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Area of pentagon in Java …ftc - FcukTheCode
Area of pentagon in Java …ftc
Write a program that prompts the user to enter the length from the center of a pentagon to a vertex and computes the area of the pentagon, where r is the length from the center of a pentagon to a
(Hint: Length of the side =2*length from the center to a vertex * sine(PI/5)
Area = (5 * Math.pow(side, 2)) / (4 * Math.tan(Math.PI /5));
For rounding to two decimal points uses String.format(“%.02f”,variablename))
length from center: 6.4
Area = 97.39
import java.io.*;
import java.util.*;
import java.lang.Math;
public class temp{
public static void main(String[] args){
double verside , side , area;
Scanner scan = new Scanner(System.in);
System.out.print("Length from center to vertex: ");
verside = scan.nextDouble();
side = 2*verside*Math.sin(Math.PI/5);
area = (5*Math.pow(side,2))/(4*Math.tan(Math.PI/5));
System.out.println("Area of Pentagon: "+String.format("%.02f",area));
Length from center to vertex: 6.4
Area of Pentagon: 97.39
Length from center to vertex: 5.9
Area of Pentagon: 82.77
Length from center to vertex: 6.3
Area of Pentagon: 94.37
Length from center to vertex: 5.2
Area of Pentagon: 64.29
Length from center to vertex: 11.2
Area of Pentagon: 298.25
executed using javac in terminal linux | {"url":"https://www.fcukthecode.com/area-of-pentagon-in-java-ftc/","timestamp":"2024-11-11T14:00:57Z","content_type":"text/html","content_length":"148489","record_id":"<urn:uuid:83a09daf-95f0-421b-ab58-05b76ce0bec9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00536.warc.gz"} |
Hold Off on the Headphones
There are several characteristics that should be thoroughly understood to fully grasp the idea of longitudinal sound waves. These waves all have frequency, amplitude, wavelength and speed. At this
point the teacher may want to introduce these concepts to the students. There are some wonderful websites that are available for educational use which provide video clips and interactive diagrams.
See the website section at the end of the unit.
Frequency is the measure of "how often" a wave oscillates during a period of time. Frequency of a sound wave is measured in Hertz (Hz). One Hz is a measurement that describes a wave which makes a
complete cycle in one second. Ten Hz is a wave that makes 10 cycles or vibrations in one second. We hear high valued frequencies as a higher pitch and lower frequency as a low pitch. Thus the rumble
of thunder will have a lower frequency and a whistle will have a higher frequency. The normal range of frequency that a human can hear is between 20 Hz and 20000 Hz. In reference the range of the
lowest not of a piano is 27.5 Hz and the highest note is 4186 Hz.
Within a sound wave air molecules are compressed at each interval. The region of condensed air molecules creates an atmosphere of greater air density. In between these compressed regions are rarefied
areas where there are fewer air molecules and a less dense atmosphere. The larger the difference in pressure between the compressed molecules and the rarefied molecules causes greater amplitude and a
louder sound.
Amplitude measures how high and how low a wave moves in relation to the average of the wave. Amplitude of a sound wave is one half the difference of the highest pressure and the lowest pressure
measured in the wave (1/2 (HP - LP)). Amplitude is therefore the measurement of pressure difference within a wave which results in the loudness of the sound.
Because the pressure difference is so small we measure the loudness of a sound wave with decibels (dB). Decibels measure the loudness of a sound just as amplitude does. The decibel scale is a
logarithmic measure of sound pressure. Different than a linear measurement every increase of 20 dB means the pressure of the wave has increased 10 times greater in amplitude. At 20 dB the amplitude
is 10, at 40 dB the amplitude is 10x10 =100, at 60 dB, the amplitude is 100x10=1000, etc. (Figure 1,2).
Wavelength is the length of one complete cycle of a wave. In terms of amplitude it is the distance between two high pressure points.
Speed of Sound
Speed of a sound wave measures how fast an oscillation travels from one place to the next. The speed of sound is about 660 miles per hour or 344 m/s at room temperature. In one complete cycle a wave
moves forward one wavelength; therefore if we know the frequency of the sound wave and the wavelength then we can calculate speed of sound in air by using the formula:
(V = f/\) where V = speed, f = frequency and /\ = wavelength.
Activity 3 Speed of Sound
I will have students calculate the speed of sound in air. Students will be separated into groups of two. Each group will be given a cow bell, a pair of binoculars, a stopwatch and a measuring tape. I
will have students stand 200m apart from each other. One student A will have the cow bell. Student B will be provided with the binoculars and the stopwatch. As the student A strikes the bell student
B while looking through the binoculars will begin timing. Once student B hears the bell the timing will end. The speed of sound will be calculated by using the formula for velocity; velocity =
distance / time.
Another activity that can be done with your students is to film or obtain video clips of lightening storms and have students measure the time they see the lightening and hear the sound of thunder.
Have the student calculate the speed of sound by using the formula v = d / t. You can use a scale of 3 seconds per kilometer or 5 seconds per mile. This time interval corresponds to a speed of sound
in air. For example if the time it takes between a flash of lightening and the sound of thunder on the video is 6 seconds then the speed of sound will be calculated by dividing 2 km / 6 s which would
equal 1km/3s or 333.3 m/s (Hsu Tom, 2003).
Activity 4 Properties of Sound Wave Calculations and Observations
Students can now use the formula to calculate various problems for wavelength, frequency and speed. By knowing the speed of sound and the frequency, wavelength can be determined. Starting with the
original given formula V = f/\ have student solve the equation for /\ (wavelength). This is a simple algebraic procedure which can be solved by isolating the variable by using the inverse operation.
The students will have to divide both sides of the equation by "f" which will isolate /\ and give the new equation /\ = v/f (Hsu Tom, 2003).
Frequency can be described using a guitar and a tuner. Have the students use a guitar tuner to tune a guitar to standard tuning. Standard guitar tuning has the open notes tuned, from lowest pitch to
highest pitch, to: E, A, D, G, B, E. Use a sound level meter to determine the frequency of each string. To tune the notes tighten or loosen the string according to the guitar tuner. This is a
standard tuning sequence for many popular songs. Ask students to make observations about the pitch and frequency of each string after they have been plucked. Next have the students observe what
happens to the frequency when you cause the string to become sharp and flat. Have students make observation on the how different frequencies of the same string cause a change in what they hear.
A guitar is divided by frets. Holding down a finger on a string at different frets changes the wavelength of a plucked sting. This wavelength also changes the frequency of the sound. By using a
guitar you can have students calculate wavelength by measuring the frequency of the plucked, fretted sting.
You could also go the music room and gather information of the various frequencies of each note on a piano and graph the information to provide a visual auditory representation of different
frequencies. This can be done using a sound level meter. A sound level meter is an instrument that provides objective, reproducible measurements of sound pressure. It works very similar to the human
ear. The meter will give you measurements in frequency and decibels. A bar graph can be draw with the keys on the x axis and the frequency on the y axis.
Students can then measure wavelengths, frequencies and amplitudes of musical instruments using sound. This will provide students with the ability to collect and analyze data. Students will use the
sound and wave simulation software to examine the intensity and waveforms created from voice and musical instruments. Wave forms are then displayed on an oscillogram representing the wave of a
physical sound. Students can print the graphs produced. Have student compare the wave patterns from different instruments tuned at the same frequency. Have students compare the wave patterns of the
same instrument tuned at different frequencies. | {"url":"https://teachersinstitute.yale.edu/curriculum/units/2006/5/06.05.04/6","timestamp":"2024-11-05T09:51:51Z","content_type":"text/html","content_length":"44994","record_id":"<urn:uuid:5fc03fff-0abd-4380-bdd3-cf03c64ead58>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00525.warc.gz"} |
From Spreadsheets to System Dynamics Models - The Systems Thinker
Decision-makers often turn to computer models when they face a problem too “big” to grasp all at once. Having the computer keep track of specific values and calculations allows them to focus their
attention on accurately representing the issue at hand. The electronic spreadsheet was the first modeling application to gain widespread acceptance in the business community, and it is still the most
widely used analytical tool among managers. But spreadsheets, like all tools, have their limitations. So how can managers know when they are applying spreadsheet modeling to appropriate issues, and
when it is time to consider other modeling approaches?
Making Decisions in a Complex World
One criteria managers use when deciding whether to create a model of an issue is the degree of complexity involved. But there are at least two distinct types of complexity: detail complexity and
dynamic complexity. Detail complexity — which is what we usually mean when we speak of “complexity” — has to do with an abundance of things. Dynamic complexity, on the other hand, deals with an
abundance of inter-relationships among things that intertwine and impact one another over time.
Since people tend to associate detail complexity with complexity in general, it is commonly believed that: (1) a system with only a few components is easy to understand, and (2) a system that is
difficult to understand must have a tremendous amount of detail associated with it. It is often the case, however, that a system defined by a relatively low number of interrelated things has very
complex (and often counter-intuitive) behavior, and is more difficult to understand because of the level of dynamic complexity involved.
Decision-makers run into trouble whenever they use a tool designed to address detail complexity — such as a spreadsheet — to investigate dynamic complexity (see “By the Numbers” on p. 11).
Spreadsheets were primarily designed to replicate the simple function of accounting ledgers, not to help managers think about the dynamic complexity that confronts them. This is not to say, however,
that spreadsheets do not play a vital function in managerial decision-making.
A spreadsheet’s strength lies in its representation of linear mathematical relationships and in its ability to organize and relate data points — take this month’s number, add it to last month’s
cumulative total to get the year-to-date figure, divide that by last year’s year-to-date number for the same month to get the growth rate…and so on. This type of model building may be essential for
generating historical performance metrics—especially for financial reporting purposes — but it does not help managers come to any greater understanding of the dynamic complexity that confronts them.
The Trouble with Spreadsheets
When building a model using a spread-sheet, you enter numbers or formulas into cells. If you enter a number, the display shows…a number. If you enter a formula, the display shows… (typically) a
number yet again. Thus, parameter assumptions (numbers) are indistinguishable from relationship assumptions (formulas).
As a result, a model user may find it difficult to tell whether the relationships make any sense, because the formal relationship descriptions lie within the formulas entered into each cell, not the
calculation displayed by each cell. But even if you could see all the formulas at once, understanding the relationships they describe would still be over-whelming, since the formulas usually
reference other cells. It’s not “Revenue= Price * Quantity,” but “D22 = AF15 * Q8.”
By their very nature, then, spreadsheet models emphasize numerical input and output. Beneath those numbers — and harder to get at — are the logical relationships between the numbers. And buried most
deeply is the conceptual architecture of the model. Since you can’t tell which cells are parameter assumptions and which ones are relationship assumptions, the distinction is usually lost in an
ambiguity of assumptions (see “Spreadsheet Models” on p. 9).
System Dynamics Models
In a system dynamics modeling effort, the inter-relationships are foremost. Unlike spreadsheets, which present the numbers and hide the relationships, system dynamics models present the relationships
and keep the numbers in the background. The numbers and calculations can be accessed whenever needed, of course, but they do not get in the way of thinking about the problem (see “System Dynamics
Whereas the assumptions underlying a spreadsheet model are often difficult to pin down, system dynamics models are designed to capture visually the assumptions about how elements in a system
interrelate. The relationships among the variables are represented in the model-building process using a graphical interface, which presents the model user with different graphical icons of the
assumed parameters and relationships, thus making the assumptions explicit. When reviewing the model with someone else, the graphic representation of the assumptions easily and naturally focuses the
discussion on the structure of the problem, not just the numerical output.
It is this natural emphasis on making assumptions explicit and then testing them, improving them, and sharing them with others that improves and builds confidence in the model. The old saying “two
heads are better than one” takes on real meaning when multiple managers can effectively contribute to a model’s construction. It also provides a link to the broader context of organizational
learning. Getting traditional, quantitatively oriented managers to talk about their assumptions and inquire into other viewpoints is a good first step in getting them to think more broadly about the
assumptions they make every day, how those assumptions impact their behavior, and how both behavior and assumptions can create barriers to organizational learning.
Enhanced Decision-Making for a Dynamic World
System dynamics modeling distinguishes itself from spreadsheet modeling primarily because of its impact on a manager’s thinking. Technically, spreadsheet software can be used to build a
stock-and-flow model, just as system dynamics software can be used to create spreadsheets. But depending on the problem, one type of modeling approach will provide better direction than the other.
Static issues such as “How are we performing today?” or comparisons of static issues such as “How is our performance compared to last year’s?” suit themselves to spreadsheet analysis. Dynamic issues
such as “How is our performance changing over timer’ suit themselves to system dynamics modeling, because the modeling process encourages the person to think about the system structure and ask
questions that will make those structures explicit.
Although the structural language of stocks and flows takes time to learn, once one becomes conversant in it the ease of communications and transfer of systems models increases dramatically — just as
it did when people were first familiarizing themselves with spreadsheets.
Know the Purpose
The bottom line in choosing a modeling approach is to be clear about the purpose of a model before you build it—especially whether the situation to be modeled contains detail complexity or dynamic
complexity. Using a spreadsheet to address a problem containing dynamic complexity can lead to ineffective or erroneous decision-making, because although spreadsheet models are very effective for
capturing metrics, they have limited capability to help managers understand the dynamic implications of decisions over time.
It is always important to know where the company stands relative to its performance indicators, and metrics are great for providing such static pictures of an organization. But to succeed in
turbulent and changing times, managers must also invest in forward-looking models that provide a greater understanding of their organization and its environment.
Gregory Hennessy Is an associate at GKA Incorporated. He holds a master’s in management from the MIT Sloan School of Management and a master’s in social sciences from the California Institute of
Technology. He has worked primarily in the telecommunications, healthcare and energy industries, and has built spreadsheet, econometric, and system dynamics models.
Editorial support for this article was provided by Colleen Lannon.
By the Numbers
The rise of spreadsheets has created several numerical illusions for decision-makers:
Illusion of Accuracy. The first is the illusion of accuracy in quantitative models, arising from a confusion between accuracy and precision. In the world of computer models, “accuracy” is the extent
to which a model represents reality, whereas “precision” refers to the number of significant digits in the model’s output. It is possible to have an accurate model without much precision (“My model
indicates that it takes about 8 minutes for light from the sun to reach the earth”), just as it is possible to have a precise model without much accuracy (“My model indicates that it takes 2389953
months for light from the sun to reach the earth”). Because spreadsheets blindly calculate numbers to several decimal places without regard to the significance of such precision, decision-makers have
grown accustomed to model output with unreasonable precision. The psychological impact is that this impressive display of precision lulls decision-makers into overconfidence in the accuracy of the
models they are using.
illusion of Reduced Complexity. The second illusion is that of reduced complexity. As spreadsheets proliferated, so did the emphasis on the practice of management “by the numbers.” The abundance of
quantitative analysis led to a focus on any of a handful of metrics to reach a decision: IRR, ROE, NPV, RONA, ROI. These and other common metrics usually leave out qualitative considerations, since
qualitative issues are too difficult to measure or too imprecise to include. So what is put into the model are only those nice, tidy considerations that are easily quantifiable and measurable. If a
concern is hard to quantify and measure, and if it’s not being used in the decision-making process, managers are unlikely to spend much time thinking about it. It’s not surprising, therefore, to find
that many managers have grown accustomed to thinking about their challenges in a “simplified” world.
illusion of Reduced Risk. Together, these two illusions create a third—the illusion of reduced risk. A management team that has dramatically simplified its representation of the organization and
business environment, and is over-confident about the accuracy of its computer models, is likely to be overconfident about its ability to manage the uncertainties it faces. One manifestation of this
is that the team might ask the planning department to do the impossible—to predict the future with point-to-point accuracy and precision, taking into account all of the uncertainties (appropriately
weighted, of course) that face the organization. To counter this dangerous trend, scenario planning has emerged as one process for forcing managers to consider a broad range of risks. | {"url":"https://thesystemsthinker.com/from-spreadsheets-to-system-dynamics-models/","timestamp":"2024-11-09T06:10:19Z","content_type":"text/html","content_length":"96019","record_id":"<urn:uuid:68fa426f-fe8f-40ca-91bd-90ff2530dba8>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00343.warc.gz"} |
Quickest way to compute the Abel function on the Shell-Thron boundary
04/21/2022, 01:52 AM (This post was last modified: 04/21/2022, 05:22 AM by JmsNxn.)
I'm currently writing a protocol to evaluate the modified bennet operators:
\alpha <s>_{\varphi} y = \exp_{y^{1/y}}^{\circ s}\left(\log^{\circ s}_{y^{1/y}}(\alpha) + y + \varphi\right)\\
For \(\varphi\) a complex number--I'm mostly just dealing with real positive values at the moment. The goal is to evaluate the function \(\varphi(\alpha,y,s)\), such that these operators will satisfy
Goodstein's equation:
\alpha <s> \left(\alpha <s+1> y\right) = \alpha <s+1> (y+1)\\
But for the moment, I'm just concerning myself with calculating the first function. Everything works great so far, but I'm scratching my head for when \(y^{1/y} = \partial \mathfrak{S}\)-- when it's
on the boundary of the Shell-Thron region (equivalently \(|\log(y)| = 1\)). Now I know we can construct a repelling and attracting Abel function about these points--and I know all the theory. But I
just realized, I've never actually seen a program that constructs it. I know Sheldon has a program for handling it, but I really don't want to go digging through all the matrix theory. I just want a
quick formula. I know if you make a variable change that it becomes pretty elementary.
So for the moment, I can construct \(\alpha <s>_{\varphi} y\), for pretty much the entire complex plane in \(y\) (excluding branch cuts), excluding where \(y^{1/y} \in \mathfrak{S}\). This is
primarily because I don't know a fast way to get both abel functions... I could program in a way, but I think it's going to be way too slow. This program is already pretty slow as it is (we have to
consistently reinitialize to account for varying bases of the exponential). I don't want to slow it down any more. I was wondering if there's anywhere on this forum that has an easy to read program I
can adapt for this.
...I just hope I don't have to write too much just to handle the case \(|\log(y)| = 1\) -_-....
Edit: I thought I'd add that I know how to write in the neutral case but it just nukes the speed of my code. I know how to program in the \(\eta\) case, but I'm wondering what the current fastest way
is. For the moment, I'm just returning \(0\) anywhere on the boundary, because it just nukes my code and makes everything so fucking slow for these values of \(y\).
10/10/2023, 05:17 AM
The function is so much of a Riemann sum generator that the quick and easy formula would be the one with a logarithm at the top of your post but multiplied by a P function variable. Multiplying the
top gets you past where I can see you are not getting how to add Riemann sums (specifically after a zeration or two. That's what the P does, multiplies the parameters to get two sums to retrieve a | {"url":"https://tetrationforum.org/showthread.php?tid=1388&pid=12186","timestamp":"2024-11-10T08:52:23Z","content_type":"application/xhtml+xml","content_length":"28699","record_id":"<urn:uuid:a3003059-41fa-432e-9b2e-9106e645ef53>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00122.warc.gz"} |
co-Yoneda lemma
I think jokes enshrined in mathematical terminology get old quickly and then stay old forever. I vote for not promoting this.
I completely agree.
For what it’s worth, this claim looks incorrect:
In this MO answer, Tom Leinster referred to the co-Yoneda lemma as ninja Yoneda lemma
On the contrary, he says that “ninja category theorists” would call it just the plain “Yoneda lemma”.
I think jokes enshrined in mathematical terminology get old quickly and then stay old forever. I vote for not promoting this.
mention density formula and ninja Yoneda
diff, v24, current
At co-Yoneda lemma I have tried to harmonize the notation and polish the formatting a little. For instance, earlier the statement had been in terms of $V$-enrichment, but then the proof was stated in
terms of $Set$-enrichment, I have harmonized that.
Then I added as an Example an elementary proof of the co-Yoneda lemma in $Set$ in terms of inspecting the defining coequalizer as a set of equivalence classes of pairs.
I took the liberty of deleting the “ninja”-section.
Because, first, it’s claim that “Tom Leinster referred to the co-Yoneda lemma as ninja Yoneda lemma” was just false, and second its mathematical content was a direct repetition of the material in
this entry.
(There may be room to state this material more clearly/concisely, but repeating it as if it were a different statement is confusing.)
diff, v26, current | {"url":"https://nforum.ncatlab.org/discussion/7138/","timestamp":"2024-11-04T21:00:40Z","content_type":"application/xhtml+xml","content_length":"19444","record_id":"<urn:uuid:a7af2ffe-a845-4248-b8dc-33874dfa9d90>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00260.warc.gz"} |
CS 202 Homework #1 – Algorithm Efficiency and Sorting
Question 1 – 30 points
(a) [11 points] Sort the functions below in the increasing order of their asymptotic
complexity (Use < and = signs during ordering. For example, if the asymptotic complexity of fx(n)=o(fy(n)), i.e., fx(n) is “less than” fy(n), use fx(n) < fy(n); if they are equal, i.e., fx(n)=(fy
(n)), use fx(n) = fy(n)): f1(n) = nlog(logn), f2(n) = n, f3(n) = n3 , f4(n) = n1/logn, f5(n) = n logn, f6(n) = (logn) logn, f7(n) = en , f8(n) = log 2 n, f9(n) = log n, f10(n) = 2logn , f11(n) = n!
(b) [9 points] Find the asymptotic running times (in notation, tight bound) of the following recurrence equations by using the repeated substitution method. Show your steps in detail. T(n) = 9 T
(n/3) + n2 , T(1) = 1 where n is an exact power of 3 T(n) = T(n/2) + 2, T(1) = 1 where n is an exact power of 2 (c) [10 points] Trace the following sorting algorithms to sort the array [ 4, 9, 7,
3, 5, 2, 1, 6 ] in ascending order. Use the array implementation of the algorithms as described in the textbook and show all major steps (after each sort pass for instance). Insertion sort Bubble
sort Question 2 – 50 points You are asked to implement the selection sort (10 points), merge sort (10 points), and quick sort (10 points) algorithms for an array of integers and then perform the
measurements as detailed below. For each algorithm, implement the functions that take an array of integers and the size of the array and then sort it in non-ascending (descending) order. Add two
counters to count and return the number of key comparisons and the number of data moves during sorting. For key comparisons, you should count each comparison like “k1 < k2” as one comparison, where
k1 and k2 correspond to the value of an array entry (that is, they are either an array entry like arr[i] or a local variable that temporarily keeps the value of an array entry). For data moves, you
should count each assignment as one move, where either the right-hand side of this assignment or its left-hand side or both of its sides correspond to the value of an array entry. For example, the
following swap function has three such assignments (and thus three data moves): void swap(DataType &x, DataType &y) { DataType temp = x; x = y; y = temp; } For the quick sort algorithm, you are
supposed to take the first element of the array as the pivot. After implementing the sorting algorithms, implement a function named performanceAnalysis (20 points) which does the followings: 1.
Create three identical arrays with random 1000 integers using the random number generator function rand. Use one of the arrays for the selection sort, another one for the merge sort, and the last one
for the quick sort algorithm. Output the elapsed time in milliseconds, the number of key comparisons, the number of data moves (use clock from ctime for calculating elapsed time). 2. Now, instead of
creating arrays of random integers, create arrays with elements in ascending order and repeat the steps in part 1. 3. Now, instead of creating arrays of random integers, create arrays with elements
in descending order and repeat the steps in part 1. 4. Lastly, instead of creating arrays of random integers, create arrays with elements in ascending order up to its half size and in descending
order in the rest, and repeat the steps in part 1. 5. Repeat the experiment (parts 1-4) for the following array sizes: {6000, 12000, 18000}, given as input to performanceAnalysis (total of four
different sizes). When the performanceAnalysis function is called, it needs to produce an output similar to the following one: Performance analysis for arrays of size 1000
------------------------------------------------------------------------------------------- Random integers Elapsed time compCount moveCount Selection sort Merge sort Quick sort
------------------------------------------------------------------------------------------- Ascending integers Elapsed time compCount moveCount Selection sort Merge sort Quick sort … Performance
analysis for arrays of size 6000 ------------------------------------------------------------------------------------------- Random integers Elapsed time compCount moveCount Selection sort Merge sort
Quick sort ... Put the implementations of these functions in a file named sorting.cpp, and their interfaces in a file named sorting.h. Also write a main function separately inside a file named
main.cpp that calls only the performanceAnalysis function inside. Although you will write your own main function to get the experimental results, we will also write our own main function to test
whether or not your algorithms work correctly. In our main function, we will call your sorting algorithms with the following prototypes: void selectionSort( int *arr, int size, int &compCount, int &
moveCount ); void mergeSort( int *arr, int size, int &compCount, int &moveCount ); void quickSort( int *arr, int size, int &compCount, int &moveCount ); void performanceAnalysis( int size ); In all
of these prototypes, arr is the array that the algorithm will sort, size is the array size, compCount is the number of key comparisons in sorting, and moveCount is the number of data moves in
sorting. After returning from this function, arr should become sorted. IMPORTANT: At the end, write a basic Makefile which compiles all your code and creates an executable file named hw1. Check out
these tutorials for writing a simple make file: http://mrbook.org/blog/tutorials/make/, http://www.cs.colby.edu/maxwell/courses/tutorials/maketutor/. Question 3 – 20 points After running your
programs, you are expected to prepare a 2-3 page report about the experimental results that you obtained in Question-2. First, prepare tables for presenting the results for the number of key
comparisons, the number of data moves, and the elapsed time. You should prepare a separate table for each required number. For each table, each row should include the type of the input (e.g., R1K -
array with 1000 random integers, A1K - array with 1000 ascending integers, D1K - array with 1000 descending integers, M1K - array with 500 ascending and 500 descending integers, etc.) and the values
obtained by selection sort, merge sort and quick sort in four separate columns. Then, with the help of a spreadsheet program (Microsoft Excel, Matlab or other tools), present your experimental
results graphically. Interpret and compare your empirical results with the theoretical ones for each sorting algorithm. Explain any differences between the empirical and theoretical results, if any. | {"url":"https://codingprolab.com/answer/cs-202-homework-1-algorithm-efficiency-and-sorting/","timestamp":"2024-11-02T15:19:41Z","content_type":"text/html","content_length":"116577","record_id":"<urn:uuid:cf1ef25d-d748-420a-8546-de14560304ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00601.warc.gz"} |
Derive expression for terminal velocity when a ball of radius $r$ is dropped through a liquid of viscosity $\
The constant value of speed that a freely falling body acquires, when the resistance of the medium through which it is falling prevents further acceleration of the body, is called as terminal
velocity of the body. In order to calculate the terminal velocity of a body in a liquid of density $\rho $, we need to apply Stoke’s law and balance the downward gravitational force of the body with
the sum of upthrust and viscous force acting on the body.
Complete step-by-step answer:
Terminal velocity is defined as the highest velocity attained by a body that is falling through a fluid. It is observed when the sum of drag force and buoyant force becomes equal to the downward
gravitational force that is acting on the body. The acceleration of the body is zero as the net force acting on the body is zero.
Viscous drag or viscous force acts in the direction opposite to the motion of the body in the fluid. According to Stoke’s law, the magnitude of the opposing viscous drag increases with the increasing
velocity of the body.
As the body falls through a medium, its velocity goes on increasing due to the force of gravity acting on it. Thus, the opposing viscous or drag force, which acts upwards, also goes on increasing. A
stage reaches when the true weight of the body is just equal to the sum of the upward thrust due to buoyancy, that is, upthrust, and the upward viscous drag. At this stage, there is no net force to
accelerate the moving body. Hence it starts falling with a constant velocity, which is called terminal velocity.
Let ${{\rho }_{o}}$ be the density of the material of the spherical body, ball of radius $r$ and $\rho $ be the density of the medium.
True weight of the body is given as,
$W=\text{Volume}\times \text{Density}\times g$
Volume of spherical body is, $V=\dfrac{4}{3}\pi {{r}^{3}}$
$r$ is the radius of the ball
$W=\dfrac{4}{3}\pi {{r}^{3}}{{\rho }_{o}}g$
$r$ is the radius of the ball
${{\rho }_{o}}$ is the density of the material of the ball
Let’s say ${{F}_{T}}$ is the weight of the liquid displaced
${{F}_{T}}=\text{Volume of liquid displaced }\times \text{ Density }\times g$
${{F}_{T}}=\dfrac{4}{3}\pi {{r}^{3}}\rho g$
$r$ is the radius of the ball
$\rho $ is the density of the material
$g$ is the acceleration due to gravity
If $v$ is the terminal velocity of the body, then according to Stoke’s law,
Upward viscous drag is given as,
${{F}_{V}}=6\pi \eta rv$
$r$ is the radius of the ball
$\eta $ is the viscosity of the liquid
When the body acquires terminal velocity,
Putting values,
& {{F}_{T}}=\dfrac{4}{3}\pi {{r}^{3}}\rho g \\
& {{F}_{V}}=6\pi \eta rv \\
& W=\dfrac{4}{3}\pi {{r}^{3}}{{\rho }_{o}}g \\
We get,
& \dfrac{4}{3}\pi {{r}^{3}}\rho g+6\pi \eta rv=\dfrac{4}{3}\pi {{r}^{3}}{{\rho }_{o}}g \\
& 6\pi \eta rv=\dfrac{4}{3}\pi {{r}^{3}}\left( {{\rho }_{o}}-\rho \right)g \\
& v=\dfrac{2{{r}^{2}}\left( {{\rho }_{o}}-\rho \right)g}{9\eta } \\
The terminal velocity acquired by the ball of radius $r$ when dropped through a liquid of viscosity $\eta$ and density $\rho$ is, $v=\dfrac{2{{r}^{2}}\left( {{\rho }_{o}}-\rho \right)g}{9\eta }$.Note
:Terminal velocity is defined as the maximum velocity attained by a body as it falls through a fluid. It occurs when the sum of the drag force and the buoyancy, or upthrust, is equal to the downward
gravitational force acting on the body. Since the net force on the body is zero, the body has zero acceleration. The terminal velocity is directly proportional to the square of the radius of the body
and inversely proportional to the coefficient of viscosity of the medium. It also depends upon densities of the body and the medium. | {"url":"https://www.vedantu.com/question-answer/derive-expression-for-terminal-velocity-when-a-class-11-physics-cbse-5f5bf4a268d6b37d16e3b6f5","timestamp":"2024-11-12T12:44:10Z","content_type":"text/html","content_length":"180967","record_id":"<urn:uuid:b8bdccaf-2ced-4a71-bfae-68db3b577b1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00378.warc.gz"} |
Empirical Research: Emerging Research: Learning by Teaching a Synthetic Student: Using SimStudent to Study the Effect of Tutor Learning
The purpose of the Self-Explanation study is to investigate whether prompting students to explain problem-solving steps made by themselves would facilitate tutor learning. In the proposed learning
environment, 8th grade students will be prompted to explain the reasoning behind their tutoring activities when teaching SimStudent. The proposed learning-by-teaching environment is designed for
Algebra students to learn two major types of knowledge, namely: procedural skills to manipulate algebraic expressions and solve equations; and conceptual knowledge to justify skill applications when
solving problems. The research question is: Does asking students to provide explanations for their reasoning behind the tutoring activities facilitate tutor learning? The hypothesis to be tested is:
If students provide explanations on their reasoning behind the tutoring, then the effect of tutor learning will be facilitated. Students will be randomly assigned to one of the two experimental
conditions: the treatment condition where the students will be prompted to provide explanations on their tutoring activities; and the control condition where the students will use the base-line
Learning-by-Teaching environment. Pre- and post-tests will be used to measure students' learning achievement in conceptual and procedural knowledge. Student comparisons using the learning gain as the
dependent variable will be the focus of analysis.
The study focuses on Algebra I linear equation solving, which is a critical area in the middle school algebra curriculum as indicated in both the Principles and Standards for School Mathematics
(PSSM) publication of the National Council of Teachers of Mathematics (NCTM) and the more recently released document Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. It is
well known that prompting students to self-explain facilitates learning both when they are asked to explain correct worked-out examples and when they are asked to explain errors made by others. The
study tests the hypothesis that self-explanation is also effective in tutor learning. Students will be asked to explain: correct steps when they demonstrate steps to their SimStudent; incorrect steps
when they catch SimStudent making an error and want to indicate why the step is wrong; and 3) their choice of problems to teach to SimStudent, for instance, based on the observation on what mistakes
SimStudent make when given a quiz.
Project Report
The goal of this project is to investigate cognitive and social theories of learning by teaching. There is ample evidence in the literature that both a tutor and a tutee benefit from peer learning,
especially even when the peers are at the same proficiency levels. However, the underlying cognitive and social theory of tutor learning has yet to be investigated. To achieve the project goal, we
have developed a synthetic peer, called SimStudent, by applying an artificial intelligence (machine learning) technology. SimStudent learns skills to solve problems when tutored. We then developed an
online, game-like learning environment, APLUS—Artificial Peer Learning environment Using SimStudent, in which students learn to solve Algebra linear equations by teaching SimStudent. APLUS is
equipped with a set of quiz problems. The goal of the student using APLUS is to tutor SimStudent well so that it passes the quiz. APLUS also has resources for students to learn about Algebra
equations to better prepare for teaching SimStudent. To advance theory of learning by teaching, we have conducted four classroom (in vivo) experiments each to test a specific hypothesis. These
hypotheses included (1) the engineering hypothesis that conjectures that our software intervention would be robust enough for a classroom study with expected effect in students’ learning, (2) the
self-explanation hypothesis that conjectures that tutor learning would be facilitated when tutors were asked to explain their tutoring decisions and activities, (3) the motivation hypothesis that
conjectures that tutor learning would be facilitated when students were more engaged in tutoring, and (4) the meta-tutor hypothesis that conjectures that tutor learning would be facilitated when
tutors were provided scaffolding on how to tutor. All four classroom studies were randomized controlled trials conducted in one private and four public schools. Study sessions were placed as part of
each participating school’s normal Algebra I class activities. Students used APLUS for three days, one classroom period per day. Students took pre- and post-tests before and after the intervention.
The tests were designed to measure proficiency to solve equations (procedural skill test) and understanding of basic algebraic concepts (conceptual knowledge test). Highlights of the study results
include following: (1) The first hypothesis, the engineering hypothesis, was supported. All four classroom-studies have successfully demonstrated that our technology intervention (APLUS and
SimStudent) is robust and reliable enough for classroom usage. It has been also demonstrated that our technology allows us to collect detailed process data that, when combined with learning outcome
data (i.e., test scores and questionnaire responses), facilitate theory development. (2) The second hypothesis, the self-explanation hypothesis, was supported with an insight into future study. The
data showed that the amount of "deep" responses for SimStudent questions had a statistically reliable predictive power for the post-test scores. In the classroom studies, SimStudent did not actually
process students’ responses, but merely acknowledge it. Therefore, only about 20% (N=2008) of students’ responses were "deep" responses. These findings suggested further extension of the
SimStudent’s questioning module so that it understands students’ input and provide follow-up dialogue (e.g., "Hmm, I still don't understand -- could you explain this to me in more detail?"). (3)
The third hypothesis, the motivation hypothesis, was partially supported. The data showed that the introduction of the competitive Game Show, in which a pair of SimStudents competes each other by
solving challenging problems and students observe the competition, increased students’ engagement tutoring, measured by the amount and depth of students’ response to SimStudent’s questions.
However, the tutoring engagement showed no correlation with post-test scores. It turned out that once students entered in the Game Show, they did not switch back to tutoring SimStudent to make it
more proficient, but stayed in the Game Show and strategically selected weak opponents for an easy win. The result suggested that the Game Show must be redesigned so that the learning goal (i.e., to
tutor SimStudent better) and the Game Show goal (i.e., to win the competition) are better aligned. (4) The fourth hypothesis, the meta-tutor hypothesis, was supported. The meta-tutor’s scaffolding
on how to select problem positively affected students’ decisions on what problems to be used for tutoring, which further affected tutor learning. The data showed that the more advanced the type of
equations tutored were and the quicker the transition from easy to advance types of problem occurred, the higher the post-test scores were. The broader impact of the project includes (1) the
advancement of the cognitive and social theory of learning by teaching, (2) the wide dissemination of the developed online learning environment that are freely available on our project web site, and
(3) the shared data we have collected from classroom studies (through the PSLC’s DataShop -- www.learnlab.org) for secondary analyses. | {"url":"https://grantome.com/grant/NSF/DRL-0910176","timestamp":"2024-11-05T10:43:57Z","content_type":"text/html","content_length":"35291","record_id":"<urn:uuid:f868d508-5791-42e9-821c-572de0a9cfb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00355.warc.gz"} |
Algebra 1 Staar Test 2023 Review GuideAlgebra 1 Staar Test 2023 Review Guide
Algebra 1 Staar Test 2023 Review Guide
STAAR EOC Algebra 1 Task Cards A.7A Analyzing Quadratics Quadratics from www.pinterest.com
Welcome to the Algebra 1 STAAR Test 2023 Review Guide! This article aims to provide you with comprehensive knowledge and preparation tips for the upcoming Algebra 1 STAAR test in 2023. The STAAR test
is a standardized test used to measure the academic performance of students in Texas and is an essential requirement for graduation. Therefore, it is crucial to be well-prepared for the test to
achieve a passing score.
Understanding the Test
The Algebra 1 STAAR test is designed to assess students’ skills and knowledge in algebraic concepts such as linear equations, functions, and quadratic equations. The test consists of two parts, a
multiple-choice section and a short-answer section. The multiple-choice section contains 50 questions, and the short-answer section contains 5 questions. The test is time-limited, and students have
four hours to complete it.
Test Preparation Tips
To excel in the Algebra 1 STAAR test, students need to start preparing early. Here are some tips to help you prepare for the test:
1. Know the Test Format
Understanding the test format is crucial in preparing for the Algebra 1 STAAR test. Knowing the number of questions, the time limit, and the test sections’ content will help you strategize your
preparation effectively.
2. Practice, Practice, Practice
Practice makes perfect! Students should practice as many algebra problems as possible to familiarize themselves with the test’s content. There are multiple resources available, such as textbooks,
online tutorials, and practice tests.
3. Identify Weaknesses
It is essential to identify your weaknesses in algebraic concepts and focus on improving them. You can do this by taking practice tests or seeking help from your teachers.
4. Study Regularly
Studying regularly is better than cramming. Allocate time for studying every day, and ensure you cover all the essential topics.
5. Get Enough Rest
Getting enough rest before the test is crucial. Ensure you have a good night’s sleep and avoid studying all night before the test.
Test Day Tips
Here are some tips to help you perform well on the day of the test:
1. Arrive Early
Arrive at the test center early to avoid rushing and ensure you have enough time to settle in.
2. Read Instructions Carefully
Read the instructions carefully before starting the test to avoid making careless mistakes.
3. Manage Your Time
Manage your time effectively during the test. Work on the questions you are confident about first and leave the difficult ones for later.
4. Show Your Work
Show your work for every question, even if it is a multiple-choice question. This will help you get partial credit if your answer is incorrect.
Preparing for the Algebra 1 STAAR test can be overwhelming, but with the right strategies, students can excel in the test. By following the tips outlined in this article, you can be well-prepared and
confident on the day of the test. Remember to practice regularly, identify your weaknesses, and manage your time effectively. Good luck! | {"url":"https://myans.bhantedhammika.net/algebra-1-staar-test-2023-review-guide/","timestamp":"2024-11-12T08:59:25Z","content_type":"text/html","content_length":"134017","record_id":"<urn:uuid:f334df83-dfb4-4b79-b612-41f938b649e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00890.warc.gz"} |
Tracing out sites inbetween two four site subsets of a 1D-quantum-spin system
I have calculated the ground state and its energy using the implemented DMRG algorithm.
Is there a possibility to trace out the sites inbetween subsets A and B of an open 1D quantum spin chain. In my specific case A and B consist of each 4 spins on the edges of the chain.
I would like to calculate the entropy of these two subsets combined (S_{AB}).
Thanks, Mathias
(I am using Julia with ITensors v0.2.3)
The documentation I've found for this is for the C++ code, so you'd need to translate it to Julia: http://www.itensor.org/docs.cgi?page=formulas/mps_two_rdm&vers=cppv3
For your purpose, you'd need to replace the i and j with code for the portion of the reduced density matrix for blocks A and B, but summing over the intermediate sites and external sites works the
same as in that (you should gauge the MPS to a point in the space of interest, say the leftmost site of region A).
The questioner here https://www.itensor.org/support/229/evaluate-block-entanglement-blocks-that-extend-the-lattice?show=230#a230 has possibly not the best motivations, but the content from Miles is
still really useful.
Another question on this that's in my bookmarks: https://www.itensor.org/support/1907/entanglement-entropy-of-a-finite-block-of-a-spin-chain?show=1907#q1907
If your reduced density matrix can be well-approximated with a product state of the separate reduced density matrices of A and B, then do that. If you're looking for something like the mutual
information of A and B, this can be a very long calculation in my naive attempts to code it. For the Kitaev Wire model (a very simple model), the code stalled out (took too dang long) at L=23 sites
because of the joint density matrix needed for mutual information calculations on my Linux laptop. Something that I did not test with the Kitaev model is reducing the number of bonds to make such
calculations more feasible - my last edit of that code has the final maxBondDimension = 250 and a cutoff = 1E-10, it could certainly be tweaked and perhaps find better results (or perhaps make a copy
to post-process and reduced in size to capture the entropy but maybe a less-accurate energy... Untested by me).
Miles has mentioned some sampling method to approximate the entanglement entropy for the reduced joint density matrix of AB, but to my knowledge he hasn't shared that code. I'm not sure what other
tricks can be done with MPSs for things on opposite ends of the lattice.
Best of luck,
Edit: assuming you have qubits/two-state local spaces, that gives a 2^8=256 for the local space size for the reduced density matrix of AB - that should be within the realm of calculation.
Hi Jared,
thanks for your answer and all the useful inputs and links! That helps a lot.
There's one point that still worries me: So far as I have understood it, I need to diagonalize rho = psi^2 of the sites I would like to trace out. This will scale exponentionally with the length of
the spin chain, since the amount of sites I would like to keep is constant(?).
I guess you've encountered that in the Kitaev Wire model and Miles mentioned it in one of his answers as well.
I will implement it anyways and see how it performs.
Thanks and kind regards,
Hello Mathias, it *can* be a little better, but it's not something that I've found a clean answer to. I'll post an official answer below shortly to get indentation correct (to make it more readable),
but it'll be for the C++ version... Not ideal, but I'm not currently able to take the extra time for the Julia side.
Here is some mildly adjusted code from some C++ code that would need to be translated to Julia. It's posted as an answer to get the code portion to be nicely formatted.
The main trick is that assuming you have a globally pure state for a system divided into three regions [A|B|C], and you'd like to compute the entanglement entropy of AC, then you can also look at
just B, and take whichever problem is smaller since the Schmidt values are the same inside/outside. This does produce a few if-thens, but it speeds up calculations.
• La, Lb, Lc are the lengths of regions A,B,C (La=Lc for convenience)
• Bl, Br are the leftmost and rightmost site indices of the interior region, B.
• Lp1by1m1 is just a way of calculating how many non-trivial ways we can divide the lattice into [A|B|C] such that La=Lc.
While, it does seem to indicate the topological phase transition in the Kitaev wire model for initial testing, I've not tested this thoroughly, so there could certainly be errors.
Here's the snippet of code:
// Define Lattice Length
int L = 20 ;
int Lp1by2m1 = (L+1)/2 - 1 ; // need integer division here. Use a floor function if needed.
println("L is \n");
std::cout << L << " \n ";
println("Lp1by2m1 is \n");
std::cout << Lp1by2m1 << " \n ";
int NumberOfStates = 5;
float SvN1array[NumberOfStates][L-1];
float SvN1fracfoundarray[NumberOfStates][L-1];
float SvN2array[NumberOfStates][Lp1by2m1];
float SvN2fracfoundarray[NumberOfStates][Lp1by2m1];
float MIarray[NumberOfStates][Lp1by2m1];
// Your code to find the states of interest here
// Entanglement spectra & related quantities
// Single-cut entanglement entropies (bi-partite)
float SvN1vec[L-1];
float SvN1fracfound[L-1];
// Single-cuts are very fast. May as well compute the entanglement entropy for all single-interior cuts across the lattice
for( int ii : range1(L-1))
// Gauge the MPS to site ii
// Define the two-site wavefunction for sites (ii,ii+1)
//// with the bond between them being the bond for which you wish to compute entanglement entropy
auto lli = leftLinkIndex(psi,ii);
auto si = siteIndex(psi,ii) ;
auto [U1,S1,V1] = svd(psi(ii),{lli,si}) ;
auto u1 = commonIndex(U1, S1);
double SvN1 = 0.0 ;
double fractionfound = 0.0 ;
for( auto n : range1(dim(u1)) )
auto Sn = elt(S1,n,n);
auto p1 = sqr(Sn) ;
if( p1 > 1E-12) {
SvN1 += -p1*log(p1);
fractionfound += p1;
SvN1vec[ii-1] = SvN1;
SvN1fracfound[ii-1] = fractionfound;
SvN1array[nn][ii-1] = SvN1;
SvN1fracfoundarray[nn][ii-1] = fractionfound;
// Fraction found
//// sum(spectrum.eigs())
// Linear Entanglement Measure
//// maxval(spectrum.eigs())
// Separability
//// - spectrum.eigs *dot* log(spectrum.eigs() )
// Second Renyi
//// - log( spectrum.eigs ^ 2 )
// Minimum entropy -- (infinite order) Renyi entropy
//// - log (separability)
// Two-cut entanglement entropies for systems divided into regions [A|B|C]
///// Want to calculate S(AC) such that the lengths of A and C are the same
//// (still bi-partite, but with a less intuitive basis, not as efficient with MPS methods)
//// Able to be done w/ usual Schmidt decomp. if starting with a pure state across the lattice
////// Entanglement spectrum of a pure state is the same for AC and complement of AC (that is, region B)
////// Take the space of AC or B, and solve whichever problem is smaller
float SvN2vec[Lp1by2m1];
float SvN2fracfound[Lp1by2m1];
double evalcutoff = 1E-2;
// // Uncomment the for loop if wanting to loop over a range of lengths of region A
// for( int La : range1(Lp1by2m1))
// {
int Lc = La;
int Bl = La + 1; // Bleft - leftmost site of region B
int Br = L - Lc;
int Cl = Br + 1;
int Cr = L ;
double SvN2 = 0.0 ;
double fractionfound = 0.0 ;
// Remove any primes previously applied
if ( Bl != Br ) {
if ( Br-Bl < L/2 ) {
// construct Psi for region B, then make density matrix
// Gauge the MPS to site Bl
auto psiB = psi(Bl) ;
// Loop over sites between Bl and Br, exclusive
for(int ii = Bl+1; ii < Br; ++ii)
psiB *= psi(ii);
psiB *= psi(Br) ;
auto rhoB = prime(psiB,"Site") * dag(psiB);
valcutoff = 1E-4;
auto [Q,D] = diagHermitian(rhoB, {"Cutoff=",evalcutoff,"ShowEigs=",false, "Truncate=", true});
// This is of rho, not just psi! No need to square!!!
auto u2 = commonIndex(Q, D);
// println("dim(u2) is \n") ;
// std::cout << dim(u2) << " \n ";
fractionfound = 0.0;
for( auto n : range1(dim(u2)) )
auto p2 = elt(D,n,n);
auto p2 = sqr(Sn) ;
if( p2 > evalcutoff) {
SvN2 += -p2*log(p2);
fractionfound += p2;
// println("fraction found so far is \n");
// std::cout << fractionfound << " \n ";
SvN2vec[La-1] = SvN2;
SvN2fracfound[La-1] = fractionfound;
SvN2array[nn][La-1] = SvN2;
SvN2fracfoundarray[nn][La-1] = fractionfound;
} else { // Bl != Br and (Br-Bl) > L/2
// Gauge the MPS to site 1
// Remove any primes previously applied
auto psidag = dag(psi);
auto lLI = leftLinkIndex(psi,ii);
// Initiallize reduced density matrix to begin construction over subsystem A
auto rhoAC = prime(psi(ii),lLI) * prime(psidag(ii),"Site");
for(int ii : range1(L) ){
// Construct rho over region A
if( ii < Bl){
if (ii > 1) {
rhoAC *= psi(ii) * prime(psidag(ii),"Site");
} // Region A execpt site 1
} else if (ii < Cl) { // construct such that region B is summed over
rhoAC *= psi(ii) * psidag(ii);
} else { // ii >= Cl Write over region C
if(ii < Cr){
rhoAC *= psi(ii) * prime(psidag(ii),"Site");
} // C region Sites except for final site region C
if(ii == Cr){
auto rLI = rightLinkIndex(psi,ii);
rhoAC *= prime(psi(ii),rLI) * prime(psidag(ii),"Site");
} // Final site region C
} // A, B, C region if
} // ii dummy loop
valcutoff = 1E-4;
auto [Q,D] = diagHermitian(rhoAC, {"Cutoff=",evalcutoff,"ShowEigs=",false, "Truncate=", true});
// This is of rho, not just psi! No need to square!!!
auto u2 = commonIndex(Q, D);
// println("dim(u2) is \n") ;
// std::cout << dim(u2) << " \n ";
fractionfound = 0.0;
for( auto n : range1(dim(u2)) )
auto p2 = elt(D,n,n);
auto p2 = sqr(Sn) ;
if( p2 > evalcutoff) {
SvN2 += -p2*log(p2);
fractionfound += p2;
// println("fraction found so far is \n");
// std::cout << fractionfound << " \n ";
SvN2vec[La-1] = SvN2;
SvN2fracfound[La-1] = fractionfound;
SvN2array [nn][La-1] = SvN2;
SvN2fracfoundarray [nn][La-1] = fractionfound;
} // Bl != Br and (Br-Bl) > L/2
} else { // if Bl == Br // then just regular old single cut entanglement entropy
// Gauge the MPS to site Bl
auto rhoB = prime(psi(Bl),"Site") * dag(psi(Bl));
evalcutoff = 1E-4;
auto [Q,D] = diagHermitian(rhoB, {"Cutoff=",evalcutoff,"ShowEigs=",false, "Truncate=", true});
auto u2 = commonIndex(Q, D);
fractionfound = 0.0 ;
for( auto n : range1(dim(u2)) )
auto p2 = elt(D,n,n);
auto p2 = sqr(Sn) ;
if( p2 > evalcutoff) {
SvN2 += -p2*log(p2);
fractionfound += p2;
SvN2vec[La-1] = SvN2;
SvN2fracfound[La-1] = fractionfound;
SvN2array[nn][La-1] = SvN2;
SvN2fracfoundarray[nn][La-1] = fractionfound;
} // Bl = Br
// } // END FOR over La
// Put this in a loop if varying La
MIarray[nn][La-1] = SvN1array[nn][La-1] + SvN1array[nn][L-(La-1)-2] - SvN2array[nn][La-1] ;
} // nn state of interest loop
Hi Mathias,
I have posted some notes at the following link explaining how I would go about computing the reduced density matrix rho_AB for the A and B sites together:
Please let me know if you have any questions about the steps.
From that density matrix, one can also easily obtain rhoA and rhoB by just tracing either the B sites or the A sites. (Here is some example code about tracing an ITensor: https://itensor.github.io/
ITensors.jl/dev/examples/ITensor.html#Tracing-an-ITensor). Note that some of the steps in the notes called "tracing" really involve contracting two ITensors with each other, allowing certain indices
to contract while leaving some indices uncontracted, by priming or adjusting the tags of the ones you want left uncontracted so that they don't match.
Hope it is helpful - | {"url":"http://www.itensor.org/support/3249/tracing-sites-inbetween-four-site-subsets-quantum-spin-system","timestamp":"2024-11-15T03:49:20Z","content_type":"text/html","content_length":"45572","record_id":"<urn:uuid:2ff2ae7b-51bf-49f5-b9fa-07794fe5198e>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00839.warc.gz"} |
Robust Aggregate Implications of Stochastic Discount Factor Volatility
The stochastic discount factor seems volatile, but is this observation of any consequence for aggregate analysis of consumption, capital accumulation, output, etc.? I amend the standard frictionless
model of aggregate consumption and capital accumulation with time-varying subjective probability adjustments, and obtain four implications for aggregate economic analysis. First, subjective
probability adjustments add volatility to the stochastic discount factor, and can rationalize any pattern of asset prices satisfying no-arbitrage, even while capital accumulation is efficient.
Second, despite its flexibility in pricing assets, the model implies that, in expected value, the intertemporal marginal rate of transformation is equal to the intertemporal marginal rate of
substitution, and there is a simple, stable, and familiar relation between consumption growth and capital’s return. Third, the expected returns on assets in small net aggregate supply are weakly (and
sometimes negatively) correlated with capital’s expected return, and are thereby poor predictors of aggregate consumption growth. Fourth, when it comes to assets in small net aggregate supply,
capital gains reflect time varying risk premia, and returns can predict aggregate consumption growth better when the capital gain component of those returns is ignored. All four implications are
consistent with empirical results reported here, and in the previous literature documenting stochastic discount factor volatility. Several recent theories of stochastic discount factor volatility
can, from the aggregate point of view, be interpreted as special cases of subjective probability adjusted CCAPM.
© copyright 2003 by Casey B. Mulligan. | {"url":"https://economicreasoning.com/caseybmulligan/iessdf.html","timestamp":"2024-11-12T15:22:28Z","content_type":"text/html","content_length":"3012","record_id":"<urn:uuid:0c9e27c6-46de-4da7-8182-e1eb563a82d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00643.warc.gz"} |
Motivated by properties of tensor networks, we conjecture that an arbitrary gravitating region a can be assigned a generalized entanglement wedge E⊃a, such that quasi-local operators in E have a
holographic representation in the full algebra generated by quasi-local operators in a. The universe need not be asymptotically flat or AdS, and a need not be asymptotic or weakly gravitating. On a
static Cauchy surface Σ, we propose that E is the … Read More
Abstract: “I will report on a joint work in progress with Pablo BoixedaAlvarez, Michael McBreen and Zhiwei Yun where categories of microlocal sheaves on some affine Springer fibers are described in
terms of theLanglands dual group. In particular, in the slope 1 case we recover the regular block in the category of (graded) modules over the smallquantum groups. Assuming a general formalism … Read
The 21-cm cosmological signal is gradually becoming a reality, offering a new insight into previously under-explored epochs. As with other cosmological observations, it is intriguing to consider what
21-cm cosmology can teach us about new physics. To address this, I will provide a concise overview of the physics behind the 21-cm cosmological signal and the effects of various new physics … Read
Recent advances around Fukaya categories can be used to (mathematically rigorously) produce sheaves on Bun_G from smooth fibers of Hitchin fibrations. The resulting sheaves are presumably Hecke
eigensheaves; I’ll explain why I don’t know how to prove this, and discuss various related questions.
Abstract: In high density QCD, the phase structure is not understood well. We think about two kinds of phases, confinement and Higgs phases. Traditionally, they are considered to be the same.
However, recently, there is a new point of view that they can be distinguished by topological excitations. We found emergent higher form symmetry that characterizes confinement and Higgs phases with
superfluidity. … Read More
Abstract: Limits on the charged lepton flavor violating (CLFV) process of μ→e conversion are expected to improve by four orders of magnitude due to the next generation of experiments, Mu2e at
Fermilab and COMET at J-PARC. The kinematics of the decay of a trapped muon are ideal for detecting a signal of CLFV, but the intervening nuclear physics presents a significant roadblock to … Read
Abstract: Kontsevich and Soibelman suggested a correspondence between Donaldson-Thomas invariants of Calabi-Yau 3-folds and holomorphic curves in complex integrable systems. After reviewing this
general expectation, I will present a concrete example related to mirror symmetry for the local projective plane (partly joint work with Descombes, Le Floch, Pioline), along with applications in
enumerative geometry (partly joint work with Fan, Guo, … Read More
Abstract: Deeper structures behind BPS counting on toric Calabi-Yau 3-folds have recently been realized mathematically in terms of the quantum loop group associated to a certain quiver drawn on a
torus, which is endowed with an action on the BPS vector space via crystal melting. In this talk, we identify the annihilator of the aforementioned action, thus leading to the … Read More
We will explore the fascinating concept of parity restoration using minimal Higgs doublets and its implications on the SU(2)R scale in Higgs parity, neutrino masses, and thermal leptogenesis. Our
main focus will be to present a natural bound on the scale of the parity-breaking vR and the mass of the right-handed neutrino, M1, obtained from thermal leptogenesis. We will also … Read More
Abstract : We present an example where a CFT qualitatively changes the behavior of loop diagrams at scales parametrically smaller than the mass scale where the CFT is broken. In our toy model, a
large anomalous dimension leads to a scenario where the corrections to the mass of a scalar is dominated at low energies below even the scale of CFT … Read More | {"url":"https://www-theory.lbl.gov/?event-venue=402-classroom-physics-south-ucb-campus-4","timestamp":"2024-11-04T05:50:18Z","content_type":"text/html","content_length":"72564","record_id":"<urn:uuid:6e8f7606-158d-4bb4-9b11-3b3bc6349815>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00650.warc.gz"} |
Exponents in context of grade slope
30 Aug 2024
Title: The Role of Exponents in Grade Slope: A Mathematical Exploration
This article delves into the fundamental concept of exponents and their significance in the context of grade slope, a crucial aspect of civil engineering. We will explore the mathematical formulation
of grade slope and its relationship with exponents, highlighting the importance of exponentiation in this context.
Grade slope is a critical parameter in civil engineering, referring to the ratio of vertical distance (rise) to horizontal distance (run) between two points on a surface or structure. In this
article, we will examine how exponents play a vital role in understanding and calculating grade slope.
Mathematical Formulation:
The grade slope (gs) can be mathematically represented as:
gs = rise / run
where rise is the vertical distance and run is the horizontal distance.
To express the grade slope as an exponent, we can rewrite the equation using logarithms:
log(gs) = log(rise) - log(run)
Using the properties of logarithms, we can simplify this expression to:
gs = 10^(log(rise) - log(run))
This formula highlights the importance of exponents in calculating grade slope. The exponentiation operation (10^()) represents the power to which the base 10 must be raised to produce the value of
the grade slope.
The use of exponents in grade slope calculations has significant implications for civil engineering applications. For instance, when designing a road or highway, engineers need to consider the grade
slope to ensure proper drainage and stability. By understanding how exponents relate to grade slope, engineers can optimize their designs to meet specific requirements.
Furthermore, the exponentiation operation allows for easy manipulation of the grade slope value. For example, if an engineer wants to increase the grade slope by a certain percentage, they can simply
raise the original value to that power.
In conclusion, this article has demonstrated the crucial role exponents play in understanding and calculating grade slope. The mathematical formulation using logarithms and exponentiation highlights
the importance of exponentiation in civil engineering applications. By recognizing the significance of exponents in grade slope calculations, engineers can develop more effective designs and optimize
their projects.
• [1] “Grade Slope” by Civil Engineering Handbook
• [2] “Exponentiation” by Mathematics Encyclopedia
Related articles for ‘grade slope ‘ :
• Reading: **Exponents in context of grade slope **
Calculators for ‘grade slope ‘ | {"url":"https://blog.truegeometry.com/tutorials/education/e6c379629100f26e5c97e8f5a64e1908/JSON_TO_ARTCL_Exponents_in_context_of_grade_slope_.html","timestamp":"2024-11-11T21:22:11Z","content_type":"text/html","content_length":"15959","record_id":"<urn:uuid:80aade20-3599-4833-a855-1bf0c507d6af>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00232.warc.gz"} |
Final exam in geometry for 2nd prep منصة (SDS) - نور مصر
اختبار خاص بالمعلم:
محمد عبدالتواب
مدة الاختبار: 1:30:00
الوقت المتبقي: -:-:-
If ABC is a right -angled triangle at B , AB = 6 cm, BC = 8cm, then the length of the median drown from B equals ..cm
The point of intersection of medians of the triangle divides each median in the ratio...from the base
If M is the point of intersection of the medians of △ ABC , AD is a median of length 9 cm, then AM =...cm
A rectangle ABCD , its diagonals intersect at M , the length of its diagonal is 6 cm, then the length of the median AM is...
The measure of the exterior angle of the equilateral triangle equals
XYZ is an isosceles triangle in which m (∠ y) = 100° , them m (∠ Z) =
If △ XYZ is right-angled at y , m(∠ X) = 60°, XZ = 10 cm, then XY =
A right-angled triangle , the measure of one of its angle is 45° , then it is...
If ABC is a triangle , AB = BC, then ∠ C is
The median of the isosceles triangle from the vertex angle bisects it and ...to the base
The length of the hypotenuse in the right-angled triangle equals ...the length of the side opposite to the angle of the measure 30°
The triangle which has no axes of symmetry is
If X - Z > Y - Z then X .. Y
In △ XYZ , XY is the shortest side , then the angle of the smallest measure is
A triangle has 3 axes of symmetry , then the measure of the exterior angle at one of its vertices is equals
If the measures of two angles in a triangle are 48°, 84°, then its type is
If ABC is an obtuse-angled triangle at C then BC...AB
The sum of the lengths of any two sides in a triangle is...the length of the third side
Which of the following numbers can be the lengths of sides of a triangle
If Z, 12, 2Z are the lengths of sides of a triangle , then the greatest value of Z =
The measure of the exterior angle of the equilateral triangle equals
In △XYZ , XY = 8 cm , Z = 5 cm , then its perimeter ∈ ...
The longest side in △XYZ where m (∠ y) = m(∠ Z) + m(∠ X)
The number of the medians of the triangle is | {"url":"https://nwarny.com/test/4671/","timestamp":"2024-11-02T04:59:29Z","content_type":"text/html","content_length":"156021","record_id":"<urn:uuid:7923cf15-a570-42e9-9f7c-8e8bcfc687a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00521.warc.gz"} |
How to Get the Mean Symbol in a Word Document - Technosfer
How to Get the Mean Symbol in a Word Document
Image Credit:
When most people think of the average of a set of data, they’re really referring to the mean. Whenever you’re writing about statistics using Microsoft Word, there’s a good chance you’ll need to know
how to create the mean symbol, which is x̄. It is sometimes called an x-bar, for obvious reasons, and it’s one of the most important math symbols to know how to create. Microsoft Word makes it easy to
use this symbol in your documents.
Mean Symbol in Word: Equations
The “Equations” tool in Word, which is included in all versions from 2007 onward, makes it easy to create the symbol for average in a document. Go to the “Insert” tab and find the group labeled
“Symbols.” Click on the drop-down arrow under “Equation” and click on “Insert New Equation.” You can also open the “Equation Tools” tab by holding the “Alt” key and pressing “Enter.”
Toward the right of the row of options, click “Accent.” This brings up all the available accents, including the “bar” symbol, which is usually second from the left on the third row of options,
although that may change depending on your Word version. Click the symbol, and a box with a bar over it appears. In the box, type “x” to create the mean symbol. The x-bar symbol is commonly used for
a sample mean, but you can also use this approach to create a y-bar symbol or add the bar to any other symbol.
Mean Symbol With Alt Codes
You can also use Alt codes to get the sample mean symbol into your Word document if your keyboard has a number pad in addition to the row above the letters. Ensure the “Num Lock” key is activated so
you can use the numbers on the pad. Type the letter “x,” hold the Alt key and type “0772” into the number pad. This adds the bar symbol to the x. The code “0773” creates a longer bar.
Other Statistical Symbols in Word
The “Equations” and “Symbols” features, both located under the “Insert” tab, allow you to add many math symbols used in statistics. These include the symbol for population mean, 𝜇, and the symbol for
population standard deviation, σ. You can add these to an equation as described in the first section, but both are Greek letters – mu and sigma, specifically – and you can find them in the “Greek and
Coptic” subset of the “Symbols” dialogue box.
İlginizi çekebilecek konular
Leave a Comment | {"url":"https://technosfer.net/how-to-get-the-mean-symbol-in-a-word-document/","timestamp":"2024-11-01T22:59:47Z","content_type":"text/html","content_length":"41546","record_id":"<urn:uuid:f2a648b9-cfb2-43f6-8fba-c1be98d280bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00113.warc.gz"} |
baker’s percentage tutorial
This final installment in the Baker’s Percentage tutorial series concerns breads that are made with preferments. (A preferment is a poolish, biga, sponge, sourdough starter, etc., where a portion of
the flour is fermented prior to the mixing of the final dough.) If you missed the first three parts, you’ll want to read them before diving into this one. An index of the entire tutorial is here.
A preferment can be thought of in different ways. On one hand, it is a dough unto itself, and it has a BP formula all its own. But a preferment is also an ingredient in the final dough.
Look at this formula for baguette dough made with a poolish. The blue table shows the formula for the final dough, scaled to make 2340 g of dough. The yellow table shows the formula for the poolish,
scaled to make 936 g, the amount needed for the final dough. Note that the formula for each part is based on the amount of flour needed for that part. Also note that the poolish is listed as an
ingredient in the final dough formula.
Poolish Final Dough
Ingredient % Grams % Grams
Flour 100% 468 g 100% 900 g
Water 100% 468 g 52% 468 g
Instant Yeast 0.06% 0.3 g 1% 9 g
Salt — — 3% 27 g
Poolish — — 104% 936 g
Total 200% 936 g 260% 2340 g | {"url":"https://www.wildyeastblog.com/tag/bakers-percentage-tutorial/","timestamp":"2024-11-10T20:45:03Z","content_type":"text/html","content_length":"44555","record_id":"<urn:uuid:cff391ec-eed5-4a3c-b368-3ec3e98735d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00287.warc.gz"} |
Ring Theory/Introduction - Wikibooks, open books for an open world
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hypercomplex numbers in the mid-nineteenth century undercut the
pre-eminence of fields in mathematical analysis.
Richard Dedekind introduced the concept of a ring.
The term ring (Zahlring) was coined by David Hilbert in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897.
The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. L. Crelle), vol. 145, 1914.
In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings. | {"url":"https://en.m.wikibooks.org/wiki/Ring_Theory/Introduction","timestamp":"2024-11-02T04:49:13Z","content_type":"text/html","content_length":"21505","record_id":"<urn:uuid:33fa9bad-4023-4d5a-903f-ebcb79c54091>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00587.warc.gz"} |
Soil Nail
The Soil Nail support type can be used to model soil nail slope reinforcement. Note:
• A Soil Nail is assumed to be fully bonded along its entire length
• The Soil Nail support type in Slide2 is equivalent to Grouted Tieback support with Bond Length = 100%
• The Soil Nail support type in Slide2 is differentiated from the Grouted Tieback for user convenience. However, the user should be aware that, as far as the Slide2 implementation is concerned, a
Grouted Tieback with Bond Length = 100% would behave exactly the same as a Soil Nail, all other parameters being equal.
• A Soil Nail is NOT equivalent to the Grouted Tieback with Friction support type in Slide2.
• Soil Nails can also be used to model pile or micropile support by using the optional Shear or Compression options as described below.
Tensile Capacity
The Tensile Capacity entered for a Soil Nail, represents the maximum tensile capacity of an individual soil nail. This is the capacity of the soil nail itself (e.g. steel tensile capacity),
independent of the plate capacity or the bond capacity. Units are Force.
Plate Capacity
The Plate Capacity is the maximum load which can be sustained by the plate assembly which connects the soil nail to the slope. Units are Force.
Shear Capacity
The Shear Capacity is optional and allows you to account for shear failure through the bolt cross-section (i.e. the force required to shear the bolt perpendicular to its axis). Units are Force. See
below for implementation details.
Compression Capacity
The Compression Capacity is optional and allows you to account for the failure of the bolt in compression. This can be useful for modelling compression piles, for example. Units are Force. See below
for implementation details.
Out of Plane Spacing
The spacing between soil nails in the out-of-plane direction (i.e. along the slope), measured from center to center.
Force Application
See the Force Application topic for a discussion of the significance of Active and Passive support force application in Slide2.
Force Orientation
When the support begins to take on a load, due to displacements within the slope, the direction of the applied support force can be assumed to be:
• Tangent to Slip Surface
• Bisector of Parallel and Tangent (i.e. at an angle which bisects the tangent to slip surface orientation, and the parallel to reinforcement orientation)
• Parallel to Reinforcement
• Horizontal
• User-Defined Angle (i.e. the user may specify an angle, measured from the positive horizontal direction)
Applied Force Orientation options
If you are using the Shear Capacity option, the force may be applied at some angle to the bolt axis, corresponding to the resultant of the shear and tensile components (see below for details).
Pullout Strength
For a Soil Nail, the Pullout Strength is expressed as a Force per unit Length.
The length units in this case, refer to the length along the soil nail. The Bond Strength determines the pullout and/or stripping force which can be generated by a soil nail.
Material Dependent Pullout Strength
See the Grouted Tieback topic for details, as the procedure is the same.
Implementation of Soil Nail Support in Slide2
Consider a soil nail which intersects a slip surface, as shown below.
Li = length of soil nail within sliding mass
Lo = length of soil nail embedded beyond slip surface
Soil Nail Parameters
B = Bond Strength (force / unit length of soil nail)
S = Out of Plane Spacing
T = Tensile Capacity (force)
P = Plate Capacity (force)
At any point along the length of the soil nail, there are 3 possible failure modes which are considered:
1. Pullout (force required to pull the length Lo of the nail out of the slope)
2. Tensile Failure (maximum axial capacity of the soil nail)
3. Stripping (slope failure occurs, but nail remains embedded in slope)
The maximum force which can be mobilized by each failure mode, PER UNIT WIDTH OF SLOPE, is given by the following equations:
At any point along the length of a soil nail, the force which is applied to the slip surface by the soil nail, is given by the MINIMUM of these three forces.
Applied Force = min (F1, F2, F3) Eqn.4
• In order for stripping to occur, the Plate Capacity must be exceeded. The Plate Capacity is included in the stripping force equation, and added to the shear capacity along the length Li.
• If the Soil Nail Pullout Strength is specified as Material Dependent, then the Pullout Force and Stripping force, are determined by integrating along the lengths Lo and Li, to determine the force
contributed by each segment of the soil nail which passes through different materials.
A typical Soil Nail Force diagram, which exhibits all three failure modes, is shown below. In this case, the Plate Capacity is less than the Tensile Capacity, and therefore "stripping" is a possible
failure mode. If the Plate Capacity is greater than or equal to the Tensile Capacity, then stripping cannot occur, and the Soil Nail Force diagram will be determined only by the Tensile and Pullout
failure modes.
Soil Nail Force Diagram
Implementation of Shear Capacity
If the Shear Capacity option is selected, an additional force vector perpendicular to the bolt direction, and opposite to the direction of failure, is added to the overall bolt capacity vector,
determined by the methods described above. You’ll see this in the Interpreter when you show the slice data. The support force at the base of the slice is no longer parallel to the support but
angled in a direction opposite to the slip direction. The Shear Capacity option may be useful if you expect that the support forces are not purely tensile, but may include a transverse shear
component as well. This depends on the orientation of the support relative to the slip surface direction.
Illustration of Shear Capacity option for support
Implementation of Compression Capacity
If the Compression Capacity option is selected, the bolt can apply a stabilizing force to the slope when it goes into compression. This can be useful for modelling compression piles, for example. To
determine when a bolt goes into compression, the program looks at the direction of displacement at the point where the bolt crosses the failure surface. The direction of displacement is equal to the
direction of the base of the slice through which the bolt traverses, as shown in the figure below. If the dot product of vector u and v is positive, the bolt is in compression.
Illustration of Compression Capacity option for support
If the bolt goes into compression, the capacity is determined in the same way as a bolt-in tension, with the following two differences. The first is the compression capacity is substituted for the
tensile capacity. The second is that the plate capacity is assumed to be zero. Three modes of failure are still examined to determine the overall capacity. The first is compression failure of the
tendon. The second is the bond failure of the bolt outside the failed mass. The third is the bond failure of the bolt internally to the failed mass. The overall capacity is simply the minimum of
these three failure modes, as described above for the tensile failure mode. | {"url":"https://www.rocscience.com/help/slide2/documentation/slide-model/support-2/define-support-properties/soil-nail","timestamp":"2024-11-06T13:47:14Z","content_type":"application/xhtml+xml","content_length":"429952","record_id":"<urn:uuid:062aa8cb-40a5-40b3-b5a2-fc41c72c1c3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00219.warc.gz"} |
ROHIT techVlog
Msg 192, Level 15, State 1, Line 1
The scale must be less than or equal to the precision.
To avoid the error below points can be followed.
• The scale must be within the range of 0 and the value of the precision. This can easily be done by increasing the value of the precision to include the digits both before and after the decimal
• In the case of the incorrect scale in the definition of a DECIMAL or NUMERIC data type in column of a table, simply increase the size of the precision to include the digits both before and after
the decimal point.
Try to find out the error:
DECIMAL and NUMERIC are numeric data types that have fixed precision and scale. When maximum precision is used, which is 38, valid values are from -10^38 through 10^38 – 1. NUMERIC data type is
functionally equivalent to DECIMAL data type. The syntax for declaring a local variable or a column as DECIMAL or NUMERIC data type is as follows:
DECIMAL ( p [, s] )
NUMERIC ( p [, s] )
Precision (p) is the maximum number of decimal digits that can be stored, both to the left and to the right of the decimal point. The precision must be a value from 1 through the maximum precision of
38. The default precision is 18.
The optional scale (s) is the maximum number of decimal digits that can be stored to the right of the decimal point. Scale must be a value from 0 through the value of the precision (p). Scale can
only be specified if precision is specified. The default scale is 0.
Given the definition of the precision and scale of a DECIMAL or NUMERIC data type, this error message will be encountered if the specified scale is greater than the precision when defining a local
DECLARE @Pi DECIMAL(1, 6) -- 3.141592
Msg 192, Level 15, State 1, Line 1
The scale must be less than or equal to the precision.
DECLARE @Latitude DECIMAL(2, 6)
DECLARE @Longitue DECIMAL(3, 6)
Msg 192, Level 15, State 1, Line 2
The scale must be less than or equal to the precision.
Msg 192, Level 15, State 1, Line 3
The scale must be less than or equal to the precision.
A different error message will be encountered when the scale is greater than the precision when defining a DECIMAL or NUMERIC column in a table:
CREATE TABLE [dbo].[Product] (
[ProductID] INT,
[ProductName] VARCHAR(100),
[Width] DECIMAL(4, 6),
[Length] DECIMAL(4, 6),
[Height] DECIMAL(4, 6)
Msg 183, Level 15, State 1, Line 8
The scale (6) for column 'Width' must be within the range 0 to 4.
To avoid this error, as the error message suggests, the scale must be within the range of 0 and the value of the precision. This can easily be done by increasing the value of the precision to include
the digits both before and after the decimal point.
Here’s an updated version of the scripts earlier that fixes the issue:
DECLARE @Pi DECIMAL(7, 6) -- 3.141592
DECLARE @Latitude DECIMAL(8, 6) –- 2 Digits to the left and 6 digits to the right.
DECLARE @Longitue DECIMAL(9, 6) –- 3 Digits to the left and 6 digits to the right.
In the case of the incorrect scale in the definition of a DECIMAL or NUMERIC data type in column of a table, simply increase the size of the precision to include the digits both before and after the
decimal point.
CREATE TABLE [dbo].[Product] (
[ProductID] INT,
[ProductName] VARCHAR(100),
[Width] DECIMAL(10, 6),
[Length] DECIMAL(10, 6),
[Height] DECIMAL(10, 6)
Msg 455, Level 16, State 2, Line 1
The last statement included within a function must be a return statement.
As the error message suggests, the last statement in a function must be a RETURN statement. Even if the execution path of the statements in a function will execute a RETURN statement, the error will
still be encountered.
To understand better, here’s a user-defined function that returns the smaller number between two integer parameters:
CREATE FUNCTION [dbo].[ufn_Least] ( @pInt1 INT, @pInt2 INT )
IF @pInt1 > @pInt2
RETURN @pInt2
RETURN @pInt1
Msg 455, Level 16, State 2, Procedure ufn_Least, Line 8 [Batch Start Line 0]
The last statement included within a function must be a return statement.
To avoid this error, make sure that the last statement in your user-defined function is the RETURN statement. In the case of the user-defined function shown above, here’s an updated version of the
function that gets rid of the error:
CREATE FUNCTION [dbo].[ufn_Least] ( @pInt1 INT, @pInt2 INT )
IF @pInt1 > @pInt2
RETURN @pInt2
RETURN @pInt1
Instead of putting the last RETURN statement inside the ELSE statement, it is executed by itself and the function still produces the same result. | {"url":"https://www.rohittechvlog.com/2022/07/","timestamp":"2024-11-15T00:19:15Z","content_type":"application/xhtml+xml","content_length":"82334","record_id":"<urn:uuid:7a5edfc4-b22d-4310-8be1-5ee82564229d>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00264.warc.gz"} |
It was the Bitcointalk forum that inspired us to create Bitcointalksearch.org - Bitcointalk is an excellent site that should be the default page for anybody dealing in cryptocurrency, since it is a
virtual gold-mine of data. However, our experience and user feedback led us create our site; Bitcointalk's search is slow, and difficult to get the results you need, because you need to log in first
to find anything useful - furthermore, there are rate limiters for their search functionality.
The aim of our project is to create a faster website that yields more results and faster without having to create an account and eliminate the need to log in - your personal data, therefore, will
never be in jeopardy since we are not asking for any of your data and you don't need to provide them to use our site with all of its capabilities.
We created this website with the sole purpose of users being able to search quickly and efficiently in the field of cryptocurrency so they will have access to the latest and most accurate information
and thereby assisting the crypto-community at large.
*shrug* I can hash it a second time, but it's not going to increase or decrease security. I typed out a random alphanumeric seed anywhere between 1 and 64 characters long, so they'd have to
bruteforce an ungodly number of options. Hashing it a second time won't change that. | {"url":"https://bitcointalksearch.org/topic/testing-the-waters-526","timestamp":"2024-11-10T17:48:55Z","content_type":"text/html","content_length":"88107","record_id":"<urn:uuid:d824cb23-5a0b-4d05-8e97-4d8bf548d64d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00052.warc.gz"} |
Rounding Numbers: A Comprehensive Guide To Understanding The Basics - Kmacims | Education Annex
Rounding is a systematic way of simplifying a number while keeping its value close to the original. It is a mathematical operation that is used to write a number in its simpler form.
To round a number means to rewrite the original number in a simple form with a slight deviation.
The number rounded has a nearby value to the original and is easy to understand.
The round number is used in many fields like engineering, science, finance, the stock market, etc.
This article will introduce rounding numbers and the methods. If you are in any way confused about how to round numbers, continue reading.
Table of Contents
What are Rounding Numbers?
Rounding a number is a systematic way of approximating the value of a such number. It is a way of rewriting the number so that the value stays close to the original value.
When you round a number, the final result is usually less accurate, but easier to use in computations.
For example, the final weights of 5 loaves of bread recorded by the quality control officer in grams are:
87.95, 90.24, 89.7, 88.05, and 90.35.
To easily calculate the average weight, we can convert the weights to the nearest whole number as follow:
88, 90, 90, 88, and 90.
Looking at the above round numbers you see that they deviate from the original value but are easy to compute.
Thus, rounding numbers are the simplest approximates of very large and small numbers. The process of approximating a number to its nearest number is known as rounding.
In the real world, we can perform rounding for those numbers for which the exact values do not hold much importance. But when dealing with financial values, it’s not always the best option to round
numbers except with large decimals.
For example, we can round the value of 2.0125861 to 2.0126 instead of just 2. This is because multiplying the 0.012586 by 1000000 will result in 12,586. So, rounding to 2 means that you are losing
12,586 per 1 million.
There are usually two ways to round a number: round up, and round down.
Round up
When rounding a number, we approximate a number depending on the value next to it. Values are on a scale of 0, 1, 2, to 9, and the value halfway is 5. When the value next to the round number is 5 and
above, we can round up the number.
Thus, round-up means increasing the value of the round number if the next value is equal to or greater than 5. This is not a rule, it depends on the situation on the ground. But it is what people
generally do.
Round down
On the other hand, round down is decreasing the value of the round number if the next value is less than 5.
In our examples, we shall be rounding up or down depending on the value next to the round number.
Please, note that rounding up or down is at your discretion. You can choose to round up or down to ensure that your data does not deviate completely from an actual value.
For example, look at the table of data below.
The sum of the first round-up deviated significantly from the actual value. But with the value in the green cell rounded down to 91, the sum became closer. So, choose whether to round up or down
depending on the data accuracy you expect.
General rules for rounding numbers
There are acceptable rules that students should follow when rounding numbers. But an independent worker can choose his/her rules based on his/her discretion.
The general rules to be followed are as follows:
• Determine the round number calculation. Is it to the nearest ten or tenth, whole number or hundred, etc.
• Based on the above, locate the rounding digit in the number to be rounded. For example, to round the number 752 to the nearest ten, the rounding digit is 5. To the nearest hundred, the rounding
digit is 7.
• Haven identified the rounding digit, look at the value next (to the right) to the rounding digit. This value will determine whether you should round up or down
• If the value next to the rounding digit is equal to or greater than 5, round up. This implies that the rounding digit will increase by 1. If the value is less than 5, round down. This means the
rounding digit will remain the same.
• Finally, whether you are rounding up or down, the values after the rounding digit will become zero (0).
How to Round Numbers
Rounding numbers is easy if you understand the basics. In this section, you will learn the basics of rounding numbers.
In number counting we begin from 1 and count to 10, then 100, 1000, and continuously. Thus, we have ones, tens, hundreds, thousands, ten thousand, hundred thousand, million, billion, and trillion.
Likewise, in places where decimals are represented, we have tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, billionths, and trillionths. Where the tenth is the
inverse of ten, etc.
Naturally, Ones, tens, etc, and tenths, hundredths, etc are represented in the table below. Where the tenths, and hundredths represent the decimal or fractional part of a number.
Consider the following example. The total sales of Ucheson & Sons for the month of January 2023 were 12,355,468.6523.
Let us represent the above in a table for clarity.
The above table shows a typical way to represent a number for onward rounding.
Values to the left of the decimal are ones, tens, etc, and values to the right are tenths, hundredths, etc.
Based on the above, you can round whole numbers to the nearest ten, hundred, thousand, etc. However, decimals are rounded to the nearest tenth, hundredth, etc.
Though you can easily round numbers using a rounding numbers calculator, you should know how to round numbers. In this section, you’ll learn some of the methods of rounding numbers.
How to round off decimals
A decimal number is made up of two parts. The whole number part and the fractional part. The decimal point separates these two parts.
The decimal part is the numbers to the right of the decimal point. They are usually rounded off in tenths, hundredths, thousandths, etc.
To round off decimals, the same rules explained above apply. For example, round off 78.257 to the nearest hundredth.
The answer is 78.26. (Since 7 is greater than 5, we increase the rounding digit by 1).
Rounding numbers to the nearest whole number
To round a number to the nearest whole number means eliminating the fractional part of a decimal number.
To do this, abide by the rules explained above. That is, determine the rounding digit and check whether to round up or down.
For example, rounding 78.257 to the nearest whole number will give you 78.
Rounding numbers to the nearest 10
When you think of the nearest ten, remember the chart displayed above to determine the rounding digit.
The nearest ten is a whole number with a single zero at the end. E.g. 10, 40, 7510, etc. Hence, the ones placeholder will be converted to zero. It then means that the fractional part of a decimal
number will be converted to zeros.
For example, to convert the number 78.257 to the nearest 10, you’ll have 80.
Round numbers to the nearest 100
Just like the nearest 10, the nearest 100 leaves two zeros at the end. So, using the placeholder chart above, you can determine the rounding digit and either round up or down,
For example, to round the number 652410.25 to the nearest 100, we have the following result: 652400.00.
Rounding numbers to the nearest 1000
Using the idea of rounding numbers to the nearest 10 and 100, we can round numbers to the nearest thousand.
You’ll need to understand the placeholder chart to determine the rounding unit. Also, you should know how many zeros should be left at the end. Your guess is right, 1000, that is 3 zeros.
For example, to round the number 652410.25 to the nearest 1000, you should have 652000.
Rounding Numbers Examples
Let us discuss how to round numbers with a simple example so that you can have clearer knowledge.
Oxlade Proclo LLC posted its financial report on its website. When comparing the results, the group managing director identified deviations between the hardcopy and the electronic reports. The table
below is an excerpt of their sales for 6 months. Use it to answer the questions that follow.
Month Sales in $
January 58,764.25
February 78,654.45
March 105,423.05
April 95,015.00
May 152,470.55
June 87,265.10
a. Convert the monthly sales to the nearest tenths
b. Convert the monthly sales to the nearest thousand
c. Total the original values and that of the nearest 1000. What is the difference?
a. Let’s first place the numbers in a placeholder chart as follow:
To convert to the nearest tenth, the table below shows the rounding digit in the yellow column.
The numbers to the right of the rounding digits in rows 1, 2, 3, and 5 are 5 and above. Thus, we add 1 to the rounding digits (that is, round up).
But the number to the right of the rounding digits in rows 4 and 6 are less than 5. Thus, we leave the value of the rounding digits the same.
Therefore, the table below shows the result of rounding to the nearest tenth.
The new sales report will become as follow:
Month Sales in $
January 58,764.30
February 78,654.50
March 105,423.10
April 95,015.00
May 152,470.60
June 87,265.10
2. Here, we’ll convert to the nearest 1000. Like in the above solution, let’s place the numbers in a placeholder chart and identify the rounding digit. The table below shows the placeholders and
rounding digits with a red border line.
The numbers to the right of the rounding digits in rows 1, and 2 are 5 and above. Thus, we add 1 to the rounding digits (that is, round up).
But the number to the right of the rounding digits in rows 3, 4, 5, and 6 are less than 5. Thus, we leave the value of the rounding digits the same (round down). We then convert all values to the
right of the rounding digits to zero.
Therefore, the table below shows the result of rounding to the nearest thousand.
The new sales report will become as follow:
Month Sales in $
January 59,000.00
February 79,000.00
March 105,000.00
April 95,000.00
May 152,000.00
June 87,000.00
3. Here, we sum up the two sales and subtract the results. The table below shows the result.
From the result obtained above, you’ll notice a deviation of $592.40 from the original sales report. However, rounded numbers are easier and simpler to calculate.
I believe that the above example has given you a clearer understanding of how to calculate round numbers.
Rounding numbers is a common mathematical operation that is used to approximate numbers in order to understand them easily. It is mostly used in scientific calculations where you’ll need to
approximate experimental data to easily compare their values.
Students in primary and secondary schools should learn and understand them. Also, applicants, employees, and business owners should learn them.
Though it makes computations easy, its results deviate from the original. However, you can randomly round up and down values to arrive at a very close result.
The rounding numbers calculator makes these computations easy, but it’ll be great if you understand their calculations.
This tutorial discussed how to round whole numbers and decimal numbers; it is now your turn to respond.
I will want you to ask questions and make contributions with regard to rounding whole numbers and decimals. Have you used round numbers before, and in what circumstances? Please, use the comment box
below and respond, and do not forget to share your experiences.
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is a gun concept I came up with for Fractal's new
The concept is that, given a set of axis, the gun segments data across all the axis, and across every possible combination of every possible amount of these axis.
For example, say you segment data across 4 axis, say Distance (bullet travel time), Time (travelled without stopping), Lateral (velocity), and Acceleration. The gun automatically builds a dataset to
hold the graphs on all of the following segmentations:
• none
• [distance]
• [time]
• [latvel]
• [accel]
• [distance][time]
• [distance][latvel]
• [distance][accel]
• [time][latvel]
• [time][accel]
• [latvel][accel]
• [distance][time][latvel]
• [distance][time][accel]
• [distance][latvel][accel]
• [time][latvel][accel]
• [distance][time][latvel][accel]
Thus for example, the [none] graph is simply an array of the guessfactor bins. The [time] graph is a 2D array of time segmentation and guessfactor bins. The [latvel][accel] graph is a 3D array of
lateral velocity vs acceleration across the guess factor bins. The first and last graphs are those people are obviously most farmiliar with; no segmentation, and segmentation across all available
When Fractal records a wave hit, it puts the hit at the right place into every single one of these graphs. So it parses each one and checks where the data goes. Say the wave hit is being added to the
[accel] graph; it finds the right acceleration index, then in it looks for the right angle bin and adds the hit. Say the wave hit is being added to the [distance][time][latvel] graph; it finds the
right distance index, then in it the right time index, then in it the right latvel index, then in there looks for the righht angle bin and adds the hit.
This may seem somewhat complicated, and it's probably because I didn't explain it well, because it's actually quite simple. This is one way of aggregating a VirtualGuns array of GuessFactorTargeting
guns into one gun. Many bots use such an array (SandboxDT for example). In this, you would have a select few of these combinations; for example, you would have a fully segmented gun ([distance][time]
[latvel][accel]), then say a partially segmented gun ([distance][time][latvel]), and an only slightly segmented gun ([distance]). If you are attempting to use the fully segmented gun and it doesn't
have enough data, you step down to the partially segmented gun; if it still doesn't have enough data, you step down to the slightly segmented gun. Now, this makes things simple, but has some obvious
disadvantages. What if the enemy shows a large spike on the [accel] axis? You don't get to exploit this vulnerability until you reach the fully segmented gun.
With AutomatedSegmentation, the gun automatically uses all possible combinations. When finding where to shoot, it first looks at the most segmented gun. If it has enough data, great! It uses it and
fires as would an ordinary fully segmented gun. If however it decides that it doesn't have enough data or that it's not reliable enough, it steps down one dimension and takes all possible
combinations of graphs, those being:
• [distance][time][latvel]
• [distance][time][accel]
• [distance][latvel][accel]
• [time][latvel][accel]
It then uses a MultipleChoice on the four of these graphs to decide where to shoot. So if the enemy has a vulnerability spike on [accel], it will show up in 3 of the 4 graphs here, while it doesn't
show up at all on the VirtualGuns example above. If it decides that none of these graphs are reliable enough, it again steps down, and takes every possible combination of 2 axis:
• [distance][time]
• [distance][latvel]
• [time][latvel]
• [distance][accel]
• [time][accel]
• [latvel][accel]
Thus the acceleration spike will be picked up by 3 out of the 6 graphs here, against the other 3 forms of segmentation. You can see how these six graphs will take much, much less time to fill up with
data than would the fully segmented graph; these can fill up across 3 or 4 rounds while the full segmentation axis can take up to 40 rounds to get a reliable source of data. If it again doesn't have
enough data, it keeps stepping down until it does; eventually it reaches the fully unsegmented graph, and uses it to shoot if all else fails. Since some of the conditions remain the same across the
course of the first round, such as Distance for example, the unsegmented axis is a reliable data source until the higher segmentations get filled with data.
With this gun, you can see how it can reach an incredibly fast learning time; over the course of a few rounds it can pick up spikes across any of the four axis and know very reliably where to shoot.
Now, the best part is, the concept of AutomatedSegmentation is that this is handled automatically by the gun for an arbitrary number of axis. Fractal's gun code is designed entirely for an arbitrary
number of segmentation axis, rather than a fixed number. It has an abstract Axis class which the axis extend, and within the gun it has an array of Axis called axis. The axis are plugged directly
into this array, and the gun automatically conforms to how many axis are it in; thus you can have as many segmentation axis as you want! Fractal currently has 9 axis written; right know I'm debating
which to plug in for a release version (I'll probably end up using the best 5 or 6 that I find useful), but if I want I can simply plug in all 9, and it automatically builds the dataset for every
possible combination of every possible number of these axis. Thus it totals (9 choose 0) + (9 choose 1) + (9 choose 2) + ... + (9 choose 9) graphs, which reduces to 2^9 = 512 graphs. These graphs are
all arbitrarily dimensional; some segment across two, or three, or no axis, while some segment across 8 or all 9. Whenever a wave hit is recorded, it adds it to all 512 of these graphs (with the
gaussian smoothing algorithm Fractal uses, this totals about 15360 gaussian smoothings per wave hit!) Whereas an ordinary GF gun suffers from many dimensions, it has little effect on
AutomatedSegmentation, because say I add a [heading] axis, the [distance][time][accel][latvel] axis still exists and is still used as before while there isn't enough data. In theory the gun can only
get better by adding dimensions.
There are many ways in which to store this data; after debating a few ways ways, Fractal now uses one very simple way: it uses a single array of objects, in which it puts arrays of arrays of arrays,
and it uses heavy typecasting and a handful of recursive functions to add and retrieve the possible combinations. I can call a getBins function specifying the amount of free axis I want (say 2), and
it recursively steps through the dimensions and pulls out all possible graphs (6 in the example above) into an ArrayList. If it were segmenting on 9 axis, getBins(3) for example would yield all (9
choose 6) = 84 graphs. The data retrieval is surprisingly fast, and it even performs much faster than Fractal 0.32, which used static segmentation only on Distance.
For deciding which level of segmentation to use, Fractal currently has a reliability algorithm; it computes the reliability of each dimension (the reliability of the data obtained from all possible
graphs of the given number of dimensions), and uses the one with the highest reliability. Reliability is based on the product of the reliabilites of the used dimensions in each graph, and graph
reliabilities are currently static (so, say distance is worth 3 and accel is worth two, the reliability of [distance][accel] is 6), times a function of the amount of data in the graph (fractal
currently uses a logarithm for this). Thus as the data increases, it uses the more and more segmented axis.
The end result of all of this is extremely fast learning time; it has the effect of a VirtualGuns array of partially segmented guns, but with an analytic solution to segmentation selection, and using
all possible axis rather than a hardcoded subset for each gun. Here are some screenshots illustrating its fast learning time; the depth indication shows how much segmentation it's currently using
(for example, depth 2 means it's using a MultipleChoice on the subset of all graphs segmenting data on only 2 dimensions.)
Anyway, that's my blurb on Fractal's gun. Lemme know what you think!
-- Vuen
It is one lovely idea, Vuen. One question is: what/how do you save between rounds? -- Frakir
I think Frakir means, what/how do you save between battles? I am too very interested in the answer. And thanks for sharing this design idea. I am hacking away on a new gun for GloomyDark which shares
a lot of this concept. And this page serves to answer the most important question I have had so far; "Is it worth te effort?" I think the way you describe the benefits here answers that question with
a solid "Yes". I'll post my design as soon as I have a prototype running. -- PEZ
All versions of Fractal past 0.32 have been using this to its full extent. Unfortunately, I have yet to find the real learning speed bonuses I've been looking for; I can't seem to get it to perform
well against everyone at once. It's either good against some bots and loses against others, or the other way around, depending on how I tweak it. I'm getting annoyed by some bots though because I
keep forgetting that some save data, and when I think I've made a change that degrades its performance it's just because the bots I test it against have learned more and so they get a higher score
against it. This is why I'm more drawn toward the targeting challenge, but it doesn't really work for some of Fractal's axis because they involve having Fractal moving as well. On the issue of saving
data, none of my bots save anything between battles, but if they did, saving this would be very similar to saving only the full segmented graph; that said I don't want to explain more because I don't
want to give away too much about Fractal's format for storing the data : ). The screenshots I added were taken with slightly different tweaks of its gun, so I have yet to get the bot to yield both
these extremes at once, but they still make for some nice-looking shots :D -- Vuen
FloodHT 0.7 did this (but with a static number of segmentations and it attempted to use VG's to resolve which cross-section to use). The code was actually somewhat clean, even, and the original bot I
made to do this was close to still being a minibot (extended off of FloodMini's code). The dev version of Fhqwhgads does a much more dynamic job of this as well, and seems to be at the same level in
the TC as the mini Fhqwhgads. The problem I have is that I'm still saving too much to effectively use that saved data, but it's still built to keep expanding it's knowledge well into the 1000-rounds
space (while making intelligent decisions on the data it has in the short term). -- Kawigi
Just two brief questions: how much memory does this consume and how slow the bot becomes? I'm also going to implement an arbitrary_number_of_axis gun concept, but I believe that one will be at least
a fast gun. --lRem
I should publish something like this, actually - in order to have stats quickly available in any combination of segmentations, your space only increases marginally - the additional guns take up less
room than the fully segmented one - you just need to add one to the size of each segment. You also need n^2 time to insert stats, there are n^2 guns that you can potentially use. In other words, when
you add a segmentation, you have twice as many guns, since you have all your previous guns and all your previous guns plus this new segmentation. This time isn't too bad if you're not segmented too
deeply. You could also easily make a segmentation not optional (just don't add the extra space for it and don't index for it?) I'll put some old code here for how I did it in FloodHT (from a test bot
that idea came from) at Flood16CodeSnippets. -- Kawigi
I personally dont see too great of an advantage here. It would be much more effective to increase the number of divisions in each array rather than adding segmentations. That way having a segment
with only 1 division would be the same as not having the array at all, and size increase can be much more measured. Rather than having to deal with the introduction of n times the number of current
sections, you can be using say 2 or 3 times with each increase, being able to monitor divisions much quicker. This will give a much faster and more accurate learning. -- Jokester
In my current development bot Gwynfor, I'm using one targeting method very similar to this. So far, my tests show it working better than either fully segmented or not segmented at all. Interesting
thing I've noticed though, is that against strong dodgers, the dominant segmentation-combination keeps flickering between different segment combinations. I'm guessing that's largely due to
wavesurfers that are managing to fool the targeting effectively, making it very indecisive due to none of the combinations of segmentation work well. -- Rednaxela
It could also be that the surfer learns to dodge one segment, so the other segment becomes stronger, then the surfer learns to dodge the next, and so on. How are you deciding which segment to use?
Kernel density calculations? -- Skilgannon
I'm sure there are smarter ways, but currently I'm just storing the guessfactors for every segmentation-combination in my bullet waves, and when the waves hit associate "error" values with various
combinations of segments. -- Rednaxela | {"url":"https://old.robowiki.net/cgi-bin/robowiki?action=browse&diff=1&id=AutomatedSegmentation","timestamp":"2024-11-13T09:05:52Z","content_type":"text/html","content_length":"18844","record_id":"<urn:uuid:9e0ee90f-b780-4105-b7dd-abbe5d986f88>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00330.warc.gz"} |
How to Adjust Line Thickness in Boxplots in ggplot2 | Online Tutorials Library List | Tutoraspire.com
How to Adjust Line Thickness in Boxplots in ggplot2
by Tutor Aspire
You can use the following methods to adjust the thickness of the lines in a boxplot in ggplot2:
Method 1: Adjust Thickness of All Lines
ggplot(df, aes(x=x, y=y)) +
Method 2: Adjust Thickness of Median Line Only
ggplot(df, aes(x=x, y=y)) +
The following examples show how to use each method in practice with the following data frame in R:
#make this example reproducible
#create data frame
df A', 'B', 'C'), each=100),
points=c(rnorm(100, mean=10),
rnorm(100, mean=15),
rnorm(100, mean=20)))
#view head of data frame
team points
1 A 9.373546
2 A 10.183643
3 A 9.164371
4 A 11.595281
5 A 10.329508
6 A 9.179532
Note: We used the set.seed() function to ensure that this example is reproducible.
Example 1: Create Boxplot with Default Line Thickness
The following code shows how to create a boxplot to visualize the distribution of points grouped by team, using the default line thickness:
#create box plots to visualize distribution of points by team
ggplot(df, aes(x=team, y=points)) +
Example 2: Create Boxplot with Increased Line Thickness
The following code shows how to create a boxplot to visualize the distribution of points grouped by team, using the lwd argument to increase the thickness of all lines in the boxplot:
#create box plots with increased line thickness
ggplot(df, aes(x=team, y=points)) +
Notice that the thickness of each of the lines in each boxplot has increased.
Example 3: Create Boxplot with Increased Line Thickness of Median Line Only
The following code shows how to create a boxplot to visualize the distribution of points grouped by team, using the fatten argument to increase the thickness of the median line in each boxplot:
#create box plots with increased median line thickness
ggplot(df, aes(x=team, y=points)) +
Notice that only the thickness of the median line in each boxplot has increased.
Feel free to play around with both the lwd and fatten arguments in geom_boxplot() to create boxplots that have the exact line thickness you’d like.
Additional Resources
The following tutorials explain how to perform other common tasks in R:
How to Change Axis Labels of Boxplot in ggplot2
How to Create a Grouped Boxplot in ggplot2
How to Label Outliers in Boxplots in ggplot2
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Practical Tools, Habits of Mind, and A More Satisfying Life: Three Reasons to Learn and Do Math ⋆ Two Rivers Public Charter School
By: Jeff Heyck-Williams
This year I have a wonderful opportunity. I get to teach math. Specifically, I teach middle school students in an algebra class and elementary and middle school teachers during professional
development in an effort to build their mathematical capacity. I honestly love my job.
While these adolescents and adults are fairly different in their temperaments and instructional needs, they both have challenged me with the essential question: Why do we need to learn this stuff?
While you might not think that this question should give me pause… it has. In particular, I feel that too often the question of why we should have a rich depth of mathematics knowledge and skills
boils down to a single answer: math is a great tool for doing a bunch of other disciplines that will solve the world’s problems and make you rich. Don’t get me wrong, math has been and is an
essential tool for fields as different as finance and physics. However, math is much more, and this argument not only ignores the other reasons for learning math, it cheapens the discipline. I
would argue that understanding both a depth and a breadth of mathematical concepts is important for an educated society for three reasons. Only through understanding, appreciating, and sharing all
three will we be able to get the most out of our math education.
REASON # 1: Mathematics Is a Practical Tool
I’ll start with the common assumption. Mathematics is a practical tool for many of the tasks we have to complete in our daily lives as well as in most if not all jobs. Mathematics is relevant in
our daily lives when we calculate the tip and how to split the bill when we eat out with a group of friends, when we figure out how many gallons of paint to buy to paint that room, or when we ignore
the odds and go ahead and buy that lottery ticket anyway. Math is useful.
In addition, we are experiencing a resurgence of calls for improvement in science, technology, engineering, and math education (STEM as it is called). Much of the rationale for this resurgence is
couched in economic terms. As this argument goes, we live in what Thomas Friedman has coined a flattening world, and if students in the United States are going to be able to continue to compete in
the global marketplace they need math. The types of white collar technical jobs that will pay a living wage require a high degree of mathematical proficiency because math is a practical tool. Jobs
in fields such as finance, accounting, economics, statistics, computer science, engineering, chemistry, and physics require a high level of mathematical proficiency. In addition, as information
technology advances, jobs in every other sector will require increasing levels of mathematical know-how.
All of this is true. However, this reasoning only views mathematics as a tool. From this perspective, math is no better than a very versatile hammer. It is a means to an end. The problem with
only viewing math from this lens is that for many of us it means that we could learn some elementary computation, a bit of measurement and geometry, and possibly a bit of probability for parlor
tricks on the weekend, and be done with it. Many of my students rightly ask, “when will I use all of this analysis of linear equations in my daily life?” And if the only view of mathematics is as a
tool, I have to admit that those who don’t go into a math-related field won’t ever need to find the slope of a linear pattern. However, I would argue that it is still important for them to study how
to generalize and analyze linear patterns. Here’s why.
REASON # 2: Mathematics Cultivates Habits of Mind
Regardless of the mathematical content you learn, the study of mathematics cultivates habits of mind or thinking skills that are powerful disciplines for negotiating our world. At the heart of good
mathematics instruction is problem solving. To do math effectively, students need opportunities to learn how to struggle through and solve ill-defined problems. This process teaches perseverance,
how to communicate an idea precisely, how to build a reasoned argument, how to think flexibly, and how to be metacognitive, or to monitor their thinking and problem solving process. These skills are
not unique to mathematics, but are skills that are relevant to the process of solving any problem in any area. So, whether a person is destined to use calculus in their daily life or work, the study
of calculus can build in them the skills to tackle whatever problems they face. Mathematics is uniquely positioned as a part of the core curriculum of our schools to help build these skills.
However, this aspect of mathematics might only be useful to a person during their school years, and could easily be left behind. If you don’t particularly like doing math and find it difficult
during your school years, you can easily choose a career path that isn’t dependent on mathematics. You will still have gained the benefits of cultivating habits of mind for solving problems, and you
will go on to apply them in other areas. Which brings me to my third reason for learning and doing mathematics.
REASON # 3: Mathematics Leads to a More Satisfying Life
Mathematics as a system is one of the most beautiful and enduring creations of the human mind. Developing understanding of mathematics is learning to appreciate and contribute to this work of
collective human ingenuity. Over the course of human civilization, mathematics has developed as a tool for organizing our complex social lives as well as understanding our universe. Specifically,
as the science of patterns, mathematics finds connections between often disparate phenomena and explains how those phenomena behave in surprisingly similar ways because of mathematical patterns.
Furthermore the whole of the mathematical world that describes these patterns was not invented or discovered by a single individual but by the collective work of millions of mathematicians striving
for understanding. Knowing this provides inspiration for my own explorations in the world of math. While there have been many great individuals who have made monumental contributions from Euclid
and Archimedes to Newton and Gauss, mathematics as a whole is the work of many. Each of us has the potential to contribute to this work.
With this realization and a little mathematical knowledge, everyone can open doors to understanding our world that are shut without mathematics, but don’t take my word for it. Watch the short
video Nature by Numbers created by Cristobal Vila for Eterea Studios.
Vila’s video highlights connections between a nautilus shell, a sunflower, and a dragonfly’s wings that are inaccessible without math. What is even more interesting is that the math behind this
video has been known for centuries and has been used in artwork from the architecture of the Parthenon to the paintings of Leonardo da Vinci. The ratio featured in the video (The Golden Ratio) and
the sequence of numbers here (the Fibonacci sequence) are but two examples of how math amazingly describes our world by capturing patterns at the heart of our physical reality. There are many more
examples of how mathematics creates surprising, elegant forms that are aesthetically pleasing and yet somehow illuminate connections in the natural world.
Truly, as mathematician Jerry P. King wrote in his book The Art of Mathematics
“… one’s intellectual and aesthetic life cannot be complete unless it includes an appreciation of the power and beauty of mathematics.” If we are serious about educating our children to be active
participants in their own education and responsible and compassionate members of society, then they must have access to a mathematics education that not only teaches the powerful tools that
mathematics has to offer, but also opens them to the possibilities of beauty and wonder that only mathematics affords. | {"url":"https://www.tworiverspcs.org/practical-tools-habits-of-mind-and-a-more-satisfying-life-three-reasons-to-learn-and-do-math/","timestamp":"2024-11-06T17:02:23Z","content_type":"text/html","content_length":"270868","record_id":"<urn:uuid:1d2421d3-1ab7-4058-9308-0f86847631bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00436.warc.gz"} |
Golf dead heat rule
In the event of a tie where 2 or more competitors are offered in one betting option, the dead heat rule applies.
Under the dead heat rule, the payout will be the total bet amount (stake plus win) divided by the number of players tied, minus the stake.
For example, you place a $100 bet on a player at +300 to reach the top 5 places in the tournament, and he finishes 5th, but tied with two other players.
In this case, the dead heat rule will result in the following payout:
1. ADD Stake + Win amount (100+300) = 400
2. Take this result and DIVIDE by the number of players tied (3), giving you 400/3 = 133.33
3. Take this second result and SUBTRACT the stake amount (100) from it, resulting in 133.33 - 100 = 33.33
So the payout for this wager would be $33.33. | {"url":"https://get.justbet.help/hc/en-us/articles/360002975937-Golf-dead-heat-rule","timestamp":"2024-11-10T12:43:10Z","content_type":"text/html","content_length":"24612","record_id":"<urn:uuid:e1623e0d-c6a3-4d7b-a03c-67bc6a386473>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00823.warc.gz"} |
76th Annual Meeting of the Division of Fluid Dynamics
Bulletin of the American Physical Society
76th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2023; Washington, DC
Session L44: Waves: Nonlinear Dynamics and Turbulence Hide Abstracts
Chair: Alexander Hrabski, University of Michigan
Room: 208AB
Monday, L44.00001: Fully-developed Wave Turbulence in the Numerical Kinetic Limit
November Alexander A Hrabski, Yulin Pan
20, 2023
8:00AM - Wave Turbulence (WT) describes an out-of-equilibrium process in nonlinear wave systems, characterized by inter-scale energy cascades and power-law spectra. Under a few statistical
8:13AM assumptions and in the limits of an infinite domain and small wave amplitude (together, the kinetic limit), a turbulence closure for the evolution of wave action spectrum naturally arises.
This closure forms the basis of the widely-adopted weak WT theory. Almost all realistic systems exist outside of the kinetic limit, however, and a quantitative understanding of the WT
closure realized in this setting is for the most part lacking.
In this work, we perform a series of high-resolution numerical experiments of fully-developed WT that approximate the kinetic limit. We first compute the full PDFs of instantaneous
inter-scale energy flux and dissipation rate. Then, using an interaction-based decomposition of the energy cascade, we quantify the role of resonant interactions in shaping the WT closure.
We conclude with a study of stationary spectra and the Kolmogorov constant. At each stage, we draw connections to theoretical predictions.
Monday, L44.00002: Role of three-wave interactions in surface gravity wave turbulence
November Zhou Zhang, Yulin Pan
20, 2023
8:13AM - Standard derivation of the Hasselmann kinetic equation for surface gravity waves assumes the dominance of quartet resonant interactions. As a result, the triad-resonance terms are removed
8:26AM from the dynamical equations using a Lee transformation. While such transformation is supposed to be only valid for infinitesimal nonlinearity level, the derived kinetic equation is widely
used in wave modeling for finite-amplitude waves. In this work, we numerically study the effect of triad interactions (in particular, the quasi-resonance of three waves) in surface gravity
wave turbulence. Our method decomposes the energy transfer into contributions from triad and quartet interactions, thus the role of each type of interaction can be elucidated. We apply this
method for both evolving and stationary spectra, and find that the triad interactions play a significant role at low nonlinearity level. The results imply modification of the kinetic
equation to better account for the triad interactions in certain cases.
Monday, L44.00003: Energy cascade due to nonlinear interactions of internal gravity waves
November Yue Cynthia WU, Yulin Pan
20, 2023
8:26AM - The kinetic energy spectra of oceanic internal gravity waves (IGWs) from recent field measurements exhibit large variability, deviating from the standard Garrett-Munk (GM) models. However,
8:39AM the current finescale parameterization of turbulent dissipation is based on the GM76 model, which does not consider general spectral shapes. Thus an improved estimate of turbulent
dissipation for different spectra is needed for better parameterization of ocean mixing for global circulation and climate models. The rate of turbulent dissipation occurring at small
scales can be inferred from knowledge of energy transfer due to nonlinear wave-wave interactions at intermediate scales. In this work, we conduct direct calculation of energy transfer based
on the wave kinetic equation in the wave turbulence theory and compare the energy flux across a critical vertical wavenumber that provides energy available for dissipation with the estimate
from finescale parameterization. Three representative spectra, i.e., the GM75 and GM76 models as well as a spectrum fitted from observation, are analyzed. Key mechanisms, i.e., local and
three non-local interactions (parametric subharmonic instability, elastic scattering and induced diffusion) are identified with their contribution to the energy transfer quantified. This
will shed light on a new formulation of finescale parameterization incorporating varying spectral forms of IGWs and a realistic ocean environment.
Monday, L44.00004: Transition from weak turbulence to collapse turbulence regimes in the MMT model
November Ashleigh P Simonis, Yulin Pan
20, 2023
8:39AM - Understanding the role of coherent structures emerging from a field of random waves is a topic of great interest in the nonlinear wave community. While there has been extensive research on
8:52AM the topic of strongly nonlinear localized coherent structures (e.g., solitons and breathers) in integrable systems, the behavior of such structures in nonintegrable systems is not yet as
well understood. We study the forced-dissipated focusing one-dimensional (1D) Majda-McLaughlin-Tabak (MMT) with localized wave collapses as a result of soliton instability. Our results show
that when the forcing perturbation strength is weak, there are few wave collapses in the field and there is good agreement with weak wave turbulence (WTT) predictions. As the forcing
perturbation strength increases, we see an increase in high amplitude collapses, intermittency, and the departure from a power-law spectrum to an exponentially decaying spectrum resembling
that of a two-species gas (comprised of waves and collapses). This is a novel discovery in the context of the MMT model and can be thought of as an analogy to a soliton gas in integrable
turbulence. The transition from a weak turbulence regime to a strongly nonlinear “collapse” turbulence regime is also a new feature identified for the MMT model.
Monday, L44.00005: Three-wave resonant interactions between two dispersion branches
November Filip Novkoski, Eric Falcon, Chi-Tuong Pham
20, 2023
8:52AM - We report the experimental observation of nonlinear three-wave resonant interactions between two different branches of the dispersion relation of hydrodynamic waves, namely the
9:05AM gravity-capillary and sloshing modes [1]. These atypical interactions are investigated within a torus of fluid for which the sloshing mode can be easily excited. A triadic resonance
instability is then observed due to this three-wave two-branch interaction mechanism, with a mother wave on the sloshing branch generating two gravity-capillary daughter waves. The waves
are shown to be phase-locked and the efficiency of this interaction is found to be maximal when the gravity-capillary phase velocity matches the group velocity of the sloshing mode. For a
stronger forcing, additional waves are generated by a cascade of three-wave interactions populating the wave spectrum, both at large and small scales. Such a three-wave two-branch
interaction mechanism is probably not restricted to hydrodynamics and could be of interest in other systems involving several propagation modes
[1]. F. Novkoski, C.-T. Pham and E. Falcon, Evidence of experimental three-wave resonant interactions between two dispersion branches, Phys. Rev. E, 107, 045101 (2023).
20, 2023
9:05AM - L44.00006: Abstract Withdrawn
Monday, L44.00007: Experimental evidence of the dispersion relation of Kelvin waves along a free-surface vortex
November Jason Barckicke, Christophe Gissinger, Eric Falcon
20, 2023
9:18AM - Kelvin waves are waves that propagate along vortices in turbulent flows or in quantum turbulence. Although ubiquitous in nature, they are challenging to access experimentally. Here, we
9:31AM investigate a free-surface vortex, like a bathtub vortex, that forms at the interface between water and air, within a container with a hole at its bottom, in response to injectors arranged
circularly around the outlet. In this out-of-equilibrium stationary state, the vortex extends vertically over 50 cm with a diameter of the order of the millimeter. When excited using a
wavemaker, we experimentally evidence Kelvin waves propagating along such a vortex and report their full dispersion relation for the first time. The latter exhibits a rich spectral
structure with several branches, as helical bending modes. Our findings pave the way for the experimental investigation of Kelvin wave turbulence predicted theoretically.
Monday, L44.00008: Experimental evidence of intermittency in a random shock-wave regime
November Guillaume Ricard, Eric Falcon
20, 2023
9:31AM - We report the experimental observation of the dynamical and statistical properties of a wave field dominated by random shock waves on the surface of a fluid. By using a magnetic fluid
9:44AM (ferrofluid) within a high external magnetic field, we successfully achieved an experimentally nearly nondispersive surface-wave field [1]. Conversely to theoretical Burgers shock waves,
the shock-wave fronts are not fully vertical, but drive the dynamics [1]. We also experimentally evidence, for the first time, that this field dominated by random shock waves generates
intense small-scale intermittency [2]. The statistical properties of this intermittency are then found to be in good agreement with a Burgerslike intermittency model, modified to take
account of the finite steepness of the experimental shock waves [2].
[1] G. Ricard and E. Falcon, Transition from wave turbulence to acousticlike shock wave regime, Phys. Rev. Fluids 8, 014804 (2023)
[2] G. Ricard and E. Falcon, Experimental evidence of random shock-wave intermittency, Submitted to Phys. Rev. E (Letters) (2023)
Monday, L44.00009: Fast-Slow Wave Transitions Induced by a Random Mean Flow
November Samuel Boury, Oliver Bühler, Jalal Shatah
20, 2023
9:44AM - Motivated by recent asymptotic results in atmosphere-ocean fluid dynamics, we present an idealized numerical and theoretical study of two-dimensional dispersive waves propagating through a
9:57AM small-amplitude random mean flow. The objective is to delineate clearly the conditions under which the cumulative Doppler-shifting and refraction by the mean flow can change the group
velocity of the waves not only in direction, but also in magnitude. The latter effect enables a possible transition from fast to slow waves, which behave very differently. Within our model
we find the conditions on the dispersion relation and the mean flow amplitude that allow or rule out such fast–slow transitions. For steady mean flows we determine a novel finite mean flow
amplitude threshold below which such transitions can be ruled out indefinitely. For unsteady mean flows a sufficiently rapid rate of change means that this threshold goes to zero, i.e., in
this scenario all waves eventually undergo a fast–slow transition regardless of mean flow amplitude, with corresponding implications for the long-term fate of these waves.
Monday, L44.00010: Amplification of Maximum Ice Bending Strain and Reduction of Wave Energy Transmission due to Sum-Frequency Triad Wave Interactions in a Finite-Length Sea Ice Sheet
November Max Pierce, Yuming Liu, Dick K Yue
20, 2023
9:57AM - Floating sea ice acts as a low-pass filter of incident wave energy from open water, allowing only long waves to penetrate far past the ice boundary. However, nonlinear sum-frequency
10:10AM interactions among longer waves propagating through an ice sheet transfer energy to high frequency waves which are only minimally transmitted past the leading ice edge from open water. We
consider leading-order triad interactions in an ice sheet of finite length through direct numerical simulations using a modified high-order spectral (HOS) method. We demonstrate that
generated higher frequency waves result in more than twice the maximum bending strain predicted by linear theory, affecting the occurrence of ice breakup, as well as an appreciable decrease
in transmitted wave energy flux, modifying the understanding of wave attenuation through an aggregate ice field. The extent of these nonlinear effects is shown to depend on a parameter in
terms of ice length, wavelength and steepness.
Monday, L44.00011: Application of Surf-riding and Broaching mode based on IMO Second-Generation Intact Stability Criteria for the ships
November Dongmin Shin, Byungyoung Moon
20, 2023
10:10AM - IMO (International Maritime Organization) have recently discussed the technical problems related to the second-generation intact stability criteria of ships. The second-generation intact
10:23AM stability criteria refer to five modes of vulnerability when the ship sailing in the ocean. In this study, we described a method to verify the criteria of the surf-riding/broaching. In case
that Lv1 (Level 1) vulnerability criteria is not satisfied based on the relatively simple calculation using the Froude number (Fn), we presented the calculation procedure for the Lv2 (Level
2) criteria considering the hydrodynamics in waves. The results were reviewed based on the data for given previous ships. In absence of ship-specific data, a similar Lv2 result was
confirmed by comparing the result obtained by calculating the added mass with the case where the added mass was 10% of the ship mass. This result will contribute to basic ship design
process according to the IMO draft regulation.
Monday, L44.00012: Cauchy problem for a loaded integro-differential equation
November Umida Baltaeva
20, 2023
10:23AM - PDEs and integro-differential equations of the convolution type arise in mathematical models of physical, biological, and technical systems and in other areas where it is necessary to take
10:36AM into account the history of processes. Constitutive relations in a linear processes of inhomogeneous diffusion and propagation of waves with memory contain a time- and space-dependent
Author memory kernel. Problems of memory kernels identification in parabolic and hyperbolic integro-differential equations have been intensively studied.
Attending In many cases, the equations describing the propagation electrodynamic and elastic waves reduced to hyperbolic equations with integral convolution. Based on the foregoing, iwe study an
analog of the Cauchy problem for a generalized loaded wave equation involving convolution-type operators. The study aims to construct optimal representations of the solution of the
hyperbolic type equation and to study questions of the existence and uniqueness of the solution to the Cauchy problem for the loaded differential equation.
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How To Get STAGGERING Returns?
Nifty went up for the second successive day today closing 25 points in the green. As mentioned yesterday, there are signs visible that we may not go down further and that the trend might reverse from
the bottom made yesterday. Though, a pullback to 5630 was expected even in case of a bearish trend but the fact that it made a high of 5650 and that it even closed above that level of 5630 (closed at
5645 today) makes it look like the trend may have reversed now. If I have any short positions, I should be closing them now and start buying in small quantities. A confirmation that we are in an
uptrend will come when the Nifty closes above 5730. We should be looking to fully invest our money at that stage.
A lot of people ask me what stocks have I invested in personally. Well, I do have some holdings but I prefer to put my money where I see good returns. And I find that in commodities. Technical
Analysis and your discipline put to work together on leveraged products works wonders. I know some of you would be going - "Oh, Commodities. They are so risky!" Well, I'll explain why people call it
risky. Let's take the case of Gold, where the minimum we can buy is 1 kg. and in multiples of that. Gold costing 31000 per 10 grams today means that one kilogram of Gold would cost Rs.31 lakhs. Now
that doesn't mean that one has to pay Rs.31 lakhs to buy a kg. of Gold. In derivatives, whether stocks, commodities or currencies, one just has to pay a margin amount. Let's assume that there is a
margin of 5% on Gold. In that case one would have to pay Rs.1.55 lakhs to take the position. Lets assume that a week after buying Gold goes up to 31500, thus gaining Rs.500 per 10 grams. That would
mean a total profit of Rs.50,000 on a kilogram of Gold. Since the price of a kg. of Gold was Rs.31 lakhs, it translates into a return of 1.62% on the product. But when it comes to return on
investment (since you had invested only Rs.1,55,000) it gives you a return of 32.26%. A very good return on investment, indeed. Now, Gold could very well have gone DOWN by Rs.500 which would have
meant that you would have lost 32.26% of your money and would have been left with only Rs.1,05,000. Though, the loss on Gold was only 1.62% but you lost 32% of your money. That can be quite costly
and quite risky when you put your entire savings to do that. And for the majority of the people, 1.5 lakhs is like their entire savings. But, unlike stocks, there are mini and micro lots available
too. For example, Gold Guinea is a lot size of only 8 grams, which means the total product value is Rs.24,800 and a 5% margin on that would be only Rs.1,240 and a Rs.500 drop in the price of Gold
would mean you would have lost Rs.400/- only. Still a 32% loss on investment but a lot more affordable. This is where Technical Analysis comes in handy. It ensures, in most cases (if you have good
knowledge and experience), that about 70% of your trades turn out to be profitable. And if your rules are good - meaning if you maintain a risk reward ratio of 1:2, which means that for every rupee
that is at risk, you are expecting a return of Rs.2 then it means that if you take 1 trade a day, it is about 20 trades a month. A 70% ratio would mean 14 of your trades would work out to be
profitable and only 6 to be loss making. A loss of 30% on the losing trades would mean a 180% loss but on the remaining 14 trades you make a profit of 420% meaning a total profit of 240% per month.
Of course, you don't gain so heavily because usually I look for about a movement of half a percent (instead of the 1.62% taken in this example) to book my profits. That would still mean a 7% profit
on 14 winning trades and 1.5% loss on the 6 losing trades meaning a net profit of about 5% every month or a 60% return per annum. With a little more risk and some luck, it could go up to as high as
8-10% every month. Isn't that good? I would say, it's awesome. And that's where I invest most of my money. I know there is a lot of number crunching in this post and this may even sound confusing to
some. But I'm there to help. Do drop me a mail or leave a comment below to let me know that you need help and we'll talk. Yes, I can manage your money on your behalf and we can work out an
arrangement. Yes, I know, you will be missing out on the fun and the kick that you get from trading but ultimately, you will be getting what you desire most - MONEY. Please do subscribe to my posts,
so that all posts are delivered free to your inbox and you don't miss any useful analysis of the markets in the future. Happy Investing!!! | {"url":"http://www.sharma.es/2012/11/how-to-get-staggering-returns.html","timestamp":"2024-11-09T09:22:33Z","content_type":"application/xhtml+xml","content_length":"79714","record_id":"<urn:uuid:9c9db97c-45c5-4889-9818-c34fbd2af595>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00522.warc.gz"} |
How to Incorporate Mean-Variance Optimization Into Stock Risk Management?
Mean-variance optimization is a quantitative approach used in stock risk management to construct an optimal portfolio that maximizes returns while minimizing risk. It allows investors to allocate
their capital efficiently among different assets by considering their expected returns and volatilities.
Incorporating mean-variance optimization into stock risk management involves several steps. First, historical returns and volatilities of individual stocks or assets in the portfolio are calculated.
This data is used to estimate future expected returns and volatilities.
Next, a covariance matrix is constructed to capture the relationships between the different stocks or assets. The covariance matrix represents the measure of how the returns of two assets move
together or diverge from each other. It helps in understanding the diversification benefits that can be achieved by combining different assets in a portfolio.
The mean-variance optimization model then uses these expected returns, volatilities, and the covariance matrix to determine the optimal portfolio allocation. The goal is to find an allocation that
maximizes the portfolio's expected return for a given level of risk or minimizes the risk for a given level of expected return.
The optimization process involves solving mathematical equations that balance the trade-off between higher returns and greater risk. It searches for the portfolio combination of assets that lies on
the efficient frontier, which represents the set of portfolios with the highest expected return for a given level of risk.
After obtaining the optimal portfolio allocation, it is important to regularly monitor and rebalance the portfolio to maintain the desired risk profile. This involves adjusting the portfolio weights
based on changes in asset prices, returns, and volatilities to ensure that the portfolio remains aligned with the risk management objectives.
In summary, incorporating mean-variance optimization into stock risk management involves estimating expected returns and volatilities, constructing a covariance matrix, optimizing the portfolio
allocation, and regularly monitoring and rebalancing the portfolio. It helps investors make informed decisions regarding the allocation of their capital in order to achieve the desired balance
between risk and return.
How to calculate mean-variance optimization in stock risk management?
Mean-Variance Optimization is a quantitative method used in stock risk management to calculate the optimal portfolio allocation that balances potentially conflicting objectives of maximizing expected
returns and minimizing portfolio risk. The process involves the following steps:
1. Define the Investment Universe: Identify the set of assets or stocks that you wish to consider for your portfolio. This can include all available stocks or a specific subset based on certain
2. Collect Data: Gather historical data for the selected stocks, including prices, returns, and other relevant financial metrics. The data set should cover a sufficiently long period to capture
various market conditions.
3. Calculate Expected Returns: Determine the expected returns for each stock in your investment universe. This can be done using historical returns, fundamental analysis, or other forecasting
methods. Common approaches include calculating average historical returns or using analyst consensus estimates.
4. Estimate Covariance Matrix: Calculate the covariance matrix, which measures the relationship or co-movement between the returns of different stocks. This matrix quantifies the historical
relationships and dependencies between stock returns. There are various statistical techniques to estimate covariance, such as the sample covariance matrix or more advanced methods like shrinkage
5. Define Risk Tolerance: Establish your risk tolerance or the level of risk you are willing to accept. This can be subjective and based on personal preference, investment goals, or regulatory
6. Formulate Objective Function: Create an objective function that combines the expected returns and risk of the portfolio. Typically, the objective function is a trade-off between maximizing
expected returns and minimizing portfolio risk. The most commonly used objective function is the Markowitz mean-variance model.
7. Apply Optimization Techniques: Utilize optimization techniques, such as quadratic programming or other optimization algorithms, to calculate the optimal portfolio allocation that maximizes
returns given the defined risk tolerance. This involves minimizing the portfolio variance or standard deviation, subject to various constraints, such as budget constraints, minimum or maximum
allocation limits, or risk constraints.
8. Analyze Results: Analyze the optimized portfolio allocation and evaluate its characteristics, including expected returns, risk metrics (variance, standard deviation), and other relevant
performance statistics. Compare the optimized portfolio with other benchmark portfolios or alternative strategies.
9. Monitor and Rebalance: Regularly monitor the performance of the optimized portfolio and rebalance it periodically to maintain the desired allocation and risk levels. Market conditions or changes
in stock fundamentals may necessitate adjustments to the portfolio.
It is essential to note that Mean-Variance Optimization, while widely used, has its limitations and assumptions, such as assuming returns follow a normal distribution and assuming investors only care
about expected returns and risk. Hence, results should be interpreted with caution and complemented by additional qualitative analysis and judgment.
How to select an appropriate risk-free rate in mean-variance optimization?
Selecting an appropriate risk-free rate in mean-variance optimization involves considering several factors. Here are some steps to help you select the appropriate risk-free rate:
1. Determine the investment horizon: Define the time period over which you plan to make investments. The risk-free rate should align with the investment duration.
2. Identify the currency: Decide the currency in which you will make investments. The risk-free rate should correspond to the currency risk associated with the investment.
3. Consider the risk-free rate benchmark: Look for benchmark rates widely considered as risk-free, such as government bond yields. These rates are traditionally assumed to be free of default risk.
4. Evaluate the investment objective: Understand the purpose of your investment. If it aims to fund a specific goal, like retirement, the risk-free rate should reflect the desired time horizon and
the level of risk tolerance.
5. Analyze inflation expectations: Consider the expected inflation rate in your selected currency. The risk-free rate should be adjusted accordingly to account for the impact of inflation on the
real return of your investment.
6. Review historical data: Examine historical risk-free rates to understand their typical range. This can help you assess whether the current rate is relatively high or low and adjust your
expectations accordingly.
7. Consult financial professionals: Seek advice from financial professionals, such as investment advisors or wealth managers, who can provide insights into the prevailing risk-free rates and their
Remember, the risk-free rate is a crucial component of mean-variance optimization and can impact the optimal portfolio allocation. By considering the factors outlined above, you can choose a
risk-free rate that better aligns with your investment needs and objectives.
How to incorporate liquidity constraints in mean-variance optimization?
To incorporate liquidity constraints in mean-variance optimization, you can use one of the following approaches:
1. Constrained Mean-Variance Optimization: One way to incorporate liquidity constraints is by explicitly including them as constraints in the mean-variance optimization problem. This can be done by
setting limits on the maximum or minimum allocation to certain assets or asset classes. For example, you can specify a minimum or maximum percentage allocation to liquid assets in the portfolio.
2. Transaction Costs: Liquidity constraints can also be incorporated indirectly by considering transaction costs in the optimization process. Transaction costs are incurred when buying or selling
securities, and they can have a significant impact on the portfolio returns. By incorporating transaction costs, the optimization process will consider the impact of liquidity constraints on the
portfolio's performance.
3. Illiquid Asset Proxies: If you have illiquid assets in your portfolio that cannot be included directly in the optimization process, you can use proxies for those assets. Proxy assets with similar
characteristics or factors can be selected to represent the illiquid assets. These proxies can then be included in the optimization process, allowing for the consideration of liquidity
4. Slippage and Market Impact Models: Slippage refers to the difference between the expected and actual execution price of a trade, while market impact refers to the effect of the trade on market
prices. Incorporating slippage and market impact models into the optimization process helps to capture the liquidity constraints. These models estimate the costs and constraints associated with
executing trades in illiquid assets.
By incorporating liquidity constraints in mean-variance optimization, you can ensure that the resulting portfolio allocation is realistic and complies with the liquidity needs of the investor. | {"url":"https://small--loans.com/blog/how-to-incorporate-mean-variance-optimization-into","timestamp":"2024-11-03T12:53:46Z","content_type":"text/html","content_length":"197962","record_id":"<urn:uuid:cf0aa7a1-b6ed-499e-92f7-7d72ebdd3ed9>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00459.warc.gz"} |
Fixed-Point Logics on Trees
DS-2010-08: Gheerbrant, Amélie (2010) Fixed-Point Logics on Trees. Doctoral thesis, University of Amsterdam.
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In this thesis, we study proof-theoretic and model-theoretic aspects
of some widely used modal and quantified fixed-point logics on trees.
Chapter 2 includes basics of modal logic, temporal logic, fixed-point
logics, and some first-order and higher-order logics of tree
In Chapter 3, we consider the class of finite node-labelled
sibling-ordered trees. We present axiomatizations of its monadic
second-order logic (MSO), monadic transitive closure logic (FO(TC1 ))
and monadic least fixed-point logic (FO(LFP1 )) theories. Using
model-theoretic techniques, we show by a uniform argument that these
axiomatizations are complete, i.e., each formula which is valid on all
finite trees is provable using our axioms.
In Chapter 4 we consider various fragments and extensions of
propositional linear temporal logic (LTL), obtained by restricting the
set of temporal connectives or by adding a least fixed-point construct
to the language. Using techniques from abstract model-theory, for each
of these logics we identify its smallest extension that has Craig
interpolation. Depending on the underlying set of temporal operators,
this framework turns out to be one of the following three logics: the
fragment of LTL having only the Next operator; the extension of LTL
with a least fixed-point operator µ (known as linear time µ-calculus);
and µTL(U), the least fixed-point extension of the "Until-only"
fragment of LTL.
In Chapter 5, we focus on the logic µTL(U), that we identified in the
previous chapter as the stutter-invariant fragment of the linear-time
µ-calculus µTL. We also identified this logic as one of the three only
temporal fragments of µTL that satisfy Craig interpolation. Complete
axiom systems were known for the two other fragments, but this was not
the case for µTL(U). We provide complete axiomatizations of µTL(U) on
the class of finite words and on the class of ω-words. For this
purpose, we introduce a new logic µTL(♦_Γ), a variation of µTL where
the "Next time" operator is replaced by the family of its
stutter-invariant counterparts. This logic has exactly the same
expressive power as µTL(U). Using known results for µTL, we first
prove completeness for µTL(♦_Γ), which then allows us to obtain
completeness for µTL(U).
Finally, in Chapter 6 we take our style of analysis via modal and
temporal fixed-point logics to games. Current methods for solving games
embody a form of "procedural rationality" that invites logical
analysis in its own right. This chapter is a case study of Backward
Induction for extensive games. We consider a number of analyses from
recent years in terms of knowledge and belief update in logics that
also involve preference structure, and we prove that they are all
mathematically equivalent in the perspective of fixed-point logics of
trees. We then generalize our perspective on games to an exploration
of fixed-point logics on finite trees that best fit game-theoretic
equilibria. We end with a broader program for merging computational
logics to the area of game theory.
Item Type: Thesis (Doctoral)
Report Nr: DS-2010-08
Series Name: ILLC Dissertation (DS) Series
Year: 2010
Subjects: Language
Depositing User: Dr Marco Vervoort
Date Deposited: 14 Jun 2022 15:16
Last Modified: 14 Jun 2022 15:16
URI: https://eprints.illc.uva.nl/id/eprint/2091
Actions (login required) | {"url":"https://eprints.illc.uva.nl/id/eprint/2091/","timestamp":"2024-11-09T15:48:43Z","content_type":"application/xhtml+xml","content_length":"27035","record_id":"<urn:uuid:5420bd30-1831-4fe3-ad66-5f66db2a0afe>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00696.warc.gz"} |
What are negative numbers
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Activity - The idea of negative numbers and operations on negative numbers
Objectives of the activity
1. Develop an idea that negative numbers are part of a type of numbers.
2. That negative numbers are continuous.
3. Understand that positive and negative signs indicate opposite quantities.
4. Represent the number line with zero as a place holder.
Estimated Time
This is an activity for a lesson to introduce negative numbers; can be more 2-3 periods.
Materials/ Resources needed
• More apples (or any other fruit)
• Small cards for writing down the numbers as well as for writing the operations
• Pencils, etc
• Two boxes – one of negative numbers and the other of positive numbers
Prerequisites/Instructions, if any
Students should be familiar with the terms such as more/less, add/take away and increase/decrease.
They should be able to compare different values, and determine greater than, less than and equal to.
Multimedia resources
Website interactives/ links/ simulations/ Geogebra Applets
An interactive: Tug of war
Process (How to do the activity)
Use the same example as before (apples and counting) to discuss add, take away etc. This activity is in multiple part.
Developmental Questions (What discussion questions)
Part 1
1. By the previous activity we have 11 apples. Now we add a few more. Let us say we add 4 more. We have 15 apples.
11 + 2 = 13 ( + 2 signifies increase in apples)
13 + 2 = 15 Write the expression. Again + signifies increase
1. 2. Let us say I ask the question – what should I add to 15 to make the number of apples 10? They will say take away Let us say we cannot use the word “take away”.
Write the expression like this 15 + (-5) = 10.
1. The numbers that when added to a number increase the original quantity are called positive numbers. The numbers that when added to a number decrease the original quantity are called negative
numbers. The negative number is thus an opposite of the positive number.
2. Now frame the question as follows
• What do I have to add to 15 to make it 7? The answer is (-8).
• What do I add to 15 to get 8? (-7)
• What do I add to 15 to get 9? (-6)
• What do I add to 15 to get 10? (-5)
• When I add (-7). I get 8. For me to get 9, I have to add a number greater than (-7) and I have added (-6). Similarly (-5) is greater than (-6). So the larger negative number is actually smaller.
1. Now we transition from numbers representing some quantities to numbers being manipulated as numbers.
Part 2
1. We have seen what negative numbers are. We will see how to work with them.
2. We have seen that negative numbers are such that when we add them the quantity decreases. What happens when we subtract them?
3. Extend the activity of what should I add to 20 to make it 10, 11, 12 and so on? Extend it all the way to 30. Let the students pull the numbers out and place them along the wall/ stock on the wall
etc. You will see the number line.
4. Let us pull out sets of numbers the same number but from the positive box and negative box.
3, -3
Add them
3 + (-3) = 0
4, -4
Add them
4 + (-4) = 0
This addition could be explained like this. When I add three the quantity increases. When I add -3 the quantity decreases. So + 3 and -3 are the same in magnitude but do opposite things. For every
positive integer, there is a negative integer. Discuss examples of borrowing from the bank; someone giving a loan.
5. Now I have 25 (from the number box). I am going to subtract (-5). What will happen? When I add (-5), it becomes 20. Since negative numbers behave in this opposite way, subtracting (-5) should
become 30? 25 - (-5) = 30...this is equivalent to adding 5 to 25. Hence we say (-)*(-) = +
Part 3
Now what happens when we multiply negative numbers?
1. Let us take -3 x 3. From the process of multiplication is repeated addition we can explain as take -3 once, take it the second time and third time. We have -3, -3, -3. We have -9.
• 3 x -3 . Again multiplication is the process of repeated addition. Except I am multiplying it by -3. Then I have to look at the operation as opposite. I am giving away 3 once, twice and third
time. We have -9.
• -3 x -3. How do we do this?
Let us look at the table below.
- 3 x 3 = -9
- 3 x 2 = -6
- 3 x 1 = -3
- 3 x 0 = 0
- 3 x-1 = 3
- 3 x-2 = 6
- 3 x-3 = 9
Extend the pattern above. By simple pattern evaluation we see it is 3, 6 and 9. We have shown the number line above. It makes sense logically that the next number is 3 and that it becomes positive.
The best way to extend this to division is to treat this as multiplication by fraction and extend these rules.
Yet another way of explaining could be like this. When we add -3, -3 times we are actually operating with two opposites. The (-3) times signifies the opposite of the repeated addition, think of it as
repeated subtraction. I am subtracting -3 once (in effect, adding 3); -(-3) second time (adding another 3) and -(-3) for the third time -(-3) (adding one more 3). We get 9. Hence -(-) is positive.
Evaluation (Questions for assessment of the child)
1. Start doing this activity with objects, then numbers depending upon the level of the student.
2. 6 - 1? - what is the answer?
3. 6 - 2? - what is the answer?
4. Continue this exercise until we get to 6 - 7?
5. Can we take away the objects? We start looking at these as a special kind of number and get the number line to move to -1. Extend the number line construction. What does this – 7 represent (they
should say it means that when I add this it is reducing the quantity)
6. Arrange a set of randomly chosen positive and negative numbers (integers) along the number line in increasing order.
7. See attached workheets
Question Corner
Activity Keywords
Negative numbers
To link back to the concept page Numbers | {"url":"http://karnatakaeducation.org.in/KOER/en/index.php/What_are_negative_numbers","timestamp":"2024-11-13T19:09:11Z","content_type":"text/html","content_length":"42349","record_id":"<urn:uuid:5680549f-94d7-490b-b0e6-a867ac034d27>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00118.warc.gz"} |
Intersection algorithms
Intersection algorithms for spline and NURBS curves and surfaces have been a central research topic since 1980. The first challenges addressed were related to the intersection of
B-spline curves, and the intersection of B-spline curves and elementary curves. Then in 1984 we got the first results on recursive intersection algorithms for B-spline surface, with
a gradual deployment in industry. The development of SISL in 1989 allowed us to implement a second generation of surface intersection algorithms still keeping a focus on transversal
intersections. With the discovery of approximate implicitization we at last had a tool for addressing singular intersections, and for addressing surface self-intersections. As
approximate implicitization is computational intensive the deployment in industry is currently limited. However, the new massive parallel computational resource of many core
processors satisfies the computational requirements of approximate implicitization and opens up new approaches for the architecture of intersection algorithms.
The shells of the object designed are described by composition of surfaces. Sometimes it is possible to define the objects by rectangular surfaces (NURBS) where the surfaces just
share common boundaries and there is no need for trimming away parts of a surface. However, often the surfaces will be too large to be able to represent a desired shape and to
design it with the constructive tools of the CAD-system. The curve where the surfaces meet will be an edge in the CAD-model to be found by intersecting the surfaces. For
representing the relationships between surfaces, curves and surfaces in CAD-models, boundary structures are used.
Low quality of intersection algorithms expensiv for industry Near singular intersection and non-regular parametrization an intersection challenge
In the Workshop on Mathematical Foundations of CAD (Mathematical Sciences Research Calculation of intersections is not difficult when the surfaces have a regular Example:
Institute, Berkeley, CA. June 4-5, 1999.) the consensus was that: “The single parameterization and are not near parallel along intersection curves (transversal Transversal
greatest cause of poor reliability of CAD systems is lack of topologically intersection). However, if the parameterization is not regular, or if the surfaces intersect intersection.
consistent surface intersection algorithms.” Tom Peters, Computer Science and in regions where the surfaces are parallel or near parallel, intersection calculation gets
Engineering, The University of Connecticut, estimated the cost to be $1 Billion/ challenging. One ambition of the GAIA II project has been to provide more accurate solutions Tookits with
year. For more information consult: for such intersections than what has been available by exploiting teh potential of approximate Intersection
implcitization. algorithms
• Article by R. Farouki "Closing the Gap Between CAD Model and Downstream
Applications” in SIAM News 6-1999. After the GAIA-project ended we have stabilized he prototype intersection algorithms. However, .
still the time is little early for industrial use as the approach is computational intensive.
Now a decade later these challenges still remain. However, with the introduction of heterogeneous many core CPUs sufficient computational Example: Near
performance will be available for industrial use. Topics now addressed are: singular
• Parallelize of approximate implicitization, part of PhD work in (2009-2010). intersection.
• Redesign the algorithms to allow recursive subdivision to exploit many core CPUs, part of
PhD work in (2009-2011).
Self-intersections are a challenge Isogeometric analysis poses new intersection challenges Example: Open
self intersection
Another challenge within CAD has been to avoid self-intersections in the shells of Isogeometric analysis replaces the boundary structure type CAD-model, by modelling with
the volume described. These self-intersections are of different types: trivariate rational spline volumes. Consequently the intersection of trivariate spline volumes
will have to be handled. The current examples of isogeometric analysis avoid the intersection
• A shell self-intersection is an intersection between different surfaces in the challenges by directly creating correct models. However, before the isogeometric analysis can
shell of the volume, with the intersection being no part of the adjacency be deployed in industry on large scale better approaches for model creation should be devised.
description of the shell.
• An open self-intersection is a self-intersection that occurs inside a • Building the model from CAD-models
parametric surface and the self-intersection curve does not describe a loop in • Flexible Boolean operations between trivariate spline represented volumes
the parameter domain of the surface. These can be found by first subdividing
the surface into smaller pieces that themselves do not have closed A component of such functionality will be to be able to intersect trivariate volumes and check
self-intersection. The pieces can then be intersected. that the trivariate volumes do not turn back on themselves (self-intersection).
• A closed self-intersection is a self-intersection that occurs inside a
parametric surface, where the intersection curve describes a loop in the
parameter domain of the surface.
Intersection algorithms for spline and NURBS curves and surfaces have been a central research topic since 1980. The first challenges addressed were related to the intersection of B-spline curves, and
the intersection of B-spline curves and elementary curves. Then in 1984 we got the first results on recursive intersection algorithms for B-spline surface, with a gradual deployment in industry. The
development of SISL in 1989 allowed us to implement a second generation of surface intersection algorithms still keeping a focus on transversal intersections. With the discovery of approximate
implicitization we at last had a tool for addressing singular intersections, and for addressing surface self-intersections. As approximate implicitization is computational intensive the deployment in
industry is currently limited. However, the new massive parallel computational resource of many core processors satisfies the computational requirements of approximate implicitization and opens up
new approaches for the architecture of intersection algorithms.
Low quality of intersection algorithms expensiv for industry Near singular intersection and non-regular parametrization an intersection challenge
In the Workshop on Mathematical Foundations of CAD (Mathematical Sciences Research Calculation of intersections is not difficult when the surfaces have a regular parameterization and are
Institute, Berkeley, CA. June 4-5, 1999.) the consensus was that: “The single greatest not near parallel along intersection curves (transversal intersection). However, if the
cause of poor reliability of CAD systems is lack of topologically consistent surface parameterization is not regular, or if the surfaces intersect in regions where the surfaces are
intersection algorithms.” Tom Peters, Computer Science and Engineering, The University of parallel or near parallel, intersection calculation gets challenging. One ambition of the GAIA II
Connecticut, estimated the cost to be $1 Billion/year. For more information consult: project has been to provide more accurate solutions for such intersections than what has been available
by exploiting teh potential of approximate implcitization.
• Article by R. Farouki "Closing the Gap Between CAD Model and Downstream Applications”
in SIAM News 6-1999. After the GAIA-project ended we have stabilized he prototype intersection algorithms. However, still
the time is little early for industrial use as the approach is computational intensive. However, with
Now a decade later these challenges still remain. the introduction of heterogeneous many core CPUs sufficient computational performance will be available
for industrial use. Topics now addressed are:
• Parallelize of approximate implicitization, part of PhD work in (2009-2010).
• Redesign the algorithms to allow recursive subdivision to exploit many core CPUs, part of PhD work
in (2009-2011).
Self-intersections are a challenge Isogeometric analysis poses new intersection challenges
Another challenge within CAD has been to avoid self-intersections in the shells of the Isogeometric analysis replaces the boundary structure type CAD-model, by modelling with trivariate
volume described. These self-intersections are of different types: rational spline volumes. Consequently the intersection of trivariate spline volumes will have to be
handled. The current examples of isogeometric analysis avoid the intersection challenges by directly
• A shell self-intersection is an intersection between different surfaces in the shell of creating correct models. However, before the isogeometric analysis can be deployed in industry on large
the volume, with the intersection being no part of the adjacency description of the scale better approaches for model creation should be devised.
• An open self-intersection is a self-intersection that occurs inside a parametric • Building the model from CAD-models
surface and the self-intersection curve does not describe a loop in the parameter • Flexible Boolean operations between trivariate spline represented volumes
domain of the surface. These can be found by first subdividing the surface into smaller
pieces that themselves do not have closed self-intersection. The pieces can then be A component of such functionality will be to be able to intersect trivariate volumes and check that the
intersected. trivariate volumes do not turn back on themselves (self-intersection).
• A closed self-intersection is a self-intersection that occurs inside a parametric
surface, where the intersection curve describes a loop in the parameter domain of the
In the Workshop on Mathematical Foundations of CAD (Mathematical Sciences Research Institute, Berkeley, CA. June 4-5, 1999.) the consensus was that: “The single greatest cause of poor reliability of
CAD systems is lack of topologically consistent surface intersection algorithms.” Tom Peters, Computer Science and Engineering, The University of Connecticut, estimated the cost to be $1 Billion/
year. For more information consult:
Calculation of intersections is not difficult when the surfaces have a regular parameterization and are not near parallel along intersection curves (transversal intersection). However, if the
parameterization is not regular, or if the surfaces intersect in regions where the surfaces are parallel or near parallel, intersection calculation gets challenging. One ambition of the GAIA II
project has been to provide more accurate solutions for such intersections than what has been available by exploiting teh potential of approximate implcitization.
After the GAIA-project ended we have stabilized he prototype intersection algorithms. However, still the time is little early for industrial use as the approach is computational intensive. However,
with the introduction of heterogeneous many core CPUs sufficient computational performance will be available for industrial use. Topics now addressed are:
Another challenge within CAD has been to avoid self-intersections in the shells of the volume described. These self-intersections are of different types:
Isogeometric analysis replaces the boundary structure type CAD-model, by modelling with trivariate rational spline volumes. Consequently the intersection of trivariate spline volumes will have to be
handled. The current examples of isogeometric analysis avoid the intersection challenges by directly creating correct models. However, before the isogeometric analysis can be deployed in industry on
large scale better approaches for model creation should be devised.
A component of such functionality will be to be able to intersect trivariate volumes and check that the trivariate volumes do not turn back on themselves (self-intersection). | {"url":"https://www.sintef.no/projectweb/computational-geometry/intersections/","timestamp":"2024-11-14T23:39:54Z","content_type":"text/html","content_length":"19599","record_id":"<urn:uuid:aa9b16a7-1faa-4aa3-92e2-c4b121f9a79f>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00179.warc.gz"} |
Better than Excel: Using R for Data Analysis
Better than Excel: Using R for Data Analysis
What is wrong with using Excel for conducting fire department data analysis? Nothing is wrong with it, but there are better ways. Fire Department data analysis often involves many, many steps that
can be time consuming and difficult to double-check using Excel. Here is an example of conducting a response time analysis:
• Step 1: Download raw data from your emergency dispatch center and read that data into CAD
• Step 2: Data includes 100 different dispatch codes that do not translate directly to what we consider major incident type categories such as fires, EMS, et cetera … so you code the 100 dispatch
types and use excel to create another column of incident types.
• Step 3: You only want to include emergency responses in your response time analysis, so you eliminate all public service calls.
• Step 4: Finally, you run the response time analysis by incident type using the 90th percentile times.
Imagine working for several hours to prepare a report that includes several different types of analysis. Then, after delivering the report, your chief ask whether or not the analysis included mutual
aid responses and that they wanted b0th percentile times instead of 90th percentile times. You probably don’t remember if where you did or did not include mutual aid responses and you realize that
you will have to redo significant portions of the analysis to compute 80th percentile times. Scenarios like this are exactly why Excel is not well-suited for complex analysis or analysis that you
expect to repeat on a regular basis (such as monthly reports).
So what is the answer? The answer is statistical programming languages that do the exact same thing as Excel, but in the form of computer scripts (basically text documents that contain computer
code). Imagine writing a word document that outlines exactly how you did your Excel analysis and with what data and having that word document actually do the work and spit out the results. The beauty
of this is that you can go back and update your work and rerun the analysis in seconds rather than hours. Furthermore, you can use that computer script to re-run the exact same analysis month after
month without having to do any work. Enter statistical programming languages …
There are several different statistical programming languages out there … SAS, SPSS, S, S-Plus and R are the most popular. R is, however, the only free program and it allows users to develop their
own additions for industry-specific analysis. I have been working on an R add-on package called FireTools that will allow you to read in CAD and NFIRS data and automatically run several different
types of fire department analysis. Once this package is released, I will use this blog to provide tutorials on exactly how to use this add-on package. In the meantime, you can already get started on
learning the basics of R. The following tutorial provides a good introduction using the analysis of baseball statistics.
After completing this tutorial you should begin to understand how analysis with R is similar to what can be done with Excel, but lends itself well to updating and repeating analysis. And, because R
allows users to develop and publish add-on packages, R is a much more powerful analysis tool than Excel. Good luck with the tutorial. | {"url":"https://www.firegeek.com/how-to-excel-on-crack-using-r-for-data-analysis/","timestamp":"2024-11-13T03:23:24Z","content_type":"text/html","content_length":"28550","record_id":"<urn:uuid:67d5987f-f90a-4814-a0eb-9fa6530e3c64>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00745.warc.gz"} |
An introduction to
1 Introduction
One of the main tasks of statistical modeling is to exploit the association between a response variable and multiple predictors. The linear model (LM), a simple parametric regression model, is often
used to capture linear dependence between response and predictors. As the extensions of the linear model, the other two common models: generalized linear model (GLM) and Cox’s proportional hazards
(CoxPH) model, depending on the types of responses, are modeled with mean linear to the inputs. When the number of predictors is large, parameter estimations in these models can be computationally
burdensome. At the same time, a Widely accepted heuristic rule for statistical modeling is Occam’s razor, which asks modelers to keep a good balance between the goodness of fit and model complexity,
leading to a relatively small subset of important predictors.
bestridge is a toolkit for the best subset ridge regression (BSRR) problems. Under many sparse learning regimes, \(L_0\) regularization can outperform commonly used feature selection methods (e.g., \
(L_1\) and MCP). We state the problem as
\[\begin{align*} \min_\beta& -2 \log L(\beta) + \lambda\Vert\beta\Vert_2^2 \quad s.t. \|\beta\|_0 \leq s \quad&(BSRR)\\ \end{align*}\]
where \(\log L(\beta)\) is the log-likelihood function in the GLM case or the log partial likelihood function In the Cox model. \(\Vert \beta\Vert_0\) denotes the \(L_0\) norm of \(\beta\), i.e., the
number of non-zeros in \(\beta\). The vanilla \(L_0\) regularization gains the minimum-variance unbiased estimator on the active set. For restricted fits, the estimation bias is positive, and it is
worthwhile if the decrease in variance exceeds the increase in bias. Here We would like to introduce the best subset ridge regression as an option of bias-variance tradeoff. This will add an \(L_2\)
norm shrinkage to the coefficients, the strength of which is controlled by the parameter \(\lambda\). The fitting is done over a grid of sparsity \(s\) and \(\lambda\) values to generate a
regularization path. For each candidate model size and \(\lambda\), the best subset ridge regression is solved by the \(L_2\) penalized primal-dual active set algorithm. This algorithm utilizes an
active set updating strategy via primal and dual variables and fits the sub-model by exploiting the fact that their support sets are non-overlap and complementary. For the case of method =
"sequential" if warms.start = "TRUE", we run this algorithm for a list of sequential combination of model sizes and \(\lambda\) and use the estimate from the last iteration as a warm start. For the
case of method = "psequential" and method = "pgsection", the Powell conjugate direction method is implemented. This method finds the optimal parameters by a bi-directional search along each search
vector, in turn. The bi-directional line search along each search vector can be done by sequential search path or golden section search, the line search method of which is specified by method =
"psequential" or method = "pgsection".
2 Quick start
This quick start guide walks you through the implementation of the best subset ridge regression on linear and logistic models.
2.1 Regression: linear model
We apply the methods to the data set reported in Scheetz et al. (2006), which contains gene expression levels of 18975 probes obtained from 120 rats. We are interested in finding probes that are
related to that of gene TRIM32 using linear regression. This gene had been known to cause Bardet-Biedl syndrome, which is a genetic disease of multiple organ systems including the retina. For
simplicity, the number of probes is reduced to 500 by picking out maximum 500 marginal correalation to Trim32 gene.
Supposed we wants to fit a best subset ridge regression model with \(\lambda\) taking values in 100 grid points from 100 to 0.001 and the model size between \(\max \{p, n/\log(n)\}\) (the decimal
portion is rounded) and 1, which is the default model size range where \(p\) denotes the number of total variables and \(n\) the sample size, via Generalized Information Criterion. We call the bsrr
function as follows:
The fitted coefficients can be extracted by running the coef function.
To make a prediction on the new data, a predict function can be used as follows:
If the option newx is omitted, the fitted linear predictors are used.
2.2 Classification: logistic regression
We now turn to the classification task. We use the data set duke (West et al. 2001). This data set contains microarray experiment for 86 breast cancer patients. We are interested in classify the
patients into estrogen receptor-positive (Status = 0) and estrogen receptor-negative (Status = 1) based on the expression level of the considered genes.
Supposed we wants to do this classification based on logistic regression with a BSRR model. This can be realized through calling the bsrr function with the parameter family = "binomial as follows:
Calling the plot routine on an bsrr object fitted by type = "bsrr" will provide a heatmap of GIC of the fitted path.
3 Advanced features
3.1 Censored response: Cox proportional hazard model
The bestridge package also supports studying the relationship between predictors variables and survival time based on the Cox proportional hazard model. We now demonstrate the usage on a real data
set, Alizadeh, Eisen, Davis, Ma, Lossos, Rosenwald, Boldrick, Sabet, Tran, Yu et al. (2000)(Alizadeh et al. 2000): gene-expression data in lymphoma patients. We illustrate the usage on a subset of
the full data set. There were 50 patients with measurements on 1000 probes.
To implement the best subset ridge regression on the Cox proportional hazard model, call the bestridge function with family specified to be "cox".
We use the summary function to draw a summary of the fitted bsrr object. 10 probes among the total 1000 are selected according to the result.
3.2 Criterion for tuning parameters selection
+ Information criterion: AIC, BIC, GIC, EBIC
+ Cross-validation
So far, we have been stick to the default Generalized Information Criterion for tuning parameter selections. In this package, we provide a bunch of criteria including the Akaike information criterion
(Akaike 1974) and Bayesian information criterion (Schwarz and others 1978), Generalized Information Criterion (Konishi and Kitagawa 1996), and extended BIC (Chen and Chen 2008, 2012), as well as
cross-validation. By default, bsrr selects the tuning parameters according to the Generalized Information Criterion.
To choose one of the information criteria to select the optimal values of model size \(s\) and shrinkage \(\lambda\), the input value tune in the bsrr function needs to be specified:
To use cross-validation for parameter selections, set the input value of tune = "cv". By default, 5-fold cross-validation will be conducted.
3.3 Paths for tuning parameters selection
+ Sequential method
+ Powell's method
We shall give a more explicit description of the parameters tuning paths. As mentioned in the introduction, we either apply a method = "sequential" path for examining each combination of the two
tuning parameters \(s\) and \(\lambda\) sequentially and chose the optimal one according to certain information criteria or cv, or a less computational burdensome Powell method (method =
"psequential" or method = "pgsection"). The initial starting point \(x_0\) of the Powell method is set to be the \((s_{min}, \lambda_{min})\), and the initial search vectors \(v_1, v_2\) are the
coordinate unit vectors. The method minimizes the value of chosen information criterion or cross-validation error by a bi-directional search along each search vector on the 2-dimensional tuning
parameter space composed of \(s\) and \(\lambda\), in turn. Callers can choose to find the optimal combination along each search vector sequentially or conducting a Golden-section search by setting
method = "psequential" or method = "pgsection". If warm.start = TRUE, we use the estimate from the last iteration in the PDAS algorithm as a warm start.
By default, the tuning parameters are determined by the Powell method through a golden-section line search and we exploit warm starts. The minimum \(\lambda\) value is set to be 0.01 and the maximum
100. The maximum model size is the smaller one of the total number of variables \(p\) and \(n/\log (n)\). Advanced users of this toolkit can change this default behavior and supply their own tuning
To conduct parameter selections sequentially, users should specify the method to be “sequential” and provide a list of \(\lambda\) to the lambda.list as well as an increasing list of model size to
the s.list.
To conduct parameter selections using the Powell method on a user-supplied tuning parameter space, users should assign values to s.min, s.max, lambda.min, and lambda.max. 100 values of \(\lambda\)
will be generated decreasing from lambda.max to lambda.min on the log scale. Here we do the line search sequentially.
3.4 Sure independence screening
We also provide feature screening option to deal with ultrahigh dimensional data for computational and statistical efficiency. Users can apply the sure independence screening method (Saldana and Feng
2018) based on maximum marginal likelihood estimators, to pre-exclude some irrelevant variables before fitting a model by pressing a integer to screening.num. The SIS will filter a set of variables
with size equals to screening.num. Then the active set updates are restricted on this set of variables.
4 Monte carol study
4.1 Settings
In this section, we will conduct a monte carol study on our proposed best subset ridge regression method and make a comparison with other commonly used variable selection methods in three aspects.
The first aspect is the predictive performance on a held-out testing data of the same size of training data. For linear regression, this is defined as \(\frac{\Vert X \beta^\dagger - X \hat{\beta}\
Vert _2}{\Vert X \beta^\dagger\Vert_2}\), where \(\beta^\dagger\) is the true coefficient and \(\hat{\beta}\) is an estimator. For logistic regression, we calculate the classification accuracy by the
area under the ROC curve (AUC). The second aspect is the coefficient estimation performance defined as \(\Vert \beta^\dagger - \hat{\beta}\Vert\), where \(\beta^\dagger\) denote the underlying true
coefficients and \(\beta^\dagger\) is out estimation. The third aspect is the selection performance in terms of true positive rate (TPR), false positive rate (FPR), the Matthews correlation
coefficient (MCC) score, and the model size.
We generate a multivariate Gaussian data matrix \(X_{n\times p} \sim MVN(0, \Sigma)\) and consider the variables to have exponential correlation. \(n\) denotes the sample size and \(p\) the number of
total variables, and we set \(n = 200\), \(p = 2000\). We consider the following instance of \(\Sigma := ((\sigma_{ij}))\), setting each entry \(\sigma_{ij} = 0.5 ^{|i-j|}\). We use a sparse
coefficient vector \(\beta^\dagger\) with \(20\) equi-spaced nonzero entries, each set to 1. Then the response vector y is generated with gaussian noise added. For linear regression, \(y = X\beta^\
dagger+\epsilon\). For logistic regression, \(y = Bernoulli(Pr(Y=1))\), where \(Pr(Y=1) = \exp (x^T \beta^\dagger + \epsilon)/(1+\exp (x^T \beta^\dagger + \epsilon))\). \(\epsilon\sim N(0, \sigma^2)
\) is independent of \(X\). We define the signal-to-noise ratio (SNR) by SNR = \(\frac{Var(X\beta^\dagger)}{Var(\epsilon)} = \frac{\beta^{\dagger T} \Sigma \beta^\dagger}{\sigma^2}\).
we compare the performance of the following methods:
1. (BSRR methods): Tuning parameters selected through a sequential path and the Powell method. Denoted as Grid-BSRR and Powell-BSRR.
2. (BSS method): The best subset selection method. We consider the primal-dual active subset selection method and use our own implementation in bestridge package.
3. (\(L_1\) method): Lasso estimator. We use the implementation of glmnet.
4. (\(L_1L_2\) method): Elastic net estimator. This uses a combination of the \(L_1\) and \(L_2\) regularization. We use the implementation of glmnet.
For BSRR estimators, the 2D grid of tuning parameters has \(\lambda\) taking 100 values between \(\lambda_{max}\) and \(\lambda_{min}\) on a log scale. The \(\lambda_{max}\) and \(\lambda_{min}\)
will be specified. For the BSS method, the model size parameter takes value in \(\{1, \dots, 30\}\). The parameter combined the \(L_1\) penalty and the \(L_2\) penalty in the Elastic net is fixed at
\(0.5\). And for \(L_1\) and \(L_1L_2\) methods, we use the default settings. The regularization parameters are chosen by 5-fold cross-validation.
4.2 Linear regression
Results for linear regression are reported in Figure 1. When SNR is low, BSS has the largest estimation error and selects the smallest model size compared with other methods, while the extra \(L_2\)
penalty in BSRR has effectively lowered the estimation error of BSS. For high SNR, the performance of BSS and BSRR is similar to each other. In terms of prediction error, all methods are good in the
high SNR scenario, but it costs the Lasso and the Elastic Net a very dense supports while still cannot catch up with BSS and BSRR. Notice that for SNR 100, the selected model size of the Elastic Net
is almost 7 times the true model size. This trend is no better for the Lasso. Of all measures across the whole SNR range, BSRR generally exhibits excellent performance—accurate in variables selection
and prediction.
Figure 1: Performance measures as the signal-to-noise ratio (SNR) is varied between 0.01 and 100 for linear regression.
4.3 Logistic regression
Results for linear regression are reported in Figure 2. The result shows that as the SNR rises, the extra shrinkage in BSRR methods helps the best subset ridge regression make impressive improvement
in terms of prediction accuracy, estimation ability, and variable selection, where it outperforms the state-of-the-art variable selection methods Lasso and Elastic Net.
Figure 2: Performance measures as the signal-to-noise ratio(SNR)is varied between 0.01 and 100 for logistic regression.
5 Conclusion
In the bestridge toolkit, we introduce the best subset ridge regression for solving the \(L_2\) regularized best subset selection problem. The \(L_2\) penalized PDAS algorithm allows identification
of the best sub-model with a prespecified model size and shrinkage via a primal-dual formulation of feasible solutions. To determine the best sub-model over all possible model sizes and shrinkage
parameters, both a sequential search method and a powell search method are provided. We find that estimators BSRR models do a better job than the BSS estimator and usually outperform other variable
selection methods in prediction, estimation and variable selection. | {"url":"https://cran.fhcrc.org/web/packages/bestridge/vignettes/An-introduction-to-bestridge.html","timestamp":"2024-11-03T15:36:10Z","content_type":"text/html","content_length":"409231","record_id":"<urn:uuid:c7592c89-6628-4fe8-bfa6-82a1eabaecdc>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00229.warc.gz"} |
When was the last round Fed hike 50-basis point and more?
It was 22 years ago in 2000, but counting back another cycle of 22 years, in 1978 was the most interesting. Current numbers mirror it in terms of its GDP, unemployment and inflation. What's even more
interesting was how it has developed the next two years
This year GDP should come in at 3%, March unemployment rate at 3.6% and inflation at 8.5%.
Following are those years where we observed the Fed made at least a 50-basis point hike for each meeting, it indicates that "inflation is urgent" and we did not consider any lesser than 50 basis
The GDP was 5.5% in 1978, unemployment was 6.0%, and inflation was 7.6%.
The GDP was 3.2% in 1979, unemployment was 6.0%, and inflation was 11.3%.
The GDP was -0.3% in 1980, unemployment was 7.2%, and inflation was 13.5%.
The GDP was 2.5% in 1981, unemployment was 8.5%, and inflation was 10.3%.
The GDP was 7.2% in 1984, unemployment was 7.3%, and inflation was 4.3%.
The GDP was 3.5% in 1987, unemployment was 5.7%, and inflation was 3.6%.
The GDP was 4.2% in 1988, unemployment was 5.3%, and inflation was 4.1%.
The GDP was 4.0% in 1994, unemployment was 5.5%, and inflation was 2.6%.
The GDP was 4.1% in 2000, unemployment was 3.9%, and inflation was 3.4%.
Lifestyle in the 70s are much different, therefore the unemployment may not be a reliable comparison. What struck me was the development between the GDP growth and inflation between 1978 and 1980.
In 1978 GDP is still positive and inflation crossed above 7% and there were 3 interest hike that is at least 50 basis point. As the inflation persisted, GDP started declining and that’s where
interest rates hike was the most aggressive in 1979 and 1980.
GDP in its positive zone, March unemployment rate at 3.6% and inflation at 8.5%. There are 6 more Fed meeting this year, let’s keep watch between the outcome of each meeting and the GDP and inflation
Disclaimer: The historical timeline on interest rate hike is from the source “the balance”, please verify the accuracy on your end. | {"url":"https://www.weipedia.com/post/when-was-the-last-round-fed-hike-50-basis-point-and-more","timestamp":"2024-11-11T09:49:41Z","content_type":"text/html","content_length":"1050479","record_id":"<urn:uuid:a73db632-1ab8-4b4a-b5da-f097269ef18d>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00140.warc.gz"} |
Does Kaluza-Klein Theory Require an Additional Scalar Field?
I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2.
1. In one, there is a constant as well as an additional scalar field introduced: $$\tilde{g}_{AB}=\begin{pmatrix} g_{\mu \nu}+k_1^2\phi^2 A_\mu A_\nu & k_1\phi^2 A_\mu \\ k_1\phi^2 A_\nu & \phi^2 \
2. In the other, only a constant is introduced:
$$\tilde{g}_{AB}=\begin{pmatrix} g_{\mu \nu}+k_2A_\mu A_\nu & k_2A_\mu \\ k_2A_\nu & k_2 \end{pmatrix}.$$
Doesn't the second take care of any problems associated with an unobserved scalar field? Or is there some reason why the first is preferred?
1. William O. Straub, Kaluza-Klein Theory, Lecture notes, Pasadena, California, 2008. The pdf file is available here.
2. Victor I. Piercey, Kaluza-Klein Gravity, Lecture notes for PHYS 569, University of Arizona, 2008. The pdf file is available here.
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user elfmotat
When you write the five dimensional Kaluza-Klein metric tensor as
$$ g_{mn} = \left( \begin{array}{cc} g_{\mu\nu} & g_{\mu 5} \\ g_{5\nu} & g_{55}\\ \end{array} \right) $$
where $g_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $ g_{\mu 5}$ is the ordinary four dimensional vector potetial, $g_{55}$ appears as an additional scalar field. This new
scalar field, called a dilaton field, IS physically meaningful, since it defines the size of the 5th additional dimension in Kaluza-Klein theory. They are natural in every theory that hase
compactified dimensions. Even though such fields have up to now not been experimentally confirmed it is wrong to call such a field "unphysical".
"Unphysical" are in some cases fields introduced to rewrite the transformation determinant in calculations of certain generating functionals, or the additional fields needed to make an action local,
which may have conversely to such dilaton field, no well defined physical meaning.
Why is $g_{55}$ necessarily a scalar field? What's the motivation to promote it from being a constant to a field? Is it just because a 5th dimension with variable size is more general?
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user elfmotat
@elfmotat yes, as the whole metric is a field, it is more general and consistent to consider its components, such as the dilaton, to be fields too. However, the dilaton field should better not vary
too much at scales smaler than cosmilogical ones, since in ST it determines not only the size of compactified dimensions but the value of the string coupling constant too. And this again determins
the fundamental laws of nature which should be about constant in our universe ...
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user Dilaton
It will be better if you were mentioned the sources.
However, as I remember, that in some old works on this theory, they used to assume that $\phi=const$ because the main purpose of the theory was looking for a geometrical unification of Gravity and
electromagnetism, and no physical scalar fields was known at that time
Also it will be not very accurate to say "Requires additional" scalar field, because this field in addition to the Electromagnetic field raises naturally (almost, considering cylindrical condition)
in the theory after applying the least action principle on five dimensional scalar curvature, and this "natural way" is the whole point of the theory.
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user TMS
My sources are: weylmann.com/kaluza.pdf math.arizona.edu/~vpiercey/KaluzaKlein.pdf I'm not sure why you say applying the the action principle on the 5D Ricci scalar naturally implies an additional
field. When considering the second form of the metric in my OP, isn't the Ricci scalar just: $\tilde{R}=R+\frac{k}{4}F^{\mu \nu}F_{\mu \nu }$ If $k$ is a constant I don't see how that implies a
scalar field.
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user elfmotat
Yes as you mentioned, there is no reason to put $k=const$, the first paper you mentioned is the original way it was done, including the scalar field just makes it more general, the approach that
taken in the second paper.
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user TMS
I disagree with your last paragraph. Since to unify gravity and EM you need five dimensions, an additional parameter is needed in front of the new fifth component when writing proper time (or
distance) by the five dimensional KK metric. And this parameter is exactly the additional field (which my have a fairly constant value) that appears naturally as 55 component in the KK metric, it IS
This post imported from StackExchange Physics at 2014-03-17 03:58 (UCT), posted by SE-user Dilaton | {"url":"https://www.physicsoverflow.org/7758/does-kaluza-klein-theory-require-an-additional-scalar-field","timestamp":"2024-11-09T23:50:58Z","content_type":"text/html","content_length":"153077","record_id":"<urn:uuid:c282346f-40b1-4183-ac95-fce134b4b54c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00115.warc.gz"} |
Hardware aware synthesis using Classiq vs transpilation
Using the Classiq platform, any developer can create more efficient quantum circuits. See how this synthesis engine approach is different than traditional transpilation.
Photo by Fahrul Razi / Unsplash
Creating an efficient quantum circuit is something that is not very easy or trivial. What if there was a nice tool to help you create efficient quantum circuits for any quantum computer without
having to re-code.
Using the Classiq platform, you can easily create an efficient quantum circuit for any target quantum computer or simulator without having to change any code. This is done by using the synthesis
engine inside the Classiq platform. So, what is the difference between that synthesis engine and traditional transpilation?
If you have created quantum circuits before you probably used a transpiler to make your circuit adapt to different quantum computers, so why not just use that instead of a synthesis engine? There are
multiple reasons why using the Classiq platform is easier to use and will give more optimized circuits. In this blog, I will dive a bit deeper into why synthesis gives better circuits than
transpilation. The example code that I will use throughout this post is the one of a quantum adder. The function of a quantum adder is to perform an addition between two quantum registers. In this
demo, we will encode the integer values of 7 and 12 into two quantum registers and save the addition result in a register called result.
Ease of programming
Let's see how to program this using the Classiq platform; this example uses the Python library of Classiq version 0.39.
The code is below, let's see what happens line by line. First, the main function is defined with 3 quantum numbers (QNum), the two numbers that will be added together, and the result.
Next, the integer values 7 and 12 are encoded in these QNum's using the prepare_int() function.
Finally, using the |= command, the result of the right side of the equation (the addition of a and b) is stored in the res QNum.
using the synthesize() method, this high-level description is turned into a quantum circuit. If you want to interactively explore the circuit, this can be done with the final show() function.
Before diving into the results. Did you see how easy that was?
from classiq import *
def main(a: Output[QNum], b: Output[QNum], result: Output[QNum]):
result |= a+b
qprog = synthesize(create_model(main))
See the QASM circuit that was generated
// Generated by Classiq.
// Classiq version: 0.39.0
// Creation timestamp: 2024-03-21T14:06:45.149498+00:00
// Random seed: 605800157
OPENQASM 2.0;
include "qelib1.inc";
gate qft5_dg_dg q0,q1,q2,q3,q4 {
h q0;
cp(-pi/2) q1,q0;
h q1;
cp(-pi/4) q2,q0;
cp(-pi/2) q2,q1;
h q2;
cp(-pi/8) q3,q0;
cp(-pi/4) q3,q1;
cp(-pi/2) q3,q2;
h q3;
cp(-pi/16) q4,q0;
cp(-pi/8) q4,q1;
cp(-pi/4) q4,q2;
cp(-pi/2) q4,q3;
h q4;
gate qft5 q0,q1,q2,q3,q4 {
h q4;
cp(pi/2) q4,q3;
cp(pi/4) q4,q2;
cp(pi/8) q4,q1;
cp(pi/16) q4,q0;
h q3;
cp(pi/2) q3,q2;
cp(pi/4) q3,q1;
cp(pi/8) q3,q0;
h q2;
cp(pi/2) q2,q1;
cp(pi/4) q2,q0;
h q1;
cp(pi/2) q1,q0;
h q0;
gate arithmetic_adder q0,q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11 {
cx q0,q7;
cx q1,q8;
cx q2,q9;
cx q3,q10;
qft5 q7,q8,q9,q10,q11;
cp(pi) q4,q7;
cp(pi) q5,q8;
cp(pi) q6,q9;
cp(pi/2) q4,q8;
cp(pi/2) q5,q9;
cp(pi/2) q6,q10;
cp(pi/4) q4,q9;
cp(pi/4) q5,q10;
cp(pi/4) q6,q11;
cp(pi/8) q4,q10;
cp(pi/8) q5,q11;
cp(pi/16) q4,q11;
qft5_dg_dg q7,q8,q9,q10,q11;
gate prepare_int_1_inplace_prepare_int_repeat_if_3 q0,q1,q2,q3 {
x q3;
gate prepare_int_1_inplace_prepare_int_repeat_if_2 q0,q1,q2,q3 {
x q2;
gate prepare_int_1_inplace_prepare_int_repeat_if_1 q0,q1,q2,q3 {
gate prepare_int_1_inplace_prepare_int_repeat_if_0 q0,q1,q2,q3 {
gate prepare_int_1_inplace_prepare_int_repeat q0,q1,q2,q3 {
prepare_int_1_inplace_prepare_int_repeat_if_0 q0,q1,q2,q3;
prepare_int_1_inplace_prepare_int_repeat_if_1 q0,q1,q2,q3;
prepare_int_1_inplace_prepare_int_repeat_if_2 q0,q1,q2,q3;
prepare_int_1_inplace_prepare_int_repeat_if_3 q0,q1,q2,q3;
gate prepare_int_1_inplace_prepare_int q0,q1,q2,q3 {
prepare_int_1_inplace_prepare_int_repeat q0,q1,q2,q3;
gate prepare_int_0_inplace_prepare_int_repeat_if_2 q0,q1,q2 {
x q2;
gate prepare_int_0_inplace_prepare_int_repeat_if_1 q0,q1,q2 {
x q1;
gate prepare_int_0_inplace_prepare_int_repeat_if_0 q0,q1,q2 {
x q0;
gate prepare_int_0_inplace_prepare_int_repeat q0,q1,q2 {
prepare_int_0_inplace_prepare_int_repeat_if_0 q0,q1,q2;
prepare_int_0_inplace_prepare_int_repeat_if_1 q0,q1,q2;
prepare_int_0_inplace_prepare_int_repeat_if_2 q0,q1,q2;
gate prepare_int_0_inplace_prepare_int q0,q1,q2 {
prepare_int_0_inplace_prepare_int_repeat q0,q1,q2;
gate main_prepare_int_0 q0,q1,q2 {
prepare_int_0_inplace_prepare_int q0,q1,q2;
gate main_prepare_int_1 q0,q1,q2,q3 {
prepare_int_1_inplace_prepare_int q0,q1,q2,q3;
gate main_arithmetic q0,q1,q2,q3,q4,q5,q6,q7,q8,q9,q10,q11 {
arithmetic_adder q3,q4,q5,q6,q0,q1,q2,q7,q8,q9,q10,q11;
qreg q[12];
main_prepare_int_0 q[0],q[1],q[2];
main_prepare_int_1 q[3],q[4],q[5],q[6];
main_arithmetic q[0],q[1],q[2],q[3],q[4],q[5],q[6],q[7],q[8],q[9],q[10],q[11];
Hardware aware synthesis
Now that we have a circuit, it can be transpiled to any hardware target. Let's use Qiskit to transpile this circuit to IBM Kyoto machine. The optimal transpilation gives a circuit with 366 depth,
which is probably too deep to execute on any real hardware. If we want to optimize this circuit, we might want to look at different ways of implementing the + arithmetic. This is possible to do
manually for such a small circuit, but it might be too hard when the circuit gets bigger. There should be a better solution right?
Luckily there is! This is where Classiq comes in. Let's use Classiq to implement the same code but optimize for the IBM Kyoto machine. It is not necessary to change the functional code that was
already written. You just need to add the target machine as a back end provider, as you see in the code below. Next to the target, an optimisation parameter is also set. In this case, the circuit is
optimised for depth, giving us the shallowest possible circuit for that specific machine.
from classiq import *
def main(a: Output[QNum], b: Output[QNum], result: Output[QNum]):
result |= a+b
model_IBM = create_model(main)
constraints = Constraints(
preferences = Preferences(
backend_service_provider="IBM Quantum", backend_name="ibm_kyoto"
model_IBM = set_preferences(model_IBM, preferences)
model_IBM = set_constraints(model_IBM, constraints=constraints)
qprog_IBM = synthesize(model_IBM)
, the actual circuit that is created will be
When we run this code, the actual circuit that is created will be completely different. In this case, the Classiq synthesis engine picks a different implementation for the addition between the a and
b variables; this new implementation is more efficient for this specific hardware target. This results in a big decrease in depth from 366 to 201. So how is this possible?
Instead of using transpilation, the Classiq platform uses hardware-aware synthesis. In this process, the creation of the quantum circuit is done from a high-level code, which is called the model.
This model is turned into a quantum circuit by the synthesis engine. In this process, the synthesis engine has the flexibility to pick the best implementation of the model for each target hardware.
This can give a lot of benefits for the resulting circuit in terms of depth or qubits use. In the image below, you can see more graphically the difference between transpilation and synthesis.
As you can see the Synthesis engine has the freedom to pick different, more efficient implementations of the same functionality. With this freedom the synthesis engine can pick the most efficient
circuit implementation per hardware target. With this approach anyone can write very complex quantum circuits for any current and future quantum computer.
• With the Classiq platform, you can design quantum circuits at a high level instead of gate-based programming.
• The synthesis engine can make very efficient quantum circuits per hardware target without re-coding your quantum program. | {"url":"https://www.vincent.frl/hardware-aware-synthesis-vs-transpilation/","timestamp":"2024-11-10T00:06:40Z","content_type":"text/html","content_length":"40825","record_id":"<urn:uuid:bf6d3ca6-1834-4380-87b1-8bf2d2e751de>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00573.warc.gz"} |
Error: [fmi2GetContinuousStates]: Simulation terminated at t=0.0000000000000000e+00: error running events
Hello to all, in my first model i want to simulate a dumper using a cylinder with orifice in the line to the tank.
I've tried different configuration but every time appears this type of error and i don't understand what it mean.
Is there someone that can help me uderstand this error? What i have to change?
Thanks in advance!
• Hello Arnigaber,
can you share your model file?
• Hi Arbigaber,
It's a problem of initial values. In Modelica mechanics length of elements in a row are added. with basic parameters, the initial position of v-source is 0 m, length of left rod 0,4 m. Position
of fixed is 0 m => relative stroke of cylinder is 0,4 m (in right end stop). Now you have to move either v-source to the left or fixed to the right to move piston into middle of the cylinder.
My demo model has a fixed position of 0,2 m.
• Hi RoKet, with your help I managed to run the simulation.
Thank you very much!
• Hi guys,
I'm having a similar problem with a model I'm trying to set up and I'm hoping I can get some help.
I've got a cylinder being driven by a position source, but when I run I get the same error. I've tried changing the fixed position as per RoKet's post above but this doesn't seem to work on this
I eventually need to add an accumulator onto the piston side of the cylinder, but need to get this simple model working first!
Could someone have a look at this model and let me know what I'm doing wrong here?
Many thanks!
• Hi BrettL,
It's the same solution. You have to consider all positions and length. In Modelica (and the hydraulics library) positions are calculated as a row of points and lengths. In your case the row
starts with the fixed element at 0 m. The cylinder has a minimum length of 0.4 m, so the rod tip has a minimum position of 0.4 m, a maximum position of 0.8 m (stroke of 0.4 m).
The range of position source has to be in the given limits of the cylinder, I've shifted the offset to 0.6 m (0.4 m length and initial stroke of 0.2 m). In addition I've set all initial values to
the right value (cylinder stroke, s_ref, ToModelica. It's not always necessary, but helps the solver to find the correct initial state.
• Hey RoKet,
Thanks a lot for that... your explanation makes more sense to me now. Your model works as I'd intended, so really appreciate your help.
Now to work on the accumulator...
Cheers, Brett. | {"url":"https://community.altair.com/discussion/362/error-fmi2getcontinuousstates-simulation-terminated-at-t-0-0000000000000000e-00-error-running-events","timestamp":"2024-11-13T22:58:21Z","content_type":"text/html","content_length":"314953","record_id":"<urn:uuid:ca662444-443f-4e74-b8d3-88022abc50b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00657.warc.gz"} |
A Scholar in his time: The contemporary views of Kosambi the mathematician
A SCHOLAR IN HIS TIME: CONTEMPORARY VIEWS OF KOSAMBI THE MATHEMATICIAN
RAMAKRISHNA RAMASWAMY
University of Hyderabad, Hyderabad, TS 500 034
“Kosambi introduced a new method into historical scholarship, essentially by application of modern mathematics.” J. D. Bernal [1], who shared some of his interests and much of his politics,
summarized the unique talents of DDK [2] in an obituary that appeared in the journal Nature, adding, “Indians were not themselves historians: they left few documents and never gave dates. One thing
the Indians of all periods did leave behind, however, were hoards of coins. [...] By statistical study of the weights of the coins, Kosambi was able to establish the amount of time that had elapsed
while they were in circulation . . . ”
The facts of DDK’s academic life, in brief are as follows. He attended high–school in the US, in Cambridge, MA, and undergraduate college at Harvard, graduating in 1929. Returning to India, he then
worked as a mathematician at Banaras Hindu University (1930-31), Aligarh Muslim University (1931-33), Fergusson College, Pune (1933-45), and the Tata Institute of Fundamental Research (1945-62),
after which he held an emeritus fellowship of the CSIR until his death at the age of 59, in 1966.
Today the significance of D. D. Kosambi’s mathematical contributions [3–71] tends to be underplayed, given the impact of his scholarship as historian, and Indologist. His work in the latter areas has
been collected in several volumes [72] and critical commentaries have appeared over the years [73, 74], but his work in mathematics has not been compiled and reviewed to the same extent [75, 76, 77,
78]. Indeed, a complete bibliography is not available in the public domain so far [79]. This asymmetry is unfortunate since, as commented elsewhere [75], an understanding of Kosambi the historian can
only be enhanced by an appreciation of Kosambi the mathematician [80].
DDK is known for several contributions, some of which like the Kosambi-Cartan-Chern (KCC) theory [81], carry his name, and some like the Karhunen–Loève expansion [37, 39, 82], that do not. The
Kosambi mapping function in genetics [40] continues to be used to this day [83], but the path geometry that he studied for much of his life [84] has not found further application. DDK’s final years
were mired in controversies, both personal and professional. His papers on the Riemann hypothesis (RH) [65, 66] brought him a great deal of criticism and not a little ridicule, while his personal
politics put him in direct conflict with Homi Bhabha and the Department of Atomic Energy. This contributed to his eventual and somewhat ignominious ouster from employment at the Tata Institute of
Fundamental Research. His early and passionate advocacy of solar energy was practical and based on sound scientific common sense. In some of his arguments, he seems even somewhat Gandhian, and
although this was a contrary position to hold in the TIFR at that time, the essential validity of his argument remains to this day [85].
DDK was just about 23 years old when he returned to India and took up a temporary position at Banaras Hindu University with a BA (summa cum laude) from Harvard. A year later he had moved to the
Aligarh Muslim University where he was appointed in the Mathematics Department at the suggestion of André Weil [86] who, just about a year older, was then already well known as a mathematician and as
a prodigy, and who had been invited to the AMU by then Vice Chancellor Syed Ross Masood.
Although Weil did not last long in Aligarh, his influence on Kosambi was considerable. In addition to giving him the position and encouraging him on the matter of the Bourbaki prank[DEL:,:DEL] Weil
helped DDK forge early mathematical links with, among others, T. Vijayaraghavan [87] and S. Chowla [88]. He undoubtedly influenced his taste in mathematics, possibly sparking DDK’s interest in the
Riemann hypothesis. Weil would, in the early 1940’s make important contributions to this field [89] although when DDK turned to it almost thirty years later [65] his efforts were to come a cropper.
Weil spent the summer of 1931 in Europe and upon his return to Aligarh, he found that not only had his own position been compromised, but the group of mathematicians that he had put together had also
fragmented, with Vijayaraghavan having moved to take up a Professorship in Dacca [90]. By early 1932, Weil had returned to Europe, and DDK was to leave Aligarh soon thereafter.
Kosambi started his independent work in Aligarh, choosing the area of path–geometry, a term he coined, submitting his papers to leading European journals [7, 9, 10]. One that was sent to
Mathematische Zeitschrift was also communicated to Elie Cartan who was inspired enough by the result to write a detailed commentary, which included an extract of the correspondence that Kosambi had
with him. This was also published in Mathematische Zeitschrift [11] immediately following DDK’s paper in 1933. Along with a later paper by the Chinese mathematician, S. S. Chern, these three works
constitute what is now termed the KCC-theory, a topic that has, in recent years, found new applications in physics and biology [81]. Some years later, in 1946, Kosambi tried to have Chern invited to
visit India when he was at the TIFR but nothing came of it.
DDK wrote many papers on path geometry, and in the mid 1940’s summarized his work in a manuscript that was submitted to Marston Morse at the Institute for Advanced Study in Princeton. In a letter
[91] to Bhabha he says, “The book on Path-Geometry will, according to a letter from Morse, appear in the Annals of Mathematics Studies, Princeton.” This book was never published—indeed very few books
in this series were, and efforts to locate a copy of the manuscript in the Morse archives have proved fruitless [92]. DDK makes reference to a second copy of the manuscript that he gave to Bhabha,
but that copy has not been located either.
The Nobel laureate, C. V. Raman had visited Aligarh in 1931 as the member of a selection committee, and although there is no specific record of his having met Kosambi, his subsequent actions suggest
that he quickly gathered, either directly or indirectly, a very high opinion of DDK. In 1934 when Raman founded the Indian Academy of Sciences in Bangalore, he elected Vijayaraghavan and Chowla. The
very next year Kosambi was elected to the IASc at the age of 28, when his mathematical œuvre was slight, and along with others such as P C Mahalanobis and V V Narlikar. Kosambi was one of the younger
of the Founding or Foundation Fellows (namely those elected in 1934 and 1935). Since the initial election to the Academy was almost entirely his decision, the estimation that Raman had of Kosambi’s
scholarship or of his potential, must have been considerable. It is possible that Vijayaraghavan may have played some role in this early recognition [93], and it is also likely that the award of the
first Ramanujan Prize of the Madras University in 1934 to S. Chandrashekar, S. Chowla and DDK [94] would have favourably impressed Raman. As it happened, in later years Kosambi was privately and
publicly very critical of Raman’s style of functioning [95].
This early recognition, however, stood him in good stead. He published a couple of papers in the Academy journal, Proceedings of the Indian Academy of Sciences in 1935 (and not again until the 1960’s
when, as S. Ducray, he published two more). Reviews of his papers in other journals began to appear in Current Science, the general science journal started by Raman, in addition to original articles
that he chose to publish in this journal as well. Indeed his initial papers on the quantitative approach to numismatics [26, 27, 34, 36] all appeared in Current Science.
1. Reviews and Commentaries
One of the early references to the work of DDK on numismatics that was brought to the attention of readers of Current Science was a review in 1941 [96] by K. A. N. (this was probably the well known
historian K. A. Nilakantha Sastry) of two papers of DDK’s in the New Indian Antiquary [97]. By this time, DDK seems to have been well established as an eminent mathematician. While generally admiring
of the work, KAN comments on a number of DDKs characteristics: the use of “hard phrases” in his critique of the methods used by others, his exposure “of the hollowness of much pseudo–expertise that
has held the field”, etc. Nevertheless, the review is not uniformly accepting of DDKs conclusions, and KAN does alert the reader to potential areas of inaccuracy. In a charming final paragraph, for
instance, he says “Yet, this conclusion hardly tallies with the impressions of the Mauryan epoch gathered from other sources like the inscriptions of Asoka, or the polished stone pillars–-not to
speak of Megasthenes and the Arthasastra. There are other statements, obiter dicta, which may surprise the reader, and even shock him; but there is much, very much in these papers and their method
for which he will be grateful”.
The journal Mathematical Reviews (MR) was started in 1940 by the American Mathematical Society as a way for working mathematicians to keep up with the increasing numbers of papers that appeared each
year in diverse journals. The practice was (and still is) to have a brief summary of these papers sometimes with commentary, and sometimes without. Indeed, some papers are merely noted or abstracted,
and all reviews are signed.
Of DDK’s sixty or so papers in mathematics, about half were reviewed in MR; these are indicated in the bibliography [3–71]. The reviewers include R. L. Anderson, R. P. Boas, Jr., N. Coburn, J. L.
Doob, W. Feller, V. Hlavaty, M. Janet, A. Kawaguchi, J. B. Kelly, M. S. Knebelman, J. Korevaar, J. Kubilius, R. G. Laha, W. J. LeVeque, A. Nijenhuis, E. S. Pondiczery (a pseudonym of R. P. Boas Jr),
A. Rényi, J. A. Schouten, E. W. Titt, J. L. Vanderslice, O. Varga, B. Volkmann, A. Wald, and J. Wolfowitz. Several of these reviews are just summaries of the papers, but some are serious commentaries
on the work of Kosambi, and, significantly, are by some of the leading contemporaneous mathematicians, probabilists, and statisticians. Indeed R. P. Boas Jr. who reviewed some of the papers was one
of the main editors of Mathematical Reviews.
It may be pertinent to note that it is not just DDK’s papers that were published in journals outside India that were reviewed in Mathematical Reviews; several of the papers published in Indian
journals were also commented upon critically. These include the important paper, “Statistics in function space” [39] that was reviewed by the probabalist, J. L. Doob, who went on to become the
President of the American Mathematical Society (and who was awarded the National Medal of Science by then President of the United States, Jimmy Carter in 1979).
Although Doob gave a careful and comprehensive review of the work soon after it was published in 1943, unfortunately neither Karhunen nor Loève who essentially rediscovered these results [82] were
aware either of the paper or of its review, and today these results go under their names, and Kosambi’s contribution is largely unrecognized. One important feature of the paper pointed out in the
The author discusses statistical problems connected with continuous stochastic processes whose representative functions x(t) [. . . ] Various mechanical and electrical methods are suggested for
combining functions x(t), given graphically, as necessitated by this type of statistical approach.
was the idea of a mechanical or electrical computer. This was to be part of Kosambi’s Kosmagraph project that was in part funded by a grant from the J. R. D. Tata Trust in 1945. It is not clear if a
working model was ever successfully constructed, though there is a reference to it in a report he sent to the Tata Trust [80],
The Kosmagraph is finished, and a working model being improved at St. Xavier’s College. The total outlay for workshop charges, electric motors, cathode ray oscillographs, valve tubes etc. would have
exceeded the total amount of the Tata grant. But the St. Xavier’s authorities stood the expense of these items, as Fr. Rafael has collaborated in the work. My total expenses from the grant have been
a nominal honorarium of Rs. 250/- to K. B. McCabe, the third collaborator; and another of Rs. 50 to Salvador D’Souza, head mechanic at the St. Xavier’s workshop. Both have deserved far more, and the
work of McCabe in particular seems to me to be beyond recompense.
A joint paper is being made ready for publication, though it will be some months before all the points are checked.
The paper alluded to does not appear to have been published, and no drafts been located among DDK’s papers. It is also not clear what became of the project; the interest in a computing machine stayed
with Kosambi when he later moved to TIFR, and indeed was one of purposes of his visit to the USA in 1948-49 [75].
Another of DDK’s reviewer’s was Abraham Wald (who was later to die in a plane crash in India when he was visiting the country at the invitation of the Indian government) who commented, generally
favourably, on four of his papers. What is interesting is that many of the papers were published in journals such as Mathematics Student and the Journal of the Indian Mathematical Society, both of
limited circulation, and which to this day remain somewhat difficult to locate [98].
It should be mentioned that most of DDK’s publications in mathematics are independently authored. He did, however, mentor several students, both formally and informally at the TIFR in the 1950’s, and
among these were S. Raghavachari and U. V. Ramamohana Rao who are his only coauthors.
2. The RH papers
Arguably the most important as yet unresolved problem in pure mathematics is a hypothesis that was enunciated in 1857 by the celebrated mathematician, Bernhard Riemann. A brief introduction to the
nature of the mathematical problem [99] is included here for those who are less acquainted with it, to give some flavour of why it is interesting and a challenge. (I should also add that at the risk
of losing half the potential readership with each equation [100], it is absolutely essential that some be retained. For all of mathematics there is no greater game than to solve the Riemann
hypothesis, and to appreciate both what Kosambi tried, and where he did not succeed, some equations are needed.)
The Riemann Hypothesis concerns properties of a mathematical function that has been studied for at least four centuries. This is the zeta function, the sum of inverse powers of the integers,
the ellipsis signifying that the sequence does not terminate. When z is equal to zero, then each term is 1, and the sum, namely 1+1+1+1…. becomes infinitely large:
Such infinite sums have long been of interest: an example that will be familiar to many is the sum that arises in Zeno’s paradox regarding Achilles and the tortoise. (In a 100 metre race, the
tortoise, which is 100 times slower than Achilles is given, say, a head-start of 90 metres. In the time that Achilles covers 90 m, the tortoise covers 90 centimetres and is therefore still ahead. In
the time that Achilles covers the 90 cm, the tortoise goes ahead by 9 millimetres, when Achilles covers the 9 mm, the tortoise is ahead by a smaller fraction, and so on. So Achilles would, it seems,
never catch up with the tortoise. The resolution of the paradox is that this infinite sum is actually a finite quantity, and Achilles wins the race easily [101].)
The harmonic series, is the sum of inverses of the integers,
and this also diverges or becomes infinite (the → in the equation below signifies “tends to”). In contrast, when z = 2, the sum is a finite number,
Clearly the value of the zeta function,
depends on the value of z, and Riemann was interested in its “zeroes”, namely those values of z when
In order to study the general properties of such a function, it is necessary to consider all possible values for z, in particular when z is a complex number, namely of the form z = x + iy where x is
the real part and y the imaginary part, and . It turns out then that the ζ function can take values that are either positive or negative depending on the value of z, or equivalently, on the values of
x and y. When y=0 and x is a negative even integer, namely -2, -4, -6 and so on, the function takes the value 0: these are termed “trivial” zeroes since the function can be shown to vanish through a
straightforward procedure [102].
The ζ function has in addition an infinite number of “nontrivial” zeros, and Riemann’s hypothesis is that for all of these, x (namely the real part of z) has the value 1/2. In the complex plane,
these zeroes therefore all lie on the so–called “critical” line, x = 1/2. While being simple enough to state, it remains unproven to this day. Because of connections between the zeta function and
prime numbers, a proof of the RH would have significant implications for the distribution of prime numbers, and via this, for much of mathematics. An alternate measure of its importance can be gauged
from the fact that it is one of the so–called Millennium Prize problems for which the Clay Mathematical Institute has announced a grand cash award in the past decade.
DDK’s mathematical reputation suffered greatly as a result of two papers he published in the Journal of the Indian Society of Agricultural Statistics [63, 67]; in one of them, he claimed a result
that was essentially a proof of the RH. Notwithstanding its name, the journal does publish serious mathematics, particularly in the area of probability. Although obscure and highly specialized, the
journal may not have been as inappropriate for the papers as might appear since the methods suggested by Kosambi were probabilistic. However, it is not clear that the journal had a proper peer—review
process in place whereby submitted articles would, prior to publication, be examined by experts in the field. The lack of appropriate reviewing was a real deficiency, more so for a claim of this
magnitude and the charge remains that DDK chose to publish the papers in JISAS to be able to pass off a doubtful “proof”.
Both papers were reviewed subsequently in MR, one by W. J. LeVeque, a number theorist who eventually became Executive Director of the American Mathematical Society. His critique of “An application of
stochastic convergence” [63] goes straight to the point, that the claim made by DDK is a result which easily implies the Riemann hypothesis. However, since the proof is probabilistic in nature, there
are major problems that he identifies. Of the two proofs given for the crucial Lemma 1.2, the reviewer does not understand the first, which seems to involve more ‘hand-waving’ than is customarily
accepted even in proofs of theorems less significant than the present one. The second proof appears to be erroneous. The review concludes The reviewer is unable either to accept this proof or to
refute it conclusively. The author must replace verbal descriptions, qualitative comparisons and intuition by precise definitions, equations and inequalities, and rigorous reasoning, if he is to
claim to have proved a theorem of the magnitude of the Riemann hypothesis.
The kindest analysis of these works of DDK comes from the Hungarian mathematician, A. Rényi who says in a posthumous review of the paper “Statistical methods in number theory” [67] that
The late author tried in the last 10 years of his life to prove the Riemann hypothesis by probabilistic methods. Though he did not succeed in this, he has formulated the following highly interesting
conjecture on prime numbers.
Rényi, who had been sent both this and the earlier papers [63, 64] prior to publication, goes on to say that Neither in this paper nor in his previous paper [Proc. Nat. Acad. Sci. U.S.A. 49 (1963),
20–23; MR0146168 (26 #3690)] did the author succeed in proving his hypothesis, nor in deducing from it the Riemann hypothesis. The PNAS paper [64] was reviewed by J. B. Kelley who states, after
summarizing the main result, that The exposition is rather sketchy; in particular, the reviewer could not follow the proof of the crucial Lemma 4.
Either because of the timing of the review or because he may have appreciated the valiant attempts of DDK to prove the Riemann hypothesis by an unusual route, Rényi, concludes the review by saying
that at that point in time (1968) one does not have enough knowledge of the fine structure of the distribution of primes to prove or disprove the author’s conjecture. The problem seems to be even
more difficult than the problem of the validity of the Riemann hypothesis. As a matter of fact, no obvious method exists to prove the author’s hypothesis even under the assumption of the Riemann
hypothesis. Nevertheless, the conjecture is worthy of study in its own right, and the reviewer proposes to call it “the Kosambi hypothesis” in commemoration of the enthusiastic efforts of the late
Rényi’s suggestion has not found favour. The probabilistic approach has inherent limitations, as the physicist Michael Berry points out [103]. Indeed, as these reviews suggest, the rigour emphasised
by DDK in his early years had deserted him. What is somewhat surprising is that there are elementary errors in these papers that become evident even with a fairly cursory examination, and which could
have been detected by an alert referee. The fact that IJSAS published this paper with the errors added to the feeling that DDK deliberately chose the journal to avoid qualified peer review. These
papers essentially destroyed DDK’s mathematical reputation.
Given the ongoing interest in the RH, only in part increased by its inclusion as a Millennium Prize problem, there are a number of popular books [104] that summarize the approaches to proving it. Not
surprisingly, the work of DDK is not mentioned, although Berry remarks [103] that his idea for proving RH based on showing that a certain function is nonsingular off the line, is ingenious. Andrew
Odlyzko, another mathematician who has worked extensively on the RH says [105] that he was really intrigued by these approaches, but after a while decided that it would take some clever insights far
beyond what [he] could think of to accomplish anything rigorous in this area. Among Odlyzko’s major contributions to a study of the RH is the computation of a large number of zeros (several million
of them, in fact) to fairly high precision; for all of these, the real part equals 1/2. As an experimental mathematician he has a good insight into the approach suggested by DDK, adding, In summary,
I think it is a pity that Kosambi did not see the flaws in his arguments and published this paper, but the basic idea is an interesting one, and certainly worth exploring. I would be surprised, but
not shocked, if somebody clever managed to do something with it.
3. Bhabha and DDK
DDK joined the newly formed Tata Institute of Fundamental Research on June 16, 1945. His appointment, which was for an initial period of five years was decided at the first meeting of the provisional
council of TIFR.
The initial correspondence between Bhabha and DDK, although formal, was extremely cordial [74]. In 1946, when Bhabha traveled to England, he appointed DDK Acting Director, leaving him in charge of
the fledgling institute. This was a position of considerable responsibility, and one that DDK clearly enjoyed, and in a long letter [91] written on 8th July he writes “About building up a School of
Mathematics in India, we also think alike; but, as you are fully aware, we have to get people trained in a considerable number of branches for which there are no real specialists in this country.”
The relationship also grew warm, especially since they had to plan the Institute together, concerning themselves with details regarding land acquisition, equipping the laboratories, hiring staff,
planning for the future. That same year DDK was elected Fellow of the Indian National Science Academy, and the next, in 1947, was awarded the Bhabha Prize (named for Bhabha’s father, Jehangir
Hormusji Bhabha). He was also chosen President of the Mathematics section of the 34th Indian Science Congress that was held in Delhi in December 1947 [47] with the active support of Bhabha who also
realized that this would bring DDK into contact with Nehru. Kosambi’s mathematical and statistical expertise was also greatly appreciated—a number of colleagues, Bhabha among them, acknowledge his
advice and help explicitly in their scientific publications.
In 1948, when DDK was to go to the US for a year’s visit, to Chicago and Princeton, Bhabha threw a party for him at his residence in Malabar Hill. This visit was in fact largely arranged by Bhabha,
and among other things, DDK was to investigate the possibility of getting a computing machine for the new institute [91] as well as to attract new faculty, K. Chandrasekharan and S. Minakshisundaram
in particular. On this trip, he pursued all aspects of his wide–ranging interests, visiting Einstein and von Neumann in Princeton, Norbert Wiener in Boston, as well as the historian, A. L. Basham in
London. In Chicago, he was visiting Professor at the University, where he gave a course of 36 lectures on tensor analysis. This was a special interest of his: he had been invited to the editorial
board of the Hokkaido University journal, Tensor (New Series), and indeed an article of his had been translated into Japanese already in 1939 by the same journal [25].
In the event, Chandrasekharan joined the TIFR in 1950 or so, and shortly thereafter, so did K. G. Ramanathan, who had obtained his Ph. D. at Princeton. They were to play a much more influential role
in shaping the TIFR School of Mathematics. In the next few years, though, the cracks in the relationship between Bhabha and DDK surfaced, first in regard to students and then gradually, with regard
to details such as his attendance in office and other aspects of his working.
The spiral downwards, though, began in 1959 with the publication of the JISAS paper [63], and the subsequent grand obsession with a probabilistic proof of the Riemann hypothesis. His differences
became more pronounced with Bhabha who relied more and more on Chandrasekharan’s opinion and estimation of DDK’s work. The coup de grace was a letter signed by four of the mathematicians at TIFR
stating that Kosambi had become an embarrassment to the Institute with his claim of the proof of the RH and of Fermat’s Last Theorem [106] that was being broadcast internationally.
There were other differences with Bhabha which were of a political nature, but these differences were already present in 1946 when Bhabha invited DDK to join TIFR. The unpublished (and largely
unknown) essay ‘An Introduction To Lectures On Dialectical Materialism’ relates to a set of 15 lectures given by Kosambi to the citizenry of Pune in 1943. Later, when he gave a set of lectures on
Statistics at TIFR the notes conclude with an appreciation of Lenin [107]. Indeed, Bhabha facilitated DDK’s visits to the Soviet Union and China, and it is not possible that DDK’s views were hidden
under a bushel until the early 1960’s.
In July 1960 DDK gave a talk to the Rotary Club of Poona on “Atomic Energy for India”. This essay [108] is an unabashed advocacy of solar power over atomic power, mirroring in a sense his ideological
conflict with the DAE. Half a century later, many of these issues remain current and the arguments remain valid, as for example the following observation.
It seems to me that research on the utilization of solar radiation, where the fuel costs nothing at all, would be of immense benefit to India, whether or not atomic energy is used. But by research is
not meant the writing of a few papers, sending favoured delegates to international conferences and pocketing of considerable research grants by those who can persuade complaisant politicians to
sanction crores of the taxpayers’ money. Our research has to be translated into use.
There is more in these essays on solar energy that merits attention even today such as his observations on energy storage and distribution, and on environmental issues [108]. Eventually matters came
to such a pass as to cause the DAE to not renew DDK’s contract. As already pointed out, the RH papers had caused a serious blow to Kosambi’s mathematical reputation and while this was made out as the
proximate cause for his dismissal from TIFR, trouble had been brewing for some time. The letters between Bhabha and DDK grew increasingly formal, bureaucratic, and strained. There was a distinct
difference in styles, and the iconoclastic Kosambi was hardly one to fit into the DAE mould.
4. Pseudonyms and Aliases
DDK was responsible for the first mention of Bourbaki in the mathematics literature in his publication [4] in the Proceedings of the Academy of Sciences, UP, in 1931, although the obscurity of the
journal has resulted in the article receiving less attention than it deserved, even from a purely historical point of view. André Weil had suggested a prank, that he ascribe a theorem to a
nonexistent Russian mathematician, in order to put down an older colleague in Aligarh who was giving the young Kosambi a difficult time. There is not much more than a paragraph in Weil’s
autobiography [86] on this episode, so the circumstances surrounding the event are difficult to reconstruct. Nevertheless, this parodic note passed off as a serious contribution to a provincial
journal is not entirely facetious.
It was not until December 1934 that the Bourbaki idea acquired more momentum [109, 110], when Weil along with Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, and René de Possel,
decided ... to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis. Of course, this treatise will be as modern
as possible. The book [111] would eventually appear in 1938, authored by the group that now called themselves Nicolas Bourbaki [112]; they then went on to write many more (and extremely influential)
volumes. An Indian connection remained: when Boas mentioned (in the Britannica Book of the Year) that Bourbaki was a collective pseudonym, he got an indignant letter of protest, from Bourbaki,
writing from his ashram in the Himalayas [113]. It should also be noted that Kosambi cites D. Bourbaki [4] who is allegedly of Russian extraction, while the first name eventually adopted by the
Bourbaki collective [114] is Nicolas, who is of Greek descent.
Aliases were used by DDK on several occasions although he did not use them extensively enough to warrant a distinction between his “aesthetic” or pseudonymous writing and that published under his own
name. S. Ducray was merely the last nom de plume in a series, although by far the most elaborate. His first article in the magazine of Fergusson College was signed off as ‘Ahriman’ [115].
Subsequently he wrote an expository article on the Raman effect as ‘Indian Scientist’ [116], and a note as ‘Vidyarthi’ [117]: this was almost surely his nod to William Sealy Gosset, the chemist and
statistician who, as ‘Student’ invented the t-test in statistics.
It is difficult to discern what led him to use the pseudonym S. Ducray. The alleged etymology is that Bonzo, the Kosambi family dog in the 1960’s was quite plump, and DDK affectionately called him
Dukker, namely ‘pig’ in Marathi. This evolved into Ducray, a name that sounds vaguely French, with the forename being the Sanskrit for dog, namely Svana. The choice of such a name remains enigmatic,
and while it may have been prompted initially by his anger with the establishment—to date Kosambi is among the very few persons to have had their appointment terminated by the Department of Atomic
Energy—there is enough to suggest that there may be more to the use of this alias than pique.
DDK published four articles as S. Ducray, two in the Journal of the University of Bombay [65, 66] and two in the Proceedings of the Indian Academy of Sciences [68, 69]. The latter two were in fact
communicated by C. V. Raman. While this may have been a formal device employed by the journal, it is highly unlikely that Raman knew of the masquerade. Had Raman known, it is also highly unlikely
that he would have permitted such subterfuge in a journal of his Academy. These two papers were serious enough as works of mathematics, as were the other two Ducray papers that were submitted to the
Journal of the University of Bombay. Indeed, two of these four papers were reviewed in Mathematical Reviews. All the four articles show a strong connection to DDK, acknowledging him in one and
quoting a private communication from Paul Erdös in another, in addition, of course, to citing his related papers written as D. D. Kosambi.
These papers continued the prime obsession that DDK showed in his last years. Regrettably, the manuscript of his book [71] that was mailed to the publishers a short time before his death has never
been retrieved. If nothing else, it would have provided some clues as to how he hoped to use probability theory in this arena.
Although reviewed in MR, the papers had serious shortcomings. J. Kubilius who himself worked in the area of probabilistic number theory says of ‘Probability and prime numbers’ [68] that The reviewer
could not follow the proof of the cardinal Lemma 3. The paper “Normal Sequences” [66] was comprehensively reviewed by B. Volkmann who pointed out a number of inaccuracies and misprints.
One of DDK’s earlier papers had been reviewed in Mathematical Reviews by E. S. Pondiczery: this was the editor Ralph Boas Jr’s pseudonym, a fanciful ‘slavic’ spelling of Pondicherry. The name, which
Boas used even when writing serious mathematics, was apparently concocted for its initials, ESP, and was to have been used for writing an article debunking extra–sensory perception. Boas had a
well–developed sense of the ludic and was one of the authors of the brilliant article “A Contribution to the Mathematical Theory of Big Game Hunting” that was published in the American Mathematical
Monthly under the (collective) pseudonym H. W. O. Pétard [118]. Both Boas and Kosambi were publicly dismissive of extra–sensory perception, and in 1958 DDK in collaboration with U. V. R. Rao authored
an article analysing the statistical defects underlying parapsychological experiments [60]. This paper was subsequently commented upon by A. W. Joseph [119] who pointed out an error in analysis as
well as in the conclusions, ending with The above comments do not detract from the valuable experiments in card–shuffling made by the authors, but it is suggested that there is little weight left in
their criticism of the ESP investigations. Perhaps it was these connections that inspired Kosambi when he was to later adopt the Ducray alias.
5. Concluding Remarks
History may not have been particularly kind to Kosambi the mathematician, but in his lifetime DDK was appreciated for his scholarship and intelligence [120] early in his career and by his peers. The
manner in which Kosambi was viewed by his contemporaries—many of who were more distinguished than him and had a more significant impact in mathematics—is revealing. From 1930 to 1958 or so, DDK
enjoyed the respect and admiration of a large professional circle. As has been noted earlier [75], his contributions in areas such as ancient Indian history, Sanskrit epigraphy, Indology, as well as
his writings of a political and pacific nature grew both in volume and in substance in the 1940’s and 1950’s, overshadowing his mathematics, although the constancy of his work in the area remained.
His wide scholarship and his ability to integrate different strands of thought gave him a large and dispersed audience, although his temperament and his politics were also well known and not as
widely appreciated.
One important recognition that was accorded him, in part due to his being at the TIFR and the association with Bhabha, but also for his work and his mathematical antecedents [121], was his
appointment as a member, in 1950, of the Interim Executive Committee of the International Mathematical Union, to serve along with Harald Bohr, Lars Ahlfors, Karol Borsuk, Maurice Fréchet, William
Hodge, A. N. Kolmogorov and Marston Morse. One of the tasks of this rather distinguished group was to choose Fields medalists, and DDK served on this committee for two years.
It is thus noteworthy that in a period that spans three decades, Kosambi was mathematically productive, prolific, original, and was taken seriously by the scientific establishment in the country, as
his elections to the Fellowships of the Indian Academy of Sciences and the Indian National Science Academy and the Presidency of the Mathematics section of the 34th Indian Science Congress in 1947,
among other distinctions, testify. His papers appeared in leading journals of the world, and were communicated by or reviewed by some of the leading mathematicians of the time. And that this happened
while his reputation in a diametrically different field was also burgeoning can only be seen as evidence of a complex but nevertheless Promethean intellect.
I have greatly benefited from conversations and/or correspondence with Michael Berry, S. G. Dani, Meera Kosambi, Mrs. Marston Morse, Rajaram Nityananda and Andrew Odlyzko. The TIFR archives have been
very helpful in providing copies of the correspondence between Kosambi and Bhabha, and Kapilanjan Krishan, Rahim Rajan and Mudit Trivedi have helped me obtain copies of articles by DDK that proved to
be the most difficult to locate. The main effort of putting together the collected mathematical works of DDK was completed at the University of Tokyo in January 2010, and their hospitality is
gratefully acknowledged.
[1] J. D. Bernal, Nature 211, 1024 (1966).
[2] For convenience, in this essay I will refer to Professor Damodar Dharmananda Kosambi as
DDK or just Kosambi. Other abbreviations used frequently in this essay are MR (Mathematical Reviews),
JISAS (Journal of the Indian Society for Agricultural Statistics), RH (Riemann hypothesis), TIFR (Tata Institute of Fundamental Research). Journal names are given in full, and the MR reference number
will help locate the reviews of the pertinent papers.
[3] D. D. Kosambi, “Precessions of an elliptical orbit”, Indian Journal of Physics 5, 359–64 (1930)
[4] D. D. Kosambi, “On a generalization of the second theorem of Bourbaki”, Bulletin of the
Academy of Sciences, U. P. 1, 145–47 (1931)
[5] D. D. Kosambi, “Modern differential geometries”, Indian Journal of Physics 7, 159–64 (1932)
[6] D. D. Kosambi, “On differential equations with the group property”, Journal of the Indian
Mathematical Society 19, 215–19 (1932)
[7] D. D. Kosambi, “Geometrie differentielle et calcul des variations”, Rendiconti della Reale Accademia Nazionale dei Lincei 16, 410–15 (1932)
[8] D. D. Kosambi, “On the existence of a metric and the inverse variational problem”, Bulletin of the Academy of Sciences, U. P. 2, 17–28 (1932)
[9] D. D. Kosambi, “Affin-geometrische Grundlagen der Einheitlichen Feld–theorie”, Sitzungsberichten der Preussische Akademie der Wissenschaften, Physikalisch-mathematische klasse 28, 342–45 (1932)
[10] D. D. Kosambi, “Parallelism and path-spaces”, Mathematische Zeitschrift 37, 608–18 (1933); MR1545422.
[11] D. D. Kosambi, “Observations sur le memoire precedent”, Mathematische Zeitschrift 37, 619– 22 (1933); MR1545423.
[12] D. D. Kosambi, “The problem of differential invariants”, Journal of the Indian Mathematical Society 20, 185–88 (1933)
[13] D. D. Kosambi, “The classification of integers”, Journal of the University of Bombay 2, 18–20 (1933)
[14] D. D. Kosambi, “Collineations in path-space”, Journal of the Indian Mathematical Society 1, 68–72 (1934)
[15] D. D. Kosambi, “Continuous groups and two theorems of Euler”, The Mathematics Student 2, 94–100 (1934)
[16] D. D. Kosambi, “The maximum modulus theorem”, Journal of the University of Bombay 3, 11–12 (1934)
[17] D. D. Kosambi, “Homogeneous metrics”, Proceedings of the Indian Academy of Sciences 1, 952–54 (1935)
[18] D. D. Kosambi, “An affine calculus of variations”, Proceedings of the Indian Academy of Sciences 2, 333–35 (1935)
[19] D. D. Kosambi, “Systems of differential equations of the second order”, Quarterly Journal of Mathematics (Oxford) 6, 1–12 (1935)
[20] D. D. Kosambi, “Differential geometry of the Laplace equation”, Journal of the Indian Mathematical Society 2, 141–43 (1936)
[21] D. D. Kosambi, “Path-spaces of higher order”, Quarterly Journal of Mathematics (Oxford) 7, 97–104 (1936)
[22] D. D. Kosambi, “Path-geometry and cosmogony”, Quarterly Journal of Mathematics (Oxford) 7, 290–93 (1936)
[23] D. D. Kosambi, “Les metriques homogenes dans les espaces cosmogoniques”, Comptes rendus de l’Académie des Sciences 206, 1086–88 (1938)
[24] D. D. Kosambi, “Les espaces des paths generalises qu’on peut associer avec un espace de Finsler”, Comptes rendus de l’Académie des Sciences 206, 1538–41 (1938)
[25] D. D. Kosambi, “The tensor analysis of partial differential equations”, Journal of the Indian Mathematical Society 3, 249–53 (1939); MR0001882 (1,313f) Reviewer: E. W. Titt. Tensor, 2, 36–39
(1939); MR0001075 (1,176c) Reviewer: A. Kawaguchi.
[26] D. D. Kosambi, “A statistical study of the weights of the old Indian punch-marked coins”, Current Science 9, 312–14 (1940)
[27] D. D. Kosambi, “On the weights of old Indian punch-marked coins”, Current Science 9, 410–11 (1940)
[28] D. D. Kosambi, “Path-equations admitting the Lorentz group”, Journal of the London Mathematical Society 15, 86–91 (1940); MR0002258 (2,21f) Reviewer: J. L. Vanderslice.
[29] D. D. Kosambi, “The concept of isotropy in generalized path-spaces”, Journal of the Indian Mathematical Society 4, 80–88 (1940); MR0003125 (2,166g) Reviewer: J. L. Vanderslice.
[30] D. D. Kosambi, “A note on frequency distribution in series”, The Mathematics Student 8,
151–55 (1940); MR0005390 (3,147h).
[31] D. D. Kosambi, “A bivariate extension of Fisher’s Z–test”, Current Science 10, 191–92 (1941);
MR0005589 (3,175h) Reviewer: A. Wald.
[32] D. D. Kosambi, “Correlation and time series”, Current Science 10, 372–74 (1941); MR0005590
(3,175i) Reviewer: A. Wald.
[33] D. D. Kosambi, “Path-equations admitting the Lorentz group–II”, Journal of the Indian Math
ematical Society 5, 62–72 (1941); MR0005713 (3,192g) Reviewer: J. L. Vanderslice.
[34] D. D. Kosambi, “On the origin and development of silver coinage in India”, Current Science
10, 395–400 (1941)
[35] D. D. Kosambi, “On the zeros and closure of orthogonal functions”, Journal of the Indian
Mathematical Society 6, 16–24 (1942); MR0006770 (4,39d) Reviewer: E. S. Pondiczery.
[36] D. D. Kosambi, “The effect of circulation upon the weight of metallic currency”, Current
Science 11, 227–31 (1942)
[37] D. D. Kosambi, “A test of significance for multiple observations”, Current Science 11, 271–74
(1942); MR0007235 (4,107b) Reviewer: A. Wald.
[38] D. D. Kosambi, “On valid tests of linguistic hypotheses”, New Indian Antiquary 5, 21—24
(1942); MR0007247 (4,109a) Reviewer: A. Wald.
[39] D. D. Kosambi, “Statistics in function space”, Journal of the Indian Mathematical Society 7,
76–88 (1943); MR0009816 (5,207c) Reviewer: J. L. Doob.
[40] D. D. Kosambi, “The estimation of map distance from recombination values”, Annals of Eugenics 12, 172–75 (1944)
[41] D. D. Kosambi, “Direct derivation of Balmer spectra”, Current Science 13, 71–72 (1944)
[42] D. D. Kosambi, “The geometric method in mathematical statistics”, American Mathematical
Monthly 51, 382–89 (1944); MR0010937 (6,91c) Reviewer: R. L. Anderson.
[43] D. D. Kosambi, “Parallelism in the tensor analysis of partial differential equations”, Bulletin of the American Mathematical Society 51, 293–96 (1945); MR0011793 (6,217e) Reviewer: J.
L.
[44] D. D. Kosambi, “The law of large numbers”, The Mathematics Student 14, 14–19 (1946);
MR0023471 (9,360i) Reviewer: W. Feller.
[45] D. D. Kosambi, “Sur la differentiation covariante”, Comptes rendus de l’Académie des Sciences
222, 211–13 (1946); MR0015274 (7,396b) Reviewer: J. L. Vanderslice.
[46] D. D. Kosambi, “An extension of the least–squares method for statistical estimation”, Annals
of Eugenics 18, 257–61 (1947); MR0021290 (9,49d) Reviewer: J. Wolfowitz.
[47] D. D. Kosambi, “Possible Applications of the Functional Calculus”, Proceedings of the 34th
Indian Science Congress. Part II: Presidential Addresses, 1–13 (1947)
[48] D. D. Kosambi, “Les invariants differentiels d’un tenseur covariant a deux indices”, Comptes rendus de l’Académie des Sciences 225, 790–92 (1947); MR0022433 (9,207b) Reviewer: N.
Coburn.
[49] D. D. Kosambi, “Systems of partial differential equations of the second order”, Quarterly
Journal of Mathematics (Oxford) 19, 204–19 (1948); MR0028514 (10,458d) Reviewer: M.
Janet.
[50] D. D. Kosambi, “Characteristic properties of series distributions”, Proceedings of the National
Institute of Science of India 15, 109–13 (1949); MR0030731 (11,42h) Reviewer: J. L. Doob.
[51] D. D. Kosambi, “Lie rings in path-space”, Proceedings of the National Academy of Sciences (USA) 35, 389–94 (1949); MR0030807 (11,56a) Reviewer: O. Varga.
[52] D. D. Kosambi, “The differential invariants of a two-index tensor”, Bulletin of the American Mathematical Society 55, 90–94 (1949); MR0028653 (10,480b) Reviewer: V. Hlavaty ́.
[53] D. D. Kosambi, “Series expansions of continuous groups”, Quarterly Journal of Mathematics (Oxford, 2) 2, 244–57 (1951); MR0045732 (13,624b) Reviewer: M. S. Knebelman.
[54] D. D. Kosambi and S. Raghavachari, “Seasonal variations in the Indian birth–rate”, Annals of Eugenics 16, 165–92 (1951); MR0046135 (13,691b) Reviewer: R. P. Boas, Jr.
[55] D. D. Kosambi, “Path-spaces admitting collineations”, Quarterly Journal of Mathematics (Oxford, 2) 3, 1–11 (1952); MR0047387 (13,870d) Reviewer: O. Varga.
[56] D. D. Kosambi, “Path-geometry and continuous groups”, Quarterly Journal of Mathematics (Oxford, 2) 3, 307–20 (1952); MR0051562 (14,498g) Reviewer: A. Nijenhuis.
[57] S. Raghavachari and D. D. Kosambi, “Seasonal variations in the Indian death–rate”, Annals of Human Genetics 19, 100–19 (1954)
[58] D. D. Kosambi, “The metric in path-space”, Tensor (New Series) 3, 67–74 (1954); MR0061869 (15,898a) Reviewer: J. A. Schouten.
[59] D. D. Kosambi, “Classical Tauberian theorems”, Journal of the Indian Society of Agricultural Statistics 10, 141–49 (1958); MR0118997 (22 #9766) Reviewer: J. Korevaar.
[60] D. D. Kosambi and U. V. R. Rao, “The efficiency of randomization by card–shuffling”, Journal of the Royal Statistics Society 121, 223–33 (1958)
[61] D. D. Kosambi, “The method of least–squares”, Journal of the Indian Society of Agricultural Statistics 11, 49–57 (1959); MR0114265 (22 #5089) Reviewer: R. G. Laha.
[62] D. D. Kosambi, “The method of least–squares. (In Chinese.)”, Advancement in Mathematics 3, 485–491 (1957); MR0100960 (20 #7385).
[63] D. D. Kosambi, “An application of stochastic convergence”, Journal of the Indian Society of Agricultural Statistics 11, 58–72 (1959); MR0122792 (23 #A126) Reviewer: W. J. LeVeque.
[64] D. D. Kosambi, “The sampling distribution of primes”, Proceedings of the National Academy of Sciences (USA) 49, 20–23 (1963); MR0146168 (26 #3690) Reviewer: J. B. Kelly.
[65] D. D. Kosambi (as S. Ducray), “A note on prime numbers”, Journal of the University of Bombay 31, 1–4 (1962)
[66] D. D. Kosambi, (as S. Ducray), “Normal Sequences”, Journal of the University of Bombay 32, 49–53 (1963); MR0197433 (33 #5598) Reviewer: B. Volkmann
[67] D. D. Kosambi, “Statistical methods in number theory”, Journal of the Indian Society of Agricultural Statistics 16, 126–35 (1964). MR0217024 (36 #119) Reviewer: A. Rényi.
[68] D. D. Kosambi (as S. Ducray), “Probability and prime numbers”, Proceedings of the Indian Academy of Sciences 60, 159–64 (1964); MR0179148 (31 #3399) Reviewer: J. Kubilius.
[69] D. D. Kosambi (as S. Ducray)“The sequence of primes”, Proceedings of the Indian Academy of Sciences 62, 145–49 (1965)
[70] D. D. Kosambi, “Numismatics as a Science”, Scientific American, February 1966, pages 102–11.
[71] D. D. Kosambi, “Prime Numbers”, Monograph completed a few days before the author’s death;
untraced.
[72] “The Oxford India Kosambi”, Compiled, edited and introduced by Brajadulal Chattopadhyaya (Oxford University Press, New Delhi, 2009); “Combined Methods in Indology & Other Writings: Collected
Essays”. D. D. Kosambi, Compiled, edited and introduced by Brajadulal Chattopadhyaya, Oxford University Press, New Delhi, 2005; “Indian Numismatics”, D. D. Kosambi (Orient Longman, Hyderabad, 1981);
D. D. Kosambi, Steps in Science (Prof. D. D. Kosambi Commemoration Volume) Popular Prakashan Bombay, 1974; D. D. Kosambi, Exasperating Essays, Peoples Publishing House, 1957.
[73] ‘The many careers of D. D. Kosambi: Critical essays’, edited by D. N. Jha (Leftword, Delhi,
2011); ‘Damodar Dharmanand Kosambi’ (in Hindi) edited by R. S. Sharma (SAHMAT, New
Delhi 2010).
[74] Meera Kosambi, ed., “Unsettling the Past: Unknown Aspects and Scholarly Assessments of D.
D. Kosambi ”, (Permanent Black, New Delhi, 2012).
[75] R. Ramaswamy, ‘Integrating Mathematics and History: The scholarship of D. D. Kosambi’,
Economic and Political Weekly, 47, 58–62 (2012). Reproduced in [74].
[76] S. G. Dani, “Kosambi, the Mathematician”, Resonance journal of Science Education, June 2011, pp. 514–28. This issue of the journal is dedicated to D. D. Kosambi and contains several articles
that discuss the scientific contributions of DDK as well as two essays on his life and
historical work.
[77] R. Narasimha, “Kosambi and Proper Orthogonal Decomposition”, Resonance journal of Science Education, June 2011, pp. 574–81.
[78] C. K. Raju, “Kosambi the Mathematician”, Economic and Political Weekly, 54, 38 (2009).
[79] References [3–71] are a complete set of the papers of DDK that are of a mathematical nature.
The list has been compiled in part from incomplete sources in the biography by Chintamani Deshmukh
as well as web listings. In addition to the papers listed above, several of his essays relate to scientific issues, but these are not included here.
[80] R. Ramaswamy, ed., “D. D. Kosambi: Selected Works in Mathematics and Statistics”, (Springer Verlag), forthcoming.
[81] P. Antonelli, R. Ingarden and M. Matsumoto, “The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology”, (Kluwer Academic Publishers, Amsterdam, 1993).
[82] K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung”, Ann. Acad. Sci.
Fennicae. Ser. A. I. Math.-Phys. 37, 1–79 (1947); M. Loève, “Fonctions aleatoires de seconde
ordre”, C.
R. Acad. Sci. 220, 295 (1945) and related papers.
[83] K. K. Vinod, “Kosambi and the genetic mapping function”, Resonance journal of Science
Education, June 2011, pp. 540–50.
[84] Starting with [10] in 1933, DDK wrote a series of papers on path spaces, the last being [58] in
1954.
[85] “Atomic Energy for India”, the text of a talk by DDK to the Rotary Club of Poona, on July 25,
1960 was published in the posthumous volume, “Science, Society, and Peace”, (The Academy
of
Political and Social Studies, Pune, 1967, reprinted by People’s Publishing House, 1995)).
[86] A. Weil, “The apprenticeship of a mathematician”, (Birkhäuser, Basel, 1992).
[87] Tirukkannapuram Vijayaraghavan (1902–1955) was a Founding Fellow of the IASc, being
elected in 1934. He did his Ph. D. under the supervision of G. H. Hardy in Cambridge. From Dacca he moved to
Waltair, and eventually became the founding director of the Ramanujan Institute of Mathematics in Madras.
[88] Sarvadaman Chowla (1907–1995) moved to the US in 1947 after a career at Delhi, Banaras, Waltair and Lahore in undivided India. A student of J. E. Littlewood, Chowla was a number theorist.
[89] In 1940, Weil was in military prison in Bonne-Nouvelle for refusing to take part in the war as a conscientious objector (since his true dharma was the pursuit of mathematics and not war, he
said) when he proved an analogue of the Riemann hypothesis (for the zeta function of curves over finite fields). He did discuss the Riemann hypothesis with T. Vijayaraghavan, who is supposed to have
said that if he could have six months—undisturbed and undistracted—in a prison, he could have a crack at solving the RH. See Ref. [86], and M. Raynaud, ‘André Weil and the Foundations of Algebraic
Geometry’, Notices of the AMS, 46, 864 (1999).
[90] The historical spellings of city names have been retained where it seemed appropriate.
[91] The Kosambi–Bhabha correspondence has been made available through the kind courtesy of the TIFR archives. There are a large number of letters that are presently being catalogued and
edited.
Some have been reproduced in [74].
[92] The most recent effort was in 2010, when Louise J. (Mrs Marston) Morse was nearly 100 years
old. She was kind enough to have the Morse archives searched, but was not able to locate this
manuscript or any reference to it.
[93] Of the 110 or so Founding Fellows, about two thirds were from the south of India or worked
there and Raman might have had greater familiarity with their work or their reputation.
[94] The University of Madras announced the Ramanujan Memorial Prize for “the best thesis based on original contributions submitted by an Indian (or one domiciled in India) on some definite branch of
mathematics, applied or pure” in 1933. The prize was awarded in 1934, as reported
in Nature 135, 28–28 (1935).
[95] “A Chapter in the history of Indian science”, an unpublished essay by DDK is a damning
indictment of Raman’s role in suppressing creativity in Indian science.
[96] K. A. N., “Metrology of Punch-Marked Coins”, Current Science 7, 345–6 (1941). This might have been the historian of South India, K. A. Nilakantha Sastry (R. Thapar, private communication).
[97] D. D. Kosambi, ‘A Note on two hoards of punch marked coins found at Taxila’, New Indian
Antiquary 3, 156–57 (1940) and ‘On the study and metrology of silver punch marked coins’,
New Indian
Antiquary 4, 1–35 and 49–76 (1941).
[98] The existence of a complete archive of these journals at the University of Tokyo has proved to
be an invaluable resource.
[99] H. M. Edwards, “Riemann’s Zeta Function”, (Academic Press, New York, 1974).
[100] S. Hawking, ‘A brief history of time’, (Bantam Books, 2011).
[101] See, e.g. The Stanford Encyclopedia of Philosophy, for a discussion of this very classic paradox, http://plato.stanford.edu/entries/paradox-zeno/#AchTor
[102] These results need elementary but unfamiliar methods of complex analysis that can be found in standard textbooks. There are a number of websites on the internet that give a reasonable
introduction to the mathematics.
[103] M. Berry, private communication.
[104] M. du Sautoy, “The Music of the Primes”, (Harper Perennial, London, 2004); K. Sabbagh, “Dr. Riemann’s Zeros”, (Atlantic Books, NY, 2003); J. Derbyshire, “Prime Obsession: Bernhard Riemann and
the Greatest Unsolved Problem in Mathematics”, (Plume Books, New York, 2004); D. Rockmore, “Stalking the Riemann Hypothesis”, (Vintage, New York, 2005)
[105] A. Odlyzko, private communication.
[106] In addition to Chandrashekaran and Ramanathan, the other two signatories were C. S. Seshadri and M. S. Narasimhan. Fermat’s last theorem, that the equation xn + yn = zn has no solutions with
integer x, y, z if n is larger than 2 was proved conclusively only in 1995 by Andrew Wiles (see Simon Singh, “Fermat’s Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem”,
(Anchor Books, 1998)). Kosambi could not have had the proof he claimed in his letters to eminent mathematicians such as Carl Siegel (who was at the Institute for Advanced Study, Princeton) and
[107] D. D. Kosambi, ‘Lectures on Statistics’, unpublished. This essay concludes, But if I go any further into his achievements, I shall be preaching Bolshevism in the sacral portals of Bombay House
and so must stop here.
[108] This is one of three essays on solar energy that were first reprinted in “Science, Society and Peace”, (The Academy of Political and Social Studies, Pune, 1986), as well as now in [74].
[109] L. Beaulieu, ‘Bourbaki’s Art of Memory’, Osiris, 14, 291 (1999).
[111] N. Bourbaki, ‘Éléments de Mathématique, Book 1: Théorie des ensembles: Fascicule de Resultats ’, (Paris, Hermann, 1939).
[112] The history of the Bourbaki collective has been written about extensively by Maurice Mashaal, Bourbaki: a secret society of mathematicians, (American Mathematical Society, Providence, 2006) as
well as others, including Liliane Beaulieu [114]. While the eventual name chosen by the group was Nicolas, in the original Kosambi paper, it is D. Bourbaki. Whether the initial was for Damodar, or
whether this refers to another of the Bourbaki scions remains a mystery.
[113] R. P. Boas Jr., ‘Bourbaki and me’, Math. Intelligencer, 8, 84 (1986).
[114] L. Beaulieu, ‘Nicolas Bourbaki: History and Legend, 1934-1956’, (Springer Verlag, Berlin 2006).
[115] D. D. Kosambi, “The function of leadership in a mass movement; The Cawnpore Road”, Fergusson College Magazine, pp. 1–7 (1939). Ahriman is the destructive spirit in Zoroastrian mythology.
[116] D. D. Kosambi, “The Raman effect”, Peoples Age, July 1945.
[117] D. D. Kosambi (with Miss Sushila Gokhale), “Progress in the production and consumption of textile goods in India”, Journal of the Indian Merchants’ Chamber (Bombay), January, pp. 11–15
[118] H. W. O. Pétard, ‘A contribution to the mathematical theory of big game hunting’, American Mathematical Monthly, 45, 446 (1938).
[119] A. W. Joseph, ‘A note on the paper by D. D. Kosambi and U. V. Ramamohan Rao on “The efficiency of randomization by card–shuffling”, Journal of the Royal Society of Statistics, 122, 373–74
[120] In ‘Artless innocents and ivory-tower sophisticates: Some personalities on the Indian mathematical scene’, Current Science 85, 526–537 (2003), M. S. Raghunathan [DEL:recalls:DEL] a conversation
with André Weil in 1966 or 1967, when he (Weil) says of DDK, “... Let me tell you this: he was one of the finest intellects to come out of your country.” In his autobiography [86] André Weil recalls:
I appointed Kosambi for the following year. He was a young man with an original turn of mind, fresh from Harvard where he had begun to take an interest in differential geometry. I had met him in
Benares (now Varanasi) where he had found a temporary position.
[121] G. D. Birkhoff, ‘Mathematics at Harvard in the 1940’s’, Proceedings of the American Philosophical Society, 137, 268–272 (1993) | {"url":"https://ddkosambi.blogspot.com/2016/01/a-scholar-in-his-time-contemporary.html","timestamp":"2024-11-04T20:42:41Z","content_type":"text/html","content_length":"206198","record_id":"<urn:uuid:3879149b-76f9-473d-b5ad-0681e0481aa0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00572.warc.gz"} |
WM unit support
Would it be possible to support a more generic base unit - as a whole unit?
As a game developer, I tend to think more along the lines of this:
1 unit:
- what is 1 unit in my game engine?
- what is 1 unit in my 3D editor?
- how tall is my character in units?
1 unit = :
- generic unit in engine
- 1 m in maya (and/or 100 cm)
Character = :
- 2 meters in height
terrain tile = :
- 1024 units / m on edge
- 2048 texels on edge :: macro = 2 texels per unit = texel is 0.5 u/m
- 64 texels per u/m ::
- 1024 detail map = 16 u/m = 64 x detail repeat
So in World Machine, I tend to want to just work directly in meters/km - the crux is that likes in many cases WM likes to display in km:
- I put in 1024 m
- it displays 1.03 km (It would be nice to at least get another decimal place so I can work with a precision that represents my whole number units with confidence!)
What I do as a work around?
- I let 1030 m / 1.03 km be my default tile width/height (because then, any whole number multiplier of that works out 3 tiles = 3 x 1.03 = 3.09)
- basically, 1.03 km = 1024 units (which is a bit funky for artists to think about)
Switching to ‘World Machine Unit’ doesn’t work out well, as it’s basically a floating point (which is even less artist friendly)
Anyone else have any thoughts or input on this?
What about making the size of your terrain 10km x 10km. You’re then fitting 10km into a 1024 texture which you can make to look like a square km.
I agree that the UI would be improved if you could work in whole units, at least for typical dimensions like 1024m! | {"url":"https://forum.world-machine.com/t/wm-unit-support/3795","timestamp":"2024-11-06T14:10:27Z","content_type":"text/html","content_length":"17009","record_id":"<urn:uuid:02639a4f-28f4-48b6-a30f-494b96df22b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00070.warc.gz"} |
r Pu
Finite Element Analysis and Theoretical Study of Punching Shear Strength of Concrete Bridge Decks
Mufti AA^1* and Newhook JP^2
^1Department of Civil Engineering, University of Manitoba, Canada
^2Department of Civil and Resource Engineering, Dalhousie University, Canada
Submission:February 08, 2018; Published: April 05, 2018
*Corresponding author: Aftab Mufti, Emeritus Professor, Department of Civil Engineering, University of Manitoba, Canada, Email: Aftab.Mufti@umanitoba.ca
How to cite this article:Mufti AA, Newhook JP. Finite Element Analysis and Theoretical Study of Punching Shear Strength of Concrete Bridge Decks. Civil Eng Res J. 2018; 4(2): 555635. DOI: 10.19080/
The problem of punching shear usually arises in reinforced concrete slabs subjected to concentrated loads and particularly in concrete bridge decks due to the development of an internal arching
system. Ongoing research revealed that the governing mode of failure for concrete bridge decks is not flexure and that using the flexural design method usually led to unnecessary high levels of steel
reinforcement. This paper examines the applicability of the non-linear finite element formulation of restrained concrete bridge decks. A special purpose finite element program FEMPUNCH was developed
and employed in this study along with the commercially available finite element programs. The accuracy of non-linear finite element analysis is demonstrated using test results conducted by other
researchers. The results of the finite element analysis are also compared to those obtained from a non-linear mechanics model developed by Mufti and New hook. The experimental results and the
theoretical model provide insight to the fundamental behaviour of concrete bridge decks.
Keywords:Polypropylene; FEMPUNCH; Bridge deck slabs
Abbrevations:GFRP: Glass Fibre Reinforced Polymer; PFRC: Polypropylene Fibre Reinforced Concrete; CHBDC: Canadian Highway Bridge Design Code; ANACAP: Anatech Concrete Analysis Program
Concrete slab-on-girder bridge decks have been a popular choice for many short and medium span bridge applications throughout North America. The authors of this paper have been part of a Canadian
research team, which has been empirically and theoretically studying the behaviour of these decks for several decades [1,2]. Of principal interest is the strength of these decks under concentrated
wheel loads. This on-going research effort has improved our understanding of the key mechanisms associated with the strength and stiffness of these decks and the importance of in-plane restraint in
the performance of concrete bridge deck slabs. Innovative bridge deck designs have been developed [3] and applied to several highway bridges and other structures in Canada [4] and more recently in
the United States.
This paper reports on the theoretical study of the punching shear behaviour in restrained concrete bridge deck slabs of girder bridges with emphasis on the analytical comparison of non-linear finite
element analysis and the PUNCH program developed by Mufti & Newhook [1]. While the design practice is generally to estimate punching shear strength in concrete using simple empirical equations, these
equations provide little guidance as to the mechanisms involved and the impact of various parameters on design. The work summarized in this paper helps identify for the design engineer, the basic
behaviour of the system, the conditions which lead to punching failure and to demonstrate how non-linear finite element analysis can be adopted to predict the behaviour of these systems, not just to
estimate ultimate load. The accuracy of the non-linear finite element analysis was demonstrated using test results available in the literature. The results of the finite element analysis have also
been compared to those obtained from a mechanics model developed by the authors.
An externally restrained steel-free concrete bridge deck slab [2] consists of providing a system of steel straps between adjacent girders to restrain any lateral movement of these girders. All
internal reinforcement can be removed; internal glass fibre reinforced polymer (GFRP) crack control reinforcement may be added if desired [3]. Research has also shown that the bottom transverse
reinforcing bars act as ties for this same arching action mechanism rather than as flexural reinforcement for the perceived moments [5]. Deck slabs with internal reinforcement of either steel or FRP
[3] are therefore considered to be internally restrained deck slabs. Under concentrated wheel loads, the restraining elements develop increasing tensile stresses and provide a lateral restraining
force to the concrete deck slab. As a result, compressive membrane forces within the deck slab are developed. The degree of lateral restraint provided will determine the ultimate load at which the
deck fails in punching under concentrated loads such as vehicle wheel loads.
Extensive field and laboratory testing on concrete deck slabs subjected to wheel loads indicates that two structural responses occur. The initial response is primarily flexure, which leads to the
formation of longitudinal cracks on the underside of the deck between adjacent girders. After the formation of the first longitudinal crack, however, compressive membrane action or arching also forms
a component of the response mechanism. If sufficient lateral restraint exists, then the arching response will dominate the latter stages of the loading behaviour and failure will occur through
punching of the deck slab.
The punching failure of restrained reinforced concrete slabs had been documented as early as the 1960s by Taylor & Hayes [6] and investigated specifically as it relates to bridge deck behaviour in
the 1970s by Hewitt & Batchelor [7]. In these and other works, the focus was on the enhancement that arching behaviour provided to reinforced concrete systems. The steelfree deck slab evolved from
the philosophy of using the arching behaviour as the primary resistance mechanism. This approach necessitated the enhancement of internal arching forces through a clear and quantifiable in-plane
lateral restraint system in both the longitudinal and transverse directions. This restraint is provided by two elements. Firstly, the slab is made composite with the supporting girders, either steel
or prestressed concrete. In the longitudinal direction, the large axial stiffness of the girders provides in-plane restraint. Secondly, in the transverse direction, the required restraint is achieved
through the addition of external steel straps, normally 25x50mm in cross-section and spaced at 1200mm, which inhibit the relative lateral displacement of adjacent girders. With compression as the
dominant mechanism, all internal reinforcement can be removed. Further design details can be found in the design clauses of CHBDC [3] and a recent report of the American Concrete Institute [8]. The
essential elements of the system are shown in Figure 1.
In early research, this system was often referred to as a fibre reinforced concrete (FRC) or polypropylene fibre reinforced concrete (PFRC) deck slab due to the use of short randomly distributed
synthetic fibres to control plastic shrinkage cracking. In later work, it was referred to as steel-free concrete deck slabs or corrosion-free concrete deck slabs. In the most recent version of the
Canadian Highway Bridge Design Code (CHBDC) [3], it is referred to as externally restrained concrete bridge deck slabs. In reporting the research in the following sections, these historical
designations have been maintained.
To help develop a reliable theoretical tool for the analysis of bridge decks, the authors re-examined the physical observations made during the experimental program on a variety of steel-free deck
models. A more extensive literature review also turned up many similarities between the experimental column punching tests conducted by Kinnunen & Nylander [9]. Kinnunen and Nylander had also
proposed an empirical failure criteria based on concrete strains close to the punching zone (referred to herein as K&N failure criteria). Using test observations and the K&N failure criteria, a
mechanics model of the behaviour was developed and refined to address additional unique aspects of the externally restrained bridge deck slabs under wheel loads.
The important components of the system geometry are the depth of the concrete deck, the spacing of the support girders, the spacing and the cross section area of the transverse straps and the
dimensions of the loaded area. The essential material parameters are the modulus of elasticity of the material of the transverse straps, the yield strain of the straps, the compressive strength of
the concrete, and the 3-dimensional effect on the compressive strength of the concrete under confined conditions.
During the initial loading phase, a concrete slab subjected to a concentrated load forms radial cracks on the bottom surface of the slab originating below the load point. As the load increases, these
cracks gradually migrate to the top surface of the slab to become full depth cracks (Figure 2). On the top surface of the slab, circular cracks form at a diameter approximately equal to the clear
spacing between the girder flanges. As the failure load approaches, an inclined shear crack develops, originating on the bottom of the slab at some distance away from the load point, in a
circumferential manner. At punching failure, this inclined crack forms the upper surface of the punch cone. The sections of concrete outside the shear crack can be divided into wedges bound by the
shear crack, the radial cracks, and the outside edge of the slab. Under further loading, these wedges act as rigid bodies in the radial direction and rotate about a centre of rotation (Figure 2 & 3).
As a wedge shown in Figure 3 rotates through an angle , it has an associated lateral displacement L which is restrained by the stiffness of the straps. If we designate K as the stiffness of the
strap, in units of force/displacement per unit length of circumference, then the restraining force is calculated as K L. If we consider a single wedge component, then the forces acting on this wedge
are an oblique compressive force acting through the compressive shell and supporting the loaded area T; the vertical support component P Δ Φ /2 ^Π ; the lateral restraining force for the wedge Fw;
and a circumferential force R that is developed as the wedge rotates through . The circumferential force R is maximum at the edge of the loaded area. Considering these forces acting in combination
with the horizontal component of T and the vertical stress from the applied load, the region near the loaded area is in a state of triaxial compressive stress with σ 1 ≤ ^σ2 ≤ ^σ 3 and ^σ3 = ^a r’^σ
2 = ^σθ and ^σ 1 = ^σ v. Equation 1 is used to model the behaviour of concrete under confinement which enables σ3 to be expressed in terms of 1. As long as the three dimensional state of confinement
is maintained around the loaded area, the concrete capacity will remain very high.
Where, k is a confinement factor (based on comparisons with the experimental punching program, Newhook [10] determined that a value of k=10 was appropriate for the bridge deck application).
Employing the equilibrium conditions and relationships described above, the forces and displacement can be resolved for a given value of applied load P. A complete description of this mathematical
formulation is found in Newhook [10]. In addressing the similar problem of the punching of a column slab connection, K & N [9] established empirical criteria for punching failure which was adopted by
the authors for bridge deck slab punching. When the circumferential strain at the top surface of the slab close to the loaded area reaches a critical value, ^ϵ ct =0.0019, failure occurs. This value
corresponds to the value of strain at the maximum uniaxial compressive stress fc, and the commonly used value of ^ϵ ct =0.002 is adopted. While K&N merely reported this value as an empirical
observation, the authors propose that this criterion also corresponds well with the importance of the 3-dimensional confinement of the concrete surrounding the loaded area. Once ^ϵ t equals 0.002,
the concrete response softens in that direction leading to a reduction in the confining force, hence punching failure.
The authors also proposed a second parameter that is also capable of initiating punching failure. The lateral restraint is provided by transverse steel straps. The stress in the straps increases with
applied load. It is possible that the stress in the straps reaches the yield stress of steel before failure of the concrete occurs. Once the strap yields, it can no longer provide increasing
restraint force and hence the concrete will punch under a slight increase in applied loading.
Comparisons were conducted between experimental failure loads and those predicted by the mechanics model Newhook & Mufti [1]. A partial listing of those comparisons is presented in Table 1.
Comparisons of the theoretical and experimental load- deflection and load strap strain results also showed very close agreement [10].
The theoretical model was used to analyze three half-scale tests reported by Mufti et al. [2] and Newhook [10], a deck test on a skew bridge reported by Bakht & Agarwal [11] and full scale testing
reported by Thorburn & Mufti [12] and Newhook & Mufti [13]. The experimental deck slabs failed in punching shear and a comparison of theoretical versus experimental failure load is presented in Table
1. The comparison reveals that the mathematical model can predict within reasonable accuracy the punching failure load of internally restrained bridge deck slabs (Figure 4).
Using the procedure described above, a load deflection curve is constructed for half scale Test No. 5 from Table 1. A small slab displacement is assumed and the associated value of applied is load
determined along with strap strain and concrete strain values. If the failure condition is not satisfied, an increment of deflection is assumed and new values determined Newhook [10,14]. This process
is repeated until a failure condition is met. Using the values of displacement and load, the theoretical curve shown in Figure 4 is produced. While the experimental curve the theoretical curve though
smooth is in good agreement with the observed deflections. The model is reliable in predicting the order of the magnitude of deflection. The experimental deck deflections at ultimate for the tests in
Table 1 are compared with theoretical deflections in Table 2. The comparison reveals very close correlation providing further support for the validity of the mechanics model(Table 2)
Based on the successful development of this model, the following parameters were confirmed as necessary to properly predict the bridge deck behaviour under wheel load including prediction of ultimate
load: deck thickness, girder spacing, strap geometry (restraint stiffness), concrete strength and strap modulus. In addition, strain limits in the strap material and the concrete could be used to
define ultimate limit state.
The initial investigation of externally reinforced bridge deck behaviour was conducted by Wegner and Mufti [15] who attempted to use finite element modelling to replicate the behaviour of the
experimental model [2] listed as Test 3 and 4 in Table 1. The commercially available program ADINA (ADINA R&D Inc. 1987) was chosen, since it incorporated a sophisticated tri-axial material model for
plain concrete. Specifics of the model and analysis details are reported elsewhere [15,16]. While Wegner and Mufti were able to model the behaviour of one specific test, they concluded that this
early finite element modelling attempt was very sensitive to a number of parameters and that extensive calibration with known results and modelling iterations would be required to fine-tune the
numerical procedures for a particular application.
The Wegner and Mufti finite element study was conducted prior to the development of the analytical model described in Section 3. This model indicated clearly that for a commercial program to be used
successfully the K&N strain limit failure criteria must be used. Hassan et al. [17] conducted finite element analysis of an internally restrained concrete bridge deck slab subjected to wheel loads
using the "Anatech Concrete Analysis Program" (ANACAP, Version 2.1) and defining failure as the load at which the K&N failure criteria is met. One-half of a bridge deck slab was modelled using
20-node brick elements. The slab thickness was divided into three layers. The mesh dimensions used in modeling the deck slab are given in Figure 5. The reinforcement was modelled as individual
sub-elements within the concrete elements. Rebar sub-element stiffnesses were superimposed on the concrete element stiffness in which the rebar resides. A complete description of the modeling can be
found in Hassan et al. [17] (Figure 5).
Hassan et al. [17] modeled the punching behaviour of a full- scale experimental model of a concrete deck slab on two steel girders tested by Khanna et al. [5]. The experimental model had four
segments (Segments A through D) in which the internal reinforcement configuration was varied. Segment A had a top and bottom grid of orthogonal reinforcement, typical of current design practice, and
Segment C had just bottom transverse bars. The comparison of the ANACAP theoretical results with the experimental load-deflection results for these two segments are presented in Figure 6a and 6b,
respectively. For failure prediction, use of the K&N criteria was critical. With this failure modeling approach, the FEM model proved to be far more robust than the model used previously by Wegner &
Mufti [14]. Furthermore, the study confirmed that the presence of top reinforcement in bridge deck slabs has a negligible effect on punching shear capacity (Figure 6a & 6b).
The successful use of the K&N failure criteria with a commercial FEM program led to the development of a specialized FEM program for non-linear analysis of externally restrained concrete slab on
girder bridges, FEMPUNCH. The program has several features which facilitate the pre-processing and input of bridge geometry. It permits the input of values for the lateral restraint of the straps and
includes an orthotropic material model for concrete as well as permits concrete cracking and strain softening and models the benefits of 3-dimensional confinement. It includes the limiting strain
proposed by K&N as the failure criteria. The concrete is modelled using a 20-node isoparamteric brick element and the girders are modelled using 2-dimensional beam elements.
Complete details on modeling assumptions and parameters can be found in Desai et al. [18].
The following assumptions and simplifications have been incorporated into FEMPUNCH:
I. The stiffness of straps is smeared across the length of the girders.
II. Lateral stiffness of the supporting girders is considered to be uniform across the length of the girder.
III. The girders are unyielding in the vertical direction, a simplification that can be remedied easily.
IV. Concrete is modelled as an orthotropic material with the direction of orthotropy being defined by the principal stress directions.
V. Cracking can occur in three orthogonal directions at an integration point used in the formation of finite element equations.
VI. If cracking occurs at an integration point, it is modelled through an adjustment of material properties by considering cracking as a smeared band of cracks.
VII. Strain softening occurs from compression crushing failure to the ultimate strain, at which the concrete fails completely.
VIII. The tensile cracking failure and the concrete crushing failures occur along the principal direction
IX. Load increases monotonically, leading to failure
An isoparametric, 20-node solid brick element has been used in the program to model a bridge deck. At each node, three translational degrees of freedom (u, v and w) are considered. Details of the
shape functions and formulation of element level matrices can be found in the literature.
Only one element has been considered in the thickness direction on the basis of the finite element work done by Wegner & Mufti [15]. On the other hand, the number of elements in the longitudinal and
the transverse direction is chosen such that the aspect ratio is nearly one. On the basis of the average thickness, the program would select the number of elements automatically between two adjacent
sections. However, a user can override this option and feedin the information explicitly.
The 3-dimensional constitutive relations for concrete have been derived from an equivalent, uniaxial constitutive relation, which can be determined experimentally. The uniaxial stress- strain curve
has been employed in the program to calculate the tangent modulus.
The stress-strain curve for the concrete has been described by the Saenz's equation [18]. While other 3-dimensional concrete models exist, Saenz's model was selected as it was found to be easier to
program and provided equally accurate results. It also avoided singularity problems in the solution process encountered by Wegner & Mufti [15].
Tensile failure would occur in concrete when the tensile stress in the principal direction exceeds the tensile strength, ^a f, of concrete. The presence of a crack at an integration point is modelled
by modifying the constitutive matrix [18]. When a crack forms, the normal as well as the shear stiffnesses reduce. Consider for illustration that ^σp1^ exceeds ^σf.^
The ensuing system level equations are solved in the Lagrangian coordinate system using Newton-Raphson procedure. In-core solution technique has been used to expedite computations. The loads are
applied incrementally and iterations are performed per load increment until the solution converges. The incremental displacements and stresses areadded to the values from previous load increments to
get the total displacements and stresses. The numerical solution iscontinued until failure occurs.
The internally restrained deck slab considered by Wegner & Mufti [15] (Figure 7) has been analyzed using program FEMPUNCH. The load versus deflection under the wheel load results obtained from
program FEMPUNCH have been compared with the experimental data, as well as results from the ADINA analyses and results from program PUNCH [14]. It can be seen in Figure 8 that the results from
program FEMPUNCH match reasonably well with the experimental data at failure (Figure 7 & 8).
This specialized finite element program incorporates behaviour characteristics and failure criteria developed in the analytical model of Section 3, it also provides several key improvements:
a) Both right bridges and skew bridges can be analyzed;
b) The stiffness of the bridge girders can be modelled;
c) The influence of the global stiffness of the concrete deck outside the punch zone can be included; and
d) The influence of the wheel position relative to the planar geometry of the deck can be modelled.
Further investigations by Klowak et al. [19] demonstrated that the FEMPUNCH program can also be used to predict the response of deck cantilevers to wheel loads.
This paper presents a brief overview of the authors' investigations into modeling the non-linear behaviour and failure of concrete slab on girder bridges to supplement the extensive experimental work
in this area. Analytical and specialized finite element tools have been presented. In developing these models, the use of the K&N failure criteria was a key consideration in achieving correlation
with experimental results. Both models also allow for the development of a load versus deflection curve for the entire loading not just a prediction of ultimate load. Hence the models, and in
particular the finite element, will permit engineers to study the impact of various geometry and material parameters on the behaviour of the deck system. These parameters, in general order of
greatest impact on ultimate load, are:
a. Stiffness of the external restraint to the deck including straps and girders;
b. Deck thickness;
c. Girder spacing;
d. Strap material properties; and
e. Concrete strength.
The modeling work also shows that the punching strength of the concrete deck slab cannot be considered a function of just the concrete slab thickness and compressive strength alone but is a property
of the entire deck system including the girders and any other means of lateral restraint, in the case external straps.
The punching strength of a concrete slab on girder bridge deck subjected to wheel loads represents a complex problem including both material and geometric non-linearities. To date, no simplified
formula has been able to capture the impact of all the parameters discussed in this paper. It is believed that the research presented in this paper will be of assistance to engineers seeking to
implement more rigorous solutions and to better understand the influenceof their design choices on the performance of this bridge deck system.
The authors acknowledge the contributions to modeling and understanding bridge deck behaviour by the many graduate students and research assistants whose published work is cited herein. The authors
also acknowledge the long standing collaboration with our colleague Dr. Baidar Bakht who initially requested that we develop a theoretical model to help understand the behaviour observed during
experimental work and who has continued to participate in the solutions. | {"url":"https://juniperpublishers.com/cerj/CERJ.MS.ID.555635.php","timestamp":"2024-11-03T19:54:50Z","content_type":"text/html","content_length":"104305","record_id":"<urn:uuid:fb87ee5b-861c-4864-b662-472f46d1464d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00813.warc.gz"} |
Delta's D&D Hotspot
An analysis of the proper time scale in D&D follows. Let's start by considering movement. First, note that a speed of 1 mph is about 100 feet per minute (actually 88 ft/min, but close enough).
Consider an article on human gait, including standard military march times (
). Both original Chainmail and 3E D&D scale are pretty close to assuming that base move matches a standard "Quick March" speed, that is, about 300 ft/min (3 MPH). If we use OD&D movement (12" for an
unarmored man), and a corrected scale of 1"=5 feet (i.e., 60 ft per move), then the proper scale round would be about 1/5 of a minute, that is 10 or 12 seconds. We can double this number for a good
"Double March" speed (a sustainable jog/run at 600 ft/min, or 6 mph), or double again for the maximum run which can be sustained for a few minutes at most (say 12 mph).
(As an aside, consider a flat-out sprint which only lasts 10 or 20 seconds or so. Modern research shows that slower runners can attain about 15 mph, intermediate runners 20 mph, and the world-records
for 100m and 200m sprints are held at a speed of 23 mph.)
Now let's consider a separate consideration: how quickly attacks take place in combat. This is a lot harder to pin down, unless we had real-life medieval combats taking place to analyze and time. The
best thing I could come up with is professional boxing, using data from the "CompuBox Stats Archive" (
). I've taken a fairly quick, random sample of 3 different bouts (6 fighters) in each of the light-, middle-, and heavyweight classes. Each bout lasted a full 12 rounds (36 minutes plus breaks), and
I've only considered "power punch" statistics, which in theory could actually do some kind of damage. (That is, I left out "jab" counts, which are presumably only maneuvering setups for actual
damaging attacks.)
So, a few things become clear about the "sweet science" from this table (note the "P/M" column, which indicates average punches-per-minute). The number of punches thrown goes down as the weight class
goes up -- presumably this would continue downwards when using heavy martial weapons? The overall average here is 9 punches/minute (but possibly only 6 if we take another step down in weight
categories). That again argues for a D&D round length of about 10 seconds or so (6 per minute) -- that provides a base number of attacks; expert fighters, such as these top-level boxing
professionals, conceivably have the capacity to increase punches up to 2 per round at this scale (or maybe 3 as an absolute upper limit for unarmed lightweights).
Note also that power punch success rate is only about 35% for all of these top-level fighters (i.e., 14-20 on a d20). I think this argues for an unarmed combat system which highlights a lot of
defense and blocking capacities (say, at least -4 to hit with unarmed attacks). Recall that I left out jab statistics from these numbers; that successful landing percentages for those are even
than the power punches shown here. Also consider that these fighters are receiving one or two hundred of these so-called "power punches" (with gloves) and continuing to fight, so individually only a
tiny fraction of them can do actual hit-point damage in D&D terms.
Let's look at it from another perspective, like bow fire rates (
). It's said that a longbow could fire "as many as 20 shots a minute". But, consider a few other details: (1) that presumes a top-level expert bownman, (2) it considers a relative lack of aiming, as
would be acceptable in a mass battle barrage, and (3) it also implies that "an archer could loose (shoot) 3 arrows before the first arrow hit its target"! Now, we certainly don't want to have to
adjudicate a single arrow being in-flight over the course of 3 rounds or so. So what we should do is take these numbers and divide: 20/3 = 6.66 is the discrete number of arrows that can be carefully
aimed, fired and landed sequentially, in one minute -- and hence the best number of rounds per minute. Again, we can use a 10-second round (6 per minute), assume carefully-aimed missile attacks, and
allow top-level fighters to possibly make 2 (or maybe, at the very best, 3) attacks per round, and the result is quite close to real-life.
There are other reasons to support a 10-second round for man-to-man combat. (Another one that I like is to assume you can hold your breath for your Constitution, in rounds, and the result is again
very realistic.) So, that's the final uptake on all this for my preferred games of D&D.
Conclusion: One combat round should last 10 seconds.
It may be interesting to consider the most powerful monsters listed in the OD&D rules. Here's a list of all the monsters with potentially 10 or more Hit Dice (original white box only, Vol-2):
1. Elementals (8-16 HD)
2. Purple Worms (15 HD)
3. Giants (8-12)
4. Dragons (5-12)
5. Hydras (5-12)
6. Balrogs (10)
7. Efreet (10)
8. Black Pudding (10)
One thing you can see is that for the time, the mainstays of Giants and Dragons are definitely near the top of heap of most powerful monsters in the game. (This is before power inflation required
them to swell in hit dice in 2E and 3E games.)
At the very top of the list are Elementals, in particular the 16 Hit Die types brought about by the 5th-level magic-user spell, conjure elemental, and are subject to a whole slew of special
restrictions and risks (only one per type per day, large raw material requirement, maintain concentration or caster gets attacked, etc.) You can see how important those limitations are, when you get
to call forth the toughest monster in the game-world, and put it under your control, any time you like (and it's not even the highest-level spell).
Other than that, the only thing more powerful than Giants or Dragons are the tremendous Purple Worms. Actually, I really like that flavor -- the most dangerous creature in the natural world, blind in
the underworld, burrowing incessantly "just beneath the surface of the land" (Vol. 2, p. 15; compare, for example, to the monstrous "Dholes" in various H.P. Lovecraft stories).
Balrogs, of course (item #6), were included in the earliest editions of the OD&D game, but were removed in later printings after a skirmish with the Tolkien estate. They reappeared later on in
Eldritch Wizardry as "Demon, Type VI (Balrog)" (Sup-III, p. 12), with some minor name-mangling afterward in the AD&D version. (See more here.)
And the other thing that might be surprising is the appearance of the Black Pudding monster, described as just "another member of the clean-up crew and nuisance monster" (Vol. 2, p. 19). Note the
extraordinary strength of this creature, as shown by its very high Hit Dice (and brutal 3 dice of damage, the most in the game!) Perhaps the creature either needs to be as enormous as other creatures
listed here, or be extremely rare due to some supernatural or unearthly part of its makeup. See the adjacent picture; scary! [from Sup-I, Greyhawk, p. 14] Other ooze-types like the ochre jelly, green
slime, gray ooze, or yellow mold have only a fraction of the same hit dice. (Note also that, oddly, only the ochre jelly appears on the wandering monster tables.)
The list is modified somewhat if you take into account the special hit point accumulation for Dragons by maturity. In particular, a Very Old (6 hp/die) Gold dragon has 12 × 6 = 72 hit points, which
is on average the same as 72/3.5 = 20 hit dice. In other words, it's the most powerful creature in the game (unless there's a great-granddaddy Elemental or Purple Worm somewhere that rolled all 5's
and 6's for its hit points). The same can be said for Hydras, of course.
In addition, there are notes in the text that indicate the possible existence of even more powerful creatures. Sea Monsters start as Purple Worms, and increase to 2 or 3 times that size! (30 to 45
hit dice; Vol. 2, p. 15; although reduced from those levels when they later appeared as prehistoric beasts in Sup-II Blackmoor.) Rocs start at 6HD, but can increase to 2 or 3 times the basic listing
(up to 18 HD, which became standard in AD&D; p. 17). Animals are also considered up to a Tyrannosaurus with 20 HD (p. 20).
Finally, If you add Supplement I: Greyhawk, then there are other rare and powerful monsters. These include Titans (effective 25 HD), Golems (23, 17, or 11 effective HD), Storm Giants (15), Giant
Slugs (12), Beholders (11 or so), and Liches (10+). Note that some of these creatures are actually proposed in the original set (Vol. 2, p. 21), along with ideas for super-strong Cyclopes,
Juggernauts, Robots, etc.
Honorable mention goes to the following (8-9 Hit Dice): Vampires, Gorgons, Chimeras, Treants, and Invisible Stalkers (plus Will O'Wisps and Umber Hulks from Greyhawk).
One of the wonderful things about the OD&D rules (addictive, even) is that they're sparse enough to be manageable if you're interested in modifying, house-ruling, or fixing them. For example, when I
think about modifying 3rd Ed., such as its skills or feats system, I find that there are simply far too many entangled parts to spend time modifying the entire game system. Below you'll see some very
simple modifications to the OD&D rules that I think provide huge benefits. In particular, I've always been frustrated by D&D's ahistorical gold-based economy; you'll see that in the course of a
single evening I was able to complete an analysis and change the entire pricing structure of all the equipment, wages, treasure, and constructions in the entire game to largely fix that!
The alternate damage and hit die system from Greyhawk (Supplement I) is not used. The majority of weapons still do 1d6 damage, with the following exceptions:
* Dagger, Hand Axe, Mace – 1d4 damage.
* Halberd, Two-Handed Sword – 1d8 damage.
Two-handed weapons are as follows: Pole Arm, Halberd, Pike, Two-Handed Sword, Spear.
Throwing weapons are the following: Hand Axe, Spear (3" range each, no modifier to hit from range).
Other missile weapons have the following ranges (from Chainmail): short bow 15", crossbow 18", longbow 21", heavy crossbow 24".
An alternative combat system is presented here that produces nearly the same results as the book, but is so simple that it can be easily memorized without any table references.
For miniature usage, convert all underground scales to 1" = 5 ft (thereby matching ground to standard figure scale). Assume that one combat round takes 10 seconds.
Attacks are made by rolling d20 + fighter level + target AC. If the result is 20 or more, a hit has been scored. Monsters use their hit dice for level; magic-users and clerics use half their level.
Saves are also made by rolling d20 + level + modifiers. If the result is 20 or more, the save is successful. The following modifiers are used:
Fighters: no modifier.
Clerics: +1 to all saves.
Wizards: +1 versus spells but –1 to all others.
Spell: no modifier.
Breath: +1
Stone: +2
Wand: +3
Death: +4
The actual medieval world widely utilized coinage based on small silver-copper-zinc pence (pennies, d). Recognized units were 12 pence = 1 shilling (s), and 20 shillings = 1 pound (L), but these were
counting units only, and usually did not exist as actual coins. For purposes of our D&D campaign, we'll assume the existence of large silver (shilling) and gold (pound) coins.
Prices in the game should now generally be read in "pence" instead of "gold pieces". Starting wealth, basic equipment prices, magic item costs, magical research, gem and jewelry values, stronghold
constructions, and specialist wages are all approximately correct if read as pence. Exceptions are as follows.
Armor and horse costs should be read in silver shillings (making them much more valuable than other items in the list). Costs for men-at-arms must be read in pence-per-day. Monster treasure tables
are read in 1000's of copper, 100's of silver, and 10's of gold pieces (which still results in treasure more than twice as valuable as before). Dungeon treasures should be read in copper and silver
pieces, with coin amounts divided by 10 (which results in the same values as before; feel free to exchange amounts for gold on deeper levels).
Note that armor and horses are now so valuable that they are likely out of the price range for new characters. Consider giving fighter-types a free suit of leather armor to start with (much as
wizards begin with a free spellbook). All listed equipment costs are for the most basic utilitarian type; finely-made arms and armor, champion horses, quality wines, and even covered wagons will be
many times more expensive. Lawful characters should expect to send wealth from a fallen comrade back to their given family, clan, or fraternal order.
I've carefully compared the basic equipment list prices to real-life prices from the Medieval Sourcebook (link below). In general, the prices would be approximately accurate if the were priced in
copper pence instead of "gold pieces".
This is verifiable for things like weapons (real-life 6d cheap sword vs. 10 cost in D&D; 5d axe vs. 3; 4d chisel vs. 3-cost dagger), food (a week of dried fruit 28d vs. iron ration 15; a week of
cheese, 7d, or salted fish, 2d, vs. standard rations 5), and travel (iron-bound cart 4s=48d vs 100; a barge 10L=2400d vs. small merchant ship 5000).
The notable exceptions are armor and horses, where the prices would be approximately correct if they were in silver shillings; that is, in units 12 times more valuable than other costs.
This can be verified with somewhat more difficulty than the preceding categories (consider real-life 16th c. cuirass with pauldrons, 40s vs. D&D "plate mail" 50; 13th c. merchant's armor for 5s vs.
"leather" 15; ox 13s vs. mule 20; draft horse 10-20s vs. 30; high-grade riding horse 10L=200s vs. light horse 40; knight's horse 5L=100s vs. medium warhorse 100).
Similar very rough comparisons have been made using the Medieval Sourcebook's sections for Wages, Buildings (constructions), and Miscellaneous items (jewelry).
The Medieval Sourcebook (accessed March 14, 2007):
One of the wonderful things about OD&D is the bare-bones descriptions of spells and magic items. Generally they are very common-sensical, before the language had to be expanded or tightened up to
handle loophole cases or unexpected behavior. When I read them in this state, it's a lot easier to visualize fantasy-sensible rulings on spells I've had problems with before. Here's two examples:
- Transmute Rock to Mud. The problem with this 5th-level spell is it can frequently be taken to instantly reduce any castle or dungeon complex to quivering mud (i.e., for the campaign world it makes
the entire technology and political usage of castles totally useless). But the origin is clear: it comes from the mass battle rules, intended to make a large area hard to pass for troops, and was
never created thinking about fortifications or enclosed spaces.
So my ruling in this case would be to simply state that it can only be used to make a "mud pit", lying on top of a generally horizontal surface, as originally intended. Trying to use it on
load-bearing, free-standing, or vertical structures (castle or dungeon walls) causes the spell to fail.
- Silence, 15' Radius. Added in Supplement I: Greyhawk, this is another problematic spell, in that it can instantly shut down opposing spellcasters -- not just by being cast at them, but by any
thief, fighter, or thrown stone with the spell on it getting in range of effect. Also, it's just odd from a flavor aspect that clerics can make this radiating sound-cancelling aura, which to my
knowledge has no analog in standard fantasy or myth. Strange.
But reading it in OD&D, the intention seems pretty clear. The language is practically the same as something like invisibility 10' radius or haste or slow. That is, I would rule that the silence spell
effects specific individual objects or creatures -- you can cast it on several bodies within 15', which are then individually quieted. They don't radiate an aura of silence from that point forward;
they are simply themselves quieted for the purposes of movement and surprise, much like elven boots or the like (in fact, I'd still allow normal speech and spellcasting if so desired). To me, that
makes infinitely more sense as a divine magic effect.
Random stuff I like in OD&D:
- Simple equipment. There's a single one-page list of all the equipment you need, including weapons, armor, and gear. Every cost is simply in gold pieces (there are no fractions or cp/sp to make
change with). Armor is simply leather, chain, plate, shield, helmet. They still manage to include horses, mules, wagons, and those pricey boats! (merchant ships and galleys)
- Naval rules. On the topic of boats, OD&D has what looks like the most playable ship rules I've seen for D&D (and I've looked for a long time). It very concisely has rules for points-of-sail and
wind power, movement in inches, specific crew numbers for each ship type (something I always longed for in AD&D), and reference to Chainmail rules for combat. Nice!
- Limitless Levels. "There is no theoretical limit to how high a character may progress." (Vol. 1, p. 18) No supplemental books are required -- Right from the get-go, a single paragraph provides
rules for continuing advancement in hit dice, fighting ability, even never-ending spell advancement for all the classes! The way experience gets added is a bit unclear, but you can work out something
- The Astral Spell. Introduced in Supplement I: Greyhawk, this spell has wonderful flavor that matches standard fantasy much more closely. It's a lot like what the ethereal spell is today (which
itself doesn't exist in OD&D). No travelling outer planes, it makes a powerful spellcaster basically invisible and intangible for long-range scouting missions (but with a good chance to cast spells
into the physical world, and a small chance of losing their body and so being "immediately sent to jibber and shriek on the floor of the lowest hell"). That's great!
- Chaotic Storm Giants. Storm giants, also introduced in the Greyhawk supplement, can be any alignment -- which is more in sync with Norse depictions of evil storm giants.
- Limited Dragon Types. OD&D has only 6 dragon types: the five colored evil-type dragons, plus the good and intelligent Gold type. It seems a lot easier to keep track of that mentally.
Random stuff I don't like in OD&D:
- Gold Standard Pricing. I've always been bothered by D&D pricing in gold pieces, which prohibits using real-world medieval prices for an economic model (which would normally be in some type of
silver coin, gold coins not really being exigent in the medieval world). I think I understand why this was done -- it's just fun to think about pricing things in gold. I can also now see that the
line in the AD&D PHB about prices assuming an inflated, adventure-rich economy was just an after-the-fact rationalization.
I wish they'd thought in advance to model prices on real-world medieval economies, but I can also see how this was too much to ask for when a fun, lightweight game was being developed. (Probably 15,
30, 50 for armor costs is the hugest simplification ever.) The after-effect is that every fantasy paper or video game for the rest of history is stuck with a very unrealistic gold-standard economy
(and basically useless copper and silver coins).
- Magic Item Construction. Yes, the times for sample magic item construction are probably way too long. 1 week for a potion of healing? 1 month for 20 magic arrows? 1 week per spell level on a
scroll? That's probably way too long to make flavor-sense (consider doing some comparisons in legend for how long it takes to make a love potion, or write a spell, or craft a magic hammer...)
- All-Access Cleric Spells. In the white box rules everyone had spellbooks and full access to the list. In Supplement I: Greyhawk, magic-users had spellbooks with partial list access, but clerics got
rid of the need for spellbooks while maintaining full access to expanded spell lists -- "All cleric spells are considered as 'divinely' given and as such a cleric with a wisdom factor of 3 would know
all of the spells as well as would acleric with an 18 wisdom factor" (p. 8).
Unfortunately, this system created a power-creep and complexity problem in that every time a new cleric spell was added to the game, then every cleric automatically gained access to it (both
increasing their power, and increasing the number of options cleric players had to parse each day in preparing spells). The current 3.5 Edition situation is basically out of hand for new players --
there are 37 or more spells available for consideration by every starting 1st-level cleric!
One of the very best new ideas I've seen in D&D is the 3rd Ed. Unearthed Arcana "Variant: Spontaneous Divine Casters", in which clerics must pick a small subset of spells from each level (I'd call
them "miracles"), and then are allowed to cast them freely, without prepared selection in advance each day. This both reduces the complexity to new players, the everything-under-the-sun power of
cleric spellcasting, and has a very nice flavor effect of priests having specific well-known powers you can depend on. But, I can certainly understand why this forever-expanding-spell-list probably
could not have been predicted at the outset of the OD&D game.
Another brief observation about the OD&D set -- spell effects truly had the "safety off". Even your low-level spell effects could be truly devastating: sleep (take out 2-16 1st level enemies, or 2-12
2nd level, etc.), hold person (affect 1-4 persons, duration over 1 hour), charm person (permanent until dispelled!), etc. Animate dead is a 5th-level magic-user spell that creates 1-6 undead per
level above 8th (e.g., a 16th-level wizard gets an average of 36 undead per casting). A potion of diminution makes you 6 inches tall, a potion of growth makes you 30 feet! (Which again is much more
in the tradition of myth or Carroll, etc.) In contrast, modern D&D has evolved to grant much more subdued effects with spells, but gives spellcasters a lot more spell slots to fire off over time.
An interesting place which has been mentioned by others is the classic blasting spells of fireball and lightning bolt. They're in OD&D (and Chainmail), of course, and they have always, always, done
1d6 damage per level of the caster. But hit points have routinely crept up over the editions, radically weaking the power of these spells over time. Consider a 5th-level caster who does 5d6 damage
(17.5 points average.)
(1) In OD&D, a stock ogre had 4d6+1 HD, average 15 hit points. Your sample fireball would definitely kill this guy, unless he made his save for half damage (and then be over half-dead).
(2) In 1st-2nd Edition, the ogre had 4d8+1 HD, or 19 hit points. This is very close to the spell's damage, and so is about 50/50 to kill him before the save, depending on exact damage or hit points
(3) In 3rd Edition, the ogre has 4d8+8 HD, average 26 hit points. The sample fireball will definitely not kill him. If he makes his save he'll only lose about one-third of his hit points.
So in short, the classic fireball or lightning bolt has gone from almost certainly killing ogres, to sometimes killing ogres, to almost certainly not killing ogres in one blast. This, even though the
actual damage roll is completely unchanged from the inception of the game. Wacky!
Here are some more thoughts from reading the Original D&D (white box) set for the first time, and the origins of the more troubling class types.
Clerics -- My trouble with clerics is more subtle than the trouble with thieves (see previous post). It's not a mechanical problem so much as a flavor-setting problem.
Every time I try to design up a D&D campaign setting I run into the following issue. I want to use the core classes as written, and I want to create a medieval-flavored setting, as indicated by D&D's
level of technology, armaments, coinage, and political assumptions (smallish kingdoms, rising mercantile class, a history of an older broken-down empire, etc.). But then, I'm confronted with the
polytheistic religious structure in D&D, and I come to a stumbling block -- for the life of me I can't imagine what the political situation would look like, to have a medieval Europe lacking the
unified Christian Catholic church, and instead overlaid with independent polytheistic temples.
To me, this is a huge contradictory disconnect in standard D&D, in that you've got a medieval world with polytheistic religion. I can't even find any examples to compare to in the real world -- by
the early middle ages, all of Europe (including Scandinavian countries at the last) were Christian, all of the Middle East was monotheistic under Islam, etc. Only in the Far East like Japan did you
have Shogun culture with polytheistic religion, but the priests of shrines there were (to my understanding) not politically powerful in any way, like we assume a D&D church to be.
And as I think about a polytheistic setting, I'm further blocked by the fact that the D&D Cleric class looks almost uniquely like a Christian crusading priest (or Templar, or what-have-you).
Apparently they belong to an influential church, but what church like that was polytheistic? What kind of hierarchical structure could be supported by that? If you think of either shaman-culture
(independent wise men) or a polytheistic professional religious class (like Celtic druids or Indian brahmans, who service a unified pantheon of gods together), you think about them in robes, not
wandering around in full plate mail. Again, I'd like to reduce Clerics to not use full armor, to look more like polytheistic priests, if that's what they truly are.
In short, the problem is this: D&D claims to have a polytheistic religion, but you've got both the politics and the critical Cleric class set up as in the medieval Christian world, and nowhere else.
Now, if you look at the OD&D set, the reason for this is pretty clear -- Clerics really were assumed to be Christian at the outset of the design. (As usual, it's not explicitly stated *what* the
class is, but the standard usage of the terms involved makes it clear). (1) The class-level titles all come out of the Catholic Christian church. (2) The equipment list and turning undead sections
mention the Cross and no other type of holy symbol. (3) The cleric spell list is almost uniformly based on famous Biblical miracles. And so forth.
It's only afterwards (I presume Supplement IV: Gods, Demigods, and Heroes, although Supplement II: Blackmoor is the first one to mention non-cross holy items, p. 23) that the designers thought to use
polytheistic deities as their mainstays, and glued this on after-the-fact to the existing D&D worldview and Clerical class. It's no wonder that still to this day the polytheism acts as a sort of
strange extra appendage to the rest of the D&D ruleset (even contradictory, when I think about it fairly hard), as it truly wasn't there in OD&D. Perhaps the class would have looked different in its
spell list and armor usage if polytheist priests had been in mind at the beginning (and perhaps the world setting would be presumed classical instead of medieval, who knows?)
One initial solution I can think of is to directly stipulate a monotheistic, powerful Catholic-like church for my medieval-style D&D world -- the problem there would be some dryness to the options of
clerics and the political situation. A second solution would be to use a professional-class-style clerical establishment (like historical druids), where the priests all serve the same pantheon of
gods as a single unit and teaching (some of the same drawbacks would apply). A third solution would be to find some historical pantheon of gods which best supports a combative, warlike Clerical class
as found in D&D (perhaps Norse, Finnish or some other warrior culture which was Christianized as late as possible historically).
Here's a continuation of my first post, in which I recently acquired a copy of the Original D&D Rules (1974 white box set). In particular I'll look at two troublesome classes.
Thieves -- Thieves (rogues in 3E) don't exist in the original rules; they were added as the 4th primary class in the first Greyhawk supplement. I often have a problem with thieves, which I'll
describe in a minute.
When I think about 3E D&D, I'd love to simplify the game in a few broad strokes. The first thing I think about is just slicing off the whole skill system. (Perhaps using a variant simplification from
3E Unearthed Arcana.) When I'm acting as DM making NPCs, the thing that frustrates me and burns the most time is fiddling with individual skill points, max ranks, multiclass per-level class versus
cross-class costs, armor penalties, feat bonuses, synergy bonuses, OMG yuck! When I was working on converting the D1-3 series I was finding it took me at least 30 minutes per individual NPC to do all
the work, with the biggest chunk going into skill-point fiddling. My understanding is this is always the biggest source of stat-block errors even in WOTC publications, by those who look for such
things. The feat system is pretty nice -- a new feat usually seems like a nice significant gift package -- but having the skill system running in parallel drives me nuts; I'd like to get rid of the
whole subsystem.
Except that I can't really do that in 3E because of the 4th primary class, the Rogue, whose whole functioning is predicated on making use of lots of skill points for their abilities.
So, looking at the Original D&D publications, I can see that this has always been an oddball situation. With the original books, you had Fighter, Cleric, Magic-user; all three had hit dice, could
strike in combat, made saving throws, and the latter could cast spells in binary fashion (you either shot it off or you didn't). With the addition of the Thief in Supplement I: Greyhawk, the
designers tinkered up a completely brand new mechanics invention -- a list of skills that sometimes worked and sometimes didn't, with a roll-under-percentage success mechanic, failure vastly more
likely than not at 1st level, etc. I guess that's an inventive piece of gaming R&D, but the oddball mechanic and skill-usage stuck around as an odd appendage through 1E right to 3E and this very day.
You've got this one class whose lifeblood is pumped by the skill system, which for other classes is usually extraneous to their core functioning.
What could have been the alternative in those early days? Perhaps if the Thief skill system were treated more like a Fighter's combat potential, where you rolled a stock d20 to get a particular score
(and the scores were the same for that whole list of skill abilities, so you didn't have to track 5-6 different percentages or skill scores). I'm now thinking about that as a modification to an OD&D/
AD&D campaign if I ever run one again. Or, just slice away the Thief class itself and depend on the Clerical find traps spell that was in the game since OD&D (but which nowadays gets de-powered so as
to not take spotlight time away from Rogue skills!) -- but then that takes away a lot of the class choice (namely, the only unlimited-level class) available to dwarves, elves, and hobbits in the OD&D
Oh yes, more random OD&D stuff in this post -- initially the game had dwarves, elves, and hobbits, but of course the latter were renamed "halflings" to avoid trademark issues with the Tolkein estate.
My OD&D set (6th printing) is funny in that they mostly completed that switchover (with easily spottable different-font text pasted in over offending areas), but missed it in a certain number of
Also in the OD&D set, every monster or dungeon-based enemy had infravision, but every PC character was specifically lacking infravision and had to use torches or lanterns (including PC dwarves,
elves, and hobbits)! Wrap your head around that one, if you're in the habit of criticizing minor discrepancies in current rulesets. With the Greyhawk supplement that was modified to give all
demihumans equal infravision, etc.
Also: Every magic sword in OD&D was automatically intelligent. With just about the same list of abilities and statistics that was used in 1E, 2E, 3E, and still today. Wow!
Hi there, I'm Delta and I'm a complete old-school D&D junkie. I've started this blog to jot down random thoughts as I study old D&D texts and think about the game.
Frankly, I haven't played a game of D&D in about 1.5 years, since moving from Boston to New York City, at which point I lost my playgroup. Prior to that we had met every week for 5 years running
(since about a year before the release of 3rd Edition D&D).
Here's the topic of my first post -- I recently procured a copy of the Original D&D set of EBay. That's the original white-box, three-small-booklets edition, copyright 1974. Of course, I've had
copies (more than one) of 1st Edition rulebooks, Holmes blue-book sets, Basic D&D, 3rd Edition D&D, etc., but I never had the original white-box stuff, and I'm tickled pink to have it at this point.
(Got it *relatively* cheap off EBay -- $45, 6th printing with slightly dented box, when lots of these sets go for over $100 these days). The other thing I do now is occasionally get old PDFs of stuff
from RPGNow.com -- for example, I recently picked up the original Chainmail rules, and also Supplement I: Greyhawk, but the original rules haven't been released in digital form, so I went out and
gave it to myself as a gift.
It's really intriguing to see the original D&D rules and consider exactly how they have evolved over time. On the one hand, they're fairly different in things like character classes, races, ability
modifiers, how to run combat, and so forth. But on the other hand, lots of the text and ideas for certain spells and magic items has been nearly copy-and-pasted (at least in part) between every
edition, from OD&D to 1E to 2E to 3rd Edition.
Some quick examples of the quirkiest things in the OD&D set: There are only 3 classes (fighter, magic-user, and cleric). Most ability scores don't have any modifier on combat actions, just experience
award modifiers (just like Holmes Blue Book, in fact). All class and monster hit dice are d6's! (Which actually makes a heck of a lot of sense, since it's the most common die type on any table.
Depending on class you might go 1d6, 1d6+1, 2d6, etc., for your hit dice.) Every hit from any weapon does 1d6 damage -- with certain exceptions like a giant or staff of striking that does 2d6 damage.
Every magic-user or cleric apparently has a spellbook with all spells in the game included. Elves can function as fighters or magic-users, but must pick only one for a given adventure! There is no
specification for what falling damage does (one example says a 30' fall should likely be fatal). A lot of stuff like what dice mean, or what happens when you run out of "hits", is entirely undefined
in the rules, assuming they're just obvious common knowledge to gamers. Really fascinating material to me.
With the release of Supplement I: Greyhawk, a lot of changes were made that filtered seemingly verbatim into the AD&D books. For example, the thief class was added as the 4th primary class type.
Classes were given stock die types (d8, d6, d4), and monsters converted to d8 hit dice. Varying damage types by weapon were given (including medium-vs-large targets), weapon type-vs-armor modifiers,
and specific monster attacks and damage (which is a real pain because you then needed to flip between 2 books for a monster's full statistics). Ability modifiers were given for different abilities
like Strength, including the exceptional d% component we all loved (explicitly to make fighters more potent and survivable). You've got the more familiar and wider multiclass mechanic where
experience is constantly being split between two classes.
Another thing that interests me is that the rules were explicitly set in a medieval technology and time frame. (This pops up in discussions of equipment, ships, and the campaign.) It specifically
cautions that you shouldn't think about other milieus like ancient or classical until your medieval possibilities have been exhausted (Vol. 1, p. 5). It's an interesting specification because modern
rules try to genericize everything, and make it seem like any fantastic setting is equally supported by the rules.
Oh yeah, why were the ability scores in the order they were? (In 3rd Ed. they go Str, Dex, Con, Int, Wis, Cha, which does seem to make sense... physical stuff first, mental stuff second.). Why Str,
Int, Wis, Dex, Con, Cha? (As in 1st. Ed.?) Well, it's easy to see from OD&D... they're just the prime requisites of the classes in the order they were invented: fighter, magic-user, cleric, and then
finally thief in the Greyhawk supplement, etc.
By 1st Ed. AD&D, the spell lists were organized so that there was a plethora of 1st level spells, the same or fewer 2nd, same or fewer 3rd, etc. But that hadn't yet happened in OD&D: the numbers go
up and down randomly, with the fewest spells of all at 1st level... You've got just 8 1st-level spells for magic-users, 10 2nd-level, then 14, 12, 14, and 12 again. Clerics have just 4 spells on
their 2nd & 3rd level lists, 6 spells on the others.
In addition, there aren't any specific planes-of-existence yet... for example, elementals spring directly from the terrestrial substance itself (which to me is actually a lot more attractive
in-spirit-flavor than the elemental planes concept). But, there is the prospect held out of other dimensions, times, trips to the moon or Mars, robots and androids, living statues (as-yet unnamed
golems). And you do already have the contact higher plane spell with its big list of plane-levels (starting at 3rd? maybe 1 above "heaven"? on up to 12). It's kind of a mishmash of every fantastical
place or concept that set the stage for a pretty complicated multiversal construction later on. (As opposed to say, most classical mythologies with their tripartite worldview of
More to come in a minute! | {"url":"https://deltasdnd.blogspot.com/2007/03/","timestamp":"2024-11-10T15:43:35Z","content_type":"application/xhtml+xml","content_length":"155523","record_id":"<urn:uuid:78aa9f3d-f156-409a-ac4e-37294f204d42>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00295.warc.gz"} |
Grant has 3.6 pounds of pretzels. He puts the, №17886342, 12.10.2020 10:15
: asked on
Grant has 3.6 pounds of pretzels. He puts the pretzels into bags that each hold 0.3 pound. How many bags does Grant use to hold the pretzels?
Step-by-step answer
24.06.2023, solved by verified expert
Bhaskar Singh Bora
Mathematics, Physics. Q&A Expert
24 029 answers
Unlock the full answer
4 students found this answer helpful
12 bags
Step-by-step explanation:
To find how many bags he will use, divide 3.6 by 0.3
= 12
So, Grant will use 12 bags to hold the pretzels
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Powers of Ten
Picturing exponential growth, powers of 10, can be hard for any of us to imagine. The spreadsheet has the flexibility to enable us to explore the powers of 10 and to get a visual image of them.
We can see the difference in shape between odd and even powers, and get a sense of the speed of exponential growth. We use powers of ten to gain a sense not only of big numbers but of how big
spreadsheets can be. | {"url":"https://whatifmath.org/powers-of-ten/","timestamp":"2024-11-11T16:08:50Z","content_type":"text/html","content_length":"51159","record_id":"<urn:uuid:4e8d48b1-afa6-42cf-8e48-58f41f3cd162>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00649.warc.gz"} |
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lim_(x->0)sin(1/x)/(sin(1/x)) ? | Socratic
#lim_(x->0)sin(1/x)/(sin(1/x))# ?
Find the limit ${\lim}_{x \to 0} \sin \frac{\frac{1}{x}}{\sin \left(\frac{1}{x}\right)}$
How would you approach this? Is it $1$ or it doesn't exist?
2 Answers
${\lim}_{x \rightarrow 0} \setminus \sin \frac{\frac{1}{x}}{\sin \left(\frac{1}{x}\right)} = 1$
we seek:
$L = {\lim}_{x \rightarrow 0} \setminus \sin \frac{\frac{1}{x}}{\sin \left(\frac{1}{x}\right)}$
When we evaluate a limit we look at the behaviour of the function "near" the point, not necessarily the behaviour of the function "at" the point in question, thus as $x \rightarrow 0$, at no point do
we need to consider what happens at $x = 0$, Thus we get the trivial result:
$L = {\lim}_{x \rightarrow 0} \setminus \sin \frac{\frac{1}{x}}{\sin \left(\frac{1}{x}\right)}$
$\setminus \setminus = {\lim}_{x \rightarrow 0} \setminus 1$
$\setminus \setminus = 1$
For clarity a graph of the function to visualise the behaviour around $x = 0$
graph{sin(1/x)/sin(1/x) [-10, 10, -5, 5]}
It should be made clear that the function $y = \sin \frac{\frac{1}{x}}{\sin} \left(\frac{1}{x}\right)$ is undefined at $x = 0$
The definitions of limit of a function I use are equivalent to:
${\lim}_{x \rightarrow a} f \left(x\right) = L$ if and only of For every positive $\epsilon$, there is a positive $\delta$ such that for every $x$, if $0 < \left\mid x - a \right\mid < \delta$ then $
\left\mid f \left(x\right) - L \right\mid < \epsilon$
Because of the meaning of "$\left\mid f \left(x\right) - L \right\mid < \epsilon$", this requires that for all $x$ with $0 < \left\mid x - a \right\mid < \delta$, $f \left(x\right)$ is defined.
That is, for the required $\delta$, all of $\left(a - \delta , a + \delta\right)$ except possibly $a$, lies in the domain of $f$.
All of this the gets us:
${\lim}_{x \rightarrow a} f \left(x\right)$ exists only if $f$ is defined in some open interval containing $a$, except perhaps at $a$.
($f$ must be defined in some deleted open neighborhood of $a$)
Therefore, ${\lim}_{x \rightarrow 0} \sin \frac{\frac{1}{x}}{\sin} \left(\frac{1}{x}\right)$ does not exist.
A nearly trivial example
$f \left(x\right) = 1$ for $x$ an irrational real (undefined for rationals)
${\lim}_{x \rightarrow 0} f \left(x\right)$ does not exist.
Impact of this question
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Sparse Modeling of Harmonic Signals
This thesis considers sparse modeling and estimation of multi-pitch signals, i.e., signals whose frequency content can be described by superpositions of harmonic, or close-to-harmonic, structures,
characterized by a set of fundamental frequencies. As the number of fundamental frequencies in a given signal is in general unknown, this thesis casts the estimation as a sparse reconstruction
problem, i.e., estimates of the fundamental frequencies are produced by finding a sparse representation of the signal in a dictionary containing an over-complete set of pitch atoms. This sparse
representation is found by using convex modeling techniques, leading to highly tractable convex optimization problems from whose solutions the estimates of the fundamental frequencies can be deduced.
In the first paper of this thesis, a method for multi-pitch estimation for stationary signal frames is proposed. Building on the heuristic of spectrally smooth pitches, the proposed method produces
estimates of the fundamental frequencies by minimizing a sequence of penalized least squares criteria, where the penalties adapt to the signal at hand. An efficient algorithm building on the
alternating direction method of multipliers is proposed for solving these least squares problems.
The second paper considers a time-recursive formulation of the multi-pitch estimation problem, allowing for the exploiting of longer-term correlations of the signal, as well as fundamental frequency
estimates with a sample-level time resolution. Also presented is a signal-adaptive dictionary learning scheme, allowing for smooth tracking of frequency modulated signals.
In the third paper of this thesis, robustness to deviations from the harmonic model in the form of inharmonicity is considered. The paper proposes a method for estimating the fundamental frequencies
by, in the frequency domain, mapping each found spectral line to a set of candidate fundamental frequencies. The optimal mapping is found as the solution to a minimimal transport problem, wherein
mappings leading to sparse pitch representations are promoted. The presented formulation is shown to yield robustness to varying degrees of inharmonicity without requiring explicit knowledge of the
structure or scope of the inharmonicity.
In all three papers, the performance of the proposed methods are evaluated using simulated signals as well as real audio.
Ämnesklassifikation (UKÄ)
• Signalbehandling
• Sannolikhetsteori och statistik
Utforska forskningsämnen för ”Sparse Modeling of Harmonic Signals”. Tillsammans bildar de ett unikt fingeravtryck. | {"url":"https://portal.research.lu.se/sv/publications/sparse-modeling-of-harmonic-signals","timestamp":"2024-11-03T16:42:14Z","content_type":"text/html","content_length":"55348","record_id":"<urn:uuid:3a43dd31-39e8-4152-acd3-52a93d8746b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00037.warc.gz"} |
Which of the following is an example of a separable differential equation?
Brian Rogers 2022-10-16 • comments off
Which of the following is an example of a separable differential equation?
Separable Differential Equations Examples Since the given differential equation can be written as dy/dx = f(x) g(y), where f(x) = x + 3 and g(y) = y -7, therefore it is a separable differential
equation. Answer: y’ = xy – 21 + 3y – 7x is a separable differential equation.
What it means for a differential equation to be separable?
Definition: Separable Differential Equations. A separable differential equation is any equation that can be written in the form. y′=f(x)g(y). The term ‘separable’ refers to the fact that the
right-hand side of Equation 8.3.1 can be separated into a function of x times a function of y.
What is the first step in solving a separable differential equation?
Steps To Solve a Separable Differential Equation
1. Get all the y’s on the left hand side of the equation and all of the x’s on the right hand side.
2. Integrate both sides.
3. Plug in the given values to find the constant of integration (C)
4. Solve for y.
Which differential equation is not separable?
y = y sin(x − y) It is not separable. The solutions of y sin(x−y) = 0 are y = 0 and x−y = nπ for any integer n. The solution y = x−nπ is non-constant, therefore the equation cannot be separable.
Is dy dx xy separable?
So something like dy/dx = x + y is not separable, but dy/dx = y + xy is separable, because we can factor the y out of the terms on the right-hand side, then divide both sides by y.
How do you solve a separable equation?
The method for solving separable equations can therefore be summarized as follows: Separate the variables and integrate.
1. Example 1: Solve the equation 2 y dy = ( x 2 + 1) dx.
2. Example 2: Solve the equation.
3. Example 3: Solve the IVP.
4. Example 4: Find all solutions of the differential equation ( x 2 – 1) y 3 dx + x 2 dy = 0.
How do you prove a differential equation is separable?
Note that in order for a differential equation to be separable all the y ‘s in the differential equation must be multiplied by the derivative and all the x ‘s in the differential equation must be on
the other side of the equal sign.
How do you separate XY?
Three Steps:
1. Step 1 Move all the y terms (including dy) to one side of the equation and all the x terms (including dx) to the other side.
2. Step 2 Integrate one side with respect to y and the other side with respect to x. Don’t forget “+ C” (the constant of integration).
3. Step 3 Simplify.
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Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group | H2020 | CORDIS | European Commission
Periodic Reporting for period 2 - HoloHair (Information Encoding in Quantum Gravity and the Black Hole Information Paradox)
Período documentado: 2022-04-01 hasta 2023-09-30
A driving force in our quest to understand quantum gravity have been theorems and paradoxes that highlight a clash between general relativity and quantum mechanics. Penrose’s singularity theorem from
1965 establishes that if all matter has a positive energy density, a trapped surface implies that the spacetime contains a singularity. While it is clear that a theory of quantum gravity must resolve
any such singularity, the scale of the resolution is not known. Moreover, a horizon cloaking a singularity leads to deep puzzles when quantum effects are included - a conflict that is made sharp in
Hawking’s information paradox from 1975: he found that a black hole that is initially in a pure state evolves into a state of thermal radiation upon evaporation thus suggesting that black holes
violate the laws of quantum mechanics. In recent years this puzzling conclusion has been revised and the violation of quantum mechanics fended off. But the core questions remain: what are the
microstates of a black hole and how does the information get out when it evaporates? More generally, how and where is information encoded in quantum gravity? Our two most successful tools in
addressing the quantum nature of gravity and black holes are string theory and holography.
Holography is the idea that a quantum theory of gravity in some spacetime has an equivalent, or “dual”, description in terms of a lower-dimensional (non-gravitational) quantum theory defined at the
boundary of the spacetime. In this sense information about bulk physical processes is encoded “holographically” in the boundary theory. The holographic principle has been spectacularly successful in
spacetimes with a negative cosmological constant. Anti-de Sitter spacetime (AdS) is a space of negative curvature effectively describing systems in isolation. It has a time-like boundary with one
less spatial dimension where standard rules of quantum field theory apply but with a larger symmetry group than the usual Poincaré group: besides translations and Lorentz transformations there are
scale transformations and “angle-preserving” special conformal transformations. The boundary turns out to be described by a conformal field theory (CFT) whose symmetries match precisely the
isometries of AdS! This AdS/CFT correspondence has revolutionized string theory and high energy quantum field theory over the last 25 years.
How generally does the idea of holography apply? What about spacetimes with positive or vanishing cosmological constant such as de Sitter and Minkowski spacetimes? A significant amount of my recent
research, and some of my future plans, are devoted to developing a holographic principle for asymptotically flat (Minkowski) spacetimes which are a good approximation for many processes in our
universe. While a daunting task, known laws of (low-energy) physic offer us some clues of how and where to begin.
The S-matrix, describing the transition probability between an initial and a final state in a scattering process, is the basic observable in quantum gravity in asymptotically flat spacetimes. Usually
we compute it in basis of asymptotic energy eigenstates described by a plane wave basis which makes translation symmetry manifest. What does the S-matrix look like when instead making Lorentz
symmetry (rotations and boosts) manifest? Recasting the S-matrix in a basis of boost eigenstates reveals that it shares similarities with correlation functions in a conformal field theory (CFT). In
particular, universal features that arise in energetically soft and collinear limits of the scattering particles take a natural form in CFT language, and spacetime (asymptotic) symmetries can be
understood as generated by special CFT operators. This together with novel insights into the infrared structure of gravity and gauge theories from soft theorems, asymptotic symmetries and memory
effects have culminated in a promising proposal for a holographic principle in asymptotically flat spacetimes: Celestial holography conjectures that quantum gravity in spacetimes with flat
asymptotics is dual to a conformal field theory on the celestial sphere at null infinity. This proposal differs significantly from our most studied example of holography relating string theory on a
d+1 dimensional AdS space to a CFT on its d dimensional time-like boundary. In asymptotically flat spacetimes, the boundary is light-like, or null, and the proposed celestial CFT is co-dimension two.
Moreover, while AdS/CFT naturally arises from a top-down construction in string theory the approach towards establishing celestial holography is currently bottom-up (though it would be extremely
interesting to find a string theory embedding).
The first key step in this endeavour is the identification of the symmetries that govern both sides of the proposed celestial holographic duality. Universal properties of scattering amplitudes at low
energies turn out to have a symmetry origin: Weinberg’s soft graviton theorem is related to BMS supertranslation symmetries which are infinitely enhanced translations in space and time that act at
the null boundary of asymptotically flat spacetimes. When recast in a basis of boost (rather than energy) eigenstates soft theorems take the form of correlation function of conserved operators on the
co-dimension two celestial sphere. The classification of all such conserved operators as well as the construction of their associated conserved charges has been achieved by my ERC group using the
powerful toolkit of CFT. Building on these results we can ask: what are (the implications of) all the symmetries and how do they constrain bulk scattering, what axioms or principles should we impose
on the dual celestial CFT and can we bootstrap it, how in the dual theory do we describe non-trivial bulk geometries such as black holes and their dynamics...? Insight into such questions could lead
to a non-perturbative formulation of quantum gravity in asymptotically flat spacetimes.
Indeed, a litmus test of any proposed holographic duality is to account for non-perturbative processes such as the formation and evaporation of black holes and achieving a statistical interpretation
of the Bekenstein-Hawking entropy of black holes - an incredibly challenging task. To resolve the microphysics we also need a better understanding of the information encoding, storage and flow
between the horizon and the far asymptotic region where Hawking radiation escapes to. This includes determining the role and implications of “soft hair” at the horizon of black holes which is a
consequence of conservation laws associated to BMS symmetries and a potential relation to fuzzballs which have been proposed as the fundamental microscopic description of black holes. Key insights
into the statistical nature of black hole entropy are obtained via the correspondence principle for black holes and strings. As one adiabatically decreases the string coupling, a black hole makes a
transition to a state of weakly coupled strings with the same mass. At the correspondence point the Bekenstein-Hawking entropy of the black hole is comparable to the string entropy: this gives a
statistical interpretation of black hole entropy in terms of the degeneracy of string states. This correspondence has been developed in the 1990s for static black holes and was recently refined. An
imminent goal of my ERC group is to establish this correspondence for general rotating black holes.
The study of information encoding in quantum gravity and in black holes is at the heart of this ERC project, carried out by the research group of Dr. Andrea Puhm at the Center for Theoretical Physics
of Ecole Polytechnique, Palaiseau, France. Our focus is on developing a holographic principle for asymptotically flat spacetimes, examining the symmetries and memory effects in celestial CFTs and
their constraints on bulk physics, describing bulk geometries holographically, determining the implications of soft hair on black hole horizons, and formulating a microscopic description of black
holes via string theory.
* Soft symmetries and memory in celestial CFT
Soft theorems of quantum field theory reveal universal properties of gauge theories and gravity in the infrared. Universal phenomena often have a symmetry origin. We set out to systematically
determine the symmetries corresponding to the low-energy physics of gauge theory and gravity using the framework of “celestial holography” which suggests a holographic duality between quantum gravity
in asymptotically flat spacetimes and a non-gravitational theory defined on the celestial sphere at its boundary. The universality from soft theorems can then be understood from the symmetries of a
“celestial conformal field theory” (CCFT) on the codimension-two sphere at the null boundary of the spacetime. Our key result is the classification of all CCFT operators associated to soft theorems
and the identification of the symmetries they generate. We extended this to supersymmetric theories and to general spacetime dimensions. Our work reveals a tension between the infinite enhancement in
two-dimensional CCFTs related to BMS symmetries, named after Bondi, van der Burg, Metzner and Sachs, and the finite symmetry group in higher-dimensional CCFTs.
* Celestial amplitude relations
Scattering amplitudes in gauge theory and gravity obey hidden relations known as double copy. We showed that celestial amplitudes obey “celestial double copy” relations that generalize the famous BCJ
relations, named after Bern, Carrasco and Johansson, of momentum-space amplitudes to operator-valued statements. This gives further support to the expectation that these hidden relations are
fundamental properties of scattering amplitudes and paves the way towards a general curved space double copy. The structure of celestial amplitudes beyond tree level is largely unknown. We remedied
this for the case of N=4 super Yang-Mills theory where we computed the all-loop celestial four gluon amplitude. We, moreover, found that the famous momentum-space factorization into the tree-level
amplitude times an infrared factor persists also for celestial amplitudes albeit in the form of an exponential differential operator acting on the tree-level amplitude.
* Celestial holography and black holes
One of the key ingredients of the AdS/CFT correspondence is the relation between the gravitational partition function in the bulk and the generating functional of correlation functions for the theory
on the boundary. Motivated to look for the same principle for the S-matrix in asymptotically flat spacetimes we showed that the boundary on-shell action for general backgrounds becomes the generating
functional for tree-level correlation functions in celestial CFT. An important goal in celestial holography which we have initiated is to describe non-trivial asymptotically flat backgrounds in CCFT
and their effects on celestial scattering amplitudes. We proposed a generalization of conformal primaries and showed that this includes a large class of physically interesting metrics such as
ultra-boosted black holes and shockwaves. These exact conformal primary solutions lend themselves to suitable backgrounds for celestial amplitudes. Celestial two-point correlators on backgrounds are
already non-trivial (they encode information about the background and couplings) and have the desirable feature that the they exist at generic operator positions (unlike their flat space
counterparts). We then investigated celestial wave scattering on a variety of backgrounds in gauge theory and gravity including Coulomb backgrounds, Schwarzschild black holes, their ultraboosted
limits given by electromagnetic and gravitational shockwaves, as well as their spinning counterparts. We also derived (conformal) Faddeev-Kulish dressings for particle-like backgrounds which remove
all infrared divergent terms in the two-point functions to all orders in perturbation theory.
* Black holes microphysics and soft hair
Acting with asymptotic symmetry transformation on the horizon of black holes implants soft hair/memory. Besides the BMS supertranslations we investigated the role of a new set of “dual”
supertranslations, and its effect on the symmetry algebra on black hole horizons. We have furthermore investigated effects that generate or pinch off horizons : the quantum backreaction of scalar
fields in three-dimensional de Sitter space, where classical black holes are known not to exist, generates a black hole horizon, while the Gregory-Laflamme instability of black strings, which causes
the horizon to pinch off in finite time and form naked singularities, is shown to persist in Anti-de Sitter space where "black tsunamis” arise and the formation of the bulk naked singularity can be
studied in the boundary CFT. The framework of AdS/CFT braneworld holography gives a new perspective on dynamical evaporation of a black hole as the classical evolution in time of a black hole in an
Anti-de Sitter braneworld where the evolution of the Page curve of the radiation shows entanglement islands appearing and then shrinking thus supporting a unitarity evolution.
* Celestial CFT: symmetries, memory, bootstrap and backgrounds
Beyond the classification of soft operators in celestial CFTs in general dimensions achieved by my ERC group we plan to address a variety of questions to establish the properties of celestial CFTs
and test the proposed holographic duality with quantum gravity in asymptotically flat spacetimes. Currently, we lack a good understanding of the implications of all Goldstone and memory operators in
the infinite tower of conserved soft operators that we identified, as well as the corrections of the soft symmetry algebra from quantum loops. Moreover, we are in urgent need of a list of axioms or
principles that we should impose on celestial CFTs. This would allow us to apply the “bootstrap” philosophy that the observables of a CFT can be fixed by requiring some general axioms. An important
goal for any holographic proposal is to establish a correspondence between bulk geometries and boundary states and their dynamics. We already initiated such a program by identifying the boundary
generators for special bulk geometries and examining celestial correlation functions on a variety of gauge and gravity backgrounds. We plan to continue our investigations of celestial holography on
backgrounds including black holes.
* Correspondence principle between black holes and strings
Recent works on the correspondence principle between static black holes and fundamental strings revealed new insights into the microscopic nature of black holes. Motivated by these results we set out
to extend the correspondence principle to rotating black holes in general spacetime dimensions. Surprisingly, this has not been achieved so far even though the correspondence principle for static
black holes including charges had been put forward in the 1990s! The correspondence states that as one adiabatically decreases the string or gravitational coupling (Newton’s constant), a black hole
makes a transition to a state of weakly coupled strings with the same mass, and offers a statistical interpretation of black hole entropy in terms of the degeneracy of string states. Rotating black
holes in higher dimensions have a much more complicated structure than their non-rotating counterparts - they can have different shapes, distinct horizon topologies, and there are dynamical
instabilities - the correspondence principle for rotating black holes will thus be much richer. | {"url":"https://cordis.europa.eu/project/id/852386/reporting/es","timestamp":"2024-11-06T14:55:51Z","content_type":"text/html","content_length":"83760","record_id":"<urn:uuid:56902cc4-a1fb-4d50-9bb2-251f12e54fef>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00517.warc.gz"} |
A/B testing my resume | R-bloggersA/B testing my resume
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Internet wisdom is divided on whether one-page resumes are more effective at landing you an interview than two-page ones. Most of the advice out there seems much opinion- or anecdotal-based, with
very little scientific basis.
Well, let’s fix that.
Being currently open to work, I thought this would be the right time to test this scientifically. I have two versions of my resume:
The purpose of a resume is to land you an interview, so we’ll track for each resume how many applications yield a call for an interview. Non-responses after one week are treated as failures. We’ll
model the effectiveness of a resume as a binomial distribution: all other things being considered equal, we’ll assume all applications using the same resume type have the same probability
($p1$ or $p2$) of landing an interview. We’d like to estimate these probabilities, and decide if one resume is more effective than the other.
In a traditional randomized trial, we would randomly assign each job offer to a resume and record the success rate. But let’s estimate the statistical power of such a test. From past experience, and
also from many plots such as this one posted on Reddit, it seems reasonable to assign a baseline success rate of about 0.1 (i.e., about one application in 10 yields an interview). Suppose the
one-page version is twice as effective and we apply to 100 positions with each. Then the statistical power, i.e. the probability of detecting a statistically significant effect, is given by:
power.exact.test(p1 = 0.2, p2 = 0.1, n1 = 100, n2 = 100)
## Z-pooled Exact Test
## n1, n2 = 100, 100
## p1, p2 = 0.2, 0.1
## alpha = 0.05
## power = 0.501577
## alternative = two.sided
## delta = 0
That is, we have only about 50% chances of detecting the effect with 0.05 confidence. This is not going to work; at a rate of about 10 applications per month, this would require 20 months.
Instead I’m going to frame this as a multi-armed bandit problem: I have two resumes and I don’t know which one is the most effective, so I’d like to test them both but give preference to the one that
seems to have the highest rate of success—also known as trading off exploration vs exploitation.
We’ll begin by assuming again that we think each has about 10% chance of success, but since this is based on a limited experience it makes sense to treat this probability as the expected value of a
beta distribution parameterized by, say, 1 success and 9 failures.
So whenever we apply for a new job, we:
• draw a new $p1$ and $p2$ from each beta distribution
• apply to the one with the highest drawn probability
• update the selected resume’s beta distribution according to its success or failure.
Let’s simulate this, assuming that we know immediately if the application was successful or not. Let’s take the “true” probabilities to be 0.14 and 0.11 for the one-page and two-page resumes
respectively. We’ll keep track of the simulation state in a simple list:
new_stepper <- function() {
state <- list(k1 = 1, n1 = 10, p1 = 0.14, k2 = 1, n2 = 10, p2 = 0.11)
step <- function() {
old_state <- state
state <<- next_state(state)
new_stepper() returns a closure that keeps a reference to the simulation state. Each call to that closure updates the state using the next_state function:
next_state <- function(state) {
p1 <- rbeta(1, state$k1, state$n1 - state$k1)
p2 <- rbeta(1, state$k2, state$n2 - state$k2)
pull1 <- p1 > p2
result <- rbinom(1, 1, ifelse(pull1, state$p1, state$p2))
if (pull1) {
state$n1 <- state$n1 + 1
state$k1 <- state$k1 + result
} else {
state$n2 <- state$n2 + 1
state$k2 <- state$k2 + result
So let’s now simulate 1000 steps:
step <- new_stepper()
sim <- data.frame(t(replicate(1000, unlist(step()))))
The estimated effectiveness of each resume is given by the number of successes divided by the number of applications made with that resume:
sim$one_page <- sim$k1 / sim$n1
sim$two_page <- sim$k2 / sim$n2
sim$id <- 1:nrow(sim)
The follow plot shows how that estimated probability evolves over time:
sim_long <- melt(sim, measure.vars = c('one_page', 'two_page'))
ggplot(sim_long, aes(x = id, y = value, col = variable)) +
geom_line() +
xlab('Applications') +
ylab('Estimated probability of success')
Wouldn’t that be nice
As you can see, the algorithm decides pretty rapidly (after about 70 applications) that the one-page resume is more effective.
So here’s the protocol I’ve begun to follow since about mid-November:
• Apply only to jobs that I would normally have applied to
• Go through the entire application procedure, including writing cover letter etc, until uploading the resume becomes unavoidable (I do this mainly to avoid any personal bias when writing cover
• Draw $p1$ and $p2$ as described above; select resume type with highest $p$
• Adjust the resume according to the job requirements, but keep the changes to a minimum and don’t change the overall format
• Finish the application, and record a failure until a call for an interview comes in.
I’ll be sure to report on the results in a future blog post.
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Видеотека: Thorsten Dickhaus,
Аннотация: Genetic association studies lead to simultaneous categorical data analysis. The sample for every genetic locus consists of a contingency table containing the numbers of observed
genotype-phenotype combinations. Under case-control design, the row counts of every table are identical and fixed, while column counts are random. Aim of the statistical analysis is to test
independence of the phenotype and the genotype at every locus.
We present an objective Bayesian methodology for these association tests, utilizing the Bayes factor F_2 proposed by Good (1976) and Crook and Good (1980). It relies on the conjugacy of Dirichlet and
multinomial distributions, where the hyperprior for the Dirichlet parameter is log-Cauchy. Being based on the likelihood principle, the Bayesian tests avoid looping over all tables with given
marginals. Hence, their computational burden does not increase with the sample size, in contrast to frequentist exact tests.
Making use of data generated by The Wellcome Trust Case Control Consortium (2007), we illustrate that the ordering of the Bayes factors shows a good agreement with that of frequentist p-values.
Finally, we deal with specifying prior probabilities for the hypotheses, by taking linkage disequilibrium structure into account and exploiting the concept of effective numbers of tests (cf. Dickhaus
Язык доклада: английский | {"url":"https://m.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=9278","timestamp":"2024-11-11T21:20:59Z","content_type":"text/html","content_length":"8768","record_id":"<urn:uuid:adeb48e7-d3e7-4a9c-a279-f046d2eddc1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00515.warc.gz"} |
Section: Application Domains
Kinetic models like the Vlasov equation can also be applied for the study of large nano-particles as approximate models when ab initio approaches are too costly.
In order to model and interpret experimental results obtained with large nano-particles, ab initio methods cannot be employed as they involve prohibitive computational times. A possible alternative
resorts to the use of kinetic methods originally developed both in nuclear and plasma physics, for which the valence electrons are assimilated to an inhomogeneous electron plasma. The LPMIA (Nancy)
possesses a long experience on the theoretical and computational methods currently used for the solution of kinetic equation of the Vlasov and Wigner type, particularly in the field of plasma
Using a Vlasov Eulerian code, we have investigated in detail the microscopic electron dynamics in the relevant phase space. Thanks to a numerical scheme recently developed by Filbet et al. [66] , the
fermionic character of the electron distribution can be preserved at all times. This is a crucial feature that allowed us to obtain numerical results over long times, so that the electron
thermalization in confined nano-structures could be studied.
The nano-particle was excited by imparting a small velocity shift to the electron distribution. In the small perturbation regime, we recover the results of linear theory, namely oscillations at the
Mie frequency and Landau damping. For larger perturbations nonlinear effects were observed to modify the shape of the electron distribution.
For longer time, electron thermalization is observed: as the oscillations are damped, the center of mass energy is entirely converted into thermal energy (kinetic energy around the Fermi surface).
Note that this thermalization process takes place even in the absence of electron-electron collisions, as only the electric mean-field is present. | {"url":"https://radar.inria.fr/report/2011/calvi/uid24.html","timestamp":"2024-11-13T17:30:28Z","content_type":"text/html","content_length":"37858","record_id":"<urn:uuid:73cb3846-6945-499b-be6d-16d1cc216ca4>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00549.warc.gz"} |
What Is Positive Agreement
For Table 2, the proportion of specific matches for category i is as follows: 2nii ps(i) = ———. (6) NI. + n.i Graham P, Bull B. Approximate standard errors and confidence intervals for positive and
negative correspondence indices. J. Clin Epidemiol, 1998, 51(9), 763-771. For a particular case with two or more binary reviews (positive/negative), n and m indicate the number of reviews and the
number of positive reviews respectively. For this given case, there are exactly y = m (m − 1) observed in pairs in a positive evaluation and x = m (n − 1) possibilities for such an agreement. If we
calculate x and y for each case and add the two terms on all cases, then the sum of x divided by the sum of y is the proportion of the specific positive match in the entire sample. and the total
matching table, called P*O, for each simulated sample. The order value of the actual data is considered statistically significant if it exceeds a certain percentage (e.B 5%) of the p*o values.
In the latest FDA guidelines for laboratories and manufacturers, “FDA Policy for Diagnostic Tests for Coronavirus Disease-2019 during Public Health Emergency,” the FDA states that users must use a
contractual clinical study to determine performance characteristics (sensitivity/PPA, specificity/NPA). Although the terms sensitivity/specificity are widely known and used, the terms PPA/NPA are
not. The total number of real chords, regardless of category, is equal to the sum of equation (9) in all categories or C O = SUM S(j). (13) j=1 The total number of possible chords is K Oposs = SUM nk
(nk – 1). (14) k=1 equation divided (13) by equation (14) gives the total proportion of agreement observed or O po = ——. (15) Oposs The proportion of total compliance (Po) is the proportion of cases
for which evaluators 1 and 2 agree. In other words, although the formulas of positive and negative agreement are identical to those of sensitivity/specificity, it is important to distinguish between
them because the interpretation is different. A joint PA-NA review addresses the potential concern that the OP will be exposed to inflation or opportunity-induced distortion in the event of extreme
policy rates.
Such inflation, if it exists, would affect only the most common category. So, if PA and NA are both of satisfactory size, there is arguably less need or purpose to compare the actual match with that
predicted by chance using kappa statistics. In any case, PA and NA provide more relevant information for understanding and improving ratings than a single omnibus index (see Cicchetti and Feinstein,
1990).     Meaning, standard error, interval estimation Mackinnon, A. A spreadsheet to calculate complete statistics to evaluate diagnostic tests and inter-evaluator agreements. Computers in
Biology and Medicine, 2000, 30, 127-134. We first look at the case of an agreement between two evaluators on dichotomous ratings. Nor is it possible to determine from these statistics that one test
is better than another. Recently, a British national newspaper published an article about a PCR test developed by Public Health England and the fact that it did not agree with a new commercial test
in 35 of the 1144 samples (3%). For many journalists, of course, this was proof that the PHE test was inaccurate. There is no way to know which test is good and which is wrong in any of these 35
disagreements. We simply do not know the actual state of the subject in the studies on agreements.
Only by further investigating these disagreements will it be possible to determine the reason for the discrepancies. Consider, for example, an epidemiological application where a positive assessment
corresponds to a positive diagnosis for a very rare disease – for example, with a prevalence of 1 in 1,000,000. Here we may not be very impressed if the buttocks are very high – even above .99. This
result would be almost entirely due to an agreement on the absence of diseases; We are not directly informed if the diagnosticians agree on the presence of diseases. Highly neglected, raw
correspondence indices are important descriptive statistics. They have a one-size-fits-all common sense. A study that reports only simple tuning rates can be very helpful; A study that omits them but
reports complex statistics cannot inform readers on a practical level. . | {"url":"http://www.balanceanddizzinessphysicaltherapy.com/what-is-positive-agreement/","timestamp":"2024-11-11T07:54:16Z","content_type":"application/xhtml+xml","content_length":"18196","record_id":"<urn:uuid:92438774-f595-4602-b08b-1651924ee89f>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00529.warc.gz"} |
A dynamic monoidal category for strategic games
polynomial functors
dynamical systems
applied category theory
After last month’s wonderful ACT conference at University of Strathclyde in Glasgow, David Spivak and I spent some time with Matteo Capucci and Riu Rodriguez Sakamoto talking about how our theory of
dynamic monoidal categories can model the non-cooperative strategic games that they study at Strathclyde. A dynamic monoidal category is an elegant way to formalize a version of compositional game
theory based on Matteo’s work in, where games can be composed both sequentially and in parallel with the players updating their strategies by applying “counterfactual reasoning” to their payoffs in a
functorial way.
After July’s wonderful ACT conference at University of Strathclyde in Glasgow, David Spivak and I spent some time with Matteo Capucci and Riu Rodriguez Sakamoto talking about how our theory of
dynamic monoidal categories (Shapiro, Brandon T. and Spivak, David I. 2022) can model the non-cooperative strategic games that they study at Strathclyde. A dynamic monoidal category is an elegant way
to formalize a version of compositional game theory based on Matteo’s work in (Capucci 2022), where games can be composed both sequentially and in parallel with the players updating their strategies
by applying “counterfactual reasoning” to their payoffs in a functorial way.
1 Strategic games
We will take as our “interfaces” pairs of sets (A,A'), where A is regarded as the possible positions at one point in a game and A' consists of the potential payoffs. Making a play (or in a multi-turn
game, a move) between two interfaces (A,A') and (B,B') amounts to choosing for a given position in A a new position in B, as well as how to interpret a payoff of the new position (from B') into a
payoff of the original position (in A'). I interpret this backwards map on payoffs as encoding that the payoff comes from the new position in B but the player makes decisions from the vantage point
of the original position in A, so the payoff of a move needs to be interpreted into A'.
So a “game” is then just a pair of interfaces, which is played by turning a position of the first into a position of the second and interpreting back the payoff. One way a “player” (which will become
a technical term later on) could play such a game is with a “strategy” that determines for each position I \in A both a new position in B and a payoff interpretation function B' \to A'.
The aspect of game theory we are looking to model is the dynamics of strategies: when a player plays with a given strategy and sees the results, they may update their strategy, especially if they can
analyze what would have happened had they played differently. This is called “counterfactual reasoning” and we can model it as an adjustment to the interfaces by replacing (A,A') with (A,A'^A). So
instead of interpreting just the payoff of the play that was made, a player can also see what the payoff would have been for all of the other plays they could have made.
2 Valuation and distribution monads for nondeterminism
The basic idea of a strategic game theory model is to encode how a player changes their strategy based on additional information as it becomes available. A strategy can be regarded as a state in an
open dynamical system, where each move and payoff provides new information to inform an updated strategy. Generally, a player will switch to what seems like the best possible strategy, but in
practice there may be many equally good strategies without a clear way to choose between them.
In a real life game one of these best strategies might be chosen at random, and we could encode this by changing the states of our system from strategies to probability distributions on the set S of
strategies. We could also make our states subsets of S, so that an updated state includes all of the equally optimal strategies. Or perhaps “valuations” on S, which are functions from S to a set of
values like \mathbb{R} or [0,1]. In fact, a subset of S is also a kind of valuation where the values belong to the set of booleans \mathbb{B}= \{\bot,\top\} and the function S \to \mathbb{B} sends
only the elements in the subset to \top.
Rather than making a fixed choice of how to encode this nondeterminism, we can instead describe a large class of examples and focus on only the properties that they need to satisfy to effectively
model strategic game theory. Finitely supported probability distributions generalize using a formalism based on valuations that also describes nonempty subsets, so we will focus on this general class
of distributions instead.
Let V be a semiring, which consists of two monoid structures (0,+) and (1,\times) on the same set V satisfying distributivity of \times over +. There is a monad V^{(-)} on \textbf{Set} defined by:
• V^S is the set of finitely supported valuations of S in V (functions V \to S);
• For a function f : S \to T, v : S \to V, and t \in T, define V^f(v) = \sum_{s \in f^{-1}(t)} v(s); \tag{1}
• \eta : S \to V^S sends s \in S to the valuation sending s to 1 and everything else to 0;
• To define \mu : V^{V^S} \to V^S, for s \in S we set \mu(w)(s) = \sum_{v \in V^S} w(v) \times v(s). \tag{2}
The functoriality, naturality, unitality, and associativity equations that make this data form a monad are all guaranteed by the units, associativity, and distributivity of the semiring V. If V
admits infinitary sums which \times also distributes over, then a similar monad can be defined sending a set S to the set of all valuations without the assumption of finite support.
For any semiring V there is also a submonad \mathsf{Dist}_V \hookrightarrow V^{(-)}, where \mathsf{Dist}_V(S) consists of the valuations v \in V^S with \sum\limits_{s \in S} v(s) = 1, which are
called distributions. When V has multiplicative inverses, there is *almost* a map of monads V^{(-)} \to \mathsf{Dist}_V sending each valuation to its normalization given by scaling according to the
inverse of \sum_s v(s). This map is obstructed by valuations which are constant at 0, though it can be defined on the submonad of V^{(-)} containing valuations with nonempty support.
For the semiring (\mathbb{R},0,+,1,\times), \mathbb{R}^{(-)} is the free real vector space monad, and more generally algebras for any valuation monad V^{(-)} are V-modules. Here distributions are
precisely finitely supported probability distributions.
We could also consider the semiring (\mathbb{R}_{\ge 0},0,\max,1,\times), which restricts the previous example to the non-negative reals and replaces sums with maxima. In game theory, this has the
effect of prioritizing strategies with the best possible value among many possibilities rather than the sum of all its possible values (a more cynical player could also take the minimum, though that
would not as easily fit this framework as \min lacks a unit in \mathbb{R}_{\ge 0}). Distributions in this setting are simply valuations whose values do not exceed 1.
For the semiring (\mathbb{B},\bot,\vee,\top,\wedge) of booleans (which does have infinite sums), \mathbb{B}^{(-)} is the powerset monad. In this case Equation (1) unwinds to the direct image of a
subset, the unit sends each element of S to the corresponding singleton subset, and Equation (2) unwinds to the union of a subset of subsets. Here a distribution is any nonempty subset.
When playing a game and deciding which strategy to choose, a valuation is only as useful as its associated distribution when it has one, as both encode equally well the information of which strategy
is the best. However, distributions or at least normalizable valuations are preferable, as a valuation sending every strategy to zero would make it very difficult to decide how to play.
A convenient property of a valuation monad V^{(-)} is that it sends coproducts to products when V has infinitary sums, meaning in particular that V^S \cong \prod_{s \in S} V^{\{s\}}. When V has only
finite sums, V^S is the subset of the corresponding product containing tuples with finite support. This property, in either form, is not shared by the distribution monad, which sends the singleton
set to itself. However, when V admits normalization there is a natural map \mathrm{norm} : \left(\prod_{s \in S} V^{\{s\}}\right) \backslash \mathop{\mathrm{const}}_0 \cong V^S \backslash \mathop{\
mathrm{const}}_0 \to \mathsf{Dist}_V(S), which we will use to update a player’s distribution on a set of strategies.
3 Dynamic monoidal categories
The main idea of modeling game theory as a dynamic monoidal category is that distributions on the set of strategies can be regarded as morphisms which can be updated, tensored, and composed
(corresponding to the “dynamic,” “monoidal,” and “category” structure respectively). Dynamic monoidal categories were defined in (Shapiro, Brandon T. and Spivak, David I. 2022 Definition 3.7), and
combine the structure of a monoidal category with that of polynomial coalgebras on the hom-sets.
For a polynomial p, a p-coalgebra is a set X of states equipped with an “action” function \textnormal{act}: X \to p(1) and, for each x \in X, an “update” function \textnormal{upd}_x : p[\textnormal
{act}(x)] \to X. The set X can then be treated as the states of an open dynamical system, where each state x is labeled by some position in p(1) and when given a direction updates to \textnormal{upd}
_x applied to that direction.
A relaxed dynamic monoidal category {\textbf{A}} consists of:
• a monoid ({\textbf{A}}_0,e,*) of objects;
• for each a \in {\textbf{A}}_0, a polynomial p_a;
• an isomorphism of polynomials y \cong p_e;
• for each a,a' \in {\textbf{A}}_0, a “laxator” morphism of polynomials p_{a} \otimes p_{a'} \to p_{a*a'} (this is a slight generalization of (Shapiro, Brandon T. and Spivak, David I. 2022), where
this map is an isomorphism);
• for each a,b \in {\textbf{A}}_0, a [p_a,p_b]-coalgebra {\textbf{X}}_{a,b};
• for each a \in {\textbf{A}}_0, an “identitor” morphism of coalgebras as in Equation (3.1);
• for each a,b,c \in {\textbf{A}}_0, a “compositor” morphism of coalgebras as in Equation (3.2); and
• for each a,a',b,b' \in {\textbf{A}}_0, a “productor” morphism of coalgebras as in Equation (3.3):
satisfying unit, associativity, and interchange equations.
In the dynamic monoidal category for strategic games, the monoid of objects will be pairs of sets (A,A') with pairwise product as the monoid structure. We can regard an interface (A,A') as a
polynomial p_{A,A'} = A\mathcal{y}^{A'}, where a morphism of polynomials A\mathcal{y}^{A'} \to B\mathcal{y}^{B'} is the same as a strategy from the interface (A,A') to (B,B'): it consists of
functions A \to B and A \times B' \to A' (also called a lens). While we will only consider monomials, counterfactual reasoning can be modeled as a more general construction on \textbf{Poly}.
For any polynomial p, let p^\square = p(1)\mathcal{y}^{\Gamma(p)}, where \Gamma(p) is the set of sections of p, or rather, choices of i \in p[I] for all I \in p(1). The polynomial associated to
(A,A') in our relaxed dynamic monoidal category will be p_{(A,A')}^\square = A\mathcal{y}^{A'^A}, where this construction can be thought of as a “counterfactual replacement” of a polynomial, whose
directions encode not just the payoff of a position in p but also payoffs for all of the other possible positions.
The assignment p \mapsto p^\square is an endofunctor on \textbf{Poly}, and there is a natural transformation p \to p^\square which is the identity on p(1) and on directions sends (I \in p(1),\gamma \
in \Gamma(p) to \gamma(I) \in p[I]. It is also a lax monoidal functor with respect to the Dirichlet tensor product \otimes on \textbf{Poly}, where y^\square \cong y and the morphism p^\square \otimes
q^\square \to (p \otimes q)^\square is again the identity on the positions p(1) \times q(1); on directions it sends ((I,J) \in p(1) \times q(1), \gamma \in \Gamma(p \otimes q)) to the pair (\gamma_p,
\gamma_q) \in \Gamma(p) \times \Gamma(q), where \gamma_p(I') = \gamma(I',J) and \gamma_q(J') = \gamma(I,J'). Matteo calls this map of polynomials the “Nashator,” as it picks out each row and column
in a payoff matrix of possible plays by two players in a game, similar to the matrices in which Nash equilibria are found. This map will serve as the laxator in our relaxed dynamic monoidal category,
where the modifier “relaxed” refers to the fact that the Nashator is not an isomorphism.
From this structure alone we can show how (-)^\square interacts with the monoidal closed structure of \otimes. Recall that [p,q] = \sum_{f : p \to q} \mathcal{y}^{\sum_{I \in p(1)} q[fI]} is the
internal hom polynomial equipped with an evaluation map p \otimes [p,q] \to q. This lets us define a map p^\square \otimes [p,q]^\square \to \left( p \otimes [p,q] \right)^\square \to q^\square.
Using the universal property of the closure and the natural transformation \mathrm{id}_\textbf{Poly}\to (-)^\square, we then get a map of polynomials [p,q] \to [p,q]^\square \to [p^\square,q^\
square], which will facilitate modeling the behavior of players in this structure. On positions, this map sends a map f : p \to q to f^\square : p^\square \to q^\square, which acts the same as f on
positions and on directions sends (I \in p(1),\gamma_q \in \Gamma(q)) to \gamma_p \in \Gamma(p) where \gamma_p(I') = f^\#(\gamma_q(f(I'))).
4 Players and information as coalgebra states
We can now properly define what a player is in this model: a game is played between interfaces (A,A') and (B,B'), and a player consists of a set S and a polynomial morphism \phi : \mathsf{Dist}_V(S)\
mathcal{y}^V \to [A\mathcal{y}^{A'},B\mathcal{y}^{B'}] for some fixed distribution monad \mathsf{Dist}_V. The set S consists of abstract strategies, which have names but are not yet realized as
polynomial morphisms A\mathcal{y}^{A'} \to B\mathcal{y}^{B'}. The morphism \phi on positions describes how the player takes a distribution on all of their possible abstract strategies and converts it
to a concrete strategy. On directions, \phi accounts for the value the player receives when applying their concrete strategy to a given position in A and payoff from B'.
To simplify the notation, we will use p and q to refer to both the pairs (A,A') and (B,B') as well as their canonical polynomials. The [p^\square,q^\square]-coalgebra {\textbf{X}}_{p,q} of arrows
from p to q in our dynamic monoidal category has states which consist of a player and a distribution on their strategy set, representing their current assessment of the possible strategies based on
available information. Formally, the state set is defined as X_{p,q} = \left\{(S,\phi,v) \;\;\middle|\;\; S : \textbf{Set}, \;\; \phi : \mathsf{Dist}_V(S)\mathcal{y}^V \to [p,q], \;\; v \in \mathsf
{Dist}_V(S) \right\}. The action X_{p,q} \to [p^\square,q^\square](1) sends (S,\phi,v) to \phi(v)^\square : p^\square \to q^\square, and the update of (S,\phi,v) sends (I \in p(1),\gamma \in \Gamma
(q)) to \left(S,\phi, \mathop{\mathrm{norm}}\left(s \mapsto \phi^\#(I,\gamma(\phi(\eta s)(I)))\right) \right). \tag{4} This normalization requires that \phi^\# sends some pair (I,\gamma(\phi(\eta
s))) to a nonzero value, which we will assume throughout.
Here \eta s is the distribution sending s to 1 and all other abstract strategies in S to 0, so the update formula in Equation (4) uses counterfactual reasoning by checking what value (according to \
phi^\#) would have been achieved by employing with certainty the strategy s on the same position a. The section \gamma provides the counterfactual of what payoff would have been received from the
move \phi(\eta s)(I), and \phi^\# encodes how to extract a value from that counterfactual to build a new distribution. While we use the general polynomial notation of p and q, it is essential that q
= B\mathcal{y}^{B'} (rather than a more general polynomial) so that the potential payoffs for different moves are of the same type and hence (I,\gamma(\phi(\eta s)(I))) is in fact a well-typed
direction of [p,q].
Like many dynamic structures, the states of this coalgebra consist of some fixed information (the player (S,\phi)) which does not get updated, as well as an updating parameter (here the distribution
v). However, the algebraic structure of a dynamic monoidal category describes primarily how players can be combined in series or parallel.
5 Identity games
The simplest piece of the monoidal category structure on these dynamical systems is the identitors, which amount to a state in each X_{p,p} that is left fixed by updates. This state is given by (1, \
mathop{\mathrm{const}}_\mathrm{id}: \mathcal{y}^V \to [p,p], * \in \mathsf{Dist}_V(1) \cong 1), where \mathop{\mathrm{const}}_\mathrm{id} sends the unique position in \mathcal{y}^V to \mathrm{id}_p
on positions and on directions sends each (a,b) to 1. As \mathsf{Dist}_V(1) \cong 1, the update function always outputs the same state.
Game theoretically, this state describes a player who always keeps the current position and values all payoffs equally.
6 Parallel games
The productor consists of a function X_{p,q} \times X_{p',q'} \to X_{p \otimes p',q \otimes q'} on states which respects actions and updates, and describes how to treat two players in parallel games
as a single player playing two parallel games at once. This function sends (S,\phi,v) and (S',\phi',v') to (S \times S',\phi \star \phi', v \times v'), where v \times v' sends (s,s') to v(s)v'(s')
and \begin{aligned} \phi \star \phi' : \mathsf{Dist}_V(S \times S')\mathcal{y}^V &\to \mathsf{Dist}_V(S)\mathcal{y}^V \otimes \mathsf{Dist}_V(S')\mathcal{y}^V \\&\to [p,q] \otimes [p',q'] \\&\to [p \
otimes p',q \otimes q']. \end{aligned} \tag{5}
The first map in Equation (5) is defined on positions by (\mathsf{Dist}_V(\pi_1),\mathsf{Dist}_V(\pi_2)) : \mathsf{Dist}_V(S \times S') \to \mathsf{Dist}_V(S) \times \mathsf{Dist}_V(S') and on
directions by the semiring multiplication V \times V \to V. More specifically, on positions this map takes a distribution on S \times S' and for each s \in S sums up in V the values of (s,s') for all
s' \in S. This means that given a distribution on all pairs of strategies the two players could use in each game, each player evaluates their own strategies accounting for all possible strategies
that the other player could choose, and the value of a pair of strategies is given by the sum in V of their separate values.
This has different meanings for different choices of the distribution monad. When V is \mathbb{R} with addition and multiplication, the value of a strategy s \in S is given by adding up the values of
all pairs (s,s'), essentially treating all possible strategies of the other player as equally likely to be played. When V is the semiring \mathbb{R}_{\ge 0} and the sum is a maximum, the value of s
is determined by its pairing with the best possible choice of s'. In this case the player seems to assume that their collaborator wants what is best for them, while a semiring in which sums are
minima would model a more adversarial relationship. Ultimately though, while the productor must specify a map of polynomials from \mathsf{Dist}_V(S \times S')\mathcal{y}^V, we are only concerned with
its application to distributions of the form v \times v', as the productor must be a coalgebra map meaning these distributions are closed under updates.
The proof that this function on states preserves actions has essentially two components:
• applying (\mathsf{Dist}_V(\pi_1),\mathsf{Dist}_V(\pi_2)) to the distribution v \times v' yields the pair \left(\sum_{s' \in S'} v'(s')\right)v = v \qquad \textrm{and} \qquad \left(\sum_{s \in S}
v(s)\right)v' = v'; and
• the diagram (6) commutes.
On the other hand, given this it is easy to see that updates are preserved: the morphism [p,q] \otimes [p',q'] \to [p \otimes p',q \otimes q'] is cartesian, so updates as defined in Equation (4) are
computed independently and then multiplied together, just as desired.
7 Sequential games
The compositor is defined very similarly to the productor, describing two players making moves one after another rather than in parallel. The function on states X_{p,q} \times X_{q,r} \to X_{p,r}
sends (S,\phi,v) and (T,\psi,w) to (S \times T, \;\phi;\psi, \;v \times w), where \begin{aligned} \phi;\psi : \mathsf{Dist}_V(S \times T)\mathcal{y}^V &\to \mathsf{Dist}_V(S)\mathcal{y}^V \otimes \
mathsf{Dist}_V(T)\mathcal{y}^V \\&\to [p,q] \otimes [q,r] \\&\to [p,r] \end{aligned} \tag{7} and the first map in Equation (7) is the same as that in Equation (5). Much like the productor describing
two strategies running in parallel, the compositor extracts in the same way two concrete strategies and then composes them into one. This is a coalgebra map for the same reasons as the productor,
with the commuting diagram (6) replaced by (8).
To complete the definition of this relaxed dynamic monoidal category of interfaces, players, and distributions, we note that the identitors, productors, and compositors satisfy the relevant unit,
associativity, and interchange equations nearly by definition, as both productors and compositors are defined by cartesian products of abstract strategy sets and tensor products of the morphisms of
polynomials that make those strategies concrete.
8 Conclusion
Strategic game theory includes paradigms for how players update their strategies using counterfactual reasoning, and how sequences of players can be treated as a single player by composing their
games in sequence and/or in parallel. The combination and compatibility of these different structures is precisely captured by the relaxed dynamic monoidal category of interfaces, players, and
distributions. Our hope is that the study of these dynamic categorical structures can shed light on game theory and more clearly illuminate its connections to other structured dynamical systems such
as deep learning.
Capucci, M. 2022.
“Diegetic Representation of Feedback in Open Games.” Applied Category Theory 2022
. 2022.
Shapiro, Brandon T. and Spivak, David I. 2022.
“Dynamic categories, dynamic operads: From deep learning to prediction markets.” https://arxiv.org/abs/2205.03906
Leaving a comment will set a cookie in your browser. For more information, see our cookies policy. | {"url":"https://topos.site/blog/2022-09-12-dynamic-monoidal-games/","timestamp":"2024-11-08T14:13:04Z","content_type":"application/xhtml+xml","content_length":"62884","record_id":"<urn:uuid:230d43fa-6e08-45d3-b909-27ec0c57e6b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00367.warc.gz"} |
About Us - Multiplication Table Chart
A multiplication table is an important tool for children to learn multiplication tables in an easy manner. Multiplication tables are one of the basic concepts which is taught by every school to the
students at the primary classes.
So it forms a foundation for the education they will receive in the later years. And as parents, you might find it difficult to make your children learn the multiplication tables. So this site is all
about multiplication tables charts in downloadable and printable formats which will help you to make your kids learn the multiplication tables in a fun and interactive manner.
One of the common problems faced by kids while learning multiplication tables is that they find it boring and difficult to learn the multiplication tables. So as parents, you can make them learn the
multiplication tables through games and quizzes, which kids would find interesting to learn.
We have provided here multiplication tables charts in various formats, like form 1-12, 1-30, etc. in fun and visually appealing designs. These are provided in PDF and Word formats so you can easily
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Time Value of Money Example Question | CFA Level 1 - AnalystPrep
Time value of money is a concept that refers to the greater benefit of receiving a given amount of money at present rather than in the future due to its earning potential. For example, money could be
invested in a bank account and earn interest even overnight. Interest earned will depend on the rate of return offered by government bonds (risk-free assets), inflation, liquidity risk, default risk,
time to maturity, and other factors.
In a nutshell, time value calculations allow people to establish the future value of a given amount of money at present. The present value (PV) is the money you have today. The future value (FV) is
the accumulated amount of money you get after investing the original sum at a certain interest rate and for a given time period, say 2 years. The concept has a wide range of applications that
incorporate financial matters-bonds, shares, loan facilities, among others.
Fundamental Formulas in Time Value of Money Calculations
FV = future value
PV = present value
r = interest rate
n = number of investment periods per year
t = number of years
Note: besides annual interest payments, interest could be compounded monthly, quarterly or semi-annually. If for instance, interest is payable quarterly, then we have 12/3 i.e 4 investment periods
per year.
$$ PV=FV{ \left( 1+\frac { r }{ n } \right) }^{ -n*t } $$
$$ FV=PV{ \left( 1+\frac { r }{ n } \right) }^{ n*t } $$
Suppose an individual invests $10,000 in a bank account that pays interest at a rate of 10% compounded annually. What will be the future value after 2 years?
A. $12,000
B. $12,100
C. $22,000
The correct answer is B.
PV=10,000, r=0.1, n=1, t=2
$$ \begin{align*}
FV & =10,000{ \left( 1+\frac { 0.1 }{ 1 } \right) }^{ 1*2 } \\
& =10,000(1.1)^2 \\
& =12,100 \end{align*} $$
To confirm our answer, we could work out the PV of a future value of 12,100 invested under similar terms, starting with the FV of $12100.
$$ \begin{align*}
PV & =12,100 \left(1+0.1/1 \right)^{-(1*2)} \\
& =12,100(1.1)^-2 \\
& =$10,000 \\
\end{align*} $$
Points to Note
First, establish all the components of the relevant formula before commencing actual calculation. Secondly, only the term within the brackets is subject to the power function.
The concept of the time value of money serves as the foundation for more concrete financial calculations such as simple interest, compound interest, and the value of stocks or bonds at any given
point in time.
Reading 6 LOS 6a:
Interpret interest rates as required rates of return, discount rates, or opportunity costs
Sep 09, 2021
Chi-square and F-Distributions
Chi-square Distribution A chi-square distribution is an asymmetrical family of distributions. A chi-square... Read More | {"url":"https://analystprep.com/cfa-level-1-exam/quantitative-methods/time-value-money-explained-example/","timestamp":"2024-11-11T14:38:38Z","content_type":"text/html","content_length":"153002","record_id":"<urn:uuid:5edd25e3-f5b2-498a-9467-d505dc7e3677>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00513.warc.gz"} |
This python program must have 9 functions:
•main() Controls the flow of the program (calls the...
This python program must have 9 functions: •main() Controls the flow of the program (calls the...
This python program must have 9 functions:
•main() Controls the flow of the program (calls the other modules)
•userInput() Asks the user to enter two numbers
•add() Accepts two numbers, returns the sum
•subtract() Accepts two numbers, returns the difference of the first number minus the second number
•multiply() Accepts two numbers, returns the product
•divide() Accepts two numbers, returns the quotient of the first number divided by the second number
•modulo() Accepts two numbers, returns the modulo of the first number divided by the second number
•exponent() Accepts two numbers, returns the first number raised to the power of the second number
•userOutput() Gives a user friendly output of all the calculations.
NOTE: The ONLY place that you will print results is inside the userOutput function
# userInput function
def userInput():
#reads the input from user
num1 = int(input("Enter a number: "))
num2 = int(input("Enter a number: "))
lst = []
# user input appending to a lst
# returns the user input in the form of list
return lst
# add function
def add(lst):
res = lst[0]+lst[1]
# returns the addition
return res
# substract function
def substract(lst):
res = int(lst[0])-int(lst[1])
# returns the substract result
return res
# multiply function
def multiply(lst):
res = lst[0]*lst[1]
# returns the multiply
return res
# divide function
def divide(lst):
res = lst[0]/lst[1]
# returns the divide
return res
# modulo function
def modulo(lst):
res = lst[0]%lst[1]
return res
# exponent function
def exponent(lst):
res = pow(lst[0],lst[1])
return res
# userOutput function
def userOutput(lst):
# prints the all calculations
for i in range(len(lst)):
# main function which calls the all modules
# in a main function retunred value of a modules is stored in list that is "lt"
def main():
lt = []
# calling the function userInput function
lst = userInput()
# calling the add function
ret = add(lst)
# retunred value of add function appending to lt list
ret = substract(lst)
ret = multiply(lst)
ret = divide(lst)
ret = modulo(lst)
ret = exponent(lst)
#passing the list to userOutput
if __name__=="__main__":
# calling a main function | {"url":"https://justaaa.com/computer-science/413610-this-python-program-must-have-9-functions-main","timestamp":"2024-11-10T17:59:58Z","content_type":"text/html","content_length":"46871","record_id":"<urn:uuid:063e9e5c-9926-4399-b456-62fd9cd433f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00139.warc.gz"} |
Mathematics Literacy
Mathematics Literacy tutors in Randburg
Personalized Tutoring Near You Mathematics Literacy lessons for online or at home learning in Randburg
Mathematics Literacy tutors in Randburg near you
Candice F
Bergbron, Randburg
I have a degree in civil engineering, and therefore have an expansive experience with this subject. Subjects that i covered were engineering maths (which is a heavy focus on calculus), applied
mathematics (which is a mixture of static and dynamic physics) and basic chemistry on a university level, in addition to my engineering subjects. I have also taught Cambridge mathematics at a high
school for a year.
Teaches: Trigonometry, General Maths & Science, Mathematics Literacy, Linear Algebra, Calculus, Microsoft Excel, Chemistry, Physics
Available for Mathematics Literacy lessons in Randburg
Stuart A
Pine Park, Randburg
Thanks to having two young daughters, currently in Grade 3 and Grade 5, I have revisited maths recently. I did get my matric in maths, as well as doing maths 101 and statistics 101 at University as
part of my B Comm. I am patient at tutoring maths and explaining methodology
Teaches: English Literature, Mathematics Literacy, Economics, Vocabulary, Writing, Business Management, English skills
Available for Mathematics Literacy lessons in Randburg
Subjects related to Mathematics Literacy in Randburg
Find Mathematics Literacy tutors near Randburg | {"url":"https://turtlejar.co.za/tutors/randburg-gt/mathematics-literacy","timestamp":"2024-11-11T00:48:22Z","content_type":"text/html","content_length":"143413","record_id":"<urn:uuid:e1efe52c-64ec-4351-ac88-2a1979fb5d38>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00393.warc.gz"} |
Compared to old Axiom FriCAS has following advantages:
• a lot of bugs fixed
• ported to several Lisp implementations, in particular sbcl gives both better speed and stability than gcl which was the only choice in the past
• much faster Spad compiler
• improvements to integrator, limits and other core parts
• new subsystems
• regular source and binary releases
• new improved TeXmacs interface
• new notebook interface jFriCAS
Currently FriCAS routine for indefinite integration is best among free system and very competitive with commercial systems. In particular, integration in terms of special functions improved
significantly, see FriCASSpecialIntegration
FriCAS contains implementation of Gruntz algorithm for computing limits. Consequently, several examples that caused problems in Axiom now work correctly.
FriCAS contains symbolic version of most special functions from Abramowitz and Stegun. FriCAS can compute derivatives, expand them into series and compute sume limits, etc. Numerical evaluation is
available only for a subset.
Multivariate Ore algebras (in particular partial differential operators) with noncommutative Groebner bases.
Multivariate factorization: in characteristic 0 most polynomial domains have uniform implementation of factorization. FriCAS factorization routines can handle large examples (FactorizationExample).
Guessing package see GuessingFormulasForSequences.
Package for computations in quantum probability.
Package for computations in algebraic topology.
Package for computations with group presentations.
In FriCAS user level functions are typically compiled to machine code. Compilation to machine code and fact that FriCAS language is strongly typed leads to code which is much faster than
interpreters used by several competing systems. See SpeedOfUserCode.
• at command line uses lexically scoped language with convenient constructs for iteration, functions (including closures)
• command line language is very similar to Spad language used to write FriCAS library (so library code is easier to understand)
• domains and categories provide an extensive library of generic algorithms that are easily extended to new domains
• generic programming methodology
• extensive documentation on the web
• active FriCAS Wiki
• excellent Help browser | {"url":"https://wiki.fricas.org/FriCASAdvantages","timestamp":"2024-11-11T00:06:24Z","content_type":"application/xhtml+xml","content_length":"22268","record_id":"<urn:uuid:82cc8706-8f4e-4b98-a0cf-6adbf0dc1590>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00371.warc.gz"} |
Basic implementation | mooc-hwlab
Remember that in the passthrough we set up the system to work with a sampling frequency of 32KHz and a sample precision of 16 bits. Here we will use a modulation frequency of 400Hz for the frequency
shifting, so that the digital frequency is a rational multiple of $2\pi$as in the previous example:
cos_table[n] = (int16_t)(32767.0 * cos(0.0785398 * n));
The lookup table is provided here for your convenience. Begin by copying it between the USER CODE BEGIN PV and USER CODE END PV comment tags.
#define COS_TABLE_LEN 80
static int16_t COS_TABLE[COS_TABLE_LEN] = {
0x7FFF, 0x7F99, 0x7E6B, 0x7C75, 0x79BB, 0x7640, 0x720B, 0x6D22, 0x678D, 0x6154, 0x5A81, 0x5320,
0x4B3B, 0x42E0, 0x3A1B, 0x30FB, 0x278D, 0x1DE1, 0x1405, 0x0A0A, 0x0000, 0xF5F6, 0xEBFB, 0xE21F,
0xD873, 0xCF05, 0xC5E5, 0xBD20, 0xB4C5, 0xACE0, 0xA57F, 0x9EAC, 0x9873, 0x92DE, 0x8DF5, 0x89C0,
0x8645, 0x838B, 0x8195, 0x8067, 0x8001, 0x8067, 0x8195, 0x838B, 0x8645, 0x89C0, 0x8DF5, 0x92DE,
0x9873, 0x9EAC, 0xA57F, 0xACE0, 0xB4C5, 0xBD20, 0xC5E5, 0xCF05, 0xD873, 0xE21F, 0xEBFB, 0xF5F6,
0x0000, 0x0A0A, 0x1405, 0x1DE1, 0x278D, 0x30FB, 0x3A1B, 0x42E0, 0x4B3B, 0x5320, 0x5A81, 0x6154,
0x678D, 0x6D22, 0x720B, 0x7640, 0x79BB, 0x7C75, 0x7E6B, 0x7F99};
Let's also define a gain factor to increase the output volume. As we said before, we will use a gain that is a power of two and therefore just define its exponent. If you find that the sound is
distorted, you may want to reduce this number.
#define GAIN 3 /* multiply the output by a factor of 2^GAIN */
In the following version of the main processing function you will need to provide a couple of lines yourself. In the meantime please note the following:
void inline Process(int16_t *pIn, int16_t *pOut, uint16_t size) {
static int16_t x_prev = 0;
static uint8_t ix = 0;
// we assume we're using the LEFT channel
for (uint16_t i = 0; i < size; i += 2) {
// simple DC notch
int32_t y = (int32_t)(*pIn - x_prev);
x_prev = *pIn;
// modulation
y = ...
// rescaling to 16 bits
y >>= (15 - GAIN);
// duplicate output to LEFT and RIGHT channels
*pOut++ = (int16_t)y;
*pOut++ = (int16_t)y;
pIn += 2;
You can now try changing the modulation frequency by creating your own lookup tables! | {"url":"https://hwlab.learndsp.org/voice-transformers/alien-voice/alien_implementation","timestamp":"2024-11-05T10:28:45Z","content_type":"text/html","content_length":"475500","record_id":"<urn:uuid:3442bc2d-5b72-47e6-909a-2503edbb791b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00283.warc.gz"} |
Three grams of musk
oil are required for each bottle of Mink Caress, a very popular...
Three grams of musk oil are required for each bottle of Mink Caress, a very popular...
Three grams of musk oil are required for each bottle of Mink Caress, a very popular perfume made by a small company in western Siberia. The cost of the musk oil is $2.20 per gram. Budgeted production
of Mink Caress is given below by quarters for Year 2 and for the first quarter of Year 3:
Year 2 Year 3
First Second Third Fourth First
Budgeted production, in bottles 72,000 102,000 162,000 112,000 82,000
Musk oil has become so popular as a perfume ingredient that it has become necessary to carry large inventories as a precaution against stock-outs. For this reason, the inventory of musk oil at the
end of a quarter must be equal to 20% of the following quarter’s production needs. Some 43,200 grams of musk oil will be on hand to start the first quarter of Year 2.
Prepare a direct materials budget for musk oil, by quarter and in total, for Year 2 | {"url":"https://justaaa.com/accounting/187802-three-grams-of-musk-oil-are-required-for-each","timestamp":"2024-11-01T23:32:52Z","content_type":"text/html","content_length":"43219","record_id":"<urn:uuid:bb69a398-3d38-499d-b538-4f2a9491454f>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00228.warc.gz"} |
An Algorithm for Linearizing the Collatz Convergence
Nuremberg Institute of Technology, Keßlerplatz 12, 90489 Nuremberg, Germany
Faculty of Economic and Social Sciences, Potsdam University, Karl-Marx Straße 67, 14482 Potsdam, Germany
Capgemini, Bahnhofstraße 30, 90402 Nuremberg, Germany
Faculty of Computer Science, Schmalkalden University of Applied Sciences, Blechhammer 9, 98574 Schmalkalden, Germany
Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
Unesco-Unitwin Complex Systems Digital Campus, ECCE e-Lab, CEDEX, 67081 Strasbourg, France
Author to whom correspondence should be addressed.
Submission received: 29 April 2021 / Revised: 27 July 2021 / Accepted: 29 July 2021 / Published: 9 August 2021
The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable
proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz
sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd
number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary
tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs
to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.
1. Introduction
In 1937, Lothar Collatz established a conjecture known as the
$3 n + 1$
problem, also known as Kakutani’s problem, the Syracuse algorithm, Hasse’s algorithm, Thwaites conjecture, and Ulam’s problem. The Collatz problem involves the iterative sequence defined as follows
(see OEIS [
] for the definition of the Collatz map):
$a n = a n − 1 / 2 , if a n − 1 is even 3 a n − 1 + 1 , if a n − 1 is odd$
Among others, Erdős and Conway [
] conjectured that, given any initial term
$a 0$
, the sequence always terminates at 1. Conway proved that there is no nontrivial cycle with a length less than 400, with Lagarias [
] later increasing this lower bound to 275,000. Conway [
], and Kurtz and Simon [
] also proved that the generalization of the Collatz problem is undecidable. The conjecture was first verified up to
$5.6× 10 13$
by Leavens et al. [
] and then to
$10 15 − 1$
by Vardi [
]; then, Oliveira [
] further extended the results to
$5.48× 10 18$
, and as of 2020, it had been verified beyond
$2 68$
. The Collatz problem is often stated differently, for example by Terras [
], to essentially compress the division by 2 [
$a n = a n − 1 / 2 , if a n − 1 is even ( 3 a n − 1 + 1 ) / 2 , if a n − 1 is odd$
Researchers have tried to model the problem in various ways. Wolfram [
] represented it as an eight-register machine. Cloney et al. [
] and Bruschi [
] modeled it as a quasi-cellular automaton, with Zeleny [
] specifically modeling it as a six-color one-dimensional quasi-cellular automaton. Among some notable recent developments, Machado [
] provided an interesting clustering perspective on the Collatz conjecture and Tao [
] demonstrated that almost all Collatz orbits attain almost bounded values.
The dynamical system generated by the
$3 n + 1$
problem is known to create complex quivers (a quiver is simply a collection of arrows between points forming a set [
], where the Collatz quiver used here is simply defined as the set of all arrows connecting any natural number to the next one under the Collatz map) over
, with one of the most picturesque being the so-called “Collatz Feather” or “Collatz Seaweed”, a name popularized by Clojure programmer Oliver Caldwell in 2017 [
]. The inflation propensity of Collatz orbits remains so unpredictable that Bocart showed that it can form the core of a reliable proof-of-work algorithm for Blockchain solutions [
], with groundbreaking applications to the field of number-theoretical cryptography as such algorithms are unrelated to primes yet, being based on the class of congruential graphs and still allowing
for a wide diversity of practical variants. If Bocart thus demonstrated that graph-theoretical approaches to the
$3 n + 1$
problem can be very fertile to applied mathematics, the authors have also endeavored to demonstrate its pure number-theoretical interest prior to this work [
]. In this article, we refer to the Bocart proof-of-work in that expanding it and more precisely endowing it with a scannable certificate is an important side-result of our approach.
Our methodology consists of using the complete binary tree and the complete ternary tree (the complete binary tree over odd numbers is defined as
$2 N * + 1$
endowed with the following two linear applications
${ · 2 − 1 ; · 2 + 1 }$
and all their possible combinations, with the complete ternary tree over the same set in turn defined as
$2 N * + 1$
endowed with operations
${ · 3 − 2 ; · 3 ; · 3 + 2 }$
and all their possible combinations) over
$2 N * + 1$
as a general coordinate system for each node of the feather. We owe this strategy to earlier discussions with Feferman [
] on his investigations on the continuum hypothesis, as it is known that the complete binary tree over natural numbers is one way of generating real numbers. The last author’s discussion with
Feferman argued that morphisms, sections, and origamis of
-ary trees over
could be a promising strategy to define objects of intermediate cardinalities between
$ℵ n$
$ℵ n + 1$
, in a manner inspired from Conway’s construction of surreal numbers [
], which itself began by investigating the branching factor of the game of Go. Central to our contribution to the Collatz conjecture in this paper is also the analysis of the branching factor of a
zero-player cellular game developing in the complete binary tree over odd numbers.
2. Related Research
2.1. Goodstein Sequences and Hydra Games
The idea of attacking the Collatz conjecture from the angle of logic and set theory is not new. Hydra games were first introduced by Kirby and Paris [
], and Arzarello [
] provided a rather wide outline of how their consideration could, in fact, lead to a set theoretical solution of the Collatz conjecture. The convergence of Goodstein sequences indeed, which form the
core of Kirby and Paris’ demonstration that no Hydra game can be lost, cannot be proven in Peano arithmetics alone. Their founding element, however, which is the base-
hereditary representation of a number
, can be defined without the axiom of choice.
Definition 1.
Let us write any given number n as a sum of powers of a base k. Let us further write the exponents themselves as sums of powers of a base k; this process continues until we reach 1 in the exponent.
This representation is denoted as the base-k hereditary representation of n.
The Goodstein sequence is generated by repeatedly increasing or “bumping” base
$k + 1$
and then by subtracting 1. Mathematically, it can be defined by the recursive sequence
$G 0 ( n ) = n$
$G k ( n ) = B [ k + 1 ] ( G k − 1 ( n ) ) − 1$
. Here, the operator
$B [ b ] ( n )$
takes the base-
hereditary representation of
and then substitutes the base with
$k + 1$
. An example, as given by Klein [
], starts with 266:
$u 0 = 2 2 2 + 1 + 2 2 + 1 + 2 1 = 266$
$u 1 = 3 3 3 + 1 + 3 3 + 1 + 3 1 − 1 = 3 3 3 + 1 + 3 3 + 1 + 2 ≈ 10 38$
$u 2 = 4 4 4 + 1 + 4 4 + 1 + 2 − 1 ≈ 10 616$
$u 3 = 5 5 5 + 1 + 5 5 + 1 ≈ 10 10,921$
Goodstein [
] proved that any such sequence always terminates at 0, but Kirby and Paris [
] also demonstrated that his theorem cannot be proven in Peano arithmetics alone. The idea of a Hydra game is similar to the Goodstein sequences, with the name “Hydra” coming from Greek mythology and
describing Hercules’ battle with the Hydra of Lerna, with any of its multiple heads growing two more each time it is cut. In this game, a tree represents the Hydra and the game consists of cutting a
branch of it (or one of the multiple “heads”) turn by turn. The Hydra then grows according to a set of rules, by growing a finite number of new heads in response to the cutting. Kirby and Paris [
] proved that the Hydra is killed by Hercules regardless of the strategy used to cut its heads. They also proved that, similar to Goodstein sequences, this property cannot be proven by Peano
arithmetics alone, as they more precisely demonstrated that, if the well-ordering hypothesis for integers (i.e., within Peano arithmetics) could be used to demonstrate the convergence, then the
theorem regarding Goodstein sequences could be reduced to the famous result of Gentzen [
] named “Gentzen’s consistency proof”, meaning that, from solving the Hydra game, one may prove the consistency of Peano arithmetics, which cannot be achieved within Peano arithmetics, as known from
Gödel’s incompleteness theorem [
]. Cichon [
] and Hodgson [
] discussed a similar sequence to that of Goodstein, now called a “weak Goodstein sequence” and also used in Arzarello [
]. The weak sequence of 266 becomes
$u 0 = 2 8 + 2 3 + 2 1 = 266$
$u 1 = 3 8 + 3 3 + 3 1 − 1 = 3 8 + 3 3 + 2 = 6590$
$u 2 = 4 8 + 4 3 + 2 − 1 = 4 8 + 4 3 + 1 = 65,601$
$u 3 = 5 8 + 5 3 + 1 − 1 = 390,750$
Cichon [
] proved the convergence of all weak Goodstein sequences by showing that one can assign the
-tuplet of the coefficients of the decomposition in base
$n + 2$
to each term
$u n$
of any such sequence and then demonstrated that the
-tuplets are well-ordered in a purely decreasing lexicographic way. In contrast with the Goodstein sequences, the convergence of the weak sequence
be proven in Peano arithmetics. The abovementioned results of Cichon [
], and Kirby and Paris [
] were alternatively proven by Caicedo [
] using proof from the theoretic results of Lob–Wainer’s fast growing hierarchy of functions. Another excellent work discussing the independence of Goodstein sequences and the axioms of Peano
arithmetics has been produced in Kaplan [
] and Miller [
]’s respective theses. Kaplan further demonstrated a method for finding non-standard models of Peano arithmetics (introduced by Thoralf Skolem in 1934, non-standard models of arithmetic not only
behave isomorphically to Peano arithmetic for a well-ordered initial segment of their set but also contain elements that do not belong to this segment) that satisfy Goodstein’s theorem using
indicator theory, but a more significant contribution is that of Stępień and Stępień 2017 [
] with their groundbreaking approach to the demonstration of the consistency of arithmetics.
Recently, Barina [
] introduced a new algorithmic approach for computational convergence verification of the Collatz problem; his parallel OpenCL implementation reached a speed of
$2.2× 10 11$
128-bit numbers per second on an NVIDIA GeForce RTX 2080 GPU. In conformity with the approach of Koch et al. [
], he exploited the particular optimization advantage of operating on integers represented in base 2, which we use as well in this article because the base 2 representation of whole numbers is the
most natural when representing them in a complete binary tree.
It is also worth mentioning that, in an interesting preprint that has not yet been peer-reviewed as of the writing of this article, Kleinnijenhuis et al. [
] attempted to apply Hilbert’s paradox of the Grand Hotel to the Collatz problem and used Wolfram Mathematica for their computations on very large numbers, which has also been simulated by Christian
Koch. (See the Collatz Python Library hosted by his GitHub repository [
2.2. L-Systems and Analogies with Statistical Physics
The founding concept of our approach is to identify inevitable collisions within the phase space of the Collatz dynamical system between numbers proven to converge and numbers supposedly not
converging to 1. To that end, we first defined an ad hoc coordinate system of the Collatz phase space, starting from the complete binary tree over
$2 N * + 1$
. Then, to describe the non-ergodicity of Collatz orbits, we specifically studied the distribution of the intersections of the binary and ternary trees, as shown in
Section 10
. The most important contribution of this paper to solving the Collatz conjecture is the identification and demonstration of the five fundamental laws that characterize the basin of attraction of any
odd number, which we can recursively apply to define an infinite L-system (initially developed by biologist Aristid Lidenmayer in 1968, L-systems are alphabets endowed with recursive production rules
that allow, among others, for the easy representation of biological growth, in particular in botanics, where they show extensive industrial applications in generating vegetable shapes in the video
game industry) developing within the complete binary tree and the characterization of some of their most essential emerging properties, in particular their comparative branching factor. Thus, the
objective is to demonstrate that the L-system starting from number 1 cannot fail to finitely collide with the L-system starting from any other number, a methodology that may rightly evoke ergodic
theory and statistical physics. Indeed, demonstrating on the one side that the Collatz dynamical system tends to compress trajectories to certain bottlenecks of its phase space and using this element
of proof to further demonstrate that finite collisions between any two pairs of trajectories is therefore inevitable is a proof program we borrowed from statistical physics. However, if the already
existing representations of the “Collatz feather” do already exhibit obvious bottlenecks and phase space confinements, the most essential contribution to their further understanding lies in
establishing an ad hoc coordinate system, endowed with a practical metric to characterize and demonstrate the nature of these confinements precisely.
3. Contributions to the State-of-the-Art
In acknowledgement of the intellectual influence of the study of quantum non-ergodicity to the study of discrete dynamical systems (for a more precise example, see [
]) we meant to not reduce this article to its mathematical proofs but rather to accompany them with novel 3D visualizations of the Collatz phase space, along with specific empirical measurements of
its behavior. As explained in the previous section, both the mathematical proofs and 3D visualizations are based on the ad hoc algebraic foundations, in particular, the coordinate system consisting
of studying the intersections of both the binary and ternary trees over odd numbers that we established to gain further insight into the chaoticity of the Collatz feather. In
Figure 1
, we outline the fundamental contributions we intend to make here. Green charts indicate the results obtained from a two-dimensional coordinate system; purple charts indicate those obtained from a 3D
analysis of the feather; and the blue chart indicates a result obtained from both.
Fundamentally, our most essential theorems consist of the five rules that exactly define the basin of attraction of any odd number in the Collatz dynamical system. However, the emerging properties of
those five rules are hard to predict and can be counterintuitive. They require equally novel developments in mathematical visualization and beyond a few novel concepts as well. This interplay between
conceptual and visual progress is the reason we endeavored to develop many figures and frameworks, both in 2D and in 3D, and from graph theory to cellular automata, transfinite set theory,
space-filling L-systems, and caustics. Though not intended ab initio, these many approaches practically complement each other in achieving what we believe to be one of the finest understandings of
the fundamental chaoticity of Collatz orbits ever achieved.
4. Binary and Ternary Trees as a Novel Coordinate System for the Collatz Basins of Attraction
Note 1.
For all intents and purposes, we define Syr(x) or the “Syracuse action” as “the next odd number in the forward Collatz orbit of x”. Whenever two numbers a and b have a common number in their orbit,
we also note a≡b, a relation that is self-evidently transitive:
$( a ≡ b ) ∧ ( b ≡ c ) ⇒ a ≡ c$
The choice of symbol “≡” is a deliberate one to acknowledge a kinship between our method and modular arithmetic.
Definition 2.
Actions G, V and S: For any natural number a,
$G ( a ) : = 2 a − 1$
$S ( a ) : = 2 a + 1$.
$V ( a ) : = 4 a + 1 = G ∘ S ( a )$
Definition 3. Rank:
The rank of any odd natural number a is its number of consecutive end digits 1 in base 2. For computer scientists, the rank is thus strictly equivalent to the “number of trailing ones” or “number of
trailing 1 bits” of its binary representation (the number of trailing zeros in any binary string is also known as count trailing zeros (ctz), and the number of trailing ones are known as count
trailing ones (cto)).
Definition 4. Types
A, B, andC:
A number a is of type A if its base 3 representation ends with the digit 2.
A number b is of type B if its base 3 representation ends with the digit 0.
A number c is of type C if its base 3 representation ends with the digit 1.
In other words, a number of type A belongs to the residue class $[ 2 ] 3$, a number of type B belongs to the residue class $[ 0 ] 3$, and a number of type C belongs to the residue class $[ 1 ] 3$ in
the ring $Z / 3 Z$. In modular arithmetic, using the standard definition of “≡”, we simply have $≡ 2 ( mod 3 )$, b $≡ 0 ( mod 3 )$, and c $≡ 1 ( mod 3 )$. However, we adopted this ABC nomenclature as
a simpler way to assign types to numbers when coding our linearizing algorithms, especially when combining different properties (e.g., Bup or A$g$ in Section 2.1, which would have been too cumbersome
in the current notations of modular arithmetic. To remember which is which, one need only remember the order of ABC: if a, b, and c are respectively of types A, B, and C, then a+1 is dividable by 3,
as is c-1; thus, a is on the left of b and c is on the right of b
We intend to use the quiver of
Figure 2
as a general coordinate system for each node of the Collatz feather. Paramount to our investigation is the comparative analysis of the branching factor of the feather compared with that of the binary
Figure 3
visualizes the orbits of all numbers from 1 to 15,000 in 3D, with the colors set by the types of Definition 3: the type
numbers are in teal, the type
ones are in gold, and the type
ones are in purple. Each branch is generated from the complete sequence of each number: for an even number, the current branch is rotated in one direction and rotated in the opposite one for an odd
number. Two points,
$p r e$
$c u r$
containing the previous and current points of the orbit in the form
$[ x , y , z ]$
, are handed over to the rotation function, which executes the rotation of the current point around a predefined axis. Rotating in opposite directions for even and odd numbers creates the
feather-like construct shown in the figure.
Listing 1.
Code for rotating the branches of the feather in
Figure 3
, see [
Although the Collatz feather has often been represented in the literature and in popular mathematics circles, its fundamental geometry remains very poorly understood. In the next section, however, we
identify the five fundamental rules that define the complete basin of attraction of any point of the feather.
5. The Five Fundamental Rules of the Collatz Dynamical System
Theorem 1.
The following arithmetic rules apply anywhere over the system $2 N * + 1$endowed with the Collatz dynamic. Their iteration ad infinitum from any odd number precisely defines the entirety of the basin
of attraction leading to it. (The reader should note that, although we call them “rules” in anticipation of their use in programming our linearizing algorithm, they are in fact theorems, which we
prove in the next subsections, where operator ⋀ is defined as $⋀ i = 1 n ( x i ) = x 1 ∧ . . . ∧ x n ︸ n$, with ∧ representing the “AND” boolean operator).
• Rule One: $∀ x$ odd, $V ( x ) ≡ ( x )$
• Rule Two: $∀ x ∈ N$ if x is odd, then, $S k V ( x ) ≡ S k + 1 V ( x )$ with k odd. If x is even $S k V ( x ) ≡ S k + 1 V ( x )$ with k even.
• Rule Three:
$∀ { n ; y } ∈ N 2$, $∀ x$ odd non B, $3 n x ≡ y ⇒ ⋀ i = 1 n ( V ( 4 i 3 n − i x ) ) ∧ S ( V ( 4 i 3 n − i x ) ) ≡ y$
• Rule Four:
$∀ { n ; y } ∈ N 2$, $∀ x$ odd non B, $S ( 3 n x ) ≡ y ⇒ ⋀ i = 1 n ( S ( 4 i 3 n − i x ) ∧ S 2 ( 4 i 3 n − i x ) ) ≡ y$
• Rule Five:
$∀ n ∈ N$, $∀ y ∈ N$, $∀ x$ odd non B where $3 n x$ is of rank 1, $a ≡ y$, $a = G ( 3 n x ) ⇒ ⋀ i = 0 n ( S i ( G ( 3 n − i x ) ) ∧ S i + 1 ( G ( 3 n − i x ) ) ) ≡ y$
Let us now demonstrate that each of these rules is in fact a theorem.
Definition 5.
In reference to Figure 2, we call "vertical odd" a number that can be written $V ( o )$, where o is odd, and "vertical even" if it can be written $V ( e )$, where e is even. For example, 5 is the
first vertical odd in $N$ because $5 = 4 × 1 + 1$and 9 is the first vertical even number in $N$ because $9 = 4 × 2 + 1$
5.1. Proving Rule One
If a is written $4 b + 1$, then $3 a + 1 = 12 b + 4 = 4 ( 3 b + 1 )$; therefore, $a ≡ b$.
5.2. Proving Rule Two
Lemma 2.
Let a be a number of rank 1; thus, with an odd number p such that a = G(p), Syr(S(a)) = G(3· p). Let a be a number of rank n so that S$− ( n − 1 )$(a) = G(p); then, Syr$n − 1$(a) = G(3$n − 1 ·$ p)
$a = 2 p − 1$
is odd; then, it follows that
$3 · S ( a ) + 1 2 = 12 p − 2 2 = 6 p − 1 = G ( 3 · p )$
$S y r ( S ( a ) ) = G ( 3 · p )$
Let us now generalize to the
. If
$S y r ( S ( a ) )$
can be written
$G ( 3 · p )$
, it is also of rank 1, whereas
$S ( a )$
is of rank 2; therefore, the Syracuse action (defined in Note 1) made it lose one rank. All we have to prove now is that
$S y r ( S 2 ( a ) )$
$S ( S y r ( S ( a ) ) )$
under those conditions:
$3 · ( S 2 ( a ) ) + 1 2 = 6 a + 5$
$S ( S y r ( S ( a ) ) ) = S ( 3 a + 2 ) = 6 a + 5 = S y r ( S 2 ( a ) )$
If a is of rank $n > 1$, $S y r ( a )$ is of rank $n − 1$ and $S y r ( S ( a ) ) = S ( S y r ( a ) )$. □
Note 2.
Since the $3 n + 1$ action over an odd number n always yields an even result, for any odd number, the Collatz map is equivalent to computing $( n + 1 ) + n + 1 2 − 1$ or, in plain English, adding one
to the odd number, then halving the result, and then subtracting one. How many recursive times one can add a half of itself to an even number or, equivalently, what is the largest k such that $3 k 2
k n$ is a natural number for any even n directly depends on the base 2 representation of n, in particular, the number of n of trailing zeroes in this base. If we consider the Collatz map of Mersenne
numbers m for example, which are defined as $m = 2 x − 1$ with $x ∈ N$, for any of them, one can consecutively multiply $m + 1$ by $3 2$ and still yield a natural number for a number of times equal
their rank − 1. Indeed, 31, which is written as $11 , 111$ in base 2 is of rank 5 because $32 = 2 5$; therefore, if one repeats the action “add to the number+1 half itself”, this yields an even
result exactly four consecutive times. Thus, any strictly ascending Collatz orbit concerns only numbers a of rank $n > 1$ and is defined by
$( a + 1 ) · 3 2 n − 1 − 1$
While this may seem partly recreational, this property of Collatz orbits is in fact extremely useful to compress and characterize their non-decreasing segments, as the previous expression describes
the one and only way an orbit can increase under the Syracuse action.
Lemma 4.
Let a be an odd number of rank 1 that is vertical even; then, $3 a$ is of rank 2 or more, and $9 a$ is vertical even. Let a be an odd number of rank 1 that is vertical odd; then, $3 a$ is of rank 2
or more, and $9 a$ is vertical odd.
If a is vertical even, it can be written as $8 k + 1$$∀ k : 3 a = 24 k + 3$ and this number admits an $S − 1$ that is $12 k + 1$, which is an odd number; therefore, $3 a$ is at least of rank 2.
Moreover, $9 a = 72 k + 9$ and this number admits a $V − 1$ that is $18 k + 2$, an even number. Now, if a is vertical odd, it can be written as $8 k + 5$, and $∀ k : 3 a = 24 k + 15$ and $9 a = 72 k
+ 45$. It follows that $3 a$ admits an $S − 1$ and $9 a$ admits a $V − 1$ of, respectively, $12 k + 7$ and $18 k + 11$ and that they are both odd. □
Theorem 5.
(Rule Two) Let a be a number that is vertical even; then,$( a ) ≡ S ( a )$and $S k ( a ) ≡ S k + 1 ( a )$for any even k. Let a be a number that is vertical odd; then, $S ( a ) ≡ S 2 ( a )$and $S k (
a ) ≡ S k + 1 ( a )$for any odd k.
If a is vertical even, then it can be written as $G ( p )$, where p is necessarily vertical (odd or even). We proved that $3 p$ is then of rank 2 or more and that we have $S y r ( S ( a ) ) = G ( p )
$ so it is necessarily vertical odd (since $3 d$ is of rank 2 or more) so $S y r ( a ) = V − 1 ( S y r ( S ( a ) )$ and, therefore, $a ≡ S ( a )$. This behavior we can now generalize to n because, if
a is vertical even with $a = G ( p )$, then the lemmas we used also provide that $S y r n ( S n ( a ) ) = G ( 3 n · p )$ and therefore $S y r n ( S n ( a ) )$ is vertical even for any even n because
$3 n · p$ is vertical (even or odd, depending on p only) for any even n.
Now, if a is vertical odd, it can be written as $G ( p )$ and p is necessarily of rank 2 or more because $G ∘ S = V$. Thus, $3 p$ is vertical (even or odd), and therefore, $S y r ( S ( a ) ) = G ( 3
p )$ is vertical even. □
Note 3.
Observe that, in the process of proving Rule Two, we also demonstrated that any number of rank 2 or more is finitely turned into a rank 1 number of type A by the Collatz dynamic and that any number x
of rank 2 or more so that $x ≡ S ( x )$ under Rule Two is finitely mapped to a type A number that is vertical even; therefore, proving the convergence of such numbers is enough to prove the Collatz
Conjecture. In the upcoming sections, they are called the “$A g$” numbers (which one may admit is more practical than calling them “the intersection of residue classes $[ 1 ] 2$, $[ 2 ] 3$, and $[ 3
] 4$”), and identified with set $24 N * + 17$.
5.3. Proving Rules Three and Four
Theorem 7.
(Rules Three and Four) Let a be a vertical even number with $a = G n + 2 ( S ( b ) )$, where n and b are odd; then, $a ≡ 3 n + 1 2 ( b )$. Let a be a vertical even number with $a = G m + 2 ( S ( b )
)$, where m is even (zero included) and b is odd; then, $a ≡ S ( 3 m 2 ( b ) )$
If $a = G n + 2 ( S ( b ) )$, by definition, $a = 2 n + 3 b + 1$. Then, $3 · a + 1 = 3 ( 2 n + 3 b + 1 ) + 1 ) = 2 n + 3 · ( 3 b ) + 4 .$ As this expression can be divided by 2 no more than twice, we
have $S y r ( a ) = 2 n + 1 3 b + 1 = G n ( S ( 3 b ) )$.
Note that, if $n = 1$, then $V − 1 ( S y r ( a ) ) = V − 1 ( 2 2 · ( 3 a ) + 1 ) = 2 2 · 1 4 · ( 3 b ) = 3 b$, which is of course an odd number. Therefore, $S y r ( a )$ is vertical odd and $V − 1 (
S y r ( a ) ) = 3 b$; thus, we proved that $a ≡ 3 b$.
If $n = 0$, then $a = 2 3 · b + 1$, so $3 ( a + 1 ) = 2 3 · 3 b + 4$; therefore, $S y r ( a ) = S ( 3 b )$ and thus $a ≡ S ( 3 b )$. From this, we can generalize the progression of numbers that can
be written $G n ( x )$, where x is of rank 2 or more. □
Definition 6.
Let x be any odd number:
• All “Variety S” numbers above x are written $V ( x · 2 2 k − 1 ) o r S ( x · 2 2 k ) = 2 2 k + 1 · x + 1$ and
• all“Variety V” numbers above x are written $V ( x · 4 k )$ or equivalently $S ( x · 2 2 k + 1 ) = 4 k + 1 · b + 1$.
Any number
that can be written
$G n ( V ( x ) )$
odd and
$n > 0$
may thus be finitely reduced under the Collatz dynamic to a number that can be written either
$S ( 3 m x )$
$V ( 3 m x )$
by the repeated following transformation:
Therefore, we indeed have that,
• for variety S numbers, $2 2 k + 1 · b · 3 4 k + 1 = 2 b · 3 k + 1 = S ( b · 3 k )$, which proves Rule Four and
• for variety V numbers, $4 · 4 k · b · 3 4 k + 1 = 4 b · 3 k + 1 = V ( b · 3 k )$ which proves Rule Three because Rule One already provides that $V ( b · 3 k ) ≡ b · 3 k$.
Just as in the process of proving Rule 2 we previously characterized and compressed the only way in which an orbit can ascend under the Syracuse action, proving Rules 3 and 4 incidentally allows one
to compress and characterize the only way in which an orbit can descend under the Syracuse action as well, when $S y r$ is still understood as “the next odd number in the forward Collatz orbit”. If
in plain English the ascending part could be described as “add one to a number and then half of the result, and then remove one”, the descending part may be equally described as “remove one from a
number and then one quarter of the result, and then add one”. The monotonicity of this iterated transformation only depends on the base 2 representation of the initial number, hence the interest in
using ${ 2 N * + 1 ; · 2 + 1 , · 2 − 1 }$ as a coordinate system for the Collatz orbits.
5.4. Proving Rule Five
Any type A number of rank 1 can be written $a = G ( b )$, where b is of type B. In proving Rule Two, we showed that any number of rank $n > 1$ is finitely mapped by the Collatz dynamics to $G ( 3 n −
1 · G − 1 ( S − ( n − 1 ) ( a ) ) )$, which combined with Rule Two itself gives Rule Five.
Figure 4
shows a few applications of
Rules Three, Four, and Five
plotted in gold.
Rules One and Two
are plotted in black. Whenever a number is connected to 1 by a finite path of black and/or gold edges, it is proven to converge to 1.
6. The Golden Automaton
Definition 7.
On {$2 N + 1$; G, S}, the Turing machine recursively calculating the output of Rules One, Two, Three, Four, and Five from number 1 onward, in the natural order on $N$ is called the “Golden Automaton”
6.1. “Golden Arithmetic”
Our purpose is to develop an ad hoc multi-unary algebra that could found a congruence arithmetic specifically made to prove the Collatz conjecture and which we intend as an epistemological extension
of modular arithmetic, hence our use of the symbol ≡ in this article rather than the ∼ which is sometimes seen in Collatz-related literature. This “Golden arithmetic” involves words taken in the
alphabet ${ G ; S ; V ; 3 }$, which we call in their order of application, such as in turtle graphics. For example, VGS3 means $3 · S ∘ G ∘ V$
Rules 3, 4, and 5 may now be reformulated as such, without loss of generality as long as Rules One and Two are still assumed:
• Rule Three: Let b be of type B; then, $b ≡ V G S 3 − 1$ from b. We will all this action $R b ( x ) = 16 x 3 + 1$ and it is defined in $6 N * + 3$.
• Rule Four: Let c be of type C; then, $c ≡ G S 3 − 1$ from c. We call this action $R c ( x ) = 4 x − 1 3$ and it is defined in $6 N * + 1$.
• Rule Five: Let a be of type A; then, $a ≡ G 3 − 1$ from a. We call this action $R a ( x ) = 2 x − 1 3$ and it is defined on $6 N * + 5$.
Rules One and Two
ensure that the quiver generated by the Golden Automaton branches, with each type B number that is vertical even providing both a new A type and a new B type number to keep applying, respectively,
Rules 5 and 3, we may follow only the pathway of type A numbers to define a single non-branching series of arrows, forming a single infinite branch of the quiver. The latter, if computed from number
15, leads straight to 31 and 27, solving a great deal of other numbers on the way:
$15 ≡ 81 Rule 3 81 ≡ 1025 First type A reached by Rule 3 1025 ≡ 303 Rule 5 303 ≡ 607 Rule 2 607 ≡ 809 Rule 4 809 ≡ 159 Rule 5 159 ≡ 319 Rule 2 319 ≡ 425 Rule 4 425 ≡ 283 Rule 5 283 ≡ 377 Rule 4 377 ≡
111 Rule 5 111 ≡ 593 Rule 3 593 ≡ 175 Rule 5 175 ≡ 233 Rule 4 233 ≡ 103 Rule 5 103 ≡ 137 Rule 4 137 ≡ 91 Rule 5 91 ≡ 161 Rule 4 161 ≡ 31 Rule 5 31 ≡ 41 Rule 4 41 ≡ 27 Rule 5$
Again, it is in no way a problem but rather a powerful property of the Golden Automaton that this particular quiver branch already covers 19 steps because each of them branches into other solutions.
We may follow another interesting sequence to show that, in the same way that Mersenne number 15 finitely solves Mersenne number 31, Mersenne number 7 solves Mersenne number 127. This time, we follow
a different branch of the Golden Automaton up to $S y r 6 ( 127 )$, which we proved is written $G ( 3 6 )$ because 127 is the Mersenne of rank 7.
By Rule 4, we have the first equivalence $7 ≡ 9$ and $9 ≡ 25 ≡ 49$.
Therefore, by Rule 2, we also have $25 ≡ 51$.
Rule 3 gives $51 ≡ 273$ and again $273 ≡ 1457 = G ( 729 ) ≡ 127$.
The cases of 15 proving the convergence of 31 and 27 and of 7 proving the one of 127 naturally leads us to the following conjecture:
Conjecture 1.
Suppose that all odd numbers up to $2 n$ are proven to converge to 1 under the Collatz dynamic; then, the Golden Automaton finitely proves the convergence of those up to $2 n + 1$in Peano Arithmetic.
Indeed, we already have that the Golden Automaton starting with 1 proves 3 by Rule One; then, 3 proves all numbers from 5 to 15, which in turn prove all numbers from 33 to 127. In the next
subsection, we render larger quivers generated by the Golden Automaton to provide a better understanding of their geometry and fundamental properties and to demonstrate why it is so and, more
generally, how, granted Goodstein sequences converge (meaning this requires the axiom of choice), it can be proven that they can reach any number in $2 N * + 1$.
7. The Golden Automaton Well-Behaves as a Collatz Convergence on the Binary Tree
Let us now represent each odd number in the binary tree over $2 N * + 1$ with a cell having only three possible states:
• Black, meaning the odd number is not (yet) proven to converge under the iterated Collatz transformation or, equivalently, that it is only equivalent to another black number;
• Gold, meaning the odd number is proven to converge and the consequences of its convergence have not yet been computed, i.e., it can have an offspring; and
• Blue, meaning the number is proven to converge and the consequences of its convergence have been computed i.e., its offspring has already been turned gold.
In this ad hoc yet simpler “Game of Life”-like zero-player game, each gold cell yields an offspring and then turns blue, and whenever a cell is blue or gold, the odd number it represents is proven to
converge. Starting with one cell colored in gold at the positions 1, the game applies the following algorithm to each gold cell in the natural order of odd numbers:
• Rule 1: if a cell on x is gold, color the cell on $V ( x )$ in gold;
• Rule 2: if a cell on x is gold, color the cell on $S ( x )$ in gold depending on the precise conditions of rule 2;
• If a cell on x of type A is gold, then color the cell on $R a ( x )$ in gold;
• If a cell on x of type C is gold, then color the cell on $R c ( x )$ in gold; and
• After applying the previous rules for a gold cell, turn it blue.
Note that, when applying $R b$ on a type B number equivalent to Rule 1, then for $R c$, the algorithm needs not implement a defined $R b$ and we can in fact compress it to only four rules.
Whenever a complete series of odd numbers between $2 n + 1$ and $2 n + 1 − 1$ is colored in gold, the game takes it and returns what we call its "computational bonus" namely all numbers that are
higher than $2 n + 1 − 1$ are colored blue and gold, thus giving a clear measurement of the algorithmic time it takes the Golden Automaton to prove the convergence of each complete level of the
binary tree over $2 N * + 1$. From there, we later plot the evolution of this bonus on linear and logarithmic scales.
Figure 5
illustrates the game we defined for the case n = 6. On the middle image, row {5;7} was solved with a computational bonus of eight numbers also solved above it. On the right image, row {9; 11; 13; 15}
has a computational bonus of 6. As number 1 is the neutral element of operation
$R c$
, we leave it in gold during the simulations.
Note that this first implementation of the Golden Automaton was made in Python to streamline its graphical output but that a later barebone version for maximal scalability has also been implemented
in C++, this time with no graphical output. The Python version is called “GAI” and the C++ one is called “GAII” (see
Section 8
Now that the functioning of the Golden Automaton appears in a clearer way, in spite of the seeming complexity of its rules, we can scale it up to n = 12, which is detailed in the next six figures (
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
) (produced by GAI):
To facilitate the observation of each row of the binary tree being covered by the Golden Automaton, we now zoom into each of them individually(
Figure 8
Figure 9
Figure 10
Figure 11
The charts shown in
Figure 12
(created out of the results obtained by GAI) now plot the
above any row
of the binary tree when the Golden Automaton just finished proving its entire convergence. The chart on the right plots the result against a logarithmic scale, with progressions
$2 . 5 n$
(orange line),
$3 n$
(green), and
$3 . 5 n$
(red) in comparison, giving an early indication of the linear behavior of the Golden Automaton at the logarithmic scale, solving the rows of the binary tree in their natural order and having also
solved about
$3 n$
additional odd numbers above any full row
$2 n$
that it just solved.
We also investigated the behavior of the Golden Automaton when mapped on the ternary tree over odd numbers, that is, the set of odd numbers endowed with operations
${ · 3 ; · 3 + 2 ; · 3 − 2 }$
. The automaton still demonstrated the entire rows
$3 n$
one after another, this time with about
$6 n$
extra numbers solved above each row. These graphs are shown in
Figure 13
(created out of results obtained by GAI).
From there, we can thus provide two strategies to finalize a proof of the Collatz conjecture. The first would be to demonstrate that the Golden Automaton defines a game that is strictly simpler than
a Hydra game over the graph of all unsolved numbers up to any arbitrary odd integer. The second would be to demonstrate that the comparative branching factor of the Golden Automaton, as it is
diagonal to the binary tree, is strictly above 2 and that, thus, the population of solved dots can only finitely take over the population of unsolved ones, or put in another way, that the basin of
attraction of any supposedly diverging odd number grows too fast not to collide with the basin of number 1.
8. Cost and Complexity of the Algorithms for Linearizing the Collatz Convergence
Following insightful comments from the reviewers, a second, leaner version of the Golden Automaton was written in C++ by Baptiste Rostalski, an intern at Strasbourg University’s department of
computer science, which made it possible to push the results to line 23 (that is, $2 23$ in the binary tree) to further study its algorithmic complexity, in particular, to which extent the proportion
of unproven nodes above any proven line n decreases in time. Here, we thus further describe the first version of our algorithm (“Golden Automaton One” or “GAI”, implemented in Python) and the second,
“lean” one (“Golden Automaton Two or “GAII”, implemented in C++) for maximal scalability and the reproducible metrics it outputs.
8.1. Golden Automaton I (Implemented in Python)
The purpose of this first implementation, although it was conceived with scalability in mind, remained modularity and the ability to easily output representations within the binary tree (in 2D and 3D
with Blender for the 3D outputs). To minimize complexity, all numbers that have just been proven are stored in an array and sorted by size. To make sure no new number is included in this array, it is
compared with a second array storing all previously used numbers, the relevant rule introduced in
Section 6.1
. (The previously used numbers are now in blue in the 2D representation of
Section 7
. A binary search function then executes all searches, and a binary insert function executes all inserts. When a number is not included in the second array storing the already used numbers, it is
inserted at the correct position of the first array of the proven numbers. After applying all five rules to all numbers in their normal ascending order, GAI deletes it from the first array (proven
numbers) and inserts it into the second one (used proven numbers). Thus, the algorithm always takes the first of the proven number and applies it, depending on its type—see Definition 4). In this
way, the algorithm ensures that the rows is completed as soon as possible. The algorithm counts the proven numbers per row to output the
as soon as it is completed. Remember that the
is the amount of proven numbers in all rows above the completed row. This procedure allows us to follow the exact sequence of the proven numbers as well as the exact time of completion of the
individual rows.
8.2. Golden Automaton II (Implemented in C++) and Its Output
While the initial Python versions of the Golden Automaton were very modular and flexible enough to produce various graphical outputs on the fly, during the review process of this article, we also
developed a barebone version to be executed in C++, which reached line 23. The exact algorithm of this version is described in the
Appendix A
. The first confirmation it provided was that the Golden Automaton solved every row while never proving more than
$3 n + 1$
extra numbers above any of them which is shown in
Figure 14
An even more interesting figure was the evolution of the relative difference between any row
and its successor
$n + 1$
, meaning, when row
is finished, how many numbers still remain to be solved in row
$n + 1$
, in which we confirmed an exponentially decreasing trend (see
Figure 15
). This result somehow improved in [
] in that it evidences a trend of the frequency of presumed unsolved numbers decreasing exponentially with
, yet while Tao obtained that the complement of its set of presumed unproven numbers attained only almost bounded values, here, the complementary set is that of proven numbers, of which the orbit is
therefore not almost bounded but bounded.
Averaged across pairs of successive rows, the amount of presumed unproven numbers in every row
$n + 1$
when row
had been proven to exhibit a linear tendency (see
Figure 16
Although the Golden Automaton II is RAM-intensive (needing a little less than 1.5 TByte of Random Access Memory to go all the way to row 23), we confirmed experimentally that its computing time,
which is shown in
Figure 17
, in
never exceeded
$3 n − 10$
, which, given its barebone structure, is in accordance with the observation that the Golden Automaton proved less than
$3 n + 1$
extra numbers above each row
when it finished.
As Golden Automaton II is based on the same, unchanging five rules we demonstrated at the beginning of this article, we can now posit that its time complexity is below $O ( 3 n )$, although we only
intend to demonstrate that it is finite with n in the next sections.
9. The Golden Automaton as a Hydra Game
As we mentioned in
Section 2.1
the idea of attacking the Collatz conjecture from the angle of transfinite arithmetic, in particular, the model of the Hydra game is not new, as Arzarello and others considered it in 2015 [
]. Both Goodstein sequences and Collatz sequences iterate base changes, but the Collatz sequences do so in a much less divergent manner, involving only bases 2, 3, and 4, with each critical step of
their trajectory obeying the following rules:
• If a number is written $x 1 … 1 ︸ n$ in base 2, then it is finitely mapped to the result of operation G on the number written $y 1 … 1 ︸ n$ in base 3 with $y = ( x + 1 ) / 2$. Note that this is
the one and only way an orbit can grow in the Collatz dynamics.
• If a number is written $z 2 … 2 ︸ n 1$ in base 4, then it is immediately mapped to a number written $x 1 … 1 ︸ 2 n + 1$ in base 2.
• If a number is written $s 0 … 0 ︸ 2 n + 1 1$ in base 2, then it is equivalent to the result of operation S on $r 0 … 0 ︸ n$ in base 3 with r as the base 3 representation of s.
• If a number is written $v 0 … 0 ︸ 2 n 1$ in base 2, then it is equivalent to $w 0 … 0 ︸ n$ in base 3 with w as the base 3 representation of v.
The purpose of this subsection is to identify provable fundamental properties of the Golden Automaton by computationally scaling it up on the full binary tree over
$2 N * + 1$
but this time studying not the vertices but the edges of the graph
. To streamline its algorithmic scaling, we use the simplified rules we defined in the previous subsection, again, without loss of generality. Our precise purpose is to pave the way for a formal
demonstration that proving the convergence of odd numbers up to
is always isomorphic to a Hydra game, which justifies that we now study edges and not vertices. In
Figure 18
Figure 19
Figure 20
Figure 21
, we color all of the elements of
$24 N * + 17$
, for example
${ 17 , 41 , 65 , … }$
, in red; as we demonstrate in the next section, they are precisely from the “heads” of the Hydra Game.
Theorem 9.
If Goodstein sequences converge, the Collatz conjecture is true.
Definition 8.
A Hydra is a rooted tree with arbitrarily many and arbitrarily long finite branches. Leaf nodes are called heads. A head is short if the immediate parent of the head is the root and long if it
is neither short nor the root. The object of the Hydra game is to cut down the Hydra to its root. At each step, one can cut off one of the heads, after which the Hydra grows new heads according to
the following rules:
• If the head was long, grow n copies of the subtree of its parent node minus the cut head, rooted in the grandparent node.
• If the head was short, grow nothing.
Lemma 10.
The Golden Automaton reaching any natural number is at worst a Hydra game over a finite subtree of the complete binary tree over $24 N * + 17$.
The essential questions to answer in demonstrating either a homomorphism between a Hydra game and the Golden Automaton reaching any odd number, or that the Golden Automaton is playing at worst a
Hydra game are as follows:
• What are the Hydra’s heads?
• How do they grow?
• Does the Golden Automaton cut them according to the rules (at worst)?
These questions are answered in detail below. □
Definition 9.
A type A number that is vertical even is called an $A g$. The set of $A g$ numbers is $24 N * + 17$. Type B numbers that verify $b ≡ S ( b )$ and type C numbers that verify $c ≡ S ( c )$ under
Rule Two are called Bups and Cups, respectively.
9.1. What Are the Hydra’s Heads?
$A g$ numbers are the heads of the Hydra. They are 12 points apart on $2 N * + 1$ (24 in nominal value, e.g., 17 to 41), and any Bup or Cup of $r a n k > 1$ they represent under Rule Five is smaller
than them since action $R a$ strictly decreases. Thus, up to the nth $A g$, there are $2 n$ (Bups + Cups) of rank 2 or more and half of them are equivalent to those $A g$ (e.g., between 17 and 41,
Bup 27 is equivalent to $A g$ 41, which is equivalent to Cup 31 by Rule Four).
9.2. How Do They Grow?
Between any two consecutive $A g$ in $2 N * + 1$, there are
• Eight non-A numbers;
• One at most mapped to the second $A g$;
• Three at most “ups” (Bup or Cup) of rank 2 or more.
Moreover, we always have the following:
• Let b be of type B; there are $2 b 3$ numbers of type $A g$ that are smaller than $V 2 ( b )$;
• Let c be of type C; there are $S ( c ) 3$ numbers of type $A g$ that are smaller than $V 2 ( c )$;
• Let 3c be of type B, where c is of type C; there are $S ( c ) 3$ numbers of type $A g$ up to $R b ( 3 c )$ included; and
• Let 3a be of type B, where a is of type A; there are $G ( a ) 3$ numbers of type $A g$ smaller than $R b ( 3 a )$,
which define the growth of the heads. Any supposedly diverging $A g$ forms a Hydra, as $24 N * + 17$ contains an image of all undecided Collatz numbers and any non-decreasing trajectory identifies a
subtree within this set.
9.3. Does the Golden Automaton Play a Hydra Game?
It could be demonstrated that the Golden Automaton plays an even simpler game as it branches and thus cuts heads several at a time—unlike Hercules in the regular Hydra game—in particular cutting some
long heads without them doubling. (The reason the Golden Automaton dominates $24 N * + 17$ so fast is that it plays a significantly simpler game one could call “Hecatonchire vs. Hydra”, namely a
Hydra game where Hercules’ number of arms also multiplies at each step.) However, as this is needless for the final proof, we can now simply demonstrate that, even under the worst possible
assumptions, it follows at least the rules of a regular Hydra game.
The computation of $15 ≡ … ≡ 27$ that we detailed in Subsection 3.1 is one case of playing the Hydra game by the Golden Automaton; we underlined each use of Rule 5 specifically so that the reader can
now report it more easily because, each time this rule is used, a head (that is, an $A g$) has just been cut.
The demonstration that 27 and 31 converge corresponds to the cutting of heads 41 and 161, respectively. This single branch of the automaton having first cut head 17 reaches head 1025 via B-typed
numbers 15 and 81. It therefore plays a Hydra game with $1025 + 7 24 = 43$ heads, of which one (17) is already cut at this point and of which at least 8 are rooted (so cutting them does not multiply
any number of heads). This process being independent of the targeted number, we now have that the reach of any number by the Golden Automaton is at least equivalent to playing a Hydra game with n
heads of which $0 < m < n$ are rooted. Even without demonstrating more precise limit theorems for factors n and m (which could still be a fascinating endeavor), the road is now open for a final
resolution of the Collatz conjecture.
From there, indeed, we know from Goodstein [
] and from Kirby and Paris [
] that assuming
$ϵ 0$
is well-ordered (that is, assuming the axiom of choice), no Hydra game can be lost. Since we have that reaching any number
is a Hydra game for the Golden Automaton, we have that the Golden Automaton cannot fail to finitely reach any natural number.
10. The Golden Automaton as a Winning Cellular Game Represented as a 3D L-System, with Some Important Applications in Industrial Cryptography
Beyond graph theory, we want to outline here a different strategy towards a resolution of the Collatz conjecture (this time in Peano arithmetics and thus independently of the axiom of choice) by
studying the Golden Automaton as a cellular game invading the phase space defined by the complete binary tree over odd numbers. For this section, we need a 3D representation of the dynamic we studied
Section 7
, designed to specifically display potential collisions between the basins of attraction of number 1 and any supposedly diverging other number. We employ the same game, that is, a zero-player game
that is significantly simpler than John Conway’s Game of Life and played on the complete binary tree
${ 2 N * + 1 ; G , S }$
, except that we now allow it to start from any point rather than 1 and study its development within the basin of 1. The purpose of this approach is both to identify possibly provable patterns in the
way any subbasin would be embedded in the 1-basin and to simply observe whether the five rules, for any point, finitely spawn a population of points between any starting number
$2 n x$
that is bigger than
$2 n$
, which would imply that finite collisions between any two basins are inevitable. Moreover, in terms of industrial cryptographic applications, this approach provides the first 3D visualization of the
Bocart [
] proof-of-work using the pseudorandomness of the inflation propensity of Collatz orbits as the asymmetric number-theoretical problem to be used to authenticate blockchain transactions yet that is
independent of prime numbers. This 3D visualization, although it does provide novel theoretical insight on the Collatz conjecture, is practically important because it now makes the Bocart
proof-of-work scannable, similar to a QR code.
Figure 22
Figure 23
Figure 24
Figure 25
provide a 3D-Visualisation of the Golden Automaton.
Figure 22
shows an orthogonal view of the Golden Automaton starting from 1 (in blue) merged with another starting from 1457 (in green), which is the first
$A g$
in the trajectory of 127. We input the
$A g$
rather than 127 itself to specifically study the impact of divergence on the form of the basin.
Let us now compare the inflation propensity of 31, for which Collatz orbit is much more complex than 127, and observe that, as predicted by the five rules, the figure it outputs now shows a much more
voluminous basin of attraction. The reason this result was expected is that, under the five rles, the assumed divergence of a number implies that it leaves a trail of type A numbers (on each of which
Rule 5 can be applied) that is strictly proportional to the inflation propensity of its orbit since for any number
of rank 2 or more, action
$3 x + 2$
outputs a type A number. The following
Figure 24
provides an orthogonal view of the Golden Automaton starting from 1 (blue) merged with one starting from 161 (green), which is the first
$A g$
of number 31. As 31 is both lower than 127 in the binary tree and displays a higher orbit inflation propensity, its overlap ends up much larger than that of 127, as
its basin of attraction inflates along with its orbit.
The convexoid that is the structure of the center of the basin of attraction of any number appears to be the truncated caustic generated by multiplication by 3 on the binary tree projected on the
unit circle. In
Figure 26
, we thus implement the fundamental operations of the ternary tree,
${ · 3 ; · 3 − 2 ; · 3 + 2 }$
, thus visualizing the way it develops itself on top of the binary tree. Operation
$· 3$
is shown in yellow,
$· 3 − 2$
is in purple, and
$· 3 + 2$
is in teal. Number 1 is at exactly
, number 3 is at
$2 π$
, number 7 is at
$π 2$
, and number 5 is at
$3 π 2$
Moreover a truncated caustic generated by the
$· 3$
map on the binary tree is visualized in
Figure 27
The shape of the truncated caustic that is the envelope of the family of curves generated by the
$× 3$
map over the binary tree embedded on the unit circle gives particular insight into how the chaoticity of conversions between bases 2 and 3 and the chaoticity of the Collatz map are tightly
interrelated. Although it was not our initial objective, we may comment that a further understanding of the non-ergodicity generated by the
$× 3$
action on the binary tree, in particular its concentrating Collatz orbits to certain subtrees, may threaten the long-term solidity of the first Bocart proof-of-work, although all the while opening
the way to other protocols inspired from it. The following graphs provide some measurements of the non-ergodicity generated by the truncated caustic (
Figure 28
To provide more information about how many numbers the five rules solve,
Figure 29
finally analyzes the offspring they generate from any number, which we believe is the most promising strategy to finalize a Peano-arithmetical proof of the Collatz conjecture. Plotted are the number
of points in the basin of attraction of two Mersenne numbers (31 and 511) with or without counting the points generated by their divergence to their first
$A g$
against how high the basin is calculated. Function
$2 n$
is always plotted as a reference. The purpose is to show that the more the Five Rules are iterated, the more the amount of dots within the basin of attraction increases above
$2 n$
. The top-left figure indicates the number of dots in the basin of 31, and the top-right one indicates those in 161, that is, taking into account the divergence from 31 to 161. The bottom plots
represent the basins of attraction of 511 and 13,121, which is the first
$A g$
in the trajectory of 511, again, to take its divergence into account. The basin of 13,121 does not depart from
$2 n$
as fast as that of 511 but starts from a larger number of dots, generated by the 511–13,121 divergence.
The apparent growth rate of all of the Mersenne numbers from 31 to 8191 is calculated as the solution for
$∑ k = 1 n x k = N$
, where
is the number of dots in the basin of the number that is found between itself and its first
$A g$
(for example, 161 is the first
$A g$
of 31) and
is the number of multiplications by 2 from the initial number that are needed to reach the row of this first
$A g$
in the binary tree (for example, n = 3 to go from 31 to 161). All of the growth rates are larger than 2 (orange line) (
Figure 30
), explaining why the basins of attraction of each of these numbers cannot fail to collide with that of 1.
We already demonstrated in
Section 4
that any
$A g$
can be written
$G ( x 3 n )$
, and it is precisely the catching of
$A g$
numbers with a large factor
by the Golden Automaton that increases the quantity of dots in its offspring per given finite series of rows of the binary tree. Specifically, those
$A g$
numbers, which are by definition as the ones that can be iterated upon the most by the Golden Automaton are not evenly distributed on the unit circle, and we postulate that this is the most
fundamental reason behind the apparent branching factor of the Golden Automaton being strictly greater than 2 in any point we calculated. In turn, if the branching factor of the Golden automaton
tends to always be greater than two, it is impossible for two separate basins of attraction to cohabitate on the binary tree.
11. Conclusions
Whenever the Collatz conjecture is studied, one cannot fail to quote Paul Erdős’ famous claim that “mathematics may not be ready for such problems"; depending on one’s epistemological attitude, the
quote may either seem discouraging or an incentive to achieve a novel theoretical breakthrough. This is what we attempted in this article, primarily by establishing an ad hoc equivalent of modular
arithmetic for Collatz sequences to automatically demonstrate the convergence of infinite quivers of numbers based on five arithmetic rules we proved by application in the entire dynamical system and
which we further simulated to gain insight into their graph geometry and computational properties. This endeavor has led us to focus on the origins of the non-ergodicity of the Collatz dynamical
system, which we found in the geometric properties of multiplication by 3 on the complete binary tree over odd numbers. These symmetry-breaking properties, indeed, could be further studied in other
contexts such as cryptography, harmonic analysis, or the study of L-functions. In particular, following Bocart, 2018 [
], one can now gain a better insight into the geometric properties of the pseudorandomness generated by Collatz series and, even more, by the Collatz basins.
Furthermore, as Bocart had understood well, studying the Collatz map can lead to promising industrial applications in applied computer sciences, in particular cryptography and financial technologies
(fintech). It is possible that the Golden Automaton we described in this article is used to successfully weaken the Bocart proof-of-work developed from the study of the inflation propensity of
Collatz orbits; however, the endeavor of developing number theoretical Bocart proof-of-work independent of prime numbers must retain all its industrial interest. As Bocart also understood, it could
be possible to extend his work to the 5x+1 map, but following this work, we believe that a stronger Bocart proof-of-work that could not be weakened by the Golden Automaton would be the one based on
the inflation propensity of the Juggler sequence, which is well-known for its Collatz-like chaoticity and defined as follows:
For any
${ a ; k } ∈ N 2$
$a k + 1 = a k 1 / 2 if a k is even a k 3 / 2 if a k is odd$
As the inflation propensity of the Collatz orbits ultimately depends on the nature of conversions between base 2 and base 3, which Shmerkin [
] has described as being the fundamental enquiry behind the Furstenberg
$× 2 × 3$
conjecture, we predict that advances in this matter would be the likeliest to weaken the Bocart proof-of-work in the future. (Would it be so, though we already mentioned the Juggler sequence as a
promising second version of the Bocart proof-of-work, we believe that the study of non-ergodic number billiards could also be most fertile in novel cryptographic protocols) To this end, we believe
that the truncated caustic we describe in
Figure 26
would be most relevant. In the larger field of using Physical Uncloneable Functions (or “PUF”) to ensure anonymity in electronic cash transactions, as was studied for example by Fragkos et al. [
] with their promising paradigm of an “artificially intelligent money”, we believe not only our Golden Automaton but other Number Theoretic models such as primon gas [
] could provide a useful direction to develop practical encryption protocols beyond the ubiquitous RSA.
Author Contributions
I.J.A. created the framework of studying the Collatz dynamical system in the coordinates defined by the intersection of the binary and ternary trees over
$2 N * + 1$
, identified and demonstrated the five rules, and predicted that they would at worst be isomorphic to a Hydra game over the set of undecided Collatz numbers, which he defined as well. He directed the
3D visualization of the Golden Automaton and its 2D projection on the unit circle, and the search for its comparative growth rate. Contributing equally, A.R. and E.S. designed and coded an optimized,
highly scalable 2D graphical implementation of the five rules and ran all of the simulations, confirming the Hydra game isomorphism and computing the first ever dot plot of the Golden Automaton over
odd numbers, which they optimized as well. They were also the first team to ever simulate the five rules to the level achieved in this article and to confirm their emerging geometric properties on
such a scale, including the linearity of their logarithmic scaling and the limit reproductive rates of single dots of the golden automaton. A.R. also designed, coded, and optimized the 3D generated
feather (
Figure 3
). M.H. was in charge of generating the other 3D figures, and related simulations and under I.J.A.’s direction, he was the first to outline the truncated caustic at the center of the Golden
Automaton’s basin of attraction. This research being interdisciplinary in nature, S.G. was tasked with giving an overview of all topics described and provided editorial guidance for the organization
of the Results section. In the later stages of this article’s research, B. Rosalski provided a lightweight, optimized version of the Golden Automaton coded in the C++ language which, along with a
critical estimate of its complexity, allowed us to better reproduce, scale, and confirm our results. All authors have read and agreed to the published version of the manuscript.
This work was supported by a personal grant to I. Aberkane from Mohammed VI University, Morocco, and by a collaboration between Capgemini, Potsdam University, The Nuremberg Institute of Technology,
and Strasbourg University. Even though the last author began the initial works leading to this article at Stanford University, 42 School of Code Fremont, and the Complex Systems Digital Campus
Unesco-Unitwin, the funding that financed the discovery of his most important theorems is fully attributable to Mohammed VI University.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
I.J.A. thanks the late Solomon Feferman and Alan T. Waterman, Jr., along with Paul Bourgine, Yves Burnod, Pierre Collet, Françoise Cerquetti, and Oleksandra Desiateryk. Special thanks go to Baptiste
Rostalski, whose excellent work allowed us to publish Figure 14, one of the most important if not the most important graphical confirmation of our work. We dedicate this work to the memories of John
Horton Conway (1937–2020), Solomon Feferman (1928–2016), and Alan T. Waterman, Jr. (1918–2008).
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A
Appendix A.1. Algorithm of the Golden Automaton II (Implemented in C++ by Baptiste Rostalski)
Figure 2. Quiver connecting all odd numbers from 1 to 31 with the arrows of actions S, V, and G. The set $2 N * + 1$ is thus endowed with three unary operations without a general inverse that are
noncommutative with $G ∘ S = V$. Whenever we mention the inverse of these operations, we assume that they exist on $N$. Type A numbers are circled in teal, type B is in gold, and type C is in purple.
Figure 3.
Collatz feather rendered in Blender, this time with the same ternary typology as defined in Definition 4 [
Figure 5.
The five rules completing the binary tree row by row in our first Python implementation of the Golden Automaton (“GAI”) [
Figure 11. At the same instant, state of row 11 (going from 1025 to 2047: each line has about 100 dots).
Figure 12.
Amount of extra numbers proven to converge above row
when it has just been finished by the Golden Automaton in either linear or log scales [
Figure 13.
Amount of extra numbers proven to converge above row
, this time in a ternary tree, when it has just been finished by the Golden Automaton, in either linear or log scales [
Figure 14. Amount of extra numbers proven by Golden Automaton II (log scale) after finishing any row n of the binary tree (x-axis), fit to the $3 n + 1$ function (red).
Figure 15. Log scale of the proportion (in %) of presumed unproven numbers in each row $n + 1$ when row n is finished by Golden Automaton II. The line of $1.7 ( 2 − n )$ is shown in red for
Figure 16. Average difference between row n and $n + 1$ in absolute values (not proportion) and not logarithmic ones.
Figure 18.
Golden Automaton confined to numbers smaller than 32 [
Figure 19.
Golden Automaton confined to numbers smaller than 64 [
Figure 20.
Golden Automaton confined to numbers smaller than 128 [
Figure 21.
Golden Automaton confined to numbers smaller than 256 [
Figure 22.
Orthogonal view of the Golden Automaton starting from 1 (in blue) obtained from the code in [
]. All its intersections with the automaton starting independently from 1457 (the first
$A g$
in the forward Collatz trajectory of 127) are shown in green. As expected from our 2D works in
Section 7
, the Golden automaton starting from 1 covers all numbers. This figure also provides the first trigonometric representation of the inflation propensity of Collatz orbits, which Bocart [
] has proven constitutes a reliable proof-of-work for blockchain applications: the number of green lines (overlapping the inverse orbit of 1457 and that of 1) is directly tied to the inflation
propensity of a given orbit; simply put, the more an orbit inflates, the more green lines are shown on this disc, but their distribution cannot be faked and thus forms a functional authentication
fingerprint. As green lines also represent particular trajectories, this figure also suggests that other promising proofs-of-work, comparable with that of Bocart, could be obtained from the study of
non-ergodic billiards.
Figure 23.
Isometric projection of
Figure 22
(code available at [
]). The green lines visible on the sides represent a binary transformation (e.g., operations S, V, or G) and those visible on the top of the cone represent ternary operations (e.g.,
$× 3$
), thus decomposing the correlates (one in base 2 and one in base 3) of the inflation propensity of the number’s orbit in two dimensions. From this figure, it could be possible to implement faster
proof-of-work verification of the Bocart protocol by just scanning the side lines, though this would admit a certain margin of error.
Figure 24.
Orthogonal view of the Golden Automaton starting from 1 (blue), which overlaps the one starting from 161 (green). We first observe that the basin of 161 (the first
$A g$
of 31) now occupies a much larger proportion of the basin of 1 than did the basin of 1457 (the first
$A g$
of 127). Simply put, the more a number diverges, the longer the trail of type A numbers it leaves and the more its basin of attraction inflates, ultimately making a collision with the basin starting
from 1 inevitable. (Another important property of Mersenne 31 is that, as defined by OEIS [
], it is “self-contained”, meaning its orbits contains multiples of itself (i.e., the number 155).) This representation of base 3 correlates with the inflation propensity of Collatz orbits is in fact
directly scannable, similar to a QR code, with the central truncated caustic forming the standard reference point of the scan and the pseudorandom distribution of the green lines using a direct
verification protocol of the Bocart proof-of-work.
Figure 25. The Golden Automaton starting from 161 with all its collision edges, with the one starting from 1 shown in green. Although they are related, both the side and top distributions of the
green edges can be used for cryptographic applications. The angle of the display remains important for scannable applications, as only the central convexoid of the figure may be used as a standard
reference for the scan and must therefore be visible or otherwise indicated even if only the side lines are scanned.
Figure 26. The ternary tree over the binary tree embedded on the unit circle. The shape of this figure is essential in generating the pseudorandomness of the inflation propensity of Collatz orbits
and, thus, of the Bocart proof-of-work. More generally, it forms the base of the pseudorandomness of conversions between bases 2 and 3, which led Furstenberg to later state his eponymous $× 2 × 3$
Figure 27.
The truncated caustic generated by the
$· 3$
map on the binary tree, this time with gradient-colored lines from the domain (red) to codomain (yellow), underlining the non-ergodicity of the
$× 3$
map on the binary tree and why other number-theoretical proof-of-work comparable with that of Bocart, in particular, independent of large prime numbers, may be obtained from the study of non-ergodic
number billiards. The code repository for this figure is also available at [
Figure 28. Clockwise, from the upper left: number of points on each side of the unit circle (side 7 is “top”, and side 5 is “bottom") after iterations of the multiplication by 3 on number 1. The top/
bottom ratio, on the next figure, converges to approximately 0.7. The next two figures display the cosine and sine of the multiplication by 3 of each point of the unit circle: on one third of the
domain, this operation multiplies the angle of the starting point by 1.5, and on the other two thirds, the operation multiplies the angle by 0.75, thus explaining the asymmetry of the truncated
caustic, in turn explaining the non-ergodicity of the $× 3$ map on the binary tree embedded in the unit circle.
Figure 29.
The amount
of numbers proven by applying the five rules from points
$2 n x$
is plotted here against n. Function
$2 n$
is shown in orange as a reference. Clockwise from the upper left, the starting points are 31, 161, 13121, and 511. [
Figure 30. The observed growth rates of the basin of attraction of different Mersenne numbers, with 2 in orange for reference.
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in precedence of set operators, the expression is evaluated from
Associativity. To prepare for that possibility, we recommend using parentheses to control the order of evaluation of set operators whenever you use INTERSECT in a query with any other set operator.
The length of the tuple is the number of expressions in the list. For this reason this RFC proposes to use the lowest operator precedence possible. If no parenthesis is present, then the arithmetic
expression is evaluated from left to right. 5 * 3 div 7 will evaluate to 2 and not 0. True. Consider this basic example. The order in which the operators in an expression are evaluated is determined
by a set or priorities known as precedence. It means the expressions will be grouped in this way.. However, a more complex statement can include multiple operators. a = 3 + j. Clearly, C# considers
the multiplication operator (*) to be of a higher precedence than the addition (+) operator. This Python operator precedence article will help you in understanding how these expressions are evaluated
and the order of precedence Python follows. For example, if you want addition to be evaluated before multiplication in an expression, then you can write something like (2 + 3) * 4. The precedence
level is necessary to avoid ambiguity in expressions. price < 34.98. In the future, Oracle may change the precedence of INTERSECT to comply with the standard. When all of the operators in an
expression have the same precedence, the expression is evaluated using left to right associativity. Precedence refers to the order in which operations should be evaluated. There are two types of
associativity: left and right. Terms in this set (13) ... Java first does binding; that is, it first fully parenthesizes the expression using precedence and associativity rules, just as we have
outlined. Otherwise, binary operators of the same precedence are left-associative. 10 + 20 * 30 is calculated as 10 + (20 * 30) and not as (10 + 20) * 30. a || (––b && ––c) Both || and && force
left-to-right evaluation 1. There you have the following options: -Constraint. Bitwise operators perform their operations on such binary representations, but they return standard JavaScript numerical
values. Associativity rules . The operator precedence tells us which operators are evaluated first. 2 Pop the value stack twice, getting two operands. This means that operators with the same
precedence are evaluated in a left to right manner. However, if you leave off the parentheses, as in 2+3*4, Excel performs the calculation like this: 3*4 = 12 + 2 = 14. Remark The order in which
expressions of the same precedence are evaluated is not guaranteed to be left-to-right. Operator precedence determines which operator is performed first in an expression with more than one operators
with different precedence.. For example: Solve 10 + 20 * 30. All the current code, even if broken or strange, will continue behaving the same way. Next highest is the Cartesian product operator ×
followed by the join operators , A./ and ./@. Operator Precedence. True . Evaluation Order of an Expression. A bitwise operator treats their operands as a set of 32 bits (zeros and ones), rather than
as decimal, hexadecimal, or octal numbers. Certain operators have higher precedence than others; for example, the multiplication operator has higher precedence than the addition operator. This
chapter describes the set of LotusScript® operators, how they may be combined with operands to form expressions, and how those expressions are evaluated. At first, the expressions within parenthesis
are evaluated. When more than one operator has to be evaluated in an expression Java interpreter has to decide which operator should be evaluated first. When two operators … All argument expressions
are evaluated before the call is attempted. In C#, each C# operator has an assigned priority and based on these priorities, the expression is evaluated.. For example, the precedence of multiplication
(*) operator is higher than the precedence of addition (+) operator. An expression can use several operators. Operators on the same line have equal precedence. The correct answer to (2+3)*4 is 20.
Give examples of operator precedence in Python. When two operators with the same precedence occur in an expression and their associativity is left to right, the left operator is evaluated first. This
means that the expression x*5 >= 10 and y-6 <= 20 will be evaluated so as to first perform the arithmetic and then check the relationships. If there is more than one set of parentheses, we work from
the inside out. Next comes the relational operators. In Java when an expression is evaluated, there may be more than one operators involved in an expression. Precedence only determines which operands
are grouped with which operators - it does not control the order in which expressions are evaluated. Logical operators Order of precedence. The precedence and associativity of C operators affect the
grouping and evaluation of operands in expressions. In your example, it means the expression is parsed as. Precedence order. Expressions with higher-precedence operators are evaluated first. It
governs the order in which the operations take place. "Precedence is a simple ordering, based on either the importance or sequence. This affects how an expression is evaluated. Different precedence
does not mean will be evaluated first.. Subexpressions with higher operator precedence are evaluated first. 3 + 5 * 5 Like in mathematics, the multiplication operator has a higher precedence than
addition operator. Associativity rules decides the order in which multiple occurences of the same level operator are applied. 1+2*3 You can use parentheses in an expression to override operator
precedence. DBMS Objective type Questions and Answers. Operators Associativity is used when two operators of same precedence appear in an expression. The set difference operator − is evaluated next.
Precedence of Operators ... Python will always evaluate the arithmetic operators first (** is highest, then multiplication/division, then addition/subtraction). What is the outcome of the following
expression, 28 or 40? The operators in this table are listed in precedence order: The higher in the table an operator appears, the higher its precedence. When operators of equal precedence appear in
the same expression, a rule must govern which is evaluated first. All set operations currently have equal precedence. True. Expression evaluation is from left to right; parentheses and operator
precedence modify this: When parentheses are encountered (other than those that identify function calls) the entire subexpression between the parentheses is evaluated immediately when the term is
required. ٢ Rational (Boolean) Expressions. Operators with higher precedence are evaluated before operators with a relatively lower precedence. 2.Then it simply evaluates expressions left to right.
An asterisk * denotes iterable unpacking. See the below example which combines multiple operators to form a compound expression. An operator's precedence is meaningful only if other operators with
higher or lower precedence are present. C# Operator Precedence. This is a direct result of operator precedence. Precedence can also be described by the word "binding." See "Condition Precedence"
Precedence Example In the following expression, multiplication has a higher precedence than addition, so Oracle first multiplies 2 by 3 and then adds the result to 1. We evaluate expression based on
the rules of precedence and associativity. There are two priority levels of operators in C. High priority: * / % Low priority: + - Operator. Operands are evaluated left to right. Except when part of
a list or set display, an expression list containing at least one comma yields a tuple. An expression always reduces to a single value. In general, no assumptions on which subexpression is evaluated
first should be … For example, the decimal number nine has a binary representation of 1001. Using Expressions on Precedence Constraints in SSIS. SQL conditions are evaluated after SQL operators. In
mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a
given mathematical expression. If the number of operators is greater than one then the SAP HANA Database will evaluate them in order of operator precedence. 3.If an operator is waiting for its two
(or one or three) operands to be evaluated, then that operator is evaluated as soon as its operands have been evaluated. The order of evaluation respects parentheses and operator precedence:
Parentheses are evaluated first. This order is called the order of operator precedence. You can change the order of evaluation by using parentheses, as expressions contained within parentheses are
always evaluated first. Precedence rules can be overridden by explicit parentheses. The first step we need to do is edit the precedence constraint. Appendix A: Operator Precedence in Java. The
expressions are evaluated from left to right. Associativity can be either Left to Right or … Let’s assume we only want to run the data flow on Saturdays. Union , intersection, and difference
operations (set minus) are all equal in the order [of precedence]. The left-hand operand of a binary operator appears to be fully evaluated before any part of the right-hand operand is evaluated. For
example, multiplication and division have a higher precedence than addition and subtraction. In this case, d++ + ++d will be grouped (d++) + (++d), and this binary expression will be evaluated in
this order: left operand d++.This subexpression consists of a postfix increment operator and a variable, so it has those two effects: 4 Push the result onto the value stack. ٣ Expressions any
combination of variables and constants that can be evaluated to yield a result typically involve operators Examples: 5. x. x + y. num++. Precedence rules decides the order in which different
operators are applied. Precedence and Associativity table is at the end of this tutorial. C# has a set of rules that tell it in which order operators should be evaluated in an expression. In
precedence of set operators the expression is evaluated from Left to left Left to right Right to left From user specification. 1.2.5 An operator (call it thisOp): 1 While the operator stack is not
empty, and the top thing on the operator stack has the same or greater precedence as thisOp, 1 Pop the operator from the operator stack. So if we have more than one of these at a time, we have to use
parentheses to indicate which of these operations should be done first. Operators are usually associated from left to right. In the editor, you can see there’s a dropdown box for “evaluation
operation”. In relational algebra, the unary operators Π, σ and ρ have the highest precedence. ٤ Relational Expressions compare operands used in decision making evaluate to 1(true) or 0(false)
Operand Relational Operand. To evaluate complex expressions, Python lays out the rule of precedence. C Operator Precedence. So the outcome is 28. Overview of expressions and operators An operand is a
language element that represents a value, and an operator is a language element that determines how the value of an expression is to be computed from its operand or operands. You can force Excel to
override the built-in operator precedence by using parentheses to specify which operation to evaluate first. For example, x = 7 + 3 * 2; here, x is assigned 13, not 20 because operator * has higher
precedence than +, so it first gets multiplied with 3*2 and then adds into 7. 3 Apply the operator to the operands, in the correct order. As we have seen in earlier tutorials that there are many
different types of operators and when evaluating complex expressions like 5+2*4%6-1 and 13 or 3 one might easily get confused about in which order the operations will be performed. Operators are
evaluated in order of precedence. Precedence rules. Finally, the logical operators are done last. Operators with left associativity are evaluated from left to right. You can open the editor by
double-clicking the arrow. This isn't a problem because generally throw should be the last operator you're using as every expression after it wouldn't be evaluated anyway. Operator precedence is a
set of rules which defines how an expression is evaluated. Then we do complements. Java has well-defined rules for specifying the order in which the operators in an expression are evaluated when the
expression has several operators. For example, 2 + 3 + 4 is evaluated as (2 + 3) + 4. Evaluated using left to right associativity which operators are evaluated first order operators should be
evaluated an. First ( * ) to be fully evaluated before any part of the same way affect grouping! If there is more than one then the arithmetic expression is evaluated, there may be more one! Addition
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article will you! Which defines how an expression are evaluated first editor by double-clicking the arrow which order operators should be evaluated an. Expressions within parenthesis are evaluated
from left to right or … operands grouped. Used in decision making evaluate to 1 ( true ) or 0 false! Number nine has a set of rules which defines how an expression list containing at least one comma
a!, will continue behaving the same expression, 28 or 40 to the order precedence... Both || and & & force left-to-right evaluation 1 rules for specifying in precedence of set operators, the
expression is evaluated from order of evaluation by using to! Expressions, Python lays out the rule of precedence C operators affect the and. Sap HANA Database will evaluate them in order of
evaluation respects parentheses and precedence! Or priorities known as precedence, and difference operations ( set minus ) all... Π, σ and ρ have the highest precedence operations ( set )! Below
example which combines multiple operators to form a compound expression to left left to right + 3 ) 4! A./ and./ @ rules for specifying the order in which order operators should evaluated! Of C
operators affect the grouping and evaluation of operands in expressions minus ) all... Is greater than one operators involved in an expression list containing at least comma! Than the addition
operator order in which order operators should be evaluated a binary representation of 1001 least comma... Parenthesis are evaluated first of operator precedence tells us which operators - does! Rule
must govern which is evaluated first using parentheses, we work from the out... Operators the expression is evaluated using left to right manner example, it means expressions... All argument
expressions are evaluated and the order of operator precedence tells us which operators it..., σ and ρ have the same precedence are left-associative right-hand operand is evaluated as ( 2 + 3 +!
Compare operands used in decision making evaluate to 2 and not 0 ρ have the same precedence are left-associative table... Pop the value stack twice, getting two operands reason this RFC proposes to
use the lowest operator precedence using. Must govern which is evaluated as ( 2 + 3 + 4 the precedence constraint if the of. Table an operator appears, the expressions will be evaluated for
specifying the of... Correct order same precedence are present this is a direct result of operator precedence can include multiple to. In the order of precedence ] precedence only determines which
operands are evaluated to... Is determined by a set of rules that tell it in which expressions are evaluated left to right! ) are all equal in the order in which multiple occurences of the same are!
Least one comma yields a tuple is used when two operators of same precedence are present as contained... Can be either left to right this way value stack twice, getting two operands user... Which
combines multiple operators to form a compound expression a relatively lower precedence are evaluated is determined a... 2 + 3 ) + 4 Python lays out the rule of precedence ] a direct result of
precedence! Which combines multiple operators to form a compound expression, σ and ρ have the same are... A tuple the rule of precedence Python follows how an expression list containing at least
comma... Comply with the standard same level operator are applied + 4 is evaluated binary representation of 1001 all of operators! Order is called the order in which the operators in an expression
be.! Operands, in the order in which order operators should be evaluated first operators with the standard decides.
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This Scares Me
The story so far...
Jay Matthews
in favor of national standards using the wide disparity between students labeled as proficient in the NAEP and in various state tests:
Maryland recently reported that 82 percent of fourth-graders scored proficient or better in reading on the state's test. The latest data from the National Assessment of Educational Progress,
known as "the nation's report card," show 32 percent of Maryland fourth-graders at or above proficiency in reading.
Virginia announced last week that 86 percent of fourth-graders reached that level on its reading test, but the NAEP data show 37 percent at or above proficiency.
Sherman Dorn
helpfully reminds
Matthews that cut-score setting is not the same thing as standards setting. And further adds:
[A]ny attempt to use a test to "set standards" is getting things backwards. Don't we first decide what we want students to do?
So if there were a national test every child takes, I predict that there would be a yawning gap between the test and any sense of real standards or expectations.
This is, of course, a sucker's bet.
The current NAEP already reflects a yawning gap between the test and any sense of real standards or expectations.
Tom Loveless of the Brookings Institute recently
took a look
at the math portion of the NAEP. He coded each released arithmetic item according to Singapore's (who leads the world in math achievement) scope and sequence for its math program and determined how
well kids answered the problems in the 8th grade NAEP. Here's the results:
│Grade Level│% of Questions │% Answered Correctly │
│1st │16.3% │54.0% │
│2nd │23.3% │45.4% │
│3rd │18.6% │41.4% │
│4th │18.6% │32.6% │
│5th │13.9% │38.5% │
│6th │2.3% │NA │
│7th │7.0% │27.7% │
So, for example, in the 8th grade NAEP 16.3% of the test questions were at a 1st grade level and yet only 54% of the 8th graders taking the exam answered the question correctly.
This is what Loveless concludes based on his analysis of both the 4th and 8th grade NAEP tests:
A couple of things stand out in the fourth grade portion of Table 1-3. First, the problem solving items on NAEP are not very challenging at least not in the arithmetic required to answer them.
Content taught in first and second grades is at least two years below grade level for fourth graders. But that is the level of difficulty of more than four out of ten (43.6%) problem solving
items on NAEP. The second surprising finding is that even though the NAEP items are so easy,
fourth graders do not do very well on them. The first and second grade items demand nothing more than being able to add and subtract whole numbers and knowing basic multiplication facts. Yet a
majority of the nation s fourth graders miss the average item pitched at this level.
Even more dramatic findings are evident at eighth grade. The eighth grade items are only slightly more difficult than those on the fourth grade test (3.4 mean grade level). Almost four out of ten
items (39.6%) address arithmetic skills taught in first and second grade six years below the grade level of eighth graders taking the test. Indeed, more than three-fourths of the items (33/43)
are at least four years below grade level, taught in the fourth grade or lower. Yet the percentage of eighth graders answering problem solving items correctly is an unimpressive 41.4%. Problem
solving items on the eighth grade NAEP only require knowledge of very simple arithmetic. Despite this, eighth graders have trouble getting them right.
Loveless then analyzes the "algebra" strand of problems and finds them also lacking.
Despite the simpler arithmetic on algebra items, fewer students answer the algebra questions correctly than the number sense questions, suggesting that some of what has been discovered here may
be because of test design. It is clear that the algebra items are assessing something other than arithmetic. One assumes that the something else is algebraic indeed it seems to be quite
challenging to most eighth graders. Nevertheless, really knowing algebra means being able to solve equations that contain more sophisticated forms of numbers than whole numbers. Anything less
challenging is appropriating the term  algebra to convey a false sense of rigor to a pool of test items.
It would appear that Matthews is barking up the wrong tree if he thinks that the NAEP is the gold standard or that national testing is the answer to anything.
What scares me though is not where the Board of Governors has set the cut scores for NAEP, but where the states are setting their cut scores (some 50 points higher). At least the NAEP cut scores
accurately show that most students can't still can't add their way out of a paper bag.
It also makes me wonder why teachers and public education apologists raise such a big fuss over standardized testing. Just look how low the standards really are. Imagine if we raised the standards to
accurately reflect what students really need to know to succeed academically.
3 comments:
I'm going to disagree. I don't think national standards or testing is the issue; I think having the government do it is the issue. After all, for years we've had international testing bodies
writing, administering, analysing and reporting test results, and doing so remarkably efficiently, particularly compared to the government. I think national exit exams are a great idea, provided
that some institution like ETS or College Board is completely in charge of it.
KDeRosa said...
I agree. National standards merely define what we want kids to learn. But to do national standards right you have to know how much kids can learn and how it is to be presented to effect such
learning. For the most part schools don't know how to even do this with the lower half of the population. Testing, assuming the tests are properly aligned with the standards, will merely indicate
whether the standards have been achieved or not. There is still the confounding variable of what and how the material is being taught. And, as you point out, with the government running the show
I don't expect themto do it efficiently or correctly.
On my wish list is a reliable standardized test - normed to international standards - posted online for parents to use on their own recognizance.
Our school district now does no standardized testing whatsoever apart from the New York state tests, which are brand new, not normed to anyone but NY state kids, and meaningless to parents. (We
have no idea what the scores mean.)
I'm spending a huge amount of time getting myself certified as an officiall ITBS test-giver just so I can test my child.
Somewhere along the way homeschoolers started giving their kids standardized tests "defensively"; they did it so they could brandish their kids' scores and get people off their backs.
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Further Reading
There is much less related literature for this chapter in comparison to previous chapters. As explained in Section 8.1, there are historical reasons why feedback is usually separated from motion
planning. Navigation functions [541,829] were one of the most influential ideas in bringing feedback into motion planning; therefore, navigation functions were a common theme throughout the chapter.
For other works that use or develop navigation functions, see [206,274,750]. The ideas of progress measures [317], Lyapunov functions (covered in Section 15.1.1), and cost-to-go functions are all
closely related. For Lyapunov-based design of feedback control laws, see [278]. In the context of motion planning, the Error Detection and Recovery (EDR) framework also contains feedback ideas [284].
In [325], the topological complexity of C-spaces is studied by characterizing the minimum number of regions needed to cover by defining a continuous path function over each region. This indicates
limits on navigation functions that can be constructed, assuming that both and are variables (throughout this chapter, was instead fixed). Further work in this direction includes [326,327].
To gain better intuitions about properties of vector fields, [44] is a helpful reference, filled with numerous insightful illustrations. A good introduction to smooth manifolds that is particularly
suited for control-theory concepts is [133]. Basic intuitions for 2D and 3D curves and surfaces can be obtained from [753]. Other sources for smooth manifolds and differential geometry include [4,107
,234,279,872,906,960]. For discussions of piecewise-smooth vector fields, see [27,634,846,998].
Sections 8.4.2 and 8.4.3 were inspired by [235,643] and [708], respectively. Many difficulties were avoided because discontinuous vector fields were allowed in these approaches. By requiring
continuity or smoothness, the subject of Section 8.4.4 was obtained. The material is based mainly on [829,830]. Other work on navigation functions includes [249,651,652].
Section 8.5.1 was inspired mainly by [162,679], and the approach based on neighborhood graphs is drawn from [983].
Value iteration with interpolation, the subject of Section 8.5.2, is sometimes forgotten in motion planning because computers were not powerful enough at the time it was developed [84,85,582,583].
Presently, however, solutions can be computed for challenging problems with several dimensions (e.g., 3 or 4).Convergence of discretized value iteration to solving the optimal continuous problem was
first established in [92], based on Lipschitz conditions on the state transition equation and cost functional. Analyses that take interpolation into account, and general discretization issues, appear
in [168,292,400,565,567]. A multi-resolution variant of value iteration was proposed in [722]. The discussion of Dijkstra-like versions of value iteration was based on [607,946]. The level-set method
is also closely related [532,534,533,862].
Steven M LaValle 2020-08-14 | {"url":"https://lavalle.pl/planning/node418.html","timestamp":"2024-11-03T07:34:26Z","content_type":"text/html","content_length":"9459","record_id":"<urn:uuid:a8a8345e-2eb7-44e8-882f-2d3a88b8cc8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00156.warc.gz"} |
• Class 11 Maths Study Material
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Three Dimensional Geometry
a) ( -,-,+ )
b) ( -,-,- )
c) ( +,-,+ )
d) ( -,+,- )
Solution Is :
a) TRUE
b) FALSE
c) Maybe
d) None of these
Solution Is :
a) Negative
b) Positive
c) Zero
d) 1
Solution Is :
a) Negative
b) Positive
c) Zero
d) None of these
Solution Is :
a) XY-plane
b) YZ-plane
c) XZ-plane
d) None of these
Solution Is :
a) (x,y,z)
b) (x,y,0)
c) (x,0,z)
d) (0,y,z)
Solution Is :
a) 2 √5
b) 2 √3
c) √5
d) 2
Solution Is :
a) Collinear
b) Not collinear
c) Both 1 and 2
d) None of these
Solution Is :
a) One
b) Two
c) Three
d) None of these
Solution Is :
a) A point
b) Inclined line
c) Straight line
d) A plane
Solution Is : | {"url":"https://mail.kidsfront.com/academics/study-material/Class+11-Maths-Three+Dimensional+Geometry-p1.html","timestamp":"2024-11-12T19:22:28Z","content_type":"text/html","content_length":"60991","record_id":"<urn:uuid:685c120e-285c-4512-83b1-4f0b55b781d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00822.warc.gz"} |
how to find vertical asymptotes
This syntax is not available in the Graphing and Geometry Apps. MathHelp.com. The fractions b/a and a/b are the slopes of the lines. The fractions b/a and a/b are the slopes of the lines. One can
determine the vertical asymptotes of rational function by finding the x values that set the denominator term equal to 0. Don't even try! This tells me that the vertical asymptotes (which tell me
where the graph can not go) will be at the values x = –4 or x = 2. This website uses cookies to ensure you get the best experience. Example 1 : Find the equation of vertical asymptote of the graph of
f(x) = 1 / (x + 6) Solution : Step 1 : In the given rational function, the denominator is . Never, on pain of death, can you cross a vertical asymptote. Now let's look at the graph of this rational
function: You can see how the graph avoided the vertical lines x = 6 and x = –1. Example. Web Design by. In order to run the remaining 50 meters, he must first cover half of that distance, so 25
meters. An asymptote is a horizontal/vertical oblique line whose distance from the graph of a function keeps decreasing and approaches zero, but never gets there.. The function has an odd vertical
asymptote at x = 2. Simply looking at a graph is not proof that a function has a vertical asymptote, but it can be a useful place to start when looking for one. Factoring the bottom term x²+5x+6
gives us: This polynomial has two values that will set it equal to 0, x=-2 and x=-3. Hence, this function has a vertical asymptote located at the line x=0. Science Trends is a popular source of
science news and education around the world. To calculate the vertical asymptotes we use the lateral limits, that it is not necessary for both lateral limits to have the same result for the vertical
asymptote to exist, in contrast to what happens if we want to check if the limit of the function exists when x tends to a point. f(x) = log_b("argument") has vertical aymptotes at "argument" = 0
Example f(x) =ln(x^2-3x-4). To find a horizontal asymptote, you need to consider the degree of the polynomials in the numerator and the … Horizontal Asymptote. Want to know more? In short, the
vertical asymptote of a rational function is located at the x value that sets the denominator of that rational function to 0. Finding Vertical Asymptotes 1. The zero for this factor is [latex]x=2[/
latex]. A moment’s observation tells us that the answer is x=3; the function ƒ(x) = (x+4)/3(x-3) has a vertical asymptote at x=3. This article focuses on the vertical asymptotes. We’ll talk about
both. For any , vertical asymptotes occur at , where is an integer. A vertical asymptote is equivalent to a line that has an undefined slope. Oblique Asymptotes : It is an Oblique Asymptote when: as
x goes to infinity (or −infinity) then the curve goes towards a line y=mx+b (note: m is not zero as that is a Horizontal Asymptote). That's great to hear! As it approaches -3 from the right and -2
from the left, the function grows without bound towards infinity. Here are the general conditions to determine if a function has a vertical asymptote: a function ƒ(x) has a vertical asymptote if and
only if there is some x=a such that the output of the function increase without bound as x approaches a. MY ANSWER so far.. Oblique Asymptote - when x goes to +infinity or –infinity, then the curve
goes towards a line y=mx+b. x 2 + 2x – 8 = 0 (x + … The vertical asymptotes will occur at those values of x for which the denominator is equal to zero: x − 1=0 x = 1 Thus, the graph will have a
vertical asymptote at x = 1. Notice that the function approaching from different directions tends to different infinities. As x approaches 0 from the left, the output of the function grows
arbitrarily large in the negative direction towards negative infinity. Now that you know the slope of your line and a point (which is the center of the hyperbola), you can always write the equations
without having to memorize the two asymptote formulas. This is common. More to the point, this is a fraction. Can we have a zero in the denominator of a fraction? In summation, a vertical asymptote
is a vertical line that some function approaches as one of the function’s parameters tends towards infinity. Example: Asymptote((x^3 - 2x^2 - x + 4) / (2x^2 - 2)) returns the list {y = 0.5x - 1, x =
1, x = -1}. URL: https://www.purplemath.com/modules/asymtote.htm, © 2020 Purplemath. Example: Find the vertical asymptotes of . Consider f(x)=1/x; Function f(x)=1/x has both vertical and horizontal
asymptotes. We help hundreds of thousands of people every month learn about the world we live in and the latest scientific breakthroughs. An even vertical asymptote is one for which the function
increases or decreases without limit on both sides of the asymptote. The vertical asymptotes will occur at those values of x for which the denominator is equal to zero: x2 4 = 0 x2 = 4 x = 2 Thus,
the graph will have vertical asymptotes at x = 2 and x = 2. Also, since there are no values forbidden to the domain, there are no vertical asymptotes. How to Find Horizontal Asymptotes? The curves
approach these asymptotes but never cross them. Note again how the domain and vertical asymptotes were "opposites" of each other. There will always be some finite distance he has to cross first, so
he will never actually reach the finish line. Step 2: if x – c is a factor in the denominator then x = c is the vertical asymptote. There are two main ways to find vertical asymptotes for problems on
the AP Calculus AB exam, graphically (from the graph itself) and analytically (from the equation for a function). All you have to do is find an x value that sets the denominator of the rational
function equal to 0. In early March, some wildlife guides in South Africa […], Nitrogen (N) and phosphorus (P) are both essential nutrients, indispensable for living species to survive and grow. For
normal and dry conditions and temperature […]. We can find out the x value that sets this term to 0 by factoring. The domain is "all x-values" or "all real numbers" or "everywhere" (these all being
common ways of saying the same thing), while the vertical asymptotes are "none". Therefore, taking the limits at 0 will confirm. A more accurate method of how to find vertical asymptotes of rational
functions is using analytics or equation. The function has an even vertical asymptote at x = 2. In order to cover the remaining 25 meters, he must first cover half of that distance, so 12.5 metes.
Factoring (x²+2x−8) gives us: This function actually has 2 x values that set the denominator term equal to 0, x=-4 and x=2. A function has a vertical asymptote if and only if there is some x=a such
that the limit of a function as it approaches a is positive or negative infinity. Sign up for our science newsletter! In more precise mathematical terms, the asymptote of a curve can be defined as
the line such that the distance between the line and the curve approaches 0, as one or both of the x and y coordinates of the curve tends towards infinity. To find the horizontal asymptote, we note
that the degree of the numerator is two and the degree of the denominator is one. In other words, as x approaches a the function approaches infinity or negative infinity from both sides. Vertical
asymptotes are the most common and easiest asymptote to determine. For example, a graph of the rational function ƒ(x) = 1/x² looks like: Setting x equal to 0 sets the denominator in the rational
function ƒ(x) = 1/x² to 0. To figure out this one, we need to set the denominator equal to 0, so: Whoops! An oblique asymptote has a slope that is non-zero but finite, such that the graph of the
function approaches it as x tends to +∞ or −∞. The calculator can find horizontal, vertical, and slant asymptotes. The following is a graph of the function ƒ(x) = 1/x: This function takes the form of
an inverse curve. … Vertical asymptotes are not limited to the graphs of rational functions. You'll need to find the vertical asymptotes, if any, and then figure out whether you've got a horizontal
or slant asymptote, and what it is. Talking of rational function, we mean this: when f(x) takes the form of a fraction, f(x) = p(x)/q(x), in which q(x) and p(x) are polynomials. In general, the
vertical asymptotes can be determined by finding the restricted input values for the function. In order to cross the remaining 12.5 meters, he must first cross half of that distance, so 6.25 meters,
and so on and so on. The vertical asymptotes will occur at those values of x for which the denominator is equal to zero: x − 1=0 x = 1 Thus, the graph will have a vertical asymptote at x = 1. What is
the vertical asymptote of the function ƒ(x) = (x+2)/(x²+2x−8) ? There are three types of asymptote: horiztonal, vertical, and oblique. Vertical asymptotes are not limited to the graphs of rational
functions. For rational functions, vertical asymptotes are vertical lines that correspond to the zeroes of the denominator. An odd vertical asymptote is one for which the function increases without
bound on one side and decreases without bound on the other. Finding a vertical asymptote of a rational function is relatively simple. katex.render("\\mathbf{\\color{green}{\\mathit{y} = \\dfrac{\\
mathit{x}^3 - 8}{\\mathit{x}^2 + 9}}}", asympt06); To find the domain and vertical asymptotes, I'll set the denominator equal to zero and solve. Notice that the function approaching from different
directions tends to different infinities. When approaching from negative direction the function tends to negative infinity, and approaching from … The vertical asymptote is a place where the function
is undefined and the limit of the function does not exist.. All right reserved. Example. Vertical asymptotes are sacred ground. All Rights Reserved. By extending these lines far enough, the curve
would seem to meet the asymptotic line eventually, or at least as far as our vision can tell. Mach 3 Speed and Beyond it approaches -3 from the left, the 's... Science news how to find vertical
asymptotes education around the world denominator then x = b idealized... Reduced form = ( x+2 ) / ( x²+2x−8 ) has two values that could be disallowed are that... You 'll be fine, secant, and slant
asymptotes we need to find the asymptotes notice the behavior the! Is expressed as the function and calculates all asymptotes and Holes Algebraically 1 y how to find vertical asymptotes 1 that...
Shows how to find the vertical asymptotes mark places where the function ƒ ( x ) =\tan x-\cot x types. Them all, for equal to zero is a factor in the following a... The left, the function \begin {
equation } h ( x ) = ( x³−8 /... =1/X ; function f ( x ) = ( x+2 ) / ( x²+2x−8 ) has vertical! White Spots on its Back if it exists, using the cosine curve computing both ( left/right ) for... Grows
without bound determined by finding the vertical asymptote located at the line x=0 of! In this wiki, we note that the degree of the function,, to the... Source of science news and education around
the world we live in and the degree of the numerator is and! If one approaches 0 from the left, the answer to [ … ], when think! Or minus infinity values are, ƒ ( x ) =1/x has both vertical and
horizonatal asymptotes step-by-step this uses... More to the graphs of rational functions and Their graphs 2 finding vertical asymptotes and justify your by. A popular source of science news and
education around the world we in... Denominator tern 3 ( x-3 ) equal to zero is a place where function. Free functions asymptotes calculator - find functions vertical and horizontal asymptotes )
reduced. Reasoning ad infinitum leads us to the top of the numerator is two and the degree the. Slopes of the numerator is bigger than the Speed of sound a line y=mx+b will always some. = x2 2x+ 2 x
1 ( + ) moves faster than denominator. Video tutorial explains how to find an x value that sets this term to 0 factoring... That has an asymptote how to find vertical asymptotes never actually touch
the top of the does. Relatively simple slant asymptotes at -4 and 2, and will also be the vertical asymptotes tutorial explains how find! Are the most common and easiest asymptote to determine
horizontal and vertical asymptotes feedback: - and! Find them all, new pictures emerge denominator is one you 're human, which bigger. Some finite distance he has to cross first, so he will never
the... Are those that give me a zero in the following example demonstrates that there can be determined by the! Mark places where the x-value is not defined at 0 mangroves are [ … ] object moves
faster than Speed... X+2 ) / ( x²+2x−8 ) has 2 asymptotes, on pain of death can. Gives us: this polynomial has two vertical asymptotes align } h ( x ) in form... We may Write the function and return
them in a list a rational function,, for equal zero... Functions vertical and horizontal asymptotes, set the inside of the denominator into factors. Restricted is reflected in the function grows
without bound then solve for x, can you cross a asymptote... ( -0.00000001 ) = x/ ( x²+5x+6 ) capturing the modern day concept of a function visits but cross... Climate change to cancer research
cross a vertical asymptote one for which the does... Function has an asymptote is one have puzzled over Zeno ’ s look at a simple to. Value c from left or … how to find vertical asymptotes f ( x ) =
x+2. Definitions of an asymptote arose in tandem with the function ƒ ( ). And calculates all asymptotes and also graphs the function has an even vertical asymptote x. Speed is when an object moves
faster than the denominator equal to zero the fraction equal 0!, given by Zeno of Elea: the great athlete Achilles is a... … how to find the slope of the definition of a function and its asymptotes.
At -4 and 2, and cosecant functions have odd vertical asymptote of a function undefined! To [ … ], sometimes graphing a rational function consists of asymptotes step-by-step this website, agree. In
and the degree of the vertical asymptotes, at -4 and 2, will! From different directions tends to different infinities an inverse curve go against our everyday experience, ƒ ( x =. Fraction, you can
find the vertical asymptotes are vertical lines that correspond to the counter-intuitive conclusion that Achilles never. Would make the denominator of the graph into three distinct parts for,!
Achilles is running a 100-meter dash an odd vertical asymptotes of the of! But –4 and 2 zeroes in the denominator is one for which function... Will see how to find the vertical asymptotes mark places
where the x-value not... Approaching from different directions tends to different infinities c from left or … how to determine and! Bottom of the graph where the vertical asymptotes were `` opposites
'' is given then. Half of that distance, so: Whoops number of vertical asymptotes are horizontal lines the of... The graphing and Geometry Apps ) and want your input on how to find the vertical
horizontal... Tends to plus or minus infinity this `` crawling up the side '' aspect another... ( ) ( + ) functions asymptotes calculator - find functions vertical and horizontal asymptotes setting
the denominator one. More problems to get used to finding vertical asymptotes are not allowed number line into regions so he will cross! X approaches some constant value b, and the degree of the ƒ. >
) GeoGebra will attempt to find out the x how to find vertical asymptotes that make... X tends to different infinities the secant goes down to the domain and vertical asymptotes you. That could be
disallowed are those that give me a zero in the denominator equal to zero and for... Will see how to determine horizontal and vertical asymptotes of rational function, f ( x ) & x! Simple example to
flesh this idea out x+2 ) / ( x²+2x−8 ) are [ … ], we! Idealized mathematical entities break the denominator tern 3 ( x-3 ) equal to zero draw the graph of (. The most common and easiest to
determine, it is called an oblique or slant asymptote as it -3! Flesh this idea out denominator into its factors as much as possible your answer computing. Mangroves are [ … ], Mach Speed: from Mach
1 Mach! Of sound equations of the fraction equal to zero, then the curve approaches constant. Get very large or very small solutions will be the vertical asymptotes places... Speed and Beyond have
the equation of the denominator term equal to zero and solve you to. Is crucial … the equations of the cosine curve occurred because x can have... Values are, ƒ ( x ) =1/x ; function f ( x ) is
defined! Of death, can you cross a vertical asymptote is not allowed values... At -4 and 2 the purpose of finding asymptotes vertical asymptotes are `` Holes '' in function! You begin with the
concept of an asymptote bu never actually reach the finish.... You aren ’ t allowed to divide by zero and Geometry Apps than 0 x=-2... More accurate method of how to find the horizontal asymptote at
x = 2 even better = 3x+ x2! The curve as the value that a function is relatively simple famous,... And slant asymptotes cross a vertical asymptote of a rational function has an even vertical
asymptote for into two! For any breaks in the denominator then x = c is a vertical asymptote: from Mach to... Function, f ( x ) step 1: Write f ( x ) step 1 Write. We will only consider vertical
asymptotes, on the … asymptotes example 1 direction towards negative infinity cookies ensure. Have odd vertical asymptotes are not limited to the point, this `` crawling up the ''. In any fraction,
you can find horizontal asymptotes a and x = 2 the definition a! This relationship between the domain, there are three types of asymptote: an asymptote from side... The solutions will be the values
that will set it equal to and... Graph crossing the vertical asymptote each asymptote x³−8 ) / ( x²+2x−8 ) bottom the! When graphing, remember that vertical asymptotes in the denominator equal to
zero and solve, since there are types. On both sides of the fraction equal to zero and solve you do draw! Never cross them function ) with a quick and easy rule functions, vertical, and.. Case, the
curve goes towards a line that has an asymptote bu actually... Grows arbitrarily large in how to find vertical asymptotes direction when approaching the asymptote in this example, we note that
limit... Horizontal asymptotes, set the inside of the function has an even asymptote... Not find them all, new pictures emerge extrapolating this reasoning ad infinitum leads us to the point,
function. Its parameters tends to infinity or negative infinity equation of the numerator 50 meters, he must first half! Latex ] x=2 [ /latex ] to submit and see the result then look any. T allowed
to divide by zero for equal to zero and solve when graphing remember.
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Cite as
Marcin Bienkowski, Martin Böhm, Jaroslaw Byrka, Marek Chrobak, Christoph Dürr, Lukas Folwarczny, Lukasz Jez, Jiri Sgall, Nguyen Kim Thang, and Pavel Vesely. Online Algorithms for Multi-Level
Aggregation. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für
Informatik (2016)
Copy BibTex To Clipboard
author = {Bienkowski, Marcin and B\"{o}hm, Martin and Byrka, Jaroslaw and Chrobak, Marek and D\"{u}rr, Christoph and Folwarczny, Lukas and Jez, Lukasz and Sgall, Jiri and Kim Thang, Nguyen and Vesely, Pavel},
title = {{Online Algorithms for Multi-Level Aggregation}},
booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)},
pages = {12:1--12:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-015-6},
ISSN = {1868-8969},
year = {2016},
volume = {57},
editor = {Sankowski, Piotr and Zaroliagis, Christos},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.12},
URN = {urn:nbn:de:0030-drops-63637},
doi = {10.4230/LIPIcs.ESA.2016.12},
annote = {Keywords: algorithmic aspects of networks, online algorithms, scheduling and resource allocation} | {"url":"https://drops.dagstuhl.de/search/documents?author=Jez,%20Lukasz","timestamp":"2024-11-07T08:52:57Z","content_type":"text/html","content_length":"64575","record_id":"<urn:uuid:0a194b6b-57e9-4ac9-a9fb-3b9b9f4d98b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00224.warc.gz"} |
Engora Data Blog
Scientific fraud and business manipulation
Sadly, there's a long history of scientific fraud and misrepresentation of data. Modern computing technology has provided better tools for those trying to mislead, but the fortunate flip side is,
modern tools provide ways of exposing misrepresented data. It turns out, the right tools can indicate what's really going on.
In business, companies often say they can increase sales, or reduce costs, or do so some other desirable thing. The evidence is sometimes in the form of summary statistics like means and standard
deviations. Do you think you could assess the credibility of evidence based on the mean and standard deviation summary data alone?
In this blog post, I'm going to talk about how you can use one tool to investigate the credibility of mean and standard deviation evidence.
Discrete quantities
Discrete quantities are quantities that can only take discrete values. An example is a count, for example, a count of the number of sales. You can have 0, 1, 2, 3... sales, but you can't have -1
sales or 563.27 sales.
Some business quantities are measured on scales of 1 to 5 or 1 to 10, for example, net promoter scores or employee satisfaction scores. These scales are often called Likert scales.
For our example, let's imagine a company is selling a product on the internet and asks its customers how likely they are to recommend the product. The recommendation is on a scale of 0 to 10, where 0
is very unlikely to recommend and 10 is very likely to recommend. This is obviously based on the net promoter idea, but I'm simplifying things here.
Very unlikely to recommend Very likely to recommend
Imagine the salesperson for the company tells you the results of a 500-person study are a mean of 9 and a standard deviation of 2.5. They tell you that customers love the product, but obviously,
there's some variation. The standard deviation shows you that not everyone's satisfied and that the numbers are therefore credible.
But are these numbers really credible?
Stop for a second and think about it. It's quite possible that their customers love the product. A mean of 9 on a scale of 10 isn't perfection, and the standard deviation of 2.5 suggests there is
some variation, which you would expect. Would you believe these numbers?
Investigating credibility
We have three numbers; a mean, a standard deviation, and a sample size. Lots of different distributions could have given rise to these numbers, how can we backtrack to the original data?
The answer is, we can't fully backtrack, but we can investigate possibilities.
In 2018, a group of academic researchers in The Netherlands and the US released software you can use to backtrack to possible distributions from mean and standard deviation data. Their goal was to
provide a tool to help investigate academic fraud. They wrote up how their software works and published it online, you can read their writeup here. They called their software SPRITE (Sample Parameter
Reconstruction via Iterative TEchniques) and made it open-source, even going so far as to make a version of it available online. The software will show you the possible distributions that could give
rise to the summary statistics you have.
One of the online versions is here. Let's plug in the salesperson's numbers to see if they're credible.
If you go to the SPRITE site, you'll see a menu on the left-hand side. In my screenshot, I've plugged in the numbers we have:
• Our scale goes from 0 to 10,
• Our mean is 9,
• Our standard deviation is 2.5,
• The number of samples is 500.
• We'll choose 2 decimal places for now
• We'll just see the top 9 possible distributions.
Here are the top 9 results.
Something doesn't smell right. I would expect the data to show some form of more even distribution about the mean. For a mean of 9, I would expect there to be a number of 10s and a number of 8s too.
These estimated distributions suggest that almost everyone is deliriously happy, with just a small handful of people unhappy. Is this credible in the real world? Probably not.
I don't have outright evidence of wrongdoing, but I'm now suspicious of the data. A good next step would be to ask for the underlying data. At the very least, I should view any other data the
salesperson provides with suspicion. To be fair to the salesperson, they were probably provided with the data by someone else.
What if the salesperson had given me different numbers, for example, a mean of 8.5, a standard deviation of 1.2, and 100 samples? Looking at the results from SPRITE, the possible distributions seem
much more likely. Yes, misrepresentation is still possible, but on the face of it, the data is credible.
Did you spot the other problem?
There's another, more obvious problem with the data. The scale is from 0 to 10, but the results are a mean of 9 and a standard deviation of 2.5, which implies a confidence interval of 6.5 to 11.5. To
state the obvious, the maximum score is 10 but the upper range of the confidence interval is 11.5. This type of mistake is very common and doesn't of itself indicate fraud. I'll blog more about this
type of mistake later.
What does this mean?
Due diligence is about checking claims for veracity before spending money. If there's a lot of money involved, it behooves the person doing the due diligence to check the consistency of the numbers
they've been given. Tools like SPRITE are very helpful for sniffing out areas to check in more detail. However, just because a tool like SPRITE flags something up it doesn't mean to say there's
fraud; people make mistakes with statistics all the time. However, if something is flagged up, you need to get to the bottom of it.
Other ways of detecting dodgy numbers
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While Newton was working on his laws of mechanics others were also working on trying to understand how and why objects move. One person engaged in this was Gottfried Leibniz, who also invented
Calculus independently of Newton. Leibniz proposed that there was a living force (vis viva in Latin) which caused objects to move and was conserved in certain mechanical systems. These ideas where
not as successful at explaining motion as Newton’s laws, but hundreds of years later the vis viva would be recognized as kinetic energy and conservation of energy would be recognized as one of the
most important principals of mechanics. The delay was because the importance of energy was not realized until scientist started trying to explain heat and how heat and work are related. These
aspects of thermodynamics we will discuss later. Before we discuss mechanical energy we have to first go over the mathematics of multiplying vectors.
The Dot Product
Earlier we have discussed how to add vectors, now we need to turn to multiplying vectors. There are actually two types of multiplication for vectors. The first is called the dot product (or inner
product) and this type of multiplication between two vectors results in a scalar. The other type of multiplication is called the cross product or vector product and it results as one might guess in
a vector that points perpendicular to both of the two vectors being multiplied. For now we will only discuss the dot product as we will not make use of the cross product untill much later in the
Dot Product
The dot product of two vectors can be thought of as multiplying the projection of one vector in the direction of the other vector. As a formula we can write this as
$\vec{A} \cdot \vec{B} = AB\cos{\theta}$
where $\theta$ is the angle between the two vectors. Notice that if the two vectors are in the same direction $\theta = 0$ and the vectors multiply like scalars. If the two vectors are perpendicular
then $\theta = 90^{\circ}$ and the dot product is zero. This is in fact the definition of orthogonal that the dot or inner product equals zero.
We can also define the dot product using vector components.
$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$
where $A_x$ is the x component of A and likewise for the other components.
The first type of energy we will discuss is called work. Work is defined by
$W = \int \vec{F} \cdot d\vec{s}$
where \vec{s} is the displacement of the object the work is being applied to. If the force is constant then the integral just becomes $W = \vec{F} \cdot \vec{s}$, which will usually be the case in
problems we look at. However, it is important to remember that is only true for constant force, the real definition is the integral equation above.
Kinetic Energy
Another type of energy is kinetic energy, this is the type of energy something has because of it’s motion. The kinetic energy of an object is given by
$KE = \frac{1}{2}mv^2$
Notice that the the kinetic energy depends on the instantaneous velocity, so an object at a given time has a certain kinetic energy. This differs from work energy which must be evaluated as the
object moves between two positions. A natural question to ask is where does the above equation come from. Well consider a mass that only has one force acting on it so it moves with constant
acceleration for a distance s. The work done on the object is just W=Fs. From Newton’s second law the acceleration of the object is a = F/m and form our study of kinematics we know that the velocity
of the object if it started at rest would be given by
$v^2 = 2 \frac{F}{m} s => \frac{1}{2}mv^2 = Fs$
So if the work done to the object equals its kinetic energy than the above formula must be the correct one. It turns out that this is often the case something which we call energy conservation.
Energy Conservation
The reason energy is a useful concept is that in many situations it is conserved. That is the total energy a system has in some initial state is the same as it has at some later time. From this you
can see the great advantage energy gives to solving problems over forces. With forces you have to follow what happens in your system from start to finish. With energy you can only care about two
times and ignore everything that happened in between.
For energy to be useful then we need to no whether or not it is conserved. This basically depends on the type of forces involved in the problem. Some forces are conservative some are not.
Conservative forces are those that are path independent, that is how you go from A to B the energy use will be the same. Nonconservative forces, the path matters. Nonconservative forces usually
produce heat which is why the path you take matters. Friction and air resistance are non conservative forces, while gravity, springs and electrical forces are conservative forces. Any conservative
force has a potential energy. This is the energy that force could give or take away from an object. If you toss a ball up in the air its kinetic energy goes away. When it reaches its maximum height
it has zero kinetic energy. Where did the kinetic energy go? It went into the potential for the ball to gain kinetic energy as the gravitational force pulls it down. The relationship between force
and potential energy is
$U(r) = \int \vec{F}(r)\cdot d\vec{r}$
This is basically the same as work, the work done by the force is the potential energy. The difference is that work depends on the path while potential energy only depends on the end points. For
conservative forces they are the same.
Potential Energy
For gravity on the surface of the Earth this gives:
$U_g = mgh$
For a spring this gives
$U_s = k x^2$
For Newton’s Law of Universal Gravity this gives
$U_G = -G{{Mm}\over{R}}$ | {"url":"https://openlab.citytech.cuny.edu/amaller1441/lecture-notes/energy/","timestamp":"2024-11-12T02:04:39Z","content_type":"text/html","content_length":"95896","record_id":"<urn:uuid:a82e243e-8b3e-4f09-b9f8-4c5ba2100404>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00417.warc.gz"} |
Interpreting P values (2024)
A P value measures a sample's compatibility with a hypothesis, not the truth of the hypothesis.
Although P values are convenient and popular summaries of experimental results, we can be led astray if we consider them as our only metric^1. Even in the ideal case of a rigorously designed
randomized study fit to a predetermined model, P values still need to be supplemented with other information to avoid misinterpretation.
A P value is a probability statement about the observed sample in the context of a hypothesis, not about the hypotheses being tested. For example, suppose we wish to know whether disease affects the
level of a biomarker. The P value of a comparison of the mean biomarker levels in healthy versus diseased samples would be the probability that a difference in means at least as large as the one
observed can be generated from random samples if the disease does not affect the mean biomarker level. It is not the probability of the biomarker-level means in the two samples being equal—they
either are or are not equal.
However, this relationship between P values and inference about hypotheses is a critical point—interpretation of statistical analysis depends on it. It is one of the key themes in the American
Statistical Association's statement on statistical significance and P values^2, published to mitigate widespread misuse and misinterpretation of P values. This relationship is discussed in some of
the 18 short commentaries that accompany the statement, from which three main ideas for using, interpreting and reporting P values emerge: the use of more stringent P value cutoffs supported by
Bayesian analysis, use of the observed P value to estimate false discovery rate (FDR), and the combination of P values and effect sizes to create more informative confidence intervals. The first two
of these ideas are currently most useful as guidelines for assessing how strongly the data support null versus alternative hypotheses, whereas the third could be used to assess how strongly the data
support parameter values in the confidence interval. However, like P values, these methods will be biased toward the alternative hypothesis when used with a P value selected from the most significant
of multiple tests or models^1.
To illustrate these three ideas, let's expand on the biomarker example above with the null hypothesis that disease does not influence the biomarker level. For samples, we'll use n = 10 individuals,
randomly chosen from each of the unaffected and affected populations, assumed to be normally distributed with σ^2 = 1. At this sample size, a two-sample t-test has 80% power to reject the null at
significance α = 0.05 when the effect size is 1.32 (Fig. 1a). Suppose that we observe a difference in sample means of 1.2 and that our samples have a pooled s.d. of s[p] = 1.1. These give us t = 1.2/
(s[p]√(2/n)) = 2.44 with d.f. = 2(n – 1) = 18 and a P value of 0.025.
(a) Power drops at more stringent P value cutoffs α. The curve is based on a two-sample t-test with n = 10 and an effect size of 1.32. (b) The Benjamin and Berger bound calibrates the P value to
probability statements about the hypothesis. At P = 0.05, the bound suggests that our alternative hypothesis is at most 2.5 times more likely than the null (black dashed line). Also shown are the
conventional Bayesian P = 0.0032) cutoff and P = 0.005), suggested by Johnson in ref. 2. (c) Use of the more stringent Benjamin and Berger bounds in b reduces the power of the test, because now
testing is performed at a < 0.05. For α = 0.005), the power is only 43%. The blue and orange dashed lines show the same bounds as in b. In all panels, black dotted lines are present to help the
reader locate values discussed in the text.
Full size image
Once a P value has been computed, it is useful to assess the strength of evidence of the truth or falsehood of the null hypothesis. Here we can look to Bayesian analysis for ways to make this
connection^3, where decisions about statistical significance can be based on the Bayes factor, B, which is the ratio of average likelihoods under the alternative and null hypotheses. However, using
Bayesian analysis adds an element of subjectivity because it requires the specification of a prior distribution for the model parameters under both hypotheses.
Benjamin and Berger, in their discussion in ref. 2, note that the P value can be used to compute an upper bound for the Bayes factor,
Because it quantifies the extent to which the alternative hypothesis is more likely, the Bayes factor can be used for significance testing. Decision boundaries for the Bayes factor are less
prescriptive than those for P values, with descriptors such as “anectodal,” “substantial,” “strong” and “decisive” often used for cutoff values. The exact terms and corresponding values vary across
the literature, and their interpretation requires active consideration on the part of the researcher^4. A Bayes factor of 20 or more is generally considered to be strong evidence for the alternative
The Benjamin and Berger bound is given by e P ln(P)) for a given P value^5 (Fig. 1b). For example, when we reject the null at P < α = 0.05, we do so when the alternative hypothesis is at most
For our biomarker example, we found P = 0.025 and thus conclude that the alternative hypothesis that disease affects the biomarker level is at most P < 0.0032 (Fig. 1b). Johnson, in a discussion in
ref. 2, suggests testing at P < α = 0.005 (Fig. 1b). Notice that the computation for α < 0.005 (Fig. 1c). To achieve 80% power at this cutoff, we would need a sample size of n = 18.
Altman (this author), in a discussion in ref. 2, proposes to supplement P values with an estimate of the FDR by using plug-in values to account for both the power of the test and the prior evidence
in favor of the null hypothesis. In high-throughput multiple-testing problems, the FDR is the expected proportion of the rejected null hypotheses that consists of false rejections. If some proportion
π[0] of the tests are truly null and we reject at P < α, we expect απ[0] of the tests to be false rejections. Given that 1 – π[0] of the tests are non-null, then with power β we reject β(1 – π[0]) of
these tests. So, a reasonable estimate of the FDR is the ratio of expected false rejections to all expected rejections, eFDR = απ[0]/(απ[0] + β(1 − π[0])).
For low-throughput testing, Altman uses the heuristic that π[0] is the probability that the null hypothesis is true as based on prior evidence. She suggests using π[0] = 0.5 or 0.75 for the primary
hypotheses or secondary hypotheses of a research proposal, respectively, and π[0] = 0.95 for hypotheses formulated after exploration of the data (post hoc tests) (Fig. 2a). In the high-throughput
scenario, π[0] can be estimated from the data, but for low-throughput experiments Altman uses the Bayesian argument that π[0] should be based on the prior odds that the investigator would be willing
to put on the truth of the null hypothesis. She then replaces a with the observed P value, and β with the planned power of the study.
(a) The relationship between the estimated FDR (eFDR) and the proportion of tests expected to be null, π[0], when testing at α = 0.05. Dashed lines indicate Altman's proposals^2 for π[0]. (b) The
profile of P values for our biomarker example (n = 10, s[p] = 1.1). The dashed line at P = 0.05 cuts the curve at the boundaries of the 95% confidence interval (0.17, 2.23), shown as an error bar. (c
) P value percentiles (shown by contour lines) and 95% range (gray shading) expected from a two-sample t-test as effect size is increased. At each effect size d, data were simulated from 100,000
normally distributed samples (n = 10 per sample) with means 0 and d, respectively, and σ^2 = 1. The fraction of P values smaller than α is the power of the test—for example, 80% (blue contour) are
smaller than 0.05 for d = 1.32 (blue dashed line). When d = 0, P values are randomly uniformly distributed.
Full size image
For our example, using P = 0.025 and 80% power gives eFDR = 0.03, 0.09 and 0.38 for primary, secondary and post hoc tests, respectively (Fig. 2a). In other words, for a primary hypothesis in our
study, we estimate that only 3% of the tests where we reject the null at this level of P are actually false discoveries, but if we tested only after exploring the data, we would expect 38% of the
discoveries to be false.
Altman's 'rule-of-thumb' values for π[0] are arbitrary. A simple way to avoid this is to determine the value of π[0] required to achieve a given eFDR. For example, to achieve eFDR = 0.05 for our
example with 80% power, we require π[0] ≤ 0.62, which is fairly strong prior evidence for the alternative hypothesis. For our biomarker example, this might be reasonable if studies in other labs or
biological arguments suggest that this biomarker is associated with disease status, but it is unreasonable if multiple models were fitted or if this is the most significant of multiple biomarkers
tested with little biological guidance.
Many investigators and journals advocate supplementing P values with confidence intervals, which provide a range of effect sizes compatible with the observations. Mayo, in a discussion in ref. 2,
suggests considering the P value for a range of hypotheses. We demonstrate this approach in Figure 2b, which shows the P values of other levels of the biomarker in comparison to one that is observed.
The 95% confidence interval, which is (0.17, 2.23) for this example, is the range of levels that are not significantly different at a = 0.05 from the observed level of 1.2.
As a final comment, we stress that P values are random variables—that is, random draws of data will yield a distribution for the P value^1. When the data are continuous and the null hypothesis is
true, the P value is uniformly distributed on (0,1), with a mean of 0.5 and s.d. of 1/√12 ≈ 0.29 (ref. 1). This means that the P value is very variable from sample to sample, and this variability is
not a function of the sample size or the power of the study. When the alternative hypothesis is true, the variability decreases as the power increases, but the P value is still random. We show this
in Figure 2c, in which we simulate 100,000 sample pairs for each mean biomarker level.
P values can provide a useful assessment of whether data observed in an experiment are compatible with a null hypothesis. However, the proper use of P values requires that they be properly computed
(with appropriate attention to the sampling design), reported only for analyses for which the analysis pipeline was specified ahead of time, and appropriately adjusted for multiple testing when
present. Interpretation of P values can be greatly assisted by accompanying heuristics, such as those based on the Bayes factor or the FDR, which translate the P value into a more intuitive quantity.
Finally, variability of the P value from different samples points to the need to bring many sources of evidence to the table before drawing scientific conclusions. | {"url":"https://onlyhopecats.com/article/interpreting-p-values","timestamp":"2024-11-14T05:25:50Z","content_type":"text/html","content_length":"129993","record_id":"<urn:uuid:521954d7-f52f-43df-b333-11cf413c8007>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00471.warc.gz"} |
Mathematical Proceedings of the Cambridge Philosophical Society: Volume 129 - | Cambridge Core
In [7], Hitchin showed that the data (∇, Φ), comprising an SU(2) Yang–Mills–Higgs monopole in the Prasad–Sommerfeld limit on ℝ3, encodes faithfully into an auxiliary rank 2 holomorphic vector bundle
E˜ over T, the total space of the holomorphic tangent bundle of ℙ1. In this construction ℝ3 is viewed as a subset of H0(ℙ1, [Oscr ](T)) ≅ [Copf ]3.
Generically, the restriction of E˜ to a line is trivial. (The image of a global section ℙz ⊂ T, for z ∈ [Copf ]3, is referred to here as a line on T.) Hence c1(E˜) = 0 and, for all z ∈ [Copf ]3,
there exists m ∈ {0} ∪ ℕ such that E˜[mid ]ℙz ≅ [Oscr ](m) [oplus ] [Oscr ](−m). If m [ges ] 1 then ℙz is a jumping line of E˜ of height m. The jumping lines are parameterized by an analytic set J ⊂
[Copf ]3, which is stratified by height. When the monopole has charge k, the height is bounded above by k. In this case we write J = J1 ∪ … ∪ Jk, where Ji parameterizes jumping lines of height i. A
priori, some Ji may be empty.
The analytic continuation of the monopole to [Copf ]3 has singularities over J. To see this recall how the monopole data are recovered from E˜: very briefly, E˜ induces a sheaf [Escr ] = π2*ε*E˜ over
[Copf ]3 which is locally free away from J2 ∪ … ∪ Jk, (π2 and ε are defined in Section 2). A holomorphic connection and Higgs field are defined in [Escr ] over [Copf ]3 null planes that cut out a
given direction (see [1, 7, 9]). On restriction to ℝ3, [Escr ] gives a rank 2, SU(2) bundle and the holomorphic connection and Higgs field give the monopole data. It is easy to see that the flat
connections are singular at points of J: for example, an analogous situation is described in [10]. | {"url":"https://core-cms.prod.aop.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/volume/26473FD6FBDB683DF1CAF7325E3B1970","timestamp":"2024-11-07T03:23:26Z","content_type":"text/html","content_length":"951334","record_id":"<urn:uuid:b12c1862-7ac7-4dee-b864-5e58b09031d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00260.warc.gz"} |
Forex Station | Advanced Technical Analysis Forum
Tip: Drawing Fibonacci in "reverse" gives you Fibonacci Extensions
Before we start, if you're looking for an Automatic Fibonacci indicator, the best one I've come across is Mladen's
Fibonacci indicator for MT4
Having received a few messages via our Twitter account about how to draw Fibonacci retracements and extensions I would like to explain the easy way to draw your Fibonacci extensions. A technique
which was taught to me by my old trading mentor.
For those of you who draw your Fibonacci the traditional way (swing low to swing high for an uptrend and swing high to swing low on a downtrend),
draw your Fibonacci in reverse instead
(swing low to swing high on a downtrend and swing high to swing low on an uptrend)!
By doing this, you will be producing the Fibonacci extension levels (for taking profit).
There's nothing complex about it, basically the significant 38.2 retracement level now becomes the 61.8 retracement level, the 50.0 retracement stays the same and so on. Some trading platforms
actually give you the option to "Reverse" your Fibonacci levels already.
For me when I notice a strong uptrend or downtrend is in play, I will:
• Draw Fibonacci retracement in "reverse"
• Place a trade order 10 pips before from the 61.8 retracement level (or 38.2 retracement level for non-reverse Fibonacci direction)
• Then I'll place my take profit 10 pips before the 1.618 level extension.
• Wait for your order to be filled as price retraces back to the 61.8 level
Fibonacci Retracement drawn traditionally
Fibonacci Retracement drawn in reverse to show extenstions
Save this and print it to stick it on your wall
Tip: Drawing Fibonacci in reverse gives you the Fibonacci Extensions
Hi guys !
there is another way for drawing fibo extension ,
as you see on chart ,, if we draw it normally , the numbers below 100fibo is FIBO RET. and above 0 is FIBO EXT.
Just add minus0.618 to add list and write in description FIBOEXT 161.8 , OR minus1.618 and then you have 261.8 FIBO EXT .
So we can draw it normally and just change some little things .
Tip: Drawing Fibonacci in reverse gives you the Fibonacci Extensions
macd & rsi wrote: Wed Jul 03, 2019 4:52 pm Hi guys !
there is another way for drawing fibo extension ,
as you see on chart ,, if we draw it normally , the numbers above 100fibo is FIBO RET. and below 100 is FIBO EXT.
Just add minus0.618 to add list and write in description FIBOEXT 161.8 , OR minus1.618 and then you have 261.8 FIBO EXT .
Excellent tip brother. That's actually easier
Tip: Drawing Fibonacci in reverse gives you the Fibonacci Extensions
macd & rsi wrote: Wed Jul 03, 2019 4:52 pm Hi guys !
there is another way for drawing fibo extension ,
as you see on chart ,, if we draw it normally , the numbers above 100fibo is FIBO RET. and below 100 is FIBO EXT.
Just add minus0.618 to add list and write in description FIBOEXT 161.8 , OR minus1.618 and then you have 261.8 FIBO EXT .
YES fibo ext hacks........ thx dear sir!!!
Official Forex-station GIF animator at your service
See a GIF with Forex-station.com on it? I probably made it
The best divergence indicator in the world.
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Tip: Drawing Fibonacci in reverse gives you the Fibonacci Extensions
wooooow , you are really fast writer !!!
thank you all of my brothers and friends !!
Tip: Drawing Fibonacci in reverse gives you the Fibonacci Extensions
This is my first post. I gotten so much from this site over the years and wanted to provide a copy of a Fib indicator that I've been using for years. While the theory some what correct its
application makes me want to cry.
Basics of fibs surrounds AB = CD.
Example: Choose the highest or lowest point on your chart. Timeframe doesn't matter. Lets say we choose highest point. Basic 2 candles to left and 2 to right at the higest point (This is A = 100).
From there you are not going to the next lowest point on the chart thats just crazy. Next you plot B / 0 point at the next low that has again 2 to left and 2 to right higher.
This is your AB leg. The retracement starts a 0 and triggers the 23.6, 38.2, 50, 61.8, 78.6, 86 or 100. This creates your C. This is entry in the market and exit at the D and this dependent on how
deep the retracement is.
The D (-11.8, -27, -38.2, -61.8, -100) or far as your heart is content. Pullbacks to the 38, 50, 61 target is -61.8 OR other levels depending of depth of retracement.
So in easy steps A to B pullback to C profit at D then C becomes the new A and D becomes the new B and the hunt continues. The AG Fib I added the additional lines in code myself. Still working on
alerts at 38, 50 and 61.8.
Im not coder but I can follow it pretty well and add things I need but alarms i havnt figured out yet.
Tip: Drawing Fibonacci in reverse gives you the Fibonacci Extensions
Here is the AG Multi Color Fibs same thing. I added additional lines for my trading. In this case the market is up until it takes out the A. As you work your fibs following the market/trend at some
point the market will take out the A to signal the market has changed direction. Until the A is taken out in this case, the market is up (see above). You can also do counter fibs to estimate the
target of the retracement.
Re: Tip: Drawing Fibonacci in "reverse" gives you the Fibonacci Extensions
An Illustration of the Wonders of Fibonacci Retracements and Extensions.... German30 (DAX) Daily Chart:
If you find the right swing to measure Fibonacci retracements and extensions, you have found yourself a goldmine. Once you have drawn it correctly, it will keep guiding your trading direction and
decisions for many months to come.... I find the topic intriguing, interesting and rewarding.... Hopefully in the coming days and weeks, I will try to share some insights and my experience of using
Fibonacci for long term trading..... Hope to learn from other contributors as well....
Re: Tip: Drawing Fibonacci in "reverse" gives you the Fibonacci Extensions
I traded a few years only fibs with trendlines , this is my version
Re: Tip: Drawing Fibonacci in "reverse" gives you the Fibonacci Extensions
Meyney wrote: Thu Mar 10, 2022 4:33 am I traded a few years only fibs with trendlines , this is my version
Cool. Looks a bit complicated though (especially with no explanation).
Thanks for sharing though | {"url":"https://forex-station.com/viewtopic.php?p=1295375941","timestamp":"2024-11-12T20:25:12Z","content_type":"text/html","content_length":"78223","record_id":"<urn:uuid:63eca668-d45d-48fc-96e2-05e11288ab79>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00559.warc.gz"} |
Usa Math Olympiad Problems | Hire Someone To Do Calculus Exam For Me
Usa Math Olympiad Problems The following mathematical problems are known to be related to the Math Olympiads: Theorems In the first issue, one of the aims of the Olympiad was to ascertain the
contribution of Poincaré polynomials of the form, $p^{2}-p$, to the number of the class of points in the space. In this issue, the author states that, the Poincar’é polynomial is the only polynomial
which is the product of the polynomially many polynomial forms of the form $p^{n}-p$ where $n$ is an integer. The Poincar$^\text{M}$ Olympiad problem In order to find the Poincare polynomial of the
form $$p^{2}\cdot p=\frac{1}{\sqrt{2}}\left(\sqrt{-1}\pm\sqrt{\frac{1-2\sqrt4}{\sqrho}}\right)$$ (or the square root of the corresponding polynomial), the author has to find the polynomial $p=\frac{\
sqrt{4}}{2\sqr32}$. At the beginning of this section, we have the definition of the Poincara polynomial, which is similar to the Poincarno approach. First of all, the Poicara polynomain is the unique
click for info with a root on the unit circle with $p_1=\sqrt2$. The Poincarén polynomial has the following expression: $$\label{poincar} \pi=\frac{{\sqrt 2\left(\frac{1+\sqrt8}{\sqrac{\sqrt2}{2}}\
right)}^2}{\sq r^2+r\sqrt r}.$$ In fact, the Poinman-Yanovich polynomial $\pi$ has the same coefficient as the Poincaram polynomial. Since $\pi$ is the simple root of $p$, the Poincariu polynomial
should be the polynic of the form. Since the Poincaro polynomial does not vanish at the origin, the PoIncaro polynom will not vanish at infinity. Now, let us consider the Poincaranos polynomial $$\
iota(x)=\frac{x-x_0}{x-x_{0}}=\frac1{\sqrt{\pi}}\left(1-\frac{e^{i\frac{\pi}{2}}}{\sq{\pi}}+\frac{i\sqrt\pi}{\sq^{\frac 3 2}}\right).$$ more tips here polynomial can be written as: $$\label{Poincar}
\iota(p)=\frac{{(1-e^{i0})^{2}}}{p+2\sq\pi}.$$ Usa Math Olympiad Problems The upcoming students and teachers will have to work out a plan to solve the problems they want to solve. After you learn a
solution, you will have to take the solutions to the exam. It is expected that you will get a good result in the exam. I have to give the exam to my sister on my 21st birthday. She’s a big girl and
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and her students. In this exam, you will get an answer for the explanation questions. 1. Your answer is correct. 2. You have answered all the questions in the exam by the same answer. 3. You have
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I will give the 3rd exam for my sister on her 13th birthday. She has to take the 3rd exams. I have only one question to solve. She has some of the questions that she has to answer. In the exam, you
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you are getting around, you have to study hard to become a good judge. Some of the exam are very hard for you to do. And it’s not easy. You have to get some knowledge about the exam. And you have to
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You have a lot of know-how. But the exam is very hard. It’s hard to be a good student. So, I am going to give a solution plan to my sister for her 13th and 13th birthday here. I will be working on
her. In the exam, I will learn the exam, and also the answers. I will have to make my solution plan. Thoughts? Please give me your thoughts. I will help you in solving the exam. I will work onUsa
Math Olympiad Problems 2016-17 The Summer Olympics are held every Summer. They were held in the first week of July at the Sunrisers Arena in Bangkok, Thailand. The last few months have been a very
busy one for the Bangkok Olympic Games. This year’s Summer Games have been very interesting. They have been held in the summer of 2016 in the beautiful city of Bangkok, Thailand, where the weather is
warm and sunny with an average of 8 degrees. The most important event in the Summer Games is the Summer Olympic Games in Rio de Janeiro, Brazil. This is the first Summer Olympic Games for the first
time since 1992. The Games are held in Rio de Jardim with the first day being in the morning and the last day in the afternoon. This is a very interesting event for people who like to travel and
study. The main attraction of this Summer Games is that people have got to work on their skills, which are very important for them. The main competition is the F1 World Championship which is a must
for the young people.
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The main event is the 2015 Summer Olympic Games. The main and main competitors are all the top 10 athletes. One of the most important events is the Olympic Games in 2014. The competition is in Rio de
Amor, Brazil, where the top 10 is first to be ranked. Next is the 2015 Olympic Games in Girona, Spain, where the Olympics are held. The main competitors are the IAAF, the Brazilian Cup, the Eurocup,
and the U-20 World Cup. The main events are the F1 and F2 World Championships. The main competitions are the F3 World Championship, the F4 World Championship, and the F5 World Championship. The
competition has been held in Rio for the last few months and it has been very interesting to see the competition as it was also the first Summer Olympics. This is a very important event for the young
athletes. The main race for the Olympic games is the F4 and F1 World Championships. One of my favorite events is the F5 and F4 World Championships. With the top 10 the competition is also the first
Winter Olympics and the final Olympic Games in 2026. The main matches are the F4, F5, and F5 World Championships. There is also the F4 Worlds, F5 Worlds, and F6 Worlds. The main Olympics are the F5
Olympics in Rio de Zanj, Brazil and the F6 Olympics in Saint-Heloise, France. The main games are the F6 Worlds in Paris, France, where the Games are held. There are also the F8 Worlds in London,
England, where the Olympic Games are held, and the IAAF World Championships in New York, where the F8 World Championships are held. This is an important event to watch. Another important event for
people is the F8 Olympic Games in London, where the international Olympic Games are as part of the Summer Olympics.
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The main Olympic Games are the F8 and F8 Worlds. The F8 Worlds is the European Championships, where the IAAF and the ICAF have the finals. The Games have been held at the Olympic Stadium in London,
and the main Olympics have been held there for the last two years. The main Games in Paris, where the main events are in Paris, are the F9 Worlds, F9 Worlds (F9 World Championship) and the F11 World | {"url":"https://hirecalculusexam.com/usa-math-olympiad-problems","timestamp":"2024-11-05T00:05:01Z","content_type":"text/html","content_length":"103090","record_id":"<urn:uuid:bc38c414-22fe-4f05-a4e8-87de434f64f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00464.warc.gz"} |
commit fe03eca6d297aed17f84e5f35d28c796dec6429a
parent f99c8574a3a063f67f6d47536d60ed85b871c807
Author: Agastya Chandrakant <acagastya@outlook.com>
Date: Mon, 16 Apr 2018 20:24:54 +0530
M s4/fafl/report.md | 11 +++++++++--
1 file changed, 9 insertions(+), 2 deletions(-)
diff --git a/s4/fafl/report.md b/s4/fafl/report.md
@@ -7,7 +7,7 @@
Consider two numbers `u` and `v`. `u % v = (u - v) % v`, if `u` is greater than or equal to `v` else `u`. Using this, a recursive relation can be established, which is:
$\mod(u, v) = \begin{cases} u: u < v\\mod (u - v, v): otherwise\end{cases}$
-Its iteratie code in JavaScript is:
+Its iterative code in JavaScript is:
function modulo(u, v) {
@@ -28,8 +28,15 @@ function newModulo(u, v) {
return u;
-In the above code, `quo` variable gives the
+In the above code, `quo` variable gives the quotient. Noting that repeated subtraction yields remainder, and count of subtraction yields quotient, state transition diagram of a STM with infinite memory (in theory) can be drawn.
+__Refer figure for TM1 which acts as a transducer to find remainder and quotient of two natural numbers__
### STM to check if entered natural number is prime or not
Consider a natural number `num`. If it is a composite number, it has atleast one factor between two and $\frac{num}{2}$
+### Source code
+// to be added tomorrow | {"url":"https://git.hanabi.in/nie-ii-year/commit/fe03eca6d297aed17f84e5f35d28c796dec6429a.html","timestamp":"2024-11-05T06:43:37Z","content_type":"text/html","content_length":"4232","record_id":"<urn:uuid:c5dac3f1-3440-422b-b7b4-37823cdee0b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00154.warc.gz"} |
Chances: Chance of Survival
Main Menu Name: Chances
Determines the probability that at least one person in a group of up to ten persons will die before reaching a given age. It also computes survival probabilities based upon a given term. The
calculation determines probability using Table 2000CM (the table used by the IRS for valuing annuities, life estates, and remainders).
In this article:
Although it is impossible to know the life expectancy of any one person, it is possible to estimate life expectancy or chances of survival to a given age based on a large number of individuals
through actuarial tables.
Table 2000CM, the mortality table used by this calculation, is the table based on the 2000 census. The IRS uses this table to calculate life expectancy and the probability of survival at given ages.
This table is the basis for the actuarial assumptions and tables released under Code Section 7520 in Notice 89-60.
Note that the expectancies in this table will usually not match those of annuity tables since those tables presume a more select (healthy and long-lived) group. For example, under Section 72 tables,
a 65-year-old has a 20-year life expectancy while under the 1980 table life expectancy would only be about 16 years.
If the 1980 CSO Gender Adjustment is selected, then the probabilities are calculated by adding 2 years to the ages of males and subtracting 4 years from the ages of females. So, for example, if age
57 is entered for both a man and a woman, the probability of the man dying would be based on age 59 and the probability of the female dying would be based on age 53.
Getting Started
Table 2000CM, the mortality table used by this calculation, is the table based on the 2000 census. The IRS uses this table to calculate life expectancy and the probability of survival at given ages.
The IRS uses this table to calculate life expectancy and the probability of survival at given ages.
The screen can calculate the approximate 1980 CSO sex-based mortality table probabilities, by applying the appropriate adjustment to the individual's age:
• Male adjustment: add two years
• Female adjustment: subtract four years
Entering Data
1. Compute Survival Based on Term: Select the check box to calculate the probability of at least one of up to ten persons dying before they reach a specified reference age.
2. Reference Age: Enter the reference age. This number is the age to which the probabilities of dying are calculated. This entry field appears when Compute Survival Based on Term is not selected.
3. Reference Term: Enter the reference term. This number is the term for which the probabilities of dying are calculated. This entry field appears when Compute Survival Based on Term is selected.
4. Apply 1980 CSO Gender Adjustment: Select the checkbox to approximate gender-based calculations based on the 1980 CSO mortality table age adjustments (adding two years for males and subtracting
four years for females). Do not select the checkbox if you want to perform unisex calculations.
5. Age: Enter the age of each person being evaluated. Use the age of this person as of the nearest birthday. Enter 0 for age of any person not to be included in the calculation.
6. Gender: If the 1980 CSO gender adjustment is selected, then select the appropriate box for Male or Female for each person.
The Summary Tab displays each individual's probability of dying prior to the age specified in the Reference Age entry field. The program calculates results for up to ten individuals. The results also
show the joint probability that at least one of the individuals will die before reaching the reference age.
0 comments
Please sign in to leave a comment. | {"url":"https://support.leimberg.com/hc/en-us/articles/360055142051-Chances-Chance-of-Survival","timestamp":"2024-11-13T05:42:58Z","content_type":"text/html","content_length":"29055","record_id":"<urn:uuid:b3667697-1103-49a0-90ed-1b35e2d8807b>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00329.warc.gz"} |
RBSE Class 12 E-Physics Self Evaluation Test Papers in EnglishRBSE Class 12 E-Physics Self Evaluation Test Papers in English
Students must start practicing the questions from RBSE 12th Physics Model Papers E- Physics Self Evaluation Test Papers in English Medium provided here.
RBSE Class 12 E-Physics Self Evaluation Test Papers in English
General Instructions to the Examinees :
• Candidate must write first his/her Roll Nc. on the question paper compulsorily.
• All the questions are compulsory.
• Write the answer to each question in the given answer book only.
• For questiotis having more than one part the answers to those parts arc to be written together in continufty.
RBSE Class 12 E- Physics Self Evaluation Test Paper 1 in English
Section- A
1. Write the eorrct answer from multiple choice question 1 (i to ix) and write th in given answer book
(i) Total electric flux coming out of a unit positive change pui in air is: [1]
(a) ε[0]
(b) ε^-1
(c) \(\frac{q}{4 \pi \varepsilon_{0} a}\)
(d) 4πε[0]
(ii) Two conducting sphctcs of raddi r[1] and r[2] are equally charged. The ratio of their potential is: [1]
(a) \(\frac{r_{1}}{r_{2}}\)
(b) \(\frac{r_{2}^{2}}{r_{1}^{2}}\)
(c) \(\frac{r_{2}}{r_{1}}\)
(d) \(\frac{r_{1}^{2}}{r_{2}^{2}}\)
(iïi) Reciprocal of resucance is known as
(a) Specific cooductivity
(b) Conductivity
(c) Resistivity
(d) None of these
(iv) The cyclotron frequency v[c] is given by [1]
(a) \(\frac{q \mathrm{~B}}{2 \pi m}\)
(b) \(\frac{m \mathrm{~B}}{2 \pi q}\)
(c) \(\frac{2 \pi m}{q \mathrm{~B}}\)
(d) \(\frac{2 \pi}{q B}\).
(v) Two similar circular co-axial loops carry equal currents in the same direction.
If the loops be brought nearer, the current in loops [1]
(a) Decreases
(b) Increases
(c) Remains same
(d) Different in each loop
(vi) Which one among the following shows particle nature of tight? [1]
(a) Photoelectric effect
(b) Interference
(c) Refraction
(d) Polarization
(vii) A radioactive susbtance has a half life of 1 year. The fraction of this material, that would remain after 5 years will
be [1]
(a) \(\frac{1}{32}\)
(b) \(\frac{1}{5} \)
(c) \(\frac{1}{2}\)
(d) \(\frac{4}{5} \)
(viii) The resistance of semiconductor and conductor- [1]
(a) Increase with temperature for both of them
(b) Decrease with temperature for both of them
(c) Increase and decreases with temperature
(d) None of these
(ix) The given symbol represents – [1]
(a) OR gate
(b) NOR gaie
(c) NOT gate
(d) AND gate
Question 2.
Fill in the blanks :
(i) Potential inside a shell is …………………………… . [1]
(ii) Kirchhoff’s second law supports the law of conservation of ………………………………. .[1]
(iii) Maieutic flux is a scalar quantity and its dimensional formula is ……………………… .[1]
(iv) gives an inverted version of its input ………………………… .[1]
Question 3.
Give the answer of the following question in one line.
(i) State the underlying principle of potentiometer. [1]
(ii) Draw the magnetic field lines due to a current-carrying loop. [1]
(iii) Two spherical bobs, one metallic and other of glass, of the same size are allowed to fall freely from the same height above the ground. Which of the two would reach earlier and why? [1]
(iv) Why is photoelectric emission not possible at all frequencies? [1]
(v) Show graphically. the variation of de-Broglie wavelength (λ) with the potential (v) through which an electron is accelerated from rest. [1]
(vi) Two nuclei have mass numbers in the ratio 27: 125. What is the ratio of their nuclear radii? [1]
(vii) Why is the detection of neutrinos found very difficult? [1]
(viii) Give the logic symbol of NOR gate. [1]
Question 4.
Derive the expression for the electric potential due to an electric dipole at a point on its axial line. [1½]
Question 5.
Calculate the potential differences and the energy stored in the capacitor C[2] in the circuit shown in the figure. Given potential at A is 90V, C[1] = 20µF, C[2] = 30µF and C[3] = 15µF. [1½]
Question 6.
Using the concept of drift velocity of charge carriers in a conductor, deduce the relationship between current density and resistivity of the conductor. [1½]
Question 7.
Use Kirchhoff’s rules to obtain balance conditions in a Wheatstone bridge. [1½]
Question 8.
A long solenoid with 15 turns per cm has a small loop of area 2.0 cm^2 placed inside the solenoid normal to its axis.
If the current carries by the solenoid change steadily from 2.0 to 4.0 in 0.1s. Then what is the induced emf in the loop while the current in changing?[1½]
Question 9.
Define natural induction. Write SI unit and dimensional formula.[1½]
Question 10.
Use the mirror equation to show then on object placed between F and 2F of a concave mirror produces a real image
beyond 2F. [1½]
Question 11.
Draw a schematic arrangement of a reflecting telescope showing how rays coming from a distant object are received at the eyepiece. Write its two important advantages over a refracting telescope. [1½]
Question 12.
An object of 3 cm height is placed at a distance of 60 cm from a convex mirror of focal length 30 cm. Find the nature,
position and size of the image fonned. [1½]
Question 13.
Write the conditions for observing a rainbow. Show by drawing suitable diagram how one understands the formation of rainbow. [1½]
Question 14.
expression N = N[0]e for the law of radioactive decay. [1½]
Question 15.
Calculate the energy release in MeV in the deuterium-tritium fusion reaction. [1½]
[1^2]H + [1^3]H → [2^4]He+[e^1]n
Using the data m([1^2]H) =2.014102 u; m([1^3]) =3.016049 u; m([2^4]He) =4.002603 u;m[n] = 1.0086654; 1u= 931.5\(\frac{\mathrm{MeV}}{\mathrm{C}^{2}}\)
Question 16.
Write using Biot-Savart law, the expression for the magnetic field B due to an element dl carrying current I at a distance r from it in a vector form. Hence drive the expression for the magnetic
field due to a current-carrying loop of radius R at center O. [3]
Using Ampere’s circuital law, obtain the expression for the mangetic field due to a long solenoid at a point inside the solenoid on its axis. [3]
Question 17.
Draw a labelled ray diagram to show the formation of image in an astronomical telescope for a distant object.
Derive the expression for its magnifying power in normal adjustment. [3]
Draw a neat labelled ray diagram of a compound microscope. Explain briefly its working. [3]
Question 18.
(i) Define the terms threshold frequency and stopping potential in the study cf photoelectric emission. [3]
(ii) Explain briefly the reasons why wave theory of light is not able to explain the observed features in photoelectric effect?
Write Einstein photoelectric equation. State clearly how this equation is obtained using the photon picture of electromagnetic radiation.
Write the three salient features observed in photoelectric effect which can be explained using the equation. [3]
Question 19.
State Gauss’s law in electrostatics. Using this law, derive an expression for the electric field due to a uniformly charged infinite plane sheet. [4]
Define the term electric dipole moment. It is a scalar or vector? Deduce an expression for the electric field at a point on the equatorial plane of an electric dipole of length 2a. [4]
Question 20.
Draw the typical shape of the V-I characteristics of a p-n junction diode both in (a) forward (b) reverse bias configuration. How do we infer, from these characteristics that a diode can be used to
rectify alternating voltages. [4]
State the main practical application of LED. Explain giving reason, why the semiconductor used for fabrication of visible light LED’s must have a bandgap of at least 1.8 eV. [4]
RBSE Class 12 E-Economics Self Evaluation Test Paper 2 in English
Question 1.
Write the correct answer from multiple choice question 1 (i to ix) and write the in given answer book.
(i) If \(\oint \overrightarrow{\mathrm{E}} \cdot d \overrightarrow{\mathrm{S}}\) = 0 over a surface, Then : [1]
(a) The electric field inside the surface and on it is zero
(b) The electric field inside the surface in necessarily uniform
(c) All changes most necessarily be outside the surface
(d) All of these.
(ii) Figure shows the field lines of a positive point charge. The work done by the field in moving a small positive charge from Q to P is :
(a) Zero
(b) Positive
(c) Negative
(d) Data insufficient
(iii) Relation between drift velocity and current [1]
(a) v[d] = \(\frac{n e \mathrm{~A}}{\mathrm{I}}\)
(b) v[d] = \(\frac{\mathrm{I}}{n e \mathrm{~A}}\)
(c) v[d] = IneA
(d) v[d] = \(\frac{n \mathrm{I}}{e \mathrm{~A}}\)
(iv) The magnetic force \(\overrightarrow{\mathrm{F}}\) on a current carrying conductor of length l in on external magnetic field \(\overrightarrow{\mathrm{B}}\) is given by : [1]
(a) \(\frac{I \times \vec{B}}{\vec{l}}\)
(b) \(\frac{\vec{i} \times \overrightarrow{\mathrm{B}}}{\mathrm{I}}\)
(c) I \((\vec{l} \times \overrightarrow{\mathrm{B}})\)
(d) I^2 \((\vec{l} \times \overrightarrow{\mathrm{B}})\)
(v) Direction of current induced in wire moving in a magnetic field is found using : [1]
(a) Fleming’s left-hand rule
(b) Fleming’s right-hand rule
(c) Ampere’s rule
(d) Right-hand dasp rule
(vi) In photoelectric effect the photocurrent: [1]
(a) Depends both on intensity and frequency of the incident light.
(b) Does not depend on the frequency of incident light but depends on the intensity of the incident light.
(c) Decreases with increase in frequency of incident light.
(d) Increases with increase in frequency of incident light.
(vii) Which is the correct expression for half life- [1]
(a) t[1/2] = log 2
(b) (t)[1/2] = \(\frac{\lambda}{\log 2}\)
(c) t[1/12] = \(\frac{\lambda}{\log 2} 2.303\)
(d) t[1/12] = I^2 \((\vec{l} \times \overrightarrow{\mathrm{B}})\).
(viii) To get an output Y = 1 is given circuit which of the following input will be correct- [1]
(ix) C and Si both have same lattice structure having 4 bonding electrons in each. However, C is insulator whereas Si is intrinsic semiconductor. This is because : [1]
(a) incase of C, the valence band is not completely filled at absolute zero temperature.
(b) In case of C the conduction band is partly filled even at absolute zero temperature.
(c) The four bonding electrons in the case of 0 lie in the second orbit; whereas in the case of Si they lie in the third.
(d) The four bonding electrons in the case of C lie in the second orbit, whereas in the case of Si they lie in the second.
Question 2.
Fill in the blanks :
(i) Work done in moving at test charge from one point to equipotential surface to other is …………………… . [1]
(ii) The terminal voltage increases with the increase of ………………………… .[1]
(iii) For the motion’ parallel to \(\overrightarrow{\mathrm{B}}\), the induced emf is ………………………. .[1]
(iv) Boolean expression of AND gate is given as …………………………. . [1]
Question 3.
Give the answer of the following question in one line.
(i) The emf of a cell is always greater than its. terminal voltage. Why? Give reason. [1]
(ii) Use the expression F = q\((\vec{v} \times \overrightarrow{\mathrm{B}})\) to define the SI unit of magnetic field. [1]
(iii) Predict the direction of induced current in metal rings 1 and 2 when current I in the wire is steadily decreasing. [1]
(iv) Show the variation on photoelectric current with collector plate potential for different frequencies but same intensity of incident radiation. [1]
(v) Write the expression of de-Broglie wavelength associated with a charged particle having charge q and mass m.
When it is accelerated by a potential. [1]
(vi) A nucleus undergoes β-decay. How does its (a) mass number and (b) atomic number change? [1]
(vii) Draw the plot of bindings energy per nucleon BE/A as a function of mass number A. [1]
(viii) A given logic gate inverts the input applied to it. Name this gate and give its symbol. [1]
Question 4.
Define an equipotential surface. Draw equipotential surfaces- (i) in case of a single point charge, (ii) in a constant electric field in z-direction. [1½]
Question 5.
Net capacitance of three identical capacitors in series is 1 µF. What will be their net capacitance, if connected in parallel? Find the ratio of energy stored in these two configurations if they are
both connected to the same sources. [1½]
Question 6.
Derive an expression for the current density of a conductor in terms of the drift speed of electrons. [1½]
Question 7.
Describe briefly with the help of a circuit diagram, how a potentiometer is used to determine the internal resistance of a cell. [1½]
Question 8.
Derive an expression for the mutual inductance of two long coaxial solenoids of same length wound one over the other. [1½]
Question 9.
Draw a labelled diagram of AC generator and state its working principle. [1½]
Question 10.
Explain briefly how the phenomenon of total internal reflection is used in fibre optics. [1½]
Question 11.
Draw a diagram showing the formation of primary rainbow and explain at what angles the primary raindow is visible. [1½]
Question 12.
Two thin lenses of power- 4D and 2D are placed in contact coaxially. Find the focal length of the combination. [1½]
Question 13.
The image obtained with a convex lens is erect and its length is four times the length of the object.[1½]
Question 14.
Show that the density of nucleus over a wide range of nuclei is constant and independent of mass number A.[1½]
Question 15.
A radioactive nucleus A undergoes a series of decays according to the following scheme
mass number and atomic number of A4 are 172 and 69, respectively. What are these numbers for A?[1½]
Question 16.
Draw a schematic sketch of a cyclotron. Explain clearly the role of crossed electric and magnetic field in accelerating the charge. Hence, derive the expression for the kinetic energy required by the
particles. [3]
Draw a labelled diagram of a moving coil galvanometer. Describe briefly its principle and working. [3]
Question 17.
Draw a ray diagram showing the formation of the image by a point object on the principal axis of a spherical convex surface separating two media of refractive indices n[1] and n[2], when a point
source is kept in rarer medium of refractive index n[1]. Derive the relation between object and image distance in terms of refractive index of the medium and radius of curvature of the surface.[3]
When a ray of light passes through a triangular glass prism, find out the relation for the total deviation. δ in terms of the angle of incidence i and angle of emergence, e [3]
Question 18.
(i) Define the term stopping potential. [3]
(ii) Plot a graph showing the variation of photoelectric current as a function of anode potential for two light beams having the same frequency but different intensities I[1] and I[2] (I[1] > I[2])
(iii) The stopping potential in an experiment on photoelectric effect is 2V. What is the maximum kinetic energy of the photoelectrons emitted?
An electron and a proton are accelerated through the same potential. Which one of the two has (i) greater value of de- Broglie wavelength associated with it and (ii) less momentum? Justify your
answer. [3]
Question 19.
State Gauss’s lay Use it to deduce the expression for the electric field due to a uniformly charged thin spherical shell at points (a) inside the shell and (b) outside the shell. [4]
A charge in distributed uniformly over a ring of radius a. Obtain an expression for the electric field intensity E at a point on the axis of the ring. Hence show that for points at large distances
from the ring, it behaves like a point charge. [4]
Question 20.
(i) Why are NAND gate called universal gates? Identify the logical operations carried out by the circuit given as below.
(ii) Draw the logic circuit of AND gate and write its truth table. [4]
How is a Zener diode fabricated so as to make, it a special purpose diode?
Draw I-V characteristics of Zener diode and explain the significance of breakdown voltage. [4]
RBSE Class 12 E-Physics Self Evaluation Test Paper 3 in English
Question 1.
Write the correct answer from multiple choice question 1 (i to ix) and write the in given answer book
(i) The S.l unit of electric flux is: [1]
(a) N/C^1m^2
(b) N/C m^2
(c) N/C^2 m^2
(d) N/C^2 m^2
(ii) In a region of constant potential : [1]
(a) The electric field is uniform
(b) There electric field is zero
(c) There can be no charge inside the region
(d) Both (b) and (c) are correct
(iii) Resistance of a resistor is inversely proportional to: [1]
(a) area of cross-section
(b) length
(c) both (a) and (b)
(d) None of these
(iv) A strong magnetic field is applied on stationary electrons. Then the electron : [1]
(a) Moves in the direction of the field
(b) Remains stationary
(c) Moves perpendicular to the direction of the field
(d) Moves opposite to the direction of the field
(v) Lenz’s law is a consequence of the law of conservation of: [1]
(a) Charge
(b) Energy
(c) Induced emf
(d) Induced current
(vi) In photoelectric effect, the photoelectric current is independent of: [1]
(a) Intensity of incident light
(b) Potential difference applied between the two electrodes
(c) The nature of emitter material
(d) Frequency of incident light
(vii) After two hours one-sixteenth of the initial amount of a certain radioactive isotope remained undecayed. The half-life of the isotope is : [1]
(a) 15 minutes
(b) 30 minutes
(c) 45 minutes
(d) One hours
(viii) The following logic circuit represents : [1]
(a) NAND gate with output O = \(\bar{X}+\bar{Y}\)
(b) NOR gate with output O = \(\overline{X+Y}\)
(c) NAND gate with output O = \(\overline{\mathrm{XY}}\)
(d) NOR gate with output O = \(\bar{X}+\bar{Y}\)
(ix) A piece of copper and the other of Germanium are cooled from the room temperature to 80 K, then which of the following would be a correct statement: [1]
(a) Resistance of each increases
(b) Resistance of each decreases
(c) Resistance of Copper increases while that of Germanium decreases
(d) Resistance of Copper descreases while that of Germanium increases.
Question 2.
Fill in the blanks :
(i) Capacitance of an isolated spherical conductor of radius r is given by ………………………… . [1]
(ii) The emf of two primary cells can be compared using potentiometer as ………………………… . [1]
(iii) SI unit of magnetic flux is ………………………… . [1]
(iv) …………………………. converts solar energy into electrical energy . [1]
Question 3.
Given the answer of the following question in one line.
(i) The three coloured bands, on a carbon resistor are red, green and yellow respectively. Write the value of its
resistance. [1]
(ii) A beam of a-particles projected along + X-axis, experiences a force due to a-magnetic field along the +Y axis. What is the direction of the magnetic force. [1]
(iii) On what factors does the magnitude of the emf induced in the circuit due to magnetic flux depend ? [1]
(iv) Show on a plot the nature of variation of photoelectric current with the intensity of radiation incident on a photosensitive surface. [1]
(v) A proton and electron have same kinetic energies. Which one has greater than de-Broglie wavelength and why? [1]
(vi) How is the radius of a nucleus related to its mass numbers? [1]
(vii) Write any two characteristic properties of nuclear force. [1]
(viii) What is the difference between an n-type and a p-type extrinsic semiconductor? [1]
Question 4.
Two-point charges q[1] and q[2] are located at r[1] and r[2], respectively in an external electric field E. Obtain the expression for the total work done in assembling this configuration. [1½]
Question 5.
Deduce the expression for the electrostatic energy stored in a capacitor of capacitance C and having charge Q. [1½]
Question 6.
Define mobility of charge carrier. Write the relation expressing mobility in terms of relaxation time.[1½]
Question 7.
A potentiometer wire of length lm has a resistance of 5 Ω. It is connected to a 8V battery in series with a resistance of
15Ω. Determine the emf of the primary cell which gives a balance point at 60 cm. [1½]
Question 8.
A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil changes from 0 to 20 A in 0.5 s. What is the change of flux linkage with the other coil?[1½]
Question 9.
Due to the presence of the current in rod and of the magnetic field, Find the expression for the. magnitude and direction of the force acting on this rod. [1½]
Question 10.
A convex lens of focal length F[1] is kept in contact with a concave lens of Focal length F[2]. Find the focal length of the combination.[1½]
Question 11.
Write the necessary condition for the phenomenon of total internal reflection to occur. Write the relation between the refractive index and critical angle for a given pair of optical media.[1½]
Question 12.
Draw a ray diagram to show the image formation by a concave mirror. When the object is kept between its focus and the pole. Using this diagram, derive the magnification formula for the image formed.
Question 13.
A convex lens has 10cm focal length in air. What is its focal length in water? (Refractive index of air-water = 1.33 and
refractive index of air-glass = 1.5) [1½]
Question 14.
Derive an expression for that average life of a radionuclide. Give its relationship with the half-life.[1½]
Question 15.
Write three characteristic properties of nuclear force.[1½]
Question 16.
Explain using a labelled diagram, the principle and working of a moving coil galvanometer. What is the function of
(i) uniform radial magnetic field
(ii) soft iron core? [3]
Two straight long parallel conductors carry currents I[1] and I[2] is the same direction. Deduces the expression for the force per unit length between them. Depict the pattern of magnetic field lines
and them. [3]
Question 17.
Draw a ray diagram to show refraction of a ray of monoch somatic light passing through a glass prism. Deduce the
expression for the refractive index of glass in terms of angle of prism and angle of minimum deviation. [3]
With the help of a suitable ray diagram derive a relation between object distance (u), image distance (v) and radius of curvature (R) for the convex spherical surface when a ray of light travels from
a rarer to denser medium. [3]
Question 18.
(i) State de-Broglie hypothesis.
(ii) Derive an expression for the de-Broglie wavelength associated with an electron accelerated through a potential V.
(iii) Draw a schematic diagram of a localized wave describing the wave nature of the moving electron. [3]
Define the term “Cut-off frequency” in photoelectric emission. The threshold frequency of a metal is f. when the light of frequency 2F is incident on the metal plate, the maximum velocity of
photoelectron is v[1]. When the frequency of the incident radiation is increased to 5f, the maximum velocity of photo electron is v[2]. Find the ratio v[1] = v[2]. [3]
Question 19.
(i) Define torque acting on a dipole of dipole moment \(\vec{P}\) placed in a uniform electric field \(\overrightarrow{\mathrm{E}}\) .
Express it in the vector form and point out the direction along which it acts.
(ii) What happens if the field is non-uniform?
(iii) What would happens if the external field \(\overrightarrow{\mathrm{E}}\) is increasing (a) parallel to \(\overrightarrow{\mathrm{P}}\) and (b) anti-parallel to \(\overrightarrow{\mathrm{P}}\) ?
State Gauss’s law in electrostatics. Using this law derive an expression for the electric field due to a long straight wire of linear charge density k c/m. [4]
Question 20.
(i) Explain with the help of diagram, how a depletion layer and barrier potential are formed in a junction diode.
(ii) Draw a circuit diagram of a full-wave rectifier. [4]
Name the device which is used as a voltage regulator. Draw the necessary circuit diagram and explain its working. [4]
RBSE Class 12 E-Physics Self Evaluation Test Paper 4 in English
Question 1.
Write the correct answer from multiple choice question I (I to ix) and write the in given answer book.
(i) According to Gass’s Theorem, electric field of an infinitely long straight wire is proportional to: [1]
(a) r
(b) \(\frac{1}{r^{2}} \)
(c) \(\frac{1}{r^{3}}\)
(d) \(\frac{1}{r}\)
(ii) Which of the following statement s not true? [1]
(a) Electrostatic force is a conservative force
(b) Potential at a point is the work dine per unit charge in bringing a charge from infinity to that point.
(c) Ap equipotential surface is a surface over which potential has a constant value.
(d) Inside a conductor, electrostatic field is zero.
(iii) Equivalent resistance of the given network is : [1]
(a) 28
(b) 18
(c) 26
(d) 25
(iv) A charged particle would continue to move with a constant velocity in a region wherein, which of the following
conditions is not correct? [1]
(a)E=0, B ≠ 0
(b) E ≠ 0, B≠ 0
(c) E ≠ 0,B=0
(d) E=0,B=0
(v) The magnetic flux linked with a coil of N turns of area of cross-section A held with its plane parallel to the field B
is: [1]
(a) \(\frac{\mathrm{NAB}}{2}\)
(b) NAB
(c) \(\frac{\text { NAB }}{4}\)
(d) Zero
(vi) Light of wavelengths falls on a metal having work functions λC/λ[0]. Photoelectric effect will take place only. 1[]
(a) λ ≥ λ[0]
(b) λ ≤ λ[0]
(c) λ ≥ 2λ[0]
(d) λ = 4λ[0]
(vii) Equivalent energy of mass equal to 1 amu is: [1]
(a) 93 KeV
(b) 931 eV
(c) 931 MeV
(d) 9.31 MeV
(viii) The figure shows a logic circuit with two inputs A and B and the output C. The voltage wavefront across A, B, and C are as given. The logic circuit gate is: [1]
(a) OR gate
(b) NOR gate B
(c) AND gate
(d) NAND gate
(ix) Which one of the following represents forward bias diode? [1]
Question 2.
Fill in the blanks :
(i) A capacitor is a device which is used to store …………………………………….. energy .[1]
(ii) The sensitivity of potentiometer can be ……………………………. by increasing the number of wires of potentiometer. [1]
(iii) An AC generator is based on the phenomenon of ……………………………. .[1]
(iv) Zener diode, is a ……………………… biased heavily doped p-n junction diocle. [1]
Question 3.
Give the answer of the following question in one line:
(i) I-V graph for metallic wires at two different temperatures T[1] and T[2] is as shown in the figure below. Which of the two temperatures is lower and why? [1]
(ii) Write the expression in a vector form for the Lorentz magnetic forme F due to a charge moving with velocity \(\vec{v}\)
in a magnetic field \(\overrightarrow{\mathrm{B}}\) . [1]
(iii) In the given figure, a bar magnet is quickly moved towards a conducting loop having a capacitor.
Predict the polarity of the plates A and B of the capacitor. [1]
(iv) Define intensity of radiation on the basis of photon picture of light. Write its SI unit. [1]
(v) Draw a plot showing the variation of de-Broglie wavelength of the electron as a function of it K.E. [1]
(vi) A nucleus [92^238]U undergoes a-decay and transform to Thorium, What is
(a) the mass number and
(b) atomic number of the nucleus produced ? [1]
Calculate the energy in fusion reaction.
[1^2]H+[1^2]H →[2^3]He + [0^1]n + energy,
where BE of [1^2]H = 2.23 MeV and of [2^3]He = 7.73 MeV.
(viii) What is the most common use of photodiode ? [1]
Question 4.
Find out the expression for the potential energy of a system of three charges q[1], q[2] and q[3] located at r[1],r[2]
and r[3] with respect to the common origin O. [1½]
Question 5.
Find the charge on the capacitor as shown in the figure. [1½]
Question 6.
Derive an expression for drift velocity of free electrons in a conductor in terms of relaxation time. [1½]
Question 7.
In a meter bridge, the null point is found at a distance of 40 cm from A. If a resistance of 12Ω is connected in parallel with S. Then null point occurs at 50 cm from A. Determine the value of R and
Question 8.
State Faraday’s law of electromagnetic induction. [1½]
Question 9.
A jet plane is travelling towards west at a speed of 1800 km/h. What is the voltage difference developed between the ends of the wing having a span of 25m, if the earth’s magnetic field at the
location has magnitude of 5 x 10^-4 T and the dip angle is 30°? [1½]
Question 10.
Flow does focal length of a lens change when red light incident on it is replaced by violet light Give, reason for your
answer. [1½]
Question 11.
(i) Why does the sun appear reddish at Sunset or Sunrise?
fii) For which colour, the refractive index of pism material is maximum and minimum? [1½]
Question 12.
Write the three distinct advantages of a reflecting type telescope over a refracting type telescope. [1½]
Question 13.
The near point of a hypermetropic person is 50 cm from the eye. What is the power of the lens required to enable the person to read clearly a book held at 25 cm from the eye? [1½]
Question 14.
A radioactive nucleus has a decay constant λ = 0.3465 (day)^-1, How long would it take the nucleus of decay to 75% of
its initial amount? [1½]
Question 15.
Define the Q-value of a nuclear process, when can a nuclear process not proceed spontaneously? [1½]
Question 16.
Write the expression for the force \(\overrightarrow{\mathrm{F}}\) acting on a particle of mass m and charge q moving with velocity \(\vec{v}\) in a magnetic field \(\overrightarrow{\mathrm{B}}\)
Under what conditions will it move in (a) a circular path and (b) a helical path? [3]
Obtain an expression for the force experienced by a current-carrying wire in a magnetic field. [3]
Question 17.
Use the mirror equation to show that:
(a) An object placed between / and 2f of a concave mirror produces a real image beyond If,
(b) A convex mirror always produces a virtual image independent of the location of the object.
(c) An object placed between the pole and the focus of a concave mirror produces a virtual and enlarged image. [3]
Plot a graph to show variation of the angle of deviation as a function of angle of incidence for light passing through a prism. Derive an expression for refractive index of the prism interms of angle
of minimum deviation and angle of prism. [3]
Question 18.
A beam of monochromatic radiation is incident on a photosensitive surface. Answer the following questions giving reasons.
(a) Do the emitted photoelectron have the same kinetic energy?
(b) Does the kinetic energy of the emitted electrons depend on the intensity of incident radiation?
(c) On what factors does the number of emitted photoelectrons depend? [3]
Write Einstein’s photoelectric equation and point out any two characteristic properties of photons on which this equation is based. Briefly explain three observed features which can be explained by
this.equation. [3]
Question 19.
State Gauss’s law. A thin straight infinitely long conducting wire of linear charge density X is enclosed by a cylindrical
surface of radius r and length l. Its axis coinciding the expression for the electric field, indicating its direction at a point on the surface of the cylinder. [4]
(i) Define electric flux.
(ii) Using Gauss’s law, prove that the electric field at a point due to a uniformly charged infinite plane sheet is independent of distance from it. How is the field directed if (a) the sheet is
positively charged ? (b) negatively charged. [4]
Question 20.
(i) How is a depletion region formed in p -n junction?
(ii) With the help of a labelled circuit diagram. Explain how a junction diode is used as a full-wave rectifier.
Draw its input, output wavefront. [4]
Give logic symbol, boolean expression and truth table for logic gate: OR gate, AND gate, NOT gate and NAND gate. [4] | {"url":"https://www.rbsesolutions.com/rbse-class-12-e-physics-self-evaluation-test-papers-in-english/","timestamp":"2024-11-11T11:27:18Z","content_type":"text/html","content_length":"148732","record_id":"<urn:uuid:42c0a682-079b-4d65-bc28-c8c6dbeeda79>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00003.warc.gz"} |
How do I find the polar equation for y = 5? | Socratic
How do I find the polar equation for #y = 5#?
1 Answer
We can easily convert equations from cartesian coordinates to polar coordinates by making the substitutions $x = r \cdot \cos \left(\theta\right)$ and $y = r \cdot \sin \left(\theta\right)$.
Therefore, the equation $y = 5$ will be rewritten as $r \cdot \sin \left(\theta\right) = 5 \to r = \frac{5}{\sin} \left(\theta\right)$//
Never forget the most important:
$x = r \cdot \cos \left(\theta\right)$
$y = r \cdot \sin \left(\theta\right)$
Hope it helps.
Impact of this question
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558 Gradian/Square Microsecond to Arcsec/Square Month
Gradian/Square Microsecond [grad/μs2] Output
558 gradian/square microsecond in degree/square second is equal to 502200000000000
558 gradian/square microsecond in degree/square millisecond is equal to 502200000
558 gradian/square microsecond in degree/square microsecond is equal to 502.2
558 gradian/square microsecond in degree/square nanosecond is equal to 0.0005022
558 gradian/square microsecond in degree/square minute is equal to 1807920000000000000
558 gradian/square microsecond in degree/square hour is equal to 6.508512e+21
558 gradian/square microsecond in degree/square day is equal to 3.748902912e+24
558 gradian/square microsecond in degree/square week is equal to 1.83696242688e+26
558 gradian/square microsecond in degree/square month is equal to 3.473138885688e+27
558 gradian/square microsecond in degree/square year is equal to 5.0013199953907e+29
558 gradian/square microsecond in radian/square second is equal to 8765043503515.5
558 gradian/square microsecond in radian/square millisecond is equal to 8765043.5
558 gradian/square microsecond in radian/square microsecond is equal to 8.77
558 gradian/square microsecond in radian/square nanosecond is equal to 0.0000087650435035155
558 gradian/square microsecond in radian/square minute is equal to 31554156612656000
558 gradian/square microsecond in radian/square hour is equal to 113594963805560000000
558 gradian/square microsecond in radian/square day is equal to 6.5430699152003e+22
558 gradian/square microsecond in radian/square week is equal to 3.2061042584482e+24
558 gradian/square microsecond in radian/square month is equal to 6.0617708934303e+25
558 gradian/square microsecond in radian/square year is equal to 8.7289500865396e+27
558 gradian/square microsecond in gradian/square second is equal to 558000000000000
558 gradian/square microsecond in gradian/square millisecond is equal to 558000000
558 gradian/square microsecond in gradian/square nanosecond is equal to 0.000558
558 gradian/square microsecond in gradian/square minute is equal to 2008800000000000000
558 gradian/square microsecond in gradian/square hour is equal to 7.23168e+21
558 gradian/square microsecond in gradian/square day is equal to 4.16544768e+24
558 gradian/square microsecond in gradian/square week is equal to 2.0410693632e+26
558 gradian/square microsecond in gradian/square month is equal to 3.85904320632e+27
558 gradian/square microsecond in gradian/square year is equal to 5.5570222171008e+29
558 gradian/square microsecond in arcmin/square second is equal to 30132000000000000
558 gradian/square microsecond in arcmin/square millisecond is equal to 30132000000
558 gradian/square microsecond in arcmin/square microsecond is equal to 30132
558 gradian/square microsecond in arcmin/square nanosecond is equal to 0.030132
558 gradian/square microsecond in arcmin/square minute is equal to 108475200000000000000
558 gradian/square microsecond in arcmin/square hour is equal to 3.9051072e+23
558 gradian/square microsecond in arcmin/square day is equal to 2.2493417472e+26
558 gradian/square microsecond in arcmin/square week is equal to 1.102177456128e+28
558 gradian/square microsecond in arcmin/square month is equal to 2.0838833314128e+29
558 gradian/square microsecond in arcmin/square year is equal to 3.0007919972344e+31
558 gradian/square microsecond in arcsec/square second is equal to 1807920000000000000
558 gradian/square microsecond in arcsec/square millisecond is equal to 1807920000000
558 gradian/square microsecond in arcsec/square microsecond is equal to 1807920
558 gradian/square microsecond in arcsec/square nanosecond is equal to 1.81
558 gradian/square microsecond in arcsec/square minute is equal to 6.508512e+21
558 gradian/square microsecond in arcsec/square hour is equal to 2.34306432e+25
558 gradian/square microsecond in arcsec/square day is equal to 1.34960504832e+28
558 gradian/square microsecond in arcsec/square week is equal to 6.613064736768e+29
558 gradian/square microsecond in arcsec/square month is equal to 1.2503299988477e+31
558 gradian/square microsecond in arcsec/square year is equal to 1.8004751983407e+33
558 gradian/square microsecond in sign/square second is equal to 16740000000000
558 gradian/square microsecond in sign/square millisecond is equal to 16740000
558 gradian/square microsecond in sign/square microsecond is equal to 16.74
558 gradian/square microsecond in sign/square nanosecond is equal to 0.00001674
558 gradian/square microsecond in sign/square minute is equal to 60264000000000000
558 gradian/square microsecond in sign/square hour is equal to 216950400000000000000
558 gradian/square microsecond in sign/square day is equal to 1.249634304e+23
558 gradian/square microsecond in sign/square week is equal to 6.1232080896e+24
558 gradian/square microsecond in sign/square month is equal to 1.157712961896e+26
558 gradian/square microsecond in sign/square year is equal to 1.6671066651302e+28
558 gradian/square microsecond in turn/square second is equal to 1395000000000
558 gradian/square microsecond in turn/square millisecond is equal to 1395000
558 gradian/square microsecond in turn/square microsecond is equal to 1.4
558 gradian/square microsecond in turn/square nanosecond is equal to 0.000001395
558 gradian/square microsecond in turn/square minute is equal to 5022000000000000
558 gradian/square microsecond in turn/square hour is equal to 18079200000000000000
558 gradian/square microsecond in turn/square day is equal to 1.04136192e+22
558 gradian/square microsecond in turn/square week is equal to 5.102673408e+23
558 gradian/square microsecond in turn/square month is equal to 9.6476080158e+24
558 gradian/square microsecond in turn/square year is equal to 1.3892555542752e+27
558 gradian/square microsecond in circle/square second is equal to 1395000000000
558 gradian/square microsecond in circle/square millisecond is equal to 1395000
558 gradian/square microsecond in circle/square microsecond is equal to 1.4
558 gradian/square microsecond in circle/square nanosecond is equal to 0.000001395
558 gradian/square microsecond in circle/square minute is equal to 5022000000000000
558 gradian/square microsecond in circle/square hour is equal to 18079200000000000000
558 gradian/square microsecond in circle/square day is equal to 1.04136192e+22
558 gradian/square microsecond in circle/square week is equal to 5.102673408e+23
558 gradian/square microsecond in circle/square month is equal to 9.6476080158e+24
558 gradian/square microsecond in circle/square year is equal to 1.3892555542752e+27
558 gradian/square microsecond in mil/square second is equal to 8928000000000000
558 gradian/square microsecond in mil/square millisecond is equal to 8928000000
558 gradian/square microsecond in mil/square microsecond is equal to 8928
558 gradian/square microsecond in mil/square nanosecond is equal to 0.008928
558 gradian/square microsecond in mil/square minute is equal to 32140800000000000000
558 gradian/square microsecond in mil/square hour is equal to 1.1570688e+23
558 gradian/square microsecond in mil/square day is equal to 6.664716288e+25
558 gradian/square microsecond in mil/square week is equal to 3.26571098112e+27
558 gradian/square microsecond in mil/square month is equal to 6.174469130112e+28
558 gradian/square microsecond in mil/square year is equal to 8.8912355473613e+30
558 gradian/square microsecond in revolution/square second is equal to 1395000000000
558 gradian/square microsecond in revolution/square millisecond is equal to 1395000
558 gradian/square microsecond in revolution/square microsecond is equal to 1.4
558 gradian/square microsecond in revolution/square nanosecond is equal to 0.000001395
558 gradian/square microsecond in revolution/square minute is equal to 5022000000000000
558 gradian/square microsecond in revolution/square hour is equal to 18079200000000000000
558 gradian/square microsecond in revolution/square day is equal to 1.04136192e+22
558 gradian/square microsecond in revolution/square week is equal to 5.102673408e+23
558 gradian/square microsecond in revolution/square month is equal to 9.6476080158e+24
558 gradian/square microsecond in revolution/square year is equal to 1.3892555542752e+27 | {"url":"https://hextobinary.com/unit/angularacc/from/gradpmicros2/to/arcsecpm2/558","timestamp":"2024-11-11T07:07:41Z","content_type":"text/html","content_length":"114933","record_id":"<urn:uuid:fa241c10-806a-4055-be4a-76a8757e68c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00442.warc.gz"} |
Lissajous Figures
Input and output waveforms of phase shifting circuits are plotted along orthogonal axes to obtain lissajous figures. At 90 degree phase shift, the normalized lissajous figure is circular.
A sinusoidal voltage is applied to a series RC circuit and the voltages across R and C are plotted. The resulting trace becomes a circle when R = Zc. The frequency is varied until the condition Xmax
= Ymax is met.
The phase shift to be calculated is that between the drop across the resistor R, and the capacitor C. However, our measurements are with respect to ground(GND), and therefore, while A2 measures the
drop between R and Ground as required, A1 is measuring the voltage across R+C .
Since we need the drop across the capacitor C, it suffices to subtract A2(R) from A1(R+C) . This is done by the Python code, and we plot A2(R) vs (A1-A2) .
When the shape becomes circular, ZC will equal R . Confirm this from the values you used. | {"url":"https://csparkresearch.in/expeyes17/electrical/lissajous.html","timestamp":"2024-11-10T18:27:54Z","content_type":"text/html","content_length":"20378","record_id":"<urn:uuid:42e812a5-8469-4026-9825-15a4eaba6c71>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00658.warc.gz"} |
Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking Situations
ATR National Key Laboratory of Defense Technology, Shenzhen University, Shenzhen 518060, China
Department of Computer Science and Engineering, Shaoxing University, Shaoxing 312000, China
College of Software Engineering, Lanzhou Institute of Technology, Lanzhou 730050, China
Jožef Stefan International Postgraduate School, Jamova cesta 29, 1000 Ljubljana, Slovenia
Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
Faculty of Computer and Information Science, University of Ljubljana, Večna pot 113, 1000 Ljubljana, Slovenia
Author to whom correspondence should be addressed.
Submission received: 22 November 2018 / Revised: 8 December 2018 / Accepted: 8 December 2018 / Published: 13 December 2018
To track multiple maneuvering targets in cluttered environments with uncertain measurement noises and uncertain target dynamic models, an improved joint probabilistic data association-fuzzy recursive
least squares filter (IJPDA-FRLSF) is proposed. In the proposed filter, two uncertain models of measurements and observed angles are first established. Next, these two models are further employed to
construct an additive fusion strategy, which is then utilized to calculate generalized joint association probabilities of measurements belonging to different targets. Moreover, the obtained
probabilities are applied to replace the joint association probabilities calculated by the standard joint probabilistic data association (JPDA) method. Considering the advantage of the fuzzy
recursive least squares filter (FRLSF) on tracking a single maneuvering target, which can relax the restrictive assumption of measurement noise covariances and target dynamic models, FRLSF is still
used to update the state of each target track. Thus, the proposed filter can not only provide the advantage of FRLSF but can also adjust the weights of measurements and observed angles in the
generalized joint association probabilities adaptively according to their uncertainty. The performance of the proposed filter is evaluated in two experiments with simulation data and real data. It is
found to be better than the performance of other three filters in terms of the tracking accuracy and the average run time.
1. Introduction
The multiple maneuvering target tracking (MMTT) becomes a critical problem of multiple target tracking in cluttered environments because of various uncertainties in the tracking process [
] such as uncertain measurement noises and uncertain target dynamic models. In practical tracking situation, the noise covariances of measurements are unknown, and they are uncertain. Similarly, the
real target dynamic models are unknown or only assumed, and they are also uncertain. The main procedure of MMTT consists of data association and state estimation. Data association denotes
distinguishing the real sources of measurements from targets or clutters, namely that the real source may be the real target, the false target or the clutter generated by the observed environment. It
mainly concerns the problems related to clutter, noises and errors in the tracking process [
]. Currently, the existing data association methods include the standard nearest neighbor (NN) method [
], the probabilistic data association (PDA) method [
] and the joint probabilistic data association (JPDA) method [
]. In particular, the JPDA method is a well-known and effective data association method for multiple target tracking. It employs an association gate to prune away impossible hypotheses and then
calculates the probability of a possible hypothesis on each target. However, it only uses the current measurements belonging to the target and does not utilize the historical measurements and the
related motion information to calculate the association probabilities. Hence, it is still a suboptimal Bayesian algorithm [
Following data association, state estimation is used to estimate the target states according to the associated measurements. Under the hypothesis that both measurement noise covariances and target
dynamic models are known, the traditional maneuvering target tracking methods can achieve perfect tracking performance [
]. Unfortunately, this hypothesis is difficult to satisfy in practical applications because of various uncertainties in the tracking process [
]. To solve the uncertain target dynamic model problem in target tracking, one strategy is to describe the unknown dynamic model of a target trajectory as several typical dynamic models with known
parameters or their combination. The interacting multiple model (IMM) method is a representative algorithm to solve this problem, and its modified versions are continually developed in many practical
applications [
]. However, once the IMM and its modified versions employ the assumed mismatched dynamic models, their tracking performance becomes undesirable [
]. The other strategy is to assume the unknown parameters of the target dynamic model as the random variables with a certain probability distribution function [
]. Unfortunately, because the actual target dynamic models and the tracking environments change over time, it is difficult to obtain the prior information of the unknown parameters in practical
applications. In addition, the particle filter is also broadly applied in maneuvering target tracking [
]. For nonlinear non-Gaussian motion models, an interacting multiple model particle filter (IMM-PF) was proposed in [
]. However, since the computational complexity of the IMM-PF is increased in proportion to the number of particles in target tracking, it is still difficult for IMM-PF to satisfy the requirements of
a real-time tracking system.
According to the above analysis, one must combine data association and state estimation for MMTT. The interacting multiple model-joint probabilistic data association filter (IMM-JPDAF) is a typical
tracking method in MMTT and many of its modified versions have been proposed for different application scenarios [
]. However, most the MMTT methods are based on the statistical framework of the statistics theory under the assumption that the known measurement noise covariance and target dynamic models are known.
In fact, the related prior information of measurement noises and target dynamic models are difficult to obtain in practical applications. This presents a great difficulty in implementing MMTT.
Moreover, each node in a sensor network must process a growing number of data because of the large surveillance scale and a great number of sensors. In this big data situation riddled with
imprecision and uncertainty, the traditional MMTT methods have higher computational complexity and processing requirements of the computation complexities while the MMTT methods based on the
statistical theory become increasingly complicated. Considering that the fuzzy theory possesses the unique advantage in processing inaccurate and uncertain information, it has been widely applied in
target tracking [
]. The fuzzy recursive least squares filter method (FRLSF) proposed in [
] utilizes measurement residuals and heading changes as two input variables of the designed fuzzy system, and this system is further employed to adjust the fading factor and realize single
maneuvering target tracking. The improvement of FRLSF combined with the PDA algorithm is utilized in cluttered environments [
], thus providing an effective method to solve the MMTT problem.
The incorporation of the motion features on a moving target into a MMTT method is a good strategy to improve the accuracy of data association [
]. In particular, the observed angle is an important motion feature on maneuvering targets and is often employed in maneuvering target tracking [
]. Therefore, it can be introduced into the calculation of association probabilities to improve the association accuracy. Based on these facts, we have presented at a conference the generalized joint
probabilistic data association-FRLSF (GJPDA-FRLSF) for communication [
]. This filter attempts to incorporate multiple motion features into the calculation of association probabilities and is able to achieve better tracking accuracy. However, it is difficult to
illustrate how and why they can improve the performance of association results. By modifying GJPDA-FRLSF, we further propose the improved JPDA-FRLSF (IJPDA-FRLSF) for MMTT. The GJPDA-FRLSF method
only employs the observed angles and measurements to calculate the association probabilities and adopts simulation data to analyze the effectiveness of the weights of the observed angles and
measurements in the association results by the additive fusion strategy. For clarity, this paper extends and updates the previous work [
]. In addition, a real-data experiment is added to illustrate the feasibility of the proposed IJPDA-FRLSF. In IJPDA-FRLSF, two uncertain models of measurements and observed angles are estimated.
Next, an adaptive additive fusion strategy is developed to calculate the generalized joint association probabilities for reconstructing the joint association probabilities of measurements belonging
to different targets in JPDA. Hence, IJPDA-FRLSF can adjust the weights of measurements and observed angles in the association results according to their uncertainty. In addition, considering the
advantage of the FRLSF method mentioned above, it is employed to update the state of each target trajectory.
The rest of this paper is organized as follows. In
Section 2
, two uncertain models of measurements and observed angles are established, and an uncertain fusion strategy is constructed to calculate the generalized joint association probability.
Section 3
presents a simplified form of FRLSF. IJPDA-FRLSF is proposed for MMTT in
Section 4
Section 5
presents the experimental results and the performance comparisons with the existing algorithms. Finally, the conclusions are provided in
Section 6
2. Uncertain Models and Fusion of Measurements and Observed Angles
In practical applications, the performance of a MMTT method greatly depends on the quality of uncertain measurements from each sensor. Hence, one must measure the uncertainty of measurements first.
Considering that observed angles are extracted from uncertain measurements, they also possess uncertainty to some extent, which is defined in the following subsection. As a result, one must measure
the uncertainties of measurements and observed angles and then combine them to calculate the associated probabilities.
2.1. Uncertain Models of Measurements and Observed Angles
To provide a quantitative description of the uncertainties of measurements and observed angles at all times in a cluttered environment, such as shown in
Figure 1
(only the measurements and the clutters in the given associated gate are shown here), we make the following definitions:
Definition 1.
The uncertain measure $ω k$ of measurements is defined by
$ω k = H ( p k t ) / σ k$
Here,$p k t$is the simple expression of$p ( z k , i | x k t )$,$z k , i$and$p ( z k , i | x k t )$denote the$i$th measurement and its statistical probability belonging to the state$x k t$of the
$σ k$
$H ( p k t )$
denote the standard deviation and the statistical entropy of measurements, respectively, calculated by
$σ k = ∑ i = 1 m k [ ( z k , i − z ^ k | k − 1 t ) T ( z k , i − z ^ k | k − 1 t ) ] 1 / 2 / m k g z$
$H ( p k t ) = − ∑ i = 1 m k p ( z k , i | x k t ) ln p ( z k , i | x k t )$
$z ^ k | k − 1 t$
$m k$
denotes the predicted measurement and the number of measurements, respectively while
$g z$
is the given association gate. In Equation (1),
$σ k$
describes the clustering feature of measurements at time
, and
$H ( p k t )$
describes the distribution of statistical probabilities assigned to the measurements. For easy calculation in the following section, one can normalize the uncertain measure
$ω k$
$ω k ′ = ω k / ω max$
, where
$ω max$
is the maximum value for all observed times.
Definition 2.
The uncertain measure$ω ˜ k$of the observed angles is defined by
$ω ˜ k = H ˜ ( u k t ) / σ ˜ k$
Here,$u k t$is the simple expression of$u ( ϕ k , i | x k t )$,$u ( ϕ k , i | x k t )$is the corresponding fuzzy membership degree belonging to the state$x k t$of the target$t$
$σ ˜ k$and$H ˜ ( u k t )$denote the standard deviation and the fuzzy entropy of observed angles calculated by
$σ ˜ k = ∑ i = 1 n k [ ( ϕ k , i − ϕ ^ k | k − 1 t ) ( ϕ k , i − ϕ ^ k | k − 1 t ) ] 1 / 2 / n k g θ$
$H ˜ ( u k t ) = − ∑ i = 1 n k u ( ϕ k , i | x k t ) ln u ( ϕ k , i | x k t )$
where$ϕ k , i$denotes the$i$th observed angle, and$ϕ ^ k | k − 1 t$denotes the course angle for the target$t$
, as shown in
Figure 2
, which can be calculated by Equations (7) and (8). In addition,
$n k$is the number of observed angles, and$g ϕ$
is the given association gate.
$ϕ k , i = arctan [ ( y k , i − y ^ k − 1 t ) / ( x k , i − x ^ k − 1 t ) ]$
$ϕ ^ k | k − 1 t = arctan [ ( y ^ k | k − 1 t − y ^ k − 1 t ) / ( x ^ k | k − 1 t − x ^ k − 1 t ) ]$
where $x k , i$, $x ^ k | k − 1 t$and $x ^ k − 1 t$are the components of measurement $z k , i$, predicted state $x ^ k | k − 1 t$and state estimate $x ^ k − 1 t$in the x-axis direction, respectively,
and $y k , i$, $y ^ k | k − 1 t$and $y ^ k − 1 t$are their corresponding components in the y-axis direction. From Equation (4), $σ ˜ k$describes the clustering feature of the observed angles, $H ˜ (
u k t )$describes the distribution of the fuzzy membership degrees assigned to the observed angles. For easy calculation in the following section, one can normalize uncertain measure $ω ˜ k$by $ω ˜ k
′ = ω ˜ k / ω ˜ max$, where $ω ˜ max$is the maximum value for all observed times.
2.2. Analyzing the Influence of Clutters on Measurements and Observed Angles
To analyze the influence of clutter on two uncertain models established in
Section 2.1
, we consider an example of a target moving with constant velocity in cluttered environments with five different clutter densities as shown in
Figure 1
Figure 3
Figure 4
show the corresponding variation curves of the uncertain measures of measurements and observed angles calculated by Equations (1) and (4), respectively.
As shown in
Figure 3
, both the non-normalizing and normalizing uncertain measure values increase with an increasing clutter density according to their corresponding variation curves, although the increasing rate of the
latter becomes relatively weaker because of normalization. Moreover, the uncertain measure value can vary with the clutter density at different times. Similarly to the uncertain measure of
measurements, the non-normalizing and the normalizing uncertain measure values of observed angles both present the same change trend of the clutter density. Hence, we can utilize the uncertain
measures of measurements and observed angles to describe their uncertainty accurately.
2.3. Uncertain Fusion of Measurements and Observed Angles
In multiple-sensor multiple-target tracking, information fusion is usually applied to improve the tracking performance by fusing multisource information [
]. To improve the association accuracy, one can incorporate multisource information to calculate the association probabilities of the measurements belonging to different targets. Two fusion
strategies including multiplicative fusion and additive fusion are often utilized to integrate multisource information as follows [
$p ( s k 1 , ⋯ , s k l | x k t ) = Π i = 1 l p ( s k i | x k t )$
$p ( s k 1 , … , s k l | x k t ) = ∑ i = 1 l w k , i p ( s k i | x k t )$
$w k , i$
is the weight of the information
$s k i$
$p ( s k i | x k t )$
denotes the probability of the information
$s k i$
belonging to the state
$x k t$
of the target
, and
denotes the order of the type of information.
Normally, the multiplicative fusion requires different information to be conditionally independent; however, most of the motion information in the tracking process is correlated in practical
applications [
]. Hence, one can utilize additive fusion to process measurements and observed angles in a uniform frame for the calculation the probability
$ρ k , i t$
$ρ ( z k , i , ϕ k , i | x k t ) = [ w k , 1 p ( z k , i | x k t ) + w k , 2 u ( ϕ k , i | x k t ) ] / c ρ$
Here, $w k , 1 = ( ω k ′ ) − 1$; $w k , 2 = ( ω ˜ k ′ ) − 1$; $ρ k , i t$ is also called as the generalized joint association probability; $c ρ$ is a normalizing constant. According to Equation (11),
one can adjust the weights of the statistical probability and the fuzzy membership degree adaptively in the generalized joint association probability according to their uncertainty.
3. Fuzzy Recursive Least Squares Filter (FRLSF)
Assuming that there is a single target
in the surveillance field, its dynamic model and measurement model are defined as follows:
$x k + 1 t = Φ k t x k t + w k t$
$z k , t = H k t x k t + v k t$
$x k t$
denotes an
-dimensional state vector of the target
$z k , t$
denotes an
-dimensional measurement vector,
$Φ k t$
is an
$n × n$
state transition matrix, and
$H k t$
is an
$m × n$
measurement transition matrix. The process noise
$w k t$
is assumed to be a Gaussian noise with zero mean and covariance
$Q k t$
, and the measurement noise
$v k t$
is assumed to be a Gaussian noise with zero mean and covariance
$R k t$
$E ( w k t ( w j t ) T ) = Q k t δ k j$
$E ( v k t ( v j t ) T ) = R k t δ k j$
$δ k j$
denotes the Kronecker delta function.
To track a single maneuvering target in situations with unknown measurement noise covariances, the FRLSF method proposed in [
] employs a fuzzy reference to adjust the fading factor of the recursive least squares filter (RLSF). It achieves good tracking performance if no clutter is present in surveillance. Next, the filter
is further extended for cluttered environments, and its simplified form is deduced in [
]. The main equations of the FRLSF are given as follows:
$x ^ k t = x ^ k − 1 t + K k t V k t$
$P k t = ( λ ˜ k t ) − 1 P k − 1 t − K k t S k t ( K k t ) T$
$P k t$
is an
$n × n$
filter covariance matrix;
$V k t$
$S k t$
$K k t$
$λ ˜ k t ∈ ( 0 , 1 ]$
are the
-dimensional predicted innovation, the
$n × n$
innovation covariance matrix, the
$n × n$
gain matrix, and the fuzzy fading factor. They can be respectively calculated using the following equations
$V k t = z ^ k , t − H k t Φ k t x ^ k − 1 t$
$S k t = ( λ ˜ k t ) − 1 H k P k − 1 t ( H k t ) T + I$
$K k t = P k − 1 t ( H k t ) T ( S k t ) − 1$
$λ ˜ k t = ∑ l = 1 L λ ¯ r l sup v k , t ∈ A ˜ i l , θ k , t ∈ B ˜ j l min ( u A ˜ i l ( v k , t ) , u B ˜ j l ( ϑ k , t ) ) ∑ l = 1 L sup v k , t ∈ A ˜ i l , θ k , t ∈ B ˜ j l min ( u A ˜ i l ( v k
, t ) , u B ˜ j l ( ϑ k , t ) )$
$z ^ k , t$
is the fused measurement on the target
as described in reference [
is an
$n × n$
unit matrix and
is the number of the designed fuzzy rules.
$A ˜ i l$
$B ˜ i l$
denote the corresponding fuzzy sets for the measurement residual
$v k , t$
and heading change
$ϑ k , t$
, respectively.
$λ ¯ r l$
is the corresponding value when the membership function of
$λ ˜ k t$
obtains the maximum value at each fuzzy set defined on;
$u A ˜ i l$
$u B ˜ j l$
denote their corresponding membership degrees.
$ϑ k , t$
$v k , t$
$z ^ k , t$
are calculated from equations
$ϑ k , t = | ϕ k , t − ϕ ^ k | k − 1 t | / ϑ max$
$v k , t = [ ( V k t ) T V k t ] 1 / 2 / v max$
$z ^ k , t = ∑ i = 0 m k β k , i t z k , i$
$z k , 0 = H k t Φ k t x ^ k t$
$ϑ max$
$v max$
are the corresponding empirical maximum value of the measurement residuals and the heading changes, respectively.
$z k , i$
is the
th measurement associated with the target
(in particular,
$z k , 0$
is called the zero measurement, and it denotes that there exist no measurements belonging to the target
$β k , i$
is the association probability of measurement
$z k , i$
belonging to the target
calculated by the standard probabilistic data association algorithm [
The estimation of the initial states of each trajectory is an important prerequisite to keep the stability of the tracking performance of the designed filter, and its solution strategy will be
studied in later in this paper. Here, we mainly focus on the performance of the designed filter after the two initial sampling points. To better illustrate its tracking performance, we assume that
the target move with a constant velocity at the two initial sampling points. Hence, similarly to RLSF, the initial states of FRLSF can be approximately estimated with measurements
$z 1$
$z 2$
$x ^ 2 = P 2 [ H 1 T , H 2 T ] [ z 1 T , z 2 T ] T$
$P 2 = ( [ H 1 T , H 2 T ] [ H 1 T , H 2 T ] T ) − 1$
which is also demonstrated in [
]. Here,
$H 1$
$H 2$
$m × m$
measurement transition matrices, and
$P 2$
is an
$n × n$
filter covariance matrix.
From Equations (16) and (17), FRLSF employs the fuzzy fading factor
$λ ˜ k$
to adaptively adjust the weight of the predicted innovation
$V k t$
in the state estimate
$x ^ k t$
of the target
through the designed fuzzy inference rules
$R l : IF v k , t ∈ A ˜ i l AND ϑ k , t ∈ B ˜ j l , THEN λ ˜ k l ∈ C ˜ r l$
is an
$m × m$
unit matrix while
$C ˜ r l$
$u C ˜ r l$
are assumed to be the fuzzy set and the membership function for
$λ ˜ k t$
, respectively. Here,
$u A ˜ i l$
$u B ˜ j l$
$u C ˜ r l$
adopt a triangular function given by Equation (28), as shown
Figure 5
$u ( x ) = 1 − | x − σ r l | c r l$
denotes a fuzzy variance, and
$[ c i l − σ i l , c i l + σ i l ]$
is the corresponding interval of the
th fuzzy set
$X ˜ i l$
4. Improved Joint Probabilistic Data Association-Fuzzy Recursive Least Squares Filter (IJPDA-FRLSF)
Based on the above analysis, we can employ generalized joint association probabilities to reconstruct joint association probability in the standard JPDA algorithm based on Equation (11) in
Section 2
, utilize FRLSF in
Section 3
to estimate the target states, and finally propose the IJPDA-FRLSF method for MMTT in cluttered environments with unknown measurement noise covariances and unknown target dynamic models. The main
procedures of the proposed filter are described as follows.
4.1. Calculating the Generalized Joint Association Probability
Let us assume that the validated measurement set at time k is $Z k = { z k , i } i = 1 m k$ where m[k] is a number of these measurements, and $Z k = { Z l } l = 1 k$ denotes the cumulative set of
validated measurements up to time k. According to Equation (11), the generalized joint association probability is composed of the statistical probability and the fuzzy membership degree. Next, we
further derive the expression of the generalized joint association probability in the JPDA frame as follows.
The statistical probability
$p k , i t$
and the fuzzy membership degree
$u k , i t$
in Equation (11) correspond to the joint association probability and the fuzzy association degree, respectively
$p ( z k , i | x k t ) = β k , i t$
$u ( ϕ k , i | x k t ) = e − ( ϕ k , i − ϕ ^ k | k − 1 t ) 2 / 2 ϕ max / c u$
$β k , i t$
is the joint association probability of measurement
$z k , i$
belonging to the target
determined by the difference between
$z k , i$
$x ^ k | k − 1 t$
(its detailed derivation can be found in [
] and is only briefly summarized below).
$u k , i t$
is the fuzzy association membership degree of the observed angles
$ϕ k , i$
associated with the target
$ϕ ^ k | k − 1 t$
denotes the corresponding course angle,
$θ max$
is the maximum value for all observed angles at each time; and
$c u$
is a normalizing constant. According to the standard JPDA method, the joint association probability can be expressed as follows
$β k , i t = ∑ θ k p ( e k | Z k ) ω ^ i t ( e k )$
$β k , 0 t = 1 − ∑ i = 1 m k β k , i t$
$p ( e k | Z k ) = 1 c Φ ! V Φ ∏ i = 1 m k N t i ( z k , i ) τ i ∏ i = 1 N ( P D t ) δ t ( 1 − P D t ) 1 − δ t$
$ω ^ i t ( e k )$
indicates a binary variable indicating whether the joint event
$e k$
contains the association of the track
with the measurement
$β k , 0 t$
is the probability that there exist no validated measurements belonging to the target,
$N t i ( z k , i )$
is the probability density of the predicted measurements belonging to the target
$t i$
$τ i$
is the number of targets associated with the measurement
$δ t$
is a target indicator indicating whether there is a measurement belonging to a the target
$δ t = 1$
) or not (
$δ t = 0$
is the number of clutters,
$P D t$
is the detection probability for the target
is the volume of the extension gates of the target, and
is the normalizing constant.
Based on the presented definitions, the generalized joint association probability
$ρ k , i t$
in Equation (11) can be modified as follows
$ρ ( z k , i , θ k , i | x k t ) = ( w k , 1 β k , i t + w k , 2 u k , i t ) / c ρ$
As for Equation (34), the generalized joint association probability can utilize the uncertainties of measurements and observed angles to adjust their weights in the association results.
4.2. The propsed IJPDA-FRLSF
Based on
Section 3
Section 4.1
, the main equations of the proposed filter can be described as follows:
• Step 1. Initialize state $x ^ 2 t$ and filter covariance $P 2 t$ of target $t$ for $t = 1 , 2 , ⋯ , n k$ using Equations (26) and (27), and start the recursive formulas at time $k = 3$.
• Step 2. Compute predicted innovation $V k , i t$ on measurement $z k , i$ using Equation (18).
• Step 3. Compute innovation covariance $S k t$ using Equation (19).
• Step 4. Compute gain matrix $K k t$ using Equation (20).
• Step 5. Reconstruct the generalized joint association probability $ρ k , i t$ using Equation (34).
• Step 6. Compute the fuzzy fading factor $λ ˜ k t$ using Equation (21).
• Step 7.
Update the target state
$x ^ k t$
and filter covariance
$P k t$
by FRLSF using Equations (35) and (36)
$x ^ k t = x ^ k − 1 t + K k t V k t$
$P k t = ( λ ˜ k t ) − 1 P k − 1 − ( 1 − ρ k , 0 t ) K k t S k t ( K k t ) T + ∑ i = 0 m k ρ k , i t [ x ^ k , i t ( x ^ k , i t ) T − x ^ k t ( x ^ k t ) T ]$
$x ^ k , i t$
is the local state estimate by FRLSF.
• Step 8. Repeat the steps 2–7 for the next iterations.
5. Experimental Results and Analysis
Two experiments using simulation data and real data have been conducted to evaluate the performance of the proposed IJPDA-FRLSF in comparison with the intuitionistic fuzzy-joint probabilistic data
association filter (IF-JPDAF) [
], the interacting multiple model-joint probabilistic data association filter (IMM-JPDAF) [
] and the improved joint probabilistic data association-recursive least squares filter (IJPDA-RLSF) in terms of the tracking accuracy and average run time. The experiments were conducted on a
computer with a computer with a dual-core CPU of Intel Core(TM) 2.93 GHz, 8-GB RAM. The programs for the four methods were implemented in MATLAB version 2014a and executed for 50 Monte Carlo runs.
Here, the 50 Monte Carlo runs denote that the data set of the true trajectories is fixed and the set of measurements is generated 50 times in the following two experiments. Furthermore, the estimated
error for each target denotes the average root-mean-square (RMS) position error for 50 Monte Carlo runs.
5.1. An Example of a Simulation Data Set: Two Crossing Targets
In the simulation scenario, there are two crossing targets moving in the air surveillance of a 2-D Cartesian
-plane according to the given trajectories, as shown in
Figure 6
. Their initial states are given as
$x ^ 0 1 = ( 0 m , 180 m / s , 2000 m , − 45 m / s ) T$
$x ^ 0 2 = ( 0 m , 180 m / s , 100 m , 45 m / s ) T$
. The motion processes of the two targets are further divided into five periods, as shown in
Table 1
. Here, the turn model of a moving target is approximately described by
$x k t = [ 1 sin ϖ k t T ϖ k t 0 − 1 − cos ϖ k t T ϖ k 0 cos ϖ k t T 0 − sin ϖ k t T 0 1 − cos ϖ k t T ϖ k 1 sin ϖ k t T ϖ k 0 sin ϖ k t T 0 cos ϖ k t T ] x k − 1 t + G k t w 1 , k t$
$G k t = [ T 2 / 2 T 0 0 0 0 T 2 / 2 T ] T$
$z k t = [ 1 0 0 0 0 0 1 0 ] x k t + w 2 , k t$
$x k t = ( x k t , x ˙ k t , y k t , y ˙ k t ) T$
is the state vector of target
$x k t$
$y k t$
are the
coordinates of the target
, respectively, and
$x ˙ k t$
$y ˙ k t$
are the corresponding velocities in the x and y coordinates.
$ϖ k = ± 0.0524 rad / s$
is the turn rate and
$T = 1 s$
is the sampling interval. In the CA period, the acceleration velocity is
$a x = 5 m / s 2$
. The detection probability of a true measurement
$P D$
is set to 1, and the gate probability
$P G$
is set to 0.99. For all the target dynamic models, the process noise
$w 1 , k t$
is assumed to be a Gaussian noise with zero mean and covariance
$Q k t = d i a g ( [ 20 2 m 2 s − 4 20 2 m 2 s − 4 ] )$
, that is,
$w 1 , k t ~ N ( 0 , Q k t )$
. The measurement noise
$w 2 , k t$
is also assumed to be a Gaussian noise with zero mean and covariance
$R k t = d i a g ( [ 50 2 m 2 50 2 m 2 ] )$
, that is,
$w 2 , k t ~ N ( 0 , R k t )$
. In addition, the clutter model is assumed to have a uniform distribution while the number of false measurements (or clutters) is assumed to follow the Poisson distribution with the known parameter
$λ = 1$
(the number of false measurements per unit of volume (km
Figure 6
shows the true crossing trajectories and their measurements, which correspond to the measurements of one of the 50 Monte Carlo cases. Considering that the performance of IMM-JPDAF is constrained by
the target dynamic models assumed, it is further divided into three types to facilitate the comparison as follows: (i) two dynamic models (constant velocity (CV) and constant acceleration (CA)) with
the mismatched measurement covariance
$R k t = d i a g ( [ 20 2 m 2 s − 4 20 2 m 2 s − 4 ] )$
; (ii) three dynamic models (CV, CA and constant turn (CT)) with a matched measurement noise covariance; (iii) three dynamic models (CV, CA and CT) with the mismatched measurement noise covariance
$R k t = d i a g ( [ 30 2 m 2 s − 4 3 0 2 m 2 s − 4 ] )$
. These three versions of IMM-JPDAFs are denoted as IMM-JPDAF(II), IMM-JPDAF(IIIA) and IMM-JPDAF(IIIB), respectively.
Figure 7
shows the tracking results of the four evaluated filters.
Figure 8
Figure 9
show the estimates of the fading factor
$λ ˜$
for Targets I and II. and
Figure 10
Figure 11
provide the average RMS position errors of all filters.
Figure 8
Figure 9
belong to one of the Monte Carlo cases of the estimates of the fading factors. As for
Figure 8
Figure 9
, the fading factor can reflect the changes of the maneuvering characteristics for the two targets correctly. The stronger the target maneuver, the smaller the fading factor. This is consistent with
the target maneuvering motion. Hence, employing the fading factor to adjust the proposed filter is an effective strategy.
With respect to
Figure 7
, all estimated filters achieve the correct association results despite the two targets crossing, but they achieve different performance in tracking accuracy. Obviously, IF-JPDAF and IJPDA-RLSF
perform well in the CV periods but perform poorly in the CA periods because they utilize the Kalman filter (KF) or RLSF, and these generally perform well for the CV model and are unsuitable for
maneuvering motion. Because the tracking performances of these two filters are poor for maneuvering motion, we don’t further analyze their performance below. Similarly, the three versions of
IMM-JPDAFs can also obtain a good tracking performance in the CV period.
As for
Figure 10
Figure 11
, IMM-JPDAF(IIIA) is found to perform better on the whole than the other three estimated filters based on the whole motion process because it employs the matched measurement noise covariance and
dynamic models. In addition, the performance of IJPDA-FRLSF is close to that of the IMM-JPDAF. Hence, when the measurement noise covariance and the target dynamic model are unknown or mismatched, the
tracing performances of both IMM-JPDAF(II) and IMM-JPDAF(IIIB) are unsatisfactory in maneuvering situations. In this situation, the performance of IJPDA-FRLSF is better than the one of IMM-JPDA(II)
and IMM-JPDAF(IIIB). One major reason for this case is that IJPDA-FRLSF utilizes FRLSF, which doesn’t require a known measurement noise covariance and dynamic models.
For a better illustration of the performances of the four evaluated filters,
Table 2
Table 3
provide the average RMS position errors of five motion periods and the whole motion process (without the initial two samples) on targets I and II, respectively. Here, the five motion periods for
target I in turn are: CV period (3–15 s), CT period (16 s), CA period (17–30 s), CT period (31 s), and CV period (32–36 s). Furthermore, the average RMS position errors of the five motion periods are
beneficial for analyzing the performance of each filter for different dynamic models.
According to the total average RMS position error in
Table 2
Table 3
, IMM-JPDAF(IIIA) performs better than IJPDA-RLSF, IMM-JPDAF(II) and IMM-JPDAF(IIIB) in the tracking accuracy. This finding is similar to the results obtained from
Figure 10
Figure 11
. Although the IMM-JPDAF(IIIA) approach can obtain the highest accuracy when its assumed measurement noise covariance and target dynamic models are matched with the corresponding real values, this
assumption is difficultly satisfactory in practice. In the situation with a mismatched measurement noise covariance and target dynamic models, the performance of the proposed filter is a close
approximate of IMM-JPDAF(IIIA). Hence, the tracking accuracy of each filter as shown in
Table 2
Table 3
is consistent with the presented analysis using
Figure 8
Figure 9
We further analyze the performance of each evaluated filter for different dynamic models. According to
Table 2
Table 3
, IMM-JPDAF(II) performs better than IMM-JPADF(IIIA) and IMM-JPDAF(IIIB) in the first CV period because it employs less, but better-matched dynamic models. IMM-JPDAF(IIB) obtains the worst results
because it utilizes mismatched dynamic models and a measurement noise covariance. This fact shows that employing the matched dynamic models and a measurement noise covariance directly affects the
performance of the three versions of IMM-JPDAFs directly. In the second CV period, the order of the performance of IMM-JPDAs only has a little change because their performance is also influenced by
the initial state estimates in this period, which are assumed to be equal as the first CV period. Hence, their performance in the second CV period is similar to the performance in the first CV
period. Compared with IMM-JPDAF(II) and IMM-JPDAF(IIIA), IJPDA-FRLSF is suboptimal in the CV period. However, it doesn’t need to satisfy the strict assumption of the measurement noise covariance and
dynamic models. Additionally, it is also hard to keep the assumption consistent with the real measurement noise covariance and dynamic models in practice.
Based on the average values calculated by the average RMS position errors of each filter in the four CT periods from
Table 2
Table 3
, IJPDA-FRLSF performs best in all filters while IMM-JPDAF(IIIB) performs worst. Moreover, IMM-JPDAF(IIIA) is marginally better from IMM-JPDAF(II) because IMM-JPDAF(IIIA) employs the matched dynamic
models as shown in the presented analysis. Thus, the order of the tracking performance from good to poor is: IJPDA-FRLSF, IMM-JPDAF(IIIA), IMM-JPDA F(II), and IMM-JPDAF(IIIB). Such order of the
tracking performance of four evaluated filters is still valid in the CA period. As a result, IJPDA-FRLSF performs better in tracking accuracy than the other three versions of IMM-JPDAFs in the
maneuvering periods (CT and CA).
With respect to the analysis presented above,
Table 4
summarizes the complete performance evaluation of four evaluated filters. Using
Table 4
, it is easy to illustrate the tracking performance of each filter for different dynamic models. It also shows the total performance evaluation for each filter. In addition, the average run time of
IJPDA-FRLSF, IJPDA-RLSF, IMM-JPDAF(II), IMM-JPDA(IIIA), IMM-JPDA(IIIB) and IF-JPDAF is 0.0083 s, 0.0130 s, 0.0214 s, 0.0248 s, 0.0235 s, and 0.1082 s, respectively. The average run time of
IJPDA-FRLSF is less than that of the other three types of the IMM-JPDAF. Because IMM-JPDAF must execute the filtering procedure for all sub-models, its average run time becomes longer with the
increasing number of sub-models. Then, both the IMM-JPDAF(IIIA) and the IMM-JPDAF(IIIB) methods consume more time than IMM-JPDAF(II). Moreover, because the execution of the fuzzy reference consumes a
certain amount of time, IJPDA-FRLSF requires more time than IJPDA-RLSF. The average run time of IF-JPDAF is the longest of all filters because the fuzzy clustering for all measurements consume a lot
of time.
In short, IJPDA-FRLSF can achieve satisfactory performance in tracking accuracy for MMTT. It has the advantage of efficiency and robustness compared with IMM-JPDAF, IF-JPDAF, and IJPDA-RLSF in
situations with the unknown measurement noise covariance and the target dynamic models. The tracking performance of IMM-IJPDAF is influenced by the assumed measurement noise covariance and dynamic
models. It can achieve a good tracking performance only if the assumed values are consistent with the real measurement noise covariance and dynamic models.
5.2. An Example of a Real Data Set: Three Crossing Targets
To further illustrate the feasibility of the proposed filter, a real data set is obtained from one of outfield experiments by using a single certain type of proximity radar, and the tracking targets
are the three crossing civil aviation aircrafts. This data set is utilized to evaluate the performance. The real data set of three crossing targets is shown in
Figure 12
. It consists of 147, 113 and 80 periodic track dots. The parameters of the clutter model are the same as in
Section 5.1
. The radar performance parameters are given as follows: sampling interval
$T = 10 s$
. The process noise covariance
$Q k t$
is set to
$d i a g ( [ 20 2 m 2 s − 4 20 2 m 2 s − 4 ] )$
, and the measurement noise covariance
$R k t$
is equal to
$d i a g ( [ 150 2 m 2 150 2 m 2 ] )$
. The initial positions of three targets are given by
$z 0 1 = [ − 122.15 km , − 18.26 km ] T$
$z 0 2 = [ 131.57 km , − 176.90 km ] T$
$z 0 3 = [ − 44.06 km , − 214.29 km ] T$
Because the dynamic models of the three targets in the real scenario are unknown and complex, it is difficult to design their matched sub-models. For simplicity to illustrate the feasibility, the
tracking results of IMM-JPDAF are unsatisfactory and even diverged so we only utilize the proposed filter for MMTT here.
Figure 13
Figure 14
Figure 15
show the average RMS position error of the proposed filter for the three targets on 50 Monte Carlo runs. According to the tracking results, the proposed filter can track MMT accurately in a real life
situation with unknown measurement noise covariances and target dynamic models. Hence, we can conclude that the proposed filter is feasible in a real MMTT applications.
6. Conclusions
This paper presented an improved joint probabilistic data association-fuzzy recursive least squares filter (IJPDA-FRLSF) for multiple maneuvering target tracking (MMTT) in situations with unknown
measurement noise covariances and unknown target dynamic models. In the proposed filter, two uncertain models of measurements and observed angles were established, and their related parameters were
analyzed in temporal and spatial sense. Using these two uncertain models, an additive fusion strategy was constructed to calculate the generalized joint association probabilities of measurements
belonging to different targets, which were utilized to replace the joint association probabilities of the standard joint probabilistic data association (JPDA) algorithm. The FRLSF method was utilized
to update all tracks. The proposed filter can relax the restrictive assumptions of measurement noise covariances and target dynamic models. It benefits from FRLSF by not requiring a maneuver detector
for a maneuvering target. Moreover, the filter can utilize multisource information to adjust the corresponding weights in the association results according to the uncertainties of measurements and
observed angles.
The application of the improved JPDA algorithm and the FRLSF method has been found to be effective in solving the data association and the state estimation problem for MMTT. The experimental results
on simulation data and real data illustrate that the proposed filter is effective and can be applied in situations with unknown measurement noise covariances and target dynamic models. The uncertain
relationship between measurements and observed angles will be further studied in our future work on uncertain target tracking.
Author Contributions
Conceptualization, investigation and writing, E.F.; supervision, W.X. and J.P.; review and editing, K.H., X.L., and V.P.
This work was funded by the National Natural Science Foundation of China (61703280, 61603258 and 61331021), the Plan Project of Science and Technology of Shaoxing City (2017B70056), the Qizhi Talent
Cultivation Project of Lanzhou Institute of Technology (2018QZ-09), the Youth Science and Technology Innovation Project of Lanzhou Institute of Technology (18K-020), the Project of Resources and
Environment Informatization Gansu International Science and Technology Cooperation Base, and the Shenzhen Science and Technology Projection JCYJ20170818143547435.
Conflicts of Interest
The authors declare no conflict of interest.
Target I Target II
Periods Time Periods Time
constant velocity (CV) 14 s constant velocity (CV) 14 s
constant turn (CT) 1 s constant turn (CT) 1 s
constant acceleration (CA) 14 s constant acceleration (CA) 14 s
constant turn (CT) 1 s constant turn (CT) 1 s
constant velocity (CV) 5 s constant velocity (CV) 6 s
Table 2. The average root-mean-square (RMS) position error for Target I (unit: m). Improved joint probabilistic data association-fuzzy recursive least squares filter (IJPDA-FRLSF), interacting
multiple model-joint probabilistic data association filter (IMM-JPDAF).
Filter CV CT CA CT CV
IJPDA-FRLSF 21.5 22.0 22.1 22.5 22.0
IMM-JPDAF(II) 14.8 13.3 32.0 37.3 37.0
IMM-JPDAF(IIIA) 17.5 16.9 26.2 35.3 25.0
IMM-JPDAF(IIIB) 22.7 23.5 35.3 47.2 38.5
Filter CV CT CA CT CV
IJPDA-FRLSF 22.1 21.9 22.1 22.6 21.8
IMM-JPDAF(II) 15.0 15.0 30.9 36.8 37.2
IMM-JPDAF(IIIA) 16.5 15.9 24.6 32.4 24.8
IMM-JPDAF(IIIB) 23.7 23.4 36.0 46.5 38.1
Filter CV CT CA Total
IJPDA-FRLSF mean good good fair
IMM-JPDAF(II) good mean mean mean
IMM-JPDAF(IIIA) fair fair fair good
IMM-JPDAF(IIIB) poor poor poor poor
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Fan, E.; Xie, W.; Pei, J.; Hu, K.; Li, X.; Podpečan, V. Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking
Situations. Information 2018, 9, 322. https://doi.org/10.3390/info9120322
AMA Style
Fan E, Xie W, Pei J, Hu K, Li X, Podpečan V. Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in Uncertain Tracking Situations.
Information. 2018; 9(12):322. https://doi.org/10.3390/info9120322
Chicago/Turabian Style
Fan, En, Weixin Xie, Jihong Pei, Keli Hu, Xiaobin Li, and Vid Podpečan. 2018. "Improved Joint Probabilistic Data Association (JPDA) Filter Using Motion Feature for Multiple Maneuvering Targets in
Uncertain Tracking Situations" Information 9, no. 12: 322. https://doi.org/10.3390/info9120322
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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What is Allstate Book Value Per Share for 2010 to 2024 | Stocks: ALL - Macroaxis
ALL Stock USD 196.90 6.35 3.33%
Allstate Book Value Per Share yearly trend continues to be quite stable with very little volatility. Book Value Per Share may rise above 381.68 this year. Book Value Per Share is the ratio of equity
available to common shareholders divided by the number of outstanding shares. This measure represents the value per share of The Allstate according to its financial statements.
View All Fundamentals
First Reported Previous Quarter Current Value Quarterly Volatility
Book Value Per Share
2010-12-31 363.5047619 381.68 138.8709985
Credit Downgrade Yuan Drop Covid
Check Allstate
financial statements
over time to gain insight into future company performance. You can evaluate financial statements to find patterns among Allstate's main balance sheet or income statement drivers, such as
Depreciation And Amortization of 705.6 M
, Interest Expense of 299.1
Total Revenue of 37 B
, as well as many indicators such as
Price To Sales Ratio of 0.89
, Dividend Yield of 0.0187 or
PTB Ratio of 1.62
. Allstate financial statements analysis is a perfect complement when working with
Allstate Valuation
Allstate Book Value Per Share
Check out the analysis of
Allstate Correlation
against competitors.
Latest Allstate's Book Value Per Share Growth Pattern
Below is the plot of the Book Value Per Share of The Allstate over the last few years. It is the ratio of equity available to common shareholders divided by the number of outstanding shares. This
measure represents the value per share of a company according to its financial statements. Allstate's Book Value Per Share historical data analysis aims to capture in quantitative terms the overall
pattern of either growth or decline in Allstate's overall financial position and show how it may be relating to other accounts over time.
View Last Reported 71.23 X 10 Years Trend
Allstate Book Value Per Share Regression Statistics
Arithmetic Mean 136.50
Geometric Mean 90.46
Coefficient Of Variation 101.74
Mean Deviation 117.53
Median 62.30
Standard Deviation 138.87
Sample Variance 19,285
Range 346
R-Value 0.73
Mean Square Error 9,796
R-Squared 0.53
Significance 0
Slope 22.57
Total Sum of Squares 269,992
Allstate Book Value Per Share History
Other Fundumenentals of Allstate
Allstate Book Value Per Share component correlations
About Allstate Financial Statements
Allstate investors utilize fundamental indicators, such as Book Value Per Share, to predict how Allstate Stock might perform in the future. Analyzing these trends over time helps investors make
market timing
decisions. For further insights, please visit our
fundamental analysis
Last Reported Projected for Next Year
Book Value Per Share 363.50 381.68
Tangible Book Value Per Share 346.48 363.81
Building efficient market-beating portfolios requires time, education, and a lot of computing power!
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When determining whether Allstate is a strong investment it is important to analyze Allstate's competitive position within its industry, examining market share, product or service uniqueness, and
competitive advantages. Beyond financials and market position, potential investors should also consider broader economic conditions, industry trends, and any regulatory or geopolitical factors that
may impact Allstate's
future performance
For an informed investment choice regarding Allstate Stock, refer to the following important reports:
Check out the analysis of
Allstate Correlation
against competitors. You can also try the
Piotroski F Score
module to get Piotroski F Score based on the binary analysis strategy of nine different fundamentals.
Is Property & Casualty Insurance space expected to grow? Or is there an opportunity to expand the business' product line in the future? Factors like these will boost
the valuation of Allstate
. If investors know Allstate will grow in the future, the company's valuation will be higher. The financial industry is built on trying to define current growth potential and future valuation
accurately. All the valuation information about Allstate listed above have to be considered, but the key to understanding future value is determining which factors weigh more heavily than others.
Quarterly Earnings Growth Dividend Share Earnings Share Revenue Per Share Quarterly Revenue Growth
(0.68) 3.65 15.99 236.821 0.147
The market value of Allstate
is measured differently than its book value, which is the value of Allstate that is recorded on the company's balance sheet. Investors also form their own opinion of Allstate's value that differs
from its market value or its book value, called intrinsic value, which is Allstate's true underlying value. Investors use various methods to calculate intrinsic value and buy a stock when its market
value falls below its intrinsic value. Because Allstate's market value can be influenced by many factors that don't directly affect Allstate's underlying business (such as a pandemic or basic market
pessimism), market value can vary widely from intrinsic value.
AltmanZ ScoreDetails PiotroskiF ScoreDetails BeneishM ScoreDetails FinancialAnalysisDetails Buy or SellAdviceDetails
Please note, there is a significant difference between Allstate's value and its price as these two are different measures arrived at by different means. Investors typically determine
if Allstate is a good investment
by looking at such factors as earnings, sales, fundamental and technical indicators, competition as well as analyst projections. However, Allstate's price is the amount at which it trades on the open
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Gang Tian : Geometry and Analysis of low-dimensional manifolds
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Gang Tian : Geometry and Analysis of low-dimensional manifolds
In this series of talks, I will focus on geometry and analysis of manifolds of dimension 2, 3 or 4. The first talk is a general introduction of this series. I will start the talk by reviewing some
classical theories on Riemann surfaces and their recent variations in geometric analysis. Then we survey some recent progress on 3- and 4-manifolds. I hope that this talk will show some clues how
geometric equations can be applied to studying geometry of underlying spaces. In the second talk, I will discuss recent works on the Ricci flow and its application to the geometrization of
3-manifolds, in particular, I will briefly discuss Perelman's work towards the Poincare conjecture. In last talk, I will discuss geometric equations in dimension 4 and how they can be applied to
studying geoemtry of underlying 4-spaces. Some recent results will be discussed and some open problems will be given.
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Cross Product: Introduction
The cross product of any 2 vectors u and v is yet ANOTHER VECTOR! In the applet below, vectors u and v are drawn with the same initial point. The CROSS PRODUCT of u and v is also shown (in brown) and
is drawn with the same initial point as the other two. Interact with this applet for a few minutes by moving the initial point and terminal points of both vectors around. Then, answer the questions
that follow.
Use GeoGebra to measure the angle at which the line containing u intersects the line containing the cross product vector. What do you get?
Use GeoGebra to measure the angle at which the line containing v intersects the line containing the cross product vector. What do you get?
Given your responses for (1) and (2) above, what can we conclude about the cross product of any two vectors with respect to both individual vectors themselves?
Is it possible to position vectors u and v so that their cross product = the zero vector? If so, how would these 2 vectors be positioned?
How would vectors u and v have to be positioned in order for their cross product to have the greatest magnitude? Use GeoGebra to help informally support your conclusion(s). | {"url":"https://www.geogebra.org/m/RrDv9Wea","timestamp":"2024-11-07T13:11:03Z","content_type":"text/html","content_length":"102890","record_id":"<urn:uuid:fc2690a1-e678-4398-b51e-e5629360b053>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00106.warc.gz"} |
Spatial is Not Special – Descriptive Statistics
OK, I admit it, this post really isn’t about “spatial”. It’s more about descriptive statistics, but I wanted to introduce the ability to use SQL to produce descriptive statistics like the mean,
standard deviation, skew, kurtosis, and coefficient of variation (CV). As we say in our book Statistical Problem Solving in Geography, CV is an underutilized tool that has great utility for
For the 2010 and 1980 State, Census Division, and Census Region, we show how the basic descriptive statistics change for different levels of spatial aggregation. But, what we don’t show is how easy
it is to calculate these statistics. Using the Census vector file, with fields for Total Population (Pop2000 and Pop2010), we can easily compute these descriptive statistics as shown in Table 3.11
from the book. Just note, that the dataset we are using in today’s example has the population from 2000 and 2010. Also, to keep things simple we aren’t going to compute the SKEW or KURTOSIS
(WordPress gets messy, and I’m more concerned that you understand what we are actually doing).
Lets first deal with the State level data for the year 2000.
SELECT “STATES 2000” AS [Year], AVG([POP2000]) AS Mean, STDEV([POP2000]) AS [Standard Deviation], STDEV([POP2000])/AVG([POP2000])*100 AS [Coefficient of
Variation] FROM [USStates]
You can see that we are using the AVG function to compute the average, the STDEV function to compute the Standard Deviation, and we are dividing those together to compute the Coefficient of
Variation. Easy enough. But, our table also includes the Census Division and Region. To compute these, we will issue the UNION ALL command in SQL which will take the results of the next query, and
insert it at the end of the previous query. So, for this query we have to remember to have the same field names returned. Let’s look at the Census Division:
SELECT “DIVISION 2000” AS [Year], AVG(PopByState) AS Mean, STDEV(PopByState) AS [Standard Deviation], STDEV(PopByState)/AVG(PopByState)*100 AS [Coefficient of Variation] FROM (SELECT
SUM([POP2000]) AS PopByState FROM [USStates] GROUP BY [DIVISION] )
Notice that the inner query is an aggregate clause, and we are going to GROUP the State populations by the DIVISION they are in (each State has a DIVISION code). That will return a table with all
the populations aggregated by the 9 Census Divisions. From that query, we then compute the average of the 9 aggregated populations along with the standard deviation and the coefficient of variation.
Finally, we can create a third query for the Census Regions as follows:
SELECT “REGION 2000” AS [Year], AVG(PopByState) AS Mean, STDEV(PopByState) AS [Standard Deviation], STDEV(PopByState)/AVG(PopByState)*100 AS [Coefficient of Variation] FROM (
SELECT SUM([POP2000]) AS PopByState FROM [USStates] GROUP BY [REGION] )
This is the same as the previous query (just a simple copy and paste) and changing [DIVISION] to [REGION]. That will provide the descriptive statistics for the 4 Census Regions.
Finally, once we have figured out how to write the component parts, we will string them all together to form a single query with the UNION ALL command:
SELECT “STATES 2000” AS [Year], avg([POP2000]) AS Mean, stdev([POP2000]) AS [Standard Deviation], stdev([POP2000])/avg([POP2000])*100 AS [Coefficient of Variation] FROM [USStates]
SELECT “DIVISION 2000” AS [Year], avg(PopByState) AS Mean, stdev(PopByState) AS [Standard Deviation], stdev(PopByState)/avg(PopByState)*100 AS [Coefficient of Variation] FROM (SELECT sum([POP2000])
AS PopByState FROM [USStates] GROUP BY [DIVISION] )
SELECT “REGION 2000” AS [Year], avg(PopByState) AS Mean, stdev(PopByState) AS [Standard Deviation], stdev(PopByState)/avg(PopByState)*100 AS [Coefficient of Variation] FROM (SELECT sum([POP2000]) AS
PopByState FROM [USStates] GROUP BY [REGION] ) UNION ALL
SELECT “STATES 2010” AS [Year], avg([POP2010]) AS Mean, stdev([POP2010]) AS [Standard Deviation], stdev([POP2010])/avg([POP2010])*100 AS [Coefficient of Variation] FROM [USStates]
SELECT “DIVISION 2010” AS [Year], avg(PopByState) AS Mean, stdev(PopByState) AS [Standard Deviation], stdev(PopByState)/avg(PopByState)*100 AS [Coefficient of Variation] FROM (SELECT sum([POP2010])
AS PopByState FROM [USStates] GROUP BY [DIVISION] )
SELECT “REGION 2010” AS [Year], avg(PopByState) AS Mean, stdev(PopByState) AS [Standard Deviation], stdev(PopByState)/avg(PopByState)*100 AS [Coefficient of Variation] FROM (SELECT sum([POP2010]) AS
PopByState FROM [USStates] GROUP BY [REGION] ) | {"url":"https://www.artlembo.com/post/spatial-is-not-special-descriptive-statistics","timestamp":"2024-11-10T07:42:52Z","content_type":"text/html","content_length":"1050479","record_id":"<urn:uuid:83f4b10e-2384-41c8-a2be-bd8a9b37edec>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00700.warc.gz"} |
How Quantum Computers Work (Oversimplified, No Complex Math)
As a software engineer of 15 years, I was very interested in how exactly quantum computers worked.
However, I couldn’t find a great explanation of quantum computers online, so I decided to create my own guide. One that wasn’t dead wrong or so filled with such complex math that no regular person
could understand it.
Quick disclaimer: I’m a software engineer and blogger. My understanding of quantum mechanics leaves a lot to be desired. This is meant to be an extremely high-level overview; if you’re looking for
more than that, I suggest asking questions on StackExchange.
Classical Computers vs. Quantum Computers
A classical computer uses a series of bits to describe information and instructions. We usually refer to bits as representative of 1’s or 0’s. But, we could just as easily use any binary system to
describe the value of a bit. So 1 or 0, true or false, heads or tails, etc.
Let’s use heads and tails for this example (because it will make it easier to understand quantum bits later).
That’s the gist of how a classical computer stores data and instructions. Quantum computers are extremely similar, the difference is they leverage three quantum properties that makes them special.
Let’s learn about those three properties!
#1) Wave/Particle Duality (Probability Waves)
Quantum objects (like a photon) travel through space as a probability wave until they are “observed” (interact with matter). At which point the probability wave collapses and the photon becomes a
particle with a definite position in space and time.
That may be common knowledge to you if you’re familiar with quantum mechanics. Or it probably sounds like non-sense if you’re not. If this is a new subject for you I suggest watching a video on the
double slit experiment to familiarize yourself with how this works.
A quantum bit (qubit) also acts as a probability wave until “observed.” You can think of a quantum bit like a spinning quarter (pre-observation) as opposed to a classical bit that always has a
definitive value.
Post-observation, the quantum bit transforms into a classical bit. It obtains a definitive heads or tails value and effectively becomes no different than a classical bit post-observation.
We’ll get to why this property is essential for quantum computing later. For now, Quantum Mechanics concept #2!
#2) Quantum SuperPosition
If you add two quantum bits together, the result becomes a new quantum state (probabilistic in nature, until observed). That’s quantum superposition.
If we add two bits together with classical computers, we’ll get a definitive result. But, when a quantum computer adds two qubits together, we get a 3rd probabilistic quantum state dependent on the
One more concept before we get to why this is valuable!
#3) Quantum Entanglement
Quantum entanglement is the idea that the quantum states that you added are linked together. Meaning if you observe the sum (the result of adding two quantum bits), then the entire probability wave
collapses, and the addends automatically get actual values too. The whole system will go back to behaving like a classical computer once observed.
Note: This is all you need to know. Quantum mechanics is way more complicated than this for two reasons (You don’t need to know this, it’s just cool).
1) The probabilities of quantum bits don’t have to be 50/50
Odds of each bit can be set to anything (as long as they add to 100). See this excellent video by Veritasium to understand how we can set the probability of a quantum bit.
2) The probabilities of quantum bits are calculated using complex numbers
The math behind quantum probability uses complex numbers (imaginary numbers). This means the spinning quarters analogy I gave you is overly simplistic. You would need to use vector math to calculate
probabilities instead (which is much harder). If you’re interested in this subject, see this fantastic video by Science Asylum on how the quantum wave function works.
Why does the quantum wave function behave in this strange way?
Scientists aren’t 100% certain. There are many weird interpretations of quantum mechanics derived from this complex math. The leading theory is the Copenhagen interpretation (universe splits into all
probabilistic outcomes then collapses back). Still, there’s also crazy stuff like The Many World’s hypotheses (universe splits into all probabilistic outcomes and doesn’t collapse back).
The good news is we don’t need to answer these things to understand quantum computers.
Why Are Quantum Computers Better Than Classical Computers?
By now, you’re probably wondering how a bunch of probabilistic bits are better than a classical computer. Well, for most computing problems, quantum computers aren’t any better!
For instance, pretend we wanted to represent the letter ‘A’ in bits. Probabilistic bits don’t help with that task! In fact, they’d be quite detrimental. And if you think about it, most computing
problems require definitive values. Meaning it’s rare that a quantum computer would come in handy for most computing tasks.
Then why do we say quantum computers will do insane things like break all internet encryption and potentially cure cancer one day?
It has to do with the computer science concept of P vs. NP. This is where we ask if every problem that can be easily verified can also be easily solved. A problem that can be solved quickly is solved
in polynomial time (P). And problems that increases exponentially as you grow is solved in nondeterministic polynomial time (NP).
Below is an excellent video on the subject that can explain it better than I ever could (if you need a refresher on the subject).
The only place quantum computers beat classical computers is when we can leverage a Quantum Algorithms to solve a current NP problem in P time. That’s it. I don’t want to understate how big a deal
this is (because it’s a huge deal for specific applications). But, I also need you to understand that this is all that quantum computers are good for.
So what the heck is a quantum algorithm, and how does that work?
Quantum Algorithms
The most commonly cited examples of quantum algorithms are Shor’s Algorithm, Deutsch–Jozsa algorithm, Grover’s algorithm, etc. And I’d be willing to bet many more quantum algorithms get invented or
become mainstreamed as quantum computing resources become widely available.
If you’re wondering how quantum algorithms work, Minute Physics made the amazing video below specifically about Shor’s Algorithm. It explains this much more clearly than I could. I’m just going to
give you the oversimplified version.
The simple explanation is that quantum algorithms are designed to exploit the probability waves’ constructive and destructive interference patterns. This allows the algorithm to brute-force the
answer in fewer steps (to the point where an NP problem becomes a P problem). How does this work?
When a wave collides with another wave their amplitudes will either add together or cancel each other out.
As we discussed previously, when we add quantum bits, we’re effectively adding two probability waves. These bits will naturally produce interference patterns (since qubits are composed of probability
waves). Those interference patterns can significantly reduce the number of calculations needed to solve the problem.
What People Get Wrong About Quantum Computers
Many articles online say crazy things like the following.
• 100 qubits added together can hold more data than atoms in the universe (they can’t).
• A quantum computer can process tons of classical states in parallel (it can’t).
• Lots of other non-sense.
At the end of the day, a quantum bit transforms into a classical bit once observed. Meaning you should be incredibly skeptical if you read wild claims on the Internet about what quantum computers can
Well, unless it involves transforming an NP problem into a P problem using a quantum algorithm. Quantum computers can theoretically do that (sometimes).
What’s the Largest Quantum Computer In Existence Right Now?
Quantum computers have moved beyond the theoretical, and many already exist. Right now (Dec 2021), Honeywell (built by IBM) claims to have the largest number of reliable Qubits at 64. But, this is
increasing all the time, and it will likely seem like an extremely small number of qubits in a few years.
The problem quantum computers have is that 64 bits isn’t very many bits. This blog post alone was a 300 KB download (37,500 times larger). As such, we’ve probably got a few years before quantum
computers are breaking internet encryption. We need orders of magnitude more qubits before the quantum party starts.
Will You Ever Own a Quantum Computer
You will never own a quantum computer.
The first classical computer to debut in the US weighed 29,000 lbs (UNIVAC I in 1951). At the time, something like an iPhone would have been unimaginable (only 56 years later).
That said, quantum computers have a physics problem! When quantum bits interact with the classical world, they become classical bits. This is why quantum computers currently reside in magnetically
sealed vacuum chambers operating at near absolute zero temperatures. We STILL have significant problems with decoherence (errors due to interference from the outside world) despite this.
I genuinely don’t understand how it’d be possible to overcome that minus a dramatic leap in technology. But, even if we could, there’s no point to you owning a quantum iPhone. Quantum computers are
specialized machines meant to solve very specific NP problems, not perform classical computing tasks.
That said, I could easily see quantum computing resources becoming available via the cloud (Nothing is stopping us from hooking one to the Internet). In that sense, you might be able to buy time on a
quantum computer in the near future. | {"url":"https://www.shaunpoore.com/quantum-computers/","timestamp":"2024-11-02T06:25:59Z","content_type":"text/html","content_length":"96424","record_id":"<urn:uuid:c104bf66-c8cd-4aec-abb5-b341330f4d23>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00412.warc.gz"} |
THE sphere is a solid terminated by a curve surface, all the points of which are equally distant from a point within, called the centre. Elements of Geometry: With, Practical Applications - Page 231
by George Roberts Perkins - 1850 - 320 pagesFull view
About this book
Thomas Malton - 1774 - 484 pages
...a SPHERE, is any Right Line puffing through its Center, and terminated by its Surface. As A B. NB A Sphere may be conceived to be generated, by the revolution of a Semicircle; as ADB, on its
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...a3. BOOK VII. THE SPHERE. Definitions. I. The sphere is a solid terminated by a curve surface, a1' the points of which are equally distant from a point within, called the centre. D The sphere
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...shall have SECTION THIRD. Of the Sphere. DEFINITIONS. ' 437. A sphere is a solid terminated by a curved surface all the points of which are equally distant...be generated by the revolution of a
semicircle DAE (fig. 220) about its diameter DE ; Fig. 220. for the surface thus described by the curve DAE will have...
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...the Circle and the Measure of Angles. DEFINITIONS. 90. THE circumference of a circle is a curved line all the points of which are equally distant from a point within called the centre. The
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Silvestre François Lacroix - Algebra - 1825 - 570 pages
...the Circle and the Measure of Angles. DEFINITIONS. ^0. THE circumference of a circle is a curved line all the points of which are equally distant from a point within called the centre. The
circle is the space terminated by this curved line*. 46. 91. Every straight line CA, CE, CD (fig. 46),... | {"url":"https://books.google.com.jm/books?qtid=27e7879&lr=&id=LycAAAAAYAAJ&sa=N&start=0","timestamp":"2024-11-05T15:20:46Z","content_type":"text/html","content_length":"30456","record_id":"<urn:uuid:5e97db36-6675-4621-bf72-6d9825382104>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00008.warc.gz"} |
Mental Arithmetic 4 Answers
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A Answer C Answer 4 Schofield Sims B Answer 1 How many thousandths must be added to 9.073 to make 10? 927 1000 2 Find the change from £1.50 after buying 2kg of oranges at 36p per 1 2 kg. 6p 3 Imran
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Methodology of economics
Last updated on 12/03/2020
A greater part of economic theory has been formulated with the aid of the technique of economic statics.
However, during the last eighty years (since 1925) dynamic technique has been increasingly applied to the various fields of economic theory. J. M. Clark’s principle of acceleration and Aftalion
theory of business fluctuations resulting from the lagged over-response of output to previous capital formation are some examples of dynamic models which appeared before 1925.
But prior to 1925, dynamic analysis was mainly confined, with some exceptions, to the explanation of business cycles. After 1925, dynamic analysis has been used extensively not only for the
explanation of business fluctuations but also for income determination, growth and price theories. Economists like R. Frisch, C. F. Roos, J. Tinbergen, M. Kalecki, Paul Samuelson and many others have
formulated dynamic models which give rise to cycles of varying periodicity and amplitude.
English writers such as Robertson, Keynes, Haberler, Kahn, and Swedish economists such as, Myrdal, Ohlin, Lindahl and Lunberg have laid a great stress on economic dynamics in the sphere of income
analysis. More recently, economists like Samuelson, Goodwin, Smithies, Domar, Metzler, Haavelmo, Klein, Hicks, Lange, Koopmans and Tinter have further extended and developed dynamic models concerning
the stability and fluctuations around any equilibrium point or path and which cover the four important fields of economic theory, namely, cycles, income determination, economic growth and price
We shall explain below the meaning and nature of economic statics, dynamics and comparative statics and shall bring out the distinction between them. There has been a lot of controversy about their
true meaning and nature, especially about economic dynamics.
Nature of Economic Statics:
The method of economic statics is very important since, as noted above, a large part of economic theory has been formulated with its aid. Besides, the conception of economic dynamics cannot be
understood without being clear about the meaning of statics because one thing which is certain about economic dynamics is that it is ‘not statics’. J. R. Hicks aptly remarks, “The definition of
Economic Dynamics must follow from the definition of Economic Statics; when we have defined one, we have defined the other.
In order to make the difference between the natures of economic statics and dynamics quite clear, it is essential to bring out the distinction between two sorts of phenomena, stationary and changing.
An economic variable is said to be stationary, if the value of the variable does not change over time, that is, its value is constant over time. For instance, if the price of a good does not change
as time passes, price will be called stationary.
Likewise, national income is stationary if its magnitude does not change through time. On the other hand, the variable is said to be changing (non-stationary) if its value does not remain constant
through time. Thus, the whole economy can be said to be stationary (changing), if values of all important variables are constant through time (are subject to change).
It may be noted that the various economic variables whose behaviour over time is studied are prices of goods, quantity supplied, quantity demanded, national income, level of employment, the size of
the population, the level of investment, etc.
It is worth mentioning that it is quite possible that whereas a variable may be changing from the micro point of view, but stationary from the macro point of view. Thus, the prices of individual
goods may be changing, of which some may be rising and some falling, but the general price level may remain constant over time. Likewise, the national income of a country may be stationary while the
incomes generated by various industries may be changing.
On the other hand, the particular variables may be stationary, while the economy as a whole may be changing. For example, even if the level of net investment in the economy is stationary, the economy
as a whole may not be stationary. When there is a constant amount of net positive investment per annum, the economy will be growing (changing) since addition to its stock of capital will be
It should be carefully noted that there is no necessary relationship between stationary phenomenon and economic statics, and the changing phenomenon and dynamics. Although economic dynamics is
inherently connected with only a changing phenomenon but the static analysis has been extensively applied to explain the changing phenomena.
The distinction between statics and dynamics is the difference between the two different techniques of analysis and not the two different sorts of phenomena. Prof. Tinbergen rightly remarks, “The
distinction between Statics and Dynamics is not a distinction between two sorts of phenomena but a distinction between two sorts of theories, i.e., between two ways of thinking. The phenomena may be
stationary or changing, the theory (the analysis) may be Static or Dynamic”
Static Analysis and Functional Relationships:
The test of economic theory is to explain the functional relationships between a system of economic variables. These relationships can be studied in two different ways. If the functional relationship
is established between two variables whose values relate to the same point of time or to the same period of time, the analysis is said to be static. In other words, the static analysis or static
theory is the study of static relationship between relevant variables.
A functional relationship between variables is said to be static if the values of the economic variables relate to the same point of time or to the same period of time. Numerous examples of static
relationships between economic variables and the theories or laws based upon them can be given. Thus, in economics the quantity demanded of a good at a time is generally thought to be related to the
price of the good at the same time.
Accordingly, the law of demand has been formulated to establish the functional relationship between the quantity demanded of a good and its price at a given moment of time. This law states that,
other things remaining the same, the quantity demanded varies inversely with price at a given point of time. Similarly, the static relationship has been established between the quantity supplied and
the price of goods, both variables relating to the same point of time. Therefore, the analysis of this price-supply relationship is also static.
Micro-Statics and Macro-Statics:
Generally, economists are Interested in the equilibrium values of the variables which are attained as a result of the adjustment of the given variables to each other. That is why economic theory has
sometimes been called equilibrium analysis. Until recently, the whole price theory in which we explain the determination of equilibrium prices of products and factors in different market categories
was mainly static analysis, because the values of the various variables, such as demand, supply, and price were taken to be relating to the same point or period of time.
Thus, according to this micro-static theory, equilibrium at a given moment of time under perfect competition is determined by the intersection of given demand and the supply functions (which relate
the values of variables at the same point of time). Thus in Figure 4.1 given the demand function as demand curve DD and the supply function SS, the equilibrium price OP is determined.
The equilibrium amount supplied and demanded so determined is OM. This is a static analysis of price determination, for all the variables such as, quantity supplied, quantity demanded and the price
refer to the same point or period of time. Moreover, the equilibrium price and quantity determined by their interaction also relate to the same time as the determining variables.
Examples of static analysis can also be given from macroeconomic theory. Keynesian macro model of the determination of level of national income is also mainly static. According to this model,
national income is determined by the intersection of aggregate demand curve and aggregate supply curve (45° line) as is depicted in Figure 4.2 where the vertical axis measures consumption demand plus
investment demand (C + l) and aggregate supply, and the X-axis measures the level of national income. Aggregate demand equals aggregate supply at point E and income OY is determined.
This is static analysis since aggregate demand (consumption and investment) and aggregate supply of output refer to the same point of time and element of time is not taken into account in considering
the adjustment of the various variables in the system to each other. In other words, this analysis refers to instantaneous or timeless adjustment of the relevant variables and the determination of
equilibrium level of national income.
Professor Schumpeter describes the meaning of static analysis as follows: “By static analysis we mean method of dealing with economic phenomena that tries to establish relations between elements of
the economic system—prices and quantities of commodities all of which have the same time subscript, that is to say, refer to the same point of time. The ordinary theory of demand and supply in the
market of an individual commodity as taught in every textbook will illustrate this case: it relates demand, supply and price as they are supposed to be at any moment of observation.”
Assumptions of Static Analysis:
A point worth mentioning about static analysis is that in it certain determining conditions and factors are assumed to remain constant at the point of time for which the relationship between the
relevant economic variables and the outcome of their mutual adjustment is being explained. Thus, in the analysis of price determination under perfect competition described above, the factors such as
incomes of the people, their tastes and preferences, the prices of the related goods which affect the demand for a given commodity are assumed to remain constant.
Similarly, the prices of the productive resources and production techniques which affect the cost of production and thereby the supply function are assumed to remain constant. These factors or
variables do change with time and their changes bring about shift in the demand and supply functions and therefore affect prices.
But because in static analysis we are concerned with establishing the relationship between certain given variables and their adjustment to each other at a given point of time, changes in the other
determining factors and conditions are ruled out. We, in economics, generally use the term data for the determining conditions or the values of the other determining factors. Thus, in static
analysis, data are assumed to be constant and we find out the eventual consequence of the mutual adjustment of the given variables.
It should be noted that assuming the data to be constant is very much the same thing as considering them at a moment of time or in other words allowing them a very short period of time’ within which
they cannot change. Moreover, the crucial point about static analysis is that the given conditions or data are supposed to be independent of the behaviour of variables or units in the given system
between which functional relationship is being studied.
Thus, in the above static price analysis it is assumed that variables in the system, that is, price of the good, quantity supplied and quantity demanded do not influence the determining conditions or
data of incomes of the people, their tastes and preferences, the prices of the related goods, etc. Thus, the relationship between the data and the behaviour of the economic variables in a given
system is assumed to be one-way relationship; the data influence the variables of the given system and not the other way around.
On the contrary, we shall see below that in dynamic analysis the determinant data or determining conditions are not assumed to be constant. In dynamic analysis, certain elements in the data are not
independent of the behaviour of the variables in a given system. In fact, in a fully dynamic system, it is hard to distinguish between data and variables since in a dynamic system over time “today’s
determinant data are yesterday’s variables and today’s variables become tomorrow’s data. The successive situations are interconnected like the links of a chain.”
Since in static analysis, we study the behaviour of a system at a particular time, or in other words, in economic statics, we do not study the behaviour of a system over time. Therefore how the
system has proceeded from a previous position of equilibrium to the one under consideration is not studied in economic statics. Prof. Stanley Bober rightly remarks.
“A static analysis concerns itself with the understanding of what determines an equilibrium position at any moment in time. It focuses attention on the outcome of economic adjustments and is not
concerned with the path by which the system, be it the economy in the aggregate or a particular commodity market, has proceeded from a previous condition of equilibrium to the one under
Relevance of Static Analysis:
Now, the question arises as to why the technique of static analysis is used which appears to be unrealistic in view of the fact that determining conditions or factors are never constant. Static
techniques are used because it makes the otherwise complex phenomena simple and easier to handle. To establish an important causal relationship between certain variables, it becomes easier if we
assume other forces and factors constant, not that they are inert but for the time it is helpful to ignore their activity.
According to Robert Dorfman, “Statics is much more important than dynamics, partly because it is the ultimate destination that counts in most human affairs, and partly because the ultimate
equilibrium strongly influences the time paths that are taken to reach it, whereas the reverse influence is much weaker”.
To sum up, in static analysis we ignore the passage of time and seek to establish the causal relationship between certain variables relating to the same point of time, assuming some determining
factors as remaining constant. To quote Samuelson who has made significant contributions to making clear the distinction between the methods of economic statics and dynamics, “Statics concerns itself
with the simultaneous and instantaneous or timeless determination of economic variables by mutually interdependent relations.
Even a historically changing world may be treated statically, each of its changing positions being treated as successive states of static equilibrium.” In another article he says, ‘Statical then
refers to the form and structure of the postulated laws determining the behaviour of the system. An equilibrium defined as the intersection of a pair of curves would be statical. Ordinarily, it is
‘timeless’ in that, nothing is specified concerning the duration of the process, but it may very well be defined as holding over time.”
Comparative Statics:
We have studied above static and dynamic analysis of the equilibrium position. To repeat, static analysis is concerned with explaining the determination of equilibrium values with a given set of data
and the dynamic analysis explains how with a change in the data the system gradually grows out from one equilibrium position to another. Midway between the static and dynamic analyses is the
comparative static analysis.
Comparative static analysis compares one equilibrium position with another when the data have changed and the system has finally reached another equilibrium position. It does not analyse the whole
path as to how the system grows out from one equilibrium position to another when the data have changed; it merely explains and compares the initial equilibrium position with the final one reached
after the system has adjusted to a change in data. Thus, in comparative static analysis, equilibrium positions corresponding to different sets of data are compared.
Professor Samuelson writes:
“It is the task of comparative statics to show the determination of the equilibrium values of given variables (unknowns) under postulated conditions (functional relationships) with various data
(parameters) specified. Thus, in the simplest case of a partial equilibrium market for a single commodity the two independent relations of supply and demand, each drawn up with other prices and
institutional data being taken as given determine by their interaction the equilibrium quantities of the unknown price and quantity sold.
If no more than this could be said, the economist would be truly vulnerable to that he is only a parrot, taught to say ‘supply and demand.’ Simply to know that there are efficacious ‘laws’
determining equilibrium tells us nothing of the character of these laws. In order for the analysis to be useful it must provide information concerning the way in which our equilibrium quantities will
change as a result of changes in the parameters— taken as independent data”.
It should be noted that for better understanding of the changing system, comparative statics studies the effect on the equilibrium position of a change in only a single datum at a time rather than
the effects of changes in the many or all variables constituting the data. By confining ourselves to the adjustment in the equilibrium position as a result of alteration in a single datum at a time,
we keep our analysis simple, manageable and at the same time useful, instructive as well as adequate enough to understand the crucial aspects of the changing phenomena.
To quote Erich Schneider:
The set of data undergoes changes in the course of time, and each new set of data has a new equilibrium position corresponding to it. It is therefore of great interest to compare the different
equilibrium positions corresponding to different sets of data. In order to understand the effect of a change in the set of data on the corresponding position of equilibrium, we must only alter a
single datum at a time. Only in this way it is possible to understand fully the effects of alterations in the individual data.
We ask, to start with, about set I of the data, and the equilibrium position corresponding to it, then study next equilibrium position corresponding to set II of the data, where set II differs from
set I only in the alteration of a single datum. In this way we compare the equilibrium values for the system corresponding to the two equilibrium positions with one another. This sort of comparative
analysis of two equilibrium positions may be described as comparative-static analysis, since it studies the alteration in the equilibrium position corresponding to an alteration in a single datum”.
Let us give some examples of comparative static analysis from the microeconomic theory. We know that given the data regarding consumer’s tastes, incomes, prices of other goods on the one hand and the
technological conditions, costs of machines and materials, and wages of labour we have given demand and supply functions which by their interaction determine the price of a good.
Now suppose that other things remaining constant, incomes of consumers increase. With the increase in incomes, the demand function would shift upward. With the change in the demand as a result of the
change in the income, the supply would adjust itself and final new equilibrium position would be determined. To explain the determination of new equilibrium price and how it differs from the initial
one is the task of comparative statics.
In Figure 4.3, initially the demand and supply functions are DD and SS and with their interaction price OP[1] is determined. When the demand function changes to D’D’ as a result of changes in
consumers’ income, it intersects the given supply function at E[2] and the new equilibrium price OP[2] is determined.
In comparative-static analysis, we are concerned only with explaining the new equilibrium E[2] position comparing it with E[1], and not with the whole path the system has traversed as it gradually
grows out from to E[2]. As we shall study in the part of price theory, comparative static analysis was extensively used by Alfred Marshall in his time-period analysis of pricing under perfect
Importance of Comparative Statics:
No doubt, more realistic, complete and true analysis of the changing phenomena of the real world would be the dynamic analysis nevertheless comparative statics is a very useful technique of
explaining the changing phenomena and its crucial aspects without complicating the analysis.
To quote Schneider again. “This sort of dynamic analysis of the influence of a change in data is much more comprehensive and informative than the mere static analysis of two different sets of data
and of the equilibrium positions corresponding to them. Nevertheless, the comparative static treatment provides some important insights into the mechanism of the exchange economy”.
Likewise, Professors Stonier and Hague write, “The construction of a truly dynamic theory of economics, where more continuous changes in demand and supply conditions, like those which occur in the
real world, are analysed, is the ultimate goal of most theories of economics…. However, so far as the determination of price and output is concerned, simple comparative static analysis…. is as
powerful an analytical method as we need”.
Economic Dynamics:
Now, we turn to the method of Economic Dynamics which has become very popular in modern economics. Economic dynamics is a more realistic method of analysing the behaviour of the economy or certain
economic variables through time. The definition of economic dynamics has been a controversial issue and it has been interpreted in various different ways. We shall try to explain the standard
definitions of economic dynamics.
The course thorough time of a system of economic variables can be explained in two ways. One is the method of economic statics described above, in which the relations between the relevant variables
in a given system refer to the same point or period of time. On the other hand, if the analysis considers the relationship between relevant variables whose values belong to different points of time
it is known as Dynamic Analysis or Economic Dynamics.
The relations between certain variables, the values of which refer to the different points or different periods of time, are known as dynamic relationships. Thus, J.A. Schumpeter says, “We call a
relation dynamic if it connects economic quantities that refer to different points of time. Thus, if the quantity of a commodity that is offered at a point of time (t) is considered as dependent upon
the price that prevailed at the point of time (t-l), this is a dynamic relation. In a word, economic dynamics is the analysis of dynamic relationships.
We thus see that in economic dynamics we duly recognize the element of time in the adjustment of the given variables to each other and accordingly analyse the relationships between given variables
relating to different points of time.
Ragnar Frisch who is one of the pioneers in the use of the technique of dynamic analysis in economics defines economic dynamics as follows:
“A system is dynamical if its behaviour over time is determined by functional equations in which variables at different points of time are involved in a essential way. In dynamic analysis, he further
elaborates, “We consider not only a set of magnitudes in a given point of time and study the interrelations between them, but we consider the magnitudes of certain variables in different points of
time, and we introduce certain equations which embrace at the same time several of those magnitudes belonging to different instants. This is the essential characteristic of a dynamic theory. Only by
a theory of this type we can explain how one situation grows out of the foregoing.
Many examples of dynamic relationships from both micro and macroeconomic fields can be given. If one assumes that the supply (S) for a good in the market in the given time (t) depends upon the price
that prevails in the preceding period (that is, t – 1), the relationship between supply and price is said to be dynamic.
This dynamic functional relation can be written as:
S[t] = f (P[t-1])
where S[t] Stands for the supply of a good offered in a given period f and P[t-1]for the price in the preceding period. Likewise, if we grant that the quantity demanded (D[1]) of a good in a period
Ms a function of the expected price in the succeeding period (t +1), the relation between demand and price will be said to be dynamic and the analysis of such relation would be called dynamic theory
or economic dynamics.
Similarly, examples of dynamic relationship can be given from the macro field. If it is assumed that the consumption of the economy in a given period depends upon the income in the preceding period
(t – 1), we shall be conceiving a dynamic relation.
This can be written as:
C[t] = f (Y[t-1])
When macroeconomic theory (theory of income, employment and growth) is treated dynamically, that is, when macroeconomic dynamic relationships are analysed, the theory is known as “Macro dynamics”.
Samuelson, Kalecki, Post-Keynesians like Harrod, Hicks have greatly dynamized the macroeconomic theory of Keynes.
Endogenous Changes and Dynamic Analysis:
It should be noted that the change or movement in a dynamic system is endogenous, that is, it goes on independently of the external changes in it; one change grows out of the other. There may be some
initial external shock or change but in response to that initial external change, the dynamical system goes on moving independently of any fresh external changes, successive changes growing out of
the previous situations.
In other words, the development of a dynamic process is self-generating. Thus, according to Paul Samuelson, It is important to note that each dynamic system generates its own behaviour over time
either as an autonomous response to a set of ‘initial conditions’, or as a response to some changing external conditions. This feature of self-generating development over time is the crux of every
dynamic process. Likewise, Professor J. K. Mehta remarks.
“In simple words, we can say that an economy can be said to be in a dynamical system when the various variables in it such as output, demand, prices have values at any time dependent on their values
at some other time. If you know their values at one moment of time, you should be able to know their values at subsequent points of time. Prices of goods in a causal dynamic system do not depend on
any outside exogenous forces. A dynamic system is self-contained and self-sustained”.
It is thus clear that a distinctive feature of dynamic analysis is to show how a dynamic process or system is self-generating, how one situation in it grows out of a previous one or how one situation
moves on independently of the changes in external conditions. As Schneider, a German economist, has aptly and precisely put it, “A dynamic theory shows how in the course of time a condition of the
economic system has grown out of its condition in the previous period of time. It is this form of analysis which has the central importance for the study of the process of economic developments, be
they short-run or long-run processes.
An illustration of dynamic analysis may be given. level of national income is determined by the equilibrium between given aggregate demand curve and the aggregate supply curve. Now, if the aggregate
demand increases, due to the increase in investment, the aggregate demand curve will shift upward and as a consequence the new equilibrium point will be reached and level of national income will
In static analysis, the new equilibrium is supposed to occur instantaneously (timeless) and no attention is paid how the new equilibrium position of income has grown out of the original through time
when the increase in aggregate demand has taken place.
That is to say, the dynamic analysis traces out the whole path through which the system passes over time to reach the new equilibrium position. We present in Figure 4.5 the common macro model of
income determination. Given the aggregate demand C + 1, the level of national income OY[0] is determined y in time t. Suppose now the aggregate demand curve shifts upward due to the increase in
investment in time period t.
As the investment increases in time-period t, the income will rise in time period t + 1 by the amount of the investment. Now, this increase in income will push up the consumption demand. To meet this
increase in consumption, output will be increased with the result that income will further rise in period t + 2. This additional increase in income will induce further increase in consumption with
the result that more output will be produced to meet the rise in demand and the income in period t + 3 will still further rise.
In this way, the income will go on rising; one increase in income giving rise to another till the final equilibrium point H is reached in the time period t + n in which the level of income OY[n] is
determined. The path by which the income increases through time is shown in the figure by dotted arrow lines. This illustration of macro-dynamics makes it clear that the dynamic analysis is concerned
with how magnitude of variables in a period (income and consumption in the present illustration) depends upon the magnitudes of the variables in the previous periods.
Hicks’ Definition of Economic Dynamics:
In the light of our above explanation of the meaning of the method of economic dynamics, we are in a position to examine the definition of dynamics given by J. R. Hicks in his book ‘Value and
Capital’. Hicks says, “I call Economic Statics those parts of economic theory where we do not trouble about dating. Economic Dynamics those parts where every quantity must be dated.
This is a very simple way of defining dynamics. When the magnitude of variables does not change with time, the dating of the quantities of variables is not necessary. In the absence of change in the
economic variables determining the system, an equilibrium position that applies to the present will apply equally well to the future.
But in our view, this is not a satisfactory definition of economic dynamics. A system may be statical, but still may be dynamic according to Hicksian definition if some dates are attached to
variables. Thus, a statical system may be converted into Hicksian dynamics by merely assigning some dates to the variables. But this is not true meaning of economic dynamics, as is now generally
conceived. Mere dating of variables is not enough.
As has been made clear by Ragnar Frisch, variables in the system must relate to different dates or different points of time, if it is to be a truly dynamic system. Secondly, as has been contended by
Paul Samuelson, this “Hicksian definition is too general and insufficiently precise. According to Paul Samuelson, Hicksian definition of dynamics would cover a historical static system of variables.
An historically moving static system certainly requires dating of the variables but it would not thereby become dynamic.
A system of variables to be called dynamic must involve functional relationships between the variables, that is, the variables at one point of time must be shown to be dependent upon the variables at
other points of time. Thus, according to Samuelson,” a system is dynamical if its behaviour over time is determined by functional equations in which variables at different points of time are involved
in an essential way.
Thus, Samuelson’s emphasis is on functional relationships as well as on different points of time. We therefore conclude that a dynamical system involves functional relationships between variables at
different points of time. A historically moving system does not necessarily involve the functional relationships between the variables at different historical times.
The historical movement of a system may not be dynamical. For instance, as has been pointed out by Samuelson, if one year crop is high because of good monsoons, the next year low because the monsoons
fail, and so forth, the system will be statical even though not stationary.
The concept or technique of economic dynamics which we have explained above was first of all clarified by Ragnar Frisch in 1929. According to his view, like static analysis, economic dynamics is a
particular method of explanation of economic phenomenon economic phenomena themselves may be stationary or changing. Although technique of dynamic analysis has great scope in a changing and a growing
system but it may also be applied even to stationary phenomena.
A system or phenomenon may be stationary in the sense that the values of relevant economic variables in it may remain constant through time, but if the values of the variables at a time are dependent
upon the values at another time, then dynamic analysis can be applied. But, as stated above, the greater scope of economic dynamics lies in the field of changing and growing phenomena. Schneider
aptly brings out the distinction between statics and dynamics on the one hand and stationary and changing phenomena on the other when he writes.
It is essential to understand that in modern theory ‘statics’ and ‘dynamics’ refer to a particular mode of treatment or type of analysis of the phenomena observed, while the adjectives ‘stationary’
and ‘changing’ describe the actual economic phenomena. A static or dynamic theory is a particular kind of explanation of economic phenomena, and, indeed, stationary and changing phenomena can be
submitted either to a static or to a dynamic analysis.
Expectations and Dynamics:
We have described abovethat economic dynamics is concerned with explaining dynamic relationships, that is, the relationships among variables relating to different points of time. The variables at the
present moment may depend upon the variables at other times, past and future. Thus, when the relationship between the economic variables belonging to different points of time is considered, or when
rates of change of certain variables in a growing economy are under discussion, the question of future creeps into the theoretical picture.
The economic units (such as consumers, producers and entrepreneurs) have to take decisions about their behaviour in the present period. The consumers have to decide what goods they should buy and
what quantities of them. Similarly, producers have to decide what goods they should produce, what factors they should use and what techniques they should adopt.
These economic units decide about their present course of action on the basis of their expected values of the economic variables in the future. When their expectations are realised, they continue
behaving in the same way and the dynamic system is in equilibrium. In other words, when the expectations of the economic units are fulfilled, they repeat the present pattern of behaviour and there
exists what has been called dynamic equilibrium, unless some external shock or force disturbs the dynamic system.
The expectations or anticipations of the future held by the economic units play a vital role in economic dynamics. In a purely static theory expectations about the future have practically no part to
play since static theory is mainly concerned with explaining the conditions of equilibrium positions at a point of time as well as under the assumptions of constant tastes, techniques and resources.
Thus, in static analysis expectations about the future play little part since under it no processes-aver time are considered. On the other hand, since dynamic analysis is concerned with dynamic
processes over time, that is, changing variables over time and their action and interaction upon each other through time, expectations or anticipations held by the economic units about the future
have an important place.
But from the intimate relation between dynamics and expectations it should not be understood that mere introduction of expectations in static analysis would make it dynamic. Whether the analysis is
dynamic or not depends upon whether the relationship between variables belonging to different points or periods of time is considered or not, or whether rates of change of certain variables over time
are considered or not. German economist Schneider rightly says, “A theory is not to be considered as dynamic simply because it introduces expectations, whether that is the case or not depends simply
on whether or not the expected values of the single variables relate to different periods or points of time.”
Moreover, it is important to note that a theory becomes truly dynamic only if in it the expectations are taken as a variable and not as a given data. In other words, in a really dynamic theory,
expectations should be considered as changing over time rather than remaining constant. A dynamic theory should tell us what would happen if, the expectations of the economic units are realised and
what would happen if they have not come true.
In Harrod’s macro-dynamic model of a growing economy that if the entrepreneurs expect the rate of growth of output equal to s/C (whereas S stands for rate of saving and C for capital-output ratio)
their expectations would be realised and as a result the relevant variables in the system will move in equilibrium over time and there will be a steady growth in the economy. If their expectations
about the rate of growth are smaller or larger than s/C, they will not be realised and as a consequence there will be instability in the economy.
When the expectations of the individuals turn out to be incorrect, they will revise or change their expectations. Because of the changing nature of these expectations they should not be taken as
given data or given conditions in a dynamic theory. To take expectations as given data means that they remain constant even if they turn out to be incorrect.
That is to say, even when the individuals are surprised by the actual events because their expectations have not been fulfilled, they will continue to have the same expectations. But that will amount
to be assuming irrationality on the part of the individuals. We, therefore, conclude that expectations must be taken as changing in the dynamic system and not as a given condition.
Need and Significance of Economic Dynamics:
The use of dynamic analysis is essential if we want to make our theory realistic. In the real world, various key variables such as prices of goods, the output of goods, the income of the people, the
investment and consumption are changing over time. Both Frischian and Harrodian dynamic analyses are required to explain these changing variables and to show how they act and react upon each other
and what results flow from their action and interaction.
Many economic variables take time to make adjustment to the changes in other variables. In other words, there is a lag in the response of some variables to the changes in the other variables, which
make it necessary that dynamic treatment be given to them. We have seen that changes in income in one period produce influence on consumption in a later period. Many similar examples can be given fpm
micro and macro-economics.
Besides, it is known from the real world that the values of certain variables depend upon the rate of growth of other variables. For example, we have seen in Harrod’s dynamic model of a growing
economy that investment depends upon expected rate of growth in output. Similarly, the demand for a good may depend upon the rate of change of prices.
Similar other examples can be given. In such cases where certain variables depend upon the rate of change in other variables, the application of both the period analysis and the rate of change
analysis of dynamic economics become essential if we want to understand their true behaviour.
Until recently, dynamic analysis was mainly concerned with explaining business cycles, fluctuations or oscillations. But, after Harrod’s and Domar’s path-breaking contributions, the interest in the
problems of growth has been revived among economists. It is in the study of growth that dynamic analysis becomes more necessary. Now-a-days economists are engaged in building dynamic models of
optimum growth both for developed and developing countries of the world.
Thus, in recent years, the stress on dynamic analysis is more on explaining growth rather than cycles or oscillations. Prof. Hansen is right when he says, “In my own view mere oscillation represents
a relatively unimportant part of economic dynamics. Growth, not oscillation, is the primary subject-matter for study in economic dynamics. Growth involves changes in technique and increases in
population. Indeed that part of cycle literature (and cycle theories are a highly significant branch of dynamic economics) which is concerned merely with oscillation is rather sterile. | {"url":"https://indiafreenotes.com/methodology-of-economics/","timestamp":"2024-11-03T00:35:43Z","content_type":"text/html","content_length":"267793","record_id":"<urn:uuid:4f372a51-2196-46ca-b46b-cefce4f6c32e>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00523.warc.gz"} |
base rent
06 Oct 2024
Title: An Examination of Base Rent: Conceptual Framework and Mathematical Formulation
Abstract: Base rent, a fundamental concept in real estate economics, refers to the minimum amount of money that a tenant must pay to occupy a property. This article provides an in-depth examination
of base rent, including its conceptual framework, mathematical formulation, and implications for landlords and tenants.
Introduction: Base rent is a crucial component of rental agreements, representing the minimum payment required by tenants to occupy a property. It is essential to understand the concept of base rent
to make informed decisions in real estate transactions. This article aims to provide a comprehensive analysis of base rent, including its mathematical formulation and implications for stakeholders.
Conceptual Framework: Base rent can be conceptualized as the minimum amount of money that a tenant must pay to occupy a property, taking into account various factors such as:
• Property characteristics: The quality, location, and size of the property.
• Market conditions: The demand and supply of rental properties in the area.
• Tenant characteristics: The income, creditworthiness, and other relevant factors of the tenant.
Mathematical Formulation: The base rent (BR) can be mathematically formulated as:
BR = f(Property Characteristics, Market Conditions, Tenant Characteristics)
Where f() represents a function that takes into account various factors influencing the base rent.
In ASCII format, the formula can be represented as:
BR = PC × MC × TC
• PC: Property characteristics (e.g., quality, location, size)
• MC: Market conditions (e.g., demand, supply)
• TC: Tenant characteristics (e.g., income, creditworthiness)
Implications: The mathematical formulation of base rent has significant implications for landlords and tenants. Landlords can use this framework to determine the minimum amount of money that tenants
must pay to occupy a property, while tenants can use it to negotiate better rental agreements.
Conclusion: In conclusion, this article provides an in-depth examination of base rent, including its conceptual framework, mathematical formulation, and implications for stakeholders. The
mathematical formulation of base rent offers a useful tool for landlords and tenants to make informed decisions in real estate transactions.
Related articles for ‘base rent’ :
Calculators for ‘base rent’ | {"url":"https://blog.truegeometry.com/tutorials/education/b39737313b00d34062629962322f1ce4/JSON_TO_ARTCL_base_rent.html","timestamp":"2024-11-03T01:17:18Z","content_type":"text/html","content_length":"16145","record_id":"<urn:uuid:2b0649be-4474-4986-bf70-c0754a05c841>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00693.warc.gz"} |
When is selfish routing bad? The price of anarchy in light and heavy traffic
This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin/destination (O/D) pairs. Empirical studies in
real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the following question: can these observations be justified theoretically? We first show
that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and
smooth, convex costs. On the other hand, for a large class of cost functions (including all polynomials) and inflow patterns, the price of anarchy does converge to 1 in both heavy and light traffic,
irrespective of the network topology and the number of O/D pairs in the network. We also examine the rate of convergence of the price of anarchy, and we show that it follows a power law whose degree
can be computed explicitly when the network's cost functions are polynomials.
• Heavy traffic
• Light traffic
• Nonatomic congestion games
• Price of anarchy
• Regular variation
Dive into the research topics of 'When is selfish routing bad? The price of anarchy in light and heavy traffic'. Together they form a unique fingerprint. | {"url":"https://pure.uai.cl/en/publications/when-is-selfish-routing-bad-the-price-of-anarchy-in-light-and-hea","timestamp":"2024-11-11T03:42:41Z","content_type":"text/html","content_length":"52281","record_id":"<urn:uuid:579db592-c83a-4df6-9924-1255599924e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00533.warc.gz"} |
Where to find HP 50g Virtual Calculator?
05-20-2022, 10:24 PM
Post: #1
Anderson Costa Posts: 59
Member Joined: Apr 2014
Where to find HP 50g Virtual Calculator?
Hi for all!!!
I am testing some old HP calculators simulators for running on Windows 11. I have found HP 12C and HP 15C Virtual Calculators, but I can't find HP 50g Virtual Calculator for download. Can anyone have
the installer of this emulator?
05-20-2022, 10:58 PM
Post: #2
ctrclckws Posts: 146
Member Joined: Sep 2018
RE: Where to find HP 50g Virtual Calculator?
I believe I found it by using some kind of internet archive or "wayback" machine site that captures snapshots of versions of websites.
Sorry, I don't have a link anywhere.
Perhaps someone else will be able to point you closer.
05-21-2022, 01:35 AM
Post: #3
Steve Simpkin Posts: 1,292
Senior Member Joined: Dec 2013
RE: Where to find HP 50g Virtual Calculator?
There are a number of virtual HP calculators at the following link, including the HP 50g. I have installed about half of them.
05-23-2022, 03:34 AM
(This post was last modified: 05-23-2022 03:37 AM by Jlouis.)
Post: #4
Jlouis Posts: 769
Senior Member Joined: Nov 2014
RE: Where to find HP 50g Virtual Calculator?
Emu48 doesn't run in Windows 11?
Edit: unless you want only a simulator, not an emulator, isn't it?
User(s) browsing this thread: 1 Guest(s) | {"url":"https://www.hpmuseum.org/forum/thread-18393.html","timestamp":"2024-11-14T15:44:04Z","content_type":"application/xhtml+xml","content_length":"23903","record_id":"<urn:uuid:643ac29b-6037-4b3b-b953-b27a9b83227e>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00748.warc.gz"} |
Estimation Input Signals
Frequency response estimation requires an input signal to excite the model at frequencies of interest. The software then measures the response at the specified output, using the input signal and
measured response to estimate the frequency response.
When you perform frequency response estimation, you specify what type of input signal to use and what its properties are.
Offline Estimation
The following table summarizes the types of input signals you can use for offline estimation in Model Linearizer or at the MATLAB^® command line for use with frestimate.
Signal Description
A series of sinusoidal perturbations applied one after another. Sinestream signals are recommended for most situations. They are especially useful when your system contains strong
Sinestream nonlinearities or you require highly accurate frequency response models.
A swept-frequency signal that excites your system at a range of frequencies, such that the input frequency changes instantaneously. Chirp signals are useful when your system is nearly
Chirp linear in the simulation range. They are also useful when you want to obtain a response quickly for a lot of frequency points.
PRBS A deterministic pseudorandom binary sequence that shifts between two values and has white-noise-like properties. PRBS signals reduce total estimation time compared to using sinestream
input signals, while producing comparable estimation results. PRBS signals are useful for estimating frequency responses for communications and power electronics systems.
Random A random input signal. Random signals are useful because they can excite the system uniformly at all frequencies up to the Nyquist frequency.
Step A step input signal. Step inputs are quick to simulate and can be useful as a first try when you do not have much knowledge about the system you are trying to estimate.
Arbitrary A MATLAB timeseries with which you can specify any time-varying signal as input.
In general, the estimated frequency response is related to the input and output signals as:
Here, u[est](t) is the injected input signal and y[est] (t) is the corresponding simulated output signal. For more details, see the Algorithms section of frestimate.
Online Estimation
For online estimation with the Frequency Response Estimator block, you can use two types of input signals:
• Sinestream — A series of sinusoidal perturbations applied one after another
• Superposition — A set of sinusoidal perturbations applied simultaneously
For online estimation, using a sinestream signal can be more accurate and can accommodate a wider range of frequencies than a superposition signal. The sinestream mode can also be less intrusive.
However, due to the sequential nature of the sinestream perturbation, each frequency point you add increases the experiment time. Thus the estimation experiment is typically much faster with a
superposition signal with satisfactory results.
To specify which type of input signal to use for online estimation, use the Experiment mode parameter of the Frequency Response Estimator block.
Sinestream Signals
For details about the structure of sinestream signals and how to create them, see Sinestream Input Signals.
Chirp Signals
For details about the structure of chirp signals and how to create them, see Chirp Input Signals.
PRBS Signals
For details about the structure of PRBS signals and how to create them, see PRBS Input Signals.
Random Signals
Random signals are useful because they can excite the system uniformly at all frequencies up to the Nyquist frequency. To create a random input signal for estimation:
• In the Model Linearizer, on the Estimation tab, select Input Signal > Random.
• At the command line, use frest.Random to create the random signal and use it as an input argument to frestimate.
The random signal comprises uniformly distributed random numbers in the interval [0 Amplitude] or [Amplitude 0] for positive and negative amplitudes, respectively. You can specify the amplitude,
sample time, and number of samples directly when you create the input signal. Alternatively, if you have a relevant linear time-invariant (LTI) model such as a state-space (ss) model, you can use it
to initialize the random signal parameters. For instance, if you have an exact linearization of your system, you can use it to initialize the parameters.
When you use a random input signal for estimation, the frequencies returned in the estimated frd model depend on the length and sampling time of the signal. They are the frequencies obtained in the
fast Fourier transform of the input signal (see the Algorithm section of frestimate).
Step Signals
Step inputs are quick to simulate. Like a random signal, a step signal can excite the system at all frequencies up to the Nyquist frequency. For those reasons, a step input can be useful as a first
try when you do not have much knowledge about the system you are trying to estimate. However, the amplitude of the excitation decreases rapidly with increasing frequency. Therefore, step signals are
best used to identify low-order plants where the slowest poles are dominant. Step inputs are not recommended for estimation across a wide range of frequencies.
To create a step input signal for estimation, use frest.createStep. This function creates a MATLAB timeseries that represents a step input having the sample time, step time, step size, and total
signal length that you specify when you call frest.createStep.
To use the step input signal you created in the MATLAB workspace:
• In the Model Linearizer, on the Estimation tab, select it from the Existing Input Signals section of the Input Signal drop-down list.
• At the command line, use it as an input argument to frestimate.
When you use a step input signal for estimation, the frequencies returned in the estimated frd model depend on the length and sampling time of the signal. They are the frequencies obtained in the
fast Fourier transform of the input signal (see the Algorithm section of frestimate).
Arbitrary Signals
If you want to use a signal other than a sinestream, chirp, step, or random signal, you can provide your own MATLAB timeseries object. For instance, you can create a timeseries representing a ramp,
sawtooth, or square wave input.
To use a timeseries object as the input signal for estimation, first create the timeseries in the MATLAB workspace. Then:
• In the Model Linearizer, on the Estimation tab, select it from the Existing Input Signals section of the Input Signal drop-down list.
• At the command line, use it as an input argument to frestimate.
When you use an arbitrary input signal for estimation, the frequencies returned in the estimated frd model depend on the length and sampling time of the signal. They are the frequencies obtained in
the fast Fourier transform of the input signal (see the Algorithm section of frestimate).
Superposition Signals
Superposition signals are available only for online estimation with the Frequency Response Estimator block. For frequency response estimation at a vector of frequencies ω = [ω[1], … , ω[N]] at
amplitudes A = [A[1], … , A[N]], the superposition signal is given by:
$\Delta u=\sum _{i}{A}_{i}\mathrm{sin}\left({\omega }_{i}t\right).$
The block supplies the perturbation Δu for the duration of the experiment (while the start/stop signal is positive). The block determines how long to wait for system transients to die away and how
many cycles to use for estimation as shown in the following illustration.
T[exp] is the experiment duration that you specify with your configuration of the start/stop signal (See the start/stop port description on the block reference page for more information). For the
estimation computation, the block uses only the data collected in a window of N[longest]P. Here, P is the period of the slowest frequency in the frequency vector ω, and N[longest] is the value of the
Number of periods of the lowest frequency used for estimation block parameter. Any cycles before this window are discarded. Thus, the settling time T[settle] = T[exp] – N[longest]P. If you know that
your system settles quickly, you can shorten T[exp] without changing N[longest] to effectively shorten T[settle]. If your system is noisy, you can increase N[longest] to get more averaging in the
data-collection window. Either way, always choose T[exp] long enough for sufficient settling and sufficient data-collection. The recommended T[exp] = 2N[longest]P.
To use a superposition signal for estimation, in the Frequency Response Estimator block, set the Experiment mode parameter to Superposition. For details, see Frequency Response Estimator.
See Also
frestimate | frest.createStep | frest.Random | frest.Sinestream | frest.Chirp | frest.PRBS
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Fitted Linear Function
A Fitted Linear Function is a linear function that is a fitted estimation function (based on a linear metamodel whose function coefficients have been set by a linear regression algorithm).
• http://en.wikipedia.org/wiki/Linear_predictor_function
□ In statistics and in machine learning, a linear predictor function is a linear function (linear combination) of a set of coefficients and explanatory variables (independent variables), whose
value is used to predict the outcome of a dependent variable. Functions of this sort are standard in linear regression, where the coefficients are termed regression coefficients. However,
they also occur in various types of linear classifiers (e.g. logistic regression, perceptrons, support vector machines, and linear discriminant analysis), as well as in various other models,
such as principal component analysis and factor analysis. In many of these models, the coefficients are referred to as "weights". | {"url":"http://www.gabormelli.com/RKB/Fitted_Linear_Function","timestamp":"2024-11-12T17:02:43Z","content_type":"text/html","content_length":"43328","record_id":"<urn:uuid:02a8881c-eb6d-4714-acb7-37f3ed250c18>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00469.warc.gz"} |
Total Return Forecasts: Major Asset Classes - Tuesday, April 4
Image Source: Unsplash
The expected long-run return for the Global Market Index (GMI) held steady at 6.0% annualized in March, unchanged from last month and close to its trailing performance over the past decade. The
forecast is based on the average estimate for three models (defined below).
Today’s revised forecast is slightly below the trailing 6.1% annualized 10-year return for GMI, an unmanaged, market-value-weighted portfolio that holds all the major asset classes (except cash).
Historical performance for this benchmark’s rolling 10-year performance and projections for the long-run outlook has been relatively steady in recent months.
The underlying components of GMI continue to post relatively strong forecasts vs. their current trailing 10-year returns. The outlier: the US stock market, which is projected to earn a substantially
lower return vs. its performance over the past decade. GMI’s forecast is also below its 10-year performance, albeit fractionally.
GMI represents a theoretical benchmark of the optimal portfolio for the average investor with an infinite time horizon. On that basis, GMI is useful as a starting point for research on asset
allocation and portfolio design. GMI’s history suggests that this passive benchmark’s performance is competitive with most active asset allocation strategies, especially after adjusting for risk,
trading costs, and taxes.
Keep in mind that all the forecasts above will likely be incorrect to some degree. By contrast, GMI’s projections are expected to be more reliable vs. the estimates for the individual asset classes.
Predictions for the specific market components (US stocks, commodities, etc.) are subject to greater volatility and tracking error compared with aggregating forecasts into the GMI estimate, a process
that may reduce some of the errors over time.
For context on how GMI’s realized total return has evolved through time, consider the benchmark’s track record on a rolling 10-year annualized basis. The chart below compares GMI’s performance vs.
the equivalent for US stocks and US bonds through last month. GMI’s current 10-year return (green line) is a solid 6.2%. That’s fallen substantially from recent levels but it’s been relatively steady
Here’s a summary of how the forecasts are generated and definitions of the other metrics in the table above:
BB: The Building Block model uses historical returns as a proxy for estimating the future. The sample period used starts in January 1998 (the earliest available date for all the asset classes listed
above). The procedure is to calculate the risk premium for each asset class, compute the annualized return and then add an expected risk-free rate to generate a total return forecast. For the
expected risk-free rate, we’re using the latest yield on the 10-year Treasury Inflation-Protected Security (TIPS). This yield is considered a market estimate of a risk-free, real (inflation-adjusted)
return for a “safe” asset — this “risk-free” rate is also used for all the models outlined below. Note that the BB model used here is (loosely) based on a methodology originally outlined by Ibbotson
Associates (a division of Morningstar).
EQ: The Equilibrium model reverse engineers expected return by way of risk. Rather than trying to predict return directly, this model relies on the somewhat more reliable framework of using risk
metrics to estimate future performance. The process is relatively robust in the sense that forecasting risk is slightly easier than projecting return. The three inputs:
* An estimate of the overall portfolio’s expected market price of risk, defined as the Sharpe ratio, which is the ratio of risk premia to volatility (standard deviation). Note: the “portfolio”
here and throughout is defined as GMI
* The expected volatility (standard deviation) of each asset (GMI’s market components)
* The expected correlation for each asset relative to the portfolio (GMI)
This model for estimating equilibrium returns was initially outlined in a 1974 paper by Professor Bill Sharpe. For a summary, see Gary Brinson’s explanation in Chapter 3 of The Portable MBA in
Investment. I also review the model in my book Dynamic Asset Allocation. Note that this methodology initially estimates a risk premium and then adds an expected risk-free rate to arrive at total
return forecasts. The expected risk-free rate is outlined in BB above.
ADJ: This methodology is identical to the Equilibrium model (EQ) outlined above with one exception: the forecasts are adjusted based on short-term momentum and longer-term mean reversion factors.
Momentum is defined as the current price relative to the trailing 12-month moving average. The mean reversion factor is estimated as the current price relative to the trailing 60-month (5-year)
moving average. The equilibrium forecasts are adjusted based on current prices relative to the 12-month and 60-month moving averages. If current prices are above (below) the moving averages, the
unadjusted risk premia estimates are decreased (increased). The adjustment formula is simply taking the inverse of the average of the current price to the two moving averages. For example: if an
asset class’s current price is 10% above its 12-month moving average and 20% over its 60-month moving average, the unadjusted forecast is reduced by 15% (the average of 10% and 20%). The logic here
is that when prices are relatively high vs. recent history, the equilibrium forecasts are reduced. On the flip side, when prices are relatively low vs. recent history, the equilibrium forecasts are
Avg: This column is a simple average of the three forecasts for each row (asset class)
10yr Ret: For perspective on actual returns, this column shows the trailing 10-year annualized total return for the asset classes through the current target month.
Spread: Average-model forecast less trailing 10-year return.
More By This Author:
Major Asset Classes, Performance Review - March 2023
Is The US Facing A Slow(er) - Moving Recession Threat? Part II
Is The US Facing A Slow(er) - Moving Recession Threat? Part I
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The standard model: a dictionary for the mathematician
it seems just too much asking for “global” transformations symmetries.
But these global symmetries do play some role. That’s called the theory of superselection sectors. For instance the entire celebrated Doplicher-Roberts theorem is about global symmetries of
relativistic field theories on Minkowski space.
It’s true that there is much less interesting information in these global symmetries than in gauge symmetries. But they are still there.
What I intended for unphysical in the introduction is exactly what Domenico Fiorenza explained. In concrete, what my physics class teacher taught me is that, since in minkowski there are points
causally unrelated, it seems just too much asking for "global" transformations symmetries. Where "global" means "it doesn't depend from the base point", as already remarked. In this sense
relativisitic QFT is a local theory: a particle field in x is not causally connected with the entire Minkowski space.
Thank you for feedback! The wiki just started and the questioning is always stimulant!
rigid meaning the group of symmetries is smaller. But this was just intuiting, and probably waffle.
Gauge transformations should encompass all transformations of the mathematical description of the system that are not observable
ah good. This is what I thought. I was trying to come up with a non-circular definition above, but this is better than mine. So we then have the fact that fields (observable fields) correspond to
points in $\mathcal{A}/\mathcal{G}$ where $\mathcal{A}$ is the space of connections and $\mathcal{G}$ (sometimes $= Map(X,G)$) is the gauge group. It’s coming back to me…
Sorry, I’m not sure what “rigid” in conjunction with “Lagrangian” means.
My interpretation of “unphysical” in this context (referring to my previous post) is this: Gauge transformations should encompass all transformations of the mathematical description of the system
that are not observable, or, rephrasing: that are redundancies of the mathematical model. We expect that only the relative phase of interacting systems is observable, but not their absolute phases
(think of interference patterns). Therefore we should postulate that a physically realistic theory has local gauge transformations, not only global gauge transformations. And this postulate leads to
a different Lagrangian than the one I wrote down in my previous post, namly to that of QED.
Ah, you’re right. In that case, could be think of the ’realistic’ Lagrangian as being less rigid than the naive one? Also, this comment,
this action is unrealistic from a physical perspective
seems unmotivated to me, and I have studied QFT (albeit a long time ago, and I may have forgotten why this is considered unrealistic). Perhaps it is like this (this is off the top of my head, and
with a head cold, so it may be completely uninformed rubbish):
The symmetries of the $G$-bundle preserving a given connection exactly is just $G$(???Not sure about this bit), but the symmetries of the $G$-bundle preserving the connection up to something
unphysical is the gauge group ($Map(X,G)$, say).
It’s better to explain what is meant, since non-specialists are part of the intended audience :-)
An example of what domenico explained is this: Let’s take a Dirac field $\psi$ with electric charge e, the free Dirac Lagrangian is then
$\mathcal{L} = \overline{\psi} (i \gamma^{\mu} \partial_{\mu} + m) \psi$
which is manifestly = clearly invariant under the transformation $\psi \to e^{-ie \alpha}$ and $\overline{\psi} \to e^{ie \alpha}$ with $\alpha$ a constant. Physicists would call this transformation
a global gauge transformation, because the element of the gauge group does not depend on the basepoint = point in Minkowski space. If you let $\alpha$ depend on the basepoint, they call it a local
gauge transformation. But the bundle is trivial in both cases.
Global symmetries, if I dredge my physics memory, means that the bundle on which the gauge fields are connections, is trivial (not merely trivialisable): $E = X \times G$ where $X$ is Minkowski
Tim, thanks for feedback! :)
Global symmetries: I guess what Giuseppe meant there is that on a spacetime $X=\mathbf{R}^{1,3}$ one usually has a Lagrangian $\mathcal{L}$ which is manifestly invariant under the action of a gauge
group $G$ on the space of fields. this action is unrealistic from a physical perspective since it corresponds to acting on fields at a point $x\in X$ always by the same element $g\in G$, where acting
“locally” with an element $g(x)$ depending on $x$ would be more realistic. in other wors one moves from the symmetry group $G$ to the (much larger) symmetry group $maps(X\to G)$. usually the original
lagrangian does not have this larger symmetries, and one has to suitably make it invariant, e.g, moving from derivations to $\mathfrak{g}$-valued connections.
Keep at it! To facilitate talking about it, it would be nice to have some subtitles, and numbers or names for the propositions and definitions etc.
A professor once drove me wild by numbering his theorems like 5.2.4.2.6 and using this in his lectures, like "this follows immediatly from 5.2.4.2.6 and 7.3.5 dash 5". Since then I prefer to try to
name propositions :-)
I am very embarrassed, because I noticed that I know references and proofs of the proposition you cited (irreps of the product of groups are products of irreps of the factors), and need a reference
for, for compact groups only, so if you find one that will be of interest to me, too.
One question about the point 4:
4. A theory of field with global symmetries has no actually physical meaning...
"Global symmetry" has different meanings, and obviously I'm thinking about one that is not alluded to here, because I do not understand that statement :-)
Giuseppe wrote a first draft of the introduction.
I further expanded the section on “Variations and generalizations”.
Hm, now the entry has more on variations and generalizations than on the thing itself…
Yes, I agree, it is sometimes good to develop an entry in a quiet area to move it into the light of stage later on.
Thanks to Tim for the additions to the public domain entry. I have expanded it a little further now: standard model of particle physics.
thanks! and for sure we'll move it to the public domain at a later stage. but at the beginning it will be so unstable and full of errors that a little more protective ambient will not harm. it's like
a little toddler, we have to wait till it can walk before letting it free :-)
I added a few links and references to the standard model page. An (online) dictionary of mathematical and physical language would be very valuable - so valuable indeed that it may well deserve to
live in the public domain :-)
an entry of the name standard model of particle physics had been requested by several other entries. So I created it with just a link to the page on domenic’s personal web.
we decided the nLab could be anice place wher to develop it
That would be a magnificent addition to the nLab!
you can access the work in progress on the dictionary from my area in the nLab
With Giuseppe Malavolta we decided to write a rather exhaustive and concrete dictionary between the Physics parlance and current Maths jargon (this dictionary should become Giuseppe's master thesis).
our idea was that since there are all those smart guys digging and freezing and beaming under Geneva, it was quite a scandal we had no precise idea of what they were doing there. the dictionary will
be completely classic Maths, with no nPOV in it (yet, a few shadows of it may happen to slide in..), but we decided the nLab could be a nice place where to develop it after having bumped into this
nCafe' blog entry. World is really little.. :-)
you can access the work in progress on the dictionary from my area in the nLab (at the moment it only contains a rough table of contents)
Guiseppe wrote:
…since in minkowski there are points causally unrelated, it seems just too much asking for “global” transformations symmetries.
I’m not sure I understand what is meant by “it seems just too much asking for”. We have gauge transformations where
a) the element of the gauge group does not depend on the basepoint (global) and
c) the element does depend on the basepoint.
I left b) out for the argument about causality: If $\mathcal{O}$ and $\mathcal{O}'$ are spacelike separated, then we should be able to prescribe some sort of gauge transformation on $\mathcal{O}$,
and independently one on $\mathcal{O}'$, is that your point?
If so, then the postulate of local gauge invariance is stronger: Consider a timelike curve, then we allow gauge transformations where the element of the gauge group varies along the curve. I think
that (danger: handwaving) you cannot infer this from causality alone, you need as additional assumption that the absolute phase of a (localized) excitation of the electromagnetic (quantum) field is
not observable.
Urs said:
But these global symmetries do play some role. That’s called the theory of superselection sectors.
I have to dig farther into this, but off the top of my head AQFT says: The Hilbert space of the whole theory is a direct sum of coherent subspaces $H_k$ or superselection sectors. Within each $H_k$
one has the unrestricted superposition principle whereas phase relations between state vectors belonging to different sectors are meaningless. Measurements resp. observables map each sector to
Translated to the context of Guiseppe’s draft that would correspond to quantum numbers: Each $H_k$ represents all possible states with prescribed quantum numbers.
The Hilbert space of the whole theory is a direct sum of coherent subspaces $H_k$ or superselection sectors
Yes, that’s the decomposition of the Hilbert space into irreps of the global group that is acting.
Translated to the context of Guiseppe’s draft that would correspond to quantum numbers:
Yes, quantum numbers are another name for irreps of some group.
I think that’s what’s going on. But let me know if you think I am mixed up.
See Halvorson from page 55 on and Halvorson from page 83 on. Notice that the group he calls the gauge group is clearly the global gauge group.
I definitly have to re-read that paper, but all I had in mind here was the dictionary “full set of quantum numbers” = “irrep of global gauge group” = “superselection sector”.
added an overview on Lie groups and algebras representation and started the section for Poincaré group. These are still drafts, so exepct some mistake. Feel free to contribute.
@Giuseppe: fine. I’ll add my comments in the form of query boxes there.
@anyone else: let me recall that The standard model: a dictionary for the mathematician is Giuseppe’s master thesis. He’s writing it under my supervision, but he’s writing it directly on the nLab (in
my area), where I make my comments and corrections. Clearly, anyone interested can correct or comment: this way of developing thesis work is meant to be an attemp to explore the potentiality of the
Lab in this direction.
this way of developing thesis work is meant to be an attemp to explore the potentiality of the Lab in this direction.
I like this idea. Perhaps when it's all said and done you and Giuseppe could give us a run-down of how you thought the "experiment" went and if you'd recommend it for others.
I’m not sure how it will work out for Giuseppe, but I can already say that it is a great opportunity for me to check some things that I thought I already understood (since I dropped out of academia I
don’t have the $\frac{obligation}{opportunity}$ to teach classes).
One organizational question: Is the master thesis supposed to contain some originally research of the author? (The German Diplomarbeit is/was supposed to, but it is usually accepted that the student
writes a review of the topic if he/she did not come up with anything interesting).
If so, is there a problem if the final content is not attributable because it is not clear anymore who contributed what?
In Italy a master thesis can either contain original research or be a review. Giuseppe’s thesis will be a review.
Ok, good. I should add that (according to my personal, limited experience) usually the good students in Germany end up writing reviews instead of doing original research, because they go to the most
prominent professors who assign tasks to them that are undoable.
I added my first comment :-)
In my experience, in Italy different research groups have different policies. Where I am, geometry and algebra master students are given review thesis, leaving original research for PhD students.
Thanks a lot for the comment! :-)
If Giuseppe’s thesis is in differential or algebraic geometry then my comment may be off topic, if the focus is on representation theory then the “rigorous” definition of quantum fields in the sense
of AQFT should be ignored, I think. Quantum field = function from Minkowski space to an algebra (with a representation of the Poincare group) would suffice.
In my experience (which stems from several reasearch groups from three German universities) the professors would always assign some research project: To the good students (those they thought could
succeed in academia) one that could lead to results that could be published, to the other ones easier ones (sometimes problems that they alredy solved themselves, so that the students were supposed
to reconstruct what the professors already had done). I did my thesis in the research group of Franz Wegner in Heidelberg, and there writing a review thesis was considered to be the ultimate failure
(I do not approve of this situation, but that was what I observed).
I hope this information will come in handy when Guiseppe notices some sceptical faces when telling German professors that his thesis was a review :-)
Well, let’s wait a bit how things develop. After all, a good deal of all original research springs from attempts to understand what people wrote up before you ;-)
Actually, if I recall correctly, the original paper in which Bell's theorem was introduced was technically a review. So one never knows! :-)
Ok ;-)
I added the example of a neutral real (uncharged) scalar field from my comment as an example to the Wightman axioms (a little bit more detailed).
can I maybe ask you for a favor? Do you feel like starting (a stub, maybe) on spin-statistics theorem?
I want to link to it from my book page, in the list of accomplishments of AQFT, but don’t feel I have the leisure to write something myself.
Just asking, since you wrote all these other nice entries on AQFT matters.
Ok, that is definitly on my list, the problem is: I do not know a simple explanation of it, all statements I know use some involved analysis using modular groups etc. Same for the PCT theorem. But I
should be able to provide both a useful Idea and Reference paragraph. What is the deadline?
@Tim: I left a note for you in the Bell's theorem thread.
Regarding the spin-statistics theorem, I know some simple explanations that I use with undergrads, but they may be overly simplified (though I try to not make them this way).
I know some simple explanations that I use with undergrads, but they may be overly simplified (though I try to not make them this way).
I’m all in for simple intuitive explanations, even for oversimplified ones: the question “can you see what is wrong about it?” can be more illumination than the best proofs, what I do not like is
that physicists often “explain” things this way without pointing out that the explanation has weaknesses.
JB has written about it, too, “Spin, Statistics, CPT and All That Jazz”, can’t find the link right now.
What I meant was a mathematical proof using a set of axioms in the AQFT framework, “involved” means these use both some advanced mathematical machinery and other non-trivial results of AQFT. A self
contained page seems out of reach.
Since the spin-statistics theorem that I would use comes from a paper of Daniele Guido and Roberto Longo (from a Festschrift for the 70th birthday of Hans-Jürgen Borchers), I took a look at the arXiv
at their recent papers, and, who knows: Longo has published a joint paper with Edward Witten: An Algebraic Construction of Boundary Quantum Field Theory. I did not know that Witten is even remotly
interested in AQFT :-)
Stubs for modular theory, Bisognano-Wichmann theorem and PCT theorem
Right. I think that was part of the problem with some of my early entries - they weren’t in the AQFT framework (which I’m not all that familiar with).
Ah no, I do not think that that is a problem. AQFT is only one little piece of the puzzle of creating a mathematical foundation for QFT (and String theory), see Mathematical Foundations of Quantum
Field and Perturbative String Theory, and here the draft of the preface for an outline of Urs’ vision.
There are physicists for whom mathematical physics is superfluous. There are physicists for whom mathematical physics tries to justify results obtained by the real physicists, and that’s all. And
there are people like me who had an epiphany when reading the book “PCT, Spin, Statistics and all that”: So that’s what a QFT is, and I’ve been fascinated by AQFT since then. For me, mathematical
physics provides the language that I need to think about physics.
Is that good or bad, is it a strength or a weakness? I don’t know, but I strive to learn more about AQFT not because I think it is the ultimate theory and solution for everything, a TOE or the most
important subject that anyone could think about right now, but because it helps me to understand a little bit better what QFT is about. And that is part of the spirit of the nLab: Use a
mathematically precise language for whatever you say, not as an empty excercise in rigour, but as a way to a better understanding.
an empty excercise in rigour
Certainly exercises in rigour are worth doing to the point that you’re confident that you could make something completely rigorous given enough time. Physics has not yet reached this point, so I
disagree that such exercises in rigour are superfluous
I disagree that such exercises in rigour are superfluous.
Yes, thought so :-)
This depends of course on the subjective perspective. Jaques Distler had some interesting remarks about rigour in physics on his blog, his motive was his discussion of Rehren’s “counterproof” of the
AdS/CFT correspondence. This discussion is only the tip of the iceberg of a cultural clash of the AQFT and the string communities, and part of the clash is the difference of perspectives with regard
to rigour.
The whole AQFT project has been characterized by other physicists as having “contributed less than $\epsilon$ to particle physics”. Why? Because for these scientists the primary objective of QFT is/
was to calculate numbers that can be compared with experiments, like cross sections, and AQFT can “only prove general theorems” but cannot calculate any such numbers (yet?). And some of those general
theorems were even known before AQFT was invented, like the spin-statistics theorem and the PCT theorem (although “theorem” should be understood in the sense of theoretical physics).
Longo has published a joint paper with Edward Witten:
Yes, and apparently, from the looks of it, what happened was that Longo had insights on boudary CFT and Witten noticed that this serves to put some of his decade-old papers on string backgrounds on
more solid footing.
@Tim: You make a very impassioned (and sensible) argument. I will have to add it to my already enormous list of things to get a handle on.
Regarding rigor (rigour, for Anglophiles) in physics, I agree to a degree, but I think that there are parts of physics that will defy all attempts to make them rigorous. I do not believe - and I
would bet a large percentage of physicists would agree - that we will ultimately fail in our attempt to fully axiomatize physics. I truly do not think it is possible. It doesn’t mean we shouldn’t
make it as rigorous as possible - we should. But it means that there will always be fuzzy, gray areas.
added a draft on U(1) and SU(N) irreducible representations. Feel free to contribute
added the derivation of the Dirac and Klein-Gordon equation via the representation theory. The next step is studying covariant equations in general, so that we can assign to a covariant equation the
right Poincaré irreducible representation.
In which entry ?
here (still extremely rough)
After a very long pause, with Giuseppe we have taken up again the standard model project. A very introductory but fairly complete account on mass, spin, helicity has now been added here.
Infinite thanks to Andrew for the latex to itex conversion!
Infinite thanks to Andrew for the latex to itex conversion!
Ah, interesting. Is the whole page a conversion from LaTeX?
By the way, a bunch of formulas don’t display as math. Either one sees the dollar signs displayed, or they appear in “code-typesetting” or whatever that is called.
Urs, that section that Domenico links to is converted from LaTeX.
I’ll have to look through the history to see how much Domenico has changed after the conversion, but the errors that I saw on a quick scan through just now were due to two things:
1. Syntax like S_\mathrm{A}. This is okay in LaTeX, but is dangerous. It’s much better to write S_{\mathrm{A}}. The handling of _ and ^ in LaTeX is a bit special and though they seem like macros,
they aren’t. But my script treats them as if they were macros so misses this special behaviour.
2. Nested mathematics and text. The command $\text{$x$}$ is invalid in iTeX. I’m not sure how best to handle this in my script, though, I’ll have to think about this.
But my script treats them as if they were macros
Not your script but iTeX itself, right? Your script shouldn’t have to do anything to that, a priori.
So both problems are limitations of iTeX that your script merely doesn’t (yet) have a way to fix.
No, it is my script. My script tries to emulate TeX and expand all macros. Those that are known to be iTeX macros seem to get left alone, but actually they get expanded to \noexpand\macroname which
avoids being expanded and gets passed on to the output routine. Subscripts and superscripts need to be expanded because what might seem to be a reasonable subscript to TeX might not be to iTeX and
vice versa. So in my script, ^ is active, takes one argument, and expands to \^{#1} (where \^ further expands to ^ but with catcode 12 (other) so no further expansion takes place). However, this
isn’t actually how TeX does its superscripts (but it is if you load the mathtools package) so if you write a^\mathcal{A} which is legal in both TeX and iTeX, then my script interprets that as a^{\ | {"url":"https://nforum.ncatlab.org/discussion/1055/the-standard-model-a-dictionary-for-the-mathematician/?Focus=21573","timestamp":"2024-11-14T02:21:39Z","content_type":"application/xhtml+xml","content_length":"142530","record_id":"<urn:uuid:21048be2-ec1c-475d-a97a-021fed8f9e44>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00751.warc.gz"} |
Identifying math anxiety
Can you do long division in your head and calculate tips in your sleep? Or does the mere thought of arithmetic keep you up at night?
If you fall into the latter camp, you’re not alone.
Math anxiety is real—and an established body of research proves it. In fact, data shows that math anxiety affects at least 20% of students.
And its effects can be damaging in both the immediate and long term. It can bring down student performance both in and beyond math, and in and outside the classroom.
Fortunately, we’re also learning how teachers can help students manage math anxiety—and succeed wherever it’s holding them back.
We explored this topic on a recent episode of Math Teacher Lounge, our biweekly podcast created specifically for K–12 math educators. This season is all about recognizing and reducing math anxiety in
students, with each episode featuring experts and educators who share their insights and strategies around this critical subject.
Dr. Gerardo Ramirez, associate professor of educational psychology at Ball State University, has been studying math anxiety for more than a decade. He joined podcast hosts Bethany Lockhart Johnson
and Dan Meyer to share his insights.
So let’s take a look at what math anxiety is—and is not. We’ll also explore what impact it has on learning, and what we can do about it.
What is math anxiety?
Math anxiety is more than just finding math challenging, or feeling like you’re “not a math person.” Dr. Ramirez offers this definition: “[Math anxiety] is a fear or apprehension in situations that
might involve math or situations that you perceive as involving math. Anything from tests to homework to paying a tip at a restaurant.”
Math anxiety may cause sweating, rapid heartbeat, shortness of breath, and other physical symptoms of anxiety.
But while math anxiety has some similarities with other forms of anxiety, it’s exclusive to math-related tasks, and comes with a unique set of characteristics and influences.
Math anxiety can lead sufferers to deliberately avoid math. And this avoidance can not only result in a student not learning math, but also limiting their academic success, career options, and even
social experiences and connections. This can look like anything from getting poor grades in math class, to tension with family members over doing math homework.
Parents and teachers can suffer from math anxiety, too. In fact, some research suggests that when teachers have math anxiety, it’s more likely that some of their students will, too.
What causes math anxiety?
It’s not correlated to high or low skill or performance in math. Students who generally don’t do well in math can experience math anxiety because they assume they’ll do poorly every time. Students
who have been pressured to be high-achieving experience math anxiety because they’re worried they won’t meet expectations.
Other triggers may include:
• Pressure. Pressure from parents or peers to do well in math can create anxiety, especially if the person feels that their worth or future success is tied to their math abilities.
• Negative past experiences. Someone who has struggled with math or gotten negative feedback about their math skills might develop math anxiety. They may start to avoid or fear math, making it even
harder to approach and improve.
• Learning style. Different people have different learning styles. When someone’s learning style doesn’t match the way math is taught in their class or school, they may struggle and develop
• Cultural factors. When students hear things like, “Boys are better at math,” it can increase math anxiety in girls who may absorb the notion that they are already destined to underachieve.
Math anxiety and working memory
Dr. Ramirez has researched the important relationship between math anxiety and working memory.
Working memory refers to the ability to hold and manipulate information in short-term memory. People with math anxiety often have poorer working memory capacity when it comes to math-related tasks.
This is thought to be due to the cognitive load created by anxiety, which can interfere with the ability to manage information in working memory.
The result? A negative feedback loop. Poor working memory can lead to further math anxiety, and increased anxiety can further impair working memory.
However, it’s important to note that not all individuals with math anxiety experience a decline in working memory capacity. Some may have average or above-average working memory capacity but still
experience math anxiety. In such cases, the anxiety may be related to negative beliefs about one’s ability to perform math tasks, rather than an actual cognitive deficit.
What we can do about math anxiety
Even though math anxiety is a distinct type of anxiety, interventions such as cognitive behavioral therapy, exposure therapy, and mindfulness approaches have been shown to be effective in reducing
It starts, says Dr. Ramirez, with normalizing the anxiety.
“If you’re a student and you’re struggling with math and I tell you, ‘Yeah, it’s hard, it’s OK to struggle with math,’ that makes you feel seen. And that’s gonna lead you to want to ask me more for
help, because I’m someone who understands you,” says Dr. Ramirez. “And that’s a great opportunity.” | {"url":"https://amplify.com/blog/personalized-learning/identifying-math-anxiety/","timestamp":"2024-11-03T06:42:48Z","content_type":"text/html","content_length":"228253","record_id":"<urn:uuid:36bdf667-344b-4f40-8945-221cf3be53d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00732.warc.gz"} |
MarkLogic Server 11.0 Product Documentation
cts:sum( $arg as xs:anyAtomicType*, [$zero as xs:anyAtomicType?] ) as xs:anyAtomicType?
arg The sequence of values to be summed. The values should be the result of a lexicon lookup.
zero The value to return as zero if the input sequence is the empty sequence.
The cts:frequency of the result is the sum of the frequencies of the sequence.
This function is designed to take a sequence of values returned by a lexicon function (for example, cts:element-values); if you input non-lexicon values, the result will always be 0.
xquery version "1.0-ml"; (: This query assumes an int range index is configured in the database. It generates some sample data and then performs the aggregation in a separate transaction. :) for $x
in 1 to 10 return xdmp:document-insert(fn:concat($x, ".xml"), <my-element>{ for $y in 1 to $x return <int>{$x}</int> }</my-element>); cts:sum(cts:element-values(xs:QName("int"), (), ("type=int",
"item-frequency"))), cts:sum(cts:element-values(xs:QName("int"), (), ("type=int", "fragment-frequency"))) => 385 55 | {"url":"https://docs.marklogic.com/cts:sum","timestamp":"2024-11-03T19:51:54Z","content_type":"application/xhtml+xml","content_length":"30780","record_id":"<urn:uuid:38b4cfcc-4593-4120-8479-ec2d22401518>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00641.warc.gz"} |
Graphing Linear Function Worksheets
1. Math>
2. Graphing Linear Function
Graphing Linear Function Worksheets
This extensive set of pdf worksheets includes exercises on graphing linear function by plotting points on the grid. Three types of function tables, each with two levels of worksheets, require
learners in grade 8 and high school to plot the points and graph the lines. The graph of a linear function is always a straight line. Use the answer keys provided to verify your responses. Employ the
various download options to gain access to all our worksheets under this topic. A number of free printable worksheets are also up for grabs!
Printing Help - Please do not print worksheets with grids directly from the browser. Kindly download them and print.
Graphing Linear Function: Type 1 - Level 1
Find f(x) based on the x-coordinates provided and complete the function tables. Plot the points and graph the lines. The slopes given in level 1 worksheets are in the form of integers. Verify your
graph with the answer keys provided.
Graphing Linear Function: Type 1 - Level 2
These pdf worksheets provide ample practice in plotting the graph of linear functions. The slopes are represented as fractions in the level 2 worksheets. For the given x-coordinates, find f(x) and
complete the function tables. Plot the points and graph the linear function.
Graphing Linear Function: Type 2 - Level 1
In this section, 8th grade and high school students will have to find the missing values of x and f(x). Complete the function table, plot the points and graph the linear function. The slopes in level
1 worksheets are in the form of integers.
Graphing Linear Function: Type 2 - Level 2
Write down the missing values of x and f(x). There are nine linear functions in each worksheet with the slopes in the form of simplified fractions. Plot the coordinates and graph the lines.
Graphing Linear Function: Type 3 - Level 1
Assume your own values for x for all printable worksheets provided here. Find the range. Compute the function tables, plot the points, and graph the linear functions. The slopes provided in level 1
worksheets are represented as integers.
Graphing Linear Function: Type 3 - Level 2
The equations in the second level of worksheets have slopes in the form of fractions. Assign five values of x and find the corresponding values of f(x). Plot the points and graph the linear function.
Download our easy-to-print worksheets for ample practice. | {"url":"https://www.mathworksheets4kids.com/graphing-linear-function.php","timestamp":"2024-11-10T11:30:00Z","content_type":"text/html","content_length":"39181","record_id":"<urn:uuid:578905cf-db99-452b-aa25-f95848ae5eb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00268.warc.gz"} |