: We have that $\overline{\varphi}(i)$ and $\overline{\varphi}(j)$ satisfy the relations of $Q_8$. Now every element of $Q_8$ can be written uniquely in the form $i^aj^b$ for some $0 \leq a < 4$
and $0 \leq b < 2$; define $\varphi(i^aj^b) = \overline{\varphi}(i)^a \overline{\varphi}(j)^b$. It is straightforward to see that $\varphi$ is in fact a homomorphism. To see injectivity, let $0 \leq
a,c < 4$ and $0 \leq b,d < 2$ and suppose $\varphi(i^aj^b) = \varphi(i^cj^d)$ but that $i^aj^b \neq i^cj^d$. Then either $a \neq c$ or $b \neq d$. If $b \neq d$, then without loss of generality we
have $b = 1$ and $d = 0$. Thus $\overline{\varphi}(j) = \overline{\varphi}(i)^{c-a}$, a contradiction. If $b = d$ and $a \neq c$, then we have $\overline{\varphi}(i)^a = \overline{\varphi}(i)^c$, a
contradiction. Thus $\varphi$ is injective. $\blacksquare$
By the lemma, it suffices to show that (1) $A^4 = B^4 = I$, (2) $A^2 = B^2$, (3) $BA = A^3B$, (4) $B \neq A^k$ for all integers $k$, and (5) $A^k$ are distinct for $k \in \{ 0,1,2,3 \}$. All but (4)
are established by a simple calculation. To see (4), note that A is diagonal; so all powers of $A$ are diagonal. But $B$ is not diagonal. Thus the mapping $\varphi : Q_8 \rightarrow GL_2(\mathbb{C})$
is an injective homomorphism.
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A Nice Little Routine
How I'm starting class this year
I added a new piece to my class routine this year. I still start each class with a five-question Do Now. It’s fast, I aim for three minutes flat. Then I do a few other quick things.1
Next comes the routine. Every day I have students grab a mini whiteboard and a marker. I ask them a few questions, have students write their answers and hold them up for me to see, spend time on
topics that students are struggling with, make a note of specific students who I should find time to support, and often do a mini-lesson on something. This serves a bunch of different purposes,
depending on the day. It might seem simple, but keeping the routine consistent helps us to be efficient and helps students know what to expect, and I can use this routine for lots of different goals.
Here are a few of those goals:
Reinforcing a question students struggle with on the Do Now. Before when a lot of students got a Do Now question wrong I did something that didn't work very well. "I'm noticing that a lot of you said
that 8 * 1/2 is 16. Remember, multiplying by one half is the same as dividing by 2." Students would dutifully correct their answer to 4... and then make the same mistake the next day. Instead, I note
the mistake and add a clarification and a few quick problems to our whiteboard work. It's way more likely to stick with a little bit of targeted practice.
Reviewing content from previous days. Review is so important. But what's really boring is waiting until the end of a unit and then reviewing everything all at once. Instead, I mix in review problems
in small chunks each day. If students have forgotten something, I realize it early and can give a reminder and add some extra practice. The small chunks spaced over time are a lot more effective than
reviewing everything all at once, and I spend more time on topics that are a high priority.
Checking for understanding from the day before. Exit tickets are a nice way of getting some quick formative assessment data. If lots of students struggle on an exit ticket, I know I need to reteach
that. But exit tickets can also give false positives. If we spend a class working on a skill, it's pretty likely that students can do it by remembering what they were doing a few minutes before.
Asking them at the start of the next class is harder, and gives me a better sense of if that knowledge will stick. This has the added benefit of reminding students of what we learned the day before
so we can build on it in the day's lesson.
Checking for understanding of prerequisite knowledge. Lots of skills involve some prerequisite knowledge that students might not have seen recently. Students need to know how to find perfect squares
to find the area of a circle. They need to know the names of basic shapes to describe cross-sections of 3-d figures. This is a great time to check for that prerequisite knowledge and consolidate if
students are shaky on it.
Previewing topics to come. Building off that last piece, there's lots of knowledge that requires a bit of practice for students to be confident. For instance, knowing the meaning of the four
inequality symbols is a 6th grade skill, but by the time we get to inequalities in February they have often forgotten the difference between < and >. If I try to do a quick prerequisite knowledge
check the day we start inequalities it will be a mess. Instead, I will start including inequality questions two weeks before that unit. It's a mini-lesson, then a question or two a day on whiteboards
and a few mixed in to our Do Nows. The spaced practice helps that knowledge stick way better than if I try to do it all the day of.
Reinforcing key foundational knowledge. Finally, there's some stuff that's just good to keep practicing. Multiplication facts. One-step equations. I don't want to take up a ton of class time for it,
but dropping in a question here and there is helpful, especially with mini whiteboards where I can adjust and respond to what students know and don’t know.2
I don't do all of these in one lesson. That’s important. The time I spend on this varies. Sometimes it's 2 minutes, sometimes it's 10 minutes. The time stretches and shrinks depending on how much I
have planned for the rest of class. It’s also flexible; since we’re working on mini whiteboards I can spend more time on a topic if students are struggling, or keep moving if they know something
well. Some lessons I focus on checking for understanding from the day before and consolidating prerequisite knowledge, prioritizing that day’s lesson. Other days, maybe when we’re doing some practice
and a quiz, I have more time to preview topics to come and do mixed review. It all depends. I map out a rough progression of what I want to focus on at the beginning of each week, but I often adjust
that as the week goes on.
One way this has changed how I look at teaching is that I see lessons less as discrete chunks to teach and then move on. Each lesson blends into the lesson before and the lesson after.
Instead of this:
It's this:
One important note: not every topic is the same size. Some require more lead up, some have more connections to what I'm doing the next day, and some stand more on their own. There's no magic formula
where every day looks the same.
Anyway, I'm loving this change. It's hard to imagine going back. I'm doing all the things I always mean to spend time on but never get to. I'm spending time revisiting the previous day's lesson and
previewing what's to come. I’m checking for understanding and then responding when I need to. The connections between topics are more clear. I’m checking for prerequisite knowledge. Quick chunks of
regular review mean I spend less time reteaching old stuff. All that takes time, but it means the time I spend teaching new content goes more smoothly and sticks for students.
If you’re curious about my full beginning-of-class routine we do an optional number puzzle for students who finish the Do Now with time to spare, a quick thinking routine that’s designed to get
students talking and participating early on in the lesson, and I give any announcements or reminders I need to.
Astute readers will remember a bunch of these ideas from my post a few months ago about the Direct Instruction program. This routine was inspired in part by learning about DI. I’ve adapted a lot of
things to my context so this isn’t really what DI does, but I want to acknowledge that influence.
Mastery is in the pursuit, not the arrival.
Expand full comment
I'm trying to figure out the second image and how it represents the change to a "traditional" lesson progression in which I assume each lesson takes the same amount of time and there is little
spiraling (hence the lack of overlap).
I'm intrigued by the practice you describe and I might see how I can adapt an equivalent practice for my Pre-Calculus students. Thanks for sharing!
Expand full comment
3 more comments...
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Vasicek model | Bis 2 Information (2024)
The formula used to determine the regulatory capital is commonly referred to as the Vasicek model. The purpose of this model is to determine the expected loss (EL) and unexpected loss (UL) for a
counterparty, as explained in the previous section. The first step in this model is to determine the expected loss. This is the average credit loss. There is a 50% change of realizing this loss or
less. The expected loss is determined using three main ingredients:
PD: Probability of default, the average probability of default over a full economic cycle;
EAD: Exposure at default, the amount of money owed at the moment a counterparty goes into default;
LGD: Percentage of the EAD lost during a default.
The expected loss (EL) is equal to the PD times the LGD times the EAD:
EL = PD X LGD X EAD<
The expected loss is half the work of the model. The EL determines (roughly) the amount of provisions which should be taken (the essence of any provision is to save money for losses you expect in the
future). The second half of the work is to determine the Unexpected Loss (UL). The UL is the additional loss in atypical circumstances, for which tier capital should be retained. The Vasicek model
estimates the UL by determining the PD in a downturn situation. The model assumes that the EAD and LGD are not affected by dire circumstances. Both parameters are considered constant for a company.
The model calculates the loss during a downturn situation (for instance an exceptionally bad economy) by multiplying the downturn PD times the LGD times the EAD. The UL is calculated by subtracting
the expected loss from the loss during a downturn situation. In formula’s this equates to:
UL = (PDdownturn X LGD X EAD) – (PD X LGD X EAD),
which is equal to:
UL = (PDdownturn – PD) X LGD X EAD<
The PD in a downturn situation is determined using the average (through the cycle) PD. At this point Vasicek uses two different models. First it uses the Merton model. This model states that a
counterparty defaults because it cannot meet its obligations at a fixed assessment horizon, because the value of its assets is lower than its due amount. Basically it states that the value of assets
serve to pay off debt. The value of a company’s assets vary through time. If the asset value drops below the total debt, the company is considered in default. This logic allows credit risk to be
modelled as a distribution of asset values with a certain cut-off point (called a default threshold), beyond which the company is in default. The area under the normal distribution of the asset value
below the debt level of a company therefore represents the PD. The following figure shows a normal distribution of the assets values. The current asset value of this example is €1,000,000, the
standard deviation is €200,000 and the total debt is €700,000. The probability of the asset value falling below €700,000 (the total debt level and therefore the default threshold) is equal to the
area red area in the graph. As a company is considered in default if the asset value drops below the total debt, this probability is equal to the PD. In our Example the red area (PD) is 6.68%.
The logic used by Merton (shown in the graph above) can also be reversed. In Vasicek a PD (for instance calculated with a scorecard) is given as input. Instead of taking the default threshold (debt
value) and inferring the PD as Merton does, Vasicek takes the PD and infers the default threshold. Vasicek does this using a standard normal distribution. This is a distribution with an average of
zero and a standard deviation of one. This way the model measures how many standard deviations the current asset value is higher than the current debt level. In other words it measures the distance
to default. The graph below shows that a PD of 6.68% means that the company is currently 1.5 standard deviations of its asset value away from default. By using the standard normal distribution the
actual asset value, standard deviation and debt level becomes irrelevant. It is only necessary to know a PD and the distance to default can be determined.
Now that the PD has been transformed to a distance to default the second step of the model comes into play. In this step Vasicek uses the Gordy model. The distance to default is a through the cycle
distance, because the PD used is through the cycle. In other words it is an average distance to default in an average situation. This distance to default (-1.5 in our example) will have to be
transformed into a distance to default during an economic downturn. To do this a single factor model is used. It is assumed that the asset value of a company is correlated to a single factor. In
other words, if the factor goes up the asset value goes up, if the factor goes down the asset value goes down. This factor is often referred to as the economy. This is done because it is intuitively
logical that the asset value of a company is correlated to the economy. We will follow this tradition; however the factor is merely conceptual. It is assumed that there is a single common factor
(whatever it may be) to which the asset value of all companies show some correlation. The common risk factor (the economy) is also assumed to be a standard normal distribution.
To recap we have a standard normal distribution representing the possible asset values, a default threshold inferred using the PD (-1.5 in our example), a standard normal distribution representing
the economy to which the asset value is correlated and a correlation between the economy and the asset value. Using the correlation it is possible to determine the asset value distribution given a
certain level of the economy. If the economy degrades the expected asset value will also decrease shifting the asset value distribution to the left. Furthermore the standard deviation will also
decrease. In other words an asset value distribution given a certain level of the economy can be calculated using the correlation between the asset value and the economy. The following graphs give an
example of how the asset value distribution can change as the economy level decreases.
As the asset value distribution shifts the distance to default also shifts (decreases). The graphs below show the effect on the PD. The increase in the red area (and decrease in the distance to
default) represents the increase in the PD due to adverse economic conditions.
The degree in which the asset value distribution is deformed depends on the level of the economy which is assumed. The level of the economy is measured as the number of standard deviations the
economy is from the average economy. For instance the economic level with a probability of 99.9% of occurring or better has a distance of 3.09 standard deviations from the average economy.
The new distance to default can be calculated by taking the average of the distance of the level of the economy (used to determine the downturn PD) and the distance to default, weighted by the
correlation. In formula’s this equates to:
DistanceToDefaultDownturn = (1-r)^-0.5 X DistanceToDefault+ (r/(1-r))^0.5 X DistanceFromEconomy.
In our example the PD was 6.68% and the distance to default was -1.5. Now assume a counterparty has a 9% correlation to the economy. Secondly determine that the economic downturn level is the 99.9%
worst possible economic level (used in BIS II). At this level the distance between the downturn level and the average economy is 3.09. In our equation the new distance to default (given the 99.9%
worst economy) is:
-0.6 = (1-9%)^-0.5 X -1.5 + (9%/(1-9%))^0.5 X 3.09
In other words the -1.5 distance to default decreases to a distance to default of -0.6. The new PD associated with a distance to default of -0.6 is 27.4%.
Now the Vasicek model has finished its job. In short it has accomplished the following tasks:
• It has determined the loss during normal circumstances (Expected Loss) using EL = PD X LGD X EAD. Where the PD is an average PD.
• It has determined the downturn PD using DistanceToDefaultDownturn = (1-r)^-0.5 X DistanceToDefault+ (r/(1-r))^0.5 X DistanceFromEconomy.
• It has determined the Unexpected Loss using UL = (PDdownturn – PD) X LGD X EAD<
Author: Muller, J.J.<
‹ Credit Loss Distribution up Probability of Default (PD) ›
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Connected Mathematics Project 4 Updates
We are proud to announce the pending release of Connected Mathematics4!
Read below to learn about the history of Connected Mathematics Project, updates made in CMP4, and how you can get Connected Mathematics4 at your school!
Get to Know Connected Mathematics 4
We invite you to take a look at the Connected Mathematics 4 Program Overiew digital booklet today.
In it you will see program features of CMP4 and see sample pages from the curriculum.
STEM Problem Format
Rather than using conventional numbering and lettering (e.g., A1, A2, B1, B2, B3, etc.) that resemble worksheets the CMP STEM Problem Format includes three parts: Initial Challenge, What If…? and Now
What Do You Know?
The format promotes learning in an environment that more closely resembles the work of STEM professionals.
Mathematical Reflection
The Mathematical Reflection for each unit consists of one overarching question that guides the development of the big mathematical idea(s) in the unit.
Arc of Learning
The CMP Arc of Learning is a teacher professional resource that makes explicit the intentions of the curriculum designers about how students engage in the learning of mathematics over time. It
characterizes deeply grounded and connected learning from informal to more sophisticated understandings that is different than the prevalent focus on student learning as passively watching and
imitating isolated skills.
Attending to Individual Learning Needs
The CMP Framework for Attending to the Individual Learning Needs characterizes five essential classroom elements for creating an environment in which teachers can support students’ development of
mathematical identities.
About Connected Mathematics Project
Since 1990, the goal of CMP is to help students and teachers develop mathematical knowledge, understanding, and skill along with an awareness of and appreciation for the rich connections among
mathematical strands and between mathematics and other disciplines. A single principle has guided the development of CMP:
All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representation, materials, tools,
techniques, and intellectual methods of the discipline of mathematics, including the ability to define and solve problems with reason, insight, inventiveness, and technical proficiency.
Quotes From a Field-Testing Teachers
“Today in class, one of my students said, "I really like this curriculum...This has been my strongest math year yet."”
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Rebar Cost Estimate
The Rebar Cost Estimate calculator computes the total cost (TC) of rebar based on the length being purchased(TL), length of the individual pieces being bought (uL) and unit price (uC) of one piece
of rebar.
INSTRUCTIONS: Choose units and enter the following:
• (TL) Total Length of Rebar being purchased
• (PO) Percent Overage to account for waste
• (uL) Length of pieces of rebar (stick) being purchased (e.g. 20 ft, 40 ft or 60 ft)
• (uC) Cost of one piece (stick) of rebar. (Default is current price of 20 ft size 4 stick)
Cost of Rebar (TC): The calculator returns the following:
• total cost of the rebar in U.S. Dollars
• number of rebar sticks, and
• total length accounting for overage in feet.
Values with units can be automatically converted to compatible units (e.g. dollars to pesos) via the pull-down menu.
The Math / Science
The following are price estimates of rebar products based on the current market price of steel and a reasonable markup from raw steel to retail rebar product.
You have computed that you need 840 feet of 1/2" (size #4) grade 60 rebar using the Rebar Calc at vCalc. You've shopped around and found that you can buy rebar for $13.1 USD per 20 foot piece at a
rebar factory store. You can now use this calculator to compute total cost of rebar.
You enter:
• TL = 840 feet
• uL = 20 feet. This is the length of the individual pieces of rebar you can buy.
• uC = $13.1 USD. This is a unit cost of $13.1 USD for #4 20' grade 60 pieces.
Cost of Rebar (TC) is returned at current market price for that length in U.S. dollars.
General Rebar Information
Rebar is short for reinforcing bar. Rebar is a roughly circular steel bar with ribs used to provide added tensile strength to concrete structures. Rebar is put in place before concrete is poured.
When the concrete has hardened, the concrete around the rebar ribs keep the rebar in place. Rebar and concrete expand similarly with temperature variations. This all has the net result of
substantially added tensile strength when rebar is part of the concrete form. Carbon steel is the most commonly used material for rebar, which may also be coated with zinc or epoxy resin.
Rebar is laid out in grids, crisscrossed patterns of rebar, tied at the intersections where runs of rebar touch. The grids have spacing between the rebar rows, and they are placed within the concrete
form by a specified inset from the edge of the concrete. Multiple parallel grids, at uniform space intervals, are referred to as rebar mats.
Rebar Terms
• Rebar - reinforcing steel bar.
• Stick - one length of rebar. In the U.S., the most common lengths of rebar sticks are 20', 40' and 60'.
• Lapping - when two sticks of rebar are overlapped and bound together.
• Lapping Factor - the multiple of a rebar diameter used to specify appropriate rebar lapping length.
• Mat - a crisscross grid of rebar sticks. There may be more than one mat with space in between mats.
• Size - the indicator of the diameter of rebar sticks. Note: guage is not a correct term for rebar.
Rebar Size
In the United States, rebar sizes are in increments of 1/8^th inches in diameter. Therefore, size 4 is 4/8^th of an inch, which is 1/2", and size 8 is a full inch in diameter. Based on this and the
density of steel used in rebar, the Rebar Size Table contains reasonably accurate specifications of rebar linear weight and lateral (face) area based on rebar size.
Rebar Lapping
The most common lengths of pre-cut rebar in the United States are 20`, 40' and 60'. These are known as rebar sticks. When the dimensions of a slab, wall or other form exceed the length of a single
stick of rebar, it is required to overlap and tie rebar pieces to create the added length. This process is called lapping, and the length of the overlapping rebar is the rebar lapping length. The
length of the lap is specified by a "Lapping Factor (LF)" which is often 40 or 60 times the diameter of the rebar. Engineering specifications of a lapping factor should always be applied.
Rebar Tools
A class of rebar tools, both powered and manual, have been developed to aid construction workers in working with rebar. These include the following:
• Rebar Cutters are used to cleanly and safely cut sections of rebar.
• Rebar Benders are used to bend rebar sticks precisely to fit into concrete forms.
• Rebar Tiers are used to tie rebar grid intersections and for rebar lapping.
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Right Triangle Trigonometry Word Problems Angle of Elevation
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Right Triangle Trigonometry Word Problems
Angle of Elevation- Angle created by a horizontal surface and the hypotenuse. Ex. The angle of elevation of the top of a pole measures 48° from a point on the ground 18 ft away from its base. Find
the height of the flagpole.
Example 1 A 6 ft man casts a shadow. If the angle of elevation to the sun is 33°, how long is the shadow?
Ex 2: Length of a Building’s Shadow An 850 -foot tall building casts a shadow of length L when the sun is θ° above the horizon. Write L as a function of θ. Complete the table below. θ L 10° 20° 30°
40° 50°
Ex 3: Drive-in Movie You are 50 feet from the screen at a drive-in movie. Your eye is on a horizontal line with the bottom of the screen and the angle of elevation to the top of the screen is 58°.
How tall is the screen?
Ex. 4 Depth of a Submarine The sonar of a navy cruiser detects a nuclear submarine that is 4000 feet from the cruiser. The angle between the water level and the submarine is 32°. How deep is the
Ex 5: Airplane Ascent When an airplane leaves the runway, its angle of climb is 18° and its speed is 275 feet per second. Find the plane’s altitude after 1 minute.
Ex 6: Skyscraper You are a block away from a skyscraper that is 780 ft tall. Your friend is between the skyscraper and yourself. The angle of elevation from your position to the top of the skyscraper
is 42°. The angle of elevation from your friend’s position to the top of the skyscraper is 71°. To the nearest foot, how far are you from your friend?
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Digital Math Resources
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Definition--Circle Concepts--Radius of a Circle
Radius of a Circle
The radius of a circle is a line segment from the center to any point on the circle.
The radius is a fundamental concept in geometry, representing the distance from the center of the circle to any point on its circumference. It is widely used in fields such as engineering, design,
and manufacturing, where precise measurements of circular objects are required. The radius is half the diameter, highlighting its relationship with other circle properties. In mathematics education,
understanding the radius is essential for students as it provides a basis for exploring more complex topics like circumference and area, and helps in developing spatial reasoning and problem-solving
For a complete collection of terms related to Circles click on this link: Circles Collection.
A circle is the locus of points equidistant from a given point, called the center. The "locus of points" are what you see as the circle.
The center is the one point not on the circular form. The distance from the center to any point on the circle is constant. This is part of the "locus of points" definition. The segment from the
center to the circle is called the radius.
Since there are an infinite number of points that define the circle, then there are also an infinite number of radii (the plural of radius). This is shown below:
By contrast there is only one center point to the circle. A line segment that crosses the center and crosses the center is called a diameter. Think of a diameter as two collinear radii.
There is also the case of a line segment that intersects the circle at two points but does not cross the center. This is called a chord.
The diameter of a circle is a special type of chord. Do you see how the diameter also meets the definition of a chord?
When a line crosses a circle at two points, then it is a secant.
When a line intersects a circle at just one point, then it is called a tangent line.
A central angle is formed by two radii. The vertex of the angle is at the center of the circle.
An inscribed angle is formed by two chords that intersect at the circle. Do you see where the vertex of the circle is located?
There is an important relationship between inscribed angles and central angles that share two points on the circle. In this case the inscribed angle is half the measurement of the central angle.
Using what we know about inscribed angles and diameters, then any inscribed angle of a diameter is a right angle. To see this, think of the diameter as a central angle of 180°.
Finally, a line tangent to a circle is perpendicular to the radius of the circle at that point.
Common Core Standards CCSS.MATH.CONTENT.HSG.C.A.2, CCSS.MATH.CONTENT.HSG.C.A.1, CCSS.MATH.CONTENT.HSG.C.A.3, CCSS.MATH.CONTENT.HSG.C.A.4, CCSS.MATH.CONTENT.4.MD.C.5.A, CCSS.MATH.CONTENT.7.G.B.4
Grade Range 4 - 8
Curriculum Nodes • Circles
• Definition of a Circle
Copyright Year 2014
Keywords defnitions, geometry, circle
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1 Introduction2 Site and location3 Instrumentation3.1 Sensors4 Methodology4.1 Database4.2 Solar irradiance models4.3 Correlation models between
4.4 Statistical indicators5 Results5.1 Atmospheric variables5.2 Irradiance variables5.2.1 Seasonal variation of hourly values5.2.2 Seasonal variation of daily values5.3 Irradiance empirical models5.3.1 Monthly-averaged daily horizontal global (MADHG) and diffuse (MADHD) irradiation models5.3.2 Daily-averaged horizontal global (DAHG) and diffuse (DAHD) irradiation models5.3.3 Hourly horizontal global (HHG) and diffuse (HHD) irradiation models5.3.4 Hourly diffuse correlation (HDC) irradiance models6 Discussions6.1 Atmospheric variables6.2 Irradiance variables6.3 Irradiance models7 ConclusionData availability statementAuthor contributionsFundingConflict of interestPublisher’s noteReferencesGlossary
Solar radiation serves as the primary energy source driving surface-atmosphere interactions, influencing a wide array of physical, chemical, and biological processes within Earth’s atmospheric and
oceanic systems (Munner, 2004a; Arya, 2005). Understanding the components of solar radiation, namely global ( E G ) , diffuse ( E D F ) , and direct ( E D R ) radiation at the surface, is
indispensable for various applications. These include identifying regions suitable for solar power generation (Janjai et al., 2009; Della-Ceca et al., 2019; Barragán-Escandón et al., 2022), assessing
energy consumption in buildings (Rodríguez-Hidalgo et al., 2012; Albarracin, 2017), supporting ecophysiological studies (Woodward and Sheeh, 1983; Monteith and Unsworth, 1990) estimating crop
evapotranspiration (Supit and Van Kappel, 1998), and facilitating urban planning (Redweik et al., 2013).
Since 1990, global organizations have emphasized the environmental risks posed by fossil fuels, promoting renewable energy. However, global energy demand is projected to rise by over 50% by 2030
(Sayigh, 2020). In recent decades, there has been a notable surge in investment and development of alternative technologies aimed at producing clean energy from renewable sources. These technologies
offer lower environmental impacts compared to traditional ones, garnering significant attention worldwide (Ellabban et al., 2014). The “Renewable Energy Policy Network for the 21st Century Report”
highlights a growing interest in renewable energies, with a global push towards achieving net-zero emissions by 2050 (REN21, 2023). Currently, solar and wind power are among the most promising and
feasible renewable resources. The implementation of photovoltaic systems ( P V ) has particularly seen a rapid expansion, with growth rates reaching 40% in recent years (Jager Waldau, 2019).
In this context, observations of solar radiation are commonly focused on global radiation. However, many applications require data on irradiation on sloping surfaces. Therefore, it is often necessary
to divide global radiation into its beam and diffuse components to derive the specific data needed from the available global radiation measurements. A number of diffuse fraction models are available
for averaging times ranging from 1 month to 1 hour or even less. Different diffuse fraction models generally require varying input data. The model proposed by Erbs et al. (1982) only needs the hourly
clearness index. The models of Maxwell (1987) and Skartveit and Olseth (1987) require both the hourly clearness index and solar elevation and the model of Skartveit et al. (1998) which added
hour-to-hour variability index and regional surface albedo.
Furthermore, the model by Perez et al. (1992) further includes ambient dew-point temperature and an hour-to-hour variability index. From a large multiclimatic database, they derived a computationally
efficient model using a four-dimensional look-up table consisting of a 6 × 6 × 5 × 7 matrix of numerical constants. Moreover, Ridley et al. (2010) developed a multiple predictor model that uses
hourly clearness index, daily clearness index, solar altitude, apparent solar time and a measure of persistence of global radiation level as predictors. This model performs marginally better than
currently used models for locations in the Northern Hemisphere and substantially better for Southern Hemisphere locations.
In general, South America lacks sufficient radiometric stations equipped to measure various components of solar radiation, typically only measuring global radiation ( E G ) and sunshine hours ( S ) .
However, comprehensive data on diffuse radiation ( E D F ) incident on tilted surfaces is essential for various applications, including hydrological, architectural (thermal comfort), urban planning,
and micro-meteorological studies. Moreover, such data supports the design of solar energy systems, aiding in the optimization of solar collector configurations and the determination of optimal tilt
angles and panel orientations to maximize energy conversion efficiency. This problem is particularly serious in Andean regions, due to the lack of high quality radiometric sensors like BSRN stations
Driemel et al. (2018). Actually, there are studies which compare global irradiance data with numerical irradiance models for the south-central region of Chile (Álvarez et al., 2011), Argentina
(Podestá et al., 2004; Ceballos et al., 2022) and spectral irradiance measurements in the high Andes of Peru (Hastenrath, 1997).
Moreover, compared to 1980, projections indicate a decrease in ultraviolet UV-B irradiance by 5%–20% at mid to high latitudes by the end of the 21st century, while a slight increase of 2%–3% is
expected in low latitudes. The tropics (25 ° N–25 ° S) exhibit low seasonal variability, with naturally low ozone levels around 250 Dobson Units contributing to high UV radiation. Notably, there have
been no significant changes in total ozone column over this period in this region (McKenzie et al., 2011). Additionally, factors contributing to heightened UV radiation levels include the altitude of
mountainous regions, clear skies, and low aerosol concentrations, notably observed in the Andean mountains between 12 ° S and 23 ° S, which boast over 100 peaks exceeding 6000 m above sea level
(Blumthaler et al., 1997; Cordero et al., 2014). This phenomenon has been investigated in various locations, including the Chilean Andes (Piazena, 1996), Bolivian Andes (Zaratti et al., 2003), and
Argentinian Andes (Cede et al., 2002; Luccini et al., 2006).
Particularly, UV solar irradiance measurements were conducted by Suárez Salas et al. (2017) in the Peruvian central Andes, at the HYGO, between January 2003 and December 2006, using a GUV-511
multi-channel filter radiometer. Data analysis revealed daily, monthly, and annual cycles of UV solar irradiance at four wavelengths (305, 320, 340, and 380 nm). Clear sky and all sky conditions were
distinguished, with February showing peak values. The highest hourly mean UV Index at noon reached 18.8 under clear sky conditions and 15.5 under all sky conditions, with outlier peaks close to 28.
Cloud cover increased spectral irradiance at 340 nm by up to 20%, indicating exceptionally high levels of UV radiation in the tropical central Andes. However, to date, there are no studies aimed at
studying solar radiation in the central Peruvian Andes with high quality data.
On the other hand, in Brazil, several studies have investigated solar radiation patterns for different sites using high quality radiometric sensors like BSRN stations (Driemel et al., 2018). For
instance, Oliveira et al. (2002b) examines seasonal variations in E G and E D R diurnal patterns using surface data from São Paulo. Codato et al. (2008) conducted a comparative analysis of solar
radiation fields between São Paulo and Botucatu. Ferreira et al. (2012) focused on São Paulo’s radiation balance, highlighting atmospheric factors’ role in E G seasonal variability. Furthermore,
Pereira et al. (2006) and Martins et al. (2007) utilized SOLAS network data to validate a satellite-derived model, shedding light on Brazil’s solar energy potential. In a related study in Rio de
Janeiro City, Marques Filho et al. (2016) obtained daily maximum values of E G during summer and minimums in winter, showing higher values compared to a similar analysis in São Paulo City. This
difference, despite both cities being within the same latitude range, can be attributed to the influence of cloudiness and marine aerosols in Rio de Janeiro City, which affect the components and
balance of solar radiation at the surface. Additionally, De Souza et al. (2016) presented the seasonal variation of E G in Alagoas city, while Gomes et al. (2022) analyzed the seasonal variation of
both E G and E D F in San Salvador city.
The present study aims to comprehensively analyze solar irradiance patterns in the western Mantaro Valley, utilizing data from BSRN sensors at the Huancayo Geophysical Observatory (HYGO), spanning
2017 to 2022, the research investigates seasonal variations and trends in surface solar irradiance components. Specifically, the study explores diurnal and seasonal fluctuations of diffuse ( E D F )
, direct ( E D R ) , and global ( E G ) irradiance. In addition, the research evaluates various irradiation models to establish correlations between solar irradiance parameters, aiming to predict E G
and E D F accurately across different time scales. The description of the site is shown in Section 2, sensors and instrumentation are described in Section 3, database and methodologies used in the
study are presented in Section 4. Section 5 summarized the results of the research, including observational characterization and empirical models of global, direct and diffuse solar irradiances,
Section 6 discuss the main contributions of the research and finally, Section 7 concludes the paper.
The measurements in this study were conducted at the Huancayo Geophysical Observatory (HYGO), situated at coordinates 12.04 ° S latitude and 75.32 ° W longitude, with an elevation of 3350 m above sea
level (m.a.s.l). HYGO is operated by the Geophysical Institute of Peru (IGP). It is located within the Mantaro River Basin (MRB) within the central Peruvian Andes, covering a vast drainage area
spanning 34,550 square kilometers (Figure 1A). The MRB encompasses various regions of the central Andes, including Junin, Ayacucho, Huancavelica, and Pasco, with elevations ranging from 500 to 5,300
m.a.s.l and a mean altitude of approximately 3870 m.a.s.l. (Figure 1B). Furthermore, HYGO is situated within the non-irrigated agricultural expanse of the Mantaro Valley (MV), positioned at a
distance of 7 km from the Mantaro River and 12 km from the city of Huancayo. The observatory is nestled between the Western Andes and the Huaytapallana cordillera to the east, (Figure 1A).
(A) The location of the HYGO (12.05 ° S, 75.32 ° W, 3313 m asl) of the Geophysical Institute of Peru, inside the domain of the Mantaro valley and the Mantaro basin. (B) Topography around the Mantaro
valley, with resolution of 0.5 km and higher altitudes close to 5,200 m asl. Longitudes, latitudes and altitudes are indicated.
The climatological data spanning 48 years (1965–2013) from HYGO reveals a predominant unimodal seasonal pattern in precipitation. This pattern distinctly delineates a dry season extending from April
to August, followed by a rainy season spanning from September to March. Notably, the latter part of August may witness intense rainfall events, with precipitation steadily increasing until it reaches
its zenith during the austral summer, specifically in January through March, as substantiated by prior studies (Silva et al., 2008; Espinoza-Villar et al., 2009). Following this period, a noteworthy
decline in precipitation is observed in April. Consequently, approximately 83% of the annual accumulated rainfall occurs during the rainy season, as documented in earlier research (Silva et al.,
Moreover, Flores-Rojas et al. (2019a) shows that the components of the energy budget exhibit both seasonal and daily variations, with the partitioning of net irradiance ( Q N ) into turbulent
sensible ( Q H ) , turbulent latent ( Q E ) heat fluxes, and soil heat flux ( Q G ) being influenced by the dynamic interplay between the soil and the atmosphere’s heat transport capabilities, as
well as the physical attributes of the surface. At solar noon, the mean monthly Q N attains its peak in November, registering at 660 W m − 2 , while reaching its nadir in July at 450 W m − 2 .
During the fall and winter months, the mean monthly Q H , peaking around 300 W m − 2 at noon, surpasses the mean monthly Q E , which reaches its maximum of approximately 100 W m − 2 at the same time.
This discrepancy can be attributed to the limited soil moisture availability during this period. Conversely, in the spring and summer months, the situation reverses, with the mean monthly Q E ,
reaching its zenith of close to 300 W m − 2 at noon, outpacing the mean monthly Q H , which reaches a maximum of approximately 220 W m − 2 at noon. This shift is a consequence of elevated
precipitation levels during this period, which enhances soil moisture availability in the Mantaro valley. Furthermore, the replenishment of nocturnal Q N loss is notably more effective through Q G
than turbulent fluxes. This distinction becomes more pronounced in the winter months, where Q E is almost negligible during the night, and the atmosphere exhibits stratification due to low surface
temperatures and diminished soil moisture levels (Flores-Rojas et al., 2019a).
In this study, we evaluate the primary atmospheric variables on the HYGO by analyzing measurements collected from a set of automatic sensors belonging to the surface weather station as illustrated in
Figure 2D and the solar irradiance data by analyzing measurements from the BSRN station showed in Figures 2A, B. Both systems are operated by the Geophysical Institute of Peru (IGP). The details of
these data sources are provided below:
1. Values of air temperature ( ° C), relative humidity (%), precipitation (mm m i n − 1 ) and water mixing ratio (g kg − 1 ) with 1 minute resolution carried out at the surface weather station at 2 m
height located on the HYGO between May 2017 and December 2022 (68 months and 2,100 days approximately).
2. Values of global, diffuse and direct solar irradiance (W m − 2 ) with 1 minute resolution carried out at the BSRN station at 6 m height located on the HYGO between May 2017 and December 2022 (68
months and 2,100 days approximately).
In meteorological measurement systems, the recording of data often encounters challenges. In the case of radiometers, measurement errors typically arise from several factors, including sensor
misalignment or tilting, interference from nearby objects causing shadowing and reflections, and the accumulation of dust and moisture on the sensor dome (Bacher et al., 2013; Vuilleumier et al.,
2014). Additionally, the behavior of solar radiation components is influenced by various factors such as cloudiness patterns, aerosol optical depth, surface albedo, cloud type, among others (Perez et
al., 1990; Gueymard, 2005), making the development of universal models a complex task. For this work, we conducted distinct data quality checks to identify and rectify missing data, data points that
clearly deviated from physical constraints, and extreme data outliers.
In cases where data were confirmed as ‘erroneous’ or ‘missing,’ the corresponding data fields were filled with a specific key sequence of numbers unique to the reporting location, serving as a clear
indicator of the problematic observations. Any data associated with flagged ‘bad’ or ‘missing’ data was subsequently excluded from the dataset. Furthermore, a secondary filter was applied to
eliminate hours featuring observations that violated fundamental physical principles or conservation laws. This included the removal of hours where reported values exhibited anomalies such as
negative radiation values, diffuse fractions exceeding 1, beam radiation surpassing extraterrestrial beam radiation levels, and instances where the dew point temperature exceeded the dry bulb
temperature (Reindl et al., 1990).
Additionally, we conducted a rigorous visual quality control process on the dataset, aiming to identify and rectify inconsistencies and spikes that could be attributed to electronic malfunctions
within the data acquisition system. Furthermore, we applied the methodology introduced by Younes et al. (2005) and Journee and Cedric (2010) to analyze the time series data. Data points were
considered valid if they met specific criteria, including a solar elevation angle ( α ) greater than 2°, as well as satisfying the following ratios: E G E T < 1.2 , E D F E T < 0.8 , E D R E T < 1.0
Here, E T represents the extraterrestrial solar radiation incident on a horizontal surface. This value was estimated analytically according to Iqbal (1983), using a value for the solar constant set
at 1367 W m − 2 (Frohlich and Lean, 1998). These conditions (Equation 1) are suitable for stations that measure direct and diffuse components, independently Younes et al. (2005). A more lenient
criterion, E G / E T < 1.2, led to the removal of 8.15% of the dataset, with no observable dependence on atmospheric turbidity or seasonality.
However, the most stringent criterion, E D F / E T < 0.8, resulted in the exclusion of 9.58% of the dataset, predominantly during the summer months (characterized by increased rainfall) (Marques
Filho et al., 2016). Lastly, the criterion E D R / E T < 1.0 led to the exclusion of 7.89% of the dataset. Moreover, approximately 3.0% of the dataset was excluded based on the criterion of α > 2 ° .
This primarily corresponded to measurements taken during sunrise and sunset periods characterized by elevated atmospheric turbidity, notably during the winter and spring seasons.
Following rigorous data quality control procedures, we identified and selected data equivalent to 1850 days (44 400 h) to effectively capture the seasonal variations in solar irradiance across the
region. This dataset encompasses approximately 88% of the entire observational period. Subsequently, comprehensive statistical analyses were conducted on various measurements, including those of E G
, E D F and E D R and other relevant meteorological variables.
The present study employs two key indexes to analyze atmospheric radiometric properties and develop empirical and correlation models: the clearness index ( K T ) , calculated as the ratio of E G to E
T , and the diffuse fraction ( K D ) , expressed as the ratio of E D F to E G , as defined in Liu and Jordan (1960). Under clear sky conditions, a substantial portion of extraterrestrial radiation
reaches the Earth’s surface, resulting in E G tending to E T , K T approaching 1 and K D close to 0. Conversely, during cloudy conditions, E G approaching to E D F , leading to K T nearing 0 and K D
approaching 1. The principal advantage of utilizing K T and K D lies in their ability to eliminate astronomical dependencies while preserving essential information concerning the influence of clouds,
moisture levels, and aerosol concentrations on radiometric properties. This approach results in a more universally applicable description of these properties, enabling their use in regions with
similar climatic characteristics.
To develop the regression models of this section, the filtered dataset from May 2017 to December 2022 (1850 days or 44 400 h) as described in Section 4.1 was divided into two segments. Sixty percent
(60%) of the total filtered dataset chosen randomly (1,110 days or 26 640 h), were used to construct the regression models, while the remaining forty percent (40%) of the total filtered dataset
chosen randomly (740 days or 17 760 h) were reserved for rigorous statistical tests to evaluate model performance and robustness.
In early modeling efforts worldwide, the core focus was on linking daily horizontal global irradiation to bright sunshine duration. This phase involved creating regression equations using
monthly-averaged data as the basis, providing foundational insights for solar energy prediction and utilization studies. The original Angstrom (1924) regression equation established a connection
between monthly-averaged daily irradiation and irradiation on clear days. However, this approach presents challenges in precisely defining what constitutes a clear day. In response to this issue,
several researchers (Garg and Garg, 1985; Turton, 1987), have devised alternative relationships, exemplified by the subsequent equation: E G ̄E T ̄= a + b n N where E G ̄and E T ̄are the monthly-averaged
daily terrestrial and extraterrestrial global irradiance on a horizontal surface, n is the average daily hours of bright sunshine and N is the day length, obtained by: ω s = cos − 1 − tan L A T tan
D E C , N = 2 ω s / 15 where ω s is expressed in degrees, LAT is the latitude of the HYGO (12 ° S) and DEC is the solar declination angle. The ratio n/N is known as fractional possible sunshine. The
extraterrestrial irradiation, E may be calculated analytically according to Munner (2004a). Moreover, the initial development of a regression model between monthly-averaged values of diffuse and
global irradiances can be attributed to Liu and Jordan (1960), presented as E D ̄/ E G ̄in relation to K ̄T = E G ̄/ E T ̄, where E D ̄represents the monthly-average daily diffuse radiation incident on a
horizontal surface. This pioneering approach has garnered international attention, with numerous researchers confirming its applicability worldwide. However, it has been noted that observed data
often deviates from predictions made using the Liu–Jordan model, raising questions regarding its universality and generality. The equation to estimate the monthly-averaged diffuse irradiance can be
parameterized by: E D ̄E G ̄= a − b K ̄T
Several values for the coefficients a and b have been proposed worldwide. For instance, Hawas and Muneer (1984) obtained a = 1.35 and b = 1.61 for the India subcontinent and Page (1977) obtained a =
1.0 and b = 1.13 for eight United Kingdom and nine worldwide locations.
On the other hand, Cowley (1978) derived a series of linear regression equations linking daily global irradiance to the duration of bright sunshine at ten stations across Great Britain. These
equations offer a valuable means to estimate daily incident radiation, a more granular measure, as opposed to monthly-averaged values. Cowley’s equation is given as: E G d E T d = d a + b n N + 1 − d
a ′ where E G d and E T d are the daily global and extraterrestrial irradiances, respectively. The ratio n/N is the daily sunshine. The parameter d satisfies this condition d = 0 if n = 0, otherwise
d = 1 if n > 0, and a’ is equal to the average ratio of E G d / E T d for overcasts days. On the other hand, in the seminal study by Liu and Jordan (1960), they initially formulated a regression
equation connecting the diffuse fraction of daily global irradiance ( K D = E D d / E G d ), referred to as the diffuse ratio with the ratio of daily global to extraterrestrial irradiation ( K T )
referred to as clearness index. Several contributions have used a third degree polynomial to estimate the diffuse ratio (Le Baron and Dirmhirn, 1983; Smietana et al., 1984; Saluja et al., 1988;
Munner, 2004b), which can be expressed as: E D d / E G d = a + b K T − c K T 2 + d K T 3 , K T > = 0.2 E D d / E G d = 0.98 , K T < 0.2
In general, the recommended coefficients of a global model for the diffuse ratio are: a = 0.962, b = 0.779, c = 4.375 and d = 2.716 according to Munner (2004a). Furthermore, the use of hourly
irradiation data significantly enhances the precision of modeling solar energy processes. However, considering that daily irradiation measurements are more widely available across various sites
compared to their hourly counterparts, it is imperative to explore the correlation between these two temporal scales. Many meteorological stations routinely report their data in the form of
monthly-averaged values of daily global irradiance, making it a crucial point of investigation.
The pioneering work in this field is attributed to Whillier (1956), whose research laid the foundation. Building upon Whillier’s findings, Liu and Jordan (1960) extended and refined the framework.
They developed a series of regression curves, which take into account the impact of temporal displacement from solar noon and day length on the hourly-to-daily global irradiation ratio ( r G ).
Collares-Pereira and Rabl (1979) subsequently reaffirmed the accuracy of Liu and Jordan’s plots. Employing a least-squares fitting approach, they further refined the models, yielding: r G = π 24 a ′
+ b ′ cos ω cos ω − cos ω s sin ω s − ω s cos ω s where ω represents the solar hour angle, 15 ° for each hour displaced from the true solar noon, ω s is given by the Equation 3. The
coefficients a’ and b’ are given by: a ′ = r + s sin ω s − 1.047 b ′ = p + q sin ω s − 1.047
The coefficients are given by: r = 0.409, s = 0.5016, p = 0.6609 q = 0.4767. Another authors have used similar equations with different coefficients (Saluja and Robertson, 1983; Hawas and Muneer,
1984). In this work, we find the best coefficients r, s, p, and q, by fitting the Equation 7 to the observed global irradiance data. Moreover, to calculate long-term hourly diffuse irradiation
averages, one can derive them from monthly-average daily diffuse irradiation values, provided that the hourly-to-daily diffuse irradiation ratio, denoted as r D is known. For the present work, we
used a generalized equation based on the formula introduced by Liu and Jordan (1960) but with coefficients similar to Equation 7, as: r D = π 24 c ′ + d ′ cos ω cos ω − cos ω s sin ω s − ω s
cos ω s where ω ir the same that the Equation 7 and the coefficients c’ and d’ are: c ′ = t + u sin ω s − 1.047 d ′ = v + w sin ω s − 1.047
Similar to the previous case of the Equation 7, we find the best coefficients t, u, v, and w, by fitting the Equation 9 to the observed diffuse irradiance data.
For the present study, we employed a diverse range of logistic functions to establish correlations between the hourly diffuse fraction ( K D h ) and the hourly clearness index ( K T h ) . These
encompassed sigmoid logistic functions (Boland and Ridley, 2008; Marques Filho et al., 2016), a fourth-degree polynomial function (Oliveira et al., 2002a), and a three-degree polynomial function
(Jacovides et al., 2006). As mentioned previously, to develop the regression models in this section, the filtered dataset from May 2017 to December 2022 (1850 days or 44,400 h), as described in
Section 4.1, was divided into two segments. Sixty percent (60%) of the total filtered dataset, chosen randomly (1,110 days or 26,640 h), was used to construct the regression models. The remaining
forty percent (40%), chosen randomly (740 days or 17,760 h), was reserved for rigorous statistical tests to evaluate model performance and robustness.
To evaluate the performance of different solar irradiance models, we used the traditional statistical indicators: R-squared ( R 2 ) which indicates the percentage of the variance in the dependent
variable that the independent variables explain collectively, R 2 measures the strength of the relationship between your model and the dependent variable on a convenient 0% – 100% scale. In addition,
the mean squared error (MSE) assesses the average squared difference between the observed and predicted values. When a model exhibits zero error, its MSE becomes zero as well. As the model’s error
increases, so does the MSE value.
On the other hand, in the case of correlation models involving K D h and K T h , we employed the Akaike’s Information Criterion (AIC) (Marques Filho et al., 2016). This criterion serves as a means to
assess the performance of various correlation models, and it is defined as follows: A I C = ln ∑ y i − μ i 2 n + 2 k n
In this equation, y i represents the observed data, μ i 335 signifies the modeled data, n stands for the sample size, and k denotes the number of model parameters (Motulski and Christopoulos, 2003).
The second term on the right side of this equation accounts for a penalty based on the number of parameters within the model. However, to gauge a model’s performance in comparison to others, it
becomes essential to compute the difference of AIC: Δ i A I C = A I C i − A I C min
In this context, A I C i represents the AIC value corresponding to model i, and A I C min denotes the minimum A I C value calculated among various models. This transformation is designed to ensure
that the top-performing model registers a Δ i A I C of 0, while all other models exhibit positive values (Burnham and Anderson, 2004).
In this work, the main climate features of the west region of the Mantaro valley are assessed based on measurements carried at the HYGO conventional meteorological station and the short-term
meteorological variables from an automatic station of the Geophysical Institute of Peru (IGP) at the HYGO, as described below:
1. Three times a day values of air temperature (T), relative humidity (RH), accumulate precipitation and cloudiness carried out at the surface weather station located in the HYGO from January 1981 to
December 2020 (40-year climate normal) (Giráldez et al., 2020).
2. One-minute average values of T, RH and precipitation measured from May 2017 to December 2022 at IGP platform in the HYGO. More information about the instruments installed in the automatic station
of HYGO can be found in a recent contribution (Flores-Rojas et al., 2021).
According to Köppen-Geiger (Peel et al., 2007) and considering climatological observations of atmospheric variables carried out on the HYGO (Figure 3) is classified as Cwb. In consequence, the MV can
be considered as temperate with dry winter (June-August) and warm summer (December-February). The criteria that define this climate zone are: mean temperature of the hottest month higher than 10 ° C,
for the HYGO is close to 14 ° C, the mean temperature of the coldest month is between 0 ° C and 18 ° C, for the HYGO is close to 10 ° C (Figure 3A). Also, the number of months where the mean
temperature is above 10 is greater than 4. Moreover, the annual amplitude of the monthly average air temperature ( T ( 1981 − 2020 ) ) presents minimums in February with values around 12 ° C and
maximums in July with values around 20 ° C (Figure 3A).
Seasonal variation of (A) mean, maximum and minimum of air temperature ( ° C), (B) mean monthly accumulated rainfall (mm m o n t h − 1 ), (C) mean, maximum, minimum, percentile 10% and percentile 90%
of daily accumulated precipitation (mm d a y − 1 ). All variables were calculated between 1981 and 2020 on the HYGO.
Another important criteria are: the precipitation of the driest month in summer is close to 90 mm m o n t h − 1 , the precipitation of the driest month in winter is close to 6 mm m o n t h − 1 , the
precipitation of the wettest month in summer is close to 130 mm m o n t h − 1 and the precipitation of the wettest month in winter is close to 12 mm m o n t h − 1 . The accumulate precipitation in
the summer is equal to 340 mm (Dec-Feb), with maximum of 130 mm in February. In the winter (Jun-Aug), the accumulate precipitation is 28 mm (Jun-Aug), with minimum of 6 mm in July (Figure 3B).
Moreover, the maximum daily accumulated precipitation is observed during May and December with values around 45 mm d a y − 1 and the minimum during June with values close to 15 mm d a y − 1 (Figure
All instances of intense rainfall events on the HYGO exhibited discernible thermal meso-scale circulations linked to the South American Low-Level Jet (SALLJ). These circulations facilitated the
transportation of moisture fluxes originating from both the Amazon basin in the east—traversing the passes with gentle slopes along the Andes—and the Pacific Ocean in the west. This atmospheric
phenomenon unfolded in the hours leading up to the occurrence of intense rainfall events within the MV. Furthermore, our investigations revealed two primary regions on the eastern side of the Andes
where moisture influxes penetrated the central Andes: one situated in the north-western region (Blue Cordillera) and the other in the south-eastern region of the Mantaro Basin.
On the western side of the Andes, several small passes with gradual slopes served as conduits for moisture fluxes originating from the Pacific Ocean (Flores-Rojas et al., 2019b). The impact of these
meso-scale circulations became particularly pronounced during intense rainfall events occurring between 14 LT and 23 LT. The trajectory of these moisture flows into specific regions within the MV
hinged on their interaction with circulations at medium and high atmospheric levels. Within this framework, we identified two distinct sets of atmospheric circulations that gave rise to severe
rainfall events above the HYGO: the EC (East Circulation) and WC (West Circulation) events. These were characterized by the prevailing atmospheric circulations at high and medium levels, respectively
(Flores-Rojas et al., 2020).
To validate the micro-climate data recorded at the HYGO automatic station (2017–2022), we compare seasonal variations in air temperature and daily accumulated precipitation with a 40-year climate
normal (1981–2020) from the conventional weather station of the HYGO. This analysis ensures the representativeness and reliability of the observed micro-climate trends. In general, comparatively to
the climate normal, the observations in the HYGO automatic station indicate that the climate on the HYGO during the period of 2017–2022 are very similar to the climatology values of temperature
(Figure 3A), with mean maximums close to 22 ° C in November and mean minimums close to −3 ° C in July, with minimum diurnal thermal amplitude close to 12 ° C in March and maximum close to 23 ° C in
July (Figure 4A).
Seasonal variation of (A) air temperature ( ° C), (B) Relative humidity (%), (C) water mixing ratio (g kg − 1 ), (D) sunshine hours measured by the heliograph installed at 1.5 m height and (E) daily
accumulated precipitation (mm day − 1 ) on the HYGO. The graphics show the minimum, mean and maximum values of the variables calculated between May 2017 and December 2022.
Furthermore, Figure 4E illustrates the seasonal dynamics of daily accumulated precipitation from 2017 to 2022, exhibiting a pattern consistent with the 40-year precipitation climate normal depicted
in Figure 3C. During this period, the maximum daily accumulated precipitation peaks in March, reaching approximately 56 mm d a y − 1 , while the minimum values, around 4 mm day − 1 , are observed in
July. Additionally, Figure 4B portrays the seasonal fluctuations in relative humidity (RH) from 2017 to 2022. Notably, the smallest amplitude of RH occurs in March, with values close to 52%, while
the maximum amplitude is observed in January, reaching approximately 68%. The seasonal variation of water vapor mixing ratio between 2017 and 2022 is depicted in Figure 4C. Mean maximum values are
observed during in January and February with values close to 8.5 g kg − 1 and minimum values around to 2 g kg − 1 in July.
Finally, Figure 4C shows the seasonal variation of sunshine hours from 2017 to 2022 registered by the Heliograph installed on the HYGO (Figure 2C). Maximum values of sunshine hours are observed in
June and July with values close to 10.5. It is important to note that October also has high maximum sunshine hours around to 10. This set of data is important because initial modeling work carried
out around the world was involved in relating daily horizontal global irradiation to duration of bright sunshine hours.
An alternative approach to assess the representativeness of measurements obtained from the HYGO automatic station involves utilizing the psychrometric diagram for characterizing the seasonal climate
conditions in the area (Gaffen and Ross, 1999). Figures 5A, B illustrate the psychrometric diagram for daily and monthly average values of specific humidity (q) (g k g − 1 ) and temperature (T) ( °
C), respectively. Both figures reveal a moderate correlation between specific humidity (q) and temperature (T), with values reaching approximately 60% for mean diurnal averages and 75% for mean
monthly averages. The seasonal average daily values (Figure 5A) of q range from 1.5 g k g − 1 on a few spring days to 11 g k g − 1 on certain days in fall and summer. The colored lines in the figure
represent the values of q as a function of T for different relative humidity (RH) levels, with atmospheric pressure set to 675.6 hPa (mean pressure at HYGO).
The psychrometric diagram for (A) daily and (B) monthly average values of specific humidity (g k g − 1 ) versus temperature average values ( ° C). Summer are indicated by red; winter by blue; fall by
green and spring by magenta. The dashed curves are the calculated relations for relative humidity (RH) of 20%, 40%, 60%, 80%, and 100%, with atmospheric pressure equal to 675.6 hPa.
Throughout all seasons, the range of daily average RH values fluctuates between 20% and 90%. The increased dispersion in the monthly average values recorded at the HYGO automatic station,
particularly notable during the summer and spring, appears to be linked to climatic variability. This variability is indicative of warmer and drier conditions experienced during these seasons in
recent years. Additionally, the seasonal average monthly values (Figure 5B) of q vary between 4 g k g − 1 in winter and 10 g k g − 1 in fall and summer. Similar to the daily values, the colored lines
in the figure depict the relationship between q and T for different RH levels, considering the atmospheric pressure of 675.6 hPa (mean pressure at HYGO). Monthly average RH values consistently fall
within the range of 40% and 75%, as demonstrated in Figure 4B.
Hence, based on the climate analysis conducted herein, it can be inferred that the radiation measurements recorded between 2017 and 2022 on the HYGO automatic station might be influenced by the fact
that the HYGO region experienced cooler and drier conditions during this period compared to the climatological norm. Despite these variations, as detailed in Section 5.2, it will be demonstrated that
the mean behavior of E G at the surface aligns with estimates derived from normal values of sunshine hours. This alignment suggests that it can be deemed representative of the climate in the MV
region. On the other hand, the E D F and E D R components of solar radiation, measured at the HYGO automatic station, appear to exhibit a greater sensitivity to local climate and land-use
In this section, we delineate the time-integrated values of surface solar radiation components denoted by E X Y , expressed in megajoules per unit area per hour (MJ m 2 h − 1 ). The subscripts X (T,
G, DF, and DR) represent extraterrestrial, global, diffuse, and direct solar radiation, respectively. All irradiance variables E D R , E D F , E G components observed at the surface. The superscripts
Y (h and d) signify time intervals for integration, with “h” representing 1 hour and “d” representing 1 day.
The seasonal fluctuation in the diurnal evolution of E D R h (Figure 6A) is derived from the matrix of monthly average E D R h values, which are interpolated using the cubic spline method (Boor,
2001). The monthly average of E D R h at noon reaches a peak close to 2.5 MJ m 2 h − 1 during winter (July) and dipping to a minimum close to 1.2 MJ m 2 h − 1 in summer (February). Moreover, the
monthly average of E D F h (Figure 6B) at noon reaches a peak around 1.80 MJ m 2 h − 1 during summer (February) and dipping to a minimum around 0.45 MJ m 2 h − 1 in winter (July). Notably, the
longest and shortest duration of sunshine hours occur in July (9 h) and February (5.5 h), respectively.
Seasonal and hourly cycles of average (A) direct ( E D R ) , (B) diffuse ( E D F ) and (C) global ( E G ) solar irradiances ( MJ m − 2 h − 1 ) on the HYGO ( 12.0 o S, 73.5 o W), from July, 2017 to
September, 2022. The graphics show the mean values of the variables calculated between May 2017 and December 2022.
On the contrary, the monthly average of E G h (Figure 6C) at noon reaches a peak around 3.6 MJ m 2 h − 1 during spring (October) and dipping to a minimum close to 2.7 MJ m 2 h − 1 in winter (July).
The highest E G h in spring months is associated to a combination of astronomical factor, cloudiness and aerosol concentration patterns observed during 2017–2022 in the HYGO. During this period, the
daily accumulated precipitation reach maximum and mean values around 24 mm d a y − 1 and 5 mm d a y − 1 , respectively. The maximum value is below the climate normal in October with maximums close to
37 mm d a y − 1 and the mean value is close to the climate mean value around 4 mm d a y − 1 . In addition, the sunshine hours during October reach a maximum close to 10 h (Figure 4). The combination
of these factors indicates that the reduction in the precipitation and the moderate presence of aerosols seems to be associated to the reduction in the cloud cover that favored large values of E G h
The diurnal evolution of monthly average of E T h , E G h , E D F h and E D R h , during summer, fall, winter and spring months are shown in Figures 7A–D, respectively. In addition, Table 1 shows the
specific values for the diurnal cycle of these irradiance variables. All components of solar irradiance exhibit a clearly defined diurnal cycle, reaching their maximum intensity at noon. The minimal
standard error of the mean, denoted by the small vertical bars, signifies that the monthly average values of solar radiation components recorded at the surface within the HYGO accurately reflect the
mean conditions observed in the MV.
Seasonal and diurnal variation of E T h , E G h , E D F h and E D R h in MJ m − 2 h − 1 for the (A) summer, (B) fall, (C) winter and (D) spring. The standard error of the mean bars correspond to 95%
confidence interval.
Seasonally average hourly values of solar irradiance components: E T h , E G h , E D F h and E D R h , clearness index ( K T ) and diffuse fraction ( K D ) observed in the HYGO. The hourly values
correspond to noontime (12 LT).
Irradiances(MJ m − 2 h − 1 ) Summer Fall Winter Spring
E T 4.83 ± 0.14 4.33 ± 0.16 3.94 ± 0.11 4.74 ± 0.17
E G 3.11 ± 0.16 3.02 ± 0.18 3.08 ± 0.14 3.32 ± 0.22
E D F 1.67 ± 0.22 1.18 ± 0.17 0.64 ± 0.16 1.32 ± 0.24
E D R 1.37 ± 0.21 1.65 ± 0.23 2.27 ± 0.19 1.92 ± 0.27
Indexes Summer Fall Winter Spring
K T 0.64 ± 0.04 0.70 ± 0.04 0.78 ± 0.04 0.70 ± 0.05
K D 0.61 ± 0.05 0.45 ± 0.06 0.25 ± 0.07 0.47 ± 0.06
During summer at noon, E G h (3.11 ± 0.16 MJ m − 2 h − 1 ) constitutes 64.4% of E T h (4.83 ± 0.14 MJ m − 2 h − 1 ). Additionally, E D F h (1.67 ± 0.22 MJ m − 2 h − 1 ) and E D R h (1.37 ± 0.21 MJ m
− 2 h − 1 ) represent 53.7% and 44.10% of E G h (3.11 ± 0.16 MJ m − 2 h − 1 ), respectively. In contrast, during winter at noon, E G h (3.08 ± 0.14 MJ m − 2 h − 1 ) accounts for 78.2% of E T h (3.94
± 0.11 MJ m − 2 h − 1 ). Moreover, E D F h (0.64 ± 0.16 MJ m − 2 h − 1 ) and E D R h (2.27 ± 0.19 MJ m − 2 h − 1 ) represent 20.8% and 73.7% of E G h , respectively.
On the other hand, during fall at noon, E G h (3.02 ± 0.18 MJ m − 2 h − 1 ) constitutes 69.7% of E T h (4.33 ± 0.14 MJ m − 2 h − 1 ). Furthermore, E D F h (1.18 ± 0.17 MJ m − 2 h − 1 ) and E D R h
(1.65 ± 0.23 MJ m − 2 h − 1 ) represent 39.1% and 54.6% of E G h , respectively. Finally, it is noteworthy that during spring at noon, E G h (3.32 ± 0.22 MJ m − 2 h − 1 ) surpasses the mean values in
summer, constituting 70.0% of E T h (4.74 ± 0.17 MJ m − 2 h − 1 ). Additionally, E D F h (1.32 ± 0.24 MJ m − 2 h − 1 ) and E D R h (1.92 ± 0.27 MJ m − 2 h − 1 ) represent 39.8% and 57.8% of E G h ,
It is important to highlight that for all seasons the sum of E D F h and E D R h do not contribute 100% to the total E G h . We consider that these small differences are caused by the filters showed
in Section 4.1 that removes several data of E D F h and E D R h mainly in hours close to noon, caused by fails of the sun-tracker 2AP (Kipp & Zonen) that move the pyrheliometer CHP1 (Kipp & Zonen) to
measure E D R h and the small black sphere to measure E D F h . These problems are more evident during winter and fall seasons and will be corrected by the acquisition of a new sun-tracker.
The examination of K T and K D provides a more comprehensive insight into the diurnal pattern of atmospheric transmittance, as it eliminates the influence of astronomical factors on E T , E G , and E
D F . Consequently, K T and K D emerge as crucial indicators for discerning the scattering and absorption processes facilitated by the presence of clouds and aerosol loads in the atmosphere. The
diurnal evolution of K T h shows a maximum close to 0.64 ± 0.04 at noon during summer (Figure 8A), around 0.71 ± 0.06 at 13 LT during fall (Figure 8B), close to 0.78 ± 0.04 at noon during winter
(Figure 8C) and around 0.72 ± 0.04 at 11 LT in spring (Figure 8D). In general, K D h has an inverse behavior in comparison with K T h . The diurnal evolution of K D h presents maximum values at noon
also during summer close to 0.61 ± 0.05 and minimum during winter around 0.25 ± 0.07. The amplitude of seasonally average values between K T h and K D h for the HYGO at noontime reach its minimum
value in summer (0.03) and its maximum in winter (0.53) with intermediate values during fall (0.25) and spring (0.23) (Table 1).
Diurnal variation of K T h and K D h for the (A) summer, (B) fall, (C) winter and (D) spring. The standard error of the mean bars correspond to 95% confidence interval.
The seasonal variation of monthly average daily values of the solar radiation components at the surface of the HYGO is presented in Figure 9A; Table 2. The highest values of E G d are observed in
October (spring) around 24.14 ± 2.10 MJ m − 2 d a y − 1 and the lowest values of E G d are observed in March (autumn) around 19.50 ± 2.15 MJ m − 2 d a y − 1 . Moreover, the highest values of E D R d
are observed during August (winter) around 14.97 ± 1.82 MJ m − 2 d a y − 1 but also are observed high values of E D R d in September and October (spring) probably partially associated with high
values of aerosol optical depth at 440 nm occurring during these months (Figure 9C). The lowest values of E D R d are observed during March (autumn) with values close to 6.59 ± 1.89 MJ m − 2 d a y −
1 . On the contrary, the highest values of E D F d are observed during January (summer), around 13.03 ± 1.91 MJ m − 2 d a y − 1 and the lowest values are observed during July (winter) around 4.57 ±
1.0 MJ m − 2 d a y − 1 . This behavior of E D F d can be explained by the strongly seasonal variation of clouds and precipitation on the HYGO, previously analyzed (Figure 4E).
Monthly variation of (A) the solar irradiance components: E T d , E G d , E D F d and E D R d , the (B) clearness index ( K T d ) and diffuse fraction ( K D d ) and (C) Angstrom turbidity coefficient
(AOD) observed in the HYGO. The standard error of the mean bars correspond to 95% confidence interval.
Seasonally average daily values of solar irradiance components: E T d , E G d , E D F d and E D R d , clearness index ( K T ) and diffuse fraction ( K D ) observed in the HYGO. Also are shown the
standard deviations for each components.
Irradiances(MJ m − 2 d a y − 1 ) Indexes
E T E G E D F E D R K T K D
January 39.94 ± 0.74 21.39 ± 1.60 13.03 ± 1.91 7.15 ± 2.40 0.54 ± 0.04 0.65 ± 0.08
February 38.84 ± 2.30 20.95 ± 1.94 12.97 ± 1.94 6.73 ± 2.30 0.54 ± 0.05 0.66 ± 0.08
March 34.91 ± 4.52 19.50 ± 2.15 11.38 ± 1.61 6.59 ± 1.89 0.56 ± 0.05 0.61 ± 0.06
April 32.98 ± 2.69 20.87 ± 1.82 8.71 ± 1.42 9.61 ± 2.81 0.63 ± 0.05 0.47 ± 0.11
May 29.51 ± 2.18 20.06 ± 1.92 6.48 ± 1.57 10.59 ± 2.04 0.68 ± 0.05 0.36 ± 0.11
June 27.96 ± 1.58 20.33 ± 1.63 5.05 ± 1.61 12.75 ± 1.72 0.73 ± 0.05 0.28 ± 0.12
July 28.96 ± 0.46 21.03 ± 1.59 4.57 ± 1.00 13.91 ± 1.56 0.73 ± 0.06 0.24 ± 0.10
August 31.48 ± 2.26 22.48 ± 1.72 5.75 ± 1.44 14.97 ± 1.82 0.71 ± 0.04 0.28 ± 0.09
September 35.16 ± 1.88 22.45 ± 1.48 8.65 ± 1.25 12.64 ± 1.43 0.64 ± 0.04 0.42 ± 0.07
October 38.29 ± 0.60 24.14 ± 1.98 9.64 ± 2.31 13.20 ± 2.56 0.63 ± 0.05 0.43 ± 0.09
November 39.26 ± 1.56 23.14 ± 2.10 12.40 ± 2.99 9.51 ± 2.99 0.59 ± 0.05 0.57 ± 0.10
December 39.76 ± 1.04 21.92 ± 2.16 12.29 ± 2.55 8.87 ± 3.30 0.55 ± 0.05 0.60 ± 0.11
Furthermore, the seasonal evolution of K T d shows maximums close to 0.73 ± 0.06 in June and July (winter) and minimums around 0.54 ± 0.04 during January and February (summer). In general, the
seasonal evolution of K D d has an inverse behavior in comparison with K T d with maximums in February (summer) close to 0.66 ± 0.08 and minimums during July (winter) with values around 0.24 ± 0.10
(Table 2). The amplitude of monthly average values between K T d and K D d for the HYGO reach its minimum value during March (0.05) and November (0.02) and reach maximum values in July (0.48) (Figure
To elucidate the seasonal variations in the average daily values of solar irradiance components ( E G d , E D F d , and E D R d ), Figure 10 present the monthly mean values of aerosol volumetric size
distributions. These distributions are derived from measurements taken with a CIMEL CE-318T sun photometer, part of the AERONET network (Holben et al., 1998; 2001). The data exhibit a generally
bimodal behavior in the monthly mean size distributions, with a predominance of the coarse mode, as shown by the average size distribution curve for the analyzed period (dashed line in all graphs).
This bimodal pattern is also evident in the seasonal mean values (dotted lines), where larger aerosols dominate in all seasons except spring (Figure 10D). Specifically, coarse-mode aerosols are
predominant in summer (Figure 10A), fall (Figure 10B), and winter (Figure 10C). These coarse-mode aerosols, with average radii around 5.0613 μ m , account for 56.3% of the cases and are primarily
associated with marine and desert sources, with lesser contributions from continental aerosols. In contrast, fine-mode aerosols, with average radii around 0.1482 μ m (43.7%), are mainly produced by
the incomplete combustion of fossil fuels and biomass burning (Estevan et al., 2019).
Monthly mean values of the aerosols volume-size distribution ( μ m 3 μ m − 2 ) , for the seasons: (A) summer, (B) autumn, (C) winter and (D) spring, during the period 2018–2022.
During the spring season, there is a clear predominance of fine-mode aerosols, as illustrated in Figure 10D. This trend is evident in the seasonal mean (dotted line) and the monthly mean values,
particularly in September and November. In September, the fine mode dominates, primarily due to aerosols generated by biomass burning in the Peruvian Amazon, which is associated with high Aerosol
Optical Depth (AOD) values measured by the sun photometer at HYGO. Previous studies indicate that starting in July, AOD values rise due to biomass burning aerosols, peaking in September (Estevan et
al., 2019). This increase aligns with the predominance of f ine-mode aerosols in August (Figure 10C) and throughout the spring months (Figure 10D).
In an atmosphere with aerosols, more scattered energy reaches the ground due to increased forward scattering (Iqbal, 1983). During spring, the pronounced presence of both fine-mode (0.1482 m) and
coarse-mode (5.0613 m) aerosols, combined with longer daily sunshine hours and increased E T d , leads to a rise in E D F d , ultimately resulting in higher E G d . This effect is particularly
noticeable in October (Figure 9A), which still experiences many days with high sunshine hours (Figure 4D), reaching a maximum E G d intensity of 24.14 MJ m − 2 d a y − 1 .
Moreover, during the study period, several high aerosol concentration events were recorded by the AERONET station’s sun photometer, primarily in September. Notably, on 24 November 2020, at 21:23 UTC,
the sun photometer recorded the highest AOD values since its installation on 19 March 2015, at HYGO, with an AOD 440 nm value of 1.23. These elevated AOD levels are linked to biomass-type aerosols.
To determine if these aerosols were related to biomass burning, the HYSPLIT trajectory model was employed. Using NCEP reanalysis meteorological data, 120-h back-trajectories were calculated at three
altitude levels: 500, 1,500, and 3000 m, from the sun photometer’s location. Figure 11 presents these back-trajectories along with fire hotspots detected by the MODIS and VIIRS satellites, indicated
as red dots. By applying a coincidence criterion of a 4 km radius and 1 km height around the trajectory, two coincident fire hotspots were identified. These hotspots were located 281.2 km and 282.6
km from the sun photometer, at altitudes of 355.4 m and 343.1 m, respectively, where the 500 m back-trajectory passed. This evidence strongly suggests that the biomass-type aerosols observed on the
specified date likely originated from these fire hotspots.
At the top, the red dots represent the fire hotspots associated with biomass burning. The yellow dot represents the position of the sun photometer at the HYGO, and the lines represent the back
trajectories at 500 m (red line), 1,500 m (blue line), and 3000 m (green line) above the sun photometer at zero time. At the bottom, the back trajectories at different altitudes over the 120-h model
run are shown.
Initial modeling efforts in numerous countries focused on establishing a relationship between daily horizontal global irradiation and the duration of bright sunshine. The initial phase of this
endeavor entailed the formulation of regression equations based on monthly-averaged data. However, subsequent advancements have led to the development of equations utilizing data recorded at daily
intervals. This progression allows for a more comprehensive understanding by connecting the discussed relationships with the daily variation between horizontal diffuse and global irradiation. This
sections will present an analysis enabling the estimation of diurnal horizontal global and diffuse irradiation at different time scales.
In this section, we introduce the MADHG irradiation model, designed to establish a connection between the monthly-averaged daily clearness index ( K T = E G / E T ) and the monthly mean daily
sunshine fraction, as defined by Equations 2, 3. Figure 12A showcases a scatter plot for Equation 2, utilizing irradiance data from the BSRN station on the HYGO spanning from May 2017 to December
2022. Notably, a robust correlation between the variables is apparent, characterized by an R 2 value of 0.76 and a root mean squared error (RMSE) of approximately 4.1%. The coefficients ‘a’ and ‘b’
in Equation 2 are illustrated in Figure 12A; Table 3 as 0.33 and 0.50, respectively. It is noteworthy that analogous plots, depicting strong correlations with comparable ‘a’ and ‘b’ values, have been
presented by various researchers for diverse sites worldwide (Munner, 2004b).
(A) Relationship between monthly-averaged clearness index ( K T = E G / E T ) and sunshine fraction and (B) variation of monthly-averaged diffuse ratio ( E D / E G ) against clearness index. All
irradiance data were measured by the BSRN station between May 2017 and December 2022.
Coefficients and statistical parameters for the empirical irradiance models. The forty percent (40%) of the total filtered dataset chosen randomly (740 days or 17 760 h) were reserved for rigorous
statistical tests to evaluate model performance and robustness. The RMSE was calculated in percentage and in MJ m − 2 d a y − 1 .
Irradiance Models Coefficients Statistics
a b c d r 2 RMSE (%) RMSE (MJ)
Model 01: MADHG 0.33 0.50 - - 0.76 4.1 0.88
Model 02: MADHD 1.63 −1.91 - - 0.91 5.0 1.08
Model 03: DAHG 0.32 0.51 - - 0.85 5.0 1.08
Model 04: DAHD 0.75 2.45 7.15 3.92 0.86 9.3 2.0
Furthermore, we introduced the MADHD irradiation model, designed to establish a relationship between the monthly-averaged daily diffuse ratio ( E D / E G ) and the monthly-averaged daily clearness
index ( K T ) , as defined by Equation 4. Figure 12B illustrates the scatter plot for Equation 4, utilizing irradiance data from the BSRN station on the HYGO spanning from May 2017 to December 2022.
Similar to the previous model, a robust correlation is evident between the discussed variables, characterized by an R 2 value of 0.91 and a root mean squared error (RMSE) of approximately 5%. The
coefficients ‘a’ and ‘b’ in Equation 4 are depicted in Figure 12B; Table 3 as 1.63 and −1.91, respectively. It is noteworthy that analogous plots, demonstrating strong correlations with comparable
‘a’ and ‘b’ values, have been presented by various researchers for numerous sites worldwide (Munner, 2004c).
In this section we present the irradiation model DAHG, which connect the daily clearness index ( K T = E G / E T ) with the daily sunshine fraction, according to the Equations 3, 5. Moreover, Figure
12B, shows the scatter plot for the Eq.Equation 5 using the irradiance data of the BSRN station on the HYGO between May 2017 and December 2022. A strong correlation between the two quantities under
discussion is evident with R 2 equal to 0.85 and RMSE around 5%. Figure 13A also shows the coefficients ‘a’ and ‘b’ of the Eq. Equation 5; Table 3, equals to 0.32 and 0.51, respectively. In general,
the relationships for daily values and monthly-averaged daily values, represented by Eqs Equations 2, 5, respectively, are different, according to Munner (2004d).
(A) Relationship between daily averaged clearness index ( K T = E G / E T ) and sunshine fraction and (B) variation of daily averaged diffuse ratio ( E D / E G ) against clearness index. All
irradiance data were measured by the BSRN station between May 2017 and December 2022.
In addition, we introduced the DAHD irradiation model, designed to establish a connection between the daily diffuse ratio ( E D / E G ) and the daily clearness index ( K T = E G / E T ), as expressed
by Equation 4. Figure 13B displays the scatter plot for Equation 6, utilizing irradiance data from the BSRN station on the HYGO between May 2017 and December 2022. Mirroring the pattern observed in
the previous model, a robust correlation is apparent between the two discussed variables, with an R 2 value of 0.86 and a mean squared error (MSE) of approximately 9.3%. Figure 12B and Table 3 also
presents the coefficients ‘a,’ ‘b,’ ‘c,’ and ‘d’ of Equation 6 as 0.75, 2.45, 7.15, and 3.92, respectively. Notably, analogous plots, demonstrating strong correlations with comparable values for ‘a,’
‘b,’ ‘c,’ and ‘d,’ have been presented by various researchers for numerous sites worldwide (Muneer and Hawas, 1984; Saluja et al., 1988).
In this section, we introduce the HHG irradiation model, which connect the ratio between the hourly global irradiation and the daily global irradiation ( r G ) with the sunset hour angle expressed as
radians from solar noon, according to the Equations 7, 8. Figure 14 illustrates the effect of the displacement of the hour from solar noon, and the daylength, on the ratio of hourly to daily global
irradiation ( r G ) . There are six time deviations occurring before solar noon (−0.5 h, −1.5 h, −2.5 h, −3.5 h, −4.5 h, −5.5 h), depicted in Figure 14A, and six deviations occurring after solar noon
(+0.5 h, +1.5 h, +2.5 h, +3.5 h, +4.5 h, +5.5 h), displayed in Figure 14B. The values of r G range from around 0.15 at ± 0.5 h to 0.01 at ± 5.5 h and the values of sunset hour angle range from 1.48
(84.8 ° ) at ± 0.5 h to 1.65 (94.5 ° ) at ± 5.5 h. This narrow range of these values is due to the latitude location of the HYGO (12 ° ).
Ratio of hourly to daily global irradiation ( r G ) against the sunset hour angle expressed in radians from solar noon (Equations 7, 8) between (A) 07 and 12 LT and between (B) 13 and 18 LT.
Unlike certain previous studies (Liu and Jordan, 1960; Iqbal, 1983), which utilized a least-squares fit to determine the coefficients of Equations 7, 8, we employed a least-squares fit approach to
ascertain these coefficients individually for each hour preceding and following solar noon. First column of Table 4 shows fitted values for the coefficients of r G and statistical indicators for the
hourly horizontal global (HHG) irradiation model fitting for each hour preceding and following solar noon. However, the values of R 2 range from 0.12 for +4.5 h to 0.83 for +2.5, and the values of
RMSE range from 5.4% to 8.4%. It is important to note that these coefficient values within −4.5 and +1.5 h, are similar to those found by Iqbal (1983) testing the applicability of the model of Liu
and Jordan (1960).
Coefficients of r G and r D and statistical indicators for the hourly horizontal global (HHG) and diffuse (HHD) irradiation models fitting for each hour preceding and following solar noon. The forty
percent (40%) of the total filtered dataset chosen randomly (740 days or 17 760 h) were reserved for rigorous statistical tests to evaluate model performance and robustness. The RMSE was calculated
in percentage and in MJ m − 2 hou r − 1 .
r G r D
Hours from solar noon Coefficients Statistics Coefficients Statistics
r s p q R 2 RMSE (%) RMSE (MJ) t u v w R 2 RMSE (%) RMSE (MJ)
−5.5 −0.02 0.94 0.53 −0.44 0.46 7.0 0.22 0.99 5.88 15.92 −2.01 0.19 8.9 0.28
−4.5 0.38 0.53 0.55 −0.49 0.32 7.7 0.24 9.02 −1.30 15.74 4.04 0.11 8.9 0.28
−3.5 0.41 0.69 0.54 −0.38 0.17 8.4 0.26 10.53 2.01 14.22 3.57 0.11 7.0 0.22
−2.5 0.36 0.84 0.49 −0.23 0.13 7.0 0.22 5.79 4.61 14.97 −2.89 0.69 7.0 0.22
−1.5 0.34 0.87 0.45 −0.16 0.12 6.3 0.20 5.44 5.35 14.86 −2.81 0.58 8.0 0.25
−0.5 0.33 0.87 0.43 −0.13 0.11 7.0 0.22 5.92 6.11 14.67 −1.62 0.43 8.9 0.28
+0.5 0.44 0.61 0.54 −0.39 0.53 6.3 0.20 7.49 6.06 14.27 0.19 0.40 7.0 0.22
+1.5 0.54 0.39 0.63 −0.60 0.59 7.0 0.22 11.86 4.27 12.42 5.02 0.15 6.3 0.20
+2.5 0.69 0.02 0.75 −0.88 0.83 5.4 0.17 0.09 −0.01 0.03 0.07 0.87 6.3 0.20
+3.5 0.72 −0.09 0.73 −0.86 0.16 8.3 0.26 0.02 0.02 −0.03 0.08 0.89 6.3 0.20
+4.5 0.85 −0.51 0.73 −0.89 0.12 8.3 0.26 1.12 −0.94 0.74 1.17 0.75 7.0 0.22
+5.5 1.28 −1.48 0.70 −0.76 0.15 6.3 0.20 0.10 −0.09 0.51 0.12 0.18 6.3 0.20
On the other hand, we also introduce the HHD irradiation model, which connect the ratio between the hourly diffuse irradiation and the daily diffuse irradiation ( r D ) with the sunset hour angle
expressed as radians from solar noon, according to the Equations 8, 10. Figure 15 illustrates the effect of the displacement of the hour from solar noon, and the daylength, on the ratio of hourly to
daily diffuse irradiation ( r D ) . As in the case of th global diffuse irradiation, there are six time deviations occurring before solar noon (−0.5, −1.5 h, −2.5 h, −3.5 h, −4.5 h, −5.5 h), depicted
in Figure 15A, and six deviations occurring after solar noon (+0.5, +1.5 h, +2.5 h, +3.5 h, +4.5 h, +5.5 h), displayed in Figure 14B. The values of r D range from around 0.12 at ± 0.5 h to 0.015 at ±
1.5 h and the values of sunset hour angle range from 1.48 (84.8 ° ) at ± 1.5 h to 1.66 (95.1 ° ) at ± 5.5 h. This narrow range of these values is due to the latitude location of the HYGO (12 ° ).
Ratio of hourly to daily diffuse irradiation ( r D ) against the sunset hour angle expressed in radians from solar noon (Equations 9 and 10) between (A) 07 and 12 LT and between (B) 13 and 18 LT.
Second column of Table 4 shows fitted values for the coefficients of r D and statistical indicators for the hourly horizontal diffuse (HHD) irradiation model fitting for each hour preceding and
following solar noon. However, the values of R 2 range from 0.11 for −3.5 h to 0.89 for +3.5, and the values of RMSE range from 6.3% to 8.9%. It is important to note that there is a wide variety of
values for the coefficients depending on the hours from solar noon, which indicates that the Equations 8, 10 may not be suitable for representing these variables.
In this study, the sigmoid logistic function is employed to depict the correlation between K D h and K T h . In comparison to alternative logistic functions Boland and Ridley (2008); Ridley et al.
(2010), it demonstrates superior capability in capturing the behavior of K D h across all K T h values, particularly when K T h approaches unity. To construct the regression model, the total dataset
spanning the years 2017–2022 is partitioned randomly into two segments, the first one for regression model development and the second one for conducting statistical tests.
For the performance evaluation of regression models, two key statistical parameters are employed: i) the coefficient of determination ( R 2 ) and ii) root mean square error (RMSE). Figure 16 shows
the scatter plot of K D h versus K T h (Liu-Jordan Diagram). The sigmoid function proposed in this work is compared with models developed for regions locate in South Hemisphere (Oliveira et al.,
2002a; Boland and Ridley, 2008; Marques Filho et al., 2016). It is important to highlight that this is the first study to utilize high-quality solar radiation data to develop empirical models of
solar radiation in the central Andes of Peru.
K T h versus K D h for the irradiance data measured on the HYGO. The orange solid squares represent the block average and vertical lines the standard deviation. The different solid lines represent
the correlation models presented in this study. Model 01: Sigmoid type 01, Model 02: Sigmoid type 02, Model 03: 4 t h polynomial, Model 04: 3 r d polynomial.
The superiority of the sigmoid type 01 function becomes apparent when comparing its conformity to the block-averaged experimental curve against other correlation models (Oliveira et al., 2002a;
Jacovides et al., 2006; Boland and Ridley, 2008). Notably, while the logistic function adjusted to the HYGO dataset enhanced the statistical performance of that particular model, the sigmoid function
consistently outperforms all alternatives, including 3 r d and 4 t h polynomial functions.
The statistical parameters outlined in Table 5 indicate that all models present a coefficient of determination higher than 83%. Notably, the proposed sigmoid function fitting exhibits the best
statistical performance, closely followed by the adjusted logistic and 4 t h polynomial functions, which show lower RMSE values. Nevertheless, all correlation models yield predictions that are
statistically significant at a confidence level of 95%. These results closely resemble those reported by Marques Filho et al. (2016) for the city of Rio de Janeiro in Brazil. However, Marques Filho
et al. (2016) did not make adjustments for polynomial functions, as they relied on the results of the study conducted by Oliveira et al. (2002a). This decision was due to the minimal divergence in
climate conditions between Rio de Janeiro and São Paulo.
Equations fitted for K D h in function of K T h and statistical performance of the models for calculating the hourly averaged diffuse fraction K D h . The forty percent (40%) of the total filtered
dataset chosen randomly (740 days or 17 760 h) were reserved for rigorous statistical tests to evaluate model performance and robustness. The RMSE was calculated in percentage and in MJ m − 2 hou r −
1 .
Models Equation r 2 RMSE (%) RMSE (MJ)
Model 01: Sigmoid 0 ≤ K T ≤ 1.0 K D = 0.1 + 0.83 1.0 1 + exp − 6.29 + 9.84 K T 0.831 14.9 0.47
Model 02: Logistic 0 ≤ K T ≤ 1.0 K D = 1.0 1 + exp − 4.52 + 7.20 K T 0.826 15.2 0.48
Model 03: 4 t h polynomial K T ≤ 0.15 0.1 < K T ≤ 0.85 K T ≥ 0.85 K D = 0.96 K D = 1.3 − 3.9 K T + 14.7 K T 2 − 24.6 K T 3 + 12.7 K T 4 K D = 0.15 0.813 15.8 0.49
Model 04: 3 r d polynomial K T ≤ 0.18 0.18 < K T ≤ 0.85 K T ≥ 0.85 K D = 0.94 K D = 0.74 + 1.62 K T − 4.26 K T 2 + 1.74 K T 3 K D = 0.12 0.809 16.4 0.51
On the other hand, the AIC statistical method, assessed using Equations 11, 12, identifies the sigmoid function for HYGO as the best model (see Table 6). However, all correlation models are
statistically relevant ( Δ i AIC <2) (Burnham and Anderson, 2004) and can effectively reproduce the relationship between K D h and K T h . Although the logistic function proposed by Boland and Ridley
(2008) relies on only two parameters, its performance when fitted to the HYGO dataset is not optimal. This is likely due to the curve of points ( K D h , K T h ) simulated by Boland and Ridley (2008)
diverging significantly from the mean values (see Figure 16).
Statistical performance of the sigmoid function model in comparison with the other empirical irradiance models for the HYGO.
Models Number of parameters AIC Δ i AIC
Model 01: Sigmoid 4 −3.655 0.000
Model 02: Logistic 2 −3.642 −0.013
Model 03: 4 t h polynomial 7 −3.647 −0.008
Model 04: 3 r d polynomial 6 −3.639 −0.016
The first goal of the present study was to evaluate the climate characteristics of the western Mantaro Valley, using data from two meteorological stations: the HYGO conventional station and the IGP
automatic station. It includes surface weather measurements from HYGO spanning 1981 to 2020 and 1-min averages from the IGP platform between 2017 and 2022. Details on the station instrumentation are
provided in recent publications (Flores-Rojas et al., 2019a; 2021; 2020).
Based on Köppen-Geiger classification (Peel et al., 2007) and data from the HYGO station, the Mantaro Valley is classified as Cwb, indicating a temperate climate with dry winters and warm summers.
The criteria include a mean temperature of the hottest month above 10 ° C and the mean temperature of the coldest month between 0 ° C and 18 ° C. Precipitation patterns show the driest and wettest
months receiving approximately 90 mm and 130 mm of rainfall respectively. Summer accumulation is 340 mm with a peak in February, while winter accumulation is 28 mm with a minimum in July. Maximum
daily precipitation occurs in May and December, while the minimum is observed in June (Figure 3).
An alternative method for evaluating data from the HYGO automatic station involves using psychrometric diagrams to characterize seasonal climate conditions. There are moderate correlations between
specific humidity (q) and temperature (T), with approximately 60% for mean diurnal averages and 75% for mean monthly averages. Daily q values range from 1.5 g k g − 1 in spring to 11 g k g − 1 in
fall and summer, with RH fluctuating between 20% and 90%. Monthly average q varies from 4 g k g − 1 in winter to 10 g k g − 1 in fall and summer, with RH consistently between 40% and 75%. These
analyses suggest that radiation measurements from the HYGO automatic station between 2017 and 2022 may reflect cooler and drier conditions compared to the climatological norm, though mean surface
solar irradiance ( E G ) remains representative of the MV region’s climate. However, components like E D F and E D R appear more sensitive to local climate and land-use factors.
The seasonal variation in E G h , especially at noon, exhibits a low amplitude, ranging from 3.02 ± 0.18 MJ m − 2 h − 1 in fall to 3.32 ± 0.22 MJ m − 2 h − 1 in spring, resulting in a seasonal
amplitude close to 0.30 MJ m − 2 h − 1 . This amplitude is notably lower than the amplitude variation for E T h , which is close to 0.90 MJ m − 2 h − 1 . This pattern can be attributed to the
opposing seasonal variations of the solar irradiance components E D F h and E D R h , which attenuate the seasonal variation of E G h . Specifically, E D F h peaks during summer and reaches its
minimum in winter, while E D R h peaks in winter and reaches its minimum in summer.
During the spring, moderate values of E D F h and E D R h lead to the highest values of E G h during this season. This behavior is associated with reduced precipitation, moderate cloud cover, and the
presence of aerosols, possibly from biomass burning in the HYGO during spring months, which was analyzed by Estevan et al. (2019). Moreover, diurnal patterns of K T h show maximum values at noon in
winter and minimums in summer, whereas K D h displays an inverse behavior showing maximums at noon in summer and minimums in winter. The amplitude of seasonally average values between K T h and K D h
is minimal in summer and maximal in winter, with intermediate values in fall and spring.
In addition, the monthly average daily solar radiation components at the HYGO corroborate the seasonal patterns observed in hourly irradiance variables. Peak values of E G d are evident in October
(spring), contrasting with lowest values observed in March (autumn). Similarly, peak values of E D R d manifest in August (winter) and begins to decrease continuously in the following months
attributable to heightened aerosol optical depth (AOD) and to the increase of cloudiness during this period (Estevan et al., 2019). In contrast, E D F d peaks in January (summer) and reaches its
minimums in July (winter), mirroring the pronounced seasonal fluctuations of aerosols, clouds and precipitation in the HYGO (Giráldez et al., 2020).
Besides, the increase in AOD during September significantly raises K D d from 0.25 in August to 0.42 in September and October. Consequently, there is a reduction in E D R d and an increase in E D F d
, similar to the findings of Huaping et al. (2021) in Wuhan, China. However, in October, E D F d slightly increases while E D R d remains almost constant compared to September, which results in a
slight increase in E G d during October, reaching its highest value of the year. This behavior is attributed to specific biomass burning events near HYGO, which cause a sudden increase in AOD and
alter the solar irradiance components E D F d and E D R d (Estevan et al., 2019).
Moreover, the monthly mean aerosol size distributions, measured by a CIMEL CE-318T sun photometer (AERONET network) (Holben et al., 1998; 2001) reveal a bimodal distribution, with coarse-mode
aerosols (5.0613 μ m ) predominating in summer, fall, and winter, accounting for 56.3% of the cases. These aerosols are mainly from marine and desert sources. Fine-mode aerosols (0.1482 μ m ), which
make up 43.7%, are primarily produced by fossil fuel combustion and biomass burning (Estevan et al., 2019). During spring, fine-mode aerosols predominate, particularly in September and November. This
is primarily due to biomass burning in the Peruvian Amazon, leading to high Aerosol Optical Depth (AOD) values measured at HYGO. Increased forward scattering in an aerosol-rich atmosphere results in
more ground-reaching energy (Iqbal, 1983). In spring, the presence of a large quantity of aerosols, coupled with longer sunshine hours, cloudiness and higher E T d , raises E D F d , leading to
increased E G d . This behavior is especially evident in October, which experiences high sunshine hours and reaches a maximum values of E G d .
The MADHG irradiation model establishes a connection between the monthly-averaged daily clearness index ( K T ) and the monthly mean daily sunshine fraction. Utilizing irradiance data from the BSRN
station on the HYGO, a strong correlation, with R 2 = 0.76, and RMSE of 4.1% are observed. Additionally, the DAHG irradiation model links the daily clearness index ( K T = E G / E T ) to the daily
sunshine fraction, revealing a strong correlation between the two variables, with R 2 equal to 0.85 and RMSE around 5.0%. It is noted that the relationships derived from the MADHG and DAHG models
exhibit slight differences, as reported by Munner (2004d).
Furthermore, the MADHD irradiation model is designed to establish a relationship between the monthly-averaged daily diffuse ratio ( E D / E G ) and the monthly-averaged daily clearness index ( K T )
. In this case, a more robust correlation is evident between the discussed variables, characterized by an R 2 value of 0.91 and RMSE of approximately 5.0%. In addition, the DAHD irradiation model,
designed to establish a connection between the daily diffuse ratio ( E D / E G ) and the daily clearness index ( K T ), mirroring the pattern observed in the previous model, a robust correlation is
apparent between the two discussed variables, with an R 2 value of 0.86 and RMSE of approximately 9.3%.
On the other hand, the HHG irradiation model links the ratio of hourly global irradiation to daily global irradiation ( r G ) with the sunset hour angle expressed in radians from solar noon.
Statistical indicators for this model reveal R 2 values ranging from 0.12 for +4.5 h from solar noon to 0.83 for +2.5 h from solar noon, with RMSE values spanning from 5.4% to 8.4%. Interestingly,
coefficient values within the −4.5 h and +1.5 h range closely resemble those from prior studies Iqbal (1983); Liu and Jordan (1960).
Similarly, the HHD irradiation model links the ratio of hourly diffuse irradiation to daily diffuse irradiation ( r D ) with the sunset hour angle in radians from solar noon. Its statistical
indicators demonstrate R 2 values ranging from 0.11 for −3.5 h from solar noon to 0.89 for +3.5 h from solar noon, and RMSE values ranging from 6.3% to 8.9%. Notably, a wide range of coefficient
values across hours from solar noon suggests that the mathematical expressions for this model may not comprehensively represent these variables.
The correlation models for hourly diffuse irradiance utilizes several function to correlate K D h and K T h . The present contribution shows that the sigmoid logistic function demonstrates superior
performance, particularly when K T h approaches unity, compared to alternative functions. Evaluation metrics include R 2 and RMSE. The sigmoid function is compared favorably to models developed for
the Southern Hemisphere, showing its first application in developing empirical solar radiation models for Peru’s central Andes. Superior performance of the sigmoid function is evident when compared
to other models, including the logistic function adjusted to HYGO dataset and polynomial functions. All models exhibit R 2 values over 83%, with the sigmoid function showing the best statistical
performance. The AIC method identifies the sigmoid function as the best model for HYGO, though all models effectively capture the K D h and K T h relationship.
It is important to highlight that the clearness index ( K T h ) is not independent of the zenith angle ( θ Z ) . A given K T h value represents significantly different conditions depending on whether
the sun is near the zenith or the horizon. This approach inherently carries an error. Better performance can be achieved by using another variable, independent of θ Z , to characterize insolation
conditions. Another well-known limitation of K T h is that, for a given K T h value within a specific range of solar elevation, the atmospheric conditions can vary significantly in terms of direct
and diffuse content. Additionally, other independent atmospheric variables can be incorporated if they provide relevant information about direct irradiance transmission. (Perez et al., 1990). For
instance, surface dew-point temperature ( T d ) serves as a reliable estimator of atmospheric precipitable water, which significantly influences both absorption and aerosol growth. This, in turn,
affects the balance between direct and diffuse irradiance, as well as the scattering and direct-to-diffuse ratio (Perez et al., 1992).
For the present study, we propose models that use the hourly clearness index ( K T h ) to estimate ( K D h ) (Erbs et al., 1982), due to their simplicity and their status as the first proposed
models. However, several other models, which perform marginally better than the currently used models for various global locations, will be implemented in future research. These models incorporate
additional atmospheric variables such as the hourly clearness index and solar elevation (Maxwell, 1987; Skartveit and Olseth, 1987), hour-to-hour variability index and regional surface albedo
(Skartveit and Olseth, 1987), dew-point temperature and hour-to-hour variability index (Perez et al., 1992), and apparent solar time with a measure of global radiation persistence (Ridley et al.,
2010). These enhanced models will be incorporated in future studies to improve performance.
The present study evaluates the climatic characteristics of the western Mantaro Valley using data from the HYGO conventional station (1981–2020) and the IGP automatic station (2017–2022). The Mantaro
Valley is classified as Cwb according to the Köppen-Geiger system, indicating a temperate climate with dry winters and warm summers. Summer sees peak precipitation, reaching 340 mm mont h − 1 in
February, while winter experiences minimal precipitation, with 28 mm mont h − 1 in July. The analysis suggests that radiation measurements accurately represent the valley’s climate, while factors
like E D F and E D R are more influenced by local climate and land-use, indicating cooler and drier conditions compared to the regional climatic norms.
The analysis of diurnal and seasonal variations in E D R , E D F , and E G in the western Mantaro Valley shows distinct patterns. At noontime, E D R h peaks during winter and decreases in summer.
Conversely, E D F h reaches its maximum in summer and declines in winter. Additionally, E G h peaks in spring and decreases in winter, influenced by astronomical factors, cloudiness, and aerosol
concentrations observed from 2017 to 2022. Seasonal variations in daily solar radiation components at the HYGO surface reveal that E D R d peaks in winter, notably in August, with the lowest values
in March. Conversely, peak E D F d values occur in summer, particularly in January, with the lowest values observed in winter, especially in July.
Notably, peak values of E G d occurred during spring, reaching its highest recorded value in October (24.14 MJ m − 2 d a y − 1 ) and its lowest in March (19.50 MJ m − 2 d a y − 1 ). This seasonal
variation in E G d correlates with periods of biomass burning, which are associated with elevated aerosol optical depth (AOD) levels in the Mantaro Valley region. Biomass burning events typically
occur from July to October annually, with September exhibiting peak AOD values. These months coincide with increased forest fire activity, both locally in Peru and in neighboring countries like
Brazil and Bolivia. The influx of biomass-burning aerosols contributes to higher AOD levels in September, affecting K D d and consequently reducing E D R d while increasing E D F d . In October, E D
F d shows a slight increase while E D R d remains constant, resulting in the peak of E G d for the year. This behavior is attributed to biomass burning events near the HYGO station.
This study investigates irradiation models that establish robust correlations among various solar radiation parameters. The MADHG and DAHG models relate the monthly-averaged daily clearness index ( K
T ) to daily sunshine fraction, while the MADHD and DAHD models connect the monthly-averaged daily diffuse ratio ( E D / E G ) to K T . Furthermore, the HHG and HHD models correlate the ratio of
hourly global or diffuse irradiation to daily values with the sunset hour angle. All these models demonstrated acceptable accuracy in predicting irradiance variables. Moreover, the sigmoid logistic
function emerges as the most effective in correlating K D h and K T h , demonstrating superior performance compared to alternative functions and exhibiting strong statistical significance. The AIC
method supports the superiority of the sigmoid function, emphasizing its efficacy in capturing the relationship between solar radiation components. It effectively reproduces the behavior of K D h as
a function of K T h , demonstrating superior statistical performance compared to other correlation models.
It is important to emphasize that this is the first study aimed at the observational characterization and empirical modeling of global and diffuse solar irradiances in the central Peruvian Andes,
using high-quality radiation data from sensors belonging to the BSRN network. In the future, this new model will be tested with E D F , and E G measurements collected in other regions of central
Andes and will be proposed better empirical models, physical parametric broadband models, perceptron neural-network techniques to estimate hourly values of the diffuse solar irradiance and machine
learning methods for solar radiation forecasting. However, the empirical models presented here can easily be used to forecast solar irradiance components with acceptable accuracy, just like the
proposals made for other South American cities. Future research should incorporate models that use additional atmospheric and solar variables for improved performance, such as solar elevation,
hour-to-hour variability index, regional surface albedo, dew-point temperature and apparent solar time with a measure of global radiation persistence.
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Kepler's Third Law
Kepler's Third Law
Kepler's third law states the relationship between a planet's average distance from the sun and the time required for the planet to complete one orbit
search_insights Statistics
Kepler's third law states
the relationship between a planet's average from the sun and the required for the planet to complete orbit
drag and drop the selected option to the right place or type it instead
Correct Answer
The time for one cycle of the sun is called a planet's ______ of revolution
Correct Answer
The following is Kepler's third law as a formula
Correct Answer
Newton's laws explain Kepler's third law
Planets at greater distances experience force from the sun and, if they are to orbit at that distance, they must be moving more than planet closer to the sun.
drag and drop the selected option to the right place or type it instead
Correct Answer
Define the period of a motion
Correct Answer
Do planets at greater distances from the sun have longer or shorter periods of revolution?
Correct Answer
sin cos tan
sin^-1 cos^-1 tan^-1 π e
x^y x^3 x^2 e^x 10^x
^y√x ^3√x √x ln log
( ) 1/x % n!
7 8 9 + MS
4 5 6 – M+
1 2 3 × M-
0 . EXP ÷ MR
± RND C = MC
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Rohlin properties for ℤ<sup>d</sup> actions on the cantor set
We study the space H(d) of continuous ℤ^d-actions on the Cantor set, particularly questions related to density of isomorphism classes. For d = 1, Kechris and Rosendal showed that there is a residual
conjugacy class. We show, in contrast, that for d ≥ 2 every conjugacy class in H(d) is meager, and that while there are actions with dense conjugacy class and the effective actions are dense, no
effective action has dense conjugacy class. Thus, the action by the group homeomorphisms on the space of actions is topologically transitive but one cannot construct a transitive point. Finally, we
show that in the spaces of transitive and minimal actions the effective action s are nowhere dense, and in particular there are minimal actions that are not approximable by minimal shifts of finite
Dive into the research topics of 'Rohlin properties for ℤ^d actions on the cantor set'. Together they form a unique fingerprint.
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Explorer, Inc. is considering a new 4-year project that requires
an initial fixed asset (equipment) investment...
Explorer, Inc. is considering a new 4-year project that requires an initial fixed asset (equipment) investment...
Explorer, Inc. is considering a new 4-year project that requires an initial fixed asset (equipment) investment of $200,000. The fixed asset is three-year MACRS property for tax purposes. In four
years, the equipment will be worth about half of what we paid for it. The project is estimated to generate $500,000 in annual sales, with costs of $400,000. The firm has to invest $100,000 in net
working capital at the start. After that, net working capital requirements will be 25 percent of sales. The tax rate is 21 percent. What is the incremental cash flow in year 1?"
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Placement Papers with Solutions
If I am given a formula and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?
Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations.
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A New Biological Constant?
Earlier, I gave
for a surprising relationship between the amount of G+C (guanine plus cytosine) in DNA and the amount of "purine loading" on the message strand in coding regions. The fact that message strands are
often purine-rich is not new, of course; it's called Szybalski's Rule. What's new and unexpected is that the amount of G+C in the genome lets you
the amount of purine loading. Also, Szybalski's rule is not always right.
Genome A+T content versus message-strand purine content (A+G) for 260 bacterial genera. Chargaff's second parity rule predicts a horizontal line at Y = 0.50. (Szybalski's rule says that all points
should lie at or above 0.50.) Surprisingly, as A+T approaches 1.0, A/T approaches the Golden Ratio.
When you look at coding regions from many different bacterial species, you find that if a species has DNA with a G+C content below about 68%, it tends to have more purines than pyrimidines on the
message strand (thus purine-rich mRNA). On the other hand, if an organism has extremely GC-rich DNA (G+C > 68%), a gene's message strand tends to have
more pyrimidines than purines
. What it means is that Szybalski's Rule is correct
only for organisms with genome G+C content less than 68%
. And Chargaff's second parity rule (which says that A=T an G=C even within a
single strand
of DNA) is flat-out wrong all the time, except at the 68% G+C point, where Chargaff is right now and then by chance.
Since the last time I wrote on this subject, I've had the chance to look at more than 1,000 additional genomes. What I've found is that the relationship between purine loading and G+C content applies
not only to bacteria (and archaea) and eukaryotes, but to mitochondrial DNA, chloroplast DNA, and virus genomes (plant, animal, phage), as well.
The accompanying graphs tell the story, but I should explain a change in the way these graphs are prepared versus the graphs in my earlier posts.
, I plotted G+C along the X-axis and purine/pyrmidine ratio on the Y-axis. I now plot A+T on the X-axis instead of G+C, in order to convert an inverse relationship to a
relationship. Also, I now plot A+G (purines, as a mole fraction) on the Y-axis. Thus, X- and Y-axes are now
expressed in mole fractions, hence both are normalized to the unit interval (i.e., all values range from 0..1).
The graph above shows the relationship between genome A+T content and purine content of message strands in genomes for 260 bacterial genera. The straight line is regression-fitted to minimize the sum
of squared absolute error. (Software by
.) The line conforms to:
y = a + bx
a = 0.45544384965539358
b = 0.14454244707261443
The line predicts that if a genome were to consist entirely of G+C (guanine and cytosine), it would be 45.54% guanine, whereas if (in some mythical creature) the genome were to consist entirely of
A+T (adenine and thymine), adenine would comprise 59.99% of the DNA. Interestingly, the 95% confidence interval permits a value of 0.61803 at X = 1.0, which would mean that as guanine and cytosine
diminish to zero, A/T approaches the
Golden Ratio
Do the most primitive bacteria (Archaea) also obey this relationship? Yes, they do. In preparing the graph below, I analyzed codon usage in 122 Archaeal genera to obtain A, G, T, and C relative
proportions in coding regions of genes. As you can see, the same basic relationship exists between purine content and A+T in Archaea as in Eubacteria. Regression analysis yielded a line with a slope
of 0.16911 and a vertical offset 0.45865. So again, it's possible (or maybe it's just a
strange coincidence) that A/T approaches the Golden Ratio as A+T approaches unity.
Analysis of coding regions in 122 Archaea reveals that the same relationship exists between A+T content and purine mole-fraction (A+G) as exists in eubacteria.
For the graph below, I analyzed 114 eukaryotic genomes (everything from fungi and protists to insects, fish, worms, flowering and non-flowering plants, mosses, algae, and sundry warm- and
cold-blooded animals). The slope of the generated regression line is 0.11567 and the vertical offset is 0.46116.
Eukaryotic organisms (N=114).
Mitochondria and chloroplasts (see the two graphs below) show a good bit more scatter in the data, but regression analysis still comes back with positive slopes (0.06702 and .13188, respectively) for
the line of least squared absolute error.
Mitochondrial DNA (N=203).
To see if this same fundamental relationship might hold even for
genetic material, I looked at codon usage in 229 varieties of bacteriophage and 536 plant and animal viruses ranging in size from 3Kb to over 200 kilobases. Interestingly enough, the relationship
between A+T and message-strand purine loading does indeed apply to viruses, despite the absence of dedicated protein-making machinery in a virion.
Plant and animal viruses (N=536).
For the 536 plant and animal viruses (above left), the regression line has a slope of 0.23707 and meets the Y-axis at 0.62337 when X = 1.0. For bacteriophage (above right), the line's slope is
0.13733 and the vertical offset is 0.46395. (When inspecting the graphs, take note that the vertical-axis scaling is not the same for each graph. Hence the slopes are deceptive.) The Y-intercept at X
= 1.0 is 0.60128. So again, it's possible A/T approaches the golden ratio as A+T approaches 100%.
The fact that viral nucleic acids follow the same purine trajectories as their hosts perhaps shouldn't come as a surprise, because viral genetic material is (in general) highly adapted to host
machinery. Purine loading appropriate to the A+T milieu is just another adaptation.
It's striking that so many genomes, from so many diverse organisms (eubacteria, archaea, eukaryotes, viruses, bacteriophages, plus organelles), follow the same basic law of approximately
A+G = 0.46 + 0.14 * (A+T)
The above law is as universal a law of biology as I've ever seen. The only question is what to call the slope term. It's clearly a biological constant of considerable significance. Its physical
interpretation is clear: It's the rate at which purines are accumulated in mRNA as genome A+T content increases. It says that a 1% increase in A+T content (or a 1%
in genome G+C content) is worth a 0.14% increase in purine content in message strands. Maybe it should be called the purine rise rate? The purine amelioration rate?
Biologists, please feel free to get in touch to discuss. I'm interested in hearing your ideas. Reach out to me on LinkedIn, or simply leave a comment below.
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Nikola Šerman,
prof. emeritus
Linear Theory
Transfer Function Concept Digest
Closed-loop dynamics is optimized to achieve the closed-loop stability with a satisfactory transients damping and a transients duration to be as short as possible. In terms of the Bode diagram
stability margins this means that the stability margins should be:
1. positive (provides stability as such),
2. sufficiently large (provides adequate transients damping),
3. within relevant frequency range (provides faster transients vanishing).
To comply with #3 the shaded area in fig. 11.1 should be as far as its achievable to the right side.
Figure 11.1 Stability margins and relevant frequency range (shaded area) in an open-loop frequency characteristics Bode plot.
Frequency characteristics of a serial connection in Bode diagram are made by a simple summation of the respective block characteristics. This offers a rational ground for qualitatively analyzing of
what are the effects of additional block inclusion into the open loop.
Figure 11.2 Closed-loop system under consideration – open loop W(s) = H(s) H[0](s) encircled by negative unity feed-back
Transfer function H(s) in fig 11.2 is an additional block. Fig 11.3 shows what would be the effects if that block is a static gain while fig 11.4 shows the same in a case of an integrator.
Figure 11.3 Effect of adding static gain H(s) = K to the component H[0](s) when K > 1.
Adding a static gain to the existing component H[0](s) results in an amplitude frequency characteristic vertical translation. The transition is upwards for K > 1 and downwards for K < 1, thus
directly influencing the stability margins. The red arrows are the stability margins after the inclusion of H(s).
Figure 11.4 Adding of an integrator s H[0](s)
Adding an integrator to the existing component H[0](s) rotates the amplitude frequency characteristic around its point at frequency ω[0]. The phase characteristic is shifted 90° downwards.
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Concrete Mix Recipes
Although there are many mix recipes for quality concrete here are some of our favorites. With any new mix design you should always test yourself before using on a project! Fishstone offers the most
cost effective way to produce high performance concrete and the knowledge to back it up.
How to calculate the amount of material needed for your pour
To obtain the cubic amount of mateial needed to fill a mold start by getting the measurements L x W x H (Length x With x Height) to come up with the total number of cubic inches needed. So for
example if we were making a countertop that is 48" long by 25.5" wide and a thickness of 2" the total cubic inches of material needed is 2,448 (48 x 25.5 x 2 = 2,448).
Now that we have the cubic inches we need to convert to cubic feet. So divide the cubic inches by 1,728(this is the number of cubic inches in one cubic ft) 2,448/1728=1.41 Cubic Feet of total
material needed. Assuming we may be slightly off or that we will have some concrete that spills on the floor I am going to add an additional .25cu ft. to my total batch (1.41 + .25 = 1.66). If we end
up with left over material it can use it to make some decorative stepping stones or the like.
We now know we need 1.66 cubic feet of concrete, which we will round out to 1.7, and can determine the exact amount of material we need to make enough mix to fill the molds. After choosing a mix
design(1 cu. Ft.) take the amount for each ingredient in the mix recipe and multiply each by the 1.7 that was calculated earlier resulting in the exact amount of each ingredient needed.
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Regressing Meaning: Definition, Examples, Uses
You can use regression analysis in many professional fields, and understanding this type of analysis technique can expand your ability to explore relationships between variables and make accurate
predictions and informed decisions. Discover the meaning of regression analysis, foundational concepts, the advantages and disadvantages of this method, and more.
Read more: What Is Linear Regression? (Types, Examples, Careers)
What is regression analysis?
Regression analysis is a statistical methodology that explores the relationship between a dependent variable and one or more independent variables. The letter “Y” generally denotes the dependent
variable, and the independent variable is an “X.”
In simpler terms, you can think of regression as a way to predict a future outcome based on what has happened in similar scenarios (i.e., based on our existing data sets). We can use this
mathematical model to predict the outcome (the dependent variable) based on the input or changes in the other variables (the independent variables). In a linear regression model, you would have a
continuous outcome and create a line equation to predict future outcome values. In a logistic regression model, your outcome is a fixed categorical event (e.g., yes/no or pass/fail), and you predict
the probability your outcome will be in a certain category.
Researchers also use regression analysis to determine which independent variables affect the dependent variable. If you suspect that a set of variables is impacting your outcome, you can use
regression analysis to determine which variables are the most important in your model and have the biggest impact on your outcome.
Foundational concepts
When you perform regression analysis, you will work with a certain key set of concepts. Understanding these concepts supports the design and application of regression analysis: independent variables,
dependent variables, correlation, and causation.
1. Independent variables
In a regression model, the independent variables, or the explanatory variable, are the factors that you believe will impact the outcome of the variable you are interested in understanding or
predicting. As the name suggests, they are independent, and the researcher can manipulate them to observe the corresponding changes in the dependent variable.
For example, if you are trying to predict someone’s likelihood of developing a disease, your independent variables might be their age, health status, activity level, and biological metrics.
2. Dependent variables
The dependent variable, or response variable, is the outcome in a regression model. This is the variable you aim to understand, predict, or explain. Its value is dependent on the changes in the
independent variables.
For example, in a business scenario, the dependent variable could be sales, which might depend on independent variables like marketing budget, pricing, and competition.
3. Correlation
Correlation is a statistical measurement representing the magnitude of the relationship between several variables. You can represent this measure as a correlation coefficient (denoted as “r”),
ranging from negative to one.
If r is positive, the correlation is positive, meaning both variables increase or decrease together. If r is negative, one variable decreases as the other increases. If r equals zero, it represents
no correlation between the variables. It is important to note that correlation does not equal causation.
Read more: Correlation vs. Causation: What’s the Difference?
5. Causation
While correlation can signal a relationship between variables, it does not infer causation. Correlation means two variables relate and change with one another. Causation means that a change in one
variable causes a change in the other. If a causal relationship between variables is present, regression analysis can predict the outcome of our dependent variable based on changes to the independent
Saying changes in one or multiple variables definitively cause an outcome is a stronger claim than saying two variables relate. Because of this, determining causal relationships requires much
stricter assumptions and analysis than correlation.
How does regression analysis work?
Regression analysis begins with data—or information about the variables you would like to assess. Using this data, you can create a mathematical model, typically a line or curve, that best
illustrates the relationship between the dependent and independent variables.
Once you have your estimate or prediction from your model, you can look at the standard error of the prediction to see whether your prediction is strong or weak. This tells you how much you can trust
your model and helps you build a confidence interval to better represent your true regression coefficient.
You can also examine statistical metrics to see how each independent variable you include affects your model. This can show you how important each variable is and help you decide which independent
variables to include in your model to predict the value of your response variable most accurately.
Origins of regression analysis
The term “regression” dates back to the 19th century. Sir Francis Galton, a British scientist, pioneered the method in his studies of heredity. He noted that extreme traits in parents often
“regressed” toward the average in their offspring, leading to the term “regression.”
Examples of regression analysis
Regression analysis is present in almost every professional field. Usually, an investigator in a certain industry shows interest in the causal effect of certain variables on another. These variables
are changed by industry, but the principle of the regression analysis is the same. Some ways in which you might use regression analysis include the following:
• Economics: Predicting a family’s spending patterns based on their location, number of children, and income
• Health care: Predicting whether someone will recover from an illness based on their age, blood pressure, weight, and medical history
Benefits of using regression analysis
One of the primary benefits of regression analysis is its ability to quantify and model the relationship between different variables, allowing you to make predictions. It allows us to estimate the
strength and direction of one or more independent variables' impact on the dependent variable.
Regression analysis is flexible and can deal with more than one independent variable simultaneously. This enables professionals to consider complex, multifactor scenarios. For instance, a company
might use regression analysis to understand how pricing, advertisement spending, and market competition collectively impact sales.
Drawbacks of regression analysis
Despite its many strengths, regression analysis does come with a set of limitations. For one, while regression analysis can suggest relationships and correlations, it does not prove causation. It may
provide evidence of a causal effect under certain conditions, which require careful study design to meet in practice.
In addition to this, regression analysis can include pitfalls such as:
• Nonconstant variances (heteroskedasticity): Nonconstant variance happens when the error term is not constant, preventing the model from being well-defined. This can cause prediction intervals to
be too wide or too narrow with new information.
Getting started with Coursera
Regression analysis allows you to explore the relationship between a dependent and independent variable to predict future outcomes. With regression analysis, you can observe how strongly each
independent variable influences the model. This helps professionals across industries make informed decisions, predict future outcomes, and explore how modifying certain practices can change
short--and long-term outcomes.
You can build your regression toolkit by taking courses or Specializations on Coursera. With beginner courses, such as Linear Regression and Modeling by Duke University. You can learn key
fundamentals like linear regression, multiple regression, and more.
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CTAN Update: pst-math
Date: December 14, 2018 5:52:08 PM CET
Herbert Voß submitted an update to the pst-math package. Version: 0.64 2018-12-13 License: lppl Summary description: Enhancement of PostScript math operators to use with PSTricks Announcement text:
pst-math is a PSTricks related package. This version adds some macros for random numbers. The package’s Catalogue entry can be viewed at
The package’s files themselves can be inspected at
Thanks for the upload. For the CTAN Team Petra Rübe-Pugliese We are supported by the TeX users groups. Please join a users group; see
pst-math – Enhancement of PostScript math operators to use with PSTricks
PostScript lacks a lot of basic operators such as tan, acos, asin, cosh, sinh, tanh, acosh, asinh, atanh, exp (with e base). Also (oddly) cos and sin use arguments in degrees.
Pst-math provides all those operators in a header file pst-math.pro with wrappers pst-math.sty and pst-math.tex.
In addition, sinc, gauss, gammaln and bessel are implemented (only partially for the latter). The package is designed essentially to work with pst-plot but can be used in whatever PS code (such as
PSTricks SpecialCoor "!", which is useful for placing labels).
The package also provides a routine SIMPSON for numerical integration and a solver of linear equation systems.
Package pst-math
Version 0.67 2023-07-03
Copyright 2023 Herbert Voss
Maintainer Herbert Voß
Christophe Jorssen (inactive)
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Retirement and Immortality
If people could live forever, then would they still be able to retire or would pension schemes become bankrupt from having to pay out never ending streams of money to eternally youthful pensioners?
This might sound like an overly academic question, however I've noticed a glut of articles recently about the prospects of
extending the human lifespan, for example:
And a feature in this week's economist:
Many of these articles raise the point that extending the human lifespan would have an effect on the affordability of retirement. Claiming that retirement as we know it would come to an end.
Is this correct though? Most of the articles that I found simply state that retirement would now be impossible without explaining precisely why. I thought I'd do some calculations and see how much
of an effect increased lifespans would have on pension scheme funding, It turns out that the situation is not as bad as it seems.
First we need to introduce an actuarial technique that allows us to calculate the cost of paying someone a pension into the future.
Suppose that we have a payment stream of amount $£1$, payable annually, and paid forever. Graphically we are talking about the following situation::
Can we place a value on this series of payments?
You might suppose that since we are paying out an infinite amount of money that the value of the perpetuity (which is the name of a payment stream that continues indefinitely) should be infinite.
However finance uses a concept called ‘net present value’ or NPV, to assign a value to this stream, and this value turns out to be finite.
Time value of money
First we need some extra information, let's assume that we have access to a risk-free bank account that pays out an interest rate of $i$% per annum. (For example, $i$ might be $5$). So that if we
invest $1$ at time $t=0$, it will be worth $1*(1+i)$ at time $t=1$, and worth $1*(1+i)*(1+i)$ at time $t=2$.
Let's first solve a simplified problem and just consider the value of the first payment of $1$ at time $1$. If we wish to invest an amount of money now, so that we can pay the $1$ due at time $t =
1$ with the money we have invested, then if we put $1*(1+i)^{-1}$ in the bank account now, this will be worth $1$ at time $1$. Similarly to invest an amount now so that we will be able to pay the
amount $1$ at time $n$, we need to invest ${1/(1+i)}^{-n}.$
Going back to the original problem, we can use this result to calculate the amount we should invest now so that we can pay all the future payments. It will be:
$\sum_{k=1}^{\infty} {1/(1+i)}^{-k} $
This is just a geometric series, which sums to $1/ (1- (1+i)) = 1/i.$
Therefore the amount we need to invest is $1/i$. So, if $i=5%$ as we had earlier, then we actually only need to invest $20$ now in order to pay able to pay someone $1$ every year, for ever!
Checking the Result
Let's check that this makes sense. Suppose that we do invest this amount at $t=0$, then at $t=1$ we will have $(20)(1.05) = 1 + 20$.
Which gives us $1$ to pay the perpetuity, and leaves enough to invest the original amount again. We can see that if nothing else changes, this system can continue indefinitely.
Returning to our previous question, would it be possible to pay a pension to someone who will live forever? The answer is yes. We can even calculate the amount that we would need to invest now to
pay the pension.
Increasing pensions
This system we have derived is not very realistic though. Most pensions increase over time, for example the state pension in the UK increases by the minimum of CPI, $2.5%$ and the average annual
wage growth. Given that a pension that increases over time will not just pay out an infinite amount, but will also grow to be infinitely big. Would such a pension still work in a scenario where
pensioners live forever?
It turns out in fact that yes, such a system is still sustainable under certain conditions.
Let’s suppose that our perpetuity increases at a rate of $g%$ per year, so here we might assume that $g=2.5$. Then if we derive the amount that we will need to invest now (at an investment rate of
$i%$) we find that we will need $((1+g)/(1+i))^n$. Summing all of these values gives the following initial value:
$\sum_{k=1}^{\infty} {(1+g)/(1+i)}^{-k} $this is another geometric sum, however now we need to consider convergence, this sum will converge iff $((1+g)*(1+i)) <1$. That is, the sum will converge iff
we can find an investment that grows faster than the perpetuity we have promised to pay.
If the sum converges then it will converge to $(1/(i-g))$.
So using the example we had earlier, where $i=5%$, $g=2.5%$ we would need to invest $ 1/0.025 = 40$. Which is substantially more than the non-increasing annuity but still finite.
So we see that even when the pension is increasing, we can still afford to set it up.
Mortality Premium
Can’t we still make the argument that pensions will become unaffordable due to the fact that these perpetuities will still cost a lot more.
Let's compare the value of an annuity assuming the pensioner will live to the average life expectancy with one where the pensioner is assumed to live forever.
Looking in my orange tables (the name for the Formula and Tables for Actuarial Exams) I see that the cost of a non-increasing pension of $1$ per year, paid to a $65$ year old male at a discount rate
of $4%$ is $12.66$. Compare this to the value of $25$ to pay the pension forever, we see that the cost of the pension is roughly double. This amount is much higher, but it's interesting to note that
the increase in cost is similar as that between increasing and non-increasing pensions!
Can we actually live forever?
We should also consider the fact that mortality will never actually be $0$ even if ageing is eliminated. Presumably accidents, and possibly even illness unrelated to age would still exist and for
this reason we would not expect to have to pay these pensions forever. If you live long enough, then the probability of all non-impossible events occurring at some point should eventually reach $1$.
This means that eventually even the most unlikely accidents would eventually happen if you lived long enough.
Your comment will be posted after it is approved.
Leave a Reply.
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A Body moves a distance of 10 m along a straight line under the action
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Coriolis Effect In A Rotating Space HabitatThe Oikofuge: Coriolis Effect In A Rotating Space Habitat
Coriolis Effect In A Rotating Space Habitat
In a previous post describing the Coriolis effect, I mentioned its relevance to space travel—if a rotating habitat is being used to generate simulated gravity, Coriolis deflection can interfere with
the performance of simple tasks and, at the extreme, generate motion sickness.
As an example of the sort of effect you could expect to encounter, I posted the following pair of diagrams:outside a rotating habitat (direction of rotation marked by the blue arrow), with the ball
and the experimenter marked at four successive positions they occupy while the ball falls. The ball moves in a straight line, with a velocity determined by the rotation speed of the habitat at its
point of release. The floor (and the experimenter) meanwhile move in a curved path, and they travel a little farther than the ball does during its time in flight.
The result, as observed by the experimenter rotating with the habitat, is shown in the second image—the ball appears to be deflected to the right as it falls. To explain this deflection, the
experimenter invokes the Coriolis pseudo-force, which I explained in much more detail last time. This rightward deflection of moving objects occurs in all counterclockwise-rotating reference frames
(leftward in clockwise-rotating frames).
Having prepared those two diagrams, I got to thinking about the range of possible trajectories one might encounter, while chucking a ball around in a rotating centrifuge. Out of curiosity, I put
together some code to sketch the resultant trajectories for objects launched at any angle, with any speed. The results I’ll show are fairly generalizable—it turns out the trajectory depends only on
the launch velocity as a proportion of the rotation speed of the habitat. Interesting things happen when the velocity is comparable to the speed of rotation of the habitat floor—at higher velocities
trajectories become progressively flatter (and for our purposes, more boring).
First, a bit of terminology. Back in 1970, Larry Niven coined two useful words in his science fiction novel Ringworld, which dealt with a (very large!) rotating space habitat. Niven called the
direction in which his habitat rotated spinward, and the opposite direction antispinward. So in the case of an object that’s simply dropped within the habitat, as in the situation diagrammed above,
we can say that the object will always hit the floor antispinward of its release point.
Which means you need to impart a little spinward velocity to an object to get it to hit the floor directly below its launch point. Here’s a set of spinward trajectories, as observed in the rotating
reference frame of the habitat, with each object being launched “horizontally” (that is, parallel to the part of the curved floor on which our experimenter is standing):
The curve labelled “0” is a launch with no horizontal velocity—just a simple drop, as previously illustrated. The curve labelled “1” is the trajectory of an object that has been thrown with an
additional velocity equal to the speed of the habitat’s rotation at the launch height. The red curves up to “5” are objects thrown with twice, three times, four times and five times the local
rotation velocity, and the blue curves subdivide the span from “0” to “1” into ten equal increments. At higher velocities, the object falls to the floor in a curve that doesn’t seem too
counterintuitive compared to a standard gravitational field.
But if our experimenter turns the other way and throws objects to antispinward, more interesting stuff happens:
The curve labelled “0” is the same trajectory as before. The blue lines are the same increments in launch speed as in the previous diagram, but in this direction the rightward deflection of Coriolis
is serving to lift each trajectory, so the object flies farther and swoops around the curve of the habitat before it strikes the floor. The green trajectory, with a launch velocity of magnitude 0.9
times the local rotation speed, is remarkable. It doesn’t just sweep out of sight to antispinward, it reappears from spinward and makes more than a complete circuit of the habitat before it hits the
For clarity, I saved the red trajectories, with velocities from 1 to 5, for another diagram:
The trajectory labelled “1” simply stays at the same height constantly, in principle going round and round the habitat for ever at the same speed, buoyed up by Coriolis force. (If there’s any air in
our habitat, of course, it would actually slow down and fall to the floor because of air resistance.) Trajectories with higher launch velocities become progressively flatter, but still exhibit upward
So what’s going on with Trajectory 1? That object has been launched with a velocity that exactly cancels the rotation speed of the habitat. To an outside observer, such an object just hangs in space,
stationary, while the habitat rotates around it, carrying the experimenter past the object repeatedly, once per rotation. To the same outside observer, all objects with blue or green antispinward
trajectories in my diagram are actually floating slowly to spinward, having had some, but not all, of their rotation speed removed—but because the habitat and experimenter move faster to spinward,
objects on these slow trajectories recede to antispinward in the rotating reference frame.
A diagram may help illustrate this. Here’s how a non-rotating observer sees the situation, when an object is thrown antispinward with a velocity less than the local rotation speed:
Now, back to something I mentioned earlier. To make an object land on the floor directly below its launch point, it needs to be given a little nudge to spinward as it’s released. The closer our
experimenter is to the axis of the habitat (the higher above its floor), the more of a nudge the object needs, and the wider the curved trajectory it follows. Here are trajectories for objects
launched from a variety of heights within the habitat:
Each of them curves steadily to the right, moving initially spinward and then returning antispinward.
The same thing happens if you launch an object “vertically” (that is, aiming directly towards the spin axis). For each height above the floor of the habitat, there is a unique launch velocity that
will allow Coriolis to curve the trajectory around so that it strikes the floor directly below the launch point:
Interestingly, the launch velocity required in this situation initially increases as our experimenter climbs closer to the spin axis, but then decreases again at radii less than about 0.3 times the
radius of the floor. But, as before, the trajectories get progressively wider as the experimenter climbs closer to the spin axis.
A corollary to all these spinward curves is that, if you want to throw an object up and catch it, you need to throw it a little antispinward of vertical. Its trajectory curves right on the way up and
on the way down, and will return to your hand in a closed loop if you have thrown it correctly. The more speed you impart, the more antispinward you need to direct your throw, so we have a family of
possible curves that will carry the tossed object back to its starting point:
If you get it wrong, and throw your object too far to antispinward, then the overhead loop may still occur, but the object won’t return to your hand, as in the green trajectory below:
All the trajectories in this diagram have the same launch speed, but different launch directions. The blue trajectory is the perfect throw-and-catch loop. The green trajectory still loops, but the
object falls to antispinward. The red trajectory corresponds to a critical launch angle, at which the loop just disappears, leaving the object momentarily stationary in the rotating reference frame,
just at the peak of its trajectory. At launch angles flatter than the critical angle, we get something like the black trajectory, in which the object simply rises and then falls again, without any
fancy embellishments. It’s important to note that all of these trajectories involve objects that have been launched with antispinward velocities of lower magnitude than the local rotation speed at
the launch point. To a non-rotating outside observer, they’re therefore still moving spinward, but more slowly than the habitat and experimenter are rotating, so they are moving antispinward in the
rotating reference frame of the experimenter and the habitat.
What’s happening with the red trajectory is that the experimenter, by choosing an upward trajectory, has propelled the object to a small radial distance within the habitat, to a point where the
slower rotation speed exactly matches the object’s slow spinward velocity. So as it passes through this point, the object is momentarily stationary relative to the rotating habitat.
In the green trajectory, the object is thrown higher, and its slow spinward velocity now exceeds the rotation speed in a region close to the spin axis. So although it moves antispinward relative to
the rotating habitat when it’s close to the floor, it moves spinward relative to the habitat when it’s close to the axis—hence the looping trajectory in the rotating habitat frame.
That’s maybe a bit difficult to visualize, so here’s a picture of what the distribution of rotation speeds looks like in the rotating habitat:
So if the experimenter throws something upwards, it travels into regions that have a lower rotation speed because they’re nearer to the spin axis.
And, again, as with the simple horizontal throws, the trajectory of a thrown object is determined by summing the experimenter’s rotation speed and the launch velocity, like this:
In this case, the rotating experimenter throws an object up and antispinward, but the resulting velocity in the non-rotating frame is pointed up and spinward. Note that the experimenter is initially
moving spinward faster than the thrown object is, so will see it recede to antispinward. But to an external, non-rotating observer, the situation looks like this:
At the peak of its trajectory, the object is able to outpace the habitat’s rotation, and so briefly moves towards the experimenter again, creating the loop that we saw appear in the rotating
reference frame.
So that’s the theory. But would these trajectories be observable in any plausible rotating space habitats?
They would. My diagrams are actually roughly to scale for the small Discovery centrifuge that featured in the novel and film 2001: A Space Odyssey.
As I discussed in a previous post about centrifugal force, that structure, 35 feet across, is probably about as small as a space centrifuge could be without causing serious motion sickness in its
inhabitants because of Coriolis effect. In Arthur C. Clarke’s novel, it rotated at 6rpm, to produce the centrifugal equivalent of lunar gravity. In Stanley Kubrick’s film it was necessarily depicted
generating the equivalent of Earth’s gravity, which would require it to rotate at about 13rpm. But the rotation speed turns out not to matter, because the centrifugal and Coriolis effects scale
equally with angular velocity, so trajectories stay the same. If one of the Discovery astronauts dropped an object from a metre above the floor of the centrifuge, it would travel along a curve like
the one I’ve illustrated above, landing about 75 centimetres antispinward of its release point. The only difference would be that it would fall more slowly in a centrifuge that was rotating slowly.
The effect is also immune to changes in linear scale—if we make the centrifuge twice as large and drop the object from twice the height, the shape of its trajectory will be the same, and it will land
twice as far to antispinward.
This constancy with scaling also applies to trajectories that involve throwing an object—so long as the launch velocity keeps the same proportion to the rotation speed, the trajectory will be the
same shape.
For the Discovery centrifuge, the rotation speed at floor level is 3.2 m/s (seven miles per hour) in the version that appears in the novel, and 7.2 m/s (16 mph) for the film version. So the
astronauts could very easily throw objects into the various trajectories I’ve shown. For 2001‘s larger space station, shown in the image at the head of this post, the rim speed is 15 m/s (34 mph) for
the lunar-gravity version in the novel, and 37 m/s (84 mph) for the 1g version in the film. So it would take a fairly strong wrist, or a hand catapult, to launch an object so that it curved out of
sight down the long circumferential corridor that featured in the film. If you dropped an object from a metre up in that environment, it would fall a mere eight centimetres to antispinward. *
And there’s the problem—as the habitat gets larger, the human scale becomes proportionally smaller, so the Coriolis effects become less noticeable. On the scale of an O’Neill habitat, kilometres in
diameter, the Coriolis deflection in a fall of one metre at the rim amounts to only a centimetre, and begins to get difficult to see; and the rotation speed at the rim is measured in hundreds of
metres per second, so launching objects on interesting trajectories becomes problematic. In these large-scale habitats, the interesting stuff happens only near the axis (where rotation velocities are
low), or on large scales (for instance, if an object falls from a great height).
Sadly, then, Tye-Yan “George” Yeh’s beautiful Coriolis fountain can only ever grace the smallest of rotating habitats.
Postscript: In response to some queries I’ve received, I’ve written a supplementary article discussing what happens to objects that are moving parallel to the habitat’s rotation axis, and also
describing the effect of Coriolis on objects that are rolling along a surface, rather than thrown through the air. You can find that here.
Post-postscript: If you’re the sort of person who finds this post interesting, you might also be interested in my posts about Human Exposure to Vacuum, Parts One (theory) and Two (experimental
Note: Just to bring my points of reference into the 21st century, I’ll point out that the centrifuges that featured in the Endurance spacecraft from the film Interstellar (2014) and the Hermes from
The Martian (2015) are intermediate in size between the two centrifugal habitats used in 2001: A Space Odyssey, which I’ve been using as examples. So we could expect Coriolis effects to feature
reasonably prominently in either environment.
All these fictional centrifuges are to some extent unrealistic, at least in the short term, because they involve a lot of mass which would need to be moved to orbit and then moved around in space.
Where centrifuges are proposed for prolonged space missions, as in Robert Zubrin’s Mars Direct project, they involve whirling a small habitat around on the end of a long tether—usually with a radius
of gyration at least as large as the large Space Station V from 2001: A Space Odyssey. Coriolis deflection would therefore be potentially observable (for instance in dropping objects or throwing and
catching), but there simply wouldn’t be room for the longer trajectories I’ve described here.
* The deflection in the trajectory of a dropped object is a useful parameter to estimate the significance of Coriolis in a given habitat. As discussed in the text, it’s unaffected by rotation rate,
and scales with the size of the habitat. If we drop an object from radius r, and it lands at radius R, the magnitude of the antispinward deflection is given by:
\small R\left [ \sqrt{\left ( \frac{R^{2}}{r^{2}}-1 \right )} -\arccos \left ( \frac{r}{R} \right )\right ]
(Note that the arccos term needs to be in radians, not degrees.)
10 thoughts on “Coriolis Effect In A Rotating Space Habitat”
1. Fascinating pair of Coriolis articles. In a space-related college class, we’re looking at the use of a horizontal rotating torus (producing about 1 g) on the surface of the Moon, where we would
have a perpendicular g/6 at right angles to the motions in 2D that you describe. Are you aware of any such problem being tackled? Have you done it yourself?
Also, would it be possible to have access to the code you wrote for this article? My intuition is that this would be superimposed on your solutions.
Looking forward to your answers here.
2. I’m glad you found it interesting.
Years ago, I designed a lunar “Paraboloid Park” to address the situation you describe—a paraboloid of rotation set vertically, apex down, and rotating around its axis of symmetry. Gravity at the
axis (through which one could enter the park) is entirely lunar. Standing there, you would see the huge paraboloid landscape rotating around and above you. Stepping out on to the rotating surface
of the paraboloid, the effective gravity vector would always be at right angles to the surface, comprised of the vector sum of lunar gravity and centrifugal gravity. Walking away from the axis,
and up the curvature of the paraboloid, you’d move into areas of higher gravity, eventually reaching a zone in which the combined effect of lunar gravity and centrifugal gravity was 1g—the
environment of your much more compact torus.
Of course, the consequences of mechanical failure in such an object would be catastrophic, but it was good fun to think about!
I haven’t ever done the Coriolis maths on such an environment, but it sounds like fun.
I’m afraid the code for these diagrams is a mess of custom routines and spreadsheets—the project grew rather haphazardly—so there’s nothing I could usefully pass on to you. My approach was fairly
artisanal, anyway. I didn’t calculate any pseudoforces at all. I simply calculated the initial velocity vector in the non-rotating frame, and then calculated the position in the rotating frame at
time intervals short enough to allow me to plot a smooth curve. I imagine your project will be aiming for something more elegant!
3. Excellent article. I’m trying to calculate the deflection of an object dropped from a height of 1m in a torus or radius 900m. So I assume r=899 and R=900. This gives a deflection of 2388m. Surely
that’s not right. What am I doing wrong?
1. Thanks for the kind words.
I’d guess you’ve calculated using the arccos in degrees rather than radians, which would account for the large value you’ve derived. (Sorry that wasn’t clear in my original text – I’ve now
added a note to make that requirement explicit.)
Using your figures, and expressing the arccos in radians, I’m getting a deflection of just over 3cm.
4. “But the rotation speed turns out not to matter, because the centrifugal and Coriolis effects scale equally with angular velocity, so trajectories stay the same.”
Actually, that’s not correct. “Centrifugal” / centripetal acceleration and force scale with the square of the angular velocity (A_cent = Ω²·R) whereas Coriolis acceleration and force scale with
the angular velocity not squared (A_Cor = 2·Ω·v). It’s true that dropped balls with zero initial relative velocity v will follow the same relative trajectory independent of the angular velocity
or “gravity” level, but for thrown balls with non-zero initial v, the shape of the trajectory depends on the ratio of the relative velocity v and the environment’s tangential velocity V. This is
illustrated in figures 9-13 in this paper: http://www.artificial-gravity.com/AIAA-2006-7321.pdf
For more mathematical analysis, see: http://www.artificial-gravity.com/AIAA-99-4524.pdf
1. Thanks for the comment, but if you check my diagrams, you’ll see that they are deliberately generated for stipulated ratios of launch velocity in the rotating frame and rotation speed at the
launch position in the inertial frame, to address the problem you mention. So, in accordance with what you point out, the depicted trajectories are independent of the rotation speed of the
environment. If you spin the toy habitat in my diagram twice as fast, you must launch your ball twice as fast, and it follows the same trajectory twice as fast.
As I wrote shortly after the section you quoted, “This constancy with scaling also applies to trajectories that involve throwing an object—–so long as the launch velocity keeps the same
proportion to the rotation speed, the trajectory will be the same shape.”(My bold.)
5. Sorry, I should have read more carefully before commenting. I missed the stipulation that “the trajectory depends only on the launch velocity *as a proportion of the rotation speed* of the
With the stipulation that launch velocity is a constant proportion of the rotation speed, so that v = Ω·k, then the Coriolis effect 2·Ω·v = 2·Ω²·k and it scales equally with the centrifugal
effect Ω²·R, for any value of Ω, assuming that k and R are both constants or also scale together.
1. Thanks for coming back to resolve the issue—I appreciate your taking the time to do so.
6. Just to run on a little with Ted’s parameter k, above:
If we make R the radius of the habitat at which our projectile is launched, and use c to designate the relative velocity parameter with which I’ve been labelling my trajectory plots, then k =
R·c. So the Coriolis pseudoforce on the projectile in flight is 2·Ω·v = 2·Ω²·k = 2·Ω²·R·c, which compares to the centrifugal pseudoforce at the launch radius R of Ω²·R. If we set c=1, then the
coriolis pseudoforce is exactly twice the centrifugal force at the moment of launch.
If we launch “horizontally” to antispinward, Coriolis opposes centrifugal. The factor of 2 means that Coriolis undoes the floorward acceleration cause by centrifugal pseudoforce, and introduces a
ceilingward acceleration that is exactly equal and opposite. This centripetal force is enough to keep the antispinward projectile with velocity parameter 1 floating along at the same distance
above the floor, describing a circle around the habitat at constant radius R. Which is exactly what you see in one of my diagrams above.
In the inertial, non-rotating reference frame, we can understand this because the projectile has been launched with a velocity that exactly counters its rotational speed, so that (barring air
resistance) it merely hovers in place while the habitat revolves around it.
7. Thank you very much for this article!
I’ve been creating a ring world for my novel. For now it has 1500 km in diameter. Creatures with intellect there are birds having a collective intelligence. My conversation with GPT gave me some
information that flying in the direction of rotation will be more difficult than in other way. This should be heavier in one way and easier in another. I want to build a plot around that. Birds
in their pagan ages would call it Calling and Whispering as metaphorical emanations of evolution and entropy.
Can we talk online, please? If it is possible, I’d very appreciate it.
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EViews Help: An Illustration
An Illustration
Suppose we produce a dynamic forecast using EQ01 over the sample 1959M01 to 1996M01. The forecast values will be placed in the series HSF, and EViews will display a graph of the forecasts and the
plus and minus two standard error bands, as well as a forecast evaluation:
This is a dynamic forecast for the period from 1959M01 through 1996M01. For every period, the previously forecasted values for HS(-1) are used in forming a forecast of the subsequent value of HS. As
noted in the output, the forecast values are saved in the series HSF. Since HSF is a standard EViews series, you may examine your forecasts using all of the standard tools for working with series
For example, we may examine the actual versus fitted values by creating a group containing HS and HSF, and plotting the two series. Select HS and HSF in the workfile window, then right-mouse click
and select . Then select and select Line & Symbolin the page to display a graph of the two series:
Note the considerable difference between this actual and fitted graph and the depicted above.
To perform a series of one-step ahead forecasts, click on on the equation toolbar, and select Static forecast. Make certain that the forecast sample is set to “1959m01 1995m06”. Click on . EViews
will display the forecast results:
We may also compare the actual and fitted values from the static forecast by examining a line graph of a group containing HS and the new HSF.
The one-step ahead static forecasts are more accurate than the dynamic forecasts since, for each period, the actual value of HS(‑1) is used in forming the forecast of HS. These one-step ahead static
forecasts are the same forecasts used in the displayed above.
Lastly, we construct a dynamic forecast beginning in 1990M02 (the first period following the estimation sample) and ending in 1996M01. Keep in mind that data are available for SP for this entire
period. The plot of the actual and the forecast values for 1989M01 to 1996M01 is given by:
Since we use the default settings for out-of-forecast sample values, EViews backfills the forecast series prior to the forecast sample (up through 1990M01), then dynamically forecasts HS for each
subsequent period through 1996M01. This is the forecast that you would have constructed if, in 1990M01, you predicted values of HS from 1990M02 through 1996M01, given knowledge about the entire path
of SP over that period.
The corresponding static forecast is displayed below:
Again, EViews backfills the values of the forecast series, HSF1, through 1990M01. This forecast is the one you would have constructed if, in 1990M01, you used all available data to estimate a model,
and then used this estimated model to perform one-step ahead forecasts every month for the next six years.
The remainder of this chapter focuses on the details associated with the construction of these forecasts, the corresponding forecast evaluations, and forecasting in more complex settings involving
equations with expressions or auto-updating series.
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System of Equation Applications (Word Problems)
System of Equation Applications (word problems)
1. Cut out each box below. 2. Paste on a new sheet in your notebook. 3. Title your page
Define 2 variables and set up system of equations. Solve, showing all steps (below the problems). Label your answers using words.
1. Kerry asked a bank teller to cash a $390 check using $20 and $50 bills. If the teller gave her a
total of 15 bills, how many of each type of bill did she receive?
Define your variable system of equations answer
2. Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales
totaled $2280 and 128 people attended. How many tickets of each type were sold?
Define your variable system of equations answer
3. On Friday, the With-It Clothiers sold some jeans for $25 a pair and some shirts at $18 each.
Receipts for the day totaled $441. On Saturday the store priced both items at $20, sold exactly
the same number of each item, and had receipts of $420. How many pairs of jeans and how
many shirts were sold each day?
Define your variable system of equations answer
4. A grain storage warehouse has a total of 30 bins. Some hold 20 tons of grain each, and the rest
hold 15 tons each. How many of each type of bin are there if the capacity of the warehouse
is 510 tons?
Define your variable system of equations answer
Name ______period ______
Use same directions from previous sheet.
5. A caterer’s total cost for catering a party includes a fixed cost, which is the same for every party.
In addition the caterer charges a certain amount for each guest. If it costs $300 to serve 25 guests
and $420 to serve 40 guests, find the fixed cost and the cost per guest.
Define your variable system of equations answer
6. A financial planner wants to invest $8000, some in stocks earning 15% annually and the rest in
bonds earning 6% annually. How much should be invested at each rate to get a return of $930
annually from the two investments?
Define your variable system of equations answer
7. For a recent job, a plumber earned $28/h, and the plumber’s apprentice earned $15/h. The
plumber worked 3 hours more than the apprentice. If together they were paid $213, how much did
each earn?
Define your variable system of equations answer
8. Davis Rent-A-Car charges a fixed amount per weekly rental plus a charge for each mile driven. A
one-week trip of 520 miles costs $250, and a two-week trip of 800 miles costs $440. Find the
weekly charge and the charge for each mile driven.
Define your variable system of equations answer
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06 ounces to grams
Convert 6 Ounces to Grams (oz to gm) with our conversion calculator. 6 ounces to grams equals 170.097148151214 oz.
Enter ounces to convert to grams.
Formula for Converting Ounces to Grams (Oz to Gm):
grams = ounces * 28.3495
By multiplying the number of grams by 28.3495, you can easily obtain the equivalent weight in grams from ounces.
Converting ounces to grams is a common task that many people encounter, especially when dealing with recipes, scientific measurements, or everyday tasks. Understanding the conversion factor is
essential for accurate measurements. In this case, the conversion factor from ounces to grams is 28.3495. This means that one ounce is equivalent to approximately 28.3495 grams.
To convert ounces to grams, you can use the following formula:
Grams = Ounces × 28.3495
Let’s break down the conversion of 06 ounces to grams step-by-step:
1. Start with the number of ounces you want to convert: 06 ounces.
2. Use the conversion factor: 28.3495 grams per ounce.
3. Multiply the number of ounces by the conversion factor: 06 ounces × 28.3495 grams/ounce.
4. Perform the calculation: 06 × 28.3495 = 170.097.
5. Round the result to two decimal places for practical use: 170.10 grams.
This means that 06 ounces is equal to approximately 170.10 grams.
Understanding how to convert ounces to grams is crucial, especially in bridging the gap between the metric and imperial systems. Many recipes, particularly those from different countries, may list
ingredients in grams, while others may use ounces. Being able to convert between these units ensures that you can follow any recipe accurately, leading to better cooking results.
Practical examples of where this conversion might be useful include:
• Cooking: When following a recipe that lists ingredients in grams, knowing how to convert ounces can help you measure accurately, ensuring your dish turns out as intended.
• Scientific Measurements: In laboratories, precise measurements are critical. Converting ounces to grams can help scientists and researchers maintain accuracy in their experiments.
• Everyday Use: Whether you’re weighing food for portion control or measuring out ingredients for a DIY project, being able to convert ounces to grams can simplify your tasks.
In conclusion, converting 06 ounces to grams is a straightforward process that can enhance your cooking, scientific endeavors, and daily activities. With the conversion factor of 28.3495, you can
easily make this conversion and ensure accuracy in your measurements.
Here are 10 items that weigh close to 06 ounces to grams –
• Medium Avocado
Shape: Oval
Dimensions: Approximately 4-5 inches long
Usage: Perfect for salads, guacamole, or as a healthy snack.
Fact: Avocados are technically a fruit and are rich in healthy fats.
• Baseball
Shape: Spherical
Dimensions: 9 inches in circumference
Usage: Used in the sport of baseball for pitching, hitting, and catching.
Fact: A standard baseball weighs about 5 ounces, making it a close match.
• Standard Coffee Mug
Shape: Cylindrical
Dimensions: 4 inches tall, 3 inches in diameter
Usage: Ideal for enjoying your favorite hot beverages.
Fact: A typical coffee mug can hold about 8-12 ounces of liquid.
• Small Bag of Flour
Shape: Rectangular
Dimensions: 7 inches tall, 5 inches wide
Usage: Commonly used in baking and cooking.
Fact: A cup of all-purpose flour weighs about 4.5 ounces.
• Medium-Sized Apple
Shape: Round
Dimensions: Approximately 3 inches in diameter
Usage: Great for snacking, baking, or making cider.
Fact: Apples float in water because 25% of their volume is air.
• Small Potted Plant
Shape: Cylindrical
Dimensions: 6 inches tall, 4 inches in diameter
Usage: Adds greenery to your home or office space.
Fact: Indoor plants can improve air quality and boost mood.
• Standard Deck of Playing Cards
Shape: Rectangular
Dimensions: 2.5 inches by 3.5 inches
Usage: Used for various card games and magic tricks.
Fact: A full deck of cards weighs about 3.5 ounces, making it a lightweight option.
• Small Water Bottle
Shape: Cylindrical
Dimensions: 8 inches tall, 2.5 inches in diameter
Usage: Convenient for staying hydrated on the go.
Fact: Many reusable water bottles are designed to hold around 16 ounces.
• Bar of Soap
Shape: Rectangular
Dimensions: 3 inches by 2 inches by 1 inch
Usage: Used for personal hygiene and cleaning.
Fact: The average bar of soap weighs about 4-6 ounces.
• Small Bag of Rice
Shape: Rectangular
Dimensions: 8 inches tall, 5 inches wide
Usage: Commonly used as a staple food in many cultures.
Fact: One cup of uncooked rice weighs about 6.5 ounces.
Other Oz <-> Gm Conversions –
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Kilometers per Liter to Miles per Gallon (US) Conversion
Kilometers per Liter to Miles per Gallon (US) Converter
Enter the fuel economy in kilometers per liter below to convert it to miles per gallon (US).
Result in Miles per Gallon (US):
1 km/L = 2.352145 mpg (US)
Do you want to convert miles per gallon (US) to kilometers per liter?
How to Convert Kilometers per Liter to Miles per Gallon (US)
To convert a measurement in kilometers per liter to a measurement in miles per gallon (US), multiply the fuel economy by the following conversion ratio: 2.352145 miles per gallon (US)/kilometer per
Since one kilometer per liter is equal to 2.352145 miles per gallon (US), you can use this simple formula to convert:
miles per gallon (US) = kilometers per liter × 2.352145
The fuel economy in miles per gallon (US) is equal to the fuel economy in kilometers per liter multiplied by 2.352145.
For example,
here's how to convert 5 kilometers per liter to miles per gallon (US) using the formula above.
miles per gallon (US) = (5 km/L × 2.352145) = 11.760725 mpg (US)
Kilometers per liter and miles per gallon (US) are both units used to measure fuel economy. Keep reading to learn more about each unit of measure.
What Are Kilometers per Liter?
Kilometers per liter is the distance that can be traveled in kilometers using one liter of fuel.
Kilometers per liter can be abbreviated as km/L, and are also sometimes abbreviated as kpl or kmpl. For example, 1 kilometer per liter can be written as 1 km/L, 1 kpl, or 1 kmpl.
In the expressions of units, the slash, or solidus (/), is used to express a change in one or more units relative to a change in one or more other units.
Learn more about kilometers per liter.
What Are Miles per Gallon (US)?
Miles per gallon is the distance traveled in miles consuming just one gallon of fuel.
When evaluating fuel efficiency, the more miles per gallon, the more fuel efficient a vehicle is and the less fuel consumed. Thus, the higher the mpg rating of a vehicle the less fuel it will
Miles per gallon (US) can be abbreviated as mpg (US), and are also sometimes abbreviated as mpg. For example, 1 mile per gallon (US) can be written as 1 mpg (US) or 1 mpg.
Learn more about miles per gallon (US).
Kilometer per Liter to Mile per Gallon (US) Conversion Table
Table showing various kilometer per liter
measurements converted to miles per gallon
Kilometers Per Liter Miles Per Gallon (US)
1 km/L 2.3521 mpg (US)
2 km/L 4.7043 mpg (US)
3 km/L 7.0564 mpg (US)
4 km/L 9.4086 mpg (US)
5 km/L 11.76 mpg (US)
6 km/L 14.11 mpg (US)
7 km/L 16.47 mpg (US)
8 km/L 18.82 mpg (US)
9 km/L 21.17 mpg (US)
10 km/L 23.52 mpg (US)
11 km/L 25.87 mpg (US)
12 km/L 28.23 mpg (US)
13 km/L 30.58 mpg (US)
14 km/L 32.93 mpg (US)
15 km/L 35.28 mpg (US)
16 km/L 37.63 mpg (US)
17 km/L 39.99 mpg (US)
18 km/L 42.34 mpg (US)
19 km/L 44.69 mpg (US)
20 km/L 47.04 mpg (US)
21 km/L 49.4 mpg (US)
22 km/L 51.75 mpg (US)
23 km/L 54.1 mpg (US)
24 km/L 56.45 mpg (US)
25 km/L 58.8 mpg (US)
26 km/L 61.16 mpg (US)
27 km/L 63.51 mpg (US)
28 km/L 65.86 mpg (US)
29 km/L 68.21 mpg (US)
30 km/L 70.56 mpg (US)
31 km/L 72.92 mpg (US)
32 km/L 75.27 mpg (US)
33 km/L 77.62 mpg (US)
34 km/L 79.97 mpg (US)
35 km/L 82.33 mpg (US)
36 km/L 84.68 mpg (US)
37 km/L 87.03 mpg (US)
38 km/L 89.38 mpg (US)
39 km/L 91.73 mpg (US)
40 km/L 94.09 mpg (US)
More Kilometer per Liter & Mile per Gallon (US) Conversions
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Modeling Scenario
We offer an opportunity to build a mathematical model using Newton's Law of Cooling for a closed plastic baggie of liquid inside a liquid container.
Article or Presentation
We discuss the use of the FREE system dynamics software Insightmaker (https://insightmaker.com/) in a first course in Ordinary Differential Equations (with a modeling emphasis).
Modeling Scenario
We build the infinite set of first order differential equations for modeling a stochastic process, the so-called birth and death equations. We will only need to use integrating factor solution
strategy or DSolve in Mathematica for success.
Modeling Scenario
The goals of this project are to compare a conceptual one-dimensional groundwater flow model to observations made in a laboratory setting, and to discuss the differences.
Modeling Scenario
This modeling scenario examines the motion of a ripcord-powered toy with the goal of using real data to estimate parameters in a first-order model of the velocity of the toy. Students may conduct
experiments or use videos to collect data.
Modeling Scenario
We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the
Modeling Scenario
We present a system of nonlinear differential equations to model the control of energy flow into producing workers or reproducers in an insect colony, using a set of given parameters and a number of
different energy functions.
Modeling Scenario
We offer students an opportunity to generate data for their team on a death and immigration model using 12 and 20 sided dice and then pass on the data to another student team for analysis with a
model they built. The key is to recover the...
Modeling Scenario
This activity offers analysis of a toy pull-back car: solution of a differential equation from model; data collection and parameter estimation; and adapting the model to predict maximum speed and
distance traveled for a new pull-back distance.
Modeling Scenario
We conduct a simulation of death and immigration, using a small set of "individuals", m&m candies or any two sided object (coin, chips), in which upon tossing a set of individuals we cause some to
die and others then to immigrate. Modeling ensues!
Modeling Scenario
We offer an opportunity to build a mathematical model using Newton's Second Law of Motion and a Free Body Diagram to analyze the forces acting on the rocket of changing mass in its upward flight
under power and then without power followed by its...
Modeling Scenario
We offer a mathematical modeling experience using differential equations to model the volume of an intraocular gas bubble used by ophthalmologists to aid the healing of a surgically repaired region
of the retina.
Modeling Scenario
This paper presents real-world data, a problem statement, and discussion of a common approach to modeling that data, including student responses. In particular, we provide time-series data on the
number of boys bedridden due to an outbreak of...
Technique Narrative
This material teaches the basics of numerical methods for first order differential equations by following graphical and numerical approaches. We discuss the order of accuracy of the methods and
compare their CPU times.
Modeling Scenario
The goal of this project is for students to develop, analyze, and compare three different models for the flight of a sponge dart moving under the influences of gravity and air resistance.
Modeling Scenario
We offer data on the sublimation of dry ice (carbon dioxide) collected in a classroom setting so that students can model the rate of change in the mass of a small solid carbon dioxide block with a
differential equation model, solve the...
Modeling Scenario
We offer raw data collected from a webcam and a thermometer for evaluating the strength of steeping tea. We ask students to build a mathematical model using the data to predict how long the tea
should steep before essentially reaching saturation.
Modeling Scenario
We examine plots on the spread of technologies and ask students to estimate and extract data from the plots and then model several of these spread of technologies phenomena with a logistic
differential equation model.
Modeling Scenario
Using a grid and m&m candies, we simulate the spread of disease. Students conduct the simulation and collect data to estimate parameters (in several ways) in a differential equation model for the
spread of the disease.
Modeling Scenario
We help students see the connection between college level chemistry course work and their differential equations coursework. We do this through modeling kinetics, or rates of chemical reaction.
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Do it without diagram!
In triangle $ABC$ we have $AB = BC$ , $\angle B = 20^\circ$ . Point $M$ on $AC$ is such that $AM : MC = 1 : 2$ , point $H$ is the projection of $C$ to $BM$ . Find $\angle AHB$ in degrees.
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Daward72's Shop - TES
Daward72's Shop
I have been teaching MATHEMATICS (11-16) since 1990. I have regularly been commended on my classroom displays and the quality of my resources as I feel this enriches the student experience. As a
self-taught graphic designer I now produce professional quality materials for our academy/academy chain across all departments including posters/banners and promotional materials. I am currently
working on updating some older resources as well as developing new ones!
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Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. (i) What is the probability that the card is the queen? (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?
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NCERT Solutions for Class 10 Maths Chapter 15
Important NCERT Questions
NCERT Books for Session 2022-2023
CBSE Board and UP Board Others state Board
EXERCISE 9.1
Page No:309
Questions No:15
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Outline of a lesson on the world around us (junior group): Summary of GCD according to the Federal State Educational Standard in the second junior group "Trip to the farm"
Summary of the GED according to the Federal State Educational Standard in the second junior group “Trip to the farm” outline of a lesson on the world around us (junior group)
FEMP classes. Junior group. September
Lesson 1
Program content
• Strengthen the ability to distinguish and name a ball (ball) and a cube (cube), regardless of the color and size of the figures.
Didactic visual material
Demonstration material. Large and small red balls, large and small green cubes; 2 boxes of red and green colors; toys: bear, truck.
Handout. Little red balls, little green cubes.
Part I. The teacher brings a truck into the group, in the back of which there is a bear, balls and cubes, and asks: “Who came to us? (Children look at the bear.) What did the bear bring in the truck?
The teacher invites the children to find a ball (gives the concept of a ball): “What did you find? What color is the ball?
The teacher asks to show what can be done with the ball. (Ride.)
Children perform similar tasks with a cube. (Actions with a cube are indicated by the word put.)
Part II. Game exercise “Hide the cube (ball).”
The teacher invites one of the children to take a ball in one hand and a cube in the other and hide one of the figures behind their back. The rest of the children must guess what the child hid and
what is left in his hand.
Part III . The teacher asks the children to help the bear put balls and cubes into boxes: the balls should be put in the red box, and the cubes in the green box.
While completing the task, the teacher asks the children: “What did you put in the box? How many balls (cubes)? Are they the same color? How else are balls and cubes different? (Big and small.)
Mishka thanks the children for their help and says goodbye to them.
Lesson 2
Program content
• Strengthen the ability to distinguish objects of contrasting size, using the words big and small.
Didactic visual material
Demonstration material. Large and small dolls, 2 beds of different sizes; 3 - 4 large cubes.
Handout. Small cubes (3 - 4 pieces for each child).
Part I. Two dolls come to visit the children. The children, together with the teacher, examine them, find out that one doll is large and the other small, and give them names.
Then the teacher draws the children’s attention to the cribs: “Are the cribs the same size? Show me the big crib. And now the little one. Where is the bed for a large doll, and where for a small one?
Put the dolls to sleep. Let's sing them a lullaby “Tired toys sleep.”
Part II. Game exercise "Let's build turrets."
The teacher places large and small cubes on the table,
invites children to compare them in size and then build towers. The teacher builds a tower from large cubes on the carpet, and the children build towers from small cubes. At the end of the work,
everyone looks at the buildings together and shows a large (small) tower.
FEMP classes. Junior group. November
Lesson 1
Program content
• Learn to compare two objects by length and denote the result of the comparison with the words long - short, longer - shorter.
• Improve the ability to compose a group of objects from individual objects and select one object from the group, denote aggregates with the words one, many, none.
Didactic visual material
Demonstration material. Two cardboard tracks of the same color but different lengths, two baskets with large and small balls.
Handout. Large and small balls (one ball for each child).
Game situation “We are funny guys.”
The lesson is held in the gym.
Part I. There are two cardboard tracks of different lengths on the floor.
The teacher asks the children what can be said about the length of the paths. Shows how this can be learned using overlays and track applications. Then he asks the children to show the long (short)
path, walk along the long (short) path. Specifies the length of the tracks.
Part II . The teacher draws the children’s attention to the baskets with balls: “What can you say about the size of the balls? How many big balls? (A lot.) Take one large ball at a time. How many
balls did each of you take? (One.) How many big balls are in the basket now? (None.) Let's try to roll the balls along a long track. How can we make sure there are a lot of balls in the basket again?
” (Children's answers.)
Children put balls in a basket. The teacher clarifies how many balls each child put in the basket and how many there are.
Children perform a similar exercise with small balls. Children roll them along a short track, put them in a basket with large balls and answer the teacher’s questions: “How many small balls? (A lot.)
How many big balls? (A lot.) How many big and small balls are there together? (Even more, a lot.)
Ill part. Outdoor game “Catch the ball”.
The teacher pours balls out of the basket and invites the children to catch up and take one ball at a time. (“How many balls did you catch?”) The children put the collected balls back into the
basket, and the teacher finds out: “How many balls did you put in the basket? How many balls are in the basket?
The game is repeated several times.
Lesson 2
Program content
• Learn to find one and many objects in a specially created environment, answer the question “how many?” using the words one, many.
• Continue to teach how to compare two objects in length using the methods of superimposition and application, to indicate the results of comparison with the words long - short, longer - shorter.
Didactic visual material
Demonstration material. Four to five groups of toys, 2 boxes of different sizes, 2 ribbons of the same color of different lengths.
Handout. Ribbons of the same color, but of different lengths (2 pieces for each child).
Game situation “Toy Store”.
Part I. The teacher invites the children to visit the toy store. Toys are laid out on chairs and tables: one at a time and several at a time. Children, together with the teacher, look at the objects
and find out what toys are sold in the store and how many there are. At the direction of the teacher, children “buy” one or many toys. The adult asks: “What toys did you buy? How many toys did you
Part II. The teacher invites the children to choose ribbons for the toy boxes.
Children look at the boxes, and the teacher finds out: “How are the boxes different? Are the boxes the same size? Show the big (small) box. How can we bandage the boxes?”
The teacher asks to compare the ribbons: “What can you say about the length of the ribbons? How can you find out? How to compare ribbons by length? (By overlay or attachment.)
Children compare ribbons using the methods of application or application, show a long (short) ribbon, the results of the comparison are denoted by the words long - short, longer - shorter.
Children put ribbons in boxes: long ones in the big one, short ones in the small one.
Part III. Game exercise “Tie the boxes with ribbons.”
The teacher, together with the children, finds out what length of ribbon can be used to tie a large (small) box. First, they compare the ribbons by length, find the long (short) ribbon and tie the
Lesson 3
Program content
• Continue to teach how to find one and many objects in a specially created environment, to designate collections with the words one, many.
• Introduce the square, teach to distinguish between a circle and a square.
Didactic visual material
Demonstration material. “Parcel” with toys (cars, nesting dolls, pyramid, ball); a square and a circle of the same color (the length of the sides of the square and the diameter of the circle are 14
Handout. Circles and squares of the same color (the length of the sides of the square and the diameter of the circle are 8 cm).
Game situation “The postman brought a parcel.”
Part I. The teacher tells the children that the postman brought them a package. The teacher invites the children to see what they have been sent. He takes toys out of the box one by one, asks the
children to name them, finds out the number of objects: “How many pyramids did they send us? How many cars (matryoshka dolls, balls) are in the package? What toys were sent to us a lot? What kind of
toys, one at a time?
Part II . The teacher takes a circle out of the package and puts it on the flannelgraph: “What figure is this? (Circle.) What color is the circle?” The teacher invites the children to trace a circle
along the contour with their hand.
Then he takes out a square, places it next to the circle, names the figure, shows the sides and corners of the square and asks the children: “What does a square have? How many sides does a square
have? How many angles does a square have?
The teacher asks the children to circle the square and show its sides (corners).
Part III. Game exercise “Show and ride”.
The children have circles and squares on their tables. The teacher invites the children to take a circle, name the figure and trace it with their hand.
Similar actions are performed with a square.
Then the teacher asks the children to try to roll a circle around the table, and then a square, and finds out: “Can I roll a square? What’s stopping the square?” (Angles.)
Lesson 4
Program content
• To consolidate the ability to find one or many objects in a specially created environment, to denote aggregates with the words one, many.
• Continue to learn to distinguish and name a circle and a square.
Didactic visual material
Demonstration material. The group setting is used - a play corner (dolls, chairs, cups, etc.; table, bear, teapot, etc.), a natural corner (plants, aquarium, watering can, cage, etc.), a book corner
( books, pictures; shelf, book stand, etc.); garage (several small cars, one large car); silhouette of a steam locomotive, sheets of colored paper (cars).
Handout. Circles and squares of the same color (side length of the square is 8 cm, circle diameter is 8 cm; one for each child).
Part I. Children travel in a group to the music “Blue Car” (music and lyrics by V. Ya. Shainsky). First stop is the doll corner.
Educator. What's in the doll corner? What kind of toys are there a lot? Of which toys only one?
Then the children stop in a book corner, a nature corner, a garage and answer similar questions from the teacher.
Part II . Didactic game "Fix the train."
Circles and squares are laid out on the table. The teacher asks the children to find the circles and finds out: “What color are the circles? What can you do with them? (Ride.)
Then, on the instructions of the teacher, the children find squares, name the shapes and try to roll them on the table. The teacher reminds that the corners interfere with the square and asks the
children to show them.
At the end of the lesson, the children “fix” the train, commenting on their actions: they put circles instead of wheels, squares instead of windows.
Summary of the book by I. Pomoraeva and V. Pozina
The book is a summary and curriculum for teachers of children 3-4 years old, distributed over the entire school year, which involves 1 lesson per week. The authors took into account the inevitable
period of adaptation of each child after entering kindergarten, so the first lesson, according to the plan, is held after September 15.
FEMP in the 2nd junior group
Plan for the autumn period
The main directions of classes in the first quarter of the educational year, according to the book co-authored with Pozina, are focused on:
• Introduce two geometric shapes: circle and square. In the first lesson, children study a circle, its shape features, after which they move on to its volumetric analogue - a ball. Later, a square
shape is added with a transition to a cube. The result of the study is the unmistakable choice of a circle and a square among other figures, as well as their comparison according to the main
• Clarification of quantitative concepts of objects, such as “one”, “none”, “many”. At the very beginning, children learn to define two quantities: “one” and “many”; after which the value “none” is
added, which children should be able to use when the teacher asks the question “how many objects?”
• Consolidation of comparative skills, first limited to such concepts as “big” and “small”, after which additional elements of paired comparison are introduced, answering questions about which
object is larger and which is smaller. The concept of length is also consolidated, starting with the statement “long” and “short”, and ending with comparisons - “longer”, “shorter”.
Defining a square among other shapes
Winter plan
In the second quarter, according to Pomoraeva’s FEMP notes, the 2nd junior group should study:
• The meanings of the words “evenly” and “many” in relation to the compared groups of objects. By the end of the second quarter, children should operate with additional meanings “little” - “many”.
• Orientation in your body, in relation to the right and left sides. The study begins with the hands and continues with other paired parts of the body.
• In classes on comparing objects, concepts such as “wide”, “narrow”, as well as “wider” - “narrower”, “equal in width” are added.
• A new geometric figure - a triangle. After analyzing its features, children learn to compare it with a square and name its distinctive features.
• Spatial arrangements of objects using the meanings “above” and “below”. After mastering these concepts, an analysis of the sizes of figures is added according to the criterion of “above”,
“below”, “high”, “low”.
Where are there more carrots and where are there fewer?
Plan for the spring period
The summary continues the description of FEMP classes, 2nd junior group - Pomoraeva recommends:
• Introduce the concept of time of day. The formation of classes begins with an introduction to the contrasting concepts of “children” and “night”, later they teach “morning” and “evening”.
• Compare not only the visual number of objects, but also determine the number of sounds by ear, limiting oneself to two concepts “one” and “many”. After consolidating this skill, they begin to
reproduce the number of sounds heard, but the numeral is not called, the children only hear the sequence of sounds, after which they try to repeat the combination.
• Improve spatial skills, to which the following definitions are added: “in front”, “behind”, “left”, “right”, “above” and “below”.
Non-traditional drawing technique for the middle group of kindergarten
Important! Each subsequent task not only introduces novelty and introduces terms. It is recommended to build mathematical classes on the basis of consolidating previous material. Therefore, before
introducing new concepts, the teacher must consolidate previously covered material in each lesson.
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Renewable Energy Archives - copypasteearth
Transitioning to renewable energy sources like solar, wind, and geothermal power is crucial for mitigating climate change and achieving energy independence. Take the example of Denmark, which has set
an ambitious goal of generating 100% of its electricity from renewable sources by 2050. By investing heavily in wind power, Denmark has already achieved over 40% of its electricity production from
wind turbines, with a single offshore wind farm capable of powering 600,000 homes.
Solar power is another promising renewable energy source, with the cost of solar panels dropping by over 70% in the last decade. In sunny regions like California, solar power is now cheaper than
electricity from fossil fuels. Geothermal energy, which harnesses heat from the earth’s core, is also becoming increasingly viable, with countries like Iceland and the Philippines already relying
heavily on geothermal power.
Transitioning to renewable energy requires significant investment in infrastructure, such as building new wind farms, solar arrays, and geothermal plants. However, the long-term benefits are clear:
reduced greenhouse gas emissions, improved air quality, and greater energy security. By embracing renewable energy, we can create a cleaner, more sustainable future for generations to come.
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1. all factors of 72
1 Answer
Getting problem in finding the factor of the given number. Then, you are lacking the concept of "factor"
Factors of given number are the sets of those numbers which exactly divide the given number.
For example, the factors of 2 is 1 & 2
Similarly, factors of 9 are 1, 3 & 9.
So, all the factors of 72 are 1, 2,3,4,6,8,9,12,18,24,36 & 72
Note: Remember! the factor of any number can be obtained by L.C.M. method
and 1 is the factor of all the numbers.
Top contributors in Mathematics category
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Question: You are a group of n friends. Each of you thinks of an integer number from 0 to 9 (assume equal p… | Assignment Writing Service: EssayNICE
Show transcribed image text You are a group of n friends. Each of you thinks of an integer number from 0 to 9 (assume equal probabilities for all these numbers). Let X be the total sum. Assuming that
each of you makes an independent choice what is the probability that the sum is 9? (Hint: let X be number the i-th person has thought: X Xi+ +Xn and think of n flavors of ice-cream)
You are a group of n friends. Each of you thinks of an integer number from 0 to 9 (assume equal probabilities for all these numbers). Let X be the total sum. Assuming that each of you makes an
independent choice what is the probability that the sum is 9? (Hint: let X be number the i-th person has thought: X Xi+ +Xn and think of n flavors of ice-cream)
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Question #3e2ee | Socratic
Question #3e2ee
1 Answer
The formula for density is Density = $\frac{M a s s}{V o l u m e}$, which you can abbreviate to D = $\frac{M}{V}$.
Here, we are given the Density (0.714 $\frac{g}{L}$) and the Volume (25.0L). Plugging into the equation we have:
$0.714$$\frac{g}{L}$ = $\frac{M}{25.0 L}$
To solve for the unknown (in this case, M), multiply both sides of the equation by $25.0 L$ (notice that the units of L cancels out on both sides, and we are left with the units we want: grams).
We are then left with:
17.85g = M, so our mass is 17.85 grams.
Impact of this question
2026 views around the world
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How does Book Report estimate borrows?
Book Report includes estimated borrow numbers. Put simply: The estimated number of borrows is the smallest possible number of borrows that could result in the given number of page reads.
Kindle Unlimited allows readers to borrow books, but the number of times a book has been borrowed is not reported to authors. Book Report estimates borrows based on two other values: The number of
page reads a book has received, and the number of pages in that book.
The estimated borrow count is the number of page reads a book has received, divided by the number of pages in that book. If a book is 200 pages long and has received 10,000 page reads, Book Report
will estimate the book has been borrowed 50 times. 10,000/200 = 50.
Because each borrow can at most result in 100% of the pages being read, the estimate the lower bound on how many people actually downloaded the book through Kindle Unlimited. The actual number of
borrows is higher than the number estimated.
Further details
• To access your borrow estimates, edit the Details section by clicking the pencil at the top right. Include the ~Borrows column.
• If you just started using Book Report and the estimates are way off, give it some time. The KENPC crawl takes some time. Wait for the spinner in the top left to disappear.
• Book Report uses the official Kindle Edition Normalized Page Count (KENPC) for these estimates.
• Once a week, Book Report will confirm the KENPC for each of your books. If it changes, the borrow estimate will use the new value from that date forward, but continue using the old value for page
reads from before that date.
• By default, one estimated borrow means the number of page reads equals the KENPC, but the average borrower probably doesn't read 100% of the book. So on the Settings tab, you can set a different
value for "What percent of KENPC counts as an estimated borrow?" If you set it to 50%, then 10,000 page reads on a 200 page book would result in 100 estimated borrows. That is because it assumes
each borrow resulted in 100 (50% of the KENPC) page reads.
• Book Report doesn't do any of the estimations or calculations for data from before July 1st, 2015. Before that date KDP provided authors with their borrow numbers and Book Report still displays
those numbers directly.
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Simple Harmonic Motion (8 of 16): Hooke's Law, Example Problems | Video Summary and Q&A | Glasp
Simple Harmonic Motion (8 of 16): Hooke's Law, Example Problems | Summary and Q&A
Simple Harmonic Motion (8 of 16): Hooke's Law, Example Problems
This video explains Hooke's Law, which states that the force needed to extend or compress a spring is directly proportional to the distance it is extended or compressed. It also demonstrates how to
graphically determine the spring constant using a simulation and how to use that information to determine the mass of unknown masses.
Key Insights
• 🌸 Hooke's Law states that the force applied to a spring is directly proportional to the change in length of the spring.
• 🌸 The spring constant measures the stiffness of a spring and is expressed in newtons per meter.
• 🌸 The spring constant can be determined by graphing the force applied to a spring versus the change in length.
in today's video we are going to go over an example problem for hooke's law now before we get started please don't forget to subscribe to my channel if i look at my analytics i see that more than 90
of people who watch my videos have not subscribed please subscribe to my channel support my channel step by step science get all my excellent physics c... Read More
Questions & Answers
Q: What is Hooke's Law?
Hooke's Law states that the force needed to extend or compress a spring is directly proportional to the distance it is extended or compressed. It is expressed as fs = -kx, where fs is the force on
the spring, k is the spring constant, and x is the change in length.
Q: How is the spring constant determined?
The spring constant can be determined by graphing the force applied to a spring versus the change in length. The slope of the line on the graph represents the spring constant.
Q: What does the spring constant indicate?
The spring constant measures the stiffness of a spring. A higher spring constant indicates a stiffer spring, while a lower spring constant indicates a softer spring.
Q: How can the mass of unknown masses be determined using the spring constant?
Once the spring constant is known, the mass of unknown masses can be determined by measuring the change in length for each mass and using the relationship between force, mass, and acceleration due to
gravity (F = mg).
Summary & Key Takeaways
• The video starts with an introduction to Hooke's Law, which states that the force exerted on a spring is equal to the negative of the spring constant multiplied by the change in length of the
• It explains that the spring constant measures the stiffness of a spring and is expressed in newtons per meter.
• The video then demonstrates how to determine the spring constant by graphing the force applied to a spring versus the change in length, and how to use that spring constant to calculate the mass
of unknown masses.
Explore More Summaries from Step by Step Science 📚
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Hierarchical Clustering in Python
Introduction to Hierarchical Clustering
Hierarchical clustering is another unsupervised learning algorithm that is used to group together the unlabeled data points having similar characteristics. Hierarchical clustering algorithms falls
into following two categories −
Agglomerative hierarchical algorithms − In agglomerative hierarchical algorithms, each data point is treated as a single cluster and then successively merge or agglomerate (bottom-up approach) the
pairs of clusters. The hierarchy of the clusters is represented as a dendrogram or tree structure.
Divisive hierarchical algorithms − On the other hand, in divisive hierarchical algorithms, all the data points are treated as one big cluster and the process of clustering involves dividing (Top-down
approach) the one big cluster into various small clusters.
Steps to Perform Agglomerative Hierarchical Clustering
We are going to explain the most used and important Hierarchical clustering i.e. agglomerative. The steps to perform the same is as follows −
• Step 1 − Treat each data point as single cluster. Hence, we will be having, say K clusters at start. The number of data points will also be K at start.
• Step 2 − Now, in this step we need to form a big cluster by joining two closet datapoints. This will result in total of K-1 clusters.
• Step 3 − Now, to form more clusters we need to join two closet clusters. This will result in total of K-2 clusters.
• Step 4 − Now, to form one big cluster repeat the above three steps until K would become 0 i.e. no more data points left to join.
• Step 5 − At last, after making one single big cluster, dendrograms will be used to divide into multiple clusters depending upon the problem.
Role of Dendrograms in Agglomerative Hierarchical Clustering
As we discussed in the last step, the role of dendrogram starts once the big cluster is formed. Dendrogram will be used to split the clusters into multiple cluster of related data points depending
upon our problem. It can be understood with the help of following example −
Example 1
To understand, let us start with importing the required libraries as follows −
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
Next, we will be plotting the datapoints we have taken for this example −
X = np.array([[7,8],[12,20],[17,19],[26,15],[32,37],[87,75],[73,85], [62,80],[73,60],[87,96],])
labels = range(1, 11)
plt.figure(figsize=(10, 7))
plt.scatter(X[:,0],X[:,1], label='True Position')
for label, x, y in zip(labels, X[:, 0], X[:, 1]):
plt.annotate(label,xy=(x, y), xytext=(-3, 3),textcoords='offset points', ha='right', va='bottom')
From the above diagram, it is very easy to see that we have two clusters in out datapoints but in the real world data, there can be thousands of clusters. Next, we will be plotting the dendrograms of
our datapoints by using Scipy library −
from scipy.cluster.hierarchy import dendrogram, linkage
from matplotlib import pyplot as plt
linked = linkage(X, 'single')
labelList = range(1, 11)
plt.figure(figsize=(10, 7))
dendrogram(linked, orientation='top',labels=labelList, distance_sort='descending',show_leaf_counts=True)
Now, once the big cluster is formed, the longest vertical distance is selected. A vertical line is then drawn through it as shown in the following diagram. As the horizontal line crosses the blue
line at two points, the number of clusters would be two.
Next, we need to import the class for clustering and call its fit_predict method to predict the cluster. We are importing AgglomerativeClustering class of sklearn.cluster library −
from sklearn.cluster import AgglomerativeClustering
cluster = AgglomerativeClustering(n_clusters=2, affinity='euclidean', linkage='ward')
Next, plot the cluster with the help of following code −
plt.scatter(X[:,0],X[:,1], c=cluster.labels_, cmap='rainbow')
The above diagram shows the two clusters from our datapoints.
As we understood the concept of dendrograms from the simple example discussed above, let us move to another example in which we are creating clusters of the data point in Pima Indian Diabetes Dataset
by using hierarchical clustering −
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline
import numpy as np
from pandas import read_csv
path = r"C:\pima-indians-diabetes.csv"
headernames = ['preg', 'plas', 'pres', 'skin', 'test', 'mass', 'pedi', 'age', 'class']
data = read_csv(path, names=headernames)
array = data.values
X = array[:,0:8]
Y = array[:,8]
(768, 9)
slno. preg Plas Pres skin test mass pedi age class
0 6 148 72 35 0 33.6 0.627 50 1
1 1 85 66 29 0 26.6 0.351 31 0
2 8 183 64 0 0 23.3 0.672 32 1
3 1 89 66 23 94 28.1 0.167 21 0
4 0 137 40 35 168 43.1 2.288 33 1
patient_data = data.iloc[:, 3:5].values
import scipy.cluster.hierarchy as shc
plt.figure(figsize=(10, 7))
plt.title("Patient Dendograms")
dend = shc.dendrogram(shc.linkage(data, method='ward'))
from sklearn.cluster import AgglomerativeClustering
cluster = AgglomerativeClustering(n_clusters=4, affinity='euclidean', linkage='ward')
plt.figure(figsize=(10, 7))
plt.scatter(patient_data[:,0], patient_data[:,1], c=cluster.labels_, cmap='rainbow')
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Convert Inches to Feet: Go Math Grade 5 Explanation
How to convert inches to feet in math?
The question of 90 in. = ______ ft ______ in. The answer is 7 ft 6 in.
When converting inches to feet, it's important to remember that 1 foot is equal to 12 inches. In the given question, we are asked to convert 90 inches to feet.
To convert 90 inches to feet, we first need to divide 90 by 12 because there are 12 inches in a foot. This division will give us the whole number of feet in 90 inches.
90 inches = 7 feet (since 12 goes into 90 evenly 7 times).
After determining the whole number of feet, we need to find the remaining inches. In this case, we have 90 inches total and we have already accounted for 84 inches in the 7 feet.
The remaining inches are 6 inches (90 - 84 = 6 inches).
Therefore, 90 inches is equivalent to 7 feet and 6 inches, which is represented as 7 ft 6 in.
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Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior
Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya str. 16a, Kazan 420008, Russia
Author to whom correspondence should be addressed.
Submission received: 6 February 2020 / Revised: 2 March 2020 / Accepted: 6 March 2020 / Published: 13 March 2020
(This article belongs to the Special Issue
We consider the magnetic monopole in the axionic dark matter environment (axionic dyon) in the framework of the Reissner-Nordström-de Sitter model. Our aim is to study the distribution of the
pseudoscalar (axion) and electric fields near the so-called folds, which are characterized by the profiles with the central minimum, the barrier on the left, and the maximum on the right of this
minimum. The electric field in the fold-like zones is shown to change the sign twice, i.e., the electric structure of the near zone of the axionic dyon contains the domain similar to a double
electric layer. We have shown that the described fold-like structures in the profile of the gravitational potential, and in the profiles of the electric and axion fields can exist, when the value of
the dyon mass belongs to the interval enclosed between two critical masses, which depend on the cosmological constant.
04.20.Jb; 04.40.Nr; 14.80.Hv
1. Introduction
Global and local phenomena in our Universe are interrelated. When we speak about the cosmological constant
, first of all we think about the global structure of the Universe and about the rate of its expansion [
]. On the other hand, the cosmological constant is associated with one of the models of the dark energy [
], and the influence of the
term on the structure of compact objects, black holes and wormholes can be interpreted as the dark energy fingerprints [
]. Specific details of the causal structure of spherically symmetric objects, the number and types of the horizons can also be associated with the cosmological constant. For instance, in the
framework of non-minimal extensions of the Einstein–Maxwell and Einstein–Yang–Mills models with non-vanishing cosmological constant the solutions regular at the center appear, if the parameters of
the non-minimal coupling are linked by specific relationships with the cosmological constant [
]. In other words, the regularity at the center is connected with the appropriate asymptotic behavior.
The dark matter, the second key element of all modern cosmological scenaria [
], can play the unique role in the problem of identification of the relativistic compact objects with strong electromagnetic fields. When one studies the properties of the gravitational field of the
object, the standard theoretical tool is based on the analysis of the dynamics of test particles; usually, one studies the trajectories of massive and massless particles reconstructing the fine
details of the gravitational fields. When we deal with the axions, the light pseudo-bosons [
], as the possible representatives of the dark matter [
], we can not monitor the motion of an individual particle, and the analysis of the dark matter halos comes to the fore.
The dark matter particles are not yet identified. There are few candidates, which can be classified as WIMPs and non-WIMPs (Weekly Interacting Massive Particles); the sets of candidates include light
bosons and fermions; these candidates can compose the systems indicated as cold, warm and hot dark matter components. In fact, one can assume that there are a few different fractions, which are
united by a common term dark matter. We analyze the axionic fraction of the dark matter, and below we use the short terms axionic dark matter and axions. For the description of the axionic dark
matter, we use the master equations for the interacting pseudoscalar and electromagnetic field [
]; over the last thirty years, they have been known as the equations of axion electrodynamics [
]. Furthermore, we follow the idea that the axionic fraction of the dark matter behaves as a correlated system, in particular, the axions can form the Bose–Einstein condensate [
The dark matter forms specific cosmological units: halos, sub-halos, filaments, walls, the structure of which admits the recognition of the type of the corresponding central elements. We assume that
compact relativistic objects with strong gravitational and electromagnetic fields distort the axionic fraction of the dark matter halos, which surround them (see, e.g., [
] for details). For instance, when the magnetic field of the star possesses the dipolar component, the axionic halo is no longer spherically symmetric and is characterized by dipole, quadruple, etc.
moments. In other words, the modeling of the halo profiles for the magnetic stars can be useful in the procedure of identification of these objects, as well as, in the detailing of their structures.
Our goal is to analyze specific details of the axionic dark matter profiles, which can be formed near the folds in the profile of the gravitational potential. The axion distribution near the folds is
non-monotonic, thus the electric field induced by the strong magnetic field in the axionic environment can signal about the appearance of the inverted layers analogous to the ones in the axionically
active plasma [
]. We show that, when the spacetime is characterized by the non-vanishing cosmological constant, the folds in the profiles of the gravitational potential appear, the solutions to the equations for
the pseudoscalar (axion) field inherit the fold-like behavior, and the axionically induced electric field changes the sign in these domains in analogy with the phenomena of stratification in the
axionically active plasma [
The paper is organized as follows. In
Section 2
, we discuss mathematical details, describing the magneto-electro-statics of the axionic dyons. In
Section 3
, we introduce and mathematically describe the idea of folds in the profile of the gravitational potential. In
Section 4
, we analyze the solutions to the key equations of the model and illustrate the appearance of the fold-like structures in the profiles of axion and electric fields.
Section 5
contains discussion.
2. Description of the Axionic Dark Matter Profiles
2.1. The Total Action Functional
We consider the model, which can be described by the total action functional
$S ( total ) = S ( EH ) + S ( BMF ) + S ( AE ) .$
$S ( EH )$
is the Einstein–Hilbert functional with the Ricci scalar
and cosmological constant
$S ( EH ) = ∫ d 4 x − g ( R + 2 Λ ) 2 κ ,$
$S ( BMF )$
describes the contribution of a matter and/or fields, which form the background spacetime. The action functional of the axiono-electromagnetic subsystem is represented in the form
$S ( AE ) = ∫ d 4 x − g 1 4 F m n F m n + ϕ F m n * + 1 2 Ψ 0 2 V − ∇ m ϕ ∇ m ϕ ,$
$F m n$
is the Maxwell tensor and
$F m n *$
is its dual tensor;
denotes the dimensionless pseudoscalar (axion) field,
is the potential of the pseudoscalar field, and
$Ψ 0 = 1 g A γ γ$
is the parameter reciprocal to the constant of the axion-photon coupling
$g A γ γ$
2.2. Background State
We follow the hierarchical approach, according to which the background gravitational field is considered to be fixed, and the axionic dark matter is distributed in this given spacetime. Why we do it?
The relativistic objects of the neutron star type are compact, but the mass density inside these objects is very high, about
$ρ n ∝ 10 15 g / cm 3$
. The average mass density of the dark matter is known to be estimated as
$ρ DM ∝ 10 − 24 g / cm 3$
, but, in contrast to the dense objects, the dark matter is distributed quasi-uniformly in the whole Universe. Thus, the gravitational field in the vicinity of the dense magnetic stars is
predetermined by the baryonic matter and by the magnetic field with very high energy. From the mathematical point of view, in order to describe the background state we use only two elements of the
total action functional (
), namely,
$S ( EH ) + S ( BMF )$
, and consider the known solutions to the corresponding master equations. As for the axionic and electric subsystems, we obtain the master equations and analyze their solutions with the assumption
that the gravity field is already found.
2.3. Master Equations of the Axion Electrodynamics
In this work we assume that the potential of the axion field is of the form
$V ( ϕ ) = m A 2 ϕ 2$
. More sophisticated periodic potential is considered in the papers [
]. Mention should be made, that we use the system of units, in which
$c = 1$
$ℏ = 1$
$G = 1$
; in this case the dimensionality of the axion mass
$m A$
coincides with the one of the inverse length. In the standard system of units we have to replace
$m A$
$m A c ℏ$
. Variation of the action functional (
), with respect to pseudoscalar field
gives the known master equation
$∇ m ∇ m ϕ + m A 2 ϕ = − 1 4 Ψ 0 2 F m n * F m n .$
Variation procedure associated with the electromagnetic potential
$A i$
gives the equation
$∇ k F i k + ϕ F * i k = 0 .$
This equation, being supplemented by the equation
can, as usual, be transformed into
$∇ k F i k = − F * i k ∇ k ϕ .$
The Equations (
), (
) and (
) are known as the master equations of the axion electrodynamics [
2.4. Static Spacetime with Spherical Symmetry
We assume that the background spacetime is static and spherically symmetric, and is described by the metric
$d s 2 = N ( r ) d t 2 − d r 2 N ( r ) − 1 r 2 d θ 2 + sin 2 θ d φ 2 .$
When the pseudoscalar and electromagnetic fields inherit the spacetime symmetry, we obtain that
is the function of the radial variable only,
$ϕ ( r )$
, and the potential of the electromagnetic field can be presented in the form
$A i = δ i 0 A 0 + δ i φ A φ .$
When one deals with the magnetic monopole of the Dirac type, the azimuthal component of the potential is considered to be chosen in the form
$A φ = Q m ( 1 − cos θ ) ,$
$Q m$
is the magnetic charge (see, e.g., [
]). The Equations (
) and (
) can be reduced to one equation only, which contains the electrostatic potential
$A 0 ( r )$
$d d r r 2 d A 0 d r + Q m ϕ = 0 .$
The Equation (
) takes now the form
$1 r 2 d d r r 2 N d ϕ d r − m A 2 ϕ = − Q m Ψ 0 2 r 2 d A 0 d r .$
Integration of the Equation (
) gives
$d A 0 d r = Q ( r ) r 2 , Q ( r ) ≡ K − Q m ϕ ,$
is the constant of integration. The function
$Q ( r ) = K − Q m ϕ ( r )$
plays here the role of an effective electric charge, which is virtually distributed around the object at the presence of the axion field. This idea allows us to use the term axionically induced
electric field.
As the next step, we replace the term
$d A 0 d r$
in the right-hand side of (
) with (
), and obtain the master equation for the pseudoscalar (axion) field
$1 r 2 d d r r 2 N d ϕ d r = m A 2 + Q m 2 Ψ 0 2 r 4 ϕ − Q m K Ψ 0 2 r 4 .$
Below, we analyze the solutions to the Equation (
) for models, in which the known metric function
$N ( r )$
contains the non-vanishing cosmological constant.
3. On the Features of the Exact Solution Describing the Dirac Magnetic Monopole in the Spacetime with Cosmological Constant
3.1. Geometrical Aspects and Definition of the Fold
The magnetic monopole forms the background spacetime with the well-known metric
$N = 1 − 2 M r + Q m 2 r 2 − 1 3 Λ r 2 .$
Since we work with the units with
$c = 1$
$G = 1$
, the asymptotic mass
and the magnetic charge of the monopole
$Q m$
have the formal dimensionality of the length. The metric (
) covers the following exact solutions.
1. When
$Q m = 0$
$M = 0$
, we obtain the de Sitter metric in the so-called
$N ( r ) = 1 − Λ 3 r 2 .$
It is well-known that using the coordinate transformations
$t = τ − 1 2 3 Λ ln 1 − Λ 3 R 2 · e 2 Λ 3 τ , r = e Λ 3 τ · R ,$
one can obtain the de Sitter metric in the
$d s 2 = d τ 2 − e 2 Λ 3 τ d R 2 + R 2 d θ 2 + sin 2 θ d φ 2 .$
When the cosmological constant is positive,
$Λ > 0$
, the coordinate transformation (
) is defined for the domain, in which the argument of the logarithm is positive, i.e., when
$r < r H ≡ 3 Λ$
, in other words,
$r H$
indicates the cosmological horizon. When
$Λ < 0$
, we deal with the anti-de Sitter spacetime, which has no horizons.
2. When
$Q m = 0$
, we obtain the Schwarzschild-de Sitter metric
$N ( r ) = 1 − 2 M r − Λ 3 r 2 .$
When $Λ > 0$, depending on the value of the mass M, the spacetime can have two, one or zero horizons. When $Λ < 0$, there is one horizon. When $Λ$ = 0, we deal with the Schwarzschild model, which is
characterized by one event horizon at r = 2M.
3. General case: The Schwarzschild–Reissner–Nordström–de Sitter solution. Searching for the horizons from the equation $N ( r ) = 0$, we happen to be faced with the algebraic equation of the fourth
order, and this spacetime is known to be equipped by three horizons as maximum. But our goal is more detailed: we would like to find the sets of parameters M, $Q m$ and $Λ$, for which the profile $N
( r )$ contains folds. We define the fold as the domain, which is characterized by the following two features:
the profile $N ( r )$ has the central minimum, the barrier on the left of the minimum, and the maximum on the right;
this domain is inside the cosmological horizon, but it is not hidden inside the event horizon, i.e., $N ( r ) > 0$ in this domain.
We have to stress, that when $Λ$ = 0, the metric function $N ( r )$ can have the barrier on the left, the minimum, but there is no maximum on the right-hand side, there is only the monotonic curve,
which tends asymptotically to the horizontal line $N = 1$. Furthermore, one can imagine the folds of the second kind, for which the barrier is on the right-hand side of the minimum, and the maximum
on the left-hand side, respectively. In this paper we consider the first variant of the fold only.
3.2. Horizons
3.2.1. Auxiliary Function Indicating the Number of Horizons
The analysis of the causal structure and of the folds appearance is based on the following approach (this approach was successfully applied in [
] for the case with the equation of the sixth order). First, we consider that
$Λ > 0$
, and rewrite the equation
$N ( r ) = 0$
in the form
$M = 1 2 f ( r ) , f ( r ) ≡ r + Q m 2 r − Λ 3 r 3 .$
The auxiliary function $f ( r )$ starts from infinity at $r = 0$ and tends to minus infinity at $r → ∞$; it can possess two or zero local extrema depending on the value of the dimensionless guiding
parameter $Λ Q m 2$.
3.2.2. The Case $Λ Q m 2 < 1 4$
$Λ Q m 2 < 1 4$
, the function
$f ( r )$
has minimum and maximum, respectively, on the spheres, indicated as
$r 1 = 1 2 Λ 1 − 1 − 4 Λ Q m 2 , r 2 = 1 2 Λ 1 + 1 − 4 Λ Q m 2 .$
The sketch of this function is depicted on the
Figure 1
a. The number of horizons is predetermined by the number of the intersection points of the horizontal mass line
$y = M$
and of the graph of the function
$y = 1 2 f ( r )$
According to the sketch
$( a )$
, the values
$r 1$
$r 2$
define two critical values of the mass:
$M 1 , 2 = 1 2 f ( r 1 , 2 ) = 1 ∓ 1 − 4 Λ Q m 2 + 4 Λ Q m 2 18 Λ 1 ∓ 1 − 4 Λ Q m 2 .$
Let us analyze the ratio
$H ≡ M 1 | Q m |$
as a function of the dimensionless parameter
$z = 4 Λ Q m 2$
$H ( z ) = 2 3 z 2 1 − 1 − z 2 + z 2 1 + 1 − z 2 , H ′ ( z ) = − 2 z 1 − z 2 1 + 1 − z 2 .$
Clearly, we deal with defined on the interval $0 ≤ z < 1$ monotonic function, which takes it maximal value $H ( 0 ) = 1$ at $z = 0$, and tends to $H ( 1 ) = 2 2 3 < 1$ at $z = 1$. In other words,
when $0 < Λ Q m 2 < 1 4$, the ratio $M 1 | Q m |$ does not exceed one, and the critical mass $M 1$ belongs to the interval $2 2 3 | Q m | < M 1 < | Q m |$. In the standard units we have to replace $M
1 → G M c 2$ and $Q m → G Q m 4 π ε 0 c 2$, where $ε 0$ is the vacuum permittivity; thus, the condition $M 1 < | Q m |$ reads $M 1 < | Q m | 4 π ε 0 G$.
For different values of the asymptotic mass of the object, M, we obtain the following results.
• When $M < M 1$, the mass line crosses the indicated graph in one point, i.e., there is only one (cosmological) horizon.
• When $M = M 1$, the mass line is the tangent one with respect to the minimum of this graph, thus, there are two horizons: the double event horizon and the simple cosmological one.
• When $M 1 < M < M 2$, there are three intersection points, thus, there are three horizons: the inner and outer event horizons and the cosmological one.
• When $M = M 2$ there are two horizons: the simple event horizon and the double cosmological one.
• When
$M > M 2$
, there is only one horizon; but in contrast to the case
$M < M 1$
, it is the specific case, when all the apparent Universe is inside the event horizon (see, e.g., [
] for some analogy).
3.2.3. The Case $Λ Q m 2 = 1 4$
For this case the values
$r 1$
$r 2$
$r 1 = r 2 = 1 2 Λ$
; the critical masses also coincide,
$M 1 = M 2 = 1 3 2 Λ$
. Now, according to the panel
$( b )$
Figure 1
, there are simple cosmological horizons for every
$M ≠ 1 3 2 Λ$
, and the event horizons are absent. In the case
$M = 1 3 2 Λ$
the graph of the function
$f ( r )$
has the cubic inflexion point, when the minimum and maximum coincide; now one has triple horizon, i.e., the inner, outer horizons coincide with the cosmological one.
3.2.4. The Case $Λ Q m 2 > 1 4$
For this case the values
$r 1$
$r 2$
are complex, thus, the extrema of the auxiliary function
$f ( r )$
are absent, and according to the panel
$( c )$
Figure 1
. for every mass
there is only one intersection point corresponding to the cosmological horizon.
Short Resume
If we search for the models with the cosmological horizon, but without the event horizons, we can choose one of the following conditions:
$Λ Q m 2 < 1 4$, $M < M 1$;
$Λ Q m 2 = 1 4$, $M < 1 3 2 Λ$;
$Λ Q m 2 > 1 4$.
We consider below the first case.
3.3. Folds
The next point of our analysis is the study of the folds. We consider now the derivative of the metric function
$N ′ ( r ) = 2 M r 2 − 2 Q m 2 r 3 − 2 Λ 3 r ,$
and rewrite the equation
$N ′ ( r ) = 0$
as follows:
$M = f ˜ ( r ) , f ˜ ( r ) ≡ Q m 2 r + Λ 3 r 3 .$
When we consider the extrema of the auxiliary function
$f ˜ ( r )$
, we obtain that there exists the minimum at
$r = r * = Q m 2 Λ 1 4$
, which corresponds to the critical value of the mass
$M c = 4 3 | Q m | ( Λ Q m 2 ) 1 4 .$
$4 Λ Q 2 ≤ 1$
, it is simply to show that
$M 1 ≥ M c$
. Indeed, the ratio
$M 1 M c$
can be rewritten as follows:
$M 1 M c = μ ( z ) = 1 + z 2 − 1 − z 2 2 z 3 2 1 − 1 − z 2 ,$
$z = 4 Λ Q m 2$
. This ratio is equal to one, when
$z = 1$
, and tends monotonically to infinity at
$z → 0$
, in other words, the graph of the function
$μ ( z )$
locates above the horizontal line
$y = 1$
, when
$0 < z < 1$
Taking into account these features, we can state the following.
• When $M > M c$, the horizontal mass line $y = M$ crosses the graph of the function $f ˜ ( r )$ twice; this means that there are two extrema: the minimum and maximum.
• When $M = M c$, two extrema coincide forming the cubic inflexion point.
• When $M < M c$, the profile $N ( r )$ is monotonic.
We are interested to study the case $M > M c$, since only this case relates to the presence of the fold, which we search for. At the same time we hope to find the solution without event horizons
(only the cosmological horizon can exist). It is possible when $M 1 > M > M c$.
3.4. Final Remarks about the Features of the Spacetime Geometry
3.4.1. The Choice of the Appropriate Scale for the Radial Variable
In order to clarify the possibility of the folds existence, we rewrite the basic equations using the replacement
$r = x | Q m | ( Λ Q m 2 ) 1 4 .$
Keeping in mind that that there are two specific radii: first, the quantity
$r Λ = 1 Λ$
, which relates to the cosmological scale, second,
$r Q = | Q m |$
, which is the Reissner-Nordström radius, we see that (
) can be rewritten using the geometric mean value
$r = x r Λ r Q$
. In terms of the new dimensionless variable
we obtain
$N ( x ) = 1 + Λ Q m 2 1 x 2 − x 2 3 − 8 3 x M M c .$
This representation of the Reissner-Nordström-de Sitter metric shows explicitly that there are, in fact, two dimensionless guiding parameters of the model: the parameter
$Λ Q m 2 = r Q r Λ$
and the reduced mass
$M M c$
. In
Figure 2
, we present three illustrations of the folds in the profile of the metric function
$N ( x )$
, which has no event horizons, but possesses the cosmological horizon.
3.4.2. On the Problem of Naked Singularity and Cosmic Censorship Conjecture
The point
$r = 0$
is singular for the metric with the coefficient (
). Since we consider the spacetime without event horizons, this point should be indicated as the naked singularity, which is typical for the Reissner–Nordström metric with dominating charge
$Q m 2 > M 2$
(the so-called super-extremal case). The discussions about physical sense of such solutions were activated by Roger Penrose, who has formulated the cosmic censorship conjecture [
]. Active debates concerning the physical status of the naked singularity continue until now, and there are three groups of disputants. The representatives of the first group insist that the
solutions describing the naked singularity are non-physical. Scientists from the second group admit that such naked singularities exist in the nature, that it is still open problem, and they propose
the tests to verify this hypothesis (see, e.g., [
] and references therein). We belong to the third group, and we believe that the paradigm of the non-minimal coupling of the photons with the spacetime curvature removes this problem at all. In the
paper [
] we have shown, that in the model of non-minimal Dirac monopole an additional horizon, formed due to the coupling of the photons to the curvature, appears, so that the point
$r = 0$
inevitably becomes hidden inside this non-minimal horizon; in other words, the non-minimal interaction plays the role of the cosmic censor. Then in the work [
] we have found the exact solution to the non-minimally extended Einstein equations, which is regular at the center. For that solution the metric coefficient
$N ( r )$
has the form
$N ( r ) = 1 + r 4 r 4 + 2 q Q m 2 − 2 M r + Q m 2 r 2 − 1 3 Λ r 2 ,$
is the non-minimal coupling parameter. The solution (
) with
$Λ = 0$
was obtained in [
], and then we studied it in [
]. Clearly, for the solution (
) we obtain
$N ( 0 ) = 1$
, i.e., the solution is regular at the center, and the question about the naked singularity disappears. To be more precise, we can attract attention of the Reader to the curve III in Figure 2
presented in the paper [
]: one can see the analog of the fold, but the function
$N ( r )$
is regular at the center. Why in the presented paper we considered the non-regular metric, if we have the example of the regular one? The explanation is very simple: the non-minimal scale associated
with the coupling parameter
is estimated to be extremely small (of the order of the Compton radius of electron). This means that the folds, which we search for in the presented paper, are arranged rather far from the
non-minimal zone, and the metric (
) gives the appropriate approximation for (
) in the fold zone.
4. Analysis of Solutions to the Key Equation of the Axion Field
4.1. The Profile of the Axion Field Distribution
In terms of the variable
the key equation for the axion field (
) takes the form
$ϕ ″ ( x ) + ϕ ′ ( x ) x 2 N ( x ) ′ x 2 N ( x ) = ϕ | Q m | Λ Ψ 0 2 x 4 N ( x ) m A 2 Ψ 0 2 x 4 + Λ − K Λ sgn Q m Ψ 0 2 x 4 N ( x ) .$
Keeping in mind that the function
$x 2 N ( x )$
is the polynomial of the fourth order, and that it has no zeros in the domain inside the cosmological horizon,
$x < x *$
, we can see that this equation has only two singular points,
$x = 0$
$x = x *$
, which are situated on the edges of the admissible interval
$0 < x < x *$
. Equation (
) belongs to the class of the Fuchs equations [
$Λ > 0$
, we can not prolong the values of the radial variable to the infinity; we have to stop the analysis on the cosmological horizon
$x = x *$
. In other words, when we speak about the far zone, we mean the requirement
$x → x *$
, and
$N ( x * ) = 0$
. One can see, that in this limit the axion field
tends to the value
$ϕ ∞$
given by
$ϕ ∞ = K Λ Q m m A 2 Ψ 0 2 x * 4 + Λ .$
The results for the near zone can be obtained numerically. Numerical simulation includes the variation of the following set of guiding parameters: first, the parameter
$Λ Q m 2$
(it already appeared in the analysis of the function
$N ( x )$
); second, the parameter
$m A 2 Q m 2$
; third, the parameter
$K Q m$
, fourth, the coupling constant
$Ψ 0$
. The analysis has shown that the graphs
$ϕ ( x )$
inherit the fold-like structure of the gravitational potential; for illustration, we presented three graphs in
Figure 3
4.2. The Profile of the Energy-Density of the Axionic Dark Matter
The scalar of the axion field energy density is standardly defined as
$W = U i U k T i k ( axion )$
, where
$U i$
is the global four-velocity vector, coinciding with the normalized time-like Killing vector
$ξ i = δ 0 i$
. Normalization of this Killing four-vector gives
$U i = ξ i ξ s ξ s = 1 N δ 0 i$
. The quantity
$T i k ( axion )$
is the stress-energy tensor of the pseudoscalar axion field. The energy density scalar can be written as
$W = U i U k Ψ 0 2 ∇ i ϕ ∇ k ϕ + 1 2 g i k m A 2 ϕ 2 − ∇ n ϕ ∇ n ϕ .$
In the static spherically symmetric case we obtain
$W ( r ) = 1 2 Ψ 0 2 N ( r ) ϕ ′ 2 ( r ) + m A 2 ϕ 2 .$
Using the profiles of the functions
$N ( x )$
$ϕ ( x )$
we can illustrate the typical behavior of the profile of the function
$W ( x )$
Figure 4
). These profiles happen to be more sophisticated than the fold-like profile of the function
$N ( x )$
, since the extrema of the functions
$N ( x )$
do not coincide.
4.3. Profiles of the Axionically Induced Electric Field
When the profile of the axion field
$ϕ ( x )$
is found, we can reconstruct the profile of the axionically induced electric field using the formula
$E ( x ) = Λ Q m 2 x 2 Q m K Q m − ϕ ( x ) .$
The electric field changes the sign on the surfaces
$x = x j$
, for which
$ϕ ( x j ) = K Q m$
. The typical profiles of the electric field are presented on
Figure 5
; these profiles look like the inverted fold-like structure (the fold-like structure will be recovered, if we change the sign of the magnetic charge
$Q m$
and the parameter
simultaneously). On these profiles one can see two values of
$x j$
, in which the electric field changes the sign. Near the cosmological horizon, the behavior of this electric field is of the Coulombian type, i.e.,
$E ∝ 1 x 2$
5. Discussion
We described an example of a new specific substructure, which can appear in the outer zone of the axionic dyon; we indicated it as a fold. The fold is presented in the profile of the metric function
$N ( r )$
as a specific zone, which contains the minimum, the barrier on the left, and the maximum on the right of this minimum. The fold is entirely located in the outer zone, i.e., it can not be harbored by
the event horizon. The necessary condition of the fold appearance is the inequality
$M 1 > M > M c$
, where
is the asymptotic mass of the dyon, and
$M 1$
$M c$
are the critical masses given by the formulas
$M 1 = 1 − 1 − 4 Λ Q m 2 + 4 Λ Q m 2 18 Λ 1 − 1 − 4 Λ Q m 2 , M c = 4 3 | Q m | ( Λ Q m 2 ) 1 4 .$
Both of the critical masses
$M 1$
$M c$
contain only two model parameters: the cosmological constant
and the magnetic charge
$Q m$
. When
$Λ = 0$
the fold can not be formed, since the maximum on the right of the central minimum disappears. On the fold bottom the derivative of the gravitational potential vanishes, thus there the massive
particle does not feel the gravitational force and can be at rest. The width and depth of the fold are regulated, according to the Formula (
), by two dimensionless parameters
$Λ Q m 2$
$M M c$
Then we analyzed the solution to the key equation for the axion field (
), and have found that the profile of the axion field reveals the substructure of the same type. To be more precise, the fold-like zones are found in the profile of the function
$ϕ ( x )$
, and in the energy density profile
$W ( x )$
Figure 3
Figure 4
, respectively). In other words, the axionic distribution bears the imprint of the fold in the gravitational potential.
Finally, we studied the profile of the electric field, induced by the magnetic field in the axionic environment. Again, the fold-like structure has been found in this profile (see
Figure 5
). In more detail, we have seen, that the electric field changes the sign twice in this zone; such a behavior is typical for the double electric layers. Similar results are obtained in the paper [
], where the change of the electric field direction was associated with the stratification of plasma in the axionic dyon magnetosphere. We think that development of this idea can be interesting for
the procedure of identification of the magnetars based on the fine spectroscopic analysis of obtained data. Why do we think so? The magnetars possess huge magnetic field of the order
$10 13$
$10 15 G$
; this magnetic field produces spectroscopic effects, such as Zeeman effect [
]. If the dark matter has the axionic nature, the axionically induced electric field appears in the vicinity of magnetic star. The corresponding coefficient of transformation is estimated to be less
$10 − 8$
; however, the axionically induced electric field near magnetars is able to produce the quite distinguishable Stark effect. When one has both effects in the magnetic and electric fields (parallel and
/or crossed), then the possibility appears to combine the well-elaborated methods and to organize an extended diagnostics [
]. Clearly, the standard Coulombian type radial electric field can be produced in many standard charged astrophysical objects, but if we hope to identify, say, the axionic dyon, we have to find some
very specific detail distinguishing this object. In this sense the fold-like structure of the profile of the electric field of the axionic dyon, which is described in our work, gives just such a
specific detail. Of course, this idea needs detailed estimations and description of the corresponding diagnostics, however, the discussion of such questions goes beyond the scope of this work.
Author Contributions
Conceptualization, A.B.; formal analysis, A.B. and D.G.; investigation, A.B. and D.G.; writing–original draft preparation, A.B.; writing–review and editing, A.B.; visualization, D.G. All authors have
read and agreed to the published version of the manuscript.
This research was funded by Russian Science Foundation grant number 16-12-10401.
The work was supported by Russian Science Foundation (Project No. 16-12-10401), and, partially, by the Program of Competitive Growth of Kazan Federal University.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1.
Typical sketches of the auxiliary function (
), which illustrate the number of horizons depending on the values of the guiding parameter
$Λ Q m 2$
and of the asymptotic mass
. Panel
$( a )$
illustrates the case
$Λ Q m 2 < 1 4$
; panel
$( b )$
relates to the case
$Λ Q m 2 = 1 4$
, and panel
$( c )$
corresponds to
$Λ Q m 2 > 1 4$
Figure 2.
Folds in the profiles of the metric function
$N ( x )$
). There exists the infinite barrier on the left-hand side, the central minimum, and the maximum on the right-hand side. The fold is situated in the domain with positive
$N ( x )$
and is not harbored by the event horizon. The dimensionless guiding parameters
$Λ Q m 2$
$M M c$
are presented near the graphs in the box.
Figure 3.
Axion field profiles
$ϕ ( x )$
, as the solutions to the master Equation (
). The guiding parameters of the model:
$Λ Q m 2$
, and
$M M c$
are fixed in the box near the graphs; for the simplicity of illustration we put
$Ψ 0 = 1$
$m A 2 = 0.1$
in the chosen system of units. The vertical line symbolizes the delimiter associated with the boundary of the solid body of the object, and its intersection with the graph
$ϕ ( x )$
defines the boundary value
$ϕ ( x 0 )$
. In the far zone, the graph of the function
$ϕ ( x )$
tends to the horizontal asymptotic line, which corresponds to
$ϕ ∞$
given by (
). The profiles of the axion field distribution inherit the fold-like structure of the profiles of the metric function
$N ( x )$
Figure 4.
Typical profiles of the energy density scalar of the axion field (
). The basic profile has the typical fold-like structure: the minimum, the barrier on the left of the minimum, the maximum on its right-hand side. In the far zone the axion energy density tends to a
constant, and the graphs have the horizontal asymptotes. The vertical line relates to the object boundary and defines the corresponding boundary value
$W ( x 0 )$
Figure 5. Typical profiles of the axionically induced electric field. The profiles have inverted fold-like structure. The electric field changes sign twice; its profile tends to the Coulombian curve
in the far zone near the cosmological horizon. The vertical line relates to the boundary of the solid body of the object; the dot relates to the boundary value of the electric field.
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
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Balakin, A.; Groshev, D. Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior. Symmetry 2020, 12, 455. https://doi.org/10.3390/
AMA Style
Balakin A, Groshev D. Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior. Symmetry. 2020; 12(3):455. https://doi.org/10.3390/
Chicago/Turabian Style
Balakin, Alexander, and Dmitry Groshev. 2020. "Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior" Symmetry 12, no. 3: 455.
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Acoustic Wave Correlation Tomography of Time-Varying Disordered Structures
Archives of Acoustics, 43, 4, pp. 647–655, 2018
Acoustic Wave Correlation Tomography of Time-Varying Disordered Structures
An original model based on first principles is constructed for the temporal correlation of acoustic waves propagating in random scattering media. The model describes the dynamics of wave fields in a
previously unexplored, moderately strong (mesoscopic) scattering regime, intermediate between those of weak scattering, on the one hand, and diffusing waves, on the other. It is shown that by
considering the wave vector as a free parameter that can vary at will, one can provide an additional dimension to the data, resulting in a tomographic-type reconstruction of the full space-time
dynamics of a complex structure, instead of a plain spectroscopic technique. In Fourier space, the problem is reduced to a spherical mean transform defined for a family of spheres containing the
origin, and therefore is easily invertible. The results may be useful in probing the statistical structure of various random media with both spatial and temporal resolution.
Keywords: wave scattering; random media; time correlation; tomographic reconstruction
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).
Bergmann P.G. (1946), The wave equation in a medium with a variable index of refraction, The Journal of the Acoustical Society of America, 17, 4, 329–333.
Berne B.J., Pecora R. (1976), Dynamic light scattering, Wiley, New York.
Carminati R., Elaloufi R., Greffet J.-J. (2004), Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light, Physical Review Letters, 92, 21, 213903.
Cerbino R., Trappe V. (2008), Differential dynamic microscopy: Probing wave vector dependent dynamics with a microscope, Physical Review Letters, 100, 18, 188102.
Cormack A.M., Quinto E.T. (1980), A Radon transform on spheres through the origin in Rn and applications to the Darboux equation, Transactions of the American Mathematical Society, 260, 2, 575–581.
Cowan M.L., Page J.H., Norisuye T., Weitz D.A. (2016), Dynamic sound scattering: Field fluctuation spectroscopy with singly scattered ultrasound in the near and far fields, The Journal of the
Acoustical Society of America, 140, 3, 1992–2001.
Cowan M.L., Page J.H., Weitz D.A. (2000), Velocity fluctuations in fluidized suspensions probed by ultrasonic correlation spectroscopy, Physical Review Letters, 85, 2, 453–456.
Devaney A.J. (2012), Mathematical foundations of imaging, tomography and wavefield inversion, Cambridge University Press, Cambridge.
Giavazzi F., Brogioli D., Trappe V., Bellini T., Cerbino R. (2009), Scattering information obtained by optical microscopy: Differential dynamic microscopy and beyond, Physical Review E, 80, 3,
Gibson A.P., Hebden J.C., Arridge S.R. (2005), Recent advances in diffuse optical imaging, Physics in Medicine and Biology, 50, 4, R1–R43.
Igarashi K., Norisuye T., Kobayashi K., Sugita K., Nakanishi H., Tran-Cong-Miyata Q. (2014), Dynamics of submicron microsphere suspensions observed by dynamic ultrasound scattering techniques in the
frequency-domain, Journal of Applied Physics, 115, 20, 203506.
Ishimaru A. (1978), Wave propagation and scattering in random media, Academic, New York.
Kohyama M., Norisuye T., Tran-Cong-Miyata Q. (2009), Dynamics of microsphere suspensions probed by high-frequency dynamic ultrasound scattering, Macromolecules, 42, 3, 752–759.
Konno T., Norisuye T., Sugita K., Nakanishi H., Tran-Cong-Miyata Q. (2016), Dynamics of micronsized particles in dilute and concentrated suspensions probed by dynamic ultrasound scattering
techniques, Ultrasonics, 65, 59–68.
Ladd A.J.C., Gang H., Zhu J.X., Weitz D.A. (1995), Time-dependent collective diffusion of colloidal particles, Physical Review Letters, 74, 2, 318–321.
Lahiri M., Wolf E., Fischer D.G., Shirai T. (2009), Determination of correlation functions of scattering potentials of stochastic media, Physical Review Letters, 102, 12, 123901.
Li J. et al. (2005), Noninvasive detection of functional brain activity with near-infrared diffusing-wave spectroscopy, Journal of Biomedical Optics, 10, 044002.
MacKintosh F.C., John S. (1989), Diffusing-wave spectroscopy and multiple scattering of light in correlated random media, Physical Review B, 40, 4, 2383–2406.
Maret G., Wolf P.E. (1987), Multiple light scattering from disordered media. The effect of Brownian motion of scatterers, Zeitschrift f¨ur Physik B Condensed Matter, 65, 4, 409–413.
Natterer F., W¨ubbeling F. (2001), Mathematical methods in image reconstruction, SIAM, Philadelphia.
Nilsen S.J., Gast A.P. (1994), The influence of structure on diffusion in screened Coulombic suspensions, Journal of Chemical Physics, 101, 6, 4975–4985.
Pine D.J., Weitz D.A., Chaikin P.M., Herboltzheimer E. (1988), Diffusing wave spectroscopy, Physical Review Letters, 60, 12, 1134–1137.
Reufer M., Martinez V.A., Schurtenberger P., Poon W.C.K. (2012), Differential dynamic microscopy for anisotropic colloidal dynamics, Langmuir, 28, 10, 4618–4624.
Rytov S.M., Kravtsov Y.A., Tatarskii V.I. (1989), Principles of statistical radiophysics, Springer, Berlin.
Samelsohn G. (2009), Diffuse time tomography of strongly scattering random structures, Waves Random Complex Media, 19, 1, 11–27.
Samelsohn G. (2016), Transmission tomography of forward-scattering structures, Journal of the Optical Society of America, 33, 6, 1181–1192.
Samelsohn G. (2017), Invertible propagator for plane wave illumination of forward-scattering structures, Applied Optics, 56, 14, 4029–4038.
Samelsohn G. (2018), Scattering of directed waves as an invertible Radon-to-Helmholtz mapping, Optik, 171, 384–392.
Samelsohn G., Freilikher V. (2003), Spectral coherence of wave fields in random media, [in:] Wave Scattering in Complex Media: From Theory to Applications, B. van Tiggelen and S. Skipetrov [Eds.],
pp. 189–201, Kluwer, Dordrecht.
Samelsohn G., Freilikher V. (2004), Localization of classical waves in weakly scattering two-dimensional media with anisotropic disorder, Physical Review E, 70, 4, 046612.
Samelsohn G., Freilikher V., Haridim M. (2008), Spectral coherence and time-domain transport of waves in random media, Physical Review E, 78, 6, 066602.
Samelsohn G., Gredeskul S.A., Mazar R. (1999), Resonances and localization of classical waves in random systems with correlated disorder, Physical Review E, 60, 5, 6081–6090.
Samelsohn G., Mazar R. (1996), Path-integral analysis of scalar wave propagation in multiple-scattering random media, Physical Review E, 54, 5, 5697–5706.
Snieder R. (2006), The theory of coda wave interferometry, Pure and Applied Geophysics, 163, 2–3, 455–473.
Stephen M.J. (1988), Temporal fluctuations in wave propagation in random media, Physical Review B, 37, 1, 1–5.
Weitz D.A., Pine D.J. (1993), Diffusing wave spectroscopy, [in:] Dynamic light scattering: the method and some applications, W. Brown [Ed.], Oxford University Press, New York.
Wolf E. (1996), Principles and development of diffraction tomography, [in:] Trends in Optics, A. Consortini [Ed.], Academic, San Diego, pp. 83–110.
Yagle A.E. (1992), Inversion of spherical means using geometric inversion and Radon transforms, Inverse Problems, 8, 6, 949–964.
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Dynamic Programming
Overview of ALATE's Work on Dynamic Programming
This is an overview of some of the work done by my students and collaborators in the area of dynamic programming. It includes work by the faculty members Quentin F. Stout, Janis Hardwick and Marilynn
Livingston, and various undergraduate and graduate students, as well as former students.
Our work on dynamic programming has emphasized finite state spaces and computational approaches that yield exact optimizations and analyses. There are two main application areas: adaptive
(sequential) designs for random responses, and determining properties of graphs. There is also some effort on optimizing parallel implementations of dynamic programming. I've also listed a few open
questions related to our work.
For an extensive collection of pointers to work on the subject throughout the world, visit the dynamic programming component of the World-Wide Web for Operations Research and Management Science
(WORMS) pages. The material there is interesting, and includes a fun collection of graphics.
For papers, Paper.ps means a Postscript version of the paper, and Paper.pdf means a PDF version. Each Abstract is in HTML.
Adaptive Designs for Random Responses
Much of the work on adaptive designs has been motivated by problems in allocating patients to treatments in a clinical trial or in industrial applications of estimation or fault tolerance, though
other areas are also being worked on. In the simplest clinical setting, each patient is assigned to one of two treatments, and the outcome (success or failure) is observed before the next patient
arrives. If we are trying to maximize the number of successes, then the problem is the well-known two-armed bandit problem, which can be solved via dynamic programming if one has a Bayesian model.
(The bandit terminology comes from considering slot machines, which are sometimes called bandits. In this view, the two different treatments are two different arms on the slot machine, and allocating
a patient to a treatment corresponds to pulling an arm. For the bandit, the problem is to maximize the expected payout.) Bandit problems are a specific form of stochastic optimization problems, and
are part of the general areas of control theory, dynamical systems, optimization theory, and discrete applied mathematics.
However, we are also interested in other criteria such as the probability of correctly identifying the better treatment at the end of the trial, or the number of times treatments are switched, which
introduces new optimization constraints or forces us to evaluate an allocation procedure multiple times. We developed the technique of path induction to reduce the time required for such repeated
evaluations. (Note that we had previously called path induction "forward induction", but that term proved confusing as it had been previously used.)
Here is a more more extensive explanation of adaptive designs. A survey paper which discusses the algorithms is:
• J. Hardwick and Q.F. Stout, ``Flexible algorithms for creating and analyzing adaptive designs'', In New Developments and Applications in Experimental Design, N. Flournoy, W.F. Rosenberger, and
W.K. Wong, eds., IMS Lecture Notes--Monograph Series vol. 34, 1998, pp. 91-105.
Keywords: adaptive design, stochastic optimization, dynamic programming, bandit problems, algorithms, high-performance computing, few-stage allocation, group sampling, switching, robustness, path
Abstract Paper.ps Paper.pdf
Some additional papers in this area are:
One important variation on this is when one has to make allocation decisions in stages, deciding allocations for several patients at one time. This is highly desirable from a practical viewpoint, but
it apparently greatly increases the time and space required to fully optimize the allocation procedure - see
• J. Hardwick and Q.F. Stout, ``Optimal few-stage designs'', J. Statistical Planning and Inference 104 (2002), pp. 121-145.
Keywords: sequential analysis, clinical trials, two-stage, three-stage, experimental design, group allocation, adaptive sampling, selection, estimation, bandit problems, product of means, dynamic
programming, algorithms.
Abstract Paper.ps Paper.pdf
A closely related variation is where one tries to minimize switching between the alternatives, since switching can be quite costly in some settings. See
• J. Hardwick and Q.F. Stout, ``Response adaptive designs that incorporate switching costs and constraings'', J. Statistical Planning and Inference 137 (2007), pp. 2654-2665
Keywords: adaptive sampling, switching costs, constraints, bandit, estimation, dynamic programming, optimal tradeoffs.
Abstract Paper.pdf
Another variation occurs when one is going to allocate an equal number of patients to each treatment (to maximize the probability of correctly selecting the better treatment), but will stop as soon
as the outcome is known. Dynamic programming can be used to minimize the expected value of other criteria, such as the number of failures or the sample size - see
• J. Hardwick and Q.F. Stout, ``Optimal adaptive equal allocation rules'', Computing Science and Statistics 24 (1992), pp. 597-601.
Keywords: sequential allocation, dynamic programming, curtailment, probability of correct selection, equal allocation, vector at a time.
Abstract Paper.ps Paper.pdf
Some of our recent results require the use of parallel computers, due to the very large memory and computation requirements of the problems.
Many properties of graphs, such as domination numbers, covering numbers, number of codewords, and packings, can be easily determined by dynamic programming in time linear in the number of nodes, if
the graph has a simple structure such as a tree or directed acyclic graph (dag). Similarly, it was fairly well-known that for certain families of graphs, such as k x n meshes (grids) or complete k-
ary trees of height n, where k is fixed, the same properties can be computed by dynamic programming in time linear in n (though exponential in k).
We showed that for these families these properties could actually be computed in constant time (though still exponential in k) by showing that they have a closed form. We exploit the fact that the
dynamic programming is essentially determining minimal weight paths of length n in a finite state space, and must ultimately reach a periodic solution (which can be detected). See
• M.L. Livingston and Q.F. Stout, ``Constant time computation of minimum dominating sets'', Congresses Numerantium 105 (1994), pp. 116-128.
Keywords: codes, covering, domination, packing, matching, perfect domination, grid graph, product graph, mesh, torus, dynamic programming, finite state space.
Abstract Paper.ps Paper.pdf
A related use of dynamic programming concerns evaluating the fault tolerance of allocation systems for parallel computers. For example, suppose we have an n x n mesh, where n is even, which we view
as containing four n/2 x n/2 quadrants. Vertices represent processors in the parallel computer, and suppose initially all processors are functioning correctly. If processors fail independently, what
is the expected number of processors such that no quadrant is fault-free? (We may have a program that needs an n/2 x n/2 submesh, but the allocation system may be only able to allocate the quadrants,
not arbitrary submeshes. Obviously the fewest faults that destroy all quadrants is 4, and the most that leave one quadrant fault-free is 3n^2/4.) This problem, and more interesting situations where
the allocable submeshes overlap, can be solved by dynamic programming - see
• M.L. Livingston and Q.F. Stout, ``Fault tolerance of allocation schemes in massively parallel computers'', Proc. 2nd Symp. Frontiers of Massively Parallel Computation (1988), pp. 491-494.
Keywords: fault tolerance, allocation, hypercube computer, mesh, torus, buddy system.
• M.L. Livingston and Q.F. Stout, ``Parallel allocation algorithms for hypercubes and meshes'', Proc. 4th Conf. on Hypercubes, Concurrent Computers, and Applications (1989), pp. 59-66.
Keywords: subcube allocation, buddy system, cyclic buddy system, submesh allocation, torus.
• M.L. Livingston and Q.F. Stout, ``Fault tolerance of the cyclic buddy subcube location scheme in hypercubes'', Proc. 6th Distributed Memory Computing Conf. (1991), IEEE, pp. 34-41.
Keywords: fault tolerance, subcube location, subcube allocation, hypercube computer, buddy system, gray-coding, folded hypercube, projective hypercube
Unfortunately, due to page constraints, many of the details of the dynamic programming involved were not given in these papers.
Because dynamic programming often involves large state spaces and extensive computations, many people have implemented it on parallel computers. Our interest is in achieving very high performance in
solving DP problems such as occur when creating the adaptive designs discussed above. The high performance is needed because we want to push the limits of the problem sizes that can be solved. To do
so, we must obtain optimal or near-optimal load balance, while minimizing communication and retaining all of the serial optimizations. The high dimensionality encountered means that the array rapidly
decreases as the dynamic programming proceeds, which forces processors to continually shift the index ranges that they are working on. For various formulations of the problem, the optimal work
assignment is itself a dynamic programming problem, though one that is far simpler than the original. Fortunately, some heuristics work extremely well.
A recent result in this area is the ability to optimally design a clinical trial for 200 patients, where there are 3 different treatments available. Other researchers have only been able to handle a
handful of patients when 3 treatments were considered, or they had to resort to suboptimal designs, because the time and space required grow as O(n^6), where n is the number of patients. One of the
things we will now be able to do is determine exactly how suboptimal their designs were, and we will be able to consider better tradeoffs such as statistical power versus failures. With simple
changes, this program can also handle 2 treatments where there are 3 possible outcomes (instead of only 2, as we normally assume). We know of no previous work which achieves optimal solutions for
this problem. With less simple changes, this program can handle some models of 2 treatments with delayed responses, i.e., you must make new allocations before all of the results of previous
allocations are known. Here too we know of no previous optimal designs.
Bob Oehmke is the graduate student working on implementing this on an IBM SP2. He is also working on programs for several other aspects of adaptive designs, and more generally on the problem of
producing efficient highly parallel programs for dynamic programming. Our most recent paper on this work is
• J. Hardwick, R. Oehmke and Q.F. Stout, ``A program for sequential allocation of three Bernoulli populations'', Computational Statistics and Data Analysis 31 (1999), pp. 397-416.
Keywords: adaptive allocation, sequential sampling, bandit, dynamic programming, design of experiments, path induction, stochastic optimization, parallel computing, IBM SP2, MPI, machine
learning, parallel algorithm, multi-arm bandit, clinical trial, high-performance computing.
Abstract Paper.ps Paper.pdf
which builds upon work first reported in
• J. Hardwick, R. Oehmke and Q.F. Stout, ``A parallel program for 3-arm bandits'', Computing Science and Statistics 29 (1997), pp. 390-395.
Keywords: sequential allocation, dynamic programming, parallel computing, bandits.
Abstract Paper.pdf
Several of the above papers mention specific open questions. Here are some of the more general ones. Feel free to contact me if you want (or can give) some more information.
For all of the variations mentioned in adaptive designs, there is a common open question:
Is the given dynamic programming algorithm the fastest way to determine the optimal answer?
That is, dynamic programming finds the optimal answer, but often requires a great deal of time and space. Can the optimal answer be found more quickly? This question applies to a great many other
problems as well.
Related to the work on fault tolerance of allocation systems in parallel computers, the following problem gives the limiting behavior, as the system becomes arbitrarily large, of being able to find a
fault-free square submesh of 1/4 the size of the full mesh.
Suppose points are place uniformly and independently in the unit square, one at a time, until there is no point-free subsquare of edgelength 1/2, where the subsquare must have its sides parallel
to the sides of the square. What is the expected number of points placed?
(It is possible that the solution will not involve dynamic programming.) It is also interesting to know the solution if the original square has both pairs of opposite sides connected, creating a
Here are some conjectures and open problems in the area of finding properties of families of graphs.
Copyright © 2005-2017 Quentin F. Stout
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RFC 5832: GOST R 34.10-2001: Digital Signature Algorithm
Independent Submission V. Dolmatov, Ed.
Request for Comments: 5832 Cryptocom, Ltd.
Category: Informational March 2010
ISSN: 2070-1721
GOST R 34.10-2001:
Digital Signature Algorithm
This document is intended to be a source of information about the
Russian Federal standard for digital signatures (GOST R 34.10-2001),
which is one of the Russian cryptographic standard algorithms (called
GOST algorithms). Recently, Russian cryptography is being used in
Internet applications, and this document has been created as
information for developers and users of GOST R 34.10-2001 for digital
signature generation and verification.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This is a contribution to the RFC Series, independently of any other
RFC stream. The RFC Editor has chosen to publish this document at
its discretion and makes no statement about its value for
implementation or deployment. Documents approved for publication by
the RFC Editor are not a candidate for any level of Internet
Standard; see Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
Dolmatov Informational [Page 1]
RFC 5832 GOST R 34.10-2001 March 2010
Copyright Notice
Copyright (c) 2010 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document.
This document may not be modified, and derivative works of it may not
be created, except to format it for publication as an RFC or to
translate it into languages other than English.
Table of Contents
1. Introduction ....................................................3
1.1. General Information ........................................3
1.2. The Purpose of GOST R 34.10-2001 ...........................3
2. Applicability ...................................................4
3. Definitions and Notations .......................................4
3.1. Definitions ................................................4
3.2. Notations ..................................................6
4. General Statements ..............................................7
5. Mathematical Conventions ........................................8
5.1. Mathematical Definitions ...................................9
5.2. Digital Signature Parameters ..............................10
5.3. Binary Vectors ............................................11
6. Main Processes .................................................12
6.1. Digital Signature Generation Process ......................12
6.2. Digital Signature Verification ............................13
7. Test Examples (Appendix to GOST R 34.10-2001) ..................14
7.1. The Digital Signature Scheme Parameters ...................14
7.2. Digital Signature Process (Algorithm I) ...................16
7.3. Verification Process of Digital Signature (Algorithm II) ..17
8. Security Considerations ........................................19
9. References .....................................................19
9.1. Normative References ......................................19
9.2. Informative References ....................................19
Appendix A. Extra Terms in the Digital Signature Area .............21
Appendix B. Contributors ..........................................22
Dolmatov Informational [Page 2]
RFC 5832 GOST R 34.10-2001 March 2010
1. Introduction
1.1. General Information
1. GOST R 34.10-2001 [GOST3410] was developed by the Federal Agency
for Government Communication and Information under the President
of the Russian Federation with the participation of the All-Russia
Scientific and Research Institute of Standardization.
GOST R 34.10-2001 was submitted by Federal Agency for Government
Communication and Information at President of the Russian
2. GOST R 34.10-2001 was accepted and activated by the Act 380-st of
12.09.2001 issued by the Government Committee of Russia for
3. GOST R 34.10-2001 was developed in accordance with the terminology
and concepts of international standards ISO 2382-2:1976 "Data
processing - Vocabulary - Part 2: Arithmetic and logic
operations"; ISO/IEC 9796:1991 "Information technology -- Security
techniques -- Digital signature schemes giving message recovery";
ISO/IEC 14888 "Information technology - Security techniques -
Digital signatures with appendix"; and ISO/IEC 10118 "Information
technology - Security techniques - Hash-functions".
4. GOST R 34.10-2001 replaces GOST R 34.10-94.
1.2. The Purpose of GOST R 34.10-2001
GOST R 34.10-2001 describes the generation and verification processes
for digital signatures, based on operations with an elliptic curve
points group, defined over a prime finite field.
GOST R 34.10-2001 has been developed to replace GOST R 34.10-94.
Necessity for this development is caused by the need to increase
digital signature security against unauthorized modification.
Digital signature security is based on the complexity of discrete
logarithm calculation in an elliptic curve points group and also on
the security of the hash function used (according to [GOST3411]).
Terminologically and conceptually, GOST R 34.10-2001 is in accordance
with international standards ISO 2382-2 [ISO2382-2], ISO/IEC 9796
[ISO9796-1991], ISO/IEC 14888 Parts 1-3 [ISO14888-1]-[ISO14888-3],
and ISO/IEC 10118 Parts 1-4 [ISO10118-1]-[ISO10118-4].
Note: the main part of GOST R 34.10-2001 is supplemented with three
Dolmatov Informational [Page 3]
RFC 5832 GOST R 34.10-2001 March 2010
"Extra Terms in the Digital Signature Area" (Appendix A of this
"Test Examples" (Section 7 of this memo);
"A Bibliography in the Digital Signature Area" (Section 9.2 of
this memo).
2. Applicability
GOST R 34.10-2001 defines an electronic digital signature (or simply
digital signature) scheme, digital signature generation and
verification processes for a given message (document), meant for
transmission via insecure public telecommunication channels in data
processing systems of different purposes.
Use of a digital signature based on GOST R 34.10-2001 makes
transmitted messages more resistant to forgery and loss of integrity,
in comparison with the digital signature scheme prescribed by the
previous standard.
GOST R 34.10-2001 is obligatory to use in the Russian Federation in
all data processing systems providing public services.
3. Definitions and Notations
3.1. Definitions
The following terms are used in the standard:
Appendix: Bit string, formed by a digital signature and by the
arbitrary text field [ISO14888-1].
Signature key: Element of secret data, specific to the subject and
used only by this subject during the signature generation process
Verification key: Element of data mathematically linked to the
signature key data element, used by the verifier during the digital
signature verification process [ISO14888-1].
Domain parameter: Element of data that is common for all the subjects
of the digital signature scheme, known or accessible to all the
subjects [ISO14888-1].
Signed message: A set of data elements, which consists of the message
and the appendix, which is a part of the message.
Dolmatov Informational [Page 4]
RFC 5832 GOST R 34.10-2001 March 2010
Pseudo-random number sequence: A sequence of numbers, which is
obtained during some arithmetic (calculation) process, used in a
specific case instead of a true random number sequence [ISO2382-2].
Random number sequence: A sequence of numbers none of which can be
predicted (calculated) using only the preceding numbers of the same
sequence [ISO2382-2].
Verification process: A process that uses the signed message, the
verification key, and the digital signature scheme parameters as
initial data and that gives the conclusion about digital signature
validity or invalidity as a result [ISO14888-1].
Signature generation process: A process that uses the message, the
signature key, and the digital signature scheme parameters as initial
data and that generates the digital signature as the result
Witness: Element of data (resulting from the verification process)
that states to the verifier whether the digital signature is valid or
invalid [ISO148881-1]).
Random number: A number chosen from the definite number set in such a
way that every number from the set can be chosen with equal
probability [ISO2382-2].
Message: String of bits of a limited length [ISO9796-1991].
Hash code: String of bits that is a result of the hash function
Hash function: The function, mapping bit strings onto bit strings of
fixed length observing the following properties:
1) it is difficult to calculate the input data, that is the pre-
image of the given function value;
2) it is difficult to find another input data that is the pre-
image of the same function value as is the given input data;
3) it is difficult to find a pair of different input data,
producing the same hash function value.
Note: Property 1 in the context of the digital signature area means
that it is impossible to recover the initial message using the
digital signature; property 2 means that it is difficult to find
another (falsified) message that produces the same digital signature
Dolmatov Informational [Page 5]
RFC 5832 GOST R 34.10-2001 March 2010
as a given message; property 3 means that it is difficult to find
some pair of different messages, which both produce the same
(Electronic) Digital signature: String of bits obtained as a result
of the signature generation process. This string has an internal
structure, depending on the specific signature generation mechanism.
Note: In GOST R 34.10-2001 terms, "Digital signature" and "Electronic
digital signature" are synonymous to save terminological succession
to native legal documents currently in force and scientific
3.2. Notations
In GOST R 34.10-2001, the following notations are used:
V256 - set of all binary vectors of a 256-bit length
V_all - set of all binary vectors of an arbitrary finite length
Z - set of all integers
p - prime number, p > 3
GF(p) - finite prime field represented by a set of integers
{0, 1, ..., p - 1}
b (mod p) - minimal non-negative number, congruent to b modulo p
M - user's message, M belongs to V_all
(H1 || H2 ) - concatenation of two binary vectors
a,b - elliptic curve coefficients
m - points of the elliptic curve group order
q - subgroup order of group of points of the elliptic curve
O - zero point of the elliptic curve
P - elliptic curve point of order q
d - integer - a signature key
Q - elliptic curve point - a verification key
Dolmatov Informational [Page 6]
RFC 5832 GOST R 34.10-2001 March 2010
^ - the power operator
/= - non-equality
sqrt - square root
zeta - digital signature for the message M
4. General Statements
A commonly accepted digital signature scheme (model) (see Section 6
of [ISO/IEC14888-1]) consists of three processes:
- generation of a pair of keys (for signature generation and for
signature verification);
- signature generation;
- signature verification.
In GOST R 34.10-2001, a process for generating a pair of keys (for
signature and verification) is not defined. Characteristics and ways
of the process realization are defined by involved subjects, who
determine corresponding parameters by their agreement.
The digital signature mechanism is defined by the realization of two
main processes (see Section 7):
- signature generation (see Section 6.1) and
- signature verification (see Section 6.2).
The digital signature is meant for the authentication of the
signatory of the electronic message. Besides, digital signature
usage gives an opportunity to provide the following properties during
signed message transmission:
- realization of control of the transmitted signed message
- proof of the authorship of the signatory of the message,
- protection of the message against possible forgery.
A schematic representation of the signed message is shown in
Figure 1.
Dolmatov Informational [Page 7]
RFC 5832 GOST R 34.10-2001 March 2010
| |
+-----------+ +------------------------+- - - +
| message M |---| digital signature zeta | text |
+-----------+ +------------------------+- - - +
Figure 1: Signed message scheme
The field "digital signature" is supplemented by the field "text"
(see Figure 1), that can contain, for example, identifiers of the
signatory of the message and/or time label.
The digital signature scheme determined in GOST R 34.10-2001 must be
implemented using operations of the elliptic curve points group,
defined over a finite prime field, and also with the use of hash
The cryptographic security of the digital signature scheme is based
on the complexity of solving the problem of the calculation of the
discrete logarithm in the elliptic curve points group and also on the
security of the hash function used. The hash function calculation
algorithm is determined in [GOST3411].
The digital signature scheme parameters needed for signature
generation and verification are determined in Section 5.2.
GOST R 34.10-2001 does not determine the process of generating
parameters needed for the digital signature scheme. Possible sets of
these parameters are defined, for example, in [RFC4357].
The digital signature represented as a binary vector of a 512-bit
length must be calculated using a definite set of rules, as stated in
Section 6.1.
The digital signature of the received message is accepted or denied
in accordance with the set of rules, as stated in Section 6.2.
5. Mathematical Conventions
To define a digital signature scheme, it is necessary to describe
basic mathematical objects used in the signature generation and
verification processes. This section lays out basic mathematical
definitions and requirements for the parameters of the digital
signature scheme.
Dolmatov Informational [Page 8]
RFC 5832 GOST R 34.10-2001 March 2010
5.1. Mathematical Definitions
Suppose a prime number p > 3 is given. Then, an elliptic curve E,
defined over a finite prime field GF(p), is the set of number pairs
(x,y), x, y belong to Fp, satisfying the identity:
y^2 = x^3 + a*x + b (mod p), (1)
where a, b belong to GF(p) and 4*a^3 + 27*b^2 is not congruent to
zero modulo p.
An invariant of the elliptic curve is the value J(E), satisfying the
J(E) = 1728 * ------------ (mod p) (2)
Elliptic curve E coefficients a,b are defined in the following way
using the invariant J(E):
| a=3*k (mod p)
| J(E)
| b=2*k (mod p), where k = ----------- (mod p), J(E) /= 0 or 1728 (3)
1728 - J(E)
The pairs (x,y) satisfying the identity (1) are called the elliptic
curve E points; x and y are called x- and y-coordinates of the point,
We will denote elliptic curve points as Q(x,y) or just Q. Two
elliptic curve points are equal if their x- and y-coordinates are
On the set of all elliptic curve E points, we will define the
addition operation, denoted by "+". For two arbitrary elliptic curve
E points Q1 (x1, y1) and Q2 (x2, y2), we will consider several
Suppose coordinates of points Q1 and Q2 satisfy the condition x1 /=
x2. In this case, their sum is defined as a point Q3 (x3,y3), with
coordinates defined by congruencies:
| x3=lambda^2-x1-x2 (mod p), y1-y2
| where lambda= ------- (mod p). (4)
| y3=lambda*(x1-x3)-y1 (mod p), x1-x2
Dolmatov Informational [Page 9]
RFC 5832 GOST R 34.10-2001 March 2010
If x1 = x2 and y1 = y2 /= 0, then we will define point Q3 coordinates
in the following way:
| x3=lambda^2-x1*2 (mod p), 3*x1^2+a
| where lambda= --------- (mod p) (5)
| y3=lambda*(x1-x3)-y1 (mod p), y1*2
If x1 = x2 and y1 = - y2 (mod p), then the sum of points Q1 and Q2 is
called a zero point O, without determination of its x- and y-
coordinates. In this case, point Q2 is called a negative of point
Q1. For the zero point, the equalities hold:
O+Q=Q+O=Q, (6)
where Q is an arbitrary point of elliptic curve E.
A set of all points of elliptic curve E, including zero point, forms
a finite abelian (commutative) group of order m regarding the
introduced addition operation. For m, the following inequalities
p + 1 - 2*sqrt(p) =< m =< p + 1 + 2*sqrt(p). (7)
The point Q is called a point of multiplicity k, or just a multiple
point of the elliptic curve E, if for some point P the following
equality holds:
Q = P + ... + P = k*P. (8)
5.2. Digital Signature Parameters
The digital signature parameters are:
- prime number p is an elliptic curve modulus, satisfying the
inequality p > 2^255. The upper bound for this number must be
determined for the specific realization of the digital signature
- elliptic curve E, defined by its invariant J(E) or by
coefficients a, b belonging to GF(p).
- integer m is an elliptic curve E points group order;
- prime number q is an order of a cyclic subgroup of the elliptic
curve E points group, which satisfies the following conditions:
Dolmatov Informational [Page 10]
RFC 5832 GOST R 34.10-2001 March 2010
| m = nq, n belongs to Z , n>=1
| (9)
| 2^254 < q < 2^256
- point P /= O of an elliptic curve E, with coordinates (x_p,
y_p), satisfying the equality q*P=O.
- hash function h(.):V_all -> V256, which maps the messages
represented as binary vectors of arbitrary finite length onto
binary vectors of a 256-bit length. The hash function is
determined in [GOST3411].
Every user of the digital signature scheme must have its personal
- signature key, which is an integer d, satisfying the inequality
0 < d < q;
- verification key, which is an elliptic curve point Q with
coordinates (x_q, y_q), satisfying the equality d*P=Q.
The previously introduced digital signature parameters must satisfy
the following requirements:
- it is necessary that the condition p^t/= 1 (mod q ) holds for
all integers t = 1, 2, ... B where B satisfies the inequality B
>= 31;
- it is necessary that the inequality m /= p holds;
- the curve invariant must satisfy the condition J(E) /= 0, 1728.
5.3. Binary Vectors
To determine the digital signature generation and verification
processes, it is necessary to map the set of integers onto the set of
binary vectors of a 256-bit length.
Consider the following binary vector of a 256-bit length where low-
order bits are placed on the right, and high-order ones are placed on
the left:
H = (alpha[255], ... , alpha[0]), H belongs to V256 (10)
where alpha[i], i = 0, ... , 255 are equal to 1 or to 0. We will say
that the number alpha belonging to Z is mapped onto the binary vector
h, if the equality holds:
Dolmatov Informational [Page 11]
RFC 5832 GOST R 34.10-2001 March 2010
alpha = alpha[0]*2^0 + alpha[1]*2^1 + ... + alpha[255]*2^255 (11)
For two binary vectors H1 and H2, which correspond to integers alpha
and beta, we define a concatenation (union) operation in the
following way. If:
H1 = (alpha[255], ... , alpha[0]),
H2 = (beta[255], ..., beta[0]),
then their union is
H1||H2 = (alpha[255], ... , alpha[0], beta[255], ..., beta[0])
that is a binary vector of 512-bit length, consisting of coefficients
of the vectors H1 and H2.
On the other hand, the introduced formulae define a way to divide a
binary vector H of 512-bit length into two binary vectors of 256-bit
length, where H is the concatenation of the two.
6. Main Processes
In this section, the digital signature generation and verification
processes of user's message are defined.
For the realization of the processes, it is necessary that all users
know the digital signature scheme parameters, which satisfy the
requirements of Section 5.2.
Besides, every user must have the signature key d and the
verification key Q(x[q], y[q]), which also must satisfy the
requirements of Section 5.2.
6.1. Digital Signature Generation Process
It is necessary to perform the following actions (steps) according to
Algorithm I to obtain the digital signature for the message M
belonging to V_all:
Step 1 - calculate the message hash code M: H = h(M). (14)
Step 2 - calculate an integer alpha, binary representation of which
is the vector H, and determine e = alpha (mod q ). (15)
If e = 0, then assign e = 1.
Dolmatov Informational [Page 12]
RFC 5832 GOST R 34.10-2001 March 2010
Step 3 - generate a random (pseudorandom) integer k, satisfying the
0 < k < q. (16)
Step 4 - calculate the elliptic curve point C = k*P and determine if:
r = x_C (mod q), (17)
where x_C is x-coordinate of the point C. If r = 0, return to
step 3.
Step 5 - calculate the value:
s = (r*d + k*e) (mod q). (18)
If s = 0, return to step 3.
Step 6 - calculate the binary vectors R and S, corresponding to r
and s, and determine the digital signature zeta = (R || S) as a
concatenation of these two binary vectors.
The initial data of this process are the signature key d and the
message M to be signed. The output result is the digital signature
6.2. Digital Signature Verification
To verify digital signatures for the received message M belonging to
V_all, it is necessary to perform the following actions (steps)
according to Algorithm II:
Step 1 - calculate the integers r and s using the received signature
zeta. If the inequalities 0 < r < q, 0 < s < q hold, go to the next
step. Otherwise, the signature is invalid.
Step 2 - calculate the hash code of the received message M:
H = h(M). (19)
Step 3 - calculate the integer alpha, the binary representation of
which is the vector H, and determine if:
e = alpha (mod q). (20)
If e = 0, then assign e = 1.
Step 4 - calculate the value v = e^(-1) (mod q). (21)
Dolmatov Informational [Page 13]
RFC 5832 GOST R 34.10-2001 March 2010
Step 5 - calculate the values:
z1 = s*v (mod q), z2 = -r*v (mod q). (22)
Step 6 - calculate the elliptic curve point C = z1*P + z2*Q and
determine if:
R = x_C (mod q), (23)
where x_C is x-coordinate of the point.
Step 7 - if the equality R = r holds, then the signature is accepted.
Otherwise, the signature is invalid.
The input data of the process are the signed message M, the digital
signature zeta, and the verification key Q. The output result is the
witness of the signature validity or invalidity.
7. Test Examples (Appendix to GOST R 34.10-2001)
This section is included in GOST R 34.10-2001 as a reference appendix
but is not officially mentioned as a part of the standard.
The values given here for the parameters p, a, b, m, q, P, the
signature key d, and the verification key Q are recommended only for
testing the correctness of actual realizations of the algorithms
described in GOST R 34.10-2001.
All numerical values are introduced in decimal and hexadecimal
notations. The numbers beginning with 0x are in hexadecimal
notation. The symbol "\\" denotes a hyphenation of a number to the
next line. For example, the notation:
represents 1234567890 in decimal and hexadecimal number systems,
7.1. The Digital Signature Scheme Parameters
The following parameters must be used for the digital signature
generation and verification (see Section 5.2).
Dolmatov Informational [Page 14]
RFC 5832 GOST R 34.10-2001 March 2010
7.1.1. Elliptic Curve Modulus
The following value is assigned to parameter p in this example:
p= 57896044618658097711785492504343953926\\
p = 0x8000000000000000000000000000\\
7.1.2. Elliptic Curve Coefficients
Parameters a and b take the following values in this example:
a = 7
a = 0x7
b = 43308876546767276905765904595650931995\\
b = 0x5FBFF498AA938CE739B8E022FBAFEF40563\\
7.1.3. Elliptic Curve Points Group Order
Parameter m takes the following value in this example:
m = 5789604461865809771178549250434395392\\
m = 0x80000000000000000000000000000\\
7.1.4. Order of Cyclic Subgroup of Elliptic Curve Points Group
Parameter q takes the following value in this example:
q = 5789604461865809771178549250434395392\\
q = 0x80000000000000000000000000000001\\
Dolmatov Informational [Page 15]
RFC 5832 GOST R 34.10-2001 March 2010
7.1.5. Elliptic Curve Point Coordinates
Point P coordinates take the following values in this example:
x_p = 2
x_p = 0x2
y_p = 40189740565390375033354494229370597\\
y_p = 0x8E2A8A0E65147D4BD6316030E16D19\\
7.1.6. Signature Key
It is supposed, in this example, that the user has the following
signature key d:
d = 554411960653632461263556241303241831\\
d = 0x7A929ADE789BB9BE10ED359DD39A72C\\
7.1.7. Verification Key
It is supposed, in this example, that the user has the verification
key Q with the following coordinate values:
x_q = 57520216126176808443631405023338071\\
x_q = 0x7F2B49E270DB6D90D8595BEC458B5\\
y_q = 17614944419213781543809391949654080\\
y_q = 0x26F1B489D6701DD185C8413A977B3\\
7.2. Digital Signature Process (Algorithm I)
Suppose that after steps 1-3, according to Algorithm I (Section 6.1),
are performed, the following numerical values are obtained:
e = 2079889367447645201713406156150827013\\
Dolmatov Informational [Page 16]
RFC 5832 GOST R 34.10-2001 March 2010
e = 0x2DFBC1B372D89A1188C09C52E0EE\\
k = 538541376773484637314038411479966192\\
k = 0x77105C9B20BCD3122823C8CF6FCC\\
And the multiple point C = k * P has the coordinates:
x_C = 297009809158179528743712049839382569\\
x_C = 0x41AA28D2F1AB148280CD9ED56FED\\
y[C] = 328425352786846634770946653225170845\\
y[C] = 0x489C375A9941A3049E33B34361DD\\
Parameter r = x_C(mod q) takes the value:
r = 297009809158179528743712049839382569\\
r = 0x41AA28D2F1AB148280CD9ED56FED\\
Parameter s = (r*d + k*e)(mod q) takes the value:
s = 57497340027008465417892531001914703\\
s = 0x1456C64BA4642A1653C235A98A602\\
7.3. Verification Process of Digital Signature (Algorithm II)
Suppose that after steps 1-3, according to Algorithm II (Section
6.2), are performed, the following numerical value is obtained:
e = 2079889367447645201713406156150827013\\
Dolmatov Informational [Page 17]
RFC 5832 GOST R 34.10-2001 March 2010
e = 0x2DFBC1B372D89A1188C09C52E0EE\\
And the parameter v = e^(-1) (mod q) takes the value:
v = 176866836059344686773017138249002685\\
v = 0x271A4EE429F84EBC423E388964555BB\\
The parameters z1 = s*v(mod q) and z2 = -r*v(mod q) take the values:
z1 = 376991675009019385568410572935126561\\
z1 = 0x5358F8FFB38F7C09ABC782A2DF2A\\
z2 = 141719984273434721125159179695007657\\
z2 = 0x3221B4FBBF6D101074EC14AFAC2D4F7\\
The point C = z1*P + z2*Q has the coordinates:
x_C = 2970098091581795287437120498393825699\\
x_C = 0x41AA28D2F1AB148280CD9ED56FED\\
y[C] = 3284253527868466347709466532251708450\\
y[C] = 0x489C375A9941A3049E33B34361DD\\
Then the parameter R = x_C (mod q) takes the value:
R = 2970098091581795287437120498393825699\\
R = 0x41AA28D2F1AB148280CD9ED56FED\\
Since the equality R = r holds, the digital signature is accepted.
Dolmatov Informational [Page 18]
RFC 5832 GOST R 34.10-2001 March 2010
8. Security Considerations
This entire document is about security considerations.
Current cryptographic resistance of GOST R 34.10-2001 digital
signature algorithm is estimated as 2^128 operations of multiple
elliptic curve point computations on prime modulus of order 2^256.
9. References
9.1. Normative References
[GOST3410] "Information technology. Cryptographic data
security. Signature and verification processes of
[electronic] digital signature.", GOST R 34.10-2001,
Gosudarstvennyi Standard of Russian Federation,
Government Committee of Russia for Standards, 2001.
(In Russian)
[GOST3411] "Information technology. Cryptographic Data
Security. Hashing function.", GOST R 34.10-94,
Gosudarstvennyi Standard of Russian Federation,
Government Committee of Russia for Standards, 1994.
(In Russian)
[RFC4357] Popov, V., Kurepkin, I., and S. Leontiev,
"Additional Cryptographic Algorithms for Use with
GOST 28147-89, GOST R 34.10-94, GOST R 34.10-2001,
and GOST R 34.11-94 Algorithms", RFC 4357, January
9.2. Informative References
[ISO2382-2] ISO 2382-2 (1976), "Data processing - Vocabulary -
Part 2: Arithmetic and logic operations".
[ISO9796-1991] ISO/IEC 9796:1991, "Information technology --
Security techniques -- Digital signature schemes
giving message recovery."
[ISO14888-1] ISO/IEC 14888-1 (1998), "Information technology -
Security techniques - Digital signatures with
appendix - Part 1: General".
[ISO14888-2] ISO/IEC 14888-2 (1999), "Information technology -
Security techniques - Digital signatures with
appendix - Part 2: Identity-based mechanisms".
Dolmatov Informational [Page 19]
RFC 5832 GOST R 34.10-2001 March 2010
[ISO14888-3] ISO/IEC 14888-3 (1998), "Information technology -
Security techniques - Digital signatures with
appendix - Part 3: Certificate-based mechanisms".
[ISO10118-1] ISO/IEC 10118-1 (2000), "Information technology -
Security techniques - Hash-functions - Part 1:
[ISO10118-2] ISO/IEC 10118-2 (2000), "Information technology -
Security techniques - Hash-functions - Part 2: Hash-
functions using an n-bit block cipher algorithm".
[ISO10118-3] ISO/IEC 10118-3 (2004), "Information technology -
Security techniques - Hash-functions - Part 3:
Dedicated hash-functions".
[ISO10118-4] ISO/IEC 10118-4 (1998), "Information technology -
Security techniques - Hash-functions - Part 4: Hash-
functions using modular arithmetic".
Dolmatov Informational [Page 20]
RFC 5832 GOST R 34.10-2001 March 2010
Appendix A. Extra Terms in the Digital Signature Area
The appendix gives extra international terms applied in the
considered and allied areas.
1. Padding: Extending a data string with extra bits [ISO10118-1].
2. Identification data: A list of data elements, including specific
object identifier, that belongs to the object and is used for its
denotation [ISO14888-1].
3. Signature equation: An equation, defined by the digital signature
function [ISO14888-1].
4. Verification function: A verification process function, defined by
the verification key, which outputs a witness of the signature
authenticity [ISO14888-1].
5. Signature function: A function within a signature generation
process, defined by the signature key and by the digital signature
scheme parameters. This function inputs a part of initial data
and, probably, a pseudo-random number sequence generator
(randomizer), and outputs the second part of the digital
Dolmatov Informational [Page 21]
RFC 5832 GOST R 34.10-2001 March 2010
Appendix B. Contributors
Dmitry Kabelev
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: kdb@cryptocom.ru
Igor Ustinov
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: igus@cryptocom.ru
Sergey Vyshensky
Moscow State University
Leninskie gory, 1
Moscow, 119991
Russian Federation
EMail: svysh@pn.sinp.msu.ru
Author's Address
Vasily Dolmatov, Ed.
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: dol@cryptocom.ru
Dolmatov Informational [Page 22]
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|
s Magnitude that Lies in Another Vector
The Amount of One Vector’s Magnitude that Lies in Another Vector’s Direction
A car’s speedometer typically works by measuring the rotational speed of the wheels. The car may not be moving directly forward (it may be skidding sideways, for example) in which case part of the
motion will not be in the direction the speedometer can measure. The magnitude of an object’s rigidbody.velocity vector will give the speed in its direction of overall motion but to isolate the speed
in the forward direction, you should use the dot product:-
var fwdSpeed = Vector3.Dot(rigidbody.velocity, transform.forward);
Naturally, the direction can be anything you like but the direction vector must always be normalized for this calculation. Not only is the result more correct than the magnitude of the velocity, it
also avoids the slow square root operation involved in finding the magnitude.
|
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|
Two Deductions: (1) from the Totality to Quantum Information Conservation; (2) from the Latter to Dark Matter and Dark Energy
Download PDFOpen PDF in browser
Two Deductions: (1) from the Totality to Quantum Information Conservation; (2) from the Latter to Dark Matter and Dark Energy
EasyChair Preprint 4409
47 pages•Date: October 16, 2020
The paper discusses the origin of dark matter and dark energy from the concepts of time and the totality in the final analysis. Though both, and especially the latter, seem to be rather
philosophical, nonetheless they are postulated axiomatically and interpreted physically, and the corresponding philosophical transcendentalism serves heuristically. The exposition of the article
means to outline the “forest for the trees”, however, in an absolutely rigorous mathematical way, which to be explicated in detail in a future paper. The “two deductions” are two successive stage of
a single conclusion mentioned above. The concept of “transcendental invariance” meaning ontologically and physically interpreting the mathematical equivalence of the axiom of choice and the
well-ordering “theorem” is utilized again. Then, time arrow is a corollary from that transcendental invariance, and in turn, it implies quantum information conservation as the Noether correlate of
the linear “increase of time” after time arrow. Quantum information conservation implies a few fundamental corollaries such as the “conservation of energy conservation” in quantum mechanics from
reasons quite different from those in classical mechanics and physics as well as the “absence of hidden variables” (versus Einstein’s conjecture) in it.
Keyphrases: General Relativity, confinement, dark energy, dark matter, entanglement, physical and mathematical transcendentalism, quantum information, the standard model, transcendental invariance
Links: https://easychair.org/publications/preprint/lkRr
Download PDFOpen PDF in browser
|
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|
Planarity and Colorings
Definition 9.6.1. Planar Graph/Plane Graph.
A graph is planar if it can be drawn in a plane so that no edges cross. A drawing of a graph on the plane such that there are no edge crossings is called a planar embedding of the graph, or a plane
graph for short.
A \(K_5\) has 10 edges. If a \(K_5\) is planar, the number of regions into which the plane is divided must be 7, by Euler’s formala (\(5+7-10=2\)). If we re-count the edges of the graph by counting
the number edges bordering the regions we get a count of at least \(7 \times 3=21\text{.}\) But we’ve counted each edge twice this way and the count must be even. This implies that the number of
edges is at least 11, which a contradiction.
1. 4
2. 3
3. 3
4. 3
5. 2
6. 4
The chromatic number is \(n\) since every vertex is connected to every other vertex.
Suppose that \(G'\) is not connected. Then \(G'\) is made up of 2 components that are planar graphs with less than \(k\) edges, \(G_1\) and \(G_2\text{.}\) For \(i=1,2\) let \(v_i,r_i, \text{and} e_i
\) be the number of vertices, regions and edges in \(G_i\text{.}\) By the induction hypothesis, \(v_i+r_i-e_i=2\) for \(i=1,2\text{.}\)
One of the regions, the infinite one, is common to both graphs. Therefore, when we add edge \(e\) back to the graph, we have \(r=r_1+r_2-1\text{,}\) \(v=v_1+v_2\text{,}\) and \(e=e_1+e_2+1\text{.}\)
\begin{equation*} \begin{split} v+r-e &=\left(v_1+v_2\right)+\left(r_1+r_2-1\right)-\left(e_1+e_2+1\right)\\ &=\left(v_1+r_1-e_1\right)+\left(v_2+r_2-e_2\right)-2\\ &=2 + 2 -2\\ &=2 \end{split} \end
Since \(\left| E\right| +\left| E^c \right|=\frac{n(n-1)}{2}\text{,}\) either \(E \text{ or } E^c\) has at least \(\frac{n(n-1)}{4}\) elements. Assume that it is \(E\) that is larger. Since \(\frac{n
(n-1)}{4}\) is greater than \(3n-6\text{ }\text{for}\text{ }n\geqslant 11\text{,}\) \(G\) would be nonplanar. Of course, if \(E^c\) is larger, then \(G'\) would be nonplanar by the same reasoning.
Can you find a graph with ten vertices such that it is planar and its complement is also planar?
Suppose that \((V,E)\) is bipartite (with colors red and blue), \(\left| E\right|\) is odd, and \(\left(v_1,v_2,\ldots ,v_{2n+1},v_1\right)\) is a Hamiltonian circuit. If \(v_1\) is red, then \(v_
{2n+1}\) would also be red. But then \(\left\{v_{2n+1},v_1\right\}\) would not be in \(E\text{,}\) a contradiction.
Draw a graph with one vertex for each edge, If two edges in the original graph meet at the same vertex, then draw an edge connecting the corresponding vertices in the new graph.
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Finite Dimensional Algebras and Quantum Groupssearch
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Finite Dimensional Algebras and Quantum Groups
Bangming Deng : Beijing Normal University, Beijing, People’s Republic of China
Jie Du : University of New South Wales, Sydney, Australia
Jianpan Wang : East China Normal University, Shanghai, People’s Republic of China
Hardcover ISBN: 978-0-8218-4186-0
Product Code: SURV/150
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN: 978-1-4704-1377-4
Product Code: SURV/150.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN: 978-0-8218-4186-0
eBook: ISBN: 978-1-4704-1377-4
Product Code: SURV/150.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
Click above image for expanded view
Finite Dimensional Algebras and Quantum Groups
Bangming Deng : Beijing Normal University, Beijing, People’s Republic of China
Jie Du : University of New South Wales, Sydney, Australia
Jianpan Wang : East China Normal University, Shanghai, People’s Republic of China
Hardcover ISBN: 978-0-8218-4186-0
Product Code: SURV/150
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN: 978-1-4704-1377-4
Product Code: SURV/150.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN: 978-0-8218-4186-0
eBook ISBN: 978-1-4704-1377-4
Product Code: SURV/150.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
• Mathematical Surveys and Monographs
Volume: 150; 2008; 759 pp
MSC: Primary 05; 16; 17; 20
The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines,
for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel–Hall algebra realization for the positive part of a
quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson–Lusztig–MacPherson realization for the entire quantum \(\mathfrak {gl}_n\).
The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of
quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac–Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of
Ringel–Hall algebras. Cartan matrices can also be used to define an important class of groups—Coxeter groups—and their associated Hecke algebras. Hecke algebras associated with symmetric groups
give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum \(\
mathfrak{gl}_n\) through a completion process of a limit algebra (the Beilinson–Lusztig–MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a
series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature.
Graduate students and research mathematicians interested in quantum groups and finite-dimensional algebras.
□ Chapters
□ 0. Getting started
□ 1. Representations of quivers
□ 2. Algebras with Frobenius morphisms
□ 3. Quivers with automorphisms
□ 4. Coxeter groups and Hecke algebras
□ 5. Hopf algebras and universal enveloping algebras
□ 6. Quantum enveloping algebras
□ 7. Kazhdan-Lusztig combinatorics for Hecke algebras
□ 8. Cells and representations of symmetric groups
□ 9. The integral theory of quantum Schur algebras
□ 10. Ringel-Hall algebras
□ 11. Bases of quantum enveloping algebras of finite type
□ 12. Green’s theorem
□ 13. Serre relations in quantum Schur algebras
□ 14. Constructing quantum $\mathrm {gl}_n$ via quantum Schur algebras
□ ...prove[s] to be a valuable reference to researchers working in the field. It contains and collects many results which have not appeared before in book form.
Mathematical Reviews
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 150; 2008; 759 pp
MSC: Primary 05; 16; 17; 20
The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for
the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel–Hall algebra realization for the positive part of a quantum
enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson–Lusztig–MacPherson realization for the entire quantum \(\mathfrak {gl}_n\).
The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers
with automorphisms and the theory of quantum enveloping algebras associated with Kac–Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel–Hall
algebras. Cartan matrices can also be used to define an important class of groups—Coxeter groups—and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an
interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum \(\mathfrak{gl}_n\)
through a completion process of a limit algebra (the Beilinson–Lusztig–MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises,
ranging from the routine to sketches of proofs of recent results from the current literature.
Graduate students and research mathematicians interested in quantum groups and finite-dimensional algebras.
• Chapters
• 0. Getting started
• 1. Representations of quivers
• 2. Algebras with Frobenius morphisms
• 3. Quivers with automorphisms
• 4. Coxeter groups and Hecke algebras
• 5. Hopf algebras and universal enveloping algebras
• 6. Quantum enveloping algebras
• 7. Kazhdan-Lusztig combinatorics for Hecke algebras
• 8. Cells and representations of symmetric groups
• 9. The integral theory of quantum Schur algebras
• 10. Ringel-Hall algebras
• 11. Bases of quantum enveloping algebras of finite type
• 12. Green’s theorem
• 13. Serre relations in quantum Schur algebras
• 14. Constructing quantum $\mathrm {gl}_n$ via quantum Schur algebras
• ...prove[s] to be a valuable reference to researchers working in the field. It contains and collects many results which have not appeared before in book form.
Mathematical Reviews
Permission – for use of book, eBook, or Journal content
You may be interested in...
Please select which format for which you are requesting permissions.
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Long Term CICR (Cross index currency referance) trading
Dyekid217, did you try the paid service fx education group ? how is it ? is it good ?
Would recommend it , specially the Price reaction and candle confluence.
Really, you use on the daily ? well... sounds good....
I prefer daily tf too, much more relax and free stress trading ...
but I suppose that you put an eye on H4 tf too.... maybe just to refine your entry... and see what is happening in more detail.
yup but also remember we pay attention to the weekly and monthly candles as well
Would recommend it , specially the Price reaction and candle confluence.
thanks cucufx
Hello People,
Who told you divergence don't work on CICR. It a leading indicator and it's use on the index is like a power horse, you'd be a fool to miss out on this one. Though I've seen people here use
different method to calculate it, it works. Read up thoroughly on divergence on Babypips and thank your self when you see the results. Hope you get the drift.
If you are trying to trade divergence, you will find more success trading hidden divergences.
In my experience, divergences aren't consistently reliable enough to use them for trade entries. Sure you can cherry-pick great examples - and i expect at the end of a trend you will find a
divergence there more often than not but i have an awesome indicator that basically finds every divergence on a chart - and what you find is that most of them do not produce a good tradeable move.
Divergence by itself is therefore not a high profitability entry technique.
Enter Signature
Dyekid217, did you try the paid service fx education group ? how is it ? is it good ?
I recommend it, you will learn a lot !!
Hi, Dyekid217.
I just tried to analyze your todays positions.
For example, GBPCHF. Sell was opened almost at the top. But look at the picture below -- on GBPLFX chart:
- red arrow shows an entering bar of GBPCHF
- I don't know what was shown by CICRAlerter at that time, but I see that MACD and cross of filter_line with SMA were not yet occured
- LaguerreRSI was at the top
So, I'm wondering why did you opened Sell position when you opened it, but not now when cross actually occured?
Attached Image
Sarythsaya: Certain definite things about divergence is that it can't be instantly traded from the right side of the chart, it needs at least 2 - 3 candles to confirm itself. They will not work well
on shorter TF's like H1 or less. At least h4 min. They will work wonders if identified correctly on the Index, that is much much better than on the pairs. I would disagree when you say they would not
produce good moves. The indicator that you are talking about, is it FX5 Macd divergence? If different, please post it here, I would like to check the reason why you make that statement of not good
moves. Moreover, i've seen it being plotted wrongly here in this thread as well as other CICR threads, even the main one and also on the FxMech site.
DK: So you are using the h4 macd filter cross, D1 for TD BO's, both on the index. Am i right here? Of course there are plenty other things, but is this the pre qualifier sort of thing?
I - No my friend. That's not the CICR. That's Lawgirl System and even that has some pre qualifiers. I gave the link to one of my posts, try to read that as the PDF is a little incoherent at places.
There are clear pre-qualifiers before one starts to consider the arrows.
II - In an earlier post I mentioned about the importance of entry price, should be neither too late nor too early. That's an art and even I don't get it right a lot of times. But with the CICR, it's
made a lot of difference and I'm able to pick better entry price. As far as the SL or TP is concerned, it's more a part of MM than anything else. A right entry price should have an SL which is behind
a recent or historic S&R level. One- the Cicr will help you find a good entry price, Two- That price gauged with the help of CICR will invariably have a relevant S&R level nearby, IMO. Already
discussed divergence above. If done well, it very powerful on the Index. Practice it correctly and thank yourself, ... again.
Regards, M.
There is no such thing as LUCK ! Chance favors the prepared mind.
If you are trying to trade divergence, you will find more success trading hidden divergences.
In my experience, divergences aren't consistently reliable enough to use them for trade entries. Sure you can cherry-pick great examples - and i expect at the end of a trend you will find a
divergence there more often than not but i have an awesome indicator that basically finds every divergence on a chart - and what you find is that most of them do not produce a good tradeable
move. Divergence by itself is therefore not a high profitability entry technique....
Any chance of posting that divergence indi you mentioned Sarathsaya?
Maybe this was allready mentioned, but you can also compare the strenght of currencies here:
http://www.fxmri.com/ (have to register).
Without registration: http://www.forexstrategiesresources....trength-index/
Thoughts are things
Hi guys if you're wondering about my positions I've taken them based off different factors.
I highly recommend you just test this system yourself as I've stated there are MANY ways you may analyze an entry.
Just play around and see what works
AUDx turned red for the week.
Laguerre is turning over.
MACD is red as well.
USDx has bounced from its monthly support.
wonder if AUDUSD is a good sell from here on in?
AUDUSD @ 1.0465 is kinda very near strong support 1.0450
What is your criteria for exiting a position?
Just the StopLoss of 100 pips as you mentioned?
Not Support/Resistance levels?
What about change of index strength? ie, from bullish to bearish?
Hi traders,
I'm sorry if I change the whole context of this thread but I want to explain how I trade this system with a small modification implemented.
First I check the daily chart of an index and mainly looking at the macd bars . If they are increasing in size you know what's next, I search for a currency that has the opposite macd daily bars
value. I know that DK watches the weekly bars but I think this prevents one from entering earlier and thus missing the perfect time since daily bar is one timeframe larger than the one that we look
at the pairs chart.TMHO the best time you can possibly exploit such a position is watching carefully when a daily macd bar is moving from a positive to negative territory or vice versa. When such
index is found I wait for the perfect index to couple with. I don't like basket trading in one exception - EUR/USD and EUR/JPY from one side and USD/CHF and EUR/USD from another since USD and JPY
almost 80% of the time move in one direction and USD and CHF move in opposite one. The second index that is waited to be found can last a bit but it's better to wait a bit than jumping into risky
waters. Next I open the future potential pairs charts e.g.: USD/pair1,USD/pair2,USD/pair3 and if after 2,3 bars I found that USD is my first pair the second one is moving in the expected direction i
take the trade. But there's also something more, I combine this pair move with a channel or a level break. If there's no such one I would quite consider the validity of this position. Laguerre on a
daily pair chart when approaching 0.15 or 0.75 level is also another alert to look for a potential trade. Secondary but not less important factor is the type of the candle. If I see a hanging man
crossing a certain level according to the rule mentioned above I would quite consider its validity too.Almost no attention in my trading is taken when considering the filter lines on the chart and
the mtf Rsi indicator because the first ones quite often cross each other when a trade is just retracing for a few bars later it still proceeds further into profit and the second one quite often
doesn't take any role in my plans because it often gives 5 one-colour arrows and still this is a wrong decision to take based on obvious loss of interest in such pair.
Next, I agree that a move has to be given a little time to start giving profits but I assume I'm wrong when I see strong move in the opposite direction (whipsaw) and especially if 4h macd pair bar
start changing its value after shortly it's been moving according to my expectations. I also start getting cautious if I find divergence and move my stoploss right below/above the recent low/high.
There's another solution to this: part of the trade is taken at the recent low/high and another part is stopped at the entry level in the worst scenario.
The last trade I took according to this method was EUR/USD. I missed the EUR/JPY.
What I saw first was the USD daily macd bar changing direction at 16am on 18th(GMT+1). If you look at the 4h chart you'll see the divergence between the 9 and 13.01 tops and the macd. Except that,
USD index was moving in channel in the last 10 days and the top line of it was hit so I expected a downmove anyway at least to the lower channel line. The opposite can be stated for the EUR index
except that it moved in channel from 29.12 till 12.01 and after that it broke the upper channel line, retraced to it with the 13.01 downmove and then moved upward.
So my choice went for the EUR/USD and my target level initially was the 17.10-16.01 upper channel line hit on the EUR index but now I think that price can go further if it doesn't move too lower
after it shows lack of strength and divergence with macd on the 4h pair chart.
I moved my stoploss right below the recent lower high and if price moves upwards tomorrow ,meets resistance in the range between 1.3000-1.3050 and then falls below the lower channel line I'll close
my position. Please comment. I want to know what you think about that. Appologies for my English but it's not my native one.
BTW look at the AUD index divergence on a daily chart. Expect some loss of interest these days or the beginning of a downtrend.
CAD also made a break of a long trendline and a retest of it from the other side. Downmove can be expected.
I also use MACD crossover for signal.
It is a lagging indicator no doubt and following a crossover is often delayed. Trying to use it on lower time frame may give me earlier signal however, is prone to "noise" / false signal caused by
It is therefore a trade-off using MACD on lower time frame vs higher time frame. I suggest using MACD on lower time frame with another indicator for confirmation to avoid noise/false signals.
As for AUD, I also see possible fall as well.
same here.. but with usd falling as well for now audusd might not be the best pair to trade right now.
I'm looking at NZDUSD for now.
I noticed you took a LONG AUDUSD.
Why? Was it because of a bullish AUDx, bearish USD on the last 4H candle?
hey guys still trading this with success..
Just added a UC short to my positions with all other positions doing relatively well.
Tried using CICR on an intraday basis but was having mixed results.. .Will stick to swing trading from here on out.. much less stressful and I only have to check my charts once-twice a day
Hi Dyekid,
This method looks very good. Looking foward to see your posts and signals to share with us bros here.
Pips Collector EA, No Martingale, No Grid
looking at the equity report i guess this isn't doing so well now. shame it was a logical idea
This thread seems to be dead.
Dyekid seems also to be now interested by another trading system.
The question which remains asked to him: Is CICR remains profitable for him ? Not too as It seems to be
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equations Archives -
Quadratic Equations Questions and Answers Set 1 Hi students, welcome to Amans Maths Blogs (AMB). On this post, you will get the Quadratic Equation Questions and Answers Set 1 is the collection of
some important questions. Practice these questions for SSC CGL CHSL CAT NTSE exams etc. It will help you
Read More
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!!!!!!!I NEED HELP WITH THESE PROBLEMS. THANK YOU!!!!!1. Mike's salary is $6,500 per month. He saves 1/5 of his monthly salary and spends the rest.How much money does Mike save every month?$ ?How much money does Mike spend every month?$ ?2. Fill in the missing values that makes the equation true.−2/5×30= ?/2 x -4 = ?3. Alex takes 1 hour and 30 minutes to get to the office.How many hours does Alex spend getting to the office over 4 days?? hours4.Fill in the missing values that make the equation true.−4/7×49= ?/6 x -14 = ?5. Drag the equivalent expressions into the appropriate columns. 5(1/5+3c) 1/4(1/2 - 4c)(5+15c), (1+15c), (1/8 - c), (1/8 - 4/4c), (-1/8 + -c), (5/5 + 15c)6. Drag the equivalent expressions into the appropriate columns. a(b + c) -x(m - n)(ab - ac), (ab + ac), -(xm - xn), (-xm + -xn)7. Part 118 × ? = 1Part 2Select all the statements that explain why the missing value you chose for Part A makes the equation true.Answer ChoicesA. The product of a rational number and its multiplicative inverse is always 1.B. The sum of a rational number and its additive inverse is always 0.C. The product of two positive rational numbers is always positive.D. The product of any two rational numbers is always 1.E. It is the multiplicative inverse of the multiplier.F. It is the additive inverse of the multiplier.8. This is an equation.(-7)(-17)=1 Select all of the statements that explain why this equation is true.Answer ChoicesA. The fraction -17 is the multiplicative inverse of -71 .B. The fraction -17 is the additive inverse of -71 .C. The product of any two rational numbers is always 1.D. The product of two negative rational numbers is always positive.E. The product of a rational number and its multiplicative inverse is always 1.F. The sum of a rational number and its additive inverse is always zero.
1. Home
2. General
3. !!!!!!!I NEED HELP WITH THESE PROBLEMS. THANK YOU!!!!!1. Mike's salary is $6,500 per month. He saves...
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Free Printable Multiplication Table 1 10 Chart Template PDF Best | Multiplication Chart Printable
Free Printable Multiplication Table 1 10 Chart Template PDF Best
Free Printable Multiplication Table 1 10 Chart Template PDF Best
Free Printable Multiplication Table 1 10 Chart Template PDF Best – A Multiplication Chart is a valuable tool for children to learn how to increase, divide, and discover the smallest number. There are
several usages for a Multiplication Chart. These convenient tools aid kids comprehend the procedure behind multiplication by using colored courses and also filling in the missing items. These charts
are cost-free to download and publish.
What is Multiplication Chart Printable?
A multiplication chart can be made use of to assist youngsters learn their multiplication realities. Multiplication charts been available in several kinds, from full web page times tables to single
web page ones. While individual tables serve for offering chunks of info, a complete web page chart makes it easier to examine truths that have already been mastered.
The multiplication chart will typically include a leading row and also a left column. The leading row will certainly have a listing of items. When you wish to discover the item of 2 numbers, pick the
initial number from the left column as well as the 2nd number from the top row. When you have these numbers, move them along the row or down the column up until you reach the square where the two
numbers satisfy. You will after that have your product.
Multiplication charts are handy discovering tools for both youngsters and also grownups. Multiplication Chart Printable 1-10 are available on the Internet and also can be printed out and laminated
for toughness.
Why Do We Use a Multiplication Chart?
A multiplication chart is a diagram that demonstrates how to increase two numbers. It commonly contains a top row and also a left column. Each row has a number standing for the product of both
numbers. You choose the very first number in the left column, relocate down the column, and afterwards pick the second number from the top row. The product will certainly be the square where the
numbers fulfill.
Multiplication charts are useful for several reasons, including assisting kids discover just how to separate as well as streamline portions. They can also assist children discover exactly how to
select a reliable common denominator. Multiplication charts can additionally be useful as workdesk resources since they work as a continuous reminder of the pupil’s development. These tools assist us
develop independent learners that understand the standard principles of multiplication.
Multiplication charts are likewise valuable for assisting students remember their times tables. As with any kind of skill, remembering multiplication tables takes time and also method.
Multiplication Chart Printable 1-10
Multiplication Table 1 10 Printable Blank Multiplication Table 1 10
Free Printable Multiplication Table Chart 1 To 10 Template PDF
Multiplication Chart Printable 1-10
If you’re looking for Multiplication Chart Printable 1-10, you’ve come to the best location. Multiplication charts are offered in various formats, including complete dimension, half dimension, and
also a variety of adorable layouts.
Multiplication charts and tables are important tools for kids’s education. These charts are terrific for use in homeschool mathematics binders or as class posters.
A Multiplication Chart Printable 1-10 is a helpful tool to reinforce math facts and also can assist a kid discover multiplication swiftly. It’s additionally a fantastic tool for skip checking and
also discovering the times tables.
Related For Multiplication Chart Printable 1-10
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650 Nanometers to Meters
650 nm to m conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in the
United Kingdom) and as a fraction (exact result). Every display form has its own advantages and in different situations particular form is more convenient than another. For example usage of
scientific notation when working with big numbers is recommended due to easier reading and comprehension. Usage of fractions is recommended when more precision is needed.
If we want to calculate how many Meters are 650 Nanometers we have to multiply 650 by 1 and divide the product by 1000000000. So for 650 we have: (650 × 1) ÷ 1000000000 = 650 ÷ 1000000000 = 6.5E-7
So finally 650 nm = 0.00000065 m
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What Is 1 Divided By 3
What Is 1 Divided By 3 - For negative numbers insert a leading negative or minus. = 3 4 ∗ 9 + 1 10 ; Web to divide 3 by 1/5 we can either multiply the first fraction by its reciprocal or we could
find a common denominator for both fractions. Web add, subtract, multiply and divide decimal numbers with this calculator. 3 goes into 10 3 times, with a remainder of 1, so 3 into 10 again. Web learn
for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. For example, 22 divided by 15 = 1.466 when calculated to 3 decimal places
because you. Web express 1 divided by 3 as a fraction. Web convert 1/3 divided by 3 to decimal here's a little bonus calculation for you to easily work out the decimal format of the fraction we
calculated. You enter the whole number in the first box, then the answer you want in the second box, you click.
Web here is the answer to questions like: I mean, do we really need to explain how this calculator works? 1 = whole number 1 = numerator 3 = denominator to make it a fraction form answer, you
multiply the whole number by the. Web learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Web convert 1/3 divided by 3
to decimal here's a little bonus calculation for you to easily work out the decimal format of the fraction we calculated. Web add, subtract, multiply and divide decimal numbers with this calculator.
Divide each digit of the dividend with the divisor starting from left to right. Once you have your final fraction,. Web when a number is divided by 2.75, the quotient is 8.5. 3 4 10 = reminder 1. Web
note that this is not the same as rounding to a specific number of decimal places. So, 3 4 ∗ 9 + 1 10 = rem 3 , as 3 36. It is not possible to represent one third using a finite decimal positional
notation. Web the fraction calculator is easy to use. 1 = numerator 3 = denominator 3 = whole number to. The fraction calculator will reduce a fraction to its simplest form. Px≥40=0.5 we know that x
is distributed…. Divide 2 numbers and find the quotient. Explore all the episodes of taj: You enter the whole number in the first box, then the answer you want in the second box, you click.
Related Post:
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Lingua Mathematica—A deep dive into the languages of models
By Paul Meixler
Models are ubiquitous in actuarial work. The following is a synopsis of the languages of models—in particular, the complex metalanguages and the structured rule-based object languages of models,
along with mappings between them, are described.
Actuaries maintain a high standard of professional conduct relevant to their practice domain through literacy in the Code of Professional Conduct and the actuarial standards of practice (ASOPs). The
Code and the ASOPs provide guidance in the English language that reflect the professional and ethical standards of actuarial practices in the United States. English is used in the Code and ASOPs as
the metalanguage. ASOP No. 1, Introductory Standard of Practice, provides, for example, foundational metalanguage definitions used by other ASOPs, including definitions of the words must and should.
Awareness of the difference in these words has an important connection to professionalism.
ASOP No. 56, Modeling, became effective October 1, 2020, and “provides guidance to actuaries when performing actuarial services with respect to designing, developing, selecting, modifying, using,
reviewing, or evaluating models.” It is written with defined terms and principles in a metalanguage. An implemented model is written in an object language, which is generally a computer language.
Excel, for example, is an object language that provides the function Present Value (PV) that models the current value of an expected future stream of cash flow from the data inputs: rate, period, and
payments. Excel’s PV function is part of the object language and maps to the definition of the term present value in a metalanguage.
Analogous to an actuary learning the ASOPs, when young children start school, they may encounter unknown standards of conduct, and they gradually learn to adjust by learning new skills. The children
learn, for example, to raise their hand to request a turn to speak. Their educational setting strives to assist them in learning linguistic and analytic skills along with existence skills. During a
beginning English reading class, for example, the children are required to communicate in English about the linguistic activities with words such as page, line, space, and comma. The language used
for this communication is the metalanguage. The children’s awareness of the metalanguage of literacy is connected to their subsequent performance in reading and writing.^[1]
A similar linguistic situation arises when students learn a second language. Their mother tongue, along with grammatical terms and rules, forms the metalanguage. The second language is the object
language. English speakers may use a French grammar book written in English to learn French. Polyglots—persons who read, speak, or understand many languages—have awareness of the metalanguage for
many natural languages. They easily compare the metalanguages of the individual natural languages. The different natural languages use different combinations of phonemes or letters for a word; a word
of type noun, for example, is a name for the same kinds of things. The challenge for polyglots is to map related names between the ontologies underlying the different natural languages.
Ontology is the study of the collections of individual things that exist or may exist in some domain. It is the study of existence, of all kinds of things—abstract and concrete—that make up a domain.
Two sources of determining ontological collections are observation and reasoning. Observations provide knowledge of the physical world and reasoning makes sense of the observations using language.
The choice of the ontological collections is the first step in designing a scientific model. The selection of the collections determines everything that may be determined from the model.^[2]
When domain experts interact among themselves and refer to different choices of the ontological collections, there is ontological uncertainty. It not only involves referring to different collections
but also referring to differences of what and how the collections interact with each other. In contrast to ontological uncertainty, semantic uncertainty is a result of the differences between the
meanings given by different domain experts to the words used in their metalanguage, or vague and ambiguous definitions used by the domain expert. A domain expert uses a natural language as the
metalanguage for the purposes of communicating about their domain.
A detailed discussion of metalanguage and object language is provided in Introduction to Symbolic Logic and its Applications by Rudolf Carnap. He states, “A natural language is given by historical
fact, hence its description is based on empirical investigation. In contrast, an artificial language is given by the construction of rules for it. The rules of an object language, as well as theorems
based on these rules, are formulated in the metalanguage.”^[3]
The goal of professional standards is to reduce the ontological and semantic uncertainty in the communication of information. The ASOPs’ purpose is to address these difficulties. The ontology and
semantics of the ASOPs standardizes terminology used to classify and find information. As an example, let’s look at a couple of definitions from Actuarial Standard of Practice No. 56, Modeling.
• Model—A simplified representation of relationships among real world variables, entities, or events using statistical, financial, economic, mathematical, non-quantitative, or scientific concepts
and equations. A model consists of three components: an information input component, which delivers data and assumptions to the model; a processing component, which transforms input into output;
and a results component, which translates the output into useful business information.
• Data—Facts or information that are either direct input to a model or inform the selection of input. Data may be collected from sources such as records, experience, experiments, surveys,
observations, benefit plan or policy provisions, or output from other models.
From the Excel example mentioned before, a cell with a formula in Excel meets the definition of a model. The object language formula “=PV(0.05, 5, 1)” in cell A1 has a input component “0.05, 5, 1”
(rate, period, and payments) including the assumption (discount rate of 5%) and data (periodic payments of 1 for 5 years). It also has a processing component “PV()”, and a results component show as “
($4.33)“. The formula “=A1*10” in cell A2 has the result component “($43.29)”. The assumptions and data of the model in cell A2 uses the output from the model in cell A1. (See Figure 1.)
The metalanguage provides the translation of the information to and from the object language. The translation of output of the Excel PV() function is “Returns a present value of an investment: the
total amount of a series of future payments is worth now.” The formulas are written in a structured rule-based object language, Excel, modeling the actuarial concept of “present value” in a complex
metalanguage using English.
As illustrated by this Excel example and what many computer programmers say, the complexity of designing a computer program (in our case models) is not manipulating a known object language, it is the
understanding and translating the metalanguage into the object language, translating the ontology and semantics of a specific domain. Understanding “the total amount of a series of future payments is
worth now” is simple for a trained actuary; however, it is quite complex for a person without financial literacy.
From the computation viewpoint, “The purpose of a program describes a computational process that consumes some information and produces new information. For a program to process information, it must
turn it into some form of data in the programming language, then it processes the data; and once it is finished, it turns the resulting data into information again.”^[4] Translating information into
data is called “to represent.” Translating data into information is called “to interpret.”
Using the Excel example, information is at the level of the metalanguage and data is at the level of the object language. The data of the object language, Excel’s PV() function, is “0.05, 5, 1”,
which represents information related to discount rate assumption, period, and payments. Based on this data, the information provides an answer or is interpreted as “what the total amount of a series
of future payments is worth now” or “has the value now”. In this example, the information that the discount rate is 5% and periodic payments of 1 for 5 years is used to produce more information—the
present value of this information has a value now of ($4.33).
Another way to represent data is with a relational database schema. A relational database has tables each consisting of a collection of columns and a collection of rows. Each database table is like
an Excel worksheet; however, each database table also has a primary key generally in the leftmost column and the cells in the other columns have foreign keys. The foreign key columns link one table
to another. In a completely normalized relational database, all the tables are mathematical sets; i.e., each element of the ontological collection is unique. Using mathematical graphs, the conceptual
layout of data represents the schema.^[5] The graphs can visually reflect complex layers of data relationships.
Figure 2 is an example of a database instance and schema.^[6] It includes two business rules: (1) every employee works in the same department as his or her manager, and (2) every department’s
secretary works in that department. The instance is on the left and the corresponding schema is on the right.
Similarly, the object language consists of complex systems of sublanguages with a stratified design. The “complex systems should be structured as a sequence of levels that are described using a
sequence of languages. Each level is constructed by combining parts that are regarded as primitive at that level, and the parts that constructed at each level are used as primitives at the next
level. The language used at each level has primitives, means of combinations, and means of abstraction appropriate to that level of detail.”^[7]
The task of the programmer (in our case the modeler) is to go back-in-forth translating between the metalanguage and the object language, and to eliminate the ontological and semantic uncertainty. In
other words, the task is to use a precise and testable translation of the model languages at all the data and language levels. A mathematically rigorous approach is available to address the
ontological and semantic uncertainty. Ontological collections in the metalanguage represent data by mathematical sets—a set semantics. Applying this rigor opens a warehouse of mathematical tools
available through Applied Category Theory. Relatively recent developments regarding this approach are introduced below.
In 1945, in General Theory of Natural Equivalences, Samuel Eilenberg and Saunders MacLane introduced the concepts of category, functor, and natural transformation. In the past 70 years, category
theory has flourished. “Category theory has found applications in a wide range of disciplines outside of pure mathematics—even beyond the closely related fields of computer science and quantum
physics. These disciplines include chemistry, neuroscience, systems biology, natural language processing, causality, network theory, dynamical systems, and database theory to name a few. And what do
they all have in common? … In other words, the techniques, tools, and ideas of category theory are being used to identify recurring themes across these various disciplines with the purpose of making
them a little more formal.”^[8]
Using metalanguage, a category is defined as a collection of individual things that have related characteristics. The definition of a category is separated into two items and two rules. The items are
(1) collection of individual things and (2) a type of relationship between pairs of individual things. The rules are (1) every individual thing is related to itself by simply being itself, and (2) if
one individual thing is related to another and the second is related to a third, then the first is related to the third.^[9] In the object language, category theory, the individual things are called
objects and the relationships are called morphisms.
Let’s look again at the definitions of model and data from ASOP No. 56, this time with bolding judiciously applied:
• Model—A simplified representation of relationships among real world variables, entities, or events using statistical, financial, economic, mathematical, non-quantitative, or scientific concepts
and equations. A model consists of three components: an information input component, which delivers data and assumptions to the model; a processing component, which transforms input into output;
and a results component, which translates the output into useful business information.
• Data—Facts or information that are either direct input to a model or inform the selection of input. Data may be collected from sources such as records, experience, experiments, surveys,
observations, benefit plan or policy provisions, or output from other models.
These definitions remind me of Rule 2 of a category. In the definition of Model bolded above, the processing component that transforms input into output looks like a morphism. In the definition of
Data bolded above, the input may be output from other models looks like the output from the first morphism is the input for the second morphism. From the Excel example, the output from the formula “=
PV(0.05, 5, 1)” in cell A1 is used as the input for the formula “=A1*10” in cell A2. In other words, each cell looks like a morphism and the morphisms compose. The input components in the ASOP No. 56
definition of model would then be the representation of information in the metalanguage with data (and assumption) in the object language. The result components would be the interpretation of data
(and assumption) in the object language with information in the metalanguage.
A category can be visually thought of as a reflexive directed graph with vertices and arrows. In a graph, every arrow points from a vertex to a vertex, just as in a category, every morphism points
from an object to an object. There may be many arrows between two vertices a and b, like many morphisms may be between objects a and b in a category—or there may be none. There is at least one arrow
from a to itself like the identity morphism in a category that is required for each object. In the graph representation of a category, the composed morphisms are generally not shown on the graph to
avoid the clutter of too many arrows. Categories are distinguished from graphs by the ability to declare an equivalence relation on the set of paths. (See Figure 3.)
In Category Theory for the Sciences, David Spivak introduces category theory with examples from different disciplines. One discipline of interest is database theory using a set semantics. He develops
a formal metalanguage using natural language and category theory to define data definitions. The data definitions use “ontology logs” or “olog” for short. Ologs are designed to record the results of
a study of existence, of all kinds of things—abstract and concrete—that make up the world in a particular domain. An olog represents a worldview, discrepancies between worldviews reflect different
realities, and different ologs would be created for different worldviews. The rules to create an olog are enforced to ensure that an olog is structurally sound, rather than it “correctly reflects
In creating the ologs, the ologs are the object language used to define the ontological collections. English as used by the domain expert, for our purposes, is the metalanguage. In creating the
database theory schema, the ologs then become the metalanguage and category theory is object language. In creating the actual database instance, the category theory used for the schema is the
metalanguage and the data in the instance is the object language. As with database theory, the data used in a model is a complex system of sublanguages with a stratified design. Ologs enable the
modeler to think with an ontological and semantic viewpoint about the phenomena in different ways—ontological primitives, means of combinations, and means of abstraction that can be well suited to a
particular phenomenon.
“A basic olog is a category in which the objects and arrows (morphisms) have been labeled by English-language phrases that indicate their intended meaning. The objects represent types of things, the
arrows represent functional relationships (also known as aspects, attributes, or observables), and the commutative diagrams represent facts.”^[10] The objects are types, collections of individual
things, placed in a text box and labeled with a singular indefinite noun phrase.
The morphisms of this category are aspects, ways of viewing or measuring it, shown with arrows. It is drawn as a labeled arrow from an object to a “set of result values.” For example, an employee can
be regarded as a person. That is, “being a person” is an aspect of an employee.
The aspects from the database instance and schema above are shown in Figure 4.
The arrows above are then composed to form the schema shown in Figure 5.
The schema is loaded with mathematical sets of data to form an instance. In a completely normalized relational database, all the tables are mathematical sets; i.e., each element of the ontological
collection is unique. For this schema, data from Figure 6 are used in the database tables in the instances above.
Each type is assigned a mathematical set of instances. In other words, the elements of the set have unique elements. In the FirstNameString set, the name “Bertrand” only occurs once without any
For ologs, the types in a text box are ontological collections of individual things. They are mathematical sets. What makes an aspect or arrow “valid” is that it must be a mathematical function. A
mathematical function is a mapping that sends each element of the domain to an element of the co-domain; that is, for every element x of the domain, there is exactly one map emanating from x, but for
an element y of the co-domain, there may be several or no maps pointing to it. The two arrows shown below in Figure 7 are not mathematical functions. The first arrow is invalid because a person may
have no or many children. The second arrow is invalid because a mechanical pencil may have many pieces of lead that it uses.
An aspect is read as an English phrase. Each aspect is read by first reading the domain text box, then the arrow, and finally the co-domain text box. Paths of aspects are read by inserting the word
“which” after each intermediate text box. For example, the following path is read “an employee is a person, which has as a birthday a date which includes a year. (See Figure 8.)
Let’s illustrate using the original example, where the object language formula “=PV(0.05, 5, 1)” in cell A1 has a input component “0.05, 5, 1” (rate, period, and payments) including the assumption
(discount rate of 5%) and data (periodic payments of 1 for 5 years), and the formula “=A1*10” in cell A2. Our original metalanguage was “Returns a present value of an investment: the total amount of
a series of future payments is worth now.” Another version of this metalanguage: The function Present Value (PV) models the current value of an expected future stream of cash flow from the data
inputs: rate, period, and payments. The model for the data in terms of an olog is shown in Figure 9.
A fact in terms of an olog is shown in Figure 10.
This olog illustrates the fact that the present value of 1 times 10 produces the same result at the present value of 10. In a category (an olog in this example), there may be an equivalence relation
on the set of paths. Paths of morphisms in a category that result in the same result are said to be commutative. In ologs, they are facts.
Another consideration of a model not discussed in this article is the logic used in the object language. Even with standard terminology, computational side effects of the model may cause some results
to be different in different computer systems. Even with standard terminology and no side effects, results may be different due to different variations of logic. Classical mathematics, for example,
is done in the topos of Set (Set Semantics) with its internal logic the “ordinary” logic.^[11] It includes the principle of excluded middle and the full axiom of choice, and methods such as proof by
contradiction. In contrast, constructive mathematics is done without these in different topos. Ologs use a Set semantics with “ordinary” logic.
As a concluding remark, this article was tailored for an audience of actuaries in the American Academy of Actuaries who may find ologs pleasing to their visual cortices. The application of these
concepts can be adopted to other domains and could be tailored for audiences of other scientific organizations. Scientists have a pressing need to adopt metalanguages “that organize their
experiments, their data, their results, and their conclusions into a framework such that this work is reusable, transferable, and comparable with the work of other scientists.”^[12] Ologs also
provide the intermediate language—first as an object language from which the scientists can make concepts in their metalanguage precise, and then as the metalanguage for translation of those concepts
into a computational or mathematical object language. The “ontology log” or olog has the possibility for such a framework. The olog metalanguage can be used to represent models and data with
mathematical sets and functions—set semantics. Applying this rigor opens a warehouse of mathematical tools available through Applied Category Theory.
The Cambridge Encyclopedia of Language
; David Crystal; 2010; page 254. [2]
Knowledge Representation: Logical, Philosophical, and Computational Foundations
; John F. Sowa; 2000; page 51. [3]
Introduction to Symbolic Logic and its Applications
; Rudolf Carnap; 1958; page 79. [4]
How to Design Programs (Second Edition)
; Matthias Felleisen, Robert Bruce Findler, Matthew Flatt, Shriram Krishnamurthi; 2018; pages 78–80. [5]
Category Theory for the Sciences
; David Spivak; 2014; pages 184–188. [6]
Ologs: A Categorical Framework for Knowledge Representation;
Spivak & Kent; 2011. [7]
Structure and Interpretation of Computer Programs
; Harold Abelson, Gerald Jay Sussman with Julie Sussman; 1996; pages 140 & 359. [8] nLab (Wiki), “applied category theory”; last updated Oct. 2022. [9]
Category Theory for the Sciences
; David Spivak; 2014. Material for the rest of the article comes from page 204 (Section 2.3) and page 208 (Section 4.5). [10]
; op. cit. [11] nLab (Wiki); “internal logic”; last updated March 2023. [12]
; op. cit.
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math/physics question (cc:@johncarlosbaez) Take the group SL(2,C), the double cover of the identity-connected component of the Lorentz group. As a manifold this group is homeomorphic to R3 × S3,
where R3 are Lorentz boosts and S3 the group SU(2) (double cover of rotations). You can also view the R3 as a Riemannian hyperbolic space H3. Now H3 has a natural Lorentz invariant measure, and S3
has a natural rotation invariant measure. So the left-invariant Haar measure on SL(2,C) should be some positive function F times the measures on H3 and S3. Is there an explicit expression for this
Haar measure in term of some/any coordinates on H3 and S3? (Asking here because I’m tired of the condescension on math/stack overflow towards explicit examples) Should this correspond to (possibly a
multiple of) the Killing form, which in this case will have signature (—-+++)? Choose your favourite coordinates on SL(2,C) and calculate the Killing form. You get a metric from which you can read
off the measure. I have done this explicitly for SL(2,R) but I expect it would be similar for SL(2,C).
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Compare Multi-Digit Numbers
This number and operations in base ten lesson teaches students how to compare the value of multi-digit numbers. The lesson includes research-based strategies and strategic questions that prepare
students for assessments. In this lesson, students determine which of a pair of multi-digit numbers is larger. The focus of this lesson is reinforcing differences between numbers based on place
Log in to view this lesson. If you do not have an account, subscribe now and begin your free 30 day trial.
Log in to view this lesson. If you do not have an account, subscribe now and begin your free 30 day trial.
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Nonuniqueness of weak solutions to the Navier-Stokes equation
For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we
prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous dissipative weak
solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.
• Convex integration
• Euler equations
• Intermittency
• Inviscid limit
• Navier-Stokes
• Turbulence
• Weak solutions
ASJC Scopus subject areas
• Statistics and Probability
• Statistics, Probability and Uncertainty
Dive into the research topics of 'Nonuniqueness of weak solutions to the Navier-Stokes equation'. Together they form a unique fingerprint.
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Calculate the Zero Point Energy of ElectronSuppose an electron is confined in a cube of length L. Calculate the zero point energy of an electron. Discuss the situation when a. walls are removed and b. one of the wall is elongated
Calculate the Zero Point Energy of Electron
Suppose an electron is confined in a cube of length L. Calculate the zero point energy of an electron. Discuss the situation when
a. walls are removed and
b. one of the wall is elongated
We know that the total energy of an electron in a cubical box of side length 'L' is-
E = E[x] + E[y] + E[z] = [h^2 / 8mL^2](n^2[x] + n^2[y] + n^2[z]) -----Equation-1
Although the zero value of n[x], n[y] or n[z] is possible, but not acceptable because the ψ function then becomes zero but an electron is assumed to be already present inside the box. Therefore, the
lowest kinetic energy permissible to electron in a cubical box is one with n[x] = n[y] = n[z] = 1. This lowest kinetic energy is called zero point energy, which is given as-
E[zero] = 3h^2 / 8mL^2
It shows that the electron inside the box is not at rest even at 0°K. Therefore the position of the electron cannot be precisely known. Since only the mean value of kinetic energy is known, the
momentum of electron is also not precisely known. The occurrence of zero point energy is therefore in accordance with Heisenberg's uncertainty principle.
A. When Walls are Removed
The energy of an electron confined between two infinitely large walls. of a distance 'L' along x-axis and supposed to have zero potentia energy is given by-
E[x] = n^2[x] h^2 / 8mL^2
where n[x] is a quantum number which can only be positive integer excluding zero. Therefore, a bound electron has only quantised energy levels with values Ex[1], Ex[2] and Ex[3]... with n[x] = 1, 2,
3 ... respectively i.e. the energy of a bound electron is not continuous, rather discrete or quantised. If the walls of the box are removed and an electron is free to move without any restriction in
a field whose potential energy may be assumed to be zero then Schrödinger equation and its solution are given by-
δ^2X/δX^2 + K^2[x]X = 0
where K^2[x] = [8π^2m/h^2] E[x]
A = A cos k[x] X + B sin k[x] X
The arbitrary constant A, B and k^2[x] can now have any value one chooses to given them.
E[x] = k^2[x] h^2 / 8π^2m
The energy is therefore not quantised in this case. A free electron has a continuous energy spectrum. It can have any value of energy what so every possible. This quantitatively explains the
occurrence of continuum in the atomic or molecular spectra on ionisation because electron lost by an atom or molecule is a free electron which can move without any restriction.
B. One of the Wall is Elongated
The occurrence of the three quantum number (n[x], n[y] and n[z]) in the energy expression of an electron in equation (1) enclosed in a cube shows that each state is characterised by three quantum
numbers and several states of identical energy are possible e.g. there are three different states having quantum number (2, 1, 1), (1, 2, 1) and (1, 1, 2) for (n[x], n[y] and n[z]) each with the same
energy 6h^2 / 8mL^2. Such a level is said to be the three fold degenerate or triply degenerate. The wave function of these three triply degenerate states are different.
Let one of the walls of the cube are elongated along x-axis by dL i.e. the cube is distorted slightly. For the state (2, 1, 1) the energy of electron in the undistorted cube-
E = E[x] + E[y] + E[z]
The new energy on distribution along x-axis is given by-
E + dE = E[x] + dE[x] + E[y] + E[z]
Whereas the new energy for other states i.e. (1,2,1) and (1,1,2) is given by-
Thus, the initial three fold degnerate levels are split on distortion of the cube into a non-degenerate level and doubly degenerate levels.That electron degeneracy is either reduced or removed on
slight distortion of the system is a common phenomenon (John-Teller Distortion).
Very Important Question for B.Sc. and M.Sc. Exams.
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Set of tools to compute various climate indices
Why should I use it?
The package contains a set of tools to compute metrics and indices for climate analysis. The package provides functions to compute extreme indices, evaluate the agreement between models and combine
these models into an ensemble. Multi-model time series of climate indices can be computed either after averaging the 2-D fields from different models provided they share a common grid or by combining
time series computed on the model’s native grid. Indices can be assigned and/or combined to construct new indices.
How do I get it?
To install and load the package you can run the next lines in your R session:
How do I use it?
The main functionalities are presented in four different vignettes with step by step examples:
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'What made it possible for evolution to produce Euclid?'
Biological Foundations For Mathematics
Centre for Computational Biology Seminar
Wednesday 30th November 2016 (12:00-13:00)
Seminar details
One of the greatest achievements of human minds was Euclid's "Elements", which includes discoveries made by earlier mathematicians, building on ancient mechanisms of perception and reasoning, some of
them shared with other species.
Many of the mathematial discoveries made 2,500 years ago or earlier are still in regular use by scientists, mathematicians and engineers.
How could biological evolution, starting with a cloud of dust, or a chemical soup, produce minds capable of making those discoveries in geometry, topology and arithmetic, that are still beyond the
reach of the most sophisticated Artificial Intelligence systems?
It is widely believed that one aspect of human thinking that computers are best able to mimic is mathematical thinking.
But that's true only of a subset of types of mathematical thinking, e.g. using logic, arithmetic, statistics, and algebra.
There are mathematical discoveries made thousands of years ago by humans that current AI systems don't seem to be able to replicate, in particular the discoveries in topology and geometry using
spatial reasoning made by Euclid and others. (I'll give examples.)
A pre-verbal human toddler seems to be able to make discoveries in 3D topology that no current robot can make, illustrated here:
It seems that related discoveries can be made and put to practical use by non-human species and pre-verbal human toddlers, though they are not aware of what they are doing, unlike adult human
A Protoplanetary Dust Cloud?
[NASA artist's impression of a protoplanetary disk, from WikiMedia]
That's one of the questions addressed in the Turing-inspired Meta-Morphogenesis project:
I hope that computational biology can help to contribute to an answer to that question and many others.
I am aware that Computational Biology has two different but overlapping aspects (ignoring simply using computation for data-processing):
1. Using computers to model a wide variety of biological mechanisms and processes, e.g. formation, transport and interactions of chemicals within organisms, or variations of populations in
2. Identifying and trying to understand varieties of information processing in living systems, including their evolution, reproduction, development, and interactions with their environment, inert
and living.
It's the second aspect I hope to gain from, though of course, it can also make use of computer modelling.
In particular I'll focus on abilities to acquire, manipulate and use information about a spatial environment.
One of the questions I'll raise is whether current computing systems can replicate or model all the forms of information processing found in living things.
In particular:
Do some brains use chemical information processing mechanisms that cannot be replicated in digital computers?
Or have researchers in AI, cognitive science and neuroscience simply not yet understood what needs to be replicated, and how to replicate it in new virtual machines running on digital computers?
Is the topic of the latest Nobel prize for chemistry relevant?
The role of mathematical mechanisms and mathematical "discoveries" used by evolution is a strand in the Turing-inspired Meta-morphogenesis project -- a very long term project.
I hope the project can benefit from related work in Computational Biology at Birmingham:
(INCOMPLETE DRAFT: Liable to change) (Comments and criticisms welcome)
This document is
A partial index of discussion notes is in
Installed: 29 Nov 2016
Updated: 1 Dec 2016
(Contents list to be added, with additional pointers.)
Types of foundation for mathematics
A closely related seminar (17 Nov 2016)
There is considerable disagreement about the nature of mathematics.
Many think of mathematics as essentially something created by human minds -- since humans develop and organise sets of axioms, definitions, proofs, constructions and theorems.
But I claim that long before humans existed biological evolution made use of many mathematical discoveries that had nothing to do with humans.
In particular, evolution implicitly made many mathematical discoveries involving physical structures and processes
Alan Turing showed, in his 1952 morphogenesis paper that a wide variety of physical patterns can be produced on the surfaces of living things by combination of reaction and diffusion processes
involving two chemicals and in recent years mathematicians and physicists and have explored a wide variety of biological and non-biological examples.
The meta-morphogenesis project attempts to investigate evolution of information processing functions and mechanisms and the ways in which they changed the mechanisms of evolution and therefore the
range of possible products of evolution.
Hence the "meta-" in "The meta-morphogenesis project".
The importance of construction kits
For more details see:
The mechanisms involved in construction of an organism can be thought of as a construction kit, or collection of construction kits. Some components of the kit are parts of the organism and are used
throughout the life of the mechanism, e.g. cell-assembly mechanisms used for growth and repair. Construction kits used for building information-processing mechanisms may continue being used and
extended long after birth as discussed in the section on epigenesis below. All of the construction kits must ultimately come from the Fundamental Construction kit (FCK) provided by physics and
Figure FCK: The Fundamental Construction Kit
A crude representation of the Fundamental Construction Kit (FCK) (on left) and (on right) a collection of trajectories from the FCK through the space of possible trajectories to increasingly complex
The Fundamental Construction Kit (FCK) provided by the physical universe made possible all the forms of life that have so far evolved on earth, and also possible but still unrealised forms of life,
in possible types of physical environment. Fig. 1 shows (crudely) how a common initial construction kit can generate many possible trajectories, in which components of the kit are assembled to
produce new instances (living or non-living).
The space of possible trajectories for combining basic constituents is enormous, but routes can be shortened and search spaces shrunk by building derived construction kits (DCKs), that assemble
larger structures in fewer steps^7, as indicated in Fig. 2.
Figure DCK: Derived Construction Kits
Further transitions: a fundamental construction kit (FCK) on left gives rise to new evolved "derived" construction kits, such as the DCK on the right, from which new trajectories can begin, rapidly
producing new more complex designs, e.g. organisms with new morphologies and new information-processing mechanisms. The shapes and colours (crudely) indicate qualitative differences between
components of old and new construction kits, and related trajectories. A DCK trajectory uses larger components and is therefore much shorter than the equivalent FCK trajectory.
The importance of layers of construction kits
Different construction kits have different mathematical properties:
E.g. compare Meccano, Fischer-teknik, Lego, Tinker toys, plasticine, blocks, sand, mud, paper+scissors+glue, various electronic devices, etc.
Bringing separately developed construction kits together can produce sudden changes in what can be built and what sorts of behaviours the constructions can produce.
There are many processes that produce new developments in construction kits. A particularly important one is abstraction by parametrisation -- only possible for some sorts of construction kit.
Parameters can be simple measures or complex structures -- e.g. a perceived obstacle as a parameter for procedures for coping with obstacles, or a perceived gap in an incomplete construction to be
filled by an object found in the environment.
Construction kits for information processing
As organisms become capable of more varied and more complex actions and develop a wider variety of needs -- for different kinds of food, for shelter, for mates, for care of offspring, they depend on
increasingly complex and varied types of information, put to increasingly complex and varied uses, as crudely indicated in this sequence:
Simplest organisms merely react to chemicals in contact with their surrounding membrane.
More complex organisms may be able to sense features of their environment and produce motion -- random motion at first, then later on directed motion (e.g. following gradients).
Later still, some organisms can alter objects in their environment in order to make tools, shelters, etc.
A whole variety of different actions in different contexts will depend on grasping spatial structures and relationships.
Many AI systems are trained to respond to perceived configurations with actions selected from a fixed repertoire.
In contrast intelligent animals can construct new types of action to solve new problems.
It is widely believed that this involves learning statistical correlations: but there is something much deeper and more powerful -- a generalisation of Gibson's view of perception.
Theories about functions of perception
• Sensory pattern recognition
• Learning responses to sensory patterns
• Acquiring information about the contents of the environment
(exosomatic information contrasted with somatic information -- about what's happening in the body).
• Learning to think about possibilities and constraints on possibilities
Compare Gibson's theory perception provides information about affordances: possible actions by the perceiver and their consequences.
This can be generalised to perceiving possibilities for change, e.g. an object can move left, move right, move up, move down, come nearer, move further, rotate in various directions, etc.
Each of the possibilities, if realised, will produce additional changes.
Gibson restricted his theory of affordances to perception of possible actions by the perceiver with good or bad consequences for the perceiver. (With some exceptions, e.g. affordances for
But we can generalise this to perception of possibilities for change that are not necessarily produced by the perceiver and understanding of effects of such changes, which may or may not be
relevant to the perceiver's needs.
I call these "proto affordances"
• The perception of proto-affodances can lead on to understanding a wide range of possibilities and impossibilities connected with topological and geometrical properties and relationships.
• In humans and other animals this can lead to processes of reasoning about how to produce or avoid various effects via complex actions: chained affordances.
Conjecture from affordances to mathematics
I suspect that whereas many intelligent animals evolved abilities to perceive and make use of possibilities and constraints on possibilities -- humans somehow developed additional meta-cognitive
abilities to notice what they are doing, form generalisations and test them.
This could also lead to meta-meta-cognitive processes comparing effects in different contexts, leading to new generalisations about what does and does not work in various conditions.
The need for collaboration and the benefits of passing on acquired competences to offspring could have used additional abilities to perceive or infer knowledge states or competences of others and to
take action to improve others, e.g. teaching.
Late this might have led to challenges and responses, i.e. something like construction of proofs.
Topology and Geometry vs Logic and arithmetic
Toplogical discoveries include noticing the consequences of combining two possibilites for 2-D closed curves:
• Curve A completely encloses curve B
• Curve B completely encloses some object C
What brain mechanisms make it obvious that in every such case
• Curve A completely encloses object C
Other examples
Closed 2D shapes can be formed of straight lines chained together.
The smallest number of lines forming a closed space is 3: a triangle.
If three lines are connected to enclose a space there will be three corners.
By changing relative lengths of lines the shapes of the triangles including the angles at the corners can be varied.
But whatever the angles they must add up to a straight line.
The "standard" proofs of the "Triangle Sum Theorem"
Two "standard" proofs of the triangle sum theorem using parallel lines, and the
Euclidean theorems
stated above, are shown below in Figure Ang1:
Figure Ang1:
Warning: I have found some online proofs of theorems in Euclidean geometry with bugs
apparently due to carelessness, so it is important to check every such proof found
online. The fact that individual thinkers can check such a proof is in part of what
needs to be explained.
Mary Pardoe's proof of the Triangle Sum Theorem
Many years ago at Sussex university I was visited by a former student Mary Pardoe
(nee Ensor), who had been teaching mathematics in schools. She told me that her
pupils had found the standard proof of the triangle sum theorem hard to take in and
remember, but that she had found an alternative proof, which was more memorable, and
easier for her pupils to understand.
Her proof just involves rotating a single directed line segment (or arrow, or pencil,
or ...) through each of the angles in turn at the corners of the triangle, which must
result in its ending up in its initial location pointing in the opposite direction,
without ever crossing over itself.
So the total rotation angle is equivalent to a straight line, or half rotation, i.e.
180 degrees, using the convention that a full rotation is 360 degrees.
The proof is illustrated below :
It may be best to think of the proof not as a static diagram but as a process, with
stages represented from left to right in Figure Ang3.
What kind of brain mechanism makes such spatial proofs work
Can we give computers similar powers?
--- To perceive possibilities and impossibilities/constraints
What did biological evolution have to do to give brains these powers?
Can we replicate them in digital computers?
More on construction kits
There are many processes that produce new developments in construction kits. A particularly important one is abstraction by parametrisation -- only possible for some sorts of construction kit.
The importance of layers of construction kits
Construction kits for virtual machinery
This was not well understood a hundred years ago despite many advances in physics and chemistry.
However, since about the 1950s or 1960 increasingly complex and varied virtual machines have been developed providing new types of abstract construction kit.
I conjecture that evolution discovered the importance of virtual machines long before humans did -- not least because human minds (and othher animal minds) require very sophisticated forms of virtual
2.3 Construction kits built during development (epigenesis)
Some new construction kits are products of evolution of a species and are initially shared only between a few members of the species (barring genetic abnormalities), alongside cross-species
construction kits shared between species, such as those used in mechanisms of reproduction and growth in related species. Evolution also discovered the benefits of "meta-construction-kits":
mechanisms that allow members of a species to build new construction kits during their own development.
Examples include mechanisms for learning that are initially generic mechanisms shared across individuals, and developed by individuals on the basis of their own previously encountered learning
experiences, which may be different in different environments for members of the same species. Human language learning is a striking example: things learnt at earlier stages make new things learnable
that might not be learnable by an individual transferred from a different environment, part way through learning a different language. This contrast between genetically specified and individually
built capabilities for learning and development was labelled a difference between "pre-configured" and "meta-configured" competences in [Chappell Sloman 2007], summarised in Fig. 3. The
meta-configured competences are partly specified in the genome but those partial specifications are combined with information abstracted from individual experiences in various domains, of increasing
abstraction and increasing complexity.
Mathematical development in humans seems to be a special case of growth of such meta-configured competences. Related ideas are in [Karmiloff-Smith 1992].
Figure 3: Figure EVO-DEVO:
A particular collection of construction kits specified in a genome can give rise to very different individuals in different contexts if the genome interacts with the environment in
increasingly complex ways during development, allowing enormously varied developmental trajectories. Precocial species use only the downward routes on the left, producing only "preconfigured"
competences. Competences of members of "altricial" species, using staggered development, may be far more varied within a species. Results of using earlier competences interact with the
genome, producing "meta-configured" competences shown on the right. This is a modified version of a figure in [Chappell Sloman 2007].
Construction kits used for assembly of new organisms that start as a seed or an egg enable many different processes in which components are assembled in parallel, using abilities of the different
sub-processes to constrain one another. Nobody knows the full variety of ways in which parallel construction processes can exercise mutual control in developing organisms. One implication is that
there are not simple correlations between genes and organism features.
Foundations for Mathematics
There are several very different things that could be described as "foundations for mathematics".
(a) Neo-Kantian cognitive foundation
This is the kind of foundation that Immanuel Kant tried to provide by describing features of minds that make it possible for them to understand mathematical concepts and discover and make use of
mathematical theorems and proofs. He claimed that the knowledge obtained in this manner was non-empirical, included necessary truths, and was synthetic - i.e. not derivable from definitions using
only logical inferences. This three-fold characterisation combines a theory of the nature of mathematical truths with a theory of the features of (natural and artificial) minds that enable them to
discover mathematical truths.
I don't know whether Kant thought it possible that some kind of non-human mind could in principle exist with the mechanisms required for making mathematical discoveries. A good theory of this sort
should be applicable to a variety of types of mind, including artificial minds designed and built by humans.
(b) Mathematical foundation: a mathematical foundation for mathematics
This is the kind of foundation that many mathematicians, logicians and philosophers have attempted to find in the last two centuries or so: a (preferably finite?) subset of mathematics from which
everything else can be derived mathematically. In this context, logic is usually regarded as a part of mathematics.
(c) Biological/evolutionary foundations
What made it possible for mathematical discoveries to be made and used by products of biological evolution, and later reasoned about and discussed?
(d) Physical/chemical foundations
How does the physical/chemical universe constrain the kinds of mathematics required for its description and how does it make possible the production, by evolution or engineering, of types of machines
(including organisms) with abilities to discover, make use of, and in some cases reason about the mathematical features.
(e) Metaphysical/Ontological foundations?
This is an attempt to answer the question: what makes it the case that there are mathematical truths, some or all of which can be discovered and used, whether by human mathematicians or anything
else, e.g. biological evolution or its products.
In principle (e) could be further split into different sorts of foundation or grounding. For example, there might be a world whose physical/chemical properties could not support evolution of
organisms with brains capable of making and organising mathematical discoveries of certain kinds. Could there, for example, be universes in which brains could evolve that are capable of making
discoveries in geometry and topology but not the arithmetical discoveries that depend on the use of mathematical induction, or proof by contradiction?
Perhaps there are aspects of the kind of mathematics discoverable and in some cases usable in this universe that would be applicable to all possible physical universes, and some aspects of
mathematics that are restricted to a subset of possible universes with special properties.
For example, could there be a kind of universe that does not support the physical mechanisms (e.g. brain mechanisms) required for discovery or invention of Euclidean geometry or its alternatives?
There may be even more limited physical universes in which it is not possible for physical information processing mechanisms to exist that can support the discovery of the full set of natural
numbers, even if some subsets are found to be useful. In that sort of universe no brain mechanisms could ever construct even the thought that the natural numbers "go on indefinitely". It is not
obvious what sort of brain could grasp the usefulness of counting using a fixed list of counting noises in connection with a wide range of tasks, and be incapable of having the thought: there is no
largest collection of objects that can be counted.
Even in our universe not all brains seem to have that capability, and it is not clear at what stage of development human children are able to comprehend such thoughts, nor how their brains need to
change during development to give them such abilities.
David Deutsch (in Deutsch(2011)) seems to think that (c) is solved already because physics allows the implementation of universal computers and they suffice for everything.
He doesn't seem to be aware that he is talking about a kind of universality that is relative to a specified space of computations and there may be other computations that are not covered, e.g.
the types of information processing used in the ancient mathematical discoveries of type (a).
It seems that brains of intelligent animals like squirrels, crows, elephants, etc. that use currently unknown types of computation lack some of the abilities required for mathematical
meta-cognition. Likewise pre-verbal human toddlers.
What counts as a possible universe is not clear. E.g. the "No-space" world containing only coexisting sounds and sound sequences, considered in Strawson(1959) may be too causally impoverished to
be capable of supporting information-processing mechanisms or biological evolution, in which case minds could neither exist nor evolve. Perhaps there would be numerical features and regularities
in the sound patterns that do exist if such a world is possible, even if it contains no minds that could discover such phenomena. I am not inclined to take that kind of speculation seriously,
though I admire other features of Strawson's book.
More interestingly, physicists discuss alternative theories about the fundamental nature of the physical world, and attempt to evaluate those theories in terms of their ability to explain the
observations of the physical sciences, e.g. physics, astrophysics, and chemistry. They could also be evaluated in terms of their ability to explain the possibility of the sorts of information
processing mechanisms that meet the requirements of animals produced by biological evolution, including humans capable of making mathematical discoveries. This information processing requirement
may turn out to add significantly to the physical requirements. An example of such a challenge is discussed in Schrödinger (1944).
I shall now elaborate on each of the three main types of foundation in a little more detail, and relate them to their roles in biological evolution.
1. Neo-Kantian cognitive foundation for mathematics
This is a modified version of Kant's attempt to describe basic features of mathematical minds. They are able to have certain sorts of experience on the basis of which they can discover and prove
various kinds of mathematical truth, including truths of arithmetic, topology and geometry. Kant lived before it was possible to specify minds in computational terms, though it seems to me that he
was describing requirements for such a specification and moving towards such a specification, though his version of computation, as far as I know, was not restricted to
computation like modern computational foundations. For example, his claim that we can discover that there are non-superimposable 3-D structures, such as a right-handed and a left-handed helix does
not specify a form of computation, but this intuitive discovery does not seem to make use of discrete operations on discrete symbols. However he was not very clear about the alternatives available,
and neither has anyone else been, as far as I know. However I have been trying to assemble candidate examples, e.g. http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangle-sum.html http://
www.cs.bham.ac.uk/research/projects/cogaff/misc/shirt.html http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html http://www.cs.bham.ac.uk/research/projects/cogaff/misc/triangles.html
One important modification is a Lakatos-inspired alteration from trying to explain how it is possible know that space (or experienced space) is Euclidean to trying to explain how it is possible to
know that there are at least three distinct ways of experiencing space: Euclidean, elliptical and hyperbolic (all easily illustrated in 2-D surfaces) with a common core and one axiom different, and
to derive many theorems common to all, and some true only in a subset. (I suspect that if Kant had known about non-Euclidean geometries he would have modified some, but not all, of his examples.)
Another requirement is to explain how a mathematical mind (like Archimedes' mind?) can discover a simple extension to Euclidean geometry, the neusis construction, that makes it easy to trisect an
arbitrary triangle. http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html A feature of (a) will be accommodating the evidence from Lakatos that although the mathematical discoveries are
non-empirical, mistakes of various kinds are possible, and may need to be repaired.
2. Mathematical foundations for mathematics
This what meta-mathematicians and some philosophers of mathematics have been seeking in the last two centuries or so, building on the work of thinkers such as Peano, Frege, Dedekind, Cantor, Russell,
Brouwer (out on a limb?) and possibly others.
Their goal is to discover or construct a mathematical foundation for mathematics, which may or may not be possible, namely some well-defined subset of mathematics (including logic and set-theory if
needed) that suffices as a basis (in some mathematical sense) for all the rest of mathematics.
It seems that there cannot be such a finite foundation that enables any mathematical truth to be proved in a finite number of steps. If there were, all mathematical truths could be enumerated, which
Cantor showed to be impossible.
An infinite foundation is trivially possible -- just combine all mathematical results into one system.
However, it is not obvious whether there can be a minimal non-finite foundation namely a part of mathematics from which everything else can be derived, but which loses that power if anything is
removed from it.
In principle there could be a unique minimal foundation or there might be several alternative minimal foundations, each of which would have to be capable of proving all the others. (Compare the
equivalence of Turing machines, Lambda calculus, Recursive function theory, and Post's production systems?)
3. Metaphysical/physical foundations for mathematics
[I don't know whether this is being investigated by anyone else, though Turing seemed to start on something like this shortly before he died.]
This kind of foundation for mathematics would take the form of a fundamental construction kit provided by the physical universe, from which many different construction kits and forms of scaffolding
can be derived which together suffice to generate not only all the mathematically specifiable structures and processes in the physical/ chemical universe, but also all the required mechanisms,
including mechanisms of biological evolution with all the forms of information processing that produce, and use, new kinds of mathematical structures and processes.
[which provide re-factoring [NB Alan]],
-- the mathematical features of quantum mechanics that Schrödinger showed in 1944 enabled long multi-stable molecules to have properties required for genetically encoded information transferable
across several generations) allowing for *huge* amounts of discretely controlled variability. (Anticipating Shannon?) Schrödinger(1944)
-- the ability to grow physical structures partly under control of a genome that specifies structures with all sorts of mathematical properties, e.g. many kinds of symmetry, fibonacci sequences,
various kinds of stability, various kinds of repetitive behaviours, various kinds of self-replication, etc. (as described by D'Arcy Thompson, Brian Goodwin, Stuart Kauffman, and others ...?)
-- including the ability to produce new physical/chemical construction kits with the mathematical properties required to produce a wide variety of biologically useful parametrisable properties (far
more than human-designed construction kits can, e.g. meccano, lego, tinker toys, mud, plasticine, etc.!)
Things produced by new (derived) construction kits include cell membranes, intra-cell mechanisms of various sorts (eg. microtubules), skin, muscle, molecular energy stores, tendons, hairs, bone,
cartilage, many kinds of woody material, capillaries, silk, eggshells, spores, feathers, digestive juices, immune mechanisms, tissue damage detection and repair mechanisms, neurones, a wide variety
of sensory detectors, e.g. for light, pressure, temperature, torsion, sound, chemicals, and thousands (millions?) more useful components, many shared across species in parametrised forms because of
mathematical commonalities.
[[These make use of implicit mathematical discoveries made by evolution concerning what's possible using available construction kits. New kits change what's possible in particular contexts]]
-- e.g. the physical/chemical mechanisms that encoded homeostatic control based on negative feedback loops and other mathematically useful forms of control, including some that modify feedback loops,
e.g. damping mechanisms, or accelerator/decelerator mechanisms.
-- Discovery of mathematical forms that allow physical mechanisms to provide required properties with great economy, e.g. use of triangles or more complex shapes that produce rigidity.
-- various uses of parametrised design to allow for variation within an individual (e.g. during growth), across individuals in a species, and between species,
-- and which eventually produced brains of mathematicians able to make and discuss many deep mathematical discoveries (e.g. those in Euclid's Elements) using still unknown brain mechanisms, which we
don't yet know how to implement/model on computers. ====
All this is partly a reaction against claims that Einstein proved Kant wrong.
It is also partly a reaction against claims that mathematics is/are created by humans, e.g. Wittgenstein: mathematics is an anthropological phenomenon.
It's the other way round: the existence of humans is a result of deep mathematical features of the universe, including increasingly many new features derived from old ones by evolution and its
construction kits over billions of years (possibility theorems with existence proofs and implicit impossibility theorems -- e.g. the possibility of a Euclid-like mathematician could not have been
proved by evolution within a short time after formation of the planet: far too many lemmas/intermediate theorems were needed),
I sense that all this is more consistent with what Mumford wrote than with most of what I've read about the nature of mathematics, though there are still many missing steps.
I welcome pointers to anything relevant, supporting, contradicting, or improving these ideas.
• David Deutsch, 2011 The Beginning of Infinity: Explanations That Transform the World,
Allen Lane and Penguin Books, London,
• Tibor Ganti, (2003) The Principles of Life,
Eds. Eors Szathmary and James Griesemer,
Translation of the 1971 Hungarian edition, with notes, OUP, New York
Usefully summarised in http://wasdarwinwrong.com/korthof66.htm
• Erwin Schrödinger (1944) What is life? CUP, Cambridge.
Some extracts from that book, with comments, can be found here:
• Aaron Sloman, 1965, `Necessary', `A Priori' and `Analytic', Analysis, 26, 1, pp. 12--16, http://www.cs.bham.ac.uk/research/projects/cogaff/62-80.html#1965-02
Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham
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[Solved] 5) A large group of students took a test | SolutionInn
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5) A large group of students took a test in Finite Math where the grades had a mean of 72 and a standard deviation of
5) A large group of students took a test in Finite Math where the grades had a mean of 72 and a standard deviation of 4. Assume that the distribution of these grades is approximated by a normal
distribution and that passing the test is a 65 a) What percent of students scored higher an 81 or higher? b) What percent of students failed the test? C) What percent of students scored between a 70
and 75? d) What happens when you try to find the percent of students that scored less than a 40
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Pi Day Archives - Stuck at the AirportFree pie at PIE Airport to celebrate Pi Day - Stuck at the Airport
St. Pete-Clearwater International Airport (PIE) is celebrating Pi Day a day early this year by handing out free individual-sized strawberry and Key Lime pies on Friday, March 13 from 1:30 to 3:14
Pie will be served both curbside (drive-by pie) and inside the terminal.
Why free pie?
The Greek letter “π” (Pi), is the symbol that represents the mathematical constant that is the ratio of the circumference of a circle to its diameter and has been calculated to over a trillion
digits, the first of which are 3.14.
And, of course, free pie is a great way to celebrate PIE airport, which gets its identifier code from its original name, Pinellas International Airport.
PIE airport celebrates Pi Day with free pie
March 14 is Pi Day, a day set aside to celebrate the mathematical constant that is the ratio of a circle’s circumference to its diameter and is approximately equal to 3.14159… (it has been calculated
to 1 trillion digits).
Pi Day will be also celebrated this year at Florida’s St. Pete-Clearwater International Airport, which has the airport code PIE.
From 1 p.m. until 3:14 p.m. the airport will be handing out pieces of strawberry and Key Lime pie to passengers, meeters and greeters and anyone who happens to be passing by.
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When Does the Time-Linkage Property Help Optimization by Evolutionary Algorithms?
Recent theoretical works show that the time-linkage property challenges evolutionary algorithms to optimize. Here we consider three positive circumstances and give the first runtime analyses to show
that the time-linkage property can also help the optimization of evolutionary algorithms. The problem is easier to optimize if the time-linkage property changes the optimal function value to an
easy-to-reach one. We construct a time-linkage variant of the CLIFF[d] problem with this feature and prove that conditional on an event that happens with Ω(1) probability, the (1+1) EA reaches the
optimum in expected O(nlnn) iterations. It is much better than the expected runtime of Θ(n^d) for the original CLIFF[d]. If the time-linkage property does not change the optimal function value but
enlarges the optimal solution set, the problem is also possible to be easier to optimize. We construct another time-linkage variant of the CLIFF[d] problem with this feature, and also prove an
expected runtime of O(nlnn) (conditional on an event happening with Ω(1) probability), compared with the expected runtime of Ω(n^d-2) for the corresponding problem without the time-linkage property.
Even if the time-linkage property neither changes the optimal function value nor the optimal solution set, it is still possible to ease this problem if the intermediate solution, from which the
optimum is easier to reach, is more prone to be maintained. We construct a time-linkage variant of the Jump problem, and proved that the expected runtime is reduced from O(n^k) to O(n^k-1). Our
experiments also verify the above theoretical findings.
Original language English
Title of host publication Parallel Problem Solving from Nature – PPSN XVIII: 18th International Conference, PPSN 2024, Hagenberg, Austria, September 14-18, 2024, Proceedings, Part III
Editors Michael AFFENZELLER, Stephan M. WINKLER, Anna V. KONONOVA, Thomas BÄCK, Heike TRAUTMANN, Tea TUŠAR, Penousal MACHADO
Publisher Springer Science and Business Media Deutschland GmbH
Pages 280-294
Number of pages 15
ISBN (Electronic) 9783031700712
ISBN (Print) 9783031700705
Publication status Published - 2024
Event 18th International Conference on Parallel Problem Solving from Nature, PPSN 2024 - Hagenberg, Austria
Duration: 14 Sept 2024 → 18 Sept 2024
Publication series
Name Lecture Notes in Computer Science
Publisher Springer
Volume 15150
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 18th International Conference on Parallel Problem Solving from Nature, PPSN 2024
Country/Territory Austria
City Hagenberg
Period 14/09/24 → 18/09/24
Bibliographical note
Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
• Evolutionary algorithms
• Time-linkage property
Dive into the research topics of 'When Does the Time-Linkage Property Help Optimization by Evolutionary Algorithms?'. Together they form a unique fingerprint.
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Tag Archives: algorithm
Today, I would like to talk to you about another upcoming feature of the next version of SPMF (2.60), which will be released soon. It will be a Workflow Editor that will allow the user to select
multiple algorithms from … Continue reading
Posted in Data Mining, open-source, spmf Tagged algorithm, association rule, data, data mining, graph mining, gui, itemset, pattern, pattern mining, sequential pattern mining, spmf Leave a comment
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Plotting Trend Lines in Excel - dummies
Excel provides a robust toolset for illustrating trends. You can do this by plotting trendlines in your Excel charts to offer a visual of your data. Here, you discover how to plot logarithmic
trendlines, power trend lines, and polynomial trend lines in Excel.
Plotting a logarithmic trend line in Excel
A logarithmic trend is one in which the data rises or falls very quickly at the beginning but then slows down and levels off over time. An example of a logarithmic trend is the sales pattern of a
highly anticipated new product, which typically sells in large quantities for a short time and then levels off.
To visualize such a trend, you can plot a logarithmic trend line. This is a curved line through the data points where the differences between the points on one side of the line and those on the other
side of the line cancel each other out.
Here are the steps to follow to plot a logarithmic trend line in Excel:
1. Click the chart to select it.
2. If your chart has multiple data series, click the series you want to analyze.
3. Choose Design→Add Chart Element→Trendline→More Trendline Options.
The Format Trendline pane appears.
4. Click the Trendline Options tab.
5. Select the Logarithmic radio button.
Excel plots the logarithmic trend line.
6. (Optional) Select the Display Equation on Chart check box.
If you just want to see the trend line, feel free to pass over Steps 6 and 7.
7. (Optional) Select the Display R-Squared Value on Chart check box.
8. Click Close.
Excel displays the regression equation and the R^2 value. This image shows a chart with the plotted exponential trend line, the regression equation, and the R^2 value.
When the best-fit trend line is a logarithmic curve, the regression equation takes the following general form:
y = m * ln(x) + b
y is the dependent variable; x is the independent variable; b and m are constants; and ln is the natural logarithm, for which you can use the Excel function LN.
Excel doesn't have a function that calculates the values of b and m directly. However, you can use the LINEST function if you “straighten out” the logarithmic curve by using a logarithmic scale for
the independent values:
{=LINEST(<em>known_ys</em>, LN(<em>known_xs</em>), <em>const</em>, <em>stats</em>)}
Plotting a power trend line in Excel
In many cases of regression analysis, the best fit is provided by a power trend, in which the data increases or decreases steadily. Such a trend is clearly not exponential or logarithmic, both of
which imply extreme behavior, either at the end of the trend (in the case of exponential) or at the beginning of the trend (in the case of logarithmic). Examples of power trends include revenues,
profits, and margins in successful companies, all of which show steady increases in the rate of growth year after year.
A power trend sounds linear, but plotting the power trend line shows a curved best-fit line through the data points. In your analysis of such data, it's usually best to try a linear trend line first.
If that doesn’t give a good fit, switch to a power trend line.
Follow these steps to plot a power trend line in Excel:
1. Click the chart to select it.
2. If your chart has multiple data series, click the series you want to analyze.
3. Choose Design→Add Chart Element→Trendline→More Trendline Options.
The Format Trendline pane appears.
4. Click the Trendline Options tab.
5. Select the Power radio button.
Excel plots the power trend line.
6. (Optional) Select the Display Equation on Chart check box.
If you just want to see the trend line, skip Steps 6 and 7.
7. (Optional) Select the Display R-Squared Value on Chart check box.
8. Click Close.
Excel displays the regression equation and the R^2 value (described below). The following image shows a chart with the plotted power trend line, the regression equation, and the R^2 value.
When the best-fit trend line is a power curve, the regression equation takes the following general form:
y = m * x<sup>b</sup>
y is the dependent variable; x is the independent variable; and b and m are constants.
There's no worksheet function available to directly calculate the values of b and m. However, you can use the LINEST function if you “straighten out” the power curve by applying a logarithmic scale
to the dependent and independent values:
{=LINEST(LN(<em>known_ys</em>), LN(<em>known_xs</em>), <em>const</em>, <em>stats</em>)}
Plotting a polynomial trend line in Excel
In many real-world scenarios, the relationship between the dependent and independent variables doesn't move in a single direction. That would be too easy. For example, rather than constantly rising —
uniformly, as in a linear trend, sharply, as in an exponential or logarithmic trend, or steadily, as in a power trend — data such as unit sales, profits, and costs might move up and down.
To visualize such a trend, you can plot a polynomial trend line, which is a best-fit line of multiple curves derived using an equation that uses multiple powers of x. The number of powers of x is the
order of the polynomial equation. Generally, the higher the order, the tighter the curve fits your existing data, but the more unpredictable your forecasted values are.
If you have a chart already, follow these steps to add a polynomial trend line in Excel:
1. Click the chart to select it.
2. If your chart has multiple data series, click the series you want to analyze.
3. Choose Design→Add Chart Element→Trendline→More Trendline Options.
The Format Trendline pane appears.
4. Click the Trendline Options tab.
5. Select the Polynomial radio button.
6. Click the Order spin button arrows to set the order of the polynomial equation you want.
Excel plots the polynomial trend line.
7. (Optional) Select the Display Equation on Chart check box.
If you just want to see the trend line, bypass Steps 7 and 8.
8. (Optional) Select the Display R-Squared Value on Chart check box.
9. Click Close.
Excel displays the regression equation and the R^2 value. The image below shows a chart with the plotted polynomial trend line, the regression equation, and the R^2 value.
When the best-fit trend line is a polynomial curve, the regression equation takes the following form:
y = m[n]x^n + … + m[2]x^2 + m[1]x + b
y is the dependent variable; x is the independent variable; and b and m<em>n</em> through m1 are constants.
To calculate the values b and mn through m1, you can use LINEST if you raise the <em>known_xs</em> values to the powers from 1 to n for an nth-order polynomial:
{=LINEST(<em>known_ys</em>, <em>known_xs</em> ^ {1,2,…,<em>n</em>}, <em>const</em>, <em>stats</em>)}
Alternatively, you can use the TREND function:
{=TREND(<em>known_ys</em>, <em>known_xs</em> ^ {1,2,…,<em>n</em>}, <em>new_xs</em>, <em>const</em>)}
About This Article
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Martin J. Gander
:: Teaching Interests
My teaching interests are both in Mathematics and Computer Science: in addition to undergraduate courses in both areas, I am interested and qualified to teach at the graduate level Scientific
Computing, Numerical Differential Equations, Matrix Computations, Differential Equations, Parallel Computing, Numerical Dynamical Systems, Algorithms and Data Structures and Object Oriented
:: Courses I teach this year in Geneva
Analyse numérique des équations aux dérivées partielles:
Beaucoup de phénomènes physiques peuvent être modélisés par des équations aux dérivées partielles, par exemple le flux d'un fluide dans un tuyau, les variations de température à l'intérieur d'un
appartement ou la cuisson dans un four à micro-ondes. Mais la résolution de ces équations est souvent difficile et les méthodes analytiques suffisent rarement pour obtenir les résultats désirés.
Ce cours est une introduction aux méthodes numériques modernes pour la résolution des équations aux dérivées partielles. Nous utiliserons Matlab pour développer des codes modèles simples, et Maple
pour nous assister dans les calculs symboliques.
:: Teaching Experience
Differential Equations (McGill 2004): Introduction to differential equations: first order and linear second order equations, higher order equations, solution by series, Laplace transform and simple
numerical methods.
Ordinary Differential Equations (McGill, 2003): Introduction to ordinary differential equations: first and second order equations, linear equations, series solutions, Frobenius method, Laplace
transforms and applications.
Introduction à l'analyse numérique I (Genève, 2003): A first undergraduate course introducing students from mathematics and computer science to numerical integration, interpolation and
approximations, numerical ordinary differential equations and linear systems. (40 students, 2 hours lecture, 1 hour exercises and 2 hours Fortran exercises)
Introduction à l'analyse numérique II (Genève 2003): A second undergraduate course introducing students from mathematics and computer science to more advances topics in numerical analysis: iterative
methods, eigenvalue and eigenvector computations and nonlinear systems. (40 students, 2 hours lecture, 1 hour exercises and 2 hours Fortran exercises)
Numerical Differential Equations (McGill, 2000-2002): A graduate course in numerical methods for ordinary and partial differential equations, including Runge-Kutta, Linear Multistep and adaptive
methods, finite elements, finite differences, finite volumes and spectral methods, fast solvers. (20 students, 4 hours lecture, exercises with Matlab and Maple)
Numerical Analysis (McGill, 1999-2001): A graduate course in numerical methods, with linear and nonlinear systems, iterative methods, eigencomputations, quadrature, approximation. (20 students, 4
hours lecture, exercises with Matlab and Maple)
Parallel Computing using Domain Decomposition (Summer course TU Denmark 1999): An introduction to domain decomposition methods. (25 students, 1 week lectures with exercises)
Introduction to Scientific Computing (Stanford, 1996-1997): An advanced undergraduate, early graduate course in scientific computing using Maple and Matlab. (30 students, 3 hours lecture and
Programming Paradigms (Solothurn, 1993): An introduction to the different programming paradigms using Object Pascal, Lisp and Prolog. (20 students, 2 hours lecture with exercises)
Introduction to Computer Science II (Solothurn, 1990-1992): Algorithms and datastructures in Pascal. (18 students, 2 hours lecture with exercises)
Introduction to Computer Science I (Solothurn, 1990-1992): An introduction to basic algorithms and programming in Pascal. (20 students, 2 hours lecture with exercises)
:: Fun Problems and Talks
• A survey talk I gave at the university of Geneva for a large group of visiting students from Greece.
• Think about an ice cube in a glass of water. Mark how high the water is before and after the ice cube has melted. What do you observe?
• The couch moving problem
• Suppose you are playing billiard on a circular billiard table. If there are two balls, what direction do you have to kick the first ball so it impacts on the boundary of the table exactly once
before hitting the other ball ? What does this have to do with the heart shape you see in your coffee mug ? See our paper in SIAM: Circular Billiard.
• Equilateral triangle circumscribing a given triangle (in German)
• Square circumscribing a given quadrilateral (in German)
• Have you ever noticed that whenever you look at a table of numbers, more of the numbers start with the digit 1 or 2 than with bigger digits. If not, take your daily newspaper, open it in the
business section and check a few tables. Why is that so ?
• A calculus problem solved with Maple
• A geometrical proof of the theorem of Pytagoras (by my friend Antonio Steiner)
• A patrol vessel has to get within a specified distance of a target vessel. Both vessels are traveling at constant speed on a plane. What is the minimum time needed for the patrol vessel to get
there ? See my note in SIAM: Note on the Optimal Intercept Time of Vessels to a Nonzero Range
• Into how many pieces can you cut a doughnut (a torus) with three planar cuts, without moving the donut ? Is the following solution correct ?
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